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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C60: Buckminsterfullerene. Nature, 318(6042), 162-163.

[3] Iijima, S., & Ichlhashi, T. (1993). Single-shell carbon nanotube of 1-nm diameter.

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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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