RHEOLOGY, STRUCTURE, AND STABILITY OF CARBON NANOTUBE ? 
UNSATURATED POLYESTER RESIN DISPERSIONS  
 
Except where reference is made to the work of others, the work described in this thesis is 
my own or was done in collaboration with my advisory committee. This thesis does not 
include proprietary or classified information. 
 
 
___________________________________________ 
Matthew Jay Kayatin 
 
Certificate of Approval: 
 
 
___________________________ ___________________________ 
Steve R. Duke Virginia A. Davis, Chair 
Associate Professor Assistant Professor 
Chemical Engineering Chemical Engineering 
 
 
 
___________________________ ___________________________ 
W. Robert Ashurst George T. Flowers 
Assistant Professor Dean 
Chemical Engineering Graduate School 
 
 
 
 
RHEOLOGY, STRUCTURE, AND STABILITY OF CARBON NANOTUBE ? 
UNSATURATED POLYESTER RESIN DISPERSIONS 
Matthew Jay Kayatin 
 
A Thesis 
Submitted to 
the Graduate Faculty of 
Auburn University 
in Partial Fulfillment of the 
Requirements for the 
Degree of 
Master of Science 
 
 
 
 
 
Auburn, Alabama 
December 19, 2008 
iii 
RHEOLOGY, STRUCTURE, AND STABILITY OF CARBON NANOTUBE ? 
UNSATURATED POLYESTER RESIN DISPERSIONS 
 
 
Matthew Jay Kayatin 
 
Permission is granted to Auburn University to make copies of this thesis at its discretion, 
upon request of individuals or institutions and at their expense. The author reserves all 
publication rights. 
 
 
 
 ________________________ 
 Signature of Author 
 
 
 ________________________ 
 Date of Graduation
iv 
THESIS ABSTRACT 
RHEOLOGY, STRUCTURE, AND STABILITY OF CARBON NANOTUBE ? 
UNSATURATED POLYESTER RESIN DISPERSIONS 
 
Matthew Jay Kayatin 
Master of Science, December 19, 2008 
 (B.S., West Virginia University, 2005) 
 
279 Typed Pages 
Directed by Virginia A. Davis 
 
 In this research, the first detailed study characterizing the dispersion of carbon 
nanotubes into an isophthalic unsaturated polyester resin was performed.  To eludicate 
the potential for mechanical property enhancement, dispersions of various carbon 
nanotubes were studied prior to cure by means of bulk rheological behavior.  The greatest 
enhancement in viscoelastic response was found upon the incorporation of single-walled 
carbon nanotubes over both acid oxidized single-walled carbon nanotubes and vapor 
grown carbon fibers.  The application of high shear mixing was found to be effective in 
reducing initial nanotube aggregate size, but unable to exfoliate individual single-walled 
carbon nanotubes alone.   
 For single-walled carbon nanotube dispersions, an intriguing concentration 
dependant linear viscoelastic response was observed; this resulted from elastic nanotube 
v 
network formation, through the percolation threshold.  Between nanotube loadings of 
0.025% vol. and 0.250% vol., the dependence of concentration on viscoelastic response 
was removed through colloidal scaling; this revealed insight into the network 
development through mastercurves.  Over the range of concentrations from 0.0030% vol. 
to 0.010% vol., the dispersions behaved as a viscous liquid.  Studying the reduced 
complex viscosity, non-Brownian behavior was observed in response to applied shear 
stresses, as predicted by the rotational Peclet number.   
For dilute dispersions of single-walled carbon nanotubes (0.0050% vol.), low 
shear rate induced aggregation phenomena was observed.  The aggregation driving force 
was determined to be chemical in nature.  Surface analysis indicated aggregation was 
avoided in the presence of hydroxyl/phenolic functionalities, as these groups could 
hydrogen bond with the isophthalic polyester.  However, no sample dispersions were 
found to be thermodynamically stable.  This observation was correlated with estimations 
of the enthalpy of mixing, indicating a positive Gibbs free energy upon mixing.        
The potential use of oxidized carbon nanotubes was investigated but unsuccessful 
for all as-produced samples.  The presence of carboxylic acid functionalities was 
confirmed after these treatments.  Acid oxidized nanotubes treated with ethanol did show 
some degree of mixing, but could not compare in performance to pristine single-walled 
carbon nanotube dispersions.     
Finally, the discovery of a lyophilization based method for creating aligned self-
assembled films from aqueous, oxidized nanotube dispersions was reported.  The 
nanotube self-assembly was believed to occur during the freezing step.  This method 
appears to be facile and shows great promise as a significant advancement in the field.  
vi 
ACKNOWLEDGEMENTS 
 I would like to thank my committee members for providing both their time and 
knowledge in the preparation and critical review of this manuscript.  Additionally, I 
would like to thank Dr. Virginia Davis for allowing me to chase my curiosities within this 
research.  Also, thanks are due to Dr. Mark Byrne for often providing advice unrelated to 
research during my stay in Auburn.  I would also like to thank Dr. Steve Duke for 
providing me with my first opportunity at teaching; a truly invaluable experience.   
 Acknowledgement and thanks are also extended to Amogh Karwa and Vinod 
Radhakrishnan for their time on spent on SEM.  Additionally, I would like to thank Dr. 
Michael Bozack, Dr. Ram Gupta, and Dr. Anne Gorden for useful discussions related to 
this work. 
vii 
The bibliography was prepared in the style of CARBON.  Microsoft Word, Microsoft 
Excel, Microsoft PowerPoint, and MDL ISIS/DRAW were used to prepare this 
manuscript. 
viii 
TABLE OF CONTENTS
LIST OF FIGURES .......................................................................................................... xii 
LIST OF TABLES........................................................................................................ xxvii 
1. INTRODUCTION ........................................................................................................ 1 
2. BACKGROUND .......................................................................................................... 4 
Polymers and Their Composites:............................................................................ 4 
Polymeric Carbon Nanotube Composites:.............................................................. 5 
Commercial Applications: ...................................................................................... 7 
Materials: ................................................................................................................ 9 
Unsaturated Polyester Resins:........................................................................... 9 
Vapor Grown Carbon Nanofibers:.................................................................. 13 
Carbon Nanotubes:.......................................................................................... 15 
Characteristics of Carbon Nanotubes: .................................................................. 15 
Molecular Structure: ....................................................................................... 15 
Mechanical Properties:.................................................................................... 20 
Electronic and Thermal Properties: ................................................................ 20 
Surface Area: .................................................................................................. 21 
Colloidal Interactions:........................................................................................... 22 
van der Waals Forces:..................................................................................... 22 
DLVO Theory:................................................................................................ 25 
ix 
Dispersion of Carbon Nanotubes:......................................................................... 27 
Binding Energy of SWNTs:............................................................................ 27 
Nanotube Dispersion in Polymers: ................................................................. 30 
Methods for Probing Dispersion State:........................................................... 35 
Shear Aggregation in CNT Dispersions: ........................................................ 39 
Modifying the Nanotube Interface via Oxidation:................................................ 42 
Introduction:.................................................................................................... 42 
Theory of Organic Acids: ............................................................................... 44 
Oxidative Treatment of Nanotubes:................................................................ 46 
Mechanism of Oxidation: ............................................................................... 50 
Etching Mode and Rate:.................................................................................. 52 
Fourier Transform Infrared Spectroscopy of Oxidized CNTs:............................. 53 
Raman Spectroscopy of SWNTs: ......................................................................... 55 
Raman Features of SWNTs: ........................................................................... 56 
X-ray Photoelectron Spectroscopy of CNTs: ....................................................... 58 
High Resolution Analysis of CNTs: ............................................................... 60 
Thermodynamics of SWNT Solutions:................................................................. 60 
Introduction:.................................................................................................... 60 
Motivation for Rod-like Particle Theory: ....................................................... 61 
Theory of Solutions for Rod-like Particles:.................................................... 63 
Entropy of Mixing: ......................................................................................... 66 
Enthalpy of Mixing:........................................................................................ 66 
Free Energy of Mixing:................................................................................... 71 
x 
Rheology Theory and Measurements: .................................................................. 73 
Introduction:.................................................................................................... 73 
Linear Viscoelasticity: .................................................................................... 74 
Oscillatory Flow: ............................................................................................ 76 
Simple Shear:.................................................................................................. 82 
Cone and Plate Rheometer:............................................................................. 86 
Parallel Plate Rheometer:................................................................................ 89 
Rheology of Particulate Suspensions:............................................................. 92 
Rheology of Nonspherical Particle Suspensions: ........................................... 93 
3. EXPERIMENTAL DETAILS .................................................................................. 101 
Microscopy: ........................................................................................................ 101 
Raw Materials:.................................................................................................... 102 
SWNT Purification: ............................................................................................ 106 
Thermal Gravimetric Analysis:........................................................................... 109 
Methods of Nanotube Oxidation:........................................................................ 111 
Spectroscopy:...................................................................................................... 119 
Dispersion Preparation:....................................................................................... 122 
Dispersion Techniques:....................................................................................... 123 
Rheology:............................................................................................................ 127 
Surface Tension: ................................................................................................. 131 
4. RESULTS AND DISCUSSION............................................................................... 132 
Characterization of Dispersion Methods: ........................................................... 132 
Syringe Dispersion:....................................................................................... 132 
xi 
Bath Sonication:............................................................................................ 136 
High Shear Mixing:....................................................................................... 137 
Viscosity of Neat UPR:....................................................................................... 146 
Rheology and Characterization of Acid Oxidized SWNT Dispersions:............. 146 
Identifying Experimental Artifacts: .................................................................... 159 
Rheological Behavior of CNTs:.......................................................................... 162 
Viscoelasticity of SWNT Dispersions: ............................................................... 166 
Determination of Rheological Percolation: ........................................................ 176 
Percolation from Crossover Modulus:................................................................ 183 
Morphology of SWNT Dispersions:................................................................... 184 
Rheology of Dilute SWNT Dispersions: ............................................................ 185 
Shear-induced Aggregation: ............................................................................... 190 
Effect of Surface Chemistry: .............................................................................. 201 
Proposed Chemical Functionalization: ............................................................... 210 
Dispersion Thermodynamics: ............................................................................. 211 
Self-assembly of SWNT Films:.......................................................................... 216 
5. CONCLUSIONS AND RECOMMENDATIONS ................................................... 231 
REFERENCES ............................................................................................................... 235 
APPENDIX A ? DERIVATION FOR ENTHALPY OF MIXING ............................... 246 
APPENDIX B ? ALTERNATE PERCOLATION GRAPHS........................................ 250 
APPENDIX C ? FTIR OF SWNT 187.3........................................................................ 252 
xii 
LIST OF FIGURES 
Figure 1: Thermoplastic (a) and thermoset (b) polymer structures.  Image reproduced 
from Kumar and Gupta (2003) [3]...................................................................................... 4 
Figure 2: Idealized ?baseline? chemical structure of a 1:1 isophthalic polyester 
constructed from equimolar ratios of isophthalic acid and maleic anhydride.  Isophthalic 
polyesters are known for their corrosion resistance, superior mechanical properties, and 
higher heat distortion temperatures with respect to other types of thermoset polyester.  
Reproduced from Mallick (1997) [18].............................................................................. 10 
Figure 3: The chemical structure of isophthalic acid.  In contrast to other unsaturated 
acids used in polyesters, there is no anhydride form of isophthalic acid.  Reproduced from 
Mallick (1997) [18]........................................................................................................... 11 
Figure 4: The chemical structure of styrene monomer.  Crosslinking between UPR 
oligomers propagates through the ?CH=CH
2
 ?tail.? ........................................................ 11 
Figure 5: High resolution images of (a) bamboo structured VGCFs and (b) stacked cup 
structured VGCF.  Images reproduced from Merkulov et al. (2000) and Endo et al. 
(2003), respectively [33, 34]............................................................................................. 14 
Figure 6: C
h
 is defined on the graphene lattice by unit vectors a
1
 and a
2
 and the chiral 
angle ? with respect to the zigzag axis (? = 0
o
).  The lattice vector of the 1-dimensional 
unit cell is defined by T.  The rotation angle ? and the translation ? constitute the basic 
xiii 
symmetry operation R = (?| ?) for the carbon nanotube.  The diagram is constructed for 
(n,m) = (4,2).  Reproduced from Dresselhaus et al. (1995) [32]....................................... 17 
Figure 7: Possible vectors specified by (n,m) for general CNTs, including zigzag, 
armchair, and chiral tubes.  Below each integer pair is listed the number of distinct caps 
that can be joined continuously to the CNT denoted by (n,m).  Encircled dots denote 
metallic structure and small dots are for semiconducting tubes.  Reproduced from 
Dresselhaus et al. (1995) [32]. .......................................................................................... 18 
Figure 8: Schematic theoretical model for a SWNT with the tube axis normal to: (a) the ? 
= 30
o
 direction with (n,m) = (5,5) (an ?armchair? tubule), (b) the ? = 0
o
 direction with 
(n,m) = (9,0) (a ?zigzag? tubule), and (c) a general direction 0 < ? < 30
o
 with (n,m) = 
(10,5) ( a ?chiral? tubule).  Reproduced from Dresselhaus et al. (1995) [32]. ................. 19 
Figure 9: Schematic representation of the interaction potential for dispersion forces 
(vdW), electrostatic repulsion, and the total interaction potential showing metaseable 
behavior.  The primary (1
o
) and secondary (2
o
) minima are highlighted. ........................ 26 
Figure 10: A single SWNT rope made up of ~100 SWNTs as it bends through the image 
plane of the microscope, showing uniform diameter and triangular packing of the tubes 
within the rope.  The diameter of the individual SWNTs were determined to be ~1.38 nm.  
Reproduced from Thess et al. (1996) [54]........................................................................ 28 
Figure 11: Schematic of an oxidized SWNT.  Carboxylic groups can be seen terminating 
the tube end and on sidewall defect site.  The presence of the acidic hydrogen is 
indicative of low pH conditions.  Image reproduced from Hirsch et al (2002) [113]. ..... 43 
Figure 12: Generic depiction of a carboxyl functional group.  The atom A is a general 
representation but typically signifies an aromatic or amorphous carbon in this work. .... 44 
xiv 
Figure 13: The evolution of MWNT alignment during the oxidation process.  Here 10 mg 
MWNT/mL of 60% nitric acid was used.  Image taken from Rosca et al. (2005) [124].. 49 
Figure 14: Schematic representation of the RBM showing vibration of the carbon atoms 
is in the radial direction as if the tube were ?breathing? and the G-band showing 
tangential vibration in the circumferential direction and atomic displacements along the 
axial direction.  Image taken from Jorio et al (2003) [145]. ............................................. 57 
Figure 15: A rigid-rod in the Flory lattice tilted at angle ? to the director.  The rod length 
is taken to be xd so that x plays the role of the aspect ratio where d is the characteristic 
dimension of the cubic lattice cell.  Reproduced from Beris and Edwards (1994), Cifferi 
(1994), & Wang and Zhou (2004) [171, 174, 175]........................................................... 64 
Figure 16: A rod divided into y
i
 submolecules having x/ y
i
 segments per submolecule and 
tilted at angle ? to the director.  The term y
i
d represents the height with respect to the 
director with units of lattice cell number.  Reproduced from Beris and Edwards (1994), 
Cifferi (1994), & Wang and Zhou (2004) [171, 174, 175]............................................... 64 
Figure 17: Illustration of important terms used in describing the shear strain of a fluid 
between two parallel sliding plates.  Reproduced from Mezger (2002) [180]. ................ 74 
Figure 18: Illustration of a stress relaxation experiment.  (a) The step increase in strain 
and (b) the stress vs. time response (relaxation) of various fluid types.  Reproduced from 
Macosko (1994) [67]......................................................................................................... 75 
Figure 19: The imposed strain wave and resulting steady-state stress wave under simple 
shear flow.  The stress wave is in frequency with the strain but shifted in phase by angle 
?.  Reproduced from Ferry (1980) [181]........................................................................... 77 
xv 
Figure 20: Trigonometric deconvolution of the steady-state stress wave in to its in-phase 
and out-of-phase components.  Reproduced from Ferry (1980) [181]. ............................ 78 
Figure 21: The prototypical linear viscoelastic response in terms of frequency dependent 
storage and loss moduli G? and G? for (a) a ?solid-like? fluid and (b) a ?liquid-like? fluid.  
Adapted from Larson (1998) [179]................................................................................... 79 
Figure 22: The sliding plate model and relative coordinate system for simple shear.  
Adapted from Gupta (2000) and Mezger (2002) [178, 180]. ........................................... 82 
Figure 23: Schematic diagram showing the notation for normal stresses in a shear field.  
Adapted from Gupta (2000) [178]. ................................................................................... 85 
Figure 24: Coordinate system and geometric parameters used to describe the cone and 
plate measuring geometry.  Adapted from Macosko (1994) [67]..................................... 87 
Figure 25: Coordinate system and geometric parameters used to describe the parallel 
plate measuring geometry.  Adapted from Macosko (1994) [67]..................................... 90 
Figure 26: Concentration regimes used to describe rod-like particles.  Adapted from 
Larson (1999) & Doi and Edwards (1986) [179, 185]...................................................... 96 
Figure 27: SEM Image of the as-received VGCF.  The measured fiber diameter is 
approximately 110 nm across.  A loose network structure can be seen from this image.
......................................................................................................................................... 103 
Figure 28: SEM images showing the morphology of the as-received (un-purified) SWNT 
183.6 imaged at (left) 4,000x magnification and (right) 35,000x magnification. .......... 104 
Figure 29: SEM images showing the morphology of the SWNT 183.6 Pure imaged at 
(left) 4,000x magnification and (right) 35,000x magnification. ..................................... 104 
xvi 
Figure 30: SEM images showing the morphology of the SWNT 187.3 imaged at (left) 
4,000x magnification and (right) 35,000x magnification. .............................................. 105 
Figure 31: SEM images showing the morphology of the Unidym SWNTs imaged at (left) 
4,000x magnification and (right) 35,000x magnification. .............................................. 106 
Figure 32: TEM Image of SWNT batch 183.6 Unpure as-received from Rice University.  
The dark sports are residual catalyst particles.  The scale bar is 200 nm in this image.. 107 
Figure 33: Soxhlet extraction apparatus used to purify the as-received SWNT 183.6 to 
high purity....................................................................................................................... 108 
Figure 34: TEM Image of SWNT 183.6 Pure after Soxhlet extraction.......................... 109 
Figure 35: TGA curves for the primary SWNTs used in this research.  The oven 
atmosphere was compressed air...................................................................................... 110 
Figure 36: Typical vacuum filtration apparatus used in this research. ........................... 112 
Figure 37: (left) Acid oxidized SWNT (SWNT-Ox) dispersed in ethanol (right) and the 
resulting film (SWNT-Ox Film) cast from an ethanol dispersion. ................................. 113 
Figure 38: Qualitative proof of successful oxidation. (left) The lack of phase separation in 
absence of stirring is indication of a stable colloidal suspension. (right) Final aqueous 
dispersion of neutral, acid oxidized SWNTs.  The sample shows excellent dispersion after 
washing. .......................................................................................................................... 118 
Figure 39: Image of the Renishaw inVia Raman microscope used in this research....... 120 
Figure 40: Image of the mixing vessel used in this research. ......................................... 126 
Figure 41: Image of the Anton Paar Physica MCR 301 rheometer with the parallel plate 
measuring geometry affixed to the P-PTD200 device and the H-PTD200 peltier 
controlled upper oven in place........................................................................................ 127 
xvii 
Figure 42: Rheological characterization of the transition to a well dispersed state for 
0.100% vol. VGCF in UPR processed by tip-to-tip syringe extrusion........................... 133 
Figure 43:  Image of a 0.100% vol. VGCF-UPR dispersion processed at 100 cycles 
syringe extrusion.  The sample was imaged at 20x with no cover glass in place........... 134 
Figure 44:  Image of a 0.100% vol. VGCF-UPR dispersion processed at 400 syringe 
extrusion cycles.  The sample was imaged at 20x with no cover glass in place............. 135 
Figure 45: Image (10x) of a 0.100% vol. SWNT-UPR dispersion processed by 24h bath 
sonication alone. ............................................................................................................. 136 
Figure 46: Images and dimensions for the custom shaped mixing impeller (left) and 
mixer flask (right) used in this work............................................................................... 138 
Figure 47: Cylindrical coordinate system used to estimate the shear stress of the mixer.
......................................................................................................................................... 138 
Figure 48: Schematic of the impeller and mixing flask wall with respect to the stirrer 
center line (C
L
).  The radial digestion of the mixing flask is represented by R and the 
dimension of the impeller by ?R..................................................................................... 140 
Figure 49: 10x image of a 0.100% vol. VGCF-UPR dispersion processed (~three days) 
by paddle-type mixing impeller.  No cover glass used on sample.................................. 143 
Figure 50:  Image of a 0.100% vol. VGCF-UPR dispersion processed (~three days) by 
the custom cut crescent-type mixing impeller.  The sample was imaged at 20x with 2x 
magnification in front of the camera.  No cover glass was used on the sample. ............ 143 
Figure 51: Optical microscopy image a 0.050% vol. dispersion of SWNT in UPR.  Image 
taken through cover slip using an oil immersion 60x (1.4 NA) objective with 2x 
magnification in front of the camera............................................................................... 145 
xviii 
Figure 52: Typical image (10x) of a 0.100% vol. SWNT-UPR dispersion processed by 
three day high shear mixing.  Inset showing 10x2 magnification. ................................. 145 
Figure 53:  Steady-state viscosity and complex viscosity for neat UPR. ....................... 146 
Figure 54: Morphology of the nitric acid oxidized (10M) SWNTs as seen by SEM at 
(left) 4,000x magnification and (right) 35,000x magnification. ..................................... 147 
Figure 55: Morphology of the 48h nitric acid (2.6 M) reflux oxidized SWNTs by SEM at 
(left) 4,000x magnification and (right) 35,000x magnification. ..................................... 148 
Figure 56: SEM images highlighting the morphology of the SWNT-Ox film SWNTs at 
(top) 4,000x magnification and (bottom) 35,000x magnification................................... 149 
Figure 57: Image (10x2) of the 0.100% vol. SWNT-Ox Film dispersion processed from 
three day shear mixing.................................................................................................... 149 
Figure 58: Comparison of the effect of final treatment of oxidized SWNT on dispersion.  
Both samples were acid oxidized for 1.5 h and dispersed at 0.100% vol. loading......... 151 
Figure 59: Comparison of the effect of final treatment of oxidized SWNT on dispersion 
by means of viscoelastic moduli.  Both samples are at 0.100% vol. loading. ................ 151 
Figure 60: Image of (left) lyophilized 6h oxidized SWNT 183.6 in UPR and (right) a 
SWNT 183.6 Pure dispersion after the same mixing time.  Both samples are at 0.005% 
vol. loading. .................................................................................................................... 152 
Figure 61: The lack of dispersion in the 0.005% vol. lyophilized 6h acid oxidized 
SWNTs.  Image taken with 20x objective and 2x magnification in front of camera (40x).
......................................................................................................................................... 153 
Figure 62: The lack of dispersion in the 0.005% vol. 48 h nitric acid reflux oxidized 
SWNTs.  Image taken with 20x objective. ..................................................................... 153 
xix 
Figure 63: A 6h oxidized sample as it appeared from the freeze dryer.  The sample mass 
was 15 mg before treatment............................................................................................ 154 
Figure 64: The FTIR spectra 6 h acid oxidized SWNTs (183.6 Pure) after lypholization.  
The sample was re-dispersed in water to saturation and dropped on a ZnSe ATR crystal.  
The sample spectra is representative of 5 film layers..................................................... 155 
Figure 65: The XPS survey spectrum for 6 h acid oxidized SWNT (183.6 Pure) after 
lyophilization.  The main surface constituents were primarily C and O.  Trace amounts of 
S and Si were observed................................................................................................... 157 
Figure 66:  Deconvolution of the high resolution C1s curve for the oxidized SWNT into 
four separate peaks.  The black jagged curve is the raw data and the blue line intersecting 
this curve represents the sum of the smaller peaks......................................................... 158 
Figure 67: The effect of sample history on the complex viscosity of a 0.025% vol. SWNT 
183.6 Pure-UPR dispersion.  The 50 mm diameter cone and plate geometry was used. 160 
Figure 68: The effect of sample history on the storage modulus of 0.025% vol. SWNT 
183.6 Pure-UPR dispersion............................................................................................. 161 
Figure 69: Comparison of material dependant storage moduli at 0.100% vol. loading.  
The samples were subjected to a controlled linear viscoelastic strain while the angular 
frequency was varied. ..................................................................................................... 162 
Figure 70: Comparison of material dependant loss moduli at 0.100% vol. loading....... 164 
Figure 71: Comparison of material dependant complex viscosity at 0.100% vol. loading.
......................................................................................................................................... 164 
Figure 72: Comparison of material dependant shear viscosity at 0.100% vol. loading.. 165 
xx 
Figure 73: Structural stability of SWNT 183.6 Pure-UPR dispersions as measured by 
ramping strain amplitude at constant shear frequency.  The circles indicate the critical 
strain at the edge of the linear viscoelastic threshold marked by non-linear response in 
storage modulus. ............................................................................................................. 167 
Figure 74: Characterization of the linear viscoelastic regime and transition to non-linear 
strain response for a 0.250% vol. SWNT dispersion...................................................... 168 
Figure 75: Loss moduli of SWNT 183.6 Pure-UPR dispersions as measured by ramping 
strain amplitude at constant shear frequency.................................................................. 169 
Figure 76: The linear viscoelastic response of the SWNT-UPR dispersions as a function 
of both SWNT 183.6 Pure loading and the shear frequency.  Small square symbols 
represent G? and small triangle symbols represent G?. .................................................. 170 
Figure 77: Viscoelastic dependence on concentration removed by means of colloidal 
scaling [199].  The loss modulus is the raw value of G? for the neat UPR.  Its frequency 
was not shifted. ............................................................................................................... 171 
Figure 78: Plot of the parameters used to scale G? and G? to master curves. ................ 172 
Figure 79:  The storage modulus response as a function of angular frequency and 
increasing nanotube loading for purified SWNT 183.6 dispersed in UPR..................... 173 
Figure 80: The loss modulus response as a function of angular frequency and increasing 
nanotube loading for purified SWNT 183.6 dispersed in UPR. ..................................... 174 
Figure 81:  The complex viscosity with increasing nanotube loading for purified SWNT 
183.6 dispersed in UPR................................................................................................... 175 
Figure 82: Image (20x2) of a 0.400% vol. SWNT-UPR dispersion processed shear 
mixing. ............................................................................................................................ 176 
xxi 
Figure 83: Divergence plot showing the complex viscosity as a function of complex 
modulus [200].  The arrows point to the value of G* that is taken to be the yield stress.
......................................................................................................................................... 178 
Figure 84: The divergent value of the complex modulus from Figure 85 plotted against its 
corresponding loading..................................................................................................... 179 
Figure 85: The storage modulus as a function of SWNT loading for fixed angular 
frequencies on a log-log axis. ......................................................................................... 180 
Figure 86: Power law scaling of the storage modulus as a function of the reduced volume 
fraction at 0.1 rad/s. ........................................................................................................ 181 
Figure 87: Power law scaling of the storage modulus as a function of the reduced volume 
fraction at 0.01 rad/s showing similar behavior as the trend at 0.1 rad/s........................ 182 
Figure 88:  High frequency regression of the scaled moduli after removal of 0.025% vol. 
data set. ........................................................................................................................... 183 
Figure 89:  Plot of the crossover modulus and frequency on a log-log axis................... 184 
Figure 90:  TEM image of the SWNT-UPR dispersion (0.050% vol.) after acetone 
washing.  A mixture of bundled SWNT fibers ranging in lengths and diameters can be 
seen. ................................................................................................................................ 185 
Figure 91:  Image of the 0.010% vol. SWNT-UPR dispersion processed shear mixing.  
The image was taken through cover glass with a 60x oil immersed objective and 2x 
magnification in from of the camera (120x)................................................................... 186 
Figure 92:  Structural stability of SWNT 183.6 Pure-UPR dispersions following 
successive dilutions.  The moduli were measured by ramping strain amplitude at constant 
angular frequency of 10 rad/s. ........................................................................................ 187 
xxii 
Figure 93:  Storage and Loss Moduli from successive dilutions of SWNT 183.6 Pure 
dispersed in UPR.  The magnitude of G? clearly increases over all frequencies with 
increasing loading. .......................................................................................................... 188 
Figure 94:  Complex viscosity from successive dilutions of SWNT 183.6 Pure dispersed 
in UPR............................................................................................................................. 188 
Figure 95:  Reduced complex viscosity from successive dilutions of SWNT 183.6 Pure 
dispersed in UPR............................................................................................................. 189 
Figure 96: Shear rate dependant start-up of flow behavior for a 0.005% vol. SWNT 
dispersion. ....................................................................................................................... 191 
Figure 97: Images showing the transparency of 0.005% vol. purified SWNT 183.6 
dispersion after treatment with a steady low shear rate of 0.01 1/s.  The sample gap was 
constant at 0.685 mm. ..................................................................................................... 192 
Figure 98: Sample from Figure 99 carefully transferred from the rheometer in its 
aggregated state to preserve the structure. Image taken with a 10x objective and 2x 
magnification. ................................................................................................................. 193 
Figure 99: A second image taken at 10x2 magnification showing highly striated structure.
......................................................................................................................................... 193 
Figure 100: A closer image of the sample showing some diffuse structure.  The image 
was taken with a 20x objective and 2x magnification in front of the camera. ............... 194 
Figure 101: Reduced complex viscosity from successive dilutions of SWNT 187.3 
dispersed in UPR............................................................................................................. 195 
xxiii 
Figure 102: Comparison of start-up flow curves for different dilute CNT samples 
dispersed in UPR at 0.005% vol.  The shear rate was constant at 0.01 1/s for all samples.
......................................................................................................................................... 196 
Figure 103:  Comparison of start-up flow curves for different shear rates on dilute SWNT 
183.6 Pure samples dispersed in UPR at 0.005% vol..................................................... 197 
Figure 104: From left to right: 183.6 Unpure, 183.6 Pure, 187.3, and Unidym SWNTs.  
Only the purified 183.6 SWNT sample did not phase separate when left undisturbed.  
Timescale of observation was on the order of weeks. .................................................... 199 
Figure 105: TEM image of a large SWNT bundle taken from a 250 ppm dispersion and 
diluted many times with acetone.  The darker areas and low resolution is a result of 
adsorbed UPR remaining after the wash......................................................................... 200 
Figure 106:  TEM image showing good removal of the UPR from the bundles revealing 
the interstitial space where solvent may reside.  UPR removal was achieved by diluting 
the sample in acetone followed by 5 min. bath sonication. ............................................ 201 
Figure 107: The XPS survey spectrum for SWNT 183.6 Pure.  The main surface 
constituents were primarily C with little O content.  Trace amounts of S was also 
observed. ......................................................................................................................... 202 
Figure 108: The XPS survey spectrum for the as-received SWNT 183.6 Unpure.  The 
main surface constituents were primarily C and O.  Trace amounts of Si was observed.
......................................................................................................................................... 203 
Figure 109: The XPS survey spectrum for SWNT 187.3.  The main surface constituents 
were C and O but trace amounts of the elements Si, S, and Sn were also observed. ..... 203 
xxiv 
Figure 110: The XPS survey spectrum for Unidym SWNTs.  The main surface 
constituents were primarily C with little O content.  Trace amounts of the elements Si, S, 
and Sn were also observed.............................................................................................. 204 
Figure 111: High resolution C1s curves for similar structured SWNTs......................... 205 
Figure 112: High resolution C1s curves for SWNTs with significant amount of surface 
bound oxygen as determined from the overall scans...................................................... 206 
Figure 113: Raman Spectra at 514 nm and 785 nm excitation of the partially purified as-
received SWNT 183.6 Unpure........................................................................................ 208 
Figure 114: Raman Spectra at 514 nm and 785 nm excitation of the SWNT 183.6 Pure.
......................................................................................................................................... 209 
Figure 115: Raman Spectra at 514 nm and 785 nm excitation of as-received Unidym 
SWNT. ............................................................................................................................ 209 
Figure 116: Raman Spectra at 514 nm and 785 nm excitation of as-received SWNT 
187.3................................................................................................................................ 210 
Figure 117: Progression of oxidation reproduced from Yue et. al (1999) [159]. ........... 211 
Figure 118: A lyophilized film of the 6h acid oxidized SWNTs.................................... 217 
Figure 119: Close up view on the edge of the film shown in Figure 118.  A high order of 
alignment can be seen even on the nanometer scale....................................................... 218 
Figure 120: Optical microscopy image of a piece of lyophilized 6 h acid oxidized SWNT 
film.  The film was held between a cover glass and imaged at 10x magnification. ....... 219 
Figure 121: Optical image of a bulk 6 h acid oxidized and lyophilized SWNT film.  The 
sample was affixed to double sided adhesive tape and the image was taken with a 20x 
objective and 2x magnification in front of the camera. .................................................. 219 
xxv 
Figure 122 : SEM image of a 2 h acid oxidized SWNTs after lyophilization................ 221 
Figure 123: Close up image of the 2 h acid oxidized SWNTs after lyopholization. ...... 222 
Figure 124: Raman Spectra of SWNTs after 6 h acid oxidation and lyophilization. ..... 223 
Figure 125: Raman Spectra of SWNTs after 2 h acid oxidation and lyophilization. ..... 223 
Figure 126: Image of (left) SWNT 183.6 after aggressive tip sonication in water for 30 
min. pulsed at 5 s on 1 s off and (right) dilute aqueous solution of lyophilized 6 h 
oxidized SWNT 183.6 re-dispersed in water by bath sonication.................................... 224 
Figure 127: The UV-visible spectra for 6 h acid oxidized SWNTs after lypholization and 
re-dispersion in water and variation of solution pH........................................................ 225 
Figure 128: Drop dried image of lyophilized 6 h acid oxidized SWNTs that were re-
dispersed in water.  Both SWNTs and amorphous carbon generated from the oxidation 
can be seen. ..................................................................................................................... 226 
Figure 129: Drop dried image of lyophilized 6 h acid oxidized SWNTs that were re-
dispersed in water and centrifuged at 17k x g for 90 min............................................... 227 
Figure 130: High resolution image of crack bridging from a drop dried and centrifuged 
aqueous dispersion of lyophilized 6 h acid oxidized SWNTs. ....................................... 228 
Figure 131: Drop dried image of the 6 h oxidized dispersion after centrifugation and drop 
drying showing crack propagation/bridging phenomena................................................ 229 
Figure 132: Comparison of the Raman Spectra at 514 nm excitation of SWNT 183.6 Pure 
before and after (183.6 Ox) acid oxidation showing small diameter digestion.............. 230 
Figure B1: The storage modulus as a function of SWNT loading for fixed angular 
frequencies on a lin-log axis. .......................................................................................... 250 
Figure B2: Plot of the crossover modulus and frequency on a lin-log axis.................... 251 
xxvi 
Figure C1: FTIR spectra for SWNT 187.3.  The sample was dispersed in 1,2-
dichlorobenzene and dropped on a ZnSe ATR crystal.  The peaks are representative of 
sp
2
 hybridized carbons. ................................................................................................... 252 
xxvii 
LIST OF TABLES 
Table 1: Purity of the SWNT samples as determined by TGA residual weight............. 111 
Table 2: Volume percentages and the corresponding mass percentages for the SWNT 
183.6 Pure-UPR dispersions. .......................................................................................... 123 
 
1 
1. INTRODUCTION
 Since the discovery of carbon nanotubes (CNTs) in the early 1990s there has been 
a growing interest in nanotube based research as indicated by reported trends in published 
articles and patents.  Recently, a similar trend has been observed for carbon nanotube-
polymer composites [1].  Without question, this is a result of the unique combination of 
physical properties possessed by carbon nanotubes and the desire to translate these 
characteristics to polymers.  Furthermore, the high surface area and large aspect ratio 
(length to diameter ratio) of this material, along with low mass density, make carbon 
nanotubes an ideal reinforcement material for composite systems.  However, as a result 
of the nanoscopic dimensions and high aspect ratio of CNTs, deviations from expected 
physical behavior for traditional composite systems are found.  This behavior is 
indicative of the increased interfacial interaction between nanotubes with themselves as 
well as the polymer matrix and must be addressed to exploit the full potential of carbon 
nanocomposite systems [2].   
 A significant hurdle in the efficient preparation of polymer - carbon nanotube 
composites lies in effectively dispersing the CNTs throughout the matrix phase.  Due to 
the smooth and highly polarizable sidewalls, single-walled carbon nanotubes (SWNTs) 
often exist in entangled crystalline ropes or bundles with a characteristic binding energy 
of ~0.5 eV/nm of parallel contact.  Given that nanotube length is often on the order of 
hundreds to thousands of nanometers, separating nanotubes from bundles is a significant 
 
2 
undertaking.  In light of this, a variety of chemical and physical techniques have been 
employed in the literature to process nanotubes as individuals.  Often the successful 
processing route is material or application dependant with solution based processing 
being highly effective for thermoset plastics.   
 In this work, the first detailed study on the shear dispersion of carbon nanotubes 
into an isophthalic unsaturated polyester resin (UPR) is reported.  The effect of 
incorporating pristine SWNTs, chemically oxidized SWNTS, or vapor grown carbon 
fibers into UPR was studied.  The resulting complex fluid was studied prior to cure as a 
means of identifying what physical phenomena may influence the final composite 
properties.  Specifically, bulk rheological properties of the system were measured 
revealing the unique viscoelastic behavior of this new class of material.  It was found that 
the nanotubes formed an elastic network above a critical concentration and show non-
Brownian behavior under imposed shear.  Additionally, an interesting low shear 
aggregation phenomena was discovered.  Comparative analysis with respect to varying 
nanotube surface chemistry was performed in order to determine what controlling factors 
are responsible for dispersion stability.   
 The results of this work are intended to motivate increased interest and 
understanding with respect to producing commercially viable carbon nanotube - UPR 
based composite materials, as well as open the door for possible new material 
technologies and applications.  Also, en route to developing unique methods for 
dispersing chemically oxidized nanotubes a new technique enabling the self - assembly of 
aligned and transparent nanotube films though lyophilization was discovered.  
 
3 
 Finally, this manuscript was prepared with the intention of providing both new 
and current carbon nanotube-polymer composite researchers with a thorough review of 
the important underlying fundamental science related to the preparation of 
nanocomposite systems through solution based methods.  To this end, a critical analysis 
of the current state of nanotube solution theory appearing in the literature was performed.
 
4 
2. BACKGROUND 
Polymers and Their Composites: 
 Polymers, in general, can be classified into two distinct classes based on their 
response to heat [3].  Thermoplastic polymers, in the simplest sense, melt when heated 
and re-solidify when cooled.  It is this inherent phase cycle of thermoplastics which 
allows them to be repeatedly processed and reprocessed.  Conversely, thermosetting 
plastics cure with the onset of heating and therefore do not melt when reheated but 
significantly degrade.  A simple explanation of this phenomenon can be visualized by 
examining the polymer chain structures in Figure 1.          
 
Figure 1: Thermoplastic (a) and thermoset (b) polymer structures.  Image reproduced from Kumar 
and Gupta (2003) [3]. 
 
Figure 1a displays a slightly branched thermoplastic structure in which the different 
polymer molecules associate through secondary associative forces (i.e. van der Waals).  
When energy is supplied to the structure the secondary forces are overcome and the 
thermoplastic polymer melts.  In contrast, Figure 1b displays a thermoset polymer post-
cure.  In this structure, the polymer molecules associate into a three-dimensional network 
supported by covalent bonds or cross-links between chains.  As opposed to van der Waals 
association found in thermoplastics, the energy required to break these covalent cross-
links is typically greater then the energy required to degrade the polymer itself [4].  By 
a) b) 
 
5 
this analysis one can argue why thermoset plastics are advantageous for high temperature 
applications; thermosets hold their shape at temperatures which thermoplastics cannot.   
 As a result of their high cross-link density thermoset plastics show increased 
resistance to heat but are typically hard and brittle, limiting their potential for load 
bearing applications [4].  These shortfalls can be overcome by processing thermoset 
plastics as composite materials, a technique that has been used in common practice for 
some time and discussed in later sections.  Polymer composites are a class of materials in 
which some reinforcement with desired properties, for example a fiber, is incorporated 
into a polymeric matrix.  Traditional fiber reinforcement materials include, but are not 
limited to, glass, carbon, and Kevlar fibers.  The properties of the reinforcing material are 
translated to the matrix material and realized in the resulting composite.  Some common 
examples of mechanical properties that composite processing may enhance include 
tensile strength, impact strength, and toughness [5].    
 
Polymeric Carbon Nanotube Composites: 
 In recent years there has been increasing research attention towards processing 
polymeric nanocomposites [6-8].  Similar to traditional polymer composites, a 
nanocomposite still comprises of a two-part system of a filler and matrix phase.  
However, one dimension of the reinforcement material exists between 0.1 ? 100 nm, an 
order of magnitude greater than its atomic constituents.  This ?nano-size? yields a unique 
dependence on material properties and high surface areas not found in traditional 
composite systems [8].  In fact, it is the very high surface area per unit volume ratio that 
makes nanomaterials such as carbon nanotubes attractive as reinforcement material for 
 
6 
polymer matrices.  As the interfacial surface area between the polymer matrix and 
nanomaterial increases, the macroscale properties of the bulk composite become 
dominated by the interface [8].  Moreover, since the interfacial area is quite large per unit 
volume one would expect that low loadings of nanomaterial reinforcement show marked 
property enhancement.  In summary, the exceptionally high aspect ratio, strength, and 
stiffness of carbon nanotubes, in combination with a low density, make nanotubes an 
ideal candidate for reinforcement in polymeric materials [9]. 
 While many thermoplastic polymers have been used to construct carbon 
nanotube-polymer composites, the majority of work with thermoset polymers has largely 
focused on epoxy based resins [1, 6, 8, 10-13].  However, vinyl ester (VE) resins have 
recently emerged in growing popularity as nanocomposite matrix polymers [8, 14-16].  
VE resins were first commercialized in 1965 by Shell Chemical under the trade name 
Epocryl and show excellent corrosion resistance [17].  An example synthesis scheme of a 
VE monomer from an epoxy resin can be found in Thostenson et al. (2008) [15].  
Essentially, vinyl esters are methacrylated epoxies which can be cured in the same 
manner as unsaturated polyester resins (UPR) [18].  Neat unsaturated polyester resins and 
VE resins are alike in that their high viscosity requires dilution with styrene monomer for 
processing and cure.  But, in contrast to Epoxy and VE resins, unsaturated polyester 
resins have rarely been used in the literature for nanocomposite research.  In fact, Seyhan 
et al. (2007) claim to be the first to study the processing and properties of CNT-UPR 
composite systems using an isophthalic polyester resin [19].  Most recently, Battisti et al. 
(2008) have also studied the ?on-line? dispersion of CNT using an isophthalic polyester 
resin [20].  It is important to note that the common thread between the thermoset research 
 
7 
to date has been the preference of multiwalled carbon nanotubes (MWNTs) in the 
majority of all work.  Presumably this is because of the relative ease of dispersing 
MWNTs over single-walled carbon nanotubes (SWNTs).   
 
Commercial Applications: 
 In spite of intense research interest CNT based nanocomposites are just starting to 
become available commercially [1].  Statically dissipative plastics have been brought to 
market by Hyperion Catalysis that possess improved toughness while retaining surface 
appeal.  Additionally, Unidym has developed CNT based transparent conductive films for 
use in touch panel, solar, and display applications.  Additionally, in the sporting goods 
business nanocomposites containing CNT have emerged in several products.  For 
example, Easton has developed a resin containing CNTs for use in the manufacture of 
bicycle components and baseball bats.  The resin is used fill in void spaces between 
traditional carbon fibers which the company claims has ?radically improved strength and 
toughness? in these areas [21].  Likewise, Babolat has used CNT to stiffen areas of their 
tennis racquets.  The company claims a 20% increase in torsion resistance with more 
power and stability [22].  Stultz golf has developed golf club shafts using the carbon 
fiber, CNT, and resin technology as well [23].  It is important to point out that the 
improvements enjoyed by the consumer in product performance are also supplemented 
with a reduction in weight, as discussed in later sections.   
 The promise of producing composite materials with improved performance from 
lightweight components has potential applications in a variety of industries.  Of particular 
interest is the ground and aerospace transportation industry, as well as applications in 
 
8 
defense.  The motivation is simple to understand by considering the work required to 
propel a jet engine.  The force necessary to move the engine over a distance could be 
directly reduced if the weight of the engine components were reduced.  In turn, if the 
work requirement is reduced the amount of energy needed to propel the jet has decreased 
leading to an increase in fuel efficiency.  A perfect example of the use of composites for 
aerospace applications is the Boeing 787-3 Dreamliner.  This aircraft has a capacity of 
290-330 passengers [24].  In order to safely compensate for the weight of this number of 
passengers lower fuel consumption was required.  The only expendable variable was the 
weight of the airliner components themselves.  To this end, Boeing used approximately 
50% wt. composite materials in their design [25].  Specifically, 35,000 kg of the 
traditional carbon fiber reinforced plastic, the primary composite material in the aircraft, 
was used.  With respect to performance this translates to a weight reduction of 30,000-
40,000 lbs over the comparably sized Airbus A330-200 and allowing for a 45% wt. 
increase in cargo carrying capacity.  Additionally, a 20% reduction in both fuel 
consumption and emissions was achieved [24].  The use of composite materials to 
construct the fuselage allowed the elimination of 1,500 aluminum sheets and 40,000-
50,000 fasteners.  This reduction and replacement of parts with corrosion resistant 
materials will improve reliability and maintenance costs in the long run. 
 Using the 787-3 Dreamliner as an example of how traditional composite materials 
can enable improved performance in lightweight applications the question remains as to 
what can be achieved from nanocomposite materials and to what extent the limits of 
application may be realized.  It is for this reason that there is interest in nanocomposite 
 
9 
research, to design the next class high performance materials with the hope of developing 
enabling technologies yet to be seen.            
 
Materials: 
 In this research it was desired to study the mechanical property enhancement 
achieved by the addition of carbon nanocylinders into an unsaturated polyester resin.  
Specifically, an isophthalic polyester was used in conjunction with two different 
nanocylinder materials of varying dimension so as to provide a range of material aspect 
ratios.  The carbon nanocylinders used in this work were vapor grown carbon nanofibers 
(VGCFs) and single-walled carbon nanotubes (SWNTs).  The following sections are 
intended to give an exhaustive description of these materials. 
 
Unsaturated Polyester Resins: 
 Unsaturated polyester resins (UPRs) are thermoset pre-polymers comprised of 
linear short chain oligomers.  These oligomers are formed from the condensation 
polymerization of either saturated/unsaturated acids or acid anhydrides with difunctional 
alcohols or oxides [17].  The result is a viscous liquid or brittle solid depending on the 
degree of polymerization when initially terminated.   Typical UPR molecular weights 
range from 1,200-3,000 g/mol represented by only a few repeat units (n = 3,4) in Figure 2 
[17, 26].  UPR is readily soluble in styrene monomer and can be found in commercial 
solutions as high as 40% by weight styrene.  The styrene serves both to lower the 
polymer viscosity for processing as well as act as a cross-linking agent between 
 
10 
unsaturation sites on adjacent oligimer chains to form a three-dimensional network 
structure.   
O
*
O
O
O
O
O
O
O
O
O
OH
O
n
 
 
Figure 2: Idealized ?baseline? chemical structure of a 1:1 isophthalic polyester constructed from 
equimolar ratios of isophthalic acid and maleic anhydride.  Isophthalic polyesters are known for 
their corrosion resistance, superior mechanical properties, and higher heat distortion temperatures 
with respect to other types of thermoset polyester.  Reproduced from Mallick (1997) [18].    
 
It is the formation of a network that distinguishes thermoset polyester resins from other 
linear thermoplastic polyesters such as poly(ethylene terephthalate) [27].  Typical acids 
used in UPR production are phthalic anhydride, a saturated acid, or an unsaturated 
dicarboxylic acid such as maleic anhydride.  Generally a diol such as propylene glycol or 
diethylene glycol is also used.  Depending on the desired resin properties, the UPR recipe 
can be adjusted.  For example, the addition of a saturated acid, such as isophthalic acid, 
decreases crosslink density upon cure which also serves to decrease brittleness, increase 
tensile properties, but lower thermal stability.  Figure 3 displays the chemical structure of 
isophthalic acid.  Also, fumaric acid is commonly used in place of maleic acid to increase 
impact resistance.  If the resin is meant to have a shelf life an amount of a free radical 
scavenging inhibitor such as benzoquinone is added to prevent any premature 
polymerization.  More detailed recipes for tailoring UPR properties can be found in 
 
11 
Mallick (1993) [4].  However, in general UPR can be formulated such that the final 
product is either hard and brittle or soft and flexible.    
OHO
O
OH
 
Figure 3: The chemical structure of isophthalic acid.  In contrast to other unsaturated acids used 
in polyesters, there is no anhydride form of isophthalic acid.  Reproduced from Mallick (1997) 
[18].   
 
 Curing is initiated by the addition of some type of organic peroxide and 
crosslinking propagates via a free radical reaction mechanism across the styrene 
molecule.  Generally, the peroxide is cleaved creating a free radical which attacks the 
unsaturated vinyl-carbon bonds in styrene, whose structure is shown in Figure 4.   
 
Figure 4: The chemical structure of styrene monomer.  Crosslinking between UPR oligomers 
propagates through the ?CH=CH
2
 ?tail.? 
  
Styrene radicals, in turn, attack unsaturation sites on the polyester oligomers and 
crosslink them by forming a bridge between adjacent molecules [4]. Typically for low 
temperature curing methyl-ethyl-ketone peroxide (MEKP) is used.  To help facilitate 
curing cobalt based accelerator is added which aids in cleavage of the peroxide groups.  
 
12 
The cure time and conditions as well as the mechanical properties of UPRs vary with 
both the oligomer constituents, the degree of cross-linking, and the amount of 
condensation product trapped in the polymeric network.  The fact that UPR can be cured 
at room temperature makes its application favorable for manufacture of large parts such 
as boat hulls or automobile bodies [26].   
 From a manufacturing standpoint, UPR is advantageous choice for its low 
viscosity, fast cure time, and low cost.  However, its mechanical properties are generally 
not as favorable as those of epoxy resins.  For cast UPR, a tensile strength of 34.5-103.5 
MPa is expected with a tensile modulus of 3.1-3.45 GPa.  On the other hand, cast epoxy 
resin can show a tensile strength and modulus as high as 130 MPa and 4.1 GPa, 
respectively.  Conversely, the thermal properties of UPR show better high temperature 
performance compared to epoxy resins.  The continuous working temperature for an 
epoxy resin is typically 150
o
C or less.  UPR shows heat deflection temperature as high as 
205
o
C making it a better choice for high temperature applications [4].  In addition, the 
mechanical disadvantages of UPR can be improved upon if processed as a composite 
material.  In practice, the peroxide or curing catalyst is first added to the resin, blended, 
than the material is processed.  Commonly UPR is compression molded into parts but can 
be resin-transfer molded for intricate geometries which are allowed to cure before 
removal [27].  If not molded, UPR is typically applied as a glass-fiber laminate.  The high 
strength to weight ratio of polyester-glass laminates, microwave transparency, and 
corrosion resistance have led to use in many air transport applications [26].  One of the 
most common polyester laminates goes by the trade name Fiberglas
?
 (Owens Corning).  
Fiberglas
?
 is a glass-fiber laminate material which became popular in to 1940s and is 
 
13 
widely used in the automotive, marine, construction, and aerospace industries.  
Fiberglas
?
 found its way as an alternative to heavy porcelain shower stalls and bathtubs 
but made a memorable impact in the construction of automotive parts when in 1953 
Chevrolet produced the first Corvette with a Fiberglas
?
 body [27, 28].  
 In the year 2000, UPR sales of 681 million kg were reported in the U.S. alone, 
ranking 3
rd
 among thermoset plastics behind phenolic resins and polyurethane.  This was 
over twice the amount of epoxy resin sold that year but less than 2% of the total market 
for plastics.  It is interesting to note that thermosets were responsible for less than 14% of 
the total sales of ~42,900 million kg of polymer sold in 2000 [29].                
           
Vapor Grown Carbon Nanofibers: 
 Vapor grown carbon nanofibers (VGCFs) are hollow shelled carbon fibers having 
various morphologies and outer diameters typically ranging from 50-200 nm [30].  
Lengths of the as grown fibers can range from 10 ?m to over 100 ?m.  The production of 
VGCFs is commonly a two step process where the fiber is grown catalytically in the axial 
direction and subsequently thickened by deposition on the outer diameter [31, 32].  Two 
common morphologies are the ?bamboo? and ?stacked cup? structured VGCFs as shown 
in Figure 5.  The bamboo structured VGCFs are grown directly on a catalyst particle and 
show surface defects [33].  Stacked-cup VGCFs consist of truncated conical graphene 
layers that have a central hollow core.  The inner and outer edges of stacked-cup VGCFs 
consist of exposed reactive graphene layers.  The specific surface area of VGCFs is on 
the order of 10
2
 m
2
/g [34].  Some commercial available grades of VGCFs have been 
reported to contain both morphologies [30].   
 
14 
 
Figure 5: High resolution images of (a) bamboo structured VGCFs and (b) stacked cup structured 
VGCF.  Images reproduced from Merkulov et al. (2000) and Endo et al. (2003), respectively [33, 
34].     
 
The tensile strength and tensile modulus of VGCFs has been measured experimentally as 
2.92 GPa and 237 GPa, respectively, and are comparable to commercially available 
poly(acrylonitrile) fibers [35].  However, reported tensile moduli have been wide ranging 
(a) 
(b) 
 
15 
in the literature.  Uchida et al. (2006) theoretically determined the modulus for the 
various VGCF morphologies to explain these discrepancies reporting a range starting as 
low as 50 GPa and as high as 775 GPa [30].  The thermal conductivity of VGCFs has 
been estimated as 1260 W/m-K [36].  The electrical resistivity of heat treated VGCFs has 
been reported as low as 0.1-1 ??-cm due to crystallization of the tube exterior [37]. 
 
Carbon Nanotubes: 
 Carbon nanotubes were first discovered by Iijima (1991) as needle-like structures 
deposited on an arc-discharge electrode used for fullerene production [38].  These 
needles appeared as coaxial tubes of graphite and were actually multiwalled nanotubes 
(MWNTs).  Soon after, Ebbesen and Ajayan (1992) first produced MWNTs in gram 
quantities [39].  It was not until 1993 that single walled nanotubes (SWNTs) were 
intentionally produced [40, 41].  Dyke and Tour (2004) have described SWNTs as ladder 
polymers of carbon where more than one bond needs to be broken to cleave the 
backbone.  However, SWNTs are far superior to typical ladder polymers since 
approximately 10-20 highly ordered carbon-carbon bonds per repeat unit need be cleaved 
for tube scission to occur.  Since the sp
2
 hybridized carbon-carbon bond is among the 
strongest known it is unlikely that a more robust material will ever be discovered [42].    
      
Characteristics of Carbon Nanotubes: 
Molecular Structure: 
 In general, SWNTs can be considered as seamless hollow cylinders of rolled 
graphene sheets having a diameter on the order of 1 nm and lengths typically no larger 
 
16 
than a micron.  In some cases, isolated SWNTs have been grown as long as 4 cm [43].  
Similarly, multi-walled carbon nanotubes (MWNTs) and double-walled carbon 
nanotubes (DWNTs) consist of coaxially arranged, nested cylinders of graphene.  Each 
concentric layer is spaced 0.34 nm apart and the MWNTs have typical lengths on the 
order of microns with outer diameters ranging from 2-20 nm [1, 32].  Thus, SWNTs and 
MWNTs have aspect ratios of approximately 10
2
-10
3
 [32].  However, these parameters 
vary within, and between, samples and batches from various laboratories or 
manufacturers [1].  In absence of defects, the carbon atoms that comprise the graphene 
sheet are covalently joined to one another via sp
2
 hybridized bonds.  However, the strain 
associated with out of plane bending of the surface increases with decreasing tube radius 
and increases the sp
3
 hybrid character of the surface [44-46].  Thus, small diameter CNTs 
are the most reactive.  For this reason the larger diameter MWNTs show weak sp
3
 
character unless the surfaces are facteted [46].      
 All known preparations of SWNTs give mixtures of diameters, lengths, and 
chiralities [42].  The chirality of a CNT refers to the number of ways a graphene sheet 
may be rolled upon itself.  Specifically, tube chirality is identified with a pair of integers 
(n,m) which in turn define the circumferential chiral vector from which the graphene is 
rolled upon.  Equation 2.1 defines the chiral vector, C
h
. 
 
21
amanC
h
rr
r
+=
 (2.1) 
Figure 6 displays the geometrical constructs of C
h
 on a graphene lattice.  The length of 
C
h
 is represented by vector OA, the nanotube circumference.  Orthogonal to C
h
 lays the 
translation vector T which serves to define a unit cell.  The chiral indices (n,m) describe 
the number of steps along the hexagonal unit vectors a
1
 and a
2
 as displayed by Figure 7. 
 
17 
 
Figure 6: C
h
 is defined on the graphene lattice by unit vectors a
1
 and a
2
 and the chiral angle ? 
with respect to the zigzag axis (? = 0
o
).  The lattice vector of the 1-dimensional unit cell is 
defined by T.  The rotation angle ? and the translation ? constitute the basic symmetry operation 
R = (?| ?) for the carbon nanotube.  The diagram is constructed for (n,m) = (4,2).  Reproduced 
from Dresselhaus et al. (1995) [32]. 
 
The magnitude of the unit vectors a
i
 can easily be determined geometrically.  The 
distance between adjacent vertices on a single hexagon is given by the carbon-carbon 
bond length (L
C-C
), approximately 1.42 nm for graphite [32].  Considering that a hexagon 
can be divided into 6 equilateral triangles, each unit vector will bisect two equilateral 
triangles along a length of distance
CC
L
?
?3
, defining the lattice constant (0.246 nm). 
Likewise, the length of vector C
h 
shown by Equation 2.1 can be easily derived from 
Figure 7 using the lattice constant.   
 
18 
 
Figure 7: Possible vectors specified by (n,m) for general CNTs, including zigzag, armchair, and 
chiral tubes.  Below each integer pair is listed the number of distinct caps that can be joined 
continuously to the CNT denoted by (n,m).  Encircled dots denote metallic structure and small 
dots are for semiconducting tubes.  Reproduced from Dresselhaus et al. (1995) [32].  
 
Equation 2.2 displays this relationship between tube diameter and the magnitude of C
h
. 
 
( )
??
h
CC
C
nmnmL
d
r
=
++
=
?
2/1
22
3
 (2.2) 
The division by ? is necessary to relate the tube circumference to the diameter.  As 
displayed by Figure 7 the indicies (n,m) and therefore C
h
 determine the tube?s electronic 
structure.  Despite the structural similarity to a single sheet of graphite, which is a 
semiconductor with zero band gap, SWNTs may be either metallic or semiconducting 
[47].  Specifically, if the difference n ? m is nonzero and divisible by three, the nanotube 
is considered semimetallic (or simply metallic) with a bandgap on the order of meV.  If 
the difference is equal to zero the nanotube is metallic (ballistic conductor) with a 
bandgap of zero.  In all other cases where the difference n ? m is nonzero and not 
 
19 
divisible by three the nanotube is semiconducting with a band gap ranging from 
approximately 0.5 to 1 eV [42].  Structurally the vectors (n,0) and (0,m) denote zig-zag 
tubes and the vectors having indices that match denote armchair tubes, named by 
examining a ring of carbon atoms around the circumference.  Both armchair and zig-zag 
tubes are achiral since they have a mirror plane whereas all other vectors where n is not 
equal to m or zero are chiral (twisted).  Figure 8 displays a schematic of each type. 
 
Figure 8: Schematic theoretical model for a SWNT with the tube axis normal to: (a) the ? = 30
o
 
direction with (n,m) = (5,5) (an ?armchair? tubule), (b) the ? = 0
o
 direction with (n,m) = (9,0) (a 
?zigzag? tubule), and (c) a general direction 0 < ? < 30
o
 with (n,m) = (10,5) ( a ?chiral? tubule).  
Reproduced from Dresselhaus et al. (1995) [32].   
 
(a) 
(b)
(c) 
 
20 
Mechanical Properties: 
 The mechanical properties of carbon nanotubes (CNT) have been determined by 
both experiment and theory.  Gao et al. (1998) theoretically determined the Young?s 
modulus of an individual SWNT to be approximately 0.64 TPa [48].  Additionally, 
average experimental values for individual SWNTs have been reported as 1.25 TPa [49].  
This is also consistent with experimental values for SWNT ropes ranging from 0.32-1.47 
TPa [50].  Since small diameter SWNT ropes have been extended elastically to 
approximately 5.8 % the SWNT tensile strength can be calculated as approximately 37 
GPa [51].  Taking into account the low density of the hollow carbon shells comprising 
CNT the modulus and strength can be normalized revealing remarkable comparisons with 
structural materials.  For typical SWNTs the density normalized modulus is 
approximately 19 times that of steel and 2.4 times that of silicon carbide nanorods.  
Likewise, the density normalized tensile strength is approximately 56 times that of steel 
wire and approximately 1.7 times greater than silicon carbide [47].  For these reasons 
there is great interest in constructing macroscopic CNT materials while retaining the 
outstanding properties of individual tubes.  
 
Electronic and Thermal Properties: 
 The electronic properties of MWNTs and SWNTs are quite similar in perfectly 
structured CNTs [47].  Because of their nearly one-dimensional structure electronic 
transport occurs ballistically in the axial direction [52].  Therefore high currents can be 
carried with resistivity on the range of 0.1-200 ??-cm [53, 54].  The thermal conductivity 
for an individual MWNT has been reported in excess of 3000 W/m-K [55].  Likewise, the 
 
21 
thermal conductivity for SWNTs was found to approach or exceed that of diamond, the 
previous benchmark material [56].  Additionally, superconductivity has been observed in 
SWNTs at extremely low temperatures [57, 58].  Thermal stability has been reported at 
temperatures as high as 2800
o
C in vacuum and 750
o
C under air [59]. 
 
Surface Area: 
 It is this complement of excellent material properties coupled with low mass 
density and high aspect ratio (~10
2 
- 10
3
 for SWNTs) that make nanotubes desirable 
alternatives for traditional micron to millimeter scale fillers in polymer composite 
manufacturing [1].   In particular, high aspect ratio nanocylinders are geometrically 
unique in that their surface area per unit volume provides an enormous means of 
communication between the polymer matrix and dispersed phase; giving rise to great 
opportunity for effective load transfer [2].  For comparison to traditional composite 
systems, reducing the fiber diameter from the order of microns to nanometers increases 
the surface area per unit volume by three orders of magnitude [8].  Specifically for 
SWNTs this value is as large as 10
3 
- 10
4
 m
2
/ml [2].  But, in order to take advantage of 
the excellent properties of CNTs they first must be separated from themselves.  It is the 
poor solubility characteristics of SWNT that have hindered their chemical manipulation 
and thus their use in applications [60].   
 
 
22 
Colloidal Interactions: 
van der Waals Forces: 
 The same forces which allow the ladybug to crawl upon ceilings or permit the 
gecko to scale horizontal walls is also responsible for the binding together of nanotubes 
in ropes [61].  Indeed, such phenomenon are a result of the same van der Waals force 
long understood to show negative deviations from Boyle?s law of ideal gases.  For a 
closed system, this is interpreted as an attractive force between particles or atoms which 
reduces the force exerted on the container in a manner inversely proportional to the 
square of the system volume [62].  These forces allow condensed phases to form and are 
stronger within solids and liquids than gases.  But, what is important to note is that the 
observance of the van der Waals pressure indicates that even electrically neutral bodies 
attract [63].   
 The empty space between two point charges or atoms is actually a constant omni-
directional inundation of electromagnetic waves.  This is a result of continuous jostling of 
charges in matter leading to spontaneous and transient electromagnetic fields.  
Additionally, the charges of matter both influence and respond to other currents and 
fields.  The additive effects of coordinated interactions of electromagnetic stimuli are 
what create the van der Waals force.  Consider an atom surrounded by electromagnetic 
waves traveling in all directions.  Now allow the presence of a neighboring atom to 
interrupt fluctuations between the two bodies.  The net result is that the particles are 
pushed together.  An excellent analogy was given by Parsegian (2005) in which a 
rowboat on rough water is heading to shore.  Once the boat is close enough to the dock to 
quell the waves between the two, the surrounding wake will push the boat in [63].   
 
23 
 In order to extend this idea and describe larger bodies than atoms molecules 
Hamaker (1937) introduced the concept of pairwise-summation for particles [64].  The 
coefficient of interaction between objects is known as the Hamaker constant A and is a 
function of the material.  The Hamaker constant is typically on the order of 10
-19
 to 10
-21
 J 
[64, 65].  Equation 2.3 gives the expression for A.   
 
21
2
??? CA =
 (2.3) 
Here C is the coefficient in the atom-atom pair potential and ?
1
 is the number of atoms 
per unit volume in Body 1.  The number of atoms per unit volume of Body 2 is 
represented by ?
2 
[65].  The atom-atom pair potential is given by Equation 2.4. 
 
6
D
C
W
?
=  (2.4) 
The potential W is the work required to bring the two atoms from infinite separation to D 
distance apart and is negative due to the sign convention used for attraction.  The 
treatment and correlation between attractive forces of curved or nonlinear surfaces with 
planar geometries was handled by Derjaguin (1934) [66].  This idea was termed as the 
?Derjaguin approximation? and took the attractive potential for planar surfaces and 
adapted it to describe curved surfaces at small separation.  In this research, interactions 
between cylindrical CNTs are of interest so only these shapes are considered.  For the 
non-retarded van der Waals interaction free energy between parallel cylinders of different 
radii the interaction potential is described by Equation 2.5 [65].   
 
2/1
21
21
2/3
212
?
?
?
?
?
?
?
?
+
?
=
RR
RR
D
AL
W
 (2.5) 
 
24 
This expression for parallel cylinders yields the energy per unit length L.  The distance of 
separation between their surfaces is represented by D and the cylinders have radii of R
1
 
and R
2
.  The attractive force between the cylinders can be found by taking the derivative 
of the work with respect to separation distance as shown by Equation 2.6.  
 
dD
dW
F
?
=
 (2.6) 
In general for parallel cylinders the force is inversely proportional to D
5/2
 power as shown 
by Equation 2.7, where the terms for the particle size were omitted to highlight the trend. 
 
2/5?
?? ALDF  (2.7) 
If the cylinders are crossed instead of parallel the interaction potential is in purely 
energetic units with no length dependence.  Equation 2.8 gives the interaction energy 
between perpendicular cylinders.  
 
D
RRA
W
6
21
?
=
 (2.8) 
Considering the trend with separation the force is inversely proportional to the square of 
the separation distance as displayed by Equation 2.9.  
 
2?
?? ADF  (2.9) 
In contrast to the treatment of pairwise atom interactions the Hamaker constants are 
typically determined from experimental data.  In this manner, the properties of the bulk 
material are averaged rather then examining individual constituent atoms or molecules.  
As a consequence of the empirical determination of A, values appearing in the literature 
were initially inconsistent in orders of magnitude [63].  This disparity was eventually 
treated by the introduction of new theories.  Regardless of the inconsistency of estimating 
 
25 
A the van der Waals potential functions displayed by Equation 2.5 and 2.8 are useful for 
examining the attractive potential as a function of the radial dependence and separation 
distance.  Some difficulties arise with determining A when using these expressions to 
describe CNTs as is discussed in later sections.   
 
DLVO Theory:   
 If van der Waals forces acted alone in Nature one might expect all dispersed 
particles to crash from solution immediately.  Fortunately, this is not the case since 
particles suspended in media are often charged and stabilized electrostatically [65].  
Derjaguin, Landau, Verwey, and Overbeek (DLVO) considered particle and collidal 
interactions as a function of two separable forces; the long range van der Waals attractive 
force and the repulsive electrostatic force.  The total interaction potential for 
electrostatically stabilized particles is represented as the sum of the electrostatic repulsive 
force and van der Waals attractive force.  Although the shape of the sum will vary with 
the magnitude of the contributions, Figure 9 displays one possible net energy curve as a 
function of interparticle separation. In this representation the total interaction potential 
shows metastable behavior with a secondary energy minimum (2
o
) at intermediate 
separation and a deep primary minimum (1
o
) or energy well at close distances.  The shape 
of the primary minimum is not highlighted by Figure 9 but in general hard sphere 
behavior is shown at particle contact with the potential energy diverging to infinity 
consistent with the Pauli exclusion principle for atomic orbitals.  The attractive behavior 
dominates at both small and large distances away from 2
o
.  On the other hand, at small 
distances the van der Waals (vdW) attraction must always dominate due to the power law 
 
26 
nature shown by Equation 2.3 which is not matched by any double-layer interaction [63, 
65].   
 
Figure 9: Schematic representation of the interaction potential for dispersion forces (vdW), 
electrostatic repulsion, and the total interaction potential showing metaseable behavior.  The 
primary (1
o
) and secondary (2
o
) minima are highlighted.     
 
If 2
o
 is accompanied by a large enough potential energy barrier (the peak of the total 
energy curve), particles are kept from falling to 1
o
 and forming irreversible aggregates.  
Typically, the energy at 2
o
 is quite low, estimated as only a few k
B
T, and Brownian 
motion is enough to breakup particle flocculation [67].       
 The electrostatic force was introduced by the theory of an electric double layer 
which results from the charging of particle surfaces.  Intuitively this repulsive force 
would explain why all particles in solution did not coagulate and precipitate out.  
Generally, the charging of a surface results from the adsorption of ions from solution to a 
Interparticle Distance 
Interparticle 
Potential 
0 
2
o
1
o
electrostatic 
vdW 
total 
 
27 
neutral surface or oppositely charged surface or from the ionization/dissociation of 
surface groups [65].  A common example, and one of specific interest to this research, is 
the pH dependant dissociation of carboxylic groups (?COOH) to a negatively charged 
carboxylate ions (?COO
-
).  The result of an adsorbed surface charge is an oppositely 
charged region of counterions around the particle.  In close proximity to the surface a 
narrow layer of bound ions develops known as the Stern layer.  Surrounding the Stern 
layer is an atmosphere of loosely associated ions known as the diffuse double layer.  To 
this end, similarly charged particles or surfaces will repel each other electrostatically in 
solution.   
 An additional important type of force that can contribute to suspension stability is 
that of polymeric or steric forces.  These are short range forces produced when polymer 
layers that have absorbed to the particle surface overlap and avoid mixing.  However, if 
the polymer remains in solution rather than adsorbing to the surface they can induce 
flocculation instead.  If the interparticle separation approaches the dimension of the 
polymer molecules the space between the particles is depleted of polymer due to entropic 
effects.  Basically, the polymer molecules gain entropy by removing themselves from the 
constraint of the interparticle dimension.  Consequently, the particles are forced together 
by depletion interactions [67].  
 
Dispersion of Carbon Nanotubes: 
Binding Energy of SWNTs: 
 Thess et al. (1996) showed that in their natural state as produced SWNTs self-
organize into crystalline ropes 0.5-20 nm in diameter and tens to hundreds of microns in 
 
28 
length.  As shown by Figure 10, the van der Waals intertube bonding holds the individual 
tubes in a triangular lattice with a lattice constant of 1.7 nm.  The parallel alignment seen 
in Figure 10 is a consequence of the highly polarizable and pristine nature of the sp
2
 
hybridized carbon interface of SWNTs [68]. 
 
Figure 10: A single SWNT rope made up of ~100 SWNTs as it bends through the image plane of 
the microscope, showing uniform diameter and triangular packing of the tubes within the rope.  
The diameter of the individual SWNTs were determined to be ~1.38 nm.  Reproduced from Thess 
et al. (1996) [54]. 
 
The pairwise interaction potential has been modeled for parallel carbon nanotubes 
revealing a deep potential energy well at equilibrium separation on the order of ~10
2
 
meV/nm [69, 70].  Due to the high SWNT aspect ratio the van der Waals attractive force 
becomes the substantial obstacle for achieving stable dispersions as tubes tend to exist in 
ropes or arrangements of bundles.  For tube - tube interactions the binding energy of 
SWNTs has been commonly accepted on the order of ~ 0.5 eV/nm, making them 
inherently very difficult to process as individuals [54, 71].  It is important to note that this 
value is smaller, but on similar order of magnitude, than what is estimated by simulation 
[69].  This discrepancy is likely due to the unavoidable presence of defects on the 
 
29 
sidewalls of ?real? tubes as well as solvent interactions.  Additionally, this lower value 
may have been meant to reflect the peeling of an outermost CNT from a bundle rather 
then a tube centered in a rope.   
 In order to calculate the intertube interaction potential, Girifalco et al. (2000) used 
a continuum approach for nanotube pairs [69].  By this method they determined the 
interaction energy over an effective area occupied by a carbon atom.  The intertube 
potential for two structures was then calculated by integrating the Lennard-Jones 
potential over the surface of the tubes [72].  From this model the potential energy well 
depth was characterized at equilibrium spacing.  Considering hexagonal symmetry, the 
cohesive or binding energy was calculated for a tube centered in a rope interacting with 
six nearest neighbors as three times the potential energy well depth [73].  From these 
calculations conclusions can be drawn as to how the van der Waals force in ropes or tube 
pairs scale with tube diameter.  For a (10,10) SWNT with diameter of 1.35 nm the 
binding energy of a tube centered in a rope (1.67 nm lattice spacing) is ~2.855 eV/nm and 
the intertube interaction potential is 951.6 meV/nm.  Whereas, increasing the tube 
diameter to 1.89 nm a (28,28) tube in a rope (4.162 nm lattice spacing) has a binding 
energy of ~4.862 eV/nm and the intertube interaction potential is 1,620.6 meV/nm [69].  
The increase in binding energies with increasing diameter and separation distance may 
seem somewhat counterintuitive from the perspective of colloidal interactions modeled 
via Hamaker functions.  However, it is important to note that the larger diameter tube 
(28,28) has more carbon atoms interacting per unit length then the (10,10).  Thus, even 
though the separation between the (28,28) tubes (0.3168 nm) is larger that for (10,10) 
tubes (0.3153 nm) the models by Girifalco et al. account for the presence of more carbon 
 
30 
atoms whose extra pull seems to outweigh the separation.  In the same manner, one must 
also adjust the Hamaker constant to account for this or incorrect conclusions will be 
drawn.  In fact, considering the separation distance, tube radius, and attractive potential 
for SWNTs modeled by Girifalco et al. the Hamaker constants for parallel alignment can 
be extracted to be -7.86 x10
-19
 J for two (10,10) tubes and -8.05 x 10
-19
 J for two (28,28) 
tubes.  Therefore, the van der Waals models based on Hamaker are of little utility in 
comparing interactions between nanotubes of different diameter/chirality a priori.                
 In order to determine in what manner nanotubes can be effectively separated from 
one another, Coffin et al. (2006) compared two modes of separating individual tubes 
from ropes; radial displacement (dilation) and array peeling.  Through theoretical 
arguments their results show that peeling of individual tubes from the outer diameter of a 
rope requires much smaller forces then dilation.  Therefore, the likely mode of separation 
is array peeling as it can be accomplished by delivering much lower intensity energy 
density to the CNT [72, 74].  For this reason it is assumed that this is the mode likely 
separating tubes from bundles by high shear.  Huang and Terentjev (2008) have 
confirmed this assumption with both theoretical calculation and experiment [75].  
Although not primarily used in this research, the authors also provide detailed analysis on 
the energetics involved in dispersion by sonication techniques.          
 
Nanotube Dispersion in Polymers: 
 It is well known that the potential for mechanical property enhancement is 
determined by both the degree of nanomaterial dispersion throughout the composite 
system and adhesion between the polymer matrix and dispersed phase [76, 77].  In spite 
 
31 
of this fact, mechanical property improvements have typically been lower than predicted 
or reported inconsistently in literature with respect to theoretical expectation [9, 78, 79].  
This phenomena is not only a consequence of a less than ideal dispersion state, but also 
the dominance of interfacial interactions as the defining length scales of the filler and 
polymer converge.  Therefore these interactions must be given careful consideration [80].  
A truly ?ideal dispersion? state is classified by complete exfoliation of the individual 
CNTs from bundles or clusters, but typically in the literature an ?ideal dispersion? is one 
in which dispersed tubes coexist with small aggregates [81].  Solution based methods 
followed by in situ polymerization have been favored and effective for achieving this 
dispersion state.  Sandler et al. (1999) developed a pioneering method for dispersion of 
CNTs in volatile solvents [82].  For typical thermoset systems the nanotubes are first 
dispersed in a low viscosity solvent and/or surfactant using ultrasonication before 
magnetically stirring the monomer and hardener in while evaporating the solvent [82-84].  
The effectiveness of solution based methods is a result of the low viscosity solutions the 
CNTs are initially dispersed in.  This is in contrast to the high viscosities of the polymer 
or resin itself.  Likewise, solution based methods have been adopted for thermoplastic 
systems using more exotic solvents capable of dissolving polyolefins and acrylics [85, 
86].  It is important to note that these methods typically employ high powered 
ultrasonication which results in a decrease in the nanotube aspect ratio over long 
exposure times and therefore limits mechanical property enhancement [71, 87].  
However, it was found that first homogenizing the nanotube solutions via high shear 
reduces the required sonication time and circumvents extreme reduction in length [88].      
 
32 
 In addition to solute dispersion, improved performance has been found by using 
functionalized nanotubes to improve the CNT dispersion state and CNT ? polymer 
interactions [9, 78, 89].  To this end, a variety of methods have been developed to 
increase nanotube-solvent compatibility including both covalent and non-covalent 
functionalization techniques.  Common non-covalent techniques involve the use of 
surfactants and biomolecules.  The interplay between the quality of the dispersion state 
and the interfacial interaction between the CNTs and solvent/polymer is a delicate 
balance.  An excellent illustration was seen in the work of Gojny et al. (2004) who 
studied the performance of nanocomposites fabricated from DWNTs and amine 
functionalized DWNTs in a epoxy resin matrix [9].  For the purpose of benchmarking 
performance, the group was able to extract the expected maximum theoretical Young?s 
modulus for mixtures of pure components.  Excellent agreement between predicted and 
experimental Young?s modulus were found at low nanotube loadings of 0.1 % wt. using 
the amine functionalized DWNTs whereas the pristine DWNTs underperformed.  This 
result was an effect of favorable interactions between the amino groups and the epoxy 
resin/amine hardener.  However, deviations were found when the loading was increased 
by an order of magnitude to 1.0 % wt. as explained by the non-ideal dispersion state not 
accounted for in the model.  Essentially, sample aggregation resulted in a decreased 
separation distance between MWNTs.  For an exhaustive review on CNT 
functionalization routes the reader should refer to Dyke et al. (2004), Banerjee et al. 
(2006), and Tasis et al. (2006) [7, 42, 90].   
 Another common dispersion method for CNT-thermoset polymer systems is the 
application of high shear.  This strategy is favorable since it is generally believed that this 
 
33 
technique can preserve the intrinsic electric and mechanical properties of an isolated CNT 
[75].  Additionally, a shear based dispersion technique is industrially scaleable.  The only 
fundamental requirement for successful shear dispersion is that the shear energy density 
be sufficiently high to overcome the van der Waals attractive forces.  However, Huang et 
al (2006) performed a systematic study of the shear dispersion state of MWNTs in 
viscous poly(dimethylsiloxane) (PDMS) revealing a long characteristic mixing time, on 
the order of days, required to achieve a well dispersed state [91].  The authors argue that 
this characteristic time was so extensive that many previous shear based dispersions 
studies should be revisited.  It is likely that this long mixing time is a result of the time 
required to expose the entire volume of the fluid to the high shear areas of the mixer as 
well as the slow kinetics required for array peeling to complete. 
 In spite of the effect of mixing time, a number of groups have used high shear 
strategies to disperse CNT in polymers.  Sandler et al. (2003) recognized the relationship 
between high matrix viscosity and high shear stresses [92].  To exploit this effect the 
group mixed MWNTs at high shear (1000 rpm; 2h) using dry ice to lower the viscosity of 
the epoxy resin [92, 93].  Likewise, Rahatekar et al. (2006) dispersed aligned MWNT 
carpets using a high shear protocol (1000 rpm, 2h) but noted that the initial aligned state 
of their tubes made dispersion much easier as compared to highly entangled MWNTs 
[94].  Gojny et al. (2004) first used a calendaring (three-roll mill) process to 
homogeneously disperse DWNTs in to an epoxy resin with very short mixing times on 
the order of minutes [9].  The authors note that this method is an attractive technique that 
can be readily scaled up in industry since calendaring is a common industrial process.  
Shortly after, Gojny et al. (2005) used this method to compare the dispersion of various 
 
34 
MWNTs and SWNTs in to an epoxy resin matrix [95].  From this work, two important 
conclusions were drawn.  The first was that while SWNTs possess the highest potential 
for mechanical property enhancement the resulting nanocomposites could not outperform 
the amine-functionalized MWNTs due to the poorer SWNT dispersion state.  The second 
conclusion was that the three-roll mill process could not produce composite material with 
a MWNT loading above 0.3 % wt..  This limitation was later addressed by Wichmann et 
al. (2006) who developed a post-calendaring method by means of a vacuum dissolver to 
construct MWNT epoxy composites at loading up to 2 % wt. [96].  Seyhan et al. (2007) 
achieved dispersion of MWNTs and amine-functionalized MWNTs in a VE/UPR resin at 
short mixing times by means of a three-roll milling process [14].  Similarly, the technique 
was later used for processing MWNT-UPR dispersions [19].    
 Even though the successes of the aforementioned shear techniques are promising 
for achieving homogenous polymer nanocomposites on an industrial scale they all have 
an inherent difficulty when processing VE or UPR resins.  This difficulty arises from the 
presence of the volatile styrene monomer required to both lower the viscosity and cure 
the resin systems.  Seyhan et al. (2007) recognized that the milling process essentially 
casts a thin film over the rollers from which much of the styrene can escape [14].  To 
bypass this difficulty they obtained a viscous styrene free UPR in which the initial 
dispersions were blended.  After dispersion the group added a VE resin as well as styrene 
monomer prior to cure.  The same conclusion was reached by Thostenson et al (2008) 
who synthesized a VE resin to have a styrene free starting material [15].  Similarly, the 
use of three-roll milling with commercial isophthalic UPR failed due to evaporative loss.  
Additionally, undesired sample polymerization was found to be initiated by the high heat 
 
35 
generation in the mill.  As a result Seyhan et al. (2007) were unable to determine the final 
concentration of their dispersion and instead switched to a styrene free UPR [19].  Most 
recently, a clever combination of shear mixing and horn ultrasonication was employed by 
Battisti et al. (2008) to disperse MWNTs in a commercial UPR.  Recognizing the 
problems associated with the heat generated during sonication and subsequent styrene 
evaporation the group affixed a condenser to the vessel.  On a separate note, another 
difficulty was encountered by Gojny et al. (2004) who discovered the initial CNT-
polymer solution was too thin to cling to the rollers [9].  Therefore, a viscosity modifier 
had to be added pre-calendering to prevent this effect.        
 
Methods for Probing Dispersion State: 
 It would be beneficial to be able to probe the nanotube dispersion state directly as 
a means of monitoring composite processing in situ.  MWNT suspensions have been 
studied using optical methods as well as light scattering techniques to probe aggregation 
modes [94, 97, 98].  However, problems arise due to the high optical absorbance of 
CNTs.  Therefore,  the applicability of small-angle light scattering (SALS) is limited to 
dilute regimes and moderate length scales [99, 100].  Difficulties are further exacerbated 
when working with SWNTs due to their inherently small size.  Thus, the applicability of 
optical microscopy is removed and smaller incident scattering wavelengths are required.  
In this manner, small-angle X-ray scattering (SAXS) has been used to study clusters or 
aggregates of tubes [101].  Likewise, small-angle neutron scattering (SANS) has been 
used to resolve these objects further and even resolve scattering from individual tubes 
revealing various power-law scattering profiles from fractals [102].  What is apparent is 
 
36 
that as the dispersion state of the CNT is improved towards individuals the ability to 
observe the sample optically tends to rely on more complex techniques.  In general, all 
optical methods are ineffective below a length scale of ~ 0.2 - 0.5 ?m and electron 
microscopy techniques are limited to the sample surface.  Furthermore, the small 
wavelength techniques suffer from the unavoidable difficulty of working in reciprocal 
space and the difficulty associated with data analysis [91].  An excellent alternative to 
monitoring dispersion state optically is to monitor its rheological response.  Thus, 
rheological techniques have emerged as an efficient and promising method for probing 
and monitoring the dispersion state and microstructure of carbon nanotube composites 
and dispersions. 
 If efficient and economically viable bulk processing of nanotube-polymer 
composites is to be realized, a well developed understanding of responses to simple 
steady-state shear flow is required [91].  Furthermore, from processing and application 
points of view, the mechanical and rheological properties of nanocomposites are very 
important. These properties are related to the material microstructure, the state of 
nanotube dispersion, the aspect ratio and orientation of nanotubes, and the interactions 
between nanotubes and polymer chains [103].  The rheological response of CNT has been 
studied both in polymer solutions, melts, and solvents.  In spite of the differences in the 
solvent type used remarkable parallel behavior has been seen.  Davis et al. (2004) 
observed Brownian rigid-rod behavior for highly individualized SWNT dispersions using 
sidewall protonation via superacids revealing a shear viscosity dependence on aspect 
ratio [104].  Rai et al. (2007) studied the effect of functionalization on SWNT-superacid 
solubility which unveiled rheological dependence on CNT dispersion states [105].  
 
37 
Hough et al (2004) studied concentrated surfactant stabilized suspensions under 
oscillatory flow displaying the viscoelastic nature of associating, rigid-rod SWNT 
networks [106].  The rheological behavior of the dispersions used in the aforementioned 
studies serve as a benchmark to which less than ideal dispersion states can be compared. 
 Less than ideal polymer solvents have also been studied rheologically to reveal 
unique behavior in their own.  Hobbie et al. (2007) examined non-Brownian MWNT-
poly(isobutylene) suspensions were under the influence of shear flow and stress.  This 
type of fluid was found to exhibit yielding behavior manifested via an elastic network.  
Interestingly, both linear and non-linear responses showed a concentration dependant 
scaling [107].  Huang et al. (2006) systematically studied dispersions of MWNTs in 
viscous poly(dimethylsiloxane) to reveal the dependence of shear mixing time on the 
emergence of network elasticity; this work identified a long characteristic mixing time for 
a well dispersed state [91].  Similar elasticity was discovered by Kinloch et al. (2002) in 
which concentrated and entangled oxidized MWNT aqueous suspensions showed 
Bingham behavior [108].  Fan et al. (2007) produced a detailed study regarding MWNT-
epoxy suspensions which revealed the variation in rheological behavior upon loading, 
aspect ratio, and initial dispersion quality supporting the utility of oscillatory shear for 
probing dispersion state [81].  Similarly, direct connections were made by Rahatekar et 
al. (2006) between rheological response and microstucture for MWNT-epoxy 
suspensions by optical observation under shear [94].   
 In contrast to epoxy resin, only limited rheological data has been reported in the 
literature for UPR matrix dispersions.  Seyhan et al. (2007) have studied the response of 
MWNT and amine-functionalized MWNT dispersions to both oscillatory non-linear 
 
38 
shear [14].  However, their data suggests a poor dispersion state for multiple reasons.  
First, the amine-functionalized MWNTs showed lower degree of mechanical property 
enhancement as compared to pristine MWNTs in the low angular frequency regime.  Of 
course, this could also be a result of less favorable interactions after functionalization.  
However, the authors report an increase in the glass transition temperature for the 
functionalized tubes.  This is an indication of the CNTs successfully impeding polymer 
motion.  Additionally, by comparison of measurements reported by Fan et al. (2006) who 
used MWNTs almost five times shorter (10 - 15 ?m) than what Seyhan et al (2007) used 
(50 ?m), while having similar matrix viscosities, obvious discrepancies can be observed.  
The response of the MWNT-epoxy dispersions reported by Fan et al. (2006) is almost 
independent of frequency at concentrations of 0.2% wt. whereas at 0.3% wt. elastic 
behavior was just beginning to be seen in the dispersions of MWNT-UPR.  This would 
most likely indicate a less than ideal dispersion state and undeveloped network in this 
work.  
 Battisti et al. (2008) have only reported the non-linear viscosity of MWNT-UPR 
dispersions at 0.25% wt. [20].  However, their results show improved steady-shear 
viscosity enhancement over the 0.3% wt. dispersions constructed by Seyhan et al (2007).  
Thus, there is opportunity for a detailed study of CNT-UPR dispersions available in the 
literature.  Furthermore, a strategy able to process commercial grade UPR would be 
beneficial industrially.  Specifically, it would be advantageous if the final CNT loading 
was repeatable and controlled after processing so that ?on-line? monitoring techniques 
such as bulk rheological characterization of the sample could be used.      
 
 
39 
Shear Aggregation in CNT Dispersions: 
 Schueler et al. (1997) first studied the shear induced agglomeration of carbon 
black dispersed in an epoxy resin [109].  They argued that the particles were stable in 
absence of shear but formed aggregates under certain conditions.  This was attributed to 
the dispersion existing in a metastable state where a potential energy barrier existed 
providing kinetic stability from the aggregated state.  The barrier could be surmounted by 
external shear forces or by increasing the ionic concentration resulting in an aggregated 
state which was concluded to be energetically more favorable than the dispersed state.  
Similarly, Martin et al. (2004) observed shear aggregation in MWNT-epoxy dispersions 
but only after the addition of the amine based hardener [93].  Their dispersions were 
created by high shear and were stable for months in storage.  The samples were found to 
be electrostatically stable in this state due to the electric double-layer.  The repulsive 
potential V between spheres was approximated using Equation 2.10. 
 ( )DV ??? exp  (2.10) 
Here, D is the interparticle distance and ? is the Debye screening length.  Since the 
screening length is proportional to the ionic conductivity the screening length decreases 
with increasing ionic concentration in the dispersion.  This, in turn, results in a reduction 
in the stability.  Thus, the authors concluded that while high shear forces can create a 
homogeneous dispersion, the addition of the amine hardener, the application of small 
shear, and an increase in temperature for curing promoted aggregation by a reduction in 
the mutual repulsion between the tubes.  The high temperature was determined to 
increase the particle mobility by reducing the resin viscosity.  Likewise, the use of shorter 
nanotubes was found to increase the rate of agglomeration due to their increased mobility 
 
40 
compared to longer MWNTs.  However, both samples were found to agglomerate after 
the addition of the hardener and application of low shear.     
 Shear aggregation has also been identified through rheological response coupled 
with optical imaging.  Lin-Gibson et al. (2004) studied the flow-induced clustering of 
semi-dilute non-Brownian MWNT-poly(isobutylene) suspensions [84].  The samples 
were primarily prepared at 0.17% vol. and measurements were made at constant shear 
with a variable gap height between the parallel plates.  Samples were found to associate 
into diffuse networks after the application of low shear (0.03 s
-1
) for ~ 30 min with a 50 
?m sample gap.  The gap size was observed to dictate the length scale of cluster growth 
from confined to bulk areas.  The MWNTs could be redispersed by application of a shear 
rate of 10 s
-1
 and oriented with the flow.  Over long steady shearing of ~ 24 hours the 
samples were found to ?quench? under the weak shear.  This aggregation behavior was 
identified by means of a unique rheological signature where the shear viscosity exhibits a 
maximum after approximately one hour of shearing and decays thereafter.  Tracking the 
first-normal stress difference over this period showed that a negative value was observed 
at the viscosity maxima which trended down with the viscosity curve.  The negative first-
normal stress difference was concluded to result from compressibility of the clusters 
where the contracting aggregate domains exhibit a collective pull on the parallel plates. 
 It is important to note that not all observed shear aggregation is a result of 
interparticle attractive forces.  In absence of interparticle attraction Schmid et al. (2000) 
showed through simulation that flow-induced aggregation can occur in non-Brownian 
fiber suspensions as a result of fiber friction alone [110].  Specifically, low 
concentrations of particles having high stiffness flocculated under the application of low 
 
41 
shear stress.  The strongest flocculation was found to occur in suspension of stiff fibers 
with irregular shapes.  The flocs were predicted to have at least three or more contact 
points and were held together by stored elastic energy in the fibers.   
 Finally, shear induced aggregation has also been linked to the non-linear 
rheological response of fluids.  Rahatekar et al. (2006) optically observed low shear 
aggregation in MWNT-epoxy suspensions under a confined sample gap of 0.3 mm [94].  
They found that the aggregate size in the dispersions was sensitive to the shear rate 
applied and a high shear rate could readily homogenize the dispersion.  This phenomena 
was concluded to be the origin of shear thinning in the sample.  The application of a low 
shear rate caused a buildup of aggregates that persisted after cessation of flow.  However, 
at low concentrations macroscopic aggregates did not form and bulk viscosity 
enhancement was not seen below 0.1% wt..  It was concluded that the high viscosity of 
the epoxy (10 Pa-s) limited the diffusion of the MWNTs through the samples at low 
concentrations where particles were further apart.  Thus, it was concluded that above a 
certain loading of MWNTs the low shear viscosity enhancement was a result of 
?mechanical? aggregate formation.  Brownian motion was found to play little to no part 
in the dynamics of aggregation. 
 In summary, it is important to realize that the shear induced aggregation of 
particles or fibers can have different origins between samples.  Aggregation has been 
found to be a result of dispersion destabilization, an increase in mobility, and interparticle 
friction alone.          
 
 
42 
Modifying the Nanotube Interface via Oxidation: 
Introduction: 
 Oxidation of carbon nanotubes is a common means of chemical modification as it 
provides a precursor for many further functionalization reactions.  These include but are 
not limited to amidation reactions, esterfication reactions, biomolecular attachment, 
polymer grafting, and coordination chemistry [7, 90].  Additionally, oxidation reactions 
typically produce a wide range of functionalities such as carboxylic acids, alcohols, 
aldehydes, ethers, lactones, and epoxides depending on the type of nanotube and 
oxidizing agent used [111].  However, the most synthetically useful group is the 
carboxylic acid as not only can it participate in many subsequent reactions but it serves to 
increase hydrophilicity.  Furthermore, the introduction of functionalities serves to 
interrupt the van der Waals attractive forces between nanotubes.  Common purification 
techniques that use acid oxidation remove tube endcaps and introduce holes into the tube 
sidewalls.  Typically, the tube ends and, depending on the strength of the oxidizing agent 
used, the sidewalls are decorated with predominately carbonyl and carboxylic groups [7].   
Figure 11 displays a SWNT after oxidative treatment.   Although this treatment can 
increase dispersion properties in aqueous solvents the length is often reduced to 
submicron dimensions and the ??? conjugated structure is damaged affecting the 
mechanical and electrical properties [7, 112].    
 
 
43 
 
Figure 11: Schematic of an oxidized SWNT.  Carboxylic groups can be seen terminating the tube 
end and on sidewall defect site.  The presence of the acidic hydrogen is indicative of low pH 
conditions.  Image reproduced from Hirsch et al (2002) [113].  
 
 In general, it is believed that chemical attack on CNT originates at end caps and 
defect sites where sp
3
 character may be present in contrast to the sp
2
 conjugated carbons 
of the pristine sidewall.  The curvature of the endcaps leaves them unstable and strained 
resulting in a higher reactivity [46, 114].  Additionally, the sidewalls contain defect sites 
such as pentagon-heptagon pairs (Stone-Wales defects), sp
3
 hybridized defects, and 
lattice vacancies [90].  Such defect sites are chemically active due to disruption of the 
delocalized electron density [46].  Interestingly, even localized bending or twisting of the 
CNT renders them more susceptible to chemical attack.  Furthermore, under certain 
oxidative conditions additional defect sites are created by attack on the graphene 
sidewalls and subsequently oxidized [114].    
 
44 
 
Theory of Organic Acids: 
 The dissociation of an organic acid follows a reversible reaction scheme dictated 
by the value of its acid dissociation constant K
a
.  A larger value of K
a
 is indicative of a 
stronger acid.  However, it is more practical to describe the extent of dissociation on a 
logarithmic scale, and in this manner the pK
a
 is expressed by Equation 2.11.   
 
aa
KpK log?=  (2.11) 
By this convention a smaller pK
a
 value indicates a stronger acid and is typically 
determined using water as the base.  Its interpretation is meant to indicate how readily a 
compound gives up a proton whereas the pH indicates the acidity of a solution.  
Carboxylic acids have pKa values ranging from 2.5 to 5 and therefore are considered 
moderately strong acids [112, 115].  Figure 12 displays what is referred to as a carboxylic 
(carboxyl) functional group.  
A
O
OH
 
Figure 12: Generic depiction of a carboxyl functional group.  The atom A is a general 
representation but typically signifies an aromatic or amorphous carbon in this work.    
 
In a carboxylic acid or carboxyl group, the oxygen atom in the C=O bond is 
electronegative and pulls the electrons bonded in the -OH group (hydroxyl) away from 
the proton.  Thus, the proton is easier to donate as indicated by the relatively low pKa for 
carboxylic acids.  Additionally, the conjugate base of the carboxylic acid (carboxylate 
 
45 
ion) is resonance stabilized sharing the delocalized electrons between oxygen atoms 
[115].   
 Whether or not a compound will exist in its acidic form or as a conjugate base is 
dependant on both the pKa of the acid and the pH of the solution.  These two quantities 
are related by means of Equation 2.12, commonly known as the Henderson-Hasselbalch 
equation.    
 
[ ]
[]acid
base
pHpK
a
log+=  (2.12) 
Here [base] represents the molarity of the compound in its conjugate form and [acid] 
represents the molarity of the compound in its acidic form.  As indicated by the 
logarithmic component of Equation 2.12, the acid and conjugate base coexist in various 
proportions over a range of pH.  Generally, compounds exist in their acidic form in 
solutions with pH lower than the compound?s pKa and vice versa.    
 As the pH is raised above pKa the carboxylic acid dissociates into the carboxylate 
ion (?COO
-
).  Basically, the carboxyl group depicted in Figure 12 loses a proton 
(hydrogen atom) leaving behind a negative charge.  This effect can induce electrostatic 
repulsion to disperse particles, such as CNTs, and keep them stable in suspension.  
However, it is not recommended to neutralize the pH with caustic salts as the ions in 
solution will compress the thickness of the electrical double layer and eventually may 
cause the particles to crash out of solution [99, 112].   
 
 
46 
Oxidative Treatment of Nanotubes: 
 Although various accounts in the literature claim to be the first to treat CNTs with 
a mixture of concentrated nitric and sulfuric acids the initial report was actually made by 
Esumi et al (1994).  The mixture of strong acids used in this experiment was originally 
found to intercalate and swell graphite [116].  Esumi et al. refluxed (140
o
C) MWNTs in a 
3:1 mixture of concentrated sulfuric to nitric acid (1g MWNT/40 mL) and found that the 
concentration of acidic sites was greater after only 20 min of exposure than when the 
same MWNTs were refluxed for 4 h in concentrated nitric acid alone.  It was concluded 
that the effect of concentrated nitric treatment was less pronounced then the acid mixture 
due to the difference in oxidative power between the two methods.  Additionally, a 
negative zeta potential was found to trend with increasing pH confirming the presence of 
acid sites and their increasing occurrence with acid exposure time [117].  For point of 
reference, the nitric acid treatment used by Esumi et al. was originally developed to open 
the endcaps of CNTs for filling of metal oxides by Tsang et al (1994) [118].  
Subsequently, Lago et al. (1995) adopted this nitric acid treatment for intertube insertion 
of palladium and were the first to suggest the introduction of predominately carboxylic  
(?COOH) and phenolic (-OH) groups to the MWNTs by means of titration to identify 
acidic site concentration [119].  Additionally, Lago et al. were the first to report that the 
use of high energy ultrasound (sonication) readily creates local defects in, or destroys, 
CNTs and the number of acidic sites introduced to the tubes can be increased with 
ultrasonic pretreatment.    
 Similar chemical treatments used for MWNTs were successfully performed with 
SWNTs soon after their discovery.  Liu et al. (1998) were the first to study the ability of 
 
47 
the sulfuric/nitric acid mixture to cut highly tangled SWNT ropes into short ?fullerene 
pipes.? [120].  The cut tubes were found to have a length of approximately 300 nm.  By 
their method a 3:1 mixture of 98% sulfuric to 70% nitric acid (10 mg SWNT/40 mL) was 
bath sonicated for 24 h at 35 - 40
o
C, diluted, collected via filtration, and washed with 
10M NaOH.  To remove amorphous carbon generated by the cutting the final product 
was then polished in a 4:1 ratio of 98% sulfuric and a 30% aqueous hydrogen peroxide 
solution (piranha) at 70
o
C for 30 min.  The open tube ends were assumed to be 
terminated with many carboxylic groups.  However, they found that the final product 
readily flocculated in aqueous solution and had to be suspended with aid of a surfactant.  
The instability of these solutions was addressed by Shaffer and Windle (1998) who used 
the method developed by Esumi et al. to oxidize MWNTs.  Fourier transform infrared 
spectroscopy (FTIR) showed the oxidative treatment introduces oxygen rich surface 
groups that were predominately phenol, carboxyl, or lactone functionalities.  The author?s 
note that the addition of ions destabilizes the suspension in accordance with a reduction 
in the Debye screening length.  Thus, the solutions were not neutralized with a strong 
base like NaOH and after dilution the nanotubes formed a well dispersed, electrostatically 
stabilized colloid in water [99].  This was a result of the ionization of the acidic groups 
present on the MWNT surface.  
 The use of nitric acid alone was also found to be effective to oxidize CNTs.  Jia et 
al. (1999) created short and straight MWNTs from an entangled starting material by 
oxidation in boiling concentrated nitric acid.  The authors suggested the final nanotube 
length was inversely related to the treatment time.  FTIR spectroscopy was used to 
identify functional groups present.  Because of the presence of many -OH, ?COOH, and 
 
48 
C=O groups the pH of the final solution could not approach 7 upon washing.  The 
oxidized tubes were assumed to be attracted to water molecules by hydrogen bonds 
between -OH groups on the nanotube surface and the water molecules [121].  Similar 
conclusions were made by Saito et al. who used 22 h bath sonication (50
o
C) to create 
shortened oxidized MWNTs by sulfuric/nitric treatment (1 mg MWNT/2 mL) [122].  The 
final product was found to have a length of ~338 nm.  The original length was not 
determined but was shown to be noticeably longer.     
 CNT oxidation has also been found to serve in disentangling the nanotubes and 
increasing their mobility.  Wang et al (2003) showed that while physical separation 
methods such as ball milling, shearing, and ultrasonication can break down large 
agglomerates of MWNTs into smaller clusters only treatment with sulfuric/nitric acids 
(140
o
C, 0.5 h) could disentangle the tubes.  This was achieved by effectively severing the 
entanglements and therefore damage was unavoidable.  However, if the tubes were first 
annealed at high temperature the defects introduced to the tube walls could be decreased 
and this method was suitable for purification [123].  Rosca et al. (2005) dispersed 
MWNTs in nitric acid by bath sonication (30 min) followed by reflux for up to 48 h 
[124].  As shown by Figure 13, after 24 h of treatment time the MWNT alignment has 
significantly increased.  This alignment was thought to be a result of tube 
disentanglement and the increase in mobility as a result of the functionalization.  
Continued treatment can increase the cutting and these smaller fragments can align 
themselves further as seen after 48 h of treatment.  Benefits from oxidation on 
mechanical properties of CNT films have also been reported. 
 
49 
 
Figure 13: The evolution of MWNT alignment during the oxidation process.  Here 10 mg 
MWNT/mL of 60% nitric acid was used.  Image taken from Rosca et al. (2005) [124]. 
 
 Zhang et al. (2004) compared the mechanical strength of buckypapers produced 
from various concentrated (3,6,10 M) of nitric acid solutions by sonication (2 h) and 
refluxing (2 h) [125].  Selective degradation of small diameter tubes was confirmed via 
Raman spectroscopy.  This resulted in a narrower tube diameter distribution, which 
increased the shear modulus of the tube bundles, resulting in higher film modulus while 
the degradation product can fill voids [126].  The mechanical properties of the films 
approached that of engineering plastics with those treated by 10 M acid showing the 
strongest mechanical properties.   
 
50 
 Recently, Rasheed et al. (2007) critically compared the performance of some of 
the more common oxidation methods on VGCFs [111].  The sulfuric/nitric acid mix (50 
mg VGCF/150 mL) was found to be the most severe treatment (24 h sonication at 45
o
C) 
and generated the most amorphous carbon.  Treatment with RuO
4
 and KMnO
4
 were 
compared with a nitric acid (200 mg VGCF/40 mL) reflux at 120
o
C for 20 h.  The group 
was able to determine that treatment by sulfuric/nitric acid produced an order of 
magnitude higher functional groups (1.05 x 10
21
 acidic sites/g VGCF) than the other 
treatments.  In addition, they found that the nitric acid was least severe but provided the 
lowest amount functional groups.  
 Although the focus thus far has been on acid oxidation methods other methods 
have been successfully performed.  By treatment with sec-butyllithium and carbon 
dioxide both alkyl and carboxyl groups were introduced to the sidewalls of SWNTs 
[127].  Additionally, a variety of strong oxidants such as K
2
Cr
2
O
7
, KMnO
4
, H
2
O
2
, OsCl
3
, 
RuCl
3
 were studied by Hwang (1995) for their ability to open nanotube endcaps [128].  
Similarly, Hernadi et al. (2001) studied KMnO
4
, H
2
O
2
, HClO
4
, and O
3
 to gauge their 
effectiveness in removing amorphous carbon by oxidation [129].  Interestingly, no 
correlation was found by this group between the oxidation rate and electrochemical 
oxidation potential.    
 
Mechanism of Oxidation: 
 As previously mentioned, Lago et al. were the first to report that the use of high 
energy ultrasound (sonication) readily creates local defects in, or destroys, CNT [119].  
Additionally, they noted that the number of acidic sites introduced to the tubes can be 
 
51 
increased with ultrasonic pretreatment.  Liu et al. (1998) argued that simultaneous 
exposure of CNTs to oxidizing agents and ultrasonication is an effective way to etch the 
CNTs.  Under the influence of ultrasonication microscopic high temperature domains are 
created by the collapse of cavitation of bubbles.  These hot areas induce attack the surface 
of SWNTs leaving a hole in the sidewall.  Subsequent attack by strong oxidants quickly 
cuts this area completely cleaving the CNT.  Because of the moderate temperatures the 
open tube ends were assumed to be continuously etched away in a time-linear fashion 
(fuse burning) by the oxidizing agent unless quenched [120].  Essentially this is a two 
step process of introducing defects then cutting the sites.  Their results show that the 
mixture of sulfuric and nitric acids can complete both steps.  Ziegler et al. (2005), on the 
other hand, used hot piranha to complete the cutting step because they found it was less 
destructive to the sidewalls.  Basically, they concluded that if the cutting was by ?fuse 
burning? from the ends much of the nanotube material is wasted in etching to a desired 
length.  The authors circumvent this with the piranha, only cutting at defect sites already 
created and thus preserving electrical properties of the sidewall [130].  Likewise, over 
time SWNT exposed to Piranha solutions were found to be stable against damaging the 
tube structure [131].        
 A mechanism for the oxidation of SWNTs was proposed by Zhang et al. (2003) 
who systematically investigated the effect of sulfuric/nitric (60 mg  SWNT/50 mL),  
2.6 M nitric, and KMnO
4
 oxidants on the structure of SWNTs [114].  They concluded 
that the oxidation process begins by attacking defect sites already present from CNT 
production and is additionally accompanied by a two step process.  First, in a defect-
generating step the oxidants attack the graphene structure via electrophilic addition and 
 
52 
generate -OH and C=O groups.  For various treatments, the effectiveness of this step 
depends on the oxidants ability to generate the ?OH groups initially and transform them 
to C=O.  Specifically, for the sulfuric/nitric acid mixture evolves very aggressive NO
2
+
 
ions which attack the graphene structure to generate active sites.  Similarly, OH
+
 ions are 
generated in piranha.  Next, during the defect consuming step the graphene structure is 
destroyed by oxidation of the active sites generated previously.  This extent of this step 
depends on the ability of the oxidant to etch the graphite like structure around the already 
generated C=O and neighboring groups.  It was found that dilute nitric could purify tubes 
by etching carbonaceous particles but not generate abundant functional groups.  FTIR 
spectroscopy indicated that even at treatment times up to 96 h dilute nitric could not 
generate new ?COOH groups.  Instead, it consumed already existent defects but could 
not etch into the aromatic rings to generate more defect sites.  Similarly, KMnO
4 
treatment was ineffective in the defect consuming step. 
     
Etching Mode and Rate: 
 Generally, there are two proposed mechanisms for the shortening of carbon 
nanotubes as a result of oxidation.  The first is ?fuse burning? taking place form the tube 
ends and the second is the cutting at induced defect sites in the nanotube walls.  Of 
course, these modes can operate simultaneously as well.  Early descriptions of tube 
etching under sonication in sulfuric/nitric acids described the mechanism where cut and 
open tube ends were slowly etched away by the ?fuse burning? process [120].    Ziegler et 
al. (2005) also agreed that ?fuse burning? was to blame for etching in sulfuric/nitric acid 
mixtures seen previously but argued that was not so in Piranah solution [130].  Marshall 
 
53 
et al. (2006) later correlated the treatment/sonication time with the degree of cutting and 
oxidation (2 mg SWNT/1 mL) after exposure from 2 h to 14 h [132].  This group 
concluded that two hours of treatment increases the acidic group content by 1%.  
Additionally, the found that at short time scales the amount of observed length reduction 
indicated that ?fuse burning? alone could not account for this.  Furthermore, the 
substantial increase in carboxylic acid groups even at shorter times indicates that cutting 
at sidewalls is necessary.  Additionally, a nonlinear increase in acidic sites was seen by 
Zhang et al. (2003).  Finally, Forrest and Alexander (2007) developed a working 
relationship between oxidation time and nanotube length for SWNT in nitric and sulfuric 
acid showing the nonlinear relationship that could not be explained by ?fuse burning? 
alone [133].    
 
Fourier Transform Infrared Spectroscopy of Oxidized CNTs:       
 Fourier transform infrared spectroscopy has been the most widely used technique 
for identifying oxidation products [99, 111, 114, 121, 122, 125, 134, 135].  In the infrared 
region the energy levels involved in the transitions of molecules are typically a result of 
vibration and rotation.  To this end, identification by means of characteristic group 
frequencies have been successful in identifying surface functionalities.  For an exhaustive 
source of tabulated spectra for both IR and Raman refer to Socrates (2004) [136].  
However, several groups have had difficulties directly resolving any differences between 
the spectra of the starting material and treated samples.  Attempts by Rasheed et al. to 
observe carboxylic functionalities were unsuccessful by FTIR unless the product was 
further derivatized with octadecylamine.  The authors hypothesize that low 
 
54 
concentrations of acidic sites may be the cause of this.  Likewise, to examine their 
oxidation product Saito et al. treated the cut tubes with alkyldiamines prior to FTIR 
analysis [122].  Additionally, difficulties have been identified with detection and 
contamination.  Li et al. (2007) were unable to detect carboxylic groups on MWNTs after 
treatment with hydrogen peroxide.  This group found difficulties with contamination by 
absorbed water manifested by peaks at 3444 cm
-1
 and 1636 cm
-1
 [112].  Similar artifacts 
were found previously by Shaffer et al. (1998) as well [99].   
 Although a few studies have indicated the presence of ether functionalities in the 
oxidized tubes using various X-ray techniques, these functional groups have generally not 
been observed via FTIR [137, 138].  Likewise, Wang et al. (2005) used a microwave 
reactor to carboxylate SWNTs in a sulfuric/nitric acid mixture and discovered various 
sulfonate functionalities were present that have not been identified elsewhere [139].  
Therefore, it is unlikely that ether or sulfonate groups typically are produced by 
oxidation.  In fact, Zhang et al. (2003) have performed theoretical calculations and 
molecular simulation to conclude that after strong oxidative treatment only ?COOH,  
-OH, and C=O functionalities can arise.  Additionally, they concluded that all groups 
must be directly connected with the aromatic rings of the CNT while C=O can only lie in 
the plane of the ring or on the bridging carbon atom of intersected rings [114].    
 Tzavalas et al. (2006) treated MWNTs with the sulfuric/nitric acid mix  
(1g MWNT/40 mL) by reflux at 140
o
C for 0.5, 2, and 4 h.  Although they note that the 
C=O stretch was hard to resolve in some cases, the C=O stretch of carboxylic acids was 
identified at 1717-1723 cm
-1
.  Additionally, a peak between 1354 - 1366 cm
-1
 was 
assigned to the -OH deformation mode.  Finally, the -OH stretch of carboxylic acid 
 
55 
monomer was found at 3180 cm
-1
 and between 3329 - 3335 cm
-1
 for hydrogen bonded 
carboxylic acids.  The authors note that due to high symmetry of C-C bonds in the tubes 
themselves they are generally inactive in the infrared [134].  Similar carboxylic peaks 
were identified by Zhang et al. (2003) for SWNT treated by the sulfuric/nitric acid mix.  
A peak at approximately 1735 cm
-1
 was assigned to the C=O stretching mode in the  
?COOH group of carboxylic acid.  The authors note that carboxylic peaks at 1740 cm
-1
 
are generally assigned to ?COOH in the free state.  This peak shifted from 1737 cm
-1
 to 
1720 cm
-1
 as treatment time progressed suggesting the presence of carboxylic groups not 
only increases with time but the hydrogen bonding between the groups become more 
effective.  After treatment with NaOH peak at 1540 - 1615 cm
-1
 for the carboxylate ion 
appeared which is proof of ?COO
-
 charging.   
  
Raman Spectroscopy of SWNTs: 
 Despite probing a similar range of incident wavenumbers, the origin of Raman 
spectra is different than that of FTIR.  FTIR measures the adsorption or transmission of 
IR light by the sample as a function of frequency.  On the other hand, with Raman 
Spectroscopy the sample irritated by intense UV-visible lasers and the scattered light is 
collected.  Raman excites the molecule to what is known as a ?virtual state;? called so 
because it is not a true quantum state rather a short lived distortion of the electron cloud 
[140].  A sample is IR-active if the molecular vibration changes its dipole moment and 
Raman-active only if the molecular vibration changes the polarizability [141].  For 
example, in a C=C bond the polarization changes significantly with a vibration associated 
with a C=C stretch.  As a result, a C=C vibration is strong toward Raman scattering 
 
56 
whereas a C=O stretch is not [140].  Thus, Raman has the capability to measure more of 
the molecular backbone of CNTs whereas FTIR readily measures the functional groups.    
 The Raman scattered light consists of two parts; the Rayleigh scattering (elastic) 
of the same frequency as incident laser and the Raman scattering (inelastic) of weak 
intensity and probability.  The Raman scattering is shifted at a frequency plus (anti-
Stokes) or minus (Stokes) the vibrational frequency of the sample molecule.  Since, the 
Stokes side of the spectrum is much stronger (it gains energy from the shift) this is 
measured [141].  As a result of this phenomenon, the vibrational frequency of the 
molecule is measured as a ?Raman shift? from the incident beam. 
 
Raman Features of SWNTs: 
 Typically the strongest features in SWNT Raman spectra are the 1
st
 order features  
of the G-band at ~1580 cm
-1
 (E
g
 mode) and the low frequency radial breathing modes 
(RBMs).  In fact, with respect to other graphitic materials the RBMs (A
g
 mode) are 
unique to SWNTs and provide direct evidence for their presence [142, 143].  The RBMs 
are located from approximately 100 - 500 cm
-1
 and are a result of out of plane bond-
stretching where all the atoms radially move coherently.  Essentially, the stretching is 
along the tube diameter equally in all directions.  The frequency of these peaks are 
inversely proportional to the tube diameter but the exact relationship varies between 
individuals and bundles.  In general, at lower Raman frequencies the vibrations can only 
serve as a probe of the nanotube surface and at higher frequencies reflect the local sp
2
 
bond structure for graphite [143].  The G-band feature in SWNTs has two main 
components; G
+
 peaks at ~1590 cm
-1
 and G
-
 at ~1570 cm
-1
.  The G
+
 vibration is 
 
57 
associated with carbon vibrations along the nanotube axis and its frequency upshifts and 
downshifts with charge transfer from dopant additions.  For example, Davis observed that 
SWNTs in superacids showed an upshift as a result of protonation [144].  G
-
 is associated 
with vibrations of carbon atoms along the circumferential direction of the tube and its 
frequency and lineshape are dependant on diameter and the electronic structure (metallic 
or semiconducting).  Figure 14 displays a schematic of the various atomic vibrations 
associated with the 1
st
 order Raman modes. 
 
Figure 14: Schematic representation of the RBM showing vibration of the carbon atoms is in the 
radial direction as if the tube were ?breathing? and the G-band showing tangential vibration in the 
circumferential direction and atomic displacements along the axial direction.  Image taken from 
Jorio et al (2003) [145].    
 
Two 2
nd
 order Raman scattering features are also present in SWNT spectra, the D and G? 
band.  These features are present in graphite as well.  The D-band is located at ~ 1350 
cm
-1
 and is dispersive; it shifts with the incident laser energy.  The G? band is located at 
approximately 2700 cm
-1
 and is an overtone of the D mode.  This peak is highly 
dispersive [146].  The D-band stems from the disorder induced mode in graphite with the 
 
58 
same name [142].  It origin is a result of scattering from a defect that breaks the basic 
symmetry of a graphene sheet and thus it is observed in sp
2
 carbons containing vacancies, 
impurities, or other symmetry-breaking defects [146].  Ferrari and Robertson (2000) 
showed a relationship between the G-peak position, the ratio of D to G peak intensity 
(I(D)/I(G)), and sp
3
 carbon (amorphous) content [147].  In general, a large I(D) with 
respect to I(G) in SWNT bundles usually indicated the presence of amorphous carbon 
[145].  Additionally, the relative intensity of the D-mode can provide direct evidence of 
covalent modification to the nanotube framework [148, 149].  Broadening of the D-band 
has been previously observed as a result sidewall functionalization by means of ozone 
oxidation in absence of significant amorphous carbon generation [150].  Thus, the 
observation of the D-band is related to either the presence of defects in the tube walls 
(vacancies, 7-5 pairs, dopants, tube ends) or to the presence of amorphous carbon 
material in the sample [151].  Furthermore, the D to G peak intensity ratio can also be 
correlated with tube length at certain wavelengths [152].  For shorter tubes the end effects 
become more important and therefore the intensity ratio was found to be higher.  But, this 
ratio did not approach zero for longer tubes.  Instead, finite residual values for I(D)/I(G) 
were observed as a result of defect induced scattering within the tube walls, confirming 
the dual nature of the existence of D-band scattering.      
 
X-ray Photoelectron Spectroscopy of CNTs: 
 X-ray photoelectron spectroscopy (XPS) measures the emission and energy of 
core level photoelectrons from a solid surface resulting from an incident X-ray beam.  
Depending on the size of the material under study the surface atoms may be in small 
 
59 
proportion and therefore detection must be both sensitive and free of atmospheric 
interference.  Thus, XPS analysis is conducted under ultra-high vacuum.  The observed 
photoelectrons are often described by means of their quantum numbers and have 
characteristic energies which reflect the atomic binding energy.   The intensity at a 
certain observed binding energy indicates the concentration present of a certain bonding 
type.  The binding energy is simply the incident X-ray energy less both the kinetic energy 
of the ejected electron and the work function of the instrument.  The binding energy 
identifies the electron specifically and is a function of the imposed X-ray energy (h?), the 
kinetic energy of the ejected electron E
k
, and the spectrometer work function.  What is 
recorded experimentally is the photoelectron spectrum as intensity vs. binding energy and 
this accurately represents the quantum electronic structure of the material.  Essentially, 
the photoelectrons have a kinetic energy distribution which represents the shell form of 
the electronic structure [153].  Only excited photoelectrons that are elastically scattered 
contribute peaks in the spectrum.  Once a photoelectron is emitted the ionized atom must 
relax by emission (fluorescence) of an X-ray photon [154].  Since the electrons in the 
sample may be loosely bound or tightly bound, when excited with the same energy, the 
photoelectrons produced may be of various kinetic energies.  Thus, even though the 
incident radiation may be monochromatic the observed spectrum will yield a 
polychromatic photoemission [153].  The depth of analysis will vary with the kinetic 
energy of the incident beam, but it is generally only a few nanometers.  This is dependant 
not on the sample penetration which is actually deeper but rather the depth from which 
photoelectrons can escape of approximately 2 nm [153].     
 
60 
High Resolution Analysis of CNTs: 
 For typical carbon based materials the main peaks present on the surface spectra 
result from the photoemission of electrons from the carbon 1s core orbital (C1s) at 
~284.6 eV and the 1s core orbital of oxygen (O1s) at ~531 eV [155].  Within high 
resolution C1s analysis characteristic binding energies are representative of a graphitic 
peak (aromatic and aliphatic) at 284.6 - 285 eV and four main oxide peaks.  The exact 
location of the oxide peaks vary slightly in the literature and are found at ~286.1 - 286.3 
eV for ?C-OH (hydroxyl, phenol and enol-keto) and ?C-O-C? (ether); 287.3 - 287.7 eV 
for ?C=O (carbonyl and quinone); 289.1 - 289.4 eV for ?COOH and ?COOR (carboxyl), 
and 290.6 eV for ?COO? (carbonate and/or absorbed CO, CO
2
) and ?-?
*
 shake up 
satellite peaks [155-159].  The shake-up satellite peaks are a result of ?-?
*
 transitions in 
aromatic rings [160].  For SWNTs specifically binding energies for the carboxylic groups 
have been fit as low as 288.6 - 288.8 eV and as high as 289.54 eV [161-163].  
 
Thermodynamics of SWNT Solutions: 
Introduction: 
 Rigorous thermodynamic analysis of SWNT dispersions and solutions are still 
rare in the literature.  To date, currently only two exhaustive studies exist that attempt to 
describe the free energy of mixing with each work taking a different approach.  The first 
thermodynamic description of nanotube solutions was treated by Sabba and Thomas 
[164].  Specifically, an approach based on Onsager?s theory for anisotropic particles was 
used to describe the Hemholtz energy of mixing.  The authors conclude that due to the 
large van der Waals attraction and poor solvents generally available for dispersion, a 
 
61 
negative value for the free energy of mixing may only be obtained at low nanotube 
number density.  Therefore, according to their analysis only extremely dilute solutions of 
nanotubes are thermodynamically favorable.   
 A later attempt was made by Bergin et al. who first formulated an appropriate 
description for the Gibbs free energy of mixing and then attempted to justify their 
theoretical treatment by experimentally measuring individual thermodynamic variables in 
a ?good? solvent [165].  A ?good? solvent is considered as one in which there is no 
enthalpy of mixing.  Solutions with zero enthalpy of mixing are considered athermal and 
therefore there is no liberation of absorption of heat between components.  An approach 
similar to that of Bergin et al. is taken in this section to describe the important 
thermodynamic aspects of CNT solutions and draw general conclusions based on these 
arguments.         
 
Motivation for Rod-like Particle Theory: 
 Flory (1955) was the first to consider that the role of polymer chain inflexibility 
or stiffness plays an integral part in polymer crystallization [166].  In motivating this idea 
the filling of given volume with a large fraction of disordered chains that resist bending 
was considered.  It was argued that if one were to imagine placing long inflexible 
molecules one at a time (while preserving disorder) into the volume after a certain 
fraction was added it would be impossible to find space for additional molecules without 
relaxing the previous requirement for chain stiffness.  By this reasoning it was 
hypothesized that only through the development of order (parallel alignment of chains) 
could the remaining, or additional, molecules be inserted.  It is important to note that no 
 
62 
weight has yet been given to the intermolecular forces that may be present in formulating 
this idea.  Similar conclusions have also been made previously by Onsager (1949) and 
Isihara (1951) who theoretically argued that separation of an ordered phase highly 
asymmetric particles will occur even in the absence of repulsive intermolecular forces 
[167-169].  However, Flory was critical of the applicability to these theories outside of 
low particle concentration; a criticism Flory would later addressed by extending his 
theory for semi-flexible polymers to rod-like particles [170].  On the other hand, it has 
been argued that Flory?s lattice approach suffers when describing liquid-crystalline 
ordering due to restrictions placed on the chain flexibility.  In general, Onsager?s theory 
is more appropriate for studying liquid-crystalline order while Flory?s theory is more 
useful for describing concentrated solutions and melts [171].  
 Flory?s semi-flexible model considered polymer chains having many segments 
arranged on a two-dimensional lattice.  As a result of the constraint of chain stiffness 
chain segments had to placed co-lineary with adjacent segments of the molecule.  Also, 
the chains are assumed to exist in a configurational arrangement that is energetically 
favorable over others, for example the trans isomer of polymethylene (-CH
2
-), but in a 
discrete manner defined by the lattice.  This treatment restricts the placement of 
subsequent chain segments.  With the geometrical constructs in place and 
considering j molecules already inserted into the lattice the next step was to estimate the 
expected number of vacant sites accessible as the first segment of the 1+j chain was 
added.  Such an exercise in combinatory mathematics was independently developed by 
Flory and Huggins for a two component liquid mixture of polymeric solute in solvent 
assuming no change in volume upon mixing [166, 170, 172, 173].  Flory then extended 
 
63 
this treatment for semi-flexible polymers by including the degree of chain bending, f .  
The result was an expression for total number of possible lattice configurations and 
subsequently the partition function.   
 To address the issue of the solute behaving as a rigid-rod ( 0=f ) with no 
preferred orientation, the semiflexible model?s partition function was modified by taking 
the flexibility and interaction parameters to zero.  The partition function for athermal 
solutions of rigid rods at two extremes was then extracted.  At one extreme the system 
existed in complete disorder and the other in the rods existed in parallel arrangement.  
However, neither expression was thought appropriate to describe intermediate 
arrangements with the isotropic extreme found to be only appropriate in the dilute regime 
or for short rods.  What was needed was an intermediate expression that would expand 
the theory past what was done previously for highly asymmetric particles.                    
 
Theory of Solutions for Rod-like Particles:      
 In developing the expectation for the intermediate orientations a new approach 
was adopted to the lattice model to overcome its inability to accommodate a rod in a 
continuously varying range of orientations [170].  Figure 15 displays molecule i inclined 
at an angle
i
? with respect to the horizontal axis of orientation, the principal axis of the 
lattice [170].  Instead of treatment as a continuous rod as shown by Figure 15, Flory 
divided each molecule into a series of submolecules having parallel arranged segments in 
a manner consistent with Figure 16 [174, 175].     
 
 
64 
 
Figure 15: A rigid-rod in the Flory lattice tilted at angle ? to the director.  The rod length is taken 
to be xd so that x plays the role of the aspect ratio where d is the characteristic dimension of the 
cubic lattice cell.  Reproduced from Beris and Edwards (1994), Cifferi (1994), & Wang and Zhou 
(2004) [171, 174, 175].  
 
 
Figure 16: A rod divided into y
i
 submolecules having x/ y
i
 segments per submolecule and tilted at 
angle ? to the director.  The term y
i
d represents the height with respect to the director with units 
of lattice cell number.  Reproduced from Beris and Edwards (1994), Cifferi (1994), & Wang and 
Zhou (2004) [171, 174, 175].  
   
Introduction of a new parameter y
i
 known as the disorder index was defined by Equation 
2.13.  
 ( )?= sinxy
i
 (2.13) 
For Equation 2.13 to be valid it is required to take x as the rod aspect ratio with a 
diameter equivalent to the lattice size.  Therefore, a total of y
i
 submolecules were taken to 
have x/y
i 
segments per submolecule,
 
aligned along the principal axis with each 
xd 
y
i
d 
? 
x/y
i 
? 
y
i
d 
 
65 
submolecule occupying one lattice site.  The omission of the index on the rod length of x 
indicates that all solute molecules are taken to be of uniform length.  In this case, with j 
molecules already in place, the number of situations accessible to an additional j+1 rod 
inserted into the lattice must be considered along with its orientation of ?
j+1
.  This was 
achieved by estimating the probability of finding a vacant lattice site while inserting each 
of the  xsin?
j+1
 submolecules one at a time, row by row.  A conditional probability 
allowing for the necessary lattice sites in adjacent rows, a result of the inflexibility, must 
also be upheld.  What results is the partition function for disordered rods in an athermal 
solution with the solvent.  Both extremes of Flory?s semi-flexible theory can also be 
recovered from this treatment.  In perfect alignment the angle of inclination would 
disappear and there would be only one submolecule per rod.  In other words, for perfect 
order the disorder index is unity and approaches x for complete disorder.  The partition 
function (Z) for rods of random disorder is displayed by Equation 2.14 where Stirling?s 
approximation has been used to treat the logarithm of combinatory factorials [171]. 
 
() ?ln1lnlnln
xx
x
xss
nxn
x
v
nvnZ ??+
?
?
?
?
?
?
+=?
 (2.14) 
In Equation 2.14 n
s
 and n
x
 represent the number of solvent and solute molecules with v
s
 
and v
x
 represent the volume fraction of solvent and solute molecules, respectively.  The 
rod aspect ratio is taken to be x and ? manifests itself as the rotational degeneracy factor, 
an arbitrary constant.   
 
 
66 
Entropy of Mixing: 
 Equation 2.14 can be related to the nanotube-solvent system by using more 
convenient factors suitable for experiment.  The nanotube loading is an independent 
variable defined by the volume fraction,? , and the aspect ratio is taken as ?.  With these 
conventions, and combining like terms, Equation 2.14 can be written in a more useful 
form as show by Equation 2.15. 
 
() ()1ln1lnln ?+
?
?
?
?
?
?
+?=? ?
??
?
?
NTNTs
nnnZ
 (2.15) 
This expression in itself contains the statistical configuration information for the isotropic 
array of nanotubes approximated as a rigid-rods in an athermal solution.  The change in 
entropy upon mixing
mix
S? can then be obtained by invoking the Boltzman? relation as 
shown by Equation 2.16 [27].   
 () ()
?
?
?
?
?
?
?+
?
?
?
?
?
?
+??==? 1ln1lnln ?
??
?
?
NTNTsBmix
nnnkZkS  (2.16) 
Here, k
B
 represents Boltzmann?s constant. 
 
Enthalpy of Mixing: 
 In describing the enthalpy of mixing it is first necessary to consider the existence 
of nanotubes in their natural state.  Evidence of crystalline nanotube ropes was shown by 
Thess et al. with hexagonal close packing [54].  Taking nanotubes having density ?
NT
 
existing in bundles of radius R
1
 and length L
1 
the number of bundles N
1 
in a fixed mass of 
nanotubes can be defined by Equation 2.17.  
 
1
2
1
1
LR
m
N
NT
NT
??
=
 (2.17) 
 
67 
 
Bergin et al. considered such a system and determined the enthalpy of mixing for a 
solvent-nanotube dispersion by calculating the energy required to separate all molecules 
to infinity less the energy required to bring them back arranged in a dispersed phase [165, 
176].  Here a dispersed phase is considered one in which the final solute bundle size has 
been reduced with respect to the bulk starting material.  This was treated by dividing the 
contributions into five energetic components.  The first component being the energy 
required to create the surfaces associated with the individual nanotubes,
NT
E
1
.  This is a 
direct function of the intertube attractive potential (energy per surface area).  A second 
enthaplic component was derived considering the energy required to remove all solvent 
molecules to infinity,
Sol
E
1
.  This was taken as the difference between the solvent energy 
of cohesion (per volume) less the solvent interfacial surface energy and is by definition a 
function of the pure solvent only.  In order to arrange the nanotubes in a dispersed state in 
the solvent a second term
NT
E
2
was used in the exact same fashion as
NT
E
1
but differing in 
the bundle final size, R
2
.  Presumably R
2
 will be much smaller than the original bundle 
size.  The fourth term
Sol
E
2
 takes the solvent molecules from infinite separation to a 
molecular packing appropriate for the liquid state specific volume while leaving voids for 
the newly dispersed nanotube bundles.  This term
Sol
E
2
is identical to
Sol
E
1
less the solvent 
interfacial surface energy interacting with the surface of the dispersed bundles.  Finally 
the interfacial energy associated with placing the nanotubes in the voids
SolNT
E
?
2
is 
 
68 
estimated as twice the solvent-nanotube binding energy per surface area
SolNT
Inter
E
?
.  
Combining all equations and rearranging like terms yields Equation 2.18 [176].             
 )(
22211
SolNTSolNTSolNT
mix
EEEEEH
?
++?+=?  (2.18) 
In this analysis, the magnitude of attraction was taken to be positive by convention.  
Bergin et al. then assumed the solvent external surface area was the same before and after 
mixing and that the bundle size was much reduced compared to the original state leading 
to an approximate solution as displayed by Equation 2.19 [165, 176]. 
 
[ ]
SolNT
Inter
Sol
Sur
NT
Sur
NT
NT
mix
EEE
R
m
H
?
?+??
?
2
2
 (2.19) 
Rigorous derivation of Equation 2.19 using identical assumptions can be found in 
Appendix A1.  Next, the solute-solvent binding energy can be approximated by Equation 
2.20 with the identity of Equation 2.21 [176].   
 ( )
2/1
Sol
Sur
NT
Sur
SolNT
Inter
EEE ?
?
 (2.20) 
 ( )
2/1
i
Suri
E??  (2.21) 
By relating the nanotube loading to the total solution volume V
Mix 
by Equation 2.22 the 
energy of mixing can be made of further experimental utility as displayed by Equation 
2.23 [176]. 
 
MixNT
NT
V
m
?
? =
 (2.22) 
 
()
2
2
2
SolNT
Mix
mix
RV
H
??
?
??
?
 (2.23) 
According to Equation 2.23, in order to minimize the heat of mixing the surface solvent 
parameters for the nanotubes and solvent should match.  The origin of R
2
 in the 
 
69 
denominator is a result of estimating the tube/bundle number density N
2
 in the final state 
from the SWNT tube density and bundle size (see Appendix A1).  It is important to note 
that these parameters are not identical to the regular solubility parameters used in the well 
known Scratchard-Hildebrand solution theory.  In Equation 2.23 the solute and solvent 
parameters are a function of the surface energy whereas Scratchard-Hildebrand uses the 
cohesive energy density [62].  Hence, from here on the parameters defined by Equation 
2.21 are referred to as ?surface-solubility? parameters.  It is important to point out that 
approximation introduced by Equation 2.20 allows the surface solubility parameters ?
i
 to 
be estimated from the pure component interaction energy using Equation 2.21, as 
discussed in Chapter 4.  For the nanotube ?surface solubility? parameter ?
NT
 both 
experiment and theory have produced approximate values for the intertube attractive 
potential
NT
Sur
E .  Additionally, the pure solvent surface energy has been shown to be 
related to the readily measurable solvent surface tension.  To this end, a surface tension 
criterion for minimizing the enthalpy of mixing has been estimated for solvent as  
~40 mJm
-2
 by screening several common nanotube solvents [165].        
 Alternatively, the enthalpy of mixing can be defined using the Flory-Huggins 
parameter, ?.  The Flory interaction parameter is dimensionless and describes the 
interaction energy between solute-solute, solute-solvent segment, and solvent-solvent 
segments [62].  For ideal solutions ? is zero and can take positive or negative values for 
real solutions.  For chemically similar mixtures ? is small compared to unity [62].  
Additionally, ? is inversely related to the temperature but independent of solute 
concentration [27, 177].  Equation 2.24 expresses the enthalpy of mixing as a function of 
 
70 
? [27].  This expression was originally adapted from the van Laar heat of mixing by Flory 
[177].     
 ??
smix
RTnH =?  (2.24) 
In the case for the nanotube-solvent system it is more useful to write n
1
 in terms of the 
solvent volume fraction.  Equation 2.25 introduces the solvent molar volume
s
V to link 
these terms. 
 
( )
s
Mix
s
V
V
n
??
=
1
 (2.25) 
Inserting the expression for n
s
 into Equation 2.24 yields Equation 2.26 upon subsequent 
rearrangement. 
 
()??? ?=
?
1
sMix
mix
V
RT
V
H
 (2.26) 
By taking advantage of Equation 2.26 the heat mixing can be measured more accurately 
than by Equation 2.23 and this was used by Bergin et al. to validate the estimation for the 
enthalpy of mixing [165].  In order to minimize the enthalpy of mixing ? should be small 
or negative [62].  Additionally, ? can be directly related to the solubility parameters and 
the solvent molar volume by Equation 2.27 [62]. 
 
()
21
SolNT
RT
v
??? ?=
 (2.27) 
Equation 2.27 again supports the minimization of ?, and therefore
Mix
H? , by matching ?
i
 
for the solute and solvent which agrees with the theory behind Equation 2.23.  However, 
the form of Equation 2.23, Equation 2.27, and any solubility parameter based expression 
contains a shortcoming for estimating the heat of mixing in a ?real? solution.  The square 
 
71 
of the difference in these equations cannot yield a heat of mixing less than zero [27].  
Thus, exothermic mixing cannot be effectively predicted or described.   
 Finally, in absence of tabulated data the estimation of the enthalpy of mixing 
using the Flory-Huggins parameter can be achieved via the second viral coefficient.  By 
differentiating the free energy of mixing with respect to solution volume Bergin et al. 
related the osmotic pressure to the second virial coefficient using a virial expansion based 
off the nanotube density [165, 176].  Subsequently, the osmotic pressure was related to 
the Rayleigh ratio via light scattering which therefore enabled the determination of the 
virial coefficient, and finally ? [176].          
 
Free Energy of Mixing: 
 The driving force behind dissolution of a solute into a solvent can be defined by 
the Gibbs free energy.  Equation 2.28 gives the free energy of mixing at constant 
temperature and pressure [3, 27]. 
 
MixMixMix
STHG ???=?  (2.28) 
In the absence of metastabilities, dissolution into a single phase occurs only if the change 
in free energy upon mixing ?G
Mix
 can be decreased.  Criteria for dissolution can be 
displayed by Equation 2.29. 
 
0<???=?
MixMixMix
STHG  (2.29) 
In the same manner, dissolution can be viewed as competition between the enthalpy and 
entropy of mixing as shown by Equation 2.30 [165]. 
 
MixMix
STH ?<?  (2.30) 
 
72 
From Equation 2.30 it can be deduced that at constant temperature and pressure 
minimizing
Mix
H? and maximizing
Mix
S? will help drive solute solubility.  For typical 
small molecule solutions the enthalpy of mixing,
Mix
H? , is small and positive whereas the 
entropy of mixing,
Mix
S? , is of a large magnitude resulting in an overall negative free 
energy of mixing, driving dissolution in the solvent.  As a result of their inherent 
properties and structure, carbon nanotubes behave differently than such an ideal small 
molecule system.  It has been generalized that in most polymer-nanotube mixtures the 
large intertube van der Waals potential is expected to result a positive enthalpy of mixing 
[165].  This is manifested in a contribution to
Mix
H? via the
NT
E
2
term of Equation 2.18 
(
NT
Sur
E
in Equation 2.16 holds the same meaning).  Furthermore, the nanotube stiffness 
and high molecular weight generally results in small, positive entropy of mixing [165].  
By comparison, Flory-Huggins theory for a system of low molecular weight solute (small 
?) and solvent to that of a low molecular weight solvent and high molecular weight 
polymer (large ?) one finds the entropy of mixing has become smaller as the solute aspect 
ratio increases (Equation 2.16).  This is a result of the loss in conformational entropy 
resulting from the linked repeat units in a polymer chain compared to the less ordered and 
unassociated low molecular weight solute-solvent system [27].  Examining the 
dependence of 
Mix
S?
(Equation 2.16) variables ? and n
NT
 supports this argument for 
nanotube-solvent systems.  Therefore, a positive free energy of mixing results in 
nanotube-polymer systems, preventing spontaneous dissolution.  Bergin et al. confirm 
this hypothesis by demonstrating that the solvent N-methyl-pyrrolidone provides an 
enthalpy of mixing of approximately zero.  The calculated result was a negative free 
 
73 
energy of mixing and this was confirmed by the spontaneous exfoliation of nanotubes in 
solution, leading to a dynamic equilibrium between individual tubes and small bundles. 
 
Rheology Theory and Measurements: 
Introduction: 
 In general, rheology is the field of science that studies the flow and deformation 
of matter [178].  Of particular interest to rheologists is the study of complex fluids.  
These types of fluids show mechanical behavior intermediate between classical solids 
and liquids.  Classical fluids show what is termed prototypical ?liquid-like? and ?solid-
like? behavior, as will be discussed in later sections.  For point of reference, a classical 
solid will maintain its shape indefinitely whereas a classical liquid will take the shape of 
its container [179].  Commonly, many complex fluids will show intermediate behavior in 
that they will maintain their shape for a certain time and eventually flow.  A fluid of this 
type is termed viscoelastic as it behaves like an elastic solid over short timescales and a 
viscous liquid over long timescales.  Interestingly, a material does not have to be out of 
the ordinary to be considered complex.  In fact, many complex fluids and materials are 
used everyday.  Common examples of everyday complex fluids are mayonnaise and ice 
cream (emulsions), biological fluids like blood (suspensions), and molten plastics 
(polymers).  Often consumer products are engineered to have complex rheological 
behavior such as the yield stress of toothpaste or ketchup [179].                   
 A variety of instruments are available to measure rheological properties of fluids.  
Commonplace to industry and quality control are simple flow indexers or capillary 
viscometers.  On the other hand, research grade rotational rheometers are available 
 
74 
capable of measuring both steady-state and transient flow properties of materials under 
controlled stresses and/or strains.  Additionally, temperature and pressure controlling 
capabilities are available on many instruments [178]. 
 
Linear Viscoelasticity: 
 A classic example illustrating how the behavior of complex and ideal fluids differ 
in response under applied strain is stress relaxation.  The shear strain (or simply strain) is 
a measure of the sample deformation due to an applied force, F.  Figure 17 illustrates a 
sliding plate model where the sample is placed between the gap of distance h apart [180].   
 
 
Figure 17: Illustration of important terms used in describing the shear strain of a fluid between 
two parallel sliding plates.  Reproduced from Mezger (2002) [180].   
 
In Figure 17 the lower plate is stationary and the deflection of the upper plate is shown by 
the distance s.  Equation 2.31 defines the strain as a ratio of distances.   
 
h
s
=?
 (2.31) 
h
s
? 
Area 
F 
 
75 
The shear strain is a dimensionless quantity by definition and is equivalent to the tangent 
of the deflection angle ?.  For point of reference a deflection angle of 45
o
 (?/4 radians) 
corresponds to a strain of 1 (100%).  Often the strain set-points are quite small (< 5%) 
and therefore the deflection is measured in units of milliradians.  While operating the 
rheometer in strain controlled mode the user sets the strain and therefore the deflection 
angle.  The deflection angle is then fixed by the instrument throughout the test.  
 Figure 18 illustrates various response types when a step increase in strain ? is 
imposed on a fluid and the dissipation or relaxation of the stress ? is measured.   
 
      
Figure 18: Illustration of a stress relaxation experiment.  (a) The step increase in strain and (b) the 
stress vs. time response (relaxation) of various fluid types.  Reproduced from Macosko (1994) 
[67]. 
 
Figure 18a shows the step strain imposed on the fluids.  Figure 18b shows the typical 
response for various types of fluids.  The Newtonian fluid relaxes as soon as the strain 
reaches a constant value characteristic of a prototypical viscous liquid.  Conversely, the 
Newtonian 
Hookean 
VE liquid 
Time 
Time 
VE solid 
Strain 
?  
Stress 
?   
(a) (b) 
 
76 
Hookean solid shows prototypical elastic behavior with no stress relaxation.  Intermediate 
behavior is seen for both the viscoelastic liquid (VE Liquid) and solid (VE Solid) where 
the stress relaxes over some time.  The difference being the VE liquid will eventually 
relax to zero stress whereas the solid approaches an equilibrium value [67].   
 The stress relaxation data can be easily converted into a modulus by taking the 
stress to strain ratio.  In this manner the stress relaxation modulus is defined by Equation 
2.32.   
 
()
( )
?
??
?
,
,
t
tG =
 (2.32) 
Typically, for imposed step strains larger than unity the stress is both a function of time 
and the imposed strain.  This behavior is known as nonlinear viscoelasticity.  On the 
other hand, for small strains, typically less than unity, log-log plots of the stress 
relaxation modulus superimpose.  This region of strain showing independent stress 
modulus is known as the linear viscoelastic regime.  Equation 2.33 defines the linear 
relaxation modulus.     
 
()
( )
?
? t
tG =
 (2.33) 
Both G and ? are only functions of time in the linear viscoelastic regime.  This is a 
fundamental observation and can be exploited to further characterize the material.   
 
Oscillatory Flow: 
 In the linear viscoelastic regime all measurements made at small strains less than 
the critical strain ?
c
 of that material are related to one another through Equation 2.34.  To 
this end small-amplitude oscillatory shear measurements provide important structural 
 
77 
information without significantly deforming the fluid?s microstructure.  Furthermore, by 
changing the frequency of strain one can vary the timescale of observation.  Typically a 
sinusoidal wave in stress or strain is imposed on the fluid with the frequency measured in 
cycles per second or commonly as angular frequency, ?, in units of radians per second.  
The strain can then be represented by equation 2.34 where ?
0
 represents the strain 
amplitude.       
 t??? sin
0
=  (2.34) 
Additionally, the strain rate is found by taking the time-derivative of Equation 2.34, the 
result displayed by Equation 2.35. 
 t???? cos
0
=&  (2.35) 
At steady state the stress will also oscillate sinusoidally with the same frequency but 
shifted by phase angle ? in regards to the strain wave, shown in Equation 2.36 [67].   
 ( )???? += tsin
0
 (2.36) 
Figure 19 displays the applied strain and resulting stress wave, graphically.  The stress 
will be out of phase with the strain rate unless the fluid is Newtonian.  In that case ? will 
shift the stress by ?/4 rad and the trigonometric functions will be of equal value.   
 
Figure 19: The imposed strain wave and resulting steady-state stress wave under simple shear 
flow.  The stress wave is in frequency with the strain but shifted in phase by angle ?.  Reproduced 
from Ferry (1980) [181]. 
 
78 
 
The stress wave can be decomposed further by applying the sum-difference identity 
leading to Equation 2.37. 
 
()( )ttt ???????? cossinsincossin
00
+=+
 (2.37) 
The first term in Equation 2.37 represents the stress component in phase with the strain 
wave (sin ?t) and the second represents the stress component out of phase with the strain 
(cos ?t).  Both components oscillate on the same frequency but do not necessarily have to 
be of the same magnitude.  Equation 2.37 holds true regardless of the deformation mode 
(e.g. shear, extension, flexure) provided that the strain amplitude does not exceed the 
linear viscoelastic threshold [178].  Graphically the components are displayed by Figure 
20. 
 
Figure 20: Trigonometric deconvolution of the steady-state stress wave in to its in-phase and out-
of-phase components.  Reproduced from Ferry (1980) [181]. 
 
Taking the ratio of Equation 2.37 to the strain amplitude, two independent moduli can 
than be defined by Equation 2.38 and Equation 2.39 [181].   
 
?
?
?
cos'
0
0
?
?
?
?
?
?
?
?
=G
 (2.38) 
 
?
?
?
sin''
0
0
?
?
?
?
?
?
?
?
=G
 (2.39) 
 
79 
Equation 2.38 displays the in-phase modulus, commonly referred to as the storage 
modulus, representing the storage of elastic energy in the sample.  Equation 2.39 
describes the out-of-phase, viscous, or what is commonly referred to as the loss modulus.  
The loss modulus represents the viscous dissipation of energy through the sample [179].  
For a perfectly elastic solid, the phase angle is close to zero radians and all energy is 
stored in the material as indicated by G? having no value.  For a perfectly viscous liquid, 
the phase lag is close to ?/4 radians and all energy is dissipated through the sample as 
heat with G? having magnitude of zero.  Figure 21a displays the linear viscoelastic 
response of a prototypical ?solid-like? fluid and Figure 21b shows the response of a 
prototypical ?liquid-like? fluid.   
       
Figure 21: The prototypical linear viscoelastic response in terms of frequency dependent storage 
and loss moduli G? and G? for (a) a ?solid-like? fluid and (b) a ?liquid-like? fluid.  Adapted from 
Larson (1998) [179]. 
 
For the ?solid-like? response in Figure 21a, G? is much greater than G? in magnitude and 
G? is also nearly or completely frequency independent.  G? does not necessarily have to 
log ? 
log G? 
log G? 
G? 
G? 
log ? 
log G?  
log G? 
G? 
G? 
  (a)   (b) 
 
80 
be frequency independent and may show some upturn at low frequency as a result of 
aging, as predicted for glassy substances [182].  For the ?liquid-like? response shown in 
Figure 21b G? dominates in magnitude and both moduli scale with frequency in a power-
law fashion.  As mentioned previously, real complex fluids are expected to show some 
behavior intermediate to these two ideal responses.      
 Taking advantage of Equation 2.38 and Equation 2.39 the expression for the stress 
wave (Equation 2.37) can be written in the more convenient form given by Equation 
2.40. 
 
( ) ( )[ ]tGtG ???? cos''sin'
0
+=
 (2.40) 
 Another useful parameter is defined by taking the ratio of the loss modulus to the 
storage modulus as displayed by Equation 2.41.   
 
'
''
tan
G
G
=?
 (2.41) 
The result is referred to as the loss tangent, damping factor, or by ?tan delta.?  Its 
magnitude is greater than unity for samples behaving ?liquid-like? and less than unity for 
?solid-like? samples.  Alternatively, liquid or solid behavior can be inferred by what 
modulus is dominant in magnitude at a given frequency.  Tan delta is simply another 
means to measure energy dissipation in viscoelastic materials [178].    
 The complex modulus G
*
 is defined by considering the amplitude of the stress 
waveform as a whole and taking its ratio to the strain amplitude [179].  Its magnitude is 
defined by Equation 2.42. 
 
( )
0
0
2/1
22*
'''
?
?
=+= GGG
 (2.42) 
 
81 
The magnitude of G
*
 can be interpreted as a measure of the overall resistance to 
deformation.  This measurement takes no consideration in to the how the energy is 
dissipated via G? and G? individually.  Finally, dividing the complex modulus by the 
angular frequency gives the magnitude of the complex viscosity ?
* 
as displayed by 
Equation 2.43. 
 
2/1
22
*
*
"'
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+?
?
?
?
?
?
==
???
?
GG
G
 (2.43) 
The complex viscosity is a measure of the fluid?s resistance to flow under oscillatory 
shear.   
 The complex viscosity can be related to the shear or apparent viscosity by means 
of the ?Cox-Merz rule.?  This relationship is strictly empirical but often holds true at high 
shear rates.  Cox-Merz simply states that shear viscosity of a fluid at steady state is 
equivalent to the complex viscosity of that fluid in the linear viscoelastic regime provided 
that they are measured at equivalent magnitudes of rotational shear rate and angular 
frequency [67].  Mathematically this can be expressed by Equation 2.44 with the 
condition of Equation 2.45.    
 
( ) ( )????
*
=
&
 (2.44) 
 
?? =
&
 (2.45) 
Again, Equation 2.44 holds true only if the conditions of matching shear rate and angular 
frequency shown by Equation 2.45 are met.  It is important to note that
?
&
used in Equation 
2.45 is intended to represent the shear rate but its meaning still holds true as the rate of 
deformation.  The difference being that under steady rotational shear the shear strain and 
 
82 
therefore shear rate is held constant whereas under oscillatory flow it remains a function 
of time.   
     
Simple Shear: 
 Although the geometries of various rheological measuring systems may vary all 
are similar in that they impose a drag flow by controlled motion along a system 
boundary.  For point of reference, refer to Figure 22 showing a simple sliding plate 
geometry.  All motion is along one coordinate direction (x
1
) with velocity varying in a 
second coordinate (x
2
) orthogonal to the shear geometry [178].  This type of flow is 
called viscometric flow.   
 
 
Figure 22: The sliding plate model and relative coordinate system for simple shear.  Adapted 
from Gupta (2000) and Mezger (2002) [178, 180]. 
 
The upper plate in Figure 22 moves along the positive x
1
 direction with velocity v while 
the sample remains between the gap of distance h.  Mathematically viscometric flow 
h
Area 
F, v 
x
3 
x
1 
x
2 
 
83 
translates to there being only one non-zero velocity component v
1
 and therefore one non-
zero component to the velocity gradient tensor [67].  This non-diminishing component is 
known as the shear rate?
&
and can be represented by Equation 2.46. 
 
2
1
x
v
?
?
??&
 (2.46) 
It is noteworthy to indicate that the shear rate is simply the velocity gradient and is often 
alternatively referred to as rate of deformation or strain rate as will be discussed in later 
sections.  For sliding-plate type geometries the magnitude of the shear rate is simply the 
velocity divided by the gap as shown by Equation 2.47. 
 
h
v
=?&
 (2.47) 
 The force that the flowing fluid is exerting on the surface, per unit area of that 
surface, in the direction parallel to flow is known as the shear stress, ?.  After imposing a 
steady shear to a fluid from a state of rest the shear stress does not reach steady state 
immediately.  Rather, the stress goes through a transient start-up period where it varies 
with time.  The ratio of the instantaneous stress ?( t,?
&
) to the imposed steady shear rate 
gives the transient start-up viscosity ?( t,?
&
) as shown by Equation 2.48. 
 
()
( )
?
??
??
&
&
&
t
t
,
, =
 (2.48) 
After a period of time has elapsed in which a steady shear has been applied the shear 
stress often comes to a steady state.  Thus, the shear stress is no longer a function of time 
and the shear viscosity is defined by Equation 2.49.  
 
84 
 
()
( )
?
??
??
&
&
& =
 (2.49) 
The behavior of the shear stress as a function of various shear rates is one way to define 
the type of fluid.  The simplest response is that of a Newtonian fluid in which the shear 
stress is directly proportional (linear) to the shear rate and the viscosity is therefore 
constant.  Deviation from this behavior indicates a shear rate dependence on viscosity and 
materials exhibiting this type of response are classified as non-Newtonian.   
 A common phenomena seen with some complex fluids is that of a yield stress.  
Often fluids with some ?solid-like? character will not initially flow upon the inception of 
shear.  Instead the stress grows linearly with shear rate up to a critical value above which 
the material ?yields? or flows at a constant shear stress.  It is noteworthy to point out that 
yielding is often indicated by the absence of flow but this term is truly intended to 
describe the point at which flow begins not resists; this is a common point of confusion 
[179].            
 In general, a fluid is a material that cannot support a shear stress without flowing 
[67].  Consider the fluid element displayed by Figure 23.  If at rest, all surface stresses on 
that element act orthogonal to the surface (normal stresses) and are not only equal in 
magnitude but independent of orientation of the element.  This type of normal stress is 
known as the hydrostatic pressure and is compressive in nature.  The absence of motion 
within the element also implies the absence of shear stresses.  Thus, for a fluid in motion 
it follows that their must be either shear stresses present or normal stresses that are 
unequal.  These shear stresses are known as viscous or extra stresses because they act on 
a fluid in addition to the fluid pressure.   
 
85 
 
 
Figure 23: Schematic diagram showing the notation for normal stresses in a shear field.  Adapted 
from Gupta (2000) [178]. 
 
The tensor T
ii
 contains the contributions of the pressure and the extra stress as shown by 
Equation 2.50. 
 
iiii
pT ?+?=
 (2.50) 
As a consequence of most liquids being considered incompressible the application of 
equal normal stresses (pressure) along the three coordinate directions shown in Figure 23 
does not lead to deformation or a change in volume.  Therefore, it is only unequal normal 
stresses that may cause motion or strain in liquids.  This means that one cannot determine 
the absolute value of the normal stresses from controlled strain experiments but only 
normal stress differences because the value of p alone is indeterminate.  Equation 2.51 
gives the expression for the first normal stress difference N
1
 and Equation 2.52 gives the 
expression for N
2
. 
 
F 
F 
11
T
22
T
33
T
 
86 
 
221122111
?? ?=?= TTN
 (2.51) 
 
332233222
?? ?=?= TTN
 (2.52) 
Subtraction of normal stresses eliminates the pressure term leaving only the difference 
between the extra stress.  A negative first normal stress difference tends to force the 
plates together and the positive first normal stress difference tends to push them apart.  
For a detailed explanation see Kumar and Gupta (2003) [3].  On the other hand the 
second normal stress difference tends to bulge the fluid at the edge of the plates.  The 
most common instance of a positive first normal stress difference is the die swell 
experienced by a melt extruded polymer emerging into the atmosphere.  But in general, if 
the fluid being sheared is polymeric the normal stresses become unequal in magnitude 
and if the fluid is Newtonian their magnitude remains the same [178].             
 
Cone and Plate Rheometer: 
 The geometrical features of the cone and plate rheometer is displayed by Figure 
24 utilizing a spherical coordinate system( )??,,r .  A flat, circular plate of radius R is 
opposed by a linearly concentric upper cone.  The tip of the cone is truncated so that no 
physical contact is made with the lower plate.  The cone angle ? is typically only a few 
degrees and the cone is rotated at an angular speed of ?.  Rigorous derivation for the 
velocity profile, shear stress, first normal stress, and finally the torque can be found in 
Macosko (1994) starting from the equations of motion in spherical coordinates [67, 183].   
 
 
 
87 
 
Figure 24: Coordinate system and geometric parameters used to describe the cone and plate 
measuring geometry.  Adapted from Macosko (1994) [67]. 
 
Assuming that the velocity is a function of r and ? only and the only non-diminishing 
component is in the? direction (
?
v ) the shear stress can be measured from the raw torque 
by Equation 2.53.  
 
3
2
3
R
M
?
? =
 (2.53) 
During an experiment the rheometer measures the raw torque, from a torque transducer, 
required to maintain the set test variable which is commonly the shear rate or the angular 
frequency at set strain amplitude.  By means of Equation 2.54 the shear stress is 
calculated and vice versa for stress controlled measurements.  It is important to note that 
there is no ? angular dependence on the shear stress due to the small magnitude of ?.  
Thus, the shear stress imposed on the fluid is essentially constant and uniform and 
therefore the shear strain and shear rate are uniform as well [67].  Analogous to Equation 
2.31 for the sliding plate, the shear strain in cone and plate geometry is measured by 
F
z
h 
R
?
?
M
r
?
,?
 
88 
introducing the angular displacement ?
d
.  Equation 2.54 represents the angular 
displacement under steady shear. 
 t
d
?=?  (2.54) 
By this definition the shear strain is displayed by Equation 2.55. 
 
?
?
? =
 (2.55) 
Likewise, the equivalent expression to Equation 2.47 for the shear rate in a cone and plate 
geometry is given by Equation 2.56. 
 
??
?
?
?
?
=
?
=
tanh
R
&
 (2.56) 
Here the angular speed of the cone is given by ? and the cone angle is represented by ?.  
The product of the angular speed and the radius R gives the tangential velocity at the 
edge.  The first normal stress difference is measured directly in cone and plate geometry 
by the total thrust on the plate.  Equation 2.57 gives the value of N
1
 as well as its 
interpretation.      
 
2
1
2
R
F
TTN
z
?
????
=?=
 (2.57) 
Here N
1
 is taken to be the difference between the normal stress in the direction of flow 
and the direction orthogonal to the lower plate.  The quantity F
z
 is the normal force 
measured by the rheometer force transducer taken to be the force required to hold the 
apex of the cone at the center of the lower plate [178, 179].  Similar expressions for 
Equation 2.57 can be found in various textbooks with all measuring the downward force 
[67, 178, 179].  As displayed by Figure 24, the rheometer used in this study measures the 
 
89 
normal force in the opposite direction.  Thus, the magnitude of Equation 2.57 must be 
adjusted for the coordinate system as displayed by Equation 2.58. 
 
2
1
2
R
F
N
z
?
?
=
 (2.58) 
 
Parallel Plate Rheometer: 
 Given that the cone and plate rheometer provides a uniform stress it is the 
preferred geometry for many rheologists.  However, if suspensions of particles or filled 
polymer melts have dimensions comparable to the small gap between the cone and plate a 
parallel plate geometry must be used.  In many ways these two devices are similar with 
the striking difference being that the flow between the parallel plates is not homogeneous 
but a function of the radius r.  Figure 25 displays the geometrical constructs parallel plate 
rheometer.  A cylindrical coordinate system( )zr ,,?  has been used to describe the plates. 
Two identical and coaxial flat parallel disks are rotated to impose flow.  Typically the 
upper plate is rotated with an angular speed of ? in the direction of ?.  The gap between 
the plates h is held constant and can be adjusted throughout testing.  An adjustable gap is 
advantageous for correcting sample edge defects or checking for sample slip whereas 
with the cone and plate rheometer the gap is fixed and requires reloading upon edge 
failure.  Rigorous derivation for the velocity profile, shear stress, first normal stress, and 
finally the torque can be found in Macosko (1994) starting from the equations of motion 
in cylindrical coordinates [67, 183].   
 
90 
 
Figure 25: Coordinate system and geometric parameters used to describe the parallel plate 
measuring geometry.  Adapted from Macosko (1994) [67]. 
 
Assuming that the velocity is a function of r and z only and the only non-diminishing 
component is in the ? direction (
?
v ) the shear stress can be measured from the raw torque 
by Equation 2.59.  
 
()
?
?
?
?
?
?
+=
R
d
Md
R
M
r
?
?
?
&
ln
ln
3
2
3  (2.59) 
Examining 2.59 it should be apparent that the derivative of the natural log of the torque 
with respect to the natural log of the plate edge shear rate
R
?
&
cannot be determined 
accurately without sufficient data to correlate the two variables.  However, if the fluid is 
Newtonian the derivative equals unity and the shear stress is defined as the apparent shear 
stress ?
a
.  Equation 2.60 gives the expression for the apparent shear stress. 
 
()
3
2
R
M
r
a
?
? =
 (2.60) 
F
z
h 
R
?
M
r
,?
 
91 
Often the apparent shear stress is simply used to calculate an apparent viscosity since 
only the raw torque measurement is required.  Obvious inaccuracy may be contained 
within measurements but this if often not addressed in the literature.  However, in the 
case of sinusoidal varying measurements within the linear viscoelastic regime the 
derivative in Equation 2.59 also diminishes to unity and the apparent stress given by 
Equation 2.60 is exact [67].  In both cases Equation 2.59 and Equation 2.60 are function 
of the radius of observation.  This is a result of the shear rate being a function of the 
angular speed which varies with radial distance from the center of the plate.  Equation 
2.61 displays the shear rate for the parallel plate geometry.     
 
()
h
r
r
?
=?
&
 (2.61) 
For sake of completeness the shear rate at the plate edge introduced previously is defined 
by Equation 2.62. 
 
h
R
R
?
=?
&
 (2.62) 
Likewise the shear strain also varies with radial position and can be calculated from 
Equation 2.63.  
 
()
h
r
r
d
?
? =
 (2.63) 
The angular displacement ?
d
 was introduced previously by Equation 2.54.  To evaluate 
visoelastic moduli the strain should be measured at the plate edge.  Equation 2.64 gives 
the expression for the edge strain. 
 
h
R
d
R
?
? =
 (2.64) 
 
92 
In contrast to the cone and plate rheometer the first normal stress difference cannot be 
measured alone for a parallel plate system.  Instead what is obtained is the difference 
between the first and second normal stress differences as shown by Equation 2.65. 
 
()
?
?
?
?
?
?
+=?
R
zz
d
Fd
R
F
NN
R
?
?
?
&
&
ln
ln
2
2
21
 (2.65) 
The normal force is represented by F
z
 and is taken to be positive consistent with Figure 
25.  Again if one were to compute Equation 2.65, consideration for the coordinate system 
of the instrument must be accounted for.  In order to completely decompose the 
difference between N
1
 and N
2
 measurements must be made in both parallel and cone and 
plate geometries and compared.  Since N
2
 is often of little magnitude the parallel plate is 
commonly used alone to estimate N
1
. 
 
Rheology of Particulate Suspensions: 
 Suspensions consist of discrete particles dispersed in a liquid medium [67].  The 
hydrodynamic disturbance resulting from the presence of particles was first determined 
by Einstein (1906 & 1911).  The presence of particulates can not only drastically change 
the measured viscosity but can also introduce non-Newtonian behavior to the fluid under 
study.  Essentially, the particle will modify the velocity distribution in a flowing liquid.  
Extra energy dissipation will arise because of this and will be reflected in a proportional 
increase in viscosity.  The stability of these types of fluids are of primary concern if one 
is to measure the rheological properties of the dispersion.  An exhaustive list of important 
assumption and criteria that need be met for accurate rheological measurements can be 
 
93 
found in the classic text by Macosko [67].  In general, for reproducible results the fluid 
structure or dispersion state should not change over the duration of the experiment.   
 Particle settling is the primary cause for concern as particles larger than 
approximately 1 ?m tend to settle under the influence of gravity alone [179].  This can be 
circumvented by either matching the density of the suspending medium to that of the 
particle or by using a medium that is very viscous.  By matching the densities of the 
particles and fluid the buoyant force can be exploited to support particles in suspension.  
Likewise, if the particles are Brownian, the ratio of gravitational to Brownian forces 
should be less than unity indicating stability from thermal motion.  Brownian motion is 
the name given to the irregular movement of pollen in water as observed by Robert 
Brown (1827) [184].  If the particle dimensions are small (generally sub-micron) then the 
random thermal motions of solvent molecules can transfer momentum to the particle.  By 
this analysis a density difference of approximately 1 g/mL leads to settling of particles 
bigger than 2 ?m in diameter [179].  On the other hand, if the medium is viscous the rate 
of particle sedimentation can be slowed.  Depending on the viscosity of the solvent this 
effect may be enough to allow for stable measurements to be taken over the duration of 
the rheological testing.  An approximate criterion for neglecting the effect of settling in 
spherical particles can be obtained from Stoke?s law as shown by Macosko (1994) [67]. 
 
Rheology of Nonspherical Particle Suspensions: 
 Generally, the rheology of nonspherical suspensions depend on the particle 
loading, the aspect ratio and its distribution, as well as the rheology of the suspending 
fluid [178].  To begin to understand how elongated particles flow single non-interacting 
 
94 
particles in dilute suspensions were considered.  At these conditions the suspension can 
be essentially thought of under the constraints of an ideal gas.  Jeffery (1922) considered 
non-interacting particles dispersed in a Newtonian fluid in the absence of Brownian 
motion.  Solving the appropriate equations of motion he concluded that axisymmetric 
non-Brownian particles (rods, prolate spheroids) not only translate but rotate indefinitely 
under the influence of a shearing flow [178, 179].  Interestingly the rotation was 
discovered to be periodic with time in what is now known as a ?Jeffery Orbit.?  Particles 
with large aspect ratios were predicted to rotate rapidly when their longest dimension was 
orthogonal to flow and very slowly when parallel.  The consequence of this type of 
behavior is that the viscosity, which depends on the dispersion microstructure, is also 
periodic and cycles with time [178].          
 If Brownian motion cannot be neglected stochastic deviations from the Jeffery 
Orbits for particles is observed and must be dealt with probabilistically.  In absence of 
flow an important Brownian phenomena known as the dilute-solution rotary diffusivity 
D
r0
 can be estimated.  The rotary diffusivity is simple a measure of the rate at which a 
particle reorients by Brownian motion.  For rods of length L and diameter d, D
r0 
can be 
estimated from the theory of Doi and Edwards and that of Kirkwood and Auer [179, 185, 
186].  Equation 2.66 gives the expression for the rotary diffusivity.    
 
( )( )
3
0
8.0/ln3
L
dLTk
D
s
B
r
??
?
=
 (2.66) 
From Equation 2.66 one can see the logarithmic dependence on the length to diameter 
ratio known as the aspect ratio of the rods.  The viscosity of the suspending fluid or 
solvent is represented by ?
s
, the temperature T, and Boltzmann?s constant by k
B
.    If the 
 
95 
solvent is non-Newtonian the viscosity should be taken in the limit of zero-shear as ?
0
.  
The rotary diffusivity at higher concentrations has also been predicted by Doi and 
Edwards but is not discussed here [185]. 
 Under the application of shear the crossover from Brownian to non-Brownian 
behavior for a flowing suspension can be calculated by the magnitude of the rotational 
Peclet number [179].  Equation 2.67 displays the dimensionless Peclet number Pe as a 
ratio of what is essentially two independent shear rates. 
 
r
D
Pe
?
&
?
 (2.67) 
The shear rate in the numerator should be taken as the lowest value imposed in 
experiment in order to characterize the lower limit of the transition.  Likewise the rotary 
diffusivity in the denominator can then be taken as the dilute-solution rotary diffusivity.  
A value of Pe greater then unity indicates a dominating contribution by the imposed shear 
rate and thus non-Brownian behavior.  On the contrary, a value of Pe less than unity 
indicates that the rotary diffusivity dominates and thus Brownian motion is relevant in the 
imposed flow field.  Identifying Brownian and non-Brownian behavior is important for 
computational modeling and interpretation of particle dispersions as the stress tensor 
contains an elastic contribution from Brownian motion [179].  In absence of Brownian 
motion this contribution is simply neglected.   
 Relaxing the criteria for dilute dispersions brings in some additional 
considerations in describing particulate suspensions.  To this end the rheology of rodlike 
particles is defined by unique concentration regimes as displayed by Figure 26.     
 
96 
 
Figure 26: Concentration regimes used to describe rod-like particles.  Adapted from Larson 
(1999) & Doi and Edwards (1986) [179, 185]. 
 
Here, the major assumption is that the rods are rigid and inflexible with the absence of 
attractive interactions [185].  In the dilute regime the rods are able to rotate freely without 
interacting with neighboring rods.  Considering a rod of length L rotating about its center 
of mass there is a volume of approximately L
3
 that must be free of nearest neighbors.  
Equation 2.68 introduces the relationship between the number concentration ? and 
volume fraction? of rods.  
 
4
2
??
?
Ld
=
 (2.68) 
From Equation 2.68 one can define transitions between each regime.  Intuitively rod-rod 
interactions can be assumed to occur when the number density approaches L
-3
 but 
experiment has shown disagreement with this prediction.  The volume fraction which 
delineates semidilute from dilute behavior is approximated by Equation 2.69. 
 Increasing Rod Volume Fraction (    )  
Semidilute: Rotation Inhibited
Concentrated Isotropic: 
Rotation & Translation 
Inhibited 
?
Dilute: No interactions 
Area swept by rotation
 
97 
 
2
0
24 ?
?
?
?
?
?
?
L
d
?
 (2.69) 
As shown by Figure 26 the semidilute regime is characterized by having neighboring 
rods impeding the rotation of one another but still allowing translation through the 
suspending fluid.  Therefore at rod concentrations somewhat above
0
? a rod may be 
inserted with negligible probability of intersecting another and therefore excluded volume 
is not a factor [179].  The effect of the rod volume is indicated in Figure 26 by the size of 
the lines used to represent each rod.  Increasing concentration in the semidilute regime 
there lays a point in which the rods have trouble isotropically arranging themselves due to 
excluded volume interactions.  This condition delineates the semidilute from the 
concentrated isotropic regime as approximated by Equation 2.70. 
 
L
d
4
'
0
?
? ?
 (2.70) 
In practice, for both rigid and flexible rod-like particles or polymers, these naming 
conventions for various concentration regimes are used.  The logic being that in the case 
of flexibility the swept area would be smaller.  Therefore, the transitions predicted for 
rigid rods would be conservative estimates on the higher end of each concentration 
regime.  For non-Brownian rods the interpretation of when the crossover from dilute to 
semidilute occurs is physically taken to be the point when the rods begin to influence 
each other hydrodynamically.  It is important to note that this crossover is not necessarily 
at the same concentration that Brownian motion of one rod would be affected by its 
nearest neighbors [179].   
 
98 
 Other deviations between Brownian and non-Brownian behavior can be seen in 
the shear viscosity of suspensions.  At low shear rates the theory of Doi and Edwards 
predicts a large increase and domination of the stress contribution from Brownian motion 
as the concentration is taken from dilute to semidilute [179].  However in non-Brownian 
solutions this is not the case and the same dramatic viscosity increase in not expected.  
The modest particle contribution to the viscosity was experimentally and theoretically 
confirmed even for fibers with large aspect ratios by Yamane et al. (1994) and Bibbo 
(1987) [187, 188].  This occurs because in absence of Brownian motion the fibers readily 
rotate to align with the shear flow contributing little to the shear viscosity which is 
especially true at large Pe.  Hence, the shear history must be considered when measuring 
the rheological properties of non-Brownian suspensions and is of utmost importance for 
reproducible results.  It may also be necessary to allow sheared suspensions to relax back 
to isotropic orientation before characterization. 
 As the concentration of the particles is increased through the semidilute regime 
there is a critical point where three-dimensional network formation occurs known as 
percolation.  This onset of network formation is interpreted differently depending on the 
mode of percolation.  Rheological percolation occurs when the particle network is 
sufficient to impede the motion of the solvent.  If the solvent is a high molecular weight 
polymer percolation occurs when the size of the polymer chain is large compared to the 
separation between the rods or particles [189].  Depending on the length of the polymer 
percolation may be a result of entanglement instead of intertube contacts.  This is initially 
characterized by a ?solid-like? viscoelastic response over short timescales.  A second 
mode of percolation exists in which electrical conductivity can be realized.  However, the 
 
99 
formation of conductive networks occurs on a different length scale.  It has been shown 
that physical contact does not need occur in nanotube systems since the intertube 
distances only need be close enough for elections to ?hop? from one tube to another.  
This distance has been estimated to be less then ~5 nm [103, 189].  Being that CNTs are 
comprised of some non-metallic tubes not all particles contribute to network 
conductivity.  Thus, in CNT systems electrical percolation can be assumed to occur at 
higher loading than rheological percolation.  Furthermore, CNT with sidewall defects 
contribute little to the conductivity.  This has been shown experimentally by Hu et al. 
(2006) for MWNTs in poly(ethylene terephthalate) where ?solid-like? viscoelasticity was 
found at 0.6 % wt. loading followed by the onset of electrical conductivity at 0.9 % wt. 
[189].  Likewise, Du et al. found rheological percolation at 0.12 % wt. and electrical 
percolation at 0.39% wt. for SWNT-poly(methyl methacrylate) composites [103].  These 
results are consistent with theory modeling the geometrical percolation for overlapping 
needles (no excluded volume) with aspect ratios from 10
2
-10
3
 to be in the range of 0.1-
1% vol. [190].  Differences between the results of Hu et al. and Du et al. can be attributed 
to differences in aspect ratio and dispersion state between the two composite systems.  
For this reason the percolation threshold for high viscosity thermoplastic melts can be 
expected to be larger than that of low viscosity thermoset solutions [189].  In general, the 
greater the aspect ratio, the less the packing density of the particles and the lower one 
would expect to find the critical volume concentration [191].             
 Evidence of fiber interactions can also be seen from the fact that a steady state is 
ever reached in steady shearing flow.  In the absence of such interactions the original 
fiber orientation would be revisited continuously over a period consistent to that 
 
100 
predicted by Jeffery.  In fact, even very weak hydrodynamic interactions incurred in the 
semidilute regime are enough to disturb the Jeffery orbits and produce a steady-state 
distribution of orientations such that the viscosity reaches a steady value [179].     
 
101 
3. EXPERIMENTAL DETAILS
Microscopy: 
Scanning Electron Microscopy: 
 In order to characterize sample surface morphology SEM images were taken. 
Samples were prepared by applying double sided carbon tape to a JEOL Specimen Mount 
(Ted Pella, Inc.).  Materials of interest were then pressed on the surface of the carbon 
tape and examined using a JEOL 7000-F Field Emission-Scanning Electron Microscope. 
 
Transmission Electron Microscopy: 
 Samples were prepared for TEM by first diluting the SWNT-UPR dispersion in 
acetone directly in the shear mixer.  Only low shear was applied so as to wash the sample 
but not further separate the SWNTs.  This solution was then dropped onto a 30 mesh 
copper grid coated with a formvar carbon support film (Electron Microscopy Sciences).  
After air dying the grid was dried under vacuum at ~105
o
C to remove residual solvents.  
A Zeiss EM 10C 10CR Transmission Electron Microscope was used to study the 
samples. 
 
Optical Microscopy: 
 At various points in the mixing process dispersions were characterized by optical 
microscopy using a Nikon (Melville, NY) Eclipse 80i optical microscope.  Samples were 
 
102 
illuminated by either transmission or reflected light and images were recorded digitally.  
Both 10x (0.3 NA) and 20x (0.45 NA) ?Luminous Universal Plan Fluor? Nikon 
objectives were used without oil immersion.  For high magnification optical images a 
Nikon 60x (1.40 NA) ?Plan Apo VC? oil immersion objective was used.  Either 1x or 2x 
magnification was also applied before the camera giving a maximum effective 
magnification of 120x.  Images were collected and processed in both Nikon Elements and 
Nikon ImagePro software.  Scalebars were calibrated and applied by means of a 
manufacturer supplied optical micrometer. 
 
Raw Materials: 
Unsaturated Polyester Resin: 
 Polylite
?
 X31003-00 base resin (Lot# 4508-38) was used as-received from 
Reichhold (Research Triangle Park, NC).  This resin is classified as an isophthalic base 
resin having 40% wt. styrene monomer and 60 % wt. proprietary polyester resin.  This 
resin contained no additional additives or catalyst. 
  
Vapor Grown Carbon Fiber: 
 Pyrograf III grade PR-19 (PR-19 XT PS LD) was obtained from Applied Sciences 
Inc. (Cedarville, OH) and used as-received.  Figure 27 displays a SEM micrograph 
showing the morphology and fiber diameter of the VGCFs.  Lengths of the VGCFs range 
from 10 ?m to over 100 ?m, as indicated by the manufacturer. 
 
103 
 
Figure 27: SEM Image of the as-received VGCF.  The measured fiber diameter is approximately 
110 nm across.  A loose network structure can be seen from this image. 
 
Single-walled Carbon Nanotubes, Batch 183.6: 
 SWNTs (Batch 183.6) produced by the high-pressure carbon monoxide (HiPco) 
process were received from Rice University (Houston, TX) partially purified following a 
deactivated thermal oxidation process as described by Xu et al. (2005) [192].  The 
following images show the overall sample morphology and network structure for both the 
as-received (183.6 Unpure) and purified (183.6 Pure) sample.  Figure 28 shows both low 
and high magnification SEM images of the as-received SWNT 183.6.  Likewise, Figure 
29 shows the structure of the sample after purification.  At low magnification both 
materials appear to have a similar overall morphology of aggregates and some ribbon-like 
structure.  But, the high magnification image of the 183.6 Pure shows an increase in rope 
size.  These ropes also appear very uniform and defect free as compared to the high 
 
104 
magnification image of the 183.6 Unpure sample.  This is likely a result of the removal of 
both residual catalyst and amorphous carbon from the as-received material.    
 
Figure 28: SEM images showing the morphology of the as-received (un-purified) SWNT 183.6 
imaged at (left) 4,000x magnification and (right) 35,000x magnification. 
 
Figure 29: SEM images showing the morphology of the SWNT 183.6 Pure imaged at (left) 
4,000x magnification and (right) 35,000x magnification. 
 
Single-walled Carbon Nanotubes, Batch 187.3: 
 SWNTs (Batch 187.3) produced by the high-pressure carbon monoxide (HiPco) 
process were received from Rice University (Houston, TX) fully purified and used as-
 
105 
received.  Figure 30 shows the overall sample morphology under low magnification and 
network structure under high magnification.  At low magnification the bulk sample 
appears in aggregate form.  Higher magnification images reveal non-uniform and loosely 
packed, entangled ropes or bundles. 
 
Figure 30: SEM images showing the morphology of the SWNT 187.3 imaged at (left) 4,000x 
magnification and (right) 35,000x magnification. 
 
Single-walled Carbon Nanotubes, Unidym: 
      SWNTs (Lot # PO355) produced by the high-pressure carbon monoxide (HiPco) 
process were used as-received from Unidym, Inc. (Menlo Park, CA).  This sample is 
referred to as SWNT Unidym throughout this work.  Figure 31 shows SEM images of  
the overall sample morphology and network structure, respectively.  The sample has the 
largest bulk density of all the SWNTs as can be seen from the tight and aggregated 
network structure in both images.  This was apparent when weighing out a certain 
amount of the tubes relative to the other batches (i.e. the Unidym batch appeared to have 
less volume for a given weight). 
 
106 
 
Figure 31: SEM images showing the morphology of the Unidym SWNTs imaged at (left) 4,000x 
magnification and (right) 35,000x magnification. 
 
SWNT Purification: 
 SWNTs (Batch 183.6) produced by the high-pressure carbon monoxide (HiPco) 
process were received from Rice University (Houston, TX) partially purified following a 
deactivated thermal oxidation process as described by Xu et al. (2005) [192].  This 
process exposed the iron catalyst used to produce the SWNTs and deactivated the metal 
oxide to a metal fluoride so as to not further oxidize the SWNT upon furnace treatment.  
Figure 32 shows a TEM image of the as received SWNTs where the presence of residual 
catalyst can be clearly seen.   
 
107 
 
Figure 32: TEM Image of SWNT batch 183.6 Unpure as-received from Rice University.  The 
dark sports are residual catalyst particles.  The scale bar is 200 nm in this image.  
 
The as-received tubes were then further processed in our lab to remove the exposed iron 
fluorides (catalyst) by two main steps; an HCl soak and a Soxhlet extraction.  The HCl 
soak was done in 200 mg batches of SWNTs.  In a typical batch, 160 ml of a 50% vol. 
aqueous solution of concentrated HCl acid was added to 200 mg of as received SWNT.  
This was heated just below 100
o
C and magnetically stirred overnight.  The solution was 
then transferred to a separation funnel and extracted with 5 ml of hexane.  The aqueous 
phase was drained and washings were repeated with deionized water approximately three 
times.  Next, the sample was transferred to a glass thimble and placed in the Soxhlet 
extractor.  Figure 33 displays the Soxhlet apparatus used in this process. 
 
108 
 
Figure 33: Soxhlet extraction apparatus used to purify the as-received SWNT 183.6 to high 
purity. 
 
A 40 ml aqueous solution of 50% vol. concentrated HCl acid was added to the Soxhlet 
flask.  This was heated to 90
o
C and cool tap water was passed through the condenser 
tube.  The Soxhlet is designed such that hot vapor exits the flask and bypasses the 
thimble.  The vapor then condenses and falls onto the sample.  After a certain amount of 
time, enough condensate collects in the thimble, building hydrostatic pressure, to drain 
out of the elevated exit tube in turn creating a siphon effect an draining the thimble.  A 
total of three 12 h cycles were completed with the HCl solution and two 12 h cycles with 
deionized water.  Finally, the acidic product was placed in a separation funnel and 
extracted one last time with hexane followed by subsequent washings with water as 
Thimble with  
Acid solution or water 
Cooling water 
Hot vapor 
sample 
 
109 
previously stated.  The final neutral, washed product was then dried at 100
o
C under 
vacuum for 24 h.  Figure 34 shows the final purified SWNT product.   
 
Figure 34: TEM Image of SWNT 183.6 Pure after Soxhlet extraction. 
 
In contrast to Figure 32 showing the as received SWNTs, no catalyst can be seen after 
Soxhlet treatment in Figure 34.  The extent of the purification was further examined by 
thermal gravimetric analysis.     
 
Thermal Gravimetric Analysis: 
 Nanotube purity was measured using a TA Instruments (New Castle, DE) Q500 
Thermal Gravimetric Analyzer (TGA).  Only samples vacuum dried overnight in an oven 
were used.  Samples were loaded into a tared platinum weigh pan and the sample 
chamber was purged with compressed air at a constant flow rate of 60 cm
3
/min.  The 
temperature was then held at 120
o
C for 15 - 20 minutes to drive off residual water.  Next, 
 
110 
the temperature was ramped at a rate of 10
o
C/min until a temperature of 800
o
C was 
reached.  Finally, the temperature was held constant for 45 minutes so a constant final 
weight could be reached.  Figure 35 shows the thermal stability of the primary SWNTs 
used in this research. 
0
10
20
30
40
50
60
70
80
90
100
100 200 300 400 500 600 700 800
Temperature (
o
C)
W
e
i
ght %
183.6 Unpure
183.6 Pure
187.3
Unidym
 
Figure 35: TGA curves for the primary SWNTs used in this research.  The oven atmosphere was 
compressed air. 
 
The final weight W
f
 was compared to the starting sample weight W
s
 after the 120
o
C hold 
and a residual mass fraction W
c
 of catalyst was determined by Equation 3.1. 
 
100% ?=
s
f
c
W
W
W
 (3.1) 
The CNT purity was then taken as the balance of W
c
%.  Table 1 summarizes the purity of 
the SWNTs under air.  Residual impurities were assumed to be catalyst particles. 
 
 
111 
SWNT Batch SWNT Purity (% wt.) 
183.6 Unpure 65 % 
183.6 Pure 98 - 99 % 
187.3 92 - 93 % 
Unidym 91 - 92 % 
 
Table 1: Purity of the SWNT samples as determined by TGA residual weight. 
 
Methods of Nanotube Oxidation: 
Oxidiation by Acid Mixture: 
 In a typical experiment, an amount of SWNT 183.6 Pure was added to a 125 mL 
Erlenmeyer flask at a ratio of 1 mg SWNT per 1.5 - 2.0 mL of concentrated acid solution.  
The acid solution was carefully constructed using a 3:1 (vol.) mixture of concentrated 
sulfuric (98% vol.) to nitric acid (70% vol.), respectively.  The flask was then sealed with 
a few layers of laboratory film and suspended over a Cole Parmer ultrasonic cleaner  
(15 W, 55 kHz) so as to not contact the bottom.  The experiment start time was taken to 
be the moment the bath sonicator was switched on.  Approximately 1 - 2 times an hour 
the flask was removed from the bath and manually swirled to put any noticeable 
aggregates back into solution.  After 1.5 h of treatment time, the acid-SWNT solution 
was diluted by a factor of 10 using deionized water.  A stirbar was added and the solution 
was covered while allowed to mix for 45 min.  This solution was then transferred to a 
 
112 
vacuum filtration apparatus and allowed to collect on a pre-wetted (organic solvent) 0.2 
?m PTFE coated (Whatman, polypropylene backed) 47 mm membrane filter.  Figure 36 
shows a typical apparatus used in this step.  When the acid-water solution was almost 
completely drained vacuum was released and 500 mL of deionized water was added to 
further wash the filtrate.  The vacuum was released leaving a moist black filtrate of 
oxidized nanotubes.  The final product was left on the filter paper as a cake and set aside 
to air dry.  The resulting air-dried sample was in the form of an entangled bucky paper 
and placed into a dessicator overnight.  The sample was then peeled from the filter paper 
and placed into a glass scintillation vial covered with pierced aluminum foil followed by 
overnight drying in a vacuum oven at ~105
o
C.  Samples prepared using this treatment are 
referred to as SWNT-Ox. 
 
Figure 36: Typical vacuum filtration apparatus used in this research. 
 
113 
Oxidation with Ethanol Finish:                                  
 The method described previously for producing SWNT-Ox was followed through 
to the point of filtration and washing.  In this treatment when the acid-water solution was 
almost completely drained vacuum was released and 500 mL of deionized water was 
added to further wash the filtrate.  When the water was almost completely drained the 
sample was then washed with approximately 100 mL of ethanol.  The vacuum was 
released leaving a moist black filtrate of ethanol finished oxidized nanotubes (SWNT-Ox 
Film).  The sample was then scraped from the filter paper and rinsed into a 20 mL 
scintillation vial with more ethanol, capped, and bath sonicated for approximately 30 min 
to disperse.  The image on the left of Figure 37 shows the sample after this step. 
    
Figure 37: (left) Acid oxidized SWNT (SWNT-Ox) dispersed in ethanol (right) and the resulting 
film (SWNT-Ox Film) cast from an ethanol dispersion.  
 
After re-dispersion was complete the sample was transferred to a watch glass and allowed 
to air-dry.  After a uniform film was cast on the glass, the sample was scraped off with a 
spatula leaving ribbons of oxidized SWNT film.  The image on the right of Figure 37 
displays a typical dry film while still attached to the watch glass.  These ribbons were 
placed into a glass scintillation vial and covered with pierced aluminum foil.  The vial 
 
114 
was then placed in a dessicator overnight or was dried overnight under vacuum at 
~105
o
C.   
 
Nitric Acid Oxidized SWNT Films: 
 To produce oxidized SWNT films the method developed by Zhang et al. (2004) 
was modified [125].  Approximately 27 mg of SWNT 183.6 Pure was added to 40 mL of 
10M nitric acid and bath sonicated until well dispersed (~3 h).  The sample was then 
covered and heated on a hot plate to 70
o
C for one hour followed by a second hour at 
90
o
C.  The hot solution was then bath sonicated for 1h to break up any remaining 
aggregates.  The sample was left overnight (~18 h) to cool and sonicated for an additional 
hour before dilution with 100 mL of deionized water.  This solution was then transferred 
to a vacuum filtration apparatus and allowed to collect on a pre-wetted (organic solvent) 
0.2 ?m PTFE coated (Whatman, polypropylene backed) 47mm membrane filter.  When 
the nitric acid-water solution was almost completely drained vacuum was released and 
500 mL of deionized water was added to wash the filtrate.  This was allowed to pull 
through the filter under vacuum followed by a second 500 mL washing.  The vacuum was 
released leaving a moist black filtrate of which was scraped from the filter paper while 
still wet and transferred to a 20 mL glass scintillation vial.  Approximately 5 mL of 
deionized water was added and the capped sample was bath sonicated for 5 minutes in a 
Cole Parmer ultrasonic cleaner (15 W, 55 kHz).  This solution was allowed to air dry and 
subsequently covered with pierced aluminum foil followed by overnight drying in a 
vacuum oven at ~105
o
C.  The increased amount of bath sonication in this method was 
required to break up and aggregates present in the raw material.  This was a result of the 
 
115 
inability of the nitric acid to intercalate the SWNTs which, in turn, required the addition 
of additional energy by sonication with respect to the acid mixture.   
 
Dilute Nitric Acid Reflux:  
 To produce oxidized SWNTs using a dilute nitric acid reflux approximately 6 mg 
of SWNT 183.6 Pure was added to 20 mL of 2.6 M nitric acid in a round bottom three-
neck flask.  The original concentrated of nitric acid (70%, 15.8 M) was diluted with 
deionized water to 2.6 M.  This solution was then suspended over a Cole Parmer 
ultrasonic cleaner (15 W, 55 kHz) so as to not contact the bottom and sonicated for 
approximately 1 hour to break up large aggregates.  During this time a silicone oil bath 
was prepared to an internal temperature of 125
o
C into which the round bottom flask was 
securely lowered.  A reflux condenser was added to the center flask neck and securely 
clamped in place.  The side flask ports were plugged and sealed.  The top of the reflux 
condenser was plugged but not sealed in case of over-pressurization.  With all 
components safely in place the cooling water was slowly switched on from outside the 
lab hood in case of water-oil contact.  It is important to note that at high temperatures the 
heating oil will react violently when exposed to cooling water.  The solution was then 
allowed to reflux for 24 h.  After which another 20 ml of 2.6 M nitric acid make-up was 
added and allowed to reflux for another 24 h.  After 48 h total time the oil bath was 
removed and the sample was allowed to cool.  The SWNT-dilute acid solution was then 
quenched by dilution in 200 mL of deionized water.  This solution was then transferred to 
a vacuum filtration apparatus and allowed to collect on a pre-wetted (organic solvent)  
0.2 ?m PTFE coated (Whatman, polypropylene backed) 47 mm membrane filter.  When 
 
116 
the acid-water solution was almost completely drained vacuum was released and 500 mL 
of deionized water was added to further wash the filtrate.  This was allowed to pull 
through the filter under vacuum followed by a second 500 mL washing.  When the water 
was almost completely drained the vacuum was released leaving a moist black filtrate of 
oxidized nanotubes.  The final product was left on the filter paper and set aside to air dry.  
The resulting entangled bucky paper was placed into a dessicator overnight.  The sample 
was then peeled from the filter paper and placed into a glass scintillation vial covered 
with pierced aluminum foil followed by overnight drying in a vacuum oven at ~105
o
C.   
 
Hot Acid Mixture Reflux: 
 The method of Esumi et al. (1996) was strictly followed [117].  Approximately 
100 mg of SWNT 186.6 was lightly ground in mortar and pestle and refluxed at 140
o
C in 
a 3:1 (vol) mixture of concentrated sulfuric (98%) to nitric acid (70%).  The sample was 
treated for 1.5 h over which all tubes appeared to be completely digested. 
 
Lypholization Following Oxidation of SWNTs: 
 In order to examine the effect of final sample morphology after oxidative 
treatment samples were oxidized, washed, dispersed in water, and lyophilized (freeze-
dried) to prevent sample aggregation.  The method used in this treatment was developed 
specifically for this research.  In a typical experiment an amount of SWNT 183.6 was 
added to a 125 mL Erlenmeyer flask at a ratio of 1 mg SWNT/4 mL concentrated acid 
solution.  The acid solution was carefully measured in a 3:1 (vol.) ratio of sulfuric acid  
(98% vol.) to nitric acid (70% vol., 15.8 M), respectively.  The flask was then sealed with 
 
117 
a few layers of laboratory film and suspended over a Cole Parmer ultrasonic cleaner  
(15 W, 55 kHz) so as to not contact the bottom.  The experiment start time was taken to 
be the moment the bath sonicator was switched on.  Approximately 1 - 2 times an hour 
the flask was removed from the bath and manually swirled to put any noticeable 
aggregates back into solution.  During the bath sonication the temperature of the SWNT-
acid solution was ~50
o
C.  At the end of the desired treatment time the acid-SWNT 
solution was diluted by a factor of 10 using deionized water.  The image on the left of 
Figure 38 shows the diluted acid-nanotube solution.  A stirbar was added and the solution 
was covered while allowed to mix overnight (typically 18 h).  The resulting suspension 
appeared homogeneously black with no phase separation and qualitatively confirmed 
oxidation via water solubility.  This mixture was then transferred to a vacuum filtration 
apparatus and allowed to collect on a pre-wetted (organic solvent) 0.2 ?m PTFE coated 
(Whatman, polypropylene backed) 47 mm membrane filter.  When the acid-water 
solution was almost completely drained vacuum was released and 500 mL of deionized 
water was added to further wash the filtrate.  This was allowed to pull through the filter 
under vacuum followed by a second 500 mL washing.  If the pressure drop was too high 
the sample was washed from the filter and a new membrane was used.  When the water 
was almost completely drained the vacuum was released leaving a moist black filtrate of 
oxidized nanotubes.  The removal of the oxidized SWNTs from the paper was achieved 
by removing the filtration funnel and base from the filtration flask and washing the 
sample into a beaker with a deionized water bottle.  Again, the resulting solution should 
be a homogeneously black aqueous dispersion (typically ~20 mL) at almost neutral pH 
showing no phase separation.  The solution was then transferred to a 20 mL scintillation 
 
118 
vial, capped, and bath sonicated for 30 min.  The image on the right of Figure 38 shows 
the sample at this point in the process.  The sonicated nanotube solution was then 
transferred to a 50 mL disposable polypropylene centrifuge tube, capped, and placed in a 
freezer.  The resulting frozen dispersion was then placed inside a lypholization apparatus 
(cap replaced with pierced laboratory film) and pumped down for at least 24 h.  Due to 
the very low bulk density of the product care should be taken when handling the freeze-
dried oxidized SWNTs to prevent accidental exposure from the airborne sample.  In a 
final step the centrifuge tube was placed inside a dessicator overnight and transferred to a 
large glass vial covered with pierced aluminum foil followed by overnight drying in a 
vacuum oven at ~105
o
C.  
      
Figure 38: Qualitative proof of successful oxidation. (left) The lack of phase separation in 
absence of stirring is indication of a stable colloidal suspension. (right) Final aqueous dispersion 
of neutral, acid oxidized SWNTs.  The sample shows excellent dispersion after washing.   
 
 
 
 
 
119 
Spectroscopy: 
UV-visible Spectroscopy: 
 A Varian Cary 3E UV-Vis spectrophotometer was used to probe pH dependant 
dispersion fluorescence of the acid oxidized and lyophilized SWNTs.  Samples were 
placed in a Starna semi-micro quartz cell and compared to a background of pure water.  
Typical scans were taken over a range on wavelengths from 800 - 200 nm.  A saturated 
solution of 6h acid oxidized and lyophilized SWNTs was created by dissolving the CNT 
in water using bath sonication and centrifuging the dispersion at 17,000 x g for 1.5 hours.  
The supernatant was then collected for analysis at its natural pH of ~6.  A dilute solution 
of NaOH (~0.1 - 1.0 M) was added dropwise to raise the pH to 10.   
 
Fourier Transform Infrared Spectroscopy: 
 In order to identify surface functional groups after oxidation infrared spectra were 
collected.  A Perkin Elmer (Waltham, MA) Spectrum 2000 FTIR Spectrometer was used 
in conjunction with a liquid nitrogen cooled MCT detector.  A 15-bounce zinc selenide 
attenuated total reflectance (ATR) crystal was used to analyze the sample.  Measurements 
were carried out at room temperature over a wavenumber range of 4000 - 600 cm
-1
 
having a 4 cm
-1
 resolution with 1,024 total scans.   
 The 6 h acid oxidized lyophilized sample was dispersed to saturation in water 
using bath sonication and dropped on the crystal.  The crystal was then held under 
vacuum at 60
o
C to cast a thin film.  This process was repeated and the spectrum was 
checked until no change in absorbance was seen indicating that only the SWNT and not 
the atmosphere above the crystal was being sampled.   
 
120 
 SWNT 187.3 was dispersed in 1,2-dichlorobenzene by means of bath sonication 
and deposited on the ATR crystal.  The crystal was held at ~100
o
C under vacuum to 
remove the solvent.    
 
Raman Spectroscopy: 
 To characterize the extent of sidewall damage on the various SWNT samples 
Raman spectra were obtained using a Renishaw (Hoffman Estates, IL) inVia Raman 
microscope and a Leica (Wetzlar, Germany) optical microscope used to focus on the 
surface of interest.  Figure 39 shows the instrument used for all experiments.   
 
Figure 39: Image of the Renishaw inVia Raman microscope used in this research. 
 
A Leica 50x (0.75 NA) ?N-plan? objective was always used for measurements.  For 514 
nm wavelength irradiation a Spectra-Physics (Irvine, CA) 263C air-cooled ion laser was 
used.  For 785 nm irradiation an SDL (San Jose, Ca) 8530 high power laser was used.  
The laser spot areas were 0.548 ?m
2
 and 1.28 ?m
2
 for the 514 nm and 785 nm lasers, 
respectively.  All measurements were taken under ambient conditions.  A total of 10 
 
121 
scans from a Raman shift of 3200 - 100 cm
-1
 were collected with a 10 sec sample 
exposure time.  The incident laser beams were polarized by the microscope.   
 
X-ray Photoelectron Spectroscopy: 
 The identification and quantification of surface functional groups was obtained 
via XPS.  For XPS analysis the SWNT samples were first placed into an acetone rinsed  
8 mL scintillation vial, covered with pierced aluminum foil, and dried for 18 h under 
vacuum at ~105
o
C.  The dry sample was then allowed to cool under vacuum before being 
capped, sealed, and stored in a dessicator.  Samples were pressed into an adhesive carbon 
tape using stereo microscopy to ensure complete surface coverage by the SWNTs.  
Photoemission experiments were performed in a custom Kratos Analytical XSAM 800 
surface analysis system using unmonochromatized MgK? (1253.6 eV) x-ray radiation 
and a hemispherical energy analyzer.  The base pressure of the system was 1x10
-10
 torr.  
An exhaustive list of calibration and XPS analyzer details for the exact instrument used in 
this research can be found in Bozack (1993) and Bozack et al. (1994) [193, 194].  The 
analyzer was operated such that electrons were collected from a 5 mm
2
 sample area.  Due 
to specimen charging during x-ray irradiation of the powdered organic specimens, the 
energy axis of each XPS spectra was shifted to make the C1s binding energy line equal to 
the standard hydrocarbon energy (C-H and C-C bonds) of 284.6 eV.  The uncertainty of 
the recorded binding energies was +/- 0.1 eV. 
 All specimens were first examined by means of an overall scan of binding 
energies from 0 - 600 eV.  This was done to identify the elements present on the surface 
as well as calculate the surface composition (atom %).  These compositions were 
 
122 
calculated using peak backgrounds determined by the iterative Shirley technique and 
instrument specific corrections were applied.  Next, high resolution scans over the C1s 
and O1s regions were performed to yield the overall line shapes of these areas.  C1s 
peaks were further deconvoluted in accordance with reported binding energies for 
oxidized SWNTs.  From this analysis surface composition of functional groups 
covalently bonded to the SWNT sample was relatively determined.     
 
Dispersion Preparation: 
 CNT-UPR dispersions were constructed by mass fraction based on actual 
measured component weights but converted to volume fraction using the specific 
material density.  The UPR was taken to have a density of 1.09 g/mL as provided by the 
manufacturer.  All SWNTs were assumed to have a mass density of 1.45 g/mL.  VGCFs 
were assumed to have a density of 1.95 g/mL as provided by the manufacturer.  The 
volume fraction
?
can be calculated from the mass fraction of the nanotubes ?
CNT
 by 
means of Equation 3.2. 
 
( )
()
NTrelative
NTrelative
??
??
?
11 ?+
=
 (3.2) 
Here, the relative density is defined as the ratio of the solvent density to that of the 
nanotubes.  Alternatively, the volume fraction can be determined directly from the 
component mass used to construct the samples.   
 In order to develop a reproducible mixing protocol dispersions of VGCFs were 
constructed at 0.100% vol., corresponding to 0.1788% wt. loading.  The volume and 
mass concentrations of the SWNT dispersions used in this research are given in Table 2. 
 
123 
Volume Percent (%) Mass Percent (%)
0.2500 0.3323 
0.2000 0.2659 
0.1825 0.2426 
0.1472 0.1957 
0.1000 0.1330 
0.0500 0.0665 
0.0250 0.0330 
0.0100 0.0133 
0.0075 0.0100 
0.0050 0.0067 
0.0030 0.0040 
  
Table 2: Volume percentages and the corresponding mass percentages for the SWNT 183.6 Pure-
UPR dispersions. 
 
Dispersion Techniques: 
Bath Sonication: 
 The effectiveness of bath sonication alone on dispersing SWNTs directly into 
UPR was determined by optical microscopy.  A 0.100% vol. SWNT 183.6 Pure-UPR 
mixture was placed into a 20 mL scintillation vial and processed for 24 h using a Cole 
Parmer ultrasonic cleaner (15 W, 55 kHz).   
 
 
124 
Syringe Dispersion: 
 A controlled amount of shear can be applied to the entire sample using a syringe 
dispersion method.  To this end, syringe dispersion has been used previously in the 
literature to successfully disperse MWNTs [195].  In this research, this technique was 
used to study the transition of a raw CNT-UPR mixture to a dispersed state.  Two 
disposable 10 mL BD syringes (Franklin Lakes, OH) were used with the appropriate 
amount of each material weighed into the cavity of one syringe.  The Luer-Lok threaded 
syringe tips were removed using a razor and a short section of 1/8? inner diameter 
Nalgene
?
 PVC tubing was used to connect the syringes.  After inserting the plunger into 
the syringe containing the sample and removing excess air the syringes were 
interconnected tip to tip.  The material was then forced to the adjacent syringe by 
depressing the plunger and this was counted as one cycle.  Samples were removed after 
100, 250, and 400 cycles and characterized by optical microscopy and oscillatory 
rheology.  A time-sweep test was conducted on the samples using a 50 mm diameter cone 
and plate measuring geometry described in subsequent sections.  Controlled frequency 
(10 s
-1
) and strain measurements were taken and both the storage and loss moduli were 
measured as a function of time.  In order to not disturb the sample a small strain of 0.1% 
was imposed on the sample throughout the test.  Well not determined beforehand, it was 
assumed that this small strain would not exceed the linear viscoelastic strain for the 
sample.   
 
 
 
 
125 
High Shear Mixer Design: 
 Shear mixer components were purchased from Ace Glass (Vineland, NJ).  
The flask head used in this work featured three top joints for access to the flask contents 
and a Teflon coated lower joint.  The flask bottom was a round bottom type with a 
capacity of 50 mL.  A 10 mm Trubore glass bearing and a 10 mm diameter Trubore 
Teflon coated, steel stirring shaft was used.  A stock 48 mm wide Teflon stirring blade 
was purchased (Ace Glass) and hand cut to size.  This was then affixed to the shaft using 
nylon hardware and offset with the center of the shaft.  The offset geometry prevented 
settling on the bottom of the flask and provided sufficient ?pumping? as to apply shear to 
the entire sample over time.  Additionally, a ?paddle-fan? shaped mixing impeller was 
purchased and used as received.  The stirring shaft was driven using a ? hp solid state 
laboratory stirrer (Ace Glass) with a maximum output speed of 4,000 rpm.  The motor 
was mounted upside down and connected to a 36? flexible stirring shaft linked via a 
nylon chuck.  In a typical mixer loading, the CNTs were measured and carefully added to 
the flask.  However, in the case of SWNT 183.6 Pure the sample was always first placed 
in a vial and lightly ground by hand using a stainless steel spatula to facilitate better 
mixing.  The flask was then tared and the appropriate mass of resin was added.  The 
mixer was then assembled and purged with ultra high purity Argon (Airgas) before being 
sealed to prevent humidity from interfering with the sample.  Alignment between the 
bearing and shaft was ensured before sealing in order to prevent damaging the 
components.  Misalignment of the components often generated excess friction and 
subsequently failed the 20A fuse filament.  A final adjustment was made to the mixing 
shaft after alignment where the chuck was loosened and a downward force was applied so 
 
126 
that contact between the mixing blade and flask sidewall was maintained.  The flask and 
contents were then submerged in a room temperature water bath with volume 
approximately 8 - 10 times the sample volume.  This was simply used to absorb excess 
heat although its effect was not studied.  Typically, the mixer was run at approximately 
500 - 1,000 rpm for 72 hours with intermittent stops (approximately 3) to allow the 
sample to settle off the flask wall.  Figure 40 shows an image of the assembled mixing 
flask, stir shaft, and impeller. 
 
Figure 40: Image of the mixing vessel used in this research.  
 
Sample Extraction and Loading: 
 Above a CNT loading of 0.010% vol. aliquots of the sample were extracted using 
a bulb and pipette having a wide inner diameter of 7 mm.  This wide cross section was 
necessary in order to prevent shear aligning the CNT and preserve isotropic orientation.  
Dilute samples of approximately 0.01% vol. and below were drawn using a 12 gauge 
 
127 
syringe (Sigma-Aldrich).  The 10? syringe was cut to a length of approximately 7.5? and 
had an inner diameter of 2.159 mm.  Small air bubbles could be observed within the 
dilute samples as a result of the intense shear mixing.  The syringe was used not only to 
ensure solvent evaporation was avoided but to degas the dilute samples as well.  Syringe 
extracted samples were always placed in a vial and allowed to relax for 12 - 18 hours.   
 
Rheology: 
 All rheological experiments were performed using a Anton Paar (GmbH) Physica 
MCR 301 series rheometer displayed by Figure 41.  
 
Figure 41: Image of the Anton Paar Physica MCR 301 rheometer with the parallel plate 
measuring geometry affixed to the P-PTD200 device and the H-PTD200 peltier controlled upper 
oven in place. 
 
This model rheometer can operate in controlled stress (CSS), controlled shear rate (CSR), 
or controlled shear strain (CSD) mode.  The motor is driven using a permanent magnet 
 
128 
synchronous drive operating on an air bearing.  The torque range for this instrument is 
from 0.1 ?Nm to 200 mNm with a torque resolution down to 0.001 ?Nm.  Normal force 
is measured using a capacitive probe with a range of -50 N to +50 N and a sensitivity of 
0.002 N.  The sign convention for a positive normal force was taken by the manufacturer 
to be exerted in the direction of the upper plate.   
 When measuring in cone and plate geometry the temperature was controlled using 
the P-PTD200 peltier temperature control device located below the lower shear plate.  
Surrounding the upper plate and the sample was the H-PTD200 peltier controlled upper 
oven.  For a detailed explanation on the principle of operation see Sato and Breedveld 
(2005) [196]. This allowed for controlled atmospheric temperature as well as introduction 
of inert gas.  The temperature range for both peltier devices was from -40
o
C to +200
o
C. 
Both peltier controlled devices were counter cooled with a Julabo F12 (GmbH) water 
recirculation bath maintained at +20
o
C.  This was plumbed in series starting with the H-
PTD200 and passing through the P-PTD200 before emptying to the bath.   
 For cone and plate rheometry the Anton Paar CP50-2 fixture was used.  Unless 
otherwise noted the cone radius used in all experiments was 24.9985 mm with a cone 
angle ? of 2.018
o
.  The cone truncation and distance between the center of the upper and 
lower surfaces was constant at 52 ?m for all measurements.  At these distances the 
required sample volume was 1.14 mL and the gap distance at the edge h was calculated to 
be 0.933 mm. 
 In order to monitor the fluid structure optically rheological measurements were 
also taken using the Anton Paar small angle light scattering (SALS) system.  This 
measuring system was of parallel plate geometry constructed out of transparent quartz 
 
129 
glass.  The SALS setup was used simple to observe the sample optically and no scattering 
experiments were conducted.  Unless otherwise noted the plate radius used in all 
experiments was 21.4600 mm with construction of transparent quartz glass.  The gap h 
was set between 0.4 and 1.0 mm for all measurements.  At a gap of 1.0 mm the required 
sample volume was 1.5 mL.  The SALS system is not equipped with any temperature 
controlling device except that of the cooling water circulation.  However, the water 
temperature at the device inlet was monitored via the SALS system.         
 All rheology measurements were taken just below room temperature at 20
o
C in 
order to limit styrene evaporation.  To load the instrument an appropriate volume of 
sample was placed on the lower plate and the upper measuring surface was lowered to the 
proper gap.  The gap-setting velocity of the upper plate while was set to 500 ?m/s on 
close approach so as to not destroy the sample structure.  If necessary, excess sample was 
trimmed from the edge using a rubber policeman in order to eliminate error from edge 
effects.  After which the normal force was zeroed and the sample was given 10 min to 
achieve thermal equilibrium. 
 In order to characterize the flow regime in these experiments the Reynolds 
number Re was estimated as shown by Equation 3.18. 
 
s
z
?
??
2
Re
&
=
 (3.18) 
Here, the strain rate is represented by?
&
and z is the average distance between the plates.  
The solvent mass density and viscosity are shown by ? and ?
s 
,respectively [91, 107].  Re 
was less than unity for all experiments indicating that assumptions for viscometric flow 
were valid.    
 
130 
Amplitude Sweep Test: 
 Amplitude sweep tests (AS) were first performed to probe the structural stability 
of the sample and the range of linear viscoelastic strain.  A controlled strain was imposed 
on the sample at a fixed frequency of 10 s
-1
 (rad/s) and the resulting storage and loss 
moduli were measured.  When the storage modulus showed nonlinear behavior the test 
was stopped and repeated to ensure the critical strain was correctly identified.  Typically 
the strain ranged from 0.01 - 100% but the maximum strain was often reduced to 10% so 
as to not destroy the structure of the samples.   
 
Frequency Sweep Test: 
 Frequency sweep (FS) experiments were conducted at controlled strain but 
variable angular frequency.  The strain set point was determined from an AS test and 
chosen to be inside the linear viscoelastic regime.  The typical measured frequency range 
was from 100 - 0.01 rad/s.  For efficiency all FS experiments were run from high to low 
frequency so that the long measuring times associated with the lowest frequencies could 
be terminated if the data was outside the measurable range of the torque transducer.  As a 
rule of thumb the time required to acquire a steady state data point is inversely 
proportional to the shear frequency.   
 
Step Rate Test: 
 The start-up of flow was characterized by conducting step-rate tests.  Here a fixed 
shear rate was set and the time of observation was varied logarithmically.  From this 
experiment the time required to achieve steady-state flow can be determined.   
 
131 
Surface Tension: 
 For thermodynamic analysis the surface tension of UPR was measured using a 
KSV Instruments (Finland) Sigma702 Tensiometer.  A DuNoy ring was physically 
immersed and extracted from the sample to collect experimental data.  This instrument 
had a tension resolution of 0.001 mN/m.  A total of 10 runs were collected and both the 
average surface tension and standard deviation was reported.  
 
132 
4. RESULTS AND DISCUSSION
Characterization of Dispersion Methods: 
Syringe Dispersion: 
 In order to estimate the magnitude of the shear stress applied to the sample the 
pressure drop across the syringe tip was calculated.  A syringe was filled with water, 
evacuated of excess air, and connected to a Psitronix (Tulane, CA) digital gauge.  The 
syringe plunger was depressed manually in the same fashion as used dispersion 
experiments and a pressure of approximately 25 - 30 psi (172 - 207 kPa) was observed.  
Taking this to be the pressure drop over the tip p
c
 the tip wall shear stress ?
w
 was 
estimated using a model developed for a capillary viscometer [67].  Equation 3.3 displays 
the wall shear stress as a function of nozzle radius r and length L.   
 
L
rp
c
w
2
=?
 (3.3) 
The geometry of the syringe was taken to have an internal tip radius of 0.875 mm and 
length of approximately 1 mm.  This resulted in an estimated shear stress of 10
4
 Pa and 
was most likely conservatively large based on the assumption for the tip pressure drop.  
Chen et al. (2000) have used the same model for a time-pressure fluid dispensing process 
as well as developed a rigorous model for the frictional pressure drop by considering the 
conservation of mechanical energy [197].   
 
133 
 The dispersion of VGCFs into UPR at a loading 0.100% vol. was studied by the 
syringe extrusion method with the intention of characterizing the transition to a well 
dispersed state.  Figure 42 shows the response of the fluid under small strain oscillatory 
shear after 100, 250, and 400 syringe cycles. 
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
0 100 200 300 400
Time (s)
Modulus (
P
a)
G' 100 Cycles G'' 100 Cycles
G' 250 Cycles G'' 250 Cycles
G' 400 Cycles G'' 400 Cycles
 
Figure 42: Rheological characterization of the transition to a well dispersed state for 0.100% vol. 
VGCF in UPR processed by tip-to-tip syringe extrusion.   
 
Examining Figure 42, the response after only 100 cycles shows elastic behavior 
dominated by G?.  Processing the fluid to 250 cycles the sample behavior has transitioned 
to a viscous liquid dominated by G?.  After 400 cycles the viscous behavior is more 
pronounced showing a decrease in G? by an order of magnitude accompanied a slight 
 
134 
increase in G?.  To understand the physical origin of this transition in rheological 
behavior optical images were analyzed.  Figure 43 shows the sample for 100 cycles. 
 
Figure 43:  Image of a 0.100% vol. VGCF-UPR dispersion processed at 100 cycles syringe 
extrusion.  The sample was imaged at 20x with no cover glass in place.  
 
The presence of a large aggregate can be seen in Figure 43 on the order of ~50 ?m. 
It is important to note that this large cluster size is comparable to the 52 ?m gap between 
the cone and plate measuring geometry.  For this reason it is believed that the solid-like 
rheological response seen in Figure 42 was a result of ?jamming.?  Thus, the direct 
probing of the solid aggregate leads to a false and large magnitude of G? as well as large 
but inaccurate complex viscosity of the fluid.  The observation of a false solid-like 
response at short mixing times is consistent with the erratic behavior seen by Huang et al. 
(2006) for MWNT-PDMS dispersions [91].  Figure 44 shows an image of the sample 
after 400 cycles.  
100 ?m 
 
135 
 
Figure 44:  Image of a 0.100% vol. VGCF-UPR dispersion processed at 400 syringe extrusion 
cycles.  The sample was imaged at 20x with no cover glass in place.   
 
The absence of large aggregates in Figure 44 is believed to be the origin of the liquid-like 
response seen at 250 cycle time in Figure 42.  The increase in liquid-like behavior 
between 250 cycles and 400 cycles is attributed to the development of a homogeneous 
dispersion state.  The development of a well-dispersed state also supports the existence of 
a minimum critical mixing time similar to that found by Huang et al. previously. 
Similarly, a transition in rheological response was found in MWNT-Epoxy dispersions by 
Fan and Advani (2007) [81].   
 Although syringe dispersion appears to be an excellent means to deliver a 
controlled stress to the sample it suffers from being both arduous to complete by hand 
and unreliable due to a weakening of the syringe material over time from exposure to the 
styrene.  Furthermore, as the material becomes better dispersed with each cycle large 
increases in the viscosity are observed making the extrusion more difficult.  However, if 
100 ?m 
 
136 
an appropriate apparatus were constructed this method may show promise.  
Unfortunately, the device used in this preliminary study was unable to successfully 
disperse SWNTs.   
 
Bath Sonication: 
 Fan and Advani (2007) created MWNT-Epoxy dispersions by means of bath 
sonication alone [81].  Specifically, after 24 hours the authors reported no change in 
rheological response indicative of a ?saturated? network structure.  Figure 45 displays the 
morphology of a 0.100% vol. SWNT 183.6 Pure-UPR dispersion. 
 
Figure 45: Image (10x) of a 0.100% vol. SWNT-UPR dispersion processed by 24h bath 
sonication alone.   
 
In contrast to the successful results reported for MWNT-epoxy dispersions large SWNT 
aggregates persisted in the UPR.  This was likely due to the higher viscosity of the neat 
UPR (2 Pa-s) with respect to the Epoxy (0.59 Pa-s).  Furthermore, as the treatment time 
100 ?m 
 
137 
progressed the final UPR viscosity increased as a result of styrene evaporation from 
sample heating.  Since it was required that the final dispersions show repeatable 
rheological behavior methods inducing styrene loss, such as sonication, were not 
employed.  
 
High Shear Mixing: 
 In order to approximate the shear stress delivered to the fluid the velocity profile 
between the annular gap of the stirring blade and the flask wall was derived.  The 
maximum shear stress was assumed to be delivered where the blade tip speed was the 
greatest.  This also happened to be the area where some contact between the impeller and 
flask wall was made; thus, minimizing the annular gap and maximizing the slope of the 
velocity profile.  Therefore, the area of the blade with the largest radius was used to 
describe the mixer geometry in estimating the shear as shown by leftmost image in Figure 
46.  The radius of the impeller r
min
, measured from the center of the stirring rod to the 
edge, was taken to be 1.85 cm.  The mixer flask radius was measured by water 
displacement and the flask radius R was taken as the distance from the center-line to the 
inner wall, measuring 2.13 cm.  On the microscale, these geometries were approximated 
as two small sections of concentric cylinders using the cylindrical coordinate system  
(r, ?, z) as displayed by Figure 47.   
 
 
138 
 
Figure 46: Images and dimensions for the custom shaped mixing impeller (left) and mixer flask 
(right) used in this work.   
 
Figure 47: Cylindrical coordinate system used to estimate the shear stress of the mixer. 
 
In this manner the velocity component in the ?-direction was assumed to depend only on 
the radius of observation, r.  This is displayed mathematically by Equation 3.4.  Thus, the 
components in the r-direction and z-direction were assumed to be zero as shown by 
Equation 3.5.    
 
)(r
??
?? =
 (3.4) 
 
zr
?? == 0  (3.5) 
R = 2.13 cm 
r
min
 = 1.85 cm 
?
r
 
139 
Assuming the working fluid as the Newtonian UPR with constant mass density the 
equation of continuity (EOC) reduces in accordance with Equation 3.6. 
 ( ) 0=?? v  (3.6) 
Specifically, considering the assumptions made by Equations 3.4 and 3.5 we can expand 
the EOC to yield Equation 3.7. 
 
0=
?
?
?
?
?
 (3.7) 
Equation 3.7 indicates that ?
?
 is constant in the direction of flow.  Next, the ?-component 
of the equation of motion was reduced in accordance with Equations 3.4 and 3.5 as well 
as the assumption of constant density and viscosity as shown by Equation 3.8. 
 ()
?
?
?
?
?
?
=
?
?r
rd
d
rdr
d 1
0  (3.8) 
Since, we have assumed that ?
?
 
is only a function of r the derivatives in Equation are no 
longer partial.  Equation 3.8 is a 2
nd
 order differential equation but can be separated and 
integrated twice to yield the velocity profile as shown by Equation 3.9.   
 
r
BAr
+=
2
?
?  (3.9) 
Here, A and B are simply the constants of integration and can be evaluated with two 
boundary conditions.  In order to reduce the number of independent variables the velocity 
was evaluated at the flask wall R and at the edge of the impeller ?R as displayed by 
Figure 48. 
 
140 
 
Figure 48: Schematic of the impeller and mixing flask wall with respect to the stirrer center line 
(C
L
).  The radial digestion of the mixing flask is represented by R and the dimension of the 
impeller by ?R. 
  
Here, ? is simply the fraction of R occupied by r
min
 and has a magnitude of 0.868 in this 
mixer.  At the flask wall R the velocity is assumed to be zero and at the impeller tip ?R 
the velocity as taken as the product of the radius and the angular frequency ?.  Equation 
3.10 and Equation 3.11 show the boundary conditions. 
 
( ) 0=R
?
?
 (3.10) 
 
( ) RR ???
?
?=
 (3.11) 
Solving for the constants the velocity profile can be represented by Equation 3.12. 
 
()?
?
?
?
?
?
?
?
?
?
?
?
??
=
1
R
r
r
R
R
 (3.12) 
 
In order to relate the velocity profile to the shear stress ?
r?
 Newton?s law of viscosity is 
introduced as shown in Equation 3.13. 
Wall 
C
L
R 
?R 
?
 
141 
 
?
?
?
?
?
?
?
?
?=
rr
r
r
?
?
?
??
 (3.13) 
By insertion of Equation 3.13, and pulling out constants from the partial derivative, 
Equation 3.14 can be derived. 
 
()
?
?
?
?
?
?
?
?
?
?
??
=
R
r
R
r
Rr
r
1
1
2
?
?
??
?
?
 (3.14) 
where, 
 
32
21
r
R
Rr
R
r
?=
?
?
?
?
?
?
?
?
?
 (3.15) 
Finally, the expression for the shear stress is shown by Equation 3.16. 
 
()
2
1
2
?
?
?
?
?
?
?
?
=
r
R
r
?
?
??
?
?  (3.16) 
The resin viscosity was measured to be approximately 2 Pa-s.  Equation 3.16 can be 
further reduced to represent the physical case by introducing the numerical value of ?.  
Based of the measured mixer component radii ? has a value of ~0.868.  One important 
observation is that as a result of the impeller being mounted offset to the stirring shaft, 
the manual application of the downward force to the blade, and asymmetry of the flask 
itself, the impeller actually contacts the flask wall at times.  However, this is not 
accounted for in the model since as ? approaches unity its value increases asymptotically.  
Furthermore, the proposed boundary conditions would not be valid in this extreme.  
Equation 3.17 displays the shear stress at the true measured value of ?. 
 
2
,
6 ?
?
?
?
?
?
??
r
R
Max
r
??
?  (3.17) 
 
142 
Given the dimensions of the mixer when the angular frequency is approximately 100 
rad/s (1,000 rpm) the maximum shear stress delivered to the resin is on the order of 2 
kPa.  This result is in excellent agreement with a similar shear mixer used by Huang et al. 
(2006) who estimated a shear stress of 3 kPa [75, 91].  However, the estimate is 
conservative since no estimation has been made for large ?.  For point of reference when 
? has a value of 0.95 the impeller is approximately 3 mm from the wall and the shear 
stress approaches a value of 5 kPa.  Thus, it can safely be concluded that the original 
estimate is conservatively low.  The critical shear energy required to disperse SWNTs has 
been roughly estimated on the order of 10
4
 J/m
2
 (Pa) [91].  Therefore, whether or not the 
mixer shear stress actually reaches this critical estimate cannot be determined from this 
analysis. 
 Attempts at reproducing the well dispersed state achieved by the syringe method 
were made using the high-shear mixer at 0.100% vol. loading.  A paddle-type impeller 
was first used to process the dispersion.  Figure 49 shows an optical image of the 
resulting mixture after approximately 3 days of processing time. The poor dispersion 
shown by Figure 49 is believed to be a result of the insufficient ?pumping? (convection) 
of the sample around the mixing flask.  This prevented much of the sample from seeing 
the high shear zones produced by the paddle.  Similar to the response after just 100 
syringe cycles, samples processed in this manner showed erratic and uncorrelated 
rheological behavior.  Although, longer mixing times beyond three days were studied, 
this resulted in partial loss of the solvent through evaporation.  Therefore, the paddle 
mixer design was abandoned and a new impeller style was considered.  The performance 
 
143 
of the custom cut mixing impeller on processing a 0.100% vol. dispersion is displayed by 
Figure 50.         
 
Figure 49: 10x image of a 0.100% vol. VGCF-UPR dispersion processed (~three days) by paddle-
type mixing impeller.  No cover glass used on sample.     
 
Figure 50:  Image of a 0.100% vol. VGCF-UPR dispersion processed (~three days) by the custom 
cut crescent-type mixing impeller.  The sample was imaged at 20x with 2x magnification in front 
of the camera.  No cover glass was used on the sample.   
50 ?m 
200 ?m 
 
144 
The image was taken after approximately three days mixing time.  Here, the visible 
aggregates have been completely broken down and only fibers and amorphous carbon 
appear in a homogenously dispersion.  The offset impeller blade eliminated ?dead-zones? 
in the mixer and exposed the sample to the high shear areas between the paddle and flask 
wall.  The rheological response was markedly different from the sample processed by the 
paddle-type impeller.  Furthermore, the results were found to be repeatable confirming a 
successful mixing protocol.  In passing it is noteworthy that by studying the rheological 
response after various mixing times, a unique characteristic showing transition to a 
dispersed state was found in the linear viscoelastic critical strain.  This strain limit was 
barely existent for samples with large VGCF clusters but widened significantly over 
poorly dispersed samples.   
 The three day shear mixing protocol developed for VGCFs was extended for use 
in SWNT dispersions.  Figure 51 displays an optical image under high magnification for 
a typical SWNT 186.3 Pure dispersion at 0.050% loading.  Clearly, only small aggregates 
and fibers can be seen.  Likewise, Figure 52 shows a 0.100% vol. dispersion with the 
absence of any large aggregates.  
 
145 
 
Figure 51: Optical microscopy image a 0.050% vol. dispersion of SWNT in UPR.  Image taken 
through cover slip using an oil immersion 60x (1.4 NA) objective with 2x magnification in front 
of the camera.  
 
 
Figure 52: Typical image (10x) of a 0.100% vol. SWNT-UPR dispersion processed by three day 
high shear mixing.  Inset showing 10x2 magnification.  
100 ?m 
100 ?m  
 
146 
Viscosity of Neat UPR: 
 The viscosity of the neat UPR used throughout this work was determined from 
measurement by both oscillatory and steady shear.  Figure 53 displays this result. 
1.0E+00
1.0E+01
0.1 1 10 100
Shear Rate, (1/s) or (rad/s)
Visc
osity (
P
a
-
s)
 
Figure 53:  Steady-state viscosity and complex viscosity for neat UPR.   
 
The resin is Newtonian with a viscosity of ~2 Pa-s and adheres to the Cox-Merz rule.  
The perfect agreement between the two viscosities is characteristic of a Newtonian fluid 
and a result of negligible phase lag between the viscoelastic moduli.  Thus, the response 
under oscillatory shear has a negligible component of G? and the magnitude of G? is 
derived only from viscous response of the fluid.   
 
Rheology and Characterization of Acid Oxidized SWNT Dispersions: 
 A consistent observation throughout this research was the inability of the oxidized 
nanotubes, from any of the oxidative treatments attempted, to form a dispersion state 
 
147 
comparable to that of the pristine SWNTs.  One hypothesis for the origin of this behavior 
was the final sample morphology after oxidation.  Filtered nanotubes often form ropes as 
a result of the high frequency of collisions encountered as the sample volume is 
decreased upon filtration [198].  The driving force is enthalpic and barriers preventing a 
tube from sliding along a rope axially are small, indicating this is a facile process.  Figure 
54 shows the sample morphology for the nitric acid (10 M) oxidized SWNTs.  At low 
magnification the sample appears in a uniformly aggregated state and under high 
magnification clear evidence of roping can be seen.  These ropes appear to be of smaller 
diameter and more abundant than present in the SWNT 183.6 Pure starting material. 
 
Figure 54: Morphology of the nitric acid oxidized (10M) SWNTs as seen by SEM at (left) 4,000x 
magnification and (right) 35,000x magnification. 
 
Conversely, Figure 55 shows the morphology of the 48h nitric acid oxidized SWNTs 
where no sample structure is apparent.  Presumably, this is a result of the long oxidation 
time and generation of a significant amount of amorphous carbon at elevated 
temperatures.  The samples displayed in both Figure 54 and Figure 55 failed to disperse 
 
148 
in UPR.  On the other hand, the SWNT-Ox Film sample did show a dispersion of 
reasonable quality, as shown later in the resin.  Figure 56 displays the SWNT-Ox Film 
sample.  Examination of the sample morphology after this treatment shows an 
intermediate structure from what was seen in Figure 54 and Figure 55.  A combination of 
smooth unstructured material and small ropes can be seen in the high magnification 
image of Figure 56.  Therefore clear no correlation between sample morphology and 
dispersion ability was apparent for these oxidized samples.        
 
Figure 55: Morphology of the 48h nitric acid (2.6 M) reflux oxidized SWNTs by SEM at (left) 
4,000x magnification and (right) 35,000x magnification. 
 
 
 
149 
 
Figure 56: SEM images highlighting the morphology of the SWNT-Ox film SWNTs at (top) 
4,000x magnification and (bottom) 35,000x magnification.   
 
The aforementioned success achieved by treatment of the acid oxidized SWNTs 
with ethanol and casting as a film can be seen below.  Figure 57 displays the optical 
quality of a 0.100% vol. SWNT-Ox Film dispersion.   
 
Figure 57: Image (10x2) of the 0.100% vol. SWNT-Ox Film dispersion processed from three day 
shear mixing.   
100 ?m 
 
150 
Compared to the inset image of 0.100% vol. SWNT 183.6 in Figure 52 the dispersion of 
the oxidized film is poor, consisting of many aggregates.  The effect of ethanol treatment 
was also studied via rheology by comparing dispersions of 1.5 h oxidized SWNTs with 
(SWNT-Ox Film) and without (SWNT-Ox) ethanol treatment.  Figure 58 shows the 
complex viscosity of these fluids as measured with the 50 mm cone and plate geometry.  
The complex viscosity of the two samples clearly shows the same trend in behavior but 
offset vertically in magnitude.  To eludicate the origin of this behavior the linear 
viscoelastic moduli were examined as seen in Figure 59.  In Figure 59 both G? and G? 
were plotted against one another to highlight the increase in magnitude of the oxidized 
film response as well as the variation in crossover point.  The crossover point is defined 
where G? and G? are of equal magnitude.  This behavior is indicative of a viscoelastic 
fluid since at low frequency the samples resemble an elastic solid and at high frequency 
viscous liquid response is displayed.  In this case, the SWNT-Ox Film displayed both an 
increased crossover modulus and an increased crossover frequency indicating the greater 
reinforcing potential of the dispersion state.      
 
 
151 
1.0E+00
1.0E+01
1.0E+02
1.0E+03
0.01 0.1 1 10 100
Angular Frequency, ? (rad/s)
Comple
x Visc
osity, |
?
*| (Pa
-
s) .
SWNT-Ox Film
SWNT-Ox
 
Figure 58: Comparison of the effect of final treatment of oxidized SWNT on dispersion.  Both 
samples were acid oxidized for 1.5 h and dispersed at 0.100% vol. loading. 
 
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
0.01 0.1 1 10 100
Angular Frequency ? (rad/s)
Modulus (Pa)
G' SWNT-Ox Film
G'' SWNT-Ox Film
G' SWNT-Ox
G'' SWNT O
 
Figure 59: Comparison of the effect of final treatment of oxidized SWNT on dispersion by means 
of viscoelastic moduli.  Both samples are at 0.100% vol. loading. 
 
152 
 Qualitative evidence for the lack of dispersion in various oxidized SWNTs can be 
seen in the following figures.  Figure 60 shows a comparison of the 6h acid oxidized and 
lyophilized SWNTs and a dispersion of SWNT 183.6 Pure.  The SWNT 183.6 Pure 
dispersion is opaque whereas the lyophilized sample is transparent.  Figure 61 shows an 
optical micrograph of the lyophilized dispersion.  As can be seen from Figure 61, the 
dispersion consists of small pieces of the freeze dried film even after intense 3 day shear 
mixing.  Figure 62 shows another poor dispersion consisting of the 48h nitric acid 
refluxed SWNTs. 
 
   
Figure 60: Image of (left) lyophilized 6h oxidized SWNT 183.6 in UPR and (right) a SWNT 
183.6 Pure dispersion after the same mixing time.  Both samples are at 0.005% vol. loading.   
 
153 
 
Figure 61: The lack of dispersion in the 0.005% vol. lyophilized 6h acid oxidized SWNTs.  Image 
taken with 20x objective and 2x magnification in front of camera (40x).   
 
 
Figure 62: The lack of dispersion in the 0.005% vol. 48 h nitric acid reflux oxidized SWNTs.  
Image taken with 20x objective.   
 
50 ?m 
 
154 
Even with this extreme 48 h oxidative treatment time the dispersion state is comparable 
to the 6h lyophilized sample.  Out of all the oxidation treatments the two performed on 
the samples in these images were the most severe and therefore productive at introducing 
functional surface groups.  To confirm that the poor dispersion states are a result of 
surface chemistry and not a result of entanglement from processing, the surface chemistry 
of the 6h oxidized and lyophilized sample was rigorously studied.  Figure 63 shows the 
?fluffy? appearance of the lyophilized product.  The physical appearance is a result of the 
low bulk density of the sample after processing.  In fact, the same mass of SWNT 183.6 
is approximately 5 times more compact. 
 
Figure 63: A 6h oxidized sample as it appeared from the freeze dryer.  The sample mass was 15 
mg before treatment.   
 
Figure 64 shows the FTIR spectra of a 6h acid oxidized lyophilized SWNT film cast from 
a reconstituted dispersion in water.  In order to remove artifacts from surface bound water 
or atmosphere the film casting process was repeated, and the spectrum was checked, until 
 
155 
no change in absorbance was observed.  This indicated that only the SWNT and not the 
atmosphere above the crystal was being sampled. 
0.10
0.14
0.18
0.22
0.26
0.30
5001000150020002500300035004000
Wavenumber (cm
-1
)
0.10
0.14
0.18
0.22
0.26
0.30
Absorbance
3400
1739
1614
1358
1014
2860
2930
 
Figure 64: The FTIR spectra 6 h acid oxidized SWNTs (183.6 Pure) after lypholization.  The 
sample was re-dispersed in water to saturation and dropped on a ZnSe ATR crystal.  The sample 
spectra is representative of 5 film layers. 
 
The spectrum shown in Figure 64 is after 5 layers of deposition and one hour under 
vacuum to remove unbound water.  The presence of the peak at 1739 cm
-1
 is a clear 
indication of carboxyl groups existing in the free state [114].  Unfortunately, spectra of 
lower wavenumber are multiple overlapping peaks.  This is likely a result of the existence 
of carboxylic acids, carboxylate groups, and bonding.  The broad peak at 3400 cm
-1
 is 
generally indicative of the -OH stretch from a carboxylic acid.  The shoulder at lower 
wavenumber is likely representative the acid monomer and at higher wavenumber for the 
 
156 
hydrogen bonded species.  The presence of the shoulder on its peak differentiates this 
from what would be seen for contaminated water.  Furthermore, the absence of a clear 
water contamination peak at 1636 cm
-1
 supports this argument [99].  The peak at 1358 
cm
-1
 represents the -OH bend deformation.  At higher wavenumber this peak is broad, 
suggesting the presence of carboxylate groups.  The peak at 1614 cm
-1
 appears to be 
structured showing the carbonyl groups (C=O) and/or carboxylate ions.  The peak at 1014 
cm
-1
 is likely a C-O stretch.  Finally, the peaks around 2930 cm
-1
 and 2860 cm
-1
 are from 
sp
2
 hybridized carbons, CH
2
 and CH, respectively.            
 The effect of oxidation was also studied by XPS.  Figure 65 shows an overall 
binding energy scan showing relevant atomic oxygen surface composition of 19%.  This 
was a substantial increase from the starting material of SWNT 183.6 Pure which had a 
surface oxygen content of only 1%, as discussed in later sections.  In order to further 
confirm the functional groups present, high resolution scans of C1s orbital photoelectron 
emission were recorded.  This was further analyzed by peak deconvolution as shown in 
Figure 66.  Peaks were fit at binding energies of 284.6 eV (aromatic carbon), 286.2 eV 
(hydroxyl or phenolic groups), 287.7 eV (carbonyl carbon), and 289.1 eV (carboxyl 
groups).  In accordance with the FTIR data, the XPS clearly confirms the presence of 
carboxylic functionalities.   
 
 
 
157 
0100200300400500600
Binding Energy (eV)
I
n
te
nsity (
a
.u.)
C1s
O1s
Atom %
C = 81
O = 19
 
Figure 65: The XPS survey spectrum for 6 h acid oxidized SWNT (183.6 Pure) after 
lyophilization.  The main surface constituents were primarily C and O.  Trace amounts of S and 
Si were observed.  
 
 
 
158 
278280282284286288290292294
Binding Energy (eV)
I
n
te
nsity (
a
.u.)
 
Figure 66:  Deconvolution of the high resolution C1s curve for the oxidized SWNT into four 
separate peaks.  The black jagged curve is the raw data and the blue line intersecting this curve 
represents the sum of the smaller peaks.   
 
Marginal success with the dispersion of oxidized nanotubes was enjoyed only 
when the sample was post-treated with ethanol.  It is important to note that these ethanol 
treated samples were not oven dried, as will be discussed in detail in later sections.  In 
absence of this ethanol treatment no degree of mixing was observed in oxidized samples.  
The fact that comparable morphology was seen between oxidized samples, and the fact 
that dispersion was impossible without the presence of adsorbed solvent, provided 
compelling evidence to suggest that the effect of surface chemistry is to blame.  Thus, it 
can also be concluded that extreme oxidative treatments were not the correct 
functionalization route for improving SWNT-UPR dispersions.  In fact, examining Figure 
 
159 
60, the oxidative treatment appears to have had a detrimental effect towards creating 
homogeneous dispersions. 
 
Identifying Experimental Artifacts: 
 During the processing of some SWNT 183.6 dispersions it was discovered that 
miniscule air bubbles were incorporated into the sample, effectively creating a ?foam? of 
entrapped air in resin.  This was only seen at loadings of 0.025% and below.  The 
elimination of this artifact was studied experimentally by examining various treatments 
and was essential for understanding a calibrated rheological response for the dispersion 
state.  For point of reference the linear viscoelastic response of the as-produced foam was 
measured.  Additionally, the effectiveness of bath sonication to remove the entrapped air 
was examined.  The agitation and heat produced lowered the sample viscosity and 
effectively de-gassed the mixture, even after only 5 - 10 min.  Finally, extrusion through 
a long and low gauge syringe was used to remove the air bubbles and the effect of 
allowing sample relaxation was studied.  The large pressure drop across the needle length 
achieved by means of vacuum driven flow degassed the sample at ambient temperatures.  
The results of this study are displayed in Figure 67.  By comparison, the bath sonicated 
sample shows the largest increase in complex viscosity whereas the extruded sample 
shows the lowest enhancement.  This is a result of the solvent loss associated with the 
heating of the sample during sonication but may also indicate an increased dispersion 
state.  The extruded sample, on the other hand, suffers from shear alignment of the 
SWNT which in turn decreases their contribution to the viscosity.  However, if this 
sample is allowed to relax overnight to an isotropic state the sample shows increased 
 
160 
resistance to flow.  It is interesting to note that in the high-shear limit the complex 
viscosity of the foam was higher than that of the extruded samples. 
1.0E+00
1.0E+01
1.0E+02
0.01 0.1 1 10 100
Angular Frequency, ? (rad/s)
Comple
x Visc
osity, |
?
*| (Pa
-
s)
Sonication
Relaxation
Foam
Extrusion
 
Figure 67: The effect of sample history on the complex viscosity of a 0.025% vol. SWNT 183.6 
Pure-UPR dispersion.  The 50 mm diameter cone and plate geometry was used.   
 
 In order to provide more insight into physical situation the response of G? for 
these dispersions was also compared.  Figure 68 highlights this behavior.  Here the two 
extremes of solvent loss from sonication and shear alignment of the dispersion can be 
seen.  A lower slope can be seen for the sample allowed to relax after extrusion than that 
of the foam.  In the low frequency limit, the inability of the entrapped air to store elastic 
energy can be seen with respect to the relaxed sample. 
 
 
161 
1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
0.01 0.1 1 10 100
Angular Frequency, ? (rad/s)
Mod
u
lus (Pa)
Sonication
Relaxation
Foam
Extrusion
 
Figure 68: The effect of sample history on the storage modulus of 0.025% vol. SWNT 183.6 
Pure-UPR dispersion. 
 
 It was assumed that the solvent evaporation was the primary contribution to the 
high viscosity of the sonicated sample and should be avoided.  This was reinforced by the 
identification of the percolation threshold for SWNT 183.6 Pure dispersions, as discussed 
in later sections.  Essentially, the concentration in the present case is too low to produce 
an interconnected network without solvent loss which essentially concentrated the 
sample.  Also, allowing the extruded sample to rest overnight recovered the resistance to 
flow at low shear rates.  Therefore, all dilute samples were collected via syringe extrusion 
and allowed to rest overnight.  On a final note, the relaxation of the sample viscosity after 
resting overnight is good indication that the dispersions of SWNT 183.6 do not settle out 
of suspension.  This serves to validate the measurements made over the experimental 
duration in this research.        
 
162 
Rheological Behavior of CNTs:  
 The effect of material type on the mechanical property enhancement was studied 
at constant a loading of 0.100% vol. of nanotubes.  This was measured by the bulk 
rheological response of the fluid prior to cure.  Using the 50 mm cone and plate geometry 
the behavior of VGCFs, SWNT 183.6 Pure, and ethanol finished acid oxidized SWNTs 
were compared to neat UPR.  The linear viscoelastic response of each material was 
characterized by an FS test.  Figure 69 displays the storage modulus of each dispersion. 
1.0E-03
1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
0.01 0.1 1 10 100
Angular Frequency, ? (rad/s)
Storage Modulus, G' (
P
a) 
SWNT 183.6 Pure SWNT-Ox Film
VGCF UPR
 
Figure 69: Comparison of material dependant storage moduli at 0.100% vol. loading.  The 
samples were subjected to a controlled linear viscoelastic strain while the angular frequency was 
varied. 
 
The response of the neat UPR is barely existent at low frequency and not measurable at 
higher frequencies.  This is consistent with a Newtonian fluid where little to no energy is 
stored in the sample.  Below 0.1 s
-1
 the torque falls below instrument sensitivity.  In 
 
163 
comparison, an increased slope is seen with the inclusion if VGCFs but the decreasing 
trend appears consistent with the resin.  A markedly different behavior is seen with the 
SWNT dispersions.  For both the pristine and oxidized film samples the value of G? 
shows development of an elastic behavior where the modulus is nearly independent of 
frequency at long timescales.  Additionally, the value of G? for SWNT 183.6 Pure 
appears to terminate at almost an order of magnitude higher than the SWNT-Ox Film 
indicating better reinforcement.  The elastic behavior is indicative of SWNT interactions 
through the formation of a three-dimensional network structure.  However, without 
comparing the effect of concentration no more conclusions can be drawn in regards to the 
network state.  Similarly, the corresponding loss moduli for the 0.100% vol. nanotube 
dispersion are shown in Figure 70.  The behavior of G? for both the neat UPR and the 
VGCF dispersions are representative of a viscous liquid over all frequencies of 
observation.  In contrast, the SWNT dispersions show deviation from the linear and 
terminal behavior of the VGCFs at low frequency.  Again, this is indicative of SWNT 
interaction.  The variation in the slope of G? for the pristine and oxidized SWNTs 
indicates a different dispersion state between the samples.  From the values of G? and G? 
the complex viscosity can be determined.  As seen in  
Figure 71, the viscosity enhancement is minimal for the VGCFs where as a three order of 
magnitude increase can be seen in the zero-shear viscosity for SWNT 183.6 Pure.  In 
comparison, for the oxidized SWNT sample both a decreased slope and higher infinite 
shear viscosity can be seen.  Examining the dispersion state via microscopy it can be 
concluded that this type of behavior can be expected for a slightly aggregated vs. 
homogenous dispersion state.   
 
164 
 
1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
0.01 0.1 1 10 100
Angular Frequency, ? (rad/s)
Loss Modulus, G" (Pa) 
SWNT 183.6 Pure
SWNT-Ox Film
VGCF
UPR
 
Figure 70: Comparison of material dependant loss moduli at 0.100% vol. loading.    
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
0.01 0.1 1 10 100
Angular Frequency, ? (rad/s)
Comple
x Vis
c
osity, |
?
*| (Pa-s) 
SWNT 183.6 Pure
SWNT-Ox Film
VGCF
UPR
 
Figure 71: Comparison of material dependant complex viscosity at 0.100% vol. loading. 
 
165 
 Finally, the non-linear rheological response of each dispersion to steady shear was 
characterized as shown in Figure 72.  Analogous behavior to the complex viscosity in  
Figure 71 is seen for all materials.  While little viscosity enhancement is seen for VGCFs, 
nearly a two order of magnitude increase in the low shear viscosity of SWNT 183.6 Pure 
is observed.  Again, the oxidized SWNTs shows less enhancement at low shear rates but 
has a significantly higher signal in the limit of high shear.  This confirms the observation 
of varying dispersion state between pristine and oxidized SWNTs was not due to 
experimental artifact.  It is noteworthy that the exact mode responsible for shear-thinning 
behavior observed with the nanotube dispersions has not been fully identified in this 
work.  But, it is assumed that this is a combination of both aggregate breakdown and 
shear alignment of the nanotube dispersions.   
1.0E+00
1.0E+01
1.0E+02
1.0E+03
0.01 0.1 1 10 100
Shear Rate, ?? (1/s)
Visc
o
s
ity, 
?
 (P
a-s)
SWNT 183.6 Pure
SWNT-Ox Film
VGCF
UPR
 
Figure 72: Comparison of material dependant shear viscosity at 0.100% vol. loading. 
 
166 
 From this material based study it is clear that the greatest viscosity enhancement 
was found when the pristine SWNTs were used as the reinforcement material.  This is a 
result of the large aspect ratio, high specific surface area, and strength of association 
through formation of a three-dimensional network structure in SWNTs.  The benefit of 
using SWNTs over VGCFs is also reflected in the lower mass density of the SWNTs.  As 
a result, great potential exists for producing a lightweight composite materials.  However, 
the effect of SWNT loading on the behavior of these dispersions must still be studied in 
order to properly characterize this material for processing.           
 
Viscoelasticity of SWNT Dispersions: 
 The reinforcing potential of SWNT 183.6 Pure was shown to have the greatest 
promise for property enhancement based on rheological measurements for various 
materials.  But, for efficient processing, the response of the fluid to stress and 
deformation would be beneficial to characterize.  To this end, dispersions were 
constructed over a range of concentrations spanning almost two orders of magnitude.      
 The structural stability of the SWNT 183.6 Pure-UPR dispersions was probed by 
amplitude sweep testing.  In this manner the linear viscoelastic regime was characterized 
and the critical strain was revealed.  Figure 73 displays the response of the fluids to 
controlled frequency, variable strain experiments.     
 
167 
1.0E+00
1.0E+01
1.0E+02
1.0E+03
0.01 0.1 1 10 100
Strain (%)
Stora
g
e Modulus (Pa)
0.2500%
0.2000%
0.1825%
0.1472%
0.1000%
0.0500%
0.0250%
 
Figure 73: Structural stability of SWNT 183.6 Pure-UPR dispersions as measured by ramping 
strain amplitude at constant shear frequency.  The circles indicate the critical strain at the edge of 
the linear viscoelastic threshold marked by non-linear response in storage modulus.   
 
The critical strain for the breakdown of linear viscoelasticity is highlighted by the small 
circles in Figure 73.  The value for the critical strain (or stress) decreases with increasing 
loading reflecting the developing complexity of the network structure, in agreement with 
Hough et al (2004) [106].  Measurements were halted after the onset of non-linear 
response so as to preserve the sample structure for FS characterization.  Figure 74 shows 
the behavior over a larger range of strain, moving though the critical linear viscoelastic 
strain. 
 
168 
1.0E+02
1.0E+03
0.01 0.1 1 10
Strain (%)
Mod
u
lus (Pa)
G'
G''
 
Figure 74: Characterization of the linear viscoelastic regime and transition to non-linear strain 
response for a 0.250% vol. SWNT dispersion.   
 
As shown in the above figure, the linear response of G? breaks down at approximately 
0.15% strain followed by an upturn in G?.  This behavior is characteristic of strain 
induced fluidization [106].  The crossover between G? and G? is characteristic of the 
yield strain.  As stated previously, typical measurements were not allowed to progress 
through the non-linear regime as the time required for the sample to relax was found to be 
long enough to allow for solvent evaporation.  Additionally, the response may not fully 
recover if aggregates were to form upon fluidization. 
 Figure 75 highlights the corresponding response in loss modulus to the storage 
moduli displayed by Figure 73.  In contrast to G?, the behavior of G? in Figure 75 does 
not exhibit a drastic non-linear response.  This is expected for most fluids as the storage 
modulus typically is used to characterize the critical strain. 
 
169 
1.0E+01
1.0E+02
1.0E+03
0.01 0.1 1 10 100
Strain (%)
Loss Modulus (Pa)
0.2500%
0.2000%
0.1825%
0.1472%
0.1000%
0.0500%
0.0250%
 
Figure 75: Loss moduli of SWNT 183.6 Pure-UPR dispersions as measured by ramping strain 
amplitude at constant shear frequency.   
 
 With knowledge of the extent of the linear viscoelastic regime for each 
concentration, the frequency dependent shear moduli were characterized by performing 
FS testing.  Figure 76 displays and overlap of both moduli for each concentration.  The 
purpose of displaying Figure 76 is as a point of reference for applying colloidal scaling 
theory.  It should be pointed out that the both the magnitude of the crossover modulus 
and the crossover frequency increase with SWNT loading.  Trappe and Weitz (2000) 
proposed a scaling behavior for weakly attractive colloidal particles.  This was achieved 
using the critical crossover parameters to normalize the crossover point to unity in moduli 
and frequency.  Through this technique, the authors suggest the concentration 
dependence on the viscoelastic moduli can be removed resulting in master curves as 
displayed by Figure 77 
 
170 
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02
Angular Frequency, ? (rad/s)
Modulus (Pa
)
 
G' 0.2500% G'' 0.2500%
G' 0.2000% G'' 0.2000%
G' 0.1825% G'' 0.1825%
G' 0.1472% G'' 0.1472%
G' 0.1000% G'' 0.1000%
G' 0.0500% G'' 0.0500%
G' 0.0250% G'' 0.0250%
 
Figure 76: The linear viscoelastic response of the SWNT-UPR dispersions as a function of both 
SWNT 183.6 Pure loading and the shear frequency.  Small square symbols represent G? and 
small triangle symbols represent G?.   
 
 
171 
1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E-04 1.0E-02 1.0E+00 1.0E+02 1.0E+04
Reduced Frequency (?/?
c
)
G'
/G
c
 and G''/G
c 
 
Figure 77: Viscoelastic dependence on concentration removed by means of colloidal scaling 
[199].  The loss modulus is the raw value of G? for the neat UPR.  Its frequency was not shifted.   
 
At high reduced frequency a small discrepancy can be seen between the data 
corresponding to the lower end of the concentration spectrum.  This was predicted by for 
more dilute loadings as the properties of the solvent are displayed [199].  As a result, the 
fluid network microstructure is not developed and character resembling the UPR resin is 
seen by the increase in slope of G?/G
c
 and the decreased magnitude of G?/G
c
 (compare 
orange line to blue line).  This observation is feasible since this concentration was found 
to be well below the percolation threshold, as discussed later.  On the other hand, at long 
reduced timescales slight deviations can be seen in both reduced moduli.  The variation 
 
172 
between G?/G
c
 is attributed to the development of the percolated network structure and its 
increasing complexity.  The upturn in G?/G
c
 can be expected in accordance with the 
aging of the network as observed experimentally for laponite suspensions by Baghdedi et 
al. (2008).  Specifically, the upturn in G? arises when local energy barriers become 
surmountable due to local arrangements throughout the sample allowing for particles to 
interact [182].  As can be seen for the lowest reduced frequency the behavior converges 
in G? and G? for concentrations of 0.1825% - 0.2500% vol. indicating the network is self 
similar at these concentrations.  The value of the plateau modulus of G?/G
c
 at low 
frequency reflects the elasticity of the network.   
 Figure 78 shows the crossover modulus, as a function of SWNT loading, used to 
scale the y-axis in Figure 77.  The trend in G
c
 appears to be linear with SWNT loading 
even though the scaling plot in Figure 77 indicates deviation from scaling behavior at low 
loading.  
1.0E-01
1.0E+01
1.0E+03
0.01 0.1 1
Volume Percent (%)
Crossover Modulus, Gc (Pa
)
 
 
Figure 78: Plot of the parameters used to scale G? and G? to master curves. 
 
 
173 
 By breaking down the viscoelastic response into its separate moduli conclusions 
on the development of the SWNT network can be drawn.  Figure 79 displays the storage 
modulus as a function of concentration.  As the concentration of SWNT is increased the 
magnitude of G? increases indicative of a more ?solid-like? response.  Furthermore, the 
frequency dependence on the magnitude of G? is removed with increasing concentration 
as indicated by the decrease in G? slope.  At highest concentrations the fluid shows 
response of an elastic solid showing little change in G? with frequency.  The apparent 
jump in low frequency modulus from 0.025% vol. to 0.100% vol. is indicative of a 
percolation event [103, 106].   
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02
Angular Frequency, ? (rad/s)
S
t
orage Modul
u
s (P
a) 
0.2500% 0.2000%
0.1825% 0.1472%
0.1000% 0.0500%
0.0250%
 
Figure 79:  The storage modulus response as a function of angular frequency and increasing 
nanotube loading for purified SWNT 183.6 dispersed in UPR.   
 
 
174 
 For additional insight into the network development, Figure 80 shows the 
corresponding loss modulus as a function of SWNT loading.  As the concentration of 
SWNT is raised the magnitude of G? increases and is evidence of increased viscous 
dissipation of elastic energy.  Furthermore, over long timescales the frequency 
dependence on the magnitude of G? is removed with increasing concentration as 
indicated by the decrease in G? slope.  The fact that the slope in G? is still present at these 
concentrations indicates that the network structure still may not be fully saturated [81].  
Similar to Figure 79 there is an apparent jump in G? from 0.025% vol. to 0.100% vol. at 
low frequency and this is taken to indicate percolation.   
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02
Angular Frequency, ? (rad/s)
Loss Modulus (Pa) 
0.2500% 0.2000%
0.1825% 0.1472%
0.1000% 0.0500%
0.0250%
 
Figure 80: The loss modulus response as a function of angular frequency and increasing nanotube 
loading for purified SWNT 183.6 dispersed in UPR.   
 
 
175 
 Finally, the over all resistance to flow is shown by the complex viscosity 
displayed in Figure 81, as a function of SWNT loading.  The samples show shear 
thinning over a wider range of frequencies as the loading is increased, representative of 
the increasing complexity of the fluid microstructure.  In the infinite shear limit the 
viscosity increases with loading.  Between concentrations of 0.025% vol. and 0.100% 
vol. there is a sudden increase in the degree of shear thinning and a jump in zero shear 
viscosity.  Again, this was taken as an indication of network formation.        
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02
Angular Frequency, ? (rad/s)
Complex V
i
scosity,
 |
?
*| (Pa-s) 
0.2500%
0.2000%
0.1825%
0.1472%
0.1000%
0.0500%
0.0250%
 
Figure 81:  The complex viscosity with increasing nanotube loading for purified SWNT 183.6 
dispersed in UPR.   
 
 It is important to note that the range of concentrations studied was limited by the 
method of mixing in this work.  Figure 82 shows an image of the highest concentration 
attempted for dispersion; 0.400% vol.  The extremely high viscosity that developed at 
 
176 
this loading prevented sufficient pumping in the mixer leaving large aggregates of SWNT 
in the sample after processing.  Therefore this concentration was not studied. 
 
Figure 82: Image (20x2) of a 0.400% vol. SWNT-UPR dispersion processed shear mixing.   
 
Determination of Rheological Percolation: 
 While linear viscoelastic data suggested network formation, methods exist to 
extract the onset of percolation and characteristics of the network interaction.  Recently, 
Mitchell and Krishnamoorti used a ?divergence? plot to elucidate the evolution of a yield 
stress in SWNT-poly(?-caprolactone) dispersions [200].  Figure 83 shows such a plot 
where the complex viscosity is plotted against the complex modulus.  Essentially, this 
plot displays the resistance to flow vs. the resistance to deformation.  The divergent 
behavior at fixed complex modulus is indicative of a finite yield stress.  The vertical 
arrows show the value of G* as the complex viscosity tends to infinity.  Open symbols 
indicate uncertainty with the divergent behavior as these data points were close to the 
instrument torque sensitivity on 0.1 ?Nm.  The G* and the corresponding loading can 
 50 ?m 
 
177 
then be used to correlate rheological percolation.  Figure 84 shows such a plot.  Linear 
regression was performed with the set of three, four, and five highest concentrations to 
determine the x-intercepts (x
int
).  A good linear fit can be seen for the highest 3 - 4 data 
points.  These x-intercepts were taken to be the represent the onset of geometric 
percolation from approximately 0.12 - 0.13% vol..  Garboczi et al. (1995) calculated the 
percolation threshold of ellipsoids with no excluded volume interaction as a function of 
aspect ratio [190].  From this model an effective anisotropy for the dispersion of SWNTs 
in UPR was found to be ~450 - 470.  But, as seen in both Figure 83 and Figure 84 
yielding behavior is apparent at 0.100% vol. loading.  A linear fit to the five highest 
concentrations shows a percolation threshold of ~0.100% vol. with L/d of ~550.  For 
comparison, a more classical approach to determining percolation threshold was also 
performed.    
 
 
 
178 
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04
Complex Modulus, G* (Pa)
Complex V
i
s
c
os
ity, |
?
*| (Pa-s
)
 
0.2500%
0.2000%
0.1825%
0.1472%
0.1000%
0.0500%
0.0250%
 
Figure 83: Divergence plot showing the complex viscosity as a function of complex modulus 
[200].  The arrows point to the value of G* that is taken to be the yield stress.    
 
 
179 
y
3
 = 2,168.6x - 292.04
y
4
 = 2,021.5x - 259.13
y
5
 = 1,610x - 172.98
0
50
100
150
200
250
300
0 0.50.10.150.20.250.3
Volume Percent (%)
G*
| 
?
*|
     
 
?
x
int4
 = 0.1281
x
int3
 = 0.1346
x
int4
 = 0.1074
 
Figure 84: The divergent value of the complex modulus from Figure 85 plotted against its 
corresponding loading.   
 
 Figure 85 displays the trend in low frequency storage modulus, plotted for three 
constant angular frequencies, as a function of loading. The two order of magnitude jump 
in G? that occurs between 0.050 % and 0.100 % vol. (gray shading) was interpreted to be 
an acceptable range for the onset of rheological percolation.  A similar plot was used by 
Hough et al. (2004) used to determine an acceptable range of threshold values.  From this 
data a power-law can be fit and the percolation threshold and scaling exponent can be 
determined.  Figure 86 displays the power-law scaling at 0.1 rad/s.  The method of Du et 
al. (2004) was arbitrarily chosen to render the x-axis of reasonable magnitude, centered 
around unity.   
 
180 
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
0.01 0.1 1
SWNT Loading (% vol.)
Storage
 Modulus (Pa)
0.0100 rad/s
0.0398 rad/s
0.1000 rad/s
 
Figure 85: The storage modulus as a function of SWNT loading for fixed angular frequencies on 
a log-log axis.   
 
The onset of rheological percolation was found to occur at ~0.100% vol. by varying 
c
? so 
as to maximize the coefficient of regression (R
2
).  This is in general agreement, being 
slightly lower, with the estimation of the percolation threshold from the divergent 
complex modulus but cannot be assumed to be a measure of individual SWNTs.  For 
point of reference a monodisperse system of individual tubes would show the onset of 
percolation at the theoretical upper limit for the isotropic concentrated regime.  The 
determined slope (critical exponent) of 1.55 is less than expected for chemical gels, with 
a predicted value of ~2.1, where bonds resist stretching but are free to rotate under strain 
[201].  Experimentally, a critical exponent of ~2.3 was extracted by Hough et al. (2004) 
for aqueous surfactant stabilized individual SWNT suspensions. 
 
181 
 
Figure 86: Power law scaling of the storage modulus as a function of the reduced volume fraction 
at 0.1 rad/s.   
 
This value was interpreted to be a result of central force percolation described previously 
by Sahimi and Arabi (1993) [201].  Davis (2006) extracted an exponent of ~1.6 for 
SWNTs in superacid and argued that the lower value was a result of weaker associations 
between protonated SWNTs than in the surfactant used by Hough et al. [144].  Hu et al. 
(2006) found a critical exponent of 1.5 in PET-MWNT melts but provided no explanation 
to the physical origin of this behavior.  In the present case, the value of approximately 1.6 
is most likely a result of the less than ideal dispersion state and the absence of rigid-rod 
behavior rather than representative of the tube-tube association at contact points.  For 
sake of repeatability, Figure 87 shows the power-law scaling for data recorded at 0.01 
rad/s. Again, similar behavior was found with regard to critical exponent and 
1.0E+01
1.0E+02
1.0E+03
0.1 1 10
S
t
or
ag
e M
o
du
l
u
s
 (
P
a
)
( )
c
c
?
???
slope = 1.55 
= 0.0098 
c
?
 
182 
concentration.  It is noteworthy that upon addition of the 5
th
 lowest concentration point 
the percolation threshold became non-physical and extended well into the dilute regime. 
 
Figure 87: Power law scaling of the storage modulus as a function of the reduced volume fraction 
at 0.01 rad/s showing similar behavior as the trend at 0.1 rad/s.   
 
 Reconsidering the colloidal scaling, the lowest concentration of 0.025% vol. can 
be removed since this was below the percolation threshold.  Figure 88 displays the high 
frequency behavior of the scaled moduli.  The high frequency regions were fit to a power 
law in order to extract the frequency dependant exponents.  According to this trend, 
G?/G
c 
? ?
0.85
 and G?/G
c 
? ?
0.63
.  For comparison, the reduced loss modulus G?
sc
 was 
found to scale with a power of 0.75 in filamentous F-actin networks and solution of 
tightly entangled semiflexible polymers by theory and experiment [202, 203].  Hobbie et 
al (2007) found power-law scaling of G?
sc 
? ?
0.7
 and G?
sc
 ? ?
0.9
 for shear aggregated, 
1.0E+01
1.0E+02
1.0E+03
0.1 1 10
S
t
or
ag
e Mod
u
l
u
s (
P
a
)
( )
c
c
?
???
slope = 1.60 
= 0.0095 
c
?
 
183 
fractal MWNT networks [107].  For both reduced moduli the observed behavior for 
SWNT-UPR dispersions is weaker.       
G'/Gc = 0.523x
0.6342
R
2
 = 0.9982
G"/Gc = 0.6658x
0.8468
R
2
 = 0.9997
1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E-04 1.0E-02 1.0E+00 1.0E+02 1.0E+04
Reduced Frequency (?/?
c
)
G'/G
c
 and G''/G
c 
 
Figure 88:  High frequency regression of the scaled moduli after removal of 0.025% vol. data set.   
 
Percolation from Crossover Modulus: 
 An interesting trend was observed when examining a plot of the crossover 
parameters shown by Figure 89.  There is a similar two order of magnitude increase in 
the crossover modulus from 0.025% vol. - 0.100% vol. consistent with the data in Figure 
85.  This behavior can be explained, in part by the fact that the crossover modulus does 
not differ much from the low viscosity plateau modulus as a result of the frequency 
independent network elasticity.  Additionally, there increasing trend is crossover 
frequency follows SWNT concentration with the increased reinforcement.  Therefore it 
may be reasonable for one to obtain the same information regarding rheological 
 
184 
percolation at higher torques as compared to a tracking of low angular frequency G?.  
This would be valuable for researchers studying low viscosity solutions of particles like 
SWNTs in superacids.  
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02
Crossover Frequency, ?
c
 (rad/s)
Crossover Modulus, 
G
c
 (Pa) 
0.2500%
0.2000%
0.1825%
0.1472%
0.1000%
0.0500%
0.0250%
 
Figure 89:  Plot of the crossover modulus and frequency on a log-log axis.   
 
Morphology of SWNT Dispersions: 
 To address the interpretation of the predicted effective hydrodynamic aspect ratio 
and the scaling found from the critical exponent, TEM images of the SWNT dispersions 
were gathered.  Figure 90 shows a range of fiber sizes from an acetone washed 
dispersion.  Clearly, by examining the image, the effective aspect ratio of ~450 - 470 
reflects the micron length SWNT bundles or fibers instead of individual tubes.   
 
 
185 
 
Figure 90:  TEM image of the SWNT-UPR dispersion (0.050% vol.) after acetone washing.  A 
mixture of bundled SWNT fibers ranging in lengths and diameters can be seen.   
 
Rheology of Dilute SWNT Dispersions: 
 Dispersions ranging in concentration from 0.030% vol. - 0.010% vol. were 
characterized by means of rheology and microscopy through successive dilution 
experiments.  Figure 91 shows a 0.010% vol. image prior to dilution.  AS testing was first 
conducted in order to determine the extent of the linear viscoelastic regime.  Figure 92 
displays the response of dilute SWNT 183.6 Pure dispersions to shear strain.  While both 
G? and G? increase with loading there is not much variation in the critical linear 
viscoelastic strain due to the dilute and non-interacting nature at these concentrations.  
Thus, over the range of concentrations in Figure 92 the fluid behaves as a viscous liquid 
 
186 
at all times.  For reference the moduli of neat UPR is shown as well.  At low strains G? 
shows some noise for the neat UPR as it is close to the torque sensitivity of the 
instrument.      
 
Figure 91:  Image of the 0.010% vol. SWNT-UPR dispersion processed shear mixing.  The image 
was taken through cover glass with a 60x oil immersed objective and 2x magnification in from of 
the camera (120x).  
 
 
187 
1.0E-02
1.0E-01
1.0E+00
1.0E+01
1.0E+02
0.1 1 10 100
Strain (%)
Modulus (P
a)
G' 0.0100% G'' 0.0100% G' 0.0075% G'' 0.0075%
G' 0.0050% G'' 0.0050% G' 0.0030% G'' 0.0030%
G' UPR G'' UPR
 
Figure 92:  Structural stability of SWNT 183.6 Pure-UPR dispersions following successive 
dilutions.  The moduli were measured by ramping strain amplitude at constant angular frequency 
of 10 rad/s.   
 
 After AS characterization the samples were subjected to FS testing in order to 
determine the response to controlled strain oscillatory shear.  Figure 93 presents the 
results of this test.  The FS data clearly shows the an increase in both moduli with loading 
but liquid-like behavior was observed for all concentrations.  Additionally, Figure 94 
shows the corresponding complex viscosity for the dilute SWNT dispersions. 
 
 
188 
1.0E-03
1.0E-01
1.0E+01
1.0E+03
0.1 1 10 100
Angular Frequency, ? (rad/s)
Modulus (Pa)
G' 0.0100% G'' 0.0100%
G' 0.0075% G'' 0.0075%
G' 0.0050% G'' 0.0050%
G' 0.0030% G'' 0.0030%
 
Figure 93:  Storage and Loss Moduli from successive dilutions of SWNT 183.6 Pure dispersed in 
UPR.  The magnitude of G? clearly increases over all frequencies with increasing loading.   
1.0E+00
1.0E+01
0.1 1 10 100
Angular Frequency, ? (rad/s)
Comple
x V
i
sc
osity, |
?
*| (Pa
-
s)
0.0100%
0.0075%
0.0050%
0.0030%
 
Figure 94:  Complex viscosity from successive dilutions of SWNT 183.6 Pure dispersed in UPR.     
 
 
189 
Similar conclusions to Figure 93 can be drawn.  The viscosity enhancement is minimal 
but consistently increases with SWNT concentration.  Since the samples were created in 
succession their viscosities should be related directly to SWNT concentration.  Equation 
4.1 defines the reduced complex viscosity,
*
red
? . 
 
s
s
red
??
??
?
?
=
*
*
 (4.1) 
Here, the neat solvent viscosity
s
? is subtracted from measured complex viscosity *? .  
and this quantity is then divided by the product of the solvent viscosity and volume 
fraction.    
 Figure 95 shows the reduced complex viscosity based on the data measured in 
Figure 94.  For all measurements a solvent viscosity of 2 Pa-s was taken. 
y = 3200.8x
-0.1611
1.0E+03
1.0E+04
0.1 1 10 100
Angular Frequency, ? (rad/s)
Red. Complex Viscosity, |
?
*|
re
d
 (P
a-s)
0.0100%
0.0075%
0.0050%
0.0030%
 
Figure 95:  Reduced complex viscosity from successive dilutions of SWNT 183.6 Pure dispersed 
in UPR.   
 
190 
 The slope of the reduced viscosity shown in Figure 95 was fit by regression.  The low 
slope of the line indicates deviation from Brownian rigid-rod behavior [104].  Angular 
frequencies higher than 100 rad/s and lower than 0.1 rad/s were also examined to ensure 
that there were no transitions in the slope.  Non-Brownian behavior can be confirmed by 
estimating the magnitude of the rotational Peclet number.  Using Equation 2.65 for the 
dilute rotary diffusivity the Peclet number can be described by Equation 4.2.    
 
()8.0/ln3
3
?
=
dLTk
L
Pe
B
s
???&
 (4.2) 
Even with a conservatively small estimate for the mean contour length of only 1 ?m, and 
using the hydrodynamic aspect ratio estimated from the divergence plot, the order of Pe 
is on the order of 10
1
 at minimum.  Thus, the imposed hydrodynamic forces of the 
rheometer are dominant and the sample shows non-Brownian behavior.  This agrees with 
experimental evidence supporting this claim; ease of fiber shear alignment, long 
relaxation times, large dependence on sample history. 
 
Shear-induced Aggregation:   
 It was originally desired to characterize the shear-rate dependent reduced 
viscosity for the dilute SWNT 183.6 Pure dispersions.  However, upon start-up of flow 
from small shear rates a peculiar rheological response was observed.  Figure 96 shows 
the shear rate dependant start-up curves for a 0.005% vol. dispersion.  The viscosity 
trends up asymptotically when held at a shear rate of 10
-3
 1/s over long shear duration.  
As a rule of thumb, the time required for start-up of flow under steady shear is 
approximately the reciprocal of the shear rate.  For 10
-3
 and 10
-2
 1/s the samples were 
 
191 
found to not reach a steady state over this time period.  This behavior was assumed to be 
a result of sample aggregation.  In addition, the application of higher shear rates appeared 
to breakdown any structure formed in the fluid.  The existence of a critical shear rate or 
corresponding shear stress for dispersion of aggregates has been observed experimentally 
for a variety of systems.  This critical stress is necessary to overcome the effects of both 
mechanical entanglement and particle attractive forces [107].  Chaouche and Koch (2000) 
first described this phenomenon in non-Brownian rigid fiber suspensions by means of a 
characteristic dimensionless number, taking the ratio of the viscous shear force to the 
adhesive contact force [204].  Indeed, this appears consistent with the current 
measurements.  Additionally, shear aggregation has been seen previously for non-
Brownian MWNT dispersions [84, 94].        
0
5
10
15
20
25
30
35
40
0.01 0.1 1 10 100 1000 10000
Time (s)
Viscosity (Pa-s)
0.01 1/s
0.1 1/s
1 1/s
100 1/s
0.001 1/s
 
Figure 96: Shear rate dependant start-up of flow behavior for a 0.005% vol. SWNT dispersion.   
 
 
192 
 Figure 97 clearly shows proof of the shear induced aggregation for a SWNT 183.6 
Pure dispersion.  The fact that this occurs on a length scale able to be seen by the naked 
eye was impressive.  The samples were originally homogeneously black.  Sufficient 
aggregation has occurred to create UPR rich areas.  It is interesting to note that this 
specific sample is stable during storage in absence of shear, as discussed later. 
    
Figure 97: Images showing the transparency of 0.005% vol. purified SWNT 183.6 dispersion 
after treatment with a steady low shear rate of 0.01 1/s.  The sample gap was constant at 0.685 
mm.   
 The samples shown in Figure 97 were carefully extracted so as to not ruin the 
tenuous structure of the material.  These were further characterized by optical microscopy 
as seen by Figure 98 - Figure 100.  The samples show areas of varying concentration in a 
striated manner.  The lighter parts of the image are thought to be SWNT depleted areas. 
 
193 
 
Figure 98: Sample from Figure 99 carefully transferred from the rheometer in its aggregated state 
to preserve the structure. Image taken with a 10x objective and 2x magnification.   
 
Figure 99: A second image taken at 10x2 magnification showing highly striated structure. 
100 ?m 
100 ?m 
 
194 
 
Figure 100: A closer image of the sample showing some diffuse structure.  The image was taken 
with a 20x objective and 2x magnification in front of the camera.   
 
 Schmid et al. (2000) showed through simulation that flow-induced aggregation 
can occur in non-Brownian fiber suspensions as a result of fiber friction alone [110].  In 
order to determine if shear aggregation was a result of the initial dispersion state alone, 
the reduced viscosity of a separate SWNT sample was characterized.  Figure 101 shows 
the reduced complex viscosity for a SWNT 187.3 dispersion.  In comparison to the 
SWNT 183.6 Pure reduced complex viscosity shown by Figure 95, shear thinning was 
slightly less pronounced (lower slope) in this sample.  Small discrepancies between the 
two SWNT dispersions may also be a result of error on initial concentration [104].  
Regardless, the behavior of both samples are clearly not drastically different.  Again, the 
data appears not to be in agreement with the theory for Brownian rigid-rods and serves to 
confirm non-Brownian behavior experimentally.  Brownian rods show stronger shear 
thinning behavior as a result of tube buckling at higher elastic forces [104].       
50 ?m 
 
195 
y = 2481.2x
-0.1237
1.0E+03
1.0E+04
0.1 1 10 100
Angular Frequency, ? (rad/s)
Red. Complex Viscosity, |
?
*|
re
d
 (P
a
-
s)
0.0100%
0.0075%
0.0050%
0.0030%
 
Figure 101: Reduced complex viscosity from successive dilutions of SWNT 187.3 dispersed in 
UPR.   
 
 In order to reveal the nature of the aggregation behavior a variety of SWNTs were 
studied.  Figure 102 shows the start-up of flow for 0.005% dispersions of SWNTs at a 
constant shear rate of 10
-2
 1/s.  The rheometer geometry for all samples was a 50 mm 
cone and plate except where indicated for 183.6 Pure (PP43) where a 43 mm parallel 
plate was used.  The sample gap was constant at 0.65 mm in the parallel plate geometry 
which was larger than the average gap in the cone and plate of 0.493 mm.  In the cone 
and plate geometry both SWNT 183.6 and SWNT Unidym show indication of 
aggregation behavior.  Although the kinetics involved in these two studies appears 
different a detailed study has not been performed.  It is important to note that sample 
 
196 
SWNT 183.6 Unpure and sample SWNT 187.3 show no aggregation behavior.  Since the 
comparison of the reduced complex viscosity failed to show marked differences between 
a sample which aggregates (183.6 Pure) and one which does not (SWNT 187.3) 
explanation was sought elsewhere.  Specifically, the attention was turned towards 
characterizing the SWNT surface chemistry in each sample, as discussed in subsequent 
sections.     
1.0E+00
1.0E+01
1.0E+02
1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05
Time (s)
Vis
c
osity (
P
a
-
s)
183.6 Pure PP43
183.6 Pure CP50
183.6 UnPure
183.7
Unidym
 
Figure 102: Comparison of start-up flow curves for different dilute CNT samples dispersed in 
UPR at 0.005% vol.  The shear rate was constant at 0.01 1/s for all samples.   
 
 Figure 103 shows the effect of sample history on the start-up of flow for SWNT 
183.6 Pure dispersions.  The tests were run in the cone and plate geometry and the shear 
rate was carefully increased from low to high.  It appears that in all cases the shear stress 
is below critical shear stress for dispersion of the aggregates since the final viscosity of 
 
197 
the previous run remains.  In comparison to Figure 96, viscosity breakdown was seen at 
0.1 s
-1
 previously.  This is evidence of a ?shear annealing? effect on the viscosity which 
again displays the importance on sample history for these dispersions. 
1.0E+00
1.0E+01
1.0E+02
1.0E-01 1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04
Time (s)
Vis
c
osity (
P
a
-
s)
0.005 1/s
0.010 1/s
0.050 1/s
0.100 1/s
 
Figure 103:  Comparison of start-up flow curves for different shear rates on dilute SWNT 183.6 
Pure samples dispersed in UPR at 0.005% vol.   
  
 In addition to the cone and plate characterization, the SWNT 183.6 Pure 
dispersion was studied with variable sample gaps as well.  These ranged from 0.674 ? 
0.344 mm in the SALS unit which has transparent quartz plates.  While asymptotic 
rheological behavior consistent with sample ?183.6 Pure PP43? in Figure 102 was 
observed for all samples, at the smaller gap of 0.344 mm evidence of an aggregated state 
was not visible with the naked eye.  This suggested behavior similar to that of Lin-Gibson 
 
198 
et al. (2003) where the gap size was found to control the sample domain size [84].  
Finally, to rule out edge drying as the cause for this viscosity rise a sample was loading at 
~1/2 the required volume for the gap size.  This prevented any sample from contacting 
the plate edges, while preserving the air/sample interface, but obviously lowered the 
measured viscosity signal since not all of the plate surface area was used.  Again, the 
asymptotic behavior was seen consistent with other loadings. 
 
Behavior at Rest: 
 The four SWNT samples used in Figure 104 were left to rest for weeks with our 
disturbance.  Revisiting the samples showed an interesting behavior.  Figure 106 displays 
an image of the sample vials after this rest period.  What appears to be phase separation 
clearly appeared in three samples.  Thus, in absence of shear the samples are only 
kinetically stable with exception to SWNT 183.6; the sample showing the strongest 
aggregation behavior.  Conversely, and confusingly the two samples showing the most 
phase separation (183.6 Unpure & 187.3) appeared to be stable under the prolonged 
application of low shear, as shown by from Figure 102.  One possible explanation for the 
stability of SWNT 183.6 Pure was revealed through TEM.  After diluting this dispersion 
with acetone, applying low shear only, and allowing it to soak for a few days the sample 
dropped onto a TEM grid.  Figure 105 shows a typical image observed.  What is shown 
by the image is a large SWNT bundle coated with UPR even after washing.  Therefore, it 
is possible that through depletion interactions, absorbed surface polymer is acting to keep 
the SWNT 183.6 sample stable over long time durations in storage.  Absorption of 
 
199 
surface polymer is feasible since favorable ?- ? interactions between SWNTs and 
unsaturated bonds in UPR may exist.   
 
 
Figure 104: From left to right: 183.6 Unpure, 183.6 Pure, 187.3, and Unidym SWNTs.  Only the 
purified 183.6 SWNT sample did not phase separate when left undisturbed.  Timescale of 
observation was on the order of weeks.   
 
 
200 
 
Figure 105: TEM image of a large SWNT bundle taken from a 250 ppm dispersion and diluted 
many times with acetone.  The darker areas and low resolution is a result of adsorbed UPR 
remaining after the wash.   
 
 If, however, the sample was bath sonicated in acetone it was found that the 
polymer could be removed completely.  Figure 106 shows such an image where the 
interstitial spaces in the SWNT bundles where solvent or polymer may have existed can 
be seen. 
 
201 
 
Figure 106:  TEM image showing good removal of the UPR from the bundles revealing the 
interstitial space where solvent may reside.  UPR removal was achieved by diluting the sample in 
acetone followed by 5 min. bath sonication. 
 
Effect of Surface Chemistry: 
 In order to characterize the surface chemistry of the samples XPS analysis was 
carried out.  Overall binding energy scans were first performed to reveal the relative 
atomic concentration on the SWNT surfaces.  Figure 107 - Figure 110 display the results 
of these scans for samples 183.6 Pure, 183.6 Unpure, 187.3 and Unidym.  All samples 
showed some trace contaminants such as sulfur, silicon, or tin.  These were present in 
negligible amounts.  The results of the overall scans reveal in intriguing trend in surface 
chemistry with shear aggregation behavior.  The two samples which showed significant 
 
202 
aggregation behavior in start-up flow experiments were SWNT 183.6 Pure and SWNT 
Unidym.  Both of these samples showed low surface oxygen content of ~1%.  On the 
other hand, the two samples there were stable under low shear flow were SWNT 187.3 
and SWNT 183.6 Unpure.  Both samples showed significant surface oxygen content of 
5% and 8%, respectively.     
0100200300400500600
Binding Energy (eV)
I
n
te
nsity (
a
.u.)
C1s
O1s
Atom %
C = 99
O = 1
 
Figure 107: The XPS survey spectrum for SWNT 183.6 Pure.  The main surface constituents 
were primarily C with little O content.  Trace amounts of S was also observed.  
 
203 
0100200300400500600
Binding Energy (eV)
I
n
te
nsity (
a
.u.)
C1s
O1s
Atom %
C = 92
O = 8
 
Figure 108: The XPS survey spectrum for the as-received SWNT 183.6 Unpure.  The main 
surface constituents were primarily C and O.  Trace amounts of Si was observed.  
 
0100200300400500600
Binding Energy (eV)
I
n
te
nsity (
a
.u.)
C1s
O1s
Atom %
C = 95
O = 5
 
Figure 109: The XPS survey spectrum for SWNT 187.3.  The main surface constituents were C 
and O but trace amounts of the elements Si, S, and Sn were also observed.  
 
204 
0100200300400500600
Binding Energy (eV)
I
n
te
n
s
ity
 (
a
.
u
.
)
C1s
O1s
Atom %
C = 99
O = 1
 
Figure 110: The XPS survey spectrum for Unidym SWNTs.  The main surface constituents were 
primarily C with little O content.  Trace amounts of the elements Si, S, and Sn were also 
observed. 
 
 To elucidate more detail about the types of surface functionalities were present in 
the samples, the high resolution C1s core election photoemission curves were compared.  
Figure 111 displays the C1s curves for similarly structured materials.  Although overall 
XPS scans indicated sample 183.6 Unpure had a significant amount of surface bound 
oxygen, the C1s structure on the high energy side of the curve does not reflect this 
oxygen as bonded to carbon.  This can be seen while examining the curves for SWNT 
183.6 Pure and Unidym.  Thus, it was concluded for SWNT 183.6 Unpure that the 
majority of the bound oxygen resides on the high percentage of residual catalyst (35% 
wt.) found in the sample by TGA and TEM experiments, described earlier.  All samples 
could be deconvoluted (not shown in detail) to reveal a major relative concentration of 
 
205 
aromatic or aliphatic carbon at 294.6 eV and a minor peak at 286.3 eV representative of 
carbon bonded to a singular oxygen atom as in various hydroxl and phenolic groups.  
280282284286288290292294
Binding Energy (eV)
I
n
te
nsity (
a
.u)
183.6 Pure
183.6 Unpure
Unidym
286.3
284.
 
Figure 111: High resolution C1s curves for similar structured SWNTs.   
 
 Figure 112 displays the C1s curves for the samples found to have significant 
surface oxygen functionalities.  Additionally, the curve for the 6h acid oxidized and 
lyophilized SWNTs was shown for contrast.  Here, the high energy side (> 284.6 eV) of 
the C1s curve shows structure representative of oxygen containing functional groups 
bound to carbon.  Both the oxidized SWNT and the as received SWNT 187.3 show 
hydroxyl and phenol groups (286.3 eV) as well as carbonyl carbons (287.7 eV).  
Additionally, SWNT 187.3 shows a significant shake-up satellite peak at 290.6 eV and 
 
206 
the oxidixed tubes show carboxylic functionalities at 289.1 eV.  On the other hand, the 
high energy side of the unpurified SWNT shows little structure supporting the previous 
conclusion of the catalyst bound oxygen.  While the purification method used on SWNT 
187.3 was unknown, evidence suggests it was treated by a weak oxidizing agent such as 
nitric acid since hydroxyl and carbonyl groups are present but not carboxylic groups.  
280282284286288290292
Binding Energy (eV)
183.6 Ox
183.6 UnPure
187.3
289.1 287.7 286.3
284.6
290.6
 
Figure 112: High resolution C1s curves for SWNTs with significant amount of surface bound 
oxygen as determined from the overall scans.   
 
 In summary, the stability of SWNT dispersions to shear aggregation appears to be 
correlated with the surface chemistry.  Specifically, the type of oxygen containing 
functional groups appears to be important.  In absence of significant amounts of surface 
 
207 
oxygen SWNT 183.6 and SWNT Unidym showed strong aggregation behavior.  SWNT 
183.6 Unpure showed stability but this was not attributed to surface bound SWNT 
oxygen.  Instead, this was attributed to impurities such as residual catalyst and 
amorphous carbon material in the as-received sample.  On the other hand, SWNT 187.3 
appeared to benefit from surface bound hydroxyl and phenolic groups.  These groups can 
be singled out because of the failure to form SWNT dispersions with carboxylic 
functionalized SWNTs shown previously.  Additionally, the apparent compatibility of 
ethanol treated SWNTs with UPR, even in spite of previous oxidative treatment, supports 
the benefit from singular bound oxygen functionalities.   
 As a final means of supporting some of these conclusions Raman spectroscopy of 
the samples was carried out in order to determine the relative D-band to G-band peak 
intensity, an indication of sidewall defects or amorphous carbon.  Figure 113 - Figure 116 
display the sample spectra.  The value of I(D)/I(G) for each laser wavelength is displayed 
above each representative laser spectra.  From the Raman analysis, the fact that the value 
of I(D)/I(G) is nearly identical in SWNT 183.6 before and after purification confirms the 
hypothesis that surface bound oxygen identified in the unpurified sample was not SWNT 
bound.  For the Unidym SWNT significant surface defects are present, but determined 
from XPS not terminated with oxygen functionalities.  It is interesting to note that the 
magnitude of I(D)/I(G) was comparable between SWNT Unidym and SWNT 187.3, even 
though the two samples show different shear aggregation behavior.  As listed in Table 1, 
Unidym SWNT showed a purity of 91 - 92% wt. while SWNT 187.3 showed 
approximately 92 - 93% wt. carbon.  Additionally, comparision of the degredation curves 
in Figure 35 between the samples shows a markedly similar behavior.  Therefore, this is 
 
208 
compelling evidence to rule out effects outside of surface chemistry, such as van der 
Waals retardation, in providing stability from shear aggregation.  This also indicates for 
the Unidym sample, possible aggressive sonication as part of the purification process.  
On the other hand, for the SWNT 187.3, the value of I(D)/I(G) was supports chemical 
functionalization proposed previously from XPS analysis. 
0
100,000
200,000
300,000
400,000
500,000
600,000
700,000
800,000
900,000
0 400 800 1200 1600 2000 2400 2800 3200
Raman Shift (cm
-1
)
I
n
te
nsity (
a
.u.)
-50000
0
50000
100000
150000
200000
250000
300000
350000
400000
785 nm
514 nm
0.03 - 0.04
0.05 - 0.07
 
Figure 113: Raman Spectra at 514 nm and 785 nm excitation of the partially purified as-received 
SWNT 183.6 Unpure.  
 
209 
0 400 800 1200 1600 2000 2400 2800 3200
Raman Shift (cm
-1
)
Intensity (a.u.)
785 nm
514 nm
0.06 - 0.07
0.03 - 0.05
 
Figure 114: Raman Spectra at 514 nm and 785 nm excitation of the SWNT 183.6 Pure.  
0
50000
100000
150000
200000
250000
300000
350000
400000
450000
500000
0 400 800 1200 1600 2000 2400 2800 3200
Raman Shift (cm
-1
)
I
n
te
n
s
ity (
a
.u.)
-50000
-30000
-10000
10000
30000
50000
70000
785 nm
514 nm
0.15 - 0.20
0.07
 
Figure 115: Raman Spectra at 514 nm and 785 nm excitation of as-received Unidym SWNT. 
 
 
210 
0 400 800 1200 1600 2000 2400 2800 3200
Raman Shift (cm
-1
)
I
n
te
nsity (
a
.u.)
785 nm
514 nm
0.12
0.15 - 0.17
 
Figure 116: Raman Spectra at 514 nm and 785 nm excitation of as-received SWNT 187.3.  
 
Proposed Chemical Functionalization: 
 Based on the results of the XPS and Raman studies, as well as on the dispersion 
behavior of carboxyl terminated SWNTs, a potential future functinalization strategy can 
be proposed.  Figure 117 displays a general schematic for the progressive oxidation of 
CNTs.  In accordance with the chemical analysis for various tubes, a successful oxidative 
treatment should be one in which significant singular bound oxygen groups are generated 
such as hydroxyl or phenolic functionalities.  Furthermore, the oxidizing agent should not 
be strong enough to generate new carboxyl groups.  Thus, in accordance with the theory 
of Zhang et al. (2003), the oxidizing agent should be effective in the defect consuming 
step but not in generation of new structures.   
 
 
 
211 
 
Figure 117: Progression of oxidation reproduced from Yue et. al (1999) [159]. 
 
 Also, a hypothesis for the physical aggregation behavior between chemical 
functionalities can be proposed.  The inability of the carboxylic terminated SWNTs to 
form dispersions in UPR is likely a result of the conjugate base lacking the ability to 
associate with the isophthalic resin backbone through hydrogen bonds.  The fact that 
SWNTs are added in dilute concentrations helps facilitate dissociation of acidic 
hydrogens to the bulk solvent.  Conversely, phenolic and hydroxyl groups retain their 
acidic hydrogen and may form associative bonds along the isophthalic and ester linkages.     
 
Dispersion Thermodynamics: 
 In order to determine whether dissolution of SWNTs in UPR is 
thermodynamically favorable the enthalpy of mixing per unit volume was first estimated 
ed by Equation 4.3.   
 
()
2
2
2
SolNT
Mix
mix
RV
H
??
?
??
?
 (4.3) 
As stated in Chapter 1, the surface solubility parameters are represented by
i
?
, the 
volume fraction of CNT by? , and the bundle radius by R
2
.  The component surface 
energy
i
Sur
E
was used to find
i
?
, as defined by Equation 4.4. 
C-H C-OH C=O COOH 
 
212 
 ( )
2/1
i
Suri
E??  (4.4) 
Here the surface interaction energy was approximated by Equation 4.5. 
 ( )
2/1
Sol
Sur
NT
Sur
SolNT
Inter
EEE ?
?
 (4.5) 
The approximation introduced by Equation 4.5 allows the surface solubility parameters ?
i
 
to be estimated from the pure component interaction energy using Equation 4.5.  
Specifically this is due to formulating the binomial term of Equation 4.3 which has 
reduced any dependence on
SolNT
Inter
E
?
; an approximation yet to be addressed in literature 
for nanotube-solvent systems.  This is similar to how van Laar and Lorenz (1925) 
approximated the interaction between dissimilar molecules using the geometric mean.  
Hildebrand and Scott (1950) subsequently argued that this approximation fails to describe 
interactions outside of nonpolar molecules [205].  In the present case this introduces 
serious consequences into the approximation.  First, calculating the surface interacting 
energy by Equation 4.5 results in multiple answers; a positive and negative interaction 
energy representing an attractive and repulsive term, respectively.  However, introduction 
of Equation 4.4 to calculate the surface solubility parameter requires the attractive 
(positive) term be chosen as calculation of
SolNT?
?
 since the repulsive (negative) term is 
undefined.  Taking the attractive term alone, and thus neglecting any repulsive 
interactions, limits the ability of Equation 4.3 to account for specific interactions between 
dissimilar molecules.  Furthermore, the general form of Equation 4.3 prevents 
determination of exothermic heats of mixing.  Specifically, this neglects the subtle 
nuances between dissimilar molecules as predicted by what is know as the ?Taft and 
Hamlet? scale for hydrogen bond donor ability [206].  Ausman et al. (2000) have used 
 
213 
the Taft and Hamlet scale to investigate SWNT solvents and concluded that the best 
solvents for SWNTs are non-hydrogen-bonding Lewis bases [207].  This conclusion 
indicates that good solvents have the ability to donate an electron pair to the SWNTs but 
cannot associate through hydrogen bonding; two factors that are entirely dependant on 
solute?solvent interaction.  In this work, the effects of subtle differences in nanotube 
surface chemistry were also experimentally shown.  Thus, the kinetics of the shear 
aggregation behavior could be correlated with the magnitude of the Gibbs free energy of 
mixing using this theory.  Lastly, it is important to note the original geometric 
approximation by van Laar was for molecules of comparable size.  With a nanotube-
solvent system there are certainly many solvent molecules interacting with a single tube.                  
 Neglecting the aforementioned shortcomings, the solvent surface energy
Sol
Sur
E  can 
be related to the solvent surface tension ? by means of Equation 4.6 [208]. 
 
Sol
Sur
Sol
Sur
TSE ?=?  (4.6) 
Here the T is the thermodynamic temperature and
Sol
Sur
S
 represents the solvent surface 
entropy.  It has been argued that the solvent surface entropy is a generic property of 
liquids and can be approximated as 0.1 mJ/m
2
-K [176, 208].  Thus, the value of
Sol
Sur
E
 can 
be approximated by Equation 4.6.  For UPR the surface tension was directly measured to 
be 34.85 mJ/m
2
.  Over a range of temperatures from 293?298 K the value of
Sol
Sur
E
 for 
UPR was calculated to be 64.4 mJ/m
2
.  This is on the same order as the graphitic surface 
energy estimated to be 70 mJ/m
2
 but no criteria has been established as to how closely 
these values should match in ?good? solvents [176].  By means of Equation 4.4 the 
surface solubility parameter
Sol
?
 can be calculated to be 8.02 mJ
1/2
/m.  To calculate the 
 
214 
nanotube surface energy
NT
Sur
E
 the widely accepted binding energy for ?real? SWNTs of 
~0.5 eV/nm was used [54, 71].  However,
NT
Sur
E
is the binding energy per surface area.  An 
approximation was made to translate the energy acting over a length into the 
corresponding surface area.  Considering the binding energy over a 1.0 nm diameter tube 
with 1.0 nm length the value of
NT
Sur
E
 was calculated to be ~51 mJ/m
2
.  The surface area 
for only one tube was calculated (1.57x10
-19
 m
2
) since the interactions in a bundle have 
been previously restricted to nearest neighbor interactions [69, 72, 73].  Again, by means 
of Equation 4.4 the surface solubility parameter for SWNT
NT
?
was calculated to be  
7.14 mJ
1/2
/m.  In comparison, if the intertube binding energy theoretically calculated for 
two (10,10) tubes of 0.9516 eV/nm is used, the surface solubility parameter
NT
?
can be as 
high as 9.84 mJ
1/2
/m [69].  The heat of mixing per unit volume can then be estimated 
from equation 4.3.  To estimate this quantity conservatively the volume fraction was 
taken as one of the lowest used in experiments, 5.0 x 10
-5
 (0.005 % vol.).  The bundle 
radius R
2 
was estimated from TEM images was taken to be on the order of 10
1
 nm.  
Solving Equation 4.3 the heat of mixing per unit volume for SWNTs having a binding 
energy of ~0.5 eV/nm was approximately +15.5 kJ/m
3
.  In comparison, Bergin et al. 
experimentally determined SWNTs in NMP, a ?good? solvent, to have a heat of mixing 
on the order of -1,882 kJ/m
3
.  It is assumed that with increasing concentration the bundle 
size will either remain the same or increase, dependant on the mixer.  Thus, the enthalpy 
of mixing will also increase with the same order of magnitude that as an increase in 
volume fraction.  Therefore, at one of the highest volume fraction used in this research of 
0.200% vol the heat of mixing can be expected to increase as high as  
 
215 
+620 kJ/m
3
.  The positive heat of mixing is in agreement with what has been generalized 
for most polymer-nanotube mixtures [165].  Morawetz (1983) argued that endothermic 
mixing high molecular weight polymers with solvents is possible only because of the 
conformational entropy gained by flexible chain molecules in the process of dilution 
[205].  It can be expected that this statement would be more critical when SWNTs are 
substituted for polymers since their rigidity decreases chain conformation.  The 
calculated heat of mixing would agree with this statement.  Re-examining the criteria for 
dissolution based on the Gibbs free energy of mixing some insight to the magnitude of 
the entropy of mixing can be gained.  Rearranging Equation 2.29 the criteria for the 
entropy of mixing upon dissolution is shown by Equation 4.7. 
 
T
H
S
Mix
Mix
?
>?
 (4.7) 
For room temperature of approximately 295 K the entropy of mixing at 0.05% vol. must 
actually be less than 52.5 J/m
3
 since experimentally thermodynamic dissolution is not 
likely to occur.  This small positive value for the entropy of mixing is in agreement with 
experimental dispersion observations and therefore is not large enough to drive 
dissolution itself.  Therefore, although the theory may have made gross assumptions 
regarding interaction, thermodynamic analysis of SWNT-UPR systems is in agreement 
with experimental observation.  The dissolution of SWNTs into UPR is only kinetically 
stable with de-mixing driven by the positive Gibbs free energy of mixing.  This is a 
consequence of the dissimilarities in the UPR and SWNT surface energies but can be 
tuned by adjusting the SWNT interface through functionalization.       
 
216 
 Finally, if the heat of mixing is examined without making any assumptions 
towards the solute-solvent interaction energy some general conclusions about successful 
solvents can be drawn.  Equation 4.8 is displayed below. 
 
[ ]
SolNT
Inter
Sol
Sur
NT
Sur
NT
NT
mix
EEE
R
m
H
?
?+??
?
2
2
 (4.8) 
Clearly, the heat of mixing can be minimized if the functionalization results in an 
increased association between the SWNTs and the solvent, which would represent an 
increased value of
SolNT
Inter
E
?
.  Also, the effect of functionalization on the intertube van der 
Waals potential is reflected by
NT
Inter
E .  This should be reduced to minimize the heat of 
mixing.  Research in estimating the contribtion of the solute?solvent interaction would be 
extremenly beneficial to the field.    
 
Self-assembly of SWNT Films: 
 The intent of studying the effect of oxidation and lyophilization on SWNT 183.6 
Pure was to eliminate the bundled sample morphology caused by filtration as well as 
provide a chemically characterized sample.  While this technique was successful in 
satisfying these tasks, the low bulk density shown previously in Figure 63 is misleading if 
only examined by the naked eye.  In fact, when studied under SEM the 6h acid oxidized 
and lyophilized SWNTs reveal a remarkable aligned morphology.  Figure 118 shows a 
low magnification image of the sample simply collected from the bulk product shown in 
Figure 63.  This sample was not intentionally cast as a film but self?assembled during 
lyophilization.  The film appears to be transparent to electrons. The edge of this film in 
Figure 118 was further examined under higher magnification to reveal the discovery of a 
 
217 
high degree of nanotube alignment shown in Figure 119.  The black arrow is intended to 
represent the director of alignment.  At the film edge bundles or individual SWNTs can 
be seen extending with the director.  What is import to grasp is the order of magnitude 
this alignment occurs on.  According to the scalebar the alignent is on tens of nanometers, 
which is truly remarkable in comparison to the current state of the art of aligned SNWT 
films.   
 
Figure 118: A lyophilized film of the 6h acid oxidized SWNTs.  
 
 
218 
 
Figure 119: Close up view on the edge of the film shown in Figure 118.  A high order of 
alignment can be seen even on the nanometer scale.   
 
 The as-produced films were also examined under optical microscopy indicating 
their transparency to light.  Figure 120 shows a piece of the film where light appears to 
transmit in certain areas.  Examining Figure 121 the ability of the entire film to transmit 
some light as can be seen by the increase in opacity on the folded corner.  It is important 
to note that no attempt has been made to template these structures.   
 
 
 
219 
 
Figure 120: Optical microscopy image of a piece of lyophilized 6 h acid oxidized SWNT film.  
The film was held between a cover glass and imaged at 10x magnification.    
 
Figure 121: Optical image of a bulk 6 h acid oxidized and lyophilized SWNT film.  The sample 
was affixed to double sided adhesive tape and the image was taken with a 20x objective and 2x 
magnification in front of the camera.     
 
200 ?m 
50 ?m 
 
220 
 The effect of oxidation treatment time was also studied.  Figure 122 shows the 
resulting samples after only 2 h of oxidation.  Even at low magnification the sample does 
not appear to be as finely structured as the 6 h treated sample.  Figure 123 shows a higher 
magnification of the film edge displayed in Figure 122.  In comparison to the 6 h 
oxidized sample, the 2 h sample appears highly bundled and isotropic.  Furthermore, 
these bundles exist on approximately an order of magnitude larger length scale than the 6 
h sample.  A transition from a bundled to a more aligned state was previously described 
by Rosca et al (2005), as mentioned in Chapter 1 [124].  This was described by an 
increase in sample mobility by tube disentanglement and cutting, which is also true in the 
preset case.  Additionally, the alignment can be explained by the increase in functional 
groups over the course of the reaction as well as an increase in mobility from digestion of 
small diameter tubes.   
 
 
221 
 
Figure 122 : SEM image of a 2 h acid oxidized SWNTs after lyophilization.   
 
 
222 
 
Figure 123: Close up image of the 2 h acid oxidized SWNTs after lyopholization.   
 
 To correlate the degree of functionalization and cutting with time, Raman 
spectroscopy on both samples was carried out as shown by Figure 124 and Figure 125. 
The results of the Raman study indicate an increase in I(D)/I(G) with treatment time of 
almost three fold indicating the presence of significant defect generation to the sidewalls, 
tube end defects, or the generation of amorphous carbon. 
  
 
223 
0
20000
40000
60000
80000
100000
120000
140000
160000
0 400 800 1200 1600 2000 2400 2800 3200
Raman Shift (cm
-1
)
I
n
te
nsity (
a
.u.)
-50000
-30000
-10000
10000
30000
50000
785 nm
514 nm
0.16 - 0.19
0.21 - 0.22
 
Figure 124: Raman Spectra of SWNTs after 6 h acid oxidation and lyophilization.  
0 400 800 1200 1600 2000 2400 2800 3200
Raman Shift (cm
-1
)
I
n
te
nsity (
a
.u.)
785 nm
514 nm
0.07 - 0.08
0.04 - 0.07
 
Figure 125: Raman Spectra of SWNTs after 2 h acid oxidation and lyophilization. 
  
 
224 
 The final oxidized and lyophilized product was found to still be water soluble.  
Figure 126 shows a comparison of the SWNT 183.6 Pure sample in water after 
aggressive tip sonication against that of the 6 h oxidized sample dispersed by bath 
sonication. 
 
Figure 126: Image of (left) SWNT 183.6 after aggressive tip sonication in water for 30 min. 
pulsed at 5 s on 1 s off and (right) dilute aqueous solution of lyophilized 6 h oxidized SWNT 
183.6 re-dispersed in water by bath sonication.  
 
The insolubility of pristine SWNT 183.6 is water is apparent.  For the 6 h oxidized 
sample dissolution in the aqueous phase is believed to be controlled by more favorable 
solute-solvent interactions induced by the oxygen rich groups on the surface as well as 
the presence of carboxylate ions.  Figure 127 shows the result of a pH dependant UV-
visible spectra to determine the presence of SWNT charging.  The as-produced sample 
had a pH of 6 and was raised to pH 10 by addition of NaOH.  Examining Figure 127, the 
peak near 650 nm is attributed to the 1
st
 Van Hove transition of metallic SWNTs [209].  
The effect of slight dilution with the addition of base appears to be negated by an increase 
 
225 
in absorbance and feature definition.  The increase in bandgap fluorescence is believed to 
be a result of tube charging from the formation of carboxylate ions and thus again proves 
carboxylic functionalization.  This absorbance behavior is in agreement with photon 
quenching observed in poor SWNT dispersions by Graff et al. (2005) [210].   
0.18
0.20
0.23
0.25
0.28
0.30
400 450 500 550 600 650 700 750 800
Wavelength (nm)
Absorbance
pH 6
pH 10
1st Van Hove transistion of 
Metallic SWNTs
 
Figure 127: The UV-visible spectra for 6 h acid oxidized SWNTs after lypholization and re-
dispersion in water and variation of solution pH.   
 
 The aqueous solution from the 6 h oxidized sample was also dropped directly 
onto an SEM stub for imaging both before and after centrifugation.  The sample without 
centrifugation in Figure 128 appears to have a significant amount of amorphous carbon 
generated from the oxidation and sonication treatment.  After centrifugation the sample 
appears markedly different as seen by Figure 129.  Here, the amorphous carbon is not 
noticeable and ?crack bridging? can be seen over distances approximately 600-800 nm, 
presumably longer than the SWNT contour length.  Higher magnification images across 
 
226 
the cracks can be seen in Figure 130.  The SWNT film appears to be mechanically robust 
as indicated by the intertube-interfacial adhesion allowing for crack bridging.  
Furthermore, this can be seen from examining Figure 131 showing the leading edge of a 
crack in the dried drop.  Here, the crack propagation, created by capillary forces as the 
solvent evaporated, appears to have been halted from progression by crack bridging of 
the SWNTs.   
 
 
Figure 128: Drop dried image of lyophilized 6 h acid oxidized SWNTs that were re-dispersed in 
water.  Both SWNTs and amorphous carbon generated from the oxidation can be seen.   
 
 
227 
 
 
Figure 129: Drop dried image of lyophilized 6 h acid oxidized SWNTs that were re-dispersed in 
water and centrifuged at 17k x g for 90 min.   
 
 
228 
 
Figure 130: High resolution image of crack bridging from a drop dried and centrifuged aqueous 
dispersion of lyophilized 6 h acid oxidized SWNTs.   
 
 
 
 
 
 
 
229 
 
Figure 131: Drop dried image of the 6 h oxidized dispersion after centrifugation and drop drying 
showing crack propagation/bridging phenomena.  
 
 The apparent mechanical robustness of the sample after oxidation can be 
explained by examining the Raman spectra before and after oxidation.  Specifically, by 
comparing the diameter dependant radial breathing modes as shown in Figure 132.  The 
disappearance of the peaks at 247 nm and 261 nm indicates the consumption of small 
diameter metallic SWNTs during treatment.  This resulted in a narrower tube diameter 
distribution, which increased the shear modulus of the tube bundles, resulting in higher 
film modulus.  Furthermore, the amorphous carbon generated can fill voids in the films 
[126]. 
 
230 
100 150 200 250 300
Raman Shift (cm
-1
)
I
n
te
nsity (
a
.u.)
SWNT 183.6 Pure
SWNT-Ox
 
Figure 132: Comparison of the Raman Spectra at 514 nm excitation of SWNT 183.6 Pure before 
and after (183.6 Ox) acid oxidation showing small diameter digestion.  
 
 In summary, this method for creating aligned films is attractive for a number of 
reasons.  First, the water is used instead of an exotic organic solvent.  Second, the 
oxidation method described for this process can replace the purification step in raw tubes, 
which may reduce the amount of damage introduced onto the nanotube sidewalls.  Early 
attempts to template macroscale films have also been successful by freezing aqueous 
dispersions of the 6 h oxidized tubes inside a Petri dish.  Through these experiments the 
mode of self-assembly has been partially identified.  It appears that the act of freezing 
itself initiates the process, not the subsequent treatment under high vacuum in the 
lyophilization chamber.  This may be a result of a change in solvent strength upon phase 
change. 
 
231 
5. CONCLUSIONS AND RECOMMENDATIONS
 The intention of this work was to provide a detailed analysis on the critical 
aspects related to the dispersion of carbon nanotubes in a viscous unsaturated polyester 
resin.  The results of various nanotube dispersion techniques such as bath sonication, 
syringe extrusion, and high shear mixing were characterized.  Only high shear mixing 
was successful in dispersing single-walled carbon nanotubes, but was unable to exfoliate 
individual tubes alone.  Bulk rheological measurements were made pre-cure in order to 
probe fluid microstructure and draw conclusions on the effect of nanotube specific 
viscoelastic property enhancement.  It was determined that pristine single-walled carbon 
nanotubes (SWNT 183.6 Pure) showed the greatest reinforcing potential over acid 
oxidized nanotubes (SWNT-Ox Film) and vapor grown carbon fibers.  Using primarily 
controlled strain oscillatory shear measurements, two major types of concentration 
dependant behavior were identified for SWNT 183.6 Pure dispersions.   
 First, over the range of concentrations from 0.025% - 0.250% vol., viscoelastic 
behavior was found.  Colloidal scaling was used to remove the dependence of 
concentration on the viscoelastic response while providing insight into network 
development.  The resulting mastercurve showed two separate subtypes of rheological 
response.  At concentrations near 0.025% vol., a developing nanotube network structure 
was observed.  At concentrations near and above 0.100% vol. a saturated and elastic 
network structure began to form.  These observations were correlated with percolation.  
 
232 
Common methods for examining percolation such as low frequency storage modulus 
response and power-law scaling were employed.  Additionally, a recently developed 
method tracking the network yield stress was compared.  It was determined that the onset 
of rheological percolation occurred at ~0.100% vol. loading in all methods.  However, 
the network strength was weaker than predicted for bond force percolation theory.  This 
was attributed for the bundled dispersion state of the nanotubes as indicated by the 
effective hydrodynamic aspect ratio and TEM analysis.  In addition, parallels between the 
onset of percolation from storage modulus data and the trend in crossover parameters 
were found.  
 In the second type of rheological behavior, for the range of concentrations from 
0.0030% - 0.010% vol., the dispersions were found to behave as a viscous liquid.  
Through the linear viscoelastic response, no indication of nanotube interaction or network 
development was observed over this concentration regime.  Using a reduced complex 
viscosity, the concentration dependence on rheological response was removed.  The slope 
of the shear thinning data revealed non-Brownian behavior.  This was supported by 
theoretical calculations for the rotational Peclet number, based on a conservative 
estimation of the rotary diffusivity.  As a result of the bundled nanotube dispersion state, 
experimentally imposed hydrodynamic forces were dominant.  Thus, the SWNT 183.6 
Pure samples were extremely sensitive to shear history.        
 Through the application of small shear rates over long time scales, shear 
dependant aggregation behavior was observed in dilute nanotube dispersions.  
Experiments using various single-walled carbon nanotubes indicated the driving force for 
shear aggregation was chemical in nature, as opposed to being a function of the 
 
233 
dispersion state.  Surface analysis and comparative measurements indicated oxygen rich 
functionalities were capable of providing dispersion stability.  Specifically, experiments 
suggest hydroxyl and phenolic groups helped prevent nanotubes from re-aggregation.  
Physically, the ability for these functional groups to hydrogen bond with the isophthalic 
polyester structure supports this finding.  
 All attempts at dispersing acid oxidized nanotubes failed without prior treatment 
in ethanol (SWNT-Ox Film).  It was concluded that carboxylic functional groups were 
not compatible with unsaturated polyester resin.  The presence of adsorbed ethanol was 
believed to aid in increasing the compatibility between oxidized nanotubes and the resin.  
The specific nature of the chemical interaction in the presence of ethanol is unknown.  
From this study, it is recommended that future experiments be carried out with a 
treatment method capable of oxidizing nanotubes to primarily alcohol based 
functionalities (hydroxyl, phenolic), rather than from carbonyl to carboxylic groups.           
 The current state of nanotube solution thermodynamics was critically analyzed 
and used to estimate the heat of mixing in SWNT-UPR dispersions.  Calculations support 
experimental evidence showing the enthalpy of mixing to be large and positive.  
Estimates of the Gibbs free energy indicate the dispersions were only kinetically stable.  
The observation of low shear aggregation and dispersion phase separation over prolonged 
periods of storage support this conclusion.        
 Finally, the discovery of a lyophilization based method for creating aligned self-
assembled films from aqueous, oxidized nanotube dispersions was reported.  In future 
applications, it is recommended that this method be used to replace typical nanotube 
 
234 
purification steps, thus minimizing nanotube damage from overprocessing.  The nanotube 
self-assembly was believed to occur during the freezing of the dispersion.   
In summary, the state of the art in commercial thermoset polymer composites 
reflects an interest in utilizing high performance and lightweight materials to push the 
limits of design and application.  With this in mind, the intent of this work was to 
characterize a potential next generation lightweight thermoset composite material with 
the hope of motivating new applications for older technologies, such as unsaturated 
polyester resins.  This research provides a significant advancement the in the scientific 
understanding of dispersion and physical behavior of carbon nanotube-unsaturated 
polyester resin systems. 
  
 
235 
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246 
APPENDIX A ? DERIVATION FOR ENTHALPY OF MIXING 
 )(
22211
SolNTSolNTSolNT
mix
EEEEEH
?
++?+=?  (A.1) 
With: 
 
NT
Sur
NT
ELRrl
lr
LR
NE
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
11
2
1
2
1
11
22 ??
?
?
 (A.2) 
 
NT
Sur
NT
ELRrl
lr
LR
NE
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
22
2
2
2
2
22
22 ??
?
?
 (A.3) 
 
2
2
2
2
LR
M
N
NT
NT
??
=
 (A.4) 
 
Sol
Sur
SolSol
CohSol
Sol
EAEVE
11
?=
 (A.5) 
 
Sol
Sur
SolNT
Inter
Sol
Sur
SolSol
CohSol
Sol
EAEAEVE
?
??=
22
 (A.6) 
 
SolNT
Inter
SolNT
Inter
SolNT
Inter
SolNT
ELRNEAE
????
==
2222
222 ?
 (A.7) 
 
 
 
 
247 
 
()
?
?
?
?
?
?
?
?
+??+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=?
?? SolNT
Inter
Sol
Sur
SolNT
Inter
Sol
Sur
SolSol
CohSol
NT
Sur
Sol
Sur
SolSol
CohSol
NT
Surmix
ELRNEAEAEVELRrl
lr
LR
N
EAEVELRrl
lr
LR
NH
222222
2
2
2
2
2
111
2
1
2
1
1
2222
22
???
?
?
??
?
?
 (A.8) 
Assumption:
SolSol
AA
12
? at low loading, therefore: 
 
?
?
?
?
?
?
?
?
++
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=?
?? SolNT
Inter
Sol
Sur
SolNT
Inter
NT
Sur
NT
Surmix
ELRNEAELRrl
lr
LR
N
ELRrl
lr
LR
NH
22222
2
2
2
2
2
11
2
1
2
1
1
2222
22
???
?
?
??
?
?
 (A.9) 
 
222
2 LRNA
SolNT
Inter
?=
?
 (A.10) 
()
?
?
?
?
?
?
?
?
?+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=?
? Sol
Sur
SolNT
Inter
NT
Sur
NT
Surmix
EELRNELRrl
lr
LR
NELRrl
lr
LR
NH 222222
22222
2
2
2
2
211
2
1
2
1
1
???
?
?
??
?
?
 (A.11) 
 
 
248 
 
()
Sol
Sur
SolNT
Inter
NT
Sur
NT
NT
NT
Sur
NT
NT
mix
EELRNELRrl
lr
LR
LR
M
ELRrl
lr
LR
LR
M
H
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=?
?
2222
22
22222
2
2
2
2
2
2
2
11
2
1
2
1
1
2
1
???
?
?
??
??
?
?
??
 (A.12) 
 
()
SolNT
Inter
Sol
Sur
NT
NTNT
Sur
NT
NTNT
Sur
NT
NT
mix
EE
R
M
E
Rr
M
E
Rr
M
H
?
?+
?
?
?
?
?
?
??
?
?
?
?
?
?
?=? 2
21212
212
?
?
?
?
?
 (A.13) 
 
()
SolNT
Inter
Sol
Sur
NT
NTNT
Sur
NT
NT
mix
EE
R
M
E
RR
M
H
?
?+
?
?
?
?
?
?
?=? 2
2112
212
??
 (A.14) 
Assumption:  If R
1
>>R
2,
 then R
1
-1
 negligible.  
 
( )
SolNT
Inter
Sol
Sur
NT
Sur
NT
NT
mix
EEE
R
M
H
?
?+=? 2
2
2
?
 (A.15) 
 
MixNT
NT
V
M
?
? =
 (A.16) 
 
( )
SolNT
Inter
Sol
Sur
NT
Sur
Mix
mix
EEE
RV
H
?
?+?
?
2
2
2
?
 (A.17) 
 
 
249 
 
If this is true:   
()
2/1
Sol
Inter
NT
Inter
SolNT
Inter
EEE ?
?
 then; 
 
( )[ ]
Sol
Sur
Sol
Inter
NT
Inter
NT
Sur
Mix
mix
EEEE
RV
H
+??
? 2/1
2
2
2?
 (A.18) 
with, 
 
( )
2/1
i
Suri
E??
 (A.19) 
 
[ ] ()
2/1
2
22
2
2
2
2
SolNTSolSolNTNT
Mix
mix
RRV
H
??
?
????
?
?=+??
?
 (A.20) 
To determine
Sol
Sur
E
 from the measured surface tension ?, given that in general: 
12
1.0
??
= KmJmS
Sol
Sur
 
 
Sol
Sur
Sol
Sur
TSE ?=?
 (A.21) 
250 
APPENDIX B ? ALTERNATE PERCOLATION GRAPHS
0.0E+00
5.0E+01
1.0E+02
1.5E+02
2.0E+02
2.5E+02
3.0E+02
0.01 0.1 1
SWNT Loading (% vol.)
Storage Modulus (Pa)
0.0100 rad/s
0.0398 rad/s
0.1000 rad/s
 
Figure B1: The storage modulus as a function of SWNT loading for fixed angular frequencies on 
a lin-log axis. 
251 
0.0E+00
2.0E+02
4.0E+02
6.0E+02
8.0E+02
1.0E+03
1.0E-02 1.0E-01 1.0E+00 1.0E+01 1.0E+02
Crossover Frequency, ?c (rad/s)
Crossover 
Modulus, Gc (Pa) 
0.2500%
0.2000%
0.1825%
0.1472%
0.1000%
0.0500%
0.0250%
 
Figure B2: Plot of the crossover modulus and frequency on a lin-log axis.   
 
252 
APPENDIX C ? FTIR OF SWNT 187.3 
5001000150020002500300035004000
Wavenumber (cm
-1
)
0
0.2
0.4
0.6
0.8
Absorbance
2914
2845
  
Figure C1: FTIR spectra for SWNT 187.3.  The sample was dispersed in 1,2-dichlorobenzene and 
dropped on a ZnSe ATR crystal.  The peaks are representative of sp
2
 hybridized carbons.