ON THE EFFECTIVE THERMAL CONDUCTIVITY OF CARBON NANOTUBE REINFORCED POLYMER COMPOSITES The members of the Committee approve the master’s thesis of Aniruddha Bagchi Seiichi Nomura Supervising Professor Kent L. Lawrence Bo Ping Wang
ON THE EFFECTIVE THERMAL CONDUCTIVITY OF CARBON
NANOTUBE REINFORCED POLYMER COMPOSITES
The members of the Committee approve the master’sthesis of Aniruddha Bagchi
Seiichi Nomura
Supervising Professor
Kent L. Lawrence
Bo Ping Wang
To my parents, Mrs. Shyamali Bagchi and Mr. Tapas Kumar Bagchi, who have made
me what I am today.
ON THE EFFECTIVE THERMAL CONDUCTIVITY OF CARBON
NANOTUBE REINFORCED POLYMER COMPOSITES
by
ANIRUDDHA BAGCHI
Presented to the Faculty of the Graduate School of
The University of Texas at Arlington in Partial Fulfillment
of the Requirements
for the Degree of
MASTER OF SCIENCE IN MECHANICAL ENGINEERING
THE UNIVERSITY OF TEXAS AT ARLINGTON
May 2005
ACKNOWLEDGEMENTS
I would like to thank my supervising professor Dr. Seiichi Nomura for his constant
encouragement and motivation and for his invaluable advice and help throughout the
entire course of this thesis. I would also like to thank Dr. Kent L. Lawrence and Dr.
Bo Ping Wang for taking the time to serve on my thesis committee and for critically
reviewing and evaluating my thesis.
I would like to thank the Dean of the Graduate School and the Department of Me-
chanical and Aerospace Engineering, The University of Texas at Arlington, for providing
me with financial support during the course of my master’s degree. I am grateful to the
teachers who have taught me during all the years that I spent in school, first in India
and then in the United States. In particular, I would like to thank Dr. Arindam Rana
for being a great friend and teacher and for encouraging me to pursue graduate studies.
Finally, I would like to thank my parents for supporting me throughout my aca-
demic career and giving me the freedom of choice. I would also like to thank my sister
and her family for the support they have provided me with during my entire stay in the
United States. Thanks must also be given to all my friends in UTA and elsewhere who
have always been by my side and have been a constant source of inspiration to me. And
of course I am forever indebted to Mr. Manoranjan Bagchi, my late grandfather, who
was and will be the best teacher I ever had.
April 14, 2005
iv
ABSTRACT
ON THE EFFECTIVE THERMAL CONDUCTIVITY OF CARBON NANOTUBE
REINFORCED POLYMER COMPOSITES
Publication No.
Aniruddha Bagchi, M.S.
The University of Texas at Arlington, 2005
Supervising Professor: Seiichi Nomura
The focus of the present thesis is to develop a fundamental understanding of the
heat conduction process in carbon nanotube reinforced polymer composites and to elu-
cidate the contribution of the various factors which affect it at the nanometer level. To
this end, a theoretical model has been developed for predicting the effective longitudi-
nal thermal conductivity of an aligned multi-walled nanotube/polymer composite. This
model is based on an effective medium theory that has been developed for predicting the
thermal conductivity of short fiber composites. To incorporate the multi-walled nanotube
structure into this theory, a continuum model of the nanotube geometry is developed by
considering its structure and the mechanism of heat conduction through it. Results show
that the effective conductivity will be much lower than expected due to the fact that the
outer nanotube layer carries the bulk of the heat flowing through the nanotube while the
contribution of the inner layers to heat flow is negligible. Also, the effective conductivity
has been found to be particularly sensitive to the multi-walled nanotube diameter. In
addition, it has been found that the high interfacial resistance between the nanotube
v
and polymer matrix is not the principal factor which affects the flow of heat in carbon
nanotube composites. Theoretical predictions are found to be very close to published
experimental results.
vi
TABLE OF CONTENTS
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Chapter
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Why Carbon Nanotube Composites? . . . . . . . . . . . . . . . . . . . . 3
1.3 Thermal Properties of CNT/Polymer Composites . . . . . . . . . . . . . 5
2. CARBON NANOTUBE POLYMER COMPOSITES . . . . . . . . . . . . . . 9
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Carbon Nanotubes and their Properties . . . . . . . . . . . . . . . . . . . 9
2.3 Structure and Morphology of Carbon Nanotubes . . . . . . . . . . . . . . 13
2.4 Nanotube Synthesis and Processing . . . . . . . . . . . . . . . . . . . . . 15
2.5 Synthesis and Processing of CNT/Polymer Composites . . . . . . . . . . 18
3. MATHEMATICAL MODEL FOR THE EFFECTIVE CONDUCTIVITY . . . 21
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 General Theory of Heat Conduction in Composite Materials . . . . . . . 21
3.2.1 Composites with Perfect Thermal Contactbetween the Constituents . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.2 Composites with Imperfect Thermal Contactbetween the Constituents . . . . . . . . . . . . . . . . . . . . . . . 27
3.3 Modeling of an MWNT Inclusion . . . . . . . . . . . . . . . . . . . . . . 31
3.3.1 Equivalent Continuum Model and Effective Solid Fiber . . . . . . 32
vii
3.3.2 The Prolate Spheroidal Inclusion . . . . . . . . . . . . . . . . . . 36
3.4 Solution of the Auxiliary Problem . . . . . . . . . . . . . . . . . . . . . . 38
4. RESULTS AND DISCUSSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Numerical Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 Comparison With Existing Theoretical Modelsand Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.4 Evaluation of Factors Affecting the Effective Conductivity . . . . . . . . 55
4.4.1 Contribution of Individual NanotubeLayers to Heat Conduction . . . . . . . . . . . . . . . . . . . . . . 55
4.4.2 Influence of Nanotube Length andDiameter on the Conductivity . . . . . . . . . . . . . . . . . . . . 57
4.4.3 Influence of the Interfacial Resistanceon the Effective Conductivity . . . . . . . . . . . . . . . . . . . . 59
5. CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK . . . . . . . . 62
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Appendix
A. INTEGRAL RELATIONSHIPS . . . . . . . . . . . . . . . . . . . . . . . . . . 64
B. DERIVATION OF THE INTERIOR AND EXTERIOR HARMONICS . . . . 67
C. MATHEMATICA PROGRAM . . . . . . . . . . . . . . . . . . . . . . . . . . 71
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
BIOGRAPHICAL STATEMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
viii
LIST OF FIGURES
Figure Page
2.1 Buckyball - A C60 molecule containing 60 carbonatoms arranged in a closed convex structure of20 hexagons and 12 pentagons . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 a) Schematic diagram showing how a graphene sheet isrolled up to form an SWNT; b) SEM micrographs ofrows of aligned SWNTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 a) Schematic representation of an MWNT; b) SEMmicrographs showing mats of aligned MWNTs . . . . . . . . . . . . . . . . 12
2.4 Nomenclature of a carbon nanotube . . . . . . . . . . . . . . . . . . . . . 13
2.5 Schematic diagram illustrating the 3 different types ofCNTs: a) Armchair nanotubes; b) Zig-Zag nanotubesand c) Chiral nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1 Schematic representation of an arbitrary two phasecomposite showing the various notations used . . . . . . . . . . . . . . . . 23
3.2 Schematic representation of a three phase continuum . . . . . . . . . . . . 29
3.3 Development of a continuum model for an MWNT.a) Schematic diagram of an MWNT showing con-centric graphene layers; b) Equivalent continuummodel; c) Effective solid fiber, and d) A prolatespheroidal inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 A prolate spheroidal coordinate system. The parametersξ, µ and ψ refer to 3 sets of orthogonal surfaces . . . . . . . . . . . . . . . 38
4.1 Variation of the effective thermal conductivity k∗33
with nanotube volume fraction . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2 Variation of the effective thermal conductivity k∗33
with nanotube length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3 Variation of the effective thermal conductivity k∗33
with nanotube diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
ix
4.4 Variation of the effective thermal conductivity k∗33
with change in interfacial conductance . . . . . . . . . . . . . . . . . . . . 59
x
CHAPTER 1
INTRODUCTION
1.1 Background
In the past few decades, the use of composite materials in structural components
has been increasing rapidly and they are now gradually replacing traditional metallic
materials in many other applications on account of the superior physical properties they
offer at only a fraction of the weight. Composites belong to a class of heterogeneous
materials which, loosely speaking, consist of at least two different components blended
together in such a way that the properties of the material produced are greatly different
from those of its constituents. One of the constituents forms a continuous phase and is
called the matrix while the other, the discontinuous phase, is uniformly dispersed within
the matrix and is called the reinforcement phase. This reinforcement phase, also known
as the filler material, may be in the form of fibers or particles. The filler material provides
the composite with its specific physical properties while the function of the matrix is to
hold the fillers together. Generally fibers are the most commonly used fillers and they
may be either continuous or chopped (short fibers).
There are many advantages that composites have as compared to traditional en-
gineering materials. Not only do they offer very high strength to weight ratios, they
provide other advantages like superior resistance to corrosion, low density, low thermal
expansion and favorable fatigue life. Another great advantage that composites have is
the ability to have tailored physical properties in a specific direction, thereby providing
great flexibility in design. As a result their use has been rapidly increasing, especially
over the last two decades. Traditionally, the use of composites has been restricted to
1
2
the aerospace industry because of the immense costs involved in manufacturing them
on a large scale. However with improvements in manufacturing techniques, production
costs have come down substantially and this has led to the widespread use of composite
materials in a host of applications ranging from military and commercial aircrafts to au-
tomobiles, sports goods, electronic chips and biological applications to name a few. With
the demand for light weight materials having superior physical properties increasing, the
use of composites will keep growing.
Most of the composite materials in use today use polymers as the matrix material
and Kevlar, boron or graphite fibers as fillers. Use of metals and ceramics as matrix ma-
terials and various metallic and non-metallic particulates as fillers is gradually increasing.
Metal and ceramic matrix composites have the advantage of being suitable to rigorous
environments, especially high temperature applications, as compared to polymer matrix
composites. Yet fiber reinforced polymer composites still constitute the majority of com-
posite materials produced due to their light weight and ease of fabrication. However,
as the use of composites increases, there will be a need for producing newer composite
materials that stand up to the test of demanding applications while being light weight
and durable. With the recent advances in material science, various new materials have
been identified which have potential as fillers in producing advanced composite materials
with polymers as the the matrix material. The most promising among them are carbon
nanotubes. Discovered a decade and a half ago [1], their outstanding physical properties
have only recently come to light [6-12]. Also their use as fillers for polymers has only been
recently appreciated [5]. As such, there is a considerable amount of research currently
going on in this field to exploit their properties for producing superior composites, as
there is a widespread belief that nanotube composites could greatly outweigh traditional
composites in terms of physical properties and performance in various environments.
3
1.2 Why Carbon Nanotube Composites?
Over the past decade or so, carbon nanotubes (CNTs) have received a significant
amount of attention within the scientific and engineering communities on account of
their outstanding physical properties, which have been found to vastly exceed those of
any currently known and available material. With a reported elastic modulus of about 1
TPa and a tensile strength of about 200 GPa [6, 7], carbon nanotubes have emerged as
the strongest available material. They have been found to possess a thermal conductivity
of about 3000 W/m K for multi-walled nanotubes (MWNTs) [9] and about 6000 W/m
K for single walled nanotubes (SWNTs) [10], making them the best known thermal
conductors available. Similar outstanding values for the electrical conductivity have also
been reported [11, 12]. This has led a large number of researchers to believe that the
properties of carbon nanotubes, if properly harnessed, can produce the next generation of
materials with hitherto inconceivable physical properties and wide ranging applications.
In view of their outstanding properties, the applications for which CNTs can be
prospective candidates are immense. Indeed many potential applications have been pro-
posed for them including conductive and high strength polymer composites, nanobear-
ings, nanoropes, energy storage and energy conversion devices, sensors and field emission
displays [13]. As can be seen, the applications encompass broad and diverse areas of
research. From the point of view of structural and electronic applications, CNT/polymer
composites hold great promise. CNTs can be used as filler materials in polymer matri-
ces to greatly improve mechanical properties, thermal transport and electric transport.
Their fiber like structure makes them particularly attractive for such applications. It is
widely believed that due to their vastly superior properties, the increase in strength and
conductivity of polymer matrices achieved with CNT fillers may be more than an order of
magnitude higher than that provided by traditional carbon fibers. As such a significant
amount of current research is aimed at addressing the various issues associated with the
4
fabrication and analysis of CNT composites so that the superior properties of CNTs can
be realized in their composites.
The development of CNT/polymer composites is quite recent and is the first real-
ized major commercial application of CNTs. In 1994, Ajayan and his colleagues [5] found
that CNTs could be aligned by embedding them in a polymer matrix, thus producing
a CNT/polymer composite for the first time. Incorporation of CNTs in plastics can
provide structural materials with dramatically increased elastic modulus and strength
as suggested by theoretical predictions. A few studies have reported to have observed
significant toughening of polymer matrices when loaded with CNTs. Ruan et al. [14], for
example, have reported that a loading of 1% volume fraction of MWNTs in an ultra-high
molecular weight polyethylene film, increased its strain energy density by about 150%
and the ductility by 140%. Similarly, Weisenberger et al. [15] have reported, that a 1.8%
volume fraction MWNT/PAN (polyacrylo nitrile) composite showed an 80% increase in
toughness as compared to the pristine epoxy. With respect to the thermal properties,
it is expected that CNT/polymer composites will have a high thermal conductivity and
this has been confirmed by experimental measurments. Choi et al. [17] reported that
a random dispersion of MWNTs in an organic fluid increased its conductivity by more
than 2.5 times at approximately 1% volume fraction of nanotubes. Biercuk et al. [18]
found that samples loaded with 1% weight fraction SWNTs, showed a 70% increase in
the thermal conductivity at 40 K rising to about 125% at room temperature. They also
found that electrical conductivity data showed a percolation threshold between 0.1 and
0.2% weight fraction SWNT loading, which is in accordance with theoretical estimates.
All these point to the fact that inclusion of CNTs in polymers can greatly enhance their
physical properties, even at extremely low CNT concentrations. As such it has been
widely accepted that CNT composites may become the next generation of composite
5
materials having both superior mechanical and conductive properties, thus making them
suitable for highly demanding applications.
1.3 Thermal Properties of CNT/Polymer Composites
The thermal properties of CNT/polymer composites are of particular interest in
many applications like conductive polymer films and nano-electronic components. Due
to the high thermal conductivity of both SWNTs and MWNTs, and their extremely high
aspect ratios, the thermal conductivity of CNT/polymer composites is also expected to
be very high. Significant enhancements in the matrix conductivity on being loaded with
CNTs have been experimentally measured, as described in the previous section. These
results though encouraging, however, fall well short of simple theoretical predictions.
For example, for a randomly dispersed MWNT/polymer composite, the rule of mixtures
predicts the ratio of the overall conductivity to the matrix conductivity to be nearly
50 at 1% volume fraction of nanotubes while experimental results show this ratio to be
only about 2.6 [17]. Other studies have reported even lesser increases [18]. This marked
discrepancy between theoretical predictions and experimental results have been explained
by considering various factors which affect heat flow in CNT/polymer composites.
The presence of an interfacial resistance between the nanotube and the matrix
material has been cited as the principal factor affecting heat flow in CNT/polymer com-
posites. A boundary resistance between the two phases acts as a barrier to the heat
flow and thus decreases the overall conductivity. Shenogin et al. [21] have given a com-
prehensive overview of the role played by the boundary resistance in determining the
thermal properties of CNT/polymer composites. Using molecular dynamics simulations,
they show that there is a weak coupling between the phonon spectra of the CNTs and the
polymer matrix. Since phonons dominate heat transport in CNTs, this weak coupling
produces a backscattering of phonons at the interface causing a drop in temperature there
6
and giving rise to a boundary resistance. In an earlier work, Huxtable et al. [22] used
molecular dynamics simulations to calculate the interfacial conductance for a nanotube
suspension in an organic fluid. Their results suggest that due to an exceptionally small
interfacial conductance of about 12 MW/m2 K, the overall conductivity of the nanotube
suspension will be much lower than expected.
Apart from the interfacial resistance, a non-uniform dispersion of CNTs in the
polymer matrix may also significantly affect overall behavior. Most current CNT synthe-
sis techniques produce samples in which the nanotubes are entangled with one another,
making their uniform dispersion in the matrix difficult. This is further compounded by
the non-reactive nanotube surface. Obtaining a uniform dispersion during fabrication has
been identified as a key process limitation and a significant amount of current research
is being devoted to achieving this using different techniques, such as the use of surfac-
tants or by chemical functionalization of the CNT surface [16, 26]. In addition, there
is also the problem of obtaining a nanotube sample having a uniform nanotube length
and diameter. Presently, it almost always, impossible to achieve this and most samples
usually contain a distribution of nanotube lengths and diameters. Thostenson and Chou
[23], have shown this to be a significant factor affecting the overall elastic properties.
Due to mathematical analogy between the elasticity and heat conduction problems, it
can be assumed that this might significantly affect the thermal properties as well. This
might also be understood theoretically by considering the fact that a variation in length
or diameter of the nanotube changes its aspect ratio in the composite, which causes a
variation in the overall conductivity.
In order to accurately model the overall thermal behavior, it is essential to have a
theory of heat conduction that accounts for all the factors affecting heat flow. However,
the heat conduction problem in CNT/polmer composites has not been properly stud-
ied theoretically unlike the elasticity problem, which has been thoroughly investigated
7
[23, 24]. At present there seems to be only one existing theoretical model for predicting
the overall thermal properties of CNT/polymer composites. This model, proposed be Nan
et al. [25], is based on a Maxwell-Garnett effective medium approximation. The model
however assumes a perfect contact between the constituents and ignores any interfacial
resistance, which results in a significant overestimation of the effective conductivity. Also
it neglects the nanotube structure and assumes the nanotube to act like a solid fiber,
which seems to be incorrect, since continuum assumptions are no longer valid at the
nanometer level. Although the theory provides a rough estimate of the overall conduc-
tivity, it fails to give a detailed insight into heat conduction process in CNT/polymer
composites and the parameters which affect it at the nanometer level.
In this thesis, the heat conduction phenomenon in CNT/polymer composites is
studied theoretically and a mathematical model is developed for predicting the effective
thermal conductivity of an aligned MWNT/polymer composite. This model is based on
an effective medium theory that has been developed by Benveniste and Miloh [29], for
predicting the overall thermal properties of composites containing spheroidal inclusions
with imperfect interfaces. The theory has been modified here to account for the non-
continuum effects at the nanometer level by developing a continuum model of the MWNT
filler that is suitable for mathematical analysis. This is done by taking into consideration
the unique MWNT geometry and the mechanism of heat transfer through it. The theory
takes into account the boundary resistance at the CNT/polymer interface which, as
stated before, has been experimentally proven to be a significant factor affecting heat
flow in CNT/polymer composites. Also, this method is able to take into account the
unique structure of MWNTs. In addition, by considering the mechanism of heat transfer
through an individual nanotube, it has been proved here that this is also a significant
factor affecting the overall conductivity. The obtained results have been compared with
published experimental results and very good agreement has been found.
8
The thesis is divided into the following chapters. In Chapter II, the structure,
properties and synthesis of carbon nanotubes and their composites are discussed in detail.
In Chapter III, the mathematical model for predicting the effective longitudinal thermal
conductivity has been derived. Chapter IV gives the obtained results and its analysis
and Chapter V gives the conclusions and recommendations.
CHAPTER 2
CARBON NANOTUBE POLYMER COMPOSITES
2.1 Introduction
This chapter deals with the structure, properties, synthesis and processing of carbon
nanotubes and their polymer composites. These parameters, especially, the structure
and properties of carbon nanotubes greatly affect the properties of their composites.
Also the synthesis techniques of both the nanotubes as well as their composites have a
significant effect on the overall composite properties. It is thus vital to have a thorough
understanding of these parameters in order to obtain a complete theory of heat conduction
in CNT/polymer composites.
2.2 Carbon Nanotubes and their Properties
The structure of carbon nanotubes is very similar to those of fullerenes. Discovered
in the mid-eighties by Smalley and his co-workers at Rice University [2], fullerenes are
geometric cage-like structures having pentagonal and hexagonal faces with carbon atoms
at the corners of the pentagons and hexagons. Found to be structurally similar to the
geodesic domes designed by the revolutionary and futuristic architect, R. Buckminster
Fuller, they were named fullerenes in his honor. The C60 molecule was the first closed
convex structure that was formed and it consisted of 60 carbon atoms arranged in 20
hexagonal and 12 pentagonal faces to form a sphere. This structure is very similar to
that of a soccer ball and hence they were called Buckyballs.
A few years later in 1991, a Japanese electron microscopist S. Iijima, reported
the discovery of carbon nanotubes [1]. These newly discovered substances were found to
9
10
Figure 2.1 Buckyball - A C60 molecule containing 60 Carbon atoms arranged in aclosed convex structure of 20 hexagons and 12 pentagons.
resemble in structure, long slender fullerenes. However unlike fullerenes, the walls of these
slender tubes were formed solely of hexagonal faces of carbon just as in a layer of graphite
(graphene). These tubes were a few nanometers in diameter with lengths varying from a
few hundred nanometers to a few microns. Due to their nanometer dimensions, they were
known as carbon nanotubes. The nanotubes that were first obtained by Iijima actually
consisted of multiple layers of graphene rolled up coaxially to form a hollow cylindrical
tube of nanometer dimensions. These kinds of nanotubes are known as multi-walled
nanotubes (MWNTs) and are one of the two types of nanotubes that exist. The other
type of nanotubes are the single walled nanotubes (SWNTs). The synthesis of SWNTs
was independently reported by Iijima et al. [3] and Bethune et al. [4] a few years after
the discovery of MWNTs by Iijima. A SWNT can be visualized as a sheet of graphite
(graphene) being rolled up endlessly to form a cylindrical tube. A schematic diagram of
an individual SWNT is given in figure 2.2. Often these tubes have caps at both ends.
11
Graphene Sheet SWNT
a)
b)
Roll-up
Figure 2.2 a) Schematic diagram showing how a graphene sheet is rolled up to form anSWNT; b) SEM micrographs of rows of aligned SWNTs.
Both SWNTs and MWNTs are endowed with exceptional strength and resilience as well
as very high thermal and electrical conductivities.
The exceptional physical properties of carbon nanotubes are a consequence of their
perfect defect free structure. In most materials, the actually observed material proper-
ties are in practice considerably lower than what can be estimated theoretically. This
degradation in properties occurs due to the presence of defects in their structures. For
example, the breaking strength of steel is about 1% of its theoretical breaking strength
due to the presence of defects in its microstructure. However the molecular perfection
in the structure of carbon nanotubes makes them almost free of defects and thus their
physical properties are very close to those predicted theoretically. Hence such high values
of strength and thermal and electrical conductivities can be explained theoretically by
considering their physical structure. The high stiffness and strength of carbon nanotubes
are due to their bond-structure. No other material in the periodic table bonds itself in an
12
Concentricgraphene layers
a)
b)
Figure 2.3 a) Schematic representation of an MWNT; b) SEM micrographs showingmats of aligned MWNTs [32].
extended network with the strength of the carbon-carbon bond. This coupled with their
seamless structure, gives them their amazing strength. The high thermal conductivity
can be explained by considering the large phonon mean free path for the nanotubes.
It has recently been shown that in nanotubes, phonon contribution to heat conduction
dominates [18]. When the dominant mode of heat conduction is through phonons, the
thermal conductivity is directly proportional to the phonon mean free path and the large
phonon mean free path for the nanotubes eventually leads to the high value of the ther-
mal conductivity. The high electrical conductivity is due to the fact that the delocalized
pi-electron donated by each carbon atom is free to move about the whole structure, thus
giving rise to metal-like electrical conductivity.
13
Figure 2.4 Nomenclature of a carbon nanotube.
2.3 Structure and Morphology of Carbon Nanotubes
As mentioned before, an individual SWNT consists of a graphene layer rolled up
endlessly to form a hollow cylindrical tube. An individual MWNT is simply several
SWNTs (several graphene layers) arranged co-axially and held together by interlayer
Van der Waal’s forces. The separation between the graphene layers is about 3.4 A which
is very close to the interlayer spacing in graphite which is 3.35 A. The properties of
individual nanotubes depend on their atomic structures, their diameter and length and
their morphology. To a large extent the atomic arrangement in a nanotube will determine
its physical properties. The atomic structure is described in terms of the tube chirality or
helicity which is defined by the chiral vector, ~Ch, and the chiral angle, θ. The chiral vector,
also known as the roll-up vector, specifies the vector connecting two crystallographically
equivalent carbon atoms in the planer hexagonal lattice that forms the walls of the tube.
Consider the figure 2.4. We can visualize cutting up the graphite sheet along the dotted
lines and rolling it up to form a seamless tube. The chiral vector will then be such that
when the tube is rolled up, the tip of the chiral vector touches its tail as is shown in the
figure. This vector can be defined in terms of the lattice translation indices (n, m) and
14
the base vectors, a1 and a2, of the hexagonal lattice plane (see figure) and is given by
the equation
~Ch = na1 + ma2. (2.1)
The numbers, n and m, specify the number of steps along the zig-zag carbon bonds
within the hexagonal lattice. These numbers together constitute a unique ‘name’ for a
tube. Tubes ‘named’ (n, 0) have C-C bonds that are parallel to the tube axis and form
at an open end a zig-zag pattern and are hence called zig-zag tubes. Tubes having the
‘name’ (n, n) have C-C bonds that are perpendicular to the tube axis and are often called
armchair tubes. All other tubes ‘named’ (n, m), where n and m are unequal, are called
chiral and have left and right handed variants. The chiral angle, θ, shown in the figure
specifies the amount of twist in the tubes. The two limiting cases exist when the chiral
angle is 0◦ and 30◦. These two cases correspond to the zig-zag (0◦) and armchair (30◦)
tubes.
In most respects, the properties of tubes of different types are essentially the same.
However the chirality has a significant influence on the electronic properties of the nan-
otubes. For example, all armchair tubes have been found to exhibit truly metallic elec-
trical conductivity. In contrast other tubes are intrinsically semiconducting with either
a very small band-gap of a few meV’s or a moderate band-gap of the order of 1 eV [12].
The chirality does not have a significant influence on the elastic stiffness or the tensile
strength of the nanotubes. However, armchair nanotubes under tension, can undergo a
change in structure where a structure of four hexagons changes into a structure having
two pentagons and two heptagons. This is known as the Stone-Wales transformation and
has been found to play a key role in the plastic deformation of nanotubes under tension
[32]. The thermal properties remain practically unchanged with change in tube chirality.
15
a)b)
(n, 0)(n, n)
q = 30 q = 0
(n, m)
q = 0 - 30o oo
c)b)
Figure 2.5 Schematic diagram illustrating the 3 different types of CNTs: a) Armchairnanotubes; b) Zig-Zag nanotubes, and c) Chiral nanotubes.
2.4 Nanotube Synthesis and Processing
Since the discovery of MWNTs by Iijima [1], various techniques have been developed
for synthesis of CNTs, each technique producing nanotubes with certain characteristics.
The principal techniques used for nanotube synthesis include carbon-arc discharge, laser
ablation of carbon, gas-phase catalytic growth from carbon monoxide and chemical va-
por deposition from hydrocarbons [32]. At present nanotubes, especially SWNTs, are
produced only on a small scale and are extremely expensive to manufacture. It is espe-
cially difficult to obtain high purity samples and most currently known processes used
to manufacture them result in samples with a high concentration of impurities. As such,
16
the costs involved in obtaining bulk samples of high purity SWNTs and to a smaller
extent, MWNTs, are prohibitive. A substantial amount of research is thus being cur-
rently devoted to devising improved synthesis techniques which would enable CNTs to
be produced commercially on a large scale.
The electric-arc discharge technique is the oldest technique available for producing
nanotubes and was first used by Iijima [1]. The technique basically consists of using two
high purity graphite rods as the anode and cathode immersed in a helium atmosphere. A
high voltage is applied between the two rods until a stable electric arc is obtained. The
nanotube material eventually gets deposited on the cathode. This technique produces
MWNTs. For producing SWNTs, the electrodes are doped with a small amount of
powdered metallic catalyst.
In the laser ablation technique a laser is used to vaporize a graphite target held
in a controlled atmosphere oven at a temperature of about 1200◦C. Like in the electric-
arc discharge technique, MWNTs are produced when no catalyst is used. To produce
SWNTs, the graphite target is doped with cobalt and nickel catalyst. The condensed
nanotube sample is then collected on a water-cooled target. The laser ablation technique
was initially used in the synthesis of fullerenes, but has been modified over time to
produce carbon nanotubes.
Both the arc-discharge as well as the laser ablation technique produce a small vol-
ume of nanotubes in comparison to the size of the carbon source employed, especially
when SWNTs are produced. The addition of catalysts results in producing samples with
a significant amount of impurities which occur in the form of catalyst particles, amor-
phous carbon and non-tubular fullerenes. Hence subsequent purification techniques are
necessary to separate the nanotubes from unnecessary by-products. Also the products
are normally tangled up or in poorly ordered mats. In addition, the high temperatures
that are involved in these techniques make it difficult to control the chirality and the
17
diameter of the nanotubes, so that the final sample obtained contains a mixture of arm-
chair, zig-zag and chiral nanotubes of varying lengths and diameters. Nonetheless, the
laser ablation technique has been recognized as a feasible technique for mass production
of SWNTs and a recent improvement in this technique enables the aligned growth of
nanotubes and also offers sufficient control over the nanotube lengths and diameters.
The limitations of the arc-discharge and the laser ablation technique have mo-
tivated the development of gas-phase techniques where nanotubes are formed by the
decomposition of a carbon containing gas. Nanotubes produced by such techniques have
substantially higher purities than those obtained by the methods mentioned above. Also
such techniques are very suitable for continuous production as the carbon source being
a gas, is constantly replaced by incoming gas. Most gas phase techniques use carbon
monoxide as the gas source. Smalley and his co-workers at Rice University have refined
the standard gas-phase process to produce large quantities of SWNTs with remarkable
purity. Recently Carbon Nanotechnologies Inc. (Houston, TX) has commercialized the
HiPco (High Pressure Conversion of carbon monoxide) process for producing high purity
carbon nanotubes on a large scale.
The CVD technique on the other hand uses other hydrocarbon gases as the carbon
source for the production of both single walled and multi-walled nanotubes. However, it
has been pointed out that since hydrocarbons pyrolyse readily on surfaces at temperatures
above 600-700◦C, nanotubes produced by such methods have substantial amorphous
carbon deposits on their surfaces. As a result, further purification of the nanotubes
is necessary to obtain sufficiently pure samples. However one unique aspect of the CVD
technique is its ability to synthesize aligned arrays of Carbon Nanotubes with controlled
diameter and length. Also the lower processing temperatures involved allow the synthesis
of the nanotubes on a wide various substrates including glass. The synthesis of well-
aligned, straight carbon nanotubes on a wide variety of substrates has been accomplished
18
by the use of plasma-enhanced chemical vapor deposition (PECVD). This technique
allows straight nanotubes to be grown over a large area with excellent uniformity in
diameter, length, straightness and size density. Also the dimensions of the nanotubes
can be controlled by using a catalyst. Adjusting the thickness of the catalyst controls
the diameter of the tubes.
All the process described above have their own individual characteristics in terms
of the purity, alignment, control over length and diameter and cost of the nanotubes
produced. Since the properties of the nanotube sample produced vary greatly from one
process to another, it is important to keep in mind the source of the nanotube sample
before processing of their composites in order to obtain the desirable properties.
2.5 Synthesis and Processing of CNT/Polymer Composites
Due to the nanometer scale of the reinforcement, the synthesis of CNT/polymer
composites is somewhat different from the techniques used to manufacture traditional
continuous and short fiber reinforced composites. Lacking direct manipulation, when
used as reinforcement in polymers, CNTs are first randomly dispersed in a solvent or
polymer fluid/melt followed by further processing to create the composite. One of the
significant challenges that exist during the synthesis of CNT/polymer composites is in
obtaining a uniform dispersion of nanotubes within the polymer matrix. Because of
their small size, carbon nanotubes tend to agglomerate when dispersed within a poly-
mer. This is especially severe in case of arc-discharge and CVD grown nanotubes as the
nanotubes get entangled during the nanotube growth itself. This is further compounded
by the non-reactive nanotube surface. To achieve an optimal amount of reinforcement
in the composite, it is essential to have a uniform dispersion of the nanotubes within the
polymer. Lack of a uniform nanotube dispersion in the matrix has been cited as one of
the key factors affecting the realization of the nanotube properties in their composites.
19
As such significant amount of current research is aimed at achieving this using differ-
ent techniques. The most common techniques are shear mixing or solution-evaporation
methods with high energy sonication. Further processing may be needed to obtain the
desired composite sample. Thostenson and Chou [23], for example, have obtained aligned
CNT/polymer composites by first dispersing the CNTs through shear mixing and then
extruding the resulting polymer melt through a rectangular die. With the use of shear
mixing or sonication techniques, however, the high energy added to achieve uniform dis-
persion of the CNTs often tends to break them up into shorter segments, thus decreasing
their aspect ratio in the final composite. Due to this, the nanotubes in the composite
will always have a CNT length distribution.
Another route to achieving uniform dispersion is to improve the reactivity of the
nanotube surface by some sort of chemical modification. The two most important meth-
ods that have been identified for this are the use of surfactants [16] and the oxidation
or chemical functionalization of the nanotube surface [27, 28]. Functionalization of the
nanotube surface can not only increase the dispersion of the nanotubes within the poly-
mer matrix, but it has also been found to increase the strength of the interface between
the nanotube and the polymer matrix. This thus results in attaining a much better re-
inforcement. Recently, functionalization of the CNT surface has been achieved through
the exposure of the vapor grown CNTs to a CO2/Ar plasma optimized with respect to
time, pressure, power and gas concentration [28]. Similarly, successful oxidation of the
nanotube surface using nitric acid treatment has been reported to have increased the
dispersion of nanotubes in the polymer matrix [27]. However such chemical functional-
ization may disrupt the C-C bonding within the graphene sheet and may thus affect the
properties of the CNT. Still, in spite of the various efforts to increase the dispersion of
nanotubes in the polymer matrix, achieving uniform dispersion remains a process limi-
20
tation and a considerable amount of future research in this area will be directed towards
this.
CHAPTER 3
MATHEMATICAL MODEL FOR THE EFFECTIVE CONDUCTIVITY
3.1 Introduction
This chapter develops a mathematical model that will be used to predict the ef-
fective thermal conductivity of CNT/polymer composites. This model is based on a
classical effective medium theory that has been developed by Benveniste and Miloh [29]
to predict the overall conductivity of short fiber composites. To begin with, the general
theory of heat conduction in a composite medium is reviewed. The case of composites
having perfect thermal contact between the constituents is discussed first, which is then
extended to the case of an imperfect thermal contact and a general formula for the ef-
fective conductivity is derived. A continuum model of the nanotube structure is then
developed by defining an equivalent continuum model and an effective solid fiber. This
enables the use of classical continuum theories for analyzing CNT-composites. Finally,
the temperature fields inside and outside the effective inclusion are calculated by solving
the steady state heat conduction equation and they are then used to obtain the effective
thermal conductivity of the composite.
3.2 General Theory of Heat Conduction in Composite Materials
In this section we review the general theory of heat conduction in composite mate-
rials. Composites belong to a broad group of heterogeneous materials which also contain
granular materials, porous media, suspensions, polycrystalline materials and others. The
study of heat conduction in such heterogeneous materials is, by nature, inherently more
difficult than that in homogeneous ones due to the difficulty that exists in obtaining
21
22
a clear mathematical description of the microstructure. The problem of determining
the overall conductivity of a composite material is an outstanding one in mathematical
physics and has been studied for well over a century since the seminal work of Maxwell
in 1873. Of the various theories to have emerged since then, one of the most powerful
and widely used ones is the effective medium theory. The simplicity and accuracy of
the theory make it highly attractive and it has been used successfully used to predict
the overall elastic moduli and thermal and electrical conductivities of composites. In our
present discussion, we will use this theory to determine the effective conductivity.
According to the effective medium theory, the given heterogeneous material having
discontinuous properties can be replaced, in an equivalent sense, by a homogeneous one
that gives the same average response to a given input at the macroscopic level. This
concept is known as homogenization and this assumption greatly simplifies the analysis,
as the well known results obtained for homogeneous materials can be used here directly.
The aim is then to derive the equations that describe heat transfer at the macroscopic
level from knowledge of the microstructure and constituent properties, with the assump-
tion that Fourier’s law holds at the microscopic level. The process of homogenization is
thus essentially an averaging one and two principal averaging techniques, volume averag-
ing and ensemble averaging, have been frequently used. In the present work, the volume
averaging technique will be considered and the dilute assumption will be employed, which
neglects all interactions between the fillers. As will be seen, the effective conductivity can
then be calculated from knowledge of the constituent properties and volume fractions if
the temperature fields inside and outside the inclusion phase can be obtained. For certain
geometries, the temperature field can be expressed analytically in a closed form and this
provides a direct and simple way to obtain the effective properties.
23
12
2
2
S1,2
S
V1
V2
Figure 3.1 Schematic representation of an arbitrary two phase composite showing thevarious notations used.
3.2.1 Composites with Perfect Thermal Contact between the Constituents
Consider a large two phase body of a total volume, V , and boundary surface, S,
comprising of two distinct homogeneous and isotropic phases of volumes, V1 and V2, and
conductivities, k(1) and k(2), respectively. Also, let their common interface be denoted
by S1,2. Here and in all subsequent discussions, ‘1’ will denote the matrix phase and
‘2’, the inclusion phase, respectively. Figure 3.1 shows such an arbitrary two phase
particulate composite. We consider the composite to be statistically homogeneous. Such
an assumption is valid when the fillers are uniformly dispersed within the matrix material
and it greatly simplifies the mathematical analysis.
Now by definition, the intensity, Hi, and heat flux, qi, are given as
Hi = −φ,i , (3.1)
qi = kijHj, (3.2)
24
where φ stands for the temperature field, kij denotes the conductivity tensor and a tensor
notation is used throughout (i = 1, 2, 3). Here the steady-state heat conduction process
with no heat generation will be considered. Under such circumstances the heat flux
vector is divergence free, which is denoted as
qi,i = 0. (3.3)
Let the material be subjected to a homogeneous boundary condition of the form
φ(S) = −Hoi xi, (3.4)
where φ(S) is the surface temperature, Hoi is the applied far-field constant intensity and
xi denotes the position vector at the surface S. This boundary condition is called homo-
geneous because when a homogeneous body is subjected to such a boundary condition,
the temperature field at any location within the body will be given by
φ(x) = −Hoi xi. (3.5)
Then for the homogenous body, the intensity and the heat flux fields will also be homo-
geneous i.e. they will be the same at all points within the material and will be given
by
Hi = Hoi , (3.6)
qi = koijH
oj , (3.7)
where koij denotes the conductivity tensor of the homogeneous medium.
Now, using the concept of volume averaging, the average intensity and heat flux
over the entire volume V of any material are given by
Hi =1
V
∫
V
Hi dV, (3.8)
qi =1
V
∫
V
qi dV, (3.9)
25
where the overbar signifies a volume average.
For a homogeneous body, the average intensity and heat flux are the same as the
values for these quantities at any location. Thus using (3.6) and (3.7), we can write
Hi = Hoi , (3.10)
qi = koijH
oj . (3.11)
This is not true for a heterogeneous medium though. However, if the heterogeneous
medium is considered to be statistically homogeneous, the intensity and heat flux fields
are also statistically homogeneous. In such a case, we can replace the given heterogeneous
material by an equivalent homogeneous one, which will give the same response to a given
input at the macroscopic level. For such an equivalent homogeneous material, the average
intensity and heat flux remain constant throughout and are equal to the corresponding
volume averages for a heterogeneous material. Hence, following (3.10) and (3.11), we can
say for an equivalent homogeneous medium,
Hi = Hoi , (3.12)
qi = k∗ijHoj . (3.13)
Here k∗ij denotes the effective thermal conductivity tensor of the equivalent homogeneous
medium.
Now for a two phase composite material having a perfect thermal contact between
the constituents, both the temperature and the normal component of the heat flux will
be continuous across the interface S1,2. Thus the average intensity and heat flux can be
expressed in terms of the corresponding quantities for the individual phases as
Hi = v1H(1)i + v2H
(2)i , (3.14)
qi = v1q(1)i + v2q
(2)i , (3.15)
26
where v1 and v2 denote the volume fractions of the matrix and particle phases, respec-
tively, and the individual phase volume averages of intensity and heat flux are defined
by
H(α)i =
1
Vα
∫
Vα
H(α)i dVα
q(α)i =
1
Vα
∫
Vα
q(α)i dVα
(α = 1, 2). (3.16)
Also for each of the individual phases,
q(1)i = k(1)H
(1)i (3.17)
q(2)i = k(2)H
(2)i . (3.18)
From (3.13), (3.15), (3.17) and (3.18) after appropriate substitutions, we get
k∗ijHoj = v1k
(1)H(1)i + v2k
(2)H(2)i (3.19)
and (3.12) and (3.14) give us
Hoi = v1H
(1)i + v1H
(2)i . (3.20)
Eliminating H(1)i from (3.19) and (3.20) and rearranging, we get the expression for the
effective thermal conductivity to be
k∗ijHoj = k(1)Ho
i + (k(2) − k(1))H(2)i v2. (3.21)
It is evident from (3.21) that the only unknown quantity in the equation is the average
intensity within the reinforcement phase. The intensity within the filler can be easily
calculated if the temperature field within it is known. Since we consider steady-state
heat conduction, the temperature field must satisfy the Laplace equation, whose form
will be defined by the geometry of the reinforcement phase. For certain geometries, the
temperature field can be easily obtained as a closed form solution to the Laplace equation.
This can then be used directly to determine the effective conductivity.
27
3.2.2 Composites with Imperfect Thermal Contact between the Constituents
We now consider composites which have an imperfect thermal contact between
the phases. Due to an imperfect interface, there will exist a finite contact resistance
at S1,2, which acts as a barrier to the heat flow. This generally occurs due to acoustic
mismatch between the constituents or a possible weak contact between them at the
common interface. As a result, in the presence of a normal heat flux, a temperature
drop occurs at the interface although the normal component of the heat flux remains
continuous, i.e.
φ(1)|S1,26= φ(2)|S1,2
, (3.22)
qi(1)ni|S1,2
= qi(2)ni|S1,2
, (3.23)
where ni denotes the outward normal at the interface S1,2.
Such interfaces which exhibit a discontinuity in temperature but allow for the conti-
nuity of the normal component of the heat flux, are referred to as interfaces with Kapitza
thermal resistance. For analysis of traditional composite materials, the interfacial resis-
tance is often neglected without causing significant errors in the obtained results, due to
the simplification in the analysis. However, as mentioned before, interfacial resistance is
one of the principal factors affecting heat flow in CNT/polymer composites. Hence the
consideration of the boundary resistance is one of the vital steps in the analysis. At this
point, the relationship which dictates the discontinuity of temperature at the interface
is not explicitly stated. The exact law governing this behavior will be discussed in the
next section when the temperature field within the inclusion is calculated.
Due to the discontinuity of temperature at the interface, a few relations derived in
the previous section need to be reconsidered. Specifically, the expression for the average
intensity given by (3.14) will no longer hold true. Hence the relation that will give the
average intensity in the presence of an interfacial contact resistance needs to be derived.
28
For doing this, consider a certain three phase continuum, where the volume frac-
tions of the phases are v1, v2 and v3, respectively. It is not necessary that each phase
corresponds to an actual physical phase; the definitions of the phases are completely
arbitrary. The consideration of such a three phase continuum is just for deriving the
relation for the average intensity in the presence of an interfacial resistance. For this
purpose only, it is assumed that each inclusion is surrounded by a third homogeneous
phase of constant thickness ∆x(3), the thickness being measured normal to the surface.
A schematic diagram of such a three phase continuum is shown in figure 3.2. The third
phase is thus equivalent to a uniform coating on the inclusion surface across where there
occurs a temperature drop from φ(1) to φ(2). For such a three phase continuum, the
average intensity will be given by
Hi = v1Hi(1)
+ v2Hi(2)
+ v3Hi(3)
, (3.24)
where, as before, Hi(α)
(α = 1, 2, 3) represents the average intensity within each phase
and v1 + v2 + v3 = 1.
For a two phase composite with an interfacial resistance, the form that (3.24) takes
in the limiting case when phase 3 goes to zero, is of particular interest. The desired form
is obtained by a limiting process when phase 3 is made to correspond to a vanishingly thin
sheet-like two dimensional surface i.e. in the limit when ∆x(3) → 0, phase 3 reduces to
a two-dimensional sheet like surface. This surface will then correspond to the interface
S1,2 across which there occurs a temperature jump. Now the behavior of the average
intensity field, Hi(3)
, associated with this third phase, is given by
Hi(3)
=1
∆x(3)G
(3)ij nj, (3.25)
where G(3)ij is a tensorial field associated with the two dimensional surface and nj is the
unit normal to the surface. The nature of G(3)ij is not immediately obvious and will become
29
2
1
3t
n
Figure 3.2 Schematic representation of a three phase continuum.
clear later. At present it is sufficient to state that it is a 2nd rank tensor associated with
the two dimensional region and satisfies (3.25). Also, since phase 3 is homogeneous,
H(3)i = H
(3)i . Thus, using (3.25) and the definition of volume fraction, we can write
v3Hi(3)
=V3
V
1
V3
∫
V3
H(3)i dV3 (3.26)
=1
V
∫
V3
G(3)ij nj
dV3
∆x(3). (3.27)
In the limit as ∆x(3) → 0, V3 → S1,2. Also noting that dV3 = ∆x(3) dS1,2, (3.27) can be
written as
v3Hi(3)
=1
V
∫
S1,2
G(3)ij nj dS1,2. (3.28)
Also, since Hi(3)
denotes the average intensity associated with phase 3, we can write
Hi(3)
= − 1
∆x(3)(φ(1) − φ(2))ni (3.29)
=1
∆x(3)(φ(2) − φ(1))ni. (3.30)
30
From (3.25) and (3.30) after a little manipulation, we end up with the relation
G(3)ij = (φ(2) − φ(1))δij, (3.31)
where δij is the Kronecker delta, defined as
δij =
{1 if i = j ;
0 otherwise.
Using (3.24), (3.28) and (3.31) we get the expression for the average intensity in
the presence of an imperfect interface as,
Hi = v1Hi(1)
+ v2Hi(2)
+1
V
∫
S1,2
(φ(2) − φ(1))ni dS1,2. (3.32)
It can be seen that this relation is almost the same as (3.14) except for the third term.
A simple observation reveals that this term accounts for the drop in temperature at the
interface which is clearly shown by the temperature difference of the two phases.
However, due to the continuity of the normal component of the heat flux at the
interface, the expression for the average flux given by (3.15) will be valid in this case.
From (3.12) and (3.32), we get
Hoi = v1Hi
(1)+ v2Hi
(2)+
1
V
∫
S1,2
(φ(2) − φ(1))ni dS1,2. (3.33)
Eliminating Hi(1)
from (3.19) and (3.33), we get after simplification and rearrangement
k∗ijHoj = k(1)Ho
i + (k(2) − k(1))v2Hi(2) − k(1) v2
V2
∫
S1,2
(φ(2) − φ(1))ni dS1,2. (3.34)
The above relation gives an expression for calculating the effective conductivity and
clearly accounts for the discontinuity in the temperature that exists at the interface.
This expression may be further simplified to yield a form that is suitable for calculation.
From (3.1) and (3.16), we get
H(2)i =
1
V2
∫
V2
−φ,(2)i dV2 (3.35)
= − 1
V2
∫
S1,2
φ(2)ni dS1,2, (3.36)
31
where in going from (3.35) to (3.36), use has been made of the Gauss theorem. Substi-
tuting (3.36) in (3.34), we get
k∗ijHoj = k(1)Ho
i +(k(1)−k(2))v2
V2
∫
S1,2
φ(2)ni dS1,2−k(1) v2
V2
∫
S1,2
(φ(2)−φ(1))ni dS1,2, (3.37)
which on simplification yields the following relation
k∗ijHoj = k(1)Ho
i + v2(k(2)Θ
(2)i − k(1)Θ
(1)i ), (3.38)
where
Θ(α)i = − 1
V2
∫
S1,2
φ(α)ni dS1,2. (3.39)
This expression will be used to calculate the effective conductivity. It can be seen that
unlike the previous case of a perfect thermal contact between the constituents, the tem-
perature fields both inside as well as outside the inclusion phase are unknowns in (3.38).
Once these are known, the effective conductivity can be easily calculated.
3.3 Modeling of an MWNT Inclusion
As seen in the previous section, the effective conductivity may be obtained once the
temperature fields inside and outside the inclusion phase are known. Since we consider
steady-state heat conduction with no heat generation, the temperature fields both inside
and outside the inclusion must satisfy the Laplace equation whose specific form will be
determined by the geometry of the inclusion phase. It is thus essential to clearly under-
stand and accurately model the MWNT geometry. As described before, the structure of
a single MWNT consists of several coaxially rolled graphene sheets that are made up of
interlinked carbon atoms. As the nanometer dimensions of the reinforcement phase are
at the atomic scale, the filler material can no longer be treated as a continuum. Since
the mathematical model described above assumes the inclusion phase to be continuous,
the MWNT structure cannot be directly incorporated into it. The development of a con-
32
tinuum model for the MWNT geometry is thus a vital step in the analysis process as it
allows the use of a continuum theory to determine the effective properties. This contin-
uum model of the MWNT must account for the fundamental assumption in continuum
mechanics that mass, momentum and energy can be represented in a mathematical sense
by continuous functions, that is, independent of length scale. Of course the question that
naturally arises is whether such continuum modeling is indeed valid. The best answer is
that previous attempts at using continuum modeling for determining the overall elastic
properties have yielded results which are in good agreement with experimental data. It
can thus be assumed that such a modeling can be applied for determining the thermal
properties as well and will yield acceptable results for the overall thermal conductivity.
The development of a continuum model for an MWNT inclusion is done in two
distinct steps. Firstly, an equivalent continuum model of an MWNT is developed by
taking into account the mechanism of heat conduction through an MWNT. Once this
is done, the structure and properties of the nanotube are taken into account and the
properties of an effective fiber are defined. This effective fiber is then considered to be the
inclusion phase that is embedded within the matrix material. So the composite material
to be analyzed is effectively an aligned short fiber composite, where the effective fiber
constitutes the reinforcement phase that is embedded within a polymer matrix. Our aim
is thus to predict the overall conductivity of this composite in the longitudinal direction
for which the effective medium theory, discussed in the previous section, can be directly
used.
3.3.1 Equivalent Continuum Model and Effective Solid Fiber
In developing a continuum model for an MWNT, special consideration must be
given to the contribution of the individual nanotube layers to thermal transport through
the nanotube. For electric conduction, it has been proved experimentally, that the in-
33
dividual layers have different current carrying capacities [31]. Similarly, with respect to
the elastic properties, it has been postulated that due to the weak Van der Waals forces
between the layers, there is very little load transfer between the layers and almost the
entire load is carried by the outer layer [8, 23]. The role of individual layers in thermal
transport however, has not been studied experimentally. Kim et al. [9], have postulated
that since, only the outer nanotube layer makes good thermal contact with the surround-
ing medium, its contribution to thermal transport through the nanotube will be higher
than the inner layers. Since the contribution of the individual layers is unknown, we as-
sume in the present work, that only the outer MWNT layer is involved in the conduction
of heat through the nanotube. This assumption seems to be justified considering the fact
that only this outer layer is in thermal contact with the surrounding matrix material
and should be thus responsible for the exchange of heat between the nanotube and the
matrix. It will be seen from the results that this is indeed the correct assumption as it
yields the correct value for the effective conductivity.
With this assumption, we can now define an equivalent continuum model of the the
nanotube structure. We choose a hollow cylinder having the same length and diameter as
that of the nanotube to represent an equivalent continuum model of the nanotube. The
thickness of the cylinder wall is the same as that of the outer nanotube layer (0.34 nm) and
we consider it to be made up of a homogeneous and isotropic material that has the same
physical properties as that of the nanotube. Such a continuum model of the nanotube
structure has been used before [8, 23] in determining the elastic properties of nanotube
composites. An analogous model has been developed here for determining the thermal
properties. Next, the heat carrying capacity of this hollow cylinder is applied to its entire
cross-section and the properties of an effective solid fiber are defined. Such an effective
fiber can be defined as a solid fiber that has the same length and diameter as that of
the hollow cylinder and has an identical temperature gradient across its length when the
34
same amount of heat is flowing through it. We are interested in finding the conductivity
of this effective fiber. This effective fiber thus retains the geometrical properties of the
nanotube while providing us with a continuum model of the nanotube structure that
is suitable for mathematical analysis. Thostenson and Chou [23], have used a similar
modeling technique for calculating the effective elastic modulus of an aligned MWNT
composite. Following them and noting the mathematical analogy between the elasticity
and heat conduction problems, we can write by the definition of an effective fiber
φ,3|NT = φ,3|eff , (3.40)
⇒ H3|NT = H3|eff , (3.41)
where the subscripts ‘NT’ and ‘eff’ refer to the nanotube continuum model and effective
fiber, respectively and the effective fibers are assumed to be aligned in the x3 direction,
so that we consider the gradient in this direction. Thus using (2) we can write
(q3
k33
)
NT
=
(q3
k33
)
eff
. (3.42)
Now using the definition of heat flux, we can write
(q3)eff =Q
Aeff
=Q
πd2/4, (3.43)
(q3)NT =Q
ANT
=Q
πtd, (∵ t ¿ d,ANT ≈ πtd) (3.44)
where Q denotes the amount of heat flowing though the nanotube along its length,
Aeff and ANT denote the cross sectional areas of the effective fiber and the nanotube
continuum model respectively, d represents the nanotube diameter and t the thickness of
the outer wall of the nanotube.
From (3.42), (3.43) and (3.44), we get after appropriate substitutions and simplifi-
cation
(k33)eff =4t
d(k33)NT (t/d < 0.25). (3.45)
35
a) b) c) d)
Figure 3.3 Development of a continuum model for an MWNT. a) Schematic diagram ofan MWNT showing concentric graphene layers; b) Equivalent continuum model; c)
Effective solid fiber, and d) A prolate spheroidal inclusion.
This gives an expression for the conductivity of the effective solid fiber in the
longitudinal direction. Since the transverse conductivity of an individual nanotube has
not been yet determined experimentally or theoretically, we do not know whether the
conductivity tensor for a nanotube is anisotropic or not. For simplicity of analysis, we
consider the consider the effective fiber to be isotropic, thus having a single value for the
conductivity. Nan and Shi [25] have used a similar assumption in their determination
of the effective conductivity of a random suspension of MWNTs in a fluid polymer.
Since we are interested in finding out the effective conductivity of the composite in the
longitudinal direction, such an assumption will not affect the final result. The problem
has thus been reduced to finding the effective thermal conductivity of a composite having
isotropic cylindrical short fibers as the filler, where the conductivity of the filler material
is given by (3.45) and there exists a contact resistance at the interface.
36
The aim is now to find the temperature field inside and outside a cylindrical inclu-
sion embedded in a continuous medium in the presence of an interfacial resistance. In
general, it is not easy to obtain a completely closed form solution for the temperature
field when a cylindrical inclusion is considered to be embedded in a homogeneous medium
due to the boundary conditions at the inclusion-matrix interface. For very high aspect
ratios however, the cylindrical geometry can be well approximated by a prolate spheroidal
geometry, as in the case of continuous fiber reinforcements. The advantage of using a
spheroidal geometry is that it admits a completely analytical closed form solution of the
Laplace equation for the interface boundary conditions, unlike the cylindrical inclusion.
Since nanotubes have extremely high aspect ratios (> 1000) and as the effective fiber
preserves this aspect ratio, the prolate spheroid can be used to approximate the solution
without introducing any significant errors in the obtained results. Therefore we need to
solve the Laplace equation in the spheroidal coordinate system.
3.3.2 The Prolate Spheroidal Inclusion
A prolate spheroid is the geometry obtained by revolving an ellipse about its major
axis and is denoted by the canonical equation
x21 + x2
2
a21
+x2
3
a23
= 1; a3 ≥ a2 = a1, (3.46)
for a spheroid whose major axis is aligned with the x3 direction; a1 and a3 refer to the
semi-minor and semi-major axes, respectively. Any point on the surface of the spheroid
is given by the parametric equations
x1 = c√
(ξ2 − 1)(1− µ2) cos ψ, (3.47)
x2 = c√
(ξ2 − 1)(1− µ2) sin ψ, (3.48)
x3 = cµξ, (3.49)
37
where 2c is the distance between the foci of the spheroid and (ξ, µ, ψ) represents a triply
orthogonal confocal spheroidal coordinate system such that
ξ ≥ 1, − 1 ≤ µ ≤ 1, 0 ≤ ψ ≤ 2π. (3.50)
In such a natural coordinate system, ξ = ξo(constant) would give family of confocal
prolate spheroids while µ = µo would give a family of hyperboloids of revolution. The
quantity ξo, denotes the inverse of the eccentricity, e, of the spheroid and is given by
ξo =
(1− a2
1
a23
)− 12
, (3.51)
while c is given by
c = (a23 − a2
1)12 . (3.52)
In the interior of the spheroid, ξ varies continuously from ξ = 1 on the axis of
symmetry (which is the x3 axis in this case) to ξ = ξo on the surface. The domain
exterior to the spheroid is given by ξ > ξo with ξ →∞ at infinity.
The metric scale factors for this coordinate system are given by
hξ = c
√ξ2 − µ2
ξ2 − 1, (3.53)
hµ = c
√ξ2 − µ2
1− µ2, (3.54)
hψ = c√
(ξ2 − 1)(1− µ2). (3.55)
The length of the spheroid is given by the length of the major axis i.e. l = 2a3.
The radius, defined by the perpendicular distance from any point on the surface to the
axis of symmetry, is given by
r2(x3) = (ξ2o − 1)
(1− x2
3
ξ2o
). (3.56)
38
= const.
= const.
= const.x
y
m
c
c
x
x
x1
2
3
Figure 3.4 A prolate spheroidal coordinate system. The parameters ξ, µ and ψ refer to3 sets of orthogonal surfaces.
The radius is clearly variable along the length of the spheroid. However for very slender
prolate spheroids, the radius is almost constant and is equal to the radius at the center,
which is equal to the length of the semi-minor axis a1. Thus in the present discussion, the
diameter at the center will be taken to be the diameter of the effective fiber i.e. d = 2a1.
3.4 Solution of the Auxiliary Problem
In this section, the final expression for determining the effective thermal conductiv-
ity will be obtained. The general expression for obtaining the effective conductivity was
obtained in Section 3.1 (3.38). The unknown parameters in that equation, namely Θ(1)i
and Θ(2)i , will now be determined. As shown by (3.39), these quantities can be obtained
once the temperature fields inside and outside the inclusion phase are calculated.
We now proceed to determine the temperature fields inside and outside the inclu-
sion phase. For doing this, we seek a solution to the auxiliary problem wherein a single
prolate spheroidal particle is embedded in an infinite matrix and a constant intensity,
39
Ho3 is applied on the surface at infinity in the x3 direction; we are interested in finding
the temperature field inside and outside this single inclusion. Here we neglect all inter-
actions between the inclusions and assume the composite to be sufficiently dilute. This
assumption is justified if we consider the extremely small size of the nanotubes, which
ensures that when the nanotubes are uniformly dispersed, they will be far away from
each other and there will be no interaction among them. Also, since we are interested
in determining the effective conductivity at low nanotube volume fractions, the dilute
assumption should not produce any significant errors in the obtained results. Due to
this assumption, the temperature fields inside and outside the inclusion phase will be the
same for every inclusion and it will suffice to just solve the auxiliary problem. Now for
the applied intensity Ho3 , the surface temperature given by (3.5) will be
φ(S) = −Ho3x3. (3.57)
For steady-state heat conduction with no heat generation, the temperature field
must satisfy the Laplace equation which is given by
φ(α),ii = 0 ; (α = 1, 2). (3.58)
In a confocal spheroidal coordinate system, the Laplacian operator is given by [33]
∇2 =1
hξhµhψ
[∂
∂ξ
(hµhψ
hξ
∂
∂ξ
)+
∂
∂µ
(hξhψ
hµ
∂
∂µ
)+
∂
∂ψ
(hξhµ
hψ
∂
∂ψ
)]. (3.59)
From (3.53), (3.54) and (3.55), substituting the values of the metric scale factors and
simplifying, we get the Laplace equation in the spheroidal coordinate system to be
∂
∂ξ
[(ξ2 − 1)
∂φ(α)
∂ξ
]+
∂
∂µ
[(1− µ2)
∂φ(α)
∂µ
]+
ξ2 − µ2
(ξ2 − 1)(1− µ2)
∂2φ(α)
∂ψ2= 0. (3.60)
40
Due to the symmetry of the temperature field about the major axis of the spheroid (x3
axis), the temperature field will be independent of the coordinate ψ. Thus the Laplace
equation reduces to
∂
∂ξ
[(ξ2 − 1)
∂φ(α)
∂ξ
]+
∂
∂µ
[(1− µ2)
∂φ(α)
∂µ
]= 0. (3.61)
The solution to the above equation will give the general solution for the temperature field.
In order to obtain the complete solution in each region(inside and outside the inclusion),
the following boundary conditions are applied:
i) Inside the spheroid, the temperature field must be regular i.e. the solution must
be devoid of singularities, and
ii) At infinity, the temperature field outside the spheroid must satisfy the boundary
condition at the surface (3.57).
Now, the surface boundary condition (3.57) can also be written as
φ(S) = −Ho3cµξ, (3.62)
where the value of x3 from (3.49) has been substituted in (3.57).
For a surface boundary condition of the form given by (3.62), the temperature at
any point within the material can be expressed as
φ(α) = Ho3cf(µ, ξ), (3.63)
where f(µ, ξ) is a function of µ and ξ. In order to solve (3.61), we use the method of
separation of variables. Let us assume that the solution for φ(α) is separable in the form
φ(α) = Ho3cΓ(ξ)Ψ(µ), (3.64)
where Γ(ξ) is a function of ξ only and Ψ(µ) is a function of µ only. Substituting this in
(3.61) and dividing throughout by (3.64) we get
1
Γ(ξ)
∂
∂ξ
[(ξ2 − 1)
∂Γ(ξ)
∂ξ
]= − 1
Ψ(µ)
∂
∂µ
[(1− µ2)
∂Ψ(µ)
∂µ
]= k2(constant). (3.65)
41
Thus from (3.65) we end up with the following ordinary differential equations:
d
dξ
[(1− ξ2)
dΓ(ξ)
dξ
]+ k2Γ(ξ) = 0, (3.66)
d
dµ
[(1− µ2)
dΨ(µ)
dµ
]+ k2Ψ(µ) = 0, (3.67)
which are the Legendre differential equations. The Legendre differential equation is a
second order ordinary differential equation given by
d
dx
[(1− x2)
du
dx
]+ n(n + 1)x = 0. (3.68)
This equation admits two linearly independent solutions Pn(x) and Qn(x), which are
known as the Legendre polynomials of the first and second kind, respectively and are
defined as [34]
Pn(z) =1
2nn!
d
dzn(z2 − 1)n, (3.69)
Qn(z) =1
2Pn(z) ln
z + 1
z − 1−Wn−1, (3.70)
where Wn−1(z) is given by
Wn−1(z) =1
nP0(z)Pn−1(z) +
1
n− 1P1(z)Pn−2(z) + · · ·+ Pn−1(z)P0(z). (3.71)
The definitions of the Legendre functions given by (3.69) and (3.70) are valid for
the entire complex plane and the argument, z, denotes any complex number. Thus a
complete solution to (3.68) can be obtained by a linear combination of Pn(x) and Qn(x).
Hence for k2 = n(n + 1), Γ(ξ) will be given by a linear combination of Pn(ξ) and
Qn(ξ), while Ψ(µ) will be given by a linear combination of Pn(µ) and Qn(µ). Thus for
(3.61), we have the following particular solutions [35]
φ(α) = Pn(ξ)Pn(µ);
φ(α) = Pn(ξ)Qn(µ);
φ(α) = Qn(ξ)Pn(µ);
φ(α) = Qn(ξ)Qn(µ).
42
Inside the spheroid on the x3 axis, ξ = 1 and −1 ≤ µ ≤ 1. Thus the Q-functions
go to infinity on the x3 axis and cannot be used for the temperature field inside the
inclusion φ(2), as they violate the first boundary condition. Hence in the region interior
to the spheroid the temperature field may be obtained by a linear combination of the
products Pn(µ)Pn(ξ). Thus the temperature field within the inclusion is given by
φ(2)(µ, ξ) = Ho3c
∞∑n=0
B2n+1P2n+1(µ)P2n+1(ξ); 1 ≤ ξ ≤ ξo, (3.72)
where B2n+1 is a set of constants. It may be noted that only the odd order Legendre
polynomials appear in the general solution of φ. This is due to the spheroidal geometry
of the inclusion which makes the temperature field, φ, antisymmetric with respect to the
major axis x3.
Outside the inclusion however, ξ = 1 does not occur and thus Qn(ξ) is still a
possibility though Qn(µ) is not as µ = 1 is still possible. The general solution for
the temperature field in the region outside the spheroid may be obtained by a linear
combination of of the products Pn(ξ)Pn(µ) and Pn(µ)Qn(ξ) and is given by
φ(1)(µ, ξ) = Ho3c
∞∑n=0
[A2n+1Q2n+1(ξ) + C2n+1P2n+1(ξ)
]P2n+1(µ), (3.73)
where A2n+1 and C2n+1 are two sets of constants.
Now, as ξ →∞, Q2n+1(ξ) → 0. Also, according to the second boundary condition,
at infinity, φ(1)(µ, ξ) = φ(S). Thus from (3.62) and (3.73), we get
−Ho3cµξ = Ho
3c
∞∑n=0
C2n+1P2n+1(µ)P2n+1(ξ). (3.74)
The only way in which the above equation can be satisfied is to take n = 0 which gives
P1(ξ) = ξ and P1(µ) = µ. Thus we get C1 = −1. The temperature field outside the
spheroidal inclusion will then be given by
φ(1)(µ, ξ) = −Ho3cP1(µ)P1(ξ) + Ho
3c
∞∑n=0
A2n+1P2n+1(µ)Q2n+1(ξ); ξ ≥ ξo. (3.75)
43
The two sets of unknown constants, A2n+1 and B2n+1, are determined by applying
the following boundary conditions at ξ = ξo
k(2)∂φ(2)
∂n= k(1)∂φ(1)
∂n(3.76)
= β(φ(1) − φ(2)), (3.77)
where n is the unit normal at the interface boundary defined from the inclusion to the
matrix and β is defined as the boundary conductance. This boundary conductance is the
inverse of the Kapitza boundary resistance described before and is defined as the ratio
of the normal component of the heat flux to the temperature drop at the interface. Now
at the interface,
∂
∂n=
1
hξ
∂
∂ξ. (3.78)
Therefore we can write the boundary conditions as
(k(2)
k(1)
)1
hξ
∂φ(2)
∂ξ=
1
hξ
∂φ(1)
∂ξ(3.79)
=β
k(1)(φ(2) − φ(1)). (3.80)
Thus substituting (3.72) and (3.75) into the above boundary conditions, we get
λ
∞∑n=0
B2n+1P2n+1(µ)P2n+1(ξo) = −P1(ξo)P1(µ) +∞∑
n=0
A2n+1P2n+1(µ)Q2n+1(ξo) (3.81)
= βhξ
[−P1(µ)P1(ξo) +
∞∑n=0
{A2n+1Q2n+1(ξo)−B2n+1P2n+1(ξo)
}P2n+1
], (3.82)
where β = βc/k(1), λ = k(2)/k(1), hξ = hξ/c and the dot denotes differentiation with
respect to the argument.
In order to solve for the constants, A2n+1 and B2n+1, we make use of the general
orthogonality property of the Legendre polynomials of the first kind given by
∫ 1
−1
Pn(µ)Pm(µ) dµ =2
2n + 1δnm, (3.83)
44
where δnm is the Kronecker delta.
Multiplying both sides of (3.81) by Pm(µ) and integrating the resulting equation
over (−1, 1), we get
−P2n+1(ξo)δ(n) + A2n+1Q2n+1(ξo) = λB2n+1P2n+1(ξo), (3.84)
where δ(n) is defined as
δ(n) =
{1 if n = 0 ;
0 otherwise.
The value of A2n+1 is thus obtained as
A2n+1 =P2n+1(ξo)
Q2n+1(ξo)[δ(n) + λB2n+1]. (3.85)
In deriving (3.84), the following results have been used
∞∑n=0
A2n+1Q2n+1(ξo)
∫ 1
−1
P2n+1(µ)Pm(µ) dµ =
(2
4n + 3
)A2n+1Q2n+1(ξo), (3.86)
∞∑n=0
B2n+1P2n+1(ξo)
∫ 1
−1
P2n+1(µ)Pm(µ) dµ =
(2
4n + 3
)B2n+1P2n+1(ξo), (3.87)
−P1(ξo)
∫ 1
−1
P1(µ)Pm(µ) dµ = −(
2
4n + 3
)P2n+1(ξo)δ(n), (3.88)
which can all be derived directly by using (3.83) (See Appendix A for detailed derivation).
Repeating the same process for (3.82), namely multiplying both sides by Pm(µ)
and integrating over (-1, 1), we get the following relation:
−P2n+1(ξo)δ(n) + A2n+1Q2n+1(ξo)−B2n+1P2n+1(ξo) =
[λ
β
(4n + 3
2
)] ∞∑m=0
B2m+1P2m+1(ξo)
∫ 1
−1
(ξ2o − 1
ξ2o − µ2
) 12
P2n+1(µ)P2m+1(µ) dµ. (3.89)
The above equation can be further simplified by substituting the value of the coef-
ficient A2n+1 from (3.85) and using the Wronskian relationship for the Legendre polyno-
mials of the first and second type which is given by
P sn(ξo)Q
sn(ξo)− P s
n(ξo)Qsn(ξo) = (−1)s+1 (n + s)!
(n− s)!
1
(ξ2o − 1)
. (3.90)
45
After simplification, the resulting set of infinite linear equations for the coefficients
B2n+1 are obtained from (3.89) as
δ(n) + B2n+1[1− (1− λ)(ξ2o − 1)P2n+1(ξo)Q2n+1(ξo)] =
(λ
β
) ∞∑m=0
B2m+1χnm(ξo), (3.91)
where the coefficients χnm(ξo) are defined by
χnm(ξo) =
(4n + 3
2
)(ξ2
o − 1)Q2n+1(ξo)P2m+1(ξo)
∫ 1
−1
(ξ2o − 1
ξ2o − µ2
) 12
P2n+1(µ)P2m+1(µ) dµ.
(3.92)
On solving (3.91), the values of the coefficients B2n+1 may be calculated. For
numerical calculations, however, solution of an infinite set of linear equations is not
possible and the series thus needs to be truncated at some point. In most cases, however,
only a few terms are needed to achieve convergence, which is a simplification from a
computational point of view. Once the coefficients, A2n+1 and B2n+1, are known, the
temperature fields inside and outside the inclusion may be readily obtained from (3.72)
and (3.75) respectively. The quantity Θ(α)3 can now be calculated. For the spheroidal
surface, we note that
n3 =∂x3
∂n=
µc
hξ
(3.93)
and dS1,2 = hµhψ dµ dψ. (3.94)
Thus using (3.39), we can write
Θ(α)3 = −
(1
V2
) ∫ 2π
0
∫ 1
−1
φ(α)cµhψhµ
hξ
dµ dψ, (3.95)
where the values of the metric coefficients have been calculated at the spheroidal interface
ξ = ξo. Substituting (3.53), (3.54) and (3.55) into the above equation and noting that
the volume of the spheroid is given by V2 = 43πξo(ξ
2o − 1)c3, we get
Θ(α)3 = − 3
4πξoc
∫ 2π
0
∫ 1
−1
φ(α)µ dµ dψ. (3.96)
46
For α = 2, we get (for derivation see Appendix B)
Θ(2)3 = −B1H
o3 . (3.97)
In deriving the above equation, use has been made of another integral property of the
Legendre polynomials given by
∫ 1
−1
µkPn(µ) dµ = 0, k = 0, 1, 2, . . . , n− 1. (3.98)
Similarly, substituting φ(1) in (3.39) and noting the orthogonality properties of the
Legendre polynomials, we get (See Appendix B)
Θ(1)3 = Ho
3 − A1Ho3
Q1(ξo)
ξo
(3.99)
= Ho3
[1− Q1(ξo)P1(ξo)
ξoQ1(ξo)(1 + λB1)
]. (3.100)
Thus from (3.38) we can write
k∗33Ho3 = Ho
3k(1) + v2(k
(2)Θ(2)3 − k(1)Θ
(1)3 ) (3.101)
= Ho3k
(1) + v2Ho3
[−k(2)B1 − k(1) +
Q1(ξo)P1(ξo)
ξoQ1(ξo)(1 + λB1)k
(1)
]. (3.102)
Finally the effective thermal conductivity in the longitudinal direction is given by
k∗33 = k(1) [1 + v2(1 + λB1)f(ξo)] , (3.103)
where B1 is obtained as a solution to (3.89) and the function f(ξo) is defined by
f(ξo) =Q1(ξo)P1(ξo)
ξoQ1(ξo)− 1 (3.104)
= [ξo(ξ2o − 1)Q1(ξo)]
−1 (3.105)
=
[1
2ξo(ξ
2o − 1) ln
(ξo + 1
ξo − 1
)− ξ2
o
]−1
, (3.106)
47
since the Legendre polynomials, P1(ξo) and Q1(ξo), are given by
P1(ξo) = ξo, (3.107)
Q1(ξo) =ξo
2ln
(ξo + 1
ξo − 1
)− 1, (3.108)
and
Q1(ξo) =1
2ln
(ξo + 1
ξo − 1
)− ξo
ξ2o − 1
. (3.109)
This concludes the derivation for the effective thermal conductivity in the lon-
gitudinal direction. Equation (3.103) gives the expression for calculating the effective
conductivity. It can be seen that the overall conductivity is expressed completely in
terms of the constituent conductivities, the inclusion geometry and volume fraction as
well as the interfacial conductance. This expression will now be used to obtain numerical
results for the effective conductivity.
CHAPTER 4
RESULTS AND DISCUSSIONS
4.1 Introduction
In this chapter, the results obtained by using the proposed theory will be presented.
For theoretical predictions, an aligned MWNT/polymer composite having a uniform
dispersion of nanotubes is considered and its effective thermal conductivity is calculated
using the formula derived in the previous chapter. The obtained results are compared to
other theoretical models and published experimental results to determine the accuracy
of the proposed model. The results are then analyzed to elucidate the contribution of
the various factors which govern heat conduction in CNT/polymer composites and gain
insights into the mechanism of heat transfer at the nanometer scale.
4.2 Numerical Calculations
In this section, numerical calculations will be carried out to determine the effective
longitudinal conductivity of an aligned MWNT/polymer composite. The nanotubes are
considered to be uniformly dispersed in the polymer matrix and only small nanotube
volume fractions have been considered. The necessary calculations have been carried out
using the symbolic computation software, Mathematica. The Mathematica program used
for the calculations is given in Appendix C.
Let
d− Average nanotube diameter.
l − Average nanotube length.
48
49
k(1) − conductivity of the polymer matrix
kNT − conductivity of an MWNT.
k(2) − conductivity of the effective fiber.
The formula for the effective longitudinal conductivity in the x3 direction, k∗33, was
derived in the previous chapter and is given by
k∗33 = k(1) [1 + v2(1 + λB1)f(ξo)] , (4.1)
where v2 is the volume fraction of the nanotube phase and f(ξo) and λ are defined as
f(ξo) =
[1
2ξo(ξ
2o − 1) ln
(ξo + 1
ξo − 1
)− ξ2
o
]−1
, (4.2)
λ =k(2)
k(1)(4.3)
and ξo denotes the inverse of the eccentricity of the ellipsoid. The constant B1 is obtained
as a solution to the following infinite set of linear simultaneous equations:
δ(n) + B2n+1[1− (1− λ)(ξ2o − 1)P2n+1(ξo)Q2n+1(ξo)] =
(λ
β
) ∞∑m=0
B2m+1χnm(ξo), (4.4)
where the coefficients χnm(ξo) are defined by
χnm(ξo) =
(4n + 3
2
)(ξ2
o − 1)Q2n+1(ξo)P2m+1(ξo)
∫ 1
−1
(ξ2o − 1
ξ2o − µ2
) 12
P2n+1(µ)P2m+1(µ) dµ,
(4.5)
β =βc
k(1)(β − interfacial conductance) (4.6)
and the definitions of c, Pn, Qn and δ(n) are as given in the previous chapter.
Thus, to determine the effective conductivity, the quantities ξo, λ and β need to
be calculated from the geometrical parameters of the MWNT sample and the conduc-
50
tivities of the MWNT and the polymer matrix. We consider the following data for the
MWNT/polymer composite:
d = 25 nm; k(1) = 0.2 W/m K;
l = 50 µm; kNT = 3000 W/m K.
Using the above data, the parameters ξo and c can be calculated from (3.51) and
(3.52). Thus, we get
ξo =
(1− a2
1
a23
)− 12
=
(1− d2
l2
)− 12
(4.7)
= 1.000000125. (4.8)
c = (a23 − a2
1)12
=
[(l
2
)2
−(
d
2
)2] 1
2
(4.9)
= 24999.9969 nm. (4.10)
The value of the function f(ξo) will then be given by
f(ξo) =
[1
2ξo(ξ
2o − 1) ln
(ξo + 1
ξo − 1
)− ξ2
o
]−1
= −1.000001824. (4.11)
Also, the conductivity of the effective fiber, k(2), given by (3.45), will be
k(2) =4t
dkNT (4.12)
= 163.2 W/m K, (4.13)
where the thickness, t, of the outer nanotube layer has been taken to be 0.34 nm. It
can be seen that the conductivity of the effective fiber is much lower than the intrinsic
51
conductivity of an MWNT. This is due to the fact that only the outer nanotube layer
has been assumed to be involved in the conduction of heat through the nanotube. As
will be seen next, this assumption yields the correct value for the overall conductivity
and hence this is a vital aspect of heat conduction in CNT composites which needs to be
considered carefully. The value of λ will thus be
λ =k(2)
k(1)
= 816. (4.14)
Finally we need the value of the interfacial conductance β. Here the value of β is
taken to be 12 MW/m2 K. This value of the interfacial conductance was determined using
molecular dynamics simulations for a nanotube suspension in an organic fluid by Huxtable
et al. [22]. This seems to be the only published value available for this parameter and
hence it will be used in the calculations here. Thus the value of β is obtained as
β =βc
k(1)(4.15)
= 1500. (4.16)
Using the above data, the value of the constant B1 can now be evaluated. As can
be seen from (4.4), B1 is obtained as a solution to an infinite set of linear simultaneous
equations. For numerical calculations, however, solution of an infinite set of equations
is not possible and the series thus needs to be truncated at some point. The number of
terms used will determine the accuracy of the solution obtained. In most cases, however,
the value of B1 converges to a unique value after a few terms. In the present case,
convergence is achieved after 6 terms. For greater accuracy, we have used 8 terms in the
solution which gives us 8 simultaneous equations to be solved. On solving, the value of
B1 is obtained to be −0.997889266.
52
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
MWNT Volume Fraction (%)
Effe
ctiv
e C
ondu
ctiv
ity k
33∗ (
W/m
K)
Halpin−Tsai Model
Present Model
d = 25 nml = 50 µm
Figure 4.1 Variation of the effective thermal conductivity k∗33 with nanotube volumefraction.
The effective conductivity can now be directly calculated from (4.1). For 1% vol-
ume fraction of nanotubes i.e v2 = 0.01, the effective conductivity is obtained to be
1.82656 W/m K. This means that the ratio of the effective conductivity to the matrix
conductivity will be about 9. This is a direct indication of how much increase in the
thermal conductivity can be obtained at even 1% volume fraction of nanotubes. It also
shows that despite the presence of a high interfacial resistance, an aligned arrangement of
MWNTs in the composite can give a significantly high value for the overall conductivity.
The variation of the effective conductivity with nanotube loading is linear as shown in
figure 4.1 and this is to be expected as we have neglected all interactions between the
nanotubes.
53
4.3 Comparison With Existing Theoretical Models and Experimental Data
The results obtained in the previous section will now be compared with other the-
oretical models to determine their accuracy. A simple model for calculating the effective
conductivity of a short-fiber composite is the Halpin-Tsai model. This model has been
derived empirically and is reasonably accurate over a wide range of data. According to
this model, the effective longitudinal conductivity of a short fiber composite will be given
by
k∗l = km
(1 + ξηvf
1− ηvf
), (4.17)
where
η =(kf/km)− 1
(kf/km) + ξand ξ = 2
l
d. (4.18)
Here k∗l denotes the effective longitudinal conductivity, kf and km denote the fiber and
matrix conductivities, vf denotes the fiber volume fraction and l and d denote the fiber
length and diameter, respectively. If we consider the nanotube to act like a solid fiber
with a conductivity kf = kNT , for the same MWNT/polymer composite data considered
before, the longitudinal conductivity at 1% nanotube volume fraction is obtained to be
6.567 W/m K. This translates into an approximately 33 times increase in the matrix
conductivity which is much greater than our estimate as well as significantly outside the
range of published experimental results.
Another simple estimate for the composite thermal conductivity can be obtained
by the formula given by Huxtable et al. [22], which is
Λcomp = 〈cos2 θ〉ΛfiberVfiber, (4.19)
where Λcomp is the composite thermal conductivity, Λfiber is the fiber thermal conductiv-
ity, Vfiber is the fiber volume fraction and θ is the angle between a given direction and
the fiber axis. The brackets indicate an average over all the fibers in the composite. For
54
well-aligned fibers, 〈cos2 θ〉 = 1 and for completely random orientations, 〈cos2 θ〉 = 1/3.
As before, if we consider the MWNT to act like a solid fiber with a longitudinal con-
ductivity Λfiber = kNT , the composite longitudinal conductivity at 1% nanotube volume
fraction will be
Λcomp = kNT Vfiber (4.20)
= 30 W/m K, (4.21)
which is even higher than the prediction of the Halpin-Tsai equation.
Relation (3.19) also gives us a simple way to roughly estimate the effective conduc-
tivity for a random MWNT dispersion. According to (3.19), the random conductivity
is roughly one-third of the longitudinal conductivity. Thus if we consider our nanotube
sample to be randomly dispersed in the polymer matrix instead of being aligned, the
effective conductivity will then be roughly a third of the calculated longitudinal conduc-
tivity. The random thermal conductivity will thus be approximately 0.6089 W/m K.
This translates into an approximately 3 times increase in the matrix conductivity. This
is in the same range as the experimental results of Choi et al. [17], who have reported
that a random dispersion of MWNTs in a synthetic poly α-olefin oil increased the matrix
conductivity by about 2.5 times. This can be counted as evidence to validate our as-
sumption of considering the effective fiber for analysis rather than treating the nanotube
to act like a solid fiber.
These results highlight a very important aspect of heat conduction in MWNT/
polymer composites. It can be inferred from the above results that for determining the
overall behavior, the nanotube cannot be assumed to act as a solid fiber. Such a con-
sideration leads to a significant overestimation in the effective properties that are much
higher than experimental results. Many authors [22, 21], however, have interpreted this
vast difference in theoretical predictions and experimental results to be due to a very
55
high interfacial resistance. Although the interfacial resistance is a principal factor that
gives a lower than expected conductivity in CNT-composites, it is not the only one. As
shown here, consideration of the effective fiber rather than the nanotube itself, to be
the reinforcement phase, gives results that are in the same range as those obtained ex-
perimentally. Since the mechanism of heat conduction through the nanotube is used to
develop the concept of the effective fiber, it must be given due consideration while deter-
mining the effective thermal properties of CNT-composites. A thorough enumeration of
this and the other parameters which affect heat conduction in CNT-composites is given
in the next section.
4.4 Evaluation of Factors Affecting the Effective Conductivity
In this section, the results obtained in the previous section will be analyzed and
the contribution of the various factors affecting heat conduction in MWNT/polymer
composites will be critically evaluated. There are three principal factors that have been
found to greatly influence the effective conductivity in CNT-composites and they affect
the conductivity in different ways. An evaluation of these factors will help us in bet-
ter understanding the mechanism of heat conduction in CNT-composites and how it is
different from heat conduction in traditional fiber reinforced composites.
4.4.1 Contribution of Individual Nanotube Layers to Heat Conduction
First, we consider the role of the individual nanotube layers in heat transport
through the nanotube. As mentioned in the previous chapter, the contribution of the
individual MWNT layers to heat conduction through the nanotube has not yet been
studied either experimentally or theoretically and hence it is presently not possible to say
with any degree of certainty, whether the individual layers have different heat carrying
capacities or not. In the present work, we have assumed that only the outer layer is
56
involved in heat transport and neglected any contribution from the inner layers. This
assumption is motivated by the fact that only the outer nanotube layer makes thermal
contact with the surrounding matrix and hence should be responsible for the bulk of the
heat transfer between the nanotube and the surrounding matrix.
Now, let us consider the case when all the layers are equally involved in the heat
transfer. Assuming that the inner diameter of the MWNT is di and its outer diameter
do, Aeff (cross-sectional area of the nanotube continuum model) would be
Aeff =π(d2
o − d2i )
4(4.22)
and the conductivity of the effective fiber, given by (3.42), (3.43) and (3.44), would then
be
k(2) =d2
o − d2i
d2o
kNT . (4.23)
If we assume the inner diameter to be 15 nm and the outer diameter to be 25 nm, we
get k(2) to be 1920 W/m K. Using this data, the effective conductivity is obtained to be
18.939 W/m K, which is about 10 times higher than our present estimate. Consideration
of the inner layers to be involved in the heat conduction, thus, results in a significant
overestimation of the effective conductivity.
It can thus be inferred that the bulk of the heat flowing through the tube is carried
by the outer layer only and the contribution of the inner layers is minimal and can be
neglected for all practical purposes. Since the outer MWNT layer carries the bulk of the
heat flowing through the nanotube, the MWNT conductivity in the composite will be
much lower than its intrinsic conductivity, as indicated by the low conductivity of the
effective fiber (4.10). Also, this inference is in tune with the findings that the outer layer
carries the bulk of the load [23] for mechanical loading and the current [31] for electrical
loading. We have thus proved theoretically that the different nanotube layers in the
57
Figure 4.2 Variation of the effective thermal conductivity k∗33 with nanotube length.
MWNT do not have the same heat carrying capacity and that in a composite material,
the maximum amount of heat flow is through the the outer layer.
4.4.2 Influence of Nanotube Length and Diameter on the Conductivity
Another significant factor which influences the effective conductivity is the variation
in nanotube length and diameter. As can be seen from (4.6), a change in either the
nanotube length or diameter will cause a change in the parameters ξo and c, which will
affect the values of B1 and f(ξo) and thus cause a change in the effective conductivity.
The variation in the effective conductivity with nanotube length at a constant diameter is
shown in figure 4.2. It can be seen that the effective conductivity, k∗33, is not very sensitive
to the nanotube length. As the length changes from 30 µm to 70 µm, the conductivity
only changes in the range of 1.82−1.83 W/m K. Figure 4.3 shows the variation in the
58
0 10 20 30 40 500
1
2
3
4
5
6
7
8
9
MWNT Diameter (nm)
Effe
ctiv
e C
ondu
ctiv
ity k
33∗ (
W/m
K)
l = 50 µm
Figure 4.3 Variation of the effective thermal conductivity k∗33 with nanotube diameter.
effective conductivity with nanotube diameter at a constant length. It can be seen that
the effective conductivity, however, changes drastically with change in the nanotube
diameter, dropping from over 8 W/m K at a diameter of 5 nm to just above 1 W/m K at
a diameter of 45 nm. A decrease in nanotube diameter can thus significantly increase the
overall conductivity. This finding coincides with the results of Thostenson and Chou [23]
for the elastic modulus. The authors have shown the effective elastic modulus to change
significantly with change in the nanotube diameter and considering the mathematical
analogy between the elasticity and heat conduction problems, our results are justified.
These findings also underline the importance of considering the geometrical properties of
the nanotube sample during analysis. As discussed in Chapter II, most present nanotube
synthesis techniques generally produce samples which have a distribution of nanotube
lengths and diameters. From the point of view of analysis, it is always desirable to have
a near uniform nanotube diameter distribution, so that a good estimate may be obtained
59
101
102
103
104
1.826
1.8265
1.827
1.8275
1.828
1.8285
1.829
Interfacial Conductance β (MW/m2 K)
Eff
ect
ive
Co
nd
uct
ivity
k3
3∗ (
W/m
K)
d = 25 nm l = 50 µm
β = 12 MW/m2 K
β = 30 MW/m2 K
β = 50 MW/m2 K β = 500 MW/m2 K
β = 2000 W/m2 K
Figure 4.4 Variation of the effective thermal conductivity k∗33 with change in interfacialconductance.
by considering the mean diameter. However, for a highly scattered diameter distribution,
considering the average diameter may result in significant errors in estimation of the
overall conductivity. In such a case, due consideration must be given to the contribution
of the different diameters by taking, for example, a weighted average or by considering
a diameter distribution function as has been done by Thostenson and Chou [23] for the
elastic modulus.
4.4.3 Influence of the Interfacial Resistance on the Effective Conductivity
Finally, we consider the influence of the interfacial resistance on the effective con-
ductivity. As has been mentioned before, the interfacial resistance between the nanotube
and the matrix has been cited as the principal factor for the less than expected thermal
conductivity of CNT/polymer composites. As such, in the present work, the interfacial
60
resistance has been taken into account while developing the mathematical model for pre-
dicting the effective conductivity. We now examine the influence of this parameter on the
effective conductivity. The variation of the effective conductivity with change in the value
of the interfacial conductance has been shown in figure 4.4. It can be seen, however, that
the effective conductivity is not very sensitive to change in the interfacial conductance.
As the interfacial conductance increases from 12 MW/m2 K to about 10000 MW/m2 K,
the interfacial conductance increases only slightly. It is worthwhile here to examine the
case when β → ∞. Such a case corresponds to a perfect thermal contact between the
nanotube and the matrix. Under such a condition, the value of the constant B1 becomes
B1 = −[1− (1− λ)(ξ2o − 1)Q1(ξo)]
−1, (4.24)
for which the effective conductivity will be given by (see Benveniste and Miloh [29])
k∗33
k1
= 1 + v2
[1− λ
1− (1− λ)(ξ2o − 1)Q1(ξo)
] [1
ξo(ξ2o − 1)Q1(ξo)
](4.25)
= 1− v2(1− λ)
1− (1− λ)(ξ2o − 1)Q1(ξo)
. (4.26)
Using the same data for the calculation, the value of k∗33 is obtained to be 1.82758
W/m K, which, as can be seen from figure 4.4, is the value that k∗33 converges to. Thus
it can be seen that the consideration of the interfacial resistance does not significantly
affect the effective conductivity. This result is very interesting and needs to be considered
in a bit more detail.
Firstly, it must be noted that the effective medium theory described here tends
to the case of perfect interfaces for low volume fractions. This result might be due to
this factor. However, it can also be inferred from the above result that the interfacial
resistance does not significantly affect the overall conductivity. It can thus be said that
although the interfacial resistance affects the overall heat conduction mechanism, it is
probably not the most significant factor that decreases the overall conductivity. The
61
interfacial resistance probably has a greater influence on the heat carrying capacity of
the composite rather than the effective conductivity.
This conclusion is very important as the interfacial resistance has been cited by
almost all authors as being the principal factor that decreases the overall conductivity
in CNT/polymer composites. However as shown here, this might not be actually the
case. The interfacial resistance does affect heat conduction in CNT/polymer composites,
but its influence on the effective conductivity may not be that great. Instead the heat
conduction mechanism through an individual MWNT may be a more important factor
that affects the overall conductivity.
CHAPTER 5
CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK
5.1 Conclusions
In this thesis, a theoretical model was developed to predict the overall thermal con-
ductivity of an aligned MWNT/polymer composite. This model is based on an effective
medium theory developed by Benveniste and Miloh [29] to predict the effective conduc-
tivity of short-fiber reinforced composites with imperfect interfaces. Their model was
modified here to account for the non-continuum effects at the nanometer level, by devel-
oping a continuum model for the MWNT, which was then incorporated into the effective
medium theory. The necessary calculations to determine the effective conductivity were
carried out using the symbolic computation software, Mathematica, which significantly
simplified the tedious calculations involved. Results obtained using the proposed model
were found to be in the same range as those obtained experimentally. The following
conclusions can be drawn from the obtained results:
(i) The results show that in spite of the non-continuum effects at the nanome-
ter level, satisfactory results for the effective properties may be obtained using general
continuum theories by developing a suitable continuum model of the nanotube geometry.
(ii) In order to accurately model the effective thermal properties, special consid-
eration must be given to the mechanism of heat conduction through the nanotube. It
has been proven here theoretically that only the outer MWNT layer is involved in heat
conduction through the nanotube. Due to this, the conductivity of the MWNT in the
composite will be much lower than its intrinsic conductivity, which in turn lowers the
effective conductivity.
62
63
(iii) The effective conductivity is very sensitive to the MWNT diameter. Hence
for proper theoretical estimation, due consideration must be given to distribution of
diameters in the nanotube sample.
(iv) It was found that the interfacial resistance at the MWNT/polymer boundary
is not the single most important factor affecting heat flow in CNT/polymer composites.
The influence of the interfacial resistance is probably much higher on the overall heat
carrying capacity of the composite rather than on the overall conductivity.
5.2 Recommendations
In the present theory, the dilute assumption has been employed which neglects all
interactions between fillers and gives a linear change in the effective conductivity with
change in volume fraction of the nanotube phase. However, in certain cases it has been
observed that the change in effective conductivity with nanotube volume fraction is non-
linear, indicating interactions among the nanotubes. The theory thus can be extended to
the non-dilute case to consider the effect of the nanotube interactions. Also the theory
only predicts the effective longitudinal conductivity. Generalization of the theory to the
predict the random isotropic conductivity is also worth investigating.
APPENDIX A
INTEGRAL RELATIONSHIPS
64
65
In this appendix, we derive the integral relations (3.86), (3.87) and (3.88) given in
Chapter III.
Let us consider the first integral relation
I1 =∞∑
n=0
A2n+1Q2n+1(ξo)
∫ 1
−1
P2n+1(µ)Pm(µ) dµ. (A.1)
In order to evaluate this integral, we make use of the general orthogonality property
of the Legendre polynomials given by
∫ 1
−1
Pn(µ)Pm(µ) dµ =2
2n + 1δnm, (A.2)
where δnm is the Kronecker delta.
Thus I1 is non-zero only when m = 2n + 1. Hence only the (2n + 1)th term of the
series will be non-zero, while all other terms will go to zero. For the (2n + 1)th term, the
value of the integral will be 1. Hence I1 reduces to
I1 =
[2
2(2n + 1) + 1
]A2n+1Q2n+1(ξo) (A.3)
=
(2
4n + 3
)A2n+1Q2n+1(ξo). (A.4)
Similarly, we can write
I2 =∞∑
n=0
B2n+1P2n+1(ξo)
∫ 1
−1
P2n+1(µ)Pm(µ) dµ (A.5)
=
[2
2(2n + 1) + 1
]B2n+1P2n+1(ξo) (A.6)
=
(2
4n + 3
)B2n+1P2n+1(ξo). (A.7)
We now proceed to derive the third integral relation. The third integral is given by
I3 = −P1(ξo)
∫ 1
−1
P1(µ)Pm(µ) dµ (A.8)
= −P1(ξo)
∫ 1
−1
P1(µ)P2n+1(µ) dµ (m = 2n + 1). (A.9)
66
The above equation is non-zero only when n = 0. This we can also write in the following
manner:
I3 = −[
2
2(2n + 1) + 1
]P2n+1(ξo)δ(n) (A.10)
= −(
2
4n + 3
)P2n+1(ξo)δ(n). (A.11)
For n = 0, the above relation reduces to
I3 = −2
3P1(ξo), (A.12)
which is the same as
I3 = −P1(ξo)
∫ 1
−1
P1(µ)P1(µ) dµ (A.13)
= −2
3P1(ξo). (A.14)
APPENDIX B
DERIVATION OF THE INTERIOR AND EXTERIOR HARMONICS
67
68
In this appendix, we will derive the expressions for the interior and exterior har-
monics, Θ(1)i and Θ
(2)i , that are given by (3.97) and (3.99) respectively in Chapter III.
B.1 Derivation of the Interior Harmonic
The general expression for the interior and exterior harmonic given by (3.96) is
Θ(α)3 = − 3
4πξoc
∫ 2π
0
∫ 1
−1
φ(α)µ dµ dψ. (B.1)
Substituting the value of φ(2) from (3.72) in the above relation, we get for the
interior harmonic
Θ(2)3 =− 3
4πξoc
∫ 2π
0
∫ 1
−1
Ho3c
∞∑n=0
B2n+1P2n+1(µ)P2n+1(ξo)µ dµ dψ (B.2)
= − 3Ho3
4πξo
[∫ 2π
0
∞∑n=0
B2n+1P2n+1(ξo)
( ∫ 1
−1
µP2n+1(µ) dµ
)dψ
]. (B.3)
To simplify the above expression, we first need to evaluate the following integral:
I1 =
∫ 1
−1
µP2n+1(µ) dµ. (B.4)
In order to evaluate the above integral, we make use of the following integral prop-
erty of the Legendre polynomials of the first type:∫ 1
−1
µkPn(µ) dµ = 0, k = 0, 1, 2, . . . n− 1. (B.5)
The above relation is non-zero only when k = n. In the integral I1, k = 1. Hence
the only non zero value for the integral will be for 2n + 1 = 1 or n = 0 and the series∞∑
n=0
B2n+1P2n+1(ξo)
∫ 1
−1
µP2n+1(µ) dµ
will then reduce to a single term. Noting that P1(µ) = µ and P1(ξo) = ξo, we get
Θ(2)3 = −3Ho
3B1
4π
∫ 2π
0
∫ 1
−1
µ2 dµ dψ (B.6)
= −Ho3B1
2π
∫ 2π
0
dψ (B.7)
= −Ho3B1. (B.8)
69
B.2 Derivation of the Exterior Harmonic
The derivation of the exterior harmonic, Θ(1)i , can be done in a way similar to the
derivation of the interior harmonic. For α = 1, the exterior harmonic can be written as
Θ(1)3 = − 3
4πξoc
∫ 2π
0
∫ 1
−1
[−Ho
3cP1(µ)P1(ξo) + Ho3c
∞∑n=0
A2n+1P2n+1(µ)Q2n+1(ξo)
]µ dµ dψ.
(B.9)
To simplify the above expression, we need to evaluate the following integrals:
I2 =
∫ 1
−1
−Ho3cP1(µ)P1(ξo)µ dµ, (B.10)
I3 =
∫ 1
−1
Ho3c
∞∑n=0
A2n+1P2n+1(µ)Q2n+1(ξo)µ dµ. (B.11)
Using the integral property of the Legendre polynomial given by (B.5), we can
write
I2 = −Ho3cP1(ξo)
∫ 1
−1
P1(µ)µ dµ (k = n = 1) (B.12)
= −Ho3cξo
∫ 1
−1
µ2 dµ (B.13)
=−2Ho
3cξo
3. (B.14)
The integral, I3, can be written as
I3 = Ho3c
∞∑n=0
A2n+1Q2n+1(ξo)
∫ 1
−1
P2n+1(µ)µ dµ. (B.15)
The above integral is non zero only when 2n + 1 = 1. Thus the series will reduce to just
one term. Thus, we can write
I3 = Ho3cA1Q1(ξo)
∫ 1
−1
µ2 dµ (B.16)
=2Ho
3cA1Q1(ξo)
3. (B.17)
70
Therefore, the exterior harmonic can be written as,
Θ(1)3 =
Ho3
2π
[1− A1Q1(ξo)
ξo
] ∫ 2π
0
dψ (B.18)
= Ho3 −Ho
3A1Q1(ξo)
ξo
. (B.19)
APPENDIX C
MATHEMATICA PROGRAM
71
72
The following is the Mathematica program used in the computations:
d = 25; l = 50000; knt = 3000; k1 = 0.2; t = 0.34; v2 = 0.01;
c = SetAccuracy[Sqrt[(l/2)^2 - (d/2)^2], 5];
betabar = SetAccuracy[12 c (0.001/k1), 4];
k2 = ((4t)/d) knt;
lambda = (k2/k1);
xi = SetAccuracy[(1 - (d/l)^2)^(-0.5), 10];
f=SetAccuracy[((1/2)xi(xi^2-1)Log[(xi+1)/(xi-1)]-xi^2)^(-1)];
p[n_, x_]:= LegendreP[n, x];
pp[n_, x_]:= D[LegendreP[n, x], x];
q[n_, x_]:= LegendreQ[n, x] // Re;
qp[n_, x_]:= D[LegendreQ[n, x], x] // Re;
chi[n_, m_]:=((4 n + 3)/2) (xi^2 - 1)((qp[2 n + 1,x] pp[2 m +
1,x]/.x -> xi)) NIntegrate[Sqrt[(xi^2-1)/(xi^2-mu^2)]p[2n
+ 1, mu] p[2 m + 1, mu]], {mu,-1,1}];
delta[n_] := If[n == 0, 1, 0];
left[n_] := delta[n] + b[2 n + 1] (1 - (1 - lambda) (xi^2
-1)(pp[2 n + 1, x] q[2 n + 1, x]) /. x -> xi);
right[n_, upper_]:=(lambda/betabar) Sum[b[2 m + 1]chi[n, m],
{m, 0, upper}];
terms = 8;
eq = Table[left[i] == right[i, terms - 1], {i, 0, terms -1}];
varlist = Table[b[i], {i, 1, 2*terms - 1, 2}];
Solve[eq, varlist];
k33 = k1 (1 + v2 (1 + lambda b[1] f))
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BIOGRAPHICAL STATEMENT
Aniruddha Bagchi was born in Bombay (now Mumbai), India, in 1981. He received
his B.E. degree from Visveswaraiah Technological University, India, in 2003 and his
M.S. degree from The University of Texas at Arlington in 2005, both in Mechanical
Engineering. His master’s thesis dealt with the development of a mathematical model
for predicting the effective thermal conductivity of carbon nanotube reinforced polymer
composites. His current research interests include the design and analysis of traditional
composite and nanocomposite materials, finite element analysis and applied mathematics.
He plans to pursue a Ph.D. degree in Mechanical Engineering.
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