Integrated Multiscale Chemical Product Design using Property Clustering and
Decomposition Techniques in a Reverse Problem Formulation
by
Charles C. Solvason
A dissertation submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Doctor of Philosophy
Auburn, Alabama
December 12, 2011
Keywords: Product Design, Chemical Product, Multiscale, Reverse Problem Formulation,
Property Clustering, Systems Engineering
Copyright 2011 by Charles C. Solvason
Approved by
Mario R. Eden, McMillan Associate Professor of Chemical Engineering
Ram B. Gupta, Woltosz Professor of Chemical Engineering
Virginia A. Davis, Associate Professor of Chemical Engineering
Nedret Billor, Associate Professor of Mathematics and Statistics
ii
Abstract
In recent years, chemical engineers in the Process Systems Engineering (PSE)
community have increasingly been using their skill set to solve problems in areas beyond the
chemical manufacturing processes, focusing instead on the chemical products themselves. This
trend reflects a movement within the field to meet the demands of an increasingly competitive
and global consumer products marketplace that stresses being the first to market in conjuction
with being the highest volume and lowest cost manufacturer. By definition, chemically
formulated products deliver specific attributes to the consumer by manipulating a multitude of
separate and often competing mechanisms in molecular architecture that operate over a wide
range of length and time scales. Examples of chemical products include performance chemicals,
paints, cosmetics, pharmaceuticals, proteins, semi-conductors, and foods, among others. Like
process design, computer aided chemical product design is a complex programming problem.
Generating, integrating, and managing the information, data, and knowledge at multiple length
scales for use in various types of product design problems is a significant undertaking. The
traditional approach to managing the complexity of this problem has been to compute
information at smaller length scales and pass it to models at larger length scales by removing
degrees of freedom (coarse-graining) with the objective being to predict macroscopic properties
from molecular information. While often the most accurate method for predicting properties,
this simulated approach has two limitations: (1) it has an immense computational cost due to
hierarchical nesting and (2) it utilizes a priori knowledge of the molecular architecture (i.e. the
number and types of atoms or electrons present). This dissertation covers the development of a
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novel, alternative approach that allows for the simultaneous design of a chemical product?s
molecular architecture across multiple scales using a reverse problem formulation, property
clustering, and decomposition techniques. The developed framework is specifically designed to
utilize experimental data, parameters, and models since the effectiveness of a chemical product is
most often determined by consumer attributes based on consumer preference tests.
In this work, three specific methodologies are developed. The first method, Attribute
Computer Aided Mixture (Blend) Design (aCAMbD) is an extension of Computer Aided Mixture
(Blend) Design (CAMbD) and includes experimental data and regression models, specifically,
Scheffe canonical and Cox polynomial models. Necessary adjustments to the original clustering
algorithm are identified and the design method is rewritten accordingly. The end result is a
method capable of performing a mixture design on any chemical constituent data set across
multiple length scales, as long as accurate attribute-component models can be established. A
case study mixture design of spun yarn is presented to illustrate the method.
The second method developed is Attribute Computer Aided Molecular Design (aCAMD).
It is an extension of Computer Aided Molecular Design (CAMD) to include experimental data
and regression models, while continuing to use group contribution method (GCM) based
property models. The technique uses design of experiments (DOE) to generate an attribute-
property relationship and maps the attribute information into a property domain where molecular
design can proceed. Adjustments to the property clustering algorithm are made to reflect the
new design approach. The result is a method capable of performing molecular design on any
attribute data set as long as a strong relationship between attribute and property models can be
established. A case study involving the molecular design of environmentally benign refrigerants
is presented to illustrate the method.
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The third method developed is Characterization Based Computer Aided Molecular
Design (cCAMD). It is a wholly new method that addresses the limitations of aCAMbD and
aCAMD, namely difficulty in finding suitable attribute-component and attribute-property models
for complex chemical products. The method uses characterization tools like infrared and near-
infrared spectroscopy (IR/NIR) to generate a set of data from a chemical constituent training set
and then applies decomposition algorithms like principal component analysis (PCA) to find the
underlying latent variable data structure. A parameterization of the data structure into a
characterization based group contribution method (cGCM) follows. Attribute data is then
mapped into the latent domain using a separate principal component regression (PCR) or partial
linear regression on to latent surfaces (PLS) model. A molecular design is then performed in the
latent domain. The resulting method is capable of performing structured molecular design across
multiple scales for any system of attributes whose molecular architecture can be adequately
described by characterization methods. A case study on the particle design of pharmaceutical
excipients for an acetaminophen tablet is presented to illustrate the method.
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Acknowledgments
I thank my advisor Dr. Mario Eden for introducing me to this research topic and
providing an incredible opportunity to succeed. His advice, patience, and direction have made
this job such a joy. I would like to thank my coworkers, Dr. Norman Sammons and Dr. Nishanth
Chemmangattuvalappil for providing support during my research investigation. I would also like
to acknowledge Dr. Jon Gabrielsson for providing his research data on excipients for direct
compressed acetaminophen tablets. This data set was crucial to my proof of concept case study
that tied this research together. I would like to thank Dr. Christopher Roberts for giving me the
opportunity to return to the city of Auburn which has been an outstanding experience that I will
never forget. And finally, I would like to acknowledge Dr. Virginia Davis, Dr. Ram Gupta and
Dr. Rafiqul Gani for inspiring aspects of this research.
In my personal life I would also like to especially thank my wife, Lisa Trinh. Her
encouragement to pursue my dreams is the reason I was able to return to Auburn University and
conduct this research. I also thank my parents, Jim and Darlene Solvason, for instilling in me the
principles of work and perseverance and my brother, Jonathan Solvason, for keeping up my
spirits. Finally, to my grandmother Helen Word and my late grandfathers Brig. General Charles
E. Word of the USAF (ret.) and John C. Solvason: Mission accomplished! I love you all.
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Table of Contents
Abstract ......................................................................................................................................... ii
Acknowledgments......................................................................................................................... v
List of Figures ............................................................................................................................. xii
List of Tables ............................................................................................................................ xvii
Nomenclature ............................................................................................................................. xxi
Chapter 1: Introduction to Process Systems Engineering ............................................................. 1
Chapter 2: Product Design for Chemical Engineers ................................................................... 11
2.1. An Emerging Paradigm............................................................................................ 11
2.2. Types of Chemical Products .................................................................................... 15
2.3. Chemical Product Scales .......................................................................................... 17
2.4. Other Definitions .................................................................................................... 21
2.5. General Design Approach ........................................................................................ 22
Chapter 3: Experimental Design of Chemical Products ............................................................. 32
3.1. Mixture Design using Experiments ......................................................................... 33
3.2. Experimental Design using Multivariate Data ......................................................... 49
3.2.1. Characterization Techniques ..................................................................... 51
3.2.2. Principal Component Analysis & Regression........................................... 61
3.2.3. Partial Least Squares on to Latent Surfaces (PLS) .................................. 66
3.2.4. Mixture Design and Optimization with Latent Variables ......................... 68
vii
3.3. Summary .................................................................................................................. 71
Chapter 4: Computer Aided Design of Chemical Products ....................................................... 72
4.1. Mathematical Formulation of Chemical Product Design ........................................ 73
4.1.1. Searching a Database ................................................................................ 75
4.1.2. Generating Candidate Solutions ............................................................... 78
4.2. Useful Property Prediction Techniques ................................................................... 81
4.2.1 Group Contribution Methods ..................................................................... 86
4.3. Computer Aided Mixture Design (CAMbD)............................................................ 91
4.4. Computer Aided Molecular Design (CAMD) ......................................................... 92
4.5. Summary .................................................................................................................. 94
Chapter 5: Motivation and Challenges ....................................................................................... 95
5.1. Current Limitations of Chemical Product Design .................................................. 98
5.2. Reverse Problem Formulation ............................................................................... 101
5.3. Challenges Addressed in this Dissertation ............................................................. 103
Chapter 6: Methods for Multiscale Chemical Product Design ................................................. 110
6.1. Property Clustering Algorithms ............................................................................. 110
6.2. Computer Aided Mixture Design (CAMbD) with Clusters .................................. 120
6.3. Computer Aided Molecular Design (CAMD) with Clusters ................................. 133
6.4. Summary ................................................................................................................ 146
Chapter 7: Attribute-Computer Aided Mixture Design (aCAMbD) ......................................... 148
7.1. Integrating Attribute Data & Models with Property Clustering ............................ 149
7.1.1. Canonical Models ................................................................................... 159
7.1.2. Polynomial Models ................................................................................. 160
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7.2. Attribute Clustering Algorithms. ........................................................................... 165
7.3. aCAMbD using Experimental Data and Data Driven Models. .............................. 168
7.4. Case Study: Polymer Spun Yarn Mixture Design ................................................. 181
7.5. Summary ................................................................................................................ 195
Chapter 8: Attribute-Computer Aided Molecular Design (aCAMD) ....................................... 198
8.1. Integrating Attribute Data & GCM Models with the RPF .................................... 199
8.2. Attribute & Property Clustering Algorithms. ........................................................ 206
8.3. aCAMD using Experimental Data and GCM Models. .......................................... 217
8.4. Case Study: Environmentally Benign Refrigerant Design .................................... 228
8.5. Summary ................................................................................................................ 270
Chapter 9: Characterization based Computer Aided Molecular Design (cCAMD) ................. 272
9.1. Integrating Multivariate Data & Models with the RPF ......................................... 275
9.1.1. Characterizing the System with IR/NIR Spectroscopy ........................... 278
9.1.2. Developing the Latent Property Domain ................................................ 294
9.2. Latent Property Clustering Algorithms .................................................................. 311
9.3. cCAMD using Characterization Data and cGCM Models. ................................... 325
9.4. Case Study: Excipient Design for Direct Compressed Tablets ............................. 330
9.5. Summary ................................................................................................................ 344
Chapter 10: Conclusions ........................................................................................................... 346
10.1. Achievements ....................................................................................................... 347
10.2. Challenges and Future Work ............................................................................... 349
References ............................................................................................................................... 351
Appendix A1: Additional Chemical Product Design Techniques ............................................ 375
ix
A1.1. Multi-Criteria Decision Making (MCDM) ......................................................... 375
A1.2. Ontology.............................................................................................................. 383
A1.3. Mathematical Optimization................................................................................. 385
Appendix A2: Additional Characterization Techniques .......................................................... 394
A2.1. Ultraviolet-Visible Spectroscopy ........................................................................ 394
A2.2. Nuclear Magnetic Resonance Spectroscopy ....................................................... 395
A2.3. Mass Spectrometry .............................................................................................. 397
A2.4. Chromatography (Gas, Liquid, or Gel Permeation) ............................................ 397
A2.5. Scattering and Diffraction Techniques (Light, Neutron, X-ray) ......................... 398
A2.6. Other Characterization Techniques ..................................................................... 401
Appendix A3: Additional Decomposition Tools ..................................................................... 404
A3.1. Network Component Analysis ............................................................................ 404
Appendix A4: Additional Physico-Chemical Property Models ............................................... 407
A4.1. Simulation Techniques ........................................................................................ 407
A4.2. Topological Indices ............................................................................................ 414
Appendix A5: Acetaminophen Case Study Supporting Data .................................................. 422
Appendix A6: IR & NIR Molecular Group Absorbance Frequencies ..................................... 423
A6.1. Methine CH ......................................................................................................... 423
A6.2. Methylene CH2 .................................................................................................... 423
A6.3. Hydroxyl OH (w/o H-bond) ................................................................................ 424
A6.4. Hydroxyl OH (w/ H-bond) .................................................................................. 424
A6.5. Secondary Alcohol CHOH (w/ H-bond) ............................................................. 424
A6.6. Primary Alcohol CH2OH (w/ H-bond) ............................................................... 425
x
A6.7. Alcohol CHCH2OH (w/ H-bond) ........................................................................ 425
A6.8. Aliphatic Ether O ................................................................................................ 426
A6.9. Ether CH2O ......................................................................................................... 426
A6.10. Aldehyde HCO (w/o H-bond) ........................................................................... 427
A6.11. Aldehyde HCO (w/ H-bond) ............................................................................. 427
A6.12. Saturated Aliphatic Aldehyde CH2CHO (w/o H-bond) .................................... 427
A6.14. Vinyl CH2CH .................................................................................................... 429
A6.15. Vinylidene CH2C .............................................................................................. 429
A6.16. cis-Vinylene CHCH .......................................................................................... 430
A6.17. trans-Vinylene CHCH ...................................................................................... 430
A6.18. Aliphatic Methyl CH3 ....................................................................................... 430
A6.19. Aryl Methyl CH3 ............................................................................................... 431
A6.20. Aliphatic Methoxy OCH3 .................................................................................. 431
A6.21. Aryl Methoxy OCH3 ......................................................................................... 432
A6.22. Aryl OH (w/o H-bond) ...................................................................................... 433
A6.23. Aryl OH (w/ H-bond) ........................................................................................ 433
A6.24. Tetramethyl C(CH3)3......................................................................................... 433
A6.25. Alkyl Peroxide OO............................................................................................ 434
A6.26. Saturated Aliphatic Ester COO ......................................................................... 434
A6.27. Saturated Aliphatic Methyl Ester COOCH3...................................................... 434
A6.28. Saturated Aliphatic Ethyl Ester COOCH2CH3.................................................. 435
A6.29. Acrylate Ester CH2CHCOO .............................................................................. 436
A6.30. Methacrylate Ester CH2C(CH3)COO ................................................................ 437
xi
A6.31 Other Groups ...................................................................................................... 438
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List of Figures
Figure 2.1. Multi-length Scale Biochemistry Approach to Product Design ............................... 18
Figure 2.2. An Example of a Multi-length and Time Scales for a Powdered Detergent ............ 19
Figure 2.3. Multi-length and Time Scales Associated with Computational Simulation ............ 20
Figure 2.4. Multi-scale Chemical Product Supply Chain ........................................................... 21
Figure 2.5. The Stage-Gate Development Process for Chemical Product Design...................... 28
Figure 3.1. Mixture Design Point and Response Surface Simplex Diagram .............................. 38
Figure 3.2. Parameterization of Component Design Space in the Simplex Diagram ................ 42
Figure 3.3. Overview of Characterization Techniques .............................................................. 52
Figure 3.4. The Process-Structure-Property Relationship for Alloy Steel ................................. 53
Figure 3.5. WAXS Characterization Structure of High Performance Alloy Steel .................... 53
Figure 3.6. IR Spectra of Mannitol ............................................................................................ 56
Figure 3.7. NIR Spectra of Mannitol ......................................................................................... 58
Figure 3.8. Graphical Representation of PCA ........................................................................... 63
Figure 3.9. Scree Plot of a Covariance-PCA IR and NIR Spectroscopy Data ........................... 64
Figure 3.10. Graphical Representation of PLS .......................................................................... 67
Figure 4.1. Methods to find Property Models and Parameters ................................................... 84
Figure 4.2. Property Model Development Scheme .................................................................... 85
Figure 5.1. Computational Approach to Multiscale Property Estimation .................................. 97
Figure 5.2. Streamlined Multiscale Property Estimation ............................................................ 99
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Figure 5.3. Chemical Product Design using Consumer Attributes ........................................... 100
Figure 5.4. Reverse Problem Formulation of Chemical Product Design ................................ 102
Figure 5.5. Reverse Problem Formulation for Multiscale Chemical Product Design ............. 104
Figure 6.1. Property Clustering Binary Mixing Diagram ......................................................... 114
Figure 6.2. CAMbD Property Clustering Conversion Algorithm ............................................. 117
Figure 6.3. CAMD Property Clustering Conversion Algorithm .............................................. 119
Figure 6.4. Property Clustering Ternary Mixing Algorithm..................................................... 126
Figure 6.5. CAMbD Candidate Generation Algorithm ............................................................. 130
Figure 6.6. CAMbD test on Pure Components .......................................................................... 131
Figure 6.7. CAMbD test on Binary Mixtures ............................................................................ 132
Figure 6.8. CAMbD test on Ternary+ Mixtures ........................................................................ 133
Figure 6.9. GCM in Property Clusters ...................................................................................... 136
Figure 6.10. CAMD Candidate Generation Algorithm ............................................................ 142
Figure 6.11. CAMD test on Pure Group Molecules ................................................................. 143
Figure 6.12. CAMD test on Binary Group Molecules .............................................................. 144
Figure 6.13. CAMD test on Ternary+ Mixtures ....................................................................... 145
Figure 7.1. Ternary Property Cluster Diagram of a Six Component Mixture .......................... 153
Figure 7.2. Negative Property Clusters in an Eight Component Mixture ................................. 154
Figure 7.3. Negative Property Cluster Domains ....................................................................... 156
Figure 7.4. Property Cluster Reference Comparison for a Three Component Mixture ............ 158
Figure 7.5. aCAMbD Property Clustering Conversion Algorithm ........................................... 166
Figure 7.6. Property Model Comparison Cluster Diagram ....................................................... 172
Figure 7.7. aCAMbD with Cluster Generation Algorithm ........................................................ 173
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Figure 7.8. aCAMbD test on Pure Components ........................................................................ 174
Figure 7.9. aCAMbD test on Binary Mixtures .......................................................................... 175
Figure 7.10. aCAMbD test on Ternary+ Mixtures .................................................................... 176
Figure 7.11. Conventional Mixture Diagrams for a Mixture of Spun Yarn ............................. 184
Figure 7.12. Property Clustering Diagram of Canonical Models of Spun Yarn....................... 187
Figure 7.13. Property Clustering Diagram of Polynomial Models of Spun Yarn .................... 190
Figure 7.14. Model Feasibility Region in Property Cluster Space ........................................... 194
Figure 8.1. Reverse Problem Formulation Problem for Consumer Attributes ......................... 202
Figure 8.2. Overlapping of Groups ........................................................................................... 211
Figure 8.3. aCAMD Property Clustering Conversion Algorithm ............................................. 215
Figure 8.4. aCAMD Candidate Generation Algorithm ............................................................. 225
Figure 8.5. aCAMD test on Pure Group Molecules.................................................................. 226
Figure 8.6. aCAMD test on Binary+ Group Molecules ............................................................ 227
Figure 8.7. Attribute Cluster Diagram of Refrigerant Design .................................................. 236
Figure 8.8. Predicted vs. Actual Attribute Plots ....................................................................... 241
Figure 8.9. Property Cluster Diagram of FR and MFR ............................................................ 253
Figure 8.10. Property Cluster Diagram of pFR and pMFR ...................................................... 254
Figure 8.11. Property Cluster Diagram of Additive FR............................................................ 256
Figure 8.12. Property Cluster Diagram of Additive Groups ..................................................... 264
Figure 8.13. Property Cluster Diagram of Additive Candidate Molecules ............................... 266
Figure 8.14. Predicted vs. Actual Hv Plots................................................................................ 267
Figure 8.15. Predicted vs. Actual Tc and Pc Plots ..................................................................... 268
Figure 9.1. Reverse Problem Formulation for Multiple Scales ................................................ 276
xv
Figure 9.2. Rotated Reverse Problem Formulation for Multiple Scales ................................... 276
Figure 9.3. cGCM Construction Example Highlighting Isobutanol ......................................... 290
Figure 9.4. Raw NIR Absorbances of L-Glutamic Acid .......................................................... 293
Figure 9.5. IR Transmittance and NIR Absorbance of Mannitol ............................................. 300
Figure 9.6. p-Standardized and Centered IR and NIR Responses of the Filler Training Set ... 301
Figure 9.7. Gamma Q-Q plot of filler excipients ...................................................................... 301
Figure 9.8. n,p-Standardized and Centered IR / NIR Responses of the Filler Training Set ..... 306
Figure 9.9. Latent Variable Predicted IR and NIR Responses of Mannitol ............................. 308
Figure 9.10. PCA Loading of the Filler Training Set ............................................................... 315
Figure 9.11. Clustered Loadings of the Filler Training Set ...................................................... 316
Figure 9.12. PCA Scores of the Filler Training Set .................................................................. 319
Figure 9.13. Clustered Scores of the Filler Training Set .......................................................... 320
Figure 9.14. IR Transmittance Methylene Group Response Plot ............................................. 321
Figure 9.15. cCAMD Cluster Conversion Algorithm ............................................................... 324
Figure 9.16. cCAMD Candidate Generation Algorithm ........................................................... 328
Figure 9.17. Scree Plot of Proof of Concept ............................................................................. 335
Figure 9.18. cCAMD Training Set ........................................................................................... 343
Figure A1.1. Decision Analysis Tree ........................................................................................ 382
Figure A1.2. Ontological Relationships for a Pharmaceutical Process .................................... 384
Figure A1.3. Branch-and-Bound Algorithm Example .............................................................. 387
Figure A1.4. Branch-and-Bound Algorithm Work Flow.......................................................... 388
Figure A1.5. Flowsheet of MINLP Optimization ..................................................................... 390
Figure A1.6. Genetic Algorithm Work Flow ............................................................................ 393
xvi
Figure A3.1. Network Component Analysis............................................................................. 406
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List of Tables
Table 2.1. Chemical Product Types ............................................................................................ 15
Table 3.1. Relative Band Intensities for C-H Stretch ................................................................ 59
Table 3.2. Group Structures with First Order Coupling ............................................................ 60
Table 4.1. Designed chemical structures for the partition coefficient ........................................ 77
Table 6.1. Common Physical-Chemical Properties used in Chemical Product Design............ 112
Table 6.2. CAMbD Property Clustering Conversion Algorithm............................................... 116
Table 6.3. CAMD Property Clustering Conversion Algorithm ................................................ 118
Table 6.4. CAMbD Candidate Generation Algorithm using Clusters ....................................... 128
Table 6.5. CAMD Candidate Generation Algorithm using Clusters ........................................ 140
Table 7.1. aCAMbD Property Clustering Conversion Algorithm for Canonical Models ......... 166
Table 7.2. aCAMbD Property Clustering Conversion Algorithm for Polynomial Models ....... 167
Table 7.3. aCAMbD Candidate Generation Algorithm using Clusters ..................................... 177
Table 7.4. Simplex Lattice Design MDOE for Polymer Spun Yarn ........................................ 181
Table 7.5. Scheffe and Cox Model Regression Coefficients .................................................... 182
Table 7.6. Nondimensionalized Property Operators and References ....................................... 185
Table 7.7. Property Clusters and AUP of Canonical Property Operators ................................. 186
Table 7.8. Property Clusters and AUP of Experimental Mixtures ............................................ 186
Table 7.9. Property Clusters and AUP of Feasibility Region .................................................. 187
Table 7.10. Property Clusters and AUP of Standard Reference Property Operators ................ 188
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Table 7.11. Property Clusters and AUP of Polynomial Property Operators ............................. 188
Table 7.12. Property Clusters and AUP of Pseudo Feasibility Region ..................................... 189
Table 7.13. Binary Mixtures that meet Rule 1 and 2 ................................................................ 192
Table 7.14. Ternary Mixtures that meet Rule 1, 2, and 3 ......................................................... 192
Table 8.1. aCAMD Cluster Conversion Procedure .................................................................. 215
Table 8.2. aCAMD Candidate Generation Procedure .............................................................. 223
Table 8.4. List of Potential Refrigerants for the Case Study .................................................... 230
Table 8.5. MDOE for the Case Study ....................................................................................... 232
Table 8.6. MDOE Attribute Responses .................................................................................... 234
Table 8.7. Calculated Property Responses ................................................................................ 238
Table 8.8. Informative Statistics for the Attribute-Property Relationships .............................. 239
Table 8.9. Informative Statistics for the Property-Attribute Relationships .............................. 243
Table 8.10. Bounds on the FR .................................................................................................. 243
Table 8.11. Bounds on the pFR ................................................................................................ 244
Table 8.12. Bounds on the MFR ............................................................................................... 244
Table 8.13. Bounds on the pMFR ............................................................................................. 245
Table 8.14. Pure Component Effects ........................................................................................ 245
Table 8.15. Nondimensional Property Operators of the Training Set ...................................... 246
Table 8.16. Property Clusters and AUP of the Training Set ..................................................... 248
Table 8.17. Nondimensional Property Operators of the Pure Component Effects ................... 249
Table 8.18. Clusters and AUP of the Pure Component Effects ................................................ 250
Table 8.19. Nondimensional Property Operators of the FR, MFR, pFR, and pMFR ............... 250
Table 8.20. Property Cluster bounds on the FR ........................................................................ 251
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Table 8.21. Property Cluster Bounds on the MFR.................................................................... 251
Table 8.22. Property Cluster Bounds on the pFR ..................................................................... 252
Table 8.23. Property Cluster Bounds on the pMFR ................................................................. 252
Table 8.24. Clusters and AUP of the Additive Domain ............................................................ 256
Table 8.25. Nondimensional Property Operators of the Additive FR ...................................... 257
Table 8.26. Property Data of First Order Groups ..................................................................... 258
Table 8.27. Nondimensional Property Operators of First Order Groups .................................. 259
Table 8.28. Maximum No. First Order Groups ........................................................................ 260
Table 8.29. Properties of 2nd Order Groups .............................................................................. 261
Table 8.30. Nondimensional Property Operators of Second Order Groups ............................. 262
Table 8.31. First Order Groups Clusters ................................................................................... 263
Table 8.32. Candidate Molecules.............................................................................................. 265
Table 9.1. Fundamental Vibrations ........................................................................................... 281
Table 9.2. Approximation of Transmittance IR/NIR Active Tones ......................................... 282
Table 9.3. Derived IR Absorption of Methylene Group ........................................................... 285
Table 9.4. Filler Excipients ....................................................................................................... 297
Table 9.5. SIMCA-P PCA Scores for Filler Excipients............................................................ 303
Table 9.6. Matlab and JMP PCA Scores for Filler Excipients ................................................. 305
Table 9.7. CAMD Candidate Generation with Latent Variables .............................................. 329
Table 9.8. CAMD Candidate Excipients .................................................................................. 330
Table 9.9. Property Descriptors ................................................................................................ 332
Table 9.10. Latent Variable Scores ........................................................................................... 333
Table 9.11. Latent Variable Loadings ....................................................................................... 335
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Table 9.12. Design or Experiments ........................................................................................... 337
Table 9.13. Regression Coefficients for Attribute-Latent Property Relationship .................... 338
Table 9.14. Target Attributes .................................................................................................... 338
Table 9.15. Latent Variable Targets ......................................................................................... 339
Table 9.16. Predicted Property Descriptor Targets ................................................................... 339
Table 9.17. Standardized Latent Variable Targets.................................................................... 339
Table 9.18. Groups for Molecular Design of Excipient............................................................ 341
Table 9.19. Designed Molecules ............................................................................................... 344
Table A1.1. MCDM Needs and Wants for PDK Inhibitor ....................................................... 377
Table A1.2. Scoring System for PDK Inhibitor Example ........................................................ 379
Table A1.3. MCDM Ranking for PDK Inhibitor Example ..................................................... 381
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Nomenclature
Latin Symbols
a total number of product attributes
A array of consumer attributes
AL linear algebra constant for x-axis
AUP augmented property index
AUPM augmented property index of mixture
AUPsM standard mixture AUP
AUPzM pseudo mixture AUP
AUPF feasibility region AUP
b latent property target identifier
B array of regressors
BL linear algebra constant for y-axis
c concentration in spectrophotometer
C cluster of component properties
CM cluster of mixture properties
CM cluster of molecular group properties
CiM cluster of molecule properties
CU feasibility upper limit cluster
CL feasibility lower limit cluster
CL cluster of loadings
CQ cluster of scores
CsM standard reference cluster of a mixture
Cz pseudo cluster of a component
C zM pseudo cluster of a mixture
d vector of structural descriptors
D total number of expressions
e design point or experiment identifier
E residual variable array
EY residual of response Y
EP residual of response P
f1 first order group subset identifier
f abs absorbance probability density function
F total number of first order groups
Fc total number of group combinations
F(?) F distribution critical value
FBN free bond number of molecular structure
FOBJ objective function for optimization
xxii
g group identifier (of any order)
g1 first order group identifier
g2 second order group identifier
g3 third order group identifier
G total number of second order groups
GS total number of overlapping groups
Gf standard Gibbs free energy
h Planck?s constant
h1 vector of composition constraints
h2 vector of process constraints
h3 vector of property model constraints
H UNIQUAC surface area parameter
Hf standard enthalpy of formation
Hv standard enthalpy of vaporization
Hfus standard enthalpy of fusion
i chemical constituent identifier
I attenuated energy in a spectrophotometer
Io incident energy in a spectrophotometer
j physical-chemical property identifier
J UNIQUAC volume parameter
k chemical product attribute identifier
K bond force constant
KOW octanol-water coefficient
KL linear algebra intercept constant
KLref linear algebra reference constant
l total number of latent properties
lb beam path length in spectrophotometer
l1 vector of compositional lower limits
l2 vector of architectural lower limits
l3 vector of combinatorial lower limits
L array of latent component loadings
m total number of latent variables
M atomic mass
MX maximum AUP
n total number of experiments
ng total number of groups (or any order)
nq quantum number vibrational integer
N total number of atoms
Nv number of vibrational motions
NRG total number of rings in structure
NR total number negative cluster regions
NT number of types of cluster regions
p total number of properties
P array of physical-chemical properties
Pc critical pressure
PM array of property mixtures
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Pabs spectroscopic absorbance
PT spectroscopic transmittance
q latent property identifier
qi relative molecular surface area
Q2 predicted fit of an empirical model
Q group contribution to vdW surface area
r molecular van der Waals volume
R2 fitness of an empirical model
R group contribution to vdW volume
s standard reference mixture
s2 variance
S tensor of variance and covariance
S summation of latent property loadings
Stot total sum of squares
SSres residual sum of squares
SSPRE predicted residual sum of sqaures
t total number of latent property targets
T array of latent component scores
Tm normal melting point
Tm normal boiling point
Tc critical temperature
Th total number of third order groups
TM array of latent mixture scores
T2 Hoetelling?s T2 distribution statistic
u total number of chemical constituents
u1 vector of compositional upper limits
u2 vector of architectural upper limits
u3 vector of combinatorial upper limits
U array of latent property scores
v1 dissimilar first order group identifier
Vc critical volume
w binary dissimilar constituent identifier
W array of latent property loadings
wS second order group factor
wT third order group factor
x vector of chemical compositions
xi mass fraction of component i
xa anharmonicity shift
X array of chemical constituent fractions
XCC x-axis value in Cartesian coordinates
y independent response variable (A or P)
? mean of independent response variable
Y array of attribute or property responses
YCC y-axis value in Cartesian coordinates
z design model identifier
zM total number of models in design
xxiv
Nomenclature
Greek Symbols
?? total number of atoms
?o polynomial reference regressor
?i polynomial regressor of component i
?ii nonlinear polynomial regressor of i
?iw polynomial regressor of interaction iw
??i canonical regressor of component i
??iw canonical regressor of interaction iw
?? cluster mixing fraction
E? band energy
? absorptivity
?? probability
? number of factors in a design?
??C? pure species activity coefficient
?? ternary dissimilar constituent identifier
?? total number of responses (a or p)
? linear constant??
?? wavelength
?bar? peak wavelength
?f number of overlapped 2nd order groups
?s number of overlapped 2nd order groups
?? mixture level identifier
?g group mixture level identifier
?max? maximum mixture type
? density of component
?M density of mixture
?? degree of fractionation
?? atom identifier
? integer of vibrational quantum number
v vibrational frequency
vo vibrational overtone frequency
? normalized property operator
?? normalized property operator of mixture
?ref normalized reference property operator
?Ms normalized property operator of standard
xxv
??z normalized operator of pseudo mixture
? property operator
?? property operator of mixture
?ref reference property operator
1
Chapter 1
Introduction to Process Systems Engineering
The focus of this dissertation is on the development of novel process system engineering
(PSE) techniques for the design of chemical products having molecular architectures that exist in
multiple length scales. PSE is generally defined as the research and development of systematic
procedures for the design and operation of chemical process systems, from the molecular length
scale through the supply-chain mega-length scale (Grossman and Westerberg 2000). It was first
described in a Special Volume of the AIChE Symposium Series in 1961 and is primarily focused
on the design and optimization of macro-scale process systems (Grossman and Westerberg
2000). The heart of PSE research is the discovery of concepts and models for the
computationally efficient prediction of the performance of a system of unit operations. Unit
operations (e.g. fluid flow, heat and mass transfer, thermodynamic phase behavior, reactions,
etc.) are basic steps in a process described by universal physical laws and may be used across all
chemical industries. It is common for a single process to contain multiple processing steps, and
each processing step to include multiple unit operations, resulting in a highly complex system.
As a result, PSE solution techniques demand the use of software simulators (e.g. AspenONE?,
ProMax?, etc.) and/or mathematical solvers (e.g. GAMS?, MatLab?, etc.). In the chemical
engineering curriculum, PSE concepts are usually introduced as a senior capstone course that
pulls together the unit operation topics covered in earlier courses, making it a cornerstone of
chemical engineering.
2
Recently, many of the industries that employ chemical engineers have experienced
increased global competition which has placed considerable pressure on their profit margins. For
a number of companies it has become apparent that it is no longer sustainable to manufacture
only commodity chemicals, defined as chemicals produced in bulk, having minimum variation
between manufacturers and simple molecular architecture. Traditionally, because of the
uniformity of commodity chemical products, any gains in profit margin were due to linear
improvements in process configurations and production volume. However, with the increase in
global competition, linear improvements in process design are not quick enough or large enough
to hold on to market share. As a result, successful manufacturers are making a nonlinear shift in
focus from process optimization to product innovation by diversifying their product portfolios to
include more specialty chemical products. Specialty chemicals are produced in small quantities,
having simple or complex molecular architecture and large variation between manufacturers.
Examples of specialty chemicals include pharmaceutical ingredients, biocides, intermediates, and
many other types of specialty market products. By shifting focus to manufacture specialty
chemical products for niche markets, manufacturers can dominate market share, drive up prices
for their products, and offset pressure from global competition. Further analysis of the
emergence of specialty chemical products is provided in Section 2.1 and Section 2.2.
One of the interesting aspects of the shift toward specialty chemical products is that the
relationship between the molecular architecture of a specialty chemical and its physical and
chemical properties is often more complex than for a commodity chemical. Molecular
architecture, by definition, includes information related to atoms and bonds at the molecular
length scale as well as nanoscale, microscale, mesoscale, and macroscale structure information
related to aspect ratio, crystallinity, morphology, and many more orientation specific
3
arrangements of chemical structure. A discussion of the length scales and their relationship to
specific molecular architectures of interest to chemical engineers is presented in Section 2.3.
Since the unique combination of the molecular architectural arrangements is often what
gives a specialty chemical its unique set of properties, developing a thorough understanding of
this relationship is vital. Research in this area has traditionally been performed by the natural
sciences, but a compressed time-to-market schedule has brought it under the domain of chemical
engineers who seek to integrate it with conventional chemical engineering concepts. As a result,
research in chemical engineering has broadened to include the quantification of the relationships
between physical-chemical properties and molecular architecture. Simultaneously, the scope of
research in the PSE community has broadened to include the development of design tools meant
to utilize newly developed length dependent structure-property relationships. Unlike the design
of commodity chemicals, which have a known molecular architecture and limited raw material
options from which to build an optimum process configuration, the design of specialty chemical
products does not have defined molecular architectures or raw material sources. Instead,
specialty chemicals rely on a set of customer needs or consumer attributes to define the chemical
product. Developing techniques to discover and optimize the molecular architecture that delivers
these attributes is the focus of research in the PSE community and is discussed in more detail in
Section 2.4.
Depending on the nature of the molecular architecture, physical-chemical properties,
and/or consumer attributes, the design techniques employed can be largely experimental or
computational in nature. Experimental techniques are preferred when the molecular architecture-
property relationship is poorly understood, the solution is in a validation stage, or high
4
computational costs preclude the use of computer aided methods. More detail on experimental
design techniques and their development in the PSE community is presented in Chapter 3.
Alternatively, computational design techniques are preferred when the molecular-
architecture-property relationship is well defined, the design is in an exploratory stage, or
computational costs are reasonable. The PSE developed mathematical formulation of this
problem is presented in Section 4.1 and some useful property models are discussed in Section
4.2. Two recently developed computational design techniques, computer aided mixture design
(CAMbD) and computer aided molecular design (CAMD), are presented in Section 4.3 and
Section 4.4, respectively.
Beyond the two design approaches detailed in Chapter 3 and Chapter 4, PSE approaches
to product design and development have proceeded in one of three directions: (1) the application
of macroscale systems techniques to other length scales, (2) the development of new techniques
specific to non-macroscale domains, and (3) the development of tools capable of communicating
molecular architecture and property information between scales. Many authors have conducted
research in the extension of PSE techniques to other length scales, such as the work on the
supply chain at the megascale by Grossman and Westerberg (2000) and the extension of the
conventional PSE formulation to the molecular scale by Gani (2004). Others have focused on
developing techniques specific to a particular scale, such as the prediction of crystalline structure
(Karamertzanis et al. 2009; Doherty and Green 2004) and protein structure (McAllister and
Floudas 2008) at the microscale. The third area involving the development of multiscale design
methods has only recently come under focus of the PSE community and it is in this area that a
significant contribution has been made in this dissertation.
5
In Chapter 5, the motivation for the development of a PSE technique to aid in the design
of products with molecular architecture existing at multiple scales is presented. Most authors,
such as Fermeglia and Pricl (2009), have approached this multiscale design problem in a similar
manner as the natural sciences, computing information at smaller length scales and passing it to
models at larger length scales by removing degrees of freedom (coarse-graining). While often
the most accurate method for predicting properties, this simulated approach has two limitations:
(1) it has an immense computational cost due to hierarchical nesting and (2) it utilizes a priori
knowledge of the molecular architecture (i.e. the number and types of atoms or electrons
present). The large computational cost typically prevents an accurate modeling of mesoscopic
structure such as the morphology of polymers without the use of constraints that significantly
limit the degrees of freedom in the simulation (Fermeglia and Pricl 2009). Furthermore, when
this method is integrated within the product-process design framework, the computational
intensiveness exponentially increases yet again since each projected molecular architecture must
be simulated to determine its physical-chemical properties (Hill 2004). To minimize the
computational cost in these types of problems, the property prediction simulations are typically
approximated with constitutive equations based on structure descriptor models such as property
based group contribution (GCM), quantitative structure property/activity relationships
(QSPR/QSAR) using topological indices (TI), or chemometric based models. The introduction of
structure descriptor models improves computational efficiency by avoiding nested simulation at
the cost of prediction accuracy. However, these structure descriptor models are still
computationally expensive when utilized in design algorithms because of their often highly
nonlinear nature. Finding a method that addresses these limitations is a challenge and is
discussed in more detail in Chapter 5.
6
Chapter 6 introduces a framework that addresses the multi-scale chemical product design
challenges illustrated in Chapter 5. In particular, Gani (2004) and Eden et al. (2004) have shown
that the reverse problem formulation (RPF) can be used to circumvent computational limitations
by decoupling the constitutive equations from the constraint and balance equations to identify the
property solution domain without committing to any components a priori (Gani and
Pistikopoulos 2002; Eden et al. 2003; Gani 2004). The improvement in computational efficiency
is achieved by only solving the systems of equations for chemical products with molecular
architecture that reside in this property solution domain and is discussed in more detail in Section
6.1.
In addition to the reverse problem formulation, the framework proposed in this
dissertation also makes use of the property clustering algorithm. The property clustering
algorithm transforms physical-chemical properties into conserved surrogate property clusters that
can be described by linear mixing rules, even if the operators themselves are nonlinear. At the
heart of the algorithm is the use of transformation functions to linearize the expression for the
physical-chemical property ? molecular architecture relationship so that the duality of linear
programming can be applied to solve the design problem in the lower dimensional property
domain instead of the high dimensional molecular architecture space. A detailed discussion on
how the cluster transformation algorithm works is presented in Section 6.2. Algorithms for
conducting CAMbD and CAMD with property clusters are presented in Sections 6.3 and 6.4,
respectively.
One of the most important aspects in the design of specialty chemical products is the
recognition that the effectiveness of a chemical product is determined by how well it meets the
needs of the customer or consumer, not by molecular architecture or physical or chemical
7
properties. These needs are most often parameterized as consumer attributes and quantified
using consumer preference experiments. As a result, consumer attributes tend to be poorly
defined in terms of physical and chemical properties, making the design of chemical products
using conventional approaches a difficult task. To alleviate this issue, this dissertation utilizes
experimental data, parameters, and models in the design of specialty chemical products with
molecular architecture that exists at multiple scales. The decision to use experimental data does
not, however, preclude the use of more common physical and chemical properties. While the use
of data places constraints on the application range of the solutions, it does not prevent the
developed framework from being applied to systems described by solely deterministic
relationships discussed in Appendix A4. Further discussion of data driven design of chemical
products is presented in Chapter 6.
Three methods to address the current limitations of multiscale chemical product design of
specialty chemicals based on consumer attributes are presented in this dissertation: (1) attribute
based computer aided mixture (blend) design (aCAMbD), (2) attribute computer aided molecular
design (aCAMD), and (3) characterization based computer aided molecular design (cCAMD).
Each of the developed methods utilizes the reverse problem formulation, property clustering, and
consumer attribute data. Chapter 7 discusses the development of the aCAMbD method to
address situations where the attributes of interest are not conventional physical-chemical
properties and mixtures of chemical constituents with different molecular architectures will meet
the requirements of the chemical product. A case study on spun yarn of polymer filaments for
marine applications is presented to illustrate the method.
In situations where mixture design will not suffice, but attributes remain the drivers of
product performance, molecular design methodologies are utilized. Chapter 8 presents an
8
aCAMD method that first converts attribute data to traditional physical-chemical properties
described by well-defined molecular architecture models, and then generates candidate solutions
using a cluster-based molecular design technique. The benefit of this approach is that chemical
products not present in the original training data set can be generated so long as they contain the
same, underlying molecular architecture as the training data set. A case study design for an
environmentally benign refrigerant is used to illustrate the method.
In the previously developed aCAMD method, attribute data was converted to physical-
chemical properties described by group contribution method (GCM) based property models that
ensured easy computation. However, GCM property models are limited to describing small
molecules (< 30 atoms) with structural isomerism. Chemmangattuvalappil et al. (2010)
developed a method for more complex molecular architectures using molecular signatures to
convert QSPR/QSAR models based on topological indices into group based parameters.
However, the method is constrained by the limitations of the QSPR/QSAR models, which
typically only describe molecular length scales (< 60 atoms) and occasionally macromolecular
length scales (< 60 atoms in the repeat unit of a polymer). QSPR/QSAR models describing
physical-chemical property ? molecular architecture relationships at other length scales or at
multiple length scales are usually unavailable since the physics in these domains are still being
investigated. As a result, researchers designing chemical products at these scales have relied on
data-driven approaches based on chemical characterization.
Chemical characterization consists of a variety of techniques using spectroscopic,
diffraction, and/or thermophysical techniques to determine the molecular architecture of a
chemical product. One of the most common techniques is infrared spectroscopy (IR), which is
capable of describing molecular architecture at multiple length scales. In Chapter 9, a
9
characterization based computer aided molecular design (cCAMD) framework is presented.
Using a chemical product training set, IR/NIR spectroscopic data on the molecular architecture is
first captured, and then converted to underlying latent variables using decomposition techniques
like principal component analysis (PCA). Next, an experimental design is conducted to
determine the effect of the latent variable structure on the consumer attributes of interest. Target
consumer attributes are then mapped down to the latent variable domain and mixture design of
the original training set is conducted. The same latent variable structure can be used to find new
chemical products that were not part of the original training set by using a method developed in
this dissertation called characterization based computer aided molecular design (cCAMD).
First, recognizing that the absorbance measured by IR/NIR spectroscopy are based on group
theory interactions with the incident radiation, group contributions based on characterization data
are estimated using a newly developed characterization based group contribution method
(cGCM). Characterization based groups are then converted to the same latent variable basis as
the original training set and recombined to find new chemical products. The primary benefit of
this approach is that the individual group contributions incorporate information from multiple
length scales, decoupling the length scales, and circumventing the need for nested simulation
typically found in multiscale design. A case study on the design of pharmaceutical excipients for
direct compressed acetaminophen tablets using data provided by Gabrielsson et al. (2003) is
presented in Chapter 9 to highlight the method.
This dissertation concludes with Chapter 10, where the achievements of this dissertation
are summarized and future challenges are discussed. Special attention is paid to the
contributions made by this author to the computer aided, data-driven design of chemical products
using consumer attributes and characterization based approaches. Additionally, it should be
10
noted that the aCAMbD approach described in Chapter 7 has been published in Industrial and
Engineering Chemistry Research (Solvason et al. 2008), the aCAMD approach described in
Chapter 8 has been published in Computers and Chemical Engineering (Solvason et al. 2008),
and aspects of the cCAMD approach described in Chapter 9 has been published in parts in
Computer Aided Chemical Engineering (Solvason et al. 2008, Solvason et al. 2009, Solvason et
al. 2010). Other case studies performed by the author, but not presented in this dissertation,
include the design of product pathways for the forest bio-refinery, published in Integrated
Biorefineries: Design, Analysis, and Optimization (Batsy et al 2011) and the design of self-
assembling monolayers for chemical deposition.
11
Chapter 2
Product Design for Chemical Engineers
Product design, which is defined as the conversion of a conceptual idea into a tangible,
manufactured object, is an important function of today's corporations. Increasing competition,
fluctuating markets, and global supply chains have all worked to place high importance on
delivering differentiated products to the consumer. Although consumer driven product design
has always been appreciated as a major issue in the mechanical and electrical engineering
sectors, in the chemical, pharmaceutical, and materials industries the systematic and efficient
design of new products is a recent concern (Costa et al. 2006). Furthermore, consumer attributes
are being pushed further up the development pipeline to the formulation scientists who are ill-
equipped at utilizing this new source of information. Compounding the issue are the unique
features of chemically formulated products that control the product?s properties, namely the
mechanisms that operate at a multitude of nested length scales. The effective utilization of this
immense data structure to design chemical products is the subject of cutting edge research in the
process systems engineering field and often referred to as an emerging paradigm within chemical
engineering (Costa et 2006).
2.1. An Emerging Paradigm
?Unit Operations? is considered the first and most recognized paradigm in chemical
engineering. Beginning in 1915, the concept of unit operations transformed the chemical
12
engineering discipline from a study of how to manufacture a specific commodity, to the
systematic design of a sequence of unit operations that is applicable across all commodities
(Wei, 1996). Although extremely useful, the unit operations paradigm proved inadequate for
solving problems with increased molecular complexity (Hill 2009). As a result, a second
paradigm involving the study of chemical science arose in 1950, best exemplified by Bird,
Stewart, and Lightfoot?s textbook on Transport Phenomena (Bird et al. 2002). It is focused on
developing universal mathematical systems to describe the underlying, continuum based,
chemical and physical sciences in order to solve relevant problems of interest (Hill 2009). This
is the paradigm under which most chemical engineering research is conducted today and,
although the discipline has moved into the biological sciences realm and new systems of interest
have been incorporated, the mindset has remained the same; i.e. the identification of underlying
phenomena, expression in a mathematical relationship, and application to a system of interest
(Hill 2009).
Process systems engineering (PSE) research under these two paradigms of chemical
engineering involves combining a broad knowledge of chemical products, property models,
scientific experimentation, and process simulation to design efficient manufacturing processes.
The usual design objective is to minimize the cost of manufacture of a product after its molecular
architecture has been developed by chemists; an approach that was originally developed for
commodity chemicals with simple molecular architecture produced in bulk quantities. However,
as manufacturers have shifted to developing specialty chemical products to combat global
pricing pressure, it is increasingly clear that this approach needs refinement. Chemical products
are no longer simple mixtures of chemicals, but rather complex molecular structures with
elaborate consumer interfaces that may exist at multiple length scales. This complexity means
13
that a sequential design approach using chemical and physical properties dependent upon both
chemical engineering science and unit operations (i.e. product design, then process design) is
likely to miss potential optimal configurations.
Developing systematic approaches that integrate the development chemistry with
manufacturing constraints has been suggested by the Committee on Chemical Engineering
Frontiers in 1988 as the third chemical engineering paradigm (Hill 2009). This new paradigm
does not preclude other paradigms from emerging, or even displace the previous paradigms of
unit operations and chemical engineering science, but rather it is a necessary result from
breakthroughs in molecular modeling, signal processing, and increasingly powerful
computational tools (Charpentier 2002). Harnessing the immense power of these new tools now
allows chemical engineers to solve entirely new and important classes of problems previously
unimaginable (Hill 2009). Unlike conventional process systems engineering (PSE) approaches
that are focused on the design and optimization of chemical processes based on a priori
knowledge of the products, it is now possible to investigate chemical product formulations prior
to experimentation and simulation of their manufacturing processes (Hill 2009). This change in
the ?way of thinking? about solving chemical engineering design problems represents the
beginning of a new direction in the field (Seider et al. 2009).
The emergence of chemical product design can also trace its roots in the United States
back to the 1980?s. In response to rising overseas competition from Japan in the manufacturing
sector, the U.S. manufacturing base experienced a paradigm shift toward holistic design practices
inspired by the Japanese concept of Musubi (Boznak 1993). Musubi literally refers to ?the
mutual bond of relation that makes growth possible? (Boznak 1993). In practice it shifted the
focus of research from improving attributes of individual features that would be assembled in to
14
a product, to one that optimized the final product attributes regardless of its effect on individual
features. In other words, using definitions associated more with business than engineering,
chemical product design shifted from a technology-push function to a market-pull function.
Technology-push occurs when a new technology is created and introduced to the market where
no market demand exists. It operates under the assumption that the reason no consumer demand
exists is because consumers are unaware of what is possible. Thus, simply introducing the
product should create the demand. Conversely, market-pull refers to using the consumer needs
and wants to drive the product development and by extension the technology development.
Examples of technology-push achievements in the field of chemical engineering are new
advances in polymers, drug-delivery mechanisms, and catalyst structures at the molecular- and
meso-scales and crystallizers, distillation columns, and separators at the macro-scale. In
addition, much work is at the micro- and nano-scales, specifically exploring the unique
chemistry and physics occurring at these levels. Example of market-pull products in chemical
engineering products include teeth whitening strips, post-it notes, environmentally benign
refrigerants, and targeted designer drugs. In many instances it can be argued that the emergence
of product design as a distinct paradigm in chemical engineering is partly due to the growth of a
market-pull focus within the chemical engineering field.
Finally, the emergence of chemical product design in chemical engineering can also be
attributed to the acceleration of the product development cycle (Costa et al. 2006). The increase
in speed has put strain on the product developer?s ability to maintain stability in the design
process, which is key to preventing runaway costs (Boznak 1993). As a consequence,
researchers have increasingly turned to computational tools to evaluate chemical products before
experimentation, which gives flexibility to handling changing design constraints early in the
15
development process. The effectiveness of these tools can be increased by constraining the
molecular architecture options evaluated. It is best achieved by first classifying chemical
products by their molecular architecture, chemical and physical properties, and/or scale specific
mechanisms.
2.2. Types of Chemical Products
Many options exist for classifying chemical product types. The products can be defined
by their end use, by the type of raw materials used to produce them, by molecular structure, or
even by the types of processing equipment needed to create them. One possibility is to define
products by their physical forms. Wibowo and Ng (2002) proposes the following classification
scheme:
Table 2.1: Chemical Product Types Classified by Physical Form (Wibowo and Ng 2001;
Wibowo and Ng 2002).
Physical Form Product Form Examples
Solid
Shaped
Composites
Capsules
Tablets
Solid Foams
Bar of soap
Whale oil capsule
Aspirin Tablet
Styrofoam
Bulk Powders Granules Powdered Detergent Pharmaceutical Excipient
Semi-Solid Pastes Creams Toothpaste Sunscreen
Liquid
Liquid Foams
Macromolecular Solutions
Microemulsions
Dilute emulsions and
suspensions
Solutions
Shaving Foam
Dishwashing Liquid
Hair Conditioner
Writing Ink
Perfume
Gas Aerosols Hair Spray
16
Although a chemical's physical form can differentiate product types, the molecular structure
between different forms can be identical. For instance, xylitol is both a pharmaceutical excipient
in a solid dosage form as well as a potential biochemical platform product in liquid chemical
form. The difference is the presence of strong intramolecular interactions overcoming the
kinetics of the molecular motion that creates long-range order in the solid dosage form.
An alternative classification was proposed by Westerberg and Subrahmanian (2000)
which listed three different classes of products based on the dominating characteristic:
1. Products which are chemicals, such as pharmaceuticals, cleaning fluids, proteins, and
lubricants.
2. Products requiring chemical manufacturing techniques such as semiconductors,
microfluidics, and other etching and layering technologies.
3. Products utilizing chemical properties in their functionality, such as lenses that
change color in light, or a tape that sticks to surfaces, but can be removed without
leaving a residue.
Costa et al. (2006) extends this classification scheme to include 6 categories: (1) specialty
chemicals, (2) formulated products, (3) bio-based concepts, (4), transformational devices, (5)
virtual chemical products, and (6) technology based consumer goods. Of particular note are the
separation of functional techniques into devices that carry out a physical transformation and
consumer products whose functionality is dependent on process technology. Likewise, two new
categories are introduced: bio-based concepts and virtual chemical products. Bio-based
concepts represent the recent growth of chemical engineering into the structural biology fields of
genomics, proteomics, drug design, and membrane design. Virtual chemical products represent
the software advances for chemical engineers including process design tools like ProMax?,
17
product design tools like ICAS?, and various open-source codes and databases like
ChemSpider? and KEGG?.
Conversely, Gani (2004) classifies products based on the length-scale in which the
dominant product attribute is exhibited:
1. Structured products where the micro-structural properties are related to product
functions (e.g. polymer membranes, drug delivery mechanisms, etc.).
2. Chemical, agrochemical, biochemical products where the corresponding processing
routes in the macro-scale play an important role (e.g. environmentally benign solvents
and refrigerants, etc.).
3. Formulations where the properties are enhanced by the addition of other chemical
products (e.g. solvent blends, coatings, etc.)
The use of scales to define product categories is unique, yet appropriate since the majority of
products have properties that are dependent on the physical structures at these scales. Since
these structures are typically interrelated, an understanding of how these scales are linked is
warranted.
2.3. Chemical Product Scales
Although Gani (2004) only discusses 3 length scales in his paper, many other scales exist.
In biochemistry the scales are organized in levels with increasing complexity beginning with the
simplest form of biology, the gene, and ending with the most complicated, the biological
production plant (Fig. 2.1). It has been shown that this multi-scale approach to biochemical
design can be used to better understand and control biological tools such as enzymes and
microorganisms for structured products (Charpentier 2009). Understanding the smaller scales is
18
becoming an intricate part of designing new products. As such, the inclusion of scales smaller
than the nano-scale will also be vital to new product design. For example, the physics governing
the nano- and micro-scales is often quite different from that of the macro-scale. Whereas simple
chemical mixtures have been used to meet product demands in the past, now designer
environmentally benign materials can be built with delivery mechanisms made possible through
functionalization and self-assembly at the smaller scales. As research continues in these areas,
new tools for modifying benign chemicals will be developed with multiple levels of intricacies
which ?offer the possibility to simultaneously engineer molecular, microscopic, and
macroscopic, material characteristics? (Charpentier 2009). The ultimate goal is to efficiently
identify alternative product formulations with as good or better attributes as current state of the
art chemical products.
Figure 2.1. Multi-length Scale Biochemistry Approach to Product Design (Charpentier 2009).
For chemical product design and engineering, Costa et al. (2006) suggests that products
be classified using the nano-scale, micro-scale, meso-scale, macro-scale, and mega-scales as
shown in Fig. 2.2.
19
Figure 2.2. An Example Multi-length and Time Scale (Edwards 2007; Costa et al. 2006).
Here the classes are arranged in order of integrated length and time scales described by
increasing molecular architecture complexity. In terms of the length scale, molecular surface
and interaction energies are described at the nano-scale, long range order such as crystallinity at
the micro- and meso-scales, processing characteristics at the macro-scale, and logistics, life cycle
analysis, and consumer preference at the mega-scale. Conspicuously missing from the
description offered by Costa et al. (2006) are length scales describing molecules. In terms of the
time scales, the molecular scale is based on motion: atoms in a molecule during a reaction are
measured in picoseconds and molecular vibration is measured in nanoseconds (Charpentier
2009). In this scale the importance of the molecular shape, intra- and inter-molecular forces are
controlled by functional groups and their orientations. The functional groups and geometries are
determined using statistical averages based on ab initio calculations at the quantum scale. For
example, during reactions, the orientations of functional groups change considerably as
molecules are pulled apart, stretched, and rearranged into new molecules. Even in a pure,
thermodynamically stable state, the atoms comprising molecules are undergoing constant motion
and rearrangement. The optimum, lowest energy position, is statistically the most common. It is
generally accepted as the ?normal? shape and orientation. However, in many cases the less
20
probable shape is precisely what is needed for a particular product. Therefore, it is important for
researchers to include these possibilities when conducting designing new products.
An attempt at a complete, multi-scale simulation figure detailing the smaller length and
time scales based on common computational design approaches is given in Fig. 2.3. Of note, the
nanoscale, microscale, and mesoscale have been grouped together under the molecular
information demarcation because of the similarities in the computational models between them.
Figure 2.3: The Multiple Length and Time Scales Associated with Computational Simulations.
Alternatively, Grossmann and Westerberg (2000) proposed a product based
representation based on the molecular architecture of the product, as shown in Fig. 2.4. This
figure contains most of the product types typically associated with chemical engineering, such as
the macro-scale unit operation and process design. However, it is increasingly clear that
chemical engineering research and application will move into the smaller chemical scales.
Organizing the complexity levels of the nanoscale, microscale, and mesoscale, including the
21
phenomenological relationships translating molecular architecture to control the functionality of
products through process engineering will become vital within the discipline (Charpentier,
2005).
Figure 2.4: Multi-scale Overview of the Chemical Product Supply Chain Demonstrating the
Potential Chemical Products Produced (Grossmann and Westerberg 2000).
2.4. Other Definitions
In the following chapters, a multiscale chemical product design approach will be
presented. Some of the terms used are defined below:
? Molecular Architecture: A chemical product?s molecular architecture includes
information related to atoms and bonds at the molecular length scale as well as
nanoscale, microscale, mesoscale, and macroscale structure information related to
aspect ratio, crystallinity, morphology, and many more orientation specific
arrangements of chemical structure.
22
? Physical-chemical Properties: Any measurable property that describes either the
physical?s systems state (physical property) or its change during an interaction or
reaction (chemical property).
? Consumer Needs, Preferences, or Product Attributes: A set of consumer desirable
product functionalities and properties that are difficult to quantify physically, such as
smell, feel, or product indices, etc.
? Commodities or Commodity Chemicals: Commodities have a simple, well-established
molecular architecture, are produced in bulk and have minimum variation between
manufacturers.
? Formulations, Specialty Chemicals, Formulated Chemical Products: Consumer
oriented specialty chemicals are made from several ingredients, produced in small
quantities, have simple or complex molecular architecture, and generally have
significant variation between manufacturers.
? Chemical Product Design: Chemical product design is the conversion process of a
conceptual idea into a tangible, manufactured object with a defined molecular
architecture.
? Design Space, Target Region: The design space is the physical-chemical property or
attribute domain in which candidate molecular architectures will be investigated.
2.5. General Design Approach
In chemical engineering, product design describes the design of chemical products and
may include high performance chemicals, formulated pharmaceuticals, semi-conductors,
household products, beauty or personal care products, and processed foods (Hill 2009).
Particularly, chemical product design involves the development of a product formulation, whose
23
structure is in the range of 0.1-100 ?m, complete with a set of specifications for both the product
and manufacturing process (Costa et al. 2006). It often encompasses process design, which can
be thought of as a subset of the product-design activity (Seider et al. 2009). However, product
design differs from the more traditional process design in several key areas (Westerberg and
Subrahmanian 2000). First, products have short life cycles (Westerberg and Subrahmanian
2000), measured in months as opposed to years for processes. Second, the minimization of
concept-to-market time can be much more important than the design of the most economical
solution. Westerberg and Subrahmanian (2000) quotes that being first to market often leads to
capturing and holding of a 70% of market share. My own experiences at the consumer product
company Kimberly-Clark suggest this number is closer to 55-60% of the market, which is still
significant. This shift in focus for designers in chemical engineering has generated a need for
additional skill sets and capabilities.
The design and development methods currently used in product design consist of a
mixture of talents and techniques. These techniques can be anything from loose guidelines that
guide the flow of information to intricate computational logic programs that optimize specific
molecular architectures for a consumer attribute. Westerberg and Subrahmanian (2000),
suggest chemical product design is a mixture of tools from business, fine arts, social sciences,
and traditional chemical engineering. Business tools, in particular, have a large application area
in product design. Anything from deciding which raw material to buy to deciding which product
to make are primarily business decisions that are part of an enormous supply chain optimization
problem (Westerberg and Subrahmanian 2000). These decisions ultimately decide whether or
not the product will be financially viable to manufacture. Whether or not it will sell is impacted
24
by the consumer side of the equation, which is dominated by the tools from the fine arts and
social sciences such as understanding the appeal of color and shape.
In most instances, consumer appeal is the primary driver of performance in the
marketplace. Consumer appeal is not only delivering a product with desirable attributes, which
is often the sole focus in engineering, but is also the delivery mechanism itself, often called the
consumer interface. For example, in 2001 Kimberly-Clark developed a new bath tissue product
called Roll Wipes?. The product was a traditional bathroom roll with the addition of an applied
anti-bacterial wash, resulting in a ?wet? tissue. Consumer research suggested there was a
considerable market demand for the product. It was to be a breakthrough for the company. It
was not. In fact, by most accounts, it was an abject failure. The reason was not the product
itself, but because of the consumer interface. In all of the focus group testing of potential
customers, the wet rolls were tested sans the product delivery mechanism that defined the
product interface. After introducing the product to the marketplace, the overwhelming feedback
from consumers revolved around frustration with the appearance and function of the delivery
mechanism. Unfortunately, the delivery mechanism had been designed to match the constraints
of the previously optimized product. Any change to the interface would mean a complete
redesign of the product. Without significant changes, sales of the product reached less than one
third of its projections and the product was canceled. Clearly, the importance of fine arts tools to
define the consumer interface and the understanding of the interactions of the consumers is of
paramount importance.
Last, but certainly not least, there is no substitute for the core competencies in chemical
engineering. These tools are in essence the ?chemical? part of chemical product design. Gani
(2004) stresses that in chemical product design, the final product is unknown, but there are some
25
general ideas how it should behave and that the problem is to ?find the most appropriate
chemicals that will exhibit and/or cause the desired behavior.? The definition of these chemical-
property relationships is where chemical engineers and chemists have developed strong skill-
sets, especially for fluids. However, over 50% of chemical products produced are in the solid
form (Costa et al. 2006), so skills in engineering chemical solids are also important. Clearly, as
chemical product design grows as a discipline within chemical engineering, it will need to
include all of the aspects of product design, especially those associated with the solid form.
The approaches to designing chemical products can be as varied as the products
themselves. However, the overall goal remains the same, to deliver the most efficient chemical
product that presents a set of attributes important to the customer. As such, the approaches to
designing chemical products involve the identification and transfer of pertinent information from
the customer to the product?s molecular architecture and vice versa. The approaches can be
classified as either non-linear or linear product design.
Non-linear product design is often defined as quantum leap innovation that is unexpected
and unpredictable, typically resulting from advances in technology (Boznak 1993). For example,
Schmid and Smith (2002) uses Sir Alexander Fleming to illustrate that a random connection of
research on the influenza virus in 1928 with ?contaminated? staphylococcus cultures exhibiting
stunted growth, resulted in a miracle breakthrough, penicillin. Whereas over 50 years earlier
John Tyndall had described the antibacterial properties of mold, it was Fleming who made the
non-linear leap and opened ?the gate for the discovery of antibiotics? (Schmid and Smith 2002).
Unfortunately, as vital as these non-linear breakthroughs are to the advancement of science, they
are also rare. As a result, most practical product design is often focused on only incremental or
linear advancement of products.
26
One type of product design that uses linear transformations is presented by Westerberg
and Subrahmanian (2000) and Dym and Little (2008). In this approach, product information
passes from a set of market-based goals to a set of consumer attributes controlled by fundamental
properties which are in turn controlled by molecular architecture. Selecting the most promising
molecular architecture is a function of both its attributes and the manufacturing process used to
make it. Cussler and Moggridge (2001) express this type of product design in a more objective
manner by identifying four steps that capture the product design process:
1. Needs. What needs should the product fulfill?
2. Ideas. What different products could satisfy these needs?
3. Selection. Which ideas are the most promising?
4. Manufacture. How can we make the product in commercial quantities?
Cooper and Edgett (2009) further develop the product design process to include 6 business
actions that emphasize interaction with the consumers:
1. Discovery. What new opportunities exist?
2. Scoping. What is the market for these opportunities?
3. Build Business Case. What are the potential product definitions and justifications?
4. Development. What is the actual product design and its business plans?
5. Validation. How can the product and is customer acceptance be validated?
6. Launch. Full commercialization of the product.
The first 3 steps listed by Cooper and Edgett are data gathering steps used in steps 4 and 5,
Development and Validation, respectively (Cooper and Edgett 2009). The four of the steps listed
27
by Cussler and Moggridge would fall into the Build Business Case, Development, and
Validation steps listed by Cooper and Edgett. Hence, the development of chemical products and
their validation form a major part of the design process. In fact, it is such a major part, the
tendency is to include all available information and design possibilities for the product under the
false pretense that excluding some product goals could eliminate the optimum product. To
mitigate overwhelming each successive step in the product design process with extraneous
information, a series of gates containing criteria to measure project deliverables, are used. These
criteria can be of the multi-criteria decision making (MCDM) variety, where weighted scores are
assigned to the project, or they can be more explicit such as physical cost of manufacture (Parker
and Moseley 2008). More details on MCDM is available in Appendix A1.
Combining the development steps with the screening stages, results in what is known as
the ?Stage-Gate? process for linear product design (Cooper and Edgett 2009). Shown in Fig. 2.5
is the Stage-Gate product development process as outlined by Cussler et al. (2010), Cooper and
Edgett (2009), and Seider et al. (2009). The process contains a successive series of stages, where
ideas and solutions are generated, followed by gates, where the solutions are screened to find
those with the highest potential. Seider et al. (2009) lists 5 stages of which the first three are
collectively referred to as chemical product design: concept, feasibility, development,
manufacturing, and product introduction.
28
Figure 2.5: The Stage-Gate Product Development Process (Seider et al., 2009).
The concept stage encompasses the initial data gathering for the product, including
attributes from the consumer, potential chemical structures to deliver those attributes, and a
preliminary process synthesis route for their production. The concepts are then screened in the
first gate to find those solutions with the most potential. Product designs that are chosen to enter
the feasibility stage are then prototyped and tested against consumer requirements from which a
complete business model is built including a detailed market analysis encompassing risk
assessment and health, safety, and environmental concerns (Seider et al., 2009). In addition, a
preliminary set of technology based production processes needed to make the product are
synthesized. The overall feasibility of each process is then screened again and those with a
29
combination of a large potential return and small risk are passed forward to development. The
development phase is where detailed design, profitability analysis, and optimization are carried
out. The goal of this stage is to develop controls of the variability in the product using a set of
large production runs in pilot-plants that could then be tested en masse in the marketplace.
Those products with properties that can be controlled well are then passed forward to
manufacturing and the further design of the product ceases (Seider et al., 2009). The final two
stages encompass manufacturing and logistical concerns with getting the product to market and
will not be discussed in this dissertation.
Process systems engineering (PSE) is generally only concerned with the first three stages
and gates and often tries to integrate them as much as possible in order to speed up the
development time. Since significant amounts of information are generated from different
sources, such as lab reports, consumer surveys, and detailed mathematical models, finding a
systematic method to convert the information into a common set of criteria which can be used to
provide guidance on decisions made regarding the product is paramount (Venkatasubramanian et
al. 2006). The data, information, and knowledge rendered are usually incompatible and require
several translational techniques derived from experimentation and/or mathematical
programming. For consumer attributes, it is important to incorporate the specific idiosyncrasies
of the different data sets and information sources that describe the molecular architecture that
deliver those attributes. Historically, commodity chemical products have relied on a large
amount of attribute, property, and molecular architecture information developed over years of
experimentation and their design is more of a function of process development. As
manufacturers have moved to produce specialty chemical products, less information is available,
so chemical product designers rely on models to screen for new molecular architectures. The
30
data, information, and models chosen dictate the manner in which the design problem is solved.
For instance, if reliable, accurate models do not exist, empirical trial and error approach based on
experimentation are used (Gani 2004). One of the most common techniques is design of
experiments (DOE) which uses probability to guide a series of experiments toward the desired
molecular architecture. Depending on the type of molecular architecture, this technique can
either be performed with conventional property parameters, attribute parameters, and/or latent
property parameters based on characterization data. Further discussion on these techniques can
be found in Chapter 3.
If reliable models do exist, mathematical programming or hybrid methods are preferred
(Gani 2004). The choice of model is dependent on the computational efficiency and accuracy
desired: more accurate, first-principle simulations are computationally inefficient while less
accurate, group contribution QSPR models are computationally efficient. The key is to achieve
as much accuracy as possible using as little computational resources as possible. Some attribute
models are well described by physical-chemical properties and specific molecular architectures.
For instance, the web strength of facial tissue is a function of the strength of intermolecular
hydrogen bonding between cellulosic polymers and fibers. Developing the computer aided
methods to design products with these easy to quantify attributes is often straightforward,
requiring minimum effort to translate attributes into common physical-chemical properties for
which QSPR relationships have already been established.
Other attributes, such as paper softness, are much more difficult to define in terms of
properties and molecular architecture. Fiber strength, fiber coarseness, aloe concentration,
silicone concentration, and other, undefined parameters, contribute to the attribute. For these
situations, either more complex experimentation using characterization is utilized or simulation
31
techniques are employed. The initial focus of the PSE community for this type of design has
been on the development of better simulation techniques. However, as the chemical engineering
discipline moves from '?data poor? to ?data rich? the shift will move toward data-driven
techniques (Venkatasubramanian 2009). Thus, the development of chemical product design
techniques based on data and data-driven models will become increasingly important. Further
discussion on computer aided approaches to chemical product design can be found in Chapter 4.
32
Chapter 3
Experimental Design of Chemical Products
The main objective for PSE research in chemical product design is to guide and focus
experimentation (Hill 2004). Unlike in process design, the requisite properties of importance in
the design of chemical products are usually consumer attributes. Consumer attributes are a set of
product requirements from the consumer that have been converted into a set of consumer
preferences that are empirical functions of physical-chemical properties and molecular
architecture. This emphasis on end-use application implies the control of the end-use property or
attribute as a primary requirement for designing chemical-based consumer products, which by
extension, means controlling the molecular structure, formation, rheology, and interfacial
phenomena at multiple length scales (Hill and Ng 1997; Smith and Ierapetritou 2009).
Because consumer attributes are difficult to quantify physically, the relationship between
them and the underlying fundamental physical-chemical properties and/or the molecular
architecture will most likely involve empirical relationships. Empirical models describe the
underlying phenomena?s relationship to a set of experimental data using regression analysis. For
situations where the set of experimental data is appreciably small, the phenomena are not well
understood, or the properties of interest are uncorrelated, the attribute data is often (1) directly
related to the molecular architecture or (2) related to the physical-chemical properties and then
the molecular architecture. Determining an attributes relationship to the molecular architecture
is performed by answering the following set of questions (Gani 2004):
33
1. How will the properties or attributes be obtained (calculated and/or measured)?
2. What level of detail regarding the molecular architecture will be used?
3. How will the molecular architecture be represented?
4. How will the candidate solutions be obtained?
In most cases the attributes are measured using a panel of ?expert users? who have repeatedly
demonstrated an ability to determine attribute differences in the desired products. Determining
the molecular architecture that delivers those attributes dictates the rest of the experimental
design. Due to the nature of chemistry, the smallest architectural building block that can be
experimented with is a complete, thermodynamically stable molecule. Except in rare occasions,
smaller building blocks, such a molecular groups, atoms, or protons/neutrons/electrons, are not
stable enough to measure individually. This does not preclude the use of larger building blocks
(like monomeric units of polymers, crystalline structure, particle shapes, etc.) from being
designed, but does limit all experimental designs to be based on mixtures. For the first case of
experimental design, mixtures of various molecular architectures are designed to directly deliver
specific consumer attributes.
3.1. Mixture Design of Experiments
Design of experiments (DOE) is a form of experimental design that utilizes statistical
methods to plan and execute informative experiments (Box et al. 1978). It is generally applied to
the first stage (concept generation) and gate (screening) of the chemical product design method
shown in Fig. 2.5. In some cases, process design options and constraints (feasibility and
development stages) are included in DOE, relying on computational simulation and design tools
to generate the process options followed by bench scale and pilot plant operations to test the
options. However, since computational tools rely on physical-chemical properties to define the
34
system, DOE techniques are generally only used to confirm upper and lower bounds on attributes
delivered by pilot-plant or larger processes.
The DOE procedure for generating and screening chemical product design concepts is as
follows (Cornell 2002):
1. Postulate a model that represents the attribute or physical-chemical property response.
2. Select experimental design points to test the facets of the molecular architecture of
interest.
3. Where observations can be collected, conduct experiments and fit the postulated
model.
4. Test the adequacy of the model using model fitness and predicted fitness followed by
calibration experiments
5. Repeat Steps 1-4 until a sufficient empirical model is developed.
6. Validate and optimize the chemical product?s molecular architecture using gradient or
nongradient mixture design techniques.
The most effective choice of model and location of design points is the primary focus of
the experimenter. The best set of points is chosen under the following constraints: (1) the size
and shape of the experimental region, (2) the number of desired experimental runs, and (3) the
type of model used for constructing the map of the attribute response (Kettaneh-Wold 1991).
Most often, a polynomial model is selected to represent the response surface since it can be
expanded through a Taylor series to improve accuracy (Cornell 2002). A first or second degree
model is usually chosen to represent the surface since it requires fewer observations. Third
degree or higher ordered models are seldom utilized (Maghsoodloo and Hool, 1984). The first or
second degree model may be represented as follows:
35
??? ??? ??ui iio xy 1 (3.1)
???? ???? ? ?? ? ?? u wi u iw wiiwui iio xxxy 1 (3.2)
Where y is the response variable and usually represents attributes (e.g. softness) or physical-
chemical properties (e.g. viscosity), depending on the system of interest. x is the independent
variable, usually representing the i or w molecular architecture fraction. The regressors,
?o???i,?and??iw, represent the pure component and mixture changes in y per unit change in x away
from a center point. ? is the unexplained error in the response not captured by the model. The
point estimate forms of the models (i.e. fitted regression models) are given below.
???? ui iio xy 1 ??? ?? (3.3)
? ?? ? ?? ??? u wi u iw wiiwui iio xxxy ??? ???? 1 (3.4)
where ? is the unbiased estimator of the response and ? , ? ? , and? ? ??are the point estimates of
the regression coefficients. To find the estimates of the regressors, the sum of the squares of the
error ? (estimated at E(?)=0 for uncorrelated random variables) in the model is minimized using
least squares regression. Least squares regression is a common technique in the development of
data driven models and is found in many forms, including classical least squares (LS), inverse
least squares (ILS), multivariate linear regression (MLR), and partial least squares applied to
latent surfaces (PLS). To derive the regressor solution for LS, it is convenient to switch to
matrix notation. Without it, the formulas become unmanageable when the number of
explanatory variables increases (Larsen 2003). Rewriting the point estimate model, shown in Eq.
3.3, in terms of matrix notation results in the following:
36
BXY ??? (3.5)
where ?, ? and X are matrices of the predicted responses, estimated regression coefficients, and
independent variables, respectively, and n represents the total number of experiments.
??
?
?
?
?
?
??
?
?
?
?
?
?
ny
y
y
Y
?
?
?
? 2
1
? (3.6)
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
u
B
?
?
?
?
?
?
? 1
0
? (3.7)
??
?
?
?
?
?
??
?
?
?
?
?
?
unn
u
u
xx
xx
xx
X
,1,
,21,2
,11,1
1
1
1
?
????
?
?
(3.8)
In terms of matrices, Eq. 3.5 can be rewritten as Eq. 3.9 (Larsen 2003).
BXXYX TT ?? ? (3.9)
1)(?? ?? XXYXB TT (3.10)
Solving for the best fit regressors gives Eq. 3.10 which is the central result of least squares.
Although the same procedure can be extended to second order and third order models, it is
seldom used because it involves fitting nonlinear models Also, for the inverse of the identity
matrix XTX to exist, X must have as many rows as columns. Since X has one row for each
sample and one column for each component, then it follows that there must be at least as many
37
samples n as components u to be able to compute Eq. 3.10. The rules governing regression
relating to samples and components are summarized by Geladi and Kowalski (1986) as the
following:
? For u > n, there is no unique solution unless one deletes the independent variables.
? For u = n, there is one unique solution.
? For u < n, a least-squares solution is possible.
? For u = n and u > n, the matrix inversion can cause problems.
Likewise, any linear dependence among the rows or columns of X can potentially lead to a
singular XTX matrix whose inverse does not exist (Kramer 1998). This is a key observation that
will be discussed in detail in Chapter 9. Other aspects of least squares regression can be found in
Montgomery (2005).
Mixture design of experiments (MDOE) is an extension of DOE that explicitly uses
chemical constituents as the factors in the design. By definition the constituent (or molecular
architecture) fractions must sum to one and each constituent fraction must lie between zero and
one:
11 ???ui ix (3.11)
10 ?? ix (3.12)
Eq 3.11 imposes a colinearity effect by removing the independence of the factors. While it does
not affect the utilization of the model, it does impact the interpretation of the regression
coefficients. The benefit of using the constraint, however, is that it affords the ability to
represent mixture data (e.g. design points, response surfaces, and target regions) in a simplex
38
diagram where each of the vertices represent pure chemical constituents or molecular
architectures as shown in Fig. 3.1.
Figure 3.1: Simplex Diagram of the Response Surface of a Mixture of Polyethylene (x1),
Polystyrene (x2), and Polypropylene (x3) using Experimental Data from Cornell (2002).
Using Eq. 3.11, Scheffe developed the first simplex-lattice designs which many
researchers considered to be the foundation of mixture design (Cornell 2002). To develop these
designs, Scheffe (1958, 1963) noted that the location of the response of a mixture made up of
exactly zero constituents must be identically zero meaning that the coefficient ?o is zero.
Furthermore, the use of Eq. 3.11 means that the quadratic terms can be rewritten as Eq. 3.13:
?????? ?? ??uw wii xxx 12 1 (3.13)
.1
.1
.1
.2
.2
.2
.3
.3
.3
.4
.4
.4
.5
.5
.5
.6
.6
.6
.7
.7
.7
.8
.8
.8
.9
.9
.9
X2
X1
X3
x 1 x 2 x 3
1 1.0 0.0 0.0 11.7
2 0.5 0.5 0.0 15.3
3 0.0 1.0 0.0 9.4
4 0.0 0.5 0.5 10.5
5 0.0 0.0 1.0 16.4
6 0.5 0.0 0.5 16.9
Ex perimental
De sign Points
Comp one nt Fractions Observed
Elong ation
Va lues
9
11
13
15
17
Elongation Response
Surface Contours
39
Combining these observations, Scheffe derived the canonical models in Eq. 3.14 and 3.15. For
ease of notation, the hats have been dropped from the estimates of the empirical model
parameters.
??? ui ii xy 1 *? (3.14)
? ?? ? ?? ?? u wi u iw wiiwui ii xxxy *1 * ?? (3.15)
These first and second order canonical models are postulated to represent over 66% of all
mixture designs encountered (Maghsoodloo and Hool, 1984). In situations where a higher order
cubic design is desired, the same procedure may be applied to the cubic polynomial equation to
make the cubic canonical model.
? ? ?? ?? ? ? ?? ?? ???
u
wi
u
w
u
ji wiijk
u
wi
u
iw wiiw
u
i ii xxxxxxy ? ?? ??? ,
**
1
* (3.16)
The effect of the canonical models is to remove the quadratic and higher terms from analysis and
leave behind only the modified pure component properties and interaction effects. However, it
must be noted that the modified regressors ?i* and ?iw* do not represent only pure component
properties or only interaction effects. On the contrary, because of the colinearity introduced in
the derivation of the canonical models, the modified regressors are amalgams of the pure
component, interaction, and quadratic effects, as shown in Eq. 3.17 and Eq. 3.18:
iiii ???? ??? 0* (3.17)
wwiiiwiw ???? ???* (3.18)
40
To execute a mixture design using canonical models, Scheffe (1958, 1963) proposed
using lattice design points to form a grid in the simplex diagram as shown in Fig. 3.1. Good
properties of the design points consist of orthogonality, rotatability, and symmetry about an
experimental center point (Lazic 2004). The design points for screening molecular architectures
are located where the predicted variance of the model coefficients will be minimized, also known
as minimizing D-optimality (Meyer and Nachtsheim 1995). For the case of the 6 design points
in Fig. 3.1., the points are located at the pure component vertices and halfway along each axis.
Additionally, it is common to place a point at the center of the design space, although this was
not done in the example shown in Fig. 3.1. Other designs based on Bayesian D-optimality, I-
optimality, and Bayesian I-optimality can be found in Meyer and Nachtsheim (1995). The
designs are particularly useful when a constrained region is studied and no classical design is
available (Eriksson et al. 1998).
The next step in the design is to conduct the experiments at the design points and measure
the attribute and physical-chemical property responses. Since the simplex lattice design is a
boundary design problem, with at least as many design points as regression terms, then the
estimated regression parameters for the empirical models may be taken directly from the
responses at each design point for the canonical model (Kettaneh-Wold 1992). However, since
the canonical model is colinear and not orthogonal, these responses may not represent the true
property or attribute estimates. The true property/attribute estimate, or orthogonal effect, is
found by taking the difference in the value of the response at pure and infinitely dilute solutions
while holding all other constituents constant. In addition, Cox (1971) noted that Scheffe?s
canonical models have the following drawbacks:
41
1. If two replicate experiments on the same system have the same expected responses
except for a constant difference between replicates, it is obvious that fitting separate
replicates, all of the regression coefficients will be different in the two replicates.
2. The absence of squared terms makes it meaningless to consider the direction and
magnitude of curvature of the response to a particular component.
3. The interpretation of the regression coefficients is in terms of the responses for simple
mixtures.
Likewise, Kettaneh-Wold (1991) points out that the removal of the constant term to generate the
canonical model makes it impossible to center these models which often leads to an ill-
conditioned XTX matrix. An ill-conditioned matrix is not symmetrical about the main diagonal,
which means that the order of differentiation is important which can lead to errors during
inversion. It is of the utmost importance that the constituents be independent when performing
least squares regressions. To address these issues, Cox (1971) proposed a variable
transformation with an arbitrary reference mixture that would allow for the use of existing
polynomial models with additional constraints involving a reference mixture called the standard
mixture.
42
Figure 3.2: The Parameterization of the Component Space in Terms of a Standard Reference
Mixture (s) such that the Incremental Change ?i in the Proportion of the Component i is
Indicative of its Effect on the Response (Cornell, 2002).
Shown in Fig. 3.2 is a simplex diagram published by Cornell (2002) with a standard
mixture s and a mixture x with a larger proportion of constituent xi. Noting that x lies on a line
from s to the xi vertex, then the ratio of the other u-1 constituents are in the same relative
proportions as the standard mixture.
iii sx ??? (3.19)
)1( wwwww sssx ????
(3.20)
0 .8
0 .7
0 .6
0 .5
0 .4
0 .3
0 .2
0 .1
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9
X j
X i
X k
s
x
? i
1 - s i
43
Rewriting the first and second degree models, Eq. 3.3 and 3.4, in terms of the change in
constituent i, ?i, gives equations Eq. 3.21 and 3.22, which provide a direct link between the
regressors and the position of the design points to the reference mixture (Cornell, 2002).
? ? ? ? ? ????
???
?
? ??? u
i ii
issyxy 1 ? (3.21)
? ? ? ? ? ??? ??
??????? ??????????? ??? u
i ii
iiu
i ii
i sssyxy 2211 ?? (3.22)
Where y?x) is the expected response at design point x and y?s) is the expected response at the
standard reference mixture. Hence, as Smith and Beverly (1997) point out, the gradient, or
change in response per unit change in xi along the line between Point S and Point X in Fig. 3.2 is
called the effect of xi, provided xi is free to range from 0 to 1. It has also been shown that the
response at the standard mixture is equivalent to the estimate of ?o regressor (Box et al. 1978).
Likewise, the transformed variables of the expressions in Eq. 3.21 and Eq. 3.22 can be
represented as:
?? ????
?
???
?
? ? u
i ii
u
i ii
i xs ??1
(3.23)
? ? ?? ???
??????? ? u
i iii
u
i ii
ii xxs ?? 221
(3.24)
The end result of this analysis are polynomial expressions identical to Eq 3.3. and 3.4, but with
the added benefit of the major colinearities being identified and removed for mixture design.
Hence, the regressors, ?i, are estimates of the pure component physical-chemical properties or
attributes. Furthermore, the estimated response surface generated by either the Scheffe canonical
44
models (Eq. 3.14 and 3.15) or the Cox polynomial models (Eq. 3.3 and 3.4) have been shown by
Cornell (2002) to be identical. More discussion on the interpretation of these models and their
parameter estimates can be found in Chapter 7.
Once the model parameters are determined using Eq. 3.10, the adequacy of the model is
tested and the design is iterated until a good empirical model is developed or optimum solution is
discovered. The iteration of an experimental design to find an optimum solution is among the
most complex problems for a researcher (Lazic 2004). In the context of MDOE, interpretation of
the models, model parameters, and the effects of each of the components on the properties or
attributes of the mixture is the objective of screening designs. Beyond screening designs, the
prediction of the optimum from the property or attribute models can help to focus future
experimentation and improve the interpretation of the property or attribute models. The success
of the design is judged by the ability to arrive at the feasibility region. Reaching the optimum is
more efficient if the empirical model is adequate (Lazic 2004). The F-test is most often used as
a measure of the adequacy or lack-of-fit of the model. Determining how much variance is being
described by the model can also be accomplished using R2, a measure of fit of the model to the
existing structure, and Q2, a measure of the predicted fit of the model when elements of the data
structure are removed.
to t
resto tSS SSSSR ??2 (3.25)
to t
PREto tSS SSSSQ ??2 (3.26)
where SStot is the total sum of squares, SSres is the residual sum of squares and SSPRE is the
predicted residual sum of squares. R2 and Q2 values of 80% and above are generally considered
good. Increasing the number of model parameters is usually the preferred choice to improve the
45
lack-of-fit, often resulting in quadratic, special cubic, or even special-quartic models (Brandvik
1998, Brandvik and Daling 1998). For these higher order models, transformations would need to
be applied in order to sufficiently linearize them in order to test their normality. A discussion on
linearization techniques is presented in Chapter 6.
Once the model is deemed adequate, the region of applicability needs to be determined.
Since the mixture design is performed in the local domain, then the resulting regression
coefficients of the property models express factor effects exactly in that part of the domain. If the
predicted optimum lies outside of the experimental region studied, then the experiments need to
be recentered around the optimum. Many techniques exist to achieve this repositioning
including gradient and nongradient optimization methods.
Gradient methods are based on the derivative of the response surface model and, as such,
are used only when the model is deemed to be adequate. The most common gradient method is
the method of steepest ascent which performs experiments at predetermined steps along the
vector formed by the gradient of the response surface (Lazic 2004, Brandvik 1998, Brandvik and
Daling 1998). The step size is set according to the component that has the largest effect. The
method is sufficient for single property models. To handle multiple responses, the method uses
weighting functions on the response surface property models to create an overall gradient
function (Lazic 2004, Atkinson 1992). The weighting factors and values are at the discretion of
the experimenter, which is a less than ideal situation. Also, when the number of molecular
architecture options is large, then graphical representation of the method becomes difficult. It
will be shown in Chapter 7 how using the property clustering algorithm and the reverse problem
formulation can be used to alleviate these issues. The parameters used in the gradient expression
are the regressors corresponding to each effect direction and remain unchanged as the solution is
46
marched toward the predicted optimum represented as the feasibility region. Once the solution
leaves the feasibility region, the method is stopped and a new MDOE may be conducted. In cases
where the gradient method is unable to reach the feasibility region, either the adequacy of the
model needs to be improved or a more realistic set of targets needs to be specified. If the
adequacy of the model cannot be improved enough to be deemed sufficient and, as such, is
deemed inadequate, then nongradient optimization is performed.
Nongradient optimization searches for the optimum using a step-by-step comparison of
measured property or attribute values. One of the most common types of nongradient
optimizations is the simplex self-directing method (Lazic 2004). This method works by first
conducting a simple mixture design around an initial guess. Next the lowest response property or
attribute value is dropped and a mirror-image experiment opposite of the dropped value is
conducted. The procedure continues until the algorithm repeats the same experiments, indicating
that the optimum is somewhere in the bounded region. Like the gradient method of optimization,
nongradient optimization struggles with marching toward a single solution for systems with
multiple molecular architectures that have competing properties or attributes models (i.e. the
methodology can easily become trapped in local optima). Solution methodologies to avoid such
situations are discussed in Chapter 4.
Once optimization has been conducted, the design of the chemical product by the
researcher is finished and process design and development begin. However, there are two major
limitations in designing chemical products using MDOE that should be considered. First, while
this transformation of the original Scheffe polynomials removes the primary colinearity
introduced by Eq. 3.11, it leaves the secondary colinearities such as those introduced by
constraints on the constituent ranges. Kettaneh-Wold (1992) suggests that the best solution may
47
be to refrain from interpreting the coefficients and rely on the predictions only but notes this
solution is not acceptable in practice since the interpretation of regression coefficients is a
necessity when the objective is to find component effects in screening situations. It is in this
arena that Kettaneh-Wold (1992) suggests the use of decomposition techniques like Principal
Component Regression (PCR) and Partial Least Squares on to Latent Surfaces (PLS).
In many cases because of the complexity of the interactions, especially those across
multiple scales, the datasets needed to describe the molecular architecture are large and highly
colinear. These datasets are typically generated using a variety of characterization techniques
such as infrared (IR) or near-infrared (NIR) spectroscopy, wide angle x-ray spectroscopy
(WAXS), and nuclear magnetic resonance (NMR), among others. The description of molecular
architecture captured by these techniques is typically too complex to be directly described by the
empirical models in a MDOE. Decomposition techniques utilize the colinear nature of large
datasets to consolidate the information into a set of underlying latent variables from which
mixture design can be conducted. Further discussion on experimental chemical product design
using characterization and decomposition techniques can be found in Section 3.2.
Second, consumer attributes are seldom related directly to mixtures of molecular
architecture in a product design because process design parameters typically dictate the use of
intensive physical-chemical properties to find optimum unit-operation arrangements. Often
instead, an attribute-property empirical relationship is developed to convert the attributes into
physical-chemical properties, followed by chemical product design based on those physical-
chemical properties. The attribute-property relationship can be captured by varying the
suspected important properties in a batch of product samples and then testing them using a panel
of ?expert users? if they are consumer attributes, or conventional lab equipment if they are
48
physical-chemical properties. One of the most efficient techniques for capturing the attribute-
property relationship is factorial design. Factorial design is a technique that varies quantitative
(e.g. molecular weight) or qualitative design factors (e.g. softness) over an experimental design
range and measures the quantitative or qualitative response of the system. The exact location of
the design points depends on the base of the design. A base 2 design (2?), where the ? factors
(i.e. a attributes or p properties) are observed at both the higher and lower bounds of the range, is
most common. In base 3 design (3?), center points are added, equidistant from the higher and
lower bounds. For all of the designs, the design points are located around a nexus that is as close
to the expected target as possible. The number of experiments used in the design is set according
to the type of model chosen to reflect the response. As before, it has been shown that polynomial
models are the most efficient models for developing the attribute-property relationship (Cornell
2002).
In most cases, the number of factors influencing the consumer attributes of interest are
relatively small and can be handled using full factorial design techniques. However, as the
number of factors increases beyond ??> 5, the number of required experiments to conduct the
design becomes too large to efficiently execute. For example, a screening design involving 6
factors would require 64 experimental runs without replication or center points. For these
situations, a ? fractional factorial (2???) becomes the choice in design and allows the
experimenter to increase the number of factors for the same number of experiments. The trade
off in using fractional factorial design is the confounding of factors with one another. With some
forethought, this situation is handled by maximizing the resolution of the design. Typically, a
resolution four (RIV) design is chosen that ensures that all main factors are protected from two-
way interactions. Sometimes when experimental runs are constrained, resolution three (RIII)
49
designs may be chosen which confounds some two-way interactions with the main factors. In
those cases, the experimenter may use previous knowledge to dismiss any infeasible two way
interactions prior to executing the experiment.
Concurrent with developing the attribute-property relationship, a set of validation tests
are used to not only confirm the existence of a suitable candidate product, but also to serve as a
tool for evaluating the candidate products against a benchmark. The benchmark may be simply a
comparison with a competitor?s product, or they may be more explicitly defined using
measurable attributes, like sales, price, etc. Most products, however, cannot be explicitly defined
in terms of these attributes and need to be converted to a set of physical-chemical properties first
before proceeding with design.
After the attribute-property relationship is developed and the attribute constraints are
converted into physical-chemical property constraints, then the chemical product design
proceeds in the same manner as it did for attributes using the steps outlined at the beginning of
this section. First, the property-molecular architecture relationship is developed through a series
of experiments. Then the candidate mixtures are optimized to reach a desired target. The most
promising candidates proceed to the process design and development step.
3.2. Experimental Design using Multivariate Data
Often in chemical product design the molecular architecture is complex, containing
important structural facets that exist over a multitude of length scales. To capture the important
structural data for a single configuration requires multiple characterization experiments, each
generating a large assortment of highly correlated descriptor data. Managing the complexity of
this data structure requires the use of decomposition techniques such as principal component
analysis (PCA), independent component analysis (ICA), and network component analysis (NCA)
50
to detect trends, groupings, and outliers in such systems (Wold et al. 1987; Gabrielsson et al.
2002). Decomposition techniques also afford the ability to reparameterize the system into a
lower dimensional latent domain where other chemometric techniques such as principal
component regression (PCR) and partial linear regression on to latent surfaces (PLS) can be used
to build the attribute and/or property ? molecular architecture relationships. The use of
characterization property data and decomposition techniques adds an additional step to
conventional MDOE blending experimental designs. An overview of the experimental design
method using characterization data is as follows (Gabrielsson et al. 2003, Gabrielsson et al.
2004, Gabrielsson et al. 2006):
1. Identify the important molecular architecture features of the chemical product using
characterization on a training set of similar chemical products.
2. Decompose the characterization data set to find the latent property domain.
3. Postulate a model that represents the attribute or physical-chemical property response
as a function of latent variables.
4. Conduct a factorial design on the latent property domain to establish the attribute-
latent property relationship (e.g. select the latent experimental design points to test the
facets of the molecular architecture of interest, collect observations at those design
points, and fit a latent variable model).
5. Test the adequacy of the model using model fitness and predicted fitness followed by
calibration experiments
6. Repeat steps 1-5 until a sufficient empirical model is developed.
7. Use mixture design to identify a set of candidate mixtures based on the original
training set of data that deliver the desired attributes.
51
8. Validate and optimize the molecular architecture of the candidate solutions using
gradient or nongradient mixture design techniques in the latent variable descriptor
domain.
3.2.1. Characterization Techniques
Characterization describes a class of experimental tools that gather information on not
only chemical constituency or molecular structure, but also on larger structural characteristics
describing the orientation and alignment of these molecules often called nano-, micro-, and
meso-structure. Some examples of characterization techniques include nuclear magnetic
resonance (NMR), wide angle x-ray diffraction spectroscopy (WAXS), and infrared
spectroscopy (IR). The techniques are often applied to a training set of molecules defined by an
experimental design used to explore the interesting facets of a set of property attributes.
Characterization, in general, provides large quantities of correlated data from which
useful information on molecular architecture can be extracted. This generated data is often
referred to as structural descriptors (P) of the molecular architecture (X). Fig. 3.3 has been
developed as a general guide to the available options, highlighting the interconnectivity of the
types of characterization, molecular architecture, and product attributes and properties available.
Managing the complexity of the information requires a systematic method for determining which
specific information on molecular architecture will be necessary to build appropriate models for
a specific application. It is proposed that the most efficient manner for achieving this objective
is to use the minimum number of characterizations necessary to generate the necessary structure
information.
52
Figure 3.3: An Overview of the Interconnectivity of Characterization Techniques, Molecular
Architecture, and Physical Properties and Attributes of Chemical and Material Products.
For example, Olson (1997) proposes a hierarchical structure to describe a high-performance
alloy steel based on the three-link chain of processing, structure (molecular architecture), and
properties as shown in Fig. 3.4 (Olson 1997; Tolle et al. 2009). Tolle et al. (2009) noted that the
structures identified by Olson, namely the matrix constituency, strengthening dispersion, grain-
refining dispersion, austenite dispersion, and grain-boundary chemistry could be described solely
through wide angle X-ray spectroscopy (WAXS) using a model-free analysis as shown in Fig.
3.5. Other characterization and simulation options for describing the high-performance alloy
steel system can be found in Olson (1997).
53
Figure 3.4: The Process-Structure-Property Relationships for a High Performance Alloy Steel
(Olson 1997).
Figure 3.5: WAXS Characterization of a High Performance Alloy Steel (Olson 1997).
54
The main characterization used in the design of chemical products in this dissertation is
spectroscopy. Spectroscopy is the detection and analysis of the radiated energy absorbed or
emitted by the architecture of a chemical species (Atkins 1998). Molecular spectra are generated
when the periodic motion of atomic nuclei within the molecule?s architecture change when
bombarded by radiation, exploiting the fact that molecules absorb specific frequencies that are
characteristic of their structure. Motions can be in a straight line like symmetrical and
asymmetrical stretching, and/or rotational like twisting, rocking, wagging, and scissoring
(Workman and Weyer 2008). Depending on the nature of the environment surrounding the local
molecular architecture, rotational, vibrational, and electronic energies are absorbed or emitted.
For example, a molecule with N atoms has three degrees of freedom of motion (3N) with three
axes of translation (x, y, z) and three axes of rotation (from inertia) (Workman and Weyer 2008).
Removing the six molecular motions leaves 3N-6 vibrational motions (3N-5 for linear molecules)
within the molecule which can be characterized using vibrational spectroscopy.
63 ?? NNv OR 53)( ?? NlinearN v (3.27)
The vibrational motions exist in many forms, each having a specific frequency associated with it.
When the vibrations have amplitudes of 10-15% of the average distance between atoms, they are
referred to as Harmonic or normal (Workman and Weyer 2008). The frequency at which
harmonic vibrations absorb energy is dependent on the type of vibration, not on the amplitude.
Rather, the amplitude of the absorption is determined by the absorptivity and the number of
molecules encountered within the beam path. These relationships are best quantified using the
Beer-Lambert Law where the absorbance Pabs is equated to the product of the absorptivity ??of a
molecular vibration, the concentration c of the molecules in the measurement beam, and the
55
pathlength lb of the beam within the measurement device. (Atkins 1998; Stephenson et al. 2001;
Workman and Weyer 2008).
babs clP ?? (3.28)
In terms of transmittance PT from 0.0 to 1.0, the absorbance can be rewritten as a function
of the incident energy Io on the sample, and the attenuated energy I after sample interaction
(Workman and Weyer 2008).
???????????? oTabs IIPP lo glo g (3.29)
The magnitude of absorption depends on the concentration of the sample, which can be
parameterized in terms of the number of atoms, bonds, functional groups, complete molecules,
microsctructure, and/or particle size and shape. Each deviation from the baseline is indicative of
an aspect of the molecular architecture of the chemical product. For IR and NIR spectroscopy,
the deviations from the baseline are usually a result of specific types of molecular functional
groups, although in some cases, the deviations may be the result of larger scale structure
(Socrates 2001, Abebe et al. 2008). The inherent vibrational motions of the constituent atoms
within a molecular group result in unique set of frequencies at which the group absorbs incident
radiation, often referred to as a spectrum. Shown in Fig. 3.6 is the infrared spectrum of five solid
form excipients of mannitol.
56
Figure 3.6: The Infrared (IR) Spectra of Different Solid Forms of Mannitol (Gabrielsson et al.
2003).
A molecule containing multiple groups may, in turn, contain overlaps in the spectral
features, but, various ?fingerprint? regions exist that can be used to differentiate between the
various forms of molecular architecture (Stephenson et al. 2001). The most common functional
groups and their corresponding frequencies have been thoroughly categorized by various authors
such as Socrates (2008), Atkins (1998), Workman and Weyer (2008) amongst many others.
Each functional group will often have more than one characteristic (or fundamental) absorption
band associated with it; a result of different types of vibrations of which the group is capable.
For example, the first order acyclic CH group has a frequency (between 2959-2849 cm-1) at
which the C-H bond vibrationally ?stretches? and another frequency (between 1360-1320 cm-1)
at which the bond vibrationally ?bends?. For a methylene group, 6 types of vibration include 2
57
types of stretching (symmetrical and asymmetrical) stretching, and all 4 types of bending
(wagging, scissoring, rocking, and twisting). As the number of atoms in a molecule or functional
group increases, interactions between atoms increases leading to ever increasing types of
vibrational modes. For this reason, absorbance data is usually only interpreted in terms of these
first six modes.
In addition to the primary fundamental absorptions, overtone and combination bands can
also exist. Overtone bands result when an additional quanta of energy is absorbed by the
molecule. When one quanta is absorbed, a fundamental vibration occurs. When two quanta are
absorbed the first overtone is vibrationally excited, three quanta lead to a second overtone, and
so on. Since the additional quanta are being absorbed, it follows that the overtone bands will
approximately exist at integer multiplications of the energy level (frequency), leading most
overtone bands to appear in the near-infrared (NIR) region. For instance, the first overtone band
of the stretching vibration of a C-H bond is at 5917-5698 cm-1, approximately twice the
fundamental vibration. Fig. 3.7 illustrates the overtone bands of the four solid forms of mannitol
in the near infrared region using NIR spectroscopy.
58
Figure 3.7: The Near-Infrared (NIR) Spectra of Different Solid Forms of Mannitol (Gabrielsson
et al. 2003)
A more rigorous estimation of these bands can be predicted using the Schrodinger wave
equation and a harmonic oscillator based on Hooke?s Law according to the following
expressions (Workman and Weyer 2008):
?????? ?? 21?? hvE (3.30)
???
?
???
? ??
21
112
1
MM
Kv
?
(3.31)
Where K is a bond force constant that depends on the type of bond between atom, M1 and M2 are
the masses of atoms 1 and 2, and ? is the vibration frequency, h is Planck?s constant, and ??is an
59
integer of the vibrational quantum number (0 for ground state). The most realistic estimation of
the overtone bands, however, includes a correction for anharmonic oscillation (Burns and
Ciurczak 2008).
ohvE ?? ? (3.32)
? ?xvxvv ao 21 211 ? ??? ???? (3.33)
Where vo? is the overtone frequency and x is the anharmonicity constant for a specific bond type
as given in Groh (1998). For example, the first overtone band (2v) for a C-H group contains a
0.019% shift due to anharmonicity (xa=0.00019) resulting in a band frequency at 5917-5697 cm-
1. An outline of the C-H stretch vibration is presented in Table 3.1 (Workman and Weyer 2008).
Table 3.1. Relative Band Intensities for C-H Stretch (Workman and Weyer 2008)
Band Wavelength Region (cm-1) Relative Intensity
Fundamental (v) 2959-2849 1
First Overtone (2v) 5917-5698 0.01
Second Overtone (3v) 8873-8547 0.001
Third Overtone (4v) 11,834-11,390 0.0001
Fourth Overtone (5v) 14,493-12,987 0.00005
It should be noted that although group frequencies are expected to exist within narrow
ranges, interference and perturbation may cause a shift in the characteristic bands due to the
electronegativity of neighboring groups or atoms (Socrates 2001). These shifts can be estimated
using combination bands, first order coupling, and Fermi resonance. Combination bands can
60
result from the confluence of fundamental bands. For instance, when a stretching vibration v is
combined with a bending vibration ?, a combination band v+? results (Workman and Weyer
2008). These types of bands are generally very broad and difficult to interpret directly in terms
of molecular architecture. First order coupling of vibrations occur between two or more bonds in
a molecular group when the species are of similar symmetry and energy. This results in a
degeneracy that splits the energy states into bands that absorb at slightly higher and lower
frequencies than expected (Burns and Ciurczak 2008). Similarly, Fermi resonance is the second
order coupling of vibrations and shifting of frequencies that result in an overtone band and a
combination band (Workman and Weyer 2008). Group structures that contain coupled
transitions are shown in Table 3.2.
Table 3.2. Group Structures Exhibiting First Order Coupling (Workman and Weyer 2008)
Absorbance Type Group Structure Common Examples
Cumulated Double Bond Stretching X=Y=Z C=C=N
Paired Stretching -XY2- -CH2-
Paired Triplet Stretching -XY3 -CH3
Paired Triplet Bending -XY3 -CH3
Secondary Amide Bending R-CO-NH-R? CH3-CO-NH-CH3
In Raman, additional low frequency modes are also observed. Whereas IR is adept at
discerning the structure of molecules with dipole moments and NIR can capture specific
functionality fingerprints, Raman can discern the structure of homonuclear systems (Burns and
Ciurczak 2008). Raman works slightly differently by exciting the molecular bonds to a virtual
energy state that is far above the vibrational energy levels, but is capable of describing bands
61
which are weak or inactive in the infrared, such as those due to the stretching of skeletal bonds
such as C=C, C?C, C?N, C-S, S-S, N=N, and O-O (Socrates 2001). This makes Raman ideal for
describing backbone structure, particularly the polarizability of symmetric structures that are
inactive in IR and NIR.
Regardless of the type of vibrationally active spectroscopy (i.e. IR, NIR, Raman, etc.),
two criteria form the backbone of the tools developed in this dissertation. First, it should be
noted that the intensity of any spectroscopic absorbance is not only due to the strength of the
bond in a functional group, but also to the number of times that bond or group occurs in the
system. This observation suggests that spectroscopic information could be adapted to the group
contribution method (Constantinou et al. 1993; Marrero and Gani 2001; Socrates 2001). A
method that incorporates IR and NIR spectroscopy data into a characterization based group
contribution (cGCM) approach is developed and discussed in detail in Chapter 9.
Second, to effectively use spectroscopic techniques to design chemical products it is
important to capture the important features of the molecular architecture of the system in the
initial training set. Although the exact number chemical constituents will vary according to the
desired accuracy of design (e.g. screening vs. optimization), it is generally deemed acceptable to
capture the spectroscopic characterization of approximately 30 chemical products with similar
molecular architecture to the desired chemical product (Cornell 2002). Mixtures of these
chemical constituents will form the initial candidate solutions for the chemical product design
prior to optimization.
3.2.2. Principal Component Analysis and Regression
After spectra are generated, decomposition techniques like PCA, PCR, and PLS are used
to consolidate the data, develop the underlying latent variable structure, and construct the
62
attribute (or property) ? latent variable relationships. The objective of decomposition techniques
is to describe the variation in the data using a minimum number of variables by fitting a p
property characterization, structural descriptor data set of molecular architecture information to a
lower, m-dimensional sub-property space or hyperplane. By using the variance-covariance
structure to compress the p property data to m principal component data, it ensures that the
property subspace is appropriately orthogonal and devoid of any colinearity which may exist in
the property or attribute domain. The most common decomposition uses least squares as the
fitting function and is known as principal component analysis (PCA) and is shown below
(Gabrielsson et al. 2002):
ELTP ???? (3.34)
Since decomposition techniques like PCA are susceptible to large differences in scales and
variance, it is general practice to standardize the property data prior to analysis (Johnson and
Wichern, 2007). In particular, the structure descriptor, property data matrix P, consisting of p
properties described by n molecular architectures, is mean-centered and variance scaled to unity
across the p properties. This standardization helps to ensure that the multiple sources of p
property descriptor data contain the same data structure and, therefore, can be decomposed into
meaningful eigenvectors and eigenvalues. The underlying latent property, or score matrix T,
consists of the relative distances to the projected values of each property on to the eigenvector
defined hyperplane as shown in Fig. 3.8.
63
Figure 3.8: A PCA Illustration of a Least Squares Fitting of a Lower, m Dimensional
Hyperplane to p Dimensional Property Data (Gabrielsson et al. 2002).
The eigenvalues can be considered measures of the eigenvector contrasts, or loadings L? of the
original variables. Each loading in the loadings matrix L? describes how the property values are
weighted together for each principal component and are represented as a hyperplane. Values of
loadings vary from being highly correlated with property values, scored as -1 and 1, to being
uncorrelated with values of 0. Finally, the difference between the projections and the property
values are contained in the residual matrix E. If the dimensionality of the p property domain is
the same as the m latent property subdomain, then the determinant of the residual matrix E is 0,
indicating that no error exists in the transformation. However, this condition is seldom met.
Most applications of PCA result in the first 2 or 3 principal components accounting for 80% to
90% of variation p property domain, as measured by the eigenvalues (Johnson and Wichern
2007). All other information is usually found in the non-zero residuals matrix E.
64
The procedure for determining the eigenvector/eigenvalue pairs in PCA, and hence the
principal components, is usually performed by either singular value decomposition (SVD),
spectral decomposition, or an iterative scheme like the NIPALS algorithm (Geladi and Kowalski
1986, Wold et al. 1987). New, orthogonal latent property components are fitted to the data
structure beginning with the latent properties with the highest eigenvalues first and continuing
until (1) enough variance in the data has been described (e.g. 80% or higher) or (2) the
eigenvalues no longer appreciably change. A scree plot can be used to determine when the
proposed decomposition is sufficient, as shown in Fig. 3.9.
Figure 3.9: A JMP Scree Plot of the Covariance Based PCA on IR and NIR Spectroscopic Data
of Filler Excipients for Direct Compressed Acetaminophen Tablets (Gabrielsson et al. 2003).
By definition, a scree plot is a visual depiction of the magnitude of the eigenvalues versus
their number in order of decreasing magnitude. The appropriate number of latent variables are
determined to be those above the bend in the plot and the remaining values and judged to be
relatively small and all about the same size, meaning they are not adding significant information
65
to the design. These components can be removed without much loss of information (Johnson
and Wichern 2007). For example, in Fig. 3.9 it was determined that 3 principal components exist
above the bend and contain 91% of the variation in the characterization data (not shown), thus
only m=3 latent properties will be used in the design. The resulting generalized PCA model is
then:
mxpnxmnxp LTP ??? (3.35)
Performing regression on decomposed data structures involves replacing the independent
variables in the MLR expression with the underlying and orthogonal latent variables. This
replacement spans the multidimensional space of X more efficiently when conducting
experimental design which results in stronger predictive expressions. When the decomposition
technique is PCA, the regression is known as PCR and has the following structure:
TBY? (3.36)
1)( ?? TTYTB TT (3.37)
where Y is an ux??set of attributes or physical-chemical properties and B is a mx? set of
regressors describing the attribute-latent property models. One of the greatest benefits of PCR is
that because the score matrix contains no colinearity, then the T?T inversion will be guaranteed.
At the same time, random noise is easily removed by including the top 2-3 principal score values
in the model that describe 80%-90% of the data. However, to ensure that the application range
of the model is suitable, it is wise to conduct any experimental design with the latent properties
as the design factors. A discussion on network component analysis (NCA), which utilizes a
priori knowledge of the molecular architecture to find the latent properties is presented in
Appendix A3.
66
3.2.3 Partial Least Squares on to Latent Sufaces (PLS)
PLS takes PCR one step further, as it deals with the decomposition of both descriptive
information (characterization property data) and response information (attribute of physical-
chemical property data). Fig. 3.10 shows a PLS model being generated between descriptor data
P, which could be molecular descriptors or property descriptors, and response data Y, which
could be attribute or property information. For example, suppose a PCA is performed on both
the P and Y data, represented as blue arrows in Fig. 3.10. The descriptor scores u and plotted
against the response scores t, and the resulting least squares fit between the data sources is the
PLS model. This approach ensures the best possible correlation between the two data sets, but
not necessarily the best description of the P and Y data individually.
Geladi and Kowalski (1986) favor an iterative algorithm called NIPALS to handle the
development of these relationships. The technique uses relational forms to build outer relations
(PCA) of P and Y, inner relations between U and T and a least squares regression for the mixture
of the two. It is important to recognize that to maintain orthogonality, loading weights will also
be needed. The generalized approach is as follows:
?? lxnxlnx WUY ?? (3.38)
where ? is again the point estimate of response (i.e. attributes or physical-chemical properties),
U is a nxl matrix of the latent variable scores and W is a lx? matrix of the latent variable
loadings. As with the property descriptor latent properties, the latent variable representations of
the attributes will contain a significantly lower dimensionality. Regressing the lower
dimensional Unxl attribute scores against the Tnxm property descriptor scores of Eq. 3.35, results in
Eq. 3.39.
67
Figure 3.10: A PLS Regression Performed on the P Descriptor and Y Response Variables
(Gabrielsson et al. 2002).
?? mxnxmnx BTU ? (3.39)
where B is a set of mx??regressors describing the latent attribute-latent property descriptor
relationship. Iterative algorithms like NIPALS are utilized to determine the latent attribute
variables, latent property descriptor variables, and regressors, A detailed description of how the
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method calculates the latent variables can be found in Geladi and Kowalski (1986), Wold et al.
(1987), and Muteki et al. (2006, 2007).
3.2.4 Mixture Design and Optimization with Latent Variables
Finding the optimum mixture or set of candidate mixtures that deliver a set of consumer
attributes or physical-chemical properties is the customary objective of the chemical product
designer. Muteki et al. (2007) outlines four distinct problem formulations when using
multivariate data: (1) the design of blend fractions where the molecular architecture training set
is predefined, (2) the design and selection of undefined molecular architectures using a database,
(3) the design of blend fractions with a predefined molecular architecture under process
constraints, and (4) the design and selection of molecular architectures and process parameters.
The focus of this dissertation is on improving the first two cases, leaving the latter two cases for
future work.
For Case 1, traditional mixture models like Scheffe canonical and Cox polynomial
models are usually employed to investigate the relationship between the molecular architecture
blend fraction X and the final product attributes and/or properties Y. Kettaneh-Wold (1992) and
Eriksson et al. (1998) propose using PLS regression to fit both Scheffe linear and Cox
polynomial models for mixture data and show how it can effectively deal with colinearities. The
proposed mixture models are typically linear in nature, but recent work by Muteki and
MacGregor (2007) has shown that nonlinear relationships can improve accuracy in some cases.
However, although this approach can deal with mixture colinearity and provide good estimates
of the model parameters, it is limited to describing attribute-molecular architecture directly
which limits the chemical product design to only be a mixture of the chemical constituents used
in the training set.
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Case 2 is a recent exploration within the process systems engineering field and addresses
the shortcomings found in Case 1. Gabrielsson et al. (2003), Muteki et al. (2006, 2007), Garcia-
Munoz et al. (2010), and Solvason et al. (2008, 2009, 2010), all propose using a combination of
attribute-latent property and latent property-molecular architecture relationships to design
chemical products in the latent domain. The attribute-latent property relationship can be derived
from an attribute property descriptor relationship using either PCA or PLS.
YEPfY ?? )( (3.40)
Any nonlinearity in the attribute or physical-chemical property response should be
handled in this expression (Muteki and MacGregor 2007). The latent property-molecular
architecture relationship is derived from the property descriptor-molecular architecture
relationship in the same manner:
PEXfP ?? )( (3.41)
The most common relationship is ideal linear mixing:
XPPM ?? (3.42)
which can be reduced to a mixture of the latent score variables using decomposition.
XTTM ?? (3.43)
It should be noted that the loadings (L ) are treated as transformation functions between the latent
domain (T) and the property descriptor domain (P). Other types of nonlinear mixing have been
explored by Muteki and MacGregor (2007), but are beyond the scope of this dissertation.
To solve the design problem, Gabrielsson et al (2003) suggests using a training set of
experiments to decompose the design space into latent variables first, followed by conventional
mixture design to find the training set candidate with a latent variable structure that matches the
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optimum predicted structure. Experimental gradient and non-gradient optimization techniques
are then employed to explore the design space. Recently, Muteki et al.(2006, 2007) and Garcia-
Munoz et al. (2010) proposed a similar method that searches a much larger database of potential
chemical products to find latent variable matches of the original training set latent variables that
deliver the desired attributes or physical-chemical properties. This approach makes the
assumption that the attributes and/or physical-chemical properties of the chemical products in the
database correlate well with the training sets properties. The same assumption also applies to the
molecular architecture. Both of these conditions are naturally met in industrial applications
because of the importance of blending fractions, but require verification in a laboratory setting
(Muteki et al. 2006). A general discussion on the mathematical formulation of this optimization
problem is presented in Chapter 4, and a specific example is provided in Appendix A7.
There are several advantages to using the two stage approach in Case 2 over the single
stage approach of Case 1. First, it provides a direct interpretation of the effect of the property
descriptors (P) on final product attributes and physical-chemical properties (Y), that is, it
provides information on what property structure descriptors affect what attributes, something that
traditional mixture models using only the X and Y matrices struggle doing (Muteki and
MacGregor 2007). Second, Gabrielsson et al. (2003) and Muteki et al. (2006, 2007) have shown
that the two stage approach provides better estimates of the blend attributes and/or physical-
chemical properties (Y) than the traditional mixture models because of the use of the additional
structure descriptor information (P). Third, Garcia-Munoz et al. (2010) and Muteki et al.( 2006,
2007) have shown that new chemical products that have never been used in the past can be found
by matching the latent structure of the designed mixture ( ? ) using a mixture design and
optimization framework (Muteki and MacGregor 2007).
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However, while a powerful addition to the tool set available to the chemical product
designer, the two stage method presented in Case 2 suffers from the same limitations all database
searches do; namely, the design is only as good as the chemical products available in the
database. Logically, if the database is ill-formed or incomplete, the ?optimized? set of chemical
product candidates will only be local estimates of the true, global optimum candidate. This
observation is discussed in more detail in Chapter 4.
3.3. Summary
In conclusion, experimental design approaches have traditionally been the choice of
chemical product designers when understanding of the molecular architecture, consumer
attributes, or physical-chemical properties are poor. They offer real, tangible results and instant
validation of the process needed to manufacture the chemical product. While many advances
have been made to improve the efficiency of experiment based chemical product designs,
particularly for optimization, the designed candidates will always be subject to the ability of the
training set to capture the appropriate relationships between the molecular architecture, the
attribute, and physical-chemical properties. Hence candidate solutions generated with
experimental based design and optimization techniques will be subject to some uncertainty. In
most cases this is an acceptable solution, however for some classes of problems computer based
design and optimization methods are preferred. A discussion on computer based methods can be
found in Chapter 4.
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Chapter 4
Computer Aided Design of Chemical Products
Chemical product design and development involves the conversion of a conceptual idea
into a tangible, manufactured object with a defined molecular architecture. Design is sometimes
also referred to as formulation, a term typically associated with the pharmaceutical industry that
describes the method of combining different spatial arrangements of chemical species to make a
medicinal product. It encompasses the generation of a concept, followed by the gathering of
many sources of information, and the conversion of that information into a product with detailed
performance criteria including a rigorous process design (Venkatasubramanian 2008, Zhao et al.
2006). Due to the immense amount of information needed to make a targeted, stable specialty
chemical product, and noting that this information is often inter-related, formulation is often
approached using mathematical logic programming. Many types of product formulation
programs exist, each incorporating specific idiosyncrasies of different data sets, information
sources, and length-scales. To solve these types of problems, a mixture of talents and techniques
is needed to acquire information and distil it into useful programming constructs. Historically,
performing this task has been the study of computer engineers and scientists, but increasingly, as
chemical engineering moves from a 'data poor' to a 'data rich' discipline, integrating information
flow methods into conventional chemical engineering design procedures will become
increasingly important (Venkatasubramanian 2009).
73
4.1. Mathematical Formulation of Chemical Product Design
A mathematical view of the chemical product and process design problem is presented by
Gani (2004). :
)}(m a x{ xfdCF TO B J ?? (4.1)
? ? 01 ?xh (4.2)
? ? 02 ?xh (4.3)
? ? 0,3 ?dxh (4.4)
? ? 111 uxgl ?? (4.5)
? ? 222 , udxgl ?? (4.6)
33 uCxBdl ??? (4.7)
In this formulation, the chemical product design objectives are defined by an objective function
(Eq. 4.1) and set of constraints (Eq. 4.2 ? 4.7). For convenience, the molecular architecture
information (X) has been broken into x, a vector set of continuous variables representing mixture
compositions of molecular architectures, and d, a vector set of binary integer variables
identifying the presence of molecular descriptors such as atoms, molecular groups, and other
types of product architectures. This differentiation in molecular architecture improves the
efficiency of the algorithms used to solve the formulation and is discussed in more detail in the
next section. In a similar fashion, the relationships between the product?s attributes and
physical-chemical properties (Y) are differentiated using the following scheme: h1(x) are shown
as a set of attribute or property equality constraints related to process design parameters (e.g.
pressure, reflux ratio), and explicitly described by the chemical composition, h2(x) are the set of
74
attribute or property equality constraints explicitly described in terms of process models like
mass and energy balances, h3(x, d) are a set of attribute equality constraints related to molecular
structure, composition, and other information regarding product architecture, g1(x) are a set of
compositionally dependent inequality bounds on the process design specifications, and g2(x, d)
are a set of composition and molecular architecture inequality bounds on the product. Linear
inequality constraints are described explicitly by Eq. 4.7. It should be noted that the binary
descriptor variables (d) may differ depending on where they are used in the mathematical
formulation. When used in Eq. 4.1, descriptors represent the presence or absence of unit
operations while in Eq. 4.4, the descriptors represent molecular architecture features. Finally, the
term f(x) represents a vector of linear or non-linear attribute-chemical composition objective
functions of the system. A detailed list of the variations of the formulation are listed below
(Gani 2004):
1. Satisfy only Eq. 4.6. This type of formulation represents a database search for
validated molecular structures that satisfy the property constraints.
2. Satisfy only Eq. 4.4. This type is referred to as enumeration, where all candidate
molecules are generated.
3. Satisfy Eq. 4.4 and 4.6. There are two methods for solving this type of formulation.
The first is to independently apply constraint 4.4 and then 4.6 in series in a process
known as ?generate and test.? The second type of formulation is to apply constraints
4.4 and 4.6 in parallel. In this type, only molecular architectures that meet constraint
4.6 are generated. This is known as a ?reverse problem formulation? and will be
discussed in more detail in a Chapter 6.
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4. Satisfy Eq. 4.1, 4.2, and 4.6. This type of formulation identifies the optimum
molecular architecture that satisfies the product constraints.
5. Satisfy Eq. 4.2 ? 4.7. This type of formulation finds all molecular architectures that
satisfy not only product attribute constraints, but also process model constraints
simultaneously.
6. Satisfy all equations. This formulation finds the optimum molecular architecture that
satisfies both product and process constraints simultaneously.
When the formulation only contains linear functions, the problem can be performed by
commonly available programming techniques in familiar software such as Visual Basic for
Applications (VBA), MatLab, and/or LINGO. However, when the formulation contains
nonlinear functions, the solution to the problem is more complicated and may require specialized
solvers such as those found in GAMS. Furthermore, the methods listed by Gani (2004) for
solving the chemical product design problem can be distilled into 3 basic types: searching a
database (Variation 1), generating a candidate (Variation 3 or 5), and optimization (Variation 4
or 6). The first is related to searching for candidate molecules through a multitude of databases.
4.1.1. Searching a Database
By definition, a database is a large collection of data that is organized so that its contents
can be easily searched and retrieved, usually by a computer. Ugi et al. (1993) notes that database
searches are one of the four categories of chemical product design, according to definitions used
by chemo-informaticists. The other categories, computer-assisted structure elucidation (CASE),
computer-assisted synthesis-design (CASD), and 3D structure builders, focus on generating
molecular architectures to meet consumer attribute targets, optimizing the molecular architecture,
76
and developing the graphical user interface (GUI), respectively. CASE and CASD are discussed
in detail in Section 4.1.2 and 4.1.3. The last category, 3D structure builders, will not be
discussed in this dissertation.
Database searches rely heavily on the creation of a comprehensive database containing
information on molecular architecture, physical-chemical properties, and attributes. Historically,
a chemical product?s molecular architecture has been defined using characterization techniques
like infrared (IR) and mass spectroscopy and these were the first sets of data to be combined in a
database (Zemany 1950). To search the data, Ray and Kirsch (1957) developed the first
structure search algorithm based on atom-to-atom connectivity indices. Later, Gluck and
Morgan (1965) developed the first canonical form of the connection table using both atom-by-
atom and bond-by-bond descriptions. The American Society of Testing and Materials (ASTM),
the Chemical Abstract Service (CAS), the National Institute of Health (NIH), and many more
organizations have all developed versions of spectral databases and retrieval systems.
Obviously, searching all of these databases with user specific codes would become quite
cumbersome which is why expert systems utilizing computer code to call up specific functions
have been developed. The use of domain-limiting steps within the expert system further reduces
the computational complexity by focusing in on molecules that meet certain physical criteria.
An example of a database search expert system is presented below.
Molecular transport through the cell membranes is an important biological function. In
most cases the hydrophobicity of a molecular compound determines whether or not it will
permeate a cellular membrane (Wang et al. 1997). Hydrophobicity and other biological
activities have been related to the octanol-water partition coefficient (logKow) via quantitative
structure-activity relationship (QSAR) studies. A database of physical property data, including
77
the partition coefficient, can be searched using the software BioPath.Explore (Ertl 2010). It
contains the physical property data important in biochemical pathways for over 1,175 molecular
structures and 1,545 transformation reactions. Suppose it is important that no molecules are
introduced to our system with partition coefficients larger than -1.95. The database is searched
and only 4 potential molecules are found to meet this criterion, as shown in the following table.
Table 4.1. Chemical Structures Provided by BioPath.Explore that Meet logKow Constraints (Ertl
2010).
Record Formula Name LogKow
3 C26H40N7O18P3S (2S, 3S)-3-Hydroxy-2-methylbutanoyl-CoA -6.921
9 C25H36N7O19P3S (R)-2-Methyl-3-oxopropanoyl-CoA -5.032
16 C6H11O10P2 (R)-Mevalonate-5-diphosphate -1.977
48 C3H8O2 1,2-Propanediol -2.817
The first two results are coenzymes and the third contains diphosphate. If additional constraints
are added to the solution such that only carbon, oxygen, and hydrogen are utilized in the
solution, then only 1,2-propanediol becomes acceptable.
The potential drawback to this type of database search is obvious: the solution is never
complete (Ugi et al. 1993; Ugi et al. 1994); as new chemicals and reactions are added to the
database, the old solutions can no longer be considered optimal. Vladutz (1963) and Ugi (1994)
note that the chemical horizon of these programs is limited to those parts of chemistry that are
stored in databases. This is evidenced by the fact that BioPath.Explore currently stores only 20%
of the reactions identified in the Kyoto Encyclopedia of Genes and Genomes (KEGG) database.
Although this problem is somewhat mitigated by open-source database communities working to
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network as much data as possible, like POPE (Akkisetty et al. 2009), the method would benefit
from a more systematic approach to searching.
Likewise, databases are also subject to the quality of the information that is loaded into
them. For instance, when the octanol-water coefficient, LogKow, for the 1,2-propanediol is cross-
referenced with other databases, such as ChemSpider (Williams 2007), a different property value
of -0.92 is obtained. Thus, using database searches by themselves have a large capacity to miss
suitable molecular architectures that deliver physico-chemial properties and attributes desired by
the consumer. An alternative approach is to generate the database from a smaller set of validated
property structure descriptors.
4.1.2. Generating Candidate Solutions
The use of computers to solve the chemical enumeration problem, often referred to as
synthesis or design, is a long standing tradition in chemistry. Early works by Corey and Wipke
(1969), first showed that products and their synthesis pathways could be developed using logic
schemes for the computer combined with ?information controllers?, or heuristics, in the hands of
the researcher. The logic portion involves an automated procedure of enumerating the chemical
structure using chemical knowledge (e.g. structural descriptors). The absolute minimum input
into a logic based enumerator is the knowledge of the number of atoms, with no specification
regarding the atom type, etc. For instance, there are 24 atoms in d-glucose. Using only the
knowledge that these 24 atoms may be arranged in any order, then 24! = 6.024 x 1023 molecular
structures would be feasible (Ugi et al. 1993). A significant reduction in the number of possible
atomic arrangements can be obtained by applying chemical knowledge to the problem, beginning
with rules about atomic nuclear structure (e.g. d-glucose only contains atoms of carbon, oxygen,
and hydrogen). It significantly reduces the problem such that NC!NO!NH! = 6!6!12! = 2.483 x
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1014 possible candidates remain. A further reduction in the possible candidates can be obtained
by including information on the valence shell electrons involved in bonding. According to
molecular graph theory, the total sum of valencies of all vertices in a graph equals twice the
number of edges plus rings (Trinajstic 1992). Eljack et al. (2007) defined this observation in
terms of the number of unused bonds in a generated molecule called the Free Bond Number
(FBN).
RGNnFBNnFBN ???????? ???? ?? ?? 212 11
?
? ??
?
? ?
(4.9)
Here, ? is the atom type, ? is the total number of atom types, n? is the number of atoms of type ?,
and NRG is the number of rings. A FBN of zero indicates that the electron valency shells of all
atoms in the molecule have been satisfied, which, in most cases, indicates one of the minimum
energy configurations of the atoms in the molecule. Applying this criterion and constraining the
problem to default molecular size parameters listed in ProCAMD in ICAS 13 (Gani et al. 2010),
and constraining the problem to acyclic compounds, reduces the potential chemical set to
approximately 10,000 compounds. Although this step excludes special chemical structures like
radicals, it benefits the vast majority of chemical product designs by limiting the results to only
structurally stable chemical products.
Information about chemical structure and physical-chemical properties can also be added
to the problem using physical property models. After the chemical structures are generated, a set
of property filters can be applied to the candidates such that only molecules with appropriate
properties remain in the candidate set. This type of procedure is the ?test? part of the ?generate
and test? procedure. One way of testing the candidate list is to match solutions to products that
have been previously discovered and validated. In the example, the 10,000 potential candidates
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are checked against compounds having exactly 24 atoms in the BioPath.Explore database (Ertl
2010) and results in a significantly reduced candidate list of only 14 options. A second way of
testing the candidate list is to identify the candidates that deliver a desired physical-chemical
property or attribute. For example, suppose the process or consumer requires that the product
have at least a boiling point above 673K. Using the ACD/Labs property prediction tool (de Bie,
2010), boiling points are predicted for the candidate products. Four structures have boiling
points below the minimum and are removed resulting in a 10 structure candidate list.
It should be noted that the candidate set used in the example represents all
mathematically feasible solutions reduced by a set of heuristics provided by the chemical product
designer. The initial list of feasible solutions can be significantly reduced prior to screening by
using molecular group theory to only generate chemically feasible products with complete
functional groups and group combinations. Functional group theory and the property prediction
techniques based on group theory are discussed in more detail in Section 4.2.
Integrating the generation step with the screening step within a mathematical formulation
using an objective function is known as optimization. By definition, optimization determines the
best or ?optimum? product that meets the constraints on the design in the formulation. There are
two main methods for finding the best product: multi-criteria decision making (MCDM) and
mathematical optimization. In both instances, the optimality of different products is evaluated
using formulations with the objective function present (variations 4 or 6). However, the
mathematical complexity of optimization procedures can significantly reduce the efficiency of
chemical product design problems. In addition, since chemical product designs must be
validated with experiments, it follows that the more efficient ?generate and test? procedure
should be used to screen the candidates and optimization should be used to choose the best
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candidate to validate with experiments. A detailed discussion on MCDM and mathematical
optimization is available in Appendix A1.
4.2. Useful Property Prediction Techniques
Over the past century, consumer attributes and physical-chemical properties have become
an increasingly important element of industrial and engineering chemistry (O'Connell et al.
2009). In particular, the uses of physical-chemical properties have become more sophisticated as
new chemical products with precise molecular architectures have entered the marketplace. The
multitude of competing chemical mechanisms that control these properties has required chemical
product designers to become more efficient. Using integrated, computation based design
methods based on accurate data, chemical product designers are developing and applying
complex models that reveal the conditions needed to attain desired product content and quality
(O'Connell et al. 2009). These models take many forms, from ab initio deterministic models to
purely experimental modeling techniques like multivariate linear regression. Regardless of the
type of property model used, it is essential that they are accurate, reliable, and computationally
efficient throughout the entire design domain. It is also important to recognize that one
particular modeling technique should not become the panacea for all problems, rather the ?tool
driven? approach should be replaced with a ?tool box? approach, where the solution method is
determined by the features of the problem at hand (Venkatasubramanian et al. 2006;
Venkatasubramanian 2009).
The roles of property models are intrinsically tied to their use in product design. Since
product design may involve both process and molecular engineering, the use of property models
will be utilized in different roles depending on the application. Gani and O?Connell (2001)
describe three distinctive roles of property models in design: service role, advice role, and the
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integration role. In the service role, property models provide estimates where needed for the
process simulation models. These process models use two kinds of properties: measurables,
such as temperature, pressure, and composition, as well as conceptuals, such as enthalpy,
entropy, fugacity, or chemical potential (O'Connell et al. 2009). In the advice role, the property
models also provide feedback on combinatorial and compositional feasibility, accuracy, and
reliability. In the integration role, the property models are reconfigured so that the feasibility
constraints can be an integral part of the design process, controlling the design of possible
process and product configurations.
Although property models are generally recognized for their traditional service role, the
advice and integration roles are actually more powerful (O'Connell et al. 2009). These two roles
improve design by widening or narrowing the search space when required, thus increasing
solution method efficiency (O'Connell et al. 2009). However, since the property models may be
linear or nonlinear, the forward based advice roles can become computationally cumbersome.
For example, the most common model formulation is a generalized expression, often in the form
of corresponding states where the variables are scaled by pure component critical properties and
combining rules are used for mixtures, as is the case for the Peng-Robinson (PR) equation of
state. Although one of the most expansive equations of state in use, it can become tenuous to
calculate thermodynamic properties of ternary and larger mixtures, as well as molecules
exhibiting long range order. There is also no guarantee that algorithmic programs using the PR
will converge to its optimum solution due to its nonlinear nature. Hence, there is a need to
balance accuracy and reliability against computational efficiency when identifying the
appropriate property models in advice and integration roles. Other requirements, such as
dependability, accessibility, generality, and effort will also need to be evaluated (O'Connell et al.
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2009). This presents a real challenge for chemical product designers with limited knowledge and
experience whose goal is to develop quantitative descriptions of natural phenomena that are as
varied as the phenomena they describe. In order to maximize the effectiveness of the model and
minimize the researcher?s effort, a classification scheme was developed by O?Connell et al.
(2009). Using a similar scheme to that of general product design, property models and/or their
parameters are classified as follows: (1) retrieval from computerized databases, (2) estimation
via prediction, and (3) new data generation via simulation or experimental measurement (Xin
and Whiting 2000).
For Class 1 property models (i.e. retrieval from computerized databases), large amounts
of property data have already been gathered by researchers and placed in large databases. Search
algorithms can then be used to find models and their parameters. The concern, of course, is that
the needed property and molecular architecture will not have been captured, or that an error in
consistency, tabulation, and/or omission has occurred (O'Connell et al. 2009). To mitigate this
risk, Gani and O?Connell (2001) recommend that users should always verify the reliability of
vital data by comparisons among sources and by using fundamental variations with state
variables as given by thermodynamic equations and derivatives. In most cases, however, the
incompleteness of databases makes them best suited for advice roles in chemical product design.
The development of Class 2 property models (i.e. estimation via prediction) is the
approach to which chemical engineers are most accustomed. The approach to developing
property models and their parameters is shown in Fig. 4.1.
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Figure 4.1: Types of Methods to Find Parameters for Property Models (O'Connell et al. 2009).
In general, parameters and models exist under different levels of empiricism, moving from easy
to use data dependent regressed models whose application range is small, to computationally
complex models like ab initio which can be applied for any chemical system. In the middle are
the most common methods of Group Contribution (GC) and Quantitative Structure Activity
Relationships (QSAR) and Quantitative Structure Property Relationships (QSPR) using
Topological Indices (TI). These methods balance full empiricism with basic chemical theory to
adapt the rigorous equations that describe molecular architecture into a workable form suitable
for design.
For Class 3 property models (i.e. new data generation), the controlling mechanisms of the
system being studied are not well understood and the trend is to use experimentation or
molecular simulation. A more detailed description of property models based on experimentation,
specifically the data driven models derived from experimental design, chemoinformatics, and
quantitative structure property relationships, is discussed in Chapter 3. An alternative method to
experimentation is to use molecular simulation to compute properties using assumed inter- and
intramolecular potential functions, or force fields, to estimate a statistical mechanical
relationship (O'Connell et al. 2009). Used correctly, these methods have the potential to be more
cost effective than real experiments. Plus, they can be used to obtain information on systems
with molecular architecture that would be otherwise impossible to measure in the laboratory. Of
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course, the concern is that the models require stringent application of parameters specific to the
system they are modeling, so much that practical application within design has been limited. As
such they have been relegated to provide primarily qualitative insights on molecular architecture
behavior and their impact on physical and chemical properties. A detailed discussion on
simulation based property modeling is available in Appendix A4.
Figure 4.2: Property Model Development Scheme for Chemical Product Design (O'Connell et
al. 2009).
After the appropriate property model class is chosen, the model is developed to give
accurate results efficiently, converging to correct solutions given variable constraints on the
molecular architecture. Model robustness must also be developed and sensitivity mitigated in
order to avoid situations where parameter uniqueness cannot be established (Gani 2004;
O'Connell et al. 2009). When no, or only inadequate, models exist O?Connell et al. (2009)
proposes an efficient development procedure, shown in Fig. 4.2. The first step is to specify the
86
problem type, system classification, information sources and measures for the expected outcome.
Next accuracy requirements are established including their trade-offs with user effort. Then, the
model is built, solved, and verified for internal consistency.
Based on the criteria needed for chemical product design (e.g. efficiency and moderate
accuracy for screening, etc.) the appropriate physical-chemical property models should be based
on easy-to-use molecular construction techniques. Group contribution (GC) and topological
indices (TI) are two such techniques. Both types of models utilize quantitative structure property
relationships (QSPR) that have been previously developed using empirical relationships between
the molecular architecture and the physical-chemical properties found in large databases (Gani
2004).
4.2.1. Group Contribution Methods
Group theory is one way in which a chemical system can be described by a subset of
chemical fragments. The fragments are typically a combination of atoms or functional groups
with well-described impacts on the overall behavior of the molecule. One of the first methods to
quantify the direct relationship of the contributions of group fragments on a molecule?s physical
properties was the universal functional activity coefficient (UNIFAC) method. It was built for
the prediction of activity coefficients directly from atomic and functional group structural units
rather than from corrections on molecular solution interaction parameters (Fredenslund et al.
1975). For example, in the pure species activity coefficient expression (?iC) as defined in the
universal quasichemical (UNIQUAC) method, both the relative molecular volume (ri) and the
relative molecular surface area (qi) are described by the UNIFAC group contribution terms, Rg1
and Qg1, respectively:
87
???????? ?????? iiiiiiiCi HJHJqJJ ln15ln1ln ? (4.10)
?? w wwii xrrJ (4.11)
?? w wwii xqqH (4.12)
11 1 gg igi Rnr ??
(4.13)
11 1 gg igi Qnq ??
(4.14)
Extensions of this approach to other relevant properties, collectively referred to as Group
Contribution Method (GCM), has been the subject of research across many disciplines (Benson
1968; Ambrose 1978; Ambrose 1980; Joback and Reid 1987; Horvath 1992; Constantinou and
Gani 1994; Constantinou et al. 1995; Marrero and Gani 2001). The culmination of this work has
resulted in the current state-of-the-art expression for molecular group contribution shown in Eq.
4.15) (Marrero and Gani 2001). It should be noted that the property contributions of the
individual groups are determined by the regression of large data sets. As such, the application of
the approach is limited to only similar systems. However, the advantage of this approach is that
a relatively few number of groups can be combined to form a large number of molecules within
the system of study (Smith et al. 1996).
1st Order 2nd Order 3rd Order
??? ??? 3 332 221 11 g ggTg ggSg ggM PnwPnwPnP (4.15)
88
In first order GCM, a molecule is a collection of a number of various simple groups ng1 and their
respective property contributions Pg1 that, when combined, predict the properties of the complete
molecule PM. This simplistic relational structure means that the accuracy of the basic group
contribution methods is acceptable only in the design of simple monofunctional molecules that
do not contain strong group interactions. To increase the accuracy of GCM, further levels of
molecular groups have been developed (Constantinou and Gani 1994). Beyond the basic level,
known as first order groups g1, is the next higher level called second order groups g2. They
essentially represent different types of interactions among the first order groups and the effects of
certain molecular group combinations on the property of the final molecule. As such, second
order groups are built from first order groups (shown in Chapter 8) to describe property
corrections for interactions Pg2 where appropriate. ws represents a property dependent constant
that aides in centering the second order property correction. The combined first and second
order groups can provide a better description of compounds having many functional groups
and/or isomers. However, even the addition of second order groups may not be able to correct for
poly-ring compounds and open-chain polyfunctional compounds with more than four carbon
atoms in the main chain (Marrero and Gani 2001). Therefore, a further level of molecular groups
have been identified and their property contributions have been regressed (Marrero and Gani
2001). The formation of third order groups g3 is analogous to the second order groups, but the
focus is on correcting the property estimate PM with Pg3 and wT for multi-ring compounds, fused
ring compounds, and compounds with many functional groups in the structure. As with second
order groups, third order groups also have first order groups as their building blocks. The
combined expression shown in Eq. 4.15 can represent a significant number of molecular
89
structures with a good degree of accuracy, making it a useful tool in CAMD algorithms, as
discussed in Section 4.4.
Similarly, GCM can also be extended to the prediction of the molecular backbone of
polymer systems. This application of GCM is intended to provide a method of estimation and/or
prediction of physical-chemical properties of polymers, in the solid, liquid, and dissolved states
when experimental values are not available (Van Krevelen 1990; Bicerano 1996). The
contributions of the structural and functional groups in the polymer backbone will be different
than those for the same groups in different environmental surroundings. Also, the physical-
chemical properties of the polymer described by GCM are not only intrinsic to a substance, like
properties for small molecules, but are also somewhat described by the size, shape, and
morphologies of the nano-, micro-, and meso-structures of the system. To maintain the
efficiency benefits of GCM in this situation, a few fundamental property descriptors are used to
describe a set of derived physical-chemical properties based on established modeling interactions
(Van Krevelen 1990; Bicerano 1996). For example, one set of correlations uses molecular
weight of the repeat unit, length of the repeat unit, van der Waals volume of the repeat unit,
cohesive energy, and rotational degrees of freedom of the repeat unit (Seitz 1993) to describe
other properties such as the glass transition temperature (Tg) and melting point temperature (Tm),
which, in turn, can be used to describe modes of crystallization and morphology (Van Krevelen
1990). More detail can be found in Van Krevelen (1990) and Bicerano (1996).
In order to utilize GCM to its fullest extent, one needs large databases of molecular
structure data and a large number of group parameters. In most cases, Marrero and Gani (2001)
provide group information for small molecule thermodynamic information and Van Krevelen
(1990) and Seitz (1993) have information for polymeric group information. However, situations
90
will invariably arise that will require descriptions of groups, molecules, or systems beyond the
scope of these databases. It is increasingly common for these situations to be handled by new
cyberinfrastructure tools that link multiple databases. One new web based GUI,
www.chemeo.com, has been specifically created for the construction of group contributions from
the shared databases of the Computer Aided Process Engineering Center (CAPEC) at the
Technical University of Denmark (d'Anterroches et al. 2005; d'Anterroches and Gani 2006;
d'Anterroches 2010). Options for building models for various safety and health, energy, phase
change, critical, and other base properties are included. This new integrated database approach
will likely see refinement in the coming years as new cyber infrastructure initiatives begin to
bear fruit.
In spite of its usefulness, GCM has some inherent limitations; the most important of
which is the need for a substantial body of experimental data for a given property in order to
properly derive reliable group contribution values (Bicerano 1996). An alternative to GCM is to
use topological tools to describe a chemical?s structure and then relate these topologies to
physical and/or chemical properties. For chemical systems, the topology of a system is simply
the pattern of interconnections between atoms (Bicerano 1996). Information regarding the (1)
total number of atoms, (2) number of each type of atoms, and (3) the bonding between the atoms
is utilized to describe the system in a graphical form called a molecular graph (Kier and Hall
1986). The objects in the graph are called vertices and the lines used to connect the objects are
called edges. Here, the vertices may be atoms, molecules, molecular groups etc. and the edges
may be bonds, interactions etc. (Trinajstic 1992). For simplicity, molecular graphs are generally
represented as hydrogen-suppressed graphs where only the molecular skeletons without
hydrogen and their bonds are used. Due to the nature of the construction of topological indices,
91
smaller experimental databases can be utilized with similar prediction accuracy to that of group
contribution. A detailed discussion on topological indices can be found in Appendix A4.
4.3. Computer Aided Mixture Design (CAMbD)
Computer aided mixture design is the application of a computer algorithm to the
mathematical formulation of chemical product design. In most cases, the algorithm is a linear
program (LP) or nonlinear program (NLP), whose solutions are described in Appendix A1. The
inputs to the algorithm are chosen by the researcher. First the chemical constituents used in the
design are defined and then a suitable property model is selected whose type (linear or nonlinear)
determines the solution strategy the algorithm will use to solve the problem. A suitable mixture
range for each chemical constituent is usually specified by the researcher and used to constrain
the problem. Various combinations of chemical constituents are then generated and tested
against a consumer target property range.
UjjLj PPP ?? pj? (4.16)
The maximum and minimum of each chemical constituent is then returned as the solution
to the design problem.
UiiLi xxx ?? ui? (4.17)
Obviously, as the number of chemical constituents increases, the solution to the design
problem becomes cumbersome. However, the efficiency of this approach can be significantly
enhanced by reversing the order of the design by setting property targets first, then substituting a
linear mixture model and rearranging to explicitly describe a set of upper limits on the mass
fractions as a function of each pure component property.
92
ji
UjU
ji PPx ?
(4.18)
The minimum of the UL for each i chemical constituent across all properties j ? p is taken as the
overall maximum of that chemical constituent in the system.
Uii xx ??0 (4.19)
It should be noted that nonlinear property models placed in to Eq. 4.16 result in an implicit
description of the component domain that must be solved for x1U and is discussed in more detail
in Chapter 7.
4.4. Computer Aided Molecular Design (CAMD)
Computer aided molecular design is the application of the generate-and-test procedure
outlined in Section 4.1 to the design of molecular chemical products using combinatorial
property models like GCM. The forward based solution to the design problem begins by
specifying the number and types of groups to be utilized. Next, all potential combinations are
generated. The potential combinations are then screened for molecular stability using Eq. 4.9
adapted for GCM:
RG
F
g gg
F
g g NnFBNnFBN ???
??????? ???? ??
?? 212 11 1 111 1
(4.20)
where the FBN is the free molecular bond number of the formulation, ng1 is the number of
occurrences of group g1, FBNg1 is the unique free bond number associated with group g1, NRG is
the number of rings in the formulation, and F is the total number of first order groups.
Molecules having a FBN of 0 are then screened using a set of property targets based on
consumer or customer preferences.
93
UjMjLj PPP ?? pj? (4.16)
Solutions that meet the constraints of Eq. 4.16 are deemed candidate solutions.
The efficiency of this approach can be significantly enhanced by reversing the order of
the design by setting property targets first, then substituting a first order GCM model and
rearranging to explicitly describe a set of upper and lower limits on the number of similar groups
for each property.
???
???
?
??
1
1 int jg
UjU
jg P
Pn
(4.21)
???
???
?
??
1
int1
g
LL
g PPn
(4.22)
UggLg nnn 11
1 ??
(4.23)
The minimum of the upper limit for each group across all properties is taken as the overall
maximum number of groups of that group type in any molecule. The inverse does not hold,
since it is not a requirement that all dissimilar groups be included in the molecule. Hence, the
overall minimum number of groups of a specific group type will always be set to 0. For higher
order property description, the combination of Eq. 4.15 and Eq. 4.16 results in an implicit model
that must be solved for ng1U and is discussed in more detail in Chapters 6 and 8. The procedure
then continues by building combinations and testing the FBN. This type of approach is known as
reverse property prediction and is a partial component of the reverse problem formulation (RPF)
that is discussed in Chapter 5 and Chapter 6.
94
4.5. Summary
Although the information presented in this chapter was not a complete description of
property models, it nevertheless gives a good overview of what challenges exist. Property
models can be derived from experimental data, computed from fundamental chemistry concepts,
or as a mixture of the two. It is increasingly clear that both the methods and applications of
property models are continuing to change and, as such, researchers must adapt to their use. In
particular, alternative modeling methodologies capable of making use of these new property
models using knowledge based programming and decision schemes will be paramount
(Venkatasubramanian et al. 2006; Venkatasubramanian 2009). A great opportunity exists within
the discipline to venture into this domain, although, to do so, means overcoming the high barrier
of entry regarding the mastery of knowledge representation and search techniques, algorithm
engineering, databases, and certain misconceptions about what the intellectual challenges are
(Venkatasubramanian 2009).
95
Chapter 5
Motivation and Challenges
In the preceding chapters, it has been shown that the physical-chemical properties and
attributes of chemical products are controlled by a multitude of separate and often competing
mechanisms operating over a wide range of length and time scales. Quantifying these behaviors
and incorporating them within the chemical product design problem has generally been
performed on a case-by-case basis using either experimental or simulation techniques. The
inefficiency of these methods has limited researchers to screening only a handful of products
prior to product and process development, posing great economic risk of designing non-optimum
products. Finding a more efficient approach that is capable of screening large numbers of
chemical products prior to development is the focus of this dissertation. As suggested by the
National Research Council?s Committee on Integrated Computational Materials Engineering
(NRCCICME) this can be achieved through the integration of the process, product, and property
models utilized in design (Committee on Integrated Computational Materials Engineering 2008).
Because of the immense data structure, number and types of models, and the attribute and
physical-chemical properties utilized, such integration is a ?Grand Challenge? (Committee on
Integrated Computational Materials Engineering 2008).
To achieve integration, computational models used in product design needed to be
closely coupled with manufacturing and distribution techniques that exist at multiple scales. The
process systems research community has approached this problem by shifting focus from the
96
design of the physical form, function, and aesthetics of assembled products (e.g. mixtures of
molecular architecture) to the design of chemically formulated products (e.g. a single molecular
architecture) (Hill 2004). Chemically formulated products are products designed at the
molecular level to deliver a specific desired attribute that may exist at multiples scales such as
the quantum-scale, nano-scale, meso-scale, macro-scale, and mega-scale. Examples of
formulated products include pharmaceuticals (Solvason et al. 2009), bio-inspired devices
(McAllister and Floudas 2008), semi-conductors (Chalivendra et al. 2009), processed foods (Hill
2004), personal hygiene products (Hill 2004), and many more product types. Current state-of-
the-art product design, or formulation, uses a generate-and-test approach, where the ?generation?
is experimentation, prediction, or simulation and the ?testing? is a validation that the results of
the generation meet the target criteria. With the recent improvements in computational power,
simulation is increasingly becoming a viable tool in the generation step because of its wide
application range. By using a small number of parameter estimates, simulation techniques
compute molecular architecture information at the smallest scale, passing it to progressively
larger length scales, under high accuracy as shown in Fig. 5.1. However, each simulation can
potentially take weeks to perform. For instance, when determining micro-scale dependent
property values such as polymer permeation, Satyanarayana et al. (2009) utilized a united-atom
MD simulation of polyisobutylene under relaxed configurations based on models developed by
Tsolou et al. (2008) that took six weeks to calculate using a parallel program on a super
computer. Since, C320 polyisobutylene was one of 12 possible polyolefins identified by the Van
Krevelen (1990) group contribution method, it could conceivably take the better part of a year to
find a suitable polymer for this portion of the chemical product design, which is far from ideal.
97
Figure 5.1: The Computational Tools used in Chemical Product Design for the Transfer of
Information Between Multiple Scales using Nested Simulation.
Alternatively, experimentation could be utilized, but the methods for making and testing
new polymers can be just as time consuming as simulation in addition to containing large
amounts of uncertainty. For example, as discussed in Chapter 3, it is possible to utilize
experimental characterization data to describe the molecular architecture at each scale after the
various molecular level architectures have been identified. This approach requires the researcher
to create the chemical product, providing instant validation of the designed structures and
manufacturing processes. Thus in order to capture the exact polyisobutylene structure
Satyanarayana identified, an appropriately sized system would need to be suspended in the
correct solvent, molecularly characterized using nuclear magnetic resonance (NMR),
polymerized to the appropriate size, and tested for the necessary microstructure using scanning
electron microscopy (SEM). If any of these steps cannot be performed, it can lead researchers to
prematurely dismiss an optimum chemical product.
98
Finding a way to link experimental and simulation approaches would offer some
significant advantages in chemical product design. For instance the targeting of potential
breakthroughs using enhanced algorithms would reduce the number experimental or simulation
design procedures required for validation (Fermeglia and Pricl 2009). Furthermore, the ability to
bridge domains via models could potentially allow nested, nonlinear routines to be reduced to a
single sub-problem. Discovering the appropriate linkage is one of the keys to the methods
developed in this dissertation.
5.1. Current Limitations of Chemical Product Design
The traditional method of solving a chemical product design problem has been to
compute information at smaller length scales and pass it to models at larger length scales. In this
approach the linkage of the scales would be critical as would be the choice of basis set from
which the model parameters are estimated. While often the most accurate method for predicting
properties, simulation has two limitations: (1) it has an immense computational cost due to
hierarchical nesting and (2) it utilizes a priori knowledge of the molecular architecture (i.e. the
number and types of atoms or electrons present). The large computational cost typically
prevents an accurate modeling of mesoscopic structure such as the morphology of polymers
without the use of constraints that significantly limit the degrees of freedom in the simulation
(Fermeglia and Pricl 2009). Furthermore, when this method is integrated within the product-
process design framework (i.e. using a simulation model as Eq. 4.4) the computational
intensiveness exponentially increases yet again since each projected molecular architecture must
be simulated to determine its physical-chemical properties. To minimize the computational cost
in these types of problems, the property prediction simulations are typically approximated with
constitutive equations based on structure descriptor models such as property based group
99
contribution method (GCM), quantitative structure property/activity relationships (QSPR/QSAR)
using topological indices (TI), or chemometric based models as shown in Fig. 5.2. As shown in
pathway ?b? in Fig. 5.2, the introduction of structure descriptor models discussed in Chapter 4,
improves computational efficiency by avoiding nested simulation at the cost of prediction
accuracy. However, these descriptor models are still computationally expensive when utilized in
design algorithms because of their often highly nonlinear nature.
Figure 5.2: The Computational Tools used in Chemical Product Design for the Transfer of
Information Between Multiple Scales using: (a) Nested Simulation and (b) Structure Descriptor
Models.
Mol ecu lar
Inf orm ation
Proc es s
Inf orm ation
Y e ars
Hou rs
Min ut e s
S e co n d s
Micr ose co n d s
Nan o se co n d s
Pic o se co n d s
Femt o se co n d s
1A 1 nm 1 ? m 1mm m et e rs
Q u a n t um M e ch a ni cs
Time
Length
Mol ecul ar
Mech a n i cs /
Dyn amic s
Meso sc al e
mo d eli n g
FEA / P r o ce ss
Sim ul ati o n
E n gi n e er i n g /
Un it O p e r ati o n s
Desi g n
Atom Ba se d
Si mul atio n
Macrosca le
Si mul atio n
Co
m
put
atio
nal
C
ost
.
.
.
Propert y Mo dels
Int . Prop.
Se itz Prop.
Si gn atur es
GCM
QSAR /
QSPR
Ot her
P = f (Sp)
GCM
?
T.I
.
(a)
(a)
(a)
(a)
(b) (b)
100
The second major limitation of the current design methodologies is the inability to
accurately predict attribute responses as a function of molecular architecture. Consumer
attributes are a notoriously difficult set of properties to define, and, as such, rely heavily on
empirical correlative and causal relationships based on experimental study. Since a product?s
consumer attributes are what consumers and customers use to judge the success of a chemical
product, they serve as the ultimate validation step. In the NRCCICME (2008) published report,
it was suggested that the best approach to multiscale product design was to utilize complimentary
experimental and theoretical based models since no model, as of yet, has shown the ability to
operate accurately across the diversity of length scales (Committee on Integrated Computational
Materials Engineering 2008). Since Hill (2004, 2009) further states that the goal of the PSE
community is to develop computational product design tools to guide and focus experimentation;
it follows that the relationship between the underlying fundamental physical-chemical properties,
molecular architecture, and consumer attributes will involve empirical relationships. Some
authors, such as Smith and Ierapetritou (2009), formulated a chemical product design problem
using consumer attribute derived models as a function of consumer preference tests, as shown in
Fig. 5.3.
Figure 5.3: Schematic of Chemical Product Design using Consumer Preference (Smith and
Ierapetritou, 2009)
101
In this approach, the consumer attribute models are derived as needed from the consumer
preference and chemical constituent databases. The attribute models are then utilized as
additional property models in the mathematical formulation (i.e. Eq. 4.4). As with the database
approaches discussed in Chapter 3, the biggest limitation of this design methodology is the
inability to conduct true formulation design; only products and mixtures of products already in
the database can be solutions to the design problem. An alternative formulation that circumvents
this limitation is presented in Chapter 8. It utilizes GCM models and attribute-property
relationships to design chemical products without the use of a molecular architecture database.
The capability of the formulation is extended in Chapter 9 to handle characterization-based
group contribution models (cGCM) that are capable of describing a larger variety of molecular
architectures.
5.2. Reverse Problem Formulation
The focus of the dissertation is on the development of efficient design methods and
techniques for multiscale chemical product design based on consumer attributes. One of the
tools recently developed by Eden et al. (2003) is employed to assist in this endeavor: the reverse
problem formulation (RPF). The reverse problem formulation (RPF) is a procedure for
reconfiguring the generate-and-test design strategy into a more computationally efficient scheme
(Gani and Pistikopoulos 2002, Eden et al. 2003, Gani 2004). Using the duality of linear
programming, the RPF solves the design problem in the lower dimensional physical-chemical
property and/or consumer attribute domain first, and then generates only the components that
have properties that match the solution. For example, in a conventional product design problem,
candidate molecules are first generated from a set of structural features in a molecular design and
then fed as inputs to a process design simulation as shown in Fig. 5.4a.
102
Figure 5.4: A Product and Process Design Problem using (a) a Sequential Solution Format and
(b) a Reverse Problem Format (Simultaneous) (Eden et al. 2004; Gani 2004; Eljack et al. 2007).
D i s c r e t e D e c i s i o n s
( e . g . t y p e o f c o m p o u n d )
C o n t i n u o u s D e c i s i o n s
( e . g . o p e r a t i n g c o n d i t i o n s )
M o l e c u l a r D e s i g n
G i v e n a s e t o f m o l e c u l a r
g r o u p s t o b e s c r e e n e d
( b u i l d i n g b l o c k s )
D i s c r e t e D e c i s i o n s
( e . g . s t r u c t u r a l m o d i f i c a t i o n s )
C o n t i n u o u s D e c i s i o n s
( e . g . o p e r a t i n g c o n d i t i o n s )
P r o c e s s D e s i g n
G i v e n s e t o f c o m p o n e n t s t o
b e s c r e e n e d
( e . g . r a w m a t e r i a l s , M S A ? s )
O p t i m i z e d m o l e c u l a r
s t r u c t u r e s t o m e e t g i v e n s e t
o f p r o p e r t y v a l u e s
( e . g . p h y s i c a l , c h e m i c a l )
O p t i m i z e p r o c e s s o b j e c t i v e s t o
m e e t d e s i r e d p e r f o r m a n c e
( e . g . r e c o v e r y , y i e l d , c o s t )
D i s c r e t e D e c i s i o n s
( e . g . t y p e o f c o m p o u n d )
C o n t i n u o u s D e c i s i o n s
( e . g . o p e r a t i n g c o n d i t i o n s )
M o l e c u l a r D e s i g n
G i v e n a s e t o f m o l e c u l a r
g r o u p s t o b e s c r e e n e d
( b u i l d i n g b l o c k s )
D i s c r e t e D e c i s i o n s
( e . g . s t r u c t u r a l m o d i f i c a t i o n s )
C o n t i n u o u s D e c i s i o n s
( e . g . o p e r a t i n g c o n d i t i o n s )
P r o c e s s D e s i g n
G i v e n s e t o f c o m p o n e n t s t o
b e s c r e e n e d
( e . g . r a w m a t e r i a l s , M S A ? s )
O p t i m i z e d m o l e c u l a r
s t r u c t u r e s t o m e e t g i v e n s e t
o f p r o p e r t y v a l u e s
( e . g . p h y s i c a l , c h e m i c a l )
O p t i m i z e p r o c e s s o b j e c t i v e s t o
m e e t d e s i r e d p e r f o r m a n c e
( e . g . r e c o v e r y , y i e l d , c o s t )
I n t e g r a t e d d e s i g n c a p a b l e
o f a c h i e v i n g d e s i r e d
p e r f o r m a n c e
D i s c r e t e D e c i s i o n s
( e . g . t y p e o f c o m p o u n d )
C o n t i n u o u s D e c i s i o n s
( e . g . o p e r a t i n g c o n d i t i o n s )
M o l e c u l a r D e s i g n
G i v e n a s e t o f m o l e c u l a r
g r o u p s t o b e s c r e e n e d
( b u i l d i n g b l o c k s )
D i s c r e t e D e c i s i o n s
( e . g . s t r u c t u r a l m o d i f i c a t i o n s )
C o n t i n u o u s D e c i s i o n s
( e . g . o p e r a t i n g c o n d i t i o n s )
P r o c e s s D e s i g n
D e s i r e d p r o c e s s
p e r f o r m a n c e
C o n s t r a i n t s o n p r o p e r t y
v a l u e s o b t a i n e d b y
t a r g e t i n g o p t i m u m
p r o c e s s p e r f o r m a n c e
D e s i g n e d M o l e c u l e s
( e . g . r a w m a t e r i a l s , M S A ? s )
I n t e g r a t e d d e s i g n c a p a b l e
o f a c h i e v i n g d e s i r e d
p e r f o r m a n c e
D i s c r e t e D e c i s i o n s
( e . g . t y p e o f c o m p o u n d )
C o n t i n u o u s D e c i s i o n s
( e . g . o p e r a t i n g c o n d i t i o n s )
M o l e c u l a r D e s i g n
G i v e n a s e t o f m o l e c u l a r
g r o u p s t o b e s c r e e n e d
( b u i l d i n g b l o c k s )
D i s c r e t e D e c i s i o n s
( e . g . s t r u c t u r a l m o d i f i c a t i o n s )
C o n t i n u o u s D e c i s i o n s
( e . g . o p e r a t i n g c o n d i t i o n s )
P r o c e s s D e s i g n
D e s i r e d p r o c e s s
p e r f o r m a n c e
C o n s t r a i n t s o n p r o p e r t y
v a l u e s o b t a i n e d b y
t a r g e t i n g o p t i m u m
p r o c e s s p e r f o r m a n c e
D e s i g n e d M o l e c u l e s
( e . g . r a w m a t e r i a l s , M S A ? s )
Propert ies
I n t e g r a t e d d e s i g n c a p a b l e
o f a c h i e v i n g d e s i r e d
p e r f o r m a n c e
D i s c r e t e D e c i s i o n s
( e . g . t y p e o f c o m p o u n d )
C o n t i n u o u s D e c i s i o n s
( e . g . o p e r a t i n g c o n d i t i o n s )
M o l e c u l a r D e s i g n
G i v e n a s e t o f m o l e c u l a r
g r o u p s t o b e s c r e e n e d
( b u i l d i n g b l o c k s )
D i s c r e t e D e c i s i o n s
( e . g . s t r u c t u r a l m o d i f i c a t i o n s )
C o n t i n u o u s D e c i s i o n s
( e . g . o p e r a t i n g c o n d i t i o n s )
P r o c e s s D e s i g n
D e s i r e d p r o c e s s
p e r f o r m a n c e
C o n s t r a i n t s o n p r o p e r t y
v a l u e s o b t a i n e d b y
t a r g e t i n g o p t i m u m
p r o c e s s p e r f o r m a n c e
D e s i g n e d M o l e c u l e s
( e . g . r a w m a t e r i a l s , M S A ? s )
I n t e g r a t e d d e s i g n c a p a b l e
o f a c h i e v i n g d e s i r e d
p e r f o r m a n c e
D i s c r e t e D e c i s i o n s
( e . g . t y p e o f c o m p o u n d )
C o n t i n u o u s D e c i s i o n s
( e . g . o p e r a t i n g c o n d i t i o n s )
M o l e c u l a r D e s i g n
G i v e n a s e t o f m o l e c u l a r
g r o u p s t o b e s c r e e n e d
( b u i l d i n g b l o c k s )
D i s c r e t e D e c i s i o n s
( e . g . s t r u c t u r a l m o d i f i c a t i o n s )
C o n t i n u o u s D e c i s i o n s
( e . g . o p e r a t i n g c o n d i t i o n s )
P r o c e s s D e s i g n
D e s i r e d p r o c e s s
p e r f o r m a n c e
C o n s t r a i n t s o n p r o p e r t y
v a l u e s o b t a i n e d b y
t a r g e t i n g o p t i m u m
p r o c e s s p e r f o r m a n c e
D e s i g n e d M o l e c u l e s
( e . g . r a w m a t e r i a l s , M S A ? s )
Re verse
Pr o pert y
Pr ediction
Re verse
Pr oce ss
Simula t ion
(b)
(a)
103
In most cases, the majority of candidate solutions fail to meet the test criteria and are screened
out in either the property prediction or process simulation steps. This is a highly inefficient
process since resources are wasted calculating non-target solutions.
As shown in Fig. 5.4b (i.e. a RPF), target physical-chemical properties or attributes can
be determined from the solution of the reverse simulation problem and, as long as these targets
are matched, any number of property models may be used to generate the molecular architectures
at the various scales to ensure a solution (Eden et al. 2004). This design structure allows the
RPF to efficiently handle complex MINLP formulations that utilize non-linear property models
without requiring tedious solution strategies.
5.3. Challenges Addressed in this Dissertation
One of the goals of this dissertation is to demonstrate how the RPF can be used to bridge
the length domains shown in Fig. 5.1 by creating insightful shortcuts that allow macro-scale
constraint information to be mapped down to the appropriate molecular architecture scale. This
new source of information can then be used to reduce the number of potential candidates
evaluated by moving the testing or validation function from the macroscale down to the smaller
molecular architecture scale, prior to computation. A visual description of this method is shown
in Fig. 5.5. The traditional solution approach sends information peripherally through the length
scales in a nested simulation or experimentation approach (i.e. pathway ?a?); whereas the goal of
this work is to link the scales with a bridge through a ?property domain? that will allow for a
much more efficient solution that will significantly reduce the computational time of multi-scale
design problems (i.e. pathway ?b?).
104
Figure 5.5: A Multiscale Product Design Framework Showing (a) the Traditional Approach to
Computational design and (b) a RPF Approach Linking each of the Scales via a Common
Property Domain.
The major limitation in achieving this goal is finding the appropriate property domain in
which to decouple the constitutive equations describing the molecular architecture from each
other. Many potential property domains exist. For instance, the computational chemistry
methods described in Appendix A4 were developed to model the potential energies of individual
molecules and interaction forces between molecules. In contrast, most engineering design
problems are based on chemical and material constituents that have physical-chemical properties
that can be represented as a continuum. Developing a universal link between the two methods
has been the focus of thermodynamics and statistical mechanics whose parameters may represent
another potential property domain. Furthermore, the materials that have the biggest potential for
transforming our society have properties dependent on nanostructure and microstructure in the
QM, DFT Mo d el s
Ab I nit i o Met ho ds
(Q ua ntum S cal e)
Fu n d a m e nt al Prin ci ple s o f
Chemic al E n gi n e e rin g
Proc es s D esign
(M ac ro Sca le )
Chem om etric Mo d el s
Consumer Attrib utes
(M ac ro/Met a Scal e )
MM, M D, MC Si mul ati o n s
Atomist ic M etho ds
(Atom ic Sca le )
GCM, T o p ol o gical I n dice s
and C h em om etric Mo d el s
CA PD / CAMD
(M ol ec ula r S cal e)
Coa rse - Gr ai n in g a n d
Chem om etric Mo d el s
Mi c rostruc ture
( Mes o Sca l e)
PR OPERTI ES
(a)
(a)
(a)
(a)
(a)
(b)
(b)
(b) (b)
(b)
(b)
105
meso-scale domain. The molecular architecture at this length scale is too large to be
computationally designed solely with molecular simulations like molecular mechanics (MM),
molecular dynamics (MD), and monte carlo (MC), resulting in a trend toward estimations via
hybrid models and uncommon property domains. Some preliminary studies have shown that
sequential approaches using reverse problem formulations (RPF) on multiple property domains
offer computational advantages. For example, structure based property models like group
contribution method (GCM) (Marrero and Gani 2001), GCM plus connectivity indices (GC+)
(Chemmangattuvalappil et al. 2009; Chemmangattuvalappil et al. 2010), and signature descriptor
based TI methods (Chemmangattuvalappil et al. 2009; Chemmangattuvalappil et al. 2010) have
expanded the number and types of properties they can describe, which allows them to replace a
number of simulation models. However, accuracy limitations will constrain these approaches to
short range order applications (e.g. less than 60-100 atoms). To address longer range order, a
partial RPF has recently been combined with grid technology for polymer design by
Satyanarayana et al. (2009). As discussed in the polyisobutylene example, this approach uses
inverse GCM models to target monomer precursors with desirable properties followed by
simulation of the polymer structures to evaluate their meso-scale impacts. Although an
improvement in design at the meso-scale, the method is not simultaneous, and is therefore
computationally inefficient. As a result the method has been limited to simple polymer structure
investigations and is unable to address more complex nano- and micro-structure arrangements. In
conclusion, finding a suitable property domain to perform the design remains case-specific and
subject to the researchers desire to balance efficiency vs. accuracy. Improving the objectivity of
this approach will be necessary to improve multiscale chemical product design.
106
Another goal of this dissertation is to develop a way to utilize experimental data,
parameters, and models in a RPF. The reason behind choosing to work with experimental
models comes from the nature of the intended end-use of chemical product design algorithms.
Since the effectiveness of a chemical product is most often determined by its consumer
attributes, and since the consumer attributes are most likely quantified using data from consumer
preference tests, then it follows that any framework should at least be partially based on
experimental data. This observation does not preclude using the more common physical and
chemical properties found in process design in the framework, rather, it only means that
experimental data needs to be integrated into the design in some fashion. Furthermore, while the
use of data places constraints on the application range of the tools developed for multiscale
product design, it does not affect the fundamentals of the methods developed and could be
applied to systems described by deterministic relationships (e.g. computational chemistry
methods) if desired.
In this dissertation, three methods for improving the solution strategy for solving
multiscale chemical product design problems are developed. All three methods use the newly
developed property clustering (PC) algorithm (Shelley and El-Halwagi 2000; Eden et al. 2004).
Property Clustering (PC) is a mathematical technique that decomposes the problem as a
Euclidean vector consisting of unit property clusters that describe orientation of the property in
the domain space and a scalar called the augmented property index (AUP) that describes its
magnitude. A detailed discussion on the benefits of PC and its utilization in design algorithms is
provided in Chapter 6.
Attribute-computer aided mixture design (aCAMbD) is the first of the three methods
developed to address the current limitations in multiscale chemical product design. It recognizes
107
that through the use of transformation functions, experimental design techniques and data-driven
property/attribute models can be directly utilized in the PC algorithm to solve chemical product
design problems using a RPF. Approaching the design problem in this manner significantly
reduces the computational complexity of the problem by using the duality of linear programming
to solve the design problem in the lower dimensional property or attribute domain instead of the
high dimensional component space. The method works by first solving for a target attribute or
physical-chemical property region and then generating solutions using predictive models based
on design of experiments (DOE) that have attributes that fall within that region. Furthermore,
for design of three properties, the design space and subsequent solution can be visualized on a
single diagram, regardless of the number of chemical constituents in the design, which provides
insight into the nature of the problem. A detailed discussion of this method, including how it is
built to utilize data driven property models, regressor based property clusters, and the reverse
problem formulation, as well as a case study on a spun yarn polymer blending is discussed in
Chapter 7.
Recognizing that aCAMbD is limited to only mixture designs, and that the effectiveness
of most chemical products is determined by its consumer attributes, which seldom have adequate
quantitative structure-property (QSPR) or quantitative structure-activity relationships (QSAR), a
second new method was developed to map consumer attribute data into a more conventional
physical property domain where group contribution methods (GCM) are used to find new
chemical structures. By mapping attribute information to a physical-chemical property domain
with strong QSAR/QSPR description, multiple sources of scientific knowledge, information, and
DOE data can be combined to design chemical products without the use of a database or a priori
knowledge of the molecular architecture. The method specifically uses property clustering (PC)
108
in a reverse problem formulation (RPF) to reduce the computational complexity of the problem
and provides insight into the nature of the relationships between molecular architecture and
consumer attributes. The method is described in more detail in Chapter 8 where it is applied to
the design environmentally benign refrigerants.
A third opportunity for improving multiscale chemical product design comes from the
recognition that the descriptive data used in chemical product design is length dependent; each
length scale has specific molecular architectures that dominate the chemistry at that scale.
Quantification of these architectures using characterization techniques results in large, highly
correlated data sets that describe the products molecular architecture. Rather than solving the
molecular design using this high dimensional data set, as is done using nested simulation and
experimentation, decomposition techniques can be applied to the correlated data set to find a
lower dimensional latent variable domain more conducive to chemical product design. The
relationship between the latent variable domain and the consumer attributes is then estimated
using chemometric techniques (e.g. MLR, PLS, etc.). To perform a chemical product design in
this domain requires the creation of a new characterization based group contribution method
(cGCM) that can take advantage of the additional molecular architecture information provided
by characterization. Information from the molecular scale on short range order, such as group
structure, conformation, and stereoregularity, is combined with information from the micro-scale
on long range order, such as the size, shape, and aspect ratio of particles within a solid matrix
within a single latent variable domain, which drastically improves the computational efficiency
of multiscale chemical product design. As demonstrated in Chapter 9, the characterization based
computer aided molecular design (cCAMD) method is built specifically to utilize IR/NIR
spectroscopy data, PCA decomposition techniques, MLR attribute-latent variable relationships,
109
latent variable property clusters, and the reverse problem formulation and results in a framework
that decouples multiscale design problems from their respective scales, solving them in a low
dimensional, computationally efficient manner. A constrained pharmaceutical excipient design
for direct compressed acetaminophen tablets is used to illustrate the method.
In conclusion, it is shown in Chapter 7, Chapter 8, and Chapter 9 how a hybrid approach
that combines Class 2 models (e.g. GCM, etc.) with experimental data generation and
decomposition techniques (e.g. DOE, IR/NIR spectroscopy, PCR, and PLS, etc.) can
dramatically improve the computational efficiency of multiscale chemical product design
problems. This is a central result of the multiscale chemical product design framework presented
in this dissertation. Many opportunities exist for extending this work to include other
characterization tools and process design and configuration parameters to develop a truly unique
and powerful design methodology within the PSE community.
110
Chapter 6
Methods for Multiscale Chemical Product Design
One of the most difficult tasks in multi-scale chemical product design is to balance the
efficiency of experimental methods against the accuracy of simulation techniques in the
computer-aided mixture and molecular designs. It was shown in Chapter 5 that the reverse
problem formulation (RPF) is particularly adept at achieving this goal by decomposing the
computational difficulty of the design problem into two sub-problems, reverse simulation and
reverse property prediction, and matching their solutions to achieve results for the chemical
product design (Eden et al. 2003). Another tool that aides in the solution of multiscale chemical
product design problems is property clustering (PC). Property clustering converts the attribute
and physical-chemical properties into conserved surrogate attribute or property clusters capable
of describing the design space under reduced dimensionality. Furthermore, property clustering
provides excellent visualization of the decomposition and algorithmic solution processes and
provides insights into the effects of model parameters on the design. The PC methods for
solving mixture and molecular design problems form the basis for the three methods developed
in this dissertation.
6.1. Property Clustering Algorithms
Property clustering was developed as a tool for tracking properties in a conserved manner
by Shelley and El-Halwagi (2000). It was later applied to process and product design by Eden et
111
al. (2003), Eljack et al. (2007), and Solvason et al. (2008). As a design tool, the technique uses
the RPF to challenge conventional design by solving the design problem in a property cluster
domain rather than the chemical constituent domain. The conserved surrogate property clusters
are generated from the attributes or physical-chemical properties important to the chemical
product. They are described by property operators, which have linear mixing rules, even if the
operators themselves are nonlinear. In equation Eq. 6.1, the property, y = P, is described by a
linear property operator expression:
? ? ? ??? ??
u
i iijjMjj xyy 1 ??
(6.1)
where ?j(yj)i is the property operator of the pure component property response yj for chemical
constituent i, xi is the mass (or mole) fraction of constituent i, and ?j(yj)M is the property operator
of the mixture. In many cases the property operator is simply the linear expression of a common
physical-chemical property. For example, if the property operator describes density, then to
meet the linear criteria imposed by equation Eq. 6.1 we would use specific volume as the
property operator, ignoring any interaction effects from mixing, as shown in Eq. 6.2 and 6.3.
? ? iijj y ?? 1?
(6.2)
?? ??
u
i iiM x1
11 ?? (6.3)
A list of physical-chemical properties commonly used in chemical product design are given in
Table 6.1 (Marrero and Gani 2001). Each universal parameter (e.g. Tm0, Tb0, Tc0, Pc1, Pc2, Vc0,
Gf0, Hf0, Hv0, Hfus0) and molecule specific parameters (e.g. Tm, Tb, Tc, Pc, Vc, Gf, Hf, Hv, Hfus) was
derived from the ICAS property database (Marerro and Gani 2001).
112
Table 6.1: Common Physical-Chemical Properties used in Chemical Product Design
Property (y) Property Operator (?(y))
Normal Melting Point (Tm) exp(Tm/Tm0)
Normal Boiling Point (Tb) exp(Tb/Tb0)
Critical Temperature (Tc) exp(Tc/Tc0)
Critical Pressure (Pc) (Pc ? Pc1)-0.5 ? Pc2
Critical Volume (Vc) Vc ? Vc0
Standard Gibbs Energy at 298K (Gf) Gf ? Gf0
Standard Enthalpy of Formation at 298K (Hf) Hf ? Hf0
Standard Enthalpy of Vaporization at 298K (Hv) Hv ? Hv0
Standard Enthalpy of Fusion (Hfus) Hfus ? Hfus0
It is often difficult to find adequate linear property operator expressions or models for the
attributes and physical-chemical properties used in the chemical product design and the
experimenter must search for linearization functions like those proposed by Johnson and
Wichern (2007) and Solvason et al. (2009) or use an approximation for which the benefits of the
PC+RPF method versus the loss of solution certainty must be weighed (Solvason et al. 2008).
This result often leads the experimenter to limit the application of this method to high volume
screening designs (Gate 1 in Fig. 2.5).
After the property operator equations are defined, the method is universal. As shown in
Eq. 6.4, the property operators are non-dimensionalized by dividing by a reference property
operator (Eden et al. 2003).
113
? ?? ?
refjj
ijjji yy???? (6.4)
This step serves the purpose of scaling the property design space to facilitate a better graphical
understanding of the problem or, as shown later in Chapter 7, is used to ensure a solution when
the property operator equations contain negative parameter estimates. The non-dimensionalized
properties are then summed into the Augmented Property Index (AUP) (Shelley and El-Halwagi
2000).
?? ??
p
j jiiAUP 1
(6.5)
A cluster is then defined by dividing the non-dimensionalized property by the AUP, as shown in
Eq. 6.6.
i
jiji AUPC ?? (6.6)
By definition, property clusters must vary between 0 and 1 (Shelley and El-Halwagi 2000; Eden
et al. 2003). This means that the cluster domain is capable of describing all possible property
values associated with a chemical product design, independent of molecular architecture. For
instance, the internal energy of an arbitrary system can be estimated from a path integral Monte
Carlo summation of molecular energies. Using PC, both the molecular energies of a unit cell and
the macroscale internal energy can be represented using the same cluster domain. Shown in Fig
6.1 is a ternary diagram, or simplex, of the cluster domain (C1, C2, and C3) of three properties
(P1, P2, and P3) that have been identified as key to the design of a chemical product. Each vertex
represents a single property cluster having the value of 1 meaning that the domain constrained by
the boundaries of the cluster diagram is the entire range of all current and future molecular
114
architectures. As a result, the property cluster domain can be used to independently and
efficiently evaluate a product?s molecular architecture at multiple scales. This concept is
explored in more detail in Chapter 9.
Figure 6.1: A Property Cluster Simplex Diagram with Property Clusters at the Vertices, Pure
Component Effects Represented as Independent, Discrete Stream Points, and the Product Targets
Represented as a Sink Region (Eden et al. 2003).
Also shown in Fig. 6.1 are the pure component property clusters of two chemical
products, S1 and S2, as well as a feasibility region, denoted as ?sink.? The pure component
clusters are calculated from three property values, P1, P2, and P3, of each chemical product (e.g.
S1 and S2) using Eq. 6.2, 6.4, and 6.5. The feasibility region is a result of the chemical product?s
target specifications, defined by upper (U) and lower (L) limits on the properties, and is
described by six unique clusters that maximize its size. If a chemical product falls outside the
feasibility region then it is not a valid solution to the chemical product design problem. The
0 . 8
0 . 7
0 . 6
0 . 5
0 . 4
0 . 3
0 . 2
0 . 1
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
0 . 6
0 . 7
0 . 8
0 . 9
0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9
0 . 9
C
3
C
2
C
1
S
1
S
2
S
MI X
Si n k
???
???
115
feasibility clusters are found by solving Eq. 6.5 and 6.6 using the following dimensionless
properties (Eden et al. 2003):
Pt 1: (?1L, ?2L, ?3U) ; Pt 2: (?1L, ?2U, ?3U) ; Pt 3: (?1L, ?2U, ?3L) ;
Pt 4: (?1U, ?2U, ?3L) ; Pt 5: (?1U, ?2L, ?3L) ; Pt 6: (?1U, ?2L, ?3U) (6.7)
It should be noted that for each new property added to the design, two new feasibility cluster
points are added to describe the feasibility region.
In some situations, the property clusters of the feasibility region or the chemical
constituents are negative or difficult to visualize. This is a result of the parameters used in the
linearized property models and is often difficult to avoid. To handle this situation, a property
reference algorithm was written to find the set of reference property operators that give the
largest values of the AUP while maintaining positive estimates for the pure component values:
Max ?
?
u
i iAUP1
(6.8)
s.t. 0?iAUP ui? (6.9)
Xi MAUP? ui? (6.10)
? ? 0?
refjj y?
pj? (6.11)
? ?? ?
refjj
ijjji yy????
(6.12)
?
? ??
p
j jiiAUP 1
(6.13)
116
A more detailed analysis of the reference property algorithm is available in Chapter 7. The
procedure for converting a set of chemical product design equations and parameters for use in a
computer aided mixture design CAMbD with property clusters is presented in Table 6.2.
Table 6.2. The CAMbD Property Clustering Conversion Algorithm
Step Description Equation
1 Linearize Property Prediction Models 6.1
2 Guess Property Operator Reference Values and Calculate Pure Component Property Operator Values 6.4
3 Convert UL and LL Feasibility Constraints to Property Operators 6.7
4 Calculate the AUP of the Pure Component Property Operators and the Feasibility Constraint Property Operators 6.5, 6.7
5 Calculate the Pure Component Clusters 6.6
6 Calculate the Feasibility Region Sink 6.6, 6.7
7 Run Reference Optimization Algorithm for Non-Positive Clusters 6.8-6.12
8 Convert all Clusters to Cartesian Coordinates and Plot 6.23-6.26
117
Figure 6.2: The Property Clustering Conversion Algorithm for use in Computer Aided Mixture
Design (CAMbD).
Eljack et al. (2007) extended the property clustering method to also include models based
on group contribution methods (GCM). For this situation, Eq. 6.1 is rewritten in terms of the
number of molecular groups and parameterized accordingly (Marrero and Gani 2001). If gjy)( is
the contribution to the attribute or physical-chemical property j from group g, ng is the number of
occurrences of that group in the molecule, then the molecular property operator ?jM can be
defined as the following:
g
n
g gj
MjMjMj nyy g ?? ?
? 1
)()( ?? (6.14)
118
where g is the type of group and r is the total number of first order group types. The normalized
property operator may be obtained by dividing ?jM with a reference value.
)(
)(
jrefj
jMjMj yy???? (6.15)
The Augmented Property Index AUPM, for each molecular fragment is defined as the summation
of all the j dimensionless property operators:
?? ??
p
j
MjMAUP
1
(6.16)
Finally, the molecular property cluster CjM is defined as
M
MjM
j AUPC ??
(6.17)
The first order, group-specific molecular property operators described by Marerro and Gani
(2001) in Table 6.1 can then be utilized in a computer aided molecular design (CAMD). Table
6.3 and Fig. 6.3 describe the property clustering conversion for systems described by group-
based, attribute or physical-chemical property models.
Table 6.3. The CAMD Property Clustering Conversion Algorithm
Step Description Equation
1 Linearize Group Based Property Prediction Models 6.14
2 Guess Property Operator Reference Values and Calculate Group Property Operator Values 6.15
3 Convert UL and LL Feasibility Constraints to Property Operators 6.7
4 Calculate the AUP of the Group Property Operators and the Feasibility Constraint Property Operators 6.16
119
5 Calculate the Group Clusters 6.17
6 Calculate the Feasibility Region Sink 6.7, 6.17
7 Run Reference Optimization Algorithm for Non-Positive Clusters 6.8-6.12
8 Convert all Clusters to Cartesian Coordinates and Plot 6.23-6.26
Figure 6.3: The Property Clustering Conversion Algorithm for use in Computer Aided
Molecular Design (CAMD).
It should be noted that the molecular property operators defined in the above equations have
limited applicability as they are formed considering only the contributions of first order
molecular groups. The first order groups have limited accuracy when predicting the properties of
120
polyfunctional molecules with more than four carbon atoms in the main chain and cyclic
molecules (Marrero and Gani, 2001). Also, first order groups cannot capture proximity effects or
differentiate between isomers (Kehiaian 1983; Wu and Sandler 1989; Wu and Sandler 1991). To
make this approach more robust, the molecular property operators have to be more accurate.
Fortunately, higher orders of molecular groups have been introduced into GCM by Constantinou
and Gani (1994) and by Marrero and Gani (2001). In their approach, the higher orders of
molecular groups are formed with first order groups as their building blocks and property
contributions are estimated for each group. In Chapter 8, a method utilizing first and second
order group contribution within the clustering framework is presented.
6.2. Computer Aided Mixture Design (CAMbD) with Clusters
There are two primary methods for performing chemical product design using property
clustering with the reverse problem formulation: computer aided mixture design (CAMbD) with
clusters and computer aided molecular design (CAMD) with clusters. The first method is useful
for finding mixtures of autonomous chemical products (e.g. molecules) while the second method
is capable of finding mixtures of molecular architectures within a single chemical product (e.g.
molecular groups). Shelley and El-Halwagi (2000) have shown that a property cluster is
conserved through mixing which provides an opportunity to track the attribute or property
response when different molecules or molecular architectures are mixed together. This intra-
conserved nature of property clusters allows for the fractional calculation of any mixture in the
cluster design space.
Suppose a CAMbD is being conducted with a linear property operator expression,
described by Eq. 6.1, and is used to relate properties P1, P2, and P3 to the molecular architecture
of chemical products. Using property clustering (PC), Eq. 6.1 can be rewritten as Eq. 6.18:
121
?? ??
u
i jiijM CC 1 ?
(6.18)
where the relative cluster arm ?i is defined using the proof for inter-stream conservation of
clusters given by Eden et al. (2003) and Shelley and El-Halwagi (2000). The relative cluster arm
?i is related to changes in the augmented property index (AUP) during mixing, as shown in Eq.
6.19:
M iii AUP
AUPx ??? (6.19)
where the AUPM is defined in Eq. 6.20:
?? ??
u
i iiM xA U PA U P 1
(6.20)
Each cluster arm ?i can be found visually from the relative lengths of the lever arms as shown in
Fig. 6.1. Alternatively, they can be calculated using Cartesian coordinates, as shown by Eden et
al. (2003). For the three property system described by a u=2 component system in Fig. 6.1, the
cluster arms for the point Smix are as follows:
? ? ? ?? ? ? ?
21,2,21,2,
2,2,2,2,
1
CCCCCCCC
MCCCCMCCCC
YYXX
YYXX
???
?????
(6.21)
? ? ? ?? ? ? ?
21,2,21,2,
2,1,2,1,
2
CCCCCCCC
MCCCCMCCCC
YYXX
YYXX
???
?????
(6.22)
where the Cartesian coordinates (XCC, YCC) are estimated from either the clusters (i.e. Eq. 6.23
and 6.24) or from the dimensionless property operators (Eq. 6.25 and Eq. 6.26).
122
iiiCC CCX 21, 5.0 ???
(6.23)
iiCC CY 2, ?
(6.24)
i
iiiCC A U PX 21, 5.0 ????? (6.25)
iiiCC AUPY 2,
??
(6.26)
The solution to the design problem can be found by first testing each pure component
against the feasibility region and then each binary mixture, ternary mixture, and so on until all
combinations have been generated. A successful solution must meet three criteria (Eden et al.
2003; Solvason et al. 2009).
Rule 1. Cluster value of the source (or mixture of sources) must be contained within the
feasibility region of the sink in the cluster domain.
Rule 2. The values of the Augmented Property Index (AUP) for the source or mixture of
sources must fall within the AUP range of the feasibility region.
Rule 3. The AUP of the candidate must match the AUP of the sink at the candidate point
as calculated from the sink boundary points.
In addition to these three rules, the experimenter may wish to apply supplemental constraints,
such as limiting the number of pure components tested, the number components evaluated in the
mixtures, or the types of mixtures considered. These constraints assume an optimization role in
the design and are used to further reduce the generated list of potential candidate mixtures.
The method for testing the pure components is straightforward. As long as the pure
component cluster falls within the cluster range of the sink, as defined by Eq. 6.7, and the pure
123
component AUP falls within the AUP range of the sink, then the AUP of the pure component will
match the AUP of the sink and it will be deemed a potential candidate solution for the chemical
product design. In Fig. 6.1, neither S1 nor S2 were found to meet the design criteria and are
excluded.
Any of the previously identified candidate clusters will naturally be able to form binary
mixtures with any of the other pure component cluster to meet Rule 1. The fractions of each
binary mixture are determined by the location where Rules 2 and 3 are met within the feasibility
sink. Since the number of pure component candidates are relatively small compared to the
number of potential mixtures, most candidate mixtures will involve pure component clusters that
reside outside of the feasibility region. A candidate mixture will not exist if a line segment
connecting the pure component clusters does not intersect the feasibility region sink. For
example, the three property system in Fig. 6.1 clearly shows that a mixture between S1 and S2
cannot be excluded. Rules 2 and 3 can be used to validate the existence of the candidate mixture
by calculating the mass factions of each chemical constituent. Alternatively, the Cartesian
coordinates of the clusters of both the pure components and feasibility region sink can be used to
test whether a line segment between the pure component cluster intersects the feasibility region.
For example, a line segment between S1 and S2 can be written as follows:
LLL KyBxA ?? (6.27)
1,2, CCCCL YYA ??
(6.28)
2,1, CCCCL
XXB ??
(6.29)
For each Cartesian coordinate representation of the feasibility cluster point, the constant (KL) is
calculated and compared against a reference (KLref), defined as:
124
1,2,2,1, CCCCCCCC
r e fL YXYXK ????
(6.30)
If feasibility clusters result in all KL < KLref or all KL > KLref, then the segment does not cross the
feasibility region and no mixture is possible. Otherwise, a mixture may be possible and Rules 2
and 3 will need to be validated. In the example in Fig. 6.1, a mixture between S1(0.3, 0.2) and S2
(0.65, 0.5) was found to have Kref = 0.02 and the six feasibility cluster points (Pt. 1-6) were found
to have KL = 0.07, 0.05, 0.0, -0.03, -0.01, and 0.04, respectively, so the mixture cannot be
excluded by Rule 1. To test Rule 2, the intersections of the line segment with the feasibility
region are estimated visually or calculated using linear algebra in Cartesian coordinates as shown
in Eq. 6.31, followed by estimates or calculations of the cluster arm fractions by Eq. 6.21-6.22.
1,,
1,,
1,2,
1,2,
CCMCC
CCMCC
CCCC
CCCC XX YYXX YY ????? (6.31)
The chemical constituent mass fractions of component S1 is found by rearranging Eq. 6.19 and
Eq. 6.20 to get Eq. 6.32 with the balance being S2.
112 211 )( A U PA U PA U P
A U Px
i ???
?? ? ? (6.32)
The AUP range of the mixture is found using Eq. 6.20 on each of the boundary intersections. If
the AUP meets Rules 2 and 3, then the mixture range between the intersections is deemed a
candidate solution. In situations where only one or neither of the boundary intersections meet
Rules 2 and 3, a new mixture boundary point is estimated by setting the AUPM of mixture to the
AUPF of the feasibility region at the intersection points and back calculating the cluster arms and
mixture fractions of the chemical constituents. The AUPF is chosen from the set of AUP of the
cluster feasibility region points by determining which property cluster exhibits the largest change
across the mixture and holding the remaining clusters constant. Alternatively, AUPF can be
125
determined by mixing the feasibility region points with the mixture design to determine the point
at which the mixture design satisfies the feasibility region. Once determined, the mass fraction is
calculated using Eq. 6.33.
wi
wMi A U PA U P A U PA U Px ??? (6.33)
Negative cluster arms or mixture fractions indicate an infeasible solution and the mixture is not
selected as a candidate for the chemical product design.
For ternary or larger mixtures, it is appropriate to start with the previously identified
binary candidate solution clusters and add another component to the solution. Any pure
component will be able to form a ternary mixture with a binary candidate solution so long as the
binary candidate solution intersection with the feasibility region does not exactly lie on two of
the six feasibility points, in which case a ternary mixture may not be possible. The procedure for
solving for a ternary mixture is the same as for a binary mixture, with the binary cluster points
forming a new feasibility region boundary for the ternary cluster mixtures, shown as ?????????? for
the mixture of components S1, S2, and S3 in Fig. 6.4.
126
Figure 6.4: A Cluster Diagram of Binary and Ternary Mixtures of Component Streams S1, S2,
S3, and S4.
To find the mixture range of S1, S2, and S3 that satisfies Rule 1, the intersection of a line between
S3 and a point along ?????? with the feasibility region boundary ?????????? is first determined. In
this case, the mixture ranges for points A and E have already been established. Point B can be
calculated by mixing A with S3, determining where the feasibility line ???? intersects with ???? .
Points C and D require an estimate of a point along ?????? first, and then a calculation of a binary
mixture between it and the boundary region. For example, to estimate the ternary mixture at
127
Point D, the binary mixture point S12 is first calculated from the intersection of a line between
????? and ??????, followed by a second binary mixture calculation between S12 and S3 that intersects
at Point D (or line ????). Once the mixture point is found the AUP is calculated in the same
manner as before and tested against Rules 2 and 3.
It should also be noted that although binary mixtures of S2 - S3, S3 - S4, and S2 - S4 were
excluded as candidate solutions in the example, a ternary mixture of S2, S3, and S4 would
completely encompass the feasibility region and would meet the Rule 1, making it a potential
candidate solution. To determine if the ternary mixture encompasses the feasibility region, the
maximum and minimum cluster values of the points (e.g. Eq. 6.7) that make up the feasibility
boundary are calculated from Eq. 6.34 and 6.35.
)(max jii
Uj CC ?
(6.34)
)(min jiiLj CC ?
(6.35)
In a similar fashion, a mixture cluster range is determined from the maximum and minimum
cluster values of the pure components in the ternary mixture. If the maximum cluster of the
ternary mixture is not larger than the maximum cluster of the feasibility region or the minimum
cluster of the mixture is not smaller than the minimum cluster of the feasibility region, then the
mixture does not overlap the feasibility region and is discarded. If the mixture does overlap the
feasibility region, then the solution to this type of problem is identical to that of the first
condition, only that the entirety of feasibility region boundary points will need to be calculated
using the ternary mixture procedure. As more chemical constituents are used in the design of the
chemical product (i.e. u ? 3) complete overlaps, like ??????? overlapping the feasibility region in
Fig. 6.4, become more common. Although not shown here, chemical product designs with more
than 3 attributes or properties can also be solved using property clustering. This situation
128
requires the use of an algebraic approach developed by Qin et al. (2004) and Eljack et al. (2007),
but is beyond the scope of this dissertation.
In conclusion, Table 6.4. and Fig. 6.5-6.8 present a summary of the computer aided
mixture design (CAMbD) method using property clustering. To begin using the algorithm, the
researcher must set the maximum number ?max of dissimilar chemical species ??utilized in a
mixture. This constraint helps to determine if the mixtures will be pure components, binary,
ternary, quarternary, etc. A similar constraint is used in the computer aided molecular design
(CAMD) and is discussed in more detail in Section 6.3. Next, pure component solutions are
found by testing the cluster outputs of the CAMbD conversion algorithm against Rules 1-3. Any
pure component candidates are saved into a candidate solution database. Next, binary mixtures
with at least one pure component candidate are calculated and tested against Rules 1-3, with
candidates being uploaded into the candidate solution database. This step is followed by
calculating and testing binary mixtures with no pure component candidates against Rules 1-3 and
outputting the binary mixtures to the candidate solution database. The procedure is repeated for
ternary and larger mixtures, beginning with mixtures that contain binary mixture candidates, then
those without, and outputting all candidate mixture into the database.
Table 6.4. The CAMbD Candidate Generation Algorithm using Clusters
Step Description Equation
1 Set ?max Maximum Dissimilar Species in Mixture -
2 Discard Pure Component Clusters (from CAM
bD Conversion
Algorithm) that Fail Rule 1 6.6, 6.7
3 Discard Pure Component Clusters that Fail Rule 2, Output Pure Component Candidate Solutions 6.5, 6.7
4 For Binary Mixtures with Pure Component Candidates, Calculate the Clusters at the Intersections of the Sink and the Mixtures 6.31, 6.23, 6.24
129
5 Calculate the Binary Cluster Arms 6.21, 6.22
6 Calculate the Fractions of Each Pure Component 6.32
7 Calculate the AUP of Mixtures, Output Mixtures that Meet Rules 2 and 3 to Candidate Library 6.20
8
For Mixtures that Fail Rule 2 or 3, set the AUPM at the UL or LL of
the Sink and Recalculate the Component Fractions, Discard Binary
Mixtures with Component Fractions < 0 or > 1
6.20
9 Recalculate Cluster Arms of Candidate Mixture 6.21, 6.22
10 Recalculate Candidate Cluster and Output Binary Candidate Mixtures 6.18
11 For Binary Mixtures with no Pure Component Candidates, Calculate the Reference Constant for each Mixture 6.23, 6.24, 6.30
12 Calculate the Sink Constants for each Mixture, Discard Mixtures where all Sink Constants are > or < Reference Constant 6.23, 6.24, 6.27-6.29
13 Calculate the Clusters at the Intersections of the Sink and the Mixture and Repeat Steps 5-9 6.31, 6.23, 6.24
14 For ? ? 3 Ternary+ Mixtures with ? - 1 Candidates, Calculate the ? - 1 Equivalent Clusters 6.31, 6.23, 6.24
15
Calculate the Clusters at the Intersections of the Sink and the ? ? 3
Mixture and Repeat Steps 5-9, Output ? ? 3 Mixtures that Meet
Rules 2 and 3
6.31, 6.23,
6.24
16
For Mixtures that Fail Rule 2 or 3, set the AUPM at the UL or LL of
the Sink and Recalculate the Component Fractions, Discard Mixtures
with Component Fractions less than 0 or greater than 1
6.20
17 Recalculate Cluster Arms of Candidate Mixture 6.21, 6.22
18 Recalculate Candidate Cluster and Output ? ? 3 Candidate Mixtures to Library 6.18
19 For ? ? 3 Ternary+ Mixtures with no ? - 1 Candidates, Calculate the Cluster Range of the Pure Components 6.34, 6.35, 6.6
20 Calculate the Cluster Range of the Feasibility Sink, Discard Mixtures whose Pure Component Cluster Range is > Sink Cluster Range 6.34, 6.35, 6.6, 6.7
21 Repeat Steps 14-20 until ?maxis reached -
130
Fig. 6.5 Computer Aided Mixture Design (CAMbD) Candidate Generation Algorithm using Property Clustering.
CAM
b
D
Binar y T es t
? = 2
CAM
b
D
P ur e Comp. T es t
? = 1
CAM
b
D Can dida t e
Gen er a ti on Al g or i thm
Output Can dida t e
Mi xtur es
In i ti ali z e CAM
b
D
Can dida t e Ge n.
Al g or i thm
Se t Mi xtur e T ype
? ?? ? ?? def aul t)
Di sc ar d Non
Can dida t e Mi xtur es
CAM
b
D
T ernar y+ T es t
? ? 3
Select Ma ximum
Mi xtur e T ype ?
ma x
Choos e
T es t Se t
Se t Mi xtur e T ype
?? ? ?? ? ?
Is ? >
?
ma x
?
Se t Mi xtur e T ype
?? ? ?? ? ?
End Alg ori thm
Se t Mi xtur e T ype
?? ? ?? ? ?
Is ? >
?
ma x
?
Is ? >
?
ma x
?
? = 1
? = 2
? ? 3
YE S YE S YE S
NO NO NO
131
Each of the tests shown in Fig. 6.5 can be defined separately. Fig. 6.6 describes the
CAMbD cluster test on pure components. Fig. 6.7 describes the CAMbD cluster test on binary
mixtures, starting with any pure component candidates identified in Fig. 6.6. Fig. 6.8 describes
the CAMbD cluster test on ternary and larger mixtures, starting with any ?-1 candidate mixtures
identified.
Figure 6.6: CAMbD with Clusters test on Pure Components
132
Figure 6.7: CAMbD with Clusters test on Binary Mixtures
133
Figure 6.8: CAMbD with Clusters test on Ternary and Larger Mixtures
6.3. Computer Aided Molecular Design (CAMD) with Clusters
The second method for performing chemical product design using property clustering is
computer aided molecular design (CAMD) with clusters. While CAMbD is useful for finding
134
mixtures of autonomous chemical products (e.g. molecules) that meet product specifications,
CAMD is capable of finding mixtures of molecular architectures (e.g. molecular groups) that
make single chemical products (e.g. molecules) that meet product specifications. Eden et al.
(2003) and Eljack et al. (2007) have shown that first order groups can be easily expressed
linearly in Eq. 6.14, making them ideal candidates for use in CAMD with clusters.
In order to perform CAMD with clusters, the property clusters must be conserved during
mixing. Eljack et al. (2007) has shown that first order molecular property clusters possess both
intra-stream and inter-stream conservation which allows for the tracking of attribute or property
responses during molecule generation. The group contribution parameters are regressed from the
CAPEC database (Marrero and Gani, 2001) on a group basis and represented as clusters
according to Eq. 6.15-6.17. The intra-stream molecular conservation results in Eq. 6.36.
1??
?
?
??
M
M
M
p
j
M
jp
j
M
A U P
A U P
A U PC j (6.36)
Inter-molecular conservation requires that the individual group clusters CMj be conserved during
mixing. The property operator expression of Eq. 6.14 can be non-dimesionalized by Eq. 6.15,
resulting in Eq. 6.37.
?? ??? gng MjggMji n1
1
11
(6.37)
Rewriting the molecular property operator expression in terms of clusters is done by inserting
Eq. 6.37 into Eq. 6.16 and rearranging to get Eq. 6.38-6.40.
M
i
F
g
M
g
M
jgg
M
i
F
g
M
jgg
M
ji A U P
A U PCn
A U P
n
C
?? ??
?
??
? 1 1111 11
(6.38)
135
Mi
Mgg
g AU P
AU Pn 1
1
1 ???
(6.39)
?? ?? Fg MjggMji CC 1
1 11
? (6.40)
The group based property model shown in Eq. 6.40 is parameterized as first order groups by
Eljack et. al. (2007) and demonstrates how molecular groups or fragments can be added together
analogous to inter-stream conservation. Higher order group contribution can increase the
accuracy of the predicted properties and will be discussed in more detail in Chapter 8.
The ternary diagram used for mixture design as a visualization tool for source-sink
mapping can again be used to highlight the molecular design method. In the cluster formulation
for CAMD, mixing of two sources is a straight line, i.e. the mixing operation can be optimized
using lever-arm analysis. Analogously, combining or ?mixing? two molecular fragments in the
molecular cluster domain follows a straight line as shown by a mixture of G1 and G2 in Fig. 6.9.
This result can be used to design molecules that meet Rule 1 (e.g. G1-G2-G3 and G1-G2-G3-
G4), such that the length of each cluster arm ?g can be calculated from Eq. 6.38 and the number
of that group, ng, can be found. However, before Rules 2 and 3 can be used to narrow the
candidate set of molecules in the design, an additional rule unique to molecular design needs to
be included. In particular, it is important to ensure that the designed molecule is electronically
complete, meaning that its Free Bond Number (FBN) shown in Eq. 4.9 must be 0. In terms of
first order molecular groups, the expression becomes Eq. 6.41.
RG
F
g gg
F
g g NnFBNnFBN ???
??????? ???? ??
?? 212 11 1 111 1
(6.41)
136
where the FBN is the free molecular bond number of the formulation, ng1 is the number of
occurrences of group g1, FBNg1 is the unique free bond number associated with group g1, NRG is
the number of rings in the formulation, and F is the total number of first order groups. Forcing
Eq. 6.40 to be 0 is stated as Rule 4.
Rule 4. The Free Bond Number (FBN) of the candidate molecule must be zero.
It should be noted that the location of the final formulation is independent of the order of group
addition, making it ideal for use in computational algorithms.
Figure 6.9: Group Addition on a Ternary Cluster Diagram using Four Dissimilar Molecular
Groups to Build a Molecule (Eljack et al. 2007).
As shown in Fig. 6.10, the method proceeds in a similar manner to that of CAMbD, but
with groups instead of components as the combinatorial building blocks. The maximum number
137
of dissimilar groups ?gmax is first set and then progressively larger combinations of dissimilar
groups are added until the maximum is reached. As shown in Fig. 6.11 for ?pure group?
mixtures (e.g. ?g = 1), the pure group property contributions are estimated from Eq. 6.38 for each
g1 group type. These clusters are then tested against the points making up the feasibility bounds
of a target feasibility region (FR) defined using the clustered versions of the points in Eq. 6.7.
All ?pure group? property clusters (e.g. Eq. 6.17) that meet Rule 1 are then passed on to test
against Rule 2. For group based design, Rule 2 can be created by setting Eq. 6.39 equal to 1 for
a ?pure group? and resulting in Eq. 6.42
UggL A UPA UPnA UP ??
11
(6.42)
Hence, unlike mixture designs using pure components, it is possible adjust the number of groups
in the molecular structure in order to meet the constraint of Eq. 6.42. This result can be used to
calculate the maximum and minimum number of groups of each type that can be utilized in a ?g
= 1 design.
???
???
?
??
1
int1
g
UU
g AU PAU Pn
(6.43)
???
???
?
??
1
int1
g
LL
g AU PAU Pn
(6.44)
UggLg nnn 11
1 ??
(6.45)
By default, any pure group mixtures that meet Rule 2, will also meet Rule 3. Any mixtures that
fail Rule 2 are discarded. Finally the number of groups ng1 are indexed from ng1L to ng1U and
tested against Rule 4 using Eq. 6.41. Any solution that fails Rule 4 is summarily discarded.
Molecules that meet Rule 4 are outputted as candidate solutions.
138
As shown in Fig. 6.12 for binary mixtures of dissimilar types of groups (i.e. v1 ? g1) in
non ring compounds (i.e. ?g = 2, NRG = 0, the CAMD procedure follows a similar path as that for
CAMbD, beginning first by testing mixtures of dissimilar groups that contain one pure group
candidate solution (e.g. a mixture of M1 and G4). Since the number of groups is discrete, it is
generally easier to procede in a forward manner by assuming Rule 1 is met and varying the
combinations of the number of groups, testing each mixture sequentially against Rule 2 - Rule 4,
and then validating that Rule 1 is indeed satisfied. The procedure starts by building candidate
molecules by indexing the number of groups ng1 from a minimum of 1 to a maximum defined by
Eq. 6.43 for each group g1. Then, the number of the dissimilar group nv1 is indexed and the first
step is repeated. The procedure continues until the number of a dissimilar group reaches a
maximum as defined by Eq. 6.46.
???
???
?
??
1
1 int v
UU
v AU PAU Pn
(6.46)
The identified combinations are then tested against rule Rule 2 using Eq. 6.42. By definition,
any molecules that meet Rule 2 will automatically meet Rule 3, so they are immediately passed
forward to be tested for molecular completeness using Rule 4. Combinations that fail Rule 2/3
or Rule 4 are discarded while combinations that satisfy Rule 4 are passed to the final validation
step. The validation of Rule 1 is performed by testing the clusters of the mixture solutions
against the FR clusters. Mixture combinations that fall inside the FR are outputted as candidate
solutions while those that fall outside the FR are discarded.
Since the number of pure group candidates are relatively small compared to the number
of potential molecules, most candidate mixtures will involve pure group clusters that reside
outside of the feasibility region. As with the CAMbD with clusters algorithm, a candidate
139
mixture will not exist if a line segment connecting the pure clusters does not intersect the
feasibility region sink (FR). Using the Cartesian coordinates of the clusters of the pure groups
and feasibility region, the presence of a line segment between pure groups that intersects the FR
can be tested using Eq. 6.27-6.30. If the intersection is not confirmed, like between pure groups
G1 and G2, then all molecular combinations of only those dissimilar groups are discarded. If an
intersection is confirmed, like theroretically between pure groups G1 and G3 in Fig. 6.9, the
binary candidate generation steps are repeated using Eq. 6.47.
UF
g gg
L A U PA U PnA U P ?? ?
? 11 11
(6.47)
The implicit structure of this expression prevents universally setting constrained upper and lower
limits on the number of groups formed beyond the limits proposed in Eq. 6.45. As a result the
number of groups in the potential candidate mixtures that meet rule 1 are set using Eq. 6.48:
Ugg nn 111 ?? (6.48)
Each of the candidate solutions is then tested against Rule 2 and Rule 3 using Eq. 6.47. As with
the pure group solutions, binary solutions that meet Rule 2 and 3 are then passed forward to the
structure validation step, followed by a validation step using Rule 1. Binary candidate and non-
candidate mixtures are outputted accordingly. Any repeated solutions are also discarded (e.g. a
binary mixture of G1-G2 can also be represented as G2-G1).
For ternary and sequentially larger combinations of dissimilar groups (e.g. ternary+), it is
appropriate to start with the previously identified ?g - 1 candidate solution clusters and add
another pure group to the solution. All ternary+ combinations of the set of ng1 within the bounds
of Eq. 6.48 are tested against Eq. 6.47. Ternary solutions that meet Rule 2 and 3 are then tested
for structural completeness using Rule 4 and vailidated using Rule 1.
140
Ternary+ solutions that do not contain any ?g - 1 candidate solutions can potentially
encompass the feasibility region. The maximum and minimum cluster values of these pure
component solutions are tested against the maximum and minimum cluster values of the FR,
shown in Eq. 6.34 and 6.35. If the solution does overlap the feasibility region, then the
procedure to solve this type of problem is identical to that of the binary procedure. Solutions that
fail to overlap the feasibility region are discarded as non-candidate solutions.
The procedure is repeated until a maximum number of dissimilar groups ?gmax is reached.
A summary of the method is provided in Table 6.5 and Fig. 6.10-6.14.
Table 6.5: The CAMD Candidate Generation Algorithm using Clusters
Step Description Equation
1 Set Maximum Dissimilar Groups in the Mixture -
2 Discard Pure Group Clusters (from CAMD Conversion Algorithm) that Fail Rule 1 6.17, 6.7
3 Calculate No. Pure Groups 6.45
4 Calculate Pure Group Solutions, Discard Pure Group Solutions that Fail Rule 2 and Rule 3 6.7, 6.16, 6.42
5 Discard Pure Group Clusters that Fail Rule 4, Output Pure Group Candidate Molecules 6.41
6 For Binary Solutions with Pure Group Candidates, Calculate the No. Binary Groups 6.46, 6.48
7 Calculate Solution AUPMM, Discard Solutions that Fail Rule 2 6.39, 6.47
8 Calculate the FBN, Discard Solutions that Fail Rule 4 6.41
9 Calculate Solution Clusters, Discard Solutions that Fail Rule 1, and Output Candidate Solution Clusters 6.7, 6.17
10 For Binary Solutions with no Pure Group Candidates, Calculate the Reference Constant for each Mixture 6.20
11
Calculate the Sink Constants for each Mixture, Discard Mixtures
where all Sink Constants are > or < Reference Constant, Repeat Steps
6-9
6.23-6.24,
6.27-6.29
141
12 For Ternary and Larger Solutions (?g ? 3) containing ?g - 1 Candidates, Calculate the No. Groups, Repeat Steps 7-9 6.46, 6.48
13 For ? ? 3 Ternary+ Mixtures with no ? - 1 Candidates, Calculate the Cluster Range of the Pure Components 6.34, 6.35, 6.6
14
Calculate the Cluster Range of the Feasibility Sink, Discard Mixtures
whose Pure Component Cluster Range is > Sink Cluster Range,
Repeat Step 12
6.34, 6.35,
6.6, 6.7
15 Repeat Step 12-14 until ?gmaxis reached -
142
Figure 6.10: A Flowchart of the CAMD Candidate Generation Algorithm using Clusters
143
Figure 6.11: Generation of Pure Group Candidate Molecules
144
Figure 6.12: Generation of Binary Group Candidate Molecules
145
Figure 6.13: Generation of Ternary+ Group Candidate Molecules
146
6.4. Summary
In conclusion, the complexity of chemical product design problems can be reduced by
using the reverse problem formulation and the property clustering algorithm in a number of
ways. First, by solving the design problems in the lower dimensional property domain, an
unlimited number of molecular architecture solutions can be evaluated at minimal computational
cost. Second, the multitude of property response plots normally associated with chemical
product design can be reduced to p - 2 simplex diagrams to aide visualization of the design
problem. Property clustering also consolidates the numerous effects of components or groups on
the response into distinguishable values which may be used to surmise which of the components
or groups has the largest effect and is thus most important. Both of these property clustering
attributes can be used to guide chemical product designs toward feasible solutions more quickly
than conventional techniques.
Since the objective of the research presented in this dissertation is to successfully link the
multiple length scales in a reverse problem formulation, use of the CAMbD and CAMD cluster
algorithms to consolidate the large amount of information and improve computational efficiency
will be essential. In Chapter 5, it was noted that such a design problem will invariably utilize
data due to the lack of explicit models that describe consumer preference behavior. Based on
this observation, Chapter 7 explores mapping experimentally derived models and data into the
property clustering structure and how property clustering can be used to improve the design
capability of such systems, resulting in a new method called attribute based computer aided
mixture design (aCAMbD). Although the technique developed significantly improves efficiency,
it is subject to choosing only the best candidate mixtures from a predetermined training set and
incapable of looking at new molecular structures. In Chapter 8, another new method is
147
developed to address this shortcoming by mapping the experimental data into a domain that is
described by known combinatorial property prediction methods, opening the door to true
computational molecular design. This new method is known as attribute based computer aided
molecular design (aCAMD). When good mapping functions cannot be derived or the
computational methods are too restrictive, Chapter 9 presents a work around that maps the
experimental data into a sub-domain capable of handling both experimental design, computer
aided molecular design, and structure design techniques through the use of characterization data.
The additional benefit of this technique is that all length-scales important to the chemical product
can be be mapped down to the sub-domain, representing the ultimate in combinatorial efficiency
and is the primary result of this dissertation. This technique is noted as characterization based
computer aided molecular design (cCAMD).
148
Chapter 7
Attribute-Computer Aided Mixture Design (aCAMbD)
The terms product synthesis and design designate problems involving identification and
selection of compounds or mixtures that are capable of performing certain tasks or possess
certain physical properties. Since the properties of the component or mixture of components
dictate whether or not the design is useful, the basis for solution approaches in this area should
be based on the properties themselves. However, the performance requirements for the desired
component(s) are usually dictated by the process and thus the identification of the desired
component properties should be driven by the desired process performance. Whereas numerous
contributions have been made in the areas of molecular synthesis and computer aided molecular
design (CAMD) by Harper et al. (1999), Harper and Gani (2000), Marcoulaki and Kokossis
(1998), and Eljack et al. (2007) among others, little focus has been on utilizing experimental
design techniques when attribute or physical-chemical property prediction tools are insufficient.
This chapter highlights a data driven Computer Aided Mixture Design (CAMbD) method, known
as attribute CAMbD or aCAMbD, that is capable of handling experimental data and regression
models within the property clustering framework. In Section 7.1, the constraints on the use of
attribute and property data within the property clustering framework is discussed and the two
most common regression models, canonical and polynomial, are investigated as prediction tools.
In Section 7.2 the attribute clustering technique is reviewed and new rules are added to handle
negative regressors. Section 7.3 describes the aCAMbD method using property clusters and
149
Section 7.4 highlights the method in a case study on the polymer blend of spun yarn. Section 7.5
concludes the chapter with a summary.
7.1. Integrating Attribute Data and Models with Property Clustering
Early in experimental mixture design, Scheffe (1958, 1963) and Cox (1971) developed
techniques to obtain property models while minimizing experimental runs or design points
utilizing simplex diagrams of the chemical constituent design space. However, finding the
solution in the chemical constituent design space is prone to combinatorial explosion.
Visualizing and solving the problem in the property space avoids this problem while also
offering insights into the effectiveness of the design. Thus, the overall objective of this
contribution is to integrate the property clustering framework with existing mixture design
techniques used to develop data driven property models. The two most common property-
mixture designs, Scheffe canonical models and Cox polynomial models, are evaluated. The
results of the exercise will be used to develop additional techniques for utilization of PCR and
PLS models under combinatorial explosion.
As discussed in Chapter 3, the most common approach to efficiently develop data driven
models is to use Design of Experiments (DOE). In DOE, a model is first postulated to represent
a potential property response surface. Next, experimental design points are placed in areas
where observations can be collected to which the model can be fitted. In the final step, the
adequacy of the model is tested. The procedure may require much iteration until the fitted
equation is determined by the experimenter to be sufficient (Cornell 2002). The most effective
choice of model and location of design points is the focus of the experimenter. The best set of
points is chosen under the following constraints: (1) the size and shape of the experimental
region, (2) the number of desired experimental runs, and (3) the type of model used for
150
constructing the map of the response (Kettaneh-Wold 1991; Kettaneh-Wold 1992). Most often,
the polynomial model is selected to represent the response surface since it can be expanded
through a Taylor series to improve accuracy (Cornell 2002). A first or second degree model is
usually chosen to represent the surface since it requires fewer observations. Third degree or
higher ordered models are seldom utilized. The point estimate forms of the models are listed in
Eq. 3.3 and 3.4.
???? ui iio xy 1 ?? (3.3)
? ?? ? ?? ??? u wi u iw wiiwui iio xxxy ??? 1 (3.4)
In chemical product design, a colinearity effect is usually imposed on the model by Eq.
3.11, which means that the constituent fractions must sum to one and each constituent fraction
must lie between zero and one. While this technique does not affect the utilization of the model,
it does impact the interpretation of the regression coefficients of the property models. In data-
driven approaches, this colinearity affect appears in Scheffe simplex-lattice designs that use
canonical models (Eq. 3.14 and 3.15).
??? ui ii xy 1 *? (3.14)
? ?? ? ?? ?? u wi u iw wiiwui ii xxxy *1 * ?? (3.15)
For these types of designs, the location of the response of a mixture made up of exactly zero
constituents must be identically zero meaning that the coefficient ?o is zero. Conversely, mixture
designs using polynomial models are known as Cox designs where ?o is a non-zero factor used to
center the design over the search space. Although the two model types use different parameters,
151
they predict identical responses (Cornell 2002). The reason for this result is that the regression
coefficients represent different entities in each model. In the Scheffe model, the regressed
parameters represent combined effects comprised of contributions from pure components,
colinear interaction effects, and nonlinear interaction effects. The Cox models remove some of
the colinearities in mixture design, but depend on good experimental design controls to limit the
effect of other colinearities, resulting in parameters that represent only pure component linear
and nonlinear effects. The parameters of the two model types are related according to Eq. 3.17
and 3.18.
iiii ???? ??? 0* (3.17)
wwiiiwiw ???? ???* (3.18)
The estimates of the regressors can be found using least squares regression. To better
interpret the impact of regressors on the predicted attribute or physical-chemical property
response of the design, it is beneficial to view the design within the property clustering
framework.
As discussed in Chapter 6, property clustering is a transformation technique that helps
facilitate a reverse problem formulation of a chemical product design. Utilizing property
clustering in a reverse problem solving role not only avoids combinatorial explosion, but offers
the potential for solving process, mixture, and molecular design problems simultaneously (Eden
et al. 2003, Solvason et al. 2008). In situations where models with the necessary degree of
accuracy do not yet exist, chemometric models of attributes ( ) or physical-chemical
properties ( ) like the Scheffe canonical and Cox polynomial can be utilized and incorporated
into the existing property clustering framework while providing two benefits not found in
traditional experimental design. First, the experimental design points on which the model is
152
based are mapped into the property cluster design space and checked against the targets used in
the design. This visualization technique is appropriate regardless of the number of components
investigated, thereby providing a means for ensuring the design space has been properly
explored. The procedure for evaluating the efficiency of coverage is related to how near the
optimum is to the design space. It would be unusual to measure this as most designs utilize pure
component measurements and involve interpolation. However, should they not, gradient, non-
gradient, and specialized optimization techniques can be used to find the optimum and a new
mixture design can be conducted at the predicted optimum (Lazic 2004, Brandvik 1998,
Brandvik and Daling, 1998).
Second, most experiment based models utilize regression coefficients as estimates of the
effects of each property on the response. Depending on the type of regression utilized, the
interpretation of these regression coefficients can be quite difficult. Visualizing the problem in
property space consolidates each components impact on the mixture, aiding in the ability to
screen components. For systems of up to 3 properties, the entire experimental design and all
associated regression coefficients can be represented on a single property cluster diagram. For
example, a 6 component system can be represented in a single ternary diagram in Fig. 7.1 where
the component effects are estimated according to the response model chosen (see Section 7.1.1
and 7.1.2), the feasibility region is estimated as discussed in Chapter 6, and the design points are
estimated according to Section 7.1.3. Additional properties are either represented with additional
diagrams or solved algebraically. Consolidating the numerous effects of the various components
on the response into distinguishable values may be used to surmise which of the components has
the largest effect and thus is most important.
153
Third, constraints on the applicable range of the model and the potential mixtures that are
formed can be defined using Hotelling?s T2 distribution or the Bonferroni approximation on the
predictive power of the models (Johnson and Wichern 2007). These can be used to apply a
second sink region which must also be matched simultaneously with the target feasibility sink
region (Solvason et al. 2008; Solvason et al. 2009).
Figure 7.1: The Component Effects, Experimental Design Points, and Feasibility Region of a
Six-component Mixture Design Mapped to the Property Cluster Domain.
One difference between the use of regression based property models and conventional
property models is that the flexibility of parameter estimation allows for the use of negative
regressors, especially when the parameters are highly confounded. If the coefficients only
Co m p o n ent E ffe cts
D esig n P o in t s o r
M ixture s
Feasib il ity R egi o n
Respo n s e s ( At t r i b u tes o r P r o p e r ti e s )
154
represented pure component values, then the property would always be positive, but if a
regressor is negative, then it follows that coefficient is indicative of a strong colinearity and/or
nonlinearity effect overwhelming the weak, positive linear effect. This situation can cause
problems within the traditional property clustering framework, but is not unsolvable.
Figure 7.2: A Property Cluster Simplex Diagram Showing a Seven-Component System using
Property Operators Derived from an Acetaminophen Excipient Design (Martinello et al. 2006;
Solvason et al. 2008). Clusters 1, 2, and 3 Represent the Properties Repose Angle, Water
Content, and Compressibility.
For example, in Fig. 7.2 a mixture design of seven components is shown in a ternary
cluster diagram. Components 6 and 7 both had component effects consisting of negative
regression coefficients which, when clustered and mapped using Cartesian coordinate
transformations of Eq. 6.25 and Eq. 6.26, place them outside the traditional cluster diagram.
0 .9
0 .8
0 .7
0 .6
0 .5
0 .4
0 .3
0 .2
0 .1
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
C
3
C
2
C
1
0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9
1
2
4
5
3
6
7
P os itive
C omp onen t C luste rs
N eg ative
C omp onen t C luste rs
155
Hence, it is possible that when using models comprised of regression coefficients in CAMD,
some solutions may require mixing with chemical constituents outside of the cluster space
defined by the ternary diagram. As a result, it is necessary to define the region outside of the
cluster space as negative cluster space and the region inside the ternary simplex as positive
cluster space.
Negative cluster space can consist of a multitude of regions. The total number of
negative cluster regions, NR, is a function of the number of p properties or a attributes, depending
on the system in question (Solvason et al. 2008; Solvason et al. 2009). For a property system,
the following holds:
pNR 2? or aNR 2? (7.1)
For a p = 3 or a = 3 property or attribute solution the negative cluster space is comprised of six
distinct regions as shown in in Fig. 7.3. These regions are of two types: those with clusters
greater than one are called Type I regions and those with negative clusters are called Type II
regions (Solvason et al. 2008; Solvason et al. 2009).
1?jiC or 1?kiC , Type I Regions (7.2)
0?jiC or 0?kiC , Type II Regions (7.3)
These regions are related to one another since in the derivation of property clusters, Eden et al.
(2003) notes that the clusters must sum to one, which also holds for clusters of attributes.
11 ???
p
j jiC
or 1
1 ???
a
k kiC
(7.4)
156
For the three property example Eq. 7.4 implies that two negative property clusters equate to a
third cluster with a value greater than one. Likewise, two clusters with a value greater than one
equate to a third cluster with a value less than one. The number of different types of (positive
and negative) cluster regions NT is related to the number of properties evaluated according to Eq.
7.16 (Solvason et al. 2008; Solvason et al. 2009).
Figure 7.3: Details of Negative Property Cluster Space for a Three-Property System (Solvason
et al. 2008; Solvason et al. 2009).
pNT ? or aNT? (7.5)
Property clusters in the positive cluster regions may also be estimated from three negative
clusters and three clusters greater than one, respectively. These values may only be used to
C 1 z e ro
C 3 z e ro
C 2 z e ro
T y p e I
C 2 > 1
T y p e I I
C 1 < 0 T y p e I I
C 3 < 0
T y p e I I
C 2 < 0
T y p e I
C 3 > 1
T y p e I
C 1 > 1
157
ascertain effects and may not be used in mixing without violating the monotonically increasing
rule for mixing proposed by Shelley and El-Halwagi (2000) and Eden et al. (2003).
To discuss this limitation it is noteworthy when dealing with negative property clusters to
first investigate the constraints on the Augmented Property Index (AUP). The non-
dimensionalized properties are summed to create the AUP. While a negative regression
coefficient may create a negative property operator, the advent of such constructs must be
constrained by the following rule (Solvason et al. 2008; Solvason et al. 2009):
Rule 5. All AUP values of the components must be positive.
The rule can be written as equation 7.6
0?AUP (7.6)
In order to meet Rule 4 it is often necessary to adjust the reference property values in Eq. 6.4.
Although the adjusted references give different values for the clusters, the underlying property
values are unchanged. For example, the cluster diagram shown in Fig. 7.4 is altered by changing
the reference values for the property operators. Although the clusters of the three components
reside in a different location, their relative proximity to each other and the feasibility region
remains the same. In other words, for the example shown in Fig. 7.4, component 3 will always
be closest to the feasibility region while component 1 will always be farthest from the feasibility
region regardless of the property operator reference chosen. Furthermore, when the experimental
design points A, B, C, D, and E from the adjusted cluster diagram are transformed back to
component space, the resulting solutions are identical to the design points derived using the
original references. This is an important aspect of property clustering as the flexibility to adjust
the property operator reference values allows negative regressors to be utilized. An algorithm
that satisfies Rule 4 and Eq. 7.6 was presented in Chapter 6.
158
0 .9
0 .8
0 .7
0 .6
0 .5
0 .4
0 .3
0 .2
0 .1
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
C3
C2
C10 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9
1
2
3
0 . 9
0 . 8
0 . 7
0 . 6
0 . 5
0 . 4
0 . 3
0 . 2
0 . 1
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
0 . 6
0 . 7
0 . 8
0 . 9
C
3
C
2
C
1
0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9
1
2
3
1
2
3
A
B
C D
E
0 .2
0 .3
0 .4
0 .4 0 .5 0 .6
3
A
B
C
D
E
0 .5 1
0 .5 2
0 .5 3
0 .5 4
0 .2 5 5 0 .2 6 0 .2 6 5 0 .2 7
Figure 7.4: A Three Component System of Polyethylene, Polystyrene, and Polypropylene
Mapped to Cluster Space using Different Property Operator Reference Values.
Adjusted Mixture
Original Mixture
Candidate
Mixtures
159
7.1.1. Canonical Models
The simplest and most effective regression model to use with design of experiments is the
Scheffe canonical model of Eq. 3.3. Using the regression coefficients as estimates of the effects
of a component on the overall mixture, Eq. 3.3 can be transformed into the property operator
model of Eq. 6.1 for attributes using the nondimensionalized property operator expression
shown in Eq. 7.7:
)(
*
krefk
ikik y? ???
(7.7)
The procedure for clustering the property operator expression then follows the method
introduced in Section 6.1. It should be noted that the Scheffe canonical model ensures an
efficient CAMbD computation to the detriment of the interpretation of its parameters, which may
limit its use in secondary optimization schemes that use parameter interpretation to predict new
experiments (e.g. genetic algorithm). The reason for this result is that the regression coefficients
for the Scheffe models are combinations of linear, collinear, and nonlinear effects as shown in
Eq. 3.17 and often result in negative regressors and clusters. However, as long as the AUP of
pure component effects remain positive, mixtures generated from the negative clusters will still
meet the inter-conservation and intra-conservation rules of property clustering.
160
7.1.2. Polynomial Models
The approach in Section 7.1.2 is valid for setting up regression based models for mixing
and developing lists of candidate solutions. However, if the objective is to screen constituents,
then some knowledge of the pure component effects will be needed in order to reduce additional
experimentation each time a new constituent is added to the list of candidates. Since the
objective of the screening design is to understand the effects of the constituents on the mixture,
then the component clusters need to represent the pure property values void of colinearities and
nonlinearities. One method for removing the colinearities is to utilize the Cox modifications on
the Scheffe canonical models to create a reference mixture, or standard mixture. Shown in Fig.
3.2 is a simplex diagram published by Cornell (2002) with a standard mixture s and a mixture x
with a larger proportion of constituent xi. Noting that x lies on a line from s to the xi vertex, then
the ratio of the other u-1 constituents are in the same relative proportions as the standard mixture.
As Smith and Beverly (1997) point out, the ?i gradient, or change in response per unit change in
xi, at s along the Cox-effect direction is called the effect of xi, provided xi is free to range from 0
to 1. The relationship between the mass fraction, standard mixture and gradient is written as Eq.
7.8, which, when inserted into the polynomial expression of Eq. 3.3, results in Eq. 7.9:
iii sx ??? (3.19)
? ? ? ? ? ????????? ??? ui i
i
issyxy 1 ?
(3.21)
where y?x) is the expected response at design point x and y?s) is the expected response at the
standard reference mixture. For an attribute system ( ), it has been shown that the response
of the standard mixture y(s) is equivalent to the centering function provided by the regressor ?o
(Cornell 2002):
161
)( krefk kosk y?
??? (7.8)
where ?sk is the property operator for the standard mixture. The second term in Eq. 3.21
provides a direct link between the regressors and the position of the design points to the
reference mixture (Cornell, 2002) such that
zn
i ii
n
i ii
i yxs ??????????? ? ?? ??1 (7.9)
? ? ? ? zysyxy ?? (7.10)
where y(s) is the response at the standard mixture and yz is the pseudo property value that
represents the attributes, k, contribution to the mixture. The equivalent property operator
expression for the kth attribute effect or ?pseudo? component then becomes Eq. 7.11:
? ?
refkk
kzk y? ??? (7.11)
with the normalized property operator expression of Eq. 7.10, thus results in Eq. 7.12.
zkskk ????? (7.12)
Likewise the augmented property index is rewritten as Eq. 7.13:
zs A U PA U PA U P ?? (7.13)
where the AUPs is the AUP of the standard mixture:
?? ??
a
k
sksAUP
1
(7.14)
and AUPz is the AUP of the pseudo component effect.
162
?? ??
a
k
zkzAUP
1
(7.15)
Redefining the property clusters in terms of the pseudo property cluster zkC and the standard
property cluster skC gives equations 7.16 and 7.17.
z
zkz
k AUPC ??
(7.16)
s
sks
k AUPC ??
(7.17)
The sum of these clusters will not give the true cluster, Ck, as defined in Eq. 7.4, without
correcting for the different AUP values. This is done using a set of correction factors as shown
in Eq. 7.18 and 7.19.
AUPAUPF zz ? (7.18)
AUPAUPF ss ? (7.19)
Fz is the pseudo correction factor and Fs is the standard correction factor. These are combined in
Eq. 7.20 to give the relationship between the true property cluster, the pseudo property cluster,
and the standard property cluster.
zkzsksk CFCFC ?? (7.20)
It should be noted that while the transformation of the original polynomials removes the
primary colinearity introduced by Eq. 3.11, it leaves the secondary colinearities such as those
introduced by constraints on the constituent ranges. So, although both Scheffe canonical and
Cox polynomial models provide estimates of the pure component effects of the chemical
163
constituents, they are, in fact, not the property or attribute values of the pure components.
Kettaneh-Wold (1992) suggests that the best solution maybe to refrain from interpreting the
coefficients and rely on the predictions only but notes this solution is not acceptable in practice
since the interpretation of regression coefficients is a necessity when the objective is to find
component effects in screening situations. One method to handle the estimate of the effects is to
constrain the solutions to the CAMbD within a feasibility region where the estimates of the
effects are considered valid, known as a model feasibility region (MFR). For multivariate
normal responses Np (?, ?), a component can be tested against the MFR using Hoetelling?s T2.
)()'( 12 ee YYSYYnT ??? ? (7.21)
where n is the number of experiments (i.e. mixtures), ? is a vector of estimated means of the
response (e.g. Y = A when the responses of interest are attributes and Y = P when the responses
are physical-chemical properties), and Ye is a vector of responses of a test mixture e. S is a p x p
(or a x a for attribute responses) matrix of the estimated variance and covariance between the
responses.
?? ???? ne ee YYYYnS 1 )')((11 (7.22)
For the Np (?, ?) population, the null hypothesis H0 is that the test mixture is described by the
response model and that the alternative hypothesis H1 is that it is not.
H0: ??= Ye (7.23)
H1: ??? Ye (7.24)
Hoetelling?s T2 is distributed as the following:
? ??pnpFpn pnT ??? ,2 )1(~ (7.25)
164
where Fn, n-p (?) is the single tailed F distribution. The null hypothesis is rejected if T2 calculated
from Eq. 7.21 is greater than the critical T2 calculated form Eq. 7.25, as shown in Eq. 7.26.
Reject H0 if ? ??pnpFpn pnT ???? ,2 )1( (7.26)
For the existing chemical consituents present in the training set of an aCAMbD, confidence
intervals on the mean responses can be derived from Eq. 7.21 (Johnson and Wichern). However,
since the primary objective of the CAMbD is to generate new mixtures, it is prudent to
reconstruct Eq. 7.21 in terms of the difference between the predicted response and the actual
response:
? ? ? ? ???
?
???
?
??
???
?
??
?
?
?????
?
???
?
??
???
?
?
?
ee
eeE
ee
ee
xXXx
YxBS
un
n
xXXx
YxBT
1
1'
1
2 1 (7.27)
where xe is the test set of new mixtures, X is the set of exiting mixtures, B?xe are the predicted
responses of the set of test mixtures, Ye is the measured response of the test mixtures, n is the
total number of experiments in the training set, u is the total number of chemical species in the
mixture, and SE is the unexplained variance in the fitted model. Using multiple linear regression,
the confidence intervals of each test mixture for each k attribute (or j property) can be written as
Eq. 7.28 (Johnson and Wichern 2007):
? ? ee
Eaunake xXXxSun nFaun unaY
kk
1, )(1)( )1( ??? ?????????? ??????????? ?? ??? ? (7.28)
The attribute boundaries of the MFR are calculated by applying Eq. 3.3 (or Eq. 3.14) and
Eq. 7.28 to each experiment in the training set and taking the maximum and minimum of the
responses. The cluster feasibility region is then built using the same procedure outlined in
Chapter 6. Further analysis of this region is given in Chapter 8.
165
In some situations the property, attribute, and/or chemical constituents are highly
confounded with large amounts of covariance. For this situation, Kettaneh-Wold (1992)
suggests the use of decomposition techniques like Principal Component Regression (PCR) and
Partial Least Squares on to Latent Surfaces (PLS) to find underlying, orthoganol properties or
components that can significantly reduce the complexity of the problem. These techniques were
discussed in detail in Chapter 3 and are applied to aCAMD with clusters in Chapter 9.
7.2. Attribute Clustering Algorithms
The attribute clustering algorithm for an attribute based computer aided mixture deisgn
(aCAMbD) proceeds in the same manner as CAMbD with the some additions. First, regardless of
the model type chosen, a training set of validated experimental design points (i.e. mixtures) are
available to for testing against the feasibility region. These mixtures can be calculated directly
from Eq. 6.6 and, if a = 3 can be plotted by the procedure outlined in Chapter 6. Second, if a
Cox polynomial model is chosen for design, a pseudo feasibility region must be calculated from
the feasibility region limits adjusted for the standard reference mixture by rearranging Eq. 7.12,
7.13, and 7.20. The same holds true for the MFR limits. An outline of the clustering algorithm
is shown in Fig. 7.5.
166
Figure 7.5: Clustering algorithm for Attribute based Computer Aided Molecluar Design
(aCAMbD)
The choice of regression model (canonical or polynomial) dictates the specific equations
utilized in the clustering approach as shown in Table 7.1 and 7.2.
Table 7.1. Attribute Clustering Procedure for Scheffe Canonical Models
Step Description Equation
1 Select Linearized Canonical Model 3.14, 3.15
2 Fit Model to Experiment Design Points and Responses to Determine Models Parameters 3.10
3 Guess Property Operator Reference Values and Calculate Pure Component Effect Property Operator Values 7.7
In i ti ali z e aCAM
b
D
Clus t er Con v er sion
Alg ori thm
Es ti ma t e Line ar
A t tr i bu t e Models
f or Chem i c al
Cons ti tuen ts
Calc ula t e the
Experi ment al &
P ur e Compon en t
Es ti ma t es f or
A t tr i but e
Oper a t or s
Se t A t tr i bu t e
R e f er en c e V alues
aCAM
b
D A t tr i bu t e Clus t eri ng Con v er sion Alg ori thm
Con v ert LL and UL
Cons tr ain ts t o
A t tr i but e Oper a t or s
Calc ula t e T ar g e t
A t tr i bu t e Clus t er
Sink
Does the
S y s t em
ha v e 3
A t tr i bu t es ?
Calc ula t e the A UP
of a l l A t tr i bu t e
Oper a t or s
Output All
Cl us t er V al ues
Con v ert Cl us t er s t o
Cart es i an
Coor dina t es & P l ot
Calc ula t e the Pur e
Componen t
Es ti ma t e C l us t er s
NO
YE S
Calc ula t e the
Experi ment al D a t a
Cl us t er s
Calc ula t e Model
F ea sibil i ty Cl us t er
Sink
Ar e A l l
Clus t er
V alues
P osit i v e?
NO
YE S
R un R e f er ence
Op ti miz a ti on
Alg ori thm
167
4 Convert UL and LL Feasibility Constraints to Property Operators 6.7
5 Calculate the AUP of the Pure Component Effect Property Operators and the Feasibility Constraint Property Operators 7.7, 6.7, 6.5
6 Calculate the Pure Component Effect Clusters 6.6
7 Calculate the Feasibility Region Sink 6.6, 6.7
8 Calculate the Variance, Covariance, and Confidence Intervals of the Experimental Design 7.20
9 Calculate the Model Feasibility Region using Confidence Intervals around the Responses of the Experimental Design Points 7.28, 6.6, 6.7
10 Run Reference Optimization Algorithm for Non-Positive Clusters 6.8-6.12
11 Convert all Clusters to Cartesian Coordinates and Plot 6.23-6.26
Table 7.2. Attribute Clustering Procedure for Cox Polynomial Models
Step Description Equation
1 Select Linearized Polynomial Model 3.3, 3.4
2 Fit Model to Experiment Design Points and Responses to Determine Models Parameters 3.10
3 Guess Property Operator Reference Values and Calculate Pseudo Pure Component Effect and Standard Property Operator Values 7.8, 7.11
4 Convert UL and LL Feasibility Constraints to Pseudo Property Feasibility Constraints 6.7, 7.12
5 Calculate the UL and LL of the Pseudo Feasibility Region AUP 7.15
7 Calculate the AUP of the Pseudo Pure Component Effect and Standard Reference Mixture 7.14, 7.15
8 Calculate the Pseudo Component Effect and Standard Reference Clusters 7.16, 7.17
8 Calculate the Variance, Covariance, and Confidence Intervals of the Experimental Design 7.20
9 Calculate the Model Feasibility Region using Confidence Intervals around the Responses of the Experimental Design Points 7.28, 6.6, 6.7
10 Run Reference Optimization Algorithm for Non-Positive Clusters 6.8-6.12
11 Convert all Clusters to Cartesian Coordinates and Plot 6.23-6.26
168
7.3. aCAMbD using Experimental Data and Data Driven Models
Once the attribute (or property) clusters have been calculated, aCAMbD can proceed in a
similar manner to CAMbD. Since Shelley and El-Halwagi (2000) have shown that a property
cluster is conserved through mixing, the fractional calculation of any mixture in the cluster
design space can be found. For Scheffe canonical models, the property operator expression is
identical to Eq. 6.1 and the design procedure follows the outline provided in Section 6.2 with
some additional rules. First, immediately after testing for Rule 1, the mixtures should be tested
against the MFR using Rule 6, discarding any solutions that fail the test as infeasible.
Rule 6. Cluster values of the source (or mixture of sources) must be contained within the
MFR of the sink in the cluster domain.
Likewise, following the tests on Rule 2/3, the mixture should be tested against Rules 7/8 for
suitable MFR AUP values.
Rule 7. The values of the Augmented Property Index (AUP) for the source or mixture of
sources must fall within the AUP range of the MFR.
Rule 8. The AUP of the candidate must match the AUP of the MFR sink at the candidate
point as calculated from the sink boundary points.
For Cox polynomial models, the calculation of the mixture fractions requires the
rewriting of Eq. 6.18 in terms of a mixture of pseudo properties and standard properties. First
Eq. 6.1 is written as Eq. 7.29.
i
u
i kikM x???? ?
(7.29)
169
The normalized property operator for each component effect can be written as Eq. 7.12 and
inserted into Eq. 7.29 to give Eq. 7.30
s
ki
u
i
zkikM x ?????? ? (7.30)
Dividing the expression by the AUP of the mixture and inserting standard mixture correction
factor and cluster definition gives Eq. 7.31.
sM
sM
sk
M
u
i
zkii
kM FA U PA U P
x
C ???
?
? ? (7.31)
Inserting the cluster definition of Eq. 7.17 into Eq. 7.31 and rearranging gives Eq 7.32.
M
u
i
zkii
skMsMkM A U P
x
CFC ?
?
?? (7.32)
Noting that the left hand side of Eq. 7.32 is the same as the product of the pseudo correction
factor and the pseudo property cluster of the mixture, the equation is rewritten as Eq. 7.33.
M
n
i
z
jiiz
jMzM AU P
x
CF ?
?
? (7.33)
Inserting the pseudo cluster definition to remove property operator in favor of the cluster and
rewriting the AUP of the mixture in pseudo terms gives Eq. 7.34:
zM
n
i
z
ji
z
iiz
jM A U P
CA U Px
C ?? (7.34)
170
Eq. 7.35 assumes a familiar form of the relative cluster arm using the pseudo relative cluster arm
z? as defined by Eq. 7.36.
?? ni zjizizjM CC ? (7.35)
zM
ziiz
AUPAUPxi ??
(7.36)
The pseudo relative cluster arm maintains the monotonically increasing criteria imposed
by Eden et al. (2003) provided that all of the augmented property indices used in the solution of
the problem are positive; a constraint placed on the solution by Eq. 7.6. A negative AUP would
violate this relationship and prevent the proper cluster solution from being obtained. The relative
cluster arms of the pseudo mixing rule in Eq. 7.35 are indicative of a mix involving a pseudo
feasibility region. The pseudo feasibility region is defined in the same manner as the true
feasibility region but corrects for the standard mixture property values using Eq. 7.12. Often this
means that the pseudo region is located in the negative cluster space. The resulting pseudo
relative cluster arms represent the addition of the pseudo components to move the mixture into
this region, only now each pseudo component better represents its contribution to the mixtures?
properties, void of most colinearities. For example, suppose a u = 3, p = 3 component system is
being evaluated by property clustering and that parameters for both the Cox polynomial and
Scheffe canonical models are estimated. As shown in Fig. 7.6, identical mixtures A, B, C, D,
and E are calculated using both the conventional cluster mixing approach with Scheffe models or
the pseudo clustering approach using Cox models. This is expected since the normal Scheffe and
Cox mixing designs achieve the same property response plots (Cornell 2002). It is the author?s
recommendation that for ease of use, the traditional property clustering method should be used
171
with the Scheffe property operators, while the pseudo property clustering method using Cox
property operators should be reserved for interpretation of component effects on the mixture
properties. An interesting benefit of using pseudo property clustering with Cox models over the
traditional method is the ability to visualize the components impact on the mixtures properties
simultaneously. In the traditional techniques, each response plot is mapped onto the component
space. Ignoring the combinatorial explosion issue for a moment, it can be seen that when two or
more iso-properties are parallel or conflict in direction, it can be difficult to know where to move
the mixture visually. This problem is compounded exponentially when multiple components are
evaluated; leading researchers to either limit the number of components in the experiment or use
powerful statistical techniques such as PLS. Pseudo property clustering offers a medium ground
between the two methods and in some cases can be used in conjunction with PCR and PLS to
further clarify solutions, especially when performing screening designs. Rules governing the
interpretation of the cluster points in the property cluster space are listed as follows:
Rule 9. The visual distance from the standard mixture to a component cluster point is
indicative of the relative magnitude of the components effect on the response.
Rule 10. If the constituents lie on opposite sides of a line which passes through the standard
reference mixture, then the constituents are said to be inversely related.
A flowchart of the aCAMbD method is shown in Fig. 7.7 and Table 7.3 lists the equations
used in the design.
172
0 .9
0 .8
0 .7
0 .6
0 .5
0 .4
0 .3
0 .2
0 .1
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
C3
C2
C10 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9
1
2
3
1
2
3
A
B
C D
E
0 .2
0 .3
0 .4
0 .4 0 .5 0 .6
Figure 7.6: A Three Component System Mapped to Cluster Space using both Scheffe and Cox
Property Operator Models.
1
z
2
z
3
z
C1 z e ro
C3 z e ro
C2 z e ro C2 z e ro
-3 0 0
2 0 0
7 0 0
12 00
1 7 0 0
-3 0 0 -1 0 0 1 0 0 3 0 0 5 0 0 7 0 0 9 0 0
1z
Sta n d a rd
2z
3z
Ps e u d o T a r g e t Reg i o n
T y p e I
C2 > 1
T y p e II
C1 < 0
T y p e II
C3 < 0
T y p e II
C2 < 0
T y p e I
C3 > 1
T y p e I
C 1 > 1
Mixture Using Scheffe
Property Operators
Mixture Using Cox
Property Operators
C 1 C 2 C 3 C 1 C 2 C 3 x 1 x 2 x 3 x 1 x 2 x 3
A 0 .3 0 0 0 .3 0 1 0 .3 9 9 -0.0 1 0 4 9 .5 4 8 -48 .5 3 8 0 .1 7 7 0 .5 1 0 0 .3 1 3 0 .1 7 7 0 .5 1 0 0 .3 1 3
B 0 .3 6 2 0 .2 7 5 0 .3 6 4 -0.0 1 0 5 0 .0 4 3 -49 .0 3 2 0 .1 7 1 0 .5 1 4 0 .3 1 5 0 .1 7 1 0 .5 1 4 0 .3 1 5
C 0 .3 6 4 0 .2 7 3 0 .3 6 3 0 .1 6 0 2 1 .5 9 5 -20 .7 5 5 0 .0 0 0 0 .0 7 8 0 .9 2 2 0 .0 0 0 0 .0 7 8 0 .9 2 2
D 0 .3 6 4 0 .2 7 3 0 .3 6 4 0 .1 6 3 2 0 .9 0 8 -20 .0 7 1 0 .0 0 0 0 .0 5 7 0 .9 4 3 0 .0 0 0 0 .0 5 7 0 .9 4 3
E 0 .3 0 0 0 .3 0 0 0 .4 0 0 0 .1 6 3 2 0 .5 3 4 -19 .6 9 7 0 .0 1 0 0 .0 5 0 0 .9 4 0 0 .0 1 0 0 .0 5 1 0 .9 3 9
Candida te
M ix tu re s
P rope rty Clus ters Comp onent Fra c tions
S ch e f f e P rop e rty O p e rato r Cox P rop e rty O p e rato r S ch e f f e P rop e rty O p e rato r Cox P rop e rty O p e rato r
E
3
z
S ta n d a rd
A
B
C
D
-1 0
0
10
20
30
40
50
60
-1 0 0 10 20 30
T y p e I
C2 > 1
T y p e II
C1 < 0
T y p e II
C3 < 0
T y p e II
C2 < 0
T y p e I
C3 > 1
T y p e I
C1 > 1
173
Figure 7.7: Overview of the Attribute Based Computer Aided Mixture Design (aCAMbD) Method.
174
Figure 7.8: aCAMbD with Clusters test on Pure Components
175
Figure 7.9: aCAMbD with Clusters test on Binary Mixtures
176
Figure 7.10: aCAMbD with Clusters test on Ternary and Larger (Ternary+) Mixtures
177
Table 7.3: The Attribute Based Computer Aided Mixture Design (aCAMbD) Method.
Step Description Equation
1 Set ?max Maximum Dissimilar Species in Mixture -
2 Discard Pure Component Clusters (from aCAM
bD Conversion
Algorithm) that Fail Rule 1 6.6, 6.7
3 Discard Pure Component Clusters that Fail Rule 2/3 6.5, 6.7
4 Discard Pure Component Clusters that Fail Rule 6 and reside outside the Model Feasibility Region 7.25, 7.26, 6.6, 6.7
5 Discard Pure Component Clusters that Fail Rule 7/8, Output Pure Component Candidates 6.5, 6.7, 7.25, 7.26
6 For Binary Mixtures with Pure Component Candidates, Calculate the Clusters at the Intersections of the Sink and the Mixtures 6.31, 6.23, 6.24
7 Calculate the Cluster Arms 6.21, 6.22
8 Calculate the Fractions of Each Pure Component 6.32
9 Calculate the AUP of Mixtures 6.20
10
For Mixtures that Fail Rule 2 or 3, set the AUPM at the UL or LL of
the Sink and Recalculate the Component Fractions, Discard Binary
Mixtures with Component Fractions < 0 or > 1
6.20
11 For Mixtures that Fail Rule 6, Recacluate the Mixture at the MFR Boundary, Discard Solutions that Fail Rule 1 and/or Rule 2/3 6.31, 6.21, 6.32
12
For Mixtures that Fail Rule 7/8, set the AUPM at the Minimum UL or
Maximum LL of the both the FR and MFR Sinks and Recalculate the
Component Fractions, Discard Binary Mixtures with Component
Fractions < 0 or > 1, Output Candidate Mixtures
7.25, 7.26,
6.6, 6.7,
6.31, 6.21,
6.32
13 For Binary Mixtures with no Pure Component Candidates, Calculate the Reference Constant for each Mixture 6.23 6.24, 6.30
14 Calculate the Sink Constants for each Mixture, Discard Mixtures where all Sink Constants are > or < Reference Constant 6.23-6.24, 6.27-6.29
15 Calculate the Clusters at the Intersections of the Sink and the Mixture and Repeat Steps 7-14, Output Candidate Mixtures 6.31, 6.23, 6.24
16 For ? ? 3 Ternary+ Mixtures with ? - 1 Candidates, Calculate the ? - 1 Equivalent Clusters 6.31, 6.23, 6.24
17 Calculate the Clusters at the Intersections of the Sink and the ? ? 3 Mixture and Repeat Steps 7-12, Output Candidate Mixtures 6.31, 6.23, 6.24
19 For ? ? 3 Ternary+ Mixtures with no ? - 1 Candidates, Calculate the Cluster Range of the Pure Components 6.34, 6.35, 6.6
178
20 Calculate the Cluster Range of the Feasibility Sink, Discard Mixtures whose Pure Component Cluster Range is > Sink Cluster Range 6.34, 6.35, 6.6, 6.7
21 Repeat Steps 16-20 until ?maxis reached -
After the initial aCAMbD has been conducted, it is useful to design a feedback loop to
help drive the design toward an optimal mixture. The determination of optimal conditions from
an experimental design is among the most complex problems for a researcher (Lazic 2004). In
many cases because of the complexity of the interactions, especially those across multiple scales,
deterministic models are not sufficient and the optimization occurs by analytical or by a hybrid
analytical-computational method. In the context of DOE, interpretation of the models, model
parameters, and the effects of each of the components on the properties of the mixture is the
objective of screening designs. Beyond screening designs, the prediction of the optimum from
the property models can help to focus future experimentation and improve the interpretation of
the property models. Reaching the optimum is more efficient if the obtained model is adequate
(Lazic 2004). As discussed in Section 7.2, the F-test is most often used as a measure of the
adequacy or lack-of-fit of the model. Increasing the number of model parameters is usually the
preferred choice to improve the lack-of-fit, often resulting in quadratic, specialcubic, or even
special-quartic models (Brandvik 1998, Brandvik and Daling 1998). For these higher order
models, transformations would need to be applied in order to sufficiently linearize them for use
in the property clustering algorithm. Once the model is deemed adequate, the region of
applicability needs to be determined. Since the mixture design is performed in the local domain,
then the resulting regression coefficients of the property models express factor effects exactly in
that part of the domain. If the predicted optimum lies outside of the experimental region studied,
then the experiments need to be recentered around the optimum. Many techniques to achieve this
179
repositioning, including gradient, nongradient, and other optimization methods, were discussed
in Chapter 3. Gradient methods are based on the derivative of the response surface model and, as
such, are used only when the model is deemed to be adequate. The most common gradient
method is the method of steepest ascent which performs experiments at predetermined steps
along the vector formed by the gradient of the response surface and is discussed in Chapter 3.
The two shortcomings of the method, namely weighting fractions and visual representation, can
be handled by property clustering. Since property clustering converts the properties into
conserved surrogate property clusters, the steepest ascent gradient method can be entirely
performed in the clustering domain at reduced complexity. The terms used in the gradient
expression are the regressors corresponding to each effect direction and remain unchanged as the
solution is marched toward the predicted optimum represented as the feasibility region.
Furthermore, the step size can be estimated in the property domain rather than the component
domain, which may decrease (or increase) the number of steps needed to reach the feasibility
region. For this reason, it is beneficial to estimate both step sizes, choosing the largest step
whose predicted property cluster remains within the MFR. Once the solution leaves the MFR,
the method is stopped and a new MDOE will need to be conducted.
In cases where the gradient method is unable to reach the feasibility region, either the
adequacy of the model needs to be improved or a more realistic set of targets needs to be
specified. If the adequacy of the model cannot be improved enough to be deemed sufficient and,
as such, is deemed inadequate, then nongradient optimization is performed. Nongradient
optimization searches for the optimum using a step-by-step comparison of obtained property
values. One of the most common types of nongradient optimizations is the simplex self-directing
method (Lazic 2004). This method works by first conducting a simple mixture design around an
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initial guess. Next the lowest response property value is dropped and a mirror-image experiment
opposite of the dropped value is conducted. Within the property clustering construct, the mirror
image may be calculated in the property cluster domain and then back calculated to get the
mixture fractions. Like the gradient method, this procedure has the potential to provide larger
steps toward the solution and continues until the algorithm repeats the same experiments,
indicating that the optimum is somewhere in the bounded region. However, since, by definition,
the use of the nongradient method is indicative of an inadequate model, then the terms of the
model will not adequately convert the measured properties to equivalent constituent fractions.
As such, it is this author?s recommendation that property clustering should only be used as a
visualization tool. Doing so allows the march toward a candidate solution for systems with
multiple chemical constituents and/or multiple properties to be monitored regardless of the
number of chemical constituents studied for up to three properties. For more than three
properties, the algebraic method developed by Qin et al. (2004) can be adapted and used.
Once the boundary of the MFR has been reached, a new MDOE can be performed. Good
properties of the MDOE consist of orthogonality, rotatability, and symmetry about an
experimental center point (Lazic 2004). The spacing of the experimental design points around
the target optimum in the new MDOE is performed in a manner similar to the original MDOE.
One exception, however, is the choice of defining the experimental design region in terms of the
chemical constituents or properties. Since the success of the design is judged by the ability to
arrive at the feasibility region, which is defined in terms of properties, then it would follow that
the new experimental design region should be conducted using properties of the feasibility
region mapped back to the component space. Using these mixture ratios will most likely result in
a loss of symmetry about the experimental design center point. Various optimality algorithms
181
describing the variance profiles associated with each type of design can be used to quantify the
effect of the loss of symmetry. It is the experimenter?s discretion whether to use the design
points that better describe the feasibility region or the design points that are more symmetric.
Once the experiments are conducted, the aCAMbD procedure is repeated.
7.4. Case Study: Polymer Spun Yarn Mixture Design
The case study used to highlight this methodology is the development of a polymer blend
of spun yarn for marine applications. The yarn is blended in a filament winding operation under
the constraints of classical lamination theory, which states that the properties of the wound
filament are universal throughout the yarn (Gurdal et al 1999). Three properties (e.g. Y = P and
p =3) were evaluated by researchers: thread elongation (P1), knot-strength (P2), and density
(P3). Four components (e.g. u = 4) were evaluated using a simplex-lattice design of n = 10
experiments. Data for components i = 1, 2, 3 (low-density polyethylene (LDPE), polystyrene,
and polypropylene) were compiled from Cornell (2002). Data for a fourth component i = 4,
nylon 6,6, was collected from Rodriguez et al. (2003) The resulting MDOE is shown in Table
7.4.
Table 7.4: Simplex-Lattice Design for the MDOE of Polymer Spun Yarn
Exp. Run
Chemical Constituents Response Properties
x1 x2 x3 x4 y1 y2 y3
1 0.96 0.02 0.02 0.00 11.75 9.70 1.29
2 0.50 0.48 0.02 0.00 10.69 11.20 1.14
3 0.50 0.02 0.48 0.00 13.91 10.80 1.18
182
4 0.73 0.25 0.02 0.00 11.22 10.45 1.22
5 0.50 0.25 0.25 0.00 12.30 11.00 1.16
6 0.73 0.02 0.25 0.00 12.83 10.25 1.24
7 0.50 0.25 0.00 0.25 10.82 10.66 1.20
8 0.50 0.00 0.25 0.25 12.57 10.44 1.22
9 0.25 0.25 0.25 0.25 11.99 11.26 1.14
10 0.57 0.17 0.17 0.08 12.01 10.64 1.20
Using linear regression, first-order Scheffe and Cox models were developed for the properties
thread elongation and knot strength. For the third property, density, a previously developed pure
component property operator model was utilized (Eden et al 2003). Fitting the models to the
data set in Table 7.4 resulted in significant models with all individual component terms also
significant. The regression coefficients for the two model types are found in Table 7.5.
Table 7.5: Scheffe and Cox Model Regression Coefficients
i
Scheffe Models Cox Models
?1(?i*) ?2(?i*) ?3(?i*) ?1(?i) ?2(?i) ?3(?i)
1 11.70 9.59 1.30 -0.2968 -1.041 0.104
2 9.40 12.85 0.98 -2.597 2.219 -0.216
3 16.40 11.98 1.07 4.403 1.349 -0.216
4 10.47 10.61 1.20 -1.524 -0.02433 0.00365
s - - - 12.00 10.63 1.20
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To determine the specific product targets, it is important to understand the relationship
between the end-use attributes and the measured properties. The product in this design is to be
used in the spinnaker sheets and guys on a race sailboat. One of the important properties in this
application is that the sheets and guys have some stretch so that the spinnaker will stay filled in a
gust of wind, but not so much that it loses its designed aerodynamic shape. It has been
determined that this attribute is best observed with a thread elongation between 12 and 16 kg of
force. The sheets and guys are also under an immense load and need a high breaking strength
while maintaining some flexibility. This attribute has been determined to be best represented by
a knot strength between 12 and 13 lb of force. Finally, during the setting and dousing of this
spinnaker on a race boat, the sheets and guys may contact the water surface. If they are too
dense, they may ride under the boat and foul the keel. If they are not dense enough, their
diameter could change too much when put under load. On the basis of these attributes, the
specific volume should be between 1.0 and 1.25 mL/g.
Utilizing the regression coefficients found in Table 7.5 in the models of Eq. 3.3 and Eq.
3.14 results in the response surfaces in the simplex diagrams of Figure 7.11. The experimental
design points and their resulting property values are also plotted. The feasibility region is the
region shown in green and yellow. Immediately obvious is the difficulty in the interpretation of
the overall design because of the use of multiple charts. Likewise the influence of single
chemical constituents is difficult to measure because they may have competing effects for
different properties; adversely effecting one property to the benefit of another. Furthermore,
should an additive be chosen to supplement the design, additional figures would be required to
determine its impact and relationship to the current design. All of these conditions make for a
less than ideal situation known as combinatorial explosion.
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(a)
(b)
(c)
Figure 7.11: The Various Simplex Diagrams based on First Order Models of the Responses (a)
Thread Elongation, (b) Knot Strength, and (c) Specific Volume. The Product Target Regions are
in Green and Yellow.
185
To provide an easier method of which to examine the impact of components on all the
properties simultaneously, the design is analyzed using property clustering. Using Eq. 7.7 for the
Scheffe model and Eq. 7.8 and Eq. 7.11 for the Cox model, a set of dimensionless property
operators are created with a set of references chosen to ensure a positive AUP, as shown in Table
7.6.
Table 7.6: Nondimensionalized Property Operators and References
I
Scheffe Models Cox Models
?1 ?2 ?3 ?1z ?2z ?3z
1 0.780 0.639 1.300 -0.00278 -1.176 1.19
2 0.627 0.857 0.980 -0.0243 2.508 -2.48
3 1.093 0.799 1.070 0.0413 1.525 -1.45
4 0.698 0.707 1.200 -0.0143 -0.02749 0.0419
s - - - 0.112 12.01 13.7
ref 15 15 1 107 0.8849 0.0872
The Scheffe dimensionless property operator models are then converted to clusters using
Eq. 6.6 and shown in Table 7.7. Next, the responses measured at each of the mixture design
points are converted to property clusters and shown in Table 7.8. Finally, the specified product
targets are converted to a target feasibility region using the method discussed in Chapter 6,
resulting in Table 7.9.
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Table 7.7: Property Clusters and AUPs of the Scheffe Canonical Property Operators
i
Property Clusters
AUP
C1 C2 C3
1 0.287 0.235 0.478 2.72
2 0.254 0.348 0.398 2.46
3 0.369 0.270 0.361 2.96
4 0.268 0.271 0.461 2.61
Table 7.8: Property Clusters and AUPs of the Experimental Mixtures
i
Property Clusters
AUP
C1 C2 C3
1 0.288 0.238 0.474 2.72
2 0.274 0.287 0.439 2.60
3 0.328 0.254 0.418 2.83
4 0.281 0.262 0.457 2.66
5 0.302 0.270 0.428 2.72
6 0.308 0.246 0.445 2.77
7 0.275 0.271 0.455 2.63
8 0.305 0.253 0.442 2.75
9 0.298 0.279 0.423 2.69
10 0.296 0.262 0.442 2.71
187
Table 7.9: Property Clusters and AUPs of the Feasibility Region
Feasibility Region
Property Clusters
AUPF
C1 C2 C3
Pt. 1 0.281 0.281 0.439 2.85
Pt. 2 0.274 0.297 0.429 2.92
Pt. 3 0.300 0.325 0.375 2.67
Pt. 4 0.364 0.295 0.341 2.93
Pt. 5 0.372 0.279 0.349 2.87
Pt. 6 0.342 0.257 0.401 3.12
Figure 7.12: The Experimental Design Points of the Case Study, a Four-Component Mixture of
Polyethylene, Polystyrene, Polypropylene, and Nylon 6,6 Mapped to Property Cluster Space
(Solvason et al. 2008; Solvason et al. 2009).
0 .9
0 .8
0 .7
0 .6
0 .5
0. 4
0 .3
0 .2
0 .1
0 .1
0 .2
0 .3
0 .4
0 .5
0. 6
0 .7
0 .8
0 .9
C
3
C
2
C
10 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9
2
3
1
4
2
3
1
4
0 . 3 0 . 4 0 . 5 0 . 6
0 .5
0 .4
0 .6
0 .3
0 .4
0. 3
0 .2
0 .2
0 .50 .3
0 .4
C ompone nt E f f ects
D esi g n P oi nts
Fe asi bi l i t y R eg i on
A t t ri bu t es or P rope rt i es 0 .5
188
All clusters are then plotted in property clustering diagrams shown in Fig. 7.12 with the vertices
representing each of the three properties in their cluster forms. By reducing the complexity of the
visualization of the design problem using property clustering, it is now clear that the third
chemical constituent, polypropylene, is closest to the feasibility region, followed by constituent 4
(nylon 6,6), constituent 2 (polystyrene), and constituent 1 (LDPE). Thus, the addition of
polypropylene to a hypothetical mixture will do little to change the properties of the mixture,
suggesting it should be used as a filler. Of the three remaining polymers, LDPE appears to have
the largest impact on the mixture properties when looking at Figure 7.12. However, since the
component effects were derived using Scheffe models, inherent colinearities exist.
Converting from the Scheffe canonical models to Cox polynomial models with a standard
reference mixture at location (0.574, 0.173, 0.173, 0.083) removes the primary colinearity
resulting in a better visualization of the effect of component i.
The Cox property operators in Table 7.6 are then used to calculate the pseudo cluster with
Eq. 7.16 shown in Table 7.11 and the standard cluster with Eq. 7.17 shown in Table 7.10.
Table 7.10: Property Clusters and AUPs of the Standard Reference Property Operators
i
Property Clusters
AUP
C1s C2s C3s
s 0.00435 0.465 0.531 25.85
Table 7.11: Property Clusters and AUPs of the Cox Polynomial Property Operators
i
Property Clusters
AUP
C1z C2z C3z
189
1 -0.2781 -117.6 118.9 0.0100
2 -18.50 1907 -1888 0.00131
3 0.3549 13.11 -12.47 0.0116
4 -142.8 -275.0 418.9 0.000100
Likewise, the pseudo feasibility region is also calculated by correcting the points listed in Table
7.9 to get Table 7.12.
Table 7.12: Property Clusters and AUPs of Pseudo Feasibility Region
Pseudo
Feas.
Region
Property Clusters
AUPF
C1z C2z C3z
Pt. 1 0.0040 0.4841 0.5119 28.01
Pt. 2 0.0039 0.5041 0.4921 29.14
Pt. 3 0.0043 0.5591 0.4366 26.28
Pt. 4 0.0057 0.5583 0.4360 26.31
Pt. 5 0.0060 0.5385 0.4556 25.18
Pt. 6 0.0043 0.4834 0.5112 28.05
Figure 7.13 illustrates the standard and pseudo clusters, zoomed out to include negative
cluster space. The values located along the horizontal x-axis and vertical y-axis are the Cartesian
coordinate scales. Here it is confirmed that polypropylene (3z) has the smallest effect on the
combined mixture properties. However, by removing most of the colinearity in the model, the
result now clearly shows that polystyrene (2z) has the strongest effect on the combined mixture
190
properties. It also shows that polystyrene (2z) and LDPE (1z) have inverse effects, a result not
clear in Fig. 7.11.
Figure 7.13: The Clustering Diagram for the Case Study Showing the Cox Polynomial Model in
Negative Cluster Space.
Evaluating the placement of the experimental design points in the property cluster space
also offers insights into the design. In Fig. 7.12 the experimental design points are translated to
the property cluster space. A single experiment consisting of an equimolar ternary mixture falls
within the feasibility region. Unfortunately, the rest of the design points are outside the
1
z
2
z
3
z
St andard
C1 z ero
C3 z ero
C2 z ero
4
z
-30 0
2 0 0
7 0 0
12 00
1 7 0 0
-30 0 -10 0 1 0 0 3 0 0 5 0 0 7 0 0 9 0 0
C 2 > 1
C 1 < 0
C 3 < 0
C 1 > 1
C 3 > 1
C 2 < 0
R es ponses or Propert ies
P se ud o C omp on en t E f f ec t s
S t an da r d R ef erence
191
feasibility region and none of the candidate mixtures fall within the AUP range. This inference
can also be made when investigating Fig 7.11 but with considerably more effort.
To find potential candidate solutions to the mixture design, the aCAMbD was executed in
accordance with Fig. 7.7 and Table 7.3. Since the calculation of canonical component clusters,
feasibility clusters and AUP values required fewer steps to compute, this data set was chosen as
the inputs for the aCAMbD algorithm. Alternatively, pseudo component clusters, pseudo
feasibility clusters, and AUPZ could have also been chosen.
First, a search for pure component candidate solutions was initiated, resulting in no
matches. Second, binary candidate solutions were generated. Mixtures 1-2, 1-4, and 2-4 were
found to fail Rule 1, so they were excluded prior to property calculation. Mixtures of
components 1-3, 2-3, and 3-4 were found to meet Rule 1 and Rule 2. However, when mixture 1-
3 was tested against Rule 3, it was discovered that its mixture AUPM range did not overlap the
feasibility region?s AUPF along its mixing curve and was thus discarded. The AUPM range of
mixture 3-4 also failed to overlap the feasibility region?s AUPF range along its mixing curve and
was likewise discarded. Fortunately, the AUPM range of mixture 2-3 did overlap the feasibility
region?s AUP range, and was thus back calculated by setting the AUPM to the AUPF of the
feasibility region at the intersection point. It was determined that the mixture range varied
primarily in C1 along C2U and C3L, so the corresponding mixture was determined using the AUPF
of Pt. 3 and Pt. 4, respectively. The resulting mixture fractions were calculated with Eq. 6.33 to
get 0.057 < x2 < 0.591 and 0.409 < x3 < 0.943, as shown in Table 7.13. Mixture 2-3 represents
the simplest mixture that delivers the desired attributes of the design problem.
192
Table 7.13: Binary Mixtures that met Rules 1 and 2.
Binary
Mixture
i-w
AUPM AUPF Mass Fraction i
LL UL LL UL LL UL
1-3 2.87 2.89 3.07 3.11 - -
2-3 2.62 2.94 2.71 2.93 0.057 0.592
3-4 2.73 2.93 2.96 2.97 - -
Any binary mixture candidate will also be able to form a ternary mixture candidate.
Therefore, ternary mixtures 1-2-3 and 2-3-4 will, by definition, meet Rules 1, 2 and 3. To
determine the maximum amount of component 1 that can be added to the solution, the AUPF was
first calculated by looking for a minimum ?2 along the mixing lines between component 1 and
the 2-3 binary mixture. The resulting mixtures produced maximums of x1 ? 0.012 and x1 ?
0.168, with the largest fraction being returned. The procedure was repeated for mixture 2-3-4
and the resulting mass fractions are shown in Table. 7.14.
Table 7.14: Ternary Mixtures that met Rule 1, 2, and 3.
Ternary
Mixture
i-w-r
Mass Fraction i Mass Fraction w
LL UL LL UL
1-2-3 0.00 0.168 0.048 0.592
2-3-4 0.056 0.592 0.400 0.943
It should be noted that the procedure to find the AUPF is cumbersome. An often quicker
alternative is to predict the property values at the intersection point of the feasibility region and
193
the mixture curve and march the solution along the curve until one of the properties no longer
falls within the feasibility limits. This point occurs at the location where AUPF is equal to
AUPM.
In addition to the ternary mixtures shown in Table 7.13, any binary mixture that met Rule
1, but did not meet Rules 2 or 3, may also be able to meet the constraints on the design. For
example, the ternary mixture 1-3-4 is comprised of two binary mixtures (1-3 and 3-4) that met
Rules 1 and 2, but failed to meet Rule 3. Selecting the binary mixture with the broadest mass
fraction delta that met Rule 2 (e.g. mixture 3-4) and investigating the remaining pure component
(e.g. component 1) results in an indeterminant AUPF. This result is directly attributable to all ?2
values used in the mixture being below the targets for the design. As a result, mixture 1-3-4 is
not a candidate solution. The procedure can also be applied to a tertiary mixture, but was beyond
the scope of this design.
A model feasibility region (MFR) can also be calculated for this aCAMbD. In particular,
the a = 3 and n = 10 multivariate system, a MFR was estimated at 95% confidence around each
of the design points as shown in Fig. 7.9. As shown in the figure, the boundary of the closest
candidate solution, a ternary mixture of component 1, 2, and 3 still remains outside of the MFR,
failing to meet Rule 6, meaning that the property operator models must be partially extrapolated
to estimate the candidate. This is an insufficient design. To prevent model extrapolation, the
design points should be repositioned so that they cover the entire feasibility region. The
procedure for executing the repositioning must take into consideration the increase in accuracy
of the property space at the expense of optimality of the component space. Two methods exist
for the reparameterization: (1) increase the complexity of the model to improve its fit or (2)
improve the estimates of the component effects by repositioning the effects over the optimum.
194
Since the fit of the first-order models were found to be excellent with a ?-value of <0.0001, then
the model structure is adequate and should not be changed. Instead, new estimates of the
component effects should be found by conducting a new MDOE. The center point of the new
MDOE should be at the optimum of design.
Figure 7.14: The Model Feasibility Region (MFR) and Candidate Solutions.
2
3
1
4
M ix 2 - 3M ix 1 - 2 - 3
0 . 3 0 . 4 0 . 5 0 . 6
0 .5
0 .4
0 .6
0 .3
0 .4
0. 3
0 .2
0 .2
0 .50 .3
0 .4
C ompone nt E f f ects
D esi g n P oi nts
Fe asi bi l i t y R eg i on
M od el FR
A t t ri bu t es or P rope rt i es
0 .5
195
It can be found in two ways: (1) using the existing response surface models and (2)
performing a domain search using a gradient method. For the first method, the Solver function in
Excel was used to find the optimum by maximizing a geometric mean square of the system of
response models. The optimization resulted in an optimum located at chemical constituent 3,
polypropylene. The magnitude of the factor level adjustment for the MDOE is then at the
discretion of the experimenter. For the second method, the gradient direction of the combined
response surface was found to be equal to the values regressors, which suggests a vector pointing
toward polypropylene. The choice of step size along the gradient is again at the discretion of the
experimenter. Both of these methods require additional experiments that are beyond the ability of
our laboratory to perform and as such limits further analysis of the case study. However, the
main objective regarding the use of property clustering to reduce combinatorial explosion in
screening designs has been thoroughly investigated. Continued analysis on optimization,
decomposition methods, and phenomenological models is presented in Chapter 8 and Chapter 9.
Regardless of the applicability of the response surface models, clustering the solution still
provides the benefit of viewing the design in its entirety on a single diagram, irrespective of the
number of components studied as long as the number of properties measured is three or less. In
cases when three or more properties are studied, additional diagrams may be used or algebraic
methods applied.
7.5. Summary
In this work, a systematic property-based framework for evaluation of mixture design
problems using property clustering has been presented. The recently introduced property
integration framework has been extended to include experimentally derived property operator
models: specifically first-order Scheffe canonical and Cox polynomial models. When
196
interpretation of the chemical constituent?s impact on the mixture property is warranted, Cox
derived property operator models are utilized such that the location of the pseudo chemical
constituent relative to the standard reference mixture is indicative of its impact on the mixture?s
properties. The accuracy of the design is visually observed by placing the design points in the
property design space.
A significant result of the developed methodology is that for problems that can be
satisfactorily described by just three properties, the experimental mixture design problems are
analyzed visually on a simplex diagram, irrespective of how many chemical constituents are
included in the search space. However, algebraic- and optimization-based approaches can easily
extend the application range to include more properties. In summary, the technique proposed
should be capable of extending the use of the property clustering algorithm and the reverse
problem formulation to include the utilization of data and data-driven models. This significant
contribution will allow the exploration of models capable of describing properties that depend on
molecular architectures that may exist at a variety of scales, so long as they can be linearized. In
particular, difficult to quantify chemical product attributes that rely on experimental descriptions,
such as toxicity or disintegration time, can now be simultaneously evaluated in a property cluster
diagram, providing valuable design insights. Furthermore, the use of data driven models opens
up the clustering framework to instant validation using known property values of chemical
products. However, the method outlined will be limited to pure components and mixtures of
pure components, which will limit the range of chemical products that can be designed. In order
to extend the design range, smaller building blocks that could be combined to build molecules
not present in the experimental design would be useful. One such technique is the group
contribution method (GCM) which is a QSPR technique that distills chemical property data
197
down to individual functional groups using group theory. Combining data-driven techniques
with GCM within the clustering framework and applying it to a reverse problem formulation to
investigate all chemical product architectures within a specific property range will be discussed
in detail in Chapter 8.
198
Chapter 8
Attribute-Computer Aided Molecular Design (aCAMD)
In his review of chemical product design, Gani (2004) defines the general structure of
product design problems in process systems engineering, identifying several areas that the
systems engineering community can improve through the development of novel methods and
tools. One particular need identified by Gani (2004) was for a method to enlarge the application
range of existing property models and/or to develop new property models in order to better
describe structured chemical products with large numbers of atoms or highly electronegative
behavior. In the absence of adequate models, the conventional approach to these types of
problems has been to rely on empirical or simulation models to describe the system, estimating
the boundaries where the models are adequate, and then designing the molecular architecture of
chemical products in those adequate domains. While accurate, this approach is ill-formed, time-
consuming, expensive, and overly narrows the design parameters, resulting in non-optimum
products. As shown by Duvedi and Achenie (1996), the application of conventional solvers like
mixed-integer, non-linear programs (MINLPs) to a computer aided molecular design (CAMD) is
limited to local optima solutions when the property prediction methods are non-linear and there
is no guarantee that the global optimum exists among the generated list of candidate molecules.
Although work in disjunctive programming (Grossman 1999, Sammons et al. 2009, Odjo et al.
2011) has been developed to address some of these shortcomings, the uncertainty in the design
199
space, lack of insight in to the relative impacts, similiarties, and interactions of each of the
candidate molecules and molecular groups, and forward problem orientation limit its use.
Recognizing the above limitations, this chapter describes a new technique called attribute
based computer aided molecular design (aCAMD) that combines experimental methods with
CAMD techniques in a reverse problem formulation (RPF) to obtain the enumeration of all
possible candidates and candidate mixtures. In Section 8.1, the mapping of data in the consumer
attribute domain in to a physical-chemical property domain described by group contribution
methods (GCM) is discussed followed by the identification of suitable alternatives using 1st and
2nd order GCM techniques in the property clustering framework. Section 8.2 discusses the
clustering of the pure component effects, candidate mixtures, feasibility region, and the model
feasibility region. Section 8.3 describes the aCAMD solution approach, interpretation of the
generated candidates, the model feasibility region (MFR), and the limitations of the technique
when cross-validating the results in both the property and attribute domains. Section 8.4
highlights the method in a case study on environmentaly friendly refrigerant design and Section
8.5 concludes the chapter with a summary.
8.1. Integrating Attribute Data and GCM Models with the RPF
It is well recognized that experimental methods yield probabilistic models which may be
of less accuracy than computational chemistry based deterministic models. However, several
well established techniques exist for the proper estimation of this residual error and its resulting
propagation. In addition the probabilistic models can be tailored to have a reduced non-linearity
which results in fewer local optima and improve the likelihood of finding the global optima. If
any non-linearities exist in the constitutive equations, they can be marginalized by using a
reverse problem formulation with property clusters to solve the problem in the property design
200
space by making using of the duality of linear programming (Eden et al. 2004, Eljack et al.
2007). Enumeration of all possible candidate mixtures is possible using group contribution
methods (GCM) to predict the molecular structure and by matching the result against the product
design sub-problem. This allows for the complete identification of all possible candidates and
candidate mixtures without being computationally expensive. The benefit of using GCM is that
the additives which are not part of the original data set can be predicted, reducing the number of
experiments needed to quantify the attribute-property-constituent relationships. Additionally,
insights into the molecular structures of the candidates and candidate mixtures can be obtained.
Finally, the method shown here demonstrates that mapping design information from one
property domain to another is feasible, which is an important key to solving multi-scale chemical
product design problems.
In product design the key to successfully designing a molecule that meets a set of
constraints is the ability to adequately predict the molecular structures and properties.
Traditionally, GCM has been used to predict the physical properties of a molecule based on the
additive nature of individual group fragments. Unfortunately, the method is limited by the
number of properties it is able to predict. For properties not described by GCM (i.e. not in Table
6.1), a relationship between non-GCM described properties and GCM described properties must
be obtained (Eljack et al. 2007). Often the non-GCM described properties are consumer
attributes. Consumer attributes consist of a list of product characteristics the consumer finds
desirable or undesirable. The attributes are typically modeled as a list of empirical and
theoretical equations of known and controllable physical properties. Represented
mathematically, there is a set of attributes A which are functions of physical properties P
(Solvason et al. 2009).
201
)(PfA? (8.1)
The physical properties themselves can be expressed as functions of their chemical signature.
These equations can be of the group contribution type (i.e. Eq. 8.2) or empirically derived (i.e.
Eq. 8.3) (Joback and Reid 1987; Constantinou and Gani 1994; Marrero and Gani 2001). The
expressions are often highly non-linear (Solvason et al. 2009).
),( 21 gg nnfP ? (8.2)
)(XfP? (8.3)
where the properties P are functions of a set of chemical components X. The property-
component relationship typically contributes to the non-linear part of an MINLP. As stated
earlier the solution to the MINLP is not guaranteed to be the global optimum (Duvedi and
Achenie 1996). An elegant reparameterization of the MINLP is the reverse problem formulation
presented by Eden et al. (2004) and Eljack et al. (2005). In this work, the process design
problem is separated from the molecular design problem. The reverse solution of the process
design problem generates the design targets for the molecular design problem. The molecular
design problem is solved in reverse to match the design targets from the process design. The
method can be extended to the attribute problem such that the solution of Eq. 8.1 determines the
property targets used in the reverse solution of Eq. 8.2 and Eq. 8.3. This approach is depicted in
Fig. 8.1.
For the special situations where the selection of the optimum candidate mixtures from a
known data set is deemed sufficient, the MINLP and reverse problem formulation are
unnecessary. In this case the attributes A can be directly regressed in terms of the components X
and the component mixtures (Solvason et al. 2009):
202
)(XfA? (8.4)
which is a special case of Eq. 3.5 with Y = A that was discussed thoroughly in Chapter 7 and will
not be discussed further in this chapter.
Figure 8.1: Description of the Reverse Problem Formulated Product Design with Property
Mapping to Group Contribution Described Properties.
Since the objective of most molecular generation techniques is to develop a complete set
of candidates and candidate mixtures, the use of property prediction techniques based on
molecular structure is preferred. In this approach a set of target properties is estimated from a set
of attributes subject to the following constraint (Solvason et al. 2009):
)( jk PfA ? where jk? (8.5)
QM, DFT Mo d el s
Ab I nit i o Met ho ds
(Q ua ntum S cal e)
Fu n d a m e nt al Prin ci ple s o f
Chemic al E n gi n e e rin g
Proc es s D esign
(M ac ro Sca le )
Chem om etric Mo d el s
Consumer Attrib utes
(M ac ro/Met a Scal e )
MM, M D, MC Si mul ati o n s
Atomist ic M etho ds
(Atom ic Sca le )
GCM, T o p ol o gical I n dice s
and C h em om etric Mo d el s
CA PD / CAMD
(M ol ec ula r S cal e)
Coa rse - Gr ai n in g a n d
Chem om etric Mo d el s
Mi c rostruc ture
( Mes o Sca l e)
PR OPERTI ES
P = f(X)
P = f(ng1 , ng2)
A = f(P)
203
As long as the number of attributes is greater than the number of properties, unique property
values can be calculated from the attribute data. Depending on the nature of the relationship
either a linear program (LP) or non-linear program (NLP) must be solved such that a set of
properties is estimated for each set of attributes. While a LP is preferred for its solution
efficiency, most attributes are nonlinear functions of physical-chemical properties.
Unfortunately, the use of a NLP introduces a potential to reach non-global solutions to the
candidate generation problem. An alternative method is to use inverse regression, especially in
situations where the attribute-property relationship is unknown. The use of regression to develop
this relationship can reduce the complexity of the design problem at the expense of introducing
uncertainty in to the solution. This uncertainty is best quantified using analysis of variance
(ANOVA) and it is essential that limitations be placed on the design space such that only the
well correlated regions are investigated. Similar to the attribute-based ?model? feasibility region
(MFR) described in Chapter 7, confidence intervals on the estimates of the attributes, AMkL and
AMkU, can be found using Hotelling?s T2 test (Stine 2001, Ramirez 2009). The aMFR upper and
lower limit for the kth attribute is shown in Eq. 8.6:
? ? ee
Ekkaunake xXXxSun nFaun unaY 1, )(1)( )1( ??? ?????????? ??????????? ?? ??? ? (7.27)
Where the response Y = A for cosumer attributes. The upper and lower limits estimated by this
procedure can then be used to develop an attribute ?model? feasibility range (aMFR) by
combining Eq. 7.27 with the responses at the design points and then selecting the maximum and
minimum of the responses to get Eq. 8.6:
UkkLk AMAAM ?? (8.6)
204
The aMFR is then created by inserting the bounds of Eq. 8.6 into Eq. 6.7. Any candidates or
candidate mixtures found outside the region are known to be invalid within a 1-? confidence.
Similarly, a property ?model? feasibility region (pMFR) can be created by applying Eq. 7.27 and
to the property responses (i.e. Y = P) of the design points in an experimental design, resulting in
Eq. 8.7. The pMFR upper and lower limit for the jth property are as follows:
? ? eeEjjpunpje xXXxSun nFpun unpP 1, )(1)( )1( ??? ?????????? ??????????? ?? ??? ? (8.7)
To determine the property target feasibility region (pFR), attribute targets set by the consumer
that define an attribute feasibility region (aFR) are transformed into property targets using an
inverse regression expression of Eq. 8.5 resulting in the following:
)(AfP? (8.8)
where the parameters of the fitted model are written as:
? ? PAAAB ??? ?1 (8.9)
Like the pMFR, the pFR confidence interval grows wider as the model moves farther away from
the mean of the predictor. However, since the pFR is developed from a regression model serving
a predictive role, new observations must include both the error associated with the model and
error associated with the new experimental observation. This results in pFR upper and lower
limits for the jth property as follows:
? ? ? ?
eeEjjpanpje AAAASan nFpan anpP 1, )(11)( )1( ??? ??????????? ??????????? ?? ??? ?
(8.10)
A set of attribute targets specified by the consumer, as shown in Eq. 8.11, can then be converted
into a property domain by taking the minimum and maximum property values generated using
Eq. 8.8 and 8.10.
205
UkkLk AAA ?? (8.11)
UjjLj PPP ?? (4.16)
The resulting property domain is only valid if the attributes and properties can be considered
unbiased, otherwise the NLP will need to be solved in the conventional manner (van Belle 2002).
The benefit of eliminating the need to solve the NLP is that a global optimum solution will
always be obtained, subject to a given function of the uncertainty in the regression expression.
As before, the measure of model fitness is R2 and the predicted fit Q2 can be used to infer the
accuracy of the obtained models described by Eq. 8.8; a large R2 represents a well fit model and
a large Q2 represents a greater likelihood that the candidate solutions generated by the aCAMD
algorithm will be the true global candidate solutions (Solvason et al. 2009). Alternatively,
Eq.8.8 may be fit univariately, in which case the variance would not be pooled, and each j
property would have independent R2j and Q2j that could be combined using a geometric mean
square.
? ?pp
j jp
RR
1
22 ??
(8.12)
? ?pp
j jp
QQ
1
22 ??
(8.13)
Other model fitness measures that use Wilks? Lamda, Pillai?s Trace, Hoetelling-Lawley Trace,
Roy?s Greatest Root, etc. can be utilized to to improve model fitness and prediction fitness and
can be found in Johnson and Wichern (2007).
206
8.2. Attribute and Property Clustering Algorithms
Once the property responses to the experiments are conducted and the property targets
are calculated, property clustering techniques are employed to cluster the property data as
previously outlined in Chapter 6. These techniques convert the property targets into conserved
surrogate clusters that are described by property operators, which have linear mixing rules, even
if the operators themselves are non-linear (Eden et al., 2004; Shelley and El-Halwagi, 2000).
Two model types can be used to design molecules in the property domain: mixture property
models and group contribution models. The choice of property model dictates the type of design
being conducted, a mixture property model results in a CAMbD and a molecular property model
results in a CAMD. When the mixture property model is chosen, the pure component clusters
can either be (1) calculated using the property expressions in Table 6.1 for each pure component
used in the design as shown in Chapter 6 or (2) by performing regression to develop the pure
component effects as shown in Chapter 7. The operators for option 1 are shown in Eq. 8.14.
)(
*
jrefj
ijij P? ???
(8.14)
The operators for option 2 are shown in Eq. 8.15.
? ?? ?
jrefj
ijjij PP????
(8.15)
For option 1, the procedure for finding candidate mixtures is given in Section 6.2. For option 2,
the procedure is is similar to that presented in Section 7.3, but with the non-dimensionalized
property operators replacing the non-dimensionalized attribute operators, and the FR being
replaced with a pFR which is calculated using Eq. 8.10.
207
Both of these mixing design methods are limited to finding candidates having the same
molecular architecture commonalities as the training set, and they require new experiments to
make informed predictions on potential additives which can result in combinatorial explosion.
For example, suppose there is interest in finding a new refrigerant and the intention is to select
the optimum candidate or candidate mixture from a list of refrigerants. To solve this design
problem using mixture design, the properties must first be directly estimated from the component
fractions or obtained from a database. For the case study presented in Section 8.4, it is projected
that a comprehensive list would contain approximately 2050 possible candidate molecules
(Solvason et al. 2009). An explicit model relating these candidates directly to the properties
would then require a minimum of 2050 experiments to quantify the 2050 parameters in the linear
mixture model unless a database or decomposition is used (Solvason et al. 2009). Alternatively,
suppose a deterministic, equation of state model like Peng-Robinson is used to evaluate the
thermodynamic properties of the refrigerants and that binary interaction parameters of the model
are readily available for all of the refrigerants (Brown 2007). Even in this ideal situation, the
relationship between the properties of each refrigerant and the product attributes will have to be
quantified through an experimental design. However, due to the non-linear nature of the
property-chemical constituent relationship, mixtures of constituents cannot be used to explicitly
reduce the attribute search space without sacrificing computational efficiency. Thus, at a
minimum, a mixture design including all 2050 pure components plus additional parameters for
estimating the non-linearity would be necessary unless the design search space was extensively
restricted. Since neither of these two results is acceptable, it is preferred to use models based on
group theory.
208
In the group contribution methods (GCM), the property function of a compound is
estimated as the summation of property contributions of all the molecular groups present in the
molecular structure (Ambrose, 1978, 1980; Joback & Reid, 1983):
1st Order 2nd Order 3rd Order
??? ??? 3 332 221 11 g ggTg ggSg gg
M PnwPnwPnP (4.15)
where PM is a function of property group contribution properties Pg1, Pg2, and Pg3 of the 1st, 2nd,
and 3rd order groups g1, g2, and g3 which occurs ng1 , ng2, and ng3 times (Eljack et al. 2007;
Solvason et al. 2009; Chemmangattuvalappil et al. 2010). In Chapter 6 it was shown that the
definition of the property models in GCM was similar to the property operators used in the
clustering framework for the 1st order group contribution. The estimate of the j normalized
property operator based on first order groups is shown in Eq. 6.37 (Eljack et al. 2007;
Chemmangattuvalappil et al. 2009):
?? ???
F
g
MijgigMji n
11 11
(6.37)
where, ?jg1 is the normalized property operator of first order group, g1. The resulting molecular
cluster is then written as Eq. 6.38:
Mi
F
g
M
ijgig
M
ji A U P
n
C
?
?
??
? 11 11
(6.38)
where the molecular AUP of molecule i is written as Eq. 8.16:
???
F
g
MigigMi A U PnA U P
11 11
(8.16)
209
The accuracy of this approach can be improved by including second order groups. These groups
can be incorporated into the cluster space by taking advantage of their linear additive rules. In
addition, since second order groups have first order groups as building blocks, then it follows
that second order groups can be considered as combinations of different first order groups
(Constantinou and Gani 2001). Thus, the number and type of second order groups can be
directly estimated from the number and type of first order groups, subject to the following rules:
Rule 11. Second order groups have first order groups as their building blocks.
Rule 12. A second order group is formed only if it has at a minimum all of the first order
groups in the required number. For instance, to form the second order group
CH(CH3)CH(CH3), there must be two -CH- and two (CH3) groups.
Rule 13. One first order group can be a part of more than one second order group. If,
however, one second order group is completely overlapped by another second
order group, only the contribution from larger group is considered. This is because
the interaction defined by the smaller group will be taken into account by the
larger group. For example, if NH3CHOH and CHOH groups are present, only the
contribution from NH3CHOH group is considered for property estimation.
The nig2 number of g2 second order groups in molecule i can be estimated using Eq. 8.17.
?
?
?
?
???
? ??
?
???
?
??
iF
iF
ig
igig nnM inI n tn ?? :
1
12 (8.17)
where nig1:niF are the number of each g1 first order group found in molecule i and niF and ?ig1:
?iF?is the total number of occurences of each g1 in g2 in molecule i. As Rule 12 suggests, the
minimum of these ratios will result in the actual number of second order groups present in the
210
molecule (Solvason et al. 2009, Chemmangattuvalappil et al. 2010). nig2 is then rounded down
to the nearest integer number because the number of second order groups cannot be a fractional
number. The normalized property operator for the second order property contributions ?ij2 is
then written as
MjgG
g ig
Mij n 2
12 22 ??? ??
(8.18)
where ?jg2 is the property contribution from the second order groups. Eq. 8.18 can predict the
property contribution from second order groups in most molecules as demonstrated by Cases A
and B in Fig. 8.2.
211
Figure 8.2: Different Situations Related to Overlapping of Second-Order Groups (Marrero and
Gani 2001).
In rare occasions one or more of the second order groups will partially overlap one another (e.g.
Case C) and/or completely overlap one smaller second order group (e.g. Case D) (Marrero and
Gani 2001). Two methods exist for estimtating Case C. The first method method finds the
maximum number of non-overlapped second order groups that describe the molecule, and drops
the remainder. This method is discussed in more detail in Chapter 9 for IR/NIR groups.
D. A 2
nd
or de r gr oup c omple t ely
o v erl app i ng anothe r 2
nd
or de r gr oup
H
3
C CH C OO H
NH
3
H
3
C H C=CH
C. 2
nd
or de r gr oup s parti all y o v erl app i ng
ea c h other
H
3
C
CH
OH
CH
3
CH
A. A 2
nd
or de r gr oup o v erl app i ng 1
st
or de r
gr oup s
B. 2
nd
or de r gr oup s not o v erl app i ng an y
other 2
nd
or de r gr oup s
OH
CH
2
H
3
C
CH H
3
C
H
3
C
CH CH
2
H
3
C
OH
212
Although valid, it has been shown that using the contributions from all groups gives better
accuracy, since they generally provide dissimilar information about the molecular structure
(Marrero and Gani 2001). When one molecular group completely overlaps another (i.e. Case D),
Rule 13 states that the contribution of the largest group must be chosen and the contribution from
the smaller group must be removed. However, it is important not to remove the group entirely
from the molecule as it may appear in other locations outside of the completely overlapped
group. Algebraically, this can be accomplished by removing the group from the set of G in Eq.
8.18 and then adding back only the unoverlapped number of that group using Eq. 8.19 and Eq.
8.20:
???
?
???
? ??
?
???
?
????
?
???
?
???
iF
iF
ig
ig
Gg
iG
vg
isig nnM i nnnM i nI n tn ???? ::
1
122
222
(8.19)
where niv2:niG is the number of each dissimilar v2 second order group in molecule i and
ng2v2:ng2G is the number of each dissimilar v2 second order group completely overlapped by
second order group g2 (Chemmangattuvalappil et al. 2009). If the smaller second order group
only appears in the overlapped group, the two terms in the right hand side of Eq. 8.19 will cancel
out. Otherwise, Eq. 8.19 estimates the number of occurences of each group remaining in the
molecule which are not completely overlapped. If ?*jg2 is the contribution from the smaller
overlapped second order groups, then the normalized property operator for the property
contributions from smaller second order groups, ?*ij2 can be calculated as the following
(Solvason et al. 2009, Chemmangattuvalappil et al. 2010):
?? ?? ??? ? Mjg
G
g ig
Mij U n 2
12 22
(8.20)
213
where g2:GU are the set of non-overlapped second order groups and g2:GO are the set of
overlapped second order groups.
OU GGG ??
(8.21)
Eq. 8.18 is then written as the following:
Mjg
G
g ig
Mij O n 2
12 22 ??? ??
(8.22)
The normalized property operator for molecule i can now be estimated as (Solvason et al. 2009;
Chemmangattuvalappil et al. 2010):
???????? MijMijMijMij 221 (8.23)
Thus, given a set of molecular groups, candidate chemical structures can be estimated from first
order group contributions, second order group contributions containing overlaps, and
contributions from non-overlapped occurences of second order group contributions overlapped
by larger second order overlaps. Clustering the non-dimensionalized property operators of Eq.
8.23 using Eq. 6.5 and 6.6 results in the clusters shown in Eq. 8.24-8.26.
Mi
MijM
ij AUPC 111
??
(8.24)
Mi
MijM
ij AUPC 222
??
(8.25)
*2
*2*
2 Mi
MijM
ij AUPC
??
(8.26)
In order to combine Eq. 8.24-8.26 with Eq. 8.23, the AUPM of each type of group contribution
must be corrected to the overall molecular AUPM using the following correction factors:
214
Mi
Mi
i AUPAUPCF 11 ?
(8.27)
Mi
Mi
i AUPAUPCF 22 ?
(8.28)
Mi
Mi
i AU PAU PCF
*2*
2 ?
(8.29)
Thus the cluster of the molecule is found to be the following:
*2*22211 MijiMijiMijiMij CCFCCFCCFC ?????? (8.30)
As with previous cluster algorithms, a system of three properties can be plotted on a ternary
diagram. An overview of the attribute based computer aided moleclular design (aCAMD)
conversion algorithm is presented in Fig. 8.3. The procedure for converting attribute, property,
and group data into non-dimensional property operators and property clusters for later use in a
candidate generation algorithm is presented in Table 8.1.
215
Fig. 8.3: The Attribute Based Computer Aided Molecular Design (aCAMD) Conversion
Algorithm.
Table 8.1: The aCAMD Cluster Conversion Procedure
Step Description Equation
1 Select Attribute-Property Model 8.8
2 Fit Model to Experiment Design Point Responses to Determine Model Parameters 8.9
3 Calculate the Variance, Covariance, and Confidence Intervals of the Attribute-Property Model 8.10
4 Calculate the UL and LL of the Property Feasibility Region (pFR) 6.14, 8.10
5 Guess Property Operator Reference Values and Calculate Nondimensional Property Operators of Experimental Responses 6.4, 8.4, 8.14, 8.15
In i ti ali z e aCAMD
Clus t er Con v er sion
Al g or i thm
Es ti ma t e A t tr i bu t e -
P r ope rt y Models
Calc ula t e the
Mol ecular Gr oup &
Experi ment al D a t a
P r ope rt y Oper a t or s
Se t P r ope rt y
R e f er en c e V alues
Con v ert LL and UL
P r ope rt y
Cons tr ai n ts t o
P r ope rt y Oper a t or s
Calc ula t e T ar g e t
A t tr i bu t e Clus t er
Si nk
Calc ula t e the A UP
of a l l P r ope rt y
Oper a t or s
Calc ula t e the
Mol ecular Gr oup
Clus t er s
Calc ula t e the
Experi me n t al Da t a
Clus t er s
Calc ula t e Model
F ea sibil i ty Cl us t er
Sink
Li ne ari z e P r ope rt y
Mo dels f or
Che mic al
Cons ti tuen ts
Con v ert A t tr i bu t e
Experi men t al Da t a
& Cons tr ain ts t o
P r ope rt y Da t a &
Cons tr ain ts
Does the
S y s t em
ha v e 3
A t tr i but es ?
Output All
Clus t er and
P r ope rt y
Oper a t or V alues
Con v ert Cl us t er s t o
Cart es i an
Coor dina t es & P l ot
NO
YE S
Ar e All
Cl us t er
V alues
P osit i v e ?
NO
YE S
Run R e f er en c e
Op ti mi z a ti on
Alg ori thm
aCAMD Con v er sion
Alg ori thm
216
6 Calculate the AUP of the Experimental Response Property Operators 6.16
7 Calculate the Nondimensional Property Operators of the pFR 6.14, 6.15
8 Calculate the AUP range of the pFR 6.16
9 Cacluate the Nondimensional 1
st and 2nd Order Molecular Group
Property Operators Found in the Chemical Constituent Training Set 6.15
10 Calculate the AUP of the Molecular Group Property Operators 6.16
11 Calculate the Maximum No. 1
st and 2nd Order Molecular Groups that
meet Step 7 6.43, 6.48
12 Calculate the 1
st and 2nd Order Molecular Group Clusters,
Experimental Response Clusters, and Cluster Domain of the pFR 6.17
13 Calculate the Variance, Covariance, and Confidence Intervals of the Linearized Property ? Component Property Model 8.7
14 Calculate the UL and LL of the Property Model Feasibility Regions 6.14, 8.7
15 Calculate the Nondimensional Property Operators of the pMFR 6.14, 6.15
16 Calculate the AUP range of the pMFR 6.16
17 Calculate the Cluster Domain of the pMFR 6.17
18 Run Reference Optimization Algorithm for Non-Positive AUPs 6.8-6.12
19 Convert all Clusters to Cartesian Coordinates and Plot 6.23-6.26
A list of first and second order group contributions for the properties outlined in Table 6.1 can be
found in Marrero and Gani (2001). More detail on the clustering of 2nd order and higher
expressions is provided in Section 8.3. In addition, Chemmangattuvalappil et al. (2010) has
developed third order molecular property operators for larger, more complex chemical
constituents including those that are polycyclic (Marrero and Gani, 2001). After all of the
molecular group property contributions have been obtained and converted to clusters, they can be
used to solve a CAMD in the cluster domain. However, as was shown in Section 4.4 and Section
217
6.3, the structure of molecular property operators dictates that the CAMD occurs in the property
operator domain. The details of the CAMD are discussed in Section 8.3.
8.3. aCAMD using Experimental Data and GCM Models
Like the aCAMbD approach in Chapter 7, the cluster conversion algorithm for the
aCAMD contains two feasibility regions. For the aCAMD these regions are the ?model?
feasibility region (pMFR) and the ?target? feasibility region (pFR). To be considered a potential
candidate, either a pure component or a mixture of components must fall within both regions and
meet both AUP values simultaneously. Hence, this structure of the aCAMD as a reverse
problem formulation (RPF) creates an opportunity to explore multiple levels of molecular
architecture (e.g. group changes and pure component changes) without commtting to products a
priori, an observation that is explored more thoroughly in Chapter 9.
The procedure for finding candidate mixtures has already been discussed in Chapter 6 for
known property models and in Chapter 7 for empirically derived property models. The method
of finding candidate molecules is dependent upon the estimation of the molecular AUP and
molecular clusters. For first order groups, Eq. 6.38 ? 6.40 dictate that the combination of groups
will obey linear mixing and lever arm analysis on a ternary cluster diagram (see Fig. 6.9). The
addition of second order groups to this construct introduces non-linearity to the solution since
second order groups ?correct? first order group approximations by accounting for group-to-group
binary interactions. The non-linearity of binary interactions prevent the use of lever-arm analysis
in the clustering domain, although recent progress discussed in Chapter 10 suggests that with
alteration, it may be achievable. For this situation, it has been shown by Chemmangattuvalappil
et al. (2008) that a solution can be reached by utilizing property operators directly in an algebraic
approach similar to that suggested in Section 4.4.
218
The general aCAMD statement is to generate the structures of all possible
molecules that can be built from g1 groups with p target properties. There will be one upper and
one lower bound on each property estimated through the process design problem.
UjMijLj PPP ?? pj? ; ui? (8.28)
Where i is the index of molecules and j is the index of properties. This equation can be written
in terms of normalized property operators as follows
UjMijLj ????? (8.29)
Here, ?ijM is the p normalized j property operator of molecule i. The target property ?ijM is
described by two inequality expressions, one for the lower bound and one for the upper bound
(Qin et al., 2004), resulting in 2p inequality expressions to represent all the possible solutions
(Eden et al. 2004; Chemmangattuvalappil et al. 2009; Solvason et al. 2009). This target property
sink area is the made up of the intersection of the pFR and pMFR. Although it is possible to
combine these regions into a pseudo target region by taking the minimum of the upper bounds
and the maximum of the lower bounds, it is not preferred because of the role of aCAMD is both
screening and optimization. Confounding the pFR and pMFR rejection criteria prevents the
researcher from discerning which set of models the non-candidate solution failed and, by
extension, why the non-candidate solution failed. For example, the reason a non-candidate
solution fails the pFR domain check may only be because of a poorly fitted experimental model
in its domain which could be solved by conducting validation experiments for a better fit.
However, a violation of the pMFR domain is more severe and suggests the underlying molecular
architecture is either poor or poorly fit, the latter of which is highly unlikely. In summary, the
preferred approach to generating solutions that meet both the pFR and pMFR is to first check
them against the pFR and then validate them using the pMFR, as was the case for CAMbD.
219
In order to determine which potential candidate structures to test, the reverse problem
formulation (RPF) is initiated using a similar approach suggested in Section 6.3. First the
number of the set of g1 ? F first order groups found in the molecular design are calculated using
Eq. 6.43 and placed in Eq. 6.48.
???
???
?
??
1
1 g
UU
g AU PAU PIntn
(6.43)
Ugg nn 111 ?? (6.48)
Then the number of the set of g2 ? G second order groups are calculated from the first order
groups using Eq. 8.17 and Eq. 8.19.
?
?
?
?
???
? ??
?
???
?
??
iF
iF
ig
igig nnM inI n tn ?? :
1
12 (8.17)
???
?
???
? ??
?
???
?
????
?
???
?
???
iF
iF
ig
ig
Gg
iG
vg
isig nnM i nnnM i nI n tn ???? ::
1
122
222
(8.19)
Once the sets are calculated, the property contributions are obtained from literature sources e.g.
Marerro and Gani (2001), Constantinou and Gani (1994), etc.
Once the maximum number of dissimilar groups ?gmax is set, the design is executed,
beginning with a minimum combination of dissimilar groups and then increasing sequentially
until the maximum is reached. For ?pure group? mixtures (e.g. ?g = 1), the pure group property
contributions are estimated from Eq 6.37 for each g1 group type since pure groups have no binary
interactions and, by extension, no second order groups. These clusters are then tested against the
points making up the feasibility bounds of a pFR. All ?pure group? property clusters (e.g. Eq.
6.17) that meet Rule 1 for the pFR are then passed on to test against the pMFR. ?Pure group?
clusters that meet Rule 1 for both the pFR and pMFR are then tested against Rule 2 in the same
220
order using Eq. 8.16. All non-candidate mixtures are discarded. The remaining ?pure group?
mixtures are then tested against Rule 3, 4, and 6 and and non-candidate solutions are discarded.
Since it has been shown by Marrero and Gani (2001) that the group contribution methods (GCM)
may not predict properties accurately for open-chain polyfunctional compounds with more than
four carbon atoms in the main chain, a new rule is created and the ?pure component? mixtures
are tested against it.
Rule 14. The number of identical functional groups in the backbone of a molecule is to be
limited to a maximum of four.
If ng1,BB are the number of first order groups in the backbone, then Rule 14 can be written as Eq.
8.33:
3,1 ?? BBgn (8.33)
Pure group molecules that fail Rule 14 are discarded. The remaining groups are denoted as
?pure group? candidate molecules.
For binary and larger mixtures (i.e. ?g ? 2 ) of dissimilar types of groups (i.e. v1 ? g1)
with no ring compounds (i.e. NRG = 0) , the aCAMD differs from the procedure for CAMD with
clusters. Since dissimilar groups are capable of forming second order groups which have
negative property operators that can increase the number of groups used in the design beyond the
result of Eq. 6.43, it can no longer be used as a valid bound on the design. Furthermore, an
explicit set of equations to determine the maximum number of a specific first order group to take
the place of Eq. 6.43 has not yet been developed. As a result, the aCAMD method for dissimilar
groups defaults to a forward based approach where all combinations of dissimilar groups are
generated and subsequently removed by testing against the bounds of the design. For this case
221
the maximum number of each group is set by the researcher, using Eq. 6.43 as a guide but not a
hard bound.
Beginning with structural stability, each potential candidate solution is first tested for
structural stability using Rule 4 and Rules 14-16. Solutions that fail any of these rules are
discarded. Of the remaining structurally sound molecules, those solutions that fail the non-
dimensional property operator bounds of the pFR are discarded, followed by those that fail the
pMFR. Alternatively these could be combined into a single region described by Eq. 8.29 and
tested as discussed earlier. Those solutions that remain are outputted as candidate solutions to
the design problem.
The same methodology can be followed to identify cyclic (i.e NRG = 1) compounds as
well with a few additional constraints. The group contribution method has questionable range of
accuracy for compounds containing more than one ring even with the inclusion of second order
groups (Marrero and Gani 2001; Chemmangattuvalappil et al. 2009; Solvason et al. 2009). So
the number of rings in the compound must be restricted to one. The decision on the groups to be
part of the ring must be made ahead of design. This is to account for the difference in the
property contributions of the same group to cyclic and acyclic compounds. No first order acyclic
groups should be found in the ring structures.
Rule 15. Ring structures within a potential candidate molecule should consist of only first
order cyclic groups.
Furthermore, the number of cyclic groups in a ring structure is also limited. Two-group
structures are infeasible and three-group structures are often thermodynamically unstable and
should be avoided. Therefore a new Rule 16 is created to check that stability of suitable
candidate structures.
222
Rule 16. The minimum number of functional groups in a ring structure of a molecule is
three.
If ng1,RG are the number of groups in a ring, then Rule 15 can be represented using Eq. 8.34:
3,1 ?? RGgn (8.34)
Ring compounds that fail Rule 15 and 16 are discarded. Those that meet Rule 15 and 16 as well
as Rule 1-4 are denoted candidate solutions.
In conclusion, this approach represents a significant reduction in the complexity of an
aCAMD. For example, the theoretical 2050+ parameters of the earlier aCAMbD design problem
can be reduced to only 20 parameters in an aCAMD, a 100:1 reduction. This reduction in
complexity may also improve the accuracy of the design by: (1) using the additional degrees of
freedom in the design to minimize the error and (2) ensuring each enumerated candidate
molecular structure posesses the desired values for the consumer attribute. The only caveat is
that accuracy of the aCAMD is predicated on the quality of two models (i.e. the attribute-
property relationship and the property-molecular architecture relationship), whereas the CAMbD
utilizes a single model. Combining the predicted fit Qp2 of the two models using Eq. 3.26, and
combining them using Eq. 8.35:
? ? MMM zz
z zz
QQ
1
1
22 ?
?
? (8.35)
where zM is the total number of models used in the design, and z is the model identifier. Values
of Eq. 8.35 that are close to 1.0 would indicate a completely validated molecular design
containing only global optimum candidates, whereas values less than 1.0 suggests that some
error may exist and that the data set may contain non-global optimum candidate solutions. In
general, a larger value of the QzM2 measure indicates a better chance that the estimated global
223
optimum candidate set is the true global optimum candidate set. In situations where QzM2 is of
questionable accuracy, a second validation check can be executed by directly calculating the
attributes for the candidate molecules and checking them against the bounds of the design given
in Eq. 8.11. The calculation of the candidate attributes is achieved by solving the attribute-
property model with a NLP solver of the inverse regression problem or a direction calculation
from the normal regression problem. Each additive attribute can then be plotted to verify that the
solution meets the specified consumer attributes.
Since the structure of the proposed method does not automatically require the solution of
an NLP, it can be certain that the potential additives will always be at the estimate of the global
optimum. The same cannot be said for the solution of most MINLP algorithms. Although the
MINLP will always provide a solution, it may only be a local optimum. There is no way of
knowing if the local optimum is close to the global optimum without using more rigorous and
computationally expensive solution strategies like disjunctive programming. Using the method
outlined in this chapter provides not only estimates of the global optimum, but also provides a
means by which to determine how well the estimated global optimum matches the true global
optimum, all while being computationally inexpensive.
The procedure determining finding the property targets from a set of consumer attributes
and solving for candidate mixtures and molecules that meet the design targets is summarized in
Table 8.3. and Fig. 8.4 ? Fig. 8.6.
Table 8.3: The Attribute Based Computer Aided Molecular Design (aCAMD) Method.
Step Description Equation
1 Set Maximum Dissimilar Groups in the Mixture -
224
2 Calculate Pure Group Solution Clusters 6.37, 6.16, 6.40
3 Discard Pure Group Clusters (from aCAMD Conversion Algorithm) that Fail Rule 1 for the pFR 8.28, 8.10, 8.15
4 Discard Pure Group Clusters (from aCAMD Conversion Algorithm) that Fail Rule 6 for the pMFR 8.28, 8.7, 8.15
5 Calculate No. Pure Groups 6.47, 6.48
6 Calculate Pure Group Solution AUPs, Discard Pure Group Solutions that Fail Rule 2 and Rule 3 for the pFR 8.28, 8.10, 6.16, 6.47
7 Discard Pure Group Solutions that Fail Rule 7 and Rule 8 for the pMFR 8.28, 8.7, 6.16, 6.42
8 Discard Pure Group Solutions that Fail Rule 4 and Output Candidate Solutions 6.41
9 Repeat Steps 2-7 for Ring Structures and Output Candidate Solutions -
10 For Binary+ Solutions, Generate All Group Combinations Given a Maximum Number of Each 1st Order Group 6.46, 6.48
11 Discard Binary+ Solutions that Fail Rule 4 6.41
12 Discard Binary+ Solutions that Fail Rule 14 8.33
13 Calculate the No. 2nd Order Groups 8.17, 8.19
14 Calculate the Property Operator Responses for the Candidate Solutions 8.23, 8.22, 8.20, 8.15
15 Discard Binary+ Solutions that Fall Outside the pFR 8.28, 8.10
16 Discard Binary+ that Fall Outside the pMFR, Output Candidate Solutions 8.28, 8.7
17 Repeat Steps 10-14 for Ring Structures -
18 Discard Binary+ Solutions that Fail Rule 15 -
19 Discard Binary+ Solutions that Fail Rule 16 8.34
20 Discard Binary+ Solutions that Fall Outside the pFR 8.28, 8.10
21 Discard Binary+ Solutions that Fall Outside the pMFR, Ouput Candidate Solutions 8.28, 8.7
22 Repeat Steps 10-21 until ?maxis reached -
225
Fig. 8.4: The aCAMD Candidate Generation Algorithm
226
Fig. 8.5: The aCAMD Pure Group Test Candidate Molecules
227
Fig. 8.6: The aCAMD Binary + Test for Candidate Molecules
228
8.4. Case Study: Environmentally Friendly Refrigerant Design
Since the 1930s chlorofluorocarbon (CFC) based refrigerants, such as
dichlorofluoromethane (R-12), have found use in home refrigerators, aerosol cans, fire
extinguishers, and automotive air conditioners primarily due to their non-toxic, non-flammable
nature and their high overall thermodynamic efficiency (He et al., 2005). Unfortunately, R-12
has been found to damage the ozone layer, increase greenhouse effects, damage telluric
environments, and affect human health (Zhao et al., 2004). Beginning in 1987 with the Montreal
Protocol on substances that deplete the ozone layer, CFC and hydrochlorofluorocarbon (HCFC)
refrigerants were phased out and replaced with more ozone friendly hydrofluorocarbon (HFC)
refrigerants such as 1,1,1,2-tetrafluoroethane (R-134a). Although R-134a has been easily
adapted to fit existing refrigeration cycles, it exhibits a high global warming potential (GWP)
leading to a moderately high potential environmental impact (PEI). In 1997 the Kyoto Protocol
on reducing greenhouse gases identified R-134a as contributing to global warming (Zhao et al.,
2004). Thus, the use of R-134a is only a temporary solution to the search for environmentally
friendly refrigerants capable of meeting the thermodynamic needs of refrigerant systems.
The design of these environmentally friendly replacement refrigerants has primarily
proceeded down one of two paths: either mixing existing refrigerants experimentally (Akasaka et
al., 2007; Coquelet et al., 2006; Eljack et al., 2005; Kondo et al., 2006; Zhao et al., 2004) or
computationally (Saleh and Wendland, 2006; Calero et al., 1998; Duvedi and Achenie, 1997;
Molnarne et al., 2005; Muller et al., 1996; Rachidi et al., 1997; Sahindis et al., 2003; Schroder
and Molnarne, 2005; Young and Cabezas, 1999). Using the experimental method, He et al.
(2005) have determined that using a 85/15 wt% of difluoroethane (R-152a) and
pentafluoroethane (R-125) is a potential substitute for dichlorofluoroethane (R-12) in a typical
229
home refrigerator. Others such as Kondo et al. (2006) have focused on improving the
flammability issues associated with using propane (R-290). Although these traditional
experimental techniques offer valuable information to the design of refrigerant mixtures, it is
time-consuming and expensive to test all possible candidates. Use of a mixed integer non-linear
program (MINLP) to elucidate a computer-aided molecular design (CAMD) offers a unique
approach to the candidate enumeration problem. As shown by Duvedi and Achenie (1997), the
MINLP is limited to local optima solutions when the property prediction methods are non-linear.
In both the experimental and MINLP techniques there is no guarantee the global optimum exists
among the list of candidate molecules. Recently, Sahindis et al. (2003) utilized a branch-and-
bound global optimization algorithm in a typical MINLP as a method to enumerate all feasible
candidate molecules. While a significant contribution to the improvement of the MINLP, the
method still relies on several intensive algorithms, which are computationally expensive.
Furthermore, it offers no additional insights to solution of the problem such as the relative
impacts of each of the candidate molecules and molecular groups, their similarities, and their
interactions.
In addition, in product design, consumer preference drives the value of a product. If the
consumer does not prefer the product to similarly priced products, then the product is not
selected. To measure the consumer preference, a set of consumer attributes A are defined.
Often, the relationship between the underlying properties and the consumer attribute is not
explicitly known. In these situations, design of experiments is employed to find approximate
empirical relationships between the attributes and either the chemical constituents or the
underlying properties. For refrigeration systems the attributes important to the consumer are the
Energy Index (EI), the Compressor Displacement Index (CDI), the Miscibility Index (MI), and
230
the Potential Environmental Impact (PEI). The EI is a measure of the thermodynamic efficiency
of the refrigerant in an ideal refrigeration loop with a condenser temperature of 300K and an
evaporator temperature of 233K. The CDI is measure of the compressor displacement. The MI
is a measure of the compatibility of the designed refrigerant with the existing equipment. The
PEI is the potential environmental impact of the designed refrigerant or refrigerant mixture as
calculated by the WAR algorithm (Young and Cabezas 2000). The PEI is an estimated value
consisting of weighted measures of the toxicology, ozone depletion potential (ODP), global
warming impact (GWP), and many other factors important to health and the environment. All
PEI values estimated were done using the default weighting factors provided by WAR GUI
Version 1.0.14 (Young and Cabezas 2000).
As noted earlier, R-134a is an excellent thermodynamic refrigerant, but its PEI value is
too high because of a high GWP. In this case study it is desired to find a suitable refrigerant with
a low PEI number to replace R-134a. In addition, the customer already has access to a low-cost
source of diflouromethane (R-32) and wishes to find an additive that can be added to it so that
the combined mixture can replace R-134a. Based on commercially available refrigerants and
proprietary knowledge, the customer has recommended using one of the 7 refrigerants listed in
Table 8.4 as the additive, although the use of other additives remains possible.
Table 8.4: List of Potential Refrigerant Additives
ASHRAE # Name CAS #
R-152a 1,1 ? Diflouroethane 75-37-6
R-290 Propane 74-98-6
R-600 Butane 106-97-8
231
R-125 Pentafluroethane 354-33-6
R-143 1,1,2-Trifluoroethane 430-66-0
R-22 Chlorodifluoromethane 75-45-6
R-600a Isobutane 106-97-8
A set of acceptable EI, CDI, MI, and PEI values have also been developed that are
deemed by the customer as suitable for a drop-in replacement of R-134a. They are as follows:
5540 ??EI (8.36)
2824 ?? CDI (8.37)
2.28.1 ?? MI (8.38)
Since the PEI cannot be estimated by the GCM, it will be used as an additional screening tool for
the pure component and candidate mixtures found to meet the other attribute targets such that the
candidate solution with the smallest PEI is chosen as the output of the aCAMD. Thus, the
overall objective of this case study is to generate the structures of all possible additive and
additive mixtures that can be built from Ng1 groups found in the customer?s training set that,
when combined with R-32, can be utilized as a replacement for R-134a.
Two methods exist in which these candidate pure components and mixtures can be
ascertained. The first method is to conduct a mixture design of experiment (MDOE) and fit the
attribute response data in terms of chemical contributions directly using the procedure outlined in
Chapter 7. For this method, the additives listed in Table 8.4 are combined with the refrigerants
R-12, R-134a, and R-32 and a d-optimal MDOE is conducted. Using the recommendations from
Custom Designer JMP 7.0.2, 31 design points of various mixture fractions of the 10 components
were developed as shown in Table 8.5.
232
Table 8.5: The MDOE for the Refrigerant Replacement Case Study.
Run
Mass Fractions
R-32 R-152a R-290 R-600 R-125 R-134a R-143 R-12 R-22 R-600a
1 0.00 0.00 0.00 0.00 0.01 0.99 0.00 0.00 0.00 0.00
2 0.10 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.90 0.00
3 0.00 0.05 0.04 0.00 0.91 0.00 0.00 0.00 0.00 0.00
4 0.00 0.00 0.98 0.00 0.00 0.00 0.00 0.00 0.02 0.00
5 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00
6 0.00 0.00 0.00 0.00 0.89 0.00 0.11 0.00 0.00 0.00
7 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.90 0.10 0.00
8 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
9 0.00 0.00 0.00 0.99 0.00 0.00 0.00 0.01 0.00 0.00
10 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00
11 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00
12 0.00 0.00 0.90 0.00 0.00 0.09 0.00 0.01 0.00 0.00
13 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00
14 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
233
15 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00
16 0.00 0.07 0.61 0.00 0.00 0.00 0.32 0.00 0.00 0.00
17 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
18 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00
19 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00
20 0.77 0.00 0.00 0.00 0.00 0.23 0.00 0.00 0.00 0.00
21 0.00 0.00 0.00 0.00 0.00 0.85 0.12 0.02 0.01 0.00
22 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00
23 0.00 0.00 0.00 0.10 0.00 0.90 0.00 0.00 0.00 0.00
24 0.36 0.00 0.64 0.00 0.00 0.00 0.00 0.00 0.00 0.00
25 0.00 0.00 0.00 0.02 0.00 0.00 0.98 0.00 0.00 0.00
26 0.90 0.00 0.00 0.00 0.00 0.00 0.00 0.10 0.00 0.00
27 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00
28 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00
29 0.00 0.87 0.00 0.00 0.00 0.13 0.00 0.00 0.00 0.00
30 0.10 0.00 0.00 0.00 0.90 0.00 0.00 0.00 0.00 0.00
31 0.07 0.00 0.00 0.00 0.00 0.00 0.93 0.00 0.00 0.00
234
The associated attribute responses of the 31 mixture fractions provided by the theoretical
customer are given in Table 8.6.
Table 8.6: The Attribute Responses to the Refrigerant Replacement MDOE.
Run
Attribute Responses
EI
(Measured)
CDI
(Measured)
MI
(Measured)
PEI
(Calculated)
1 51.7 26.4 1.99 0.79
2 58.9 31.4 1.76 1.34
3 56.5 23.9 1.69 1.05
4 100.0 26.3 3.65 0.87
5 55.9 30.9 1.73 1.49
6 54.4 23.6 1.58 1.13
7 37.5 26.3 1.57 2.63
8 71.1 28.8 2.82 0.05
9 66.0 24.0 4.27 1.28
10 67.6 24.6 2.05 1.24
11 69.1 23.0 4.10 1.47
12 95.9 26.2 3.54 0.86
13 69.1 23.0 4.10 1.47
14 90.1 36.0 1.96 0.00
15 35.7 25.7 1.55 2.75
16 89.5 25.9 3.13 0.92
235
17 71.1 28.7 2.82 0.05
18 69.1 23.0 4.10 1.47
19 55.9 30.9 1.73 1.49
20 80.0 33.9 1.98 0.18
21 52.8 26.2 1.99 0.89
22 66.2 24.0 4.29 1.27
23 53.7 26.2 2.20 0.83
24 97.3 29.8 2.93 0.54
25 67.5 24.6 2.09 1.24
26 83.1 35.0 1.94 0.28
27 35.7 25.7 1.55 2.75
28 66.2 24.0 4.30 1.27
29 68.9 28.5 2.72 0.14
30 57.0 24.8 1.59 1.00
31 69.2 25.4 2.04 1.16
These responses, targets, and fitted models were then non-dimesionalized using reference
property operators of 10 for EI, 3 for CDI, and 0.2 for MI and clustered using the procedure
outlined in Chapter 7, resulting in Fig. 8.7.
236
Figure 8.7: The Attribute Cluster Diagram for Refrigerant Replacement Case Study.
As shown, only R-134a meets the target attributes. No other single component meets the
necessary target attributes, a result common in refrigeration systems (Zhao et al. 2008). In
addition, although some of the designed mixtures fall within the target feasibility region, no
tested mixture satisfied Rules 1-3 simultaneously. It is also clear that no new mixtures of the
candidates suggested by the customer with R-32 will satisfy the target region when using lever-
arm analysis, unless that candidate is also mixed with R-134a. This option was deemed
unacceptable since the customer desired to fully replace R-134a and, as a result no mixture of the
components given by the customer will produce a solution to this design problem.
0 .9
0 .8
0 .7
0 .6
0 .5
0 .4
0 .3
0 .2
0 .1
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
C
3
C
2
C
1
0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9
R32
R15 2 a
R29 0
R60 0
R60 0 a
R12 5
R13 4 a
R14 3 a
R12
R22
C ompone nt A t t r i bu t e E f f ects
D esi g n P oi nt R esp on ses
A t t r i bu t e FR ( aF R )
A t t r i bu t e M od el FR ( aM FR )
A t t ri bu t es
237
Rather than attempt to augment the mixture design with additional chemical constituents,
new molecular structures were designed from the underlying molecular architecture present in
the design using the aCAMD approach discussed earlier in this chapter. In this approach the
attribute data is mapped to a property domain where group contribution models can be used to
build new molecules not present in the original training set using the training set?s functional
groups. The attribute relationship described by Eq. 8.5 must be defined such that the number of
attributes is equal or greater than number properties used in the design (e.g. a = 3 means p ? 3).
There is reason to believe that EI, CDI, and MI are functions of the following properties: critical
temperature Tc, critical pressure Pc, and the latent heat of vaporization at standard conditions Hv.
All of these properties can be described by GCM. Furthermore it has been found that EI is
proportional to Tc, Pc, and Hv, CDI to Tc and Pc, and MI to Tc, Pc, and Hv. The simplest model
that can be built from these intuitive relationships is that of first order polynomials as shown in
Eq. 8.39 ? 8.41:
vcco HPTEI ??????? 321 ???? (8.39)
cco PTC D I ????? 21 ??? (8.40)
vcco HPTMI ??????? 321 ???? (8.41)
To determine the regressors in these relationships, property data for each of the design point
mixtures was provided by the customer as shown in Table 8.7. These properties could also be
determined using equations-of-state and group contribution property estimations.
238
Table 8.7: The Calculated Property Responses to the Refrigerant Replacement MDOE (Marrero
and Gani 2001).
Run
Properties
Tc
(K)
Pc
(bar)
Hv
(kJ/mol)
1 373.8 40.5 18.1
2 367.5 50.7 15.6
3 342.9 36.7 13.5
4 369.8 42.7 14.6
5 369.3 49.9 15.8
6 340.2 36.2 13.2
7 383.4 42.2 16.6
8 386.4 45.2 18.4
9 425.1 38.0 20.8
10 346.8 38.3 13.5
11 408.2 36.5 19.1
12 370.3 42.3 14.9
13 408.2 36.5 19.1
14 351.4 57.9 14.2
15 385.0 41.3 16.7
16 363.7 41.4 14.5
17 386.4 45.2 18.4
18 408.2 36.5 19.1
19 369.3 49.9 15.8
20 356.6 53.9 15.1
21 371.2 40.4 17.6
239
22 425.3 38.0 20.8
23 379.4 40.3 18.5
24 363.1 48.1 14.4
25 348.3 38.3 13.6
26 354.8 56.3 14.4
27 385.0 41.3 16.7
28 425.4 38.0 20.8
29 384.9 44.6 18.4
30 340.6 38.2 13.2
31 347.1 39.6 13.5
The regressors found to fit the above equations were estimated using the statistical package JMP
7.0.2 and are listed in Table 8.8.
Table 8.8: The Regressors for the Refrigerant Attribute Models
Informative
Statistics
All Data Points (n = 31) Excluding high R12 (n = 28)
EI CDI MI EI CDI MI
?o -116.3887 5.1674 -12.7938 -209.2422 5.0184 -16.8315
?1 0.6467 -0.0070 0.0551 1.0933 -0.0064 0.0745
?2 1.4505 0.5726 -0.0085 1.4181 0.5720 -0.0100
?3 -7.3196 - -0.2985 -11.5491 - -0.4825
R2 0.3015 0.9902 0.5989 0.5611 0.9915 0.8723
Q2 0.1615 0.9891 0.5246 0.4434 0.9908 0.8379
R2adj 0.2239 0.9895 0.5543 0.5062 0.9908 0.8564
When performing the regression, it is apparent that a poor fit for EI is achieved with an
R2 value of 0.3015, due primarily to the experiments containing large fractions of
240
dichlorofluoromethane (R-12). It may be beneficial to remove these experiments from the
design should they be determined to be outliers. Continuing with the other attribute data, the
CDI is found to fit the data well with a R2 value of 0.9902. The MI is also found to fit the data
poorly, with an R2 value of 0.5989. Again, experiments containing large fraction of R-12 appear
to be outliers. An inspection of actual vs. predictive plots also suggests that these experiments
might be outliers. In Figure 8.7a the lines of best fit and the confidence intervals are plotted
using all experiments. Figure 8.7b shows the same information as Fig. 8.7a with the suspected
outliers removed. To determine if potential outliers should be excluded, an analysis is performed
to determine if the experiments with large amounts of R-12 overly influenced the model. In this
method, the suspected experiments are removed and the regression is repeated. The results are
compared to the prior results containing all possible solutions. If large changes in the R2 and the
regressors are observed, then the experiments are considered overly influential and removed
from the data set. In Fig. 8.7b it is found that the R2 values were significantly improved. The R2
values for EI, CDI, and MI improved to 0.5611, 0.9915, and 0.8723, respectively. The parameter
estimates also changed noticeably. Thus it is concluded that the experiments involving large
amount of R-12 were overly influential in determining the regressors and are removed from the
analysis. When removing the experiments containing large amounts of R-12 from design, the
range of the model must also be adjusted by placing boundaries around the region of measured
values only. This result is an acceptable limitation since the R-12 PEI is the highest of the
compounds evaluated.
241
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
M
I
A
c
tu
a
l
1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 4 . 5
M I P r e d ic t e d P < . 0 0 0 1
R S q = 0 . 8 7 R M S E = 0 . 3 6 0 2
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
M
I
A
c
tu
a
l
1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 4 . 5
M I P r e d ic t e d P < . 0 0 0 1
R S q = 0 . 6 0 R M S E = 0 . 6 4 6 1
30
40
50
60
70
80
90
100
110
E
I
A
c
tu
a
l
30 40 50 60 70 80 90 100 110
E I P r e d ic t e d P = 0 . 0 1 9 8
R S q = 0 . 3 0 R M S E = 1 6 . 7 8 2
50
60
70
80
90
100
110
E
I
A
c
tu
a
l
40 50 60 70 80 90 100 110
E I P r e d ic t e d P = 0 . 0 0 0 2
R S q = 0 . 5 6 R M S E = 1 1 . 7 0 8
2 2 . 5
25
2 7 . 5
30
3 2 . 5
35
3 7 . 5
C
D
I
A
c
tu
a
l
2 2 . 5 2 5 . 0 2 7 . 5 3 0 . 0 3 2 . 5 3 5 . 0 3 7 . 5
C D I P r e d ic t e d P < . 0 0 0 1
R S q = 0 . 9 9 R M S E = 0 . 3 5 7 7
2 2 . 5
25
2 7 . 5
30
3 2 . 5
35
3 7 . 5
C
D
I
A
c
tu
a
l
2 2 . 5 2 5 . 0 2 7 . 5 3 0 . 0 3 2 . 5 3 5 . 0 7 5
C D I P r e d ic t e d P < . 0 0 0 1
R S q = 0 . 9 9 R M S E = 0 . 3 6 3 7
(a) (b)
Figure 8.8: The actual vs. predicted plots of EI, CDI, and MI with (a) all of the design points (n
=31) and (b) the high level R-12 experiments removed (n=28).
242
Further improvement may also be obtained by developing a non-linear relationship for
Eq. 8.5. In this case, a non-linear solver must be used to find the optimum set of properties that
deliver the set of attributes. Unfortunately, the use of non-linear programming introduces
increased computational complexity and the potential for returning non-global optimum
solutions. An advantageous, alternate technique for building stronger attribute-property
relationships is to use inverse regression. Inverse regression reverses the dependent and
independent variables in Eq. 8.5 to arrive at Eq. 8.42:
)(AfP? (8.42)
The inverse linear regression problem is only valid when both the properties and attributes are
random variables meaning they are normally distributed. This is true for this case study since the
attributes are system responses and the properties follow a normal curve. The resulting inverse
regression expressions for Tc, Pc, and Hv are shown in Eq. 8.43-8.45:
MIC D IEIT oc ??????? 321 ???? (8.43)
MIC D IEIP oc ??????? 321 ???? (8.44)
MIEIH ov ????? 31 ??? (8.45)
The regressors for these expressions are shown in Table 8.9.
243
Table 8.9: The Regressors for the Inverse Regression Expressions.
Informative
Statistics
Excluding high R12 (n = 28)
Tc Pc HV
?o 266.3071 -5.7126 16.1679
?1 -1.1867 0.0105 -0.0834
?2 3.5923 1.7235 -
?3 35.0855 0.3916 2.2703
R2 0.9917 0.9946 0.7243
Q2 0.9892 0.9925 0.6624
R2adj 0.9906 0.9939 0.7022
Using the expressions in Eq. 8.43-8.45, the upper and lower bounds for each consumer attribute
listed in Eq. 8.36-8.38 were converted into a set of minimum and maximum limits on the
physical-chemical properties, as shown in Table 8.10.
Table 8.10: Upper and Lower Limits on the Physical-Chemical Properties Estimated.
FR Tc Pc HV
LL 350.4 36.9 15.7
UL 396.6 43.8 17.8
Since some uncertainty exists in the attribute-property models, new lower and upper bounds are
estimated from these minimum and maximum limits using Eq. 8.10. This broadening of the
feasibility range ensures that, with 95% confidence, any molecule found outside the region is a
non-candidate solution and should be discarded. The resulting target domain is known as the
property feasibility region (pFR) and shown in Table 8.10.
244
Table 8.11: The Upper and Lower Limits on the Physical-Chemical Properties in the pFR
pFR Tc Pc Hv
LL 342.2 35.2 11.1*
UL 405.7 45.6 22.8
*11.733 is the lower limit of the Marrero and Gani GCM method
Of note is the large broadening of the standard enthalpy of vaporization Hv resulting in an
infeasible lower limit. This value was replaced by the lower limit of the GCM model at 11.733
kJ/mol. Likewise a constraint on the applicability range of the property-component model
termed model feasibility region is mapped to the design space as the upper and lower bounds on
the property domain. The region is determined using the minimum and maximum of the
responses at the design points resulting in Table 8.12.
Table 8.12: The Upper and Lower Limits on the Modeled Physical-Chemical Property
Responses.
MFR Tc Pc HV
LL 340.2 36.2 13.2
UL 425.4 57.9 20.8
A pMFR can be constructed by accounting for the uncertainty in the physical-chemical property
response model. This uncertainty can either be estimated using Eq. 8.7 or by using the
information on root mean square error (RMSE) from Marrero and Gani (2001). The RMSE for
the experimental design is smaller so the first option was selected, resulting in the pMFR given
in Table 8.13.
245
Table 8.13: The Upper and Lower Limits on the Physical-Chemical Properties in the pMFR
pMFR Tc Pc Hv
LL 334.4 30.4 4.4*
UL 431.3 66.3 26.7
*11.733 is the lower limit of the Marrero and Gani GCM method
For a candidate or candidate mixture to be considered valid, it must fall within both the
feasibility region and the model feasibility region and satisfy the AUP ranges of both regions
simultaneously. The estimation of new mixtures, those not present in the original training set,
occurs through the use of the aCAMbD method which linearly mixes various pure components to
arrive at the targets in Table 8.10-8.13. The pure component effects are estimated from literature
sources and provided by the customer. They were corroborated using Perry and Green (1997)
and Sahinidis et al. (2005) resulting in Table 8.14.
Table 8.14: Pure Component Properties of the Pure Component Effects
Pure
Component
i
ASHRAE #
Pure Component Properties
Tc
(K)
Pc
(bar)
Hv
(kJ/mol)
1 R-32 351.4 57.9 11.99
2 R-152a 386.4 45.2 18.41
3 R-290 369.8 42.5 14.60
4 R-600 425.4 38.0 20.81
5 R-125 339.4 36.0 13.14
6 R-134a 374.2 40.6 18.20
7 R-143a 346.8 38.3 13.49
8 R-12 385.0 41.3 16.67
246
9 R-22 369.3 49.9 15.75
10 R-600a 408.2 36.5 19.08
All data necessary for the execution of the aCAMD has now been gathered, but requires
conversion to nondimensional property operators, clusters, and AUP values in order to facilitate
a solution. Beginning with the training set, the linearized forms of critical temperature, critical
pressure, and the standard enthalpy of vaporization listed in Table 6.1 are used to convert the
training set into property operators. The property operators are then nondimensionalized
according to Eq. 8.15 using the reference values listed in Table 8.15.
Table 8.15: Nondimensionalized Property Operators of the Training Set
Run
Nondimensional Property Operators
?1 ?2 ?3
1 1.01 1.22 0.64
2 0.98 0.81 0.36
3 0.88 1.43 0.17
4 0.99 1.12 0.29
5 0.99 0.84 0.40
6 0.87 1.46 0.14
7* 1.05 1.14 0.48
8 1.06 1.01 0.67
9 1.26 1.35 0.90
10 0.90 1.34 0.18
11 1.17 1.44 0.73
12 0.99 1.14 0.32
247
13 1.17 1.44 0.73
14 0.91 0.60 0.03
15* 1.06 1.18 0.49
16 0.96 1.18 0.28
17 1.06 1.01 0.67
18 1.17 1.44 0.73
19 0.99 0.84 0.40
20 0.94 0.71 0.17
21 1.00 1.23 0.59
22 1.26 1.35 0.91
23 1.03 1.23 0.67
24 0.96 0.90 0.19
25 0.90 1.34 0.19
26 0.93 0.64 0.07
27* 1.06 1.18 0.49
28 1.26 1.36 0.91
29 1.06 1.04 0.66
30 0.87 1.35 0.13
31 0.90 1.27 0.17
ref 5 0.05 10
*Experiments not included in the parameterization of Eq. 8.42
The resulting property operators are then be converted into property clusters using Eq. 6.6. and
shown in Table 8.16.
248
Table 8.16: Clusters and AUPs of the Training Set
Run
Property Clusters
AUP
C1 C2 C3
1 0.351 0.426 0.223 2.87
2 0.455 0.376 0.169 2.16
3 0.355 0.576 0.069 2.48
4 0.413 0.467 0.121 2.40
5 0.443 0.376 0.180 2.23
6 0.352 0.589 0.059 2.47
7* 0.392 0.427 0.181 2.68
8 0.388 0.370 0.243 2.74
9 0.358 0.385 0.257 3.52
10 0.372 0.555 0.073 2.41
11 0.349 0.431 0.220 3.35
12 0.405 0.464 0.131 2.45
13 0.349 0.431 0.220 3.35
14 0.596 0.388 0.017 1.53
15* 0.386 0.433 0.180 2.74
16 0.398 0.487 0.115 2.42
17 0.387 0.370 0.243 2.75
18 0.349 0.431 0.220 3.35
19 0.443 0.376 0.180 2.23
20 0.516 0.391 0.093 1.81
21 0.354 0.437 0.208 2.81
22 0.357 0.385 0.258 3.52
249
23 0.351 0.420 0.229 2.94
24 0.468 0.438 0.093 2.05
25 0.371 0.550 0.078 2.43
26 0.565 0.391 0.044 1.64
27* 0.386 0.433 0.180 2.74
28 0.357 0.385 0.258 3.52
29 0.383 0.376 0.241 2.76
30 0.372 0.573 0.055 2.35
31 0.385 0.544 0.071 2.33
Likewise, the nondimensionalized property operators and clusters for each pure component
effect are also calculated and shown in Table 8.17 and Table 8.18, respectively.
Table 8.17: Nondimensionalized Property Operators of the Pure Component Effects
ASHRAE #
Nondimensional Property Operators
?1 ?2 ?3
R-32 0.91 0.60 0.03
R-152a 1.06 1.01 0.67
R-290 0.99 1.13 0.29
R-600 1.26 1.36 0.91
R-125 0.87 1.47 0.14
R-134a 1.01 1.22 0.65
R-143a 0.90 1.34 0.18
R-12 1.06 1.18 0.49
R-22 0.99 0.84 0.40
250
R-600a 1.17 1.44 0.73
Table 8.18: Clusters of the Pure Component Effects
ASHRAE #
Property Clusters
AUP
C1 C2 C3
R-32 0.596 0.388 0.017 1.53
R-152a 0.387 0.370 0.243 2.75
R-290 0.412 0.469 0.119 2.40
R-600 0.357 0.385 0.258 3.52
R-125 0.350 0.594 0.057 2.48
R-134a 0.351 0.425 0.225 2.88
R-143a 0.372 0.555 0.073 2.41
R-12 0.386 0.433 0.180 2.74
R-22 0.443 0.376 0.180 2.23
R-600a 0.349 0.431 0.220 3.35
Finally, the FR, MFR, pFR, and pMFR nondimensionalized property operator regions are
calculated using Eq. 8.15 and shown in Table 8.19.
Table 8.19: The Nondimensionalized Property Operators for the Feasibility Region Types
Feasibility Region Type Bounds ?1 ?2 ?3
FR
LL 0.91 1.07 0.39
UL 1.11 1.41 0.61
MFR
LL 0.87 0.60 0.15
UL 1.26 1.46 0.91
251
pFR
LL 0.88 1.00 0.00
UL 1.16 1.52 1.11
pMFR
LL 0.85 0.40 0.00
UL 1.29 1.87 1.50
From these bounds, the cluster feasibility regions can be calculated using Eq. 6.7 resulting in
Tables 8.20-8.23.
Table 8.20: The Property Cluster Bounds on the FR
FR
Property Clusters
AUP
C1 C2 C3
Pt. 1 0.351 0.413 0.235 2.59
Pt. 2 0.310 0.482 0.207 2.94
Pt. 3 0.335 0.520 0.145 2.72
Pt. 4 0.381 0.485 0.367 2.92
Pt. 5 0.432 0.416 0.153 2.58
Pt. 6 0.398 0.384 0.218 2.79
Table 8.21: The Property Cluster Bounds on the MFR
MFR
Property Clusters
AUP
C1 C2 C3
Pt. 1 0.367 0.251 0.382 2.37
Pt. 2 0.269 0.451 0.280 3.24
Pt. 3 0.352 0.589 0.059 2.48
252
Pt. 4 0.440 0.509 0.051 2.86
Pt. 5 0.629 0.298 0.073 2.00
Pt. 6 0.456 0.216 0.329 2.76
Table 8.22: The Property Cluster Bounds on the pFR
pFR
Property Clusters
AUP
C1 C2 C3
Pt. 1 0.294 0.334 0.371 2.99
Pt. 2 0.251 0.433 0.316 3.51
Pt. 3 0.367 0.633 0.00 2.40
Pt. 4 0.432 0.568 0.00 2.67
Pt. 5 0.537 0.463 0.00 2.15
Pt. 6 0.354 0.306 0.340 3.26
Table 8.23: The Property Cluster Bounds on the pMFR
pMFR
Property Clusters
AUP
C1 C2 C3
Pt. 1 0.310 0.144 0.546 2.74
Pt. 2 0.201 0.444 0.355 4.22
Pt. 3 0.312 0.688 0.00 2.72
Pt. 4 0.408 0.592 0.00 3.16
Pt. 5 0.766 0.234 0.00 1.69
253
Pt. 6 0.405 0.124 0.470 3.18
The resulting clusters of the FR, MFR, training set responses, and the training set pure
component effects are plotted in Fig. 8.9. The clusters of the pFR, pMFR, training set
responses, and the pure components are plotted in Fig. 8.10.
Figure 8.9: The Property Cluster Diagram Highlighting the Mixture Design, Pure Chemical
Constituents, FR, and MFR.
0 .9
0 .8
0 .7
0 .6
0 .5
0 .4
0 .3
0 .2
0 .1
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
C
3
C
2
C
1
0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9
R32
R15 2 a
R29 0
R60 0
R60 0 a
R12 5
R13 4 a
R14 3 a
R12
R22
P ure C ompone nt P r op ert i es
D esi g n P oi nt R esp on ses
P r op ert y FR ( FR )
P r op ert y M od el F R ( M F R )
P hy s. - C he m. P rope rt i es
254
Figure 8.10: The Property Cluster Diagram Highlighting the Mixture Design, Pure Chemical
Constituents, pFR, and pMFR.
As shown in both Fig. 8.9 and Fig. 8.10, the model feasibility region and target feasibility
region completely overlap, meaning that all feasible mixtures estimated can be found by
interpolation of the model. However, due to a poor fit of the the Hv model, there is sizable
variability along C3, as evidenced by the variation of the shape of the FR vs. the pFR, and to a
lesser extent the MFR vs. pMFR feasibility ranges. The larger of the domains (pFR and pMFR)
are chosen to ensure that no feasible solutions are excluded from the design. Several refrigerants
fall simultaneously within both the pFR and pMFR, but none of the candidates have AUP values
that match the AUP values of the sink, with the exception of R-12. This result is expected since
0 .9
0 .8
0 .7
0 .6
0 .5
0 .4
0 .3
0 .2
0 .1
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
C
3
C
2
C
1
0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9
R32
R15 2 a
R29 0
R60 0
R60 0 a
R12 5
R13 4 a
R14 3 a
R12
R22
P ure C ompone nt P r op ert i es
D esi g n P oi nt R esp on ses
P r op ert y FR ( pF R )
P r op ert y M od el FR ( pM FR )
P h y s. - C h em. P rop erti es
255
R-134a was meant as a replacement for R-12. Since the potential environmental impact (PEI) of
R-134a is 0.784/kg and the PEI of R-12 is 2.751/kg, R-12 will not meet the objective of lowering
the PEI in this design problem. Furthermore, since no other mixture of components in the
training set met the objectives of the design, it is desirable to either find new mixtures and/or
design new refrigerants that can help meet the constraints. The remaining focus of this case
study will be on the latter.
Due to large inventories of R-32 available to the consumer, it is desirable to find a way to
use this refrigerant in the design. Since R-32 does not lie within the feasibility region, an
additive must be used. The ?additive? feasibility range must be bounded by the model feasibility
range and the target feasibility range so that the resulting mixture meets both constraints
simultaneously as shown in Fig. 8.11. To determine the combination of the two feasibility
regions, we select the boundary point clusters of the appropriate pFR and pMFR and calculate
the cluster at the pMFR intersection of a hypothetical component and R-32 that crosses the
boundary of the pFR. The ?additive domain? cluster boundary points are given in Table 8.24.
256
Figure 8.11: The Property Cluster Diagram Highlighting the Additive Feasibility Region.
The property operator upper and lower limit boundaries are then be calculated from Table 8.24
using minimization and maximization.
Table 8.24: The Additive Domain Feasibility Region Boundary Clusters and AUPs.
pMFR
Property Clusters
AUP
C1 C2 C3
Pt. 1 0.269 0.594 0.138 3.26
Pt. 2 0.178 0.393 0.429 2.15
0 .9
0 .8
0 .7
0 .6
0 .5
0 .4
0 .3
0 .2
0. 1
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0. 9
C
3
C
2
C
1
0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9
R32
R15 2 a
R29 0
R60 0
R60 0 a
R12 5
R13 4 a
R14 3 a
R12
R22
C ompone nt P r op ert y E f f ects
P r op ert y FR ( pF R )
P r op ert y M od el FR ( pM FR )
A dd i t i v e FR
P hy s. - C he m. P rope rt i es
257
Pt. 3 0.258 0.120 0.622 2.67
Pt. 4 0.322 0.110 0.568 2.72
Pt. 5 0.583 0.199 0.218 4.22
Pt. 6 0.335 0.540 0.125 3.23
Table 8.25: The Additive Domain Feasibility Region Bounds.
Additive FR ?1 ?2 ?3 AUP*
LL 0.85 0.59 0.00 2.15
UL 1.16 1.87 1.50 4.22
* Unbalanced AUP from Table 8.24.
It should be noted that since the additive region is an amalgam of the pFR, pMFR, and a
hypothethical component mixed with R-32, it is no longer balanced. In other words, a recreation
of the ?additive? feasibility region using the values on Table 8.25 and the feasibility construction
method listed in Chapter 6 would result in an overestimation of the domain. In this situation, it is
valid to use the domain boundaries listed in Table 8.24 to test Rules 1 and 6, while using the
unbalanced minimum and maximum AUP values to test Rules 2 and 7. Rules 3 and 8 must be
satisfied by matching the AUP values of the pFR, pMFR, and potential candidate solution or
mixture. For the solutions estimated using the forward approach, the unbalanced solution
domain (i.e. Table 8.24) provides an additional test for potential candidate solutions that meet the
constraints listed in Table 8.25.
After the target additive region has been identified, compounds that can satisfy it are
identified using the aCAMD method. Since the additive is a refrigerant component, its desirable
for the molecular weight not be too high and the search is limited to compounds with molecular
258
weight between 30 and 250 g/mol. The group contribution expressions for the target properties
become as shown:
? ?? ????
g sN
g
N
s vsvgvv hnhnHH 1 1 210 ?
(8.46)
? ?? ???
g sN
g
N
s cscgcc tntnTT 1 1 210 ln. ?
(8.47)
2
1 1 2121
?
? ? ?
??????? ???? ? ?g sN
g
N
s cscgccc
pnpnPPP (8.48)
???
gN
g gmnM 1 0
(8.49)
The property operators for each property can be formed according to Section 6.3 and Section 8.2.
All group contribution property values are taken from Marrero and Gani (2001). In this design,
ten different acyclic molecular groups are considered for generating the potential candidates.
These groups are typically found in refrigerants and often display less environmental impact. The
property ranges in terms of normalized operators shown in Tables 8.25 are augmented to reflect
the molecular weight constraints (e.g. 0.3 ? ?4 ? 2.5) using a reference value of 100. The first
order group contribution parameters are as follows.
Table 8.26: Property Data of Selected First Order Groups (Marrero and Gani 2001)
Group
g1 Name FBN tc1 pc1 hv1 m0
1 CH3 1 1.751 0.018615 0.217 15
2 CH2 2 1.3327 0.013547 4.910 14
259
3 CH 3 0.5960 0.007259 7.962 13
4 CH2F 1 3.3179 0.023315 8.238 33
5 CF 3 2.1633 -0.010120 6.739 31
6 CF2 2 0.8543 0.018572 1.621 50
7 CF3 1 1.7737 0.048565 7.352 69
8 CH2O 2 2.4217 0.017954 9.997 30
9 CH3O 1 3.4393 0.020084 5.783 31
10 CH2=CH 1 3.2295 0.025745 4.031 27
It should be noted that although some of these groups may be represented by smaller, first order
groups (e.g. CF may also be represented by a C and F group), in practice, better model
agreement is obtained by representing the arrangement by the group with the largest molecular
weight (Marrero and Gani 2001). In terms of first order groups, the parameters are rewritten as
non-dimensionalized property operators and AUP values as shown in Table 8.27. For ease of
comparision, the molecular weight property operator is not included in AUP of the mixture. It
will serve as an additional screening step for the design moledules.
Table 8.27: Non-dimensional Property Operators and AUPs of Selected Molecular Fragments
Group
g1 ?1g1
M ?2g1M ?3g1M AUPg1M
1 0.35 0.37 0.02 0.74
2 0.27 0.27 0.49 1.03
3 0.12 0.15 0.80 1.06
4 0.66 0.47 0.82 1.95
5 0.43 -0.20 0.67 0.90
260
6 0.17 0.37 0.16 0.70
7 0.35 0.97 0.74 2.06
8 0.48 0.36 1.00 1.84
9 0.69 0.40 0.58 1.67
10 0.65 0.51 0.40 1.56
Using Eq. 6.43, an inequality expression for each property is generated. Using only first order
group property operators, the maximum possible number of each group is calculated and shown
in Table 8.28.
Table 8.28: Non-dimensional Property Operators and AUPs of Selected Molecular Fragments
Group
g1 Formula
Calculated
ng1U
Buffered
ng1U
1 CH3 5 6
2 CH2 4 4
3 CH 3 4
4 CH2F 2 2
5 CF 4 4
6 CF2 5 5
7 CF3 2 2
8 CH2O 2 2
9 CH3O 2 2
10 CH2=CH 2 3
However, as noted in Section 8.3, these values may not be the true limits of the number of
groups because of the potential for negative second order property operators. Also, since some
261
of the first order groups listed in Table 8.27 contain smaller first order groups (i.e. Table 8.27 is
not a complete set of first order groups in the system), second order group instances containing
the smaller first order groups must also be accounted for. The second order groups with known
contributions that can be built from the first order groups listed in Table 8.27 are given in Table
8.29 (Constaninou and Gani 1994). Those groups containing smaller first order groups (i.e. C, F,
etc.) are denoted accordingly.
Table 8.29: Properties of 2nd Order Groups built from 1st Order Groups (Marrero and Gani, 2001)
Group
g2 Formula tc1 pc1 hv1
1 (CH3)2CH -0.0471 0.000473 -0.419
2 CH(CH3)CH(CH3) 0.5602 -0.003207 0.532
3 CH2=CH-CH=CH2 0.4214 0.000792 1.632
4 CH3-CH=CH2 -0.0172 -0.000101 0.064
5 CH2-CH=CH2 0.0262 -0.000815 -0.060
6 CH-CH=CH2 -0.1526 -0.000163 0.004
7 CH2-O-CH=CH2 0.2900 -0.000432 0.372
8 CH3-O-CH=CH2 0.2900 -0.000432 0.372
9* (CH3)3C -0.1778 0.000340 -0.417
10* CH(CH3)C(CH3)2 0.8994 -0.008733 0.623
11* C(CH3)2C(CH3)2 1.5535 -0.016852 5.086
12* C-CH=CH2 -0.1526 -0.000163 0.004
* 2nd order groups built from smaller first order groups
The 2nd order property contributions can then be expressed in terms of non-dimensionalized
property operators and the augmented property index as shown in Table 8.30.
262
Table 8.30: The non-dimensionalized property operators of 2nd order groups
Group
g2 ?1g2
M ?2g2M ?3g2M AUPg2M
1 -0.0094 0.0095 -0.0419 -0.042
2 0.1120 -0.0641 0.0532 0.101
3 0.0843 0.0158 0.1632 0.263
4 -0.0034 -0.0020 0.0064 0.001
5 0.0052 -0.0163 -0.0060 -0.017
6 -0.0305 -0.0033 0.0004 -0.033
7 0.0580 -0.0086 0.0372 0.087
8 0.0580 -0.0086 0.0372 0.087
9 -0.0356 0.0068 -0.0417 -0.070
10 0.1799 -0.1747 0.0623 0.068
11 0.3107 -0.3370 0.5086 0.482
12 -0.0305 -0.0033 0.0004 -0.033
The 2nd order groups with negative AUPM (i.e. groups 1, 5, 6, and 12) contain the 1st order groups
CH3, CH2, CH, and CH2=CH, meaning that a molecule containing these second order groups
could theoretically contain a higher number of the listed 1st order groups than shown in Table
8.27. To protect against this scenario, the maximum number of 1st order groups are buffered by
20% where necessary, resulting in the buffered column in Table 8.27. A buffer of 20% was
chosen because second order groups generally correct first order groups by ?10%.
Proceeding with the additive design using the aCAMD, all possible combinations of first
order and second order groups are then estimated subject to the constraints placed on the search
space shown in Table 8.25. In addition, the customer has indicated a desire to limit the number
263
of dissimilar groups to two. Following the procedure outlined in Table 8.3 and Fig. 8.4-8.6,
potential candidate molecules are generated. The clusters of the first order groups used in the
aCAMD are given in Table 8.31.
Table 8.31: First Order Group Clusters
Group
g1 C1g1
M C 2g1M C 3g1M
1 0.471 0.500 0.029
2 0.259 0.263 0.477
3 0.112 0.137 0.751
4 0.340 0.239 0.422
5 0.479 -0.224 0.745
6 0.243 0.527 0.230
7 0.172 0.471 0.357
8 0.263 0.195 0.542
9 0.412 0.241 0.347
10 0.413 0.329 0.258
As shown in Fig. 8.12, one group, CF, resides outside the positive cluster domain. Since, any
mixtures containing CF would need to reside within the positive cluster domain, any ?pure
group? mixtures of this group are automatically discarded. Of the remaining groups, only two
?pure groups?, CF2 and CH3, are capable of producing candidates that meet Rule 1. AUP values
for each of the candidates can then be tested and matched against the AUP range of the additive
feasibility region. Between 4 and 5 CF2 groups are capable of meeting this constraint and the
maximum number of groups constraint. The number of CH3 groups meeting these constraints are
between of 3 and 5. Calculating the Free Bond Number (FBN) of these molecules for non-ring
264
compounds indicates that neither set contains a thermodynamically stable molecule and all pure
groups are discarded.
Figure 8.12: 1st Order Group contributions in the Cluster Domain
For molecules having two dissimilar groups of Table 8.26, 389 potential non-ring candidate
molecules can be built. Using the procedure outlined in Fig. 8.6, each non-ring candidate
molecule was tested against Rule 4 (i.e. FBN = 0). In total, 321 molecules failed Rule 4 and
were discarded. Five more molecules failed Rule 14 and were discarded leaving 63 potential
candidate molecules. Finally, 56 molecules failed to fall within the ?additive? feasibility region
0 .9
0 .8
0 .7
0 .6
0 .5
0 .4
0 .3
0 .2
0 .1
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
C
3
C
2
C
1
0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9
CH3
CH2
CH
C H 2F
CF
CF2
CF3
CH 2 O
CH 3 O
CH 2 = CH
R32
G r ou p C on t r i bu t i on s
C ompl ete M ol ecu l es
P r op ert y FR ( pF R )
P r op ert y M od el FR ( pM FR )
A dd i t i v e FR
P h y s. - C h em. P rop erti es
265
constraints of Table 8.24, leaving 7 potential candidate molecules. Two of the seven molecules,
R-161 and Methoxymethane, fall just outside the cluster domain because it is unbalanced. The
remaining five potential candidate solutions are shown in Table 8.32 and Fig. 8.13.
Table 8.32: Candidate Molecules
ASHRAE
# Name Formula ?1
M ?2M ?3M
R-290 Propane CH3-CH2-CH3 0.967 1.016 0.534
R-272ca 2,2-Difluropropane CH3-CF2-CH3 0.871 1.116 0.206
- 2,2,3,3-Tetrafluorobutane CH3-(CF2)2-CH3 1.042 1.487 0.368
R-1243 3,3,3-Trifluoropropene CH2=CH-CF3 1.001 1.486 1.138
R-143m Methyl trifluoromethyl ether CH3O-CF3 1.043 1.373 1.314
It should be noted that only these 5 candidate molecules were fully constructed by the algorithm.
All five of the candidate molecules meet the property contraints on critical temperature, critical
pressure, enthalpy of vaporization, and molecular weight of the molecule. Of them, R-290 was
part of the training set and was discarded because its attribute values did not fall within the
consumer targets. The difference between the designed molecule and training set molecule can
be attributed to two sources of error: (1) the widening of the target domain because of a poor fit
in Eq. 8.45 and (2) a systematic underestimation of the enthalpy of vaporization by the Marrero
and Gani GCM method (2001). The first source of error could be improved by developing a
stronger relationship for Eq. 8.45. However, since the aCAMD is built to only exclude infeasible
solutions and return candidate solutions, with no promise as to the suitability of the designed
molecules, the effect of this uncertainty is limited.
266
Figure 8.13: Candidate Molecules in the Physical-Chemical Property Cluster Domain
For the second source of error, plots of the enthalpy predicted vs. actual responses of the
enthalpy of vaporization shown in Fig. 8.14a indicate a severe incongruity in the
parameterization of the terms by Marrero and Gani (2001).
0 .9
0 .8
0 .7
0 .6
0 .5
0 .4
0 .3
0 .2
0 .1
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
C
3
C
2
C
1
0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9
R 32
R29 0
R 27 2
2 ,2 ,3 ,3 -
T e traf luo ro b u ta n e
R - 1243
R - 143m
R - 161
M e th o x y m e th a n e
C ompl ete M ol ecu l es
P r op ert y FR ( pF R )
P r op ert y M od el FR ( pM FR )
A dd i t i v e FR
P hy s. - C he m. P rope rt i es
267
Figure 8.14: Predicted vs. Actual Latent Heat of Vaporization using: (a) GCM at 298K and (b)
GCM at the Boiling Point Tb.
Upon inspection, it appears that the Marrero and Gani GCM prediction is actually providing the
enthalpy of vaporization at the boiling point and not at the standard state of 298K and 1 bar as
was reported. Using the correlation in Eq. 8.50 developed by Watson (1943), the predicted
enthalpy of vaporization is corrected to the standard state and plotted in Fig. 8.14b.
38.0
1
15.2981
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
c
b
c
vbv
T
T
THH (8.50)
Although the prediction of the enthalpy of vaporization improves, a significant amount of error
over the approximately ?1.61 standard deviation reported by Marrero and Gani (2001) remains
in the system. The error is primarily a result of the contributions of the halide groups like CF2
and CF3 which also affect the critical temperature (Tc) as shown in Fig. 8.15a. The critical
pressures (Pc) in Fig. 8.15b did not appear to be affected by these groups.
10
12
14
16
18
20
22
24
10 12 14 16 18 20 22 24
Pr
e
d
ic
te
d
A c tu al
Hv
(a )
10
12
14
16
18
20
22
24
10 12 14 16 18 20 22 24
P
red
ict
ed
A ct u al
H v
(b )
268
Figure 8.15: Predicted vs. Actual Critical Properties: (a) Temperature and (b) Pressure.
These results suggest it may be beneficial to reparameterize the GCM method for only the
groups of those molecules in the training set since they are used to build the candidate molecules.
This type of approach is discussed in more detail in Chapter 9.
Of the four remaining candidate molecules, the PEI value provided by the WAR GUI
Version 1.0.17 (Young and Cabezas 2000) could only be determined for a single component, R-
1243. It was at 6.068 PEI/kg which is significantly higher than R-134a at 0.3255 and R-32 is
thus ruled out as a potential candidate Further environmental study of the 3 species without a
PEI is warranted but is beyond the scope of this dissertation.
Based on these results, a binary mixture involving R-32 could not be found to satisfy the
constraints of the design. However, noting that the PEI of R-161 is at 0.00406 and that it is close
to the feasibility region, it is quite possible that a ternary mixture involving R-32 and R-161
could be utilized. To determine which of the tested components to add, it is necessary to
evaluate the difference between the AUP of the target sink and the AUP of the R-32/R-161
mixture. It is found that the critical pressure cluster, represented by the property cluster C2, is
too low and needs to be raised. Of the components utilized in this design, R-125 has the lowest
28 0
30 0
32 0
34 0
36 0
38 0
40 0
42 0
44 0
46 0
28 0 30 0 32 0 34 0 36 0 38 0 40 0 42 0 44 0 46 0
Pr
e
d
ic
te
d
A ct u al
Tc
(a )
30
35
40
45
50
55
60
30 35 40 45 50 55 60
Pr
e
d
ic
te
d
A ct u al
Pc
(b)
269
Pc value, which corresponds to the highest C2 cluster value, which should help to balance the
binary mixture. However, R-125 also has a high PEI value of 2.158. Making the assumption of
linear mixture behavior and using lever-arm analysis, a ternary mixture of R-32/R-125/R-161 of
33/10/57 wt.% is found to simultaneously match both the AUP of the sink, cluster feasibility
region, and PEI of 0.3255, matching R-134a. Subsequent reduction of R-125 will lower the PEI
but at a cost of lowering the amount of R-32 that can be utilized in the design. In a similar
experimental study comparing the cycle performance of R-32/R-125/ R-161 against R-407c, Han
et al. (2007) found a mixture of 23/25/52 wt% to be sufficient for replacement for R-407c.
Considering that R-407c is a 15/34/51 wt% mixture of R-32, R-125, and R-134a, it is reasonable
to conclude that this mixture is similar to R-134a and can be used to corroborate these results.
Accordingly, it is found that this method predicts a R-407c mixture between 15/34/51 wt% and
32/39/29 wt% which nearly encompasses the result from Han et al. (2007). The major difference
between the two results is the amount of R-125 in the mixture and is a result of the experimental
solution not being bound by the PEI, as was done in this case study. Had this constraint been
relaxed, the R-125 fractions would have been severely reduced. There is also some indication in
the gradients of the mixture responses provided by Han et al. (2007) that the mixing may be
nonlinear. Further experimentation on this candidate solution is warranted.
In summary, the overall confidence that this solution is the global solution to the design
problem is calculated by applying Eq. 8.13 to the property-attribute relationships in Table 8.9
and the GCM models in Marrero and Gani (2001), resulting in 88.4% and 98.1% for each step.
However as noted earlier, significant errors in the calculation of Hv by Marrero and Gani (2001)
suggest that the error in the second step is much larger and could potentially reduce the
confidence by another 20-40%.
270
8.5. Summary
In conclusion, an attribute computer aided molecular design (aCAMD) technique using
property clusters and the reverse problem formulation was developed and discussed in this
chapter. The method allows for the complete and efficient solution of product design problems
that rely on regression models to describe consumer attributes and property models based on
group contribution. The use of smaller building blocks based on functional groups allows the
technique to construct molecules that are not part of the original experimental data set, a
significant contribution to the product design problem. In addition, because of the uncertainty
encountered in defining consumer attributes, the method uses relaxations of the target solution
domains in order to ensure that only infeasible candidate solutions are excluded from the design.
However, the efficiency of the method suffers in situations of high uncertainty that are often
found in the multiscale design of chemical products. In the case study, the accuracy (and
efficiency) of the design suffered from two main sources of uncertainty: (1) data driven models
in property-attribute relationship and (2) poorly fit group contributions in the property-molecular
architecture relationship. For case 1, better models may be obtained by reconfiguring the
problem in terms of underlying latent variables using decomposition techniques. Decomposition
is a mathematical technique that can determine the most valuable underlying latent variables.
Latent variables are, by definition, mathematical inferences of underlying independent variables
that control the response of the system. The ability of decomposition to handle large sets of
multivariate data suggests it could be utilized to evaluate the structural information of a chemical
product over multiple scales. The use of decomposition and latent variable models to design
chemical products at multiple scales is discussed in Chapter 9.
271
For case 2, it has been shown that it may be beneficial to parameterize the property-
molecular architecture relationship on a case-specific basis in order to improve predictive power.
This is especially true for chemical products with architecture that exists over multiple scales.
One of the most common and efficient ways to describe this architecture is to use
characterization to generate enormous multivariate data sets and derive case specific
combinatorial techniques (based on GCM) according to the type of characterization used. The
use of characterization based group contribution (cGCM) is discussed in more detail in Chapter
9.
272
Chapter 9
Characterization based Computer Aided Molecular Design (cCAMD)
As discussed throughout this dissertation, combinatorial techniques like GCM are
valuable methods for efficiently designing molecules for specific end uses (Constantinou and
Gani 1994; Marrero and Gani 2001; Gani 2004; Hill 2009). Unfortunately, in chemical product
design, the associated properties of concern are most often consumer attributes which do not
have group contribution parameters (Hill 2004; Hill 2009). This limitation was partially
addressed in Chapter 8 by mapping the consumer attribute information to a set of intensive
properties at the macroscale that can be described by group contribution (Solvason et al. 2009).
This step is often performed via chemometrics which defines an empirical relationship through
the use of design-of-experiments (DOE) and multivariate-linear-regression (MLR). Any
uncertainty in the relationship between the attributes and properties is handled by increasing the
size of the feasibility regions in the property domain (Solvason et al. 2009). However, if the
attribute-property relationship is poorly defined, then the feasibility regions are enlarged such
that any RPF efficiency gains are lost and the overall accuracy of the design is reduced. For
multiscale chemical product design this observation is especially apparent. For example, the
consumer attribute ?softness? in facial tissue is a function of molecular scale structure (e.g.
backbone flexibility), microscale structure (e.g. fiber coarseness), and mesoscale structure (e.g.
fiber orientation) of the product. Using only GCM (i.e. molecular scale) properties to build the
attribute-property relationship would fail to capture the architecture effects of the other length
273
scales on the product?s attributes, resulting in a poor attribute-property relationship, a broad
feasibility region, and many, poorly designed chemical product candidates that may or may not
be true solutions. In contrast, if the system?s attributes and properties can be described by only
molecular scale architecture (e.g. solvents), then the system can be completely modeled by GCM
(Eljack et al. 2007). It is for the former, multiscale system, that a new method called
characterization based computer aided molecular design (cCAMD) has been developed in
Chapter 9.
The cCAMD method uses characterization data to describe the molecular architecture of
the system, decomposition techniques to consolidate that information into a linearized latent
property subdomain, and chemometric techniques to relate the latent property subdomain to the
attribute domain. With this solution structure, attribute target data can be converted into latent
property targets where molecular architectures can be linearly combined to deliver them using
the property clustering algorithm. This approach differs from the multivariate mixture design
approaches discussed in Chapter 3 in the type of building blocks used in the design. Whereas
Muteki and McGregor (2006) and Garcia-Serna et al. (2011) search large databases of molecules
for pure components or combine pure components to generate candidate mixtures, this method
uses a much smaller set of molecular groups present in the original training set to build candidate
molecules and candidate mixtures. The result is a method that generates a complete set of
candidate molecules, void of database completeness limitations.
For this method to work properly, important multi-scale molecular architecture features
must be appropriately captured by various characterization techniques and converted into
molecular group-like features which can be recombined in a characterization based group
contribution method (cGCM) to generate new chemical products. Since infrared (IR) and near
274
infrared (NIR) spectroscopy techiniques are particularly adept at capturing molecular scale and
microscale features of chemical products, they were chosen to highlight the method (see Section
9.1.1).
Once the spectral features have been captured, decomposition techniques are applied to
determine the latent property subdomain in order to improve the combinatorial efficiency and
certainty of the designed candidate molecules. The use of latent property descriptors requires the
construction of an attribute-latent property relationship in order to provide a vehicle for mapping
attribute targets into the latent subdomain. The decomposition and relationship construction
techniques for the latent domain are discussed in Section 9.1.2.
In Section 9.2 the necessary changes to the property clustering method to handle latent
property descriptors of molecular architecture and targets are examined. In particular, the design
must be augmented to handle negative mixture fractions. The section concludes with the cluster
conversion algorithm for cCAMD.
In Section 9.3 the candidate generation algorithm for cCAMD is developed and
discussed. Changes to the structure generation approach and additional structure checks are
highlighted. The resulting algorithm is capable of simultaneously building candidate molecules
with multiple types of molecular architecture that exist across multiple scales.
In Section 9.4 a case study on the design of a pharmaceutical excipient formulation for
direct compressed acetaminophen tablets is used to illustrate the method. Two types of designs
are conducted, a mixture design to highlight the impact of the property clustering technique, and
a molecular design to highlight the newly developed characterization based group contribution
method (cGCM). The chapter concludes with a summary in Section 9.5.
275
9.1. Integrating Multivariate Data & Models with the RPF
In general, the objective of the cCAMD method is to enumerate all possible molecules
and molecular architectures that meet a set of target attributes by using a mapping function to
move the chemical product design problem into sub-domain with linear programming
characteristics. Identifying a proper sub-domain in which to conduct the design is paramount to
ensuring a repeatable, accurate, and physically meaningful solution occurs. A suitable sub-
domain should possess the following qualities: (1) the sub-domain latent property-molecular
architecture relationship should be linear to ensure computational efficiency and solution global
optimality, (2) the sub-domain should be capable of representing all relevant molecular
architecture important to the design, and (3) sub-domain mapping noise should be minimized by
using latent property-attribute relationships with strong predictive power (Solvason et al. 2009,
Solvason et al. 2010).
Many techniques exist that can be used to find a suitable property sub-domain. One of
the most common is the decomposition of property and molecular architecture information into
an eigenfunction. When the technique minimizes the mahalnobis distance to build the
underlying, orthogonal latent variable structure, it is called principal component analysis (PCA)
(Johnson and Wichern 2007). Alternatively, when information about the relationships between
molecular architecture and the attributes of the system is available, network component analysis
(NCA) can be used to find the underlying sub-domain using a MINLP formulation proposed by
(Tolle et al. 2009). It should be noted that in order for either of these decomposition techniques
to be effective, large amounts of heavily correlated data that describe the molecular architecture
will be required. When this requirement is combined with the second criteria of a complete
description of the system, it is clear that data-gathering techniques spanning multiple scales will
276
need to be utilized. It has been shown that characterization tools and techniques are excellent
methods of generating data and information about the system.
Figure 9.1: A Multiscale Reverse Problem Formulation. The Scales Not Blocked Out are
Capable of being Described using the Techniques Developed in this Section.
Figure 9.2: A Cross-section of the Reverse Problem Formulation Applied to a Multiscale
Chemical Product Design.
QM, DFT Mo d el s
Ab I nit i o Met ho ds
(Q ua ntum S cal e)
Fu n d a m e nt al Prin ci ple s o f
Chemic al E n gi n e e rin g
Proc es s D esign
(M ac ro Sca le )
Chem om etric Mo d el s
Consumer Attrib utes
(M ac ro/Met a Scal e )
MM, M D, MC Si mul ati o n s
Atomist ic M etho ds
(Atom ic Sca le )
GCM, T o p ol o gical I n dice s
and C h em om etric Mo d el s
CA PD / CAMD
(M ol ec ula r S cal e)
Coa rse - Gr ai n in g a n d
Chem om etric Mo d el s
Mi c rostruc ture
( Mes o Sca l e)
PR OPERTI ES
So l u t i o n
Su b - D o ma i n
C o n ve n t i o n a l
Mu l t i - sca l e
D o ma i n s
??TfA? )(TfP? )(
)(
1gnf
XfT??
277
As shown in Fig. 9.1 and 9.2, most chemical products with multiscale molecular
architecture are usually present in a solid form and, in particular, rely on microstructure at the
micro- through meso-scales as well as molecular structure at the molecular scale to deliver the
needed consumer attributes. Unlike the aCAMD method of Chapter 8 which used only
molecular scale information, the cCAMD method uses characterization property data (P) of an
experimental training set to capture molecular architecture at multiple scales followed by a
decomposition to find a underlying latent property domain (T).
)(TfP? (9.1)
This type of approach overcomes the limitation of the aCAMD method by capturing a complete
description of the relevant molecular architecture important to the product. Likewise,
documented characterization responses for specific types of molecular architectures can also be
decomposed to find a set of latent domain parameters to be used in a cGCM or linear mixture
design:
)( 1gnfT? (9.2)
)(XfT? (9.3)
where ng1 are the number of distinct first order characterization based groups, g1, and X is the
vector of pure component mass fractions in the product. Finally, the targets for the design are
obtained by developing the attribute-latent property relationship and mapping the consumer
attributes (A) into the latent domain.
)(TfA? (9.4)
With the targets defined, characterization based groups and pure component molecular
architectures can be mixed to determine the chemical product candidates. The strength of the
278
method is predicated on two main aspects, (1) the proper characterization of the system and (2)
the proper decomposition to find a suitable latent variable domain.
9.1.1. Characterizing the System with IR/NIR Spectroscopy
As reviewed in Chapter 3, infrared (IR) and near infrared (NIR) spectroscopy can be
useful characterization techniques for the determination of chemical constituency, molecular
structure, and nano-, micro-, and meso-scale orientation and alignment. The characterizations
are often performed on a training set of molecules defined by an experimental design used to
explore the interesting facets of a product?s design. Recent developments in these spectroscopic
techniques have been shown to provide better determination of overtone and combination bands
of familiar molecules and improved descriptions of bonding phenomena such as van der Waals
forces (Nesbitt 1988; Saykally 1989; Hutson 1990; Heaven 1992; Iachello and Levine 1995),
making them invaluable for the product designer.
Since, as discussed in Chapter 8, significant computational efficiency is often gained
when the design problem is approached using group theory, it is beneficial to relate the IR/NIR
characterizations to functional groups. Using the concept of symmetry, many atoms and
combinations of atoms in a symmetrical molecule can be considered to be in the same chemical
environment, and by extension, the vibrational motions and absorptions or IR and NIR spectra
will be identical. Hence, it makes more sense to parameterize specific group vibrational spectra
as opposed to atomic or complete molecular spectra as was done in Chapter 3. Groups can be
considered as the formation of atoms interacting in a molecular association. The calculation of
group frequencies is a summation of vibrational motions described by harmonic oscillation,
anharmonic oscillation, first order coupling, and Fermi resonance (second order coupling).
Various concepts regarding rotational axes, reflection planes, inversion centers, improper
279
rotational axes, and point groups can be used to mathematically justify this concept, but are
beyond the scope of this dissertation (Workman and Weyer 2008). Instead, a brief summary of
the vibrational frequencies in terms of normal coordinates in group theory is presented.
Many factors are responsible for determining the frequency of absorbance for a specific
molecular architecture. The strongest fundamental bands correlate with the strongest bonds and
inversely correlate with the mass of an atom (Socrates 2001). Overtone bands arise from
anharmonicity of the fundamental bands and can be used to differentiate between group bonds
that absorb in the same frequency. However, combination bands and coupling may cause some
of these vibrations to shift, degenerate, or become inactive in IR or NIR. The overlap of features
will need to be addressed by more sophisticated approaches (Stephenson et al. 2001). One such
technique is to utilize the group contribution method (GCM).
As discussed in Chapters 4, 6, and 8, GCM is a property prediction technique based on
the UNIFAC method of summing individual property contributions from specific functional
groups in a molecule. First order groups contain basic information and can be combined linearly
since they assume no interaction between groups. Second order groups can be estimated from
first order groups and correct for the interactions between first order groups. Third order groups
can be derived in a similar method and help to correct for polyfunctional compounds with more
than four carbon atoms in the main chain (Marrero and Gani 2001). The complete GCM
expression is as follows:
1St Order 2nd Order 3rd Order
??? ??? 3 332 221 11 g ggTg ggSg ggM PnwPnwPnP
(4.15)
The method proposed in this chapter uses characterization data from IR and NIR
spectroscopy to define the group contributions in the GCM expression in order to predict the
280
spectra of a full molecule. Additionally, molecular architecture information beyond the scope of
traditional GCM, such as group orientation and crystal polymorphism, will be included as long
as it can be ascertained from IR and NIR spectroscopy. In order to handle these special
constraints, a modified characterization based group contribution is proposed that parameterizes
the groups in a nested structure. To be effective, the dissemination of characterization data must
follow a set of rules designed to shadow those developed by Marrero and Gani (2001). First, the
characterization must be able to completely quantify each individual molecular group used in the
design. Within a group, the number of absorbance frequencies can be calculated by modifying
Eq. 3.27 to account for bonds by assuming each bond is attached to a hydrogen atom.
Rule 17. The total number of fundamental vibrations of a group will be calculated using Eq.
3.27, where each open ended bond is estimated as a hydrogen atom.
This assumption will grossly overestimate the number of fundamental modes of vibration, but
will serve as a necessary constraint. For instance, this method will estimate that a CH2 group had
9 fundamental modes, when, in fact, it only has 6 fundamental modes of vibration. It should be
noted that this overestimation is only a concern when molecular groups have a small number of
atoms with strong absorbances (e.g. C=O). For larger groups, significant degeneracy from
interactions leads many fundamental tones to become inactive. For these situations, the inactive
tones can be ignored; the use of only the physically active tones listed in various texts will be
sufficient.
Rule 18. The number of fundamental tones calculated by Rule 17 will be reduced to only
the tones that have been experimentally validated. No quantum mechanical
estimations of tones will be utilized.
281
For instance, the physically identified fundamental bands of CH2 are presented in Table 9.1
(Socrates 2001; McKenzie 2002). The intensity of absorption of each fundamental absorption is
listed in terms of a relative rates of strong (s), medium (m), medium-strong (m-s), and inactive
(i). Although it was expected that the CH2 group would have 9 fundamental tones, only 3 could
be experimentally verified: symmetrical stretching, asymmetrical stretching, and scissoring bend.
The remaining bands were found to be i inactive.
Table 9.1: Fundamental Vibrations of the CH2 Group (Socrates 2001; McKenzie 2002)
Band Wavelength Region (cm-1) Relative Intensity
Symmetrical Stretching (vs) 2870-2840 (2855) m
Asymmetrical Stretching (va) 2940-2915 (2925) m-s
Scissoring Bend (?s) 1480-1440 (1460) m
Twisting Bend (?t) 1350-1150 (1250) i
Wagging Bend (?w) 1350-1150 (1250) i
Rocking Bend (?r) ~720 i
Vibrational intensities have generally been overlooked or neglected in the analysis of vibrational
spectra (Socrates 2001). Since relative intensities are primarily functions of the atom specific
dipole changes caused by the vibration of the corresponding bonds, it follows that their size and
shape are indicators of molecular architecture. Although many texts list these intensities in
subjective terms due to instrumentation flexibility, their relation to one another can be objective
(Socrates 2001). Workman and Springsteen (1998) noted that these peaks are generally
calibrated based on a linear relation to the area under the peak or a polynomial relationship to the
282
height of the peak. Based on these observations, a quantitative description of the size and shape
of absorption bands is proposed in Table 9.2.
Table 9.2: Approximations of Transmittance of IR/NIR Active Tones (Socrates 2001; Workman
and Weyer 2008).
Relative Intensity Peak Transmittance Transmittance Shape
s 10% Normal
m-s 30% Normal
m 50% Normal
m-w 70% Normal
w 90% Normal
br n/a Inverse Beta (square)
sh n/a ?-Var. Normal
d n/a 2, ?-Var. Normal
i 100% n/a
The peak intensities are scaled from 0-100% transmittance, depending on the level of
absorbance. A w indicates 90% transmittance, m-w a 70% transmittance, and so on.
Rule 19. The peak intensities are scaled according to transmittance, as shown in Table 9.2.
The shape for the absorbance/transmittance of the tone is determined using probability theory,
where normal distributions are fitted to the mean of each transmittance, with the ranges of the
wavelength region listed in the texts marking the 95% probability interval. Thus, the ? standard
deviation for the response is calculated as:
4 LLUL ??? ??
(9.5)
283
where PRANGE is the difference between the expected regions of the upper and lower
absorbances/transmittances. The transmittance probability values are given as a normal
distribution:
2
2
2
)(
22
1 ???
??
bar
ef abs ??? (9.6)
where ? is the wavelength where the absorbance/transmittance occurs, ?bar is the peak
wavelength location, and f abs is the proability density function. These values can then be scaled
using the peak transmittances associated with the tone.
absabsp e a k
TTp e a k
absabsp e a k
TTp e a k
ff
PP
ff PP ?
??
?? ??22
(9.7)
For situations where no range is listed, such as for the rocking-bend vibration of 720cm-1 in
Table 9.1, a range of ?10cm-1 is artificially applied.
Rule 20. A normal distribution is used to approximate the shape of an absorbed tone,
spanning the range listed in the text. For situations where no range is listed, a
range of + 10cm-1 is artificially applied.
Occasionally, the transmittance (absorbance) is noted as sh sharp. For this situation the range is
narrowed by reducing the equivalent standard deviation by half. When the transmittance tone is
listed as a br broad absorbance, an inverse beta distribution, defined by two positive shape
parameters, known as ? and ? which are each set to 0.5, is utilized. For simplicity this
distribution is best approximated using a square estimation, spanning the identified range.
Occasionally, the absorbance presents as a d doublet, which can be handled by splitting the
distribution into two, half variance normal distributions centered around the doublet peaks.
284
Table 9.2 lists various approximations of the transmittance types noted by Socrates (2001) and
Workman and Weyer (2008).
Rule 21. A normal distribution with half the variance is used to approximate the shape of
sharp tone and an inverted beta distribution with ?=?=0.5 is used to approximate
the shape of a broad tone.
Once the fundamental tones have been identified, the overtones can be estimated from Workman
and Weyer (2008). As noted in Table 3.1, beyond the 2nd overtone the relative intensities of
absorptions are sufficiently small as to be negligible when combined with the fundamental tones
found in the IR frequency range.
Rule 22. Using Eq. 3.33, the locations of the 1st and 2nd overtones are calculated for the
fundamental tones identified in Rule 18.
The overtones are then cross-referenced to find the physically active tones. For the CH2
methylene group, Workman and Weyer (2008) identify a 1st overtone peak of asymmetrical
stretching (2va) and symmetrical stretching (2vs) at 5671 cm-1 and 5680cm-1, respectively (Tosi
and Pinto 1972; Buback and Harfoush 1983). The remaining scissoring bend first overtone
(2?s) in Table 9.1 was inactive according to Socrates (2001). For the second overtone region,
both stretching vibrations (3vs, 3va) occur between 8389cm-1 to 8247cm-1, depending on the
influence of any methyl groups in the sample, which is addressed through higher ordered group
contributions (Workman and Weyer 2008). The scissoring bend 2nd overtone (3?s) was inactive.
This result was expected since the 1st overtone band was also inactive, and, according to Table
3.1, the intensity was expected to be even smaller for the higher overtone. This observation is
stated as Rule 23.
285
Rule 23. Once a tone has been labeled inactive, all higher ordered tones and combination
tones utilizing the inactive tone shall also be considered inactive. This rule will
not apply to combinations involving two different bond tones, even if one of them
is inactive.
In addition, the proximity of fundamental asymmetric and symmetric stretching tones
also gives rise to a strong combination band (va + vs) at 5800cm-1 (Buback and Voegele 1993).
This combination tone has been shown to move slightly in linear aliphatic compounds due to the
influence of methyl groups, but can be accounted for using higher order contribution corrections
which are later described in Eq. 9.8 (Workman and Weyer 2008). Other combination bands
include fundamental stretching-bending combinations of vs+?s and va+?s at 4336-4332 cm-1 and
4257-4262 cm-1, respectively. Also, a doublet form a first overtone uncoupled stretch plus a
fundamental bending combination is also prevalent at 7186cm-1 and 7080 cm-1 (Murray 1987).
A number of weaker combination bands containing inactive fundamental bands can be found in
the area between the first combination and first overtones bands (Workman and Weyer 2008).
The final, combined IR & NIR methylene CH2 group spectrum is shown in Table 9.3. A list of
other groups can be found in Appendix A6.
Table 9.3: The Derived IR Absorption Spectrum for the CH2 Methylene Group.
Band Wavelength Region (cm-1) Relative Intensity & Shape
Symmetrical Stretching (vs) 2870-2840 (2855) m
Asymmetrical Stretching (va) 2940-2915 (2925) m-s
Scissoring Bend (?s) 1480-1440 (1460) m
1st Overtone Asym. Str. (2va) 5681-5661 (5671) w (in NIR)
286
1st Overtone Sym. Str. (2vs) 5690-5670 (5680) w (in NIR)
2nd Overtone Asym. Str. (3va) 8399-8379 (8389) m-w, sh (in NIR)
2nd Overtone Sym. Str. (3vs) 8257-8237 (8247) m-w, sh (in NIR)
Combination (va + vs) 5900-5700 (5800) m-w (in NIR)
Combination (vs+?s) 4336-4332 s (in NIR)
Combination (va+?s) 4262-4257 s (in NIR)
1st Overtone Comb (2v+?) 7186 - 7080 m-w (in NIR)
One of the important observations concerning vibrational spectroscopy is that the
strength of the absorption is dependent on both the type of vibration and the number of times the
vibration appears in the molecular architecture (Socrates 2001). This conclusion lends itself
easily to concept of additive absorption peaks. Unfortunately, as noted earlier, not all intensities
are directly additive; some locations of absorptions move and some intensities only change at a
fraction of what is expected. That is because the intensity of the band may also indicate the
presence of certain atoms or groups adjacent to the functional group responsible for the
absorption band (Socrates 2001). To account for this behavior, 2nd, 3rd, and other higher order
group contributions can be utilized to provide corrections to the 1st order group contributions.
Marrero and Gani (2001), amongst many others (Joback and Reid 1983; Joback and Reid 1987;
Constantinou et al. 1996), identified several 2nd and 3rd order group contributions typically found
in thermodynamic databases. As discussed in Chapter 8, a method to identify the 2nd order
groups from 1st order groups was developed by Chemmangattuvalappil et al. (2009) and
Solvason et al. (2009) to aid the product developer. These techniques can also be used in the
development of IR/NIR characterization groups. For example 2-propanol contains the following
1st order groups: 2 methyl CH3 groups, 1 methine CH group, and 1 hydroxyl OH group. It also
287
contains the 2nd order secondary alcohol CHOH group. The 2nd order group is necessary since
the hydroxyl OH group induces a strong dipole in the methine CH group and, as a result, the
location and intensities of the IR and NIR absorptions are affected.
Although an effective method of correcting for interactions between groups, simply
developing the IR/NIR parameters for only the groups identified in the GCM limits the scope of
this method to the description of molecular architecture to that of small molecules, and polymer
backbones. In order to expand the method to include other architecture information provided by
IR/NIR spectroscopy, such as information on inter- and intra-molecular hydrogen bonding
(Socrates 2001), polymorphic crystal structure (Salari and Young 1998), and many others, it is
necessary to reformulate the group contribution approach in terms of Eq. 9.7. In this
formulation, the first term is the 1st order group contribution and the second term ?i is a
combination of the corrections from all higher order contributions, including those specific to
molecular architecture existing at larger length scales.
? ?? i iii CNXf )()( ?
(9.8)
The parameters of the corrections can be estimated in two ways. In the first method, the
fundamental, combination, and overtones of each higher order as defined by
Chemmangattuvalappil et al. (2009), Marrero and Gani (2001), and/or the researcher can be
estimated using the rules outlined earlier in this section. Although this result gives accurate
results, it is quite tedious. The second method is simpler, but has greater potential for error and
can lead to degeneracy. In this method, progressively larger groups completely contain the
information of the smaller groups, but also contain corrections for 2nd order, 3rd order, structural,
and orientation effects. The specific differences between the orders of the groups are not
quantified directly, but rather different groups containing various levels of information are
288
developed. The groups are hierarchical in nature, with the groups containing the higher ordered
information preferred over the groups containing lower orders. For example, in the
determination of the IR/NIR spectra for 2-propanol, a second order CHOH group would
completely replace the CH and OH first order groups since it already contains their information;
plus it contains additional information on the induced dipole interaction.
Rule 24. A hierarchical group structure is created such that higher order groups contain
interaction information plus lower order group information.
As a result, the best model of the molecular architecture is achieved by specifying the largest
functional group first, then the second largest, and so on. In most cases, a higher order
functional group will completely overlap lower order functional groups.
Rule 25. When one group completely overlaps another group, the larger group will be
chosen for the design. As a result, the largest functional group is specified first,
the next largest second, and so on.
In some situations only partial overlaps may occur. In the GCM method proposed by Marrero
and Gani (2001) and reviewed in Chapter 8, 1st order groups are not allowed to partially overlap
while 2nd order and higher groups are allowed to partially overlap. The assumption made by
Marrero and Gani (2001) is that any error from double counting of 2nd order effects is
outweighed by the errors associated with failing to account for second order effects when they
do, in fact, exist. For the characterization based group contribution method (cGCM) proposed
here, no partial overlaps are allowed since the contributions from 1st and 2nd order and higher
groups cannot be separately adjusted without rewriting the current state-of-the-art combinatorial
algorithm. Since the method will fail to specify any corrections to the first order combination at
289
the overlap interface, groups are built such that the combinations occur across the minimum
number of C-C bonds (Marrero and Gani 2001).
Rule 26. Candidates with partial overlaps should be reconfigured to lower order groups
containing the minimum number of bond connections and maximum amount of
information.
For example, isobutanol can be built from a multitude of groups, including combinations of the
following: a methyl CH3 group, methylene CH2 group, a methine CH group, an alcohol OH
group, a secondary alcohol CHOH group, a primary alcohol CH2OH group, and a terminated
primary alcohol R1R2CH2OH. Figure 9.3 outlines 6 potential methods of combining the groups
to build isobutanol. The first method uses only 1st order groups, and has the minimum amount of
information available. The second and third methods use a combination of 1st and 2nd order
terms, introducing some interaction corrections. The 4th method incorporates some 3rd order
information, but contains two C-C bonds over which no interaction information can be obtained.
The 5th method represents the best available combination, using only one C-C bond and two 2nd
order groups. Finally, although the 6th method combines a 2nd order and 3rd order group and has
only 1 bond connection, it also possesses a partial overlap which cannot be handled with the
cGCM algorithm although it is handled in the traditional GCM algorithm (Marrero and Gani
2001).
290
Figure 9.3: Potential Combinations of IR/NIR Groups for Building Isobutanol.
Occasionally, bonds will need to be built across other atoms, like the C-O bond in an aliphatic
ether (-O-). For this reason, an additional constraint is applied when building molecules
involving this group such that it must be attached to the carbon atom in other groups.
Rule 27. Candidate molecules can also be built over aliphatic ether groups (-O-), which
contain bond information between the O atom and two C atoms. As such, the
ether groups can only be attached to the carbon atoms of other groups.
One of the benefits of using characterization based combinatorial methods is the ability to
handle information unique to the characterization. For instance, IR/NIR spectroscopy not only
291
provides information on the functional groups within the molecule, but also can provide
information on stereoregularity (Zerbi et al. 1965; Socrates 2001; Solvason et al. 2010),
hydrogen bonding (Desiraju 1996; Workman and Weyer 2008; Izutsu et al. 2009), solid particle
shape/size (O'Neil et al. 1998; Ndindayino et al. 1999; Sorensen et al. 2006; Almaya and Aburub
2008), polymorphism (Deeley and Spragg 1991; Kondo 1997; Norris et al. 1997; Salari and
Young 1998; Yu et al. 1998; Nikonenko et al. 2002; Abebe et al. 2008) and crystallinity (Zerbi
et al. 1965; Salari and Young 1998; Seyer et al. 2000; Stephenson et al. 2001). This extra
structural information can be made to enhance the understanding of the molecular architecture.
For instance, O?Neil et al.(1998) demonstrated how bulk properties such as particle size and
porosity can influence IR/NIR spectra. In agreement with Mie theory, the reflectance of the IR
spectra varied with the mean particle size, although the particle shape also had a large influence
(Ciurczak 1987; O'Neil et al. 1998). In particular, the baseline NIR reflectance elevation was
observed to decrease as the particle diameter increased (Abebe et al. 2008). Abebe et al. (2008)
also noted that the more non-spherical the particle is, the more muted the elevation of the NIR
spectra. Beyond the bulk properties of the particle size and shape is its microstructure. In
general, the particles may be of granule or crystalline form. In the granule form, multiple,
alternating layers of amorphous and crystalline regions are present. The crystalline form is
generally treated as a single polymorph. Although some progress relating specific crystallinity
bands to particular NIR was made by Zerbi (1965), the results still remain case specific. At the
present time, x-ray diffraction (XRD), wide angle x-ray spectroscopy (WAXS), and small angle
x-ray spectroscopy (SAXS) remain the preferred methods for developing the crystalline
structure.
292
As opposed to the group combinatorial approach outlined earlier, there is not a distinct
hierarchy for which to add a diameter effect, nor is there a distinct particle size at which a
particular IR effect is observed across all molecules. Rather the effect is relative to the overall
baseline reflectance-absorbance on a case-by-case basis. This is beneficial to the combinatorial
approach used in mixture and molecular design. For example, diffuse reflectance is one of the
six ways in which an incident EM wave can interact with a material; it can also be absorbed,
transmitted, diffuse transmitted (i.e. scattered), specularly reflected (i.e. regular reflected), and
retroreflected back to the source (Workman and Springsteen 1998). In most instances for IR and
NIR spectroscopy, only three of the modes are observed, transmittance, absorbance, and diffuse
reflectance. In most cases, a reflected-absorbance response (e.g. Fig. 9.4) is obtained by
applying Eq. 3.29 to the reflectance (PR) instead of transmittance (PT). It has been shown that
NIR wavelength diffuse reflectance exhibits an inverse relationship with mean particle size in
agreement with Mie theory (O'Neil et al. 1998). The reason for this is that as the particle
diameter increases, the total volume of the sample does not, meaning that void space increases in
the sample which, in turn, decreases the pathlength of the EM signal and its resulting reflected-
absorbance. In similar fashion, the increase in void space also results in fewer solid particles
reflecting the EM signal, leading to a lower diffuse reflectance. This phenomenon is sometimes
referred to as baseline elevation, which results from systematic error inherent in the
spectrophotometer, and is handled using linear regression baseline fitting. For particle size dp,
O?Neil et al. (1998) and Abebe et al. (2008) showed that a line can be fit to two regions of the
spectrum in order to calibrate the spectral response:
RRop PbPbbd 2211ln ??? (9.9)
293
where each regression coefficient is species dependent and the two centered reflectance
wavelengths (PR) were chosen by searching for the combination that provided the minimum error
of calibration for a Kubelka-Munk function (O'Neil et al. 1998). The quadratic least squares
inverse of Eq 9.9 is most useful and can provide particle diameter corrections for specific types
of molecular architecture. In particular, models can be built for complex mixtures when only
some of the constituent concentrations are known; the only requirement being the selection of the
appropriate wavelengths that correspond to the reflected-absorbances of the desired molecular
architecture (Workman and Springsteen 1998). This requirement results because the baseline
elevation of the intensity of the reflected-absorbance response is not constant across all
wavelengths as shown in Fig. 9.4 (Abebe et al. 2008).
Fig. 9.4: Raw, Unscaled NIR Reflected-Absorbances of Slurries of L-Glutamic Acid Crystals at
Different Diameters (Abebe et al. 2008)
294
As a result, corrections for particle diameter may be assigned to specific group reflected-
absorbances provided baseline elevation expressions or responses are available. Unfortunately, in
order to ensure a unique description of the inverse of Eq. 9.9, the number of samples used in the
training set, usually 30-60, would then represent the maximum number of descriptors that could
be fit. In otherwords, a full description of the spectroscopic molecular architecture would require
a probibitively large training set which, by extension, would introduce colinearity into the
solution resulting in an over-fit and poor predictive description. A better way to capture this
information is to make use of eigenvector quantitation (i.e. decomposition) to find a latent sub-
domain that contains all molecular architecture information. This approach works well within the
reverse problem formulation since the objective is to find a computationally efficient method of
systematically determining the molecular architectures that deliver a set of desired property
attributes.
9.1.2. Developing the Latent Property Domain
Although the IR/NIR characterization based group contribution can handle some
interaction effects within the group structure, the colinearity of larger data sets require the use of
domain reduction techniques. In addition, the IR/NIR characterization based group contribution
method of defining groups in terms of a hierarchical structures means that enumeration will
result in multiple candidates with identical molecular architectures with varying levels of
information. Although it has been shown that this degeneracy can be handled using a set of rules
that help select the best representation of the candidate molecule, separating the effects into
independent spectra is difficult. Furthermore, as the size of the molecule grows, increasing
amounts of structure based information, from simple hydrogen bonding, to crystal unit cell
structure and morphology, will require an exponential, and possibly, case specific rule structure.
295
In order to better handle these types of interactions, it is suggested that (1) deconvolution
algorithms be applied to better ascertain the effects on the IR spectra or (2) the combinatorial
structure be mapped into a linear sub-domain that facilitates an easier computation. This sub-
domain provides both an avenue for combinatorial efficiency and suitable variable structure from
which attribute-property models can be built.
The applicability of vibrational spectroscopy techniques are constrained, to a great extent,
by the low resolution of bands in the spectra of complex substances (Nadler et al. 1989). The
spectra available for identification, such as for the methylene CH2 group, are considerably
smaller than the actual number of individual components (Nikonenko et al. 2002).
Deconvolution algorithms serve to enhance the absorption spectra in order to better differentiate
the individual absorption bands. The technique works by removing the bandpass function
(similar to a Fourier transform) that naturally smoothes a spectra (Nadler et al. 1989). Two
methods are typically applied, signal space iterative deconvolution and frequency space
deconvolution (e.g. Fourier transform). Although a strong technique, it struggles from subjective
interpretation of spectra. In contrast, using a decomposition algorithm to develop a sub-domain
uses the colinear and convoluted nature of the absorption spectra to build a strong latent property
structure. The choice of the appropriate training set spectra is vital to building the latent variable
structure since the training set defines the molecular architecture building blocks that can be used
in the characterization based GCM (cGCM). The training set is also used to handle unusual
interactions that cannot be described by characterization based cGCM. These interactions may
include non-equilibrium thermodynamic states, instrumentation bias like temperature drift, and
many other types of systemic error. Decomposition algorithms like PCA help to neutralize these
types of interactions by using the variance-covariance structure to compress the most important
296
molecular architecture data to an underlying latent variable structure while limiting the impact of
the unusual interatciton. As a result, only the interactions identified by the cGCM structure as
estimated from the data are included in the design in the latent subdomain.
Rule 28. Only the interactions identified by the cGCM structure as estimated from the
training set data should be included in the design.
For this method to be successful, multivariate normality of the data structure must be
verified. To test the non point estimate of multivariate normality, the square of the generalized
mahalnobis distance is evaluated using the chi square distribution (Johnson and Wichern 2007).
Under sufficient conditions for normal populations, point estimates of the system can replace the
non point estimates. The distribution that best approximates this condition is Hotelling?s T2
distribution. Multivariate normality is checked using a gamma Q-Q plot of the residuals against
the quantiles of the system. When the line is straight, has a slope of 1 and, contains few outliers,
then it is said to be multivariate normal. If the system is found to not be multivariate normal,
various linearization functions must be applied, such as mean square, logarithmic, Fisher
correlation, and power transformations, among others (Johnson and Wichern 2007). Once the
system is validated as multivariate normal, it must be centered and standardized in order for the
decomposition to be at its most effective. This is accomplished by subtracting the mean of each
property and dividing by the square root of the variance of each property (Gabrielsson et al.
2002; Johnson and Wichern 2007).
Rule 29. The molecular architecture characterization data must be verified to be
multivariate normal before any statistical techniques are applied.
Rule 30. The molecular architecture characterization data should be centered and
standardized prior to analysis with decomposition.
297
Sometimes, when two different characterization data sources are combined, it is better to
standardize and center them separately. This helps to minimize the systematic noise in the
system that results from the choice of characterization equipment. Examples include IR and NIR
spectroscopy (Gabrielsson et al. 2002; Gabrielsson et al. 2003; Gabrielsson et al. 2004;
Gabrielsson et al. 2006). Dissimilar characterization methods may require a more rigorous
analysis. For example, in the design of an excipient filler for a direct compressed acetaminophen
tablet, Gabrielsson et al. (2003) evaluated 28 potential candidates using both IR and NIR
spectroscopy. The candidate fillers can be found in Table 9.4 (Gabrielsson et al. 2003).
Table 9.4: Filler Candidates for a Pharmaceutical Excipient in a Direct Compressed
Acetaminophen Tablet (Gabrielsson et al. 2003)
Fillers
i
Common
Name Type of Filler Moleucular Structure
Particle
Diameter
(?m) and
Structure*
1 Pearlitol SD200
Mannitol
200, g
2 Pearlitol 300DC 250, g
3 Pearlitol 400 DC 360, g
4 Pearlitol 500 DC 520, g
5 C Mannidex N/A, g
6 Avicel PH-101
Microcrystalline
Cellulose
50, c
7 Avicel PH-102 90, c
8 Avicel PH-112 90, c
9 Avicel PH-200 High Density
Microcrystalline
Cellulose
180, c
10 Avicel PH-301 50, c
11 Avicel PH-302 90, c
298
12 Xylitab DC
Xylitol
N/A, g
13 Xylisorb 90 90, c
14 Xylisorb 300 250, c
15 Xylisorb 700 700, c
16 Xylitab 100 200, g
17 Xylitab 200 300, g
18 Xylitab 300 150, g
19 C Xylidex CR 16055
Maltodextrin
200, g
20 Lycatab 300, g
21 C Sperse MD 01314 150, g
22 C Pharm 01980 N/A
23 C Pharm 01983 N/A
24 Maltisorb P90
Maltitol
N/A
25 Maltisorb P200 N/A
26 Promogram W Dextrin
N/A
27 Isomalt DC 100
Isomalt
N/A
28 Isomalt DC 110 90, N/A
* Structure types are g granular or c crystalline.
299
The IR and NIR spectroscopic images of the first five fillers, derivatives of Mannitol, are
provided in Figure 9.5. Transmittance IR spectroscopic data was measured between 4500 to 650
cm-1 and represented as 1997 separate wavelengths, approximate 2 cm-1 apart. Reflected-
absorbance NIR spectroscopic data was measured between 400 and 2500 nm and represented
1050 separate wavelengths, approximately 2 nm apart. A table of the IR and NIR spectroscopic
responses for all filler excipients in Table 9.4 is available in Appendix 5.
The IR and NIR data sets were then standardized and centered separately using Eq. 9.10
and 9.11 and then combined to form the characteristic spectroscopic profile shown in Fig. 9.6.
absi
abs ia vgabsjiabs
ji s PPP ,2 ,? ??
(9.10)
Ti
T iavgTjiT
ji s PPP ,2 ,? ??
(9.11)
It should be noted that number of j?p absorbance wavelengths are much larger than the number
of i?u samples and appear highly correlated, making them strong candidates for decomposition.
Testing for multivariate normality in Fig. 9.7 reveals that the system behaves in a multivariate
normal manner, except for the last two filler excipients of maltodextrin, runs 27 and 28, which
appear to be outliers in the dataset. These values may indicate a nonlinear relationship in the
data that may artificially influence the definition of latent variable structure. In spite of this
observation, investigation of the IR and NIR spectra yielded no insight into the cause. Further
analysis of the spectroscopic properties P is warranted, but in the interest of matching the results
published by Gabrielsson et al. (2003) these data points were included.
300
Figure 9.5: (a) IR Transmittance and (b) NIR Reflected-Absorbance Spectra for Mannitol
(Gabrielsson et al. 2003).
0
10
20
30
40
50
60
70
80
90
100
60080010001200140016001800200022002400260028003000320034003600380040004200440046004800
T
r
a
n
s
m
i
t
t
a
n
c
e
Fre qu e nc y c m - 1
I R S p e c t r a
Pe a rl i t o l
SD 2 0 0
Pe a rl i t o l
300DC
Pe a rl i t o l
400DC
Pe a rl i t o l
500DC
C Ma n n i d e x
0
10
20
30
40
50
60
70
80
90
100
400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600
A
b
s
o
r
b
a
n
c
e
W a v e l e ng t h nm
N I R S p e c t r a
Pe a rl i t o l
SD 2 0 0
Pe a rl i t o l
300DC
Pe a rl i t o l
400DC
Pe a rl i t o l
500DC
C Ma n n i d e x
(a)
(b)
301
Figure 9.6: p-Standardized and Centered IR Transmittance and NIR Reflected-Absorbance filler
excipient spectra (Gabrielsson et al. 2003).
Figure 9.7: The Gamma Q-Q Plot for the Filler Excipients Presented by Gabrielsson et al.
(2003)
Once the system is determined to be multivariate normal, a variety of decomposition
techniques exist to aid in the reduction of the number of j?p properties describing the system to a
-4
-3
-2
-1
0
1
2
3
4
0 50 0 10 00 15 00 20 00 25 00 30 00
S
epar
ately
M
ea
n
& V
ar
iance
Cent
er
ed
IR/NIR
Respo
n
se
s
P r o p er t y Nu mb er
IR & NIR S pec tra 1 2 3 4 5 6
7 8 9 10 11 12
13 14 15 16 17 18
19 20 21 22 23 24
25 26 27 28
302
set of t?m latent variable scores and l?m latent variable loadings. These techniques include
singular value decomposition (SVD), spectral decomposition, and the NIPALS algorithm and
can be found in a variety of commercial softwares, such as JMP, SAS, MatLab, and SIMCA-P.
Decomposition by principal component analysis (PCA) is the most common method of finding
the latent property subdomain. By definition, PCA uses the variance-covariance structure to
compress the molecular architecture data to principal component data that contains much of the
system variability. This result can improve the interpretation of the data structure by
consolidating multiple molecular architecture effects into single, underlying latent variables
which are devoid of colinearity. It is generally considered acceptable to apply decomposition
techniques to spectroscopy based characterizations because the main molecular architecture
information is collected in the larger population l loadings while the influence of that information
is located in the scores t (Naes and Isaksson 1992). If the original property data of molecular
architecture descriptors are of the same type, then the intensities of the absorbances can be
decomposed into loadings Lpxm , which represent the underlying latent structure of the data
(Johnson and Wichern 2007) and scores Tnxm, which represents the weightings of each latent
spectra (Workman and Springsteen 1998).
mxpnxmnxp LTP ?? (3.35)
As noted in Chapter 3, the first 2 or 3 principal components typically account for 80% to
90% total variance. The remaining components can be removed without much loss of
information (Johnson and Wichern, 2007). While this approach represents a significant gain in
efficiency, the drawback is that this reformulation is not unique. For instance, a decomposition
of the IR/NIR spectra of the fillers in Figure 9.6 by the SIMCA-P program is given in Table 9.5.
The program uses the NIPALS algorithm and the analysis was performed on both the covariance
303
and correlation matrices. Three principal components were found to contain 87% of the data and
are presented in Table 9.5.
Table 9.5: The Principal Components for the Filler Excipients using SIMCA-P Software on the
Covariances and Correlations (Gabrielsson et al. 2003).
Sample T1
Cov
(SIMCA-P)
T2Cov
(SIMCA-P)
T3Cov
(SIMCA-P)
T1Corr
(SIMCA-P)
T2Corr
(SIMCA-P)
T3Corr
(SIMCA-P)
1 -4.4 -42.8 17.3 -4.0 -42.4 -22.2
2 7.2 -35.9 15.5 7.5 -35.8 -19.8
3 7.5 -36.8 17.0 7.8 -36.6 -21.4
4 10.5 -35.6 15.0 10.7 -35.6 -19.3
5 1.2 -38.8 15.1 1.6 -38.4 -19.3
6 -53.6 6.1 -21.6 -52.8 3.2 21.6
7 -55.3 5.0 -24.3 -54.5 1.8 24.1
8 -57.9 -2.6 -33.1 -57.1 -6.5 32.2
9 -57.2 4.9 -21.9 -56.3 2.0 21.9
10 -48.0 7.0 -26.0 -47.3 3.6 26.0
11 -42.3 5.1 -20.2 -41.8 1.8 19.8
12 53.6 19.5 -9.5 53.0 17.3 10.7
13 53.6 9.9 -19.3 53.1 6.9 19.6
14 55.3 5.4 -14.6 54.7 2.6 14.1
15 61.8 10.9 -8.4 61.2 8.6 8.4
16 48.5 9.6 -16.4 48.1 6.9 16.6
17 49.8 11.8 -13.2 49.4 9.5 13.7
18 47.6 5.0 -23.4 47.2 1.7 23.0
19 61.8 11.9 -8.4 61.2 9.7 8.7
304
20 -26.8 20.1 19.2 -26.5 20.3 -17.9
21 -23.4 25.3 22.4 -23.2 25.8 -20.6
22 -27.9 30.1 26.4 -27.6 31.1 -23.9
23 -22.0 24.3 28.3 -21.8 25.3 -26.5
24 -16.0 -34.4 -11.2 -15.6 -37.4 6.4
25 -10.9 -32.8 -5.8 -10.6 -35.6 0.8
26 -28.1 37.5 15.8 -27.8 37.8 -12.3
27 7.1 21.1 32.4 7.1 22.8 -30.5
28 4.3 27.3 36.7 4.2 29.5 -33.9
The corresponding eigenvector loadings Lpxm for the scores Tnxm, shown in Table 9.5 are
given in Appendix A5. By convention, the eigenvector loadings capture spectral variations that
are common across the training set and the scores describe variation within the training set and
can be used to predict the molecular architecture of new molecules. However, when an
alternative, spectral decomposition is applied using Matlab and JMP softwares, a different set of
scores are found as shown in Table 9.6. One of the major differences between the
decompositions used in Table 9.5 and 9.6 is that the default analysis in SIMCA-P program finds
the underlying latent structure based on the covariance structure, while the default analysis in
MatLab and JMP uses a correlation structure. Since the two spectroscopic data sets were
standardized and centered separately, it is believed a better analysis would be achieved by
performing decomposition on the correlation data set using a normalization across the training
set molecules, as shown in Eq. 9.12:
2
,??
j
a vgjjiC O RRji sPPP ?? (9.12)
305
Table 9.6: The Principal Components for the Fillers Excipients using Matlab and JMP Softwares
(Gabrielsson et al. 2003).
Sample
T1
(MatLab)
T2
(MatLab)
T3
(MatLab)
T1
(JMP)
T2
(JMP)
T3
(JMP)
1 -4.0 42.4 22.2 -4.0 -42.4 22.2
2 7.5 35.8 19.8 7.5 -35.8 19.8
3 7.8 36.6 21.4 7.8 -36.6 21.4
4 10.7 35.6 19.3 10.7 -35.6 19.3
5 1.6 38.4 19.3 1.6 -38.4 19.3
6 -52.8 -3.2 -21.6 -52.8 3.2 -21.6
7 -54.5 -1.8 -24.1 -54.5 1.8 -24.1
8 -57.1 6.5 -32.2 -57.1 -6.5 -32.2
9 -56.3 -2.0 -21.9 -56.3 2.0 -21.9
10 -47.3 -3.6 -26.0 -47.3 3.6 -26.0
11 -41.8 -1.8 -19.8 -41.8 1.8 -19.8
12 53.0 -17.3 -10.7 53.0 17.3 -10.7
13 53.1 -6.9 -19.6 53.1 6.9 -19.6
14 54.7 -2.6 -14.1 54.7 2.6 -14.1
15 61.2 -8.6 -8.4 61.2 8.6 -8.4
16 48.1 -6.9 -16.6 48.1 6.9 -16.6
17 49.4 -9.5 -13.7 49.4 9.5 -13.7
18 47.2 -1.7 -23.0 47.2 1.7 -23.0
19 61.2 -9.7 -8.7 61.2 9.7 -8.7
20 -26.5 -20.3 17.9 -26.5 20.3 17.9
21 -23.2 -25.8 20.6 -23.2 25.8 20.6
22 -27.6 -31.1 23.9 -27.6 31.1 23.9
306
23 -21.8 -25.3 26.5 -21.8 25.3 26.5
24 -15.6 37.4 -6.4 -15.6 -37.4 -6.4
25 -10.6 35.6 -0.8 -10.6 -35.6 -0.8
26 -27.8 -37.8 12.3 -27.8 37.8 12.3
27 7.1 -22.8 30.5 7.1 22.8 30.5
28 4.2 -29.5 33.9 4.2 29.5 33.9
This operation results in the correlated standardized and centered data structure shown in Fig.
9.8.
Figure 9.8: n,p-Standardized and Centered IR Transmittance and NIR reflectance-absorbance
Filler Excipient Spectra (Gabrielsson et al. 2003).
While this transformation explains the magnitude differences between the scores in
Tables 9.5 and 9.6, it does not account for the sign orientation. The sign variation between the
- 4
- 3
- 2
- 1
0
1
2
3
4
0 500 1000 1500 2000 2500 3000
M
e
a
n
&
V
a
r
i
a
n
c
e
C
e
n
t
e
r
e
d
R
e
s
p
o
n
s
e
P rop e rt y N u m b e r
I R / N I R S p e c t r a
1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
FT - IR N I R
307
calculated scores of the correlations is a result of the algorithms used in MatLab, JMP, and
SIMCA-P. Each algorithm arbitrarily assigns direction to either the loadings or the scores at the
time of calculation. Investigations of each pair of loadings and scores in Appendix A5 reveals
that the the sign convention is equal and opposite for each m latent variable across all p property
descriptors. Thus, when the loadings and scores are recombined, they return identical values,
regardless of whether the loadings or the scores absorb the negative values. For example, the
Simca-P, MatLab, and JMP predicted IR and NIR responses using only the first three correlated
latent variable scores (Table 9.5 and Table 9.6) and loadings (Appendix A5) are matched against
the actual response plot of Pearlitol SD200 (i.e. Mannitol) in Fig. 9.9. The responses using each
of the three algorithms are identical even though the latent variable structures are not. The slight
deviation from the actual response is the result of only using three of the 3047 latent descriptors
to build the model, albeit the ones describing 87% of the response. In conclusion, although the
non-uniqueness of decomposition prevents direct interpretation of the underlying latent variable
structure, it does not prevent the interpretation of contrasts and, so long as a single set of
loadings is used to map all descriptors down to the latent variable subspace, the solutions
generated with the technique will be unique.
Rule 31. As long as a single set of loadings are chosen to map the problem into the
subdomain, then the molecular architecture design will result in a unique solution
for that given set of loadings.
308
Figure 9.9: The (a) IR and (b) NIR Response plots of the Recombined Scores and Loadings
Found using Simca-P, MatLab, and JMP.
In some cases, the decomposition may be artificially influenced by poorly standardized
data or nonlinear influences. For example, Fig. 9.7 showed that Maltodextrin samples 27 and 28
may be overly influential in the development of the underlying architecture. This observation
could have influenced the decision of Gabrielsson et al. (2003) to base the latent variable
structure on the covariance structure and not the correlation structure of the spectroscopic
0
20
40
60
80
10 0
12 0
60 016 0026 0036 0046 00
Tr
an
sm
itt
an
c
e
Fr e q u e n c y (c m - 1 )
I R Spe ct r a
Ac tu al
Pre d ic ted b y
SIM CA - P
Pre d ic ted b y
J MP
Pre d ic ted b y
Ma tL ab
(a)
0
10
20
30
40
50
60
0 50 0 10 00 15 00 20 00 25 00 30 00
A
b
sor
b
an
c
e
Wav e l e n g th (n m )
NI R Spe ct r a
Actu al
Pre d ic ted b y SIM CA - P
Pre d ic te d b y J MP
Pre d ic ted b y M at La b
(b)
309
property in order to minimize their impacts. However, the convention is to base a decomposition
analysis on correlation data structure when the scales of the characterization data are considered
dissimilar, which they were in this instance. Therefore, in this author?s opinion, this analysis
should have been based on the correlation data structure. All decompositions in the case study
reflect this opinion.
Lastly, once the latent domain is described, the latent property-attribute relationship is
constructed. This relationship serves the primary function of mapping the target domain into the
latent variable domain so that a combinatorial design using cGCM can be performed. The
relationship between the principal component scores Tnxm and the attribute properties Anxa is
developed using a multivariate linear regress (MLR) on a new design of experiment (DOE)
factorial design where the scores are varied between their high (+1) and low (-1) levels
(Gabrielsson et al. 2002; Gabrielsson et al. 2003; Gabrielsson et al. 2004; Gabrielsson et al.
2006):
mxanxmnxa BTA ? (9.13)
where Bmxa are the regressed coefficients found using MLR. The combined relationship is
typically referred to as Principal Component Regression (PCR) since the independent variable is
located in the latent sub-domain. As discussed in Chapter 3, one of the issues with PCR is that
while the decomposition is designed to ensure a strong relationship between the absorbance
properties, P, and the latent domain, it does not guarantee a strong relationship between the
attributes A and the latent domain. The alternative is to integrate the two approaches,
simultaneously calculating the structure of the latent domain and its relationship to the attributes
of interest in an approach known as or a partial linear regression on to latent surfaces (PLS).
This approach weighs spectra containing attribute-specific molecular architecture more heavily
310
in the latent variable domain, essentially compressing attribute information into the first few
loading vectors (Geladi and Kowalski 1986; Workman and Springsteen 1998). Although an
argument can be made to use a PLS model to develop the relationship between the property
descriptors and the attribute domain if a heavy correlation between attribute and absorbance
property descriptors exists, the technique was not chosen since it sacrifices some latent property
descriptor accuracy for improved attribute-latent property descriptions. This choice is discussed
in more detail in the case study in Sec. 9.4.
Regardless of the approach, the choice of model should be made by directly comparing
the model fitness and predictive power; choosing the one with better measures. Model fitness
can be measured with a variety of techniques. One of the most common is the R2 fitness measure
given in Eq. 3.25 which describes the percentage of variance explained by the model. In similar
fashion, the predictive power of the model Q2 given in Eq. 3.26 describes the percentage of
predictive variance explained by the model.
to t
resto tSS SSSSR ??2 (3.25)
to t
PREto tSS SSSSQ ??2 (3.26)
The overall fitness and predicted power of each approach can be measured using Eq. 9.14
and Eq. 9.15:
? ???
D
TOT RR 1 22
(9.14)
? ??? DD QQ 1 22 (9.15)
311
where D is the the total number of models used in the mapping function. Care must be taken to
ensure that appropriate model fitness, as measured by R2, is not sacrificed to improve the
prediction power, Q2. This can be achieved not only by selecting the appropriate latent domain
structure, but also by selecting the appropriate molecular architecture relationship. The choice of
molecular architecture relationship is partially dictated by the choice of characterization and
partially by the training set. Choosing the characterization that maximizes the prediction power
for a given number of experiments may be a procedure that can be written as a MINLP, but is
beyond the scope of this dissertation.
Rule 32. The PLS or MLR model with the highest model fit and predictive power should be
chosen to model the attribute-latent variable relationship.
Once the attribute-latent property, property descriptor-latent property, and cGCM
expressions are developed, the method is ready for use in a computer aided molecular design
algorithm. Since significant efficiency improvements have been achieved by combining
aCAMbD and aCAMD with property clusters and the reverse problem formulation, the cCAMD
method will also utilize this solution structure.
9.2. Latent Property Clustering Algorithms
As noted in Chapter 6, property clustering is a tool used to improve the interpretation of
the subspace properties by deconstructing the design problem into a Euclidean vector in the
cluster domain and a scalar called the Augmented Property Index AUP. The clusters themselves
are conserved surrogate properties described by property operators, which have linear mixing
rules, even if the operators themselves are nonlinear (Shelley and El-Halwagi 2000; Eden et al.
2003). Methods for the application of group contribution methods for molecular design with
312
experimental data using property clustering have previously been developed in Chapter 8 and a
new approach using characterization data was discussed in Section 9.1.1. To extend this
approach to utilize the efficiency gains of decomposition by performing molecular design on the
latent variables using the property clustering algorithm, it is important to recognize that the
decomposition data structure follows a linear mixing rule. For example, rearranging the
decomposition shown in Eq. 3.35 results in Eq. 9.16:
p xmn xpn xm LPT ? (9.16)
This data structure is similar to the matrix form of the mixing expression shown in Eq. 6.1.
uxpnxuMnxp X ?? ? (9.17)
If the loadings Lpxm are thought of as the pure values of the principal components, then Eq. 9.16
suggests that the scores Tnxm have behave in a similar manner to the predicted mixture properties
?Mnxp. In this situation the pj? properties for each ui? pure component, normally denoted as
?uxp in a conventional mixture design, have been replaced by a set of mq? latent properties for
each pj? property descriptor denoted as Lpxm. Likewise, the ne? mixtures of ui? pure
component fractions, usually denoted as Xnxu in a mixture design, are actually the pj? property
descriptors responses (e.g. transmittances/absorbances for IR/NIR, etc.) for each ne?
experimental design mixture, or Pnxp. Thus, the decomposition expression in Eq. 9.16 can be
treated as a linear mixture of the latent variable loadings. There is, however, a concerning
difference between the two methods: the mixture fractions Xnxu sum to one across the ui?
components and the multivariate data Pnxp do not do the same across the pj? property
descriptors. In order for latent variable models to be utilized in property clustering, it is
313
necessary to ensure this condition is met by standardizing the latent variable structure by
dividing Pnxp and Tnxm by the sum total of the property descriptors for each experimental run,
Snxn.
??
p
j iee PS
, ne? (9.18)
n x mn x nn x m TSQ 1)( ?? (9.19)
n xpn xnn xp PSR 1)( ?? (9.20)
The resulting expression for the decomposition linear mixture model becomes:
pxmnxpnxm LRQ ? (9.21)
where the new Qnxm matrix now represents standardized scores or mixtures, the loadings matrix
Lpxm remains unchanged, and the Rnxp matrix now represents fractions of loadings whose
cumulative sum is one for each run. Unfortunately, although the components sum to one, they
are sometimes negative due to the mean-centering of the multivariate property data prior to PCA.
The constraint that the fractions must be between 0 and 1 is removed with no effect on the
associated mathematics, only on their interpretation, so long as a positive AUP is maintained by
adjusting the reference property values. At this point, the loadings Lpxm are the underlying latent
variable domain subspace. In terms of property clusters, the relationship becomes:
???
p
j jqejeq LRQ 1
, ne? , pj? , mq? (9.22)
The normalized property operator of the latent variable properties may be obtained by
dividing Ljq with a reference value ?ref,q.
314
qref
jqLjq L
,???
(9.23)
The latent domain Augmented Property Index AUPL, for each latent fragment is defined as the
summation of all the u dimensionless property operators:
?? ??
m
q
LLj
qjAUP 1
(9.24)
Finally, the latent property cluster CjuL is defined as
Lj
L
Ljq AUPC jq?? (9.25)
Shown in Fig. 9.10 is are the loadings found in Appendix A5 for the decomposed IR/NIR
absorbances of the 28 filler excipients shown in Fig. 9.8 (Gabrielsson et al. 2003). For this
system of p=3047 property descriptors and n=28 training set samples, 3 latent properties were
found to contain 87% of the data. When plotted in two dimsnsions, the ellipsoid becomes a
series of ellipses that are indicative of a well explored, multivariate normal domain. As
expected, the plot of the two latent properties containing the most property descriptor
information, L1 vs. L2 in Fig. 9.10(a), has a well-defined elliptical shape while Fig. 9.10(c),
which contains latent properties with less descriptor information, is less defined.
315
Fig. 9.10: From the PCA decomposition of the Training Set Excipient Property Descriptors
shown in Fig. 9.8, Plots of the Loadings: (a) L1 vs. L2, (b) L1 vs. L3, and (c) L2 vs. L3.
- 0.0 5
- 0.0 4
- 0.0 3
- 0.0 2
- 0.0 1
0
0. 01
0. 02
0. 03
0. 04
0 . 0 5
- 0.0 3 - 0.0 2 - 0.0 1 0 0. 01 0. 02 0. 03
Loadi ngs P lot
(L 1 v s L2)
- 0.0 5
- 0.0 4
- 0.0 3
- 0.0 2
- 0.0 1
0
0. 01
0. 02
0. 03
0. 04
0. 05
0 . 0 6
- 0.0 3 - 0.0 2 - 0.0 1 0 0. 01 0. 02 0. 03
Loadi ngs P lot
(L 1 v s L3)
- 0.0 5
- 0.0 4
- 0.0 3
- 0.0 2
- 0.0 1
0
0. 01
0. 02
0. 03
0. 04
0. 05
0 . 0 6
- 0.0 5 - 0.0 4 - 0.0 3 - 0.0 2 - 0.0 1 0 0. 01 0. 02 0. 03 0. 04 0. 05
Loadi ngs P lot
(L 2 v s L3)
(a)
(b)
(c)
316
Using property clustering, these three figures are consolidated into a single chart shown
in Fig. 9.11, where the x and y axis are the Cartesian coordinates of the cluster domain. The
black circles typically used to denote pure components have been changed to pluses and minuses
to illustrate two sets of loadings: those with positive AUP and those with negative AUP. The
negative AUP can be made positive by changing the reference values, but for ease of
transparency, the reference values were all left at unity. The figure also shows a good
distribution of the loadings in each of the 8 cluster regions identified in Chapter 7 which indicate
a strong description of the latent domain.
Figure 9.11: The Clustered Loadings L
pxm Showing the Latent Structure for the Filler Excipient
Training Set shown in Table 9.6.
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5 -4 -3 -2 -1 0 1 2 3 4 5
C 1 z e ro
C 2 z e ro
C 3 z e ro
A U P -
A U P +
A U P > 0 : ( - + - )
A U P < 0 : ( + - + )
A U P > 0 : ( - + + )
A U P < 0 : ( + - - )
A U P > 0 : ( - - + )
A U P < 0 : ( + + - )
A U P > 0 : ( + - + )
A U P < 0 : ( - + - )
A U P > 0 : ( + - - )
A U P < 0 : ( - + + )
A U P > 0 : ( + + - )
A U P < 0 : ( - - + )
317
The loadings can then be mixed according to the ratios Rnxp to find the training set of
scores Qnxm as shown in Eq. 9.22. However, since each characterization experiment, particularly
IR and NIR spectroscopy, are typically conducted only on pure components, Eq. 9.22 can be
rewritten as Eq. 9.26.
???
p
j jqijiq LRQ 1
, ui? , pj? , mq? (9.26)
This expression implies that the scores for each pure component are simply varying ratios of the
underlying latent variable structure. Since the latent variable loadings are common across all
molecules and molecular architecture in the training set, then the AUPL in Eq. 9.24 can be used
to create the score clusters, CiqQ. The AUPQ of each pure component is found using Eq. 9.27.
?? ??
p
j ij
LjQi RA U PA U P
1
(9.27)
The latent property lever arm, ?ij, is then calculated using Eq. 9.28.
Q
Ljij
ij
iAU P
AU PR ??? (9.28)
Finally the mixture score cluster for each q?m latent variable is calculated.
?? ??
p
j
LijQ
jqiq CC 1?
(9.29)
Conversely the score clusters can also be directly calculated from the scores using Eq. 9.30-9.32:
qref
iqQiq Q
,???
(9.30)
318
?? ??
m
q
QQj
qjAUP 1
(9.31)
Q
Q
Qiq
j
iqAUPC ?? (9.32)
Shown in Fig. 9.12 are conventional plots of the JMP scores given in Table 9.6 that
illustrate the similarities in molecular architecture between excipients. As with the loadings
plots, it is easier to distinguish contrasts in the molecular architecture in the plots that contain
more descriptor information. Although the general influence of the loadings on the scores can be
inferred by overlaying the loadings plots on the score plots, the magnitude of that influence is not
directly apparent. In contrast, when the scores are clustered, the magnitude of each loadings
impact on the score value can be directly calculated from the overlayed loading clusters. Fig.
9.11 shows the clustered JMP scores, which can be built through lever arm analysis from the
scores listed in Fig. 9.9. Like the loading cluster diagram, two types of mixture score clusters are
shown, those with AUP that are positive (circles) and those with AUP that are negative (squares).
As demonstrated in Chapter 7, the clusters with negative AUP must be converted to clusters with
positive AUP in order to properly interpret the solution. This can be accomplished by adjusting
the reference values for the clusters using the algorithm developed in Chapter 6.
319
Fig. 9.12: From the PCA Decomposition of the Training Set Excipient Property Descriptors
Shown in Fig. 9.8, Plots of the Loadings: (a) L1 vs. L2, (b) L1 vs. L3, and (c) L2 vs. L3.
1
2
3
4
5
6
7
8
9 10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
- 50 . 0
- 40 . 0
- 30 . 0
- 20 . 0
- 10 . 0
0. 0
10 .0
20 .0
30 .0
40 .0
5 0 .0
- 80 . 0 - 60 . 0 - 40 . 0 - 20 . 0 0. 0 20 .0 40 .0 60 .0 80 .0
S c ore P lot
(t 1 v s t2)
1
2
3
45
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
- 40 . 0
- 30 . 0
- 20 . 0
- 10 . 0
0. 0
10 .0
20 .0
30 .0
4 0 .0
- 80 . 0 - 60 . 0 - 40 . 0 - 20 . 0 0. 0 20 .0 40 .0 60 .0 80 .0
S c ore P lot
(t 1 v s t3)
1 2
3
45
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
- 40 . 0
- 30 . 0
- 20 . 0
- 10 .
0. 0
10 .0
20 .0
30 .0
4 0 .0
- 50 . 0 - 40 . 0 - 30 . 0 - 20 . 0 - 10 . 0 0. 0 10 .0 20 .0 30 .0 40 .0 50 .0
S c ore P lot
(t 2 v s t3)
(a)
(b)
(c)
320
Figure 9.13: The Clustered Scores Qnxm Showing the Latent Structure for the Filler Excipient
Training Set Shown in Table 9.6.
By extension, since the score of a pure component can be constructed from ratios of its
loadings, then the score of a molecular group can as well. Because the IR/NIR pure component
descriptor data in the training set are a measure of the frequencies and intensities of group
absorbances packaged together in a complete molecule, then the loadings can be treated as
constant for any type of molecular architecture in the training set data. Hence, Eq. 9.26 can be
rewritten as Eq. 9.33:
1
2
3
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
2122
23
24
25
26
27
28
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5 -4 -3 -2 -1 0 1 2 3 4 5
AUP > 0 : ( - + - )
AUP < 0 : ( + - + )
AUP > 0 : ( - + + )
AUP < 0 : ( + - - )
AUP > 0 : ( - - + )
AUP < 0 : ( + + - )
AUP > 0 : ( + - + )
AUP < 0 : ( - + - )
AUP > 0 : ( + - - )
AUP < 0 : ( - + + )
AUP > 0 : ( + + - )
AUP < 0 : ( - - + )
321
???
p
j jqgjgq LRQ 1
, Fg? , pj? , mq? (9.33)
where the Rgj property ratios are estimated from the group specific IR/NIR responses generated
by the procedure in Section 9.1.1. For example, to calculate the equivalent scores of q=3 latent
properties for an CH2 group, absorbances and intensities listed in Appendix A6 are converted to
the IR response plot shown in Fig. 9.14.
Figure 9.14: The IR transmittance response plot for a CH2 group.
The group response data is then scaled and centered to generate PFxp, followed by standardization
using Eq. 9.34 and Eq. 9.35 to generate RFxp:
??
p
j jgg PS
, Fg? (9.34)
0
10
20
30
40
50
60
70
80
90
10 0
050
0
10
00
15
00
20
00
2
5
0
0
30
00
35
00
40
00
45
00
50
00
T
r
ansmit
t
ance
F r equ ency cm - 1
IR G roup S pec tra
322
F x pF x FF x p PSR 1)( ?? (9.35)
The standardized QFxm group scores can then be calculated using the matrix form of Eq. 9.33
shown in Eq. 9.36:
p xmFxpFxm LRQ ? (9.36)
To find the score clusters, CgqQ, the AUP of the score mixture is first calculated using Eq. 9.37,
followed by the latent property lever arm, ?gj, in Eq. 9.38.
?? ??
p
j gj
LjQg RA U PA U P
1
(9.37)
Qg
Ljgj
gj AU P
AU PR ??? (9.38)
Finally the group score cluster is calculated.
?? ??
p
j
LgqgjQ CC
gq 1?
(9.39)
Conversely the score clusters can also be directly calculated from the scores using Eq. 9.40-9.42:
qref
gqQgq Q
,???
(9.40)
?? ??
m
q
QQg
qgAUP 1
(9.41)
Q
QQ
gq
g
gqAUPC ?? (9.42)
323
The target feasibility region is constructed in the same manner. After converting a set of
attribute targets into a set of Ttxq latent property target scores (i.e. solving Eq. 9.13) using a
procedure outlined in Chapter 8, the latent property targets are then recombined with the Lpxm
loadings to generate a set of Ptxm property descriptors of the property targets.
mxptxmtxp LTP ?? (9.43)
The scores are then standardidized using Eq. 9.38 and 9.39.
??
p
j jtt PS
, tb? (9.44)
txmtxttxm TSQ 1)( ?? (9.45)
Conversely the score clusters can also be directly calculated from the scores using Eq. 9.46-9.48:
qref
gqQbq Q
,???
(9.46)
?? ??
m
q
QQb
qbAUP 1
(9.47)
Q
QQ
bq
b
bqAUPC ?? (9.48)
A summary of the data generation and conversion process for cCAMD is shown in Fig. 9.15
324
Figure 9.15: The cCAMD Cluster Conversion Flowchart.
325
9.3. cCAMD using Characterization Data and cGCM Models
It has been shown in Section 9.1 how characterization tools like IR and NIR spectroscopy
can be used to describe molecular architecture and that this molecular architecture can be
described using a new combinatorial tool called cGCM. It has also been shown how
decomposition techniques can be used to compress this data structure to a more manageable
arrangement suitable for multiscale chemical product design in Section 9.2. In order to conduct
a molecular architecture design it is important to recognize that the constraints imposed by
decomposition should also be observed for any new molecules or mixtures created. For instance,
by definition, the mixture, pure component, and group representations of the underlying latent
variable loadings are all linear mixture expressions.
p xmn xpn xm LPT ? (9.16)
In Eq. 3.43 it was shown how the latent variable scores can also be linearly mixed:
XTTM ?? (3.43)
where X is a uxn matrix of chemical constituent mass fractions and TM is a nxm matrix of
candidate mixtures. Thus, candidate mixture scores can then be treated as linear mixtures of
linear mixtures of the latent variable loadings. This observation assumes that any nonlinearity
in the system is handled by the attribute-latent property relationship (Muteki et al. 2006).
Violation of this constraint will significantly reduce the effectiveness of the cCAMD approach
and can be avoided by ensuring the system is multivariate normal.
If Eq. 3.43 is set equal to the chemical product design target scores and solved for mass
fractions of the chemical constituents in the training set, the procedure is referred to as a Case 1
326
mixture design in Section 3.2.4. In terms of property clusters, the mixture expression is first
standarndized using Eq. 9.45:
?? ??
u
i iiqeq xQQ 1
ui? , mq? (9.49)
where Qeq is also known as QM or mixture score. The AUP, lever arm, and cluster expressions
are then given by Eq. 9.50-9.52.
?? ??
u
i ie
QiQe xA U PA U P
1
(9.50)
Qe
Qiie
ie AU P
AU Px ??? (9.51)
?? ??
u
i
QqiieQeq CC
1 ?
(9.52)
Additionally, Garcia-Munoz et al. (2010) and Muteki and Macgegor (2006) have shown
that the new mixture scores can be matched against a database to find new molecular
architectures that are not part of the original training set, denoted as Case 2 in Chapter 3.
However, the candidate molecular architectures found are subject to the completeness of the
database. Using the cGCM property model on the group scores overcomes this shortcoming by
providing a means to build all potentional candidates that have the underlying latent variable
loading structure of the training set. A set of group latent variables identified in the cGCM can
then be combined to find new molecules that were not part of the original training set. The
structure of the molecular design follows that of a first order GCM expression.
?? ??
F
g igqgqi nQQ 1
(9.53)
327
The conversion of this approach to make use of the efficiency gains provided by the reverse
problem formulation and property clustering results in a technique called characterization based
computer aided molecular design (cCAMD) with clusters. The approach is similar the CAMD
method discussed in Chapter 6, only it uses the latent property scores for the individual group
contributions. Noting that the property operator expression of Eq. 9.53 can be non-
dimesionalized, the expression becomes Eq. 9.54.
?? ??? gng MQqggMQqi n1 (9.54)
Rewriting the molecular property operator expression in terms of clusters is done by inserting
Eq. 9.54 into Eq. 9.42 and rearranging to get Eq. 9.55-9.57.
MQ
i
F
g
MQ
g
MQ
qgg
MQ
i
F
g
MQ
qgg
MQ
qi A U P
A U PCn
A U P
n
C
?? ??
?
??
?
(9.55)
MQi
MQgg
g A U P
A U Pn 1
1
1 ???
(9.56)
?? ?? Fg MQqggMQqi CC 1
1 11
? (9.57)
The group based property model shown in Eq. 9.57 demonstrates how molecular groups or
fragments can be added together analogous to inter-stream conservation (i.e. Eq. 9.52).
The method proceeds in a similar manner to that of CAMD, but with additionals tests on
the completeness of the molecular structures. Groups generated using the cCAMD alogithm are
fed into the CAMD candidate generation algorithm. As with conventional CAMD, the
maximum number of dissimilar groups ?gmax is first set. Then, progressively larger combinations
of dissimilar groups are added until the maximum is reached. The outputted solutions from the
328
CAMD algorithm are then tested sequentially against Rules 25-27. Any solutions that fail Rules
25-27 are discarded. Solutions that satisfy Rules 25-27 are deemed candidate solutions. The
procedure is repeated until a maximum number of dissimilar groups ?gmax is reached. A
summary of the cCAMD candidate generation method is provided in Fig. 9.16.
Figure 9.16: A Flowchart of the cCAMD Candidate Generation Algorithm using Clusters
It should be noted that the procedure calls the CAMD candidate generation procedure discussed
in Table 6.5. For this instance, the expressions referenced in Table 6.5 have been replaced with
the equivalent latent variable expressions, shown in Table 9.7.
329
Table 9.7: CAMD Candidate Generation Procedure with Latent Variables
Step Description Equation
1 Set Maximum Dissimilar Groups in the Mixture -
2 Discard Pure Group Clusters (from CAMD Conversion Algorithm) that Fail Rule 1 6.7, 9.48, 9.32
3 Calculate No. Pure Groups 6.47
4 Calculate Pure Group Solutions, Discard Pure Group Solutions that Fail Rule 2 and Rule 3 6.7, 6.16, 6.42
5 Discard Pure Group Clusters that Fail Rule 4, Output Pure Group Candidate Molecules 6.41
6 For Binary Solutions with Pure Group Candidates, Calculate the No. Binary Groups 6.46, 6.48, 9.56
7 Calculate Solution AUPMM, Discard Solutions that Fail Rule 2 6.39, 6.47, 9.56
8 Calculate the FBN, Discard Solutions that Fail Rule 4 6.41
9 Calculate Solution Clusters, Discard Solutions that Fail Rule 1, and Output Candidate Solution Clusters 6.7, 9.32, 9.57
10 For Binary Solutions with no Pure Group Candidates, Calculate the Reference Constant for each Mixture 6.20
11
Calculate the Sink Constants for each Mixture, Discard Mixtures
where all Sink Constants are > or < Reference Constant, Repeat Steps
6-9
6.23-6.24,
6.27-6.29
12 For Ternary and Larger Solutions (?g ? 3) containing ?g - 1 Candidates, Calculate the No. Groups, Repeat Steps 7-9 6.46, 6.48
13 For ? ? 3 Ternary+ Mixtures with no ? - 1 Candidates, Calculate the Cluster Range of the Pure Components 6.34, 6.35, 6.6
14
Calculate the Cluster Range of the Feasibility Sink, Discard Mixtures
whose Pure Component Cluster Range is > Sink Cluster Range,
Repeat Step 12
6.34, 6.35,
6.6, 6.7
15 Repeat Step 12-14 until ?gmaxis reached -
330
9.4. Case Study: Excipient Design for Direct Compressed Tablets
A detailed analysis of an excipient design of a direct compressed parecetamol tablet as
discussed in Gabrielsson et al. (2002, 2003, 2004, 2006) serves as the primary case study for
illustrating the approach presented in this chapter. A limited version of this case study is
presented in this section as a proof-of-concept that demonstrates the method using components
of the filler training set referenced in Section 9.1.
In Gabrielsson et al. (2003), the structure of a direct compressed paracetamol tablet is
made up of three distinct classes of excipients (i.e. binders, disintegrants, and fillers) and the
active ingredient acetaminophen. Binders generally hold the ingredients in a tablet together,
providing mechanical strength. Disintegrants help the tablet expand and dissolve when entering
the digestive tract. Fillers increase the bulk volume of the tablet, but can also serve in both a
binding and disitegrant role as well. For this reason, the excipient design proposed by
Gabrielsson et al. (2003) was further narrowed to focus on the design of only filler excipients.
A n=24 a list of the fillers for the proof of concept are shown in Table 9.8.
Table 9.8: Excipients used in the Proof-of-Concept Case Study
Fillers
i
Common
Name Type of Filler Moleucular Structure
Particle
Diameter
(?m) and
Structure*
1 Pearlitol SD200
Mannitol
200, g
2 Pearlitol 300DC 250, g
3 Pearlitol 400 DC 360, g
4 Pearlitol 500 DC 520, g
5 C Mannidex N/A, g
331
6 Avicel PH-101
Microcrystalline
Cellulose
50, c
7 Avicel PH-102 90, c
8 Avicel PH-112 90, c
9 Avicel PH-200 High Density
Microcrystalline
Cellulose
180, c
10 Avicel PH-301 50, c
11 Avicel PH-302 90, c
12 Xylitab DC
Xylitol
N/A, g
13 Xylisorb 90 90, c
14 Xylisorb 300 250, c
15 Xylisorb 700 700, c
16 Xylitab 100 200, g
17 Xylitab 200 300, g
18 Xylitab 300 150, g
19 C Xylidex CR 16055
Maltodextrin
200, g
20 Lycatab 300, g
21 C Sperse MD 01314 150, g
22 C Pharm 01980 N/A
23 C Pharm 01983 N/A
24 Maltisorb P90 Maltitol
N/A
332
Three representative property descriptors (i.e. P1, P2, and P3) were chosen to evaluate the
IR and NIR characterizon, as shown in Table 9.9.
Table 9.9: Property Descriptors of the Proof-of-Concept
Filler
i P1 P2 P3
?? 93 74 37
? 94 78 35
? 96 80 35
?? 101 84 39
?? 102 85 38
?? 103 81 37
?? 104 83 39
?? 106 83 39
?? 107 82 38
??? 112 89 40
??? 113 88 40
??? 114 86 40
??? 116 90 43
??? 117 90 41
??? 117 91 41
??? 119 93 41
??? 120 89 40
??? 120 93 44
??? 121 95 42
333
??? 125 93 45
??? 127 96 45
??? 128 95 45
??? 131 95 46
??? 135 106 47
In order to ensure the system is multivariate normal, a natural log function was applied to
each descriptor, and the system was centered, but not standardized to create Pnxp. From this data
set, a PCA decomposition was performed on the covariance matrix, which due to the extensive
normalization, resulted in better description of the latent variable domain than that of a PCA
decomposition on the correlation matrix. JMP 7.0.2 was chosen to perform the decomposition
resulting in the Tnxm scores shown in Table 9.10.
Table 9.10: Latent Variable Scores of the Proof-of-Concept
Filler
i T1 T2 T3
?? -0.2685 0.0611 0.0004
? -0.2633 -0.0157 -0.0067
? -0.2360 -0.0341 -0.0074
?? -0.1199 0.0141 -0.0367
?? -0.1207 -0.0150 -0.0286
?? -0.1526 -0.0089 0.0170
?? -0.1060 0.0166 -0.0084
?? -0.0930 0.0135 0.0052
?? -0.1064 -0.0012 0.0279
??? -0.0066 -0.0167 -0.0072
334
??? -0.0063 -0.0114 0.0062
??? -0.0120 0.0008 0.0268
??? 0.0609 0.0280 -0.0126
??? 0.0419 -0.0109 0.0090
??? 0.0475 -0.0174 0.0021
??? 0.0702 -0.0331 0.0007
??? 0.0406 -0.0277 0.0420
??? 0.1128 0.0213 -0.0163
??? 0.1050 -0.0294 -0.0085
??? 0.1524 0.0325 0.0055
??? 0.1795 0.0111 -0.0029
??? 0.1795 0.0161 0.0092
??? 0.2068 0.0297 0.0186
??? 0.2945 -0.0232 -0.0351
335
The Lpxm loadings associated with these scores are given in Table 9.11. For these results, it was
determined that 98.5% of the response data could be explained using only the first two principal
components (i.e. m ? 2). This can be confirmed by investigating the scree plot shown in Fig.
9.17. However, for completeness, all three latent variables are used in the design.
Figure 9.17: Scree plot of the Proof-of-Concept Principal Components
Table 9.11: Latent Variable Loadings of the Proof-of-Concept
Descriptor
j L1 L2 L3
?? 0.68310 -0.15948 0.71270
? 0.51022 -0.59401 -0.62195
? 0.52254 0.78849 -0.32440
336
Gabrielsson et al (2003) also identifies 8 different attributes important to the function of
pharmaceutical excipients. These are as follows: disintegration time (DT), crushing strength
(CS), Hausner ratio, disintegration type, ejection force (EF), adhesion, compression force, and
tablet height. Disintegration time is the average time for a tablet to break down and pass through
a sieve of a predetermined size. Crushing strength is the amount of force needed to break a
tablet under compression and is a measure of a tablets strength and hardness. The Hausner ratio
is a measure of how well a solid excipient powder flows. The disintegration type could be either
bursting or erosion and could be used to describe the release profile. Ejection force is the
strength needed to overcome the friction coefficient between the die wall and the tablet and is a
function of adhesion. Adhesion is a measure of intermolecular bonding and microstructure.
Compression force is the force used to compress the tablet to the appropriate tablet height. Of
these attributes, disintegration time (DT), crushing strength (CS), and ejection force (EF) were
selected for further study since these attributes offer a well-balanced description of the properties
important to a pharmaceutical tablet.
Normally to develop the attribute-property relations (e.g. Eq. 9.13) a separate full
factorial design of experiments (DOE) with centerpoints is executed where molecules containing
the appropriate descriptor structure type are chosen to represent each of the design points.
However, in this instance, attribute values were also measured for each chemical in the training
set and shown in Table 9.12
337
Table 9.12: Design of Experiments Attribute Responses
Filler
i DT CS EF
?? 0.0288 -1.4281 1.6503
? 0.0142 -1.8934 2.0880
? 0.0090 -1.8304 1.9957
?? 0.0055 -0.5194 0.8334
?? 0.0016 -0.7767 1.0158
?? 0.0111 -1.2156 1.2004
?? 0.0089 -0.5678 0.7014
?? 0.0093 -0.5777 0.6168
?? 0.0108 -0.9042 0.7995
??? -0.0034 -0.1222 0.1559
??? -0.0007 -0.1595 0.1154
??? 0.0046 -0.2309 0.0747
??? -0.0010 0.6910 -0.6276
??? -0.0035 0.1613 -0.2528
??? -0.0059 0.1937 -0.2521
??? -0.0104 0.2479 -0.3261
??? -0.0018 -0.1553 -0.1527
??? -0.0062 1.0238 -0.9767
??? -0.0134 0.5674 -0.6080
??? -0.0039 1.2518 -1.3538
??? -0.0106 1.3364 -1.4229
??? -0.0081 1.3023 -1.4587
??? -0.0063 1.5332 -1.7526
338
??? -0.0287 2.0742 -2.0664
The three attributes DT, CS, and EF have been notoriously difficult to analyze using
traditional mixing design because of the complex and highly nonlinear nature of pharmaceutical
excipients (Martenello et al. 2008). In order to better control these attributes, they are mapped
down to a domain subspace where they can be approximated as linear combinations of molecular
and group parameters. The informative statistics of the attribute-latent property models (e.g. Eq.
9.13) are given in Table 9.13.
Table 9.13. Regression Coefficients for the Attribute-Latent Property Relationship
Informative
Statistics DT CS EF
??? -0.0676 6.9160 -7.5516
?? 0.1731 7.0561 -6.1735
?? 0.1371 -5.7299 -0.4081
For a good a molecular design it is desirable for the attributes to fall within the target ranges
given in Table 9.14. Applying Eq. 9.13 to Table 9.14 results in the target scores shown in Table
9.15.
Table 9.14. Target Attributes for the Molecular Design
Attribute
Targets DT CS EF
UL 0.1909 1.764 -1.544
LL -0.0169 1.729 -1.888
339
Table 9.15. Latent Variable Targets
Score
Targets T1 T2 T3
UL 0.75 1.00 0.50
LL 0.25 0.00 0.00
These scores are then recombined with the loadings in Table 9.11 using Eq. 9.43 to generate the
equivalent property descriptor bounds, corrected for log normal and centered behavior, as shown
in Table 9.16.
Table 9.16: Predicted Property Descriptor Targets
Property
Targets P1 P2 P3
UL 229.2 100.0 112.3
LL 229.2 52.2 46.2
In order to ensure that the domain is properly explored, the property responses at the six points
that comprise the feasibility region (e.g. Eq. 6.7) are evaluated for the standardization matrix.
The standardized latent variable targets are found by applying the standardization matrix to the
latent variable scores at each of the six points and selecting the maximum and minimum
resulting in Table 9.17.
Table 9.17. Attribute targets mapped to the descriptor sub-domain
Subspace
Targets Q1 Q2 Q3 S
UL 0.801 2.881 1.602 1.321
LL 0.539 0.00 0.00 0.312
340
The molecular groups present in the training set and identifiable by the characterization
were CH, CH2, OH, CHO, O, CH2-O, CHOH, CH2OH, CHCH2OH, CHCHO, ?-pyranose, Ocyc,
?-pyranose, and cellulose. The last two groups were only present in the micro-crystalline
cellulose excipient and, as such, were excluded from the short chain molecular design. In the
previous molecular design using these molecular groups with different target attributes (Solvason
et al. 2009), the selection of the ? stereoisomer of pyranose compounds as candidates was
preferred, while the microcrystalline cellulose (MCC), which relies on the ? stereoisomer was
notably outside the target range (Solvason et al., 2009). This was an unexpected result since
MCC is not only one of the most common excipients used in acetaminophen tablets, but also
contained the most information on long-range order when characterized. In order to better
understand this conclusion, a simple investigation of the meso-scale microstructure was
conducted by analyzing the influence of particle size (50-300 ?m) on the attribute properties.
There are three ways to illustrate the influence of particle size on the design problem.
The first is to illustrate an average IR/NIR characterization for each particle size. Although a
mathematically valid method for visualizing the particle size effect, a particle size ?IR/NIR
spectra? has no physical meaning. A more appropriate method is to observe the effect of particle
size on the individual chemical constituents in the model. Since increasing particle size results in
larger reflective-absorbances (and smaller reflectances), the primary effect was an increase in the
AUP values of the training set molecules and individual groups. A smaller, secondary effect due
to a disproportionally strong effect observed in the 4500-6000 cm-1 range for polyols (Storz and
Steffens, 2004) caused a slight alteration of the location of the training set and molecular group
clusters (Fig. 5.9). This influence suggests that a third method be employed, namely that
functional groups be parameterized for different particle sizes and then recombined to buld a list
341
of multiscale solutions simultaneously. Of the groups listed above, CHOH, CH2OH, ?-
pyranose, ?-pyranose, and cellulose exhibited changes in their property descriptors according to
the diameter of the particle. For this proof-of-concept case study the final particle size should be
in the 100-400 ?m domain. Groups parameterized at 100 ?m increments in this domain are
shown in Table 9.18.
Table 9.18: Groups for Molecular Design of Excipient
Group
g
Group
Type
Particle
Diameter
(?m)
P1 P2 P3
1 CH - 85.5 52 18
2 CH2 - 0.5102902 60 15
3 OH - 0.5225436 20 12
4? CHO? - 85.5 87 26.5
8 O - 90 24 15
9 CH2O - 90 28 15
10 CHOH 100 63.1 46.2 23.3
11 CHOH 200 64.4 47.6 23.7
12 CHOH 300 65.6 49.0 24.1
13 CHOH 400 66.8 50.5 24.5
14 CH2OH 100 78.9 106.2 26.3
15 CH2OH 200 80.1 107.6 26.7
16 CH2OH 300 81.3 109.0 27.1
17 CH2OH 400 82.6 110.5 27.5
18 CHCH2OH - 81 160 27
19 CHCHO - 135 60 30
342
20 Ocyc - 72 16 18
21 ?(pyranose) 100 67.2 78.9 31.6
22 ?(pyranose) 200 67.2 79 31.5
23 ?(pyranose) 300 67.3 79.1 31.4
24 ?(pyranose) 400 67.4 79.2 31.3
25 ?(pyranose) 100 109 141.8 31.8
26 ?(pyranose) 200 111.5 144.3 34.3
27 ?(pyranose) 300 114 146.8 36.8
28 ?(pyranose) 400 116.5 149.3 39.3
29 Cellulose 100 84.7 103.4 42.2
30 Cellulose 200 82.6 100.6 41.4
31 Cellulose 300 80.4 97.8 40.6
32 Cellulose 400 78.2 95.0 39.7
343
Figure 9.18. Design cluster diagram featuring training set particle sizes in ?m.
Following the design algorithm presented in Section 9.3, an excipient design was
conducted and the results are shown in Table 9.19. The designed structures are small diols, with
one ?-glucopyranose structure in the 200mm range. The ?? form of the glycosydic linkages are
found in the most common excipient, microcrystalline cellulose. However, as the molecule size
increases, aldehyde structures become more prevalent and the orientation preference in the
glycosydic linkage disappears. In the candidate list published by Solvason et al. (2009) using
slightly different constraints, the orientation preference is for the ?-glucopyranose over the ?-
360
250
520
200
50
90
90
50
90
180
90
700
250
150
300
200
200 90
- 0 . 5
- 0 . 4
- 0 . 3
- 0 . 2
- 0 . 1
0
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 2 1 . 3 1 . 4 1 . 5
C 1 z e r o C 2 z e r o
C 3 z e r o T a r g e t F e a si b i l i t y R e g i o n
P r o p e r t y L o a d i n g s M a n n i t o l
M C C H D M C
X y l i t o l X y l i t a b
A v g . P a r t . S i z e M a l t i t o l
P3
P1
C3
C2
C1
344
glucopyranose structure. Rather, the indication is that particle size is the primary driving force in
these structures as long as enough alcohol groups are present for sufficient intermolecular
hydrogen bonding. This is a reasonable conclusion since ejection force, crushing strength, and
disintegration time are primarily related to the strength of particle-particle interactions and only
secondarily related to the molecular interactions in pharmaceutical powders.
Table 9.19. Designed Molecules and Discrete Particle Sizes of 100, 200, 300, and 400 ?m.
Canidate Molecules Q1 Q2 Q3 Particle Size (?m)
CH-CH2OH 1.19 0.74 0.33 300
CH2OH-CH2OH 1.26 1.52 0.91 300
CH2OH-CH2OH 1.25 1.41 0.83 200
CH2OH-?(pyranose)-CH2OH 1.75 0.05 0.00 200
OH-CH2OH 1.19 0.68 0.29 200
CH2OH-CH2OH 1.24 1.31 0.77 100
OH-CH2OH 1.19 0.63 0.25 100
CH2OH-CH2OH 1.27 1.66 1.00 400
CH2OH-CHCHO-CH2OH 1.72 1.72 0.09 400
OH-CH2-O-CH2OH 1.20 0.80 0.37 400
9.5. Summary
It was proposed in this chapter that a cCAMD method be developed such that attribute and
property data are mapped into a property sub-domain, derived using decomposition techniques,
that contains a better description of the effects of molecular architecture on the product?s
properties and attributes. The constraints on this new domain were that it should have a linear
345
property-molecular architecture relationship and that it should be capable of being described by
combinatorial techniques in order to maximize the efficiency of the reverse problem formulation
(RPF) with property clustering approach. A new method called characterization based group
contribution method (cGCM) is proposed for this task and will result in molecular architecture
information at multiple length-scales being utilized in the design. The approach also allowed the
product designer the ability to customize the domain subspace to include only the molecular
architecture information important to a specific set of product attributes and captured by
characterization.
The combination of property clustering and principal component analysis offers many
insights and advantages for the deisgn of chemical products at multiple scales. In particular
CAMD problems are no longer hindered by a lack of structure information in the molecular
design. Rather, the uncertainty in predicting large molecular structures has now been removed
from the models and replaced solely with the experimenter?s ability to choose appropriate
training sets, for which many proven techniques exist. This represents a useful addition to the
existing CAMD methodology. Furthermore, the method is universal in nature and can be
extended to include many characterization techniques.
346
Chapter 10
Conclusions
In this dissertation, a multi-scale product and process design framework based on the
reverse problem formulation was developed for chemical products. The framework was
designed to utilize experimental data, parameters, and models. The reason behind choosing to
base the framework on experimental models comes from the nature of their intended end-use.
Since the effectiveness of a chemical product was most often determined by its consumer
attributes, and since the consumer attributes were most likely quantified using data from
consumer preference tests, then it followed that the framework should at least be partially based
on experimental data. This observation did not preclude using the more common physical and
chemical properties found in process design in the framework, rather, it only means that
experimental data needed to be integrated into the design in some fashion. Furthermore, while
the use of data placed constraints on the application range of the tools developed for multiscale
product design, it did not affect the framework developed and could be applied to systems
described by deterministic relationships if desired. The use of experimental data provided
insight into opportunities for improving the solution strategy for solving multiscale design
problems.
347
10.1. Achievements
The first opportunity was the recognition that through the use of transformation functions
and experimental design techniques, property and attribute models could be linearized. As a
result, the duality of linear programming could be applied to solve the design problem in the
lower dimensional property domain instead of the high dimensional component space.
Approaching the design problem in this manner significantly reduced the computational
complexity of the problem. Likewise, this opportunity was visualized in the property domain
design space using property clustering, which provided insight into the design while reducing the
computational complexity of the design problem. A new method called attribute based computer
aided mixture design (aCAMbD) with clusters was developed to take advantage of this
opportunity for data-driven chemical product design problems and applied to a polymer blending
problem. The method was built specifically to utilize data driven property models, regressor
based property clusters, and the reverse problem formulation.
Recognizing that combinations of multiple sources of scientific knowledge, information,
and data can be used in the design of chemical products, a second opportunity was identified. In
particular, it was observed that the effectiveness of most chemical products is determined by its
consumer attributes which did not have QSPR/QSAR relationships. Alternatively, the attribute
information could be mapped into a physical property domain with strong QSAR/QSPR
descriptions. As a result, a second new method called (aCAMD) with clusters was developed to
map consumer attribute data into a more conventional physical property domain where GCM
was used to find new chemical structures. The method was built to use data driven attribute-
property relationships, GCM based property models to design molecules, and property clusters in
348
a reverse problem formulation to reduce the computational complexity of the problem. The
method was applied in the design of environmentally benign refrigerants.
A third opportunity resulted from the recognition that the various sources of information
used in multiscale chemical product design come from different length scales, each with
particular molecular architectures that dominate the chemistry at that scale. Quantification of
these sources of information would be best achieved through the use of characterization
techniques which could be combined to handle complex molecular architecture descriptions.
Combining the different sets of characterization information would result in a large, highly
correlated data set that completely describes the molecular architecture of the chemical product.
Rather than solving the molecular design in this high dimensional data set, decomposition
techniques could be applied to the characterization data set to find a lower dimensional latent
variable structure. The relationship between the underlying latent variable structure and the
consumer attributes could then be estimated using chemometric techniques. In addition, a new
characterization based group contribution method (cGCM) could be developed to take advantage
of the additional molecular architecture information provided by characterization. As a result, a
third new method called characterization based computer aided molecular design (cCAMD) was
developed to map consumer attribute data to a latent variable domain where a newly developed
cGCM method was used to find candidate molecular architectures. In particular characterization
information from the molecular scale on short range order, such as group structure,
conformation, and stereoregularity, can be combined with information from the micro-scale on
long range order, such as the size, shape, and aspect ratio of particles within a solid matrix. The
method was built specifically to utilize IR/NIR spectroscopy data, PCA decomposition
techniques, MLR attribute-latent variable relationships, latent variable property clusters, and the
349
reverse problem formulation. The method results in a framework that decouples multiscale
design problems from their respective scales, solving them in a low dimensional,
computationally efficient manner. The method was applied to a constrained pharmaceutical
excipient design with only three wavelengths.
10.2. Challenges and Future Work
Immediate extensions of the work presented in this dissertation include the use of partial
least squares on to latent surfaces (PLS) to simultaneously develop the attribute-latent property
and property descriptor ? latent property relationships. This technique sacrifices some descriptor
predictive capability for improvements in the attribute-latent property predictive power. Case
studies incorporating the design of blend fractions with a predefined molecular architecture
under process constraints could be used to both highlight the PLS improvement, as well as
address the Case 3 class of problems discussed in Chapter 3. The same approach could also be
used to select and design molecular architectures and process parameters to address Case 4
classes of problems.
One of the constraints of the property clustering method using decomposition techniques
is that the developed characterization based group contribution method (cGCM) has been
constructed to utilize only first order groups. It can be shown that this method can be extended
to second groups which open the method to more common chermical engineering properties like
binary interaction parameters that are common in many equations of state. Use of the reverse
problem formulation and the developed property clustering methods, augmented for binary
interaction parameters, would help to solve flash calculations and separations of unknown
mixture fractions quickly. In particular, it is the author?s belief that this can be accomplished by
treating binary systems as systems of ternary mixtures where the third component is a binary
350
interaction parameter. The lever arms associated with the binary interaction parameter has been
shown to be a function of the normal lever arm between the two components. Hence, at worst an
iterative scheme could be developed to back calculate the lever arm at a design point.
Long term, the Euclidean vector suggests that linear mixing can extend to multiple
properties so long as the conversion between clusters and property axes are defined. In other
words, all cluster design problems should be capable of being described by Euclidean vectors.
This opens up the design problems to handle an increasing number of molecular architecture and
characterization techniques not explored in this dissertation. Likewise, since computational
chemistry techniques are capable of generating characterization parameters, it should be possible
to utilize the reverse problem formulation to redefine the manner in which computational
chemistry is performed. In particular, spectroscopy has been shown to follow time dependent
computational chemistry techniques that make use of the Hamiltonian functions. Methods to
express the eigenvalue functions of the Hamiltonian in algebraic terms could open up
computational chemistry techniques for use in chemical engineering product and process design.
351
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375
Appendix A1
Additional Chemical Product Design Techniques
This section contains other useful tools for the design of chemical products and processes,
including multicriteria decision making to aid in the stage-gate development process, ontology to
aid in data gathering and interpretation, and mathematical optimization to aid in efficient solution
derivation.
A1.1. Multi-Criteria Decision Making (MCDM)
One of simplest methods for converting data and information into workable products was
proposed by Dym and Little (2008). In this method, information is first obtained from the client,
who is either the customer or end-use consumer. The information is then transformed to a set of
goals or attributes which may cover overviews of form, function, costs, potential markets, and
safety issues. Next, these goals are weighted by importance and impact on the product. Those
goals deemed most important and carrying the highest impact are kept in the design. Products
that have these important attributes are then selected. When this process is systematically
applied, it is referred to as decision analysis (Kepner and Tregoe 1997) or multi-criteria decision
making (MCDM). Although the idea of using subjective decision analysis is far from perfect, it
has several advantages (Cussler and Moggridge 2001)
376
1. The selection of feasible products is typically a point of fierce management review
and will require the support of managers outside of the core product development
team. It can become an excellent tool for garnering support for the product.
2. The need to weight and score each idea make the consulting of experts and customers
inevitable while numerical scoring trends make it difficult for a single view to
dominate decisions.
3. The separate scoring of different criteria ensures that strengths and weaknesses or
each product are obvious.
An example MCDM process called Kepner-Tregoe Decision Analysis was used by
AstraZeneca to aid in the design of the synthesis route for the pyruvate dehydrogenase kinase
(PDK) inhibitor, AZD7545, as shown in Table A1.1. The first step of the analysis was to
develop a succinct objective, such as ?To identify routes for the long-term manufacture of the
required drug candidate, paying particular attention to critical factors (e.g., safety,
environmental, cost, intellectual property, etc.)? (Parker and Moseley 2008) Next, a list of
product criteria were generated and classified as either ?musts? or ?wants. ? Must criteria were
critical issues without which the product could not be made. Want criteria were comprised of all
other objectives. The wants are then weighted between 0 and 10, and, for this example, chemical
feasibility was given the highest value indicating it was the most important want criteria.
377
Table A1.1: An Example MCDM for the Route Identification of PDK Inhibitor, AZD7545 (Parker and Moseley 2008).
378
The next step of the process was to generate alternatives, or potential solutions to the
objectives. In the case of the example presented, different synthesis routes were identified by
expertise, scientific journal review, or computational methods, using an enzyme classification
system. Twenty-one types of routes were evaluated. It should be noted that if the product was
more complex (i.e. contained molecular architecture at multiple length scales), it would have had
a longer list of alternatives. The first two alternatives failed to meet the must criteria so they
were eliminated immediately. The 19 remaining possibilities were evaluated on their want
criteria, which was the most difficult to perform because of their highly subjective nature. In
order to minimize the subjective nature, it was important to decide on the scoring method prior to
determining which alternatives to evaluate.
As shown in Table A1.2 the alternatives were scored against 5 wants: accommodation,
chemical feasibility, costs of goods, environment, and number of stages. Additional want criteria
could have been thermodynamic and kinetic barriers, reaction yields, etc. A default score of 5
was given to each alternative for each want criteria, with subtractions made for those alternatives
with less than desirable characteristics. In some cases, it was necessary to identify adverse
consequences resulting from some of the alternatives, such as chemical feasibility and undesired
inter-dependencies.
379
Table A1.2: Scoring System for Wants in the Route Identification of PDK Inhibitor, AZD7545 (Parker and Moseley 2008)
380
Finally, the criteria were gathered for the decision analysis matrix and were calculated by
adding all of the want criteria scores together for each alternative. The alternatives with the
highest scores were chosen for further analysis, including experimental product prototyping and
algorithmic process development. The lowest scored alternatives are removed from contention
along with any criteria that failed the must criteria. In the PDK Inhibitor AZD7545 example,
laboratory scale prototyping and product synthesis was performed and the scores were adjusted
to reflect the additional source of information. In particular, route 3.3, which substituted a nitro-
group for the bromide in brombenzamide, was initially the second-best option identified.
However, no coupling reaction between thocyanate and 4-nitrobenzamide, which had a known
pathway, could be established, so it then failed the feasibility must criteria, and was removed
from consideration. The final, post-feasibility, pre-development list of options are presented in
Table A1.3.
381
Table A1.3: The final MCDM for the Route Identification of PDK Inhibitor, AZD7545 (Parker and Moseley 2008).
382
Obviously, using a subjective MDCM will have significant limitations. Agreement
cannot always be reached on how to score particular issues and not all perspectives are equally
correct. In an effort to make the MCDM process less subjective, Wu et al. (2009) proposed an
alternative representation in form of a process flow diagram (Fig. A1.1). Although this approach
helps to define how the information flows, it does not improve how the decisions are made,
which will ultimately limit the range of applicability of the MCDM methodology.
Figure A1.1: A Decision Analysis Flow Diagram for an MCDM (Wu et al. 2009)
383
A1.2. Ontology
Venkatasubramanian et al. (2006) proposes using ontology to design chemical products.
Ontology is one of several tools developed over the past 20 years to support chemical product
and process development and, in particular, it is used to explicitly describe information in terms
of conceptual domains and their inter-relationships (Gruber, 1993). As a specific form of
informatics, ontology describes classes and attributes on a case by case basis and can be used to
combine information sources from a variety of databases in multiple languages. The process
involves the development of both a syntax (i.e., structure) and the semantics (i.e., meaning) of
the system. The end result should be able to automate tasks such as acquisition, representation,
storage, manipulation, modification of, and reasoning with, data, information, and knowledge
(Venkatasubramanian 2009). Although this is a long term view of the roles of ontology, the
immediate benefit comes from the development of framework which enables optimization and
design to proceed.
It has already been shown how decision analysis (e.g. MCDM) can be used to create and
solve product design frameworks. Ontology represents a more rigorous mathematical approach
to creating and solving chemical product design problems, including the identification of the
various categories of information, followed by the construction of the information framework
using an evolutionary design process consisting of proposing, implementing, and refining
product classes, and then solving the design using automated decision analysis (Noy and
McGuinness 2001). For example, the pharmaceutical industry utilizes information from
molecular structure, biological interactions, solid dosages, batch manufacturing processes,
Federal Drug Admistration (FDA) controls, among many others. Terabytes of information are
gathered and combined from these sources of information and combined into a large chemical
384
database resulting in tens of thousands of pages of reports (Venkatasubramanian 2009). To
manage this onslaught of data Venkatasubramanian developed the Purdue Ontology for
Pharmaceutical Engineering (POPE) to facilitate the screening of information, improve the
decision-making, and create data linkages using intelligent systems engineering
(Venkatasubramanian et al. 2006). In Fig. A1.2, an example of a ontological framework is
presented using POPE. Knowledge is classified in three important categories: process
knowledge, mathematical knowledge, and process control knowledge.
Figure A1.2: Ontological Relationships for a Pharmaceutical Process (Venkatasubramanian
2009).
P r o c es s
K n o w l ed g e
P h a r m a c eu ti c a l
P r o c es s
P r o c es s
C o n tr o l
K n o w l ed g e
M a th em a ti c a l
K n o w l ed g e
M a n i p u l a t ed
V a r i a b l e
P ID
C o n t r o l l e r
C o n t r o l l e r
C o n t r o l l e d
V a r i a b l e
D ep en d en t
V a r i a b l es
P r o c es s
D i s t u r b a n c e
V a r i a b l e
M P C
C o n t r o l l er
E q u a t i o n S et
In d ep en d en t
V a r i a b l es
A s s u m p t i o n s
M a t h em a t i c a l
M o d el
M o d e l
P a r a m e t e r s
V a r i a b l es
V a l u e In p u t
U n i t
O p er a t i o n
E q u i p m en t
P o r t
S t r e a m
M a t er i a l
P r o p er t y
P h y s i c a l
S t a t e
P h a s e
S y s t e m
P a r a m et er
S u b s t a n c e
V a l u e
385
To perform a chemical product design, the relationship between mathematical knowledge and
process knowledge, specifically the components related to unit operations, material properties,
and physical constructs are mapped into an evolutionary decision framework. Candidate
solutions are then found by setting the desired values or important characteristics and executing
the automated decision analysis to find the chemical products and processes that meet those
conditions. Like all data driven models, the design is only as good as good as the database used
to develop it. However the documentation requirements for heavily regulated industries like the
pharmaceutical industry ensure enough information is captured to build a robust ontology
decision network for chemical product design.
A1.3. Mathematical Optimization
As discussed in Appendix A1.1, the simplest way to choose the final product is to select
one from a pre-screened set of options using MCDM. The MCDM method incorporates multiple
important criteria (e.g. attributes and physico-chemical properties) into the decision-making
process while providing a structured framework for the reduction of difficulty in complex
decision-making (Wang et al. 1997). In the first step, candidate molecules are generated using
combinatorial techniques and constrained using a set structure targets. Next, the potential
candidates are then uploaded into the rows or columns of the MCDM matrix. Pertinent physico-
chemical properties and/or attributes are loaded vice versa into the columns or rows of the
MCDM matrix along with any additional criteria of importance, such as technical, economic, or
environmental measures chosen using principles based on systematic analysis, consistency,
independency, measurability, and comparability (Kepner and Tregoe 1997). In some cases,
statistical tools such as predicted variance, min-max deviation, and correlation coefficients are
used to select the best criteria to use in the MCDM. After the product criteria and potential
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candidate solutions have been uploaded and normalized, a score is calculated for each candidate
by multiplying the criteria?s weighted importance by its normalized criteria value for each
candidate and then summing across all criteria. The candidate with the highest score is
considered ?optimum.? More details concerning this method are available in Appendix A1.1.
Alternatively, mathematical optimization is a completely automated method using only
the constraints provided in the formulation with no subjective weighting of solutions. In
comparison to MCDM that finds the best from a set of potential candidates, mathematical
optimization finds the best from all feasible candidates. Thus the mathematical optimization is a
more rigorous solution strategy, but comes at the cost of computational efficiency.
In most cases, the mathematical optimization involves finding the best molecular
structure, from atomic backbone to long-range order morphology, given chemical structural
constraints and objectives. The computational efficiency of a program is greatly enhanced when
all of those constraints and objectives in the formulation are linear and continuous; a condition
that is commonly referred to as Linear Programming (LP). Linear programs are solved using the
simplex algorithm which finds the global optimum by searching the vertices of a solution space
that is defined by linear target constraints (Biegler and Grossmann 1997). When binary variables
are introduced, such as in Eqs. 4.2, 4.6, and 4.7, the problem becomes a Mixed Integer Linear
Programming (MILP) problem. The simplest way to solve these problems is to use a direct
brute-force method that enumerates each potential solution of binary variables, selecting the best
as the global optimum. Unfortunately, this approach is computationally inefficient. A more
elegant method is the branch-and-bound technique which works by relaxing and reformulating
the problem so that no binary variables exist, as shown in Fig. A1.3. In this example, an
objective function (Fobj=Z) is maximized subject to a set of constraints. Under relaxation, the
387
example objective value is found to be 41.25. Next, any y variables (e.g. attributes or physico-
chemical properties) that contain non-integer values are then set, one at a time, to integer values,
creating a series of sub-problems known as nodes (Borchers and Mitchell 1994). Each sub-
problem LP is solved and the nodes that contain a higher objective function value are retained
while those containing the lower values are cut. The process continues until there is no further
improvement in the objective function (Biegler and Grossmann 1997). The performance of the
branch-and-bound algorithm depends heavily on the choice of the non-integer variable to be used
for branching and the selection of the node to be branched. As a rule, the noninteger variable to
be chosen should be the one with the largest fraction (Gurdal et al. 1999), noted as x2 in the
example reformulation. The determination of which node to branch should be the one with the
smallest objective value, which should generate a feasible design with a tighter upper bound.
Nodes with objective functions higher than the relaxed value (Node 4) are deemed infeasible (i.e.
beyond the relaxed variables solution domain). A branch-and-bound algorithm workflow, shown
in Fig. A1.4, was first proposed by (Land and Doig 1960) and later refined by Borchers (1992).
Figure A1.3: A Branch-and-Bound Example with an Optimum Solution Found Using Integer
Programming within the Boundary Domain (Borchers and Mitchell 1994).
C u t
Al l ( c o n ti n u o u s es t. )
L B , U B = ( 2 .2 5 , 3 .7 5 )
Z = 4 1 .2 5
N o d e 1 : x 2 < 3
( 3 , 3 )
Z =3 9
N o d e 2 : x 2 > 4
( 1 .8 , 4 )
Z =4 1
N o d e 3 : x 1 < 1
( 1 .8 , 4 )
Z =4 0 .5 5
N o d e 4 : x 1 > 2
In f ea s i b l e
N o d e 5 : x 2 < 4
( 1 , 4 )
Z =3 7
N o d e 6 : x 2 > 5
( 1 .8 , 4 )
Z =4 0
C u t C u t
Max Z = 5x
1
+ 8x
2
s.t . x
1
+ x
2
? 6
5x
1
+ 9x
2
? 45
x
1
, x
2
? 0 in teg er
388
Figure A1.4: A Branch-and-Bound Algorithm Workflow (Borchers 1992).
The branch-and-bound algorithm has also been adapted for Non-Linear Programming
problems (NLPs). In NLPs, the objective functions and constraints may be linear and/or
nonlinear and such problems can be solved in a variety of ways. The easiest is to reparameterize
the problem as a sequence of linear programs (Schmit and Farshi 1973), commonly referred to as
Sequential Linear Programming (SLPs), or reduced gradient methods. Using a Taylor series
expansion is the most common reformulation method in this construct, and it is solved by
standard methods such as simplex optimization or interior point methods. It is most efficient for
problems with a large number of linear constraints and well described model of nonlinear
solution topography (e.g. a response surface with many valleys). However, in many cases, a
Taylor series expansion does not match the curvature of the reformulation well enough and is,
therefore, inappropriate to use (Gockenbach and Symes 2003). In this case sequential quadratic
In i ti a l i z e B r a n c h
a n d B o u n d
A l g o r i th m
P i c k a N ew
Su b p r o b l em
So l v e th e
Su b p r o b l em
O u tp u t
NO Y E S
In f ea s i b l e
o r LB >
U B?
T r ee
E m p ty?
Fr a c ti o n a l
V a r i a b l e?
In teg er
So l u ti o n ?
D el ete
Su b p r o b l em
B r a n c h
U B =m i n ( U B , F
o b j
)
Y E S
Y E SNO
389
programming (SQP) methods, which reformulate the problem into an alternative solution space
using a Lagrangian of the objective function, are utilized. SQP is one of the most popular
algorithms for nonlinear continuous optimization because of its robustness. Within the
Lagrangian is the second partial derivative matrix, or Hessian function, which is used to develop
a series of quadratic problems that result in fewer calculations and faster solution of the problem,
especially in comparison with the reduced gradient method (Biegler and Grossman 1997,
Sammons 2009). Although both of these tools are powerful methods for solving NLPs, neither
one can guarantee that the solution they find is indeed the global solution unless the problem is
completely convex and differentiable (Biegler and Grossmann 1997; Sammons 2009).
Finally, the most complicated optimization problem is the Mixed Integer Non-Linear
Programming (MINLP) problem which utilizes both integer and non-linear programming
techniques. This approach makes use of the branch-and-bound algorithm with special controls
on the reparameterization of both the objective and constraints. The primary methods used to
solve MINLPs are Outer-Approximation (OA) and Generalized Benders Decomposition (GBD)
(Biegler and Grossman 1997, Sammons 2009). The solution strategy of both of these approaches
is presented in Fig. A1.5.
390
I n i t i a l i z a t i o n o f D e s i g n V a r i a b l e s
N L P S u b p r o b l e m
U p p e r B o u n d > L o w e r B o u n d
S t a r t
M I L P M a s t e r
P r o b l e m
U p p e r B o u n d
L o w e r B o u n d
S t o p
U p p e r B o u n d < = L o w e r B o u n d
Figure A1.5: Flowsheet of MINLP Optimization (Diwekar 2003; Sammons 2009).
After initialization, the OA method uses an alternating sequence of NLP and MILP sub-problems
with the MILP providing fixed binary variables to the NLP, which then determines new bounds
for the MILP. The process continues until no lower bound is found below the current best upper
bound. The GBD is very similar, only using the largest Lagrangian approximation obtained from
the NLP sub-problem in the MILP sub-problem (Biegler and Grossmann 1997; Sammons 2009).
However, since both methods use a NLP in the solution to the MINLP, a global optimum
solution cannot always be guaranteed. Some improvement has been made using alpha branch
and bound techniques (?-BB) (Sahinidis 2004), which heavily constrain the problems.
Nevertheless, as these techniques are applied to increasingly larger problems that begin to mimic
real world problems, the computational expense and ability of the methods come into question.
In response, the trend has been to combine these deterministic methods with tools that serve to
smooth the response surfaces of the solution domain in an effort to prevent the trapping of the
391
global optimization techniques in local minima solutions. These hybrid approaches most often
make use of stochastic tools to loosely guide the more exact deterministic techniques into an area
of solution.
Most hybrid methods incorporate one of two types of stochastic search techniques,
Simulated Annealing (SA) and Genetic Programming (GP). These tools are especially adept at
finding solutions where a global minimum is sought, like for programs involving nonlinear
functions which may have multiple local minima. Both of these algorithms mimic naturally
occurring phenomena and their implementation utilizes statistical tools associated with random
selection and probabilistic decisions (Gurdal et al. 1999). Simulated annealing is a technique
using that uses a probabilistic step function to perturb the solution in an effort to prevent the
trapping of the algorithm in a local minimum. The perturbation is usually constrained such that
the move is only adopted if the new solution has a better value for the objective function.
Genetic algorithms are techniques derived from biology that approximate the
evolutionary principles of Darwin, mimicking the mechanism of natural selection by applying
reproduction, crossover, mutation, and permutation to iterative solutions found in programming.
These methods do not work in a sequential manner, but rather jump from point to point in the
solution space, which keeps many solution points that may have the potential of being close to
minima in the pool during the search process, rather than risk converging on a single point that
may not be the global optimum (Gurdal et al. 1999). As shown in Fig. A1.6, in the first step of a
genetic algorithm, an initial population is generated using a probabilistic function meant to offer
good cross section of the design space to be investigated. The solutions in the initial population
are also measured by their fitness, which is the value of the objective function in unconstrained
problems. For constrained problems, such as in chemical product design, the fitness measure
392
must also take into account the proximity of the solution to the constraint margins. Based on the
fitness measurement of the solutions in the population, the genetic algorithm can elect to apply a
filter to make changes to the population for the next generation. For instance, if the reproduction
filter is chosen, a probability function is applied where solutions with improved values of the
objective are chosen more often than those whose objective functions are poor. A crossover
filter is the swapping of solutions of two parent populations to create two child populations in the
next generation and mutation involves randomly changing some of the solutions to be swapped.
A permutation filter, also referred to as immigration, is the adding of new solutions into the
population that have little similarity to the existing solutions in the population. Once these filters
are applied, a new generation of solutions is measured by their population fitness, and the
process is repeated. Termination of the genetic algorithm occurs when a set number of
generations have been evaluated, or an approximate solution in which no noticeable
improvement occurs in subsequent generations has been discovered (Diwekar 2003; Sammons
2009). Once these conditions are met, a constrained branch and bound deterministic
optimization can be conducted.
393
Figure A1.6: Genetic Algorithm Work Flow (Diwekar 2003; Sammons 2009).
Although stochastic search techniques can help center deterministic branch-and-bound
optimization algorithms near the global optimum, they are computationally expensive, requiring
thousands of analyses. The final computational cost is primarily the result of the size of the
design and the way constraints hinder movement in the design space (Gurdal et al. 1999). They
are also dependent on the requirements regarding reliability. It is common that each execution of
the algorithm can result in a different solution. How close the solutions consistently get to the
global optimum is the reliability measure. The goal is to have the stochastic search technique
?practically equal? to the global optimum so that the optimum found by the deterministic branch-
and-bound technique is the global optimum (Le Riche and Haftka 1995; Gurdal et al. 1999).
M o d e l
F i t t e r S o l u t i o n s
S t a r t
W a s t e
E v a l u a t e f i t n e s s o f t h e p o p u l a t i o n
( o b j e c t i v e )
S t o p
I n i t i a l
G e n e t i c
P o o l
U n f i t S o l u t i o n s
O p t i m a l ?
Y e s
N o
R e p r o d u c t i o n
F i t t e r
S o l u t i o n s
C r o s s o v e r
& M u t a t i o n
N e w
S o l u t i o n s
N e w
G e n e t i c
P o o l
R a n d o m
S o l u t i o n s
I m m i g r a t i o n
394
Appendix A2
Additional Characterization Techniques
This section contains summaries of other characterization techniques that may be used in
chemical product design to generate the P property structure descriptor data. Many useful
techniques exist for determining the molecular architecture of chemical product. Mass
spectrometry provides information on the molecular formula and size of the chemicals in the
product. Chromatography techniques provide more information on the polarity and size of the
specific components of the product. Spectroscopy such as IR, UV-Vis, and NMR provide
specific information on the presence of functional groups, information on the orbital
configurations of the electrons, and details of the carbon-hydrogen structure of the chemical
product. Other characterization techniques provide information at larger length scales.
Scattering and diffraction techniques provide information on particle size, morphology, and
crystal packing. For polymeric structures, thermomechanical techniques like torsional braid
analysis, dynamic mechanical analysis, thermogravimetric analysis, calorimetry, and differential
thermal analysis are used to investigate phase transitions and reaction conditions in polymer
chemistry. Viscoelastic properties are investigated using a variety of rheometric tools.
A2.1. Ultraviolet - Visible Spectroscopy
Ultraviolet-visible (UV-Vis) spectroscopy is only applicable to conjugated organic and
metal ion systems and, hence, is less common than other spectroscopy methods (McMurry
395
1995). It measures the amount of energy needed to promote an electron from one orbital to
another in a system of connected p-orbitals. The amount of energy needed can be directly
related to the nature of the conjugated system. Similar to IR spectroscopic systems, it uses the
Beer-Lambert law to relate the absorbance of light to the presence of specific conjugated systems
within the structural groups of the chemical product. Although it is useful in distinguishing
between enantiomers, its application is beyond the scope of this dissertation.
A2.2. Nuclear Magnetic Resonance Spectroscopy
Both vibrational and UV-vis spectroscopy provide information about functional group
structure of the molecular architecture of chemical products. An alternative spectroscopic
technique called Nuclear Magnetic Resonance (NMR) can also provide this information but is
also capable of characterizing materials in dynamical states and slightly larger length scales.
This property makes NMR extremely useful for studying structure-property relationships of
biological systems. In general, each NMR signature depends on the magnetic environment of the
NMR active nuclei, providing unique information about polymers on an atomic-length scale
(Cheng and English 2003). The signatures of the NMR spectra can be assigned to specific atoms
along the polymer backbone and side chains (Bovey 1982; Bovey 1988). The technique can be
used to monitor the extent of reaction, check the purity of polymers, identify unknown materials,
and to study polymer microstructure, dynamics, and interactions (Cheng and English 2003). The
spectra may also contain information about the polymerization mechanism, side reactions,
compositional heterogeneity, and morphology.
NMR is an absorption spectroscopy involving the absorption of radio frequency electro-
magnetic waves (Atkins 1998). Essentially, an atom is placed in an electromagnetic (EM) field
and the change in the spin state of the nucleus of the atom is measured. The nuclei must have an
396
odd number of protons and neutrons to exhibit spin. 1H, 13C, 19F, 15N, and 29Si are of the most
interest. The signal strength of 1H (or proton NMR) is approximately 50 times larger than 13C
due to the relative abundance of 1H and is therefore the preferred absorption spectra. From
quantum mechanics, when the nuclei are placed in an EM field, they adopt specific orientations
either ?parallel to? or ?antiparallel to? (Mirau 1993). The two orientations do not have the same
energy, with the parallel orientation typically having a slightly lower energy making it a slightly
favored orientation (McMurry 1995). When the nuclei are subjected to EM radiation at the right
frequency or resonance, the lower-energy state ?spin-flips? to the higher energy state. The
frequency necessary for resonance depends on both the strength of the magnetic field and the
identity of the nuclei. Resonance frequencies in terms of empirical quantities are called chemical
shifts, which are related to the difference between the resonance frequency of the nucleus and
that of a standard reference, thereby ensuring that the chemical shift is independent of the
frequency of the spectrometer (McMurry 1995). The standard typically used is tetramethylsilane
(TMS). Likewise, the NMR absorption peak for a given nucleus is also directly proportional to
the number of these atoms in the sample. Much like vibrational spectroscopy, the chemical
environments in which these nuclei exist are often the same, leading to a classification scheme
based on group theory. Developing the ?NMR groups? would be a worthwhile task, although it is
beyond the scope of this dissertation.
NMR can also be extended beyond group theory to include polymeric information. Two
major types of polymer NMR exist: solution and solid. Solution NMR evaluates the primary
chemical structure of the macromolecular chain. Solid NMR, though more difficult to achieve a
signal response, provides information on the primary structure, secondary structure (e.g. 2D
ordering such as hydrogen bonding between chains), and the tertiary structure such as globular
397
folding in proteins or the three dimensional crystal structure. Solution NMR is the preferred
method for naturally occurring polymers and most thermoplastic synthetic polymers, but is
limited to solutions, gels, dispersions, melts, etc. Solid NMR is required for most thermoset
synthetic polymers, even for primary structure information. Other modifications to the NMR
experiment include frequency domain (HR-NMR), time domain (LR-NMR), liquid
chromatography (HPLC-NMR & LC-NMR), size exclusion (SEC-NMR), supercritical (SFC-
NMR), and electrophoresis NMR (Cheng and English 2003).
A2.3. Mass Spectrometry
Mass spectrometry is an analytical technique that elucidates the molecular structure of a
molecule by measuring the mass-to-charge ratio for an individual molecule. Samples are first
ionized and transported by magnetic or electric fields to an analyzer that sorts the ions according
to their ratio.
BvEaQ
m ??????????? (A2.1)
Where m is the mass of the ion, Q is the ion charge, a is the acceleration, E is the electric field, v
x B is the vector cross product of the ion velocity and the magnetic field. The individual ratios of
the atomic and molecular ions in the sample can provide an excellent estimation of molecular
formula. To obtain more specific information of the molecular architecture, mass spectrometry
is most often used in conjunction with gas, liquid, or gel permeation chromatography.
A2.4. Chromatography (Gas, Liquid, or Gel Permeation)
Chromatography is a characterization technique that relies on the differential affinity of a
substance for a specific medium in order to indentify and quantify systems with complex
398
molecular architecture. The technique works by first introducing samples into a solvent, which
carries them through column packed with materials specific to the type of separation desired.
Separation types include observing the absorption affinities based on polarity (normal and
reversed phase), size (gel permeation), charged ion site binding (ion-exchange), and molecular
forces such as van der Waals, etc. (bioaffinity). The species are then quantified using a variety
of detection mechanisms, one of which is mass spectrometry. Comparing the resulting
chromatograph from a mass spectrometer with known samples in the same material column can
elucidate differences in the molecular architecture. Other techniques, including light scattering
could also be used.
A2.5. Scattering and Diffraction Techniques (Light, Neutron, X-ray)
Scattering is the process where some forms of radiated particles or waves are forced to
deviate from a straight trajectory by one or more localized non-uniformities. The amount of
scattering is proportional to the particle size or packing arrangement of the chemical product. In
general, scattering implies interaction of waves with spatially uncoordinated (unordered) atoms.
For small particles with wavelengths much smaller than the incident radiation, elastic light-
scattering techniques like Rayleigh scattering can be used to measure both molecular weight and
particle size. In this technique, the incident light intensity can be parameterized as a function of
the scattering angle of the reflected light. The scattering angle ? is a measure of the bending of
the radiation wavelength around a particle, which can be very useful for size characterization.
62
2
24
2
2
22122c o s1 ?????????????? ?????????? dnnRII o ???
(A2.2)
399
Where R is the distance to the particle, ? is the scattering angle, n is the refractive index of the
particle, and d is the diameter of the particle. Larger particles with wavelengths on the same
order as the incident radiation can be described by Mie scattering. Particles with size much larger
than the incident radiation wavelength are described by Geometric scattering.
Some of the most powerful scattering techniques use x-rays as the incident radiation. X-
ray scattering analyzes the oscillations of electrons and nuclei by measuring the elastic scattering
angles of incident X-rays. It is useful in obtaining information about the size, shape, and
surface-to-volume ratio of macromolecules and lamallae in the nanometer to micrometer range.
Types of x-ray scattering include small angle x-ray scattering (SAXS), wide-angle x-ray
scattering (WAXS), and x-ray reflectivity. As the names of the techniques suggests, SAXS
determine molecular architecture on the nano to micro length scales, which results in a scattering
angle near 0 degrees. WAXS evaluates angles larger than 5 degrees, determining the molecular
architecture at the molecular and nano-scales. Unlike many other characterization tools,
scattering techniques can be used to study structural features present in materials at length scales
that vary from angstroms (10-10 m) to microns (10-6 m) (Tolle et al. 2009).
Similar to scattering techniques, diffraction can also be used to determine molecular
architecture from incident x-ray bombardment. Diffraction is in fact a combination of scattering
and interference. The interference results from systems with ordered or packed periodic
molecular architecture. Hence, diffraction is almost universally used to study crystalline solids
whereas scattering can be used on liquids and amorphous solids. Diffraction characterization
measurement techniques measures both the part of the wave that strikes an object and how the
wave that passes through the object begins to occupy the space vacated by the blocked wave.
Types of x-ray diffraction techniques includes single-crystal x-ray diffraction, which determines
400
the complete structure of large crystals, powder diffraction (XRD) which determines the grain
size and preferred orientation, and thin film diffraction, used to characterize the crystallographic
structure and preferred orientation of substrate anchored films. If the molecular architecture of
the product is that of single crystals then single crystal x-ray diffraction is ideal for determining
the crystal structure (Stephenson et al. 2001). More common is that the molecular architecture is
in powder form, with many crystals, which calls for XRD methods to determine its structure.
For XRD, the wavelengths of the diffracted beams ? are related to interplanar spacings d in the
crystalline powder according to a mathematical relation called Bragg?s Law (Connolly 2007):
?? sin2dn ? (A2.3)
Where n is an integer and ? is the diffraction angle. The diffraction angle, usually reported in 2?
form, explains many elements of crystalline structure. X-rays can either scatter uniformly from
the atomic centers, which indicates atomic structure, or non-uniformly from the packed
arrangement of atoms, indicating a specific type of crystalline structure (Connolly 2009). Since
a large fraction of materials scatter isotropically (Bragg Diffraction), the scattering intensity can
be reduced to a single one-dimensional specta q as a function of the scattering vector (Tolle et al.
2009),
? ??sin4?q
(A2.4)
Areas of interaction lead to varying intensity responses that result in diffraction patterns, which
are then interpreted in terms of molecular architecture (Atkins 1998). XRD probes the orderly
arrangement of molecules in a the crystal lattice of solids and is more dependent on long range
order effects and to lesser degree, the determination of the degree of crystallinity (Stephenson et
401
al. 2001). Two primary techniques exist, one using the individual peaks, the other using the
whole pattern intensity.
In some cases, inelastic scattering occurs. Forms of inelastic scattering include Raman
scattering and, in some cases, some types of x-ray scattering and diffraction. Inelastic scattering
means that the kinetic energy of the incident radiation is not conserved, and changes in this
energy are dependent on the molecular architecture. Raman scattering measures this as a shift in
the wavelength frequency of the photon and is discussed in more detail in a previous section.
When x-rays are scattered inelastically and lose energy, they are exhibiting Compton scattering.
More information on this technique can be found in Klein (2002) and Atkins (1998).
A2.6. Other Characterization Techniques
Beyond spectroscopy related characterization are the thermo-physical tools for
characterization. These are loosely based on the concept of calorimetry which measures the
temperature change of a sample against a control under induced heat transfer, chemical reaction,
or applied physical stress. A standard solution calorimeter describes heats of reaction, mixing,
solution, and dilution. When combined with a differential scanner that increases the temperature
at a set rate, the calorimeter can provide additional information on the phase transitions of
polymers, including the glass transition temperature Tg, crystallization temperature Tc, and the
melting point temperature Tm, by measuring the changes in heat flow into the system. The glass
transition temperature is the point at which free rotation of bonds ceases and the polymer
becomes frozen in an entangled, disordered state (Rodriguez et al. 2003). Above this
temperature segmental motion begins to occur. The crystallization temperature is the point at
which the polymer frees itself enough to begin realigning itself in a lower energy crystalline
state. The melting temperature is the highest point in which a crystal lattice is stable.
402
Temperatures above Tm result in endothermic melting. Similar to the differential scanning
calorimeter (DSC) approach, differential thermal analysis (DTA) also provides information on
the long-range order of thermo-physical properties of chemical products. As opposed to DSC, it
works by administering a steady heat transfer rate and measuring a temperature difference. Both
of these techniques are useful for long range order structural differences when the structural
similarity renders diffraction techniques difficult to interpret (Atkins 1998; Rodriguez et al.
2003).
Another useful thermo-physical technique is dynamic mechanical analysis (DMA). This
technique is used to measure the mechanical properties of a wide range of materials, specifically
the viscoelastic properties and loss moduli of polymers as a function of temperature and
frequency (time). The technique works by applying sinusoidal stress (or strain) and measuring
the reciprocal strain (or stress). Likewise, the temperature of the sample is often varied, which
allows for the investigation of the phase transition temperatures. Types of viscoelastic properties
that can be described are the peak tensile stress ?, dynamic (Young?s) modulus E, often reported
as a combination of elastic storage modulus E? and the dissipated loss modulus E??, and dynamic
shear modulus G, reported as storage modulus G? and loss modulus G??. These mechanical
properties are extremely important in the physical handling of the chemical product during
manufacturing, as well as when the chemical product is used in a structural configuration.
Torsional braid analysis is another method of evaluating the thermomechanical behavior
of the chemical products that cannot support their own weight. In this technique, the chemical
product is placed into a glass braid, which is then oscillated under increasing temperatures.
Changes in the oscillation behavior can be directly related to the viscoelastic properties of the
sample. In a similar technique, Thermogravimetric analysis (TGA) measures changes in
403
molecular weight of chemical products as a function of measured temperature changes, looking
specifically for the presence of adsorbed moisture, solvents, or other entrapped contaminants.
Thus far the majority of these viscoelastic techniques discussed are looking for long-
range order structural changes of chemical products based on temperature changes. In contrast,
rheometry uses rheological properties like viscosity to estimate thermomechanical behavior such
as yield stress, modulus, creep, and recovery. Types of rheometric tools include capillary and
rotational geometries.
Finally, microscopy is one of the most common and simple techniques used in
characterization. Particularly useful are optical microscopy, scanning electron microscopy, and
transmission electron microscopy (TEM). Optical spectroscopy uses light to illuminates
structures that cannot be seen with the naked eye. It is particularly adept at determining the
phase of single crystalline structures. SEM works by scanning a fine beam of high energy
electrons over the surface of a sample which interacts to produce signal images of the surface?s
topography and composition. It typically has micron scale accuracy, making it ideal for
determining polymer morphology and particle size. TEM is similar to SEM, but works on the
smaller nanometer scale structures.
404
Appendix A3
Additional Decomposition Tools
This section contains additional decomposition tools that can be used to develop the
latent variable structure of the chemical product.
A3.1. Network Component Analysis
Conventional approaches, such as PCA, typically seek a loadings matrix L such that the
resulting scores matrix T satisfies orthogonality or independence criteria (Liao et al. 2003).
When dealing with data generated from structured networks where some information pathways
are known, and thus constrained, these decomposition techniques present two drawbacks (Liao et
al. 2003; Tolle et al. 2009). First, the implicit statistical assumptions on the data structure, i.e.
least squares regression, preclude the inclusion of physical and chemical systems understanding.
Second, the reconstructed connectivity structure is unlikely to be consistent with the known
underlying phenomenological structure. Therefore, a decomposition method is sought that makes
no assumptions on the statistical properties of the system, but rather allows proper handling of
prior knowledge on the structure characterizing a given system (Liao et al. 2003).
Tolle et al. (2009) proposes that Network Component Analysis (NCA) provides the best
opportunity for integrating this a priori knowledge. For PCA, the solution is inherently
nonunique, meaning that an infinite number of scores and loadings structures may be derived and
the best chosen through least squares analysis. NCA constrains the loading structure to known
405
phenomenological bipartite topology, to which convex optimization can be applied (Tolle et al.
2009). This effectively places zeros in the loadings matrix where no established relationships
exist, leaving non-zeros where they do. The problem can then be reformulated as an MINLP
where the NLP is solved using the IPOPT algorithm for each combination of integers (Wachter
and Biegler 2006; Tolle et al. 2009). When applied to process-structure-property relationships
found in chemical products, NCA gives the data structure shown in Fig. A3.1. For example,
Tolle et al. (2009) used network component analysis (NCA) to decompose wide angle x-ray
spectroscopy (WAXS) data into a series of network connections of material structural units that
could quantitatively describe the complex interactions occurring at this scale. The key feature of
this approach is that unique, physically meaningful solutions can be identified with little or no a
priori information regarding the system?s characteristics as long as accurate and reliable
characterization data is available (Tolle et al. 2009).
406
Figure A3.1: A NCA illustration of process-structure-property relationships for high
performance alloy steel (Olson 1997; Tolle et al. 2009).
407
Appendix A4
Additional Physical-Chemical Property Models
This section contains alternative physico-chemical property models using either ab initio
simulation techniques (e.g. quantum mechanics (QM), molecular mechanics (MM), molecular
dynamics (MD), Brownian dynamics (BD), dissipative particle dynamics (DPD), monte carlo
(MC) stochastic techniques, and topological indices, etc.) and/or less accurate but more efficient
topological indices (T.I.) combined with quantitative structure property (QSPR) models.
A4.1. Simulation Techniques
Recent years have witnessed an increase in the number of people using computational
chemistry to estimate chemical and physical properties of systems (Young 2001). The advent of
computational chemistry software has brought these powerful techniques to the forefront of
chemistry and chemical engineering. Venkatasubramanian (2009) notes that the use of these
techniques represents the coming of a ?data deluge? in the chemical engineering field. The data
is generated by either tracking the electrons (e.g. QM), molecules (e.g. MM, MD), or molecular
clusters (e.g. BD, DPD, etc.) of the system. The development of chemical architectures at
different scales will require a fundamental understanding of the hierarchical structure and
behavior of these systems and will result in multiscale modeling and simulation strategies that
provide seamless nesting and coupling (Zeng et al. 2008).
408
In most cases, traditional multi-scale design problems begin with ab initio, or ?from the
beginning? calculations. Ab initio calculations are computational chemistry tools that are
derived directly from theoretical principles and work by tracking the location probabilities of the
electrons in a molecular structure. Ab intio methods are based on a quantum mechanical (QM)
description of electrons in an eigen equation, first proposed by Schrodinger and Heisenberg
(Atkins 1998):
??? EH ?? (A4.1)
where ? is the Hamiltonian operator, ? is a Schrodinger wave function of the electron and
nuclear positions, and ? is the energy. The Born interpretation states that the square of the wave
function corresponds to a probability function describing its location (Atkins 1998). Dynamical
information is provided by the Hamiltonian operator, which describes the electronic motion of
the particles, both nuclei and electrons, in the system (Young 2001).
? ?? ????? p a r tic le sji ij jip a r tic le si ii r qqmH 2? 2 (A4.2)
where 2i? is the Laplacian operator on particle i, mi is the mass of particle i, and qi is the charge of
particle i, and rij is the distance between particles. The first term describes the kinetic energy of
the particle and the second term describes the Coulombic attraction and repulsion of particles
(Young 2001). The calculation of chemical and physical properties from this derivation is
achieved by first taking the expectation of the Hamiltonian operator and solving for the potential
energy of the system (Iachello and Levine 1995; Young 2001).
? ??? HE ?* (A4.3)
409
Other observable, chemical and physical properties can either be substituted, or calculated from
the potential energy surface using the Hellmann-Feynman theorem (Iachello and Levine 1995)
which relates the derivative of property specific energy to the specific property Hamiltonian
(Young 2001).
PHdPdE ???
? (A4.4)
It should be noted that Eq. A4.4 is the time-independent, non-relativistic version of the
Hamiltonian for the Schrodinger equation. Additional terms appear in the Hamiltonian when
either time and position relativity or electromagnetic interactions occur (Young 2001). For
instance, a more computationally friendly form of the Hamiltonian can be found using the Born-
Oppenheimer approximation. This approximation adjusts the Hamiltonian to treat the heavier,
slower moving nuclei as stationary in reference to the electrons moving around it (Atkins 1998):?
? ?? ?? ?????? e le c tr o n sji ijn u c le ii e le c tr o n sj ij ie le c tr o n si i rrZH 12? 2 (A4.5)
where the first term is the kinetic energy of the electrons only, the second term is the attraction of
the electrons to the nuclei, and the third term is the repulsion between electrons. Any motion of
nuclei can be described by considering this entire formulation as a potential energy surface on
which the nuclei can move (Young 2001).
Another common approximation is the Hartree-Fock (HF) which is the central
approximation behind self-consistent central field theory (Atkins 1998). It addresses the difficult
computation of electron-electron interactions by integrating the repulsion term to an average
effect. The overwhelming benefit of this method is that it breaks the many-electron Schrodinger
equation into many, simpler one-electron equations, which are then solved to yield individual
410
wave functions called orbitals (Young 2001). These individual orbitals represent the well-known
electronic shell structure in chemical periodicity (Atkins 1998). The mathematical functions
used to approximate the electronic wave function are called basis sets. The most common use a
linear combination of Gaussian-Type Orbitals (GTO), represented as exp(-ar2) (Young 2001).
One of the limitations of HF calculations is that they do not include electron correlation,
which means this approximation tends to overestimate surface energy. Since this estimate is
always equal to or greater than the exact energy, the HF can serve as a basis on which
improvements in the mathematical functions of the approximations can be measured. A detailed
account of other approximations can be found in various computational chemistry texts. In
general, the relative accuracy is as follows (Young 2001):
HF << MP2 < CISD ~ MP4 ~ CCSD < CCSD(T) < CCSDT < Full CI
where MP2 and MP4 are the HF corrected using second and fourth order Moller-Plesset
perturbation theory; CI and CISD are the corrections using the complete configuration
interaction and the configuration interaction with single and double-excitation only; and CCSD,
CCSD(T), and CCSD are the corrections using the coupled clusters at various orders of
expansion. It should be noted that although these and other mathematical approximations can be
used to simplify the computation, no experimental data or empirical models are used.
Approximations are more prevalent in larger, more complex molecules, but are necessary
because even the simple Hartree-Fock (HF) approximation scales to the N4 basis function sets
(Young 2001). Other approximations scale even worse, which limits the capability of this
method to small to medium organic molecules.
A slight improvement in the molecular size modeling capabilities can be achieved using
semi-empirical methods. Semi-empirical calculations utilize the structure of the HF
411
approximation with a Hamiltonian and a wave function, but replace some electron specific
estimations with experimentally derived parameters, or decoupled approximations. A list of
semi-empirical methods can be found in Young (2001). The advantage of these methods is that
they are much faster, which allows their application range to include finding the geometry and
dynamic energies of moderate sized molecules. However, semi-empirical methods give poor
results for van der Waals and dispersion intermolecular forces due to the lack of diffuse basis
functions, both of which are vital interactions in the meso-scale (Leach 2001).
Alternatively, density functional theory (DFT) has recently been shown to achieve similar
accuracy to ab initio calculations while being computationally less intensive by estimating the
energy of a molecule from its electron density instead of a wave function. However, because it
is a relatively new method, the approach requires extensive validation which can limit the scope
and size of the molecules computed.
Overall, the most severe limitation of ab initio calculations is the limited size of the
molecules that can be modeled. Molecular Mechanics (MM) is a method that limits this
computational intensiveness by calculating the energy of a molecule from a set of force field
constants which are constant across all molecular types and can be derived from ab initio
calculations or experimental data. This is best accomplished by using an atom based approach
that implicitly includes electron information. In this approach, interaction potentials and their
corresponding parameters, commonly known as the force field, describe in detail how the
particles in a system interact with each other and by extension how the potential energy of a
system depends on the particle coordinates (Zeng et al. 2008). Such a force field may be
obtained by the ab initio quantum methods, empirical methods such as the Leonard?Jones,
Morse, Born-Mayer, or quantum-empirical methods that nest quantum mechanics within
412
molecular mechanics such as the embedded atom model (Zeng et al. 2008). The criteria for
selecting a force field include the accuracy, transferability and computational speed. A typical
interaction potential U may consist of a number of bonded and nonbonded interaction terms
(Zeng et al. 2008):
?? ?? a n g l e
a n g l e
bond
bond
N
i cbaa n g lea n g le
N
i babondbondn rrriUrriUrrrU ),,,(),,(),,,( 11
?????????
?? ?? i n v e rsi o n
i n v e rsi o n
t o rsi o n
t o rsi o n
N
i dcbain v e r s io nin v e r s io n
N
i dcbato r s io nto r s io n rrrriUrrrriU ),,,,(),,,,(
????????
? ?? ? ?? ??? ? ?? 11 111 1 ),,,(),,,( Ni Nj batice le c tr o s taNi Nj bav d W rrjiUrrjiU ???? (A4.6)
The first four terms represent bonded interactions: bond stretching, bond bending, dihedral angle
torsion, inversion interaction. The last two terms are non-bonded interactions, van der Waals
energy and electrostatic energy. Since electrons are not explicitly included, electronic property
based processes cannot be modeled. The reduction in complexity afforded by MM calculations
means that both large molecules and molecules with intermolecular forces can be modeled. The
cost of this complexity reduction is that the method is extremely sensitive to parameterization,
meaning that it is important that the basis sets be based on experimental and computational
studies similar to the system being studied. Some of the most common basis set functions are as
follows: assistant model building with energy refinement (AMBER), consistent force field
(CFF), Carbohydrate hydroxyls represented by external atoms (CHEAT), generalized organic
force fields (MM1-MM4), and the universal force field (UFF). Other force field basis sets and
recommendations on when to use them can be found in readily available computational
chemistry texts. In some cases it is advantageous to suppress hydrogen, choosing to model CH2
413
as a group rather than as an sp3 carbon bonded to two hydrogen atoms. This is known as the
united atom force field and is common way to reduce the complexity of modeling large
macromolecules.
Statistical mechanics provides the means to convert the individual optimized molecular
structure of a single molecule into physical properties that describes many molecules in many
conformations and energy states. Molecular Dynamics (MD) and Monte Carlo (MC) simulations
are the most common methods for generating these relationships. MD simulations model the
time-dependent behavior of molecular systems, such as vibrational or Brownian motion. MD are
the time dependent form of molecular mechanics, solved in discrete time steps, under different
ensembles of constraints. Examples include grand canonical (?VT), microcanonical (NVE),
canonical (NVT), and isothermal-isobaric (NPT) (Zeng et al. 2008). It is important to validate
that the simulation is delivering chemical properties that exhibit normal Gaussian distributions
and that they are matched with experimental data since this type of time step simulation is highly
susceptible to systematic error propagation. MC simulations differ from MD simulations in that
the location, orientation, and geometry of a set of molecules is randomly chosen according to a
statistical distribution (Young 2001). As a result, MC simulations are less computationally
intensive than MD simulations, but are limited since they cannot yield time dependent
information. Both of these statistical mechanics simulation methods can become as
computationally expensive as ab initio calculations due to the large number of molecules being
calculated, especially for large molecular systems.
Bridging the gap between these molecular methods and true continuum behavior has been
the focus of recent research. Whereas the molecular interactions of small molecules can be
easily described by existing thermodynamic models, large molecules, particularly those with
414
nano-, micro-, and meso-scale architectures still rely on combinatorial models to describe their
behaviors. Common techniques include Brownian Dynamics (BD), Dissipative Particle
Dynamics (DPD), Lattice-Boltzmann (LB), and time-dependent Ginsburg-Landau (TDGL), and
dynamic DFT (Zeng et al. 2008). Brownian dynamics replaces the explicit definitions of small
molecule solutions with implicit continuum definitions which allows for a much larger time step
to be used. Interaction forces on individual atoms are estimated by the Langevin equation,
summing effects from conservative forces, momentums, and random Gaussian noise. It should
be noted that this approximation means that energy and momentum of the solution is no longer
conserved, meaning that the macroscopic behavior is not hydrodynamic and does not obey the
Navier-Stokes equation. Like MD and BD, DPD is a particle based method that can simulate
fluidic behavior. However, it uses a coarse-grained molecular assembly as a basic unit instead of
single atoms (Zeng et al. 2008). In this manner it parameterizes the interaction forces into
conservative, dissipative, and random forces. The nature of the forces is pairwise and steps can
be taken to ensure momentum is conserved, meaning that this method obeys Navier-Stokes. The
details of these and other techniques can be found in Zeng et al. (2008).
A4.2. Topological Indices
The representation of a molecule in the form of a graph is the first step in the
development of a topological index (Chemmangattuvalappil et al. 2010). Using the molecular
graph, a set of values for topological indices (TI) can be estimated. Various Quantitative
Structure Property Relationships (QSPR) or Quantitative Structure Activity Relationships
(QSAR) are then used to relate the TI values to a set of physical and chemical properties. The
important qualities that make a meaningful TI include direct structural interpretation, correlation
with at least one property, linearly independence, non-triviality, and a strong structural basis,
415
among others (Randic and Basak 2001). Although hundreds of topological indices can be found
in the literature, only a few of them have found wide applications. Some of the more commonly
used topological indices are connectivity indices, edge adjacency indices, shape indices, Wiener
index, and the Hosoya topological index (Chemmangattuvalappil et al. 2010). Of these, the most
widely used topological index is the connectivity index (Trinajstic 1992).
The success of connectivity indices (CI) is due to the fact that each of its fragments can
be calculated directly from the valence (Lewis) bond diagrams. Beginning with the atomic level,
the first atomic index is the simple connectivity index, ?, equal to the number of non-hydrogen
atoms to which a given non-hydrogen atom is bonded (Bicerano 1996). The second atomic
index is the valence connectivity index, ?v, which incorporates the electronic structure
information of each non-hydrogen atom and is sometimes referred to as the valence delta.
hZvv ??? (A4.7)
Here, the Zv of an atom is the count of all adjacent bonded atoms and all ? and lone pair electrons
while h is the number of hydrogen atoms bonded to that atom. When dealing with high atomic
weight atoms, the effect of non-valence electrons on the atomic size and properties must be
considered (Keir and Hall 1986). For that, the valence delta is redefined for an atom of atomic
weight Z as follows (Bicerano 1996; Chemmangattuvalappil et al. 2010):
? ?? ?1?? ?? vvv ZZ hZ? (A4.8)
If i and j are the atoms involved in the bond, then the bond indices, ? and ?v, are defined through
the pairing of atomic indices (Kier and Hall 1986; Gani et al. 2005):
ji ??? ? (A4.9)
416
vjviv ??? ? (A4.10)
An alternative method for classifying these elements is to use the adjacency matrix, J,,
which originates in graph theory, and indicates which atoms are bonded (Ugi et al. 1979). From
the adjacency matrix, the connectivity matrix, C, can be calculated. The connectivity matrix not
only indicates which atoms are connected through covalent bonds, but also the number of
covalent bonds between each atom. Further extension of this approach results in the bond-
electron, BE, matrix which represents both the covalent bond and the free valence electrons of
each atomic system (Ugi et al. 1979). Examples of these matrices for hydrogen cyanide (HCN)
are given below (Ugi et al. 1979):
? ?
??
?
?
?
??
?
?
?
?
010
101
010
H C NJ (A4.11)
? ?
??
?
?
?
??
?
?
?
?
N
C
H
H C NC
30
31
01 (A4.12)
? ?
??
?
?
?
??
?
?
?
?
230
301
010
H C NBE (A4.13)
Using the information from the atom and bond indices shown above, the calculation of the
zero?th order (atomic) connectivity indices for the entire molecule can be defined in terms of the
summation of the vertices of the hydrogen-suppressed graphs (Bicerano 1996):
? ??????? vertices ?? 10 (A4.14)
417
? ????????? vertices vv ?? 10 (A4.15)
The first order (bond) indices, 1??and 1?v , for the entire molecule are defined as the summation
over the edges of the hydrogen-suppressed graphs (Bicerano 1996; Chemmangattuvalappil et al.
2010) are as follows:
? ????????? edges ?? 11 (A4.16)
? ????????? ed g es vv ?? 11 (A4.17)
Once the indices are built, various linear and non-linear regression models can be used to relate
them to a specific physical property in QSPR and QSAR relationships. The general form of the
relationship is listed below:
)(TIf?? (A4.18)
where ? is a property function corresponding to a property P. The total number and types of
connectivity indices required for the models are dependent upon the properties dependence on
particular types of interactions. In most instances, the first and second order connectivity indices
are sufficient to describe the molecular graph. A finite number of higher-order connectivity
indices have also been created to describe molecular interactions occurring at longer lengths.
Examples of higher order connectivity indices include structural and topological information on
ring substitutions, branches, non-hydrogen clusters, among others (Bicerano 1996).
Like the van Krevelen group contribution method (van Krevelen 1990) for the design of
long range ordered systems in polymers, Bicerano (1996) has also developed similar CI methods
418
for polymeric backbones. In this method, Bicerano (1996) avoids the truncation errors
associated with variable polymer chain lengths and the resulting molecular weights by basing the
CI on an alternative representation of the monomeric graphs. Essentially, the graphs are built by
completely enveloping one bond of the monomeric unit, and ignoring the other. Strong
correlations for a variety of volumetric, thermodynamic, chemical, mechanical, optical, and
magnetic properties have been shown to be related to topological indices (Bicerano 1996).
It should be noted that CI are more difficult to use in the design of chemical structures
than the property models in the group contribution methods (GCM) (Chemmangattuvalappil et
al. 2009). In particular, GCM can be directly used in reverse problem formulations since their
property functions can be expressed as linear expressions, opening up additional computational
reduction possibilities. However, the applicability of group contribution methods is limited to a
small number of properties and molecular groups. One way to address this problem is to broaden
the application range of group contribution methods (GCM) by incorporating CI in an GC+
formulation (Chemmangattuvalappil et al. 2009). It has been proposed that the first order
connectivity indices can be written for particular groups. However, if first order connectivity
indices for the different groups are written separately, their sum will not give the CI value of the
molecule. This is because, the contribution to first order CI due to the bonds between two
separate groups are not fully represented. Therefore, an additional term for the bond between
different groups has to be included in the expression (Gani et al. 2005; Chemmangattuvalappil et
al. 2010):
? ? ? ? ?????????????????? k
mg r o u p
v
1 m g r o u p s ofo u t bondsk bonds i n t e r n a l
1 5.01 ??? (A4.19)
419
Here, k is the number of bonds inside the group for which the expression is written and m is the
number of free bonds in the group. The pure component property model proposed based on the
CI model is given by Gani et al. (2005). Here, Y is the sought property, Ai is the number of atom
i, ai is the estimated contribution of atom i, while b, c and d are adjustable parameters.
? ? ? ? ? ? dcbAaYf vvi ii ???? ? 10 2)( ?? (A4.20)
The CI method for estimating pure component properties cannot produce very accurate
results because, a large number of compounds are represented with very few parameters (Gani et
al. 2005; Chemmangattuvalappil et al. 2010). So, its applicability is limited to the prediction of
properties of the groups and/or property contributions which are not available in established
datasets (Constantinou and Gani 1994; Constantinou et al. 1995; Marrero and Gani 2001).
? ? ? ? ? ?mvmvi imimm cbAaYf 10,, 2)( ?? ??? ? (A4.21)
dYfnYf m mm ???????? ? )()( * (A4.22)
???????????????????????????? ??? t tts ssi ii CNCNYfCNYf )()( *
(A4.23)
where m is the number of different missing groups and nm is the number of times the missing
group is present in the molecule. f(Y*) is the property function of the missing group, Amj is the
property contribution of atom j, while a,b and c are the adjustable parameters estimated by Gani
et al. (2005).
An alternative representation of CI is the use of molecular signature. Molecular signature
is a molecular descriptor introduced by Visco et al. (2002) and Faulon et al. (2003a) that
represents atoms in a molecule using extended valencies of a pre-defined height. If G is a
molecular graph and x is an atom of G, the atomic signature of height h of x is a canonical
420
representation of the sub-graph of G containing all atoms that are at a distance h from x
(Chemmangattuvalappil et al. 2010). Topological indices like CI generally only look one or two
bonds away. Conversely, molecular signature can look an unlimited distance away, meaning
more structural information can be captured, leading to much improved accuracy in the design of
molecular architectures. The systematic procedure for the construction of atomic signature
developed by Visco et al. (2002) is explained in more detail by Chemmangattuvalappil et al.
(2010).
Once the atomic signatures of all atoms are estimated in this way, the molecular signature
can be represented as the linear combinations of all atomic signatures. If h?G(hXi) is a base
vector, h?i is the number of atoms having the signature of the base vector and hKG is the number
of base vectors, then the molecular signature h?(G) is represented as:
)()()(
1 i
hGK
i
hihG
Vx
hh XxG G
h
G
???? ??
??
?? (A4.24)
These signatures can produce meaningful quantitative relationship models with performances
comparable to many of the existing TIs (Faulon et al. 2003; Chemmangattuvalappil et al. 2010).
In fact, Faulon et al. (2003) showed that since both methods are based on graph theory, then one
method can be derived from the other. For example, the general relationship between a TI and
its signature has been expressed as a dot product between the vector of the occurrence number of
atomic signature of height h and the vector of TI values computed for each root of those atomic
signatures (Chemmangattuvalappil et al. 2010):
? ?? ???? hGh r o o tTIkGTI ?)( (A4.25)
Where k is a constant, h?G is the vector of the occurrence number of atomic signature of height h
and TI (root (h?)) is the vector of TI values calculated for each root of atomic signature. It has
421
been shown that different topological indices of different signature heights can be simultaneously
combined using molecular signature and parameterized in an easy to use group contribution
format (Chemmangattuvalappil et al. 2009; Chemmangattuvalappil et al. 2010).
? ?? ??? hi rootTIL (A4.26)
ii
N
i
h LTI ??
?? 1
(A4.27)
Remembering that TI can be expressed as chemical and physical properties through QSPR and
QSAR quantitative models, then a GCM molecular operator can be formulated:
i
N
i i LxP ??? 1)(?
(A4.28)
Implementing molecular signatures in this manner requires the use of additional rules for the
formation of complete structures. The rules help to ensure that the signatures selected based on
the property constraints will connect to form a complete graph without any free bonds. These
rules include the handshaking lemma, coloring function (valency of each carbon atom at each
level), and the handshaking dilemma, which are discussed in detail by Chemmangattuvalappil et
al. (2010).
422
Appendix A5
Excipient Case Study Supporting Information
This section contains supporting documentation of the acetaminophen excipient tablet
design. Due to the size of the data sets used in the case study, they cannot be displayed in the
dissertation in a conventional manner. As a result they have been converted to .mat files and can
be provided upon request by contacting Dr. Mario Eden at edenmar@auburn.edu. A list of the
data sets that were used is given below:
? Pnxp, a set of the pure component IR/NIR properties of the training set
? Rnxp, a set of the standardized IR/NIR properties of the training set
? Lmxp, a set of the PCA loadings of the training set
? Ptxp, a set of the projected IR/NIR properties of the design targets
? Rtxp, a set of the standardized IR/NIR properties of the design targets
423
Appendix A6
IR & NIR Molecular Group Absorbance Frequencies
The IR absorbance frequencies and magnitudes of the functional groups presented below
were found in Socrates (2001). The NIR frequencies were found in Workman and Weyer
(2008). Only those absorbances verified in these texts were included below. No conjugated
double bonds were included, although these can be added later to differentiate between isomers.
Also, any intensities listed as variable, were not included.
A6.1. Methine CH
Band Wavelength Region (cm-1) Relative Intensity & Shape
Bending (?) 1360-1320 (1340) w
Stretching (v) 2890-2880 (2885) w
1st Overtone Bending (2?) 4049-4029 (4039) m-w (in NIR)
Combination (vs+?) 4272-4252 (4262) m (in NIR)
Combination (va+?) 4333-4313 (4323) m-s (in NIR)
A6.2. Methylene CH2
Band Wavelength Region (cm-1) Relative Intensity & Shape
Scissoring Bend (?s) 1480-1440 (1460) m
Symmetrical Stretching (vs) 2870-2840 (2855) m
Asymmetrical Stretching (va) 2940-2915 (2925) m-s
Combination (va+?s) 4262-4257 (4260) m-s (in NIR)
Combination (vs+?s) 4336-4332 (4334) m-s (in NIR)
1st Overtone Asym. Str. (2va) 5681-5661 (5671) w (in NIR)
1st Overtone Sym. Str. (2vs) 5690-5670 (5680) w (in NIR)
Combination (va + vs) 5900-5700 (5800) m-w (in NIR)
1st Overtone Comb (2v+?) 7186-7080 (7183) m-w (in NIR)
424
2nd Overtone Sym. Str. (3vs) 8257-8237 (8247) m-w, sh (in NIR)
2nd Overtone Asym. Str. (3va) 8399-8379 (8389) m-w, sh (in NIR)
*Note* Coupling interactions and corrections will need to be added.
A6.3. Hydroxyl OH (w/o H-bond)
Band Wavelength Region (cm-1) Relative Intensity & Shape
Bending (?) 710-570 (640) m, br
Stretching (v) 3670-3580 (3625) m-w, sh
Combination (v + ?) 4550-5550 (5050) m-w, br (in NIR)
1st Overtone Str. (2v) 7160-7020 (7090) m, br (in NIR)
2nd Overtone Str. (3v) 10410-10390 (10400) w (in NIR)
A6.4. Hydroxyl OH (w/ H-bond)
Band Wavelength Region (cm-1) Relative Intensity & Shape
Bending (?) 710-570 (640) m, br
Stretching (v) 3550-3230 (3440) m-s, br
Combination (many types) 4550-5550 (5050) m-w, br (in NIR)
1st Overtone Str. (2v) 6850-6240 (6545) m, br (in NIR)
1st Overtone Str. (2v) 7200-7000 (7100) m (in NIR)
2nd Overtone Str. (3v) 9560-9540 (9550) w (in NIR)
A6.5. Secondary Alcohol CHOH (w/ H-bond)
Band Wavelength Region (cm-1) Relative Intensity & Shape
C-O OoP. Def. Bend (?? 390-330 (360) m-w
C-O IP. Def. Bend (?d?? 500-440 (470) W
O-H OoP. Def. Bending (?d) 660-600 (630) m-w, br
C-CO Stretch (v) 900-800 (850) m-w
C-O Stretch (v) 1150-1075 (1113) S
C-H Def. Bending (?d) 1350-1290 (1320) W
C-H Wag Bend (?w) 1400-1330 (1365) W
O-H + C-H2 Coup. Bend. (?c) 1430-1370 (1400) m-s, br
O-H Def. Bend. 1440-1260 (1350) m-s, br
C-H Stretching (v) 2890-2880 (2885) W
O-H Stretching (v) 3550-3230 (3440) m-s, br
C-H 1st Over. Bend. (2?) 4049-4029 (4039) m-w (in NIR)
C-H Combination (vs+?) 4272-4252 (4262) m (in NIR)
C-H Combination (va+?) 4333-4313 (4323) m-s (in NIR)
O-H Combination (many) 4550-5050 (4800) m-w, br (in NIR)
O-H Comb. (v + ?) 4970-4950 (4960) w, sh (in NIR)
425
O-H + C-H Comb. (v + v) 4720-4700 (4710) w, sh (in NIR)
C-H + OH Combo. (?+v) 6000-5080 (5090) w, sh (in NIR)
O-H 1st Overtone Str. (2v) 6850-6240 (6545) m, d (in NIR)
O-H 1st Overtone Str. (2v) 7150-7000 (7075) m, d (in NIR)
O-H + 1st C-H Comb. (v+2?) 9396-9376 (9386) m-w (in NIR)
O-H + 2nd C-O Comb. (v+3v) 9730-9710 (9720) w (in NIR)
O-H 2nd Overtone Str. (3v) 9560-9540 (9550) w (in NIR)
A6.6 Primary Alcohol CH2OH (w/ H-bond)
Band Wavelength Region (cm-1) Relative Intensity & Shape
C-O Def. Bend (?? 555-395 (475) m-w
C-O IP. Def. Bend (?d?? 500-440 (470) w
O-H OoP. Def. Bending (?d) 710-570 (640) m-w, br
C-CO Stretch (v) 900-800 (850) m
C-H2 Twist Bend (?t) 960-800 (880) m-w
C-C-O Stretch (v) 1090-1000 (1045) S
C-H2 Twist. Bending (?t) 1300-1280 (1320) m-w
C-H2 Wag Bend (?w) 1390-1280 (1335) m-w
O-H Def. Bend. 1440-1260 (1350) m-s, br
C-H2 Def Bend (?d) 1480-1410 (1445) m-w
C-H2 Sym. Stretch (vs) 2935-2840 (2888) m-w
C-H2 Asym. Stretch (va). 2990-2900 (2945) m-w
O-H Stretching (v) 3550-3230 (3440) m-s, br
C-H 1st Over. Bend. (2?) 4049-4029 (4039) m-w (in NIR)
C-H Combination (vs+?) 4272-4252 (4262) m (in NIR)
C-H Combination (va+?) 4333-4313 (4323) m-s (in NIR)
O-H Combination (many) 4550-5050 (4800) m-w, br (in NIR)
O-H Comb. (v + ?) 4970-4950 (4960) w, sh (in NIR)
O-H + C-H Comb. (v + v) 4720-4700 (4710) w, sh (in NIR)
C-H + OH Combo. (?+v) 6000-5080 (5090) w, sh (in NIR)
O-H 1st Overtone Str. (2v) 6850-6240 (6545) m, d (in NIR)
O-H 1st Overtone Str. (2v) 7200-7000 (7100) m, d (in NIR)
O-H + 1st C-H Comb. (v+2?) 9396-9376 (9386) m-w (in NIR)
O-H + 2nd C-O Comb. (v+3v) 9730-9710 (9720) w (in NIR)
O-H 2nd Overtone Str. (3v) 9560-9540 (9550) w (in NIR)
A6.7. Alcohol CHCH2OH (w/ H-bond)
Band Wavelength Region (cm-1) Relative Intensity & Shape
C-O Def. Bend (?? 555-395 (475) m-w
C-O IP. Def. Bend (?d?? 500-440 (470) w
O-H OoP. Def. Bending (?d) 710-570 (640) m-w, br
426
C-CO Stretch (v) 900-800 (850) m
C-H2 Twist Bend (?t) 960-800 (865) m-w
C-C-O Stretch (v) 1045-1025 (1035) s
C-H2 Twist. Bending (?t) 1300-1280 (1320) m-w
C-H2 Wag Bend (?w) 1390-1280 (1335) m-w
O-H Def. Bend. 1440-1260 (1350) m-s, br
C-H2 Def Bend (?d) 1480-1410 (1445) m-w
C-H2 Sym. Stretch (vs) 2935-2840 (2888) m-w
C-H2 Asym. Stretch (va). 2990-2900 (2945) m-w
O-H Stretching (v) 3550-3230 (3440) m-s, br
C-H 1st Over. Bend. (2?) 4049-4029 (4039) m-w (in NIR)
C-H Combination (vs+?) 4272-4252 (4262) m (in NIR)
C-H Combination (va+?) 4333-4313 (4323) m-s (in NIR)
O-H Combination (many) 4550-5050 (4800) m-w, br (in NIR)
O-H Comb. (v + ?) 4970-4950 (4960) w, sh (in NIR)
O-H + C-H Comb. (v + v) 4720-4700 (4710) w, sh (in NIR)
C-H + OH Combo. (?+v) 6000-5080 (5090) w, sh (in NIR)
O-H 1st Overtone Str. (2v) 6850-6240 (6545) m, d (in NIR)
O-H 1st Overtone Str. (2v) 7200-7000 (7100) m, d (in NIR)
O-H + 1st C-H Comb. (v+2?) 9396-9376 (9386) m-w (in NIR)
O-H + 2nd C-O Comb. (v+3v) 9730-9710 (9720) w (in NIR)
O-H 2nd Overtone Str. (3v) 9560-9540 (9550) w (in NIR)
A6.8. Aliphatic Ether O
Band Wavelength Region (cm-1) Relative Intensity & Shape
Bending (?s) 440-420 (430) w
Symmetrical Stretching (vs) 1140-820 (980) w
Asymmetrical Stretching (va) 1150-1060 (1105) s
*Note* Coupling interactions and corrections will need to be added.
A6.9. Ether CH2O
Band Wavelength Region (cm-1) Relative Intensity & Shape
C-O-C Def. Bending (?d) 440-420 (430) w
C-O-C Asym. Stretching (vs) 1140-1085 (1120) s
C-O-C Sym. Stretching (vs) 1140-820 (980) w
C-H2 Def. Bending (?d) 1400-1360 (1380) m
C-H2 Wag. Bending (?w) 1475-1445 (1460) m
C-H2 Sym. Stretching (vs) 2880-2835 (2858) m
C-H2 Asym. Stretching (va) 2955-2920 (2938) m
C-H + C-O-C Comb.(vs+vs) 4080-3980 (4000) s (in NIR)
C-H2 Comb. (va+?s) 4262-4257 (4260) m-s (in NIR)
427
C-H2 Comb. (vs+?s) 4336-4332 (4334) m-s (in NIR)
C-H2 1st Ov. Asym. Str. (2va) 5681-5661 (5671) w (in NIR)
C-H2 1st Ov. Sym. Str. (2vs) 5690-5670 (5680) w (in NIR)
C-H2 Comb. (va + vs) 5900-5700 (5800) m-w (in NIR)
1st Ov. Comb. (2v+?) 7186 ? 7080 (7183) m-w (in NIR)
C-H2 2nd Ov. Sym. Str. (3vs) 8257-8237 (8247) m-w, sh (in NIR)
C-H2 2nd Ov. Asym. Str. (3va) 8399-8379 (8389) m-w, sh (in NIR)
A6.10. Aldehyde HCO (w/o H-bond)
Band Wavelength Region (cm-1) Relative Intensity & Shape
C-H Def. Bending (?d) 975-780 (877.5) m-w
C-H IP Rock. Bend. (?r) 1375-1350 (1383) m-s
C-H Stretch Fermi (v+2v) 1400-1380 (1390) m-w,d
C=O Stretch (v) 1790-1710 (1750) s
C-H Stretch (v) 2745-2650 (2720) m-w
C-H Stretch (v) 2900-2800 (2850) m-w
C=O + 2nd C-H Comb.(v+2?) 4524-4504 (4514) m-w (in NIR)
C-H + C=O Combo. (?+v) 4758-4738 (4748) w (in NIR)
C=O + C-H Comb. (v+v) 4760-4445 (4603) m-w (in NIR)
2nd C=O+2nd C-H Com.(v+2?) 4898-4878 (4888) w (in NIR)
2nd C=O + C-H Com. (2v+v) 7860-7840 (7850) w (in NIR)
A6.11. Aldehyde HCO (w/ H-bond)
Band Wavelength Region (cm-1) Relative Intensity & Shape
C-H Def. Bending (?d) 975-780 (877.5) m-w
C-H IP Rock. Bend. (?r) 1375-1350 (1383) m-s
C-H Stretch Fermi (v+2v) 1400-1380 (1390) m-w,d
C=O Stretch (v) 1770-1690 (1730) s
C-H Stretch (v) 2745-2650 (2720) m-w
C-H Stretch (v) 2900-2800 (2850) m-w
C=O + 2nd C-H Comb.(v+2?) 4504-4484 (4494) m-w (in NIR)
C-H + C=O Combo. (?+v) 4738-4718 (4728) w (in NIR)
C=O + C-H Comb. (v+v) 4740-4425 (4583) m-w (in NIR)
C=O+2nd C-H Com.(v+2?) 4878-4858 (4868) w (in NIR)
1st C=O + C-H Com. (2v+v) 7860-7840 (7850) w (in NIR)
A6.12. Saturated Aliphatic Aldehyde CH2HCO (w/o H-bond)
Band Wavelength Region (cm-1) Relative Intensity & Shape
C-CO IP Def. Bend. (?d) 566-520 (543) m-s
428
C-C-CO IP Def. Bend. (?r) 695-635 (665) m-s
C-H2 Twist. Bending (?t) 1300-1280 (1320) m-w
C-H2 Wag Bend (?w) 1390-1280 (1335) m-w
C-H IP Rock. Bend. (?r) 1440-1325 (57.5) m-s
C=O Stretch (v) 1740-1720 (1730) s
C-H 1st Ov. IP Def. Bend. (?) 2740-2700 (2720) m-w
C-H Stretch (v) 2870-2800 (2835) m-w
C-H Comb. (va+?s) 4262-4257 (4260) m-s (in NIR)
C-H Comb. (vs+?s) 4336-4332 (4334) m-s (in NIR)
C=O + 2nd C-H Comb.(v+2?) 4524-4504 (4514) m-w (in NIR)
C-H + C=O Combo. (?+v) 4758-4738 (4748) w (in NIR)
C=O + C-H Comb. (v+v) 4760-4445 (4603) m-w (in NIR)
2nd C=O+2nd C-H Com(2v+2?) 4898-4878 (4888) w (in NIR)
C-H 1st Over.Asym. Str. (2va) 5681-5661 (5671) w (in NIR)
C-H 1st Over. Sym. Str. (2vs) 5690-5670 (5680) w (in NIR)
C-H Comb. (va + vs) 5900-5700 (5800) m-w (in NIR)
C-H 1st Over. Comb. (2v+?) 7186-7080 (7183) m-w (in NIR)
2nd C=O + C-H Com. (2v+v) 7860-7840 (7850) w (in NIR)
C-H 2nd Over. Sym. Str. (3vs) 8257-8237 (8247) m-w, sh (in NIR)
C-H 2nd Over. Asym. Str. (3va) 8399-8379 (8389) m-w, sh (in NIR)
A6.13. Saturated Aliphatic Aldehyde CH2HCO (w/ H-bond)
Band Wavelength Region (cm-1) Relative Intensity & Shape
C-CO IP Def. Bend. (?d) 566-520 (543) m-s
C-C-CO IP Def. Bend. (?r) 695-635 (665) m-s
C-H2 Twist. Bending (?t) 1300-1280 (1320) m-w
C-H2 Wag Bend (?w) 1390-1280 (1335) m-w
C-H IP Rock. Bend. (?r) 1440-1325 (57.5) m-s
C=O Stretch (v) 1720-1700 (1710) s
C-H 1st Ov. IP Def. Bend. (?) 2740-2700 (2720) m-w
C-H Stretch (v) 2870-2800 (2835) m-w
C-H Comb. (va+?s) 4262-4257 (4260) m-s (in NIR)
C-H Comb. (vs+?s) 4336-4332 (4334) m-s (in NIR)
C=O + 2nd C-H Comb.(v+2?) 4504-4484 (4494) m-w (in NIR)
C-H + C=O Combo. (?+v) 4738-4718 (4728) w (in NIR)
C=O + C-H Comb. (v+v) 4740-4425 (4583) m-w (in NIR)
2nd C=O+2nd C-H Com.(v+2?) 4878-4858 (4868) w (in NIR)
C-H 1st Over.Asym. Str. (2va) 5681-5661 (5671) w (in NIR)
C-H 1st Over. Sym. Str. (2vs) 5690-5670 (5680) w (in NIR)
C-H Comb. (va + vs) 5900-5700 (5800) m-w (in NIR)
C-H 1st Over. Comb. (2v+?) 7186-7080 (7183) m-w (in NIR)
2nd C=O + C-H Com. (2v+v) 7840-7820 (7830) w (in NIR)
429
C-H 2nd Over. Sym. Str. (3vs) 8257-8237 (8247) m-w, sh (in NIR)
C-H 2nd Over. Asym. Str. (3va) 8399-8379 (8389) m-w, sh (in NIR)
A6.14. Vinyl CH2CH
Band Wavelength Region (cm-1) Relative Intensity & Shape
C=C Tors. Bend (?T) 485-410 (448) m-s
C=C Eth. Twist. Bend. (?t) 600-380 (490) m-s
C=C Eth. Twist. Bend. (?t) 720-410 (565) w
C-H2 OoP Rock. Bend. (?r) 980-810 (895) s
C-H OoP Bending. (?r) 1010-940 (975) s
C-H IP Def. Bend. (8d) 1180-1010 (1095) m-w
C-H2 Def. Bend. (?d) 1330-1240 (1285) m
C-H2 Sci. Bend. (?s) 1440-1360 (1400) m
C=C Stretching (v) 1645-1640 (1643) m-w
C-H2 1st Overtone Bend (2?) 1840-1820 (1830) w or i
C-H 1st Overtone Bend (2?) 1990-1970 (1980) w or i
C-H2 Sym. Stretch (vs) 3070-2930 (3000) M
C-H Stretch (v) 3110-2980 (3045) M
C-H2 Asym. Stretch (va) 3150-3000 (3075) M
C-H2 2nd Ov. Asy. Bend. (2?a) 4492-4472 (4482) m (in NIR)
C-H2 2nd Ov. Bend. (2?) 4610-4590 (4600) w (in NIR)
C-H2 2nd Ov. Sym. Bend. (2?s) 4780-4670 (4725) m-w (in NIR)
C-H2 1st Over. Stretch (2v) 6010-5990 (6000) w (in NIR)
C-H Combo. (v+2v) 6130-6110 (6120) w (in NIR)
A6.15. Vinylidene CH2C
Band Wavelength Region (cm-1) Relative Intensity & Shape
C=C Skeletal Stretch (v) 470-435 (452) m-w
C=C Skeletal Stretch (v) 560-530 (545) s
C=C Eth. Twist. Bend. (?t) 715-680 (698) w
C-H2 OoP Rock. Bend. (?r) 895-885 (890) s
C-H2 IP Def. Bend. (?d) 1320-1290 (1305) w
C-H2 Sci. Def Bend. (?s) 1420-1405 (1413) w
C=C Stretching (v) 1675-1625 (1650) m-w
C-H2 1st Overtone Bend (2?) 1800-1750 (1775) w
C-H2 Sym. Stretch (vs) 2985-2970 (2978) m-w
C-H2 Asym. Stretch (va) 3095-3075 (3085) m-w
C-H2 2nd Ov. Asy. Bend. (2?a) 4492-4472 (4482) m (in NIR)
C-H2 2nd Ov. Bend. (2?) 4610-4590 (4600) w (in NIR)
C-H2 2nd Ov. Sym. Bend. (2?s) 4780-4670 (4725) m-w (in NIR)
C-H2 1st Over. Stretch (2v) 6010-5990 (6000) w (in NIR)
430
C-H Combo. (v+2v) 6140-6120 (6130) w (in NIR)
A6.16. cis-Vinylene CHCH
Band Wavelength Region (cm-1) Relative Intensity & Shape
C-H Tors. Bend (?T) 490-320 (405) m-s
C=C Skeletal Bend (?T) 500-460 (480) s
-C=CH Def. Bend. (?d) 590-440 (515) m-s
C=C Eth. Twist. Bend. (?t) 630-570 (600) s
C-H Wag. Bend. (?w) 790-650 (720) m-s
C-H Wag. Bend. (?w) 1000-850 (925) m-w
C-H Def. Bend. (?d) 1295-1185 (1243) w
C-H Def. Bend. (?d) 1425-1355 (1390) w
C=C Stretching (v) 1665-1630 (1648) m
C-H Stretch (v) 3040-2980 (3010) m
C-H Stretch (v) 3090-3010 (3050) m
C-H + C=C Combo. Str. (v+v) 4683-4663 (4673) w (in NIR)
C-H + C=C Combo. Str. (v+v) 4610-4590 (4600) m-w (in NIR)
C-H 1st Overtone Stretch (2v) 5973-5953 (5963) w (in NIR)
A6.17. trans-Vinylene CHCH
Band Wavelength Region (cm-1) Relative Intensity & Shape
C-H Tors. Bend (?T) 490-320 (405) m-s
C=C Skeletal Bend (?T) 500-480 (490) s
-C=CH Def. Bend. (?d) 590-440 (515) m-s
C=C Eth. Twist. Bend. (?t) 580-515 (548) m-s
C-H Wag. Bend. (?w) 850-750 (800) m-w
C-H Wag. Bend. (?w) 1000-910 (955) v (w or i)
C-H Def. Bend. (?d) 1305-1260 (1283) v (w or i)
C-H Def. Bend. (?d) 1340-1355 (1390) v (w or i)
C=C Stretching (v) 1680-1665 (1673) m-w
C-H Stretch (v) 3050-3000 (3025) m
C-H Stretch (v) 3065-3015 (3030) m
A6.18. Aliphatic Methyl CH3
Band Wavelength Region (cm-1) Relative Intensity & Shape
Sym Bend (?s) 1390-1370 (1380) m-s
Asym. Bend. (?a) 1465-1440 (1453) m
Symmetrical Stretching (vs) 2885-2865 (2875) m
Asymmetrical Stretching (va) 2975-2950 (2963) m-s
431
Comb. Sym. Bend. (?u+?u+?u) 4110-3990 (4100) s (in NIR)
Combination (v+??? 4420-4370 (4395) m-s (in NIR)
Com. Asym. Bend. (?u+?u+?u)? 4425-4375 (4400) m-s (in NIR)
1st Overtone Sym. Str. (2vs) 5901-5849 (5876) w (in NIR)
1st Overtone Asym. Str. (2va) 5925-5885 (5905) w (in NIR)
1st Overtone Comb (2v+?) 7273-7253 (7263) m (in NIR)
1st Overtone Comb (2v+?) 7365-7345 (7355) m (in NIR)
2nd Overtone Stretch (3v) 8257-8237 (8247) s (in NIR)
*Note* The bending combination (?u+?u+?u) may actually be a 2nd Overtone (3?), but that would
violate the power hierarchy rule since no 1st order bending mode was identified.
A6.19. Aryl Methyl CH3
Band Wavelength Region (cm-1) Relative Intensity & Shape
Rocking Bend. (?r) 1060-900 (965) m-w, d
Rocking Bend. (?r) 1130-1000 (1075) m-w, d
Sym Bend (?s) 1405-1355 (1380) m-s
Asym. Bend. (?a) 1470-1400 (1435) m, d
Asym. Bend. (?a) 1480-1430 (1455) m, d
1st Over. Asym. Bend. (2?a) 2830-2740 (2785) m-w
1st Over. Sym. Bend. (2?a) 2870-2860 (2865) m-w
Symmetrical Stretching (vs) 2930-2920 (2925) m-s, d
Asymmetrical Stretching (va) 2955-2935 (2963) m-s
Symmetrical Stretching (vs) 3000-2965 (2983) m-s, d
Comb. Sym. Bend. (?u+?u+?u) 4110-3990 (4100) s (in NIR)
Combination (v+??? 4420-4370 (4395) m-s (in NIR)
Com. Asym. Bend. (?u+?u+?u)? 4425-4375 (4400) m-s (in NIR)
Com. Str. 2nd O Bend (va+2?a) 5660-5640 (5650) w (in NIR)
1st Overtone Sym. Str. (2vs) 5745-5725 (5735) w (in NIR)
1st Overtone Asym. Str. (2va) 5925-5885 (5790) w (in NIR)
1st Overtone Comb (2v+?) 7273-7253 (7263) m (in NIR)
1st Overtone Comb (2v+?) 7365-7345 (7355) m (in NIR)
2nd Overtone Stretch (3v) 8257-8237 (8247) s (in NIR)
*Note* An unsaturated methyl group was used for the bending vibrations since no aryl specific
modes were identified by Socrates.
A6.20. Aliphatic Methoxy OCH3
Band Wavelength Region (cm-1) Relative Intensity & Shape
C-O Def. Bend. (?d) 580-340 (460) m-w
CH3/CO Rocking Bend (?d) 1190-1100 (1145) m-w
CH3 Rock Bend 1235-1155 (1195) m-w
CH3 Sym Bend (?s) 1460-1420 (1440) M
432
CH3 Asym. Bend. (?a) 1475-1435 (1455) m, d
CH3 Asym. Bend. (?a) 1485-1445 (1465) m, d
C-H3 Sym. Str. (vs) 2880-2815 (2848) m, sh
C-H3 Asym. Str. (va) 2985-2920 (2950) m, d
C-H Asym. Str. (va) 3030-2950 (2990) m, d
CH3 Co. S. Bend. (?u+?u+?u) 4110-3990 (4100) s (in NIR)
CH3 Combination (v+??? 4420-4370 (4395) m-s (in NIR)
CH3 Co. A. Bend. (?u+?u+?u)? 4425-4375 (4400) m-s (in NIR)
OCH3 1st Ov. Sym. Str. (2vs) 5798-5748 (5773) w (in NIR)
OCH3 1st Ov. Asym. Str. (2va) 5905-5855 (5880) w (in NIR)
OCH3 1st Ov. Comb (2v+?) 7173-7153 (7163) m (in NIR)
OCH3 1st Ov. Comb (2v+?) 7345-7325 (7335) m (in NIR)
CH3 2nd Ov. Stretch (3v) 8257-8237 (8247) m (in NIR)
*Note* A combination of groups listed in the Chapter 2 and Chapter 7 where ranges were
extended to include both ranges.
A6.21. Aryl Methoxy OCH3
Band Wavelength Region (cm-1) Relative Intensity & Shape
C-O Def. Bend. (?d) 580-340 (460) m-w
C-O Bend (?) 1050-1010 (1030) m
CH3/CO Rocking Bend (?r) 1190-1100 (1145) m-w
CH3 Rock Bend (?r) 1235-1155 (1195) m-w
C-O Bend (?) 1310-1210 (1260) m
CH3 Sym Bend (?s) 1460-1420 (1440) m
CH3 Asym. Bend. (?a) 1475-1435 (1455) m, d
CH3 Asym. Bend. (?a) 1485-1445 (1465) m, d
C-H3 Sym. Str. (vs) 2860-2815 (2850) m, sh
C-H3 Asym. Str. (va) 2985-2920 (2950) m-w, d
C-H Asym. Str. (va) 3005-2965 (2985) m-w, d
CH3 Co. S. Bend. (?u+?u+?u) 4110-3990 (4100) s (in NIR)
CH3 Combination (v+??? 4420-4370 (4395) m-s (in NIR)
CH3 Co. A. Bend. (?u+?u+?u)? 4425-4375 (4400) m-s (in NIR)
OCH3 1st Ov. Sym. Str. (2vs) 5798-5748 (5773) w (in NIR)
OCH3 1st Ov. Asym. Str. (2va) 5905-5855 (5880) w (in NIR)
OCH3 1st Ov. Comb (2v+?) 7173-7153 (7163) m (in NIR)
OCH3 1st Ov. Comb (2v+?) 7345-7325 (7335) m (in NIR)
CH3 2nd Ov. Stretch (3v) 8257-8237 (8247) m (in NIR)
*Note* A combination of groups listed in the Chapter 2 and Chapter 7 where ranges were
extended to include both ranges.
433
A6.22. Aryl OH (w/o H-bond)
Band Wavelength Region (cm-1) Relative Intensity & Shape
C-OH IP Bending (?? 450-375 (413) w
C-O Stretch (v) 1260-1150 (1205) s
O-H IP Bending (?) 1410-1300 (1355) s
O-H Stretching (v) 3620-3590 (3605) m, sh
O-H Comb. Str. (v + v) 4883-4683 (4783) m-s (in NIR)
O-H + C-O Comb. Str. (v + v) 4967-4767 (4867) m-w (in NIR)
O-H Combination (v + ?) 5049-4849 (4949) w (in NIR)
1st Overtone Str. (2v) 7140-6940 (7040) m-w, br (in NIR)
2nd Overtone Str. (3v) 10010-9990 (10000) w (in NIR)
A6.23. Aryl OH (w/ H-bond)
Band Wavelength Region (cm-1) Relative Intensity & Shape
C-OH IP Bending (?? 450-375 (413) w
O-H OoP. Def. Bending (?d) 720-600 (660) s, br
C-O Stretch (v) 1260-1180 (1220) s
O-H IP Bending (?) 1410-1310 (1360) s
O-H Stretching (v) 3250-3000 (3125) m, br
O-H Comb. Str. (v + v) 4883-4683 (4783) m-s (in NIR)
O-H + C-O Comb. Str. (v + v) 4967-4767 (4867) m-w (in NIR)
O-H Combination (v + ?) 5049-4849 (4949) w (in NIR)
O-H 1st Overtone Str. (2v) 6470-6270 (6370) w, br (in NIR)
O-H 1st Overtone Str. (2v) 7000-6700 (6850) w, br (in NIR)
2nd Overtone Str. (3v) 9920-9900 (9910) w, d (in NIR)
2nd Overtone Str. (3v) 10300-10280 (10290) w, d (in NIR)
A6.24. Tetramethyl C(CH3)3
Band Wavelength Region (cm-1) Relative Intensity & Shape
C-C Skeletal Bend (?s) 930-925 (928) m
C-C Skeletal Bend (?s) 1010-990 (1000) m-w
C-C Skeletal Bend (?s) 1225-1165 (1195) m
C-C Skeletal Bend (?s) 1255-1245 (1250) m
C-CH3 Sym. Bend. (?s) 1395-1350 (1365) m-s
C-CH3 Sym. Bend. (?s) 1420-1375 (1398) m
C-CH3 Asym. Bend. (?a) 1475-1435 (1455) m
C-H Sym. Stretching (vs) 2885-2865 (2875) m
C-H Asym. Stretching (va) 2975-2950 (2963) m-s
C-H Co. S. Bend. (?u+?u+?u) 4110-3990 (4100) s (in NIR)
434
C-H Combination (v+??? 4420-4370 (4395) m-s (in NIR)
Com. Asym. Bend. (?u+?u+?u)? 4425-4375 (4400) m-s (in NIR)
1st Overtone Sym. Str. (2vs) 5901-5849 (5876) w (in NIR)
1st Overtone Asym. Str. (2va) 5925-5885 (5905) w (in NIR)
1st Overtone Comb (2v+?) 7273-7253 (7263) m (in NIR)
1st Overtone Comb (2v+?) 7365-7345 (7355) m (in NIR)
2nd Overtone Stretch (3v) 8257-8237 (8247) s (in NIR)
A6.25. Alkyl Peroxide OO
Band Wavelength Region (cm-1) Relative Intensity & Shape
O-O Stretch (v) 900-800 (850) w
C-O Stretch (v) 1150-1030 (1090) m-s
A6.26. Saturated Aliphatic Ester COO
Band Wavelength Region (cm-1) Relative Intensity & Shape
C-O-C Sym. Stretch (vs) 1160-1050 (1105) s
C-O-C Asym. Stretch (va) 1275-1185 (1230) s
C=O Stretch (v) 1750-1725 (1738) s
C=O 1st Overtone (2vs) 3460-3440 (3450) w
A6.27. Saturated Aliphatic Methyl Ester COOCH3
Band Wavelength Region (cm-1) Relative Intensity & Shape
Unlisted 450-430 (440) m-s
CO-O Rocking Bend (?r) 530-340 (435) w
C-C-O Sym. Stretch (vs) 1160-1050 (1105) s
C-O Stretch (v) 1175-1155 (1165) s
C-C-O Asym. Stretch (va) 1275-1185 (1230) s
O-CH3 Stretch (v) 1315-1195 (1245) s
Unlisted 1370-1350 (1360) w
CH3 Sym. Def. Bend (?) 1460-1420 (1440) m-w
CH3 Asym. Def. Bend (?) 1465-1420 (1443) m-s
CH3 Asym. Def. Bend (?) 1485-1435 (1455) m
C=O Stretch (v) 1750-1725 (1738) s
CH3 Sym. Stretch (v) 3000-2860 (2930) m
CH3 Asym. Stretch (v) 3030-2950 (2990) m-w
CH3 Asym. Stretch (v)? 3050-2980 (3015) m-w
C=O 1st Overtone (2vs) 3460-3440 (3450) w
CH3 Com. S. Bend. (?u+?u+?u) 4110-3990 (4100) s (in NIR)
CH3 Comb. (v+??? 4420-4370 (4395) m-s (in NIR)
CH3 Com. A. Bend. ?u+?u+?u)? 4425-4375 (4400) m-s (in NIR)
435
Com. Str. 2nd O Bend (va+2?a) 5660-5640 (5650) w (in NIR)
OCH3 1st Ov. Sym. Str. (2vs) 5901-5725 (5803) w (in NIR)
OCH3 1st Ov. Asym. Str. (2va) 5925-5855 (5890) w (in NIR)
OCH3 1st Ov. Comb (2v+?) 7273-7153 (7253) m (in NIR)
OCH3 1st Ov. Comb (2v+?) 7365-7325 (7345) m (in NIR)
CH3 2nd Ov. Stretch (3v) 8257-8237 (8247) m-s (in NIR)
*Note* The unlisted absorptions could not be cross-referenced within Socrates, so they were left
as undefined. Also, no NIR data on ester groups was available in Workman and Weyer
(2008), so a combination of aldehydes and methoxys were used to approximate the
behavior.
A6.28. Saturated Aliphatic Ethyl Ester CH3CH2COO
Band Wavelength Region (cm-1) Relative Intensity & Shape
C-O-C Def Bend (?) 370-250 (310) m-w
C-O-C Def Bend (?) 395-305 (350) m-w
CO-O Rocking Bend (?r) 485-365 (425) m-w
CO OoP Rocking Bend (?r) 700-550 (625) w
CH2 Rocking Bend (?r) 825-775 (800) w
C-C str (v) 940-850 (895) w
CH3 Rock. Bend (?r) 1150-1080 (1115) w
C-C-O Sym. Stretch (vs) 1160-1050 (1105) s
CH3 Rock. Bend (?r) 1195-1135 (1165) w
C-C-O Asym. Stretch (va) 1275-1185 (1230) s
CH2 Twist. Bend (?T) 1340-1325 (1333) m-w
CH2 Wag. Bend (?w) 1385-1335 (1360) m-w
CH3 Sym. Def. Bend (?) 1390-1360 (1375) m-s
CH3 Asym. Def. Bend (?) 1480-1435 (1458) m
OCH2 Def. Bend. (?) 1490-1460 (1475) m-w
C=O Stretch (v) 1750-1725 (1738) s
CH3 Stretch (v) 2920-2860 (2890) w
CH3 Sym. Stretch (vs) 2930-2890 (2910) w
CH3 Asym. Stretch (va) 2995-2930 (2963) m
C=O 1st Overtone (2vs) 3460-3440 (3450) w
CH3 Com. S. Bend. (?u+?u+?u) 4110-3990 (4100) s (in NIR)
C-H Comb. (va+?s) 4262-4257 (4260) m-s (in NIR)
C-H Comb. (vs+?s) 4336-4332 (4334) m-s (in NIR)
CH3 Comb. (v+??? 4420-4370 (4395) m-s (in NIR)
CH3 Com. A. Bend. ?u+?u+?u)? 4425-4375 (4400) m-s (in NIR)
Com. Str. 2nd O Bend (va+2?a) 5660-5640 (5650) w (in NIR)
CH3 1st Ov. Sym. Str. (2vs) 5901-5725 (5803) w (in NIR)
CH3 1st Ov. Asym. Str. (2va) 5925-5855 (5890) w (in NIR)
CH3 1st Ov. Comb (2v+?) 7273-7153 (7253) m (in NIR)
436
CH3 1st Ov. Comb (2v+?) 7365-7325 (7345) m (in NIR)
CH3 2nd Ov. Stretch (3v) 8257-8237 (8247) m-s (in NIR)
*Note* The unlisted absorptions could not be cross-referenced within Socrates, so they were left
as undefined. Also, no NIR data on ester groups was available in Workman and Weyer,
so a combination of aldehydes and methoxys were used to approximate the behavior.
A6.29. Acrylate Ester CH2CHCOO
Band Wavelength Region (cm-1) Relative Intensity & Shape
C=C Tors. Bend (?T) 485-410 (448) m-s
CO-O Rocking Bend (?r) 485-365 (425) m-w
C=C Eth. Twist. Bend. (?t) 600-380 (490) m-s
C-O-C Def Bend (?) 675-660 (668) m
CO OoP Rocking Bend (?r) 700-550 (625) w
=CH2 Twist Bend (?T) 810-800 (805) m-s
CH2 Rocking Bend (?r) 825-775 (800) w
C-C str (v) 940-850 (895) w
=CH2 Wag. Bend (?w) 970-960 (965) s
C-H Def. Wag (?w)? 990-980 (985) m
C-H OoP Bending. (?r) 1010-940 (975) s
C-C Skel. Bend (?) 1070-1065 (1068) m
CH3 Rock. Bend (?r) 1150-1080 (1115) w
C-C-O Sym. Stretch (vs) 1160-1050 (1105) s
C-H IP Def. Bend. (8d) 1180-1010 (1095) m-w
unlisted 1200-1195 (1198) s
C-C-O Asym. Stretch (va) 1275-1185 (1230) s
=CH Rock. Bend (?r) 1290-1270 (1280) m
unlisted 1290-1280 (1285) s
=CH2 Def Bend (?) 1420-1400 (1410) m
C-H2 Sci. Bend. (?s) 1440-1360 (1400) m
C=C Stretch (v) 1635-1615 (1625) m
C=C Stretch (v) 1650-1630 (1640) m-s
C=O Stretch (v) 1725-1710 (1718) s
C-H2 1st Overtone Bend (2?) 1840-1820 (1830) w or i
C-H 1st Overtone Bend (2?) 1990-1970 (1980) w or i
C-H2 Sym. Stretch (vs) 3070-2930 (3000) m
C-H Stretch (v) 3110-2980 (3045) m
C-H2 Asym. Stretch (va) 3150-3000 (3075) m
C=O 1st Overtone (2vs) 3460-3440 (3450) w
C-H2 2nd Ov. Asy. Bend. (2?a) 4492-4472 (4482) m (in NIR)
C-H2 2nd Ov. Bend. (2?) 4610-4590 (4600) w (in NIR)
C-H2 2nd Ov. Sym. Bend. (2?s) 4780-4670 (4725) m-w (in NIR)
C-H2 1st Over. Stretch (2v) 6010-5990 (6000) w (in NIR)
437
C-H Combo. (v+2v) 6130-6110 (6120) w (in NIR)
A6.30. Methacrylate Ester CH2C(CH3)COO
Band Wavelength Region (cm-1) Relative Intensity & Shape
C=C Skeletal Stretch (v) 470-435 (452) m-w
C=C Skeletal Stretch (v) 560-530 (545) s
C-O-C Def Bend (?) 660-645 (653) m
C=C Eth. Twist. Bend. (?t) 715-680 (698) w
C-C Skel Bend (?) 825-805 (815) m-s
C-H2 OoP Rock. Bend. (?r) 895-885 (890) s
=CH2 Wag. Bend (?w) 950-935 (943) s
C-C Skel. Bend (?) 1010-990 (1000) m
C-C Skel. Bend (?) 1020-1000 (1010) m
C-O-C Sym. Stretch (vs) 1160-1150 (1155) s
C-O-C Asym. Stretch (va) 1275-1185 (1230) s
unlisted 1310-1290 (1300) s
C-H2 IP Def. Bend. (?d) 1320-1290 (1305) w
=CH Rock. Bend (?r) 1335-1315 (1325) m
CH3 Sym Bend (?s) 1390-1370 (1380) m-s
=CH2 Def Bend (?) 1420-1400 (1410) m
CH3 Asym. Bend. (?a) 1465-1440 (1453) m
C=C Stretch (v) 1650-1630 (1640) m
C=O Stretch (v) 1725-1710 (1718) s
C-H2 1st Overtone Bend (2?) 1800-1750 (1775) w
CH3 Sym. Stretching (vs) 2885-2865 (2875) m
C-H2 Sym. Stretch (vs) 2985-2970 (2978) m-w
CH3 Asym. Stretching (va) 2975-2950 (2963) m-s
C-H2 Asym. Stretch (va) 3095-3075 (3085) m-w
C=O 1st Overtone (2vs) 3460-3440 (3450) w
CH3 Com. S. Bend. (?u+?u+?u) 4110-3990 (4100) s (in NIR)
CH3 Combination (v+??? 4420-4370 (4395) m-s (in NIR)
CH3 Com. A. Bend.(?u+?u+?u)? 4425-4375 (4400) m-s (in NIR)
C-H2 2nd Ov. Asy. Bend. (2?a) 4492-4472 (4482) m (in NIR)
C-H2 2nd Ov. Bend. (2?) 4610-4590 (4600) w (in NIR)
C-H2 2nd Ov. Sym. Bend. (2?s) 4780-4670 (4725) m-w (in NIR)
CH3 1st Ov. Sym. Str. (2vs) 5901-5849 (5876) w (in NIR)
CH3 1st Ov. Asym. Str. (2va) 5925-5885 (5905) w (in NIR)
C-H2 1st Over. Stretch (2v) 6010-5990 (6000) w (in NIR)
C-H Combo. (v+2v) 6140-6120 (6130) w (in NIR)
CH3 1st Ov. Comb (2v+?) 7273-7253 (7263) m (in NIR)
CH3 1st Ov. Comb (2v+?) 7365-7345 (7355) m (in NIR)
CH3 2nd Ov. Stretch (3v) 8257-8237 (8247) s (in NIR)
438
A6.31. Other Groups
Additional groups available in the library, but not included in this dissertation are ?-
Pyranose and ?-Pyranose.