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
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Applied Physics, 2003, Vol 93, No 2., 793-818.
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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
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41. Chen, G., ?Ballistic-Diffusive Equations for Transient Heat Conduction From
Nano to Macroscales?, Journal of Heat Transfer, 2002, 124(2), 320-328.
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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,
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th
Conference on Molecular Technology, 1999, California.
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Nanotubes?, Solid State Commun., 1996,100, 177
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Length SWNTs", Physica B, 2002, 323, 1-4, 193-195.
47. Maruyama, S., "Molecular Dynamics Method for Microscale Heat Transfer",
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2000, 15, 12-27.
49. Beyer, G., ?Nanocomposites: a New Class of Flame Retardants for Polymers?,
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Polymer Layered Silicate Nanocomposites: Polyolefin, Epoxy, and Vinyl Ester
Resins?, Chemistry and Technology of Polymer Additives, 1999, 14, 249-265.
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54. Gilman, J. W., Jackson, C. L., Morgan, A. B., Harris, R. H., Jr., Manias, E.,
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Nanocomposites?, Chem. Mater., 2001, 13, 3774-3780.
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walled Carbon Nanotubes and Ni Catalyzed Nanofibers: Growth Mechanisms
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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.
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Conductivity of Single-walled Carbon Nanotubes?, Nanotechnology, 2001, 12,
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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
or (4.19) for b
2
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< xc.^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