MANUFACTURING AND HEAT TRANSFER ANALYSIS OF  
NANO-MICRO FIBER COMPOSITES 
 
Birg?l A?c?o?lu 
 
A Dissertation 
Submitted to  
the Graduate Faculty of 
Auburn University 
in Partial Fulfillment of the 
 Requirements for the Degree of 
Doctor of Philosophy 
 
Auburn, Alabama 
 
 
August 8, 2005 

 iii
 
 
 
 
 
VITA 
 
 
Birg?l A?c?o?lu, daughter of Sevginar A?c?o?lu (Aksoy) and Fikret A?c?o?lu was 
born in Trabzon, Turkey on September 16, 1978. She graduated from Trabzon Anatolia 
Trade High School in May 1995 and received a Bachelor of Science degree in 
Mechanical Engineering from Karadeniz Technical University in June 1999.  
She received a Master of Science degree in Mechanical Engineering from Karadeniz 
Technical University in August 2002.  She joined the Ph.D. program in Textile and 
Apparel Science in the Department of Textile Engineering, Auburn University in August 
2002.  
 
 
 
 
 
 
 
 
 
 
 iv
 
 
 
 
 
DISSERTATION ABSTRACT 
 
 
MANUFACTURING AND HEAT TRANSFER ANALYSIS OF  
NANO-MICRO FIBER COMPOSITES 
 
Birg?l A?c?o?lu 
Doctor of Philosophy, August 8, 2005 
(M.Sc., Karadeniz Technical University, Turkey, 2002) 
(B.Sc., Karadeniz Technical University, Turkey, 1999) 
 
171 Typed Pages 
Directed by Dr. Sabit Adanur 
 
 
Nano-micro composites are widely used in many areas, such as sensors, flame 
retardant materials, batteries, and filtration. In this work, thermal properties of the filler 
fiber composites are studied experimentally, analytically, and numerically.  For the 
thermal studies, the transverse thermal conductivity is discussed and calculated both 
analytically and numerically. In the analytical study, a hexagonal cell model is developed 
 v
which includes an interfacial area. The volume fraction of the filler fibers is kept between 
10-30%. From the results, it is seen that the numerical and analytical results showed 
much similarity.   
A novel device is designed to manufacture yarns continuously. Nano-micro fibers 
are manufactured and collected to form web and yarn. To collect the fibers, nonwoven 
fabrics are used, which allow easy release of fibers from the surface. Scanning electron 
microscope, thermal gravimetric analysis, and digital imaging are used to analyze the 
structures. Tensile strength, surface tension, and air permeability measurements are done.  
To transition from micro to nano is discussed in terms of modeling. It was shown 
that for the electrospun fibers whose diameters are above 200 nm, the conventional heat 
transfer modeling methods are still valid.  
 
 
 
 
 
 
 
 
 vi
 
 
 
 
 
Style manual or journal used  
             Textile Research Journal  
 
Computer software used 
 Microsoft Word, Excel, MATLAB, ANSYS 7.0, AUTO-CAD, TTS Data     
            Analysis 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 vii
 
 
 
 
 
ACKNOWLEDGEMENTS 
 
 
In this study, I had a chance to be between nano and micro areas and this was the 
help of my adviser Dr. Sabit Adanur. He guided me, advised me, helped me to see my 
potential. I am grateful to him for giving me the encouragement to take the steps inside 
the research. 
I would like to give my appreciation and sincere thanks to Dr. P. Schwartz, Dr. L. 
Slaten and Dr. R. Knight for serving on my committee and for their kind help.  
I would like to thank also Dr. M. Miller, Dr. R. Broughton, Dr. Y. Gowayed, Dr. 
H. Aglan, Dr. H. Bas, Dr. L. Gumusel, Dr. R. Farag, and Dr. M. Traore for their 
suggestions and sincere help. 
I want to thank all the members of the Textile Engineering Department for their 
help and National Textile Center for financial support of this research.  
My closest friends in Auburn?I could not have imagined living here without 
you. You are the family to me. 
And the reason for most of my tears, happiness, sadness? my dear family? I can 
not tell you how much you made me to think how lucky I am to have you. My mother; 
Sevginar, my father; Fikret, my sister; Elif Burcu, my brother; Arman, my closest friends 
Sabire, Esenc and Yucel?so much applause for you.  
 
 
 
 viii
 
 
 
 
 
TABLE OF CONTENTS 
 
 
LIST OF TABLES?????????????????????????...?..xi 
LIST OF FIGURES?????????????????????????.......xii 
 
CHAPTER 1 .......................................................................................................................1 
INTRODUCTION .............................................................................................................. 1 
1.1. References................................................................................................................ 3 
 
CHAPTER 2 .......................................................................................................................4 
LITERATURE REVIEW ................................................................................................... 4 
2.1. Nano-micro (NM) Composites ................................................................................ 4 
2.2. Interface Effect......................................................................................................... 6 
2.3. Electrospinning ...................................................................................................... 12 
2.4. Thermal Properties of Nano-Micro Composites.................................................... 12 
2.4.1. Thermal Conductivity in Macro Level ........................................................... 13 
2.4.2. Thermal Conductivity in Nano Level ............................................................. 15 
2.4.3. Flame Resistance ............................................................................................ 18 
2.5. Modeling................................................................................................................ 21 
2.6. References.............................................................................................................. 24 
 
CHAPTER 3 ..................................................................................................................... 32 
NANO-MICRO FIBER BASED FILM, WEB AND YARN MANUFACTURING ...... 32 
3.1. Materials ................................................................................................................ 32 
3.1.1. Polyvinyl Alcohol (PVA) Properties .............................................................. 32 
3.1.2. Laponite? Properties ...................................................................................... 35 
3.2. Continuous Nano-Micro Fiber Based Yarn Manufacturing .................................. 36 
 ix
3.2.1. Defining the Maximum Collected Nanofiber Web Area.................................... 39 
3.2.2. The Voltage Effect in Electrospinning ........................................................... 41 
3.2.3. Scanning Electron Microscope (SEM) Imaging............................................. 50 
3.2.4. Continuous Manufacturing Device................................................................. 51 
3.2.5. Coating............................................................................................................ 54 
3.2.6. Twisting of Manufactured Nano-micro Fibers??????????.?...55 
3.2.7. Differential Scanning Calorimetry (DSC) Studies ......................................... 56 
3.2.8. Air Permeability Measurements ..................................................................... 57 
3.2.9. Dynamic Contact Analyzer............................................................................. 57 
3.2.10. Thermal Conductivity Measurement ............................................................ 58 
3.3. References.............................................................................................................. 61 
 
CHAPTER 4 ..................................................................................................................... 63 
ANALYTICAL MODELING OF FILLER FIBER REINFORCED COMPOSITES ..... 63 
4.1. Description of the Problem .................................................................................... 65 
4.2. Modeling for Analytical Thermal Resistance........................................................ 67 
4.2.1. Thermal Resistance of the First Region, R
1a
................................................... 68 
4.2.2. Thermal Resistance of the Second Region, R
2
............................................... 69 
4.2.3. Thermal Resistance of the Third Region, R
3
.................................................. 69 
4.2.4. Thermal Resistance of the Forth Region, R
4
.................................................. 74 
4.3. Dimensionless Total Thermal Resistance of the Model, R
t
................................... 77 
4.4. Calculation of the Volume Fractions of the Model ............................................... 79 
4.5. Dimensionless Total Thermal Resistance of the Model without a Barrier............ 81 
4.6. Computer Implementation of the Analytical Model.............................................. 85 
4.7. References.............................................................................................................. 88 
 
CHAPTER 5 ..................................................................................................................... 89 
HEAT TRANSFER ANALYSIS OF NANO-MICRO FIBER COMPOSITES BY 
FINITE ELEMENT METHOD ........................................................................................ 89 
5.1. Introduction............................................................................................................ 89 
5.2. A simple FEM Model ............................................................................................ 90 
 x
5.3. Modeling with ANSYS.......................................................................................... 94 
5.4. Modeling Configurations..................................................................................... 100 
5.4.1. The Effect of Material Properties of the Filler Fiber .................................... 100 
5.4.2. Polymer?s Heat Deflection Temperature (HDT) .......................................... 103 
5.4.3. Defining the Unit Cells ................................................................................. 105 
5. 5. References........................................................................................................... 109 
 
CHAPTER 6 ................................................................................................................... 110 
RESULTS AND DISCUSSION..................................................................................... 110 
6.1. MATLAB Results................................................................................................ 112 
6.2. ANSYS Results.................................................................................................... 117 
6.3. SEM Results ........................................................................................................ 137 
6.4. Tensile Testing..................................................................................................... 146 
6.5. Air Permeability................................................................................................... 148 
6.6. Surface Tension ................................................................................................... 149 
6.7. Differential Scanning Calorimetry (DSC) ........................................................... 149 
6.8. Thermogravimetric Analysis (TGA).................................................................... 157 
6.9. Continuous Yarn Manufacturing ......................................................................... 160 
6.10. References.......................................................................................................... 164 
 
CHAPTER 7 ................................................................................................................... 165 
CONCLUSIONS AND RECOMMENDATIONS ......................................................... 165 
APPENDIX?????????????????..????????????168 
 
 
 
 
 
 
 
 
 
 
 
 
 xi
 
 
 
 
 
LIST OF TABLES 
 
 
 
3.1. Densities of PVA [3]. ???...????????????????.. 33
3.2. Nanofiber manufacturing methods [9]. ?????????????.. 38
3.3. Optimization of the experimental parameters (10% PVA water solution) 39
3.4. Effects on DSC graph areas. ?????????????????.. 59
4.1. Dimensionless length (b) values. ???????????????... 81
5.1. Thermal conductivity of some materials?????????????. 102
5.2. Some of the polymers? heat deflection temperature and melting points    
       [4]???????????????????????????? 103
5.3. Material properties and dimensionless effective thermal conductivities of   
      the hexagonal and rectangular unit cells?????????????. 106
6.1. Dimensionless Effective thermal conductivity by Rule of Mixtures. ?? 125
6.2. Fiber dimensions under SEM. ????????????????? 140
6.3. Air permeability of the samples????????????????. 148
6.4. Surface tension of the solutions. ???????????????? 149
6.5. PVA properties [4] ????????????????????... 150
6.6  Comparison of glass transition temperatures???????????.. 154
 
 
 
 
 
 
 
 
 
 
 
 xii
 
 
 
 
 
LIST OF FIGURES 
 
 
2.1 Typical structure of a composite????????????????. 6
2.2 The variance of the microcopies according to the structure size [19]??. 11
3.1 Repeating unit of PVA??????????????????. 33
3.2 The structure of Laponite? [4]????????????????? 35
3.3 Steps for continuous nano-micro fiber based web yarn manufacturing?? 36
3.4 Parts of the electrospinning setup???????????????? 37
3.5 Nano-micro fiber spread area.????????????????....... 40
3.6 Schematic shape of nano-micro fiber spread area?????????? 40
3.7 Direction of electrospinning a) The set-up of Hofman [13], b) The 
experimental set-up used in the present work???????????.. 43
3.8 Electrospinning with 15kV??????????????????.. 44
3.9 Electrospinning with 17.5 kV. ?????????????????. 45
3.10 Electrospinning with 20 kV. ?????????????????? 46
3.11 Forces acting on polymer solution [14].???????????..??. 47
3.12 Applied voltage effect on the spinning angle???????????? 48
3.13 Electrospinning direction for 15, 17.5 and 20 kV applied voltage values? 49
3.14 Image of the PVA fiber samples?...??????????????? 50
3.15 Schematic of the continuous yarn spinning made by SQNT-filled 
composite [15]???????????????????????. 51
3.16 Schematic of yarn manufacturing...???????????????.. 53
3.17 Coated fabrics. ?????????????????????......... 54
3.18 Twisting of nano fiber based web. .???????????????.. 55
3.19 Heat flow and temperature relationship in DSC [16]. ??????.......... 56
 xiii
4.1 Composite structures for the model a) Cylindrical shape, b) Rectangular 
shape???????????????????????????. 64
4.2 Hexagonal model for transverse heat conduction. A) 3-D model b) 2-D 
model (Q: heat flux.) ????????????????????? 65
4.3 Symmetric part of the hexagonal model and the region divisions???? 66
4.4 The first region???????????????????????.. 68
4.5 Schematic of the third region?????????????????? 69
4.6 Schematic of the fourth region ????????????????.? 75
4.7 Schematic of the model without barrier. ????????????...... 82
5.1 Heat conduction in a thin rod [1]??????????????..?.... 90
5.2 Number of the nodes on the element [1]?????????????... 91
5.3 The finite element division.? ?????????????????. 93
5.4 Composite structures????????????????????? 95
5.5 Radial, axial and 3-D nano-micro composite models (d: the length of the 
unit cells)?????????????????????????... 96
5.6 Necessary computer memory in modeling????????????? 97
5.7 PLANE35 2-D 6 node triangular thermal solid [3]????????.. 98
5.8 SOLID90 3-D 20 node triangular thermal solid [3]????????. 98
5.9 Mesh types of 2-D and 3-D nano-micro composite models??????.    99
5.10 Problem description?????????????????????.. 101
5.11 Mesh model of the problem??????????????????.. 101
5.12 Application of two different materials. ?????????????? 102
5.13 Temperature distribution for different materials??????????... 103
5.14 Maximum heat flux values for composite thermal stability??????. 104
5.15 Unit cells a) Rectangle unit cell, b) Hexagonal unit cell???????.. 105
5.16 Temperature distribution in the X direction for rectangular shape???.. 107
5.17 Heat flux variation in the X direction for hexagonal shape??????.. 108
5.18 Temperature distribution in the X direction for hexagonal shape????. 108
5.19 Heat flux distribution in the X direction for hexagonal shape?????.. 109
6.1. Change of dimensionless effective thermal conductivity with ? for 
 xiv
different V
d
 values, (t=0)???????????????????.. 113
6.2. Change of dimensionless effective thermal conductivity with ? for 
different V
d
 values, (t=0.1 x r
d
)????????????????? 113
6.3. Change of dimensionless effective thermal conductivity with ? for 
different V
d
 values, (t=0.2 x r
d
)????????????????? 114
6.4. Change of dimensionless effective thermal conductivity with ? for 
different V
d
 values, (t=0.3 x r
d
)????????????????? 115
6.5. Effect of the barrier thickness on the dimensionless effective thermal 
conductivity, (V
d
=0.1). ???????????????.????? 116
6.6. Effect of the barrier thickness on the dimensionless effective thermal 
conductivity, (V
d
=0.2)????????????????????.. 116
6.7. Effect of the barrier thickness on the dimensionless effective thermal 
conductivity, (V
d
=0.3)????????????????????.. 117
6.8.   k
+
e
 calculations for the hexagonal unit cell in ANSYS????????. 120
6.9. Dimensionless effective thermal conductivity (t=0). ????????... 121
6.10 Dimensionless effective thermal conductivity (t=0.1x r
d
)??????? 121
6.11 Dimensionless effective thermal conductivity (t =0.2 x r
d
). ?????? 122
6.12 Dimensionless effective thermal conductivity (t =0.3 x r
d
)?? ????. 122
6.13 Dimensionless effective thermal conductivity (V
d
=0.1)..?........................ 123
6.14 Dimensionless effective thermal conductivity (V
d
= 0.2)?? ???..?.. 123
6.15 Dimensionless effective thermal conductivity (V
d
= 0.3)??? ???? 124
6.16 Dimensionless effective thermal conductivity by the Rule of Mixtures?.. 125
6.17 Dimensionless effective thermal conductivity comparison between the 
ANSYS and Rule of Mixtures (V
d
=0.2)?????? ???????. 126
6.18 Dimensionless effective thermal conductivity comparison between the 
ANSYS and Rule of Mixtures (V
d
=0.3)???? ?????????. 126
6.19 Comparison of the dimensionless effective thermal conductivity among 
the ANSYS, MATLAB and Rule of Mixtures ( t=0 and V
d
=0.1)???? 127
6.20 Meshing for the hexagonal model???????????????? 128
6.21 Temperature distribution of the hexagonal unit cell ( t=0 and V
d
 =0.1)?.. 129
 xv
6.22 Heat flux distribution in the X direction of the hexagonal unit cell (t=0 
and V
d
=0.1)????????????????????????.. 129
6.23 Heat flux vector in the X direction of the hexagonal unit cell (t=0 and 
V
d
=0.1)??????????????????????????. 130
6.24 Thermal gradient in the X direction of the hexagonal unit cell (t =0 and 
(V
d
=0.1, k
d
=1000, k
f
=10)?? ????????????????.. 131
6.25 Heat flux in the X direction of the hexagonal unit cell (t=0, V
d
=0.1, 
k
d
=0.01, k
f
=10)???????????????????????. 132
6.26 Thermal gradient in the X direction of the hexagonal unit cell 
(t=0,V
d
=0.1, k
d
=0.01, k
f
=10)? ????????????????? 132
6.27 Thermal gradient in the X direction of the hexagonal unit cell (t=0.1 x r
d
 
and V
d
=0.1)??????????????????.??????. 134
6.28 Thermal gradient vector in the X direction of the hexagonal unit cell 
(t=0.1 x r
d
 and V
d
= 0.1)? ?????????????.????? 134
6.29 Thermal gradient in the X direction of the hexagonal unit cell (t=0.2 x r
d 
and V
d
= 0.1)????????????????????????. 135
6.30 Thermal gradient in the X direction of the hexagonal unit cell (t=0.1 x r
d
 
and V
d
=0.1)????????????????????????.. 135
6.31 Heat flux in the X direction of the hexagonal unit cell ( t=0.2 x r
d
 and 
V
d
=0.2)?...................................................................................................... 136
6.32 Temperature gradient in the X direction of the hexagonal unit cell (t=0.2 x 
r
d
 and V
d
=0.2)???.................................................................................... 136
6.33 Temperature distribution in the X direction of the hexagonal unit cell 
(t=0.2 x rd and V
d
= 0.2)??? ????????????????... 137
6.34 Web formation after 1 second of spinning????????????? 138
6.35 Web formation after 24 seconds of spinning??????????........ 138
6.36 SEM pan?????????????????????????.. 139
6.37 SEM image of nanofiber based web (10 wt % PVA, 3000x )?????. 141
6.38 SEM image of nanofiber bundle (12 wt.% PVA, 10000x)??????.. 141
6.39 SEM image of nanofibers (12 wt. % PVA, 10000x)????????.. 142
 xvi
6.40 Variety of fiber size (15 wt. % PVA, 3000x)? ??????????.. 142
6.41 Single fiber (12wt. % PVA, 10000x)? ?????????????.. 143
6.42 PVA and Laponite? under SEM (12 wt PVA + 5 wt % Laponite?, 
10000x)??????????????????????????. 144
6.43 Bead problem in electrospinning (15 wt% PVA, 1000x)??????? 145
6.44 Bead formation [2]? ????????????????????.. 145
6.45 Load-elongation curve for nano-micro fiber based yarn???????... 147
6.46 Comparison of load-elongation curves for nano-micro fiber based yarns? 147
6.47 DSC results of 12 wt % PVA?????????????????... 151
6.48 DSC results of 12 wt % PVA+ 1 wt % Laponite??????????.. 151
6.49 DSC results of 12 wt %PVA+ 2 wt % Laponite??????????? 152
6.50 DSC results of 12 wt %PVA+ 3 wt % Laponite??????????? 152
6.51 Schematic representation of unfolding of polymer chains a) Original 
polymer crystal, b) Crystal with a solvent [5]? ??????????.. 153
6.52 Glass transition temperature region for 12 wt % PVA + 1 wt % 
Laponite?..................................................................................................... 155
6.53 Glass transition temperature region for 12 wt % PVA and 2 wt % 
Laponite??????????????????????????. 155
6.54 Glass transition temperature region for 12 wt % PVA and 3 wt % 
Laponite?????...................................................................................... 156
6.55 Melting point comparison among three samples: 12 wt % PVA + 1wt % 
Laponite?, 12 wt % PVA + 2wt % Laponite?, and 12 wt % PVA + 3wt 
% Laponite?????????????????????????. 157
6.56 Weight loss comparison by TGA up to 300 ?C among three samples: Pure 
12 wt % PVA, 12 wt % PVA+ 1 wt % Laponite?, and 12 wt % PVA + 3 
wt % Laponite???.................................................................................... 158
6.57 Weight loss comparison by TGA up to 800 ?C among three samples: Pure 
12 wt % PVA, 12 wt % PVA+ 1 wt % Laponite?, and 12 wt % PVA + 3 
wt % Laponite???................................................................................... 159
6.58 Picture and schematic of the collecting mechanism????????? 161
 xvii
6.59 Aluminum pool??????????????????????? 163
6.60 Coated yarns????????????????????????.. 163
 
 1
 
 
 
 
 
CHAPTER 1 
INTRODUCTION 
 
Although the term of nano is not a new word in our lives, the usage of nano sized 
materials is pretty new as well as the attempts to model them. After discovery of carbon 
nanotubes by Iijima [1], so many new questions have arisen in the field of mechanical, 
thermal or electrical modeling. 
In terms of heat transfer modeling, it brought up a really good question: if the 
conventional way of thinking to define the heat transfer behavior is still adequate or not. 
The answer came with the definition of nano scale heat transfer and interface effect. It 
was shown that, when the free path between the molecules is small enough, the Fourier 
Law and the Boltzmann equation become inapplicable. This fact moved the researchers 
to consider and to focus on the molecular level of modeling. Beside, there is the 
importance of the interface effect. The interface is the result of a large surface area of the 
nano sized particles when these particles are dispersed inside the composite.  
When the single nano sized fiber or nanotube is considered alone, the molecular 
or atomistic modeling is a good way to solve the problem.  If one of the filler of the 
composite is nano sized, the composite is considered to be a nanocomposite and it 
becomes hard to have a common feature between nano and micro/macro size particles.  
 
 2
  Recently, so many successful application examples are seen in terms of structural, 
optical, electronics, chemical, flame retardancy properties and the like. Especially carbon 
nanotubes have been used and taken much attention during the recent years because of 
their advantage in mechanical, electrical, thermal and chemical properties in the field of 
supercapacitors, and batteries.  In opto-electronic area, oxygen manipulation is done by 
using nanotubes.  To understand the nitrogen dioxide sensing mechanism of tin dioxide 
nanoribbons, Accelrys uses carbon nanotubes [2]. In pharmaceutical industry, especially 
in drug delivery, block-copolymers help to increase the circulation time in nanoscopic 
carrier systems. In semiconductors by Motorola Inc., and Accelrys, thin films are 
developed by deposition of NO on a Si (100) surface. In automobile, electronics and 
furnishing industries, clay-polymer reinforced composites are used because of their neat 
properties. These increasing application areas highlight the nano particles embedded 
structures, which lead us to consider not only the micro size particle but also nano size 
particles.  
The thermal behavior of nano-micro composites is studied experimentally, 
analytically, and numerically. For thermal studies, thermal conductivity is chosen 
because of its direct effect on both thermal and flame resistance behavior.  Interfacial 
effects are important in nano-micro composites and can affect the composite 
performance. It is known that the interface can decrease the effective conductivity. 
Therefore, the interface area is also considered in the modeling. For analytical modeling, 
thermal resistance approach is used by considering a hexagonal unit cell to obtain the 
transverse effective thermal conductivity ratios. When  nano sized fillers is considered, 4-
5% of the volume fraction can be enough to give certain properties to the composite. In 
 3
micro sized materials, the volume fraction of filler fibers is usually more than 40%. In the 
hexagonal model, for analytical and numerical analysis, the volume fraction is considered 
to be between 10-30%.  
A novel device was designed to manufacture yarns continuously. Nano-micro 
fibers were manufactured and collected to form web and yarn. To collect the fibers, 
nonwoven fabrics were used, which allow easy release of fibers from the surface. 
Scanning electron microscope, thermal gravimetric analysis, and digital imaging are used 
to analyze the structures. Tensile strength, surface tension, and air permeability 
measurements are done.  
One of the important objectives of this study is to define the relation between 
micro and nano sized particles in terms of modeling and manufacturing. For nano scale 
structure modeling, the current studies include the molecular and atomistic modeling; for 
the micro or macro level, the conventional methods are considered. What if a nano-micro 
combination structure is used? This necessitates the discussion of the linkage between 
nano and micro levels in this study. 
 
1.1. References 
 
1. Iijima, S., ?Helical Mictotubules of Graphitic Carbon?, Nature, 1991, 354, p56. 
2. http://www.accelrys.com/chemicals (accessed in Jan 2005) 
 
 
 
 
 
 
 4
 
 
 
 
 
CHAPTER 2 
LITERATURE REVIEW 
 
2.1. Nano-micro (NM) Composites  
Wool, cotton, chocolate chip cookies, sea, ceramics, clouds, fogs, rain are all 
composites. When one looks around, it is hard to find a material which is not composite. 
Composite as a term is described by Milton as:  ?Composites are materials that have 
inhomogeneous on length scales that are much larger than the atomic scale but which are 
essentially homogenous at macroscopic length scales or at least some intermediate length 
scales [1].  
Why do we study composites? This has so many answers. For Poisson it was a 
tool to describe the magnetism by assuming the body composed of conducting spheres; 
for Maxwell, it was a tool to solve the conductivity equation in a conducting matrix; for 
Einstein it was a tool to calculate the effective shear viscosity of a suspension of rigid 
spheres in a fluid.  
A new area recently appeared in our lives is ?Nanocomposites?. The reason why 
the composites are considered nanocomposites is due to the fact that one or more of the 
components is in a dimension of nano (10
 -9 
m). Generally, nanocomposites could be 
classified into three groups: one dimensional, two dimensional, and isodimensional. The 
 5
group of the nanocomposite is contingent on the number of dimensions that the dispersed 
particles of the composite possess. For example, spherical silica nanocomposite is 
isodimensional; whereas nanotube, nanocarbon or nanofiber embedded structures are two 
dimensional. In addition, the filler in the sheet form that has a nanoscale thickness could 
be given as an example for one dimensional group [2]. 
Nanocomposites captured a lot of attention recently because of their unique 
properties. Gilman et al., investigated polymer-clay composites and found that if the 
mechanical properties are considered, by just adding 5% of silicate mass into the nylon-6, 
the mechanical properties make significant progress like 40% higher tensile strength, 
68% greater tensile modulus, 60% higher flexural strength, etc. On the other hand, heat 
distortion temperature (HDT) increases from 65 ?C to 126 ?C. Gas permeability decreases 
because of the barrier properties [3]. 
The first report related to polymer-clay composites goes back to Blumstein?s 
research in 1961 [4]. He demonstrated polymerization of vinyl monomers intercalated 
into montmorillonite (NMT) clay.  There are different methods that one could use to 
prepare nanoclay based composites such as the polymerization (in situ method), solvent-
swollen polymer (solution blending), or polymer melt (melt blending).  
Polymer-clay composites have two forms: intercalated and delaminated. 
Intercalated structures were named because of their self-assembled, well-ordered, and 
multi-layered attitude. The individual silicate layers could be 2 nm or 3 nm. In the 
laminated structure, the individual silicate layers are no longer close enough to interact 
with the adjacent layers.  
 6
Because of their unique properties, the application areas of nanocomposites are 
expanding day by day. Solar absorbers can be given as an example, in which silica-
carbon nanocomposites are used. The reason to use carbon nanocomposites inside solar 
energy absorbers is that they absorb more radiation in the solar region [5]. 
Nanocomposites are also used in automotives as barrier packaging, polyethylene pipe and 
wire/cable coatings and more [6]. 
 
2.2. Interface Effect 
Interfaces are found in almost everything. There is an interface even between an 
ocean and the air. One-dimensional, two dimensional (much common) and three 
dimensional interfaces could occur in the nature. Between the air and the ocean, one can 
see also different dimensions. The interface could occur among three material phases 
such as solid, liquid, and gas. Therefore it is apparent that it is hard to imagine a problem 
without interface effect including nanostructures (Figure 2.1) [7]. 
 
 
Figure 2.1. Typical structure of a composite 
 
 7
In nano structures, interface bears even more importance since what makes 
nanostructures unique can be mostly explained by the effect of interface among 
nanofillers such as fiber, tube, clay, matrix, and the like.   
To understand the behavior of interfaces, it is necessary to obtain either 
experimental or simulation results. Studies indicate that considerable amount of 
information could be gathered from the simulations because analytical explanations do 
not suffice. Despite the research studies conducted, there is still room for a clear 
definition of interface in nanoscale. What do we see at the interface? What kind of 
bonding could appear? What kind of topology, transport across it, deformation, chemical 
activities, and forces could appear? These are some of the questions to be addressed. 
Even if we find answers to these questions, we still may not be able to explain the effect 
of the interface sufficiently [8]. 
Van Assche and Van Mele, discussed the effect of interphase.  They  emphasized 
that macroscopic proporties of composite materials, including fiber-reinforced polymers, 
blends, and multilayer systems, were often strongly affected by the develepment of 
interphase regions with properties differing from the properties of the constituent 
materials. Interphases could arise due to preperential adsorption, catalytic influences of a 
surface, inter-diffusion, phase separation, etc. The resulting gradients in composition 
(polymer blends) or crosslink density (thermosets) lead to graidents in the microscopic 
properties [9]. 
The load transfer from the polymer to nanotubes depends on three factors: the 
interaction between the nanotubes and polymer, micro-mechanical interlocking between 
nanotubes and polymer, and the chemical bonding between nanotubes and polymer. If the 
 8
nanotubes have chemical stability, the effect of the chemical bonding could be negligible. 
Studies regarding the effect of interface in nanosize structures started in the last 10 years 
[10].  
Lopattananon et al., emphasized the importance of micromechanical test to 
characterize the mechanical properties of the interface. They used single-fiber 
fragmentation method because of its simplicity. This method applies a tensile force to a 
specimen with a single embedded fiber in a thin resin test piece. They also claim that the 
term of good interface is one which transfers the highest fraction of the applied load in 
the form of a stress to the fiber [11].  
In another study, Van Assche and Van Mele used micro-thermal analysis to study 
the interphases in a particle-filled composite based on an epoxy resin matrix. In their 
analysis, they combined the advantage of thermal analysis (characterization) and 
microscopy (visualization). They also studied spatial variations in glass transition and 
melting behavior by performing local thermal analysis. To detect the interface by micro-
TA, the first step to be taken is to map the surface to find topography and thermal 
properties. Then for characterizing the interface close to particles, local thermal analysis 
is used. Thermal probe is embedded in the composites and moved. Thus the change in the 
properties is presented [12]. 
Naslain explained the importance of the interfacial part [13]. It controls the crack 
deflection, load transfer, diffusion barrier, and residual stress relation. Naslain mentioned 
one of the classical approaches to design fiber-matrix (FM) interfacial zone in ceramic 
matrix composites (CMCs) from a processing standpoint which is through the in situ 
 9
formation of a weak interphase resulting from some chemical reaction at the FM interface 
during the high temperature step of composite processing.  
In some models, it was assumed that interface has no thickness. Some researchers 
indicated that the interface could not be homogenous and has a thickness less than 1 ?m. 
Also it is emphasized that there is a discrepancy in the interphase, and sometimes 
estimation could be used for its properties.  
In micro level, the length of the interface could be in nanoscale. In nanoparticle 
embedded composites, it may be even smaller, which makes it difficult to measure.  
There are two main functions of interfaces for ceramic matrix composites: the 
first one is to act as a mechanical fuse and to maintain a good load transfer between the 
fibers and the fabric. In addition, in very reactive systems, the interphase could act as a 
diffusion barrier [13].  
In modeling, Theocharis and Panagiotopoulos presented a novel concentric 
cylindrical model for growing phase of the interphases. They also showed the properties 
as a function of radial distance from the fiber. It is suggested that some test methods 
could be used to measure the mechanical properties; however, for thermal analysis, there 
were no direct methods until 2001. After the development of the atomic force 
microscopy, new concepts appeared in the interfacial area and measuring the nanoscale 
thermal properties of the materials at nanoscale became possible. They used thermal 
atomic force microscopy to evaluate the interface. It was also shown that, epoxy resin, 
curing agent and fiber systems are important factors for interphase properties [14].  
Interfaces are important because, during the manufacturing of composites, large 
number of cracks could appear especially at the interfaces which could cause weak 
 10
bonding. For stress distribution of these cracks, the interfaces were considered as an 
extension of the matrix. There are some studies in which these effects are investigated by 
finite element method or boundary element method [15].  
Interphase may not be as important in some situations as was shown by Ash et al.  
In this study, the effect of the interface was compared to the effect of the separation of the 
blade or other vise angle in microband test, and it was proposed that the interface is not as 
important as the vise angle [16].  
The shape of the interface is not often regular. Lipscomb and Xomeritakism 
analyzed the mass transfer in composite materials that have irregular interfaces. In some 
considerations, this irregularity is assumed linear. The irregularity gives much 
permeability to the composites [17]. 
Pegoretti et al., studied the toughness of the fiber matrix interface in nylon-6/glass 
composites experimentally.  They also used finite element modeling to measure the 
interfacial bonding [18].  
Until the 1980s, information on the sub-micrometer scale length was accessible 
only using the indirect techniques such as electron or X-ray diffraction or with electron 
microcopies which required vacuum environment and conductive materials. The 
inventing of Scanning Tunneling Microscope (STM) in 1982 was changed this. It was 
aimed to generate real-space images of surfaces with a resolution on the nanometer scale. 
The Atomic Force Microscope (AFM), known also as Scanning Force Microscope (SFM) 
came after this invention. Thus, it became possible to investigate the insulating materials 
such as polymers and biomolecules [19]. The variance of the microcopies can be seen in 
Figure 2.2. 
 11
 
Figure 2.2. The variance of the microcopies according to the structure size [19]. 
In nanosized structures, it is known that interface improves the material properties 
including structural, thermal, and mechanical resistance. Wu et al., determined that, by 
adding nanosized clays into the polymer, because of the interaction between the matrix 
and clay interface, the mechanical properties increase [20].  
Chemical characteristic of the surface and the interface of TiO
2
-muscovite 
nanocomposites were studied by Song et al. They used scanning electron microscopy 
(SEM), transmission electron microscopy (TEM) and X-ray photoelectron spectroscopy 
(XPS). In this study, it is emphasized that the crystallization of the structure started at the 
interface [21]. 
To enhance the understanding of solid-solid interface thermal conductance, 
thermal transport through highly perfect interfaces between epitaxial TiN and single 
crystal oxides were measured. Thermal conductance plays an important role in the 
thermal transport area in nano scale structures. Thermal modeling is done to find the 
thermal conductance. The thermal conductance was modeled by dividing the lock-in-
phase component of the lock-in-signal by lock-out phase component of the lock-in-signal. 
It can be seen from the calculations that lateral heat flow could be negligible but radial 
 12
heat flow can not be. The interface disorder in samples produces strong phonon scattering 
at the interface. The interface disorder in all samples is weak and transmission coefficient 
is always close to unity. Diffuse mismatch model and lateral dynamic calculations can be 
used to compare with the model [22]. 
2.3. Electrospinning 
Electrospinning, as described by Shin as ?a garden hose, whipping around the 
water squirts out one end? is not a new method [23]. It has been known since 1934 by the 
approval of several patents [24-27].  
A high voltage is used to spin an electrically charged jet of polymer solution or 
melt, which dries or solidifies to leave a polymer fiber [28-29]. One of the electrodes is 
placed into the spinning solution and the other one is attached to the collector. Capillary 
tube which contains the polymer fluid is subjected to the electrical field by its surface 
tension. This makes a charge on the surface of the liquid and the surface of the fluid is 
seen in the form of a conical shape known as Taylor cone. When the electricity in the 
field is increased, a critical value is attained, the repulsive electrostatic force overcomes 
the surface tension and a charged jet of fluid is ejected from the tip of the Taylor cone.  
2.4. Thermal Properties of Nano-Micro Composites 
  In conventional heat transfer and fluid flow concepts, fluid phase is considered 
continuous by including properties such as thermal conductivity, pressure or domain 
temperature. In the nano-scale, the continuum treatment does not hold  because the 
domain is not enormously bigger than moleculer scale and therefore the Fourier?s law is 
 13
not applicable. Thus explanation of the domain has to contain the collection of the 
molecules.  
2.4.1. Thermal Conductivity in Macro Level 
Polymers can have an interaction with different kind of fillers and these fillers can 
be in the form of fibers. One of the important properties necessary to define the behavior 
of the composites is thermal conductivity.  It is known that the thermal conductivity of a 
material has a direct relationship with the flammability of the material.  
There are theoretical studies to explain effective thermal conductivity such as 
using the two phase mixtures concept, etc.   
As a simple approach for unidirectionally aligned model, the components are 
thought to be arranged in layers and placed in serial or parallel to heat flow. Thus the 
thermal conductivity is calculated as follows: 
For the series model: 
 
??
fm
fm
c
kk
kk
k
+?
=
)1(
       (2.1) 
For the parallel model: 
k
c 
= ?k
f
 + (1-?)k
m                                                       
(2.2)  
 
where, k
c
, k
f
 and k
m
 are respectively the thermal conductivities of the composite, filler 
materials and polymer matrix, and ? is the volume fraction of fiber. 
Maxwell obtained a relationship for the conductivity of randomly distributed 
fibers as follows [30]; 
 14
                      
)k(k.2.kk
)k(k..22.kk
kk
mfmf
mfmf
mc
???+
??++
=                         (2.3) 
where ? is the volume fraction of the fiber.  
Lewis and Nielsen [31] derived a model by modification of Halpin-Tsai equation 
and including the effect of particle structure.  
There are some analytical and numerical models in macro-size composites, but 
these models mostly focus on either random short fibers or aligned short fibers. 
    
..1
.A.1
kk  
mc
????
??+
=               (2.4) 
Where; 
?
?
??
+=?
+
?
=?
2
m
m
mf
mf
1
1   and     
Akk
1kk
     (2.5) 
Where ?
m
 is the packing factor, A is a factor that depends on the shape and orientation of 
the particles.  
Hatta and Taya, by using equivalent inclusion method for steady state heat 
conduction, discussed the effective thermal conductivity, showing the importance of the 
interaction between the fiber orientation and composite. To do this, some numerical 
methods were used which considered the volume fraction, fiber aspect ratio and 
distribution as factors [32]. 
When the fibers are arbitrarily placed, Nan and Birringer introduced a method to 
determine the effective thermal conductivity (k
e
) of the composite by combining the 
interfacial contact resistance with Kapitza thermal contact resistance [33]. 
 15
Esparragozaa et al., gave a simple analytical approach to define temperature 
distribution in a cylindrical domain which consists of a single fiber inside. The principle 
of conservation of energy was used for the boundary layer [34]. 
Hasselman and Donaldson worked on the inclusion size and discussed the effect 
of thermal conductivity; the thermal barrier effect was demonstrated experimentally [35]. 
To measure the thermal conductivity of fiber reinforced composites, Sweeting and 
Liu had developed a model by applying a thermal gradient to the composite and 
measuring the heat flow [36]. 
Islam and Pramilam presented a model to predict the transverse thermal 
conductivity of composites including the effect of interface by using FEM [37]. 
 
2.4.2. Thermal Conductivity in Nano Level 
In contrast  to mechanical studies, there is not much study in the field of heat 
transfer at the nano level. The existent studies focus on thin films, carbon nanotubes and 
their derivatives.  
It is clear that the nano scale heat transfer differs from the Fourier law because of 
the boundary and interface scattering and the finite relaxation time of heat carriers. It is 
shown by Cahill et al., that there is a linear connection between interface density and 
thermal conductivity in Si and Ge superlattices [38].  
In single walled carbon nanotubes (SWCNs), thermal transport is explained by the 
phonons.  SWCNs has a thermal management in  high performance. Phonons dominate 
thermal transport at all temperatures in carbon materials. 
 The phonon thermal conductivity could be demonstrated as 
 16
?=C
p
*?
s
* l.       (2.6) 
 In this equation C
p
 is the specific heat, ?
s
 is the speed of the sound and l is the mean free 
path. This equation is applicable for both one-dimensional phonon systems and SWNTs 
[39]. 
To study the size effect in nano level, Boltzmann equation could be used with 
some limitations. These limitations are because of the geometries such as thin films and 
superlattices. The transient ballistic diffusive equation is applied to study two 
dimensional non local, phonon transport phenomena [40]. Beside this, Chen used the 
ballistic diffusive heat conduction equations under the relaxation time. The ballistic-
diffusive equations are used because of the suitability for fast heat conduction process 
and also small structures. It can be used to measure transient heat conduction [41].  
The measurement methods become more important for nanoscale metrology such 
as the 3? method, time domain thermoreflectance, sources of coherent phonons, micro-
fabricated test structures and scanning electron microscope.  
Cahill et al., summarized the thermal studies in nanostructures in a critical review 
about the thermal transport [42]. Although this work focused on electronic devices, it 
gives a brief understanding about thermal transport. Mirmira and Fletcher summarized 
most of the studies related to thermal transport in nano scale. Several studies related to 
thermal conductivity of thin films were given [43]. Several studies exist regarding the 
effective thermal conductivity modeling for thin films. When the experimental and 
theoretical models are analyzed, it becomes clear that, there are many assumptions in the 
models and they are only for some kind of specific structures or materials, and only in the 
steady state situation, neglecting other cases. There is not an available model which could 
 17
be used for different kind of structure types. Some of the leading studies in nano thermal 
areas is summarized below. 
  The first indication of thermal stability improvement in nanocomposites appears 
in the work by Blumstein [4] who studied the thermal stability of poly(methyl 
methacrylate) (PMMA) intercalated within montmorillonite. Che and Cagin studied the 
thermal conductivity of nanotubes and they found that the carbon nanotubes had very 
high thermal conductivity comparable to diamond crystal and in-plane graphite sheet. 
They also showed that nanotube bundles had similar properties to graphite crystal in 
which dramatic difference in thermal conductivities along different crystal axis was 
observed [44]. Heat capacity of carbon nanotubes was studied by Benedict and his co-
workers to find out how the heat capacity of small nanotubes could be measurably 
different than that of bulk graphite. They predicted that all single-walled tubes should 
have sufficient heat capacity [45]. The heat conduction in finite length single-walled 
carbon nanotubes (SWNTs) was simulated by the molecular dynamics method with the 
Tersoff?Brenner bond order potential; the thermal conductivity was calculated from the 
measured temperature gradient and the energy budgets in phantom molecules by Shigeo 
Maruyama [46,47]. On the other side, for finding thermal expansion and diffusion 
coefficients of carbon nanotube-polymer composites, classical molecular dynamics 
simulations are used. 
Ando studied electronic and transport properties by using experimental methods.  
He demonstrated that carbon nanotubes (CNs) are important in terms of transport 
properties and this is the result of their extraordinary structures [48]. 
 
 18
2.4.3. Flame Resistance 
To protect from fire and to survive in a place which shows huge temperature 
difference is of vital importance for life. The factors related to fire hazards can be 
explained by ignitibility, ease of extinction, flammability of the generated volatiles, 
amount and rate of heat released on burning, flame spread, smoke obscuration and smoke 
toxicity [49].  
Although the leading nano studies begin with the study of Iijima in carbon 
nanotubes [50], the flammability studies of nano scale clay embedded composites start 
with Gilman et al., after 1995 [51] and focus on experimental work and simulations.  
There are some important factors in flame resistance behaviors. 
Heat release (HR) is important because it causes a fast ignition and a flame spread.  HR 
helps to control the fire intensity and is therefore much more important than the 
ignitability. Also it is underlined by the National Institute for Standards and Technology 
(NIST) as a single important factor.  
The importance of the new developments is emphasized by Gilman et al. on 
organic treatments of montmorillonite. The flammability of thermoplastic and thermoset 
polymer layered silicate nanocomposites were investigated. It was found out that peak 
and average heat release rates (HRR) were significantly improved. The main difference 
between the pure vinyl esters and the nanocomposites is the mass loss rates. They also 
reported that heat combustion, and carbon monoxide yields do not change [51]. 
Thermogravimetric analysis (TGA) characterizes the thermal stability of a 
polymer. The mass loss because of the decomposition can also be determined as a 
function of temperature. 
 19
Cone Calorimeter (CC) is used to measure HR. It uses the relationship between 
the oxygen consumed from the air and the amount of heat released during the polymer 
degradation. Using CC allows to measure heat release rate (HRR) and carbon monoxide 
ratio. HRR is important to evaluate fire safety [52]. Five properties could be measured by 
CC: 
- Peak Heat Release (PHR) 
- Mass loss rate (MLR) 
- Specific extension area (SEA) 
- Ignition Time (T
ign
) 
- Carbon monoxide and carbon dioxide yield.  
 
One important advancement in the flammability studies is clay treatment. It affects 
the thermal stability also. Nyden and Gilman, discussed the importance of clay treatment 
in their study.   
An important factor aspect in nanocomposites is the relation between flammability 
and physical properties. It was underlined that, nano-dispersed montmorillonite causes 
non-char forming polymers. The dispersed clay helps with insulating and therefore the 
flame resistance increases. Nyden and Gilman used molecular dynamic methods to 
understand the flammability reduction. They used experimental measurements for 
comparison. The comparison was done according to mass loss which was calculated as a 
function of the distance of separation between the graphite sheets. As a molecular 
dynamic method they used MD_REACT method which is based on Hamilton?s equations 
[53]. 
 20
  In another study, experimental results showed that the rate of mass loss from 
polymer-clay nanocomposites exposed to fire-like heat fluxes is significantly reduced 
from the values observed from the immiscible composites containing the same amounts 
of polymer and clay [54]. 
Morgan et al., prepared nanocomposites of polypropylene -graft- maleic 
anhydride with organically modified clays. They studied the combustion behavior and 
observed synergy between the nanocomposite and conventional phase fire retardants 
[55]. 
Zhu et al., prepared three organically modified clays and used them to produce 
nanocomposites. They investigated the behavior using X-ray diffraction and transmission 
electron microscopy. For characterization, thermogravimetric analysis and cone 
calorimeters were used [56].  
The flame resistance of carbon nanotubes was investigated by Vander Wall and 
Hall. They used a configuration to demonstrate the flame resistance. They prepared 
intercalated, exfoliated and mixture of both forms [57].  
Carbon nanotubes have unique properties; however it is difficult to functionalize 
them because of their basal plane sites. Also this plane is not accessible by interstitial 
lattice sites for intercalation. However this is not seen as a disadvantage in nanofibers 
because they are composites of short carbon segments [58]. 
The flammability studies mostly focus on polymer clay nanocomposites. Morgan 
et al., used polystyrene clay composites to define the flammability properties in 
nanoscale. They obtained intercalated polymer-clay nanocomposites. They obtained a 
lower ratio of HRR and found that the total burn time was increased [59]. 
 21
Gilman and his co-workers used both polystyrene and polypropylene 
nanocomposites [54]. 
Nyden and Gilman, established a new processing capability for variable 
composition samples which are extruded, analyzed and burned on the same device. This 
allows fore easy measurements from one sample to define the flammability properties 
[53].  
  
2.5. Modeling 
In macroscopic problems, for heat conduction in a very thin film of solid atoms, 
the thermal energy is in the form of potential energy. Therefore the heat conduction is an 
interaction between atoms and molecules.  In a very thin film, thermal conduction 
depends on the dimension of the space domain. If the domain is large enough, one can 
see kinetic and potential interactions. The linear relationship between heat flux and 
temperature gradient is linear [60]. 
According to molecular dynamics approach, which is assumed to be the best 
approach for understanding the nano phenomena, there are three different techniques for 
measuring thermal conductivity in nano scale structures. The first one is equilubrium 
molecular dynamics which is based on Green-Kubo?s formula; the second one is non-
equilubrium formula and the last one is non-equilubrium molecular dynamics with direct 
temperature difference. 
Osman and Srivastava calculated the thermal conductivity of single-walled carbon 
nanotubes over a temperature range of 100-500 K using molecular dynamic calculations. 
 22
They used Tersoff-Brenner potential for carbon-carbon (C-C) interactions [61]. Berber et 
al., combined the equilibrium and non-equilibrium molecular dynamic simulations to find 
the unusually high thermal conductivity: 6600 W/mK [62].  
When the correctness of the Fourier?s law is discussed, the usage of this law in 
nano scale is also questioned. In molecular dynamics calculations, this law could be 
applicable only for non-equilibrium calculations. Because of the relation between the heat 
diffusion length and the time for diffusion,  Fourrier?s law could leave its place totaly to 
the molecular dynamics calculations and to the ultra short time scale studies. 
In an another study, it was shown that for the heat conduction of thin films, the 
temperature gradient increases with respect to the heat flux and the average temperature. 
The thickness and the initial temperature do not play a role [63]. By using two parameters 
such as the positional order parameter and the kinetic H-function, the equilibration of 
heat conduction simulations for thin films was investigated. This method also could be 
applied for macroscopic molecular dynamic simulations to investigate the heat transfer 
behavior.  
Nanostructures such as superlattices and quantum wires give the researchers a 
chance to develop alternative methods for electron and phonon transport processes. Some 
of the structures want lower thermal conductivity and higher electrical conductivity such 
as thermionic refrigeration and power generation operations. Also nano-clay embedded 
structures can be given as an example in which lower thermal conductivity is wanted. In a 
study by Chen et al., these structures for solid-state energy conversations were discussed 
[64]. Thermal conductivity was explained by Ren and Dow in the area of superlatices 
[65]. 
 23
It is pointed out by Cahill et al., that, when the interface density increases, the 
heat conduction decreases. This is an important point to define the difference between 
nano and macro scale structures [38].  
  Much of the early work studying the mechanical properties of nanotubes utilized 
computational methods such as molecular dynamics. These models focused primarily on 
Single Walled Carbon Nanotubes (SWNTs) because of the increase in computational 
resources necessary to model larger systems. 
  Nyden and Gilman performed molecular dynamics simulations for the thermal 
degradation of polymer nanocomposites in an attempt to explain the reduction in the 
flammability of nano-confined polypropylene as compared to the pure polymer [53]. 
Laplaze et al., studied the carbon nanotubes to understand the dynamics of synthesis 
processes. He obtained heat and mass transport in a solar reactor using ?in situ? 
measurements linked to numerical simulation [58]. 3-D finite element modeling was used 
to study the influence of nonlinear response of organic components. In another work, 
nonlinear elasto-plastic models for the organic component are applied to model the 
mechanical response of nacre (mother of pearl). Nanoscale material parameters (elastic 
modulus and hardness) were obtained using nanoindentation experiments. Fisher et al. 
predicted the modulus using effective nanotube properties. In their study they used 
ANSYS program as a finite element method. They used the micromechanical techniques 
to study the effective elastic moduli of nanotube-reinforced polymers [66]. 
To deal with nanosize, one should resort to modeling in both atomistic and 
molecular level. Moreover, in heat transfer and fluid flow treatments, as the size of the 
flow domain approaches to nano level, the continuum treatment breaks down since the 
 24
fluid consists of molecules and, when the flow domain size is no longer enormously 
greater than molecular scale, the fluid must be considered in terms of collections of 
molecules. 
As an atomistic model, Brenner developed an empirical model for nanotubes that 
depends on Tersoff?s covalent bonding [67]. Hannson et al., used the molecular dynamic 
simulations to get the electronic behavior of the carbon nanotubes [68].  
 
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 27
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 28
35. Hasselman, D. P. H., and Donaldson, Y. J., ?Role of Particle Size in the Effective 
Thermal Conductivity of Composites with an Interfacial Thermal Barrier?, 
Journal of Wide Band Gap Mat., 2000, 7(4), 306. 
36. Sweeting, R. D., and  Liu, X. L., ?Measurement of Thermal Conductivity for 
Fiber-reinforced Ccomposites? Composites Part A 2004, 35A (7-8), 933-938. 
37. Islam Md. R, M. Pramila, A., ?Thermal Conductivity of Fiber Reinforced 
Composites by the FEM?, 1999, J. Comp. Mater, 33, 1699. 
38. Cahill, D. G., Ford, W.F., Goodson, K.E., Mahan G.D., Majumdar, A., Maris, 
H.J., Merlin R., and Phillpot S.R., ?Nanoscale Thermal Transport?, Journal of 
Applied Physics, 2003, Vol 93, No 2., 793-818. 
39. Biercuk, M., Llaguno, M. C., Radosavljevic, M., Hyun, J. K., Fischer, J. E. and 
Johnson, A. T., "Carbon Nanotube Composites for Thermal Management" 
Applied Physics Lett.,2002, 80, 2767- 2769. 
40. Yang, R., and Chen, G., ?Two-dimensional Nanoscale Heat Conduction Using 
Ballistic-Diffusive Equations?, presented at IMECE 2001, New York. 
41. Chen, G., ?Ballistic-Diffusive Equations for Transient Heat Conduction From 
Nano to Macroscales?, Journal of Heat Transfer, 2002, 124(2), 320-328. 
42. Cahill, D., Goodson, K., and Majumdar, A., ?Thermometry and Thermal 
Transport in Micro/nanoscale Solid-state Devices and Structures?, Journal of 
Heat Transfer, 2002, 124, 223. 
43. Mirmira, S. R., and Fletcher, L. S., ?Review of the Thermal Conductivity of Thin 
Films?, Journal of Thermophysics and Heat Transfer, 1998, Vol.12, No.2, 121-
131. 
 29
44. Che, C., and Cagin T., ?Thermal Conductivity of Nanotubes?, Nanotechnology, 
7
th
 Conference on Molecular Technology, 1999, California.   
45. Benedict, L. X., Louie, S. T., and Cohn, M. L.,?Heat Capacity of Carbon 
Nanotubes?,  Solid State Commun., 1996,100, 177 
46. Maruyama, S., "A Molecular Dynamics Simulation of Heat Conduction of Finite 
Length SWNTs", Physica B, 2002, 323, 1-4, 193-195.  
47. Maruyama, S., "Molecular Dynamics Method for Microscale Heat Transfer", 
Advances in Numerical Heat Transfer, 2000, 2, 189-226. 
48. Ando, T., ?Theory of Transport in Carbon Nanotubes?, Semicond. Sci. Technol., 
2000, 15, 12-27. 
49. Beyer, G., ?Nanocomposites: a New Class of Flame Retardants for Polymers?, 
Plastics Additives & Compounding, 2002, 4(10), 22-28. 
50. Iijima, S., ?Helical Microtubules of Graphitic Carbon?, S Nature, 1991, 354, p56. 
51. Gilman, J. W., Kashiwagi, T., Nyden, M. R., Brown, J. E. T., Jackson, C. L., 
Lomakin, S. M., Giannelis, E. P., and  Manias, E., ?Flammability Studies of 
Polymer Layered Silicate Nanocomposites: Polyolefin, Epoxy, and Vinyl Ester 
Resins?, Chemistry and Technology of Polymer Additives, 1999, 14, 249-265. 
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Important Variable in Fire Hazard?, Fire Safety Journal, 1992, 18, 255-261. 
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Degration of Nano-confined Polypropylene?, Flammability of Polystyrene-Clay 
Nanocomposites?, Comp. Theor. Polymer Sci., 1997, 7,191-198. 
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54. Gilman, J. W., Jackson, C. L., Morgan, A. B., Harris, R. H., Jr., Manias, E., 
Giannelis, E. P., Wuthenow, M., Hilton, D., and Phillips, S. H.,?Flammability 
Properties of Polymer-layered-silicate Nanocomposites, Polypropylene and 
Polystyrene Nanocomposites?, Chem. Mater, 2000, 1866-1873.  
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Polystyrene-clay Nanocomposites?, Polymeric Materials Science & Engineering 
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Nanocomposites?, Chem. Mater., 2001, 13, 3774-3780. 
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walled Carbon Nanotubes and Ni Catalyzed Nanofibers: Growth Mechanisms 
and Consequences?, Chemical Physics Letters, 2001, 349,178-184. 
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Nanotubes: Dynamics of Synthesis Process?, Carbon,  2002, 40, 1621-1634. 
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Zhu, J., ?Flammability of Polystyrene-clay Nanocomposites?, Polymer. Mater. 
Sci. Eng., 2000, 83, 53-54. 
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Domain?, Purdue University, 2001. 
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Conductivity of Single-walled Carbon Nanotubes?, Nanotechnology, 2001, 12, 
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Carbon Nanotubes?, Physical Review of Carbon Nanotubes, 2000, 84, No20. 
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Conversion Applications?, Proc. Energy conversion and Applications, 2001, 1, 
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7639-7644.  
 
 
 
 
 
 
 
 
 
 
 32
 
 
 
 
 
CHAPTER 3 
NANO-MICRO FIBER BASED FILM, WEB AND YARN MANUFACTURING 
 
In this chapter, experimental studies are explained for nano-micro fiber based 
film, web and yarn manufacturing. Thus, electrospinning, electrical field, optimum 
properties of spinning, coating, differential scanning microscopy (DSC), thermo 
gravimetric analysis (TGA), scanning electron microscope (SEM), air permeability and 
surface tension of the fabric are discussed. 
3.1. Materials 
3.1.1. Polyvinyl Alcohol (PVA) Properties 
The reason why PVA is selected in this work is because polyvinyl alcohol based 
fibers have been used for a long time as a reinforcement for polyester resins and the 
solution preparation does not take much time [1]. Besides, PVA provides high polarity 
and hydrogen bonding. This makes PVA infusible but soluble in water. Commercially, 
PVA is classified in two groups: partially and fully hydrolyzed PVA, which depends on 
the amount of acetate groups they have.  
PVA is resistant to oils, fats, greases and it provides strong adhesion for paper and 
textiles. The chemical structure of PVA is seen in Figure 3.1 [2]. 
 33
  
Figure 3.1. Repeating unit  of PVA. 
PVA?s density may change depending on its amorphous and crystalline structure. 
As seen in Table 3.1, the density of the amorphous part of the polymer (?
a
) is around 1.26 
g/cm
3
 and the density of the crystalline part of the polymer (?
c
) is around 1.35 g/cm
3
. 
This gives a ratio of ?
c
/?
a 
=
 
1.07.  
                      
Table 3.1. Densities of PVA [3]. 
 
?
c
 (g/cm
3
) ?
a
 (g/cm
3
) ?
c
/?
a
 
PVA 1.35 1.26 1.07 
 
Thermal expansivity is an important thermodynamic property and it is described 
mostly as a ratio between the pressure (P), volume (V) and temperature (T). This is also 
important in terms of thermodynamics to give an explanation about the equilibrium state 
of a system [3]. To measure PVA?s specific thermal expansivity, the following equation 
is used; 
e
T
p
?
?
?
?
?
?
?
?
??
   (cm
3
/g?K)       (3.1) 
where e is the thermal expansivity, T is temperature, ? is the volume, and P is the 
constant pressure.   
The temperature coefficient of density: 
 34
q
T
p
?
?
?
?
?
?
?
?
??
 (g/cm
3
?K)      (3.2) 
The coefficient of thermal expansion is another important parameter in heat 
transfer. Its accuracy is important for design, especially in microfilm production.  It 
basically gives the fractional change in volume for a given unit change of temperature 
and can be given as follows: 
??
?
?
?
?
?
?
?
??
?
p
T
1
  ( ?K
-1
)       (3.3) 
The coefficient of thermal expansion can be used in two ways: as a volumetric 
thermal expansion coefficient and as a linear thermal expansion coefficient.  The linear 
coefficient of thermal expansion is used more often. It is simply described as the 
fractional change in length of a bar per degree of temperature. 
??
?
?
?
?
?
?
?
?
p
T
L
L
1
  (?K
-1
)       (3.4) 
In the present study, the PVA is provided by Sigma-Aldrich Co., MI, in the 
powder form. The average molecular weight is 70,000-100,000. The PVA was gradually 
added to the distilled water while stirring until 12-15% weight ratio was obtained. Then 
heating took place about 4 hours and it was let to cool down. By using a centrifugal 
device, the solution was rotated 5 minutes to reduce air bubbles. Another PVA, obtained 
from ACROS ORGANICS, was also used. It was 95% hydrolyzed with an average 
molecular weight of 95,000.  
 35
3.1.2. Laponite? Properties 
Laponite? is a nano clay but compared to the natural clays (hectorite, 
bentonite?), it has much smaller dimensions (Figure 3.2). In this study, the Laponite? 
was provided by DH Litter Company. Normally in a Laponite? particle there are 
between 30000-40000 unit cells. These unit cells are combination salts of sodium, 
magnesium and lithium with sodium silicate. The chemical formulation of the Laponite? 
is;  
 ()( )[]
7.0
4203.05.58
7.0 OHOLiMgSiNa
?+
      (3.5) 
The surface has a negative charge and the edges have small localized positive 
charges. Laponite? has three metal cations mostly Mg
2+
, in a half unit cell.  
 Laponite? is prepared similar to the PVA in our study. Laponite? is added to the 
PVA solution as a ratio of 1/3 wt%. It was mixed with the PVA before dissolving it 
inside the distilled water. The heating time was decreased for stirring because of high 
surface area.  
 
Figure 3.2. The structure of Laponite? [4]. 
 
 36
3.2. Continuous Nano-Micro Fiber Based Yarn Manufacturing 
 
Figure 3.3. Steps for continuous nano-micro fiber based web yarn manufacturing. 
 
Electrospinning or ??electrostatic spinning?? is a well known method for 
producing fibers with a size down to nano scale.  Although it is considered to be a new 
method for nano scale, the electrospinning method as an idea goes back to almost 60 
years. The first attempt to produce polymer filaments by using an electrostatic force was 
made in the 1930s by Formhals [5]. He used a polymer solution such as cellulose acetate 
and placed it inside the electrical field. Another milestone was achieved by Vonnegut and 
Neubauer in 1952 [6]. They managed to produce 0.1 mm in diameter fibers by 
developing a simple apparatus. They simply used a glass tube which was down to a 
capillary. This glass tube had a radius of a few tenths of millimeter. They filled the tube 
with water and the electrical wire was connected with a voltage between 5-10 kV; then 
they introduced the liquid inside this field. Another study was done by Simons in the field 
of nonwovens [7]. He managed to produce ultra thin and very light weight nonwoven by 
Polyvinyl Alcohol (PVA) 
Electrospinning 
Fiber Collection 
Web 
Twisting 
Yarn 
 37
immersing the positive charge into the solution and placing the negative charge to a belt 
where he collected the fibers. Electrospun acrylic fibers were produced by an apparatus 
of Baumgarten [8].  He managed to use a capillary tube maintained in constant size by 
adjusting the pump and he applied a high voltage.  
 
 
Figure 3.4 Parts of the electrospinning setup. 
 
In the present study, the components of the electrospinning device are shown in 
Figure 3.4. The collector is an aluminum plate and it is placed behind the nonwoven 
fabric where the fibers are collected. The distance between the needle and the collector 
can be arranged but mostly it is kept around 30 cm which is enough for collecting the 
fibers. The syringe pump speed was adjusted to have a flow rate between 0.5-1 ml/h 
depending on solution viscosity. When the viscosity is higher, the pump speed is 
decreased. When the flow rate is high, big beads were observed. 
 
Power 
 Supply 
 
Syringe 
Pump 
Electrical Field 
Metal Collector 
Polymer Solution 
 38
 It is also important to note that, electrospinning is not the only method to produce 
nano-micro sized fibers. There are numerous techniques to produce nano sized polymer 
based fibers. In Table 3.2, a short explanation of the techniques is given and compared 
with each other. 
 
Table 3.2. Nanofiber manufacturing methods [9]. 
 Explanation and Advantages Disadvantages 
Drawing 
It is almost similar to dry 
spinning method. 
It can produce very long single 
nanofibers one by one. 
Only  viscoelastic materials can 
be used because of the 
possibility of strong 
deformations 
Template 
Synthesis 
It uses a nanoporous membrane 
as a template to make 
nanofibers. Raw materials such 
as conducting polymers, metals, 
semiconductors and carbons can 
be fabricated for electronic 
applications 
It can not be used to get one-by- 
one continuous nanofibers. 
Phase Separation 
It consists of dissolution, 
gelatin, and extraction using a 
different solvent, freezing, and 
drying. 
It takes long time to transfer the 
solid polymer onto the nano-
porous foam. 
Self-Assembly 
In this process, individual 
components can organize 
themselves into the desired aim. 
It takes much time to be 
completed. 
 
 39
3.2.1. Defining the Maximum Collected Nanofiber Web Area 
To find the optimum values for manufacturing, voltage, needle, distance between 
the collector and needle and pump values are discussed.  
The applied voltage was changed between 10 kV and 20 kV. Two different needle 
sizes were used: 18 G 1 ? and 22 G 1 ?. The distance between the collector and needle 
was also changed to get the optimum length. The distance was taken as 10 cm, 15 cm, 20 
cm, 25 cm, and 30 cm from the collector. Temperature of the solution was kept at room 
temperature which was around 22 
o
C. The needle tip-target distance was investigated and 
it was found that it had no significant effect on electrospinning. Huang et al., suggested 
that a critical concentration of polymer solution has to be used to produce electrospun 
fibers as extensive chain entanglements are necessary to produce the fibers [9]. Table 3.3 
shows the optimum parameters for electrospinning.  
Table 3.3. Optimization of the experimental parameters (10% PVA water solution) 
Experiments  
1
st
  2
nd
 3
rd
  4
th
  5
th
  6
th
  7
th
  8
th
  9
th
  10
th
 
Voltage(kV) 20 20 20 20 15 20 20 20 20 15 
Needle (G 1 ?) 18 18 18 18 18 22 22 22 22 22 
Tip-needle 
distance (cm) 
30 10 15 20 25 30 10 15 20 25 
Temperature of 
the solution (?C) 
25 24 23 22 22 22 22 22 22 22 
Spinning time 
(min) 
5 5 5 5 5 5 5 5 5 5 
 
 
 40
To investigate the dimensions of the coated area, an aluminum foil was used to 
cover the coated area and the solution was placed closer to the coated area. The 
dimensions of the foil were measured. The shape was circular with two regions: the high 
density area with dimensions of 115 mm (vertically) and 90 mm (horizontally). The less 
dense area has dimensions of 147 mm (vertically) and 147 mm (horizontally) (Figures 3.5 
and 3.6). 
 
 
Figure 3.5 Nano-micro fiber spread area. 
 
Figure 3. 6. Schematic shape of the nano-micro fiber spread area. 
 41
3.2.2. The Voltage Effect in Electrospinning  
Electrospinning is a process that contains an electrical field where millimeter 
diameter fluid is pushed through. After introducing the fluid inside the electrical field, the 
Taylor cone happens at the tip of the nozzle. When the jet accelerates, it thins inside the 
field [9]. 
After many experimental studies, the most important part of the process was 
found out: the rapidly whipping fluid jet. The whipping of the jet to perform the spinning 
is so rapid; recently Hohman et al., showed that it could be captured by long exposure 
photograph technique [10]. 
Jaworek and Krupa classified the modes for the interaction of fluid and electrical 
field and divided them into four in a review study [11]. The first is dripping and it can be 
explained when Taylor cone happens; spherical droplets detach from it.  The second 
phase is spindle mode where the jet is elongated into a thin filament. The third one is 
oscillating jet mode where the drops are emitted from a twisted jet. The last one is 
precession mode in which jet breaks into droplets, when the whipping jet is emitted from 
the nozzle.  
It is shown by Fridrikh et al., that the fiber diameter in spinning can be 
controllable [12]. They presented a model to demonstrate the fluid jet inside the electrical 
field to find out the relationship between the fiber diameter and volume charge density. In 
order to do this, they changed the properties of electrospinning field such as material 
conductivity (K), dielectric permittivity (?), dynamic viscosity (?), surface tension (?), 
density, operating characteristics (flow rate (Q), applied electrical field (E) and electric 
current (I)).  
 42
They found out that,  
 
()
1
3
2
2
2
2ln 3
t
Q
h
I
??
??
??
=
??
?
??
     (3.6) 
 
where h
t
 is the terminal diameter, Q is the flow rate, I is the electrical current, ? is 
the dielectric permittivity, ? is the surface tension and ?  is the dimensionless wavelength 
of the instability. The same group also studied the stability theory to explain the 
relationship of the charged fluid jet inside a tangential electrical field by the function of 
all fluid parameters. Hohman et al., used the stability theory to develop a quantitative 
method to find out the starting point of electrospinning. Their theory predicts two 
conditions of the modes; the first emphasizes that the centerline of the jet remains straight 
but the jet radius is adjusted, for the second one, the reverse condition was done which 
involves constant centerline [13].   
The schematic of the experimental device used in the present work is shown in 
Figure 3.7. We placed the collector not on the ground but on the wall. The reason was to 
reduce the beading effect. 
 
 43
a 
b 
Figure 3.7. Direction of electrospinning a) The set-up of Hofman [13], b) The 
experimental set-up used in the present work. 
 
 Another important property is the voltage effect. When the applied voltage is less 
than 15 kV, it was seen that there is no spinning inside the electrical field, mostly because 
of the surface tension of the polymer. The electrical force is not enough to overcome the 
surface tension of the polymer. When the applied voltage is 15 kV and higher, the 
spinning starts.  
The effect of voltage over polymer solution is summarized in Figures 3.8, 3.9 and 
3.10. When the voltage is increased, the spinning starts earlier. Also it was observed that, 
the electric field strength increase the electrostatic repulsive force on the fluid jet and this 
causes thinner fiber formation.  
 
 44
 
Initial time 
(Voltage is applied.) 
10
th
 
 
12
th
 
(Solution drops.) 
 
13
th
 
(Spinning started. 
Dripping and spinning at 
the same time. ) 
14
th
 
(Spinning angle 
becomes stable.) 
15
th
 
 
22
nd
 23
rd
 24
th
 
 
25
th
 26
th
 27
th
 
Figure 3.8. Electrospinning with 15 kV. 
 
 45
Initial time 
(Voltage is applied.) 
5
th
  
(Small electrospinning 
occurs.) 
10
th
  
 
13
th
  
 
14
th
  
 
15
th
 
(Solution drops.) 
Figure 3.9. Electrospinning with 17.5 kV. 
 
 
 
 
 
 
 46
 
Initial time 
(Voltage is applied.) 
1
st
  
(Electrospinning occurs and 
spinning angle becomes 
stable.) 
2
nd
   
 
 
4
th
   
 
6
th
   
(Dripping occurs.) 
10
th
   
(Whipping occurs.) 
Figure 3.10. Electrospinning with 20 kV. 
 
The polymer drop is balanced by upward and downward forces. These forces are 
simply classified into two groups: electrical forces and non-electrical forces. The 
electrical and gravitational forces (F
grav
) are downward and the rheological (F
rheo
), inertial 
(Fin), surface tension (F
surf
) and aerodynamics forces (F
aero
) act upward, as shown in 
Figure 3.11 [14]. The dominant electrical force (F
elec
) is described as: 
 
 
F
elec 
(F) = F
rheo
(A) + F
in
(B) +F
surf
(C)
 
+ F
aero
(D) - F
grav
 (E)    (3.7) 
 47
 
Figure 3.11. Forces acting on polymer solution [14]. 
In the experimental study, when the first fiber formation occurred, the direction of 
the spinning was captured and the spinning angle was determined. For the applied 15 kV 
voltage value, the spinning angle was found between -7?, and -10?; and for 17.5 kV, it 
was observed that the angle is around -30?and for 20 kV the spinning angle is around 45?. 
After these observations, equation (3.7) can be transformed into; 
F
elec 
=f(?)         (3.8) 
where ? is the spinning angle and it has a relationship with the voltage. If it is 
assumed that; 
V(15 kV) ~ ? (?/18)  
V(17.5 kV) ~ ? (?/6)  
V(20 kV) ~ ? (?/4) 
and when these values are plotted in Excel, Figure 3.12 is obtained.  
 48
y = 7x - 94.167
0
10
20
30
40
50
0102030
Applied voltage (kV)
S
p
i
n
ni
ng a
ngl
e
 (
?
)
 
Figure 3.12. Applied voltage effect on the spinning angle. 
From the graph, the empirical relationship between the voltage and spinning angle 
is obtained as; 
Y=7x-94.167            (3.9) 
where x is the applied voltage and Y is the spinning angle (?) 
So the equation 3.9 can be written as; 
F
elec 
=f(7*[V]-94.167)         (3.10) 
  
 
The observed relationships can be seen in Figure 3.13.  
 
 
 
 
 
 
 49
 
 
 
 
Figure 3.13. Electrospinning directions for 15, 17.5 and 20 kV applied voltage values. 
 
 
 
 50
3.2.3. Scanning Electron Microscope (SEM) Imaging 
Before putting the dried sample under SEM (EMS 550X in Biology Department 
of Auburn University), it was covered with Au using EMS 550X Sputter Coating Device. 
Argon gas was applied to start the Au coating process. Since the coated sample is a 
polymer, the coating time was increased to 4 minutes in order not to have melting 
problem under the SEM.  
The voltage level was adjusted to 10 kV and was kept the same for all the 
samples. The SEM images of the PVA samples can be seen in Figure 3.14.  
 
  
A (PVA 15 wt. %, 20kv, x3000). B (PVA 10 wt.%, x10000) 
Figure 3.14. Image of the PVA fiber samples 
 
 
 
 51
3.2.4. Continuous Manufacturing Device 
The continuous nanofiber based yarn manufacturing is a challenge. The electrical 
field needs to be somewhat controlled. Optimum viscosity of the solution must be used.  
Ko et al., combined the electrospinning method and conventional yarn manufacturing 
(Figure 3.15).  
 
 
Figure 3.15. Schematic of the continuous yarn spinning made by SQNT-filled composite 
[15]. 
We designed a nanofiber web collecting mechanism which is shown in Figure 
3.16. It was found that, by using the optimum properties of electrospinning and having an 
appropriate environment for the electrospun fibers, they could be easily pulled from the 
collected surface and can be made into a yarn in a continuous manufacturing process. 
 52
Parts of the device are: 
? Nonwoven fabric covered cylinder. 
? Basic electrospinning device 
? Needle and syringe 
? Power supply 
? Syringe pump for giving stable solution flow.  
? Pulling mechanism which forwards the gathered nanofiber web to the next 
process.  
? Nonwoven fabric covered cylinder which is made of aluminum; this part must be 
attached to the main collector to gather the nanofibers. 
? Power unit to provide the desired movement for both cylinder and pulling 
mechanism.  
 
 
 
 
 
 
 
 53
 
 
Figure 3.16. Schematic of yarn manufacturing device 
After producing the nano-micro fiber based yarn, tensile strength measurements 
were done. We used Instron tensile testing device to measure the elongation and 
maximum force to break the nano-micro fiber based yarn. Tensile stress is measured as 
the force required to rapture a fiber with a unit of force/linear density. There are some 
factors which affect fiber strength: the molecular length, orientation, intermolecular 
forces and degree of crystallinity.  When the area under the stress strain curve is bigger, 
the breaking energy is maximized. 
 54
3.2.5. Coating 
Nanofibers have interesting properties which is the result of their extremely high 
surface to weight ratio compared to the other conventional fibrous structures. This makes 
these fibers ideal to be used in application areas such as filtration because of high pore 
volume, tight pore size, etc. Recently, in protective clothing, electrospun fiber webs have 
been used. 
 In this study, the fibers are collected on different kind of fabric surfaces and the 
best fabric was chosen to pull the web form easily. As seen in Figure 3.17, paper, woven, 
and nonwoven fabrics were used for coating. 
 
Figure 3.17. Coated fabrics. 
 
 55
The coating time differed from one hour to two hours. When the coating was done 
over a nonwoven fabric, the nanofibers were seen in the form of web rather than film.  
 
3.2.6. Twisting of Manufactured Nano-micro Fibers  
After getting the fibers in a web form, the fibers were gently pulled and twisted by 
hand since the web form is so sensitive. Making a yarn will help to handle the fibers 
easily. 
Figure 3.18 shows the twisting process. It is important to note that, Since PVA 
can be sticky, twisting is done effectively. 
          
Figure 3.18. Twisting of nano fiber based web. 
 
 56
3.2.7. Differential Scanning Calorimetry (DSC) Studies 
In normal conditions, in the melting region there are two processes that require 
energy; heat capacity and heat of fusion. Anything which melts very rapidly (faster than 
the modulation) will resolve the heat of the fusion into a nonreversing heat flow. In 
general, a slower heating rate allows for more cycles in the melting region. Distributing 
the heat of fusion over many cycles effectively decreases the amplitude of each cycle. 
This decrease in amplitude leads to a decrease in reversing of the heat flow at the melt as 
shown in Figure 3.19 [16].  
 
 
Figure 3.19. Heat flow and temperature relationship in DSC [16]. 
 
Increasing the sample mass increases the temperature gradient in the sample and it 
increases the time for heat transfer to occur. This translates to less melting in-phase with 
the temperature modulation.  
 57
3.2.8. Air Permeability Measurements 
Air permeability is a term to describe the breathability of the fabrics for various 
purposes such as industrial filter fabrics, tents, medical fabrics, etc. [17]. To determine 
the air permeability of micro-nano structures, air permeability test apparatus was used. 
The most commonly used methods for air permeability in the literature are ENISO 9237 
and AS 2001.2.34.  
 
The area of the orifice is; 
A
 orifice
= 2.54 *10
-6
 m
2
.         
The air permeability is calculated by using the measured penetration time: 
() ()
2
3
morificetheofareaThe/utemintimenPenetratio
)m(pressureappliedonbaseddifferenceVolume
typermeabiliAir =       (3.11) 
 
2
3
6
263
m
min
m
10*86.9
sec60
min1
sec*4.46
m10*54.2*m0003.0
typermeabiliAir
?
?
==   (3.12) 
 
3.2.9. Dynamic Contact Analyzer 
In our study, surface tensions of the PVA and PVA+Laponite? based solutions 
were measured. Laponite was taken as a ratio of 5% of the PVA solvent. It means that for 
12% PVA, the Laponite? is 5% of this PVA. The total weight of the solution was 100 gr. 
 58
3.2.10. Thermal Conductivity Measurement 
Thermal conductivity can be measured by various experimental ways such as 
DSC method or air permeability measurements. 
 
a) By MDSC (Modified Differential Scanning Calorimetry) Method 
In MDSC (almost similar to DSC), the magnitude of the measured kinetic heat 
flow is a function of time at constant temperature which is the working procedure of it. 
The general theory can be written as; 
)t,T(f
dt
dT
Cp
dt
dQ
+=         (3.13) 
where Q is the heat flux, Cp is the heat capacity, T is the temperature, and t is the time. 
Total Heat Flow (DSC) = Heat Capacity Components + Kinetic Component 
   = Heating Rate Dependent + Time Dependent 
   =MDSC Reversing + MDSC Non-reversing 
Reversing heat flow contains:    
o Glass Transition 
o Melting (most) 
Non-reversing heat flow: 
o Enthalpy relaxation 
o Evaporation 
o Crystallization 
o Decomposition 
o Cure 
 59
o Melting (some) 
The values can be summarized in Table 3.4.  
 
Table 3.4. Effects on DSC graph areas. 
 Amplitude Period Heating Rate Sample Mass 
Change + - - + 
Reversing area in 
the melt 
- - - - 
Nonreversing area 
in the melt 
+ + + + 
(+): Increasing  (-): Decreasing 
 
In thermal conductivity measurement two specimens are used: one thin (0.4 mm) 
and one thick (3 mm). The thin sample is encapsulated in a standard aluminum pan to 
measure the specific heat capacity. The thick sample is used to measure the apparent heat 
capacity without a pan [16]. 
 
PMdC
LC8
K
2
p
2
0
=          (3.14) 
where; 
K
0
; Uncorrected thermal conductivity (W/M?C) 
L; Sample length (mm) 
C; Apparent heat capacity ( mJ/?C) 
C
p
; Specific heat capacity (J/g?C) 
M; Thick sample mass (mg) 
d; Thick sample diameter (mm) 
 60
P; Period of measurement (sec) 
 
b) By Air Permeability Test 
It is known that, by using air permeability test, thermal conductivity can be 
calculated based on Darcy?s permeability equation. This method depends on the 
relationship between Darcy?s permeability equation and Fourier?s thermal conductivity 
equation. Based on this, thermal conductivity can be predicted easier compared to the 
other methods [18]. 
 
Basically, by using the following two equations; 
L
P
kq
fpfp
?
=         (3.15) 
and  
L
T
kq
etctc
?
=           (3.16) 
where, 
q
fp
; linear flow rate 
k
fp
; air flow permeability, 
?P; pressure drop 
L; sample thickness 
q
tc
; flow rate from Fourier?s law 
k
e
; effective thermal conductivity 
?T; temperature differential 
 
 61
The effective thermal conductivity can be obtained as; 
()
()
fp
tcfp
etc
qT
qPk
k
?
?
=           (3.17) 
3.3. References 
1. http://composite.about.com/library/glossary/p/bldef-p4165.htm. (accessed in April 
2005).  
2. Broughton, R. M., TXEN 6310 Lecture Notes, Auburn University, Spring 2004. 
3. Van Krevelen, D.W., ?Properties of Polymers Correlations with Chemical 
Structure?, Elsevier Publishing Company, 1972, 2.  
4. www.laponite.com (accessed in Jan 2005). 
5. Formhals, A., ?Process and Apparatus for Preparing Artificial Threads?, US 
Patent 1,975,504, 1934. 
6. Vonnegut, B., and Neubauer, R.,? Production of Monodisperse Liquid Particles 
by Electrical Atomization?, J. of Colloid Science, 1952, 7, 616-622.  
7. Simons, H. L., ?Process and Apparatus for Producing Patterned Non-woven 
Fabrics?, US patent 3,280229, 1966. 
8. Baumgarten PK., ?Electrostatic Spinning of Acylic Microfibers?, J of Colloid and 
Interface Science, 1971, 36, 71-9.  
9. Huang Z. M., Zhang, Y. Z., Kotaki, M., and Ramakrishna, S., ?A Review on 
Polymer Nanofibers by Electrospinning and Their Applications in 
Nanocomposites?, Comp. Sci. and Techn,. 2003, 63, 2223-2253. 
 62
10. Hohman, M. M., Shin, M., Rutledge, G., and Brenner, M. P., ?Electrospinning 
and Electrically Forced Jets. I. Stability Theory?, Physics of Fluids, 2001, 13, 
2201-2221.  
11. Jaworek, A., and Krupa, A., ??Classification of the Modes of End Spraying??, J. 
Aerosol Sci., 1999, 30(5), 873. 
12. Fridrikh, S. V., Yu, J., Brenner, M., and Rutledge, G., ?Controlling the Fiber 
Diameter during Electrospinning?, Physical Review Letters, 2003, 90(14), 
144502. 
13. Hohman M., Shin, M., Brenner, M., and Rutledge, G., ?Electrospinning and 
Electrically Forced Jets. II. Applications?, Physical Fluids 2001, 13(8), 2201-
2220. 
14. http://www.gpi-test.com/4110.htm. (accessed in Nov. 2004) 
15. Ko, F., Gogotsi, Y., Ali, A., Naguib, N., Ye, H., Yang, G., Li, C., and Willis, P., 
?Electrospinning of Continuous Carbon Nanotube-filled Nanofiber Yarns?, Adv. 
Mater. 2003, 15(14), 1161-1165. 
16. TA Instruments, ?DSC Operator?s Manual?, July 1998. 
17. Pamela Banks-Lee and Mohammadi, M., ?Utilization of Air Permeability in 
Predicting the Thermal Conductivity?, International Nonwoven Journal, 2004, 
13(2), 28-33. 
18. http://www.inda.org/subscrip/inj04_2/p28-33-banks.pdf (accessed in Nov. 2004) 
 
 
 
 
 
 63
 
 
 
 
 
CHAPTER 4 
ANALYTICAL MODELING OF FILLER FIBER REINFORCED COMPOSITES 
Nomenclatures: 
K Thermal conductivity   
L Length    
Q Heat flow rate   
R Thermal resistance   
T Temperature   
V Volume fraction   
T Thickness of the interface (barrier)   
A Radius of barrier with filler radius   
R Radius   
Subscripts:    
T Total   
B Barrier   
e Effective 
f Main fiber   
d Filler fiber 
- (bar) Dimensionless   
? 
(hat) Without barrier   
Greeks:    
? 
Thermal conductivity ratio between the filler fiber and the main fiber 
= k
d
/k
f
 
? 
Thermal conductivity ratio between the barrier  and the main fiber 
=k
b
/k
f
 
 
 64
Fiber-reinforced composites have had significant importance for a long time. 
Many experimental, theoretical and numerical studies have been carried out by various 
research groups in the universities and the industry. 
In general, in a composite, there are three different types of components having 
different fundamental properties. Figure 4.1 shows the basic components of the 
composite, which are matrix, interface and filler fiber, used in the model.  
In this study, the composites are assumed to have not only rectangular shape but 
also the cylindrical shape. In order to solve the problem which is showen in Figure 4.1, 
the hexagonal unit cell was developed and analyzed to find the effective thermal 
conductivity.  
 
 
(a) 
 
(b) 
Figure 4.1. Composite structures for the model. a) Cylindrical shape b) Rectangular 
shape 
 
 
 65
4.1. Description of the Problem 
Based on the analytical calculation of Zou et al., the hexagonal unit cell was 
developed and analyzed analytically [1]. The applied procedure depends on the thermal-
electrical analogy method which has been widely used in different kind of problems.    
The problem consists of two real and one imaginary component. Imaginary part is 
the thermal barrier part and compared to the filler size, it has much smaller size. The real 
parts are the main fiber as a matrix and filler fiber.   
In the present model, a hexagonal unit cell is developed as shown in Figure 4.2. 
Because of the continuous behavior, it is assumed that, 2-D modeling is sufficient to 
represent the behavior of the structure.  
 
 
(a) 
L
r
d
a
L/
2
L
(b) 
 
Figure 4.2. Hexagonal model for transverse heat conduction. a) 3-D model b) 2-D 
model. (Q: heat flux.) 
 66
The cross-section of the hexagonal unit cell was partitioned into regions and the 
thermal resistance method was applied to each region. For total thermal resistance, it is 
considered that these regions are connected parallel.  
 
?
?
?
?
?
?
+++=
4321
11111
RRRRR
at
      (4.1) 
where R
t
 is the total thermal resistance of the composite. 
R
1a 
, R
2
 , R
3
 , R
4
 : The thermal resistance of the first, second, third and fourth  regions, 
respectively (Figure 4.3). 
First Region
Second Region
Third Region
Fourth Region
30
o
 
Figure 4.3. Symmetric part of the hexagonal model and the region divisions. 
The thermal characteristics of the components are:  
 
k
f
 : thermal conductivity of main fiber, 
 67
k
d
 : thermal conductivity of filler fiber,  
k
b
 : thermal conductivity of barrier, 
 
The assumptions for the modeling are;  
a) Because of the continuity, the modeling is done in 2-D. 
b) Filler fibers are embedded in a homogenous matrix 
c) Filler fibers have cylindrical shape and have a radius of r
d
; the barrier interface 
thickness is taken as ?t? which is equal to ?a-r
d
? where a is the distance of barrier 
boundary from the center. The length of each side of the hexagonal unit cell is 
taken as ?L?.  
d) Temperature difference was applied only to the left and right sides, -1 ?C and 1 
?C, respectively.  
e) The volume fraction of the filler fibers inside the matrix is taken between 10-30%. 
The distribution of the filler fibers is considered to be uniform and parallel. 
f)  The thermal contact resistance is negligible depending on the existence of 
thermal barrier layer.     
g) The composite is considered without any cracks; it is considered not to have 
voids.  
4.2. Modeling for Analytical Thermal Resistance 
  To investigate the total thermal resistance behavior of the proposed composite 
with respect to the change of variable k
f
/k
d
, the symmetric part of the whole hexagonal 
 68
area, which is shown in Figure 4.3, is considered.  In this case, the total thermal 
resistance of the designated part is obtained as, 
f
tf
k
R
3
1
=                        (4.2) 
Then, by dividing equation (4.1) by equation (4.2), dimensionless total thermal resistance 
of the composite (
?
t
R ) is given as 
f
a
t
k
RRRR
R
3
1111
1
4321
+++
=
?
       (4.3) 
 
4.2.1. Thermal Resistance of the First Region, R
1a
 
The first region consists of only the main fiber component.  Thermal resistance of 
the 1
st
 part is calculated by considering that, it is half of a rectangle (Figure 4.4.) 
 
L/2
1a
R
1a
R
3/2 L(
R
1b
=
1b
R
=
R
1
R
3/2 L(
1
L/2
 
Figure 4.4. The first region. 
As a result, the value of R
1a
 is half of R
1
 which results in the following equation.  
 69
f
f
a
k
L
k
L
R
2
3
2
2
2
3
1
==      
3
2
1
1
f
a
k
R
=          (4.) 
4.2.2. Thermal Resistance of the Second Region, R
2
 
Thermal resistance of this region can be written as;  
L
a
L
k
R
f
2
3
21
2
?
?
?
?
?
?
?
=         (4.5) 
4.2.3. Thermal Resistance of the Third Region, R
3
 
The third region is the combination of the main fiber and the barrier region 
(Figure 4.5.).  
3/2 L(
 
 
d
r
1
y
O
dR
a
1
3
dR
3
2
Third 
Region
1
dy
 
Figure 4.5. Schematic of the third region. 
 70
The 3
rd
 part was divided into infinite divisions and  
11
33
3
2
3
21
dyk
CosaL
dyk
Cosa
dRdRdR
fb
?
?
?
?
?
?
?
?
?
+=+=
?
?
             
1fb
bf
3
dykk
kCosaL
2
3
Coska
dR
?
?
?
?
?
?
?
?
??+?
=
     (4.6) 
where; 
??
?
dady
ay
cos
sin
1
1
=
=
         (4.7) 
After putting the forms inside the equation (4.6), we get; 
 
bf
fb
kCosaLCoska
dCosakk
dR
?
?
?
?
?
?
?
?
?+
=
??
??
2
3
1
3
     (4.8) 
Before solving the above equation, it is assumed that; 
()
?
?
?
?
=
=
=
1
2
3
c
k
k
a
L
b
f
b
         (4.9) 
Then the equation 4.8 becomes; 
 71
[] []
()
()
??
??
+
?
=
+?
=
?+
=
?+
=
22
22
3
sec
11
sec1
1sec1coscos
cos
1
?
?
?
?
?
?
?
?
??
??
?
?
???
??
??
??
???
???
b
dk
b
dk
b
dk
b
dk
R
ff
ff
 
?
+
=
2
3
sec
1
?
?
?
?
bc
dk
R
f
         (4.10) 
After integrating the right side of the above equation; 
?
+
?
?
?
?
?
?
?
?=
2
3
cos2
1
?
?
?
?
?
?
cb
d
c
kb
c
k
R
ff
      (4.1) 
where;  
?  is varying form ? to 
?
90  such that 
a
r
d
=?sin  and therefore 
a
r
d1
sin
?
=?      (4.12) 
The solution of the integral depends on b and c; the first option is if b
2 
> c
2
 and 
the other option is if c
2
 >b
2 
[2-4]. Then, the solution for the third region becomes as 
follows: 
        
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?=
?>
??
2
tantan
4
tantan
2
2
1
(*)
11
22
3
22
??
?
?
cb
cb
cb
cb
cbc
kb
c
k
R
cb
ff
  
     
 72
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
+
?
?
?
?
?
?
?
?
?
?=
??
2
tantantan
2
2
1
11
22
3
?
?
?
cb
cb
cb
cb
cbc
kb
c
k
R
ff
  
           (4.13) 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
+
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
+
?
+
?
?
?
?
?
?
?
?
?=
?<
2
tan
2
tan
ln
4
tan
4
tan
ln
2
1
(**)
22
3
22
?
?
?
?
?
?
bc
bc
bc
bc
bc
bc
bc
bc
bcc
kb
c
k
R
cb
ff
  
           (4.14) 
as a next step, when tan(?/4) is equal to 1; 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
+
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
+
?
+
?
?
?
?
?
?
?
?
?=
2
tan
2
tan
ln
1
1
ln
2
1
22
3
?
?
?
?
bc
bc
bc
bc
bc
bc
bc
bc
bcc
kb
c
k
R
ff
  
           (4.15) 
The rest of the solution steps for the third region are as follows; 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??+
?
?++
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??+
?
?++
?
??
?
?
?
?
?
?=
bc
bcbc
bc
bcbc
bc
bcbc
bc
bcbc
bcc
kb
c
k
R
ff
2
tan
2
tan
lnln
2
1
22
3
?
?
?
?    
 
 73
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??+
?++
?
?
?
?
?
?
?
?
?
??+
?++
?
?
?
?
?
?
?
?
?=
2
sin
2
cos
2
sin
2
cos
lnln
2
1
22
3
??
??
?
?
bcbc
bcbc
bcbc
bcbc
bcc
kb
c
k
R
ff
           (4.16) 
As a result; 
()()
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
+
?
?
?
?
?
?
?++
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??+
?
?
?
?
?
?
?++
?
?
?
?
?
?
?
?
?=
22
2
22
2
22
3
2
sin
2
cos
2
sin
2
cos
ln
ln
2
1
??
??
?
?
bcbc
bcbc
bcbc
bcbc
bcc
kb
c
k
R
ff
  
()
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+?
?
?
?
?
?
?
?
?
?
?
?
?
?
++?
?
?
?
?
?
?
?
?
?
?+
?
?
?
?
?
?
?
?
?=
2
2
2
22
22
3
2
tan
2
tan
lnln
2
1
bcbc
bcbc
b
bcc
bcc
kb
c
k
R
ff
?
?
?
?     
 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?++
?
?
?
?
?
?
?
?
?
?+
?
?
?
?
?
?
?
?
?=
?
??
?
?
cos
sincos
lnln
2
1
2222
22
3
cb
bcbc
b
bcc
bcc
kb
c
k
R
ff
  
(4.17)  
 
When the ?ln? terms are put together; 
 74
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?++
?
?
?
?
?
?
?
?
?+
?
?
?
?
?
?
?
?
?=
?
??
?
?
cos
sincos
ln
2
1
22
22
22
3
cb
bcbc
b
bcc
bcc
kb
c
k
R
ff
   (4.18)  
 
then, the final result for the third region ( b
2
<c
2
 ) is ; 
 
( )()
( )
?
?
?
?
?
?
?
?
?++
+?+
?
?
?
?
?
?
?
?
?=
??
?
?
?
sincos
cos
ln
2
1
22
22
22
3 bcbcb
cbbcc
bcc
kb
c
k
R
ff
  (4.19) 
 
4.2.4. Thermal Resistance of the Forth Region, R
4
 
Inside the 4
th
 region, there are three components; the main fiber, the barrier and 
the filler. For infinite divisions of the each region, (Figure 4.6), the thermal resistances 
are added such that,  
 
321
444
4
dRdRdRdR ++=        (4.20) 
 
 75
Forth
dR
4
1
dR
4
3
y
dy
Region
2
r
d
2
o
a
2
4
dR
( 3/2 L
 
Figure 4.6. Schematic of the fourth region. 
222
4
cos
2
3
coscoscos
dyk
aL
dyk
ra
dyk
r
dR
fb
d
d
d
?
???
?
+
?
+=     (4.21) 
??
?
??
??
??
?
dak
aL
dak
ra
dak
r
dR
fb
d
d
d
cos
cos
2
3
cos
coscos
cos
cos
4
?
+
?
+=      
 
()
?
?
?
?
?
?
?
?
?+?+
=
????
??
cos
2
3
coscoscos
cos
1
4
aLkkrakkrkk
dakkk
dR
bdddfdfb
fbd
   
           (4.2) 
where, 
?? sinsin
2
ary
d
==   
?? dady cos
2
=  
 76
2
2
sin1sin1cos
sinsin
?
?
?
?
?
?
?
?
?=?=
=
???
??
d
d
r
a
r
a
     (4.23) 
?
?
?
?
?
?
?
?
?+
?
?
?
?
?
?
?
?
?
?
??+?
=
????
??
cos
2
3
sin1cossin1
cos
1
2
2
2
2
2
2
4
aLkk
r
a
rakk
r
a
rkk
dakkk
R
bd
d
dfd
d
dfb
fbd  (4.24) 
        
It is also known from Figure (4.6) that ?  is varying from 
?
0 to?, and ?  is varying from 
? to
?
90 . Integrating equation (4.24), 
 
?
?
?
?
?
?
?
?
?
?+
?
?
?
?
?
?
?
?
?
?
??+?
=
?
????
??
0
2
2
2
2
2
2
4
cos
2
3
sin1cossin1
cos
1
aLkk
r
a
rakk
r
a
rkk
dakkk
R
bd
d
dfd
d
dfb
fbd
 
 
The solution steps for the fourth region are as follows;  
() ()
?
+?+??
=
?
????????
????
0
2
2
2
4
cos1sin
cos
1
b
a
r
dk
R
d
f
   (4.25) 
where; 
 
f
d
k
k
=?          (4.26) 
?
+
?
?
?
?
?
?
?
?
?+?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
??
??
0
2
2
2
4
cos1
1
sin
11
cos
1
b
a
r
dk
R
d
f
 
 77
Considering the following relations lead to further mathematical manipulations: 
 
??
11
?=d  
a = r + t  
()
22
2
2
2
22
2
1
1111
?
?
?
?
?
?
+
=
?
?
?
?
?
?
+
=
+
==
?
t
r
tr
r
tr
r
aa
r
d
d
d
d
     (4.27) 
?
t  represents the dimensionless barrier thickness 
?
?
?
?
?
?
?
?
=
?
d
r
t
t  
Therefore, the last form of the integral becomes as follows; 
?
?
?
?
?
?
?
?
+
++
=
?
?
??
??
0 2
2
4
sin
1
1
cos
cos
1
t
dcb
dk
R
f
     (4.28) 
4.3. Dimensionless Total Thermal Resistance of the Model, R
t
 
Dimensionless total thermal resistance of the model is found by inserting equation 
(4.4), (4.5), and either (4.13) for the case of b
2
>c
2
 or (4.19) for b
2
<c
2
, and also (4.28) into 
equation (4.3). 
Therefore, the final form of the effective thermal conductivity becomes; 
 
?>
22
cb 
 
 78
f
f
ff
f
f
t
k
cb
cb
cb
cb
cbc
kb
t
dcb
dk
c
k
L
a
L
k
k
R
3
1
2
tantantan
2
sin
1
1
cos
cos
2
2
3
2
3
2
1
11
22
0 2
2
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
+
?
?
?
?
?
?
?
?
?
?
+
++
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?+
?
?
?
?
?
?
?
+=
??
?
? ?
?
??
??
?
?
?
 
           (4.29) 
?<
22
cb 
 
( )()
( )
f
f
ff
f
f
t
k
bcbcb
cbbcc
bcc
kb
t
dcb
dk
c
k
L
a
L
k
k
R
3
1
sincos
cos
ln
sin
1
1
cos
cos
2
2
3
2
3
2
1
22
22
22
0 2
2
?
?
?
?
?
?
?
?
?
?
?
?
?
?++
+?+
?
?
?
?
?
?
?
?
?
+
++
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?+
?
?
?
?
?
?
?
+=
?
?
?
??
?
??
??
?
?
?
 
           (4.30) 
After some minor simplifications of the both parts, dimensionless total thermal 
resistance of the model is obtained as;  
?>
22
cb 
 
 79
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
+
?
?
?
?
?
?
?
?
?
?
+
++
+
?
?
?
?
?
?
?+?=
??
?
? ?
2
tantantan
3
32
sin
1
1
cos
cos
3
3
3
3
23
3
1
1
11
22
0 2
2
?
??
??
?
?
?
cb
cb
cb
cb
cbc
b
t
dcb
d
cb
R
t
  
           (4.31) 
?<
22
cb 
 
( )
()
( )
20
2
22
22
22
3331cos
1
323 3 1
cos sin
1
cos
3
ln
3
cos sin
t
d
bc
R
bc d
t
ccb bc
b
cc b
bcb c b
?
???
?
??
?
??
?
?
??
=? + ? +
??
??
++ ?
??
+
??
??
??
+? +
??
?
?
++?
??
?
 (4.32) 
4.4. Calculation of the Volume Fractions of the Model 
The area of the hexagon is; 
2
hexagon
L
2
33
A =  
( )
2
d
2
barrier
raA ??=         (4.3) 
2
ddisk
rA ?=  
2
diskbarrier
aAA ?=+  
 80
If the thickness of the model is 1 unit length, then the volume fraction of each region is 
found as follows: 
Dimensionless volume fraction of the disk (filler fiber) 
22
2
2
2
33 33
2
dd
d
rr
L
L
??
? ==      (4.34) 
Dimensionless volume fraction of the disk and barrier 
             
22
2
2
2
33 33
2
bd
aa
L
L
??
? ?+= =       (4.35) 
To see the effect of the change of 
d
? on the value of  b, the ratio of 
a
L
 has to be 
calculated, 
()
() ()
2
2
2
2
2
33
2
33
22
33 33
bd
bd
bdbd
a
L
L
a
L
a
?
? ?
?
??
??
? ???
+=
=
+
==
++
    (4.36) 
() ()
332
0.5373
2233
bd bd
L
b
a
??
? ???
== ?
++
  (4.37) 
When ?
b
 is 0, the value of b becomes; 
0.5373
d
b
?
?
?        (4.38) 
The other b values depend on 
d
?  as given in Table 4.1.  
 81
Table 4.1. Dimensionless length (b) values. 
d
?  b 
0.2 2.129187 
0.3 1.738474 
0.5 1.346616 
0.7 1.138098 
0.8 1.064593 
0.95 0.977 
 
4.5. Dimensionless Total Thermal Resistance of the Model without a Barrier 
When there is no barrier then the model is considered to be consisting of three 
regions in a parallel connection as shown in Figure 4.7.  Thermal resistances of the first 
and second regions are as follows, respectively. 
3
2
1
1
f
a
k
R
=                   (4.39) 
L
r
L
k
R
df
2
3
21
2
?
?
?
?
?
?
?
=         (4.40) 
 82
( 3/2 L
 
 
1
y
L/2
L
R
2
1
3
dR
o
r
d
3
2
dR
R
1a
dy
1
 
Figure 4.7. Schematic of the model without barrier. 
 
Calculation of the thermal resistance of the third region requires more 
mathematical manipulations such that, 
1f
d
1d
d
33
3
dyk
CosrL
2
3
dyk
Cosr
RdRdRd
21
?
?
?
?
?
?
?
?
??
+
?
=+=
)))
           
 
1
3
2
3
dykk
kCosrLCoskr
Rd
fd
bdfd
?
?
?
?
?
?
?
?
?+
=
??
)
     (4.1) 
 83
where; 
??
?
drdy
ry
d
d
cos
sin
1
1
=
=
          (4.2) 
After putting the forms inside the equation (4.41), we get; 
 
dddf
dfd
kCosrLCosrk
dCosrkk
Rd
?
?
?
?
?
?
?
?
?+
=
??
??
2
3
1
3
)      (4.3) 
Before solving the above equation, it is assumed that; 
1
1
2
3
?=
=
?
c
r
L
b
d
)
)
         (4.44) 
After some mathematical manipulations, the equation (4.43) becomes; 
?
?
?
?
?
?
?
?
?
?+
=
2
03
2
3
1
?
??
??
dddf
dfd
kCosrLCosrk
dCosrkk
R
)
 
 
?
+
=
2
03
sec
1
?
?
?
bc
dk
R
f
)
)
)          (4.45) 
 
After integrating the right side of the above equation; 
?
+
?=
2
03
cos2
1
?
?
??
cb
d
c
kb
b
k
R
ff
)
)
)
)
))        (4.46) 
 84
where; 
?  is varying from 
?
0to 
?
90   
If similar calculations are performed to solve equation (4.46), as it is done for the barrier 
case of equation (4.11), the solution depends on whether 
22
c b
)
)
>  or 
22
c b
)
)
< . 
        
?>
22
(*) cb
)
)
      
1
22
3
2
1
tan
2
ff
kbk
bc
Rc bc
cb c
?
?
?
=?
+
?
)
)
)
))
)) )
))
     (4.7) 
?<
22
(*) cb
)
)
 
?
?
?
?
?
?
?
?
?+
?
?=
b
bcc
bcc
kb
c
k
R
ff
)
)
))
)
))
)
)
)
22
22
3
ln
2
1 ?
    (4.8) 
Total dimensionless thermal resistance of the model without barrier is calculated as; 
f
a
t
kR
RR
R
3
11111
321
?
?
?
?
?
?
?
?
++=
?
)
)
      (4.9) 
For ?>
22
cb
)
)
 
cb
cb
cbc
b
cb
R
t
)
)
)
)
)
)
)
)
)
)
)
+
?
?
?+?=
?
?
1
22
tan
3
32
3
3
23
3
1
1 ?
    (4.50) 
?<
22
cb
)
)
 
 85
?
?
?
?
?
?
?
?
?+
?
?+?=
?
b
bcc
bcc
b
cb
R
t
)
)
))
)
))
)
)
)
)
22
22
ln
3
3
3
3
23
3
1
1 ?
   (4.51) 
4.6. Computer Implementation of the Analytical Model  
MATLAB programming (given in the Appendix) is used to implement the 
equations (4.29) and (4.30). Since these equations require for solving the integral part, the 
Simpson Method was used [5]. 
In Simpson?s Rule, since the integral is the total area under the curve, the interval 
[a,b] is divided into  an even number  of n subintervals. The divisions are called x
0
=a, x
1
, 
x
2
,?.?,x
n
=b and in this division, the x
i
 can be found as 
i
x= a+i. x ?  
where  
i = 0, 1, 2,?,n  
ba
x
n
?
?=  
The approximation scheme of the method for n=6 is given as  
[])()(4)(2)(4)(2)(4)(
3
)(
6543210
xfxfxfxfxfxfxf
x
dxxf
b
a
++++++
?
?
?
 (4.52) 
Therefore, the integral part of the model is implemented such that; 
 
[
]
0123
20
2
456
cos
( ) 4() 2( ) 4( )
31
cos sin
1
2( ) 4( ) ( )
dx
f xfxfxfx
bc d
t
fx fx fx
?
??
??
?
?
?+++
++ ?
??
+
??
??
+++
?
 (4.53) 
 86
where 
  
6
x
?
?=  
0
0
=x  ,  
1
6
x
?
=  ,  
2
3
x
?
=  ,      (4.54) 
3
2
x
?
=  , 
4
2
3
x
?
=  , 
5
5
6
x
?
=  , 
6
x ?=  
This form was applied in the integral form which is shown in equation 48, then; 
()
0
2
22
cos0 1
()
11
os0 sin 0
1
1
fx
bcc d bcd
t
t
?+
==
++ ? ++
?? +
+
??
??
    (4.55) 
1
2
2
cos
6
()
1
cos sin
66
1
fx
bc d
t
?
??
?
=
++ ?
??
+
??
??
    (4.56) 
2
2
2
2
cos
6
()
21
cos sin
66
1
fx
bc d
t
?
??
?
=
++
??
+
??
??
    (4.57) 
 87
3
2
2
cos
2
()
1
cos sin
22
1
fx
bc d
t
?
??
?
=
++ ?
??
+
??
??
     (4.58) 
4
2
2
2
cos
3
()
21
cos sin
33
1
fx
bc d
t
?
??
?
=
++
??
+
??
??
     (4.59) 
5
2
2
5
cos
6
()
515
cos sin
66
1
fx
bc d
t
?
??
?
=
++
??
+
??
??
     (4.60) 
6
2
2
cos
()
1
cos sin
1
fx
bc d
t
?
??
?
=
++ ?
??
+
??
??
     (4.61) 
 
 
 88
4.7. References 
1. Zou, M. Q., Yu, B. M., and Zhang, D. M., 
?
An analytical Solution for Transverse 
Thermal Conductivities of Unidirectional Fiber Composites with Thermal 
Barrier?, J. Phys. D., 2002, 35, 1867-1874. 
2. Spiegel, M. R., Mathematical Handbook, Schaum?s Outline Series. 
3. Standard Mathematical Tables, CRC Press, 26
th
 Edition, 1981. 
4. Handbook of Mathematical, Scientific and Engineering Formulas, Tables, 
Functions, Graphs, Transforms, Research Education Association, 1994. 
5. Ciarlet, P.G., and Lions, J.L., Handbook of Numerical Analysis, Amsterdam; 
New York, 1990.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 89
 
 
 
 
 
CHAPTER 5 
HEAT TRANSFER ANALYSIS OF NANO-MICRO FIBER COMPOSITES BY 
FINITE ELEMENT METHOD 
 
5.1. Introduction 
The purpose of this study is to investigate the thermal behavior of fiber 
composites by using finite element package program ?ANSYS 7.0?.  The fiber size varies 
between nano and micro level. The fibers also show difference in terms of direction, and 
shape. The effective thermal conductivity, total heat flow, temperature difference and 
heat flux are analyzed.  
In the literature, many heat transfer problems in the area of fiber 
composites were solved by using finite element analysis. In many situations, 
calculation of heat conduction gave sufficient solution [1].  
Finite Element Method (FEM) is a numerical tool to solve numerous kinds of 
problems. The technique depends on the decomposition of a domain with a complicated 
geometry into geometrically simple elements. Once the single element solution is 
obtained, it is used to get the complete system by applying the boundary conditions. 
Basically, FEM consists of the following steps [2]; 
 
 90
a. Discretization of the domain which is a part of the process of dividing the 
geometry into smaller pieces called meshing. 
b.  Set-up element matrices,  
c. Transformation from natural coordinates into local coordinates, 
d. Assembly, 
e. Introduction of boundary conditions, 
f. Solution of the linear system of equations.  
5.2. A simple FEM Model 
Basic steady state heat transfer equation for Figure 5.1 is written as; 
0Q
dx
dT
k
dx
d
=+
?
?
?
?
?
?
      0< x<L     (5.1) 
Where x is the length of the thin rod, Q is the generated heat and T is the 
temperature [1].  
 
Figure 5.1. Heat conduction in a thin rod [1]. 
 
It is considered that the general boundary conditions are  
 91
 T= f on ?
1
 and q= ? on ?
2
 .  
 
The FEM analysis starts with choosing the appropriate element type which 
is ?the linear element? for this model. When the linear element is employed to the 
structure, Figure 5.1 transforms into Figure 5.2. 
 
Figure 5.2. Number of the nodes on the element [1]. 
 
()
()
2
h
2
h
2
h
2
xx
Nxx
e
e
e
21
2
1j
jj
?
+=
?
+
+
=?=
?
=
      (5.2) 
Here, N
1
 and N
2
 are orderly  
( )
2
1
)(N
1
??
=?   and  
( )
2
1
)(N
2
?+
=?  
with the trial function; 
()?=
?
=
2
1j
jj
NTT
?
   
When this is inserted in the main equation (5.2)  
i
e
i
e
i
qdxQNdx
dx
dN
dx
T
?
d
k ?=
??
 , i=1,2 
X
2
 
1 2 
X
X
1
 
h
e
 
e
Node 
Element 
L 
 92
[][][ ]
2
i
1
iii
Nq?Nq?Nq?q
???
+==  
 
i
e
ij
2
1j
e
i
j
qdxQNTdx
dx
dN
dx
dN
k ?=
?
?
?
?
?
?
?
?
?
=
, i=1,2 
 
[ ]
ij
2
1j
e
ijj
2
1j
e
i
j
qQdxNNTdx
dx
dN
dx
dN
k ?=
?
?
?
?
?
?
?
?
?
?
==
  ,i=1,2 
 
ij
2
1j
1
1
ijj
2
1j
1
1
i
j
qQd
d
dx
NNTd
d
dx
dx
d
d
dN
dx
d
d
dN
k ?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
=
?
 ,  i=1,2 
2
d
hdx
e
?
=  
If the above equations are summed up; 
ij
2
1j
1
1
ij
e
j
2
1j
1
1
i
j
e
qQdNN
2
h
Td
d
dN
d
dN
k
h
2
?
?
?
?
?
?
?
?=
?
?
?
?
?
?
?
??
?
?
?
?
=
?
=
?
 , i=1,2 
eeeee
qQMTK ?=  
Stiffness matrix is obtained as; 
[]
?
?
?
??
=
1
1
i
j
e
ije
d
d
dN
d
dN
k
h
2
K 
The mass matrix; 
[]
?
?
?
?
?
?
?=
?
?
1
1
ij
e
ije
dNN
2
h
M 
The element vectors are; 
 93
?
?
?
?
?
?
=
2
1
e
T
T
T   
?
?
?
?
?
?
=
2
1
e
Q
Q
Q    
?
?
?
?
?
?
=
2
1
e
q
q
q  
And finally by combining all of them; 
e
e
e
e
e
1-1k
K=
-1 1h
21h
M=
126
? ?
? ?
? ?
??
??
??
 
If the other element types are used, instead of linear element, which is the 
simplest element, the accuracy becomes higher, as shown in Figure 5.3. These elements 
are compared with the exact solution. 
 
Figure 5.3. The finite element division. 
 
It is clear that the more we divide the shape into parts, the more accurate results 
we get. But in nano size, it is not possible to divide the volume into many parts. ANSYS 
can solve and divide the problems up to 10
-60
m. It makes a distinction for nanosize.  
 
x
y 
Exact solution 
 
1 Linear element 
 
Quadratic element 
 
4 Linear elements 
 94
5.3. Modeling with ANSYS 
  The ANSYS software is evolved from a structural finite element analysis tool into 
a well developed multiphysics analysis tool and it has a capacity to be used in a broad 
range of areas including structural, thermal, fluid, acoustics, electromagnetics and 
electronic circuits more than 30 years. Especially the ANSYS/Multiphysics product can 
make a coupling between these physical areas. The size of the physical area can be down 
to the micro-system scale (10
-6
 m) and in some areas into the nano-scale (10
-9
 m) range 
[1].  
In ANSYS product family, steady-state and transient thermal analyses are 
supported by the divisions of ANSYS/Multiphysics, ANSYS/Mechanical, 
ANSYS/FLOTRAN and ANSYS/Thermal. In this study, ANSYS/Multiphysics was used 
to solve the steady-state problems.  
 
The solution procedure 
The basis for thermal analysis in ANSYS is a heat balance equation obtained from 
the principle of conservation of energy. The finite element solution performed via 
ANSYS calculates nodal temperatures, and then uses the nodal temperatures to obtain 
other thermal quantities. The ANSYS program could support five types of thermal 
boundary conditions; constant temperatures, constant heat fluxes, applied heat transfer 
(film) coefficient of convections, heat flow rates, and heat generation rates. 
 
 
 95
Step 1. Problem 
 Nano-micro fibers in the composite could be in the form of either aligned or 
random (Figure 5.4). In the modeling phase we considered our composite as aligned to 
use symmetry.   
 
 
Random fiber  composite 
 
Aligned fiber  composite 
Figure 5.4.  Composite structures 
Step 2.  Model  
In modeling, (a) fiber volume fraction and matrix volume fraction, (b) fiber aspect 
ratio, (c) fiber-matrix modulus ratio, (d) fiber spacing, (e) fiber orientation and (f) fiber 
end gap size are important. Both the matrix and fiber must be taken in nano-level in the 
modeling phase. 
 96
In Figure 5.5, the models used in ANSYS are given for both 2-D and 3-D models. 
 
Figure 5.5. Radial, axial and 3-D nano-micro composite models (d: the length of 
the unit cells). 
The model sizes inside the interface differ between nano and micro size: 
therefore, it is important to emphasize that the necessary computer memory increases as 
we decrease the size of the filler fiber, as can be seen in Figure 5.5. This comparison was 
done for 2-D models. 
 97
 
Figure 5.6. Necessary computer memory in modeling 
 
Figure 5.6 shows the necessary memory for the  modeling. When the size of the 
fiber decreases, the required memory size increases.  
 
Step 3:  ANSYS Analysis type 
Thermal analysis in the ANSYS has four different element types. The first one is 
mass element that allows us to consider melting and freezing. The others are link, solid 
and shell elements. Solid element is used if in-plane and transverse conduction is needed. 
 
 
Commonly used 2-D ANSYS elements 
The first element type used in this study is PLANE35 2-D triangular thermal 
solid. It has a 6-node triangular element. It has a triangular shape which allows meshing 
 98
in irregular structures. It has one degree of freedom, and temperature, at each node 
(Figure 5.7).  
 
Figure 5.7. PLANE35 2-D 6 node triangular thermal solid [3]. 
The other element type ?SOLID90? was used in modeling that has 20 nodes with 
a single degree of freedom (Figure 5.8).  
 
Figure 5.8. SOLID90 3-D 20 node triangular thermal solid [3]. 
 
 
 99
Step 4. Material Properties  
 
The thermal conductivity for nano-micro fibers was taken as a nanoclay thermal 
conductivity from the literature around 0.2 W/mK and for polymer or matrix around 1.2 
W/mK.  
Step 5. Meshing 
There are four main element types for thermal analysis; mass, beam, link and 
solid. Beam models are for 1-D line models, but they are simple for advanced 
calculations. Mass model is only for specific needs such as melting. Based on the element 
type, appropriate mesh is chosen for the solution.  The mesh types are shown in Figure 
5.9. 
 
 
 
 
 
 
 
Figure 5.9. Mesh types of 2-D and 3-D nano-micro composite models 
 
 100
Step 6. Boundary Conditions and Applied Loads. 
 
In ANSYS, for the thermal analysis, the following loads can be applied; 
1. Convection 
2. Radiation  
3. Heat flow rates  
4. Heat fluxes (heat flow per unit area)  
5. Heat generation rates (heat flow per unit volume)  
6. Constant temperature boundaries.  
In the modeling, only the constant temperature and constant heat flux were applied.  
5.4. Modeling Configurations 
5.4.1. The Effect of Material Properties of the Filler Fiber 
 In this configuration, the effect of fiber material properties on the temperature 
difference profiles is obtained. Eight different material properties were used to obtain the 
temperature difference. 
 The geometry is a 2-D rectangle model. The fiber, which is in the cylindrical form 
is used as filler (Figure 5.10). 
 101
 
Figure 5.10. Problem description 
Meshing is done after defining the material properties and the element type for the 
problem. Figures 5.11 and 5.12 show the meshing types for 3-D models. 
 
Figure 5.11. Mesh model of the problem 
 
X 
Y 
L
Matrix 
Fiber 
 102
 
Figure 5.12. Application of two different materials. 
Temperature difference was applied as a load to both right and left side in Figure 
5.12. The bottom and upper parts were thermally isolated.  
Materials with different thermal conductivity values were used for the fiber and 
matrix, as shown in Table 5.1.   
 
Table 5.1. Thermal conductivity of the materials. 
Materials 
Thermal 
conductivity 
(W/mK) 
PVA 2 
Acrylic 0.2 
Graphite 24 
Carbon 1.7 
Nylon 6 0.26 
Polyethylene 0.46 
Polypropylene 0.12 
Polystyrene 0.13 
 
 103
The temperature distribution of the materials is given in Figure 5.13.  
 
Figure 5.13. Temperature distribution for different materials. 
5.4.2. Polymer?s Heat Deflection Temperature (HDT) 
The polymer?s deflection temperature (HDT) can differ from one polymer to 
another polymer. If HDT is taken around 150? C, the maximum applicable heat flux was 
found for different radius of nanofiber. In Table 5.2., the HDT values of some polymers 
are given. 
Table 5.2. Some of the polymers? heat deflection temperatures and melting points [4]. 
 
Polymer Type 
Deflection 
Temperature at 
0.46 MPa (?C) 
Deflection 
Temperature at 
1.8 MPa (?C) 
Melting 
Point (?C) 
Acrylic 95 85 130 
Nylon 6 160 60 220 
Nylon 6 + 30% Glass Fiber 220 200 220 
Polyethylene, HDPE 85 60 130 
PET + 30% Glass Fiber  250 230 250 
Polypropylene + 30% Glass Fiber 170 160 170 
Polystyrene 95 85 - 
 104
Applied loads 
? Constant temperature to the rectangular shape with cylindrical filler was applied 
to the left, top and bottom sides as 10 ?C and, to the right side as 100 ?C. The 
temperature distributions are given in Chapter 6. Also to get one dimensional 
temperature distribution, temperature was only applied on the left and right sides.  
? Constant heat flux was applied to reach the heat deflection temperature (HDT).  
This is the temperature at which a standard test bar deflects 0.25 mm under a 
stated load of either 0.45 MPa or 1.82 MPa.  
? To reach the HDT values, the necessary heat flux values are given in Figure 5.14.  
 
0 
250000000 
500000000 
750000000 
800000000 
500000000 
1500000000 
1750000000 
2000000000 
Heat fl
ux 
[
W
]
 
r= 50 nm
r= 45 nm
r= 40 nm
r= 35 nm
r= 30 nm
r= 25 nm
r= 20 nm
r= 15 nm
r= 10 nm
r= 5 nm
r= 1 nm
No fiber
Fiber dimensions 
 
Figure 5.14. Maximum heat flux values for composite thermal stability. 
 
 
 105
5.4.3. Defining the Unit Cells 
Rectangular and hexagonal unit cells are compared for the nano-micro composite 
study. 
 A square which has a dimension of L x L (in the model it was taken as 2 mm) 
and a hexagon with a length of L for each side were taken as a dimension in ANSYS. By 
using different fiber volume fraction, the dimensionless effective conductivities were 
found and compared with each other to discuss which unit cell is more suitable for our 
study.  
As seen from Figure 5.15, both models have similar volume fraction, applied 
boundary conditions and material properties. As boundary conditions, -1? C and 1? C 
were applied to the left and right sides, respectively.    
 The dimensionless effective conductivity, which is explained in detail in Chapter 
6, was found by using the average thermal flux value in the X direction when Y= L/2.  
 
(a) 
  
(b) 
Figure 5.15. Unit cells a) Rectangular unit cell, b) Hexagonal unit cell 
 
 106
Table 5.3 lists the dimensionless effective thermal conductivities that were found 
depending on the material properties.  
Table 5.3. Material properties and dimensionless effective thermal conductivities of the 
hexagonal and rectangular unit cells. 
Fiber Matrix 
Effective 
Thermal 
Conductivity (k
e
) 
 
Dimensionless 
Effective 
Thermal 
Conductivity (k
e
+
) 
 
Comparison with 
Rule of Mixt. 
(%) 
Example (V
d
=0
.1
)
 
K
e1
 
(W/mK) 
K
e2
 
(W/mK) 
Rect. Hex. R H 
Rule of 
Mixtures 
 
R H 
1 0.1 10 8.24 6.53 0.824 0.653 0.8002 109.34 137.9 
2 10 10 10 10.04 1 1.004 1 100 99.60 
3 100 10 11.8 14.4 1.18 1.44 3.7 161.02 131.9 
Fiber Matrix Effective 
Thermal 
Conductivity 
(k
e
+
) 
 
Dimensionless 
Effective 
Thermal 
Conductivity (k
e
+
) 
 
Comparison with 
Rule of Mixt. 
(%) 
Example (V
d
=0
.2
)
 
K
e1
 
(W/mK) 
K
e2
 
(W/mK) 
Rect. Hex. R H 
Rule of 
Mixtures 
R H 
1 0.1 10 6.76 4.8 0.676 0.48 0.802 118.64 167.1 
2 10 10 10 10.2 1 10.02 1 100 98.04 
3 100 10 13.9 18.71 1.39 1.871 2.8 201.44 149.6 
Fiber Matrix Effective 
Thermal 
Conductivity 
(k
e
+
) 
 
Dimensionless 
Effective 
Thermal 
Conductivity (k
e
+
) 
 
Comparison with 
Rule of Mixt. 
(%) 
Example (V
d
=0
.3
)
 
K
e1
 
(W/mK) 
K
e2
 
(W/mK) 
Rect. Hex. R H 
Rule of 
Mixtures 
 
R H 
1 0.1 10 5.61 2.44 0.561 0.244 0.703 125.12 288.1 
2 10 10 10 10.07 1 1.007 1 100 99.30 
3 100 10 16.4 25.2 1.64 2.52 3.7 225.61 146.8 
 
Generally, thermal conductivity of the filler fibers has bigger values than the 
matrix and in order to use a conventional model, the difference between the thermal 
conductivities must be significant [5]. As seen from Table 5.3, compared to the 
rectangular unit cell, the hexagonal unit cell has less error with respect to the rule of 
mixtures. This leads us to use hexagonal cell for the ANSYS analysis.  
Heat flux variation and temperature distribution of the unit cells are given below. 
Figure 5.16 is the temperature difference and Figure 5.17 is the heat flux in the X 
 107
direction for rectangular shape. Figure 5.18 and 5.19 show the temperature and heat flux 
variation for the hexagonal shape.  
 
Figure 5.16. Temperature distribution in the X direction for rectangular shape. 
 
 
 
 
 
 
 108
 
Figure 5.17. Heat flux variation in the X direction for rectangular shape. 
 
Figure 5.18. Temperature distribution in the X direction for hexagonal shape. 
 
 109
 
Figure 5.19. Heat flux distribution in the X direction for hexagonal shape. 
5. 5. References 
1. Lewis, R.W., The Finite Element Method in Heat Transfer, Wiley, 1996. 
2. Lecture notes, http://www.ltm.uni-erlangen. de/ mitarbeiter/ Willner/ exercise 
Willner _061204.pdf (accessed in Jan. 2005) 
3. http://www.ansys.com/services/documentation/manuals.htm (accessed in May 
2004). 
4. http://www.matweb.com/reference/deflection-temperature.asp (accessed in March 
2005) 
5. Nan, C.-W., Shi, Z., Lin, Y., ?A Simple Model for Thermal Conductivity of 
Carbon Nanotube-based Composites?, Chemical Physics Letters, 2003, 375, 666-
669. 
 110
 
 
 
 
 
CHAPTER 6 
RESULTS AND DISCUSSION 
 
Thermal conductivity of materials may differ from one material to another and it 
is considered as one of the main heat transfer properties of the materials. The recent 
studies have shown that, the dimensionless effective thermal conductivity is important for 
the flame resistance behavior of materials.  
In this study, a heat transfer model is developed to discuss the dimensionless 
effective thermal conductivity of the nano-micro fiber  composites and a device is 
developed for continuous yarn manufacturing based on nano-micro sized fibers.  
Modeling studies consist of two parts: analytical and numerical modeling. The 
analytical heat transfer model is based on electrical analogy technique in which heat was 
considered to be applied in only one direction and therefore analytical modeling of 
transverse thermal conductivity became possible.    
 The analytical modeling was done considering the filler fiber placed inside 
another fiber which is different from the most other situations in which the composite is 
considered having fiber as reinforcement in a matrix. Here, since reinforcement and 
matrix have cylindrical shape, the hexagonal unit model was developed to study the heat 
transfer behavior.  
 111
The hexagonal model has three regions including the barrier region. The thickness of 
the barrier is normally around 10
-6
 -10
-8
 m in the previous composite studies. In the 
ANSYS and MATLAB modeling, the thickness size was taken as (0.1-0.3)*r
d
. Since the 
barrier thickness was taken as a separate region, after some value of the thickness, no 
difference in terms of k
e 
was observed.  Comparisons were done among the three 
methods: analytical modeling using MATLAB, rule of mixture which is used only when 
the barrier thickness is taken to be zero and the ANSYS modeling. 
For the analytical results, the MATLAB program was used to develop a simple model 
and then the boundary conditions were applied to it. For the ANSYS modeling, thermal 
conductivity values of the regions were given and then the total effective thermal 
conductivity was obtained by using the thermal flux values in the X direction. In the 
ANSYS modeling, since the model is not 3-D and the analysis type is steady-state, the 
whole shape was taken into consideration instead of the half part. If the symmetry 
property has to be used, there has to be symmetry in terms of physical structure, material 
properties and the applied loads. Among these, if one is not performed, the symmetry can 
not be considered. In the present model, the symmetry line is where y=0.  
Another objective of this study is to develop continuous yarn from micro-nano sized 
fibers. Electrospinning method is used to produce and collect the fibers. Pulling and 
twisting of the yarn was done as a second step.    
From the SEM results, it was seen that the radius of the fibers manufactured by 
electrospinning method varied between 200 nm and 2 ?m which is big enough to 
consider the continuum approach such as FEM for our model since it is well known that 
when the size of the particle is not bigger than the free path, the continuum approach 
 112
breaks down. Therefore, the term ?nano-micro? is used in our study to include both nano 
and micro concepts. 
 For the most of the thermal conductivity models, it is known that when the volume 
fraction of the filler is less than 20-30%, most of the models do not give correct values. In 
this research, it was observed that, once the hexagonal shape is used, the volume fraction 
of the filler can be decreased to 10%.  
6.1. MATLAB Results 
The analytical model was solved by using MATLAB to find the variation of the 
dimensionless effective thermal conductivity. As for the thermal resistance approach and 
the dimensionless attempt to simply the last form of the analytical equation, firstly, the 
equation was solved considering no barrier thickness in the model which makes the 
model consists of two divisions: matrix and the filler fiber where the filler fibers are both 
nano and micro dimensions. Then, the MATLAB program was run including a barrier 
thickness between (0.1-0.3)*r
d
.  
The dimensionless effective thermal conductivity is given in Figures 6.1-6.4, for 
volume fraction values of (0.1-0.3), when the dimensionless barrier thickness (t
+
) is  
(0.1-0.3) x r
d
. It is seen that, increasing the volume fraction of the filler fiber inside the 
matrix increases the total effective thermal conductivity values. Since the model is 
hexagonal shape, it helped us to restrict the volume fraction of the filler fiber between 0.1 
and 0.3. This is because, the first region in the analytical model does not contain the 
barrier and filler fiber regions. When ?=1, both the filler fiber and the reinforcement fiber 
have the same thermal conductivity; the dimensionless effective thermal conductivity 
 113
becomes one, which is an expected value from the model. The minimum values are 
obtained when the volume fraction value of the filler fiber is 0.3 and the ? (k
d
/k
f
) value is 
smaller than one.  
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.01 0.1 1 10 100
?
Vd=0.1
Vd=0.2
Vd=0.3
 
Figure 6.1. Change of dimensionless effective thermal conductivity with ? for different 
V
d
 values, (t=0). 
 
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.01 0.1 1 10 100
?
Vd=0.1
Vd=0.2
Vd=0.3
 
Figure 6.2. Change of dimensionless effective thermal conductivity with ? for different 
V
d
 values, (t=0.1 x r
d
). 
 114
As seen from the Figures, when the barrier thickness is increased, the 
dimensionless effective thermal conductivity value (k
e
) increases. This is because the 
barrier thermal conductivity values are taken as the average of the filler fiber and the 
reinforcement fiber?s thermal conductivities.  
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.01 0.1 1 10 10
?
Vd=0.1
Vd=0.2
Vd=0.3
 
Figure 6.3. Change of dimensionless effective thermal conductivity with ? for different 
V
d
 values, (t=0.2 x r
d
). 
 
The maximum volume fraction value is taken as 0.3; from Figure 6.4, the 
maximum dimensionless effective thermal conductivity is around 1.7. 
 115
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.01 0.1 1 10 100
?
Vd=0.1
Vd=0.2
Vd=0.3
 
Figure 6.4. Change of dimensionless effective thermal conductivity with ? for different 
V
d
 values, (t=0.3 x r
d
). 
 
Figures 6.5-6.7 show the effect of barrier thickness on k
e
. As explained earlier, the 
barrier region is taken as the third region in the hexagonal model. Since its thickness 
values do not show big difference, the k
e
 values are found to be close. 
 
 
 116
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.01 0.1 1 10 100
?
t=0
t=0.1*rd
t=0.2*rd
t=0.3*rd
 
Figure 6.5. Effect of the barrier thickness on the dimensionless effective thermal 
conductivity, (V
d
=0.1). 
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.01 0.1 1 10 100
?
t=0
t=0.1*rd
t=0.2*rd
t=0.3*rd
 
Figure 6.6. Effect of the barrier thickness on the dimensionless effective thermal 
conductivity, (V
d
=0.2). 
 117
 
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.01 0.1 1 10 100
?
y(
)
t=0
t=0.1*rd
t=0.2*rd
t=0.3*rd
 
Figure 6.7. Effect of the barrier thickness on the dimensionless effective thermal 
conductivity, (V
d
=0.3). 
 
6.2. ANSYS Results 
As reported by Islam [1], there are at least four different ways to calculate the 
dimensionless effective thermal conductivity by FEM. The first one, which is used in this 
study, is to apply temperature difference as boundary conditions to the left and right 
sides. The second is to apply both constant temperature and constant heat flux to the 
sides. The third is to apply constant heat flux to both sides and the last one is to apply 
temperature differences as boundary conditions at the horizontal boundaries.  
Hexagonal unit cell was used with the ANSYS finite element program. The 
solution was obtained in different steps. The material properties were entered first, 
 118
followed by element type, then modeling the shape, and applying the boundary 
conditions. Finally, the results were obtained which is referred to as post procedure. 
As the first step, to reduce the computer memory requirement, the symmetry 
effect was discussed. Since the problem is symmetrical around two planes of symmetry, 
only one quarter of the real volume specimen is considered in the model.  
The GLUE command was used after establishing the parts of the model to give 
the initial composite properties to the shape. The domain is discretizated into smaller 
regions both by PLANE35 and PLANE55 elements. PLANE35 has a triangle element 
and as a result it has one degree of freedom at the nodes.  PLANE55 was used to model 
the areas which have small thickness especially the barrier region. An important thing to 
be noted here is that, although the layer elements are used in mechanical modeling to take 
the advantage of the composite structure, in heat transfer modeling, there are not so many 
available layered shell element types. 
As material properties, only thermal conductivity was given and it is assumed to 
be constant and isotropic in every direction. After defining the material properties, and 
introducing to the areas, the meshing process starts.  Meshing was done mostly 
automatically. The number of division was increased between the lines to get more 
sensitive results.  
The boundary conditions for all the problems were taken similarly as applied 
constant temperatures to both left and right sides of the vertical line. For ANSYS, the 
effective thermal conductivity, when there is no heat flux applied and when the 
dimensions are 2a x 2a ,  is given for the unit cell as follows: 
 
 119
 
1
(, )
2
2
a
xx
a
x
qqayd
a
T
T
a
?
=
?
=
?
  
              
T
a2.q
k
x
e
?
?=                        (6.1) 
where,  
x
q  : average heat flux value in x direction,  
a    : length of the unit cell, 
T? : temperature gradient, 
k
e   
 : effective thermal conductivity, 
xT  : average temperature value in x direction. 
 
 
 
 
 
 
 
 
 
 
 
 120
 
 
 
Figure 6.8.  k
e
+
 calculations for the hexagonal unit cell in ANSYS 
 
For the hexagonal cell (Figure 6.8), the average heat flux in the X direction for the 
first region; 
2
0
33
,
22
L
xx
qLqLydy
??
=
??
??
?
      (6.2) 
 
The dimensionless effective thermal conductivity can be calculated as; 
k
e
+
 =
3
.
2
x
qL
T
?
?
       (6.3) 
? Temperature was applied for the left side as -1?C and for the right side as 1?C.  
? Thermal conductivity of the main fiber was taken as 10 W/mK and only the 
thermal conductivities of filler fiber and barrier were changed. 
Average heat 
flux value 
calculated 
region  
3
2
L  
  Y 
       X 
 121
? k
b
 is taken as the average of the total thermal conductivities.  
 
Figure 6.9. Dimensionless effective thermal conductivity (t=0). 
 
In Figure 6.9, the k
e
+
 value was compared when the barrier thickness is zero for 
different filler fiber volume fractions. The k
e
+
 value is also calculated when there is a 
barrier thickness and the results are shown in Figures 6.10-6.12. 
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0.001 0.01 0.1 1 10 100
?
Vd=0.1
Vd=0.2
Vd=0.3
 
Figure 6.10. Dimensionless effective thermal conductivity (t=0.1x r
d
). 
 122
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0.001 0.01 0.1 1 10 100
?
Vd=0.1
Vd=0.2
Vd=0.3
 
Figure 6.11. Dimensionless effective thermal conductivity (t =0.2 x r
d
). 
 
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0.001 0.01 0.1 1 10 100
?
Vd=0.1
Vd=0.2
Vd=0.3
 
Figure 6.12. Dimensionless effective thermal conductivity (t =0.3 x r
d
). 
 
 123
0
1
2
3
4
5
0.001 0.01 0.1 1 10 100
?
t=0
t=0.1*rd
t=0.2*rd
t=0.3*rd
 
Figure 6.13. Dimensionless effective thermal conductivity (V
d
=0.1). 
 
Next, the effect of the barrier thickness is discussed when there is a constant V
d
. 
Compared to the MATLAB results, in ANSYS, the results showed more difference 
(Figures 6.13 - 6.15). 
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0.001 0.01 0.1 1 10 100
?
t=0
t=0.1*rd
t=0.2*rd
t=0.3*rd
 
Figure 6.14. Dimensionless effective thermal conductivity (V
d
= 0.2). 
 
 124
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0.001 0.01 0.1 1 10 100
?
t=0
t=0.1*rd
t=0.2*rd
t=0.3*rd
 
Figure 6.15. Dimensionless effective thermal conductivity (V
d
= 0.3). 
 
The Rule of mixtures has been used for a long time to define the dimensionless 
effective thermal conductivity values when there is no thickness (Table 6.1 and Figure 
6.16). The Rule of Mixtures can be described as; 
k
c 
= ?k
f
 + (1-?)k
m
 
where; 
k
c
 :   composite?s thermal conductivity, 
k
f  
:   filler fiber?s thermal conductivity, 
k
m
 :  matrix?s thermal conductivity, 
?   :  volume fraction of the filler fiber. 
 
The Rule of Mixtures results were compared to ANSYS and MATLAB results in 
Figures 6.17-6.19. The MATLAB and ANSYS results showed much similarity compared 
to the Rule of Mixtures. The comparison with the Rule of Mixtures results was limited to 
 125
only when there is no barrier thickness. It was found that although the filler fiber and the 
matrix have similar thermal conductivity values, when the difference between the thermal 
conductivities increases, the difference between the dimensionless effective thermal 
conductivity values also increases. 
Table 6.1. Dimensionless effective thermal conductivity by Rule of Mixtures. 
 
Volume fraction of the 
filler fiber (V
d
) 
 ?(k
d
/k
f
) 0.1 0.2 0.3 
0.001 0.9001 0.8002 0.7003 
0.01 0.901 0.802 0.703 
0.10.910.820.73 
1111
10 1.9 2.8 3.7 
Dimensionless 
Effective 
Thermal 
Conductivity 
(k
e
+
) 
 100 10.9 20.8 30.7 
0
0.5
1
1.5
2
2.5
3
3.5
4
0.001 0.01 0.1 1 10
Vd=0.1
Vd=0.2
Vd=0.3
 
Figure 6.16. Dimensionless effective thermal conductivity by the Rule of Mixtures 
 
 
? 
 126
0
0.5
1
1.5
2
2.5
3
3.5
4
0.001 0.01 0.1 1 10
?
ANSYS
Rule of Mixtures
 
Figure 6.17. Dimensionless effective thermal conductivity comparison between the 
ANSYS and Rule of Mixtures (V
d
=0.2). 
 
0
0.5
1
1.5
2
2.5
3
3.5
4
0.001 0.01 0.1 1 10
?
ANSYS
Rule of Mixtures
 
Figure 6.18. Dimensionless effective thermal conductivity comparison between the 
ANSYS and Rule of Mixtures (V
d
=0.3). 
 
 127
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0.01 0.1 1 10
?
ANSYS
MATLAB
Rule of Mixtures
 
Figure 6.19. Comparison of the dimensionless effective thermal conductivity among the 
ANSYS, MATLAB and Rule of Mixtures ( t=0 and V
d
=0.1). 
 
When the barrier thickness is zero, the barrier region becomes part of the matrix, thus 
only two different thermal conductivity values (for matrix and filler fiber) were given for 
the model. As stated earlier, under the steady-state conditions, the boundary conditions 
were applied to the left and right side of the model as -1 ?C and 1 ?C respectively. 
  Figure 6.20 shows the meshing of the hexagonal unit cell. When the model 
becomes smaller, the dimensions of the triangle elements also become smaller. The 
different color distribution is the result of the same element type but different material 
properties. Two different material properties were introduced to the model: for cylindrical 
part and for the main fiber part. 
 128
 
Figure 6.20. Meshing for the hexagonal model 
 
Figure 6.21 shows the temperature distribution of the hexagonal when the thermal 
conductivity of the filler fiber (k
d
) is 1000 W/mK and the thermal conductivity of the 
main fiber (k
f
) is 10 W/mK. Since the boundary conditions are -1?
 
C to the left side and 
1? C to the right side and all the force was applied in the X direction, the temperature 
distribution is expected to increase closer to the filler fiber since it has bigger thermal 
conductivity.  
 
 129
 
Figure 6.21.  Temperature distribution of the hexagonal unit cell ( t=0 and V
d
 =0.1). 
 
Figure 6.22 shows the heat flux variation in the X direction under the similar 
conditions of Figure 6.21. Figure 6.23 shows the vectorel form of the heat flux variation. 
It is important to note that, there is no barrier layer for the results up to Figure 6.28. 
 
Figure 6.22.  Heat flux distribution in the X direction of the hexagonal unit cell (t=0 and 
V
d
=0.1). 
 130
 
 
Figure 6.23.  Heat flux vector in the X direction of the hexagonal unit cell (t=0 and V
d 
=0.1). 
 
In Figure 6.24, thermal gradient in the X direction is given. At the top and bottom, 
thermal gradient has maximum values, since no boundary conditions were applied to 
these edges. 
 131
 
Figure 6.24. Thermal gradient in the X direction of the hexagonal unit cell (t =0 and 
(V
d
=0.1, k
d
=1000, k
f
=10). 
 
In Figures 6.25 and 6.26, k
d
 is decreased to 0.01 W/mK and the other factors are 
kept the same. Heat flux and thermal gradient values showed difference especially closer 
to the filler fiber area. 
 132
 
Figure 6.25. Heat flux in the X direction of the hexagonal unit cell (t=0, V
d
=0.1, k
d
=0.01, 
k
f
=10). 
 
 
Figure 6.26. Thermal gradient in the X direction of the hexagonal unit cell (t=0,V
d
=0.1, 
k
d
=0.01, k
f
=10). 
 
 133
Temperature difference is the result of the boundary conditions and the thermal 
conductivity of the filler fiber and the matrix. Since the boundary conditions were applied 
in only through the X direction, the heat flux variation in Y direction is found to be much 
smaller than the X direction.  The vectors show the direction of the heat transfer inside 
the model. If it is negative, it means that the direction of the heat flow is not to the model 
direction. Also, it is important to note that since the model is 2-D, heat flux was not 
applied as boundary conditions. 
Since the interface region is, under normal conditions, the result of improper 
parameters such as voids, cracks, and the like, the barrier layer is modeled to have small 
thickness value, i.e.  0.1 x r
d 
 , 0.2 x r
d 
 and 0.3 x r
d 
 where r
d
 is the radius of the filler 
fiber. The reason why the thickness is to be taken bigger than the size which was taken 
for MATLAB is because of the sensitivity of the ANSYS. Since the main model is much 
bigger than the barrier region, the ANSYS results became closer to the case where there 
is no barrier layer. The analysis showed that, the thickness has to be considered at least 
bigger than 0.1 x r
d
.  
Figures 6.27-6.30 show the results obtained to determine the barrier effect on the 
dimensionless effective thermal conductivity. The thermal conductivity of the barrier is 
taken as the average of the filler fiber and matrix?s thermal conductivity. Thermal 
conductivity of the matrix is taken constant as 10W/mK. When the volume fraction of the 
filler fiber is increased, its effects on the thermal conductivity are shown in Figure 6.31-
6.33.  
 134
 
Figure 6.27. Thermal gradient in the X direction of the hexagonal unit cell (t=0.1 x r
d
 and 
V
d
= 0.1). 
 
 
 
 
Figure 6.28. Thermal gradient vector in the X direction of the hexagonal unit cell (t=0.1 
x r
d
 and V
d
= 0.1). 
 
 135
 
Figure 6.29. Thermal gradient in the X direction of the hexagonal unit cell (t=0.2 x r
d
 and 
V
d
= 0.1). 
 
 
Figure 6.30. Thermal gradient in the X direction of the hexagonal unit cell (t=0.1 x r
d
 and  
V
d
= 0.1). 
 
 
 136
 
Figure 6.31. Heat flux in the X direction of the hexagonal unit cell ( t=0.2 x r
d
 and 
V
d
=0.2). 
 
 
Figure 6.32. Temperature gradient in the X direction of the hexagonal unit cell (t=0.2 x r
d
 
and V
d
= 0.2). 
 
 137
 
Figure 6.33. Temperature distribution in the X direction of the hexagonal unit cell (t=0.2 
x r
d
 and V
d
= 0.2). 
 
6.3. SEM Results 
Scanning electron microscope (SEM) was used to determine the dimensions of 
the electrospun fibers. The SEM study was done after gold coating in the Biology 
Department. 
First PVA solutions were prepared with 10, 12 and 15 wt% PVA. By using the 
electrospinning method, fiber web was manufactured. The effect of electrospinning time 
was also investigated (Figures 6.34-6.35). A continuous web form developed after 1-2 
hours, which gave us enough area and thickness to pull the electrospun based fiber web 
from the collector.  
 138
 
Figure 6.34. Web formation after 1 second of spinning. 
 
 
Figure 6.35. Web formation after 24 seconds of spinning. 
 
The web was placed on a pan for coating. To save time, the stub was divided into 
four parts as shown in Figure 6.36. To avoid the melting problem that can be caused by 
SEM electron bombing, the coating time was increased to 4 minutes. 
 139
 
Figure 6.36. SEM pan 
 
The effect of concentration on the fiber size can be seen in Table 6.2. When the 
concentration is increased, the fiber size also increases.  10, 12 and 15 wt % PVA 
concentrations were used to compare the fiber sizes. When the concentration is increased, 
it also affected the uniformity of the fiber. For 10 wt % PVA concentration, the fiber 
diameter was around 410 nm, for 12 wt % PVA concentration, it was around   425 nm 
and lastly for 15 wt % PVA, it was 563 nm. 
 
 
 
 
 
 
 
 
 140
Table 6.2. Fiber dimensions under SEM. 
Fiber Diameter (nm) 
Section 1 Section 2 Section 3 Section 4 
158 100 100 176 
280 158 194 210 
310 229 229 280 
350 280 270 310 
380 320 280 520 
494 370 282 810 
700 380 350 1376 
758 458 410  
1
st
  sample 
(10 wt %) 
880 547 458  
Average 478.89 315.77 321.63 526 
Total average 410.57 
Fiber Diameter( nm) 
Section 1 Section 2 Section 3 Section 4 
490 280 490 560 
420 600 600 700 
280 450 420 200 
2
nd
 sample 
(12wt %) 
317 380 300 317 
Average 376.75 427.5 452.5 444.25 
Total average 425.25 
Fiber Diameter( nm) 
Section 1 Section 2 Section 3 Section 4 
420 420 1830 940 
310 630 700 200 
520 280 280  
350 247 560  
280 560   
560 400  
3
rd
 sample 
(15wt %) 
490    
Average 418.57 422.83 842.5 570 
Total average 563.47 
 
Figures 6.37-6.41 are from the SEM results. As seen from the figures, fibers are 
not aligned. When the magnification is increased, fiber bundles were observed which 
were originally thought as the bigger radius fibers with small magnification (Figure 6.38). 
 141
 
Figure 6.37.  SEM image of nanofiber based web (10 wt. % PVA, 3000x ). 
 
 
 
Figure 6.38. SEM image of nanofiber bundle (12 wt.% PVA, 10000x). 
 
 
 142
 
Figure 6.39. SEM image of nanofibers (12 wt. % PVA, 10000x). 
 
 
Figure 6.40. Variety of fiber size (15 wt. % PVA, 3000x). 
 
 
 143
 
Figure 6.41. Single fiber (12wt. % PVA, 10000x). 
 
After manufacturing of pure PVA fibers, the Laponite? powder was added to the 
PVA solution. Figure 6.42 shows SEM result when the Laponite? and PVA are prepared 
together. When the Laponite? is added to the solution, it may cause a slub in the fiber, as 
shown in the figure.  
 
 144
 
Figure 6.42. PVA and Laponite? under SEM (12 wt PVA + 5 wt % Laponite?, 
10000x). 
 
One of the problems to be mentioned in electrospinning is the beading problem 
(Figure 6.43). To obtain a uniform web, the beading needs to be reduced. After using 
different concentration values of PVA, the 12 wt % was decided to be used for the rest of 
the study, since it produced the least amount of beading. Beading can be affected from 
surface tension of the web form.  When the viscosity increased, it causes less beading 
problem (Figure 6.44).  
 
Laponite? 
 145
 
Figure 6.43.  Bead problem in electrospinning (15 wt% PVA, 1000x). 
 
 
Figure 6.44. Bead formation [2]. 
 
Bead 
 146
6.4. Tensile Testing 
Mechanical properties are important to measure in terms of stability. Water was 
used as a solvent which also acts as a plasticizer and causes humidity inside the material. 
This, in turn, causes the elongation and tear strength to increase. The more plasticizer 
used, the more elongation and tear strength expected. However, tensile strength is 
decreased with plasticizer. 
Tensile strength testing was done to find out the strength of the manufactured 
yarns. 12 wt % PVA was used to manufacture fibers. Then a piece of yarn was produced 
by twisting ten different bundles of fibers. The yarn pieces were made from fibers with 
similar shape and dimensions. Using the Instron machine, the breaking tensile strength of 
the yarns was measured. Before measuring, the specimen was vertically aligned to avoid 
the side loading or bending moments.  
For a representative yarn, the maximum load was 0.246 kgf and the displacement 
was around 12 mm (Figure 6.45).  
 
The properties of the yarn: 
     Linear Density:  300.0 den 
     Spec gauge len:  15.0 mm 
     Ext. gauge len:  101.6 mm 
 
 
 
 
 147
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0 5 10 15 20 25
Displacement (mm)
A
p
p
l
i
e
d
 l
oad
 (
k
gf
)
12 wt % PVA
 
Figure 6.45. Load-elongation curve for nano-micro fiber based yarn. 
 
Figure 6.46 shows the comparison of several yarns for tensile strength. 
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0 5 10 15 20 25
Displacement (mm)
M
A
x
i
mu
m lo
a
d
 (
k
g
f
)
Sample 1
Sample 2
Sample 3
Sample 4
Sample 5
Sample 6
Sample 7
Sample 8
Sample 9
Sample10
 
Figure 6.46. Comparison of load-elongation curves for nano-micro fiber based yarns 
 148
Even though the samples were prepared from the same solution, the maximum 
load showed some difference for 10 different samples. This shows that the weight and the 
fiber distribution were not uniform. 
 
6.5. Air Permeability  
Electrospun webs have less pores and high surface tension compared to the other 
nonwoven structures. Air permeability test was done to evaluate and compare different 
fabrics for breathability.  
For 12 wt. % PVA based electrospun web, 4 different samples were tested to get 
an average air permeability value. The air permeability values for all the samples were 
consistent (Table 6.3). 
Table 6.3.  Air permeability of the samples 
Sample Fabric properties 
Average 
Penetration 
time 
(sec) 
Air 
Permeability 
(  m
3
/min/m
2
) 
Average 
Thickness 
(mm) 
1 
Nanofiber web 
(sample 1) 
46.4 
(maximum 
thickness) 
9.86*10
-10
 1.34 
 
2 
Nanofiber web 
(sample 2) 
6.733 6.79*10
-9
 0.037 
3 
Nanofiber web 
(sample 3) 
7.65 5.98*10
-9
 0.034286 
4 
Nanofiber web 
(sample 4) 
6.27 7.29*10
-9
 0.0308 
 
 149
6.6. Surface Tension  
Surface tension is important to avoid beading problem. The voltage effect also 
depends on surface tension. Surface tension depends on the polymerization of the 
solution. When the polymerization is low, the surface tension is also expected to be low.  
When Laponite? clay is added into the PVA solution, surface tension increases 
from 63.07 mN/m to 69 mN/m as shown in Table 6.4. This increase is mostly the result 
of high surface area of the Laponite? particles. This is also proved by Zhang  by 
comparing different polyvinyl alcohol concentrations [3]. 
 
Table 6.4.  Surface tension of the solutions. 
 
Material 
 
Surface Tension (mN/m) 
Samples 
1
st
 
measurement
 
2
nd 
 
3
rd
 4
th
 5
th
 6
th
 Average
PVA 
(12%) 
 
67.47 
 
 
52.48 
 
 
63.85 
 
67.10 
 
64.97 
 
62.
60 
 
~ 63.07 
 
PVA + 
Laponite? 
(12%+5%) 
70.72 64.91 72.25 71.17 68.20 -- ~69.45 
 
 
6.7. Differential Scanning Calorimetry (DSC)  
DSC is a method to figure out the relationship between temperature and heat flow. 
It provides quantitative and qualitative data on endothermic and exothermic processes of 
materials which are the result of melting, oxidation, and other heat related changes. 
 150
 In this study, pure PVA and Laponite? added solutions are investigated to 
understand the glass transition and melting point differences.  
 PVA has a glass transition and melting point that depend on if it is partially or 
fully hydrolyzed (Table 6.5.). It can be assumed that, the melting point changes between 
180?-230? C. After adding Laponite? the melting point increased. 
 
Table 6.5. PVA properties [4] 
 Partially Hydrolyzed Fully Hydrolyzed 
Glass Transition 
Temperature (?C) 
58 85 
Melting point  
Temperature(?C) 
180 230 
 
The samples were prepared for measuring under modulated differential scanning 
calorimetry (MDSC). The weight was around 5-6 mg and encapsulated in the pan. Since 
a polymer is used for MDSC, the thickness is taken as small as possible to get more 
accurate results.  
Figures 6.47-50 are the MDSC results, from which we may determine the 
crystalline temperatures, glass transition temperatures, melting points, etc. The effect of 
the Laponite? addition was shown by comparing the glass transition and melting point of 
different samples. The samples are 12 wt %PVA, 12 wt % PVA + 1% wt Laponite? , 12 
wt % PVA + 2 wt % Laponite? and 12 wt % PVA + 3% wt Laponite?.  
 
 151
222.84?C
-1.863W/g
185.87?C
-0.7881W/g
119.76?C
-1.904W/g
154.95?C
2.701W/g
-2
0
2
4
6
Heat Flow (W/g)
0 50 100 150 200 250 300
Temperature (?C)
Exo Up Universal V3.8B TA Instruments
 
Figure 6.47. DSC results of 12 wt % PVA 
 
194.41?C
-0.6405W/g
112.65?C
-1.131W/g
226.75?C
-2.085W/g
165.26?C
2.480W/g
-4
-2
0
2
4
He
at
 Flow (W/
g
)
0 50 100 150 200 250 300
Temperature (?C)
Exo Up Universal V3.8B TA Instruments
 
Figure 6.48.  DSC results of 12 wt % PVA+ 1 wt % Laponite? 
 152
224.62?C
-1.331W/g
192.27?C
-0.4978W/g
117.27?C
-1.437W/g
162.41?C
2.203W/g
-2
0
2
4
Heat
 Fl
ow (
W
/
g
)
0 50 100 150 200 250 300
Temperature (?C)
Exo Up Universal V3.8B TA Instruments
 
Figure 6.49. DSC results of 12 wt %PVA+ 2 wt % Laponite? 
 
 
226.40?C
-1.411W/g
116.56?C
-1.420W/g
168.81?C
1.988W/g
140.38?C
2.460W/g
197.25?C
-0.4346W/g
-2
0
2
4
Heat Flow (W/g)
0 50 100 150 200 250 300
Temperature (?C)
Exo Up Universal V3.8B TA Instruments
 
Figure 6.50. DSC results of 12 wt %PVA+ 3 wt % Laponite? 
 153
Glass Transition Temperature 
The glass transition temperatures were found out different for pure PVA sample 
and PVA + Laponite? sample. The glass transition temperature indicates when the 
structure starts to have molecular mobility inside. At this time, enthalpic relaxation 
occurs and since this is a kinetic process, it is seen in nonreversing heat flow. As seen 
from the DSC results, the glass transition temperatures are always lower than the 
crystalline melting points. Intermolecular forces are important for the mobility of the 
molecules. When the PVA solution contains Laponite? inside, it becomes harder to have 
a molecular movement, thus the glass transition temperature increases when the solution 
has higher concentration of Laponite?. Amorphous structure and crystillation process 
can be seen in Figure 6.51. 
 
 
Figure 6.51. Schematic representation of unfolding of polymer chains a) Original 
polymer crystal, b) Crystal with a solvent [5]. 
 
 154
Glass transition temperature was determined by averaging the first sudden heat 
flow peak value and the point when the flow starts to decrease. For the additional 1 wt % 
Laponite? into the solution, the glass transition temperature was obtained as;  
 
The glass transition temperature : 
( )
09.99
2
74.10766.92
2
XX
X
21
=
+
=
+
=  ?C 
where   X
1
:heat flow value when the flow starts to show a sudden decrease 
X
2
: The first peak value 
For the different Laponite? addition, the glass transition values are listed in Table 6.6. 
 
Table 6.6. Comparison of glass transition temperatures 
Solution 
Glass Transition 
Temperature 
(?C) 
Pure 12 wt %PVA 
 
105.00 
1 wt % Laponite? +12 
wt % PVA 
107.74 
2 wt % Laponite?+12 
wt %PVA 
115.17 
3 wt % Laponite? +12 
wt %PVA 
114.28 
 
The results showed that, when the Laponite? is added in PVA solution, both the 
glass transition and the melting points increase. 3 wt% showed a little decrease compared 
to 2 wt % Laponite? addition for the glass transition temperatures; but the melting point 
showed an increase.   
 
The glass transition values are given in Figures 6.52- 6.54. 
 155
 
Figure 6.52. Glass transition temperature region for 12 wt % PVA + 1 wt % Laponite?. 
109.44?C(I)
99.65?C
115.17?C
Glass Transition Temperature 
12 wt % PVA + 2 wt % Laponite
-4
-3
-2
-1
0
1
He
at Flow (W
/g
)
40 60 80 100 120 140 160 180
Temperature (?C)
Exo Up Universal V3.8B TA Instruments
 
Figure 6.53. Glass transition temperature region for 12 wt % PVA and 2 wt % 
Laponite?. 
99.09?C(I)
92.66?C
107.74?C
Glass Transition Temperature 
12 wt % PVA  1wt % Laponite 
-3 
-2 
-1 
0
1
Heat Flow (W/g)
60 80 100 120 140 160
Temperature (?C)
Exo Up Universal V3.8B TA Instruments
 156
111.05?C(I)
106.53?C
114.28?C
Glass Transition Temperature 
12 wt % PVA + 3% Laponite
-2.0
-1.5
-1.0
-0.5
0.0
0.5
H
e
at
 Fl
ow
 (W
/
g
)
60 80 100 120 140 160 180 200
Temperature (?C)
Exo Up Universal V3.8B TA Instruments
 
Figure 6.54. Glass transition temperature region for 12 wt % PVA and 3 wt % 
Laponite?. 
 
 
Melting 
Melting only occurs in the crystalline regions of the polymer which means there is 
enough energy for polymer?s formation inside the crystalline regions. Normally, melting 
is affected by several factors: intermolecular forces, stiffness, symmetry and regularity of 
the polymer. It was observed that, by adding more Laponite? inside the PVA solution, 
the melting point showed a higher value as shown in Figure 6.55. 
 
 157
226.19?C
-1.407W/g
224.67?C
-1.332W/g
226.99?C
-1.831W/g
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
H
eat Fl
o
w
 (W/g)
190 200 210 220 230 240 250
Temperature (?C)
???????   12 wt % PVA +2 wt % Laponite
? ? ? ?       12 wt % PVA + 1wt % Laponite
????? ?      12 wt % PVA + 3 wt % Laponite
Exo Up Universal V3.8B TA Instruments
 
Figure 6.55. Melting point comparison among three samples: 12 wt % PVA + 1wt % 
Laponite?, 12 wt % PVA + 2wt % Laponite?, and 12 wt % PVA + 3wt % Laponite? 
 
6.8. Thermogravimetric Analysis (TGA) 
Thermogravimetric analysis measures the rate of weight change in a material with 
increasing temperature over time due to decomposition, oxidation, or dehydration. 
In this study, TGA is used to understand how Laponite? affects the PVA fiber 
web in terms of weight loss. Figures 6.56 and 6.57 show that by adding Laponite? inside 
the PVA solution, the barrier properties and thermal stability were improved. 
 
 
 
 158
z
z
z
z
z
z
z
z
z
z
z
z
z
?
?
?
?
? ?
?
?
?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
Neat PVA
PVA + 1% Laponite
PVA + 3% Laponite
80
85
90
95
100
105
110
Weight (%
)
0 50 100 150 200 250 300
Temperature (?C)
z??????  pva.001  PVA
? ? ? ?     PVA1lap.001  PVA1lap
????? ?    PVA3lap.001  PVA3lap
Universal V3.8B TA
 
Figure 6.56. Weight loss comparison by TGA up to 300 ?C among three samples: Pure 
12 wt % PVA, 12 wt % PVA+ 1 wt % Laponite?, and 12 wt % PVA + 3 wt % 
Laponite?. 
 
 159
z
z
z
z
z
z
z
z
z
z
z
z
z z
z z
z
?
?
?
?
?
?
?
?
?
?
?
? ? ? ? ?
?
?
?
?
?
?
?
?
?
?
? ? ? ? ?
0
20
40
60
80
100
120
W
e
ight (%
)
0 200 400 600 800
Temperature (?C)
z??????  pva.001  PVA
? ? ? ?     PVA1lap.001  PVA1lap
????? ?    PVA3lap.001  PVA3lap
Universal V3.8B TA
 
Figure 6.57. Weight loss comparison by TGA up to 800 ?C among three samples: Pure 
12 wt % PVA, 12 wt % PVA+ 1 wt % Laponite?, and 12 wt % PVA + 3 wt % 
Laponite?. 
 
 
 
 
 
 
 
 
 
 
 160
6.9. Continuous Yarn Manufacturing 
Continuous yarn manufacturing is a challenge. There are many parameters that 
have to be taken into consideration before modeling the set-up. 
There were two important parameters before designing the process: to get a linear 
voltage distribution and to get the fibers aligned in the electrospun web. The first concern 
was solved by placing the collector behind a nonwoven fabric. The second issue was 
solved by rotating the cylindrical collector and nonwoven fabric together. By rotating the 
nonwoven fabric, the pulling becomes much easier and thus continuous fiber bundle can 
be obtained. It is important to note that, when the nonwoven fabric is placed in a 
cylindrical shape pipe between the collector and the syringe, the spinning becomes 
inefficient. Another important factor is that the needle is placed at the back of a wooden 
piece which helps to contribute the linearity of the electrical forces. The picture and 
schematic of the collecting mechanism is shown in Figure 6.58.  
 
 
 
 161
 
 
Figure 6.58. Picture and schematic of the collecting mechanism 
 
 
Needle 
Aluminum 
collector 
Nonwoven 
fabric 
Hand controlled 
rotation  
 162
 
Experimental Observations 
? Circular piece of the collector is much useful for gathering the nanofibers 
together. It is also seen that, by the help of a magnet, the distribution of the fibers 
was towards the circular piece.   
? When two needles were used instead of one, spinning efficiency could be 
increased. 
? When nonwoven fabric is used as a collector, formation of film is avoided. 
Spinning time also has an effect on film forming. 
? 12 wt % PVA gave the optimum spinning conditions with reduced beading. 
? The space between the collector and needle was kept constant to have an effective 
spinning. 
? At some point magnetic resonance was seen, which naturally created a spinning 
by the help of vibration and made it easy to have aligned fibers.  
? When the area of the collector is increased, the spinning efficiency also increases. 
Therefore ?aluminum pools? were manufactured to increase the collector?s area. 
After creating these pools, we used them as a collector (Figure 6.59).  
 163
 
Figure 6.59. Aluminum pool. 
? Yarn coating is done by using different yarns. It was seen that wetting the yarn before 
coating improves the coating. Figure 6.60 shows the yarns that are  coated by 
electrospun PVA fibers. When the radius is decreased, the coating becomes much 
harder.  
  
(a) (b) 
 
Figure 6.60. Coated yarns 
 
 
Aluminum 
pipes 
 164
6.10. References 
1. Islam, Md. R., Pramila, A., ?Thermal Conductivities of Fiber Reinforced Composites 
by the FEM?, Journal of Composites, 1998, 33(18), 1699-1715. 
2. Fong, H., and Reneker, D. H., ?Beaded Nanofibers Formed During Electrospinning?, 
Polymer, 1990, 40, 4585-4558. 
3. Zhang, C., ?Study of Morphology of Electrospun Poly (vinyl alcohol) Mats?, 
European Polymer Journal, 2005, 41, 423-432. 
4. http://www.erkol.com/eng/Characteristics.htm (accessed in Jan 2005). 
5. Mallapragada S.,and Peppas, N. A., ?Dissolution Mechanism of Semicrystalline 
Poly(vinyl alcohol) in Water?, Journal of Polymer Science: Part B: Polymer Physics, 
1996, 34, 1339-1346. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 165
 
 
 
 
 
CHAPTER 7 
CONCLUSIONS AND RECOMMENDATIONS 
 
The following conclusions can be drawn from this research. 
 
- The thermal behavior of nano-micro sized fiber composites has been analyzed and 
compared experimentally, analytically, and numerically. It was found that the 
analytical modeling in which the hexagonal unit cell is used showed much 
accurate results compared to the rectangle unit cell when the thermal conductivity 
of the filler fiber had higher value compared to the main fiber. By using the 
hexagonal cell, the model was restricted for the filler fibers? volume fraction 
which is changed between 10 and 30%. 
-  In numerical modeling, ANSYS 7.0 software was used to find the effective 
thermal conductivity of the nano-micro fiber composites by using the hexagonal 
unit cell. It was found that when the barrier thickness becomes larger, the 
effective thermal conductivity value increases. 
- To manufacture a continuous nano and micro sized fibrous yarn, a system was 
designed and the optimum properties of the device were discussed. Two problems 
were solved by using this device; the first is the linear electrical force distribution 
 166
and the second is the aligning problem which is solved by rotating the nonwoven 
fabric covered cylinder. 
- Electrospinning method was used to collect the fibers in the web form. Different 
kind of fabric surfaces were used to collect the fibers. The non-woven fabrics 
were found to be the best fabric to get the fibers in the web form. 
- Electrospinning was monitored to discuss the voltage effect over polymer 
solution.  It was found that, when the voltage is increased from 15 kV to 20 kV, 
the spinning starts earlier and the total electrical force increases. 
- Scanning Electron Microscope was used to find the fiber size and alignment of 
the nano-micro web form. Beading effect, the optimum solution viscosity, 
optimum voltage effect were discussed. It was founded that, when the 10 wt. 
%PVA solution is used, the average fiber radius becomes around 450 nm and 
when the viscosity is increased to 15 wt %, the radius of the fibers increases to 
500 nm.  
- Air permeability and surface tension of the web form was measured and 
calculated to discuss the stability of the web form. It is found that by using the 
nano-micro fiber web form, very low air permeability can be obtained. 
- Maximum elongation measurements were done to find out the continuity of the 
yarn which was formed by twisting the nano-micro fiber based web form.  It was 
found that, the continuity was satisfied. 
- Differential Scanning Microscope was used to discuss the heat behavior of the 
PVA and PVA+Laponite? solutions. The heat flow was found for different ratio 
of solutions and compared each other. The glass transition temperatures were 
 167
increased by adding the additional Laponite? inside the PVA solution. Compared 
to the melting temperature value, glass transition values showed more increase. 
- Thermal Gravimetric Analysis was done to discuss the mass loss for the solutions. 
It was found that when the Laponite? was added into the PVA solutions, time for 
total mass loss increased. 
 
The following is recommended for any future work on this subject: 
- Automatic continuous manufacturing system need to be designed. Especially 
pulling mechanism needs to be designed with higher sensitivity. 
- Although thermal measuring was done to get the mass loss and heat flow, thermal 
conductivity measurements need to be done experimentally.  
- To compare the nano and micro effects in composites, apart from the micro sized 
modeling of composites, there is a need for nano scale modeling. This could be 
done by using either molecular or atomistic modeling methods. 
- Different kinds of materials need to be used for verifying the correctness of both 
analytical and numerical models. 
- Although transverse thermal conductivity has more importance and effect on 
defining the behavior of composite materials, longitudinal thermal conductivity 
needs to be discussed. 
- Flame resistance behavior of the composites needs to be discussed and modeled 
experimentally, numerically and analytically.  
 
 
 168
 
 
 
APPENDIX  
 
% FORTRAN PROGRAM for the Hexagonal Unit Cell 
 
format long e 
clear all 
clc 
%__________________________________________________________ 
%  INPUT VALUES 
% %__________________________________________________________ 
%Input Volume fraction of Fiber; 
 Radius of the disk; rd 
rd=input('Radius of the disk; rd_'); 
Radius of the barrier; a 
  a=input('Radius of the barrier; a_'); 
% ______________________ 
%     Length of the hexagonal side; L 
%     L=input('Length of the hexagonal side; L_');  
L=30; 
    % Thickness 
    t=(a-rd); 
     % Dimensionless Thickness 
    tt=t/rd; 
    % Angles; 
      %Between barrier and fiber; Phi 
      %Phi=input('Between barrier and fiber; Phi_'); 
     
XXX= asin(rd/a); 
 169
  
% Thermal conductivities 
%     % of the disk 
%    kd=input(' ke of the disk_'); 
     kd=20000; 
%    % of the barrier 
%    kb=input('ke of the barrier_'); 
     kb=10000; 
%    % of the fiber 
%    kf=input('ke of the fiber_'); 
     kf=10; 
    % Then calculate the areas and volume fractions; 
% Area of ... 
        % the disk; Ad 
    Ad= pi*rd^2; 
     
    % the barrier;Ab 
    Ab=pi*(a^2-rd^2); 
     
    % the fiber;Af 
    Af=3*(sqrt(3)/2)*L^2; 
     
% Volume fraction;  
    % Disk volume fraction; Vd 
    Vd= Ad/Af 
    % Barrier volume fraction;Vb 
    Vb= Ab/Af  
  
% Dimensionless forms; 
    % Dimensionless thermal conductivity 
 170
    
    Alfa1= kb/kf; 
%   Beta=kd/kf; % It is given in the loop. 
     
    bb=sqrt(3)*L/(2*a); 
     
%_________________________________________________________________ 
  
Beta=0.01:0.01:100; 
 JJ=Beta; 
% Alfa1=input('Alfa....'); 
 Alfa=Alfa1.*ones(1,10000); 
 d=1./Beta-1./Alfa; 
c=1./Alfa-1; 
 b=bb.*ones(1,10000); 
 t=tt.*ones(1,10000); 
 XX=XXX.*ones(1,10000); 
 % ________________________ 
% REGIONS 
%The 1st region 
% ________________________ 
S1=sqrt(3)/2-1./b+(1./c).*(pi/2-XX);  
 %The 2nd region 
% ________________________    
if  b.^2>c.^2; 
    P=sqrt(b.^2-c.^2); 
%     PP=atan(sqrt((b-c)./(b+c))-atan(sqrt((b-c)./(b+c)).*tan(XX/2)));  
    PP=atan(P./(b+c))-atan((P./(b+c)).*tan(XX/2)); 
    S2=-((b)./(c.*P)).*PP; 
    display('_____________')    
 171
 elseif b.^2<c.^2 
    Y=sqrt(c.^2-b.^2); 
    YY=log2((c+Y).*(b+c.*cos(XX))./(b.*(b+c.*cos(XX)+Y.*sin(XX)))); 
    S2=-(b./(c.*Y)).*YY; 
    display ('____________') 
 else 
        display ('Equality..!') 
end 
 %  For 3rd region: 
% Composite Simpson Method; 
% _________________________ 
  
 S30=1./(b+c+sqrt(1./(t+1).^2).*d); 
  
S31=cos(XX/6)./(b+c.*cos(XX/6)+sqrt(1./(t+1).^2-sin(XX/6).^2).*d); 
S32=cos(XX/3)./(b+c.*cos(XX/3)+sqrt(1./(t+1).^2-sin(XX/3).^2).*d); 
S33=cos(XX/2)./(b+c.*cos(XX/2)+sqrt(1./(t+1).^2-sin(XX/2).^2).*d); 
S34=cos(2*XX/3)./(b+c.*cos(2*XX/3)+sqrt(1./(t+1).^2-sin(2*XX/3).^2).*d); 
S35=cos(5*XX/6)./(b+c.*cos(5*XX/6)+sqrt(1./(t+1).^2-sin(5*XX/6).^2).*d); 
S36=cos(XX)./(b+c.*cos(XX)+sqrt(1./(t+1).^2-sin(XX).^2).*d); 
  
S3=(XX/18).*(S30+4*S31+2*S32+4*S33+2*S34+4*S35+S36); 
  
SS=(3*sqrt(3)/2)*(S1+S2+S3); 
   
xlabel('\bfBeta') 
ylabel('\bf Partial Effective Thermal Conductivity') 
title('\bfFor integral form...') 
axis ([0.01 100 0 0.001]) 
semilogx(JJ,SS)