Design Guidelines for Deploying Closed Loop Systems

DEFORMATION DEPENDENT ELECTRICAL RESISTANCE OF MWCNT LAYER AND MWCNT/PEO COMPOSITE FILMS

A Dissertation Submitted to the Faculty of Purdue University by Myounggu Park

In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

August 2007 Purdue University West Lafayette, Indiana

ii

I dedicate this dissertation to my father, Hoo-shin Park in heaven.

iii

ACKNOWLEDGMENTS

“If today were the last day of my life, would I want to do what I am about to do today?” - Steve Jobs I have been the luckiest man on earth because I could do what I wanted to do: participate in the never-ending pursuit of ultimate knowledge. A part of that journey is about to end here in Purdue University. First and most of all, I would like to say that it has been a great privilege to be a student of Dr. Hyonny Kim. Without his guidance, helpful suggestions and encouragement, this research would not have been possible. Special thanks go to my committee members, Dr. CT Sun, Dr. R. Byron Pipes and Dr. Jeffery P. Youngblood, for their invaluable comments and lectures. I thank the staff members in AAE for their unforgettable services. Also, I would like to thank Dr. Timothy S. Fisher and Dr. Thomas Siegmund in ME for a PECVD system and a small scale tensile testing machine. And I thank CML lab members, collaborators in Dr. Fisher’s lab, international and Korean friends in AAE and the people of upper room Christian fellowship, for the joy and friendship that we have shared. I gladly acknowledge that this research was supported by Purdue Research Foundation. I would like to share this great moment with my lovely wife Juyoung Yoon, who completes me, and with my beautiful son Samuel (Junghyun), who is the source of my heavenly joy. I will not forget my parents-in-law Shinoh Yoon and Philsaeng Kim for their love and prayers. I thank my sister, brother, sister-in-law and brother-in-law, Hyunju, Sanguk, Soyun, and Seokju for their support. Lastly, my deep appreciation goes to my mother, Soonja Kim, who showed me the true meaning of love.

iv

TABLE OF CONTENTS

Page LIST OF TABLES........................................................................................................... viii LIST OF FIGURES ........................................................................................................... ix ABSTRACT...................................................................................................................... xii CHAPTER 1. INTRODUCTION ....................................................................................... 1 1.1. Global Objectives ..................................................................................................... 1 1.2. Properties of Carbon Nanotubes............................................................................... 3 CHAPTER 2. ELECTRICAL PROPERTIES OF CARBON NANOTUBE...................... 8 2.1. Tight Binding Approximation .................................................................................. 8 2.2. Zone Folding Approximation ................................................................................. 10 2.3. Ballistic Conduction of a CNT ............................................................................... 12 2.4. Band Gap Change Induced by External Strain....................................................... 14 2.4.1. Model by Yang et al......................................................................................... 14 2.4.2. Model by Kleiner et al ..................................................................................... 17 2.5. Summary................................................................................................................. 20 CHAPTER 3. ELECTRICAL RESISTANCE OF MWCNT LAYER............................. 22 3.1. Objectives ............................................................................................................... 22 3.2. Growth of MWCNT Layer on Copper Substrate ................................................... 22 3.2.1. Preparation of Copper Substrate ...................................................................... 22 3.2.2. Metal Deposition Process................................................................................. 24 3.2.3. PECVD Process ............................................................................................... 25 3.2.4. Characterization of MWCNT Layer ................................................................ 27 3.3. Measurement of Electrical Resistance Induced by Compressive Strain ................ 29 3.3.1. Test Setup......................................................................................................... 29

v Page 3.3.2. Bare Cu-Cu Contact ......................................................................................... 31 3.3.3. MWCNT-Enhanced Contact............................................................................ 33 3.4. Discussions ............................................................................................................. 35 3.4.1. Contact Resistance Reduction Mechanism ...................................................... 35 3.4.2. van der Waals Interaction................................................................................. 37 3.4.3. Compaction-Based Resistance Change Model ................................................ 38 3.5. Summary................................................................................................................. 40 CHAPTER 4. FABRICATION OF MWCNT/POLYMER COMPOSITE FILM............ 42 4.1. Objectives ............................................................................................................... 42 4.2. Raw Materials......................................................................................................... 43 4.2.1. Multiwalled Carbon Nanotubes (MWCNT) .................................................... 43 4.2.2. Sodium Dodecyl Sulfate (SDS) ....................................................................... 43 4.2.3. Polymer Matrix ................................................................................................ 44 4.3. Compaction-Based in situ Polymerization ............................................................. 46 4.3.1. Experimental Procedure ................................................................................... 46 4.3.2. MWCNT/Polyvinyl Acetate Composite Film.................................................. 50 4.3.3. MWCNT/Polyvinyl Alcohol (PVA) Composite Film ..................................... 51 4.3.4. MWCNT/Polyethylene Oxide (PEO) Composite Film.................................... 52 4.4. Electrical Resistivity of Fabricated Composite Film.............................................. 54 4.4.1. DC Resistance Measurement Test Setup ......................................................... 54 4.4.2. Resistivity of MWCNT/Polymer Composite Film .......................................... 56 4.5. Discussions ............................................................................................................. 57 4.5.1. Colloid Stability ............................................................................................... 57 4.5.2. Resistivity vs. Microstructure .......................................................................... 57 4.5.3. Manufacturing Model for MWCNT/Polymer Composites .............................. 58 4.6. Summary................................................................................................................. 61 CHAPTER 5. ELECTRICAL RESISTANCE OF MWCNT/PEO COMPOSITE........... 63 5.1. Objectives ............................................................................................................... 63 5.2. Determination of Percolation Threshold ................................................................ 64

vi Page 5.2.1. Percolation Threshold Prediction..................................................................... 64 5.2.2. Experimental Percolation Threshold................................................................ 66 5.3. Measurement of Electrical Resistance induced by Tensile Strain.......................... 69 5.3.1. Test Setup......................................................................................................... 69 5.3.2. Strain Dependent Resistance............................................................................ 71 5.4. Parameters Affecting Electrical Resistance............................................................ 73 5.4.1. Microstructure Change..................................................................................... 74 5.4.2. Volume and Angle Change .............................................................................. 75 5.4.3. Electrical Property Change of CNT under Strain ............................................ 84 5.4.4. Tunneling Distance Effects .............................................................................. 85 5.5. Modeling................................................................................................................. 87 5.6. Application as Strain Sensor................................................................................... 91 5.7. Summary................................................................................................................. 92 CHAPTER 6. CONCLUSIONS AND FUTURE WORK................................................ 93 6.1. Conclusions ............................................................................................................ 93 6.2. Future Work............................................................................................................ 96 6.2.1. Electrical Resistance of a CNT inside the Matrix ............................................ 96 6.2.2. Mechanical Properties of MWCNT/PEO Composite ...................................... 96 LIST OF REFERENCES.................................................................................................. 97 APPENDICES ................................................................................................................ 113 Appendix A. ................................................................................................................ 113 A.1. Fundamentals of Band Structure ......................................................................... 113 A.2. Band Structure of Graphene Sheet ...................................................................... 118 A.3. Band Structure of CNT........................................................................................ 122 Appendix B.................................................................................................................. 126 B.1. Transport Properties of CNT ............................................................................... 126 Appendix C.................................................................................................................. 133 C.1. Model by Yang et al for Band Gap Change Due to Strain .................................. 133 Appendix D. ................................................................................................................ 140

vii Page D.1. Model by Kleiner et al for Band Gap Change Due to Strain............................... 140 VITA ............................................................................................................................... 147

viii

LIST OF TABLES

Table

Page

Table 1.1. Properties of Carbon Nanotubes ........................................................................ 7 Table 3.1. Basic Parameters for Metal Deposition ........................................................... 25 Table 3.2. Basic Parameters for PECVD Process............................................................. 27 Table 4.1. EDS Analysis Results of MWCNTs Sample................................................... 43 Table 4.2. Physical Properties of Polymers and Vendor Information .............................. 45 Table 4.3. Dimensions of MWCNT/Polymer Composite Sample.................................... 54 Table 4.4. Film Thickness – Model Prediction and Measured Value............................... 61 Table 4.5. Amount of Materials Used............................................................................... 61 Table 5.1. Theoretical Calculation of Percolation Threshold. .......................................... 66 Table 5.2. MWCNT/PEO Composite Film Details. ......................................................... 67 Table 5.3. Basic Constants................................................................................................ 87 Table 5.4. Strain Sensitivity Comparison ......................................................................... 91

ix

LIST OF FIGURES

Figure

Page

Figure 1.1. CNT-Field Emission Display (FED) Monitor.................................................. 2 Figure 1.2. Application of CNT in Bio-Engineering Field................................................. 2 Figure 1.3. Future Air Vehicles Adopting Morphing Structure.......................................... 2 Figure 1.4. Rolling Graphene Sheet to Form Single-walled Nanotube .............................. 4 Figure 1.5. Chiral Vector of Single Walled Carbon Nanotube........................................... 4 Figure 1.6. Chirality of Single Walled Carbon Nanotubes................................................. 6 Figure 1.7. Types of Carbon Nanotubes ............................................................................. 6 Figure 2.1. Band Structure of Zig-zag CNT ..................................................................... 11 Figure 2.2. Band Structure of Armchair CNT .................................................................. 12 Figure 2.3. Ballistic Transport of CNT through One Channel ......................................... 13 Figure 2.4. Electrical Conductance Increase as New Channels Open .............................. 13 Figure 2.5. Band Structure without Axial Strain: CNT (12, 0)......................................... 16 Figure 2.6. Band Structure with Axial Strain: CNT (12, 0).............................................. 16 Figure 2.7. Chirality-Dependent Band Gap of CNTs without Torsional Strain ............... 18 Figure 2.8. Chirality-Dependent Band Gap of CNTs with Torsional Strain .................... 19 Figure 2.9. Band Gap of Metallic CNT (7, 7) with Torsional Strain................................ 19 Figure 3.1. Preparation of Copper Substrate..................................................................... 23 Figure 3.2. Electron Beam Evaporator ............................................................................. 24 Figure 3.3. Metal Deposited Cu Substrate ........................................................................ 25 Figure 3.4. Microwave - Plasma Enhanced Chemical Vapor Deposition System............ 26 Figure 3.5. As-Fabricated MWCNT Layer on Cu Substrate (Unit = cm) ........................ 27 Figure 3.6. MWCNTs Layer on Copper Substrate ........................................................... 28 Figure 3.7. TEM Image of Individual MWCNT............................................................... 29

x Figure

Page

Figure 3.8. AFM Images of MWCNT Layer on Groove.................................................. 29 Figure 3.9. Overall Test Setup .......................................................................................... 30 Figure 3.10. Four Point Measurement Scheme................................................................. 31 Figure 3.11. Contact Resistance of Bare Cu-Cu Interface as Function of Probe Tip Position....................................................................................................................... 32 Figure 3.12. Contact Force of Bare Cu-Cu Interface as Function of Probe Tip Position. 32 Figure 3.13. Contact Resistance of Cu-MWCNT-Cu Interface as Function of Probe Tip34 Figure 3.14. Contact Force of Cu-MWCNT-Cu Interface as Function of Probe Tip....... 34 Figure 3.15. Comparison of Contact Resistance between Bare Cu-Cu Interface and CuMWCNT-Cu Interface ............................................................................................... 35 Figure 3.16. Classification of the Contact Surface ........................................................... 36 Figure 3.17. Contact Resistance Reduction by Parallel Contacts Created by MWCNTs. 36 Figure 3.18. Electrical Junction Formed by Two MWCNTs............................................ 37 Figure 3.19. Relative Orientation of Two CNTs at Contact Surface................................ 37 Figure 3.20. Resistance Change Based on Compaction Model ........................................ 39 Figure 3.21. MWCNTs Separated from Surface .............................................................. 40 Figure 3.22. No Growth of CNTs ..................................................................................... 41 Figure 3.23. Optical Micrograph of Sample Surfaces ...................................................... 41 Figure 4.1. Molecular Formula of SDS ............................................................................ 44 Figure 4.2. As-Purchased MWCNTs ................................................................................ 46 Figure 4.3. MWCNT/Polymer Composite Film Fabrication Procedure........................... 47 Figure 4.4. As-Purchased MWCNTs and SDS (Unit = cm) ............................................. 47 Figure 4.5. Ultrasonicator ................................................................................................. 48 Figure 4.6. As-Purchased Polymers (Unit = cm).............................................................. 49 Figure 4.7. Ultasonication of MWCNT and Polymer Mixture......................................... 49 Figure 4.8. Oven Used for Water Evaporation ................................................................. 50 Figure 4.9. As-Fabricated MWCNT/Polyvinyl Acetate Composite Film on Cu Substrate (unit: cm) .................................................................................................................... 50 Figure 4.10. MWCNT/Polyvinyl Acetate Composite Film .............................................. 51

xi Figure

Page

Figure 4.11. As-Fabricated Free Standing MWCNT/PVA Composite Film.................... 52 Figure 4.12. MWCNT/PVA Composite Film................................................................... 52 Figure 4.13. As-Fabricated Free Standing MWCNT/PEO Composite Film .................... 53 Figure 4.14. MWCNT/PEO Composite Film ................................................................... 53 Figure 4.15. Voltage Divider Circuit ................................................................................ 55 Figure 4.16. Detailed View of MWCNT/Polymer Composite Region............................. 55 Figure 4.17. Resistivity of MWCNT/Polymer Composite Films ..................................... 56 Figure 4.18. Cross-Sectional Morphologies Showing MWCNT Distribution ................. 58 Figure 4.19. MWCNT/Polymer Composite Film Casting Mold ...................................... 59 Figure 5.1. Two-Point DC Resistance Measurement Setup ............................................. 67 Figure 5.2. Electrical Conductivity vs. MWCNT in Volume Fraction............................. 69 Figure 5.3. Electrical Resistance and Strain Measurement Setup .................................... 70 Figure 5.4. Detailed View of Detached Film Free Ends................................................... 71 Figure 5.5. Tensile Test Results of Polycarbonate Substrate............................................ 72 Figure 5.6. Resistance vs. Strain for 0.56 vol% of MWCNT ........................................... 72 Figure 5.7. Resistance vs. Strain for 1.44 vol% of MWCNT ........................................... 73 Figure 5.8. Microstructure of MWCNT/PEO Composite before Straining...................... 74 Figure 5.9. Microstructure of MWCNT/PEO Composite after Straining......................... 75 Figure 5.10. Unit Volume of MWCNT/PEO Composite under Strain............................. 75 Figure 5.11. Volume Change of MWCNT/PEO Composite under Axial Strain.............. 78 Figure 5.12. Angles of MWCNT in PEO Matrix.............................................................. 80 Figure 5.13. Angle Change of MWCNT in PEO Induced by Strain ................................ 81 Figure 5.14. Electrical Resistance Change vs. Strain ....................................................... 83 Figure 5.15. Tunneling Distance Change under Strain..................................................... 86 Figure 5.16. Contact Geometry of CNTs inside Matrix ................................................... 88 Figure 5.17. Resistance Change Model vs. Experiments.................................................. 90

xii

ABSTRACT

Park, Myounggu. Ph.D., Purdue University, August, 2007. Deformation Dependent Electrical Resistance of MWCNT Layer and MWCNT/PEO Composite Films. Major Professor: Hyonny Kim.

It has been well documented that the electrical properties of a carbon naotube (CNT) can be either metallic or semiconducting depending upon the tube’s chirality. Theoretical aspects of the unique electrical properties of CNTs are reviewed. Based upon the fundamental understanding of this special feature, the deformation-dependent electrical resistance of multiwalled carbon nanotube (MWCNT) layer and MWCNT/ polyethylene oxide (PEO) composite in the macroscopic scale are investigated considering both experimental and theoretical aspects. In the first set of experiments, a MWCNT layer was grown by plasma enhanced chemical vapor deposition (PECVD) process on a surface of copper substrate and a copper probe was applied to this surface inducing compressive deformation onto the MWCNT layer. It was found that the electrical resistance of the MWCNT layer under compression was reduced by 80 percent. The possible mechanisms for electrical resistance reduction were analyzed and suggested. Also, the MWCNTenhanced surface showed a finite slope of electrical resistance as a function of contact force, thereby making possible the use of this arrangement as a small-scale force or pressure sensor. However, there is limitation on the direct use of MWCNT grown directly onto copper substrates for real applications due to the easy separation of the MWCNTs from the copper surface and the low yield of MWCNTs by the given metal deposition and PECVD system. The processing method developed for the second set of experiments uses intentional coagulation of dispersed MWCNT in polymer solution. This process is simple and

xiii effective to fabricate MWCNT-filled polymer films. MWCNT/PEO composite was selected after comparative resistivity measurement and microstructure analysis. The percolation threshold of MWCNT/PEO was determined experimentally to be between 0.14 to 0.28 vol% of MWCNT. Films having MWCNT content above the percolation threshold were conductive and exhibited repeatable values of electrical conductivity. Unique and repeatable relationships of resistance versus strain were obtained for multiple samples with different volume fractions of MWCNT. The overall pattern of electrical resistance change versus strain for the samples of each volume fraction of MWCNT consists of linear and non-linear regions. A model to describe the combination of linear and non-linear modes of electrical resistance change as a function of strain is suggested. The unique characteristics in electrical resistance change for different volume fractions implies that nanotube-based composites can be used as tunable strain sensors for application into embedded sensor systems in structures.

1

CHAPTER 1. INTRODUCTION

1.1. Global Objectives After Iijima [1] reported helical graphitic microtubules called carbon nanotubes (CNTs) in 1991, research about this novel material has increased exponentially every year [2]. It is well known that individual CNTs have the highest Young’s modulus, thermal conductivity and current density, as well as high electrical conductivity that is comparable to metal. They also have excellent strength and flexibility. A large body of research has been conducted to take advantage of the outstanding properties of CNTs. Especially, in electrical, computer science, and biomedical fields, the effort for the application of CNTs at the nano or micro-scale have been very active. For example, CNT transistors have been researched as a next generation memory device [3]. Also, due to the excellent field emission characteristics of CNT, a small clump of CNTs were successfully applied as a field emitter [4] (Figure 1.1). In bio-medical field, a team led by Dr. Hongjie Dai of Stanford University has proven that SWCNTs wrapped in a polyethylene can be used to cure tumor. Also, bio-tissues requiring bio-compatible substrate was successfully grown on the CNT paper [5, 6] (Figure 1.2). The Aerospace engineering field is not an exception. Future air vehicles are likely to adopt sensor/actuator-embedded composite structures to enhance performance and to monitor the health of the structure with the intention of reducing maintenance costs [7, 8] (Figure 1.3). Such a composite structure is usually called a “smart” structure due to its capability of sensing and responding to the surrounding environmental situation. Among the many functions being pursued to incorporate into the host composite structure, strain sensing is one of the most basic ones. The main reason for developing an embedded strain sensing system in a composite

2 structure is that it allows one to measure static and dynamic response without significant adverse effects on the host structure [9].

Figure 1.1. CNT-Field Emission Display (FED) Monitor [4]

(a) CNT Array

(b) Bio Tissue on CNT Substrate

Figure 1.2. Application of CNT in Bio-Engineering Field [5, 6]

Figure 1.3. Future Air Vehicles Adopting Morphing Structure [7]

3 One of the most common methods constructing such a smart structure is to blend CNT and polymer and measure the electrical resistance change [10]. Experimental work for this kind of research has been developing. The mechanism of strain dependent electrical resistance change is not fully understood at present. However, it is indispensable to understand the mechanisms of strain dependent electrical resistance of CNT-composites in order to design smart structures using these materials. In this research, an overarching goal is set up to obtain a fundamental understanding of strain dependent electrical resistance change of CNTs embedded in polymer. Therefore, the topic of this thesis is focused on both the measurement and physical understanding of the electrical resistance change of CNT composite films. To reach those goals, a multiwalled carbon nanotube (MWCNT)/PEO (polyethylene oxide) composite film as well as MWCNT layer on a copper surface was fabricated and their deformation dependent electrical resistance was measured and analyzed. The global objectives of this research are summarized as: (i) to understand how micro and nano-scale interactions of individual CNTs affect macro-scale electrical resistance change, (ii) to develop a model to describe the experiments, and (iii) demonstrate a real application for the CNT-filled polymer, such as a strain sensor.

1.2. Properties of Carbon Nanotubes CNTs can be considered as a seamless cylindrical rolled up graphene sheet like sp2bonded carbon atoms [11]. Rolling of a graphene sheet is schematically illustrated in [12]. The coordinate system used to define individual single walled carbon naotube is indicated in Figure 1.5 where a1 and a2 are the basis vectors [13].

4

Figure 1.4. Rolling Graphene Sheet to Form Single-walled Nanotube [12]

Figure 1.5. Chiral Vector of Single Walled Carbon Nanotube

A chiral vector is defined using a pair of integers (n, m) as c = na1 + ma2

Equation Chapter 1 Section 1(1.1)

The chiral vector c is perpendicular to the axis of the tube, and the maginituge, c , is the circumference of the tube. The diameter of tube can be expressed in terms of (n,m) and c as following:

5

d=

c

π

=

3ac −c

π

n 2 + nm + m 2

(1.2)

where ac-c (=1.421 Å) is the carbon-carbon atom distance in graphene sheet. Also, chiral angle with respect to the (m,0) direction is as following

⎛

3m ⎞ ⎟⎟ ⎝ 2n + m ⎠

α = tan −1 ⎜⎜

(1.3)

The zigzag direction is the (n,0) direction and has a chiral angle of α = 0° (Figure 1.5). The armchair direction is the (n,n) direction and has a chiral angle of α = 30° . The zigzag and armchair tubes are named due to the hexagon pattern along the tube axial direction. The hexagons in the zigzag and armchair tubes will form rings perpendicular to the tube axis (Figure 1.6 (a) and (c)) [14]. On the other hand, the hexagons of any tubes formed form the regions 0° < α < 30° will spiral around the tube axis and this tube with spiraling hexagons are called chiral tubes (Figure 1.6 (b)) [14]. Also, carbon nanotubes can be classified into two types - single walled carbon nanotubes (SWCNT) and multi walled carbon nanotubes (MWCNT) - based upon growth conditions. SWCNT and MWCNT are depicted in Figure 1.7 [14]. A MWCNT consists of multiple numbers of concentric SWCNTs. The diameter of SWCNT is about 1 nm and the diameter of MWCNT is about 20 – 150 nm.

6

(a) Zigzag: (n,m) = (15,0)

(b) Chiral:

(c) Armchair:

(n,m) = (8,4)

(n,m) = (10,10)

Figure 1.6. Chirality of Single Walled Carbon Nanotubes [14]

(a) Single Walled CNT

(b) Multiwalled CNT

Figure 1.7. Types of Carbon Nanotubes [14]

The chirality of the CNT (see Figure 1.6) describes the structure of the graphene sheet, which in turn strongly affects the electrical and thermal properties [15-24]. Table 1.1 summarizes documented electrical, mechanical, and thermal properties of CNTs. Note that the current density, Young’s modulus and thermal conductivity of CNTs are the highest among known materials. Also, CNTs can be either metallic or semi-conducting in their electrical conductivity depending on chirality. In this research the electrical property of MWCNT under deformation is mainly explored via contact of a MWCNT layer enhanced surface (CHAPTER 2) and stretching of a MWCNT/polymer composite film (CHAPTER 5).

7 Table 1.1. Properties of Carbon Nanotubes Properties

Values

Reference

Electrical Resistivity

10-6 – 10-4 Ω-cm

[15], [16]

Current Density

109 – 1010 A/cm2

[17]

Young’s Modulus

1 – 2 TPa

[18], [19]

Tensile Strength

40 – 50 GPa

[20], [21]

Thermal Conductivity

2000 – 4000 W/mK

[22], [23], [24]

8

CHAPTER 2. ELECTRICAL PROPERTIES OF CARBON NANOTUBE

Unlike conventional conductive filler like carbon fiber, a CNT can be either metallic or semiconducting according to the chirality. In addition to that, externally applied strain can distort the initial molecular structure and this can induce the band gap change resulting in electrical resistance change of CNT when it was embedded inside the polymer matrix. Therefore, it is important to understand the principles of the band gap change of CNT depending on chirality and external strain. In this chapter, these topics are summarized. The fundamental principles of the band structure of CNT (Appendix A), transport properties of CNT (Appendix B) and band gap change of CNT due to strain (Appendix C and D) are described in detail at each corresponding appendices.

2.1. Tight Binding Approximation Generally, there are two models describing the energy state of electrons inside a solid. One is the free electron model and the other is the tight binding model. In the free electron model, electrons are considered as freely moving particles in a box and this is a good model for metals. While in the tight binding model, electrons are considered to be tightly bound to the nucleus and it is suitable for describing carbon-carbon bonding. In this model, Eo is the energy of an electron on one isolated atom and t is referred to as the hopping parameter representing the probability of the transition of an electron from one atom to the neighboring atom. If t is large, hopping of electrons becomes difficult. The energy dispersion relation of a graphene sheet is approximated [25] by the tight binding model asEquation Section 2 E (k ) = E (k x , k y ) = Eo + t 1 + 4 cos 2 k y b + 4 cos k x a cos k y b

(2.1)

9 where kx and ky are the wave vectors and a and b are constants. If we put kxa = 0, then Eq. (2.1) becomes E (k ) = Eo + t (1 + 2 cos k y b)

(2.2)

For a graphene sheet to be conductive, there should be no band gap. For this, let Eq.(2.2) be equal to zero and also, let’s set Eo which can be assumed as the Fermi level to zero for mathematical convenience: t (1 + 2 cos k y b) = 0

(2.3)

Then the points satisfying above condition are obtained

(k x a, k y b) = (0,

2π ) 3

(2.4)

Because we are interested in the electrical characteristics of a graphene sheet around the Fermi level, it is useful to make an approximation that describes the regions of the energy dispersion plot around E=0. This can be done by taking a Taylor series expansion of the energy dispersion relationship of graphene sheet at the point (k x a, k y b) = (0,

2π ) where 3

the energy gap is zero

2π ⎞ ⎛ E ≈ Eo ± ho (k ) = Eo + at k + ⎜ k y − ⎟ 3b ⎠ ⎝ 2 x

2

(2.5)

10 2.2. Zone Folding Approximation Band structure of a CNT can be considered to be same as that of a graphene sheet and this approximation is called the zone folding approximation. CNTs are considered as a rolled up graphene sheet (Figure 1.4). Thus, when graphene sheets are formed to a tubular shape, the wave vector (k x , k y ) can not be continuous due to the chiral vector and they are restricted to the discrete values. For a zig-zag CNT having chirality of (n, 0), the graphene sheet is rolled up in the y (n, 0) direction (see Figure 1.5). Then ky values are restricted as

ky =

2π v 2bn

(2.6)

The energy dispersion relation can be calculated [25] as

Ev = ± at k x2 + kv2

where kv =

(2.7)

2π v 2π 2π ⎛ 2n ⎞ − = ⎜v − ⎟ 2bn 3b 2bn ⎝ 3 ⎠

From Eq.(2.45), it is clear that when n is a multiple of three, kv = 0 which means that Ev = ± atk x . Then, there is no band gap and this CNT is conductive. However, if n is not a multiple of three, Eq. (2.7) can not be zero and this CNT is called semimetallic throughout this document. Thus it becomes semiconducting CNT. Band structure of the zig-zag CNTs having the chirality of (n, m) = (12, 0) and (n, m) = (13, 0) are plotted in Figure 2.1a and Figure 2.1b [26, 27].

11

(a) Metallic: (n, m) = (12,0)

(b) Semiconducting: (n, m) = (13,0)

Figure 2.1. Band Structure of Zig-zag CNT

For an armchair CNT, the graphene sheet is rolled up in the x direction (see Figure 1.5). Then kx values are restricted to k x (2an) = 2π v

(2.8)

where a, n, and ν are integers. Note that (n, m) = (n, n) for armchair CNT. The energy dispersion relation is calculated [25] as

2π ⎞ ⎛ Ev = ± at k + ⎜ k y − ⎟ 3b ⎠ ⎝

2

2 v

From Eq. (2.9), band gap is always zero at ( kv = 0, k y =

(2.9)

2π ) and thus an armchair CNT 3b

is always metallic. Band structure of the armchair CNTs having the chirality of (n, m) = (7, 7) and (n, m) = (10, 10) are plotted in Figure 2.2 [26, 27].

12

(a) (n, m) = (7, 7)

(b) (n, m) = (10, 10)

Figure 2.2. Band Structure of Armchair CNT

2.3. Ballistic Conduction of a CNT Ballistic conduction means that there is no electron scattering over the length of the channel between source and drain. This condition can be satisfied when the phase breaking length ( lφ ) of the electron states in the CNT is greater than the momentum relaxation length ( lm ) which in turn is much greater than the length of the CNT [13]. Ballistic conductin of a metallic CNT having one accessible energy state is depicted in Figure 2.3. When the conditions for ballistic conduction is met, conductance becomes quantized in units of Go MN and total conductance G can be written [25, 28, 29] as N

G = Go M ∑ Τ j 0

Go MN , where Go ≡

2e 2 h

77.5 μ S

(2.10)

where M is number of modes of sub-bands and N is the number of available channel. Since more conduction channels are available as the applied bias (Vgate) is increased, N increase with applied bias (Figure B.3 in appendix B). If N in Eq. (2.10) is expressed as

N (Vgate ) and M is substituted by one, I (Vgate ) for a ballistic conductor is given by

13 I (Vgate ) = G (Vgate ) ⋅ Vapp = Go N (Vgate ) ⋅ Vapp

(2.11)

I vs V plot is illustrated using Eq. (2.11). Note that whenever a new channel is open, there is a quantum jump in current (Figure 2.4).

Figure 2.3. Ballistic Transport of CNT through One Channel

Figure 2.4. Electrical Conductance Increase as New Channels Open

14 2.4. Band Gap Change Induced by External Strain It was shown how electrical properties of CNT can depend on the chirality. The effect of externally applied strain on the electrical property of CNT is of interest and the two most often used models are reviewed: model of Yang et al [27] and Kleiner et al [30]. Using the model of Yang et al the complete band structure of a CNT is calculated and from the modification of hopping parameters of E-k expression band gap change can be indicated. Unlike the model of Yang et al, the model of Kleiner et al can predict the band gap change directly considering both intrinsic curvature of a CNT and externally applied strain. The expressions for band gap derived by Kleiner et al is a good guide to predict the band gap change of CNT induced by deformation.

2.4.1. Model by Yang et al Yang et al [27] modified unperturbed dispersion relation of graphene sheet derived by Wallace [31] to calculate the E-k relation of distorted graphene sheet due to deformation as following E (k ) = (t12 + t22 + t32 + 2t1t2 cos[k i(r1 − r2 )] + 2t1t2 cos[k i(r1 − r2 )]

(2.12)

+ 2t3t1 cos[k i(r3 − r1 )])1/ 2

where k = kc cˆ + kt tˆ and kc and kt are components of wave vector k, ti (i=1, 2, 3) are the hopping parameters in bond vector direction (see appendix C). The wave vector, kc of the distorted CNT is also quantized around the circumference and thus, the kc is restricted as kc (1 + ε c )ch ao = 2π q

(2.13)

15 where ao is the length of lattice vector, q is the integer, εc is the circumferential strain and ch is the circumference of undistorted CNT. Now, Eq. (2.12) can be written as using the expressions for bond vectors, ri (i=1, 2, 3) (see Appendix C) and Eq. (2.13)

E (kt ) = (t12 + t22 + t32 + 2t1t2 cos[π q

n + 2m 3 n ' 3 tan(γ ) n − kt ao − q ] 2 ch 2 ch 1 + ε c ch2

+ 2t1t3 cos[π q

2n + m 3m ' 3 tan(γ ) n + kt ao + π q ] 2 ch 2 ch 1 + ε c ch2

+ 2t2t3 cos[π q

n−m 3 n+m ' 3 tan(γ ) n + m 1/ 2 + kt ao + q ]) 2 ch 2 ch 1+ εc ch2

(2.14)

where kt' = (1 + ε t )kt and εt is the axial strain. When there is an external strain, the hopping parameters (ti: i=1, 2, 3) are changed and this can be the main source of band gap change. For example, a zig-zag CNT (12, 0) can have a band gap when there is an axial strain. E-k relations of this CNT are plotted both before and after stretching in Figure 2.5 and Figure 2.6 [26, 27]. The magnitude strain is 0.05. Therefore, the effect of the externally applied strain is clear and the strain can induce band gap change, thereby the resistance of an individual CNT.

16

Figure 2.5. Band Structure without Axial Strain: CNT (12, 0)

Figure 2.6. Band Structure with Axial Strain: CNT (12, 0)

17 2.4.2. Model by Kleiner et al In their model, Kleiner et al [30] considered the effects of both the intrinsic curvature of CNT and the deformations on metallic CNT. They approach the problem in terms of tight binding model. First, they considered the change in hopping parameter ( δ ti : i = 1, 2, 3) due to intrinsic curvature of CNT and derived the band gap change due to intrinsic curvature of CNT as

Egcurve =

toπ 2 (n − m)(2n 2 + 5nm + 2m 2 ) 5 8ch

(2.15)

where to is the initial hopping parameter, n and m are the chiral vector indices. Second, they consider the change in hopping parameter ( δ ti ) due to deformation of CNT and derived the band gap expression as

Egdeform =

aob 4ch3

3(n − m)(2n 2 + 5nm + 2m 2 )(ε c − ε t ) − 9nm(n + m)γ

(2.16)

By summing up these two expressions (Eqs (2.15) and (2.16)), total band gap is obtained ⎛t π2 a b 3 ⎞ 9a b Egtotal = ⎜ o 5 + o 3 (ε c − ε t ) ⎟ 3(n − m)(2n 2 + 5nm + 2m 2 ) − o3 nm(n + m)γ ⎜ 8c ⎟ 4ch 4ch ⎝ h ⎠

(2.17)

Eq. (2.17) can be written using R (radius of CNT) and θ (see appendix D) as following

⎛ t π2 a b 3 ⎞ ab 3 Egtotal = ⎜⎜ o 2 + o (ε c − ε t ) ⎟⎟ sin 3θ − o γ cos 3θ 2 2 ⎝ 16 R ⎠

(2.18)

18 As an example, consider CNTs having a diameter around 1.2 nm but a different chirality. Then from Eq. (2.18), it can be shown that as the chirality changes (as θ is increases), band gap change becomes larger (Figure 2.7). Also, the band gap change of a CNT due to small torsional strain (0.01) is shown in Figure 2.8. Note that the presence of small torsional strain makes a huge difference in band gap change at θ is zero (armchair CNT). Band gap of armchair CNT (7, 7) is calculated according to applied torsional strain and plotted in Figure 2.9. Therefore, even for metallic CNTs, the electrical resistance of CNT itself can change under small torsional strain.

Figure 2.7. Chirality-Dependent Band Gap of CNTs without Torsional Strain

19

Figure 2.8. Chirality-Dependent Band Gap of CNTs with Torsional Strain

Figure 2.9. Band Gap of Metallic CNT (7, 7) with Torsional Strain

20 2.5. Summary Energy dispersion relations are calculated based upon two approximations: tight binding and zone folding [32]. This approximate calculation is enough to describe the band gap change of a CNT and thus the electrical property of a CNT. Actually all CNTs have an intrinsic curvature which causes hybridization of the orthogonal π and σ orbitals. However, in these approximations, the effect of curvature of the CNT is ignored. For a CNT which has diameter larger than 1 nm, the tight binding and zone folding approximation can describe a band structure well. For example, the model developed by Yang et al [27] is based on these assumptions. The outer diameter of MWCNT used in this research is about 15 nm and therefore, it is reasonable that the band structure of CNT is not altered by the curvature. The Model developed by Kleiner et al [30] gives a direct expression for the band gap of CNT and can be used as a simple and useful guide. This model considers both the curvature of the CNT and the externally applied strain and has been quite successful to predict band gap change. As for MWCNTs consisting of multiple graphene tubes, they can have a combination of effects from band gap change in the different tubes. Both from theoretical and experimental aspects, the MWCNT become most likely to have metallic characteristics [33, 34]. Therefore, it can be assumed that the MWCNTs are not sensitive to the external strain and remain as metallic under deformation. However, as it was already shown, if semiconducting CNTs are used as a conductive filler inside the matrix, the resistance change from band gap change should be considered. Ballistic conduction of transport property of CNT, was predicted theoretically and confirmed by experiments. However, in reality, the electrical resistance can increase due to the collisions of electron with phonons and defects in CNT [35]. It was found by Kane et al [36] that the electrical resistance increased due to thermally excited phonons as temperature increased. Also, interaction between CNTs and surrounding molecules can increase electrical resistance. Therefore, for an imperfect CNT, the conductance can not be calculated simply from the band structure for a given electron energy. It needs explicit calculation of the transmission function which is beyond the scope of this study.

21 Concerning band gap change induced by strain, theoretical models are well developed. However, the relation between band gap and resistance is not yet fully developed and in need of more research.

22

CHAPTER 3. ELECTRICAL RESISTANCE OF MWCNT LAYER

3.1. Objectives In this chapter, the electrical characteristics of a multiwalled carbon nanotube (MWCNT) layer under compressive deformation are explored. A MWCNT layer is located between two copper substrates and the substrates are moved towards each other, so that the MWCNT layer is compressed. The electrical resistance of a MWCNT layer can be interpreted as electrical contact resistance. There have been attempts to use MWCNT layer for the enhancement of interfacial transport properties like electrical and thermal contact resistance [37, 38]. Tong, et. al [38] show that a vertically aligned CNT film improves interfacial electrical resistance and the CNT layer grown on a silicon wafer substrate showed the minimum resistance, varying from 1 Ω to 20 Ω. However, Tong, et. al [38] did not specifically address possible contact resistance reduction mechanisms. The objectives of the works summarized are to: (i) fabricate MWCNT layer on a copper surface and establish a proper experimental setup, (ii) identify mechanisms describing macro-scale electrical resistance change and (iii) develop a model describing electrical resistance under compression.

3.2. Growth of MWCNT Layer on Copper Substrate

3.2.1. Preparation of Copper Substrate Copper plate (Alloy 110; Electrolytic Tough Pitch Copper) was cut using water jet and precision cutter into small rectangular block having dimensions of 10x10x0.5 mm. The

23 surface of these copper blocks were mechanically ground (see Figure 3.1a) using polishing machine (Buehler, Ecomet 3) (Figure 3.1b). Mechanical grinding can provide a flat surface and remove contaminants on the copper surface. This flat and clean surface is necessary for a good adhesion of deposited metal in the next step. Note that resistivity of copper is low at 1.47x10-8 Ω-m [39] which is an attribute that is desired for accurately measuring the electrical resistance of the MWCNT layer.

(a) Polished Cu Substrate

(b) Polishing Machine Figure 3.1. Preparation of Copper Substrate

24 3.2.2. Metal Deposition Process The used metal deposition system is shown in Figure 3.2. During the evaporation process, a solid (metal) phase is transformed into a vapor phase under high vacuum environment. In this metal evaporation system, thermally excited electron beam is used as heat source. The basic parameters used for this process are summarized in Table 3.1. Three metal layers of Ti, Al, and Ni were deposited onto the copper substrate using electron-beam evaporation (see samples in Figure 3.3a). Jig used to hold copper substrate during the process is shown in Figure 3.3b. The Ti layer promotes adhesion of the MWCNT to the copper substrate. The Al layer acts as a “buffer” layer which is known to enhance the CNT growth with the Ni catalyst [37, 40, 41] that provides seed sites for CNT growth [42, 43]. Thus, the metals deposited on to the copper surface should remain stable for the next PECVD process to grow CNTs. The thickness of Ti, Al, and Ni are: 30 nm, 10 nm, and 6 nm respectively.

Figure 3.2. Electron Beam Evaporator

25 Table 3.1. Basic Parameters for Metal Deposition Elements

Initial Vacuum Electron Beam Current

Ti Al

Process Vacuum

0.08 Amps ~ 1x10-6 Torr

Ni

~ 1x10-3 Torr

0.185 Amps 0.12 Amps

(a) Samples

(b) Jig

Figure 3.3. Metal Deposited Cu Substrate

3.2.3. PECVD Process The CNTs were grown on this metal deposited copper substrate surface by a microwave plasma enhanced chemical vapor deposition (PECVD) process. The PECVD system used in this research consists of microwave generator assembly (Seki Technotron: AX 5000/5400) and microwave plasma reactor (Seki Technotron: AX 5220/5250) (see Figure 3.4a). The microwave plasma reactor area of the PECVD system is shown detailed in Figure 3.4 [44]. This system incorporates 5 kW at 2.54 GHz microwave generator to produce plasmas at high power densities [45]. The feed gases were H2 and CH4. H2 is used to generate hydrogen plasma providing reactive radicals to break the atomic bonding of CH4. The carbon atoms then rearrange on the surface of Ni (catalyst) and grow as tubular shape. It is believed that Al layer is melted during PECVD process and help form a small particle shaped Ni islands for CNTs grow. The basic parameters for PECVD

26 process are summarized in Table 3.2. An as-fabricated MWCNT layer on Cu substrate is shown in Figure 3.5. The black surface visible in the photo is the MWCNT layer.

(a) Overall System

(b) Microwave Plasma Reactor Area Figure 3.4. Microwave - Plasma Enhanced Chemical Vapor Deposition System

27 Table 3.2. Basic Parameters for PECVD Process Substrate Height

52 mm

Plasma Power

150 W

Operating Pressure

10 Torr

Substrate Temperature

800 °C

Bias

0

H2 Flow Rate

72 sccm

CH4 Flow Rate

8 sccm

Growth Time

20 min

Figure 3.5. As-Fabricated MWCNT Layer on Cu Substrate (Unit = cm)

3.2.4. Characterization of MWCNT Layer The MWCNT layer grown on the copper substrate was observed using field emission scanning electron microscopy (FE-SEM). There was no additional sample preparation needed for the observation using FE-SEM because both MWCNT layer and copper substrate are conductive. MWCNTs are observed to grow on the machining grooves of the copper surface (see Figure 3.6a and Figure 3.6b). The overall CNT layer does not show any preferred direction (see Figure 3.6c). The individual CNTs have a bamboo-like structure which is a typical feature of relatively large diameter MWCNTs (Figure 3.6d).

28 The tubular structure of MWCNT was confirmed by TEM (transmission electron microscopy) [37] (Figure 3.7). The surface height profile and roughness of the MWCNT layer was obtained using a Veeco DI 3100 atomic force microscope (AFM). Typical 3D groove shape measured by the AFM is shown in Figure 3.8a. The mean roughness value of the MWCNT layer at a peak location of groove (square region in Figure 3.8b) is 122 nm. Note that the surface roughness of a silicon on insulation (SOI) wafer is 1 to 2 nm [46] and for a polished metal surface the roughness is 800 nm. Thus the MWCNT layer is relatively rough compared with a SOI wafer but smoother than mechanically polished metal. The height profile of the MWCNT layer (Figure 3.8a) indicates that the surface also has some sharp peak features.

(a) Machining Lines on Copper Substrate

(c) Enlarged View of MWCNTs

(b) MWCNTs on Machining Line

(d) Shape of Individual MWCNT

Figure 3.6. MWCNTs Layer on Copper Substrate

29

Figure 3.7. TEM Image of Individual MWCNT

(a) MWCNT Layer on Groove

(b) MWCNT Layer at Peak of Groove

Figure 3.8. AFM Images of MWCNT Layer on Groove

3.3. Measurement of Electrical Resistance Induced by Compressive Strain

3.3.1. Test Setup A schematic of the test setup is shown in Figure 3.9. While subjecting the MWCNTenhanced Cu substrate to compressive loading using a Cu probe, electrical resistance change was monitored by a multimeter (Hewlett Packard 3478A). To precisely measure

30 small resistance changes, a four point measurement scheme was adopted (see Figure 3.10). The probe material was also chosen to be Cu in order to match the properties of the Cu substrate. The probe tip area (0.31 mm2) is much smaller in dimension than the substrate so that multiple measurements can be made with each specimen by changing the probing location. A small-scale mechanical testing machine (Bose Endura ELF 3200) was used to control the probe displacement and to measure the interaction force between the probe and MWCNT-enhanced Cu substrate surface. The position of the probe tip was adjusted toward the sample surface while monitoring the position of the probe tip through a CCD camera. Starting from this non-contacting position (infinite electrical resistance) the probe was displaced downward slowly in 1.0 μm increments until first measurable electrical resistance was observed. This location was set to be the initial position (Z = 0 μm) of the probe, and the probe tip was subsequently moved downward by 1.0 μm increments. At each step of displacement, contact resistance and force data were recorded. When the resistance displayed a trend close to a constant value, the probe descent was stopped. The probe was then moved upward (reverse direction) in 1.0 μm increments while measuring the contact resistance and force until electrical contact was lost (infinite resistance).

Figure 3.9. Overall Test Setup

31

Figure 3.10. Four Point Measurement Scheme

3.3.2. Bare Cu-Cu Contact Figure 3.11 shows the measured electrical resistance for the Cu probe contacting a bare Cu surface, plotted as a function of probe tip displacement. The first finite contact resistance measured was 300 Ω corresponding to an initial contact force of 0.006 N. This probe position was regarded as the initial position (Z = 0 μm). With the probe moving downward, resistance decreased and force increased (Figure 3.11). However, after the Cu probe passed Z = 3 μm, the resistance remained constant at a value of 20 Ω regardless of contact force. The downward movement of the Cu probe was stopped at Z = 17 μm and the probe was then moved upward at 1 μm increments with resistance and force data collected as before. The resistance did not change significantly until the probe moved upward to Z = 7 μm. At Z = 6 μm the resistance increased to 0.4 MΩ and thereafter indicated infinite resistance. Note that electrical contact was lost before the probe reached the initial position (Z = 0 μm). The contact force did not follow the same path during loading (probe moving downward) and unloading (probe moving upward). This is indicated in the forcedisplacement measurements shown in Figure 3.12. Contact force initially shows a linear tendency (initial stiffness: 0.067x106 N/m) and then non-linear behavior as the probe

32 moved downward. Lower force values at corresponding Z positions with non-linear behavior were observed as the probe moved upward.

1x108

1.E+08

∞

∞

Probe Moving Downward

7 1.E+07

Contact Resistance of Cu-Cu (Ω)

1x10

Probe Moving Upward

6 1x10 1.E+06

5 1.E+05 1x10

Initial Force Measurement Position (0.006 N)

4 1.E+04 1x10

3 1.E+03 1x10

Loss of Electrical Contact

2 1.E+02 1x10

1 1.E+01 1x10

Initial Electrical Contact Position

0 1.E+00 1x10

0

2

4

6

8

10

12

14

16

18

Relative Position of Probe Tip, Z (μm)

Figure 3.11. Contact Resistance of Bare Cu-Cu Interface as Function of Probe Tip Position

1

Probe Moving Downward Probe Moving Upward

Load (N)

0.8 Initial Electrical Contact Position

0.6

Loss of Electrical Contact


0.4

0.2

0

0 0

1

22

3

4 4

5

66

7

88

9

10 10

11

12 12

13

14 14

15

16 16

17

18 18


Figure 3.12. Contact Force of Bare Cu-Cu Interface as Function of Probe Tip Position

33 3.3.3. MWCNT-Enhanced Contact A typical contact resistance change between the Cu probe and the MWCNT enhanced Cu substrate as a function of position is shown in Figure 3.13. As the probe was lowered, resistance decreased. The resistance ranged from a maximum value of 108 Ω to a minimum value of 4 Ω. The position corresponding to the first finite resistance value is indicated in Figure 3.13 and is identified as the initial electrical contact position (Z = 0 μm). The resistance did not change significantly until the probe moved downward past Z = 7 μm. At Z = 11 μm, the first measurable reaction force was observed. The electrical resistance then reduced significantly to a steady value of 4 Ω with increased probe movement. Note that between the initial position (Z = 0 μm) and Z = 11 μm, there was no measurable force but electrical contact was still maintained (finite resistance was measured). Resistance measured while the probe moved upward (reverse process) for the first several steps (from Z = 20 μm to Z = 14 μm) showed similar or slightly higher values at corresponding positions of the downward measurement. However, the resistance did not increase to an infinite value when the probe passed the position from where contact force between two surfaces dropped to zero (Z = 13 μm). Electrical contact is maintained even past the initial position (Z = 0 μm), up to Z = -7 μm. This trend is opposite to that observed for the bare Cu-Cu contact. Also, step-like features of resistance change are evident during both downward and upward movements of the probe. These features are thought to be the result of clumping between MWCNTs due to van der Waals forces. In Figure 3.14, contact force is plotted as a function of probe tip displacement. The overall trend of force change is more linear than the control case (bare Cu-Cu contact plotted in Figure 3.12). The average stiffness during downward movement (0.173x106 N/m) is over 2.58 times higher than the initial stiffness of the bare Cu-Cu contact (0.067x106 N/m). One of the notable features is that after the probe registers a measurable force, the trends of contact resistance versus force for different tests (two Cu-MWCNTCu tests) are found to closely overlap each other, as shown in Figure 3.15.

34

1E+08 1x10

Contact Resistance of Cu-MWCNTs-Cu (Ω)

8

∞

∞

Probe Moving Downward

1E+077

1x10

Probe Moving Upward

6

1E+06 1x10

5

1E+05 1x10

4

1E+04 1x10

3

1E+03 1x10

Initial Electrical Contact Position

2

1E+02 1x10


1

1E+01 1x10

0

1x10 1E+00

0

0 -61 2 -4 4 12 13 6 14 15 8 16 10 12 20 21 14 22 23 16 24 25 18 26 27 20 28 29 22 3 4 -2 5 6 70 8 2 9 10 11 17 18 19


Figure 3.13. Contact Resistance of Cu-MWCNT-Cu Interface as Function of Probe Tip

1.4

Probe Moving Downward Probe Moving Upward

1.2

Load (N)

1

0.8 Initial Electrical Contact Position

0.6

0.4


0.2

0 0 -6

1

2 -4

3

4 -2

5

6

0

7

8

2

9

10

4

11

12

6

13

14

8

15

16

18 10 17 12

19

20 14

21

22 16

23

24

18 25 26 20 27 28 22

29


Figure 3.14. Contact Force of Cu-MWCNT-Cu Interface as Function of Probe Tip

35

50

Cu-MWCNTs-Cu Test 1; Probe Moving Downward Cu-MWCNTs-Cu Test 1; Probe Moving Upward Cu-MWCNTs-Cu Test 2; Probe Moving Downward

Contact Resistance (Ω)

40

Cu-MWCNTs-Cu Test 2; Probe Moving Upward Cu-Cu; Probe Moving Downward 30

Cu-Cu; Probe Moving Upward Cu - Cu Contact

20

10

Cu - MWCNTs - Cu Contact

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

Compressive Force (N)

Figure 3.15. Comparison of Contact Resistance between Bare Cu-Cu Interface and CuMWCNT-Cu Interface

3.4. Discussions

3.4.1. Contact Resistance Reduction Mechanism For the same apparent contact area, the Cu-MWCNT-Cu interface showed a minimum resistance of 4 Ω while the Cu-Cu interface showed a minimum resistance of 20 Ω. Therefore 80% reduction was observed. Note that only a portion of the apparent contact surface, indicated as Ac (α-spots) in Figure 3.16, participates in electrical conduction [47]. In the case of the Cu-MWCNT-Cu contact, CNTs significantly increase the size of Ac (α-spots). It can be simplified conceptually as shown in Figure 3.17. The gap between two contacting members is filled with MWCNTs thereby increasing the contact area (see Figure 3.17b) via numerous parallel electrical contact paths. Note that if the load bearing area is increased, then the force will increase accordingly. Thus it can be concluded that

36 MWCNT layer is also effective in enlarging the load bearing area, and thus the higher contact stiffness was measured. Resistance reduction is also possible though electrical junctions made between CNTs, as illustrated in Figure 3.18. The non-directionally grown MWCNTs on the substrate’s surface (see Figure 3.6) create electrical junctions among adjacent CNTs to reduce the contact resistance. Other researchers suggest that contact resistance varies widely depending upon the relative orientation of two CNT surfaces (see Figure 3.19) [48, 49]. Therefore it is believed that the ensemble of the numerous contacts and junctions created during the probe movement dictate the macroscopic contact resistance. ` Ab: Load

Aa: Apparent Contact Area

Bearing Area Ac: α-spots

Figure 3.16. Classification of the Contact Surface

(a) Bare Cu-Cu Contact

(b) Parallel Contacts by Many MWCNTs

Figure 3.17. Contact Resistance Reduction by Parallel Contacts Created by MWCNTs

37

Figure 3.18. Electrical Junction Formed by Two MWCNTs

(a) A-A Sequence: In Registry

(b) A-B Sequence: Out of Registry

(Lower Junction Resistance)

(Higher Junction Resistance)

Figure 3.19. Relative Orientation of Two CNTs at Contact Surface

3.4.2. van der Waals Interaction Due to its own molecular structure, interactions among CNTs are dominated by van der Waals forces that tend to bundle individual CNTs [50, 51]. MWCNT-enhanced Cu surfaces exhibited step-like resistance changes and maintained electrical contact over a larger distance than bare Cu-Cu contact, particularly during movement of the contacting probe away from the surface (see Figure 3.13). It is hypothesized that this is due to uprooted MWCNTs bridging the probe and substrate via van der Waals forces, thereby maintaining electrical contact during probe movement.

38 3.4.3. Compaction-Based Resistance Change Model A model is developed by H. Kim [52] and presented to predict the change in contact resistance as a function of applied compressive force, as plotted in Figure 3.15. This model is based on resistance reduction due to the compaction of the MWCNT layer beneath a probe tip of area A (0.31 mm2). The volume of MWCNTs beneath the probe has initial volume fraction vo. During the probe’s downward stroke, volume fraction v is assumed to be related to strain as

ν = (1 + ε ) νo

Equation Section (Next)(3.1)

where ε is true strain, defined as positive in compression. Compressive force F is observed to be linearly related to downward probe movement (see Figure 3.14) over a roughly 8-9 μm stroke. This linear behavior can be represented as F = Eeff Aε

(3.2)

Eeff is the effective elastic modulus of the MWCNT layer which can be found by the relationship

Eeff =

L k A

(3.3)

where k is the experimentally measured stiffness in Figure 3.14 during downward probe movement and L is chosen as 8 μm. Based on Eq. (3.3), Eeff is found to range from 3.2 to 4.5 MPa. A relationship between electrical conductance C and volume fraction is now assumed to be in the form of a power law of order n, such that as volume fraction of the MWCNT increases, the conductivity also increases beyond the starting value Co corresponding to the initial volume fraction vo.

39

⎛ν ⎞ C = Co ⎜ ⎟ ⎝ν o ⎠

n

(3.4)

Resistance is the inverse of C and can be expressed as

⎛ν ⎞ R = Ro ⎜ ⎟ ⎝ν o ⎠

−n

(3.5)

Finally, Eqs. (3.1) to (3.3) can be combined into Eq. (3.5) resulting in a relationship between contact resistance R and probe compressive force as following

⎛ F ⎞ R = Ro ⎜ 1 + ⎜ E A ⎟⎟ eff ⎝ ⎠

n

(3.6)

Parameters Ro and n in Eq. (3.6) are fitting constants that must be chosen to best match the resistance versus force data in Figure 3.15. As shown in Figure 3.20, choices of Ro = 26.1 Ω and n = 2.87 are found to give a best fit to the pooled data (Tests 1 and 2 downward stroke).

30 Cu-MWCNTs-Cu Test 1; Probe Moving Downward Cu-MWCNTs-Cu Test 2; Probe Moving Downward

25

Contact Resistance (Ω)

Model Prediction 20

15

10

5

0 0

0.2

0.4

0.6

0.8

1

1.2

Compressive Force (N)

Figure 3.20. Resistance Change Based on Compaction Model

40 3.5. Summary Experimental measurements have shown that a MWCNT layer between Cu-Cu surfaces reduces electrical contact resistance by 80 percent. It is thus an excellent interfacial material to reduce electrical resistance. Also, the MWCNT-enhanced surface showed a finite slope of electrical resistance as a function of contact force, thereby making possible the use of this arrangement as a small-scale or thin-gap force or pressure sensor. However, there are also limitations on using an as-grown MWCNT layer on these applications. Most of all, due to its weak bonding to the substrate, MWCNTs are easily separated from the substrate surface, thus the repeatability and reliability might not be guaranteed in real applications. For example, MWCNT layer fell off easily when the surface was scratched by razor blade (see Figure 3.21). Furthermore, the fabrication of a MWCNT layer onto substrates of interest using PECVD was not a practical choice of manufacturing, due to high costs and low yield (less than 10% yield success rate) of the available metal evaporation and PECVD systems (see Figure 3.22). Main reason of failure of CNT growth on substrate can be attributed to the bad adhesion of catalyst metals on copper substrate (Figure 3.23). Also, there is an argument that Al layer must not be used as a buffer layer because it hinders the growth of CNTs [53]. Thus, in-depth study about the effects of catalyst metal layer is needed. Therefore, MWCNT filled polymers were investigated. The fabrication and characterization of MWCNT/polymer composite film are detailed in the following chapter.

Figure 3.21. MWCNTs Separated from Surface

41

Figure 3.22. No Growth of CNTs

(a) Partial Growth of CNTs

(b) Abnormal Catalyst Metal Adhesion

Figure 3.23. Optical Micrograph of Sample Surfaces

42

CHAPTER 4. FABRICATION OF MWCNT/POLYMER COMPOSITE FILM

4.1. Objectives CNT composite in film form has great potential for many applications such as strain sensor, transparent conductive film, and membrane for filtering [54 -59]. Frequently used methods to fabricate CNT film are filtering method [54, 56, 57], in-situ polymerization [55], layer-by-layer (LbL) method [58] and evaporation method [59, 60]. Among these methods, evaporation method can be conducted without sophisticated equipment. This method is originated from latex polymerization [61 - 64] which is used widely in industry. Significant amount of research has been done about the fabrication of MWCNT/Polymer composite [65 - 68]. However, the methods for fabrication of MWCNT/polymer composite were different from author to author and also dictated by process parameters, thus it can not be directly applied to our work and assumed to have identical material structure as described in the literature. Depending on parameters, the CNT-filled polymer film may have various inner and surface structures. Thus it is indispensable to establish correct relationships between MWCNT/polymer composite and parameters in our fabrication method. The research summarized in this chapter has the following objectives: (i) select the appropriate matrix materials and process parameters and (ii) understand the relationship between conductivity of MWCNT/polymer composite film and fabrication parameters.

43 4.2. Raw Materials The basic properties of materials used in evaporation method of the current research are summarized in this section. Water is chosen as the evaporation medium and MWCNTs are used as conductive filler inside the matrix. Because CNT do not disperse in water, sodium dodecyl sulfate (SDS) is used to lower surface tension of water and interfacial tension between CNT and water. Water-insoluble polyvinyl acetate and two different water-soluble polymers, polyvinyl alcohol (PVA) and polyethylene oxide (PEO) are tried as polymer matrix.

4.2.1. Multiwalled Carbon Nanotubes (MWCNT) The specification of commercially-available MWCNT [69] purchased from NANO LAB is as following: length 1-5 µm; diameter 15±5 nm; purity 95% and the density is 1.8 g/cm3. Metallic residuals can be iron, cobalt and nickel. EDS (energy dispersive spectroscopy) results show that iron level is very low [69].

Table 4.1. EDS Analysis Results of MWCNTs Sample Elements

Weight %

Atomic %

C·K

98.92

99.74

S·K

0.14

0.05

Fe·K

0.94

0.20

Total

100

100

4.2.2. Sodium Dodecyl Sulfate (SDS) Surfactants usually contain both hydrophobic groups (their "tails") and hydrophilic groups (their "heads") [70]. Therefore, they are soluble in both organic solvents and water. Surfactants reduce the surface tension of water and also reduce the interfacial

44 tension between oil and water [71]. In the present research, sodium dodecyl sulfate (or sulphate) (SDS or NaDS) is used. SDS is classified as an ionic surfactant due to net negative charge at the head and the molecular formula is C12H25NaO4S (see Figure 4.1) The molecule has a tail of 12 carbon atoms, attached to a sulfate group, giving the molecule the amphiphilic properties. It is prepared by sulfonation of dodecanol (lauryl alcohol, C12H25OH) followed by neutralization with sodium carbonate. The SDS used in this research was purchased from the vendor bioworld (cat# 730103). It is particle type and has density 0.32 g/cm3 [73].

Figure 4.1. Molecular Formula of SDS [72]

4.2.3. Polymer Matrix Polyvinyl acetate [74] is a rubbery synthetic polymer and the molecular formula of the monomer is -[CH2=CHO-COCH3]-. It is synthesized by polymerization of vinyl acetate. Polyvinyl acetate is mainly used as an emulsion in water and an adhesive for porous materials. Also, it is widely used as a paint or an adhesive because it is highly soluble in solvents like ketone, ester, and alcohol. After Polyvinyl acetate is solved with solvent, the glass transition temperature becomes lower from about 38 – 40 °C down to room temperature or lower. Then it has better flowability and workability as an adhesive agent. Glass transition temperature (Tg) rises as the solvent evaporates. Polyvinyl alcohol (PVA) is a water-soluble synthetic thermoplastic polymer [75]. The molecular formula of this monomer is -[CH2CH2O]-. The hydrophilic property of PVA makes it permeable for both water and hydrated salt. PVA is synthesized by partial or complete hydrolysis of polyvinyl acetate removing acetate groups and not by polymerization of the monomer. PVA has an excellent film forming, emulsifying, and

45 adhesive properties. It is also resistant to oil, grease and solvent. As for mechanical properties it has high tensile strength and flexibility. Also, it has high oxygen and aroma barrier. However, these properties are dependent on humidity, in other words, with higher humidity more water is absorbed. The water which acts as a plasticizer will then reduce its tensile strength, but increase its elongation and tear strength. PVA decomposes rapidly above 200°C. Polyethylene oxide (PEO) is a synthetic thermoplastic polymer [76]. PEO is soluble in water, methanol, benzene, dichloromethane and is insoluble in diethyl ether and hexane. The molecular structure of PEO is HO-[CH2-CH2-O]n-H. PEO is also known as PEG (polyethylene glycol). Conventionally, PEO corresponds to longer polymer and PEG shorter one. PEO is a low-melting solid and can be even liquid depending on the molecular weights. Both are prepared by polymerization of ethylene oxide. While PEG and PEO with different molecular weights find use in different applications and have different physical properties (e.g. viscosity) due to chain length effects, their chemical properties are nearly identical. Their melting points vary depending on the formula weight of the polymer. In Table 4.2, the physical properties of these polymers and vendor information are summarized [77, 78]. Note that PVA corresponds to polyvinyl alcohol and not polyvinyl acetate in this document.

Table 4.2. Physical Properties of Polymers and Vendor Information Physical Properties

Polyvinyl Acetate

PVA

PEO

Solubility

Solvent

Water and Solvent

Water and Solvent

Density

1.18 g/cm3

1.27 - 1.31 g/cm3

1.13 g/cm3

Melting Point (Tm)

180 to 240 °C

180 to 190 °C

62 to 67 °C

Glass Transition Temperature (Tg)

38 to 40 °C

75 to 85 °C

-50 to -57 °C

Catalog Number

-

363081-500G

372781-250G

Vendor

Polyseamseal

Sigma-Aldrich

Sigma-Aldrich

46 4.3. Compaction-Based in situ Polymerization

4.3.1. Experimental Procedure As-purchased MWCNTs are observed by FE-SEM to be aggregated due to van der Waals interaction (see Figure 4.2a) and multiple walls are well observed in TEM photo (see Figure 4.2b) [69]. MWCNT is hydrophobic and they tend to aggregate as it can be seen in Figure 4.2a, thus SDS is used to lower the surface tension between MWCNT and water [79, 80]. In Figure 4.3, the processing procedure is schematically described. First, SDS solution (1 wt%) was prepared using a magnetic stirrer (2 g of SDS added to 198 g of water). MWCNTs were mixed with surfactant solution by an ultrasnonicator (Misonix Sonicator® 3000) for 10 minutes. As-purchased MWCNTs and SDS are shown in Figure 4.4. The amount of MWCNTs is selected according to the target volume (weight) fraction. Ultrasonicator used in this study is shown in Figure 4.5. Ultrasonicator converts electrical energy into mechanical vibration and the horn is vibrates at 20 KHz.

(a) FE-SEM Photo

(b) TEM Photo

Figure 4.2. As-Purchased MWCNTs

47

Figure 4.3. MWCNT/Polymer Composite Film Fabrication Procedure

Figure 4.4. As-Purchased MWCNTs and SDS (Unit = cm)

48

Figure 4.5. Ultrasonicator

Polyvinyl Acetate based commercial adhesive, polyvinyl alcohol (PVA), and polyethylene oxide (PEO) were used as the matrix. As-purchased three different polymers are shown in Figure 4.6. Each polymer (1 g) was dispersed in D.I. water (150 ml) by a magnetic stirrer for 30 min. After that, both MWCNT and polymer colloidal solutions were mixed together and then ultrasonication was conducted again for 2 hours (Figure 4.7). Finally, this solution was poured into a beaker which contains copper or release film substrate. Evaporation of water was achieved in an oven operating at 90 ºC (Figure 4.8). The D.I. water was evaporated until a MWCNT/polymer composite film was formed on the copper or release film substrate. Following cool-down, D.I. water or solvent rinse (ethanol) of the composite film was conducted to remove residual surfactant on the MWCNT film surface.

49

Figure 4.6. As-Purchased Polymers (Unit = cm)

Figure 4.7. Ultasonication of MWCNT and Polymer Mixture

50

Figure 4.8. Oven Used for Water Evaporation

4.3.2. MWCNT/Polyvinyl Acetate Composite Film The as-fabricated MWCNT/polyvinyl acetate composite film bonded to a Cu substrate is shown in Figure 4.9. The surface of MWCNT film showed a dense and porous layer which is mostly composed of MWCNTs (Figure 4.10). The cross section of the film also showed densely populated MWCNTs and porous inner structure. This film structure is very similar to that of filtered SWCNT film which was observed to be self-assembled mats of entangled SWCNTs [81]. The thickness is measured as 6.57 µm.

Figure 4.9. As-Fabricated MWCNT/Polyvinyl Acetate Composite Film on Cu Substrate (unit: cm)

51

(a) Plan View

(b) Cross-Sectional View

Figure 4.10. MWCNT/Polyvinyl Acetate Composite Film

4.3.3. MWCNT/Polyvinyl Alcohol (PVA) Composite Film The as-fabricated free-standing MWCNT/PVA composite film is shown in Figure 4.11. It showed non-porous morphology (Figure 4.12) unlike the MWCNT/polyvinyl acetate composite film. MWCNTs were distributed non-directionally. To obtain cross sectional morphology, a small piece of the MWCNT/PVA composite film was dipped into liquid nitrogen (LN2) for 1 minute and then immediately fractured. The morphology visible on the cross-section showed granular shapes implying that there was grain structure (see Figure 4.12b). It was found that the thickness of the sample was not uniform but varied from 91.3 μm to 117.4 μm. The density of MWCNT is 1.8 g/cm3 and the density of PVA is 1.3 g/cm3. Thus by simple calculation, the volume fraction of MWCNT and PVA are calculated as 7.2% and 92.8% respectively for this 10 wt% MWCNT sample.

52

Figure 4.11. As-Fabricated Free Standing MWCNT/PVA Composite Film

(a) Plan View


Figure 4.12. MWCNT/PVA Composite Film

4.3.4. MWCNT/Polyethylene Oxide (PEO) Composite Film The as-fabricated free-standing MWCNT/PEO composite film is shown in Figure 4.13. It was a non-porous film and MWCNTs were non-directionally dispersed (Figure 4.14). The morphology of the freeze-fractured cross-section was different from that of MWCNT/PVA composite film in that there were no distinctive grain boundaries (see Figure 4.14b) and the MWCNTs were observed to be well distributed. The thickness of MWCNT/PEO composite film is 96.8 μm and is uniform. The density of MWCNT used in this research is 1.8 g/cm3 and the density of PEO 1.13 g/cm3. Thus by simple

53 calculation, the volume fraction of MWCNT and PEO are calculated as 6.30 vol% and 93.7 vol% respectively for this 10 wt% MWCNT sample.

Figure 4.13. As-Fabricated Free Standing MWCNT/PEO Composite Film

(a) Plan View


Figure 4.14. MWCNT/PEO Composite Film

54 4.4. Electrical Resistivity of Fabricated Composite Film

4.4.1. DC Resistance Measurement Test Setup Direct current (DC) resistance measurement of MWCNT/polymer composite film was conducted using a voltage divider circuit [82] shown in Figure 4.15. The role of the voltage divider circuit is to lower power consumption of the MWCNT/polymer composite film sample, thereby avoiding heat damage [83]. By measuring v1 and vs, the resistance of the MWCNT/polymer composite film (Rx) can be calculated by following relationship. ⎛ν ⎞ Rx = R1 ⎜ s − 1⎟ ⎝ν 2 ⎠

Equation Section 4(4.1)

The detailed view of MWCNT/polymer composite film region is shown in Figure 4.16. Smooth jaw alligator clips were used to connect the MWCNT/polymer composite film to the circuit. Four MWCNT/polymer composite film samples were measured and their dimensions are shown in Table 4.3.

Table 4.3. Dimensions of MWCNT/Polymer Composite Sample Composite Film MWCNT/PVA MWCNT/PEO

Dimensions of sample (mm)

Sample 1

0.19 x 21.30 x 4.0

Sample 2

0.14 x 2.65 x 4.64

Sample 1

0.09 x 20.77 x 6.37

Sample 2

0.2 x 20.37 x 4.14

55

Figure 4.15. Voltage Divider Circuit

Figure 4.16. Detailed View of MWCNT/Polymer Composite Region

56 4.4.2. Resistivity of MWCNT/Polymer Composite Film Resistance of four different samples (see Table 4.3) was measured. The resistivity change as a function of clip distance is shown in Figure 4.17. The resistivity of MWCNT/PVA composite film ranges from 7.6x103 to 5x104 Ω-cm while the resistivity of MWCNT/PEO composite film ranges from 2 to 7 Ω-cm. This huge difference between these two MWCNT/polymer composite films might be attributed to the colloidal instability and polymer solidification process which is discussed in section 3.4. In comparison, neat PVA and PEO polymer (unfilled) are reported to have resistivity of 3.1x107 to 3.8x107 Ω-cm and 1013 to 1018 Ω-cm respectively. Thus, relatively small addition of CNT into these polymers yields several orders of magnitude reduction in resistivity.

Figure 4.17. Resistivity of MWCNT/Polymer Composite Films

57 4.5. Discussions

4.5.1. Colloid Stability In this work, polymer and MWCNT colloids are coagulated intentionally through evaporation of the water [84]. Thus, stability and control of coagulation of the colloid particles are very important to produce a desired MWCNT/polymer composite film. According to theory which was developed by two groups of authors independently, i.e., by Derjaguin and Landau [85] and by Verwey and Overbeek (DLVO theory) [86], the rate of aggregation is proportional to the term exp (-VT/kBT), where VT represents the free energy of interaction, kB is Boltzmann’s constant and T is the absolute temperature. In case of MWCNT/polyvinyl acetate composite film, the polyvinyl acetate particles began to form sediment almost immediately (VT =

32 3 π r + 8π r 2l + 4l 2 r < sin γ > μ 3

(5.3)

where μ is an average of sinγ which is calculated as π/4 for an isotropic system. Thus Eq. (5.3) can be written as < Ve > =

32 3 π r + 8π r 2l + π rl 2 3

(5.4)

The universal value of for the onset of percolation in a continuum is generated by Monte Carlo simulation for an object system having identical shape [95]. If we use universal values of for a capped cylinder: 1.4 for randomly distributed and 2.8 for objects in parallel, then we can find upper and lower limit of percolation threshold by the following double inequality via combining Eqs. (5.1), (5.2) and (5.4) ⎛ 1.4Vcyl ⎞ ⎛ 2.8Vcyl ⎞ 1 − exp ⎜ − ⎟ ≤ Φ c ≤ 1 − exp ⎜ − ⎟ ⎝ < Ve > ⎠ ⎝ < Ve > ⎠

(5.5)

This excluded volume method was successfully applied to explain the percolation threshold change of various micro-scale objects [96]. Based on the specification of MWCNT (length 1-5 µm; diameter 15±5 nm) the percolation threshold of the material used in this research is calculated to be in the range of 0.13% ≤ Φ c ≤ 2.59% . One of the non percolation based theories was used to calculate percolation threshold (Фc) can be also calculated by an equation derived by Helsing and Helte [97] using average field approximation (AFA). This is one of the effective medium theory (EMT) [98] methods. In this method, the average effect of randomly distributed inclusions inside a matrix is calculated by treating a whole composite sample as a homogeneous effective

66 medium. From the equation by Helsing and Helte [97], if inclusions are can be modeled as ellipsoids (long and thin oblate), the percolation threshold (Фc) is represented as Φ c = 1.18ε

(5.6)

Note that the length of the major axis is R and the two minor axes is εR where ε (ε − ε c )(cos 2 < η > −ν sin 2 < η >) ⎟ − 1 ⎜ ⎟ h ⎝ ⎠ where ε c is the critical strain, < so > is the average distance between CNTs, and < η > is the average angle between the direction of applied strain and the tunneling path. < η > is calculated from < η > = sin −1

π 4

[91]. Basic constants used are in Table 5.3. Note that it is

the average angle between two rods touching randomly and this angle is similar to the

90

angle between strain direction and the tunneling path. Best fit is obtained when < so > and

ε c are 600 Å and 0.0083 for 0.56 vol%, and 85 Å and 0.028 for 1.44 vol% as shown in Figure 5.17. Eq. (5.35) is capable of describing strain dependent resistance reasonably well except at the transition region near the critical strain. The resistance change model suggested here for MWCNT/PEO composite film is highly simplified model and used MWCNTs are wavy and not straight as assumed. Thus, the effect of MWCNT shape on the percolation threshold ( Φ c ) should be considered. In addition to that, anomalies like voids, crack, non-uniformity of film thickness should be accounted for because these affect stress distribution which can change the behavior of deformation dependent electrical resistance of the film. In this suggested model, empirically determined values ( Φ c , t) are used and therefore, the repeatability and reproducibility of the obtained values should be guaranteed.

Figure 5.17. Resistance Change Model vs. Experiments

91 5.6. Application as Strain Sensor The MWCNT/PEO composite film reported herein showed repeatable and tunable electrical resistance versus strain relationships for different volume fractions of MWCNT, thereby making it applicable as a strain sensor. Strain sensitivity is one of key performance descriptors for strain sensing materials and is defined [120] as

S=

ΔR / Ro

(5.38)

ε

where ∆R is resistance difference, Ro is the initial resistance and ε is strain. The strain sensitivity for the two tested MWCNT/PEO samples and the strain sensitivity of metal alloys [121] typically used for conventional foil type strain gages are shown in Table 5.4. In the linear region, the strain sensitivity of 0.56 and 1.44 vol% samples are comparable to the high range of conventional foil gages. In the non-linear region, the 1.44 vol% sample showed much higher strain sensitivity than the conventional foil gage. The 1.44 vol% MWCNT/PEO film has been shown to be operational over a wide strain range, beyond 0.07 (limited by polycarbonate dogbone substrate), whereas previous works [88, 122, 123] have reported measurement up to a strain of 0.04. For the 0.56 vol% case, the abrupt change in resistance suggests that it can function like a switch. A useful linear strain range for 0.56 vol% of MWCNT is up to 0.008 strain and a useful strain range for 1.44 vol% of MWCNT is beyond 0.07. Note that this can be even higher since the test was stopped due to the onset of the localized necking of the polycarbonate dogbone substrate at about 0.07 strain. Table 5.4. Strain Sensitivity Comparison MWCNT/PEO Composite Film

Metal Alloys Used for

MWCNT 0.56 vol%

MWCNT 1.44 vol%

Foil Type Strain Gage

Region I

Region II

Region I

Region II

0.74 – 5.1

3.7

-

1.6

50

92 5.7. Summary The processing method developed using intentional coagulation of a dispersed MWCNT and PEO mixture was effective for fabricating a conductive MWCNT/PEO composite film. Experimentally determined percolation threshold of MWCNT/PEO is found to be between 0.14 to 0.28 vol%. It is well known that the percolation based models shows relatively good agreement with the experimental results compared with the non-percolation based theories [89] and in the present report, the experimentally determined percolation threshold is overlapped better with the predicted range by the excluded volume method model than that by the average field approximation model (see Table 5.1). The conductivity obtained was consistent and repeatable depending on volume fraction. Unique and repeatable relationships between electrical resistance and strain for different volume fractions of MWCNT were observed. The overall pattern of electrical resistance change versus strain for the samples of each volume fraction of MWCNT consists of linear and non-linear regions. In the linear region, scaling rule prediction is reasonably well matched with experimental results but it breaks down at the non linear region. In the non-linear region, tunneling resistance can be a major parameter affecting electrical resistance of MWCNT/PEO film. A model combining scaling rule and tunneling resistance can predict the full range of electrical resistance change as a function of strain. The unique and repeatable characteristics of electrical resistance change of the MWCNT/PEO films permit these samples to be used as tunable strain sensors for application into embedded sensor systems in composite structures.

93

CHAPTER 6. CONCLUSIONS AND FUTURE WORK

6.1. Conclusions The carbon naotube is a newly emerging material with relatively short history, but its potential is enormous due to the unique physical and mechanical properties. In the current research, we used MWCNT layer grown by PECVD process on copper substrate and MWCNT/PEO composite film fabricated by compaction based in-situ polymerization to investigate deformation dependent electrical resistance. The MWCNTs grown on copper substrate was under compressive deformation and MWCNT/PEO composite film was under tensile deformation. Each deformation case has been explored at the macro-scale and the following conclusions are made. 1. The MWCNT layer grown on a copper surface was shown to be effective to reduce electrical contact resistance under compressive deformation. The relationship between compressive force and the electrical resistance of the MWCNT layer showed 80 percent reduction in electrical contact resistance compared with bare copper contact. The increase in actual contact area by the MWCNTs spanning the gap between the two copper surfaces, and the electrical junctions created by the MWCNTs during the compaction, are argued to be the main electrical resistance reduction mechanisms. The MWCNTs have high surface area due to extremely high aspect ratio and were found to be effective at filling the gap and creating current paths between the copper surfaces. Additionally, the junctions made among adjacent MWCNTs during compaction of the MWCNT layer can make additional current paths. A compaction-based

94 resistance change model showed was shown to predict closely the experimental data except at the very low force regime. 2. The MWCNT layer showed a finite slope of electrical resistance as a function of contact force, thereby making possible the use of this arrangement as a smallscale force or pressure sensor. However, there are some limitations on the direct use of as-grown MWCNT on copper substrates. The easy separation of the MWCNTs from the copper surface and low yield of MWCNTs by the given process system are major obstacles for real-world application of the MWCNT layer. The anchoring action of the Ti layer was found to be not strong enough to hold MWCNTs firmly. PECVD is used widely to grow CNTs but in the given metal deposition system, the catalyst metal layer was not stably adhered to the surface of the copper substrate, which resulted in the poor yield of MWCNTs. Thus, more research on the improvement of the anchoring action of metal layers and the increase of PECVD yield is needed. 3. A processing method developed using intentional coagulation of dispersed MWCNT in PEO polymer solution was effective for fabricating a conductive MWCNT/PEO composite film. Depending on the colloidal stability of the MWCNT and polymer particles, the microstructures of MWCNT/polymer were quite varied and MWCNT/PEO composite system was found to be most suitable for the chosen process parameters, based on the resistivity measurement results and microstructure analysis. The SDS surfactant was very effective to formation of micelle separating MWCNTs. The SDS micelle-suspended MWCNT solution remained stable during the high temperature evaporation of water. This process is a simple and effective way to fabricate MWCNT filled polymer films. Most of all, the conductivity obtained was consistent and repeatable depending only on volume fraction. 4. The deformation-dependent electrical resistance of MWCNT/PEO is unique and repeatable for different volume fractions of MWCNT. The overall pattern of electrical resistance change versus strain for the samples of each volume fraction of MWCNT consists of linear and non-linear regions. A scaling rule model based

95 on percolation theory and a tunneling resistance model based on hopping of an electron are adopted for describing the experimental results. In the linear region, the scaling rule prediction matched well with the experimental results. In the nonlinear region, tunneling resistance was determined to be a major parameter affecting the bulk electrical resistance of the MWCNT/PEO film. It is hypothesized that in the linear region, the overlapping of MWCNTs inside the matrix is maintained during stretching of the film. Thus the percolation based theory can be still applicable. However, after passing the critical strain, the overlapping of MWCNT is no longer maintained and thus the percolation-based theory breaks down. For stretching beyond the critical strain, electron hopping based tunneling resistance is hypothesized to be the major mechanism for electrical conduction through the film. Therefore, a model combining the scaling rule and tunneling resistance models was developed and shown to predict the linear and non-linear modes of electrical resistance change vs. strain. The unique and repeatable characteristics in electrical resistance change of the MWCNT/PEO films permit these samples to be used as tunable strain sensors for application into embedded sensor systems in composite structures. 5. The MWCNTs inside the matrix can be considered as the network of resistors and the resistance of an individual CNT is known to change due to the change of the band gap of the CNT. The band gap of a CNT is dependent upon the chirality and thus, many variations of band gap of a CNT can be possible, exhibiting metallic to semiconducting properties. In the present research, MWCNTs used for conductive filler can be regarded as metallic which is insensitive to the externally applied strain. Therefore, the resistance change of individual CNT is ignored. However, if semiconducting CNTs are used, the change in electrical resistance of the CNTs should also be considered.

96 6.2. Future Work

6.2.1. Electrical Resistance of a CNT inside the Matrix Axial strain of CNT inside the matrix is analyzed using shear lag model and simple symmetric 2D finite element (FE) model. Especially, the effects of the volume fraction of CNT and the modulus of the matrix on the uniformity of axial strain should be investigated. Uniform state of strain is important because most theoretical works on electromechanical properties of SWCNT assume uniform strain in the axial direction. The electrical resistance change of CNT is dictated by the band gap change of CNT and the band gap change is directly affected by the actual level of axial strain. Thus, the volume fraction of CNT and the modulus of the matrix are important parameters determining the electrical resistance of an individual CNT inside the matrix. The outcomes of the study can be related to the design of micro and macro-scale strain sensors using CNTs.

6.2.2. Mechanical Properties of MWCNT/PEO Composite Modulus, strength, and fracture mechanisms of MWCNT/PEO are important aspects for the application of MWCNT/PEO to the multifunctional structures. Uniaxial tensile test of MWCNT/PEO composite should be conducted using a small scale test machine. Modulus change according to the volume fraction of MWCNTs is measured and analyzed. Fracture mechanisms are also important when it comes to the application of the composite as a strain sensor. Localized fracture can cause abnormal and unpredictable behavior of electrical resistance change. Thus, understanding the fracture mechanism is necessary for design and failure prediction. Optical and SEM (scanning electron microscopy) observation of fracture surfaces should be conducted to find the fracture mechanisms of the composite.

97

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113 Appendix A.

In appendix A, the basic principle of band structure is summarized. In this summary, a step by step procedure of band structure calculation [25] is introduced based upon tight binding and zone folding approximations.

A.1. Fundamentals of Band Structure The one-electron Schrodinger equation (see Eq. (A.1)) is a usual starting point for describing the behavior of electrons in the conduction band assuming that the only effect of the periodic lattice is to modify electron mass [124]. Also, it allows one to calculate the energy levels for any confining potential U (r ) .

i

2 ⎞ ∂Ψ ⎛ ∂2 =⎜− + U (r ) ⎟ Ψ 2 ∂t ⎝ 2m ∂x ⎠

where Ψ is the wave function, i is

−1 ,

(A.1)

is the Plank’s constant, m is the mass of

electron and U (r ) is any confining potential. Eq. (A.1) can be solved by the method of separation of variables:

Ψ ( x, t ) = Φ ( x)ϕ (t )

(A.2)

This method turns Eq. (A.1) into two ordinary differential equations:

E =i

1 dϕ ϕ dt

d2 EΦ = (− + U )Φ = H Φ 2m dx 2

(A.3)

(A.4)

114

d2 where E is energy state and (− + U ) is defined as H (Hamiltonian). Eq. (A.4) is 2m dx 2 referred to as the time-independent Schrodinger equation. We want to find the electronic band structure and this is equivalent to solve Eq. (A.4) [125]. Rearrange Eq.(A.4) as

E ( k )Φ ( k ) = H Φ ( k )

(A.5)

where E (k ) are the eigenvalues at wave vector k, and Φ (k ) are the eigenfunctions. Note that variable x is changed to wave vector k. In tight-binding approximation, the Bloch functions ( Φ ) are written as linear combinations of the atomic wave functions ( φ ). Then the matrix form of Eq.(A.5) is written as following

E{φ} = [ H ]{φ}

(A.6)

Consider nth atom and assume a wave solution to the Eq. (A.6) as

φn = φo eik i d

n

(A.7)

where k is the wave vector and d n is the position vector of neighbor atoms. Eq. (A.6) can be written as following:

Eφn = ∑ H nmφm

(A.8)

m

By combining Eq. (A.7) and Eq. (A.8), Eq. (A.9) is obtained Eeik i dn φo = ∑ H nm eik i dm φo m

After Eq. (A.9) is simplified, following expression is obtained.

(A.9)

115 E (k ) = ∑ H nm eik i ( dm − dn )

(A.10)

m

As for Hamiltonian matrix [H], it can be constructed either by 1st principle calculation or semi-empirical values. For example, [H] can be written as Eq. (A.11) for 1D chain of atoms (Figure A.1) considering an electron hopping only to the nearest atoms and periodic boundary condition [25].

t: Hopping Parameter

Figure A.1. 1D Chain of Atoms

⎛ Eo ⎜ ⎜ t ⎜ 0 [H ] = ⎜ ⎜ 0 ⎜ 0 ⎜⎜ ⎝ t

t

0

0

0

Eo

t

0

0

t 0

Eo t

t Eo

0

0

0

0

0

0

t

t ⎞ ⎟ 0⎟ 0⎟ ⎟ 0⎟ t ⎟ ⎟ Eo ⎟⎠

(A.11)

where Eo is the energy of an electron on an isolated atom and t is the hopping parameter of an electron to the nearest atoms. Then, Eq. (A.10) for 1D chain of atoms with one basis function which is equivalent to a unit cell having one atom (see Figure A.1) can be calculated as

116

E (k ) = ∑ H mn eik i ( dm − dn ) = Eo + teika + te −ika = Eo + 2t cos ka

(A.12)

m

This relationship between E and k is called energy dispersion relation:

E (k ) = Eo + 2t cos ka

(A.13)

Similarly, Eq. (A.10) for 2D case with one basis function can be calculated as E (k ) = ∑ H mn eik i ( dn − dm ) = Eo + teik i ( − axˆ ) + teik i ( + axˆ ) + teik i ( − byˆ ) + teik i ( + byˆ ) m

= Eo + te

− ik x a

+ te

ik x a

+ te

− ik y b

+ te

ik y b

(A.14)

= Eo + 2t cos k x a + 2t cos k y b

Thus the energy dispersion relation is expressed as

E (k ) = Eo + 2t cos k x a + 2t cos k y b

(A.15)

Now, consider 1D chain of atoms with two basis functions (Figure A.2). It is equivalent to a unit cell having two atoms. The construction of [H] for such system is shown in Eq. (A.15)

Figure A.2. 1D Chain of Unit Cell Containing Two Atoms

117 ⎛ ⎛ Eo t1 ⎞ ⎛ 0 0 ⎞ 0 ⎜⎜ ⎟ ⎜ ⎟ ⎜ ⎝ t1 Eo ⎠ ⎝ t2 0 ⎠ ⎜ ⎛0 t ⎞ ⎛ E t1 ⎞ ⎛ 0 0 ⎞ o 2 ⎜ ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ [ H ] = ⎜ ⎝ 0 0 ⎠ ⎝ t1 Eo ⎠ ⎝ t2 0 ⎠ ⎜ ⎛ 0 t2 ⎞ ⎛ Eo t1 ⎞ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎜ ⎝ 0 0 ⎠ ⎝ t1 Eo ⎠ ⎜ ⎜ 0 0 ⎝

⎞ 0⎟ ⎟ ⎟ 0⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(A.16)

where Eo is the energy of an electron on an isolated atom and t1 and t2 are the hopping parameters of an electron to the nearest atoms. Eq. (A.16) can be represented as

E{φn } = ∑ [ H mn ]{φm }

(A.17)

m

Note that [H] is in matrix form. Solution to Eq. (A.17) is assumed as {φn } = {φo }eik i dn

(A.18)

By combining Eq. (A.17) and Eq. (A.18), Eq. (A.19) is obtained. E{φo } = ∑ [ H nm ] eik i ( dm − dn ) {φo } = [h(k )]{φo }

(A.19)

m

Then we can obtain following expression and it represents the band structure of 1D case with two basis functions E (k ) = [h(k )] = ∑ [ H nm ] eik i ( dm − dn ) m

Expand Eq. (A.20) as following

(A.20)

118 [ E (k )] = ∑ [ H nm ] eik i ( dm − dn ) = H nn + H ( n −1) n + H n ( n +1) m

⎛E =⎜ o ⎝ t1

t1 ⎞ ⎛ 0 t2 ⎞ −ikl ⎛ 0 e +⎜ ⎟+ Eo ⎠ ⎝⎜ 0 0 ⎟⎠ ⎝ t2

0 ⎞ ikl ⎛ Eo ⎟e = ⎜ ikl 0⎠ ⎝ t1 + t2 e

t1 + t2 e − ikl ⎞ ⎟ Eo ⎠

(A.21)

Note that the matrix of Eq. (A.21) is hermitian and a hermitian matrix is a square matrix having complex entries which is equal to its own conjugate transpose ( ai , j = a*j ,i ). Then, the energy dispersion relation for 1D chain of atoms with two basis functions is calculated as

[ E (k )] = Eo ± t1 + t2 e− ikl = Eo ± t12 + 2t1t2 cos kl + t22

(A.22)

A.2. Band Structure of Graphene Sheet In this section, band structure of a graphene sheet is calculated. A hexagonal lattice of graphene sheet and basis vectors of unit cell is depicted in Figure A.3. a1 and a2 are lattice vectors, xˆ and yˆ are unit vectors in the x and y direction.

Figure A.3. Hexagonal Lattice of Graphene Sheet

119 The chiral vector ( c ) in the graphene sheet (see Figure 1.5) can be expressed by the lattice vectors and the lattice vectors are written in terms of xˆ and yˆ .

c = na1 + ma2

(A.23)

a1 = axˆ + byˆ =

3ac −c 3ac −c xˆ + yˆ 2 2

(A.24)

a2 = axˆ − byˆ =

3ac −c 3ac −c xˆ − yˆ 2 2

(A.25)

where m, n, a, and b are the integers and ac-c is the distance between carbon atoms. Note that a is 3b . The graphene sheet is a 2D case with two basis functions and Eq. (A.26) is obtained by following a similar procedure shown in the previous section (see Eq. (A.14) and Eq. (A.20)). t ⎞ ⎛ 0 t ⎞ ik i ( − a2 ) ⎛ 0 t ⎞ ik i ( − a1 ) ⎛E +⎜ + [ E (k )] = ⎜ o ⎟+⎜ ⎟e ⎟e ⎝0 0⎠ ⎝ t Eo ⎠ ⎝ 0 0 ⎠ ⎛ 0 0 ⎞ ik i ( a2 ) ⎛ 0 0 ⎞ ik i ( a1 ) +⎜ ⎜ ⎟e ⎟e ⎝ t 0⎠ ⎝ t 0⎠

(A.26)

After Eq. (A.26) is simplified and rewritten, Eq. (A.27) is obtained as following. ⎛E [ E (k )] = ⎜ *o ⎝ ho

ho ⎞ ⎟ Eo ⎠

(A.27)

where ho = t (1 + e −ik i a1 + e − ik i a2 )

(A.28)

ho* = t (1 + eik i a1 + eik i a2 )

(A.29)

120 Note that the matrix of Eq. (A.27) is hermitian matrix and the energy dispersion relation can be calculated as following E (k ) = Eo ± ho

(A.30)

Eq. (A.28) and Eq. (A.29) are simplified as following ho (k ) = t (1 + e − ik i ( axˆ +byˆ ) + e− ik i ( axˆ −byˆ ) ) = t (1 + e − ikx i a (e

− ik y b

+e

− ik y b

))

= t (1 + 2e − ikx i a cos k y b)

ho* (k ) = t (1 + eik i ( axˆ +byˆ ) + eik i ( axˆ −byˆ ) ) = t (1 + eikx i a (e

ik y b

+e

− ik y b

))

= t (1 + 2eikx i a cos k y b)

(A.31)

(A.32)

Then, ho is calculated using Eqs. (A.31) and (A.32) as

ho = ho ho* = t (1 + 2e − ik x i a cos k y b)(1 + 2eikx i a cos k y b) = t 1 + 4 cos 2 k y b + 4 cos k x a cos k y b)

(A.33)

Thus, from Eqs. (A.30) and (A.33) the energy dispersion relation of a graphene sheet is obtained as

E (k ) = E (k x , k y ) = Eo ± ho = Eo + t 1 + 4 cos 2 k y b + 4 cos k x a cos k y b

(A.34)

For example, if we put kxa = 0, then Eq. (A.33) becomes E (k ) = Eo + t (1 + 2 cos k y b)

(A.35)

121 For a graphene sheet to be conductive, there should be no band gap. Now, let Eq.(A.35) be equal to zero and also, let’s set Eo (energy level of the Fermi level) to zero for mathematical convenience: t (1 + 2 cos k y b) = 0

(A.36)

Then, a point satisfying above condition is obtained as following

(k x a, k y b) = (0,

2π ) 3

(A.37)

Because we are interested in the electrical characteristics of a graphene sheet around the Fermi energy, it is useful to make an approximation that describes the regions of the energy dispersion plot around E=0. This can be done by taking a Taylor series expansion of Eq.(A.30) at the point (k x a, k y b) = (0,

2π ) where the energy gap is zero. First, the 3

Taylor series expansion of Eq. (A.31) is

ho (k x , k y ) ≈ 0 +

∂ho ∂h 2π (k x ) + o (k y − ) ∂k x ∂k y 3b

(A.38)

where, ∂ho 1 = 2tia 2eikx a cos k y b = 2tia (− ) = −ita ∂k x 2

(A.39)

∂ho = −2tb 2eikx a sin k y b = −tb 3 ∂k y

(A.40)

Note that (k x a, k y b) is regarded as (k x , k y ) . Eq. (A.41) is obtained by simplifying Eq. (A.38).

122 ho (k x , k y ) ≈ −ita (k x ) − tb 3(k y −

2π ) 3b

2π 2π ⎞ ⎛ = −ita ( k x ) − ta( k y − ) = −ita ⎜ k x − i( k y − ) ⎟ 3b 3b ⎠ ⎝

(A.41)

Then, using Eq. (A.30) and Eq. (A.41), the energy dispersion relation for a graphene sheet near the point of Eq. (A.37) is obtained as:

2π ⎞ ⎛ E ≈ Eo ± ho (k ) = Eo + at k + ⎜ k y − ⎟ 3b ⎠ ⎝ 2 x

2

(A.42)

A.3. Band Structure of CNT Band structure of CNT can be considered to be same as that of a graphene sheet and this approximation is called the zone folding approximation. However, due to the chiral vector, when forming tubular shape, the wave vector (k x , k y ) can not be continuous and it should have discrete values. In this section, zig-zag and arm-chair CNT are analyzed and their energy dispersion relations are derived and compared to investigate the chirality dependence of band gap change. A.3.1. Band structure of Zig-zag CNT For a zig-zag CNT, the graphene sheet is rolled up in the y direction ( (n, 0) direction in Figure 1.5 and Figure A.3). Then ky values are restricted as k y (2bn) = 2π v

(A.43)

where b, n, and ν are integers. Note that chiral vector ( c ) is in the same direction with ky. Eq. (A.43) can be rewritten as

123 ky =

2π v 2bn

(A.44)

2

2π ⎞ ⎛ 2 Eq. (A.42) becomes Eq. (A.45) if ⎜ k y − ⎟ and Eo are substituted to kv and zero b 3 ⎝ ⎠ respectively for mathematical continence.

Ev = ± at k x2 + kv2

where kv =

(A.45)

2π v 2π 2π ⎛ 2n ⎞ − = ⎜v − ⎟ 2bn 3b 2bn ⎝ 3 ⎠

From Eq.(A.45), it is clear that when n is a multiple of three, kv = 0 which means that Ev = ± atk x . Then, there is no gap in the energy dispersion relation curve between the valence and conduction band and it has a continuous energy state resulting in metallic conduction properties. Complete band structure for the a zig-zag CNT having a chirality of (n, m) = (12, 0) is plotted in Figure A.4a [26, 27]. In Figure A.4a, kt/ktmax (= kx/kxmax) is a normalized axial wave vector. Note that because n is multiple of three, there is no gap at (kx = 0, E = 0). This CNT is called semimetal in this document. However, if n is not a multiple of three, Eq. (A.41) is become Ev = ± at k x2 + kv2 and it brings a gap between valence band and conduction band at kx is zero. Therefore the CNT becomes semiconducting. Complete band structure for a zig-zag CNT having a chirality of (n, m) = (13, 0) is plotted in Figure A.4b [26, 27]. In Figure A.4, kt/ktmax (= kx/kxmax) is a normalized axial wave vector. Note that there is a gap at (kx = 0, E = 0). This is a good example showing the dependence of electrical property of zig-zag CNT on chirality

124

(a) Metallic: (n, m) = (12,0)

(b) Semiconducting: (n, m) = (13,0)

Figure A.4. Band Structure of Zig-zag CNT

A.3.2. Band Structure of Armchair CNT For an armchair CNT, the graphene sheet is rolled up in x direction (see Figure 1.5 and Figure A.3). Then kx values are restricted to k x (2an) = 2π v

(A.46)

where a, n, and ν are integers. Note that (n, m) = (n, n). Then Eq. (A.41) can be expressed as Eq. (A.46) if kx and Eo are substituted to kv and zero respectively.

2π ⎞ ⎛ Ev = ± at kv2 + ⎜ k y − ⎟ 3b ⎠ ⎝

From Eq. (A.46), Eν can be always zero at ( kv = 0, k y =

2

(A.47)

2π ) and thus, an armchair CNT 3b

is always metallic. Complete band structure of two different armchair CNTs having a chirality of (n, m) = (7, 7) and (n, m) = (10, 10) are plotted in Figure A.5 [26, 27]. In

125 Figure A.5, kt/ktmax (= ky/kymax) is normalized axial wave vector. Note that there is no gap at ( k y = ±

2π , E = 0) for both cases. 3b

(a) (n, m) = (7, 7)

(b) (n, m) = (10, 10)

Figure A.5. Band Structure of Armchair CNT

126 Appendix B.

Unlike bulk materials, transport properties of a CNT do not obey the Ohm’s law and it has been reported that it shows the ballistic conduction [13, 35, 126]. In appendix B, the ballistic transport of CNT is summarized.

B.1. Transport Properties of CNT In ballistic conduction, electrical current conducts by ‘transmission’ character and it is known that the conductance becomes a property of the specific system rather than the material [13]. It means that there is no electron scattering over the length of the sample. This it can be stated more elaborately that ballistic conduction happens when the phase breaking length ( lφ ) of the electron states in the CNT is greater than the momentum relaxation length ( lm ) which in turn is much greater than the length of the CNT. B.1.1. Accessible Energy State and Ballistic Conduction For an electron to conduct it needs accessible energy states and the net current through the nanotube is the summation of the currents through the every accessible energy level between the Fermi energies of the conductive pads. Accessible energy states of CNT can be known by calculating density of states (DOS). DOS is the number of energy states per unit energy and it depends on energy dispersion relations [25]. DOS is be calculated as following using E-k relation of a CNT [35]

n( E ) =

1 1 dE where υ ( E ) = π υ (E) dk

(B.1)

where υ ( E ) is the group velocity for electrons. The DOS of a metallic (7, 7) CNT and a semiconducting (13, 0) CNT are shown in Figure B.1b and B.2b respectively with the

127 energy dispersion curves (Figure B.1a and B.2a). Note that the metallic CNT has a finite DOS at the Fermi energy (EF = 0) and the semiconducting CNT has a zero DOS at EF = 0 [27, 28]. Therefore, metallic CNT can be intrinsic ballistic conductor.

(a) Energy Dispersion Curve

(b) DOS

Figure B.1. Density of States (DOS) of Metallic (7, 7) CNT

(a) Energy Dispersion Curve

(b) DOS

Figure B.2. Density of States (DOS) of Semiconducting (13, 0) CNT

128 This situation is shown in Figure B.3 and Figure B.4. It is assumed that a metallic armchair CNT is contacted with two conductive pads having a chemical potential μ 1 (Figure B.3). In Figure B.4, the ballistic transport of an electron through the accessible energy state of the CNT is illustrated. Note that there is only one available energy state (solid line in Figure B.4) which the current can flow through between μ 1 and μ 2 . The chemical potential change can be expressed as μ 1− eVapp (= μ 2 ) where e is the charge of an electron and this comes from the externally applied voltage. The accessible energy levels are often called ‘open conduction channels’ and it is known as a function of energy, N(E) [13].

Figure B.3. Metallic CNT Contacted with Conductive Pads

Figure B.4. Ballistic Transport of CNT through One Channel

129 The separation in energy ( Δε ) (see Figure B.4) between energy bands can be related to the chirality and diameter of CNT. Following estimations are obtained by Egger et al [127] and Sattler et al [128]

Δε m = 3to

ac −c d

(B.2)

Δε s = 2to

ac −c d

(B.3)

where to is the hopping parameter, ac-c is the distance between carbon atoms and d is the diameter of CNT. These expressions are useful for estimating what values are expected to open a conduction channel. Note that in case of MWCNT, there can be numerous concentric walls and the number of conducting walls should be considered. Also, it is reported that there are two bands crossing the Fermi level in the Brillouin zone for a CNT and this number is referred to as the number of modes per sub-band, M and is reported as two in the literatures [129, 130, 131]. However, recent experimental results [132, 28] suggest that M is one and in the next section, it follows these results. Note that in Eq. (B.12), M is chosen to be one [13]. B.1.2. Typical I vs V Curve for Ballistic Conduction The derivation of quantum conductance expression followed the work of E.D. Graugnard [13]. The total current due to the flow of electrons with energy E between μ 1 and μ 2 with the velocity υ ( E ) > 0 in Figure B.4 is written as [35, 29, 133]. I = eM ∫ n( E )υ ( E )Τ( E ) N ( E )[ f ( E −μ 1) − f ( E − μ 2 )]dE

(B.4)

130 where Τ( E ) is the transmission probability of an electron passing from one contact pad to the other pad through the open channels of the CNT, N(E) is the number of conduction channels, and f ( E − μ i ) where i = 1, 2 is the Fermi-Dirac distribution function with qusi-Fermi energy μ i . The product of the density of states ( n( E ) ) and the velocity ( υ ( E ) ) in Eq. (B.5) is calculated to be a constant from Eq. (B.1).

n( E )υ ( E ) =

1 1 1 ⋅υ ( E ) = = h π υ (E) π

(B.5)

Combining Eq. (B.5) and Eq. (B.4) following expression is obtained after putting the constants outside of the integral

I=

The pre-factor

2e M ∫ Τ( E ) N ( E )[ f ( E −μ 1) − f ( E − μ 2 )]dE h

(B.6)

e is the quantized current per conduction channel per energy, and the two h

accounts for the spin degeneracy of the electron [134]. If Τ( E ) and N ( E ) are assumed to be constants N and T, over the range of μ 1< E < μ 2 , then Eq. (B.6) can be written as:

I=

N N 2e 2e 2 ( μ 1 − μ 2 ) ( μ 1− μ 2 ) M ∑ Τ j = M ∑Τj h h e 0 0

(B.7)

where the difference in the Fermi-Dirac functions is approximated as the difference in the electrochemical potentials of the contact pads. The transmission probability for an electron to transmit through the jth channel is defined as Tj, which is summed over the number of open channels. The applied voltage Vapp can be expressed as following:

Vapp = V =

( μ 1− μ 2 ) e

(B.8)

131 Combining Eqs. (B.7) and (B.8), following equation is obtained

I=

N N 2e 2 2e 2 Vapp M ∑ Τ j = GoVapp M ∑ Τ j , where Go ≡ h h 0 0

77.5 μ S

(B.9)

where Go is the quantum of conductance. For small bias (Vgate in Figure B.3), G can be identified from the law, I = GV and the total conductance, G can be expressed as N

G = Go M ∑ Τ j

(B.10)

0

If ballistic conduction conditions described in previous section are satisfied, then the transmission coefficients Tj in Eq. (B.10) are close to one. Eq. (B.10) become as following. Note that the conductance becomes quantized in units of Go MN [135, 136] N

G = Go M ∑ Τ j

Go MN

(B.11)

0

As the applied bias ( Vgate ) is increased, more conduction channels become available. Thus N can be increased with bias. If this is expressed as N (Vgate ) and M is substituted by one, I (Vgate ) for a ballistic conductor is given by I (Vgate ) = G (Vgate ) ⋅ Vapp = Go N (Vgate ) ⋅ Vapp

(B.12)

Example of I vs V plot is illustrated using Eq. (B.12). Note that whenever new channel is open, there is quantum jump in current (Figure B.5).

132

Figure B.5. Electrical Conductance Increase as New Channels Open

133 Appendix C.

C.1. Model by Yang et al for Band Gap Change Due to Strain The model developed by Yang et al [27] can be used to calculate the complete band structure and it has been widely used. For example, this model is the backbone of the simulation code ‘CNT bands 2.0’ for generating E-k relations of CNT [26]. This model does not show the band gap change directly. However, manipulate the hopping parameters in the final E-k relations under deformation the band gap generated can be observed. C.1.1. Dispersion Relation of Graphene Sheet under Deformation The band structure of the distorted graphene sheet can be calculated as following by modifying the undistorted E-k relation derived by Wallace [31] E (k ) = (t12 + t22 + t32 + 2t1t2 cos[k i(r1 − r2 )] + 2t2t3 cos[k i(r2 − r3 )]

(C.1)

+ 2t3t1 cos[k i(r3 − r1 )])1/ 2

where k = kc cˆ + kt tˆ . The change in bond vectors can affects the hopping parameters ti (i=1, 2, 3) between carbon atoms and also, between the lattice vectors. The relationship between the bond length and the hopping parameter is known as ti = to (ro / ri ) 2 [137], where to and ro are the hopping parameter and bond length of an unstrained graphene sheet. The value of to is around 3 eV. C.1.2. Modification of Bond Vectors Using Geometric Relations

134 The axes of the chirality-dependent coordinate system are depicted in Figure C.1. The cˆ axis corresponding to (n,m) CNT are the line joining the (0,0) and (n,m) carbon atoms,

and tˆ axis is perpendicular to cˆ axis. Also, the global x-y coordinate system is indicated in Figure C.1. The chirality-dependent coordinate system can be regarded as a coordinate system rotated by θ with respect to global x-y. Note that angle θ is different from the chiral angle (α) defined in Figure 1.5. The global x-y coordinate system and chiralitydependent coordinate system are related by

cˆ = cos θ xˆ + sin θ yˆ

(C.2)

tˆ = − sin θ xˆ + cos θ yˆ

(C.3)

1 n−m sin θ = ( ) 2 ch

(C.4)

where

3 n+m ( ) 2 ch

(C.5)

ch = n 2 + m 2 + nm

(C.6)

cos θ =

135

Figure C.1. Chirality-Dependent Coordinate System

The bond vectors can be represented using Eqs. (C.2) and (C.3) as

r1 =

ao n + m a n−m ˆ cˆ − o t + δ r1 2 ch 2 3 ch

(C.7)

ao m a 2n + m ˆ cˆ + o t + δ r2 2 ch 2 3 ch

(C.8)

r2 = −

r3 = −(r1 + r2 )

(C.9)

where ao is the equilibrium lattice vector length ( a1 = a2 ), δ ri (i=1, 2, 3) represents deviation from an undistorted graphene sheet. The application of a uniaxial or torsional strain can cause the change in bond vectors as following rit = (1 + ε t )rit ( Axial Direction)

(C.10)

ric = (1 + ε c )ric (Circumferential Direction)

(C.11)

136

ric = ric + tan(γ )rit (Torsion)

(C.12)

where ε t , ε c are the strain along t and c respectively and γ is the shear strain. Eqs. (C.10) and (C.11) represent the effect of uniaxial tensile strain and Eq.(C.12) represent the effect of torsional strain. Using Eqs. (C.10) – (C.12), the lattice vectors of the distorted graphene sheet can be written as ⎛ 1 2n + m 3 m⎞ 3mˆ a1 = r1 − r3 = ao ⎜⎜ (1 + ε c ) t + tan(γ ) ⎟⎟ cˆ + ao (1 + ε t ) 2 ch 2 ch ⎠ 2 ch ⎝

(C.13)

⎛ 1 n + 2m 3 n⎞ 3 n ˆ a2 = r1 − r2 = ao ⎜⎜ (1 + ε c ) t − tan(γ ) ⎟⎟ cˆ − ao (1 + ε t ) 2 ch 2 ch ⎠ 2 ch ⎝

(C.14)

Recall that in the real-space, the displacement vectors are represented as [117] r = j1a1 + j2 a2

(C.15)

where j1 and j2 are arbitrary integers. The one dimensional (1D) unit cell displacement (T) is the shortest rt for which rc=0. It means that two lattice points r = 0 and r = j1a1 + j2 a2 have the same cˆ coordinate. This requires that the following conditions on j1 and j2 as (1 + ε c )[ j1 (2n + m) + j2 (n + 2m)] + tan(γ ) 3[ j1m − j2 n] = 0

(C.16)

Now, new 1D unit cell displacement is calculated as

T = ao (1 + ε t )

3 ( j1m − j2 n) 2 ch

(C.17)

137

If only uniaxial strain is present (γ=0), Eq. (C.16) is rewritten as (1 + ε c )[ j1 (2n + m) + j2 (n + 2m)] = 0

(C.18)

j1 (2n + m) + j2 (n + 2m) = 0

(C.19)

This means that

The smallest values of j1 and j2 satisfying Eq. (C.19) are as following

j1 =

( n + 2m) (2n + m) , j2 = − gcd(2n + m, n + 2m) gcd(2n + m, n + 2m)

(C.20)

Using Eqs. (C.17) and (C.20), the displacement (T) of 1D unit cell of an (n,m) CNT is calculated as

T=

ao ch (1 + ε t ) 3 gcd(2n + m, n + 2m)

(C.21)

Thus, under only uniaxial strain, the unit cell displacement is equal to (1 + ε t ) times the unstrained unit cell displacement. When there is only torsion strain, Eq. (C.16) can be written as j1 (2n + m) + j2 (n + 2m) + tan(γ ) 3( j1m − j2 n) = 0

(C.22)

For arbitrary values of γ, n, and m, this equation corresponds to a large T compared with only uniaxial case (Eq. (C.21)). For example, under torsion, the unit cell displacement T of an armchair tube can be much larger than ao depending on the value of γ.

138

C.1.3. Final Expression of Dispersion Relation of CNT under Deformation From Eq. (C.1) and the relations from the geometric analysis, final expressions of dispersion relation of CNT under deformation can be made. First, the circumference of the distorted sheet is expressed as (1 + ε c )ch ao

(C.23)

The wave function of the CNT is quantized around the circumference and so kc is given by kc (1 + ε c )ch ao = 2π q

(C.24)

Where q is integer. Now, Eq. (C.23) can be written as

E (kt ) = (t12 + t22 + t32 + 2t1t2 cos[π q

n + 2m 3 n ' 3 tan(γ ) n kt ao − q − ] 2 2 ch 1 + ε c ch2 ch

+ 2t1t3 cos[π q

2n + m 3m ' 3 tan(γ ) n kt ao + π q + ] 2 ch 2 ch 1 + ε c ch2

+ 2t2t3 cos[π q

n−m 3 n+m ' 3 tan(γ ) n + m 1/ 2 kt ao + π q + ]) 2 ch ch2 2 ch 1+ εc

(C.25)

where, kt' = (1 + ε t )kt . For only uniaxial strain (γ=0), the range of kt is given by −π / T < kt < π / T , where T is the 1D lattice vector displacement calculated by Eq. (C.21). When there is a strain the hopping parameters will be different and this is the main source of band gap change. The band gap is generated for a zig-zag CNT (12, 0) under axial strain (0.05) and the band structures both without applied strain and with applied stain case are shown in Figure C.2 and Figure C.3 [26, 27].

139

Figure C.2. Band Structure without Axial Strain: CNT (12, 0)

Figure C.3. Band Structure with Axial Strain: CNT (12, 0)

140 Appendix D.

D.1. Model by Kleiner et al for Band Gap Change Due to Strain In their model, Kleiner et al [30] considered the effects of both the intrinsic curvature of CNT and the deformations on metallic CNT [32]. Especially, model of Kleiner et al can predict the band gap change directly and is simple in its form. Thus, it can be used as a good guide to predict the band gap change of CNT induced by deformation. They approach the problem in terms of a symmetry breaking in the tight binding model. In a tight binding model, a graphene sheet has separation between bonding and antibonding bands as 3

2to ∑ eik ⋅ri

(D.1)

i =1

where k is the wave vector, ri (i = 1, 2, 3) is the bond vectors (see Figure C.1), to is the hopping parameter which is the nearest neighbor Hamiltonian matrix element. Now, the symmetry of carbon atoms forming hexagons in graphene sheet is broken due to the externally applied strain and hopping parameters ti depending on the direction of the bonds ri should be changed. Then, the equation for k vector is obtained as following and defines the points of zero band gap in k space as following: 3

∑t e i =1

i

ik ⋅ri

=0

(D.2)

Note that small changes in hopping parameters ( ti = to + δ ti ) can shift the band crossing location as k F' = k F + Δk where k F' is the perturbed Fermi points and k F is the unperturbed Fermi points. It is known that for metallic CNT, the Fermi points can be written using Eq. (D.2) as following

141 3

∑e

ik F ⋅ri

=0

(D.3)

i =1

Because in this section, we are using the same CNT coordinate system as shown in the appendix C (see Figure C.1), bond vectors can be represented as following

r1 =

ao ⎛ 1 ⎞ (n + m)cˆ − (n − m)tˆ ⎟ ⎜ 2ch ⎝ 3 ⎠

(D.4)

ao ⎛ 1 ⎞ −mcˆ + (2n + m)tˆ ⎟ ⎜ 2ch ⎝ 3 ⎠

(D.5)

ao ⎛ 1 ⎞ − ncˆ − (n + 2m)tˆ ⎟ ⎜ 2ch ⎝ 3 ⎠

(D.6)

r2 =

r3 = −

where ao is the length of the unit vectors and ch = n 2 + m 2 + nm is the circumference of a CNT. By combining Eq. (D.3) and Eqs. (D.4) – (D.6), the original Fermi point of graphene sheet can be written as

kF =

2π (m + 2n)kˆc + m 3kˆt 3ao ch

(

)

(D.7)

where kc and kt are corresponding to k along circumferential and axial directions, respectively. To find a new band gap is same as to find the new Fermi point nearest quantized circumferential kc values and the distance is given by ∆kc. By expanding Eq. (D.2) to linear order in the perturbations, ∆kc can be expressed as

Δk c =

2π (δ t1 (m − n) + δ t2 (2n + m) − δ t3 (n + 2m) ) ao ch to 3

(D.8)

142 Note that the distance of the quantization lines in Eq. (D.8) is not affected by the change of bond vectors, but only by the change in the hopping parameters. Also, the band gap is obtained close to the Fermi point. The dispersion relation near the Fermi point can approximated as linear. Then the band gap can be expressed as Eg = 3ao to Δkc

(D.9)

Generally, the change of the nearest-neighbor hopping parameters is expressed in terms of a bond deviation matrix D as

δ ti = ri Dri / ri 2

(D.10)

This deviation matrix scheme is useful to describing the effects of curvature and stain on the hopping parameter change. D.1.1. Band Gap Change Induced by Intrinsic Curvature of CNT Let us consider the change in hopping parameter ( δ ti ) due to intrinsic curvature of CNT. The hopping parameters are proportional to cos φ where φ is the misalignment angle between two neighboring orbitals.

1⎛ a ⎞ cos φ ≈ 1 − ⎜ c −c ⎟ 8⎝ R ⎠

2

(D.11)

where R (= aoCh/2π) is the CNT’s radius and ac-c is the carbon-carbon bond length (= ao / 3 ). Then the deformation tensor for the intrinsic curvature of CNT is expressed in the basis of cˆ and tˆ as

143

D

curve

⎛ to ⎛ ac −c ⎞ 2 ⎜ = ⎜ 8 ⎜⎝ R ⎟⎠ ⎜ 0 ⎝

⎞ 0⎟ ⎟ 0 ⎟⎠

(D.12)

Note that the bond deviation matrix D due to intrinsic curvature of CNT is indicated here as Dcurve and corresponding band gap as Egcurve (see Eq. (D.13)). By combining Eqs. (D.8) - (D.10) and Eq. (D.12), the band gap change due to intrinsic curvature of CNT is written as

Egcurve =

toπ 2 (n − m)(2n 2 + 5nm + 2m 2 ) 5 8ch

(D.13)

D.1.2. Band Gap Change Induced by Externally Applied Strain Let us consider the change in hopping parameter ( δ ti ) due to deformation of CNT. A 2D stain tensor S on the CNT surface can be written as ⎛ε S ≡⎜ c ⎝0

γ⎞ ε t ⎟⎠

(D.14)

where ε c is the uniform circumferential strain, ε t is the uniform axial strain and γ is the torsional strain. The bond deviation matrix D is represented here as Ddeform and corresponding band gap as Egdeform (see Eq. (D.16)). D deform = r bS

(D.15)

where r is the change in the magnitude of bond length(= ao / 3 ), b represents the linear change in hopping parameters (3.5 eV/Å) and S is the strain tensor. Note that hopping parameter change is calculated as ti = ti + b ri . Following similar procedure

144 shown in intrinsic curvature case, the band gap due to deformation of CNT can be calculated as

Egdeform =

aob 4ch3

3(n − m)(2n 2 + 5nm + 2m 2 )(ε c − ε t ) − 9nm(n + m)γ

(D.16)

Actually, Eq. (D.16) represents a band gap change due to a 2D linear deformation within the graphene sheet and this calculation is based upon zone folding approximation. Note that when equal strain in both c and t direction is applied, no band gap change is induced. Zig-zag CNT ((n, m) = (n, 0)) has the maximum sensitivity for uniaxial strain ( ε c and ε t ) but insensitive to torsion ( γ ). In case of armchair CNT ((n, m) = (n, n)), the opposite is true and thus, torsion is the only possible way to change band gap of armchair CNT. D.1.3. Total Band Gap Change of CNT The total band gap is expressed by summing up Eq. (D.13) and Eq. (D.16) as

⎛t π2 a b 3 ⎞ 9a b Egtotal = ⎜ o 5 + o 3 (ε c − ε t ) ⎟ 3(n − m)(2n 2 + 5nm + 2m 2 ) − o3 nm(n + m)γ ⎜ 8c ⎟ c 4 4ch h ⎝ h ⎠

(D.17)

Eq. (D.17) can be rewritten using R (radius of CNT) and θ (see Figure C.1) as following

⎛ t π2 a b 3 ⎞ ab 3 γ cos 3θ Egtotal = ⎜⎜ o 2 + o (ε c − ε t ) ⎟⎟ sin 3θ − o 16 R 2 2 ⎝ ⎠

(D.18)

As an example, consider CNTs having a diameter around 1.2 nm but a different chirality. Then, from Eq. (D.18), it can be shown that as the chirality changes (as θ is increases), band gap change becomes larger (Figure D.1). Also, the band gap change of a CNT due to small torsional strain (0.01) is shown in Figure D.2. The presence of small torsional

145 strain makes a huge difference in band gap change at θ is zero (armchair CNT). Band gap of armchair CNT (7, 7) is calculated according to applied torsional strain and plotted in Figure D.3. Therefore, even for metallic CNTs, it is possible to change the electrical resistance by applying torsional strain.

Figure D.1. Chirality-Dependent Band Gap of CNTs: without Torsional Strain

Figure D.2. Chirality-Dependent Band Gap of CNTs with Torsional Strain

146

Figure D.3. Band Gap of Metallic CNT (7, 7) with Torsional Strain

147

VITA

Myounggu Park University of Michigan, Department of Mechanical Engineering, 2250 GG Brown, 2350 Hayward, Ann Arbor, MI 48109-2125, USA Email: [email protected] or [email protected]

EDUCATION ■ PH.D., AERONAUTICS AND ASTRONAUTICS, PURDUE UNIVERSITY, AUG., 2007 ■ M.S., METALLURGICAL ENGINEERING, INHA UNIVERSITY, FEB., 1997 ■ B.S., METALLURGICAL ENGINEERING, INHA UNIVERSITY, FEB., 1995

WORK EXPERIENCE ■ POST-DOCTORAL RESEARCH FELLOW Department of Mechanical Engineering, University of Michigan Ann Arbor, 2007 - Present ■ RESEARCH ASSISTANT School of Aeronautics and Astronautics, Purdue University West Lafayette, 2003-2007 ■ TEACHING ASSISTANT School of Aeronautics and Astronautics, Purdue University West Lafayette, Spring, 2005 ■ FAILURE ANALYST Aero-technology Research Institute, Republic of Korea Air Force, 1998-2003 ■ RESEARCH ASSISTANT Department of Metallurgical Engineering, Inha University, 1995-1997

LIST OF REFERENCES

APPENDICES

VITA

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