Modeling and simulation of buckling of pristine and defective carbon nanotubes

CHAPTER 5: BUCKLING OF EMBEDDED CNTS 92 5.4 Continuum mechanics modeling of embedded CNTs 108 5.5.4 Buckling of embedded CNTs of various diameters 121 5.5.5 Buckling of embedded CNTs of

MODELING AND SIMULATION OF BUCKLING OF PRISTINE AND DEFECTIVE CARBON NANOTUBES

D D THANUJA KRISHANTHI KULATHUNGA

(B.Eng.(Hons.), University of Moratuwa, Sri Lanka)

A THESIS SUBMITTED

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF CIVIL ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2011

It is an honor for me to be guided by my co-supervisor, Professor J N Reddy, Department of Mechanical Engineering, Texas A&M University, USA I’m grateful to him for his constructive suggestions

I appreciate the financial support and the facilities provided by National University of Singapore to carry out this research

It’s a pleasure to thank my colleagues, Ms Rupika Swarnamala, Ms Wang Shasha, Dr Anastasia Maria Santoso, Ms Zhang Sufen, Dr Sai Sudha Ramesh, Dr Liu Xuemei, Dr Nguyen Dat, Dr Lado R Chandra and Dr Patria Kusumaningrum for their kind and encouraging words

Last but not least I would like to thank my parents, brothers, sister and my husband for being the best support to hurdle all the obstacles throughout this research work

2.1.2 Techniques employed in the buckling analysis 24

2.2.3 Rehybridization defect (sp3 interwall bridging) 34

CHAPTER 3: MOLECULAR DYNAMICS SIMULATIONS 37

3.1 Introduction to molecular dynamics simulations 37

3.2 Simulation procedure for CNT under axial compression 43

4.4 Analytical solution for buckling strain of SWNTs 69

4.5 Analytical solution for buckling strain of DWNTs 73

CHAPTER 5: BUCKLING OF EMBEDDED CNTS 92

5.4 Continuum mechanics modeling of embedded CNTs 108

5.5.4 Buckling of embedded CNTs of various diameters 121 5.5.5 Buckling of embedded CNTs of various lengths 125

6.3.1 Effect of vacancy defects on buckling properties of SWNTs 135 6.3.2 Effect of vacancy defects on buckling of DWNTs 149

6.3.3 Effect of vacancy defects on buckled shape of CNTs 151

SUMMARY

Carbon nanotube (CNT) is one of several nanomaterials that has attracted enormous attention of researchers within the last two decades Among the research focuses of CNTs, the buckling behavior of CNT has taken an important place in view that buckling is a major mode of structural instability of CNTs owing to their hollow tubular nature and high aspect ratio Despite the high number of studies conducted on the buckling of CNTs, there are still several issues that are not addressed sufficiently in the literature For example, there appear to be little work carried out in investigating the buckling of embedded CNTs and defective CNTs Moreover, there exist discrepancies between the results obtained from various modeling techniques and these discrepancies have not been subjected to sufficient discussion The objective of this thesis is therefore to investigate the buckling of freestanding pristine and defective CNTs as well as embedded pristine CNTs Molecular dynamics simulation (MDS) technique is employed as the main modeling tool while analytical method based on continuum mechanics is also employed wherever possible

The scope of work carried out in this study can be divided mainly into three parts In the first part of the study, improved analytical formulae are proposed for the buckling strain of single and double-walled carbon nanotubes based on Sanders and first-order shell theories It is noticed that the buckling strain values computed from existing formula in the literature based on the Donnell shell theory do not show sensitivity to aspect ratio (length/diameter ratio) of CNT The lack of sensitivity is

contrary to the results obtained from MDS where buckling strains actually show considerable sensitivity to the aspect ratio of CNT In the derivation of this widely employed formula, certain terms have been omitted assuming that the shell buckles into a large number of longitudinal waves However, MDS results suggest this assumption is not reasonable for the buckling of CNT To avoid this erroneous result, no such simplification was made in the derivation of the analytical formulae proposed in this study In addition, the proposed formula based on the first-order shell theory accounts for the effect of transverse shear deformation, which could be high in CNTs with small diameter and low aspect ratios As a result, the proposed formulae appear to generate results with improved accuracy compared to the widely employed formula existing in the literature

The second part of the thesis is focused on the buckling of embedded CNTs MDS based studies on buckling of embedded CNTs are found to be lacking in the literature As a result, the accuracy of the continuum mechanics models employed in analyzing buckling of embedded CNTs is not sufficiently verified Buckling of fibers has been identified as a major mode of failure of composites in general This scenario has been observed in CNT based composites as well It would therefore be important to investigate the buckling properties of embedded CNTs Detailed molecular dynamics study is thus carried out in this thesis to investigate the buckling

of pristine embedded CNTs In addition, the accuracy of the continuum mechanics based analytical models in predicting buckling properties of embedded CNTs is discussed The analytical models employed in this study are well-known equations for beam or shell on an elastic foundation in which the elastic foundation is modeled

as a Winkler foundation In addition to these well-known formulae, the first-order shell theory based formula proposed in the first part of the study combined with the

Winkler model is also proposed to predict the buckling properties of embedded CNTs Of the three formulae considered in this study, the proposed formula is found to produce the most accurate results irrespective of the buckling mode However, even the results obtained from the proposed formula appear to deviate considerably from the results obtained from molecular dynamics simulations

It appears from a review of the literature that the buckling of defective CNTs

is also not sufficiently examined Despite the use of very high quality techniques in the production of CNTs, it has been reported that CNTs are still being produced with defects Defects are found to create significant effect on tensile properties of CNTs Hence, it is important to know the effect of defects on the buckling properties of CNTs The third part of this thesis presents the investigation on the buckling properties of defective freestanding SWNTs and DWNTs using MDS Several types

of defects have been identified in CNTs Among them, non-reconstructed vacancy defects are the type of defects that cause the highest degradation of buckling properties Therefore in this study, various configurations of non-reconstructed vacancy defects are investigated to identify the severity of degradation of buckling properties expected in defective CNTs The effect of defects on buckling properties under various thermal environments is also examined

Figure 3.1: Graphical representation of bond energies 40

Figure 3.3: Critical buckling strains obtained using different thermostats and

Figure 3.4: Convergence study on duration of simulation 54 Figure 4.1: Deformation of a transverse normal according to various shell theories 60

Figure 4.3: Critical buckling strain versus aspect ratio for SWNTs 82 Figure 4.4: Critical buckling strain of SWNTs with varying diameters 85

Figure 4.6: Critical buckling strain of DWNT (4, 4) (9, 9) 88 Figure 4.7: Critical buckling strain of DWNTs at aspect ratio 6 89

Figure 5.2: Schematic representation of different polymer matrices 97

Figure 5.3: Schematic representation of periodic boundary condition 98

Figure 5.5: Schematic representation of vdW separation distance 101

Figure 5.7: Deformation of embedded CNT during equilibration 111

Figure 5.9: Variation of buckling stress with the increasing volume fraction 118 Figure 5.10: VdW energy at different volume fractions 119 Figure 5.11: Variation of number of matrix atoms per unit volume with volume

Figure 5.12: Buckling properties of CNTs with varying diameters 122 Figure 5.13: Buckled shapes of CNTs with varying diameters 123 Figure 5.14: Variation of buckling properties with varying lengths 126 Figure 5.15: Buckled shapes of (5, 5) CNTs with varying lengths 128 Figure 6.1: Various configurations of vacancy clusters 137 Figure 6.2: Force-displacement curves for pristine and defective (7, 7) CNTs 139 Figure 6.3: Force-displacement curves for symmetric and asymmetric vacancies 140 Figure 6.4: Force-displacement curves for pristine and defective CNTs 142 Figure 6.5: Load-displacement curves for one or two vacancy clusters 143 Figure 6.6: Dangling bonds in armchair and zigzag CNTs 146 Figure 6.7: Effect of temperature on pristine and defective CNTs 147 Figure 6.8: Buckling mode shapes of pristine and defective SWNTs 152 Figure 6.9: Buckling mode shapes for pristine and defective DWNT 153

LIST OF TABLES

Table 3.1: Functional forms of energy components included in COMPASS force

Table 4.3: Error percentages correspond to the analytical formulae 84Table 5.1: Buckling properties for embedded SWNT with periodic and non-periodic

Table 5.2: Buckling properties of embedded SWNT with periodic and non-periodic

Table 6.1: Percentage reduction of buckling properties due to vacancy defect 138Table 6.2: Effect of chirality on buckling of defective CNTs 145

Force acting on ith atom

V Total potential energy of the system

u 0 , v 0 , w 0 Displacements of a point in the middle plane of the shell in x,

N Axial compressive load per unit circumferential length

R i Radius of the ith wall

h Thickness of the shell

c i Van derWaals coefficients of ith wall

CNT

h vdw Van der Waals separation distance between CNT and matrix

0

K modified Bessel function of the second kind

to three other carbon atoms via covalent bonds Therefore, CNT can be viewed as a graphene sheet rolled up to the shape of a cylinder Depending on the geometry of CNT, the following categories of CNT can be found

Single-walled and multi-walled carbon nanotubes

CNTs as single-walled carbon nanotubes (SWNTs) or multi-walled carbon nanotubes (MWNTs) As the names imply, SWNT consists of a single CNT whereas MWNT contains several CNTs inserted within one another with an inter-wall spacing of approximately 0.34 nm (Kiang et al., 1998) Walls of MWNTs are bonded by the weak van der Waals forces which are defined as attractive or repulsive forces between atoms or molecules

Zigzag, armchair and chiral carbon nanotubes

CNTs are also categorized as chiral or achiral carbon nanotubes The latter comprises of both zigzag and armchair carbon nanotubes This categorization is done based on the orientation of carbon-carbon (C-C) bonds Orientation of C-C bonds is characterized by the chiral vector, C
h =ma1+na2

where a
1 and a
2

are the base vectors of magnitude 1.42 A0 and the integers m and n are the translation

indices (see Figure 1.1) Translation indices are also called chiral indices The chiral angle is defined as the angleθ between C
h and a1

and it varies from 00 to 300 Nanotubes having chiral angle of 00 and 300 are called zigzag and armchair, respectively, while nanotubes having chiral angles between 00 and 300 are called chiral CNTs SWNTs are uniquely defined by the integer pair (n,m) or equivalently

the corresponding chiral angle Zigzag and armchair CNTs are represented by the integer pairs (n, 0) and (n, n) respectively, while chiral CNTs are represented by

(n,m) Chiral indices n and m take values greater than or equal to 3 except for zigzag

CNTs where m takes the value of zero The diameter (in nm) and chiral angle of a

SWNT can be expressed in terms of chiral indices as shown in equations 1.1 and 1.2, respectively

Figure 1.1: Graphical representation of chirality of CNTs

Pristine and defective carbon nanotubes

It is now well known that CNTs are susceptible to defects (Ebbesen and Takada, 1995, Yuwei et al., 2005) This leads to another categorization of CNTs as either pristine or defective CNTs Defects can occur in CNTs during production or purification processes Some defects are produced in an excessive temperature environment or under higher stresses Also, defects can be deliberately introduced

by chemical treatment or irradiation (exposing to radiation) For example, in developing field-effect transistors, pentagon-heptagon defects are introduced deliberately to join two CNTs with different chiralities through intramolecular junctions Defects produced through all these means can be categorized into three main groups They are topological defects, rehybridization defects and incomplete bonding defects Topological defects occur due to the presence of carbon rings other

than hexagons Incomplete bonding defects refer to the vacancy defects which are produced due to missing carbon atoms Rehybridization defects are produced from the presence of sp3 bonds in CNT Normally, CNTs consist of sp2 carbon bonds where each carbon atom is bonded to 3 other carbon atoms Under certain circumstances, sp2 bonds can be turned into sp3 bonds where each carbon atom is bonded to 4 other carbon atoms

1.1.2 Mechanical properties of carbon nanotubes

Carbon nanotube is known to possess superior mechanical properties These properties have spurred considerable interest amongst researchers and as a result many experimental and theoretical studies have been performed so far to explore these remarkable material properties Most of the experimental based studies suggest the longitudinal Young’s modulus of CNT to be about 1 TPa (Krishnan et al., 1998, Wong et al., 1997, Dresselhaus et al., 2004, Hernandez et al., 1999) This value of Young’s modulus is five times larger than the Young’s modulus of steel However, some theoretical studies suggested the Young’s modulus of CNT to be about 5 TPa For example, Yakobson et al (1996) suggested a value of 5.5 TPa for the Young’s modulus of CNTs based on the bending stiffness value of CNTs Here, they considered the CNT thickness to be 0.066 nm Similarly, Zhou et al (2000) also obtained 5.1 TPa for the Young’s modulus and 0.074 nm for the thickness using bending strain energy obtained from electronic energy band theory In contrast, Giannopoulos et al (2008), Jin and Yuan (2003) and Chunyu and Tsu-Wei (2003) obtained Young’s modulus of about 1 TPa from their theoretical investigations However, in these studies the thickness of CNT is taken as 0.34 nm Therefore, it is clear that theoretically calculated Young’s modulus value depends on the value of

the thickness of CNT assumed Unfortunately there appears to be no agreement in the literature what the actual value of thickness should be which could not be resolved since measurement of the thickness through experiments is not easy to carry out

In addition to the remarkable Young’s modulus, CNT is known to possess over a hundred times higher tensile strength compared to that of steel Although various studies (Li et al., 2000, Yu et al., 2000a, Yu et al., 2000b, Demczyk et al., 2002) suggested various values for the tensile strength of CNT, the mostly accepted value is about 60 GPa

1.1.3 Applications of carbon nanotubes

Due to the exceptional mechanical properties and the light weight, scientists envisaged many potential applications of CNTs Some of those potential applications have now become real-world applications The followings are some of the real-world as well as potential applications of CNTs

Carbon nanotube based probe tips

This is one of the real-world applications of CNTs up to date CNT probes have been shown to improve image resolution as well as durability of AFM (atomic force microscope) tips compared to the conventional Silicon tips This high resolution provided by CNT probes enable AFM to produce better imaging of surfaces as well as biological species such as DNA

As reinforcing agents in nanocomposites

Due to the high stiffness, strength as well as high aspect ratio of CNTs compared to the conventional carbon fibers, researchers predicted CNTs as a good reinforcing agent in composites Many studies have already been devoted to investigate the applicability of CNT as reinforcing fibers in polymer, metal and ceramic matrices (Qian et al., 2000, Zhan et al., 2003) These CNT composites are expected to be useful in applications where high strength and light weight is important such as sporting goods, body armors, automobiles and aircrafts

Carbon nanotube sensors

Electrical properties of CNTs are highly sensitive to the atomic structure of CNT Owing to this characteristic, CNTs have been proposed for use as chemical or mechanical sensors (Inpil et al., 2006, Sinha et al., 2006, Endo et al., 2008)

Carbon nanotube brushes

With the invention of methods to develop aligned CNT arrays, CNT brush has also become a potential application of CNT Anyuan et al (2005) proposed aligned MWNTs as a feasible option for microscale brushes which can be used to clean nanoparticles from narrow spaces or to coat inside holes Also, Toth et al (2009) suggested CNT to be used as brushes in electric motors due to the flexibility and conductance of CNTs Figure 1.2 shows two CNT brushes constructed by Anyuan et al (2005) In these brushes, bristles are made of CNTs while the handles are made of Silicon Carbide

Figure 1.2: CNT brushes

Conductive plastics and films

Due to the higher aspect ratios and good electrical conductivity, CNTs have found applications in conductive plastics These conductive plastics are made by filling plastics with nanotubes In addition, carbon nanotubes are now tested to make conductive films which can be used in LCD’s and touch screens

1.2 Motivation

Understanding the mechanical properties and behavior of CNTs is crucial towards identifying potential applications of CNTs Furthermore, this knowledge is essential in designing CNT based devices Therefore, many researches are focused

on studying the mechanical behavior of CNT such as fracture, buckling and vibration of CNTs Owing to the hollow tubular nature, buckling has been identified

as a major mode of failure of CNTs In addition to the degradation of mechanical strength, buckling has been found to affect electrical and optical properties of CNTs

as well (Dekker, 1999, Shan and Bao, 2006) Therefore, buckling of CNTs has

on the buckling of CNTs However, there are still several areas which have not gained sufficient attention

One of such areas which have not been fully explored is to examine in depth whether continuum mechanics models can be used to predict well the buckling properties of CNTs These continuum models consist of analytical as well as numerical models Analytical models have been extensively explored due to their simplicity in application Even though the accuracy of the analytical expressions of some of the previous works has been verified for certain cases, it is noted that there are also some discrepancies between the results of many of these analytical models and the results obtained from MDS One shortcoming of the analytical shell models widely employed in the literature to calculate buckling strains of CNTs is the insensitivity to the aspect ratio of the CNT (length to diameter ratio) which is contrary to findings from MDS based studies which concluded that the buckling strain is quite sensitive to aspect ratio of CNTs even within the shell buckling region (Li and Chou, 2004, Liew et al., 2004, Sears and Batra, 2006, Wang et al., 2005, Zhang et al., 2009a) It is also noticed that as a result of this shortcoming, there is a significant error in the values calculated from existing shell models especially for CNTs with smaller diameters It is noted that the existing shell models used for the buckling analysis of CNTs are based on simplified Donnell’s Shell theory The simplification in this shell theory is based on the assumption that cylindrical shells buckles into a large number of longitudinal waves However, in contrast to this assumption, MDS based studies showed that CNTs displaying shell buckling behavior generally buckle only into one or two waves (Cao and Chen, 2006a, Shen and Zhang, 2006, Tan et al., 2007, Lu et al., 2008) Moreover, Donnell’s shell theory neglects the transverse shear which could be significant in tubes with smaller

diameters Therefore, it is likely that accounting for these two factors would improve the accuracy of the existing shell models

Another issue which has not gained sufficient attention is the buckling of embedded CNTs CNT has been identified as an excellent candidate as reinforcing fibers in nanocomposite Moreover, buckling of embedded CNTs has been observed experimentally and identified as a major mode of failure of composites (Lourie et al., 1998, Thostenson and Chou, 2004, Hadjiev et al., 2006) Hence, it is important

to know the buckling properties of CNTs embedded in matrices Having identified this importance, several studies have been devoted to the study of the buckling of embedded CNTs (Ru, 2001, Kitipornchai et al., 2005, Liew et al., 2005, Yao and Qiang, 2006, Yang and Wang, 2006, Zhang et al., 2006b) However, these studies are based on continuum mechanics modeling Moreover, these continuum mechanics models are not sufficiently verified due to the absence of MDS studies From a review of existing literature, there appears to be only one study carried out on the buckling of embedded CNTs which employs MDS (Namilae and Chandra, 2006) In this study, the matrix considered is a crystalline polymer employing non-periodic boundary conditions Commercial polymers are combination of crystalline and amorphous polymer and hence it is important to study the behavior of embedded CNT in amorphous polymer as well On the other hand, non-periodic boundary conditions may not represent the bulk nature in a composite Therefore, clearly there exists a necessity for more MDS based studies to investigate the buckling of embedded CNTs

The next issue which has not been studied enough is the buckling of defective CNTs Various types of defect has been found unavoidably in CNTs (Ebbesen and Takada, 1995, Charlier, 2002) Even with the use of high quality

production methods, it has been reported that there are defective CNTs still being produced Many studies have been focused on exploring the tensile properties and fracture of defective CNTs (Belytschko et al., 2002, Chandra et al., 2004, Sammalkorpi et al., 2004, Xiao and Hou, 2006, Wang et al., 2007c, Yuan et al.,

2007, Pozrikidis, 2009) These studies have discovered that defects do influence significantly the tensile properties of CNTs Comprehensive studies on the effect of defects on the buckling of CNTs are however very much lacking in the literature Even the few studies reported in the literature show contradicting results For example, Xin et al (2007) observed a significant effect of defects on the buckling properties of SWNTs while Wang et al (2008c) observed only a minor effect

Several methods have been developed to identify the type of defects in CNTs (Krasheninnikov and Nordlund, 2002, Hashimoto et al., 2004, Ishigami et al., 2004, Miyamoto et al., 2004, Yuwei et al., 2005) and more research efforts are currently in progress to develop advanced techniques in identifying the defects Therefore, it is possible to estimate the expected behaviour of a sample of CNTs under compression, once the defect density and the degradation of properties due to defects are known This could also be used as a measure to decide whether CNTs produced

by a certain technique is good enough for certain application Hence, exploring degradation of buckling properties caused by various types of defects can also be identified as an important research area

1.3 Objective and scope of work

It is clear from section 1.2 that there exist several gaps in the literature regarding understanding the buckling properties of CNTs despite the high number of studies conducted so far It is thus important to address these gaps through more studies The objective of this thesis is to investigate the buckling properties of pristine/defective freestanding CNTs and pristine embedded CNTs Molecular dynamics simulation (MDS) technique will be employed as the primary tool for the study supplemented with an analytical method based on continuum mechanics modeling of the CNT wherever possible

The scope of work carried out to achieve the objective of this study can be grouped into three parts In the first part of the study, improved analytical expressions for the buckling strain of pristine CNTs are obtained These expressions are derived from well-established continuum mechanics principles using both Sanders and First-order shell theories to model the buckling behavior of CNT It should be emphasized there that the Sanders shell theory solutions derived here are more accurate than existing Donnell shell theory solutions as Sanders shell theory is

a more refined thin shell theory First-order shell theory based solution accounts for the transverse shear deformation It is expected that these two factors contribute towards improving the accuracy of the proposed analytical shell model expressions for calculating the buckling strain and hence buckling stress of CNTs Analytical solutions are presented for the buckling of both SWNT and DWNTs Finally, the first part of the current study will present the results obtained from molecular dynamics simulation (MDS) using COMPASS forcefield to simulate the interaction

of atoms in the CNT Results from MDS will be used to verify the solutions generated from the proposed analytical expressions

In the second part of this study, the buckling of SWNTs embedded within a polymer matrix (embedded SWNT) is investigated using MDS The applicability of analytical models existing in the literature as well as the analytical model proposed

in the first part of the study, combined with Winkler model to account for the interaction between the matrix and CNT is explored The Winkler model has been widely adopted to account for the increment in buckling properties of the embedded CNT due to the interaction from the matrix In the Winkler model, the matrix is considered as an infinite elastic medium

In the third part of this study, attention will be paid to the investigation on the buckling of freestanding defective CNTs As mentioned previously, CNT is susceptible to several types of defects However, non-reconstructed vacancy defects can be expected to cause higher degradation in mechanical properties Therefore, the study will be focused only on CNTs with non-reconstructed vacancy defects Effects

of different configurations of vacancy defects, such as the number of missing atoms and the shape of defects, on the buckling properties of SWNTs and DWNTs are studied The cumulative effect of defects and temperature on SWNTs is also assessed

1.4 Layout of the thesis

The chapters in this thesis are organized as follows Chapter 2 presents a literature review on the buckling of CNTs and the various types of defects that can occur in CNTs Chapter 3 begins with a general introduction to MDS, the main modeling tool employed in this present work Also, this chapter explains the simulation procedure as well as the MDS parameters employed in the present study Details of the results from a parametric study carried out to select appropriate values for the MD parameters, such as thermostat, duration of simulation and displacement step, are also included in this chapter

In chapter 4, analytical expressions are derived for the buckling strains of pristine freestanding SWNTs and DWNTs based on Sanders and first-order shell theories Results obtained from these analytical expressions are verified against corresponding results obtained using MDS The accuracy of the derived analytical expressions is also discussed with respect to the results of available analytical expressions reported in the literature

In chapter 5, a detail analysis of pristine embedded SWNTs is presented where MDS technique is used to determine the buckling property of embedded SWNTs of various diameters and lengths In addition, the applicability of analytical continuum mechanics models in calculating the buckling properties of SWNTs is also discussed

Chapter 6 presents a MDS study on the buckling of defective freestanding CNTs Here, effects of different defect parameters, such as number of missing atoms, number of dangling bonds, asymmetry of defects and number of vacancy

the buckling of defective SWNTs is also discussed Finally, the effect of single vacancy defects on the buckling of DWNTs is discussed

Chapter 7 presents the concluding remarks and future research directions relevant to the present study

an important issue in the design of NEMS (Nanoelectromechanical Systems) and sensors where mechanical, optical and electrical properties of CNTs are equally important Owing to these reasons, many studies have been devoted to the study of buckling of CNTs This section presents a review of some previous work carried out

to examine the factors affecting buckling of CNTs and the various modeling techniques employed in investigating buckling of CNT

2.1.1 Factors affecting buckling

Studies conducted on the buckling of CNTs have identified several factors that affect the buckling properties as well as buckling modes of CNTs Some of these factors are described below

Dimensions

Dimensional factors such as aspect ratio and diameter are found to affect significantly the buckling properties of CNTs In general, it is known that as the aspect ratio of a cylinder increases, its buckling mode changes from shell to beam buckling This scenario has been reported to be common for CNTs as well For example, several studies including those performed by Sears and Batra (2006), Buehler et al (2004) and Wang and Mioduchowski (2007) reported that there exists a transition of buckling mode from shell-like buckling to beam-like buckling as the aspect ratio increases Moreover, Buehler et al (2004) reported wire-like buckling behavior in CNTs with extremely high aspect ratios Also, this shell to beam buckling transition has been reported in CNTs under different loading conditions such as axial compression (Buehler et al., 2004, Sears and Batra, 2006, Wang and Mioduchowski, 2007) and radial pressure (Wang and Mioduchowski, 2007) Furthermore, Buehler et al (2004) has identified 12.5 as the critical aspect ratio at which transition of shell to beam-like buckling occurs

In addition to the buckling mode transition, an increase in aspect ratio was found to decrease the buckling strain and load For example, MDS-based study conducted by Liew et al (2004) revealed a decrease in critical buckling strain of (10, 10) SWNTs as the aspect ratio increases The CNTs employed in their study are reported to have aspect ratios less than 10 CNTs having such low aspect ratios are

likely to undergo shell buckling Therefore, this is one example for buckling strain

of CNTs being sensitive to aspect ratio even within the shell buckling region Li and Chou (2004) also observed a decrease in critical buckling force for (3, 3) and (5, 0) SWNTs as the aspect ratio increases using molecular structural mechanics approach Sears and Batra (2006) studied several SWNTs as well as multi-walled carbon nanotubes (MWNTs) with wide range of aspect ratios and found that critical buckling strain decreases with the increase in aspect ratio within both shell and beam buckling ranges, however, the decrease of buckling strain was found to be higher within the beam buckling range compared to the shell buckling range Wang et al (2005) also observed the same trend for (10, 10) SWNT However, it is noted that this aspect ratio dependency of buckling strain is not captured by the existing continuum mechanics models for shell buckling

The magnitude of the diameter of CNTs is also found to have significant effect on the buckling properties of CNTs Several studies have concluded that for SWNTs, there exists an optimum diameter at which buckling load reaches a maximum (Liew et al., 2004, Wang et al., 2008a) However, this optimum diameter

is not captured by Li and Chou (2004) This could be due to the limited range of diameters considered in their study They investigated the buckling behavior of SWNTs for up to 1.2 nm in diameter only where it was found that the buckling load increases with increasing values of the diameter

For MWNTs, Wang and Mioduchowski (2007) reported that the buckling load decreases with increasing inner diameter for MWNTs with low aspect ratios which shows shell buckling mode In contrary, the buckling load was found to increase as the inner diameter increases for MWNTs with high aspect ratios which shows beam buckling mode

Number of walls

The number of walls is another geometric factor affecting the buckling properties of CNTs Ru (2000a) concluded through a continuum shell model that inserting a tube into a MWNT, doesn’t increase the buckling strain but does increase the buckling load due to the increase in cross-sectional area However, in a more detailed study using molecular mechanics approach, Chang et al (2005b) concluded that inserting tubes into a MWNT can increase the critical buckling strain of thin MWNTs but not that of thick (solid) MWNTs Moreover, studies by Liew et al (2004) and Chang et al (2005b) revealed that wrapping of a MWNT by another CNT decreases the buckling strain of MWNTs In another detailed study, Sears and Batra (2006) concluded that within the shell buckling region, critical buckling strain

of MWNTs is equal to the critical buckling strain of the largest tube constituting MWNT

Two major buckling modes of MWNTs have been reported by various studies For example, Liew et al (2004) observed progressive buckling from the outer tube to the inner tube In contrast to this, Sears and Batra (2006) concluded through molecular mechanics and continuum mechanics that all the walls of MWNTs buckle at the same time to the same shape keeping the interwall spacing same as that before buckle In the study carried out by Chang et al (2005b), both these buckling modes have been reported According to their observations, all the tubes buckle simultaneously in the case of thin MWNTs, while in thick MWNTs, the outer tube buckles first with progressive buckling of inner tubes

Chirality

Several studies have shown that buckling of carbon nanotubes is sensitive to chirality Zhang et al (2006) have revealed through MDS that the critical buckling strain of a CNT is inversely proportional to the chiral angle Furthermore, they have observed that the buckling mode is similar in both chiral and armchair CNTs but is different for zigzag CNTs Moreover, Li and Chou (2004), Chang et al (2005a) , Hu

et al (2007), Zhan and Shen (2007) and Cao and Chen (2007) have also confirmed the chirality dependence of buckling properties through their studies Furthermore, most of these studies have shown that the buckling strain/load of armchair CNT is considerably lesser than that of zigzag CNT

Defects

It is known that the occurrence of buckling is sensitive to the presence of defects (geometric imperfections) Several studies can be found in the literature with the aim of investigating the effect of defects on buckling of CNTs For example, the influence of two types of pin hole defects with 6 and 24 missing atoms on the buckling strength of SWNTs with different chiralities was studied by Hirai et al (2003) They observed a significant reduction in the buckling properties due to the presence of large pin hole defects Xin et al (2008) studied the effect of single atom vacancy defect on the buckling of SWNT and DWNT using MDS They observed significant effect of single atom vacancy on buckling load and buckling strain especially in tubes with relatively smaller diameters They further concluded that the relative positions of defects in the inner and outer walls play an important role in the buckling of DWNTs However, they employed Morse potential which accounts only for bond stretching, bond angle and bond torsion energies Wang et al (2008c) also

studied the effect of single atomic vacancy defect on the buckling of (8, 0) SWCNT using MDS and continuum beam model Their results did not find any considerable effect of defect on buckling strain of CNTs, in contrast to the results of Xin et al (2008) Zhang et al (2009c) investigated the buckling of CNTs with mono-vacancies and bi-vacancies at 300 K and 800 K At 300 K, their results showed that buckling strain of asymmetric bi-vacancy is superior to the symmetric bi-vacancy as well as to the mono-vacancy Further, they observed an increase in buckling load only in the CNT with symmetric bi-vacancy, as the temperature increased from 300

K to 800 K, while CNTs with other vacancies as well as the pristine CNT showed reduction in buckling load, as the temperature increased Therefore, it is evident that despite the few studies conducted on buckling of defective CNTs, the results of these studies are also debatable

Temperature

It has been revealed through atomistic simulations and experiments that temperature has a considerable effect on the buckling of CNTs Zhang et al (2007) found through MDS that the critical buckling pressure is lower at higher temperatures Furthermore, they concluded that the temperature effect is less significant on torsion or pressure induced buckling compared to axial buckling Shen and Zhang (2006) also revealed that temperature has a significant effect on the postbuckling behavior of CNTs under axial compression but less effect on the postbuckling behavior of CNTs under torsion Continuum shell models with temperature dependent material properties have also been used to analyze the temperature effect on the buckling and post buckling of CNTs (Shen and Zhang,

2007) In this study, they concluded that the effect of temperature on post buckling

of DWNTs is not as significant as that for SWNTs

Surrounding media

Several studies have shown that CNT is eligible to be a good reinforcing fiber in composites Consequently, the effect of surrounding media (matrix) on buckling properties of CNTs has become an interesting topic among researchers

In spite of the difficulty in conducting experiments, several experiment based studies can be found on the buckling of embedded CNTs One of such studies is the work conducted by Zhang et al (2006a) to investigate the buckling of oxidized SWNT in polycarbonate matrix They observed that buckling load of polycarbonate matrix was increased by 29.2% and 51.2% due to the addition of 1wt.-% and 2wt.-% oxidized SWNT in to the matrix respectively In another experimental study, Lourie

et al (1998) examined the buckling of CNTs embedded in epoxy resin matrix Moreover, Hadjiev et al (2006) also observed the buckling of functionalized CNTs embedded in epoxy resin They employed the Winkler model to calculate the Young’s modulus of the matrix with the help of experimentally observed stress values From this, they predicted that the functionalization could weaken the interaction between CNT and the polymer

Owing to the high computational cost involved in MDS, most of the theoretical studies on CNT composites are based on continuum mechanics modeling

Ru (2001) developed an expression for the buckling strain of DWNTs embedded in elastic medium based on Winkler model combined with continuum shell model.Liew et al (2005) and Kitipornchai et al (2005) employed a continuum mechanics shell model combined with Pasternak foundation model, which accounts for both

normal and shear stresses between the CNT and the matrix, to derive analytical expression for buckling load of embedded DWNT and TWNT (triple-walled nanotube), respectively All these analytical models predicted higher buckling stress

in embedded CNTs compared to freestanding CNTs However, Kitipornchai et al (2005) further observed that for TWNTs with higher radii, buckling stress reached to

a constant value irrespective of whether they are freestanding or embedded Zhang et

al (2006b) also derived analytical expression based on energy method for critical buckling pressure of embedded DWNTs Similarly, Qiang (2003), Yao and Qiang (2006), Wang et al.(2007d) and Wang and Yang (2007) also used analytical approach to study the torsional buckling of embedded CNTs while Yang and Wang (2006) used it to study the bending buckling of embedded CNTs

From a review of existing literature, it appears that there is only one based study on the buckling of embedded CNT This study was carried out by Namilae and Chandra (2006) They considered (10, 10) CNT of different lengths at

MDS-300 K with and without the surrounding matrix The matrix considered here was crystalline polythene with a density of about 0.8 g/cc Tersoff-Brenner potential and 12-6 Leonard-Jones potential were used to model bonded and non-bonded interactions, respectively Compressive deformation was applied on the CNT only Further, their results were compared against the results obtained from an analytical expression which was based on the Donnell shell theory and Wrinkler foundation model They observed that the increase in critical stress obtained from MDS is much lesser than the increase obtained from continuum mechanics model Furthermore, they studied the buckling of CNTs bonded chemically to the matrix by hybridization and found that these CNTs have lower critical stress compared to the CNT with no matrix This reduction in critical stress was explained as a result of initial curvature

due to hybridization However, their study has been limited to crystalline polymer matrix and the boundary condition applied was non periodic which is not capable of representing the bulk nature of a composite

In view that defective CNTs are likely to be produced especially in the case

of mass production of CNTs for composites, it would be useful to study the degradation of buckling properties resulting from the presence of defects in embedded CNTs There appear to be only one study (Montazeri and Naghdabadi, 2009) focused on this topic This study is based on multi-scale modeling

Boundary condition

It is known that both beam and shell buckling is influenced by the boundary conditions However, the effect of boundary condition on beams is significantly higher compared to the effect on shells Most of the analytical expressions for buckling properties based on continuum shell models assume simply supported boundary conditions In MDS, boundary conditions are achieved by constraining the atom positions of end atom layers during the simulations However, Cao and Chen (2007) conducted a MDS study by removing this conventional boundary condition

to see the effect of boundary condition on the buckling properties They observed that CNTs with conventional boundary conditions show lower buckling strain than CNTs without any constraints Moreover, there are several continuum mechanics based studies which investigated the effect of boundary conditions on the buckling

of CNTs Das and Wille (2002) concluded through a finite element shell model that critical buckling pressure of a simply supported CNT is half that of a clamped CNT Moreover, Tong et al (2009) imposed homogeneous (all the tubes having same boundary condition) as well as heterogeneous (different tubes have different

boundary conditions) boundary conditions on MWNTs to study the effect of free, simply supported and clamped boundary conditions They observed that tubes with lower aspect ratios are less sensitive to the boundary conditions Also, they observed that for tubes with higher aspect ratios, there is a considerable increase in the buckling property for clamped boundary condition as compared to the simply supported and free boundary conditions

2.1.2 Techniques employed in the buckling analysis

Commonly employed techniques in the study of buckling of CNTs include experimental methods, atomistic simulations and analytical as well as numerical methods based on continuum mechanics In this section, a review of previous works employing these techniques in investigating buckling of CNTs is presented

Experimental studies

Conducting experiments on CNTs is a challenging task due to the small size

of nanotubes Despite the difficulty, there exist a few experimental studies on the buckling of CNTs

One of the earliest experimental study on the buckling of CNTs was made by Iijima et al (1996) through high resolution electron microscope In 1998, an experimental study on the buckling and kinking of MWNTs embedded within a polymeric film was carried out by Lourie et al (1998) A nanoindentation test to calculate the buckling load of MWNTs was conducted by Waters et al (2005, 2004) They concluded that the buckling load values determined experimentally agree well with the elastic shell theory solution A similar observation was made by Akita et al (2006) through their experimental study and concluded that the experimental results

are agreeable with the results obtained through MDS as well as analytical solutions based on the Euler beam buckling model On the other hand, the experimental buckling load values obtained by Guduru and Xia (2007) showed a significant 40-50% larger values compared to the buckling load values predicted by theoretical models They concluded that this increase is a result of interwall bridging of MWNTs

Atomistic simulations

In the nanometer scale, atomic interactions are expected to play an important role in affecting the mechanical properties and behavior of structures Therefore, atomistic simulations have been generally agreed in the research community as a more accurate tool in investigating the mechanical properties and behavior of CNTs Three main types of atomistic simulations exist; namely ab initio, semi empirical

and molecular dynamics methods Density functional theory is an example of an ab initio method while tight binding method is an example of a semi empirical method

However, due to relatively lower computational effort compared to the ab initio and

semi empirical methods, MDS technique is the most commonly employed atomistic simulation method for the analysis of CNTs

In MDS, the evolution of an atomic system over a certain period of time is simulated using Newton’s second law Atomic interactions are governed by a selected force field A detail description of MDS will be given in chapter 2 The first MDS study on the buckling of CNT was reported by Yakobson et al (1996) They employed Tersoff-Brenner potential and studied CNT under axial compression, bending and torsion Based on their simulations conducted on (7, 7) SWNT of aspect ratio 6 only, they concluded that continuum model can be considered as a

good tool to predict the buckling strain of CNTs Followed by this pioneering work, many studies can be found using MDS to explore the buckling of CNTs These studies employed various types of force fields, including Morse potential (Xin et al., 2008), Tersoff-Brenner Potential (Cornwell and Wille, 1997, Wang et al., 2005), REBO potential (Ni et al., 2002, Shen and Zhang, 2006, Tan et al., 2007) and COMPASS potential (Cao and Chen, 2007, Wang et al., 2008c), to model the carbon-carbon bond interactions while Leonard-Jones potential is commonly used to model the van der Waals interaction between walls of MWNTs In addition to the potential function, parameters such as thermostat, ensemble, time step and duration also affect the accuracy of the results obtained from MDS These parameters will be discussed in chapter 3 in detail

Despite the higher computational time demanded by MDS as compared to continuum mechanics modeling, a considerably higher accuracy can be achieved especially in situations where the CNTs are relatively smaller in diameter or where there are defects in the CNTs or when CNTs are exposed to a high temperature environment

Continuum mechanics

Motivated by the observation of Yakobson et al (1996), many researchers have resorted to the use of continuum mechanics models to analyze CNTs Continuum mechanics-based models are either analytical or numerical The latter includes finite element method Commonly used analytical continuum models to analyze CNTs include the Euler beam model, Timoshenko beam model and various thin shell models Some researchers have extended these models further by adding