Modeling and simulation of buckling of pristine and defective carbon nanotubes

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 i ACKNOWLEDGEMENTS I would like to take this opportunity to acknowledge the guidance and support of many people without those this thesis would have not been possible. I owe my deepest gratitude to my supervisor, Associate Professor Ang Kok Keng, for his invaluable guidance, encouragement and dedicated support which helped me immensely in the completion of this research work. 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. i TABLE OF CONTENTS ACKNOWLEDGEMENT i TABLE OF CONTENTS ii SUMMARY vi LIST OF FIGURES ix LIST OF TABLES xi NOMENCLATURE xii CHAPTER 1: INTRODUCTION 1.1 Background 1 1.1.1 Types of CNTs 1.1.2 Mechanical properties of carbon nanotubes 1.1.3 Applications of carbon nanotubes 1.2 Motivation 1.3 Objective and scope of work 11 1.4 Layout of the thesis 13 CHAPTER 2: LITERATURE REVIEW 2.1 Buckling of CNTs 15 15 2.1.1 Factors affecting buckling 16 2.1.2 Techniques employed in the buckling analysis 24 2.2 Defects in CNTs 30 2.2.1 Vacancy defects 30 2.2.2 Stone-Wales defect (SW defect) 33 2.2.3 Rehybridization defect (sp3 interwall bridging) 34 ii CHAPTER 3: MOLECULAR DYNAMICS SIMULATIONS 3.1 Introduction to molecular dynamics simulations 37 37 3.1.1 Basic steps in MDS 37 3.1.2 Force fields 40 3.1.3 Ensembles 42 3.2 Simulation procedure for CNT under axial compression 43 3.3 Proposed simulation model 46 3.3.1 COMPASS force field 47 3.3.2 Simulation parameters 49 CHAPTER 4: AN IMPROVED SHELL MODEL FOR PRISTINE CNTS 55 4.1 Background 55 4.2 Introduction to shell theories 57 4.3 Governing equations 61 4.4 Analytical solution for buckling strain of SWNTs 69 4.4.1 Sanders shell theory solution 69 4.4.2 First-order shell theory solution 71 4.5 Analytical solution for buckling strain of DWNTs 73 4.5.1 Van der Waals interaction 73 4.5.2 Sanders shell theory solution 75 4.5.3 First-order shell theory solution 78 4.6 Results and discussion 80 4.6.1 Critical buckling strain of SWNTs 81 4.6.2 Critical buckling strains of DWNTs 86 4.7 Concluding remarks 89 iii CHAPTER 5: BUCKLING OF EMBEDDED CNTS 92 5.1 Background 92 5.2 Simulation model 95 5.2.1 Matrix 96 5.2.2 Boundary condition 98 5.2.3 Volume fraction 100 Simulation procedure 102 5.3 5.3.1 Packing 103 5.3.2 Equilibration 104 5.3.3 Application of boundary condition 105 5.3.4 MDS parameters 107 5.4 Continuum mechanics modeling of embedded CNTs 108 5.5 Results and discussion 110 5.5.1 Convergence study 112 5.5.2 Choice of boundary condition 113 5.5.3 Effect of volume fraction 116 5.5.4 Buckling of embedded CNTs of various diameters 121 5.5.5 Buckling of embedded CNTs of various lengths 125 5.6 Concluding remarks CHAPTER 6: BUCKLING OF DEFECTIVE CNTS 129 133 6.1 Background 133 6.2 Atomic simulation model 135 6.3 Results and discussion 135 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 iv 6.3.3 6.4 Effect of vacancy defects on buckled shape of CNTs Concluding remarks CHAPTER 7: CONCLUSIONS AND RECOMMENDATIONS 151 154 156 7.1 Conclusions 156 7.2 Recommendations for future work 163 REFERENCES 166 LIST OF PUBLICATIONS 182 v 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 not show sensitivity to aspect ratio (length/diameter ratio) of CNT. The lack of sensitivity is vi 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 vii 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. viii LIST OF FIGURES Figure 1.1: Graphical representation of chirality of CNTs Figure 1.2: CNT brushes Figure 2.1: Slits and holes resulting from large number of missing atoms 31 Figure 2.2: Various configurations of vacancy defects 32 Figure 2.3: Formation of SW defect through 900 bond rotation 34 Figure 2.4: Typical defects in SWNT 35 Figure 2.5: Rehybridization defect in DWNT 36 Figure 3.1: Graphical representation of bond energies 40 Figure 3.2: Force-displacement curves of CNT 45 Figure 3.3: Critical buckling strains obtained using different thermostats and displacement steps 51 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 4.2: Sign convention 61 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 4.5: Buckled shapes of (7, 7) SWNT 86 Figure 4.6: Critical buckling strain of DWNT (4, 4) (9, 9) 88 Figure 4.7: Critical buckling strain of DWNTs at aspect ratio 89 Figure 5.1: Simulation model 96 Figure 5.2: Schematic representation of different polymer matrices 97 ix ESTILI, M., HANSANG, K., KAWASAKI, A., SEUNGCHAN, C., TAKAGI, K., KIKUCHI, K. & KAWAI, M. 2010. 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Journal of Physics: Condensed Matter, 21, 435301 (8 pp.) 2. KULATHUNGA D. D. T. K., ANG K. K. & REDDY J.N. 2010. Molecular dynamics analysis on buckling of defective carbon nanotubes. Journal of Physics: Condensed Matter, 22, 345301 (7 pp.) 3. KULATHUNGA D. D. T. K., ANG K. K. & REDDY J.N. 2011. Modeling and simulation of buckling of embedded carbon nanotubes. International Journal of Solids and Structures, (under review) 182 183 Conference Papers 1. ANG K. K., KULATHUNGA D. D. T. K. & REDDY J.N (2008). Effect of shear deformation on buckling of axially compressed carbon nanotubes, International Conference on Multiscale Modeling and Simulation (ICMMS), 2-4 January 2008, India. 2. KULATHUNGA D. D. T. K., ANG K. K. & REDDY J.N. (2008). Axial buckling analysis of carbon nanotubes using higher-order shell theory, The Twenty-First KKCNN Symposium on Civil Engineering, 27-28 October 2008, Singapore. 3. ANG K. K., KULATHUNGA D. D. T. K. & REDDY J.N (2010). Buckling of defective carbon nanotube, International Conference on Computing in Civil and Building Engineering (ICCCBE), 30 June -02 July 2010, Nottingham, UK 184 [...]... 5.1: Buckling properties for embedded SWNT with periodic and non-periodic boundary conditions 114 Table 5.2: Buckling properties of embedded SWNT with periodic and non-periodic (fixed matrix) boundary conditions 116 Table 6.1: Percentage reduction of buckling properties due to vacancy defect 138 Table 6.2: Effect of chirality on buckling of defective CNTs 145 Table 6.3: Buckling properties of defective. .. 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, Ch = ma1 + na2 where a1 and a2 are the base vectors of magnitude 1.42 A0 and the integers m and n are the translation indices (see Figure... 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 12 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... 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... 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 15 2.1.1 Factors affecting buckling Studies conducted on the buckling of CNTs have... parameters, such as number of missing atoms, number of dangling bonds, asymmetry of defects and number of vacancy clusters, on SWNTs are discussed in detail Moreover, the effect of temperature on 13 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... 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... 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. .. 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... (1991) Carbon nanotube (CNT) is basically a cylinder with a single layer of atoms where each carbon atom is bonded 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 . i MODELING AND SIMULATION OF BUCKLING OF PRISTINE AND DEFECTIVE CARBON NANOTUBES D. D. THANUJA KRISHANTHI KULATHUNGA (B.Eng.(Hons.), University of Moratuwa, Sri. 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. Effect of volume fraction 116 5.5.4 Buckling of embedded CNTs of various diameters 121 5.5.5 Buckling of embedded CNTs of various lengths 125 5.6 Concluding remarks 129 CHAPTER 6: BUCKLING OF DEFECTIVE