Abstract of “Topics in Nanomechanics, Energy Storage Systems, and Emerging Nano- materials” by Dibakar Datta, Ph.D., Brown University, May, 2015 (Thesis Defended on January 16, 2015 ). In 1959, physicist Richard Feynman gave a historical lecture: ‘There’s Plenty of Room at the Bottom’. Even after sixty years, the exciting field of nanotechnology is explod- ing. When nanomaterials are in use, these will definitely undergo externally applied loading. In real-life situation, fracture in nanomaterials will always be complex in nature. However, most of the studies so far are on simplified systems. Hence we have studied fracture of graphene under complex loading. Nanomaterials have enormous application in lubrication industries. Friction between bilayers for three different cases: graphene/graphene, graphene/h-BN, and h-BN/h-BN have been investigated. We showed how could we tune friction by hydrogen termination. Another important application of nanomaterials is nanomedicine. We discussed how graphene could be used for drug-delivery and environmental barrier applications. Besides nanomedicine and lubrication, during the past two decades, the demand for the storage of electrical energy has mushroomed both for portable applications and for static applications. We considered two kinds of problems: Atomistic mechanism of phase boundary forma- tion during initial lithiation in crystalline silicon and 2D materials for energy storage. We discovered defective graphene would be a potential anode materials for different ion batteries and analyzed the underlying charge-transfer mechanism governing the enhanced adsorption of adatoms. However, the existing nanomaterials have several limitations. Hence in recent years, tremendous attentions have been given towards newly synthesized emerging nanomaterials. Among these, we studied surface termi- nated germanene as Topological Insulator (TI) and electronic properties of 1T/2H interface of Molybdenum Disulphide (MoS2 ). Topics in Nanomechanics, Energy Storage Systems, and Emerging Nanomaterials by Dibakar Datta A dissertation submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in the School of Engineering at Brown University Providence, Rhode Island United States of America May, 2015 (Thesis Defended on January 16, 2015 ) c Copyright 2015 by Dibakar Datta This dissertation by Dibakar Datta is accepted in its present form by the School of Engineering as satisfying the dissertation requirement for the degree of Doctor of Philosophy. Date Professor Vivek B. Shenoy, Adviser Recommended to the Graduate Council Date Professor Allan F. Bower, Reader Date Professor Pradeep Guduru, Reader Approved by the Graduate Council Date Professor Peter M. Weber Dean of the Graduate School iii Vita EDUCATIONAL QUALIFICATION • Doctorate of Philosophy (September 2010 - January 2015) * Brown University, Providence, USA, School of Engineering Major : Mechanics of Solids and Structures ; Minors : Physics and Chemistry * Visiting Scholar (January 2013 - January 2014) at the University of Penn- sylvania, Philadelphia, USA, Laboratory of the Structures of Matters (LRSM), Department of Materials Science and Engineering. • Master of Science (September 2008 - June 2010) Technical University of Catalonia, Barcelona, Spain, Ecole Centrale de Nantes, Nantes, France, and Electricite de France, Paris, France Major : Computational Mechanics • Master of Engineering (August 2006 - May 2008) Indian Institute of Science (IISc), Bangalore, India Major : Structural Engineering • Bachelor of Engineering (August 2002 - May 2006) Indian Institute of Engineering Science and Technology (IIEST), Shibpur, India Major : Civil Engineering iv PUBLICATION (During PhD Period) : The following papers appeared extensively in this thesis : • D. Datta, J. Li, V. B. Shenoy, Defective graphene as promising anode materials for Na and Ca-ion battery, ACS Applied Materials and Interfaces • D. Datta, J. Li, N.Koratker, V. B. Shenoy, Enhanced Lithiation in Defective Graphene, Carbon • S. P. Kim, D. Datta, V.B. Shenoy, Atomistic Mechanism of Phase Boundary Evolution during Initial Lithiation of Crytalline Silicon, The Journal of Phys- ical Chemistry • D. Datta, S. Nadimpalli, Y. Li, V. B. Shenoy, Graphene Fracture under Complex Loading, In Preparation • D. Datta, Y. Li, S. P. Kim, B. Guo, W. Zhang, V. B. Shenoy, Friction Between Bilayer of 2D Crystalline Nanomaterials: Graphene-Graphene, Graphene- Boron Nitride, and Boron Nitride-Boron Nitride, In Preparation • D. Datta, J. Li, V. B. Shenoy, Surface Terminated Germanene as Emerging Nanomaterials, In Preparation • D. Datta, S. P. Kim, V. B. Shenoy, Mechanics of Graphene/CNT-Polystyrene Nanocomposites, In Preparation • Y. Li, D. Datta, S.Li, Z.Li, V. B. Shenoy, Pattern Arrangement Regulated Me- chanical Properties of Hydrogenated Graphene, Computational Materials Science • Y. Li, D. Datta, Z. Li, V. B. Shenoy, Mechanical Properties of Hydrogen Func- tionalized Graphene Allotropes, Computational Materials Science v The following papers appeared PARTIALLY in this thesis : • R. Mukherjee, A. V. Thomas, D. Datta, E. Singh, J. Li, O. Eksik, V. B. Shenoy, N. Koratker, Defect-Induced Plating of Lithium Metal within Porous Graphene Networks, Nature Communications • Y. Chen, F. Guo, A. Jachak, S. P. Kim, D. Datta, J. Liu, I. Kulatos, C. Vaslet, H. D. Jang, J. Huang, A. Kane, V. B. Shenoy, R. Hurt, Aerosol-Synthesis of Cargo-Filled Graphene Nanosacks, Nano Letters Author also participated in these papers which are NOT part of this thesis at all : • J-H. Cho, D. Datta, S-Y. Park, V. B. Shenoy, D. H. Gracias, Plastic Deforma- tion Drives Wrinkling, Saddling, and Wedging of Annular Bilayer Nanostruc- tures, Nano Letters • Y. Li, D. Datta, Z. Li, Tuning anomalous strength characteristics of graphene with tilt grain boundaries by hydrogen functionalization, Submitted • F. Fan, S. Huang, H. Yang, M. Raju, D. Datta, V. B. Shenoy, A.C.T. van Duin, S. Zhang, T. Zhu, Mechanical Properties of Amorphous LixSi Alloys : A Reac- tive Force Field Study, Modeling in Materials Science and Engineering • J. Li, D. Datta, V. B. Shenoy, Methyl Terminated Germanene as Topological Insulator, In Preparation • F. Guo, G. Silverberg, S. Bowers, S. P. Kim, D. Datta, V. B. Shenoy, R. Hurt, Graphene-Based Environmental Barriers, Environ.Sci. Technol. Author co-authored one book : • D. Datta, V. B. Shenoy ; Molecular Simulations Methods in Mechanics and Physics; LAP Lambert Academic Publishing; ISBN-13:978-3-659- 56515-1. vi Acknowledgments I was born and grew up at a remote village in Northeast India. Until my High- School, I completely lacked right guidance and most importantly proper academic atmosphere. I was continuously told that I am not good for anything. Any student, no matter talented or hard-working, will lose motivation and self-confidence if he/she is always discouraged. Surrounding with smart and inspiring people is as important as being talented or working hard. During those days, I never thought, not even in my dream, that one day I would write PhD thesis at an Ivy-League university. This has been possible because right from my undergrad, I was very lucky to have smart and inspiring people at every stage of my career. Here I will acknowledge some of them : I have no proper word to thank my academic father Prof. Vivek B. Shenoy. He is undoubtedly one of the greatest living polymaths of the world. He generously gave me an opportunity to study under his guidance. Throughout my PhD, he continuously inspired me. He kept confidence in me even when I was not productive at all during first two years. Whenever I talk to Prof. Shenoy, I realize that I know nothing. He knows everything and works in almost every field. Moreover, Shenoy Research Group is truly amazing. Everyone helps each other and one will automatically get inspiration to do better work. Stimulating and intriguing discussion during group meetings really gives a lot of fundamental insight. Thank you God that I have been part of this group ! vii Brown University has a proud history of Solid Mechanics. I feel lucky to be surrounded by living legends like Prof. Huajian Gao, Prof. Kyung-Suk Kim, Prof. Pradeep Guduru, Prof. Allan F. Bower, Prof. Janeet Blume and many more. Apart from Solid Mechanics group, I feel lucky to interact with living legends from other departments : Prof. Kosterlitz, Prof. Marston, Prof. Geisser, Prof. Stratt, Prof. Debold, Prof. Menon, Prof. Pelcovits, Prof. Dorca. I feel fortunate to closely work with world-famous experimentalists : Prof. Robert Hurt, Prof. Nikhil Koratker, and Prof. David Gracias. Besides Brown, I was fortunate to work at the University of Pennsylvania, which is also an amazing place. Apart from these great professors, officials at Brown & UPenn are very efficient. I am especially grateful to my readers Prof. Pradeep Guduru and Prof. Allan F. Bower. Despite their busy schedule, they agreed to be my thesis committee members as well as my preliminary examination committee. I was fortunate to have Dr. Junwen Li and Dr. Sang-Pil Kim in our group. They are incredibly smart. I am their first PhD student - albeit unofficially. They have guided me throughout my PhD period. I am also lucky to meet Dr. Yinfeng Li and Dr. Siva Prasad Nadimpalli. I am looking forward to working with them in rest of my career. I am also grateful to our group alumni Dr. Dhananjay Tambe, Dr. Nikhil Medhekar, Dr. Maria Stournara, Dr. Rassin Grantab, Dr. Peng Chen, Dr. Akbar Bagri, Dr. Priya Johari for teaching me many things. Of course, PhD life without great friends is impossible. I was lucky to have un- believable apartment mates : Bhawani, Sujat, Malay and other great friends - Sara, viii Benjamin, Xuao, Teng, Hemant, Abhilash, Dequan, Hossein, Hailong, Xiao and many more. I dont know how to describe my lifetime experience at two Ivy-League univer- sities. Two years in Europe was really inspiring days. I was lucky to write my MSc thesis under the direction of Prof. Nicolas Moes - who is among the ten most cited mechanicians of the world. Besides, my co-adviser Dr. Paul-Emile Bernard is in- credibly smart. I am grateful to my other professors at Ecole Centrale de Nantes : Prof. Nicolas Chavageoun, Prof. Jean-Pique, Prof. Steven Le Coure and many more. During my six months internship under the direction of Prof. Patric Massin, I learnt a lot. Also at EDF, I had amazing friends: Mathieu Tanguy, Axel Caron. In Europe, I met Khalid At Said- the most helpful friend of my life. Also, I had great moments with Pubudu, Saied, Bimalda. Life at BarcelonaTech-Barcelona was the most enjoyable period of my life. I am grateful to Prof. Curiel-Sosa, Prof. Diez, Prof. Fernandez-Mendez, Prof. Vidal, Prof. Huerta and many more. There I met some great friends - Pavan, Amin and many more. I think two years at Indian Institute of Science (IISc) - Bangalore was ‘The Turn- ing Point of My Life’. IISc is undoubtedly a great place. Its amazing feeling to be surrounded by great professors like Prof. C. S. Monohar, Prof. J. M. Chandrakishen, Prof. A. Ramamswamy, Prof. D. Roy. Prof. Anindya Chatterjee. They always in- spire me (even today !). At IISc, my greatest achievement was to have great seniors: Susantada, Trisadi, Indrada, Farouqueda, Amitda, Bisuda, Suvada, Subimalda and many more. They gave me right direction of life. Moreover, at IISc, I learnt a lot from my outstanding and brilliant classmates: Fathima, Radhika, Rajendar, Srinuvasulu, Aditya, Vikas, Santosh and many more. ix My undergrad at IIEST-Shibpur was the most difficult period of my life. It was a real test of my mental strength. But during those days, I could figure out who are my real friends. I have no proper word to thank Prof. Kalyan Kumar Bhar for ‘Saving My Life’. If Prof. K. K. Bhar was not there during my undergrad, my career would have ended there. I think God has sent him to save my life. Apart from him, I am forever grateful to Prof. S. K. Roy, Prof. S. C. Dutta, Prof. S. Chakraborti, Prof. S. Manumder, Prof. D Sengupta, Prof. G. Banerjee, Prof. A. D. Ghosh and many more. Of course, I will remain forever grateful to my real friends: Gopan, Biplab, Pandu, Rambo, Ahijit and many more. I feel fortunate to have friends I met through various on-line sources : Amrita (Rumi), Debdas, and many more. They are as if part of my family. I am lucky to have a wonderful family. My parents (Mr. Sadhan Kumar Datta & Mrs. Manju Datta) have been a great mental support for me. They taught me the value of hard work and honesty. My sisters (Mrs. Mala Datta Barman & Mrs. Mousumi Datta Roy), brother-in-laws (Mr. Subodh Roy & Mr. Sanjib Barman), nephew (Mr. Somnath Roy), niece (Ms. Soumi Barman) are wonderful part of my life. My mother always tells me: ‘Life is like marathon. You may go behind for a while, but keep running and never give up ’. Definitely maa, I will keep running. During the last phase of my PhD, I met my wife Ana - Mrs. Anamika Nath Datta (married on January 29, 2015). She has been a great mental support for me. I also feel lucky to have other great family members - uncles, aunties, cousin sisters, broth- ers and all our in-laws. They are all wonderful part of my life! x DEDICATED TO MY : Parents Mr. Sadhan Kumar Datta (Baba) Mrs. Manju Datta (Maa) Wife Mrs. Anamika Nath Datta (Ana) Sisters Mrs. Mousumi Datta Roy (Bordi ) Mrs. Mala Datta Barman (Chordi ) Brother-in-Laws Mr. Subodh Roy (Badalda) Mr. Sanjib Barman (Sanjibda) Nephew and Niece Mr. Somnath Roy (Bhagne) Ms. Soumi Barman (Bhagni ) and all my true friends and other family members who have been always with me irrespective of my ecstasy and agony! Contents Signature Page iii Vita iv Acknowledgments vii Table of Contents xi List of Tables xvi List of Figures xvii 1 Motivation and Overview of the Thesis 1 1.1 The Birth of Nanotechnology . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 What is Nanotechnology ? . . . . . . . . . . . . . . . . . . . . 3 1.1.2 Application of Nanotechnology . . . . . . . . . . . . . . . . . 4 1.2 Motivation of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Fracture of Graphene under Complex Loading 18 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 Models and Methodologies . . . . . . . . . . . . . . . . . . . . . . . . 21 xi 2.2.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.1 Model-1 : Effect of Loading Angle . . . . . . . . . . . . . . . . 25 2.3.2 Model-2 : Effect of Slit Angle . . . . . . . . . . . . . . . . . . 28 2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3 Friction between Bilayer of Nanomaterials 35 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 Models and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3.1 Friction During Movement in Zigzag Direction . . . . . . . . . 44 3.3.2 Friction During Movement in Armchair Direction . . . . . . . 46 3.3.3 Effect of Surface Functionalization . . . . . . . . . . . . . . . 47 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4 Graphene for Biomedical Application 49 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.2 Computational Modeling of Graphene Nanosack Technologies . . . . 50 4.2.1 Computational Modeling of Graphene Nanosack Technologies 50 4.2.2 Interaction of Graphene Oxide and Water . . . . . . . . . . . 51 4.3 Mercury Diffusion Between Graphene Bilayer . . . . . . . . . . . . . 53 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5 Energy Research : Where Chemistry Meets Mechanics 61 5.1 Introduction : Why shall we study Energy Storage Systems ? . . . . . 61 5.2 Batteries : The most common form of storing electrical energy . . . . 62 5.2.1 The configuration of a battery . . . . . . . . . . . . . . . . . . 63 xii 5.3 Major Challenges and Opportunities in Energy Research . . . . . . . 64 5.3.1 Nanomaterials for Energy Storage . . . . . . . . . . . . . . . . 64 5.3.2 Materials with Multipurpose for Energy Storage . . . . . . . . 64 5.3.3 Theoretical Modeling of Energy Storage Systems . . . . . . . 65 5.4 Energy Storage Problems Considered in this Thesis . . . . . . . . . . 66 5.4.1 Atomistic Mechanism of Phase Boundary Motion . . . . . . . 66 5.4.2 2D Materials for Energy Storage . . . . . . . . . . . . . . . . . 67 6 Atomistic Mechanisms of Phase Boundary Evolution during Initial Lithiation of Crystalline Silicon 69 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.2 Simulation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 7 Enhanced Lithiation for Defective Graphene 86 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 7.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 7.3 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.4.1 Analysis of adsorption potential for lowest defect density . . . 93 7.4.2 Charge transfer analysis . . . . . . . . . . . . . . . . . . . . . 96 7.4.3 Adsorption potential and capacities for different defect densities 97 7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 8 Defective Graphene as a High-Capacity Na- and Ca-Ion Battery 101 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 8.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 103 xiii 8.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 9 Surface Terminated Germanene as Emerging Nanomaterials 112 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 9.2 Models and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 9.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 115 9.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 10 Electronic Properties of 1T/2H of MoS2 124 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 10.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 10.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 126 10.3.1 Schottky Barrier . . . . . . . . . . . . . . . . . . . . . . . . . 126 10.3.2 Stability of the Interface . . . . . . . . . . . . . . . . . . . . . 127 10.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 11 Summary of Contributions and Recommendation for Future Work 132 11.1 Summery of Contributions . . . . . . . . . . . . . . . . . . . . . . . . 132 11.2 Recommendation for Future Work . . . . . . . . . . . . . . . . . . . . 134 11.2.1 A. Nanomechanics . . . . . . . . . . . . . . . . . . . . . . . . 134 11.2.2 B. Energy Research . . . . . . . . . . . . . . . . . . . . . . . . 136 11.2.3 C. Emerging Nanomaterials . . . . . . . . . . . . . . . . . . . 138 A Mechanical Strength of Hydrogen Functionalized Graphene Allotropes139 A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 A.2 Models and Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 A.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 142 A.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 xiv B Plastic Fracture of Silicon Nanowire 150 B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 B.2 Simulation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 B.2.1 MD Potential and Procedure . . . . . . . . . . . . . . . . . . . 151 B.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 152 B.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 C Viscoelastic Fracture of Silly-Putty 156 C.1 What is Silly Putty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 C.2 Modeling Breaking of Silly Putty . . . . . . . . . . . . . . . . . . . . 157 D Atomic Stress Computation 162 D.1 Stress at Atomic Scale . . . . . . . . . . . . . . . . . . . . . . . . . . 162 E Supporting Figures of Complex Fracture of Graphene 166 F Supporting Information : Friction between Bilayer of Nanomaterials171 F.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 G Graphene-Polystyrene Nanocomposites 174 H Supporting Figures : Enhanced Lithiation in Defective Graphene 178 H.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Bibliography 185 xv List of Tables 6.1 Summary of Physical Properties Pertaining to the Phase Boundary Evolution during the Lithiation of a-Si and c-Si . . . . . . . . . . . . 78 8.1 Charge Transfer from Na/Ca to Graphene. . . . . . . . . . . . . . . . 105 B.1 Geometry of the Sample (All data are in Angstrom) . . . . . . . . . . 152 xvi List of Figures 1.1 There’s Plenty of Room at the Bottom - Prof. Feynman’s lecture at an American Physical Society meeting at Caltech on December 29, 1959 2 1.2 Most definitions revolve around the study and control of phenomena and materials at length scales below 100 nm . . . . . . . . . . . . . . 3 1.3 Number of nanotechnology patents published by the US Patent and Trademark Office (USPTO), European Patent Office (EPO) and Japan Patent Office (JPO) according to publication date. The drop in the number of USPTO patents in 2005 is due to the USPTO enforcing a stricter definition of nanotechnology. The decline in the number of JPO patents for 2005 and 2006 is due to the delay between the publication and granting of patents at the JPO . . . . . . . . . . . . . . . . . . . 5 1.4 Nanotechnologies - to be more specific: nanomaterials are already used in numerous products and industrial applications . . . . . . . . . . . 13 xvii 1.5 Motivation of Nanotribology : Optical and AFM images of atom- ically thin sheets of (from left to right) graphene, MoS2 , NbSe2 , and h − BN on silicon oxide. (A) Bright-field optical microscope images of thin sample flakes. The red dotted squares represent subsequent AFM scan areas. Scale bars, 10 µm. (B and C) Topographic and friction (forward scan) images measured simultaneously by AFM from the in- dicated areas. 1L, 2L, 3L, etc. indicate sheets with thicknesses of one, two, three, etc., atomic layers. BL (“bulk-like”) denotes an area with a very thick flake, and S represents an area with bare SiO2 substrate. Scale bars, 10 µm. (D) Friction on areas with different layer thick- nesses. For each sample, friction is normalized to the value obtained for the thinnest layer. Error bars represent the standard deviation of the friction signals of each area. In each chart, the same color repre- sents data from the same sample . . . . . . . . . . . . . . . . . . . . 14 1.6 Motivation of Complex Fracture : Tortuous (non-planar) crack path, mainly following metallic binder (i.e. ductile ligament bridging is operative); compromising scenario between crack deflection attempting to follow binder paths and transgranular carbide cracking, particularly for large carbides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.7 Motivation of Nanomedicine : Nanomedicine is the future of medicine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.8 Motivation of Energy Research . . . . . . . . . . . . . . . . . . 16 1.9 Motivation of Studying Materials Beyond Graphene . . . . . . . . . . 17 xviii 2.1 Two different models of graphene nanoribbons (GNR) of size 2b (b ≈ 50 ˚ A) with a slit of length 2a (a = 12 ˚ A) in the middle of GNR. (a) Model-1: the slit (or crack) is oriented in horizontal direction i.e. parallel to the fixed base, and the top end of sheet is loaded. Mixed-mode load- ing condition is obtained by varying ‘loading angle’ (b) Model-2 : the boundary conditions (i.e. loads and constraints) are unaltered. Crack is oriented at an angle θ (‘slit angle’) . . . . . . . . . . . . . . . . . . 21 2.2 (a) crack with zigzag edge (b)-(f) crack with slit angle 15◦ , 30◦ , 45◦ , 60◦ , and 75◦ respectively. (g) crack with armchair edge. Mixed-mode loading con- dition is obtained by varying ‘slit angle’. Color of atoms denotes the coordination number . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Stress Intensity Factors (SIF) for cracks with armchair and zigzag edge for a/b = 0.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4 Variation of Effective Stress Intensity Factors (Keff ) w.r.t. a/b for cracks with armchair (a-c) and zigzag (d-f) edges . . . . . . . . . . . 31 2.5 Fracture initiation and propagation for different loading angle for crack with armchair and zigzag edges . . . . . . . . . . . . . . . . . . . . . 32 2.6 Tensile strength variation with respect to a/b for different slit angle . 33 2.7 Fracture initiation and propagation for different slit angle for crack with armchair and zigzag edges . . . . . . . . . . . . . . . . . . . . . 34 3.1 Systems for investigating of frictional characteristics between bilayer of nanomaterials : (a) Graphene/Graphene, (b) h-Boron Nitride (h- BN)/h-BN, and (c) Graphene/h-BN. . . . . . . . . . . . . . . . . . . 36 xix 3.2 Hydrogen Functionalization Nanomaterials (a) Functionalization on one and (b) two sides of the underlying sheet. (a1-a6) Different per- centages of one-side functionalization : (a1)1% (a2) 5% (a3) 20% (a4) 40% (a5) 60% (a6) 80%. . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3 Frictional characteristics between Graphene/Graphene bilayer pulled in zigzag direction. Variation of (a) shear and (b) normal stress for (c) different configurations. (d) unit cell for computation . . . . . . . . . 38 3.4 Frictional characteristics between h-BN/h-BN bilayer pulled in zigzag direction. Variation of (a) shear and (b) normal stress for different configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.5 Frictional characteristics between Graphene/h-BN bilayer pulled in zigzag direction. Variation of (a) shear and (b) normal stress for dif- ferent configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.6 Frictional characteristics between Graphene/Graphene bilayer pulled in armchair direction. Variation of (a,c) shear and (b,d) normal stress for different configurations . . . . . . . . . . . . . . . . . . . . . . . . 41 3.7 Frictional characteristics between h-BN/h-BN bilayer pulled in arm- chair direction. Variation of (a,c) shear and (b,d) normal stress for different configurations . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.8 Frictional characteristics between Graphene/h-BN bilayer pulled in armchair direction. Variation of (a,c) shear and (b,d) normal stress for different configurations . . . . . . . . . . . . . . . . . . . . . . . . 43 3.9 Effect of hydrogen functionalization on friction between Graphene/Graphene bilayer. Variation of shear(a,c) and normal (b,d) stress for one and two sided functionalization respectively . . . . . . . . . . . . . . . . . . . 44 xx 3.10 Effect of hydrogen functionalization on friction between Graphene/h- BN bilayer. Variation of shear(a) and normal (b) stress for one sided functionalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.1 Series of images from MD simulations on a nanoscale graphene sheet segment interacting with a water nanodroplet. The simulation is ini- tialized by immersing graphene in the droplet (a) and the snapshots correspond to 0.80 nanoseconds (b), 1.20 ns (c), and 2.00 ns (d). Af- ter 2.00 nanoseconds the sheet is observed to spontaneously migrate to the droplet surface. The NVE-molecular dynamics calculations at 300K were performed with 26120 water molecules. The dimensions of the graphene sheet used in the simulations are 8.4 nm × 8.73 nm. . . 51 4.2 Selected images from MD simulations of a small graphene segment (8.4 nm × 8.73 nm) interacting with a water nano-droplet during drying. (a) initial state, (b) 0.04 ns, (c) 0.12 ns, (d) 0.18 ns. Shrinking of the droplet first induces curvature in the graphene, but as drying proceeds the curvature reaches a maximum value (near c) and then the graphene relaxes back to a planar configuration. This behavior is consistent with the small size of the graphene segment and the high energy penalty for the high curvature required for closure. The simulations were carried out at room temperature. Drying was achieved by removing 5% water molecules every 0.01 ns. . . . . . . . . . . . . . . . . . . . . . . . . . 57 xxi 4.3 Projected concentration distributions of the functional groups on the graphene oxide film. Hydroxyl groups (a) before and (b) after drying. Carbonyl or epoxy groups (c) before and (d) after drying. Arrows in- dicate the direction of folding. Dashed line guides the overall shape of the film. The color codes for the average density of each functional group from lowest (white) to highest (red) at the corresponding lo- cation. Some of the kink or creases regions appear to be enriched in carbonyl / epoxy groups relative to hydroxyl groups. . . . . . . . . . 58 4.4 Mechanisms of nanosack formation. (a,b) Molecular dynamics simu- lations of water droplet-actuated scrolling, folding, or collapse for (a) monolayer graphene, which scrolls through a “guide and glide” mech- anism. (b) Monolayer GO which closes and collapses by a “cling and drag” mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.5 (a) Molecular dynamics (MD) simulations of Hg-atom diffusion in graphene interlayer spaces containing water. The diffusivity of a tracer Hg-atom is seen to decrease with decreasing interlayer spacing, and Hg becomes immobile below 7 ˚ A spacing. Inset shows a graphene interlayer region with with 8 ˚ A spacing, and one mercury atom (blue) in water (red- grey beads). The dashed line gives the known diffusivity of Hg in bulk water. Inset : Mean Square Distribution at different bilayer spacing. (b) Top and Side View of Hg-atom diffusion in graphene interlayer . . 60 5.1 The power density and energy density for batteries, capacitors, and fuel cells. (Energy is the capacity to do work; power is the rate at which work is done.) . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.2 The configuration of a battery . . . . . . . . . . . . . . . . . . . . . . 63 xxii 5.3 Motivation of Modeling Phase Boundary : (a) Schematic of phase boundary formation (b-f) Experimental observation of phase boundary formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.4 2D Materials offer more promise for energy storage . . . . . . . . . . 67 5.5 Nature stores energy with Na and Ca ions, not Li ions . . . . . . . . 68 6.1 Atom and charge configurations during initial lithiation of (a) a-Si, and (b - d) c-Si. The snapshots of c-Si are viewed along the [110] direction, and a set of bonds connecting two (111) planes are highlighted by red lines. The atom colors denote the charge state. Green indicates neutral atoms, blue indicates Si atoms q < 0.5e, and red indicates Li atoms q > +0.5e. All snapshots are taken after a reaction time of 200 ps at 1200 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.2 Relative concentration profiles based on the results of Figure 6.1. Con- centration is regarded as the ratio of the number of Li (or Si) atoms to the total number of atoms in the given space. The first point of intersection from the Li side is taken as the interface (CSi = CLi = 0.50) 77 6.3 Analysis of Si local structures, based on the results of Figure 6.1. (a) Averaged partial coordination number of Si adjacent atoms within 2.8˚ A of the cutoff radius. (b) Spatial probabilities of the Si segments (5-node chains, dumbbells, boomerangs, and isolated Si atoms) are shown in different colors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.4 Partial radial distribution function g(r) of Si-Si, Li-Si, and Li-Li pairs in the phase boundary (left) and the lithiated Si region(right), for differ- ent orientations of the c-Si samples. Dashed lines indicate the reference peak positions of c − LiSi and a − Li4 Si, obtained from Chevrier and Dahn (2009) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 xxiii 6.5 Time evolution of Li concentration at the phase boundary region. The Li concentrations at different times are superimposed by fixing the initial position of the interface. Results for different steps are obtained from individual MD simulation at temperatures of (a) 600, (b) 900, and (c) 1200 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.1 Experimental observation of (a) Single Vacancies (b) Double Vacancies (c) Stone-Wales Defect. (d) Formation energy is high for defects with an under-coordinated carbon atom . . . . . . . . . . . . . . . . . . . 89 7.2 (a) Pristine graphene and graphene with DV defects: (b) 6.25, (c) 12.50, (d) 16.00, (e) 18.75, and (f) 25%. Systems shown here are 2×2 in size with periodicity in their in-plane dimensions. The super cell used in the calculation is marked in black. All systems are relaxed structure. (g) Equilibrium energy per carbon atom for different percentages of DV defect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 7.3 Graphene with SW defects: (a) 25, (b) 50, (c) 75, and (d) 100%. Systems shown here are 2 × 2 in size with periodicity in their in-plane dimensions. The super cell used in the calculation is marked in black. All systems are relaxed structure. (e) Equilibrium energy per carbon atom for different percentages of SW defect . . . . . . . . . . . . . . . 93 7.4 Graphene with (ac) 6.25% DV defect and (df) 25% SW defect: adatom (a, d) over the defect (O position), (b, e) neighborhood of defect (N position), and (c, f) away from defect (A position) . . . . . . . . . . . 94 7.5 Lithiation potential for Li adsorption on different locations: pristine graphene (inset) and graphene with DV and SW defects at the Hex and Top sites. For each site, three positions, O (blue), N (green), and A (brown), are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . 95 xxiv 7.6 Bonding charge density for Li for (a) pristine, (b, c) Stone-Wales and (d, e) divacancy systems obtained as the charge density difference be- tween the valence charge density before and after bonding. Red and blue colors indicate the electron accumulation and depletion, respec- tively. The color scale is in the units of e/Bohr3 . . . . . . . . . . . . 96 7.7 Charge-transfer vs lithiation potential . . . . . . . . . . . . . . . . . . 98 7.8 Capacity and corresponding lithiation potential for different percent- ages of (a) DV and (b) SW defects . . . . . . . . . . . . . . . . . . . 99 7.9 Maximum capacity for different DV and SW defect densities. . . . . . 100 8.1 (a) Sodiation and (b) Calciation potential for Na/Ca adsorption on different locations: pristine graphene (inset) and graphene with DV and SW defects at the Hex and Top sites. For each site, three positions, O(blue), N(green), and A(brown), are shown . . . . . . . . . . . . . . 104 8.2 Bonding charge density for Na and Ca (Top site and O position) for (a, d) pristine, (b, e) StoneWales, and (c, f) divacancy systems obtained as the charge-density difference between the valence charge density before and after the bonding. Red and blue colors indicate the electron accumulation and depletion, respectively. The color scale is in the units of e/Bohr3 . Potential vs charge transfer for (g) Na and (h) Ca adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 8.3 Sodiation potential for different percentages of Na adsorbed for differ- ent percentages of (a) DV and (c) SW defects. Top and side view of one of the (b) Na8 C26 and (d) Na6 C32 relaxed configurations . . . . . 109 8.4 Calciation potential for different percentages of Ca adsorbed for differ- ent percentages of (a) DV and (c) SW defects. Top and side view of one of the (b) Ca8 C26 and (d) Ca6 C32 relaxed configurations . . . . . 110 xxv 8.5 Maximum percentages of Na/Ca adsorbed for different percentages of DV and SW defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 9.1 Experimental observation of the formation of Germanane : Hydrogen terminated Germanene . . . . . . . . . . . . . . . . . . . . . . . . . . 113 9.2 Surface terminated germanene under consideration : (a) GeH (b) GeOH (c) GeX (X : Halogen) and (d) GeHX . . . . . . . . . . . . . . . . . . 114 9.3 Charge Transfer analysis of GeH, GeF, and GeOH . . . . . . . . . . . 116 9.4 Density of States (DOS) analysis of GeH, GeF, and GeOH . . . . . . 117 9.5 Band structure of Ge, GeH, and GeOH . . . . . . . . . . . . . . . . . 118 9.6 Band structure of Halogen and Hydrogen-Halogen terminated Ger- manene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 9.7 Band structure and Density of States (DOS) for 50% Hydrogen termi- nated Germanene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 9.8 Strain effect on band structure for 50% Hydrogen termination of Ger- manene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 9.9 Tuning band gap of Germanane with external strain . . . . . . . . . . 121 9.10 Germanane with SOC under 10% external strain acts as Topological Insulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 9.11 Hydroxyl terminated Germanene with SOC under no external strain acts as Topological Insulator . . . . . . . . . . . . . . . . . . . . . . . 123 10.1 A schematic showing two different interfaces for the case 1T-MoS2 sheet doped inside periodic 2H-MoS2 sheet. Interface at right becomes unstable after equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 128 10.2 Local Density of States (LDOS) of Mo atom at three zones: 2H-MoS2 , Interface and 1T-MoS2 in armchair direction. The Fermi level is set to zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 xxvi 10.3 Density of States (LDOS) of Mo atom at three zones: 2H-MoS2 , In- terface and 1T-MoS2 in zigzag direction. The Fermi level is set to zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 10.4 Interface Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 10.5 Band structure for MoS2 heterostructure with 1T/2H interface. Here y no. of unit cells of 1T-MoS2 are doped inside x-no of 2H-MoS2 unit cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 A.1 Optimized pristine carbon atomic structures for the examined graphene allotropes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 A.2 Stressstrain curves of hydrogen functionalized (a) graphyne (b) cyclic graphene (c) octagonal graphene and (d) biphenylene for H-coverage of 0%, 10%, 50% and 100% . . . . . . . . . . . . . . . . . . . . . . . 148 A.3 Deterioration of (a) Young’s modulus and (b) tensile strength for the investigated GAs for different H-coverage . . . . . . . . . . . . . . . . 148 A.4 Fracture in biphenylene sheet functionalized with 40% H-coverage: (a) configuration at initial stage (b) onset of bond breaking (c) crack for- mation, (d) nucleation, (e) propagation, and (f) tearing of the sheet . 149 B.1 Nanowire Stress/Strain Curve for width = 5.098 nm. Two different strain rates (109 /s and 108 /s) are considered . . . . . . . . . . . . . . 154 B.2 Stress-strain curves for nanowires loaded at a strain rate of 109 /s for different cross sectional area. Length to Width ratio is same for all cases155 B.3 Failure Mechanism of Gold Nanowire . . . . . . . . . . . . . . . . . . 155 C.1 Silly Putty subjected to (a) low and (b) high strain rate . . . . . . . 156 C.2 Modeling breaking of Silly-Putty by Morse Bond . . . . . . . . . . . . 157 xxvii C.3 Morse Bond : Variation of Energy (eV) and Force (eV ˚ A) w.r.t. to distance (˚ A) between connecting atoms . . . . . . . . . . . . . . . . . 159 C.4 Data collapse : No addition of Dashpot will results in data collapse and can’t capture Silly-Putty behavior . . . . . . . . . . . . . . . . . 160 C.5 Capturing silly-putty behavior with the inclusion of dash-pot . . . . . 161 D.1 Atomistic System and Internal Force Between Atoms . . . . . . . . . 162 D.2 Discrete Molecular System . . . . . . . . . . . . . . . . . . . . . . . . 163 E.1 GNR (with crack of length 2a) is subjected to loading (marked in red) in both (a) zigzag and (b) armchair direction . . . . . . . . . . . . . . 166 E.2 Crack surface does not matter in Classical Mechanics at continuum scale as it matters in atomic scale . . . . . . . . . . . . . . . . . . . . 167 E.3 Stress Intensity Factors (SIF) for cracks with armchair (a-c) and zigzag (d-f) edges for different a/b . . . . . . . . . . . . . . . . . . . . . . . 168 E.4 Variation of maximum normal stress for different loading angle . . . . 169 E.5 Stress along the direction perpendicular to the crack in the graphene sheet pulled in armchair direction . . . . . . . . . . . . . . . . . . . . 169 E.6 Variation of Potential Energy for crack with armchair and zigzag edges for the case of pure tension . . . . . . . . . . . . . . . . . . . . . . . . 170 E.7 Surface energy plays a critical role in Atomistic System . . . . . . . . 170 E.8 Potential energy for different slit angle for GNR pulled in zigzag direction170 F.1 Computation of equilibrium lattice constant of graphene . . . . . . . 171 F.2 Increase in friction between graphene/graphene bilayer due to the pres- ence of stone-wales defect . . . . . . . . . . . . . . . . . . . . . . . . 172 F.3 Computation of equilibrium distance between h-BN/h-BN bilayer . . 173 F.4 Computation of equilibrium distance between h-BN/graphene bilayer 173 xxviii G.1 Motivation of Nanocomposites . . . . . . . . . . . . . . . . . . . . . . 174 G.2 Structure of Polystyrene Polymer . . . . . . . . . . . . . . . . . . . . 175 G.3 Computation fo Density of Polystyrene: (a) Different number of Polystyrene monomer chains (20, 60, 100, 150). (b) Volume and (c) Density at dif- ferent MD steps (d) NVT and NPT Stages : Analogous to Squeezing and Relaxing a paper (e) Temperature fluctuation at different MD stages176 G.4 Adhesion between polystyrene and graphene nanocomposites . . . . . 177 G.5 Shear between polystyrene and graphene nanocomposites. (a)-(b) Trans- verse and Side View (c) Effect of hydrogenation (d) Shear between polystyrene and pristine graphene . . . . . . . . . . . . . . . . . . . . 177 H.1 Li adsorption on DV Hex (a-c) and Top (d-f) site at (a,d) over the defect (O position) (b,e) neighborhood of the defect (N position) and (c,f) away from the defect (N position) . . . . . . . . . . . . . . . . . 179 H.2 Li adsorption on SW Hex (a-c) and Top (d-f) site at (a,d) over the defect (O position) (b,e) neighborhood of the defect (N position) and (c,f) away from the defect (N position) . . . . . . . . . . . . . . . . . 180 H.3 Different relaxed configurations of Li11 C42 for 16.00% divacancy defect 180 H.4 Different relaxed configurations of Li5 C28 and Li6 C28 for 12.50% diva- cancy defect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 H.5 Different relaxed configurations of Li7 C26 , Li10 C26 and Li12 C26 for 18.75% divacancy defect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 H.6 Different relaxed configurations of Li7 C24 , Li12 C24 and Li18 C24 for 25.00% divacancy defect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 H.7 Different relaxed configurations of Li6 C32 for 50.00% Stone-Wales defect182 H.8 Different relaxed configurations of Li8 C32 , Li14 C32 , and Li17 C32 for 100.00% Stone-Wales defect . . . . . . . . . . . . . . . . . . . . . . . 183 xxix H.9 Different relaxed configurations of Li6 C32 , and Li8 C32 for 75.00% Stone- Wales defect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 H.10 Lithiation potential for different capacity for 25% DV defect. Here three points at each capacity corresponds to three different initial con- figurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 xxx Chapter 1 Motivation and Overview of the Thesis 1.1 The Birth of Nanotechnology Prof. Richard Feynman, the Nobel Prize winning physicist, gave a famous lecture - There’s Plenty of Room at the Bottom at an American Physical Society meeting at Caltech on December 29, 1959 (Figure 1.1 shows the original pages of this speeach in the Popular Science magazine in 1960 1 ). Prof. Feynman commented : ‘I would like to describe a field, in which little has been done, but in which an enormous amount can be done in principle. This field is not quite the same as the others in that it will not tell us much of fundamental physics (in the sense of, “What are the strange particles?”) but it is more like solid-state physics in the sense that it might tell us much of great interest about the strange phenomena that occur in complex situations. Furthermore, a point that is most important is that it would have an enormous number of technical applications.’ Feynman considered a number of interesting ramifications of a general ability to manipulate matter on an atomic scale. He was particularly interested in 1 Source : Popular Science Magazine 1 2 Figure 1.1: There’s Plenty of Room at the Bottom - Prof. Feynman’s lecture at an American Physical Society meeting at Caltech on December 29, 1959. the possibilities of denser computer circuitry, and microscopes which could see things much smaller than is possible with scanning electron microscopes. He also suggested that it should be possible, in principle, to make nanoscale machines that “arrange the atoms the way we want”, and do chemical synthesis by mechanical manipulation. However, despite his emphasis that it would have an enormous number of technical applications, the talk went unnoticed and it didn’t inspire the conceptual beginnings of the field. In the 1990s it was rediscovered and publicised as a seminal event in the field, probably to boost the history of nanotechnology with Feynman’s reputation 2 . Fifty years later, there’s still plenty of room at the bottom. The new field of Nanotechnology has exploded. At the cutting edge, researchers are successfully 2 Source : wiki 3 manufacturing everything from corporate logos to radios that are all small enough to be stacked end-to-end perhaps a million items long across the proverbial head of a pin. The advent of personal computers and smart phones has brought the power of such miniaturization into sharp focus for the general public. In a very real sense, we have all become bottom feeders 3 . 1.1.1 What is Nanotechnology ? Figure 1.2: Most definitions revolve around the study and control of phenomena and materials at length scales below 100 nm. The definition of Nanotechnology varies significantly for different person. The inaugural issue of Nature Nanotechnology asked 13 researchers from different areas about the definition. However, their response differs significantly : from enthusiastic to skeptical, reflect a variety of perspectives 4 . Most people define it in terms of scale. Their notion of nanotechnology revolve around the study and control of phenomena and materials at length scales below 100 nm and quite often they make a comparison 3 Source : Quest 4 Source : Nature Nanotechnology 4 with a human hair, which is about 80,000 nm wide. Figure 1.2 5 shows that study of Nanotechnology is primarily governed by Molecular and Quantum Mechanics. How- ever, in recent years, systems at nanoscale are also modeled by Continuum Mechanics and Finite Element Methods. Hence, study of Nanotechnology is very much interdis- ciplinary. Thats makes the perfect definition quite hard! A good definition that is practical and unconstrained by any arbitrary size limita- tions can be given as 6 : The design, characterization, production, and application of structures, devices, and systems by controlled manipulation of size and shape at the nanometer scale (atomic, molecular, and macromolecular scale) that produces struc- tures, devices, and systems with at least one novel/superior characteristic or property. Governments are spending a lot of money on funding nanotechnology research and de- velopment. Estimates for worldwide public investment into nanotechnology projects in 2008 were well over $6 billion. Figure 1.3 shows the rapid growth of research in Nanotech after 1990 7 . 1.1.2 Application of Nanotechnology Nanotechnologies, in general, nanomaterials have enormous applications as shown in Figure 1.4 8 . A closure look at the applications may appear that most of the applications are represent evolutionary developments of existing technologies: for example, the reduction in size of electronics devices. Among many, some important practical applications are (as described in Nanowerk): 5 Source : 3rd International Workshop on Physics Based Materials Models and Experimental Observations 6 Source : Protecting new ideas and inventions in nanomedicine with patents 7 Source : Trends in nanotechnology patents 8 Source : Commercial scale production of inorganic nanoparticles 5 Figure 1.3: Number of nanotechnology patents published by the US Patent and Trademark Office (USPTO), European Patent Office (EPO) and Japan Patent Office (JPO) according to publication date. The drop in the number of USPTO patents in 2005 is due to the USPTO enforcing a stricter definition of nanotechnology. The decline in the number of JPO patents for 2005 and 2006 is due to the delay between the publication and granting of patents at the JPO. • Composites : An important use of nanomaterials (particle, tubes) is in com- posites, materials where one or more separate components are combined in order to exhibit overall the best properties of each component. Graphene/CNT based nanocomposites have been extensively used in recent years 9 . • Coatings and Surfaces : An atomic look at Bird’s wing will teach us how can we use nanotechnology for coatings. With thickness designed at the nano- or atomic scale, self-cleaning window has been engineered to be highly hydrophobic (water repellent), antibacterial, and even destroy chemical agents. • Tougher and Harder Cutting Tools : Small perturbation in atomic arrange- ment sometime leads to enormous change in its macroscopic strength. Cutting tools made of nanocrystalline materials are more wear and erosion-resistant, 9 Appendix G briefly discuss about Graphene/Polystyrene nanocomposites. 6 and last longer than their conventional (large-grained) counterparts. • Lubricants : Nanomaterials have been extensively studied for lubrication pur- poses. Nanospheres of inorganic materials act as nanosized ball bearings. By controlling the shape, we can make them more durable than conventional solid lubricants and wear additives. • Food Nanotechnology : This is new emerging field with potential applica- tions in the area of functional food by engineering biological molecules. They have different functions as compared to those in nature. This opens up a whole new area of research and development. • Energy : One of the most important applications of Nanotechnology is energy storage systems. It provides the potential to enhance energy efficiency across all branches of industry. It has potential applications for different ion-batteries as well as for different anode materials. • Graphene Nanotechnology in Energy : Among many, graphene-based nanomaterials have many promising applications in energy-related sectors. Graphene improves both energy capacity and charge rate in rechargeable batteries. Ac- tivated graphene makes superior supercapacitors for energy storage. Graphene electrodes may lead to a promising approach for making solar cells that are inexpensive, lightweight and flexible. • Space Research : Nanotechnology will play an important role in future space missions. Nanosensors, dramatically improved high-performance mate- rials, or highly efficient propulsion systems are but a few examples. In addition, nanocomposites have enormous applications in space structures. 7 • Cosmetics : In recent years, the applications of nanotechnology and nanoma- terials can be found in many cosmetic products including moisturizers, hair care products, make up and sunscreen. • Construction Industry : The use of nanotechnology materials and appli- cations in the construction industry can be considered for enhancing material properties and also in the context of energy conservation. • Nanotechnology in Displays : Nanotechnology has enormous application in display technologies : Organic LEDs, electronic paper and other devices intended to show still images, and Field Emission Displays. • Nanomedicine : Nanomedicine, considered as the future of medicine, not only has the potential to change medical science dramatically but to open a new field of human enhancements that is poised to add a profound and complex set of ethical questions for health care professionals. • Nanotechnology and the Environment : Nanotechnological products, pro- cesses and applications are expected to contribute significantly to environmental and climate protection by saving raw materials, energy and water as well as by reducing greenhouse gases and hazardous wastes. • Green Nanotechnology : There is a general perception that nanotechnologies will have a significant impact on developing ‘green’ and ‘clean’ technologies with considerable environmental benefits. The best examples are the use of nanotechnology in areas ranging from water treatment to energy breakthroughs and hydrogen applications. 8 1.2 Motivation of the Thesis Section 1.1.2 gives an overview of various applications of Nanotechnology in many different disciplines. In this thesis, we will consider some selected problems which are at present burning issues. As mentioned Nanotechnology has enormous application in lubrication industries. Thus it is important to investigate frictional characteris- tics between two-dimensional nanomaterials. As shown in Figure 1.5 10 , extensive experimental research have been carried out to investigate frictional characteristics between bilayer of nanomaterials. Hence it is important to investigate frictional prop- erties theoretically. When nanomaterials are in use, these will definitely undergo externally applied loading which leads to fracture and deformation. As shown in Figure 1.6 11 , in real- life situation, fracture in nanomaterials will always be complex in nature. There have been many theoretical studies on fracture of nanomaterials. However, all these stud- ies are on simplified systems. Hence in this thesis, we consider fracture of graphene under complex loading. Among various applications of nanomaterials, as shown in Figure 1.7 12 , nanomedicine is one of the emerging fields. This is now a large industry, with sales reaching $6.8 billion in 2004, and with over 200 companies and 38 products worldwide, a minimum of $3.8 billion in nanotechnology R & D is being invested every year 13 . Problems for nanomedicine involve understanding the issues related to toxicity and environmental 10 Source : Carpick Group at The University of Pennsylvania 11 Source : MetalPowderReport 12 Source : Quantum Day 13 Source : Nanotechnology: A Gentle Introduction to the Next Big Idea 9 impact of nanoscale materials 14 . Hence we will discuss about graphene for applica- tion in drug-delivery and environmental barrier. Among the various applications, besides lubrication and nanomedicine, nanotech- nology has tremendous application in energy sector (Figure 1.8 15 ). During the past two decades, the demand for the storage of electrical energy has mushroomed both for portable applications and for static applications. As storage and power demands have increased predominantly in the form of batteries, the system has evolved. However, the present electrochemical systems are too costly to penetrate major new markets, still higher performance is required, and environmentally acceptable materials are preferred. These limitations can be overcome only by major advances in new ma- terials whose constituent elements must be available in large quantities in nature ; nanomaterials appear to have a key role to play 16 . Among the various energy storage systems, Lithium Ion Batteries (LIBs) have been ubiquitous in practical applications. Silicon anode is commonly used anode ma- terials in most practical applications. We will consider one problem (The Evolution of Phase Boundary) for Lithium-Silicon system. Since Silicon anode encounters se- vere problem of volume expansion and excessive fracture, we will turn our attention to alternatives. Among these, the celebrated 2D nanomaterials Graphene promises significant application in energy storage. We will discuss how can we do that. In ad- dition, due to scarcity of lithium, it is apparent that after few decades, we will end up in energy sortage. So we will discuss Na- and Ca- ion batteries as alternatives to LIBs. While graphene and other 2D materials have been extensively used, these have 14 Source : Wiki 15 Source : Nano Werk 16 Source : Cambridge Journal : Materials Challenges Facing Electrical Energy Storage 10 several issues with practical applications. Hence we will study some emerging nano- materials i.e. Functionalized Germanene, Molybdenum Sulphide (MoS2 ). 1.3 Organization of the Thesis The thesis is organized in the following ways : Chapter 2 : Fracture of Graphene under Complex Loading • Analysis of fracture properties (SIF, crack angle etc) of graphene for different loading and slit angle considering both armchair and zigzag directions. Chapter 3 : Friction between Bilayer of Nanomaterials • Investigation of frictional characteristics between different bilayer of 2D nano- materials (graphene-graphene, graphene-boron nitride, boron nitride-boron ni- tride). Chapter 4 : Graphene for Biomedical Application • Modeling of graphene for the application of drug delivery and environmental barrier. Chapter 5 : Energy Research : Where Chemistry Meets Mechanics • Overview of practical issues in energy research and problems considered in this thesis. Chapter 6 : Atomistic Mechanisms of Phase Boundary Evolution during Initial Lithiation of Crystalline Silicon • Modeling of phase boundary - a sharp interface in between c − Si and a − Lix Si and investigation of kinetics of Li intercalation into Si of different orientations : (100), (110), (111). 11 Chapter 7 : Enhanced Lithiation for Defective Graphene • Detail analysis of defective graphene as potential high capacity anode materials for Lithium Ion Batteries (LIBs). • Discovery of underlying charge transfer mechanism for enhanced lithiation. Chapter 8 : Defective Graphene as a High-Capacity Na- and Ca-Ion Bat- teries • Discussion on problems associated with LIBs and Na-and Ca- Ion batteries as potential alternatives. • Analysis of defective graphene as high capacity anode for Na- and Ca- ion batteries. Chapter 9 : Surface Terminated Germanene as Emerging Nanomaterials • Investigation on how can we use surface terminated germanene as topological insulator. Chapter 10 : Electronic Properties of 1T/2H of MoS2 • Analysis of electronic properties of 1T/2H interface of MoS2 . Chapter 11 : Summary of Contributions and Recommendation for Future Work • Overview of major contributions made and recommendations of further works. Appendix A : Mechanical Strength of Hydrogen Functionalized Graphene Allotropes • Brief overview of mechanical properties of graphene allotropes. 12 Appendix B : Plastic Fracture of Silicon Nanowire • Analysis of plastic fracture of Silicon nanowire. Appendix C : Viscoelastic Fracture of Silly-Putty • Analysis of visco-elastic fracture of Silly-Putty. Appendix D : Atomic Stress Computation • Theoretical framework of Virial Stress/Atomic Stress Computation. Appendix E : Supporting Figures of Complex Fracture of Graphene • Supporting figures for Chapter 2 Appendix F : Supporting Information : Friction between Bilayer of Nano- materials • Supporting figures for Chapter 3 Appendix G : Graphene-Polystyrene Nanocomposites • Brief discussion about Graphene Polystyrene Nanocomposites. Appendix H : Supporting Figures : Enhanced Lithiation in Defective Graphene • Supporting figures for Chapter 7 13 Figure 1.4: Nanotechnologies - to be more specific: nanomaterials are already used in numerous products and industrial applications. 14 Figure 1.5: Motivation of Nanotribology : Optical and AFM images of atomi- cally thin sheets of (from left to right) graphene, MoS2 , NbSe2 , and h − BN on silicon oxide. (A) Bright-field optical microscope images of thin sample flakes. The red dotted squares represent subsequent AFM scan areas. Scale bars, 10 µm. (B and C) Topographic and friction (forward scan) images measured simultaneously by AFM from the indicated areas. 1L, 2L, 3L, etc. indicate sheets with thicknesses of one, two, three, etc., atomic layers. BL (“bulk-like”) denotes an area with a very thick flake, and S represents an area with bare SiO2 substrate. Scale bars, 10 µm. (D) Friction on areas with different layer thicknesses. For each sample, friction is normal- ized to the value obtained for the thinnest layer. Error bars represent the standard deviation of the friction signals of each area. In each chart, the same color represents data from the same sample. 15 Figure 1.6: Motivation of Complex Fracture : Tortuous (non-planar) crack path, mainly following metallic binder (i.e. ductile ligament bridging is operative); compromising scenario between crack deflection attempting to follow binder paths and transgranular carbide cracking, particularly for large carbides. Figure 1.7: Motivation of Nanomedicine: Nanomedicine is the future of medicine 16 Figure 1.8: Motivation of Energy Research : Basic LIB characteristics required for different applications(DOD: Depth of Discharge, SOC: State of Charge). 17 Figure 1.9: Motivation of Studying Materials Beyond Graphene Chapter 2 Fracture of Graphene under Complex Loading 2.1 Introduction Graphene, a two-dimensional (2D) material of one-atom-thick sp2 hybridized car- bon atom, has been the cynosure of Nanotechnology ever since its discovery [Geim et al. (2004)]. Because of its extraordinary properties, graphene has been studied extensively with potential applications in fields such as electrical, thermal, and nano- electronics etc [Geim and Novoselov (2007)]. Due to its superior mechanical prop- erties, i.e. Young’s modulus ≈ 1 TPa and breaking strength ≈ 130 GPa [Lee et al. (2008)], graphene is being considered as the strengthening agent in nano-composites [Young et al. (2012)]. The strength of graphene and its allotropes, however, is gov- erned by the impurities and defects that are always present [Banhart et al. (2011), Hashimoto et al. (2004)] due to the nature of synthesizing process or processing con- ditions. In addition, effect of functionalization plays a crucial role on fracture of graphene [Pei et al. (2010)] and its allotropes [Li et al. (2014)] 1 . Vacancy type defect 1 We have discussed about it in Appendix A 18 19 significantly reduces its strength [Zandiatashbar et al. (2014)]. Similarly, defects such as pre-existing cracks will grow under external loads and cause fracture failure [Zhang et al. (2014)] and strength reduction. As a result, several studies have been focused to understand the fracture behavior of graphene 2 . Omeltchenko et al. (1997) studied crack propagation in a graphite sheet with million atom molecular-dynamics simulation based on Brenner’s reactive empiri- cal bond-order potential. They have estimated the stress intensity factor (SIF) √ √ of graphene sheet to be 4.7 MPa m. Similar values, 4.21 MPa m for zigzag and √ 3.71 MPa m for armchair cracks were reported by Xu et al. (2012) based on coupled quantum/continuum technique. In a recent experimental study, Zhang et al. (2014) √ measured the SIF of graphene to be 4.0 ± 0.6 MPa m which is in agreement with the estimates from earlier theoretical studies [Omeltchenko et al. (1997), Xu et al. (2012)]. In addition, Zhang et al. (2014) also confirmed that the classical Griffith theory can be applied to brittle fracture of graphene with reasonable accuracy. Investigation has been done on crack formation and propagation mechanism in suspended graphene sheets using transmission electron microscopy (TEM). They found that the tearing direction was predominantly aligned to the armchair and zigzag directions. Meyer et al. (2007) performed optical and scanning electron microscopy (SEM) of graphene films on substrate in order to investigate morphologies of fractured graphene. De- wapriya et al. (2013) studied the effect of temperature on the fracture of graphene using molecular dynamics (MD) simulations. They found that the toughness values decreased with the increase in temperature. Although the above studies have contributed to the understanding of fracture of 2 While grpahene undergoes brittle fracture, as discussed in Appendix B and C, Silicon Nanowire and Silly-Putty undergoes Plastic and Viscoelastic fracture respectively. 20 graphene, the important issue of mixed-mode fracture i.e. fracture of graphene under combined tensile and shear loading conditions still needs to be addressed. Graphene fracture under pure tensile loading which was the primary focus of the most of the previous studies, is an idealistic situation and rarely occurs in the service conditions. In almost all practical situations, cracks in graphene are subjected to mixed-mode loading conditions and mixed-mode fracture is inevitable during tearing of graphene [Sen et al. (2010), Moura and Marder (2013)]. Recently Zhang et al. (2012) performed computational study on this important issue. They reported stress intensity factors and direction of crack propagation at room temperature for different orientation of loading conditions. They also observed that the torn edges of fresh cracks were along either zigzag or armchair, where zigzag edges were more preferable. However, the effect of crack length on the mixed-mode fracture of graphene has not been addressed. In addition, mixed-mode fracture of graphene when loaded in armchair versus zigzag direction needs to be studied. The classical molecular dynam- ics (MD) simulations were performed on two different set of models to address these issues. Graphene sheet with pre-existent cracks of different lengths were subjected to mixed-mode loading conditions. Effect of mode-mixity on stress intensity factors was investigated. Crack kinking behavior under various loading conditions was also studied. Crack length effect on fracture strength of graphene sheet under various mixed-mode conditions was studied. 21 Figure 2.1: Two different models of graphene nanoribbons (GNR) of size 2b (b ≈ 50 ˚ A) with a slit of length 2a (a = 12 ˚A) in the middle of GNR. (a) Model-1: the slit (or crack) is oriented in horizontal direction i.e. parallel to the fixed base, and the top end of sheet is loaded. Mixed-mode loading condition is obtained by varying ‘loading angle’ (b) Model-2 : the boundary conditions (i.e. loads and constraints) are unaltered. Crack is oriented at an angle θ (‘slit angle’). 2.2 Models and Methodologies 2.2.1 Models Two different models of square shaped graphene nanoribbons (GNR) of size 2b (b ≈ 50 ˚ A) with a slit of length 2a (a = 2 − 12 ˚ A, 2a implies the two crack tips) in the middle of GNR, as shown in Figure 2.1 were considered for the analysis. In model-1 (shown in Figure 2.1a), the slit (or crack) is oriented in horizontal direction i.e. parallel to the fixed base, and the top end of sheet was loaded. Different loading conditions were obtained by varying the loading angle Φ, which is defined with respect to x -axis. For example, Φ = 0◦ and 90◦ produce pure shear and pure tensile loading conditions 22 Figure 2.2: (a) crack with zigzag edge (b)-(f) crack with slit angle 15◦ , 30◦ , 45◦ , 60◦ , and 75◦ respectively. (g) crack with armchair edge. Mixed-mode loading condition is obtained by varying ‘slit angle’. Color of atoms denotes the coordination number. respectively. The blue dotted line in Figure 2.1a represents the direction of crack propagation, which is denoted with an angle α w.r.t. the crack direction. In contrast to model-1, the boundary conditions (i.e. loads and constraints) in model-2 (Figure 2.1b) were unaltered. However, different fracture conditions were obtained here by varying the orientation of the slit i.e. varying the slit angle θ (from 0◦ to 90◦ ) as shown in Figure 2.1b. Pulling the GNR in model-2 (i.e. in Figure 2.1b) with a slit angle θ = 0◦ induces pure tensile stresses near crack tip, but an increase in θ towards 90◦ induces mixed-mode fracture conditions. GNR was loaded in both zigzag and armchair directions as shown in Figures E.1a and 23 Figure 2.3: Stress Intensity Factors (SIF) for cracks with armchair and zigzag edge for a/b = 0.12 E.1b respectively. The above models with different loading conditions were chosen to mimic various loading situations that a nanodevice or a graphene sheet encounters in service. Unlike the classical continuum systems (Figures E.2a and E.2b), the details of atomic arrangement at the surface become important due to the role played by edge stresses and surface energy. Figure 2.2a and 2.2g represent the crack tip details with zigzag and armchair edge. Figures 2.2b-f represent crack (or slit) oriented in 15◦ , 30◦ , 45◦ , 60◦ , and 75◦ respectively. For slit angle 15◦ , 45◦ , and 75◦ , crack surface is a mixture of zigzag and armchair edge with complex crack tip. Due to the crystal structure of graphene, it was possible to create a crack with only zigzag surface at θ = 30◦ and a crack with only armchair surfaces at θ = 60◦ . However, crack tip is different compared to the case of θ = 0◦ . 24 2.2.2 Methods Classical molecular dynamics (MD) simulation were performed using LAMMPS [Plimpton (1995)] package with an Adaptive Intermolecular Reactive Bond Order (AIREBO) [Tutein et al. (2000)] potential with an interaction cut-off parameter of 1.92 ˚ A [Grantab et al. (2010)]. The time step of the MD simulations was 1 fs. Ho- mogeneous strain of 0.5% was applied by displacing the simulation box followed by a relaxation for 10,000 MD steps. Until the complete failure of the graphene sheet, this procedure of relaxation and stretching was repeated and, consequently, resulted in an effective strain-rate of 0.05% ps−1 . All sets of the simulation were performed at different temperatures under NVE ensemble. For model-1, all calculations were done at 300K while for model-2, we performed calculation at 300K and 1000K. For these parameters, we have not considered any out of plan buckling. Atomic stress of individual carbon atoms in the graphene sheet was calculated using the virial theorem 3 as described by Pei et al [Pei et al. (2010)]. n ! 1 1 γ γ γ X j i σijγ = γ m vi vj + rγβ fγβ (2.1) Ω 2 β=1 where i and j denote indices in Cartesian coordinate systems; γ and β are the atomic indices; mγ and v γ denote the mass and velocity of atom γ ; rγ,β is the distance between atoms γ and β ; Ωγ is the atomic volume of atom γ. After the stress of each graphene sheet is obtained, the stress of the graphene sheet is computed by averaging over all the carbon atoms in the sheet. √ Mode-I and Mode-II stress intensity factors were evaluated as KI = σn πa and 3 Theoretical background of Virial Stress has been discussed in Appendix D 25 √ KII = σs πa, respectively. Here, σn and σs are the normal and shear stress values at the instant of first bond rupture. For fracture under mixed-mode loading, an effective q stress intensity factor was evaluated as Keff = (KI )2 + (KII )2 . 2.3 Results and Discussion 2.3.1 Model-1 : Effect of Loading Angle Figures 2.3 represents KI , KII , Keff for crack with armchair and zigzag edge for √ √ a/b = 0.12. For armchair edge, Keff first decays from 3.40 MPa m to 2.75 MPa m √ for loading angle up to 45◦ followed by increase to 3.60 MPa m for pure tension case. √ √ While for zigzag case, it increases from 2.40 MPa m to 2.75 MPa m and remains √ steady. We notice that at Φ = 45◦ , Keff is same (2.75 MPa m) for both armchair and zigzag cases. Figure 2.4 represents Keff w.r.t. a/b for Φ = 0◦ , 45◦ , 90◦ (both armchair and zigzag cases). Beyond a/b = 0.10, Keff values converge in the range of √ √ 3.10 − 3.80 MPa m and 2.60 − 3.10 MPa m for armchair and zigzag edge respec- tively. Since SIF is a material property, it has to be constant for a given material. The variation in Keff can be attributed to the anisotropic structure of Graphene. For small values of a/b, probably high stress concentration at crack tip lowers Keff . In case of zigzag edge, the converged values of Keff are less as compared to armchair case as in this case crack initiates at lower stress. Moreover, in this case, we observe differ- ent trend in SIF. The KI value increases with loading angle before getting saturated beyond loading angle of around 40◦ . Figure E.3 gives detail overview of variation of SIF w.r.t. a/b for different values of loading angle. Crack initiation and propagation direction are primarily governed by crack tip. As shown in Figure 2.2a, for crack with zigzag edge (C-Zig-0◦ ), arrangement of crack 26 tip offers a favorable situation of crack propagation. As indicated by red arrow, it can easily break the bonding between carbon atoms at crack tip like unzipping. However, for crack with armchair edge (C-Arm-0◦ ),crack tip never has bonding perpendicular to the crack direction (Figure 2.2g). As shown by violet arrows, bonding at crack tip is always inclined. Hence unlike zigzag case, unzipping mechanism is not possible here. Hence intuitively, without doing any computation, we can argue that for C- Zig-0◦ , crack will initiate at lower stress as compared to C-Arm-0◦ . Beyond loading angle ≈ 40◦ , crack tip is predominantly under tensile stress which initiates breaking of bond indicated by red arrow (Figure 2.2a). Because of this, for high loading angle, crack initiation stress and thus KI is same. The path of crack propagation is important besides the prediction of when and how far it will grow. Crack subjected to complex loading i.e. combination of Mode-I and Mode-II will generally not propagate straight ahead of crack tip unless there is initial flaw or weak plane. There are several theories at continuum scale to investi- gate the crack initiation path: (1) maximum circumferential stress [Erdogan and Sih (1963)], (2) minimum strain energy density [Sih (1974)], (3) maximum energy release rate [Wu (1978)] and (4) local symmetry [Goldstein and Salganik (1974)]. However, unlike continuum system, as explained before, study of atomistic fracture requires understanding of detailed arrangement of crack surface and especially crack tip. We notice in Figure 2.5, for zigzag case (lower panel), crack initiates at small angle even for small loading angle i.e. towards the case of tensile loading condition. Because even for loading angle ≈ 30◦ ,stress concentration around crack tip breaks bond located at crack tip (Figure 2.2a). Naturally, for very low loading angle, say for Φ = 30◦ , stress concentrates more on different bonds which leads to bond breaking 27 and crack propagation at high angle. However, for armchair case, as explained earlier, at the crack tip there is no such bonding aligned perpendicular to crack direction. Hence for different loading conditions, stress field around crack tip generates different crack propagation path as shown in Figure 2.5 (upper panel). Figure E.4 shows the normal stress σn (at the initiation of fracture propagation) for model-1 for different loading angle for different ratio of a/b. We notice that, as explained before, strength in zigzag direction is less than in armchair. Figure E.5 shows the normal stress distribution in graphene sheet. For pure tension case, max- imum tensile stress occurs at the two tip ends of the crack and stress distribution is symmetric w.r.t. the loading direction. For pure shear case, stress distribution is symmetric w.r.t. the diagonal direction of the sheet. From normal stress distribution, it is obvious that as we decrease loading angle, system is more subjected to shear. Hence the net tensile and compressive force cancels each other, which reduces the maximum normal stress. Among different major theories that explain the complex nature of fracture in materials, theories involving energy concept is widely used. According to Griffith [Griffith (1921)], the macroscopic potential energy of the system consisting of the internal stored elastic energy and the external potential energy of the applied load, varies with the size of the crack. For the atomistic systems under considerations, potential energy [Lu et al. (2011)] V can be expressed as: N X U () 4b2 − Acrack   V = U0 + + γ() [4b + Lcrack ] (2.2) | {z } | {z } i=1 | {z } V2 V3 V1 Where  is the strain of the GNR along the direction of loading angle (relative to 28 the ground state of the system), U0 is the potential energy per carbon atom at the ground state of graphene, N is the number of carbon atoms, U is the bulk strain energy density of monolayer (per unit area) and γ() is the edge energy density (per unit length of the free edges). Acrack and Lcrack are the total area and edge length of the slit/crack. Figures E.6a and E.6b show the variation of V with strain for the case of armchair and zigzag GNR respectively subjected to loading angle 90◦ i.e. pure tension. For armchair, V = −7.1 eV corresponds to failure of the pristine GNR i.e. a/b = 0 at a strain of 0.17. The failure strain for pristine GNR is in excellent agreement with previous study [Lu et al. (2011)]. For a/b = 0.04, failure strain is 0.12 with V ≈ −7.35 eV. In zigzag direction, GNR breaks at V = −7.1 eV at a strain of 0.15. Energy drops to −7.5 eV corresponding to strain of 0.09 for a/b = 0.04. We notice that presence of even low crack length lowers energy enormously compared to pristine case. However, the decay in potential energy is not significant when we compare different initial crack sizes. 2.3.2 Model-2 : Effect of Slit Angle Figure 2.6a and 2.6b show the variation of tensile strength as a function of a/b ratio for crack with armchair edges at 300K and 1000K temperatures, respectively. Figures 2.6c and 2.6d show similar variation for cracks with zigzag surfaces. It can be noted from Figure 2.6a that when the crack (or slit) is perpendicular to the loading di- rection, i.e. for θ = 0◦ , the strength of GNR decreases significantly with a/b, reaching almost 60 GPa at a/b = 0.12 from 100 GPa corresponding to a pristine GNR without flaws( a/b = 0), and decays at lower rate beyond a/b = 0.12. Although the magni- tude of tensile strength was different, similar trend of strength versus crack length was observed (Figure 2.6b) at higher temperature (1000K). For example, strength of pristine graphene at 300K was 100 GPa but it decreased to 60 GPa at 1000K. Similar 29 trends for the strength of GNR with respect to crack length was observed in Figures 2.6c and 2.6d for cracks with zigzag surfaces, and results are in good agreement with previous studies [Zhao and Aluru (2010)]. The strength variation for other crack orientation i.e. for mixed-mode loading conditions was not investigated before. For lower value of θ, similar trend of strength decay is observed but with a lower rate. We notice that strength of GNR pulled in armchair direction is less than that of GNR pulled in zigzag direction. This can be again explained with the arrangement of crack tip mentioned in Figures 2.2. It is interesting to note from Figure 2.6 that the strength of GNR for θ = 75◦ and 90◦ reduces gradually with crack length in the beginning but starts increasing beyond a/b = 0.08. This increasing with crack length is counter intuitive as the opposite is expected in a continuum system for example (Figure E.7). A possible explanation for this can be provided by looking at the variation of potential energy for zigzag case. Decrease in potential energy corresponds to a lower breaking strength. Figure E.8a shows that for θ = 0◦ drastic decrease in V at a/b = 0.08 followed by decrease at low rate. Similar trend with different rate is observed for θ = 60◦ . Figure E.8c depicts the anomalous variation i.e. decrease in V at a/b = 0.04 followed by increase. It is possible that for very low value of a/b, higher stress concentration may decrease strength. Another reason can be the effect of edge stress. Compressive edge stress along crack surface may increase overall strength of the system [Reddy et al. (2008)]. Figure 2.7 demonstrates stress distribution and direction of crack initiation and nucleation. From elementary mechanics, we know that brittle fracture will occur along the plane of maximum tensile stress. We notice that like model-1, for GNR pulled in zigzag direction, crack initiation predominantly happen along the sheet in zigzag 30 direction i.e. direction perpendicular to the loading. We observe the presence of bond described in Figure 2.2 in the neighborhood of crack tip. Tensile stress concentration at the crack tip breaks this bond initiating crack propagation along zigzag direction. Moreover, crack propagates in this direction with least resistance like unzipping. For GNR pulled in armchair direction, crack initiation due to tensile stress concentration (Figure 2.7) is not perpendicular to the loading direction. 2.4 Conclusions We have systematically investigated effect of loading and slit angle on fracture properties of graphene. In addition, effects of crack length and crack orientations (armchair and zigzag) are also considered. Our results suggest that crack with zigzag edge is more vulnerable to fracture compared to armchair case. We have verified this by computing the effective stress intensity factor (Keff ), which for zigzag case is less. In addition, we have showed how crack will propagate for different cases and how strength decays with loading directions. Given that in almost all practical situations, graphene will be subjected to complex loading, our work will bring new important insight for practical applications. It will be interesting to experimentally verify our results. 31 Figure 2.4: Variation of Effective Stress Intensity Factors (Keff ) w.r.t. a/b for cracks with armchair (a-c) and zigzag (d-f) edges. 32 Figure 2.5: Fracture initiation and propagation for different loading angle for crack with armchair and zigzag edges. 33 Figure 2.6: Tensile strength variation with respect to a/b for different slit angle. 34 Figure 2.7: Fracture initiation and propagation for different slit angle for crack with armchair and zigzag edges. Chapter 3 Friction between Bilayer of Nanomaterials 3.1 Introduction Over the last few decades, significant attentions have been given to the develop- ment of materials and structures with nanoscale features [Lee et al. (2010)]. During practical applications, bilayers of nanomaterials may come under contact. Hence, it is very important to understand the behaviors of materials in these situations. Mate- rials at nanoscale have high relative surface area which renders adhesion, friction, and wears. These are consequential for nanoscale data storage devices, nanocomposites, and nanoeletromechanical systems (NEMS). Especially, two-dimensional (2D) archi- tectures are of particular interest, because of the ease of their experimental synthesis. At nano-world, structural dimensionality is a governing factor of material behavior. For example, Quasi-0D materials (quantum dots, nanoparticles) and quasi-1D materi- als (nanowires, nanotubes) behave very differently from their 3D counterparts [Yanson et al. (1998)]. Beyond 0D and 1D, in recent years, 2D materials have drawn significant 35 36 attentions as these demonstrate very distinct and unique properties. Among the 2D materials, graphene has been the cynosure in nanoworld as it exhibits notable elec- tronic, thermal, chemical, and mechanical properties [Geim et al. (2004)]. Because of many unique properties, graphene has been a promising candidate for next-generation electronic devices, MEMS and NEMS [Novoselov et al. (2004)]. For such wide ranges of applications, it is important to understand the frictional properties of these mate- rials. Figure 3.1: Systems for investigating of frictional characteristics between bilayer of nanomaterials : (a) Graphene/Graphene, (b) h-Boron Nitride (h-BN)/h-BN, and (c) Graphene/h-BN. Meyer et al. (2007) demonstrated the possibility of characterization of free-standing graphene sheet. However, for most practical applications, we need graphene to be grown on [Mattausch and Pankratov (2007)] or transferred [Ritter and Lyding (2009)] to a supporting substrate. However, Du et al. (2008) showed that the presence of sub- strates leads to a significant reduction in electron mobility from 100, 000 cm2 V−1 s−1 to ∼ 1000 cm2 V−1 s−1 due to the charged surface states and impurities [Nomura and MacDonald (2007)], surface roughness [Ishigami et al. (2007)], and surface optical 37 Figure 3.2: Hydrogen Functionalization Nanomaterials (a) Functionalization on one and (b) two sides of the underlying sheet. (a1-a6) Different percentages of one-side functionalization : (a1) 1% (a2) 5% (a3) 20% (a4) 40% (a5) 60% (a6) 80%. phonons [Chen et al. (2008)]. To have a substrate-supported geometry as well as the quality achieved with a suspended sample, graphene / hexagonal boron nitride (h-BN) heterostructures (where h-BN serves as a dielectric layer between graphene and substrate) has been a significant alternative to reduce the substrate influences on graphene. [Cho and Fuhrer (2008)] demonstrated that h-BN has strong in-plane bonds, large band gap, and planar structure which provide an ideal flat, insulating, and inert surface, isolating the graphene from the substrate. Graphene/h-BN het- erostructure exhibits electron mobility as high as ∼ 60, 000 cm2 V−1 s−1 (an order of magnitude higher than the commonly reported substrate-supported graphene). Hence the application of graphene/h-BN heterostructure for nano-electronics has attracted enormous interest in various communities. 38 Figure 3.3: Frictional characteristics between Graphene/Graphene bilayer pulled in zigzag direction. Variation of (a) shear and (b) normal stress for (c) different configurations. (d) unit cell for computation. However, until now, most of the existing studies on graphene/h-BN heterostruc- tures focus on the electrical [Ding et al. (2009)], magnetic [Yazyev and Pasquarello (2009)], and thermal properties [Jiang et al. (2011)] and field effect transistor [Jain et al. (2013)]. In order to utilize graphene/h-BN heterostructure in graphene based nano-electronics as structural and functional components, it is important to study the mechanical properties of this heterostructure. Moreover, mechanical properties (or deformations) of heterostructures are strongly tied to its electrical performance. 39 Figure 3.4: Frictional characteristics between h-BN/h-BN bilayer pulled in zigzag direction. Variation of (a) shear and (b) normal stress for different configurations. Therefore, understanding the mechanical properties of graphene/h-BN hetertostruc- tures, where the interface plays governing roles, is crucial in enabling future high- quality and practical graphene-based applications in nano-electronics. Jiang et al. (2014) experimentally observed interfacial sliding and buckling of graphene on polymer substrates. However, the studies on mechanical modeling for interfaces in graphene/h-BN heterostructure are rarely seen. Though there has been some experimental studies on friction between h-BN layers [Lee et al. (2010)], there 40 Figure 3.5: Frictional characteristics between Graphene/h-BN bilayer pulled in zigzag direction. Variation of (a) shear and (b) normal stress for different config- urations. has been not much theoretical investigation. Besides studying pristine layers, various kinds of functionalization of nano-sheets has attracted much attention recently as a means to engineer its intrinsic properties, especially for electronic applications [Elias et al. (2009)]. Besides changing the electronic band gap, functionalization also influences friction significantly. Previous AFM measurements reveled that fluorinated graphene has much higher nanoscale friction than pristine graphene [Kwon et al. (2012)]. Be- sides fluorination, hydrogenation and oxidation of graphene were also found to en- hance friction. Despite these existing studies, there is still a great need of investigating 41 Figure 3.6: Frictional characteristics between Graphene/Graphene bilayer pulled in armchair direction. Variation of (a,c) shear and (b,d) normal stress for different configurations. frictional properties of different bilayer of nanosheets systematically. The objective of this work is to study pristine bilayer: graphene/graphene, graphene/h-BN, and h- BN/h-BN in order to understand the interaction between nanomaterials. In addition, we will systematically investigate the effect of different percentages of hydrogenation on one and both-sides. 42 Figure 3.7: Frictional characteristics between h-BN/h-BN bilayer pulled in armchair direction. Variation of (a,c) shear and (b,d) normal stress for different configurations. 3.2 Models and Methods Figure 3.1 shows the periodic systems considered here. As shown in Figure 3.2, for hydrogenation, we have considered two cases i.e. one side (Figure 3.2a and two sides Figure 3.2b). For each case, different percentages of hydrogenations have been investigated. Again for each percentage, five different statistical distributions were considered and averaged value has been reported here. Molecular Dynamics simula- tion has been performed using the newly developed ReaxFF potential with LAMMPS package [Plimpton (1995)]. We have used charge atom style with initial charge for 43 Figure 3.8: Frictional characteristics between Graphene/h-BN bilayer pulled in arm- chair direction. Variation of (a,c) shear and (b,d) normal stress for different configu- rations. Carbon, Boron, and Nitrogen as 0.0, 0.90, and -0.90 respectively. After minimization of the system, both the sheets were fixed as rigid body and one of the layers was translated. There was no dynamic calculation involved during this process. Force on each atom was computed. Total forces in three different directions were divided by the sheet area to get the respective stresses. 44 Figure 3.9: Effect of hydrogen functionalization on friction between Graphene/Graphene bilayer. Variation of shear(a,c) and normal (b,d) stress for one and two sided functionalization respectively. 3.3 Results and Discussion 3.3.1 Friction During Movement in Zigzag Direction Figures 3.3, 3.4, and 3.5 show the frictional characteristics when the sheet is pulled in zigzag direction. For all the cases, unit cell is shown in Figure 3.3d. Hence we can predict that because of periodicity, frictional characteristics in zigzag and armchair √ directions will repeat in aδ = 3 and aδ = 3 respectively, where δ is the total displace- ment of the sheet. When the system is perturbed from the initial position (Position A), shear force acts in opposite direction to prevent that movement. This shear in 45 Figure 3.10: Effect of hydrogen functionalization on friction between Graphene/h-BN bilayer. Variation of shear(a) and normal (b) stress for one sided functionalization. δ opposite direction of movement increases until a = 0.50, when it reaches its maximum δ value of ∼ 60 MPa. Opposite shear then started reducing to zero at a = 1 i.e. at position B where half of the atoms in underlying sheet are under the atoms of upper δ sheet and remaining half atoms are placed at the center of hexagon. Between a =1 δ δ and a = 2, shear force does not take a very high value. At a = 1.5, the behavior is δ reversed before shear force starts favoring the movement at a = 2. We notice that initial position A is so called AA while positions B and D are known as AB position. δ Once the sheet starts moving from a = 2, shear force favors the movement until one δ period is completed i.e. a = 3. We notice that, area under positive shear (marked as magenta) and negative one (marked as green) is same. At AB configuration i.e. 46 at B and D positions, since half of the atoms at underlying sheets are exactly under the upper sheet, maximum repulsive normal force occurs in these positions. Red and green in Figure 3.3d indicate repulsion and attraction respectively. For h-BN/h-BN configurations, a qualitative frictional characteristic is same except the fact that nor- mal force in this case is always repulsive. However, we notice that compared to the case of graphene bilayer, friction is 5 times higher in this case. This is the reason we never use h-BN for commercial lubrication. However, for the graphene/h-BN heterostructure, shear stress acts in opposite δ δ direction until a = 2 (ignoring small fluctuation around a = 1). Maximum opposing δ δ stress of 60 MPa occurs at a = 1.5. Beyond a = 2, shear stress favors the movement δ with maximum stress of 80 MPa in the direction of displacement occurs at a = 2.5. In figures 3.3 and 3.4, both positions B and D correspond to same AB configuration. However, in this case, as shown by positions B and C, two different AB configurations will be different. In one case, Boron atoms will be under Carbon and Nitrogen will be at the center of hexagon. While, in other case, reverse configuration occurs. This results in ‘breaking of symmetry’ of frictional characteristics as compared to graphene/graphene and h-BN/h-BN cases. 3.3.2 Friction During Movement in Armchair Direction Figures 3.6, 3.7, 3.8 show the frictional characteristics when the sheet is displaced in armchair direction. As shown in Figure 3.6, when the sheet is displaced starting from AA configuration, there will be no shear in zigzag direction. Because of the initial juxtaposed position, underlying atoms are still in the neighborhood of cor- responding upper atoms. Thus atoms don’t experience any force in zigzag direction resulting in no shear in that direction. Friction opposes the movement until √δ = 0.5. 3a 47 At this stage (position C), all atoms are aligned in same horizontal line at a gap of √ 3a 2 . Because of this arrangement, normal stress is maximum (150 MPa) and shear stress is zero. Beyond this point, shear favors the movement. For the initial AB con- figuration, half of the atoms in underlying sheet are placed at the middle of hexagon. Hence during movement in armchair direction, atoms experience force in zigzag di- rection. The behavior in zigzag direction is reverse. In armchair direction (the gray region), the maximum shear occurs at position C (Figure 3.6c). For the h-BN/h-BN bilayer, qualitatively the overall behavior is same except force in normal direction. However, the magnitude of shear, as in the case of pulling in zigzag direction, is 5 times more. As shown in Figure 3.8, for graphene/h-BN heterostructure, when dis- placed from initial configuration AA, unlike other two cases, shear occurs in zigzag direction. For initial configuration AB, frictional characteristics in armchair direction are qualitatively same with the difference is that the force in zigzag direction acts in opposite direction in this case. 3.3.3 Effect of Surface Functionalization Figures 3.9 and 3.10 show the friction between graphene-graphene and graphene/h- BN layer for different percentages of hydrogenation. For each percentage, we have considered 5 different distributions and the averaged value has been reported here. We notice in Figures 3.9a and 3.9c that both for hydrogenation in one side and two sides, shear stress reach maximum value of 150 MPa at hydrogen coverage of 5% followed by rapid decrease. For the case of one side hydrogenation, stress drops to a steady value of 40 MPa at 20% coverage. Stress drops to zero for the case of hydro- genation on two sides at same coverage. Normal stress increases to a steady value of 1000 MPa. However, for graphene/h-BN heterostructure, we notice a different trend i.e. shear stress keeps on increasing with hydrogen coverage. 48 This can be explained by two competing mechanisms [Dong et al. (2013)]. Upon hydrogenation, interlock between hydrogen adatoms and upper moving sheet causes enhanced friction. However, when more hydrogen atoms are in contact, the repulsive force between the upper sheet and hydrogen atoms will push the sheet upward, which leads to weakening of the interlock resulting in reduced friction. However, for the case of graphene/h-BN, because of the difference in atoms in upper and lower sheets, in- terlock becomes stronger with the addition of more hydrogens. Hence we notice shear stress keeps increasing. From Figures 3.9 and 3.10, we conclude that hydrogen can be useful in lowering the friction and can be implemented for commercial lubrication. However, for graphene/h-BN, hydrogen will not work in lowering friction. Hence this can’t be used for commercial lubrication. 3.4 Conclusions We have systematically investigated frictional characteristics between three dif- ferent kinds of bilayer of nanomaterials : graphene/graphene, h-BN/h-BN, and h- BN/graphene. Our results show that friction between h-BN/h-BN is six times higher compared to the case of graphene/graphene. For the case of graphene/h-BN, friction is a little more than the case of graphene/graphene. However, in this case, because of different atoms in upper and lower sheets, we observe frictional force in the direction perpendicular to the direction of displacement of sheet. We noticed that upon hy- drogen functionalization beyond 5%, friction drastically drops for graphene/graphene case. However, for h-BN/h-BN case, because of the dominance of interlocking mech- anism, friction keeps increasing. Hence, hydrogen functionalized graphene/graphene is a potential candidate for lubrication. Further experimental investigation for verifi- cations of our results is highly desirable. Chapter 4 Graphene for Biomedical Application 4.1 Introduction Among the low-dimensional materials, graphene has attracted enormous atten- tion since its discovery in 2004 [Novoselov et al. (2004)]. Over the last decade, it has been extensively studied for various applications [Wei et al. (2012)]. Among these graphene has extensive applications in nanomedicine [Bawa et al. (2005)] especially in drug delivery and environmental barrier. There is a particular interest in graphene wrapping for nanocomposite materials [Zhou et al. (2011)], where the wrapped com- ponent can be nanoparticles [Cassagneau and Fendler (1999)], nano- wires [Han et al. (2010)], or bacteria [Akhavan et al. (2011)]. It has been recently reported that GO folds under the action of water surface tension during aerosol microdroplet drying to form crumpled graphene nano- particles [Ma et al. (2012)]. Aerosol microdroplet drying is a simple and scalable continuous nanomanufacturing process, and one of its attractive features is the potential to use multicomponent feed solutions to fabricate 49 50 composite materials with control of stoichiometry. On the other hand, many envi- ronmental technologies rely on containment by engineered barriers that inhibit the release or transport of toxicants. Graphene is a new, atomically thin, two-dimensional sheet material, whose aspect ratio, chemical resistance, flexibility, and impermeabil- ity make it a promising candidate for inclusion in a next generation of engineered barriers. Using classical molecular dynamics simulation, we will demonstrate here how can we use graphene for drug delivery and environmental barrier. 4.2 Computational Modeling of Graphene Nano- sack Technologies 4.2.1 Computational Modeling of Graphene Nanosack Tech- nologies Figure 4.1 and 4.2 show the results of MD simulations, in which we studied the interaction of water droplets with 3D graphene layer segments. We used AIREBO potential for C-C interactions [Brenner et al. (2002)], TIP4P potential for H2O [Jor- gensen et al. (1983)], and Lennard-Jones type of pair potential for C and H2O inter- actions [Walther et al. (2004)]. Figure 4.1 shows that when graphene segments are immersed in nanodroplets, they spontaneously migrate to the outer surface, driven by hydrophobic forces. Figure 4.2 shows a series of images in which graphene of small lateral dimensions interacts with water nanodroplets during drying. The graphene segment is initially placed at the interface, and as drying shrinks the droplet, curva- ture is induced in the graphene sheet. Further drying, however, produces a relaxation back to the planar state, rather than closure and buckling. 51 Figure 4.1: Series of images from MD simulations on a nanoscale graphene sheet seg- ment interacting with a water nanodroplet. The simulation is initialized by immersing graphene in the droplet (a) and the snapshots correspond to 0.80 nanoseconds (b), 1.20 ns (c), and 2.00 ns (d). After 2.00 nanoseconds the sheet is observed to sponta- neously migrate to the droplet surface. The NVE-molecular dynamics calculations at 300K were performed with 26120 water molecules. The dimensions of the graphene sheet used in the simulations are 8.4 nm × 8.73 nm. 4.2.2 Interaction of Graphene Oxide and Water For the rational development of nanosack technologies, we would like to better understand GO-water interactions and GO buckling, collapse, and creasing. For sim- plicity, we omit the filler phase and consider the limiting case of empty nanosacks. Figure 4.3a and 4.3b show selected images from MD simulations of droplet drying in the presence of monolayer graphene and GO respectively after an initial period where they first localize at the gas-water interface (Figure 4.1). We used the LAMMPS code 52 [Plimpton (1995)] and modeled the interatomic interactions with a reactive force field (ReaxFF), which is a general bond- order-dependent potential that provides accurate descriptions of bond breaking and bond formation in hydrocarbonoxygen systems [Chenoweth et al. (2008)]. This potential has been successfully used in other studies of graphene/water systems [Medhekar et al. (2010)]. To manage the simulation size, we used 2000 H2 O molecules and a semi-2D graphene film of 130 × 15 ˚ A with peri- odic boundary conditions in the Z direction. Graphene oxide structure is prepared by thermal annealing (T = 1000 K) of C10 O2 (OH)2 structure, which is two epoxy and hydroxyl groups per 10 carbon atoms and distributed randomly on either side of the graphene basal plane [Medhekar et al. (2010)]. The size of the simulation box is 150 × 100 × 15 ˚ A, and charge transfer is performed by the charge equilibration (QEq) method [Rappe and Goddard (1991)] at every MD step. The temperature of the system is kept at 300 K, controlled by rescaling atomic velocities every 10 MD steps (each MD time step ∆t = 0.2 fs). To mimic drying, 10% of the H2 O molecules were randomly selected and removed every 5 × 104 fs. The simulations predict very different behavior for graphene and GO during droplet drying (Figure 4.3). For graphene there is a gap at the water interface, and the droplet appears to template or “guide” the graphene into a scroll structure during drying. We believe that weak van der Waals forces in the water/graphene system [Leenaerts et al. (2009)] allow graphene to slide on the droplet surface, which enables this “guide and glide” assembly mode, similar to observations from previous simulations [Patra et al. (2009)]. Graphene oxide in contrast appears to stick on the droplet volume is reduced by drying (“cling and drag” mechanism). Water-GO in- teractions are known to involve hydrogen bonding with oxygen-containing functional groups. We calculated attraction energies between single water molecular and epoxy, 53 carbonyl, and hydroxyl groups in our simulations as 0.30, 0.31, and 0.17 eV, respec- tively, typical of hydrogen-bonded interactions, and we believe these strong forces are responsible for the “cling and drag” assembly. Kinks appear early in the evolving GO structure, and the end product is a crumpled and plastically deformed particle. To elucidate the atomic level events during the folding of graphene oxide, we ex- amine the spatial distribution of functional groups and their relation to the evolving structure of the GO layer. At the beginning, the GO film has three types of func- tional groups, hydroxyl, epoxy and carbonyl (Figure 4.4a and 4.4c). During folding we observe some concentration of carboxyl and carbonyl functionalities in the “kink” regions of plastic deformation (Figure 4.4d), while hydroxyl groups show preference for the joint regions (Figure 4.4b). Oxidized graphene surfaces have been shown to fold or kink in preferred locations along fault lines associated with functional groups or defects [Schniepp et al. (2008)] and Figure 4.4 suggests these regions are relatively rich in epoxy and carbonyl (CO) functional groups. Both epoxy and carbonyl groups involve two bonds between oxygen and the graphene backbone carbon, while COH involves one, and it is likely that the high concentrations of epoxy and carbonyl are associated with larger perturbations to planarity and bonding and are the predeter- mined preferred sites for kink formation during water-actuated collapse. More work is needed to understand the atomic-level events that accompany plastic deformation during crumpling. 4.3 Mercury Diffusion Between Graphene Bilayer To rationalize the thickness dependent permeability of Hg through GO film, we set a simple model as shown in Figure 4.5b. This model is designed based on two previous works of MD simulations for G or GO with H2 O molecules. Medhekar et al. 54 (2010) showed that the interlayer spacing of GO can be controlled by H2O fraction which plays a role in formation of both extent and collective strength of interlayer hy- drogen bond networks. Recently, Nair et al. (2012) suggested a model to investigate water permeation through sub-nm graphene capillary and found that 2D network of H2 O molecule starts to form 6 ˚ A of interlayer spacing between two pristine graphene sheets. From these work, we notice that oxidized region with functional group atoms on GO basal plane play plays a role of spacers that keep graphene planes, and Hg diffusion occurs mainly through the water molecule, i.e. aqueous environment. There- fore, now we can set up the model of diffusion coefficient calculation of Hg in water. We combine set of MD potentials among the materials (Hg, H2 O, and C) by adopt- ing the potential hybrid method which enables the use of multiple potentials in one simulation [Plimpton (1995)]. For describing water interactions, SPC/E water model is employed [Berendsen et al. (1987)] and the long-range Coulombic interactions are computed in conjunction with the particle-particle particle-mesh solver. Other in- teractions such as Hg-Hg and C-C are assumed to be Van der Waals interactions described by the Lennard-Jones 12-6 potential [Chen et al. (2005)]. It is good reason to use simple LJ potential for describing the C-C interactions in this work, because all graphene sheets are assumed to be static. The standard Lorentz-Berthelot mixing rules are utilized for obtaining LJ potential parameters for the cross-interactions. The cut-off distance for all interactions is set to 10 ˚ A. The systems we considered are two graphene sheets which size is 63 ˚ A × 63 ˚ A and H2 O molecules contains one Hg atom in between the sheets. Periodic boundary condition is enforced along lateral direction to mimic the infinite width of graphene sandwich structure. The initial spacing be- tween the sheets is chosen 20 ˚ A and keeps closing the gap up to the desired interlayer distance with constant velocity. The number of H2 O molecules is determined based on both maximum allowance volume between sheets and equilibrium number density 55 of H2 O molecule in liquid phase (33.3 molecules per nm3 ). The temperature of the system except carbon atoms is kept at 300K controlled by a Berendsen thermostat, which rescales their velocities every timestep (each MD time step ∆ = 1.0 fs)[]. The mean-squared displacement (MSD) is utilized for obtaining the diffusion coefficient 1 and the relation is expressed by D = 6t < |ri (t) − ri (t)|2 >. The system is relaxed at the desired interlayer distance for 0.3 ns and, then, the MSD of a Hg atom starts to calculated for 0.5 ns under NVT ensemble. All simulations are carried out by the LAMMPS package [Plimpton (1995)]. Figure 4.5 shows the results of Hg diffusion coefficient in water for interlayer distance of graphene sheet. According to the results, Hg seems to be frozen when the spacing is less than 7 ˚ A and becomes mobile from 8 ˚ A of interlayer distance. The Hg diffusivity in bulk water at 300K is 1.88cm−5 /sec [Kuss et al. (2009)] and this value can be the upper limit of the Hg diffusivity in this simulation. Consequently, the simulation captures this trend and the value becomes 1.3cm−5 /sec for 10 ˚ A of interlayer distance. From atomic point of view, Hg is clogged by immobile H2 O molecules for the distance less than 10 ˚ A and placed at the center of the carbon ring. In such a narrow spacing, water is formed immobile monolayer structure, while double layer structure of H2 O is gradually aligned when the spacing becomes wider than 7˚ A. Eventually, double layer enables Hg atom to move through the H2O molecules by replacing their positions. These results provide an important understanding of the Hg diffusion mechanism between GO sheets. Furthermore, the critical value of interlayer distance for Hg permeation is in excellent agreement with the current experiments. 56 4.4 Conclusions The generality of the assembly principle, the flexibility in composition, the ease of nanomanufacturing, and the increasing availability of the GO precursor are strong mo- tivation for further development of graphene nanosack technologies. Graphene-based materials show great promise for a next generation nanomedicine and environmen- tal barrier materials. The excellent barrier performance exhibited for liquid water and mercury vapor are not believed to be highly chemically specific properties, and could likely be extended to a wide range of toxicants. Much more work is needed on the large-scale fabrication, permeability, and mechanical stability of graphene-based environmental barriers and drug-delivery systems. 57 Figure 4.2: Selected images from MD simulations of a small graphene segment (8.4 nm × 8.73 nm) interacting with a water nano-droplet during drying. (a) initial state, (b) 0.04 ns, (c) 0.12 ns, (d) 0.18 ns. Shrinking of the droplet first induces curvature in the graphene, but as drying proceeds the curvature reaches a maximum value (near c) and then the graphene relaxes back to a planar configuration. This behavior is consistent with the small size of the graphene segment and the high energy penalty for the high curvature required for closure. The simulations were carried out at room temperature. Drying was achieved by removing 5% water molecules every 0.01 ns. 58 Figure 4.3: Projected concentration distributions of the functional groups on the graphene oxide film. Hydroxyl groups (a) before and (b) after drying. Carbonyl or epoxy groups (c) before and (d) after drying. Arrows indicate the direction of folding. Dashed line guides the overall shape of the film. The color codes for the average density of each functional group from lowest (white) to highest (red) at the corresponding location. Some of the kink or creases regions appear to be enriched in carbonyl / epoxy groups relative to hydroxyl groups. 59 Figure 4.4: Mechanisms of nanosack formation. (a,b) Molecular dynamics sim- ulations of water droplet-actuated scrolling, folding, or collapse for (a) monolayer graphene, which scrolls through a “guide and glide” mechanism. (b) Monolayer GO which closes and collapses by a “cling and drag” mechanism. 60 Figure 4.5: (a) Molecular dynamics (MD) simulations of Hg-atom diffusion in graphene interlayer spaces containing water. The diffusivity of a tracer Hg-atom is seen to decrease with decreasing interlayer spacing, and Hg becomes immobile be- low 7 ˚A spacing. Inset shows a graphene interlayer region with with 8 ˚ A spacing, and one mercury atom (blue) in water (red-grey beads). The dashed line gives the known diffusivity of Hg in bulk water. Inset : Mean Square Distribution at different bilayer spacing. (b) Top and Side View of Hg-atom diffusion in graphene interlayer. Chapter 5 Energy Research : Where Chemistry Meets Mechanics 5.1 Introduction : Why shall we study Energy Storage Systems ? In recent decades, there has been a urgent need for energy storage of all kinds : mobile electronic devices, transportation, heavy industry applications, commercial- ization of renewable resources such as solar and wind power. We need storage size in all scales : milliwatts for small devises to multi-megawatts for heavy industry/space applications. Among various storage systems, there has been a great attention in Hy- brid Electric Vehicles (HEVs), where batteries and/or capacitors are used to capture the energy. However, with the enormous increase in demand of energy, the next gen- eration of electric vehicles will be plug-in hybrids, where larger batteries will be used and the vehicles can be recharged by plugging into the electrical power line. In all practical applications of energy storage devices, low cost and long life are essential for commercial success. However, the present chemical storage batteries and capacitor 61 62 Figure 5.1: The power density and energy density for batteries, capacitors, and fuel cells. (Energy is the capacity to do work; power is the rate at which work is done.) charge storage systems will not be sufficient enough to achieve our requirements. 5.2 Batteries : The most common form of storing electrical energy Among many energy storage systems, the most common form of storing electrical energy is battery, which ranges in size from button cells used in portable electronics to multi-megawatt applications. Rechargeable batteries have been extensively used for last decades as a efficient storage devices, with output energy generally exceeding 90% of input energy. As shown in Figure 5.1 1 , the energy obtained from any storage device is strongly depended not only on the device but also on the power output. We 1 Source : MRS Bulletin 63 Figure 5.2: The configuration of a battery. can see that compared to capacitors, we can have higher energies from batteries 2 . Hence this thesis will focus on batteries. 5.2.1 The configuration of a battery Figure 5.2 shows the basic configuration of a battery 3 . It contains one or more electrochemical cells, which can be connected in series or parallel to provide the desired voltage and power. The anode is the electropositive electrode from which electrons are generated for the external work. In a lithium cell, the anode contains lithium, commonly held within graphite or silicon in the well-known lithium-ion bat- teries. The cathode is the electronegative electrode to which positive ions migrate inside the cell and to which electrons migrate through the external electrical circuit on discharge; during charging, the ions and electrons flow in the opposite directions. The electrolyte allows the flow of the positive ions, for example, lithium ions from one electrode to another. It allows only the flow of ions and not the flow of electrons. 2 However, capacitors are high-power devices with limited energy storage applications 3 Source : Basic Research Needs for Electrical Energy Storage (Office of Basic Energy Sciences, U.S. Department of Energy, Washington, DC, 2007) 64 The commonly used electrolyte is a liquid solution containing a salt dissolved in a solvent. The electrolyte must be stable in the presence of both electrodes. The cur- rent collectors allow the transport of electrons to and from the electrodes. They are usually metals and must not react with the electrode materials. Typically, copper is used for the anode and aluminum for the cathode (the lighter weight aluminum reacts with lithium and, therefore, cannot be used for lithium based anodes). 5.3 Major Challenges and Opportunities in En- ergy Research Among many open challenges, some of the issues that we need to immediately address. [Whittingham (2008)]: 5.3.1 Nanomaterials for Energy Storage In recent years, several nanomaterials like graphene, its allotropes, boron nitride, silicene, MoS2 etc have got tremendous attentions. Among the various other ap- plications, these have a lot of importance in energy storage. However, there are many fundamental questions we need to answer : How can the different properties of nanoparticles and their composites be used to increase the power and energy ef- ficiency of battery systems? How do we control the morphology and particle size of nanomaterials to optimize transport behavior as well as packing density? We will address some of these questions in Chapter 7 and 8. 5.3.2 Materials with Multipurpose for Energy Storage As shown in Figure 5.2, batteries have many components that are critical to performance but don’t make any contribution towards energy or power density and in 65 fact reduce both the gravimetric and volumetric capacities. Hence a major challenge in energy research is to develop multipurpose materials, e.g. a conductive binder or a separator that can also serve as a cell cutoff in the event of thermal runaway. 5.3.3 Theoretical Modeling of Energy Storage Systems Besides designing novel materials for energy storage, there are many phenomena that need to be theoretically modeled : • Modeling the interface : The solid electrolyte interphase (SEI) layer formed between the electrodes and the electrolytes is very critical to the performance of electrochemical devices. Despite few recent attempts, this complex and con- tinually changing interface is not well understood yet. • Modeling kinetics and mechanism of phase changes : The mechanism by which mew phases are nucleated and translate through electrode materials is not yet properly understood. • Modeling nanoscale materials and surfaces : It is important to model the length scale at which the kinetics and thermodynamics of nanoparticles become significantly different from those of the bulk. • Design of electrode structures : Control of the microstructure of the com- plex composites making up electrodes is critical to the long-term operation of the battery. Modeling is needed to prescribe the optimum particle size and mor- phology, the optimum mixture of these particulates with the conductive diluent and binder, the optimum porosity, and the impact of electrodes swelling and contracting on reaction. 66 5.4 Energy Storage Problems Considered in this Thesis In Section 5.3, we have discussed some key issues we need to immediately ad- dressed. Because of the limitation of time, we will consider some burning issues in this thesis : 5.4.1 Atomistic Mechanism of Phase Boundary Motion Figure 5.3: Motivation of Modeling Phase Boundary : (a) Schematic of phase boundary formation (b-f) Experimental observation of phase boundary formation. We mentioned in Section 5.3.3, modeling of many interesting phenomena of energy storage is of utmost importance. Among these, as shown in Figure 5.3, a thin layer is formed during the lithiation of crystalline silicon. Chon et al. (2011) first have 67 experimentally investigated phase boundary formation for three different orientations of crystalline silicon : [100], [110], [111] as shown in Figure 5.3 d,e,f respectively. Liu et al. (2012) (Figure 5.3b-c) later experimentally discovered this as well. However, despite experimental observations, there has been no systematic theoretical study on phase boundary motion. So in Chapter 6, we will discuss about it. 5.4.2 2D Materials for Energy Storage Figure 5.4: 2D Materials offer more promise for energy storage. As discussed in Section 5.3.2, ever since different nanomaterials have been discov- ered, there is a tremendous interest about its application in energy storage. Among different nanomaterials, 2D materials offers most promise as these have several ad- vantages : • More active sites than 0D and 1D and exhibit more effective surface than other structures. • Superior properties including small weight, large surface area and a sensible distribution. 68 • Ideal frameworks for fast energy storage, which requires stability, high ac- tive surface area and open shortened path for adatom insertion/deinsertion. In Chapter 7, we will discuss how can we use celebrated 2D nanomaterial, Graphene, for high-capacity energy storage for Lithium Ion Batteries(LIBs). Chapter 8 will discuss about the different problems associated with LIBs. As a potential alternatives, Na- and Ca- ion batteries offer real promise. As shown in Figure 5.5, Na and Ca ions are abundant. Moreover, nature stores energy with Na and Ca ions, not Li ions. Figure 5.5: Nature stores energy with Na and Ca ions, not Li ions. Chapter 6 Atomistic Mechanisms of Phase Boundary Evolution during Initial Lithiation of Crystalline Silicon 6.1 Introduction Rechargeable lithium-ion batteries (LIBs) have been extensively used in portable electronics , light vehicles, and miscellaneous power devices over the past decades because of their high gravimetric energy storage and are currently being considered for use in critical applications such as heavy automotive and medical devices [Dahn et al. (1995)]. In terms of energy density, the seemingly ubiquitous LIBs exhibit su- perb performance as compared to other types of rechargeable batteries [Tarascon and Armand (2001)]. In most Li-ion batteries, the negative electrode (anode) material is graphite, which forms lithiumgraphite intercalation compounds (Li-GICs)[Dahn (1991)]. However, silicon can store approximately 10 times more Li than graphite (the gravimetric energy densities of silicon and graphite are 3579 and 372 mAh/g, 69 70 respectively). In addition, silicon and silicon-composite materials are cheap, readily available, and possess high charge capacity and low discharge potential. Thus, silicon- based materials are strong potential candidates for anode materials in LIBs [Dimov et al. (2007)]. However, the high charge capacity is associated with excessive volume swelling (∼ 370%) and structural changes during the charging/discharging process, leading to mechanical fracture, inter-particle disconnection, irreversible capacity loss, and limited electrode cycle life [Lee et al. (2001)]. Despite the advantages of Si elec- trodes in LIBs,mechanical damages induced by large volumetric strain during cycling present a major challenge in optimizing the microstructure of composite electrodes. To overcome these problems, various nano-structured architectures of Si electrodes have been proposed, with proven improvements in the mechanical durability of Si- electrode batteries [Chan et al. (2008), Zhang et al. (2010)]. The dynamics of Li in Si anodes have been investigated experimentally and the- oretically [Shenoy et al. (2010), Pharr et al. (2012)]. Using transmission electron mi- croscopy (TEM), Liu et al. (2012) directly observed the crystal-to-amorphous phase transformation (CAPT) of Si during lithiation resulting from anisotropic swelling as tensile hoop stress accumulated in the lithiated shell. Anomalous shape changes dur- ing the lithiation of single crystal Si nanopillars have been shown to depend on the orientation of the Si structure [Lee et al. (2011)]. The composition of the lithiated region and its CAPT has been revealed by nuclear magnetic resonance and by analyz- ing the LiSi interphase structures [Trill et al. (2010)]. The phase boundary formation and the stress associated with damage evolution have been characterized by real-time measurement during the initial lithiation/delithiation cycle [Chon et al. (2011)]. Liu et al. (2012) observed a sharp interface between the c-Si and an a-Lix Si alloy. The observed phase boundary by Chon et al. (2011) between the c-Si and the lithiated Si 71 region is expected to be pivotal in understanding the plastic deformation of electrodes. In contrast to the experimental studies, theoretical approaches to date have been hindered by our limited ability to interpret the experimental observations or to find physical parameters that can be integrated into large-scale simulations. Lithiation- induced shape changes and evolution of stress during the volumetric expansion of plastically deformed lithiated silicon have been investigated using continuum models [Zhao et al. (2011), Pharr et al. (2012)]. Atomic-scale theoretical studies, however, have not progressed beyond the density functional theory (DFT), which character- izes diffusion energy barriers, elastic properties, and formation energies [Chan et al. (2008), Zhao et al. (2011), Huang and Zhu (2011)]. The ab initio molecular dynamics method is a useful for exploring the dynamics during lithiation [Johari et al. (2011)] but is restricted to lengths and time scales of nanometers and picoseconds, respec- tively. An alternative (and more promising) method is classical molecular dynamics (MD), which yields information on structural and thermodynamic evolutions under various conditions. MD operates over tens of nanometers and a few nanoseconds of simulation time. However, MD requires knowledge of the interatomic potentials that accurately describe chemical reactions. Such knowledge is limited for the specific case of Li-ion batteries to the formation of the solid electrolyte interphase (SEI) [Kim et al. (2011)]. Because of lack of suitable potentials, MD simulations have not yet been imple- mented to investigate phase boundary motion in LiSi systems, despite the importance of this process to the performance of the anode. However, Fan et al. (2013) recently have employed a newly developed reactive force field (ReaxFF) to study the me- chanical properties of a − Lix Si alloys using MD simulations. This ReaxFF potential 72 provides accurate predictions of a set of fundamental properties for Lix Si alloys, such as Li composition dependent elastic modulus, open-cell voltage, and volume expan- sion. In addition, ReaxFF is capable of describing various bonding environments, essential for improving the chemical accuracy of simulated structures and properties spanning a wide range of Li compositions. The optimized force field parameters, used to predict lattice parameters and the heat of formation of selected condensed phases, closely match those of the previous DFT calculations and experiments. In this work, we have studied the dynamic evolution of the first lithiation cycle into var- ious Si structures (a-Si and c-Si with (100)-, (110)-, and (111)-oriented structures). In addition, the structural evolution of the phase boundary formation between the crystalline and lithiated silicon is systematically investigated. 6.2 Simulation Methods To obtain the a-Si structure, a diamond cubic crystal was melted followed by quenching of liquid by an NPT MD calculation [Luedtke and Landman (1989)]. The c-Si structures of different orientations were created by rotating the Si(100) structure considering the periodicity at the boundary. We focused solely on lithiation behavior, ignoring the effects of electric current density, electrolyte, and cathode materials. The sample size extended sufficiently along the axial direction for formation and evolution of lithiation to be observed during the simulation time but was restricted to slightly longer than the cutoff radius of the interatomic potential (here, 10 ˚ A) in order to maximize the computational efficiency. Given the above considerations, the size of the simulation domain was approximately 300 ˚ A × 17 ˚ A (where the lengths denote the axial and off-axial directions,respectively). The axial direction was occupied by Si atoms up to approximately 110 ˚ A; the remaining space contained a sufficient number of Li atoms (Li : Si = 12 : 7 in all cases). In total, the systems were composed of 73 about 3908 and 6253 atoms, respectively. Periodic boundary conditions were enforced along both off-axial directions. A virtual flat wall, erected at the far axial boundary, generated a perpendicular force to prevent atoms from escaping the simulation box. Motivated by the experimental [Balke et al. (2010), Pell (1960)] and theoretical [Johari et al. (2011)] work at high temperatures, the MD calculations were performed at temperatures ranging from 600 to 1500 K in order to reduce the simulation time by accelerating the reactions. This approach has been employed in several previous studies for simulating chemical reactions on time scales relevant for MD simulations. For example, Johari et al. (2011) studied different stages of lithiation of crystalline as well as a-Si anodes at 1200 K and have successfully examined the diffusion kinetics of Li and Si atoms in both crystalline and amorphous Si. The temperature was controlled by a Berendsen thermostat, which rescaled atom velocities at each step (one MD time step ∆t = 0.2 fs)[Berendsen et al. (1984)]. The charge transfer was performed by the charge equilibrium (QEq) method [Rappe and Goddard (1991)] at each step. 6.3 Results and Discussion During initial lithiation, distinct structural evolutions are observed in the lithiated Si region of different Si structures (Figure 6.1). The reaction is most active in a-Si (Figure 6.1a). As Li reacts with c-Si, different interface structures are formed. The thickness of the lithiated Si region depends on the direction of the crystal orienta- tions. The unique interphase, the so-called phase boundary, is formed between the regions of c-Si and lithiated Si and can be easily distinguished by the atom colors rep- resenting the charge states (Figure 6.1b-d). Since the atoms at the phase boundary are well ordered and evenly distributed, we infer that this region is crystalline. As 74 described by the supporting movies (uploaded in YouTube 1 ), which are taken from the simulations at 1200 K, the lithiation behavior at the phase boundary is governed by hopping diffusion between the tetrahedral sites (the most stable positions in the Si crystal for Li insertion [Zhao et al. (2011)]). The energy barrier of a Li atom hopping between two tetrahedral sites in the Si crystal structure is computed by a nudged elastic band (NEB) calculation [Henkelman and Jonsson (2000)]. Multiple NEB simulations established an energy barrier of 0.52 eV for Li diffusion, which is in good agreement with previous results calculated by DFT [Kim et al. (2010)]. Lithiation behavior at the phase boundary (both direction of Li diffusion and propagation of Si - Si bond fracture) depends heavily on crystalline orientation. Pre- vious studies have reported that Li preferentially diffuses along the [110] direction (as demonstrated in Figure 6.1) and tends to accumulate at tetrahedral sites [Lee et al. (2011)], causing Si-Si bond breakage between (111) planes. This mechanism is sim- ilar to that of half-stacking fault exhibited by H atoms [Martsinovich et al. (2003)]. Furthermore, breaking is developed by the chemical change known as charge-induced Si-bond weakening, rather than by mechanical swelling caused by the Li insertions [Zhao et al. (2011)]. Therefore, the thickness of the phase boundary is strongly re- lated to the Si - Si bond locations between the (111) planes (highlighted in red in Figure 6.1b-d). In Si(100), these bonds are oblique to the normal surface, which in- creases the time required for Li atoms to occupy all tetrahedral sites in a particular plane. Therefore, compared with the other states, the phase boundary structure of Si(100) shows a longer region of phase evolution from crystalline to amorphous struc- tures. Conversely, the bonds of Si(111) are normal to the surface, enabling CAPT to develop by layer-by-layer cleavage. Consequently, formation of a phase boundary at the Si(111) surface would be unlikely. This behavior, demonstrated in the supporting 1 Movies of lithiation into Si(100) and Si(110) 75 Figure 6.1: Atom and charge configurations during initial lithiation of (a) a-Si, and (b - d) c-Si. The snapshots of c-Si are viewed along the [110] direction, and a set of bonds connecting two (111) planes are highlighted by red lines. The atom colors denote the charge state. Green indicates neutral atoms, blue indicates Si atoms q < 0.5e, and red indicates Li atoms q > +0.5e. All snapshots are taken after a reaction time of 200 ps at 1200 K. movie 2 , has also been recently observed by in situ TEM [Liu et al. (2012)]. We now attempt to quantify the physical properties associated with phase bound- ary motion. Propagation velocities are calculated from the final thickness of the remaining Si after 300 ps MD simulation at 1200 K. The phase transition occurs in a region less than 2 nm wide. The smallest phase boundary appears in the [111] 2 Movies of lithiation into Si(111) 76 crystal, where the transformation from a-Lix Si to c-Si is abrupt. The phase bound- ary thicknesses obtained from our MD simulations and the HR TEM by Chon et al. (2011) are listed in Table 6.1, together with the simulated phase boundary velocities. The fastest propagation velocity occurs in a-Si (Table 6.1). Among the crystalline Si samples, the propagation velocity in Si(111) exceeds that in Si(100) by approximately 10%. The propagation velocity in Si(110) lies between these values. Relating these results to the phase boundary thicknesses, we infer that a thinner boundary enables Li atoms to more easily penetrate the Si phase, increasing the phase boundary prop- agation. Beyond the phase boundary, the lithiated Si region is clearly amorphous. This phenomenon is attributed to CAPT, reported in recent experimental work [Liu et al. (2012)]. The amorphous regions (Figure fig:LiSi:charge) appear to extend along the axial direction from the initial Si surface. However, this effect might result from ac- tive diffusion of Si out of the structure, which exceeds that of Li into Si. The system constraints along the off-axial directions can trigger the growth of lithiated Si along the axial direction, which is distinct from previous work using thin films [Chon et al. (2011)]. Regarding the growth mechanism, once the a-Li-Si alloy begins to expand, it is pushed away from the interface between the crystalline and amorphous phase because the yield strength of the c-Si greatly exceeds that of the a-Lix Si alloy[Lee et al. (2011)]. To investigate atomic diffusion during lithiation, we compute the concentration profile from the results of Figure 6.1 (see Figure 6.2). The Si and Li concentrations are observed to be almost equal around the interface region. Although a long, clearly 77 Figure 6.2: Relative concentration profiles based on the results of Figure 6.1. Con- centration is regarded as the ratio of the number of Li (or Si) atoms to the total number of atoms in the given space. The first point of intersection from the Li side is taken as the interface (CSi = CLi = 0.50). defined region of 1 : 1 Li : Si composition is developed in the a-Si sample, this re- gion cannot be regarded as the phase boundary because the concentration of both Si and Li gradually decreases or increases outward from the interface, signifying normal diffusion. By contrast, the phase boundary is characterized by drastic changes in con- centration, as observed in the c-Si cases. The Si(100) sample develops the thickest and most obvious phase boundary, while those of Si(110) and Si(111) are relatively vague. The concentration profile remains at approximately 0.8 Li atoms per Si atom 78 beyond the phase boundary (in the amorphous lithiated Si region). In the a-Si sam- ple, Li(Si) concentration monotonically increases (decreases) up to the pure Li region. On the basis of the concentration and the phase boundary thickness, we hypothesize that the amorphous lithiated Si region is a-Li15 Si4 (in which the atomic ratio of Li to Si is 0.79). To investigate the breakage of Si-Si network bonds across the lithiated Si region, thickness of phase boundary (nm) samples MD simulation experiment velocity (nm/ns) a-Si N/A N/A 15.5 Si(100) 1.61 ∼1.5 8.80 Si(110) 1.23 ∼1.0 9.47 Si(111) 0.59 <1.5 9.70 Table 6.1: Summary of Physical Properties Pertaining to the Phase Boundary Evo- lution during the Lithiation of a-Si and c-Si the partial coordination number of Si-Si pairs (CNSi−Si ) is computed. As shown in Figure 6.3a, the initial CNSi−Si is 4.0 (indicating a perfect diamond structure) and declines to zero (an isolated Si atom or pure Li region). The CNSi−Si in the c- Si structures changes suddenly around the interface, indicating the presence of the phase boundary. The lack of a phase boundary in the a-Si structure is also verified, since the CNSi−Si decays monotonically with constant slope. According to Figure 6.2, the phase boundaries develop at 10.00.0 ˚ A from the interface, where the CNSi−Si remains approximately 4.0. This implies that the Si-Si bond network at the phase boundary sustains its lattice structure by allowing the diffusion of Li atoms into the tetrahedral. Once the relative Li concentration reaches around 0.5,the Si-Si bonds become weaker than those of Li-Si and are eventually eroded by Li atoms. According to a recent DFT study, CAPT is triggered at a ratio of 0.3 Li atoms per Si atom 79 [Jung and Han (2012)], which is consistent with our result considering the differences between the two simulation models (Li diffusion from one side vs bulk mixing of Li-Si). We next investigate the structural evolution of the Si segments during lithiation by analyzing the local structure of the Si-Si network. Johari et al. (2011) investigated the mixing mechanism of lithiated Si and analyzed the evolution of the Si-Si bond network structure infiltrated by Li atoms at high temperature. They reported that Li atoms break the Si rings and chains to create ephemeral structures such as stars and boomerangs, which eventually transform to Si-Si dumbbells and isolated Si atoms. We extend this study to discover the spatial evolution of the Si crystal dissolution. Although we cannot consider every possible type of Si segment, important structures such as chains (herein, we consider 5-node chains only), boomerangs, dumbbells, and isolated Si atoms are extracted as representatives. The spatial distribution of these structures across the samples is shown in Figure 6.3b. Two inferences can be drawn from this study. First, the distribution of chain structures in a-Si is much broader than in c-Si, implying that the weak Si-Si bond network in a-Si encourages lithiation and that high-node Si segments such as chains can survive for extended times. A broad chain distribution is also observed in Si(110), consistent with the crystallographic effect in which lithiation along the [110] direction of the diamond cubic Si structure proceeds rapidly (as described by Lee et al. (2011)). Second, the distribution of Si segments alters along the length of the strip; high nodes (chains, boomerangs) and low nodes (dumbbells, isolated atoms) dominate at the Si and the Li side, respec- tively. Consequently, the lithiated Si region predominantly contains dumbbells and isolated Si atoms, while isolated atoms consistently appear at the outermost position of the lithiated Si region. 80 The structure of Li-Si composites at the phase boundary and within the lithiated Si region is revealed by the partial radial distribution function (RDF), g(r)(see Figure 6.4). The geometry of the phase boundary region depends on Si crystal orientation (as shown in Figure 6.1). The ratio of Li to Si atoms in the phase boundary region is assumed to be in between 0.4 and 0.6. From the concentration profiles (see Figure 6.2), we realize that the phase boundary formed between the crystalline and lithiated silicon is of thickness ∼1 nm. This is in perfect agreement with the in situ TEM study by Liu et al. (2012). The interface of the reaction front is atomically sharp, and the Li:Si ratio is almost one in this region. In the phase boundary region of Si-Si, the first concentration peak occurs at 2.4 ˚ A, followed by second and third peaks at 4.2 and 5.8 ˚ A,respectively. These peaks are located at the same position in the phase boundary regions of all cases. The presence of sharp peaks indicates that atoms are confined at their locations; in other words, the phase boundary is crystalline. The g(r) of Si-Li yields smaller peaks than those of Si-Si. In the g(r) of Li-Li,the first peak is broadened while subsequent peaks are flattened. The location of the first peak is the same at all orientations. We compute the g(r) of c-LiSi and compare its peak (marked in black) with the g(r) of the phase boundary region. The compari- son confirms that the phase boundary between the c-Si and lithiated Si regions is of c-LiSi structure and is ∼ 1 nm thick (see Figure 6.2). We then analyze the g(r) of the lithiated Si region (right panel in Figure 6.4). Because this region is amorphous, most of the g(r) peaks are blunted. Hence, to explore the structure in this region, we consider only the first (or possibly the second) peak. Comparing the g(r) results of the lithiated Si region with that of a-Li4 Si as a reference (marked in red), we confirm that the lithiated Si zone is a-Li15 Si4 . The reference g(r) has been extracted from previous DFT work [Chevrier and Dahn (2009)]. 81 We hypothesize that lithiation quantity correlates with both phase boundary mo- tion and surface density of Si in various orientations. Si atoms begin to dissociate from the lattice structure immediately following the phase boundary region, and the number of interfacial Si atoms is identical to the number beyond the interface. More- over, as revealed by the concentration profile and partial RDF, the lithiated region is consistently an amorphous phase of Li15 Si4 . Therefore, the lithiation quantity can be estimated by counting the number of Si atoms that have undergone phase boundary propagations. The surface densities of (100), (110), and (111) Si are 6.78, 9.59, and 7.83 atoms/nm2 , respectively. Because the time required for Li to cross two crystalline planes are likely rapid at all orientations, it is ignored in our analysis. Multiplying the velocities by the surface densities and normalizing with respect to (100), we obtain 1.0, 1.52, and 1.27 for (100), (110), and (111), respectively. These values qualita- tively predict the growth speed of fully lithiated Si and should be strongly correlated with orientation-dependent swelling behavior during lithiation, as reported in previ- ous studies [Lee et al. (2011)]. Finally, we reveal the formation mechanism of the phase boundary and also CAPT on an atomic scale. The evolution of the phase boundary is traced by detailed obser- vations and careful comparisons during the lithiation process. To this end, varying the temperature of the MD simulation controls the reaction speed. Both the phase boundary and CAPT appear to evolve via a three steps mechanism (see Figure 6.5). In the first step (Figure 6.5a) the Li soaks into the Si structure. During this period, the Li atoms diffuse into the tetrahedral sites of the Si lattice via a hopping mecha- nism, but the Si-Si bond network at the interface remains rigid. The arrow in Figure 6.5 indicates that the concentration of Li atoms increases toward the Si side, whereas no Si atoms diffuse out at this stage. Beyond the soaking period, the phase boundary 82 develops its territory by increasingly filling the tetrahedral sites. We refer to this step as the expansion period (Figure 6.5b). In contrast to the first step, the phase boundary expands both inward and outward from the interface. Finally, the matured phase boundary propagates, maintaining a thickness of ∼ 1 nm. As shown in Figure 6.5c, the Li concentration at this stage shifts toward the left (into the Si structure) and saturates at 0.8 on the right (beyond the phase boundary). 6.4 Conclusions We discovered the phase boundary at which Li atoms react with c-Si structures. Li atoms in the Si crystalline phase preferentially diffuse along the [110] direction, disrupting the Si-Si bonds between (111) planes. The location of the (111) plane plays a key role in the thickness of the phase boundary; a relatively thick phase boundary is developed at the (100) surface, while an atomically sharp interface of negligible thickness is formed at the (111) surface. The phase boundary of a-Si is indistinguishable, as verified by comprehensive analysis of concentration profile, local coordination numbers, and Si segment distributions. The propagation velocity of the phase boundary is most rapid for (111) surfaces and slowest for (100) surfaces. An amorphous phase of lithiated Si is developed beyond the phase boundary, in which the ratio of lithium to silicon atoms is steady at 0.8. Partial RDF studies revealed that the structures of the phase boundary and the lithiated Si region are c-LiSi and a-Li15 Si4 , respectively. We qualitatively predict that, during amorphous phase formation of lithiated Si region, the [110] surface swells the most, followed by [111] and [100]. Finally, we propose the three-step mechanism to explain both phase boundary evolution and CAPT. These findings provide important insight into the initial stages of structural evolution at the interface of Si substrates undergoing lithiation. 83 Figure 6.3: Analysis of Si local structures, based on the results of Figure 6.1. (a) Averaged partial coordination number of Si adjacent atoms within 2.8˚ A of the cut- off radius. (b) Spatial probabilities of the Si segments (5-node chains, dumbbells, boomerangs, and isolated Si atoms) are shown in different colors. 84 Figure 6.4: Partial radial distribution function g(r) of Si-Si, Li-Si, and Li-Li pairs in the phase boundary (left) and the lithiated Si region(right), for different orientations of the c-Si samples. Dashed lines indicate the reference peak positions of c − LiSi and a − Li4 Si, obtained from Chevrier and Dahn (2009). 85 Figure 6.5: Time evolution of Li concentration at the phase boundary region. The Li concentrations at different times are superimposed by fixing the initial position of the interface. Results for different steps are obtained from individual MD simulation at temperatures of (a) 600, (b) 900, and (c) 1200 K. Chapter 7 Enhanced Lithiation for Defective Graphene 7.1 Introduction Rechargeable lithium ion batteries (LIBs) are driving a renaissance in electric- vehicle development, portable electronics,light vehicles, advanced energy storage sys- tems and miscellaneous power devices [Dahn et al. (1995)]. The seemingly ubiquitous LIBs are the systems of choice, offering high energy density, flexible and lightweight design, and longer lifespan than comparable battery technologies [Tarascon and Ar- mand (2001), Kim et al. (2014)]. However, as the global energy demand increases, developing new materials with higher charge capacity and better cycling performance for LIBs are of extreme importance. Among the most intensively studied anode ma- terials in LIBs are silicon and graphite [Stournara et al. (2012)]. Graphite is the widely used anode material in LIBs due to its high coulombic efficiency (the ratio of the number of charges that enter the battery during charging compared to the number that can be extracted from the battery during discharging) and excellent 86 87 electrochemical stability [Tarascon and Armand (2001)]. However, its gravimetric capacity is relatively low (theoretical value of 372 mAh/g) leading to its limited per- formance in LIBs [Tarascon and Armand (2001), Migge et al. (2005)]. Compared to graphite, silicon offers ∼ 10 times higher gravimetric energy densities as LIB anodes. However, this extremely high capacity of silicon is associated with massive structural changes and volume expansion on the order of 300%, resulting in electrode particle fracture, disconnection between the particles, capacity loss, and thus very limited cycle life [Shenoy et al. (2010)]. Recent experimental studies show that low dimen- sional materials like graphene [Medeiros et al. (2010), Romero et al. (2011)] and its oxide[Stournara and Shenoy (2011)], carbon nanotubes [Shimoda et al. (2002), Meu- nier et al. (2002)], fullerenes [Buiel and Dahn (1999)], and silicon nanowires [Chan et al. (2008), Chockla et al. (2011)] offer higher specific capacities and could serve as a potential anode materials in LIBs. Among the low dimensional materials, graphene, an atomic layer of carbon atoms arranged in a honeycomb lattice, is actively being pursued as a materials for next gen- eration power devices since its discovery in 2004 [Novoselov et al. (2004)]. Because of its superior electrical conductivity, high charge carrier mobility (20m2 V−1 s−1 ), high surface area (2600m2 g−1 ) and a broad electrochemical window [Novoselov, Geim, Mo- rozov, Jiang, Katsnelson and Grigorieva (2005), Novoselov et al. (2007)], graphene can be particularly advantageous for its applications in energy storage devices [Wei et al. (2012)], for example in LIBs [Yoo et al. (2008), Jang et al. (2011)]. How- ever, impurities and defects (both vacancy and StoneWales) in graphene cannot be avoided [Banhart et al. (2011), Hashimoto et al. (2004)] either because of the pro- duction process being used or because of the conditions under which the graphene 88 device operates [Zandiatashbar et al. (2014)]. Recently, transmission electron mi- croscopy (TEM) [Meyer et al. (2008), Kotakoski et al. (2011)] and scanning tunnel- ing microscopy (STM) [Ugeda et al. (2010), Zandiatashbar et al. (2014)] have been used to obtain images of defective graphene with atomic resolution [Banhart et al. (2011)]. Topological defects significantly influence the electronic, optical, mechani- cal, and thermal properties of graphene [Banhart et al. (2011)]. DFT studies [Zhou et al. (2012)] suggest that the intentional creation of point defects before absorbing Li would enhance adsorption significantly. In our recent paper [Zhou et al. (2012)], we have demonstrated this in terms of gravimetric capacity and investigated defect- induced plating of lithium metal within porous graphene networks. Despite studies on Li adsorption in defective graphene, several important questions remain unclear: What will be the lithiation and corresponding capacities for different percentages of divacancy (DV) and StoneWales (SW) defects? For different defect densities, what will be the maximum possible capacities? How exactly do lithium adatoms cluster on and around the defects? Which type of defect has the strongest influence on Li adsorption? Most importantly, what is the underlying mechanism of enhanced Li adsorption on graphene with structural defects? 7.2 Methods All calculations are performed using the Vienna Ab Initio Simulation Package (VASP) [Kresse and Furthmuller (1996)] with the Projector Augmented Wave (PAW) [Kresse and Joubert (1999), Blochl (1994)] method and the Perdew-Burke-Ernzerhof (PBE) [Perdew et al. (1996)] form of the generalized gradient approximation (GGA) for exchange and correlation. For convergence studies, we determined a kinetic energy cut-off of 600 eV and a Fermi smearing width of 0.05 eV. The Brillouin zones of 4 × 4 and 5 × 5 super cell are sampled with the Γ-centered k-point grid of 9 × 9 × 1 and 89 Figure 7.1: Experimental observation of (a) Single Vacancies (b) Double Vacancies (c) Stone-Wales Defect. (d) Formation energy is high for defects with an under- coordinated carbon atom. 7 × 7 × 1 respectively. In order to avoid the spurious coupling effect between graphene layers along the z -axis, the vacuum separation in the model structure is set to 18˚ A. All atoms and cell-vectors are relaxed with a force tolerance of 0.02 eV/˚ A. The lithiation potential V is defined as [Aydinol and Ceder (1997), Stournara and Shenoy (2011)], ∆Gf V =− (7.1) z•F where F is the Faraday constant and z is the charge (in electrons) transported by lithium in the electrolyte. In most nonelectronically conducting electrolytes z = 1 for Li intercalation. The change in Gibb‘s free energy is : ∆Gf = ∆Ef + P ∆Vf − T ∆Sf (7.2) 90 Since the term P ∆Vf is of the order of 10−5 eV [Aydinol and Ceder (1997)], whereas the term T ∆Sf is of the order of the thermal energy (26 meV at room temperature), the entropy and the pressure terms can be neglected and the free energy will be approximately equal to the formation energy ∆Ef obtained from DFT calculations. The formation energy is defined as : ∆Ef = ELin G − (nELi + EG ) (7.3) where n is the number of Li atoms inserted in the computational cell, ELin G is the total energy of the Lithiated Graphene (Lin G) structure, ELi is the total energy of a single Li atom in elemental body-centered cubic Li, and EG is the total energy of a particular graphene structure. From DFT calculation, we have obtained ELi to be −1.8978 eV. If the energies are expressed in electron volts, the potential of the Lin G structures vs. Li/Li+ as a function of lithium content can be shown as [Aydinol et al. (1997)]: ∆Ef V =− (7.4) n The composition range over which Li can be reversibly intercalated determines the battery capacity. In our calculations, we have included spin polarization but not the van der Waals (VdW) interaction. The inclusion of VdW interaction is expected to affect our results based on PBE. But the charge transfer mechanism underlying the enhanced adsorption will remain unchanged. Since VdW interaction gives rise to stronger binding, its inclusion will lead to higher capacities. However, overall trend or major conclusion of this work will remain the same. 91 7.3 Models First, we discuss the models we considered i.e defective graphene systems. Single vacancies (SV) with a carbon atom missing in graphene (or in the outermost layer of graphite) have been experimentally observed using TEM [Meyer et al. (2008)] and STM[Ugeda et al. (2010)]. However, Meyer et al. (2008) showed that SV undergoes a JahnTeller distortion, which leads to the saturation of two of the three dangling bonds toward the missing atom. For reasons of geometry, one dangling bond always remains. The SV appears as a protrusion in STM images because of an increase in the local density of states at the Fermi energy, which is spatially localized on the dangling bonds [Ugeda et al. (2010)]. It is intuitively clear that the formation energy of such a defect is high because of the presence of an under-coordinated carbon atom. Hence, instead of SV defects, we have concentrated on DV defects, where no dangling bond is present. The atomic network remains coherent with minor perturbations in the bond lengths around the defect. Simulations [El-Barbary et al. (2003)] indicate that the formation energy, Ef , of a DV is of the same order as for an SV (about 8 eV). As two atoms are now missing, the energy per missing atom (4 eV per atom) is much lower than for an SV. Hence, a DV is thermodynamically favored over an SV. Moreover, DV defects are the most common type of vacancy defects observed exper- imentally [Lahiri et al. (2010)], and as mentioned before, structures with any other kind of vacancy defect with dangling bonds are not stable [Brunetto et al. (2012)]. As shown in Figure 7.2, a DV defect can be obtained by removing C-C dimers from pristine graphene. Five different percentages of defects are considered here: 6.25 (Figure 7.2b), 12.50 (Figure7.2c), 16.00 (Figure 7.2d), 18.75 (Figure 7.2e), and 25% (Figure 7.2f). All of the systems shown here are relaxed structures. Figure 7.2g shows that numerical value of equilibrium energy (i.e., the total ground-state energy 92 Figure 7.2: (a) Pristine graphene and graphene with DV defects: (b) 6.25, (c) 12.50, (d) 16.00, (e) 18.75, and (f) 25%. Systems shown here are 2 × 2 in size with periodicity in their in-plane dimensions. The super cell used in the calculation is marked in black. All systems are relaxed structure. (g) Equilibrium energy per carbon atom for different percentages of DV defect. per carbon atom) gradually decreases with the increase in DV defects. Like DV, SW defects are another common type of structural defect observed ex- perimentally [Reich et al. (2005)]. The SW (5577) defect has a formation energy Ef = 5eV [Reich et al. (2005)]. The defective structure retains the same number of atoms as pristine graphene, and no dangling bonds are introduced. As shown in Figure 7.3, we have considered four types of SW defects with different defect concen- tration: 25 (Figure 7.3a), 50 (Figure 7.3b), 75 (Figure 7.3c), and 100% (Figure 7.3d). For 100% SW defect, we have the Haeckelite structure [Terrones et al. (2000)], which is a sheet full of 5-7 rings. The equilibrium energy per carbon atom is much less in this configuration (Figure 7.3e). 93 Figure 7.3: Graphene with SW defects: (a) 25, (b) 50, (c) 75, and (d) 100%. Systems shown here are 2 × 2 in size with periodicity in their in-plane dimensions. The super cell used in the calculation is marked in black. All systems are relaxed structure. (e) Equilibrium energy per carbon atom for different percentages of SW defect. 7.4 Results and Discussion 7.4.1 Analysis of adsorption potential for lowest defect den- sity To get insight into adsorption of Li on defective structure, we first consider the lowest defect density i.e. 6.25% DV and 25% SW defect. The lattice constant of graphene is 2.46 ˚ A [Fan et al. (2012), Yang (2009)]. Two sites of high symmetry for adsorption: the sites on the top of a carbon atom (Top) and the site at the center of a hexagon (Hex) of the graphene sheet are considered for detailed analysis of adsorption potential for these lowest defect densities. For each site, we consider three positions: over the defect (O-position), neighborhood of the defect (N-position), and away from 94 the defect (A-position). Figures H.1, H.1, and 7.4 explain this concept of site and position that we have considered. Here, defect location considers the defect with maximum intensity. For example, Figure 7.4: Graphene with (ac) 6.25% DV defect and (df) 25% SW defect: adatom (a, d) over the defect (O position), (b, e) neighborhood of defect (N position), and (c, f) away from defect (A position). one may claim the site over the five-carbon ring in Figure 7.4 a as being over the de- fect. However, the site over the eight-carbon ring refers to maximum defect intensity. Similarly, we consider the O, N, and A positions at the Top site. In describing the Top site at the O position, we have a couple of options for the case of the DV defect. As shown in Figure 7.4a, we can place adatom in any of the eight carbons enclosing the octagon. However, carbon at locations 1 and 3 has the same neighborhood as well as for carbon at locations 2 and 4. Similarly, carbon in between 1 and 2 has the same surrounding as it does between 3 and 4. Hence, we conclude that we have only two different options (locations 1 and 2). Between these two locations, we select the location that has the maximum defective neighborhood (MDN). Location 1 is at the junction of an octagon, hexagon, and pentagon, whereas for location 2, it is at the junction of an octagon and two hexagons. Hence, we select location 1, which has the 95 Figure 7.5: Lithiation potential for Li adsorption on different locations: pristine graphene (inset) and graphene with DV and SW defects at the Hex and Top sites. For each site, three positions, O (blue), N (green), and A (brown), are shown. MDN. However, for graphene with StoneWale defect, as shown in Figure 7.4d, both locations 1 and 2 have the same neighborhood (i.e., they are identical). Hence, we select location 1. Figure 7.5 shows the lithiation potential for three different positions (O, N, A) at both Hex and Top sites. Here negative potential indicates that adsorption is not possible. As indicated in inset in Fig. 7.5 (red), lithiation is not possible in case of pristine graphene [Zhou et al. (2012)]. However, presence of DV defects favor lithiation. O-position (blue) has maximum potential as it has MDN. Lithiation potential decreases form O to A positions. In case of SW defect, lowest defect density (25%) does not favor absorption for any positions. However, we notice that O-position is the most favorable position. This means, if we increase the defect density, it will 96 favor adsorption. Hence we have considered higher defect densities i.e. 50%, 75%, and 100% for further calculation. Figure 7.6: Bonding charge density for Li for (a) pristine, (b, c) Stone-Wales and (d, e) divacancy systems obtained as the charge density difference between the valence charge density before and after bonding. Red and blue colors indicate the electron accumulation and depletion, respectively. The color scale is in the units of e/Bohr3 . 7.4.2 Charge transfer analysis To get deeper physical insight into absorption, we performed bonding charge- density analysis [Li et al. (2013)]. Figure 7.6 shows the bonding charge density 97 passing through the bond between Li and the nearest carbon atom. The bonding charge density is obtained as the difference between the valence charge density of strain-free grapheneLi sheet and the superposition of valence charge density of the constituent atoms. A positive value (red) indicates electron accumulation, whereas a negative value (blue) denotes electron depletion. Changes in bonding charge distri- bution after introduction of defects clearly show that enhanced charge transfer from Li to underlying graphene sheet favors adsorption of adatom. To quantify the charge distribution, we performed Bader charge analysis [Tang et al. (2009)]. Figure 7.7 shows the charge transfer values for different defect sites and positions. The potential increases with increase in charge transfer. Beyond a threshold charge transfer of ∼ 0.70e, Li adsorption is possible. 7.4.3 Adsorption potential and capacities for different de- fect densities Analyzing Figures 7.5, 7.6, and 7.7 we notice that O position of Hex site (Hex-O) is the most favorable location of adsorption. Hence for initial distribution of Li, we have mainly focused on this location.For each defect density, to have certain capacity, we need adsorption of certain amount of Li adatoms. These adatoms can be placed over defective graphene in many different ways. For example, as shown Figure H.4, for 12.50% defect, we would like to have Li5 C28 (capacity = 398.4592 mAh/g) and Li6 C28 (capacity = 478.1510 mAh/g). As shown in upper and lower panel, lithium adatoms are placed in three different configurations. In fact, we can place adatoms in many other configurations i.e. for every case; we have several options for the initial distribution. For each configuration, adsorption potential will be different. Naturally, if more adatoms are placed over and around the defect, potential will be more. If we compare configuration 1 and 2 in upper panel of Figure H.4, we notice for these 98 cases, potentials are 0.0281 and 0.0523 eV respectively. For location 2, all adatoms are placed inside octagons while for location 1; one of them is placed outside resulting in lower potential. Likewise, as shown in Figures H.3 - H.9, for every case, we consider three among many possible configurations. Figure 7.7: Charge-transfer vs lithiation potential For each percentage of defects, we have carried out DFT calculations for different percentages of Li until adsorption is no longer possible i.e. potential becomes negative. Figure 7.8 a summarizes lithiation potential for five different DV defect densities. For 99 Figure 7.8: Capacity and corresponding lithiation potential for different percentages of (a) DV and (b) SW defects highest defect density of 25%, we notice the formation of Li3 C8 at a potential of 0.84 eV, corresponding to a maximum capacity of 837 mAh/g. The favorable formation of Li3 C8 has been confirmed in our recent experimental studies [Mukherjee et al. (2014)]. Figure 7.8 b shows Li adsorption to defective graphene with SW defects. The results indicate significant Li absorption for 100% SW defects i.e. for the Haeckelite structure. Error bar of different height indicates the variation in potential range for different capacities. Figure 7.9 summarizes the maximum capacity observed for different DV and SW defect densities. For highest DV defect of 25%, the maximum capacity is as high as ∼ 1675 mAh/g, corresponding to a lithiation potential of ∼ 0.1 eV. In the case of SW defects, we observe the highest capacity of ∼ 1100 mAh/g for the Haeckelite structure. 7.5 Conclusions In summary, we have performed first-principles calculations to investigate the Li adsorption on graphene with various DV and SW defect densities. We confirm that lithiation is not possible in pristine graphene. However, presence of defect favors 100 Figure 7.9: Maximum capacity for different DV and SW defect densities. adsorption. The potential is larger when adatom is on and around the defective zone. We have demonstrated the underlying charge-transfer mechanism for enhanced adsorption. Our results provide deeper insight into adsorption and will help to create better anode materials for higher capacity and better cycling performance for LIBs. In addition, since the charge transfer occurring in both defects is identified to account for the enhanced adsorption of Lithium, we hope this work will provide the first theoretical understanding on the experimentally observed lithiation and attract more researchers’ attention on this important topic and promote the development of this field. Chapter 8 Defective Graphene as a High-Capacity Na- and Ca-Ion Battery 8.1 Introduction Rechargable lithium-ion batteries (LIBs) have been extensively used in portable electronics, light vehicles, and miscellaneous power devices over the past decade [Dahn et al. (1995)]. In terms of energy density, the seemingly ubiquitous LIBs exhibit su- perb performance as compared to other types of rechargeable batteries [Tarascon and Armand (2001)]. However, among light metals, Li is a very rare element. Its concen- tration in the upper continental crust is estimated to be 35 ppm [Teng et al. (2004)]. Hence, in recent years, there have been great concerns that available Li resources buried in the earth would not be sufficient to meet the ever increasing demands for LIBs [Palomares et al. (2012)]. These concerns have led to the active search for suit- able alternatives [Ong et al. (2011)]. Among these, sodium-ion batteries (NIBs) [Ong 101 102 et al. (2011), Kim, Seo, Ma, Ceder and Kang (2012)] and calcium-ion batteries (CIBs) [Amatucci et al. (2001)] have drawn significant attention. Although the energy density of a NIB is generally lower than that of a LIB [Palo- mares et al. (2012)], high energy density becomes less critical for battery applications in large-scale storage [Kim, Seo, Ma, Ceder and Kang (2012)]. More importantly, the abundance and low cost of Na in the earth (10 320 ppm in seawater and 28 300 ppm in the lithosphere) [Lin et al. (2013)] and low reduction potential (2.71 V vs standard hydrogen electrode (SHE)) provide a lucrative low-cost, safe, and environmentally benign alternative to Li in batteries [Mortazavi et al. (2013)]. Like NIBs, CIBs offer several benefits such as low cost, natural abundance, chemical safety, low reduction potential (2.87 V vs SHE), and lighter mass-to-charge ratio [Amatucci et al. (2001)]. The use of polyvalent cations is the key to obtaining much larger discharge capacities than those of LIBs. Moreover, nature stores energy with Na and Ca ions, not Li ions [Xu and Lavan (2008)]. Electrochemical properties of the electrode materials are the cynosure of important battery-performance characteristics such as specific capacity and operating voltage [Kim, Seo, Ma, Ceder and Kang (2012)]. Hence, the major challenge in advancing NIB and CIB technologies lies in finding better electrode materials. The best starting point is the investigation of the structure and chemistries of electrode materials that function well for Li intercalation. Graphite, the most widely used anode material for LIBs, has a relatively low gravimetric capacity.Even for NIBs and CIBs, use of graphite yields very low capacity [Divincenzo and Mele (1985)]. As mentioned in Chapter 7, recent experimental studies show that if we can lower the dimensionality of the conventional anode materials via nanotechnology, we can achieve higher capacity. 103 Among the low-dimensional materials, graphene has attracted enormous attention since its discovery in 2004 [Novoselov et al. (2004)]. We have discussed in detail in Chapter 7 about the presence of defects in graphene and how these defects enhance the Li adsorption on graphene, giving a higher gravimetric capacity. Hence, the open question of how will defects in graphene influence the adsorption of Na and Ca remains. To answer this question in detail, we have carried out the first-principles calculations based on DFT to investigate thoroughly the Na and Ca adsorption on graphene with various percentages of DV and SW defects 1 . 8.2 Results and Discussion The sodiation and calciation potentials for three different positions (O, N, and A) for both the Hex and Top sites are shown in Figure 8.1. For DV Hex (Hex site of DV defect) and DV Top (Top site of DV defect), we notice that the O position (blue), as expected, is the most favorable position for adsorption. Sodiation potential is reduced to zero or is negative from the O to A positions. For the SW defect, the lowest defect density (25%) does not favor Na adsorption for any location. However, the O position has a less negative potential compared to the N and A positions. The same procedure applies for the calculations for Ca, and a similar trend is obtained, as shown in Figure 8.1h. It is clear that adatoms tend to cluster around the defective zone. To obtain insight on the adsorption on defective sheets, we performed bonding charge-density analysis. Figure 8.2 shows the bonding charge-density passing through the bond between Na/Ca and the nearest carbon atom. The bonding charge density is 1 We have followed the same procedure (of course, different potentials for Na and Ca) as described in Section 7.2. We have computed the equilibrium energy for Na and Ca as 1.307 and 1.980 eV, respectively. 104 Figure 8.1: (a) Sodiation and (b) Calciation potential for Na/Ca adsorption on different locations: pristine graphene (inset) and graphene with DV and SW defects at the Hex and Top sites. For each site, three positions, O(blue), N(green), and A(brown), are shown. obtained as the difference between the valence charge density of strain-free graphene- Na/Ca sheet and the superposition of the valence charge density of the constituent atoms. A positive value (red) indicates electron accumulation, whereas a negative value (blue) denotes electron depletion. These changes in bonding charge distribu- tions after introduction of defects clearly show that the enhanced charge transfer from Na/Ca to graphene sheet leads to adsorption of adatoms. The charge redistribution can be quantitatively estimated by computing the charge transfer using Bader charge analysis. Table 8.1 shows the magnitude of the charge transfer for different positions. In case of the Na+ ion, charge transfer to pristine 105 graphene is 0.6617e, whereas for structures with DV and SW defects, the transferred charges are increased to 0.8848e and 0.8073e, respectively. For Ca2+ , the correspond- ing values of charge transfer are 0.8208e, 1.3727e, and1.1189e respectively. For each case, the DV defect case has more charge transfer, resulting in more adsorption of adatoms. In Figure 8.2g,h, for both Na and Ca adsorption, the potential increases with the increase in charge transfer. Any amount of charge transfer does not imply adsorption. There is a threshold of charge transfer beyond which adsorption is possi- ble. Depending on the adatom (i.e. Na/Ca in this study), the threshold is different. From Figure 8.2g,h, we can observe that charge transfer over 0.85e (approximately) favors Na adsorption, whereas for Ca, the corresponding threshold is around 1.30e. ion pristine divacancy stone-wales Na+ 0.6617e 0.8848e 0.8073e Ca2+ 0.8208e 1.3727e 1.1189e Table 8.1: Charge Transfer from Na/Ca to Graphene. From our results in Figures 8.1 and 8.2, we have discovered that the O position of the Hex site is the most favorable location of adsorption. Hence, we primarily focus on this location while initially distributing the Na/Ca adatoms. Still, for every case, there are many possibilities of initial distribution. For each case, we have considered three different initial configurations to obtain the potential range, and we reported the average values. It is obvious that at low concentration a greater possibility of initial distribution leads to a wider range of potential. For each percentage of de- fects, we have carried out DFT calculations for different Na/Ca concentrations until we crossed the maximum limit of capacity (i.e. when the potential becomes negative). 106 Figure 8.3a summarizes the sodiation potential for five different DV defect per- centages. For each defect density, the potential decreases with the increase in the Na concentration. For higher defect density, the potential is larger for a given Na concentration and the maximum percentage of adsorbed Na is increased. Beyond this limit of maximum defect density, the structure will have dangling bonds. Figure 8.3 b shows one of the configurations of Na8 C26 where Na adatoms are mainly located on and around O positions (i.e., adatoms tend to cluster around the defective zone). As shown in Figure 8.3 c, for SW defects, the percentage of adsorption is increased with the increase in defect density. Figure 8.3 d shows one of the configurations of Na6 C32 . The results for calcium adsorption are summarized in Figure 8.4. We note that the adsorption behavior of Ca in DV and SW graphene is qualitatively the same as for Na. Figure 8.5 summarizes the maximum percentage of Na/Ca adsorbed for different percentages of DV and SW defects. Capacity, C (mAh/g), can be computed from percentage of adsorption, p, as 1 h p i C= · v · F · 103 (8.1) Ac 100 Where p is the percentage of adsorption of adatoms on graphene (%), v is the valancy (Na = 1; Ca = 2), F is the Faraday constant (26.801 Ah/mole), and Ac is the atomic mass of Carbon (12.011). For the 6.25% DV defect, the maximum percentage of adsorption is 6.67%, corre- sponding to a capacity of 148.8325 mAh/g for Na and 297.6649 mAh/g for Ca. With the increase in defect density, we obtained a maximum percentage of adsorption for Na/Ca around 19, 25, 40, and 65% for 12.50, 16, 18.75, and 25% defects, respectively. Hence, for a maximum defect density of 25%, we can obtain a maximum capacity of 107 around 1450 mAh/g for Na and 2900 mAh/g for Ca. For SW defects, the maximum percentage of adsorption is around 10, 13, and 48% for 50, 75, and 100% of SW de- fects, respectively. Hence, for the 100% SW defect (i.e., structure full of 5-7 rings), we can achieve a maximum capacity of around 1071 and 2142 mAh/g for Na and Ca, respectively. We observe that for DV defects the capacity increases gradually with the increase of defect density. However, for SW defects, until the system reaches its maximum defect density (i.e. a system full of 57 rings), the capacity does not increase much. This can be attributed to the fact that for Haeckelite structure, the drop in equilibrium energy is drastic, whereas for DV defects, the drop in equilibrium energy is gradual. 8.3 Conclusions We have performed first-principles calculations to study the Na and Ca adsorp- tion on graphene with various percentages of DV and SW defects. Our results show that adsorption is not possible in pristine graphene. However, the presence of defects enhances the adsorption, and the potential is larger when the adatoms are on and around the defective zone. With the increase in defect density, the maximum capacity obtained is much higher than that of graphite. This study will help to create better anode materials that can replace graphite for higher capacity and better cycling per- formance NIBs and CIBs. It will be interesting to compare our results with future experiments. 108 Figure 8.2: Bonding charge density for Na and Ca (Top site and O position) for (a, d) pristine, (b, e) StoneWales, and (c, f) divacancy systems obtained as the charge- density difference between the valence charge density before and after the bonding. Red and blue colors indicate the electron accumulation and depletion, respectively. The color scale is in the units of e/Bohr3 . Potential vs charge transfer for (g) Na and (h) Ca adsorption. 109 Figure 8.3: Sodiation potential for different percentages of Na adsorbed for different percentages of (a) DV and (c) SW defects. Top and side view of one of the (b) Na8 C26 and (d) Na6 C32 relaxed configurations. 110 Figure 8.4: Calciation potential for different percentages of Ca adsorbed for different percentages of (a) DV and (c) SW defects. Top and side view of one of the (b) Ca8 C26 and (d) Ca6 C32 relaxed configurations. 111 Figure 8.5: Maximum percentages of Na/Ca adsorbed for different percentages of DV and SW defects. Chapter 9 Surface Terminated Germanene as Emerging Nanomaterials 9.1 Introduction In recent years, there have been many breakthroughs in the study of topological insulators (TIs) - a new class of materials with a bulk band gap and topologically protected boundary states [Hasan and Kane (2010)]. TIs will result in new device paradigms for spintronics and quantum computation as it predicts many intriguing phenomena, such as giant magnetoelectric effects [Qi et al. (2008)] and the appear- ance of Majorana fermions. In particular, two-dimensional (2D) TIs have significant advantages over three-dimensional (3D) TIs in many respects. Many materials have been theoretically predicted to be 2D TIs [Kane and Mele (2005)]. However, so far only the HgTe/CdTe [Konig et al. (2007)] and InAs/GaSb [Knez et al. (2011)] quan- tum wells have been verified by experiments, which, however, still face some major challenges. Therefore design of 2D TIs with larger gaps from the commonly used materials is of urgent need for their practical importance. 112 113 Graphene, with many superior properties ranging from mechanical [Si et al. (2012)] Figure 9.1: Experimental observation of the formation of Germanane : Hydrogen terminated Germanene. to electronics [Abergela et al. (2010)], has made remarkable progress in numerous applications. However, graphene has extremely small bulk gaps (10−3 meV)[Min et al. (2006)]. Graphene’s popularity triggered extensive research on other 2D materi- als, i.e. silicene, germanene, tin monolayer, boron nitride layers, MoS2 , and many others [Cahangirov et al. (2009)]. Among them, like graphene, silicene could be well produced; however, it’s practical applications as 2D TIs are enormously hindered be- cause of it’s small band gap of 1.55 meV [Liu et al. (2011)]). However, Germanene and tin monolayer have a larger topologically nontrivial gap [Liu et al. (2011)] but have not yet been experimentally synthesized. Very recently, germanane (GeH), a one-atom-thick sheet of hydrogenated germanene, structurally similar to graphane, has been synthesized successfully [Bianco et al. (2013)]. This is regarded as a promis- ing new star in the field of 2D nanomaterials [Koski and Cui (2013)] because of its high mobility and easier integrability into the current electronics industry [Bianco et al. (2013)]. In addition, recent interest in germanane has also stimulated the synthesis 114 Figure 9.2: Surface terminated germanene under consideration : (a) GeH (b) GeOH (c) GeX (X : Halogen) and (d) GeHX. of its cousins, such as halogene, hydroxyl terminated germanenes. In this Chapter, using first-principles calculations, we investigate the electronic and topological properties of a single layer of hydrogenated/halogenated germanene. Our results show that Germanane with 10% strain and hydroxyl terminated Ger- manene with no strain are topological insulators. 115 9.2 Models and Methods All calculations are performed within the framework of density functional theory (DFT) with ab initio pseudopotentials and plane-wave formalism as implemented in the Vienna ab initio simulation package (VASP)[Kresse and Furthmuller (1996)]. The Brillouin zone is integrated with a 18 × 18 × 1 k mesh. The plane-wave cut-off energy is set as 400eV . The system is modeled by surface terminated germanene layer and a vacuum region more than 10 ˚ A thick to eliminate the spurious interaction between neighboring slabs. The structures are relaxed until the remaining force acting on each atom is less than 0.01 eV/˚ A within the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) functional. Systems considered for this study are shown in Figure 9.2. 9.3 Results and Discussion With surface termination, there will be transfer of charge from adatom to under- lying Germanene sheet. We first perform bader charge analysis to quantify charge transfer for three selected configurations: GeH, GeF, and GeOH. As shown in Fig- ure9.3, compared to Hydrogen and Fluorine, more charge is transferred for the case of halogen. While for hydroxyl termination, we observe even more charge transfer. For deeper understanding, we perform density of states (DOS) analysis (Figure 9.4). For Germanene, Halogen (Fluorine), and Oxygen, p-orbital plays significant role. For hydrogen, only s-orbital has dominant role. Figure 9.5 and 9.6 show detail band analysis. While Germanene has zero band gap (Figure 9.5a), Germanane has a gap of about 1.8 eV, which is in good agreement with experimental observation [Bianco et al. (2013)]. However, in case of hydroxyl termination, we notice a zero band gap at Γ point. 116 Figure 9.3: Charge Transfer analysis of GeH, GeF, and GeOH. Figure 9.6 gives a detail analysis of how to tune band gap by halogen termination and combination of halogen-hydrogen termination. Among the halogen termination, only Cl and Br induce a small amount of band gap. However, addition of hydrogen in one side opens significant gap for the cases of F, Cl, and Br termination. While for partial termination i.e. hydrogenation on one side, we notice the presence of a dangling band (Figure 9.7). PDOS analysis shows the existence of spike in p- and s- orbital for Germanene and Hydrogen cases respectively. However, with the addition of externally applied strain (Figure 9.8), the dangling band disappears. For last few years, Germanene (hydrogen terminated Germanene) got tremendous interest. We notice in Figure 9.9 that band of Germanane can be tuned by external 117 Figure 9.4: Density of States (DOS) analysis of GeH, GeF, and GeOH. applied strain. For 10% strain, we have zero band gap. Comparing this case with the case of band gap for hydroxyl termination (Figure 9.5), we can predict that addition of Spin Orbit Coupling (SOC) may reveal these structures to be topological insulator. SOC calculation reveals that Germanane with 10% strain (Figure 9.10) and hydroxyl terminated Germanene with no strain (Figure 9.11) are topological insulators. 9.4 Conclusions In summary, based on first-principle calculations, we have studied the band topolo- gies in functionalized germanene, including the recently synthesized germanene and halogenated/hydroxyl terminated germanenes. Among these, we discovered that with the inclusion of Spin Orbit Coupling (SOC), Germanane acts as Topological Insulator with 10% externally applied strain. However, hydroxyl terminated Germanene acts as TI without any externally applied strain. Further investigation is required to verify 118 Figure 9.5: Band structure of Ge, GeH, and GeOH. whether other termination will act as TI and under which conditions. 119 Figure 9.6: Band structure of Halogen and Hydrogen-Halogen terminated Ger- manene. 120 Figure 9.7: Band structure and Density of States (DOS) for 50% Hydrogen termi- nated Germanene. Figure 9.8: Strain effect on band structure for 50% Hydrogen termination of Ger- manene. 121 Figure 9.9: Tuning band gap of Germanane with external strain. 122 Figure 9.10: Germanane with SOC under 10% external strain acts as Topological Insulator. 123 Figure 9.11: Hydroxyl terminated Germanene with SOC under no external strain acts as Topological Insulator. Chapter 10 Electronic Properties of 1T/2H of MoS2 10.1 Introduction Ignited by the promising physical properties of two-dimensional (2D) graphene, there have been ever increasing interest in investigating other 2D and one-atom-thick layered materials [Novoselov, Jiang, Schedin, Booth, Khotkevich, Morozov and K. (2005)]. Among the novel 2D materials under investigation, molybdenum disulfide (MoS2 ) has attracted considerable attention in recent years [Chhowalla et al. (2013)]. Unlike semimetallic graphene, monolayer (MoS2 ) is a semiconductor with large di- rect band gap [Mak et al. (2010)], which can be further manipulated by changing its thickness. The presence of a band gap opens a realm of electronic and photonic pos- sibilities [Wang et al. (2012)]. However, considering modern electronics and photonic devices, heterostructures of semiconductors and metals are even more crucial than the semiconductors themselves [Yin et al. (2013)]. Some recent studies have investi- gated MoS2 /Ti2 C, MoS2 /Ti2 CY2 (Y=F and OH) [Gan et al. (2013)] and MoS2 /Au, 124 125 MoS2 /Ti [Popov et al. (2012)]. Eda et al. (2012) recently reported chemically exfoliated MoS2 single layer that exhibits a hybrid structure of semiconducting (trigonal prismatic (2H)) and metallic (octahedral geometry (1T)) phases that are coherently bonded such that the resulting single-layer heterostructure is chemically monolithic and single crystalline. Because of the changes in the crystal symmetry, the two phases exhibit substantially different electronic structures. 2H-(MoS2 ) is a semiconductor with a band gap between the filled dz2 and empty dx2 −y2 ,xy bands while the 1T phase is metallic with Fermi level lying in the middle of degenerate dxy,yz,xz single band [Mattheiss (1973)]. Despite the experimental evidence, theoretical studies on this heterostructure have not yet been reported. Here we explore the electronic properties of 1T-2H interface of (MoS2 ) using the first principle DFT calculations. 10.2 Methodology All calculations are performed using the Vienna Ab Initio Simulations Package (VASP) [Kresse and Furthmuller (1996)] with the Projector Augmented Wave (PAW) [Kresse and Joubert (1999), Blochl (1994)] method and the Perdew- Burke-Ernzerhof (PBE) [Perdew et al. (1996)] form of the Generalized Gradient Approximation (GGA) for exchange and correlation functional. An energy cutoff of 480 eV was used in the plane wave expansion of wave functions. The Brillouin zone of super cell are sampled with Γ-centered k-point grid of 1 × 12 × ×1. In order to avoid the spurious coupling effect between periodic graphene layers along the normal direction, the vacuum sep- aration in the model structure is set to 18 ˚ A. All atoms and cell vectors are relaxed with a force tolerance of 0.02 eV/˚ A. 126 First we have done lattice relaxation using our in-house code. We used that relaxed structure for convergence studies (both energy and k-point). Using the con- verged energy and k-point, self-consistency calculation was carried out. The charge distribution information obtained in this calculation was used for further calculation i.e. density of states, bader charge analysis, band structure etc. 10.3 Results and Discussion 10.3.1 Schottky Barrier The interface considered in our study is shown in Figure 10.1. The structure reported in experimental study has one interface and infinite in extent []. We can use periodic structure for the geometrical reason. However, in this case, we have two different kinds of interfaces. It can be seen that 2H to 1T transformation can be √ realized by gliding one of the S planes in the < 2100 > direction by a/ 3. Moreover, interface at right becomes unstable after equilibrium. Because as explained by Eda et al[], the hexagonal arrangement of Mo and its orientation remains undisturbed across the boundary which was confirmed by the nearly identical Fourier Transform image of the two phases. Figure 10.2 and 10.3 show the Local Density of States (LDOS) of Mo atoms in armchair and zigzag directions. We have selected Mo atoms for LDOS analysis inside 2H-MoS2 , 1T-MoS2 sheet and at the interface. We notice that 2H-MoS2 sheet is semiconductor while 1T-MoS2 is metallic. We can determine the Schottky Barrier from LDOS which is around 0.80 eV. 127 10.3.2 Stability of the Interface As shown in Figure 10.4 (a-e), we can create heterostructure by doping 1T-MoS2 sheet inside the periodic 2H-MoS2 sheet. Here x − 2H y − 1T means y number of 1T-MoS2 unit cells are doped inside x number of 2H-MoS2 unit cells. We define the interface energy/unit cell as 1 Ef = [E2H−1T − (n2H E2H + n2H E1T )] (10.1) (n2H + n1T ) where, n2H/1T : total number 2H-MoS2 and 1T-MoS2 unit cells respectively, E2H−1T : total energy of the heterostructure, E2H : energy/unit cell of 2H-MoS2 sheet, E1T : en- ergy/unit cell of 1T-MoS2 sheet. As shown in Figure 10.4 f, interface energy/unit cell increases with increased number of 1T-MoS2 unit cell doped inside 2H-MoS2 sheet. However, after certain stage, it tends to saturate. Figure 10.5 shows different band structure for different number of 1T-MoS2 unit cell doped inside the periodic 2H-MoS2 sheet. We notice that direct band gap in pure 2H-MoS2 is gradually reduced with the increased number 1T-MoS2 doping. 10.4 Conclusions We have briefly discussed about the electronic properties of 1T doped 2H nanosheets. Our results show that 1T/2H interface acts as semi-conductor. Band gap of 2H-MoS2 can be tuned by doping of 1T-MoS2 . We notice that one side of the interfaces is not stable. Hence it will be better to investigate a nanoribbon with only the sta- ble interface and hydrogen terminated edges. In addition, other effects like strain, functionalization, topological defects will be very interesting to study. 128 Figure 10.1: A schematic showing two different interfaces for the case 1T-MoS2 sheet doped inside periodic 2H-MoS2 sheet. Interface at right becomes unstable after equilibrium. 129 Figure 10.2: Local Density of States (LDOS) of Mo atom at three zones: 2H-MoS2 , Interface and 1T-MoS2 in armchair direction. The Fermi level is set to zero. Figure 10.3: Density of States (LDOS) of Mo atom at three zones: 2H-MoS2 , Interface and 1T-MoS2 in zigzag direction. The Fermi level is set to zero. 130 Figure 10.4: (a-e) A schematic showing the 1T-MoS2 sheet doped inside 2H-MoS2 sheet. All the systems are equilibrium structures with periodicity in in-plane dimen- sion. Here x − 2H y − 1T means y no. of unit cells of 1T-MoS2 are doped inside x-no of 2H-MoS2 unit cells. (f) Interface energy/unit cell (eV) for different number of 1T-MoS2 unit cells doped in 2H-MoS2 sheet. 131 Figure 10.5: Band structure for MoS2 heterostructure with 1T/2H interface. Here y no. of unit cells of 1T-MoS2 are doped inside x-no of 2H-MoS2 unit cells. Chapter 11 Summary of Contributions and Recommendation for Future Work 11.1 Summery of Contributions 1. Fracture at Nanoscale : Analyzed fracture properties (SIF, crack angle etc) of graphene for different loading and slit angle considering both armchair and zigzag directions. This work provides deeper understanding of fracture of nanomaterials under complex loading. 2. Nanotribology : Investigated frictional characteristics between different bi- layer of 2D nanomaterials (graphene-graphene, graphene-boron nitride, boron nitride-boron nitride). 3. Nanofluidics : • Discovered that water microdroplet containing graphene oxide and a sec- ond solute spontaneously segregate into sack-cargo nanostructures upon drying. Cargo filled nanosack can be used for drug delivery. 132 133 • Investigated diffusion of mercury atom in graphene interlayer spaces con- taining water and proposed graphene can be used as environmental barrier. 4. Atomistic Mechanism of Phase Boundary Evolution : Modeled Phase Boundary - a sharp interface in between c-Si and a-Lix Si and investigated ki- netics of Li intercalation into Si of different orientations : (100), (110), and (111). 5. 2D Nanomaterials for Energy Storage : • Explored in depth the interaction between defects and Li in the context of Li-ion batteries. We explained the mechanisms by which defects (e.g. vacancies, divacancies etc.) attract Li which results in very high lithium content intercalated states leading to exceptionally high specific charge storage capacities in defective graphene electrodes. We also showed that high lithium concentration near defects can trigger the plating of lithium metal which further increases the specific capacities that are achievable. Performed Bader Charge Analysis to uncover the underlying Charge Trans- fer Mechanism • Addressed the problems associated with LIBs and requirements of Na- and Ca- ion batteries. We discovered that defective graphene can also be a potential high-capacity anode materials for batteries beyond LIBs. 6. Emerging Nanomaterials : • Discovered for functionalized germanene, proper tensile strain can induce topological phase transition. • Investigated electronic properties of 1T/2H interface of MoS2. 134 11.2 Recommendation for Future Work 11.2.1 A. Nanomechanics 1. Fracture at Nanoscale • There have been a lot of studies on graphene fracture, but there are only a couple of studies on ‘Energy Release Rate’ of graphene at atomic scale. It is important to investigate that in detail. • Condition of complex fracture is more realistic. However, very few studies have been done on it. It will be interesting to perform multiscale studies on complex fracture for different nanomaterials. • Heterostructures i.e. combination of different nanostructures (e.g. BN doped Graphene, Interface of different structures) are currently being ex- tensively used for nanoelectronics. Hence because of practical relevance, it is interesting to study fracture of heterostructures. • Fracture of battery materials will be of enormous interest for coming years with a lot of promise for industry applciations. This is the field of Chemo- mechanics - where Chemistry meets Mechanics. 2. Nanotribology • While moving the indenter, sometime because of the normal load, instabil- ity of the layer under the AFM tip may occur. Several issues may control this instability: number of interlayer, different nanostructures, functional- ization of underlying layer, applied normal load etc. • When two monolayers of different materials are brought together to form heterostructures, interlayer interactions cause mechanical deformations in 135 both layers, which significantly affects the electronic properties. A multi- scale computational method needs to be developed to predict the elastic deformation fields in bilayer heterostructures using density functional the- ory (DFT) informed continuum simulations. • Though several studies have been done on misfit angle and incommensura- bility, it will be interesting to study these for many different nanomaterials. • There will be a significant difference in frictional characteristics depending on orientation of underlying wafer i.e. Si(100) and Si(111). It will be worth studying this for different nanomaterials. • Defects are always present in nanostructure. It is important to study influence of defects in frictional characteristics e.g. localized defects in graphene results in the frictional peak. • The studies presented in Chapter 3 can be extended by incorporating wa- ter nanodroplet inside the bilayer and investigate the effect of lubrication. Number of layer of droplets, its density etc will govern the frictional char- acteristics as well. 3. Nanomedicine • Our work on graphene based drug delivery (published in Nano Letter in 2012) has been cited more than 50 times. Hence, there is a clear need for innovative technologies to improve the delivery of therapeutic and diag- nostic agents in the body. Recent breakthroughs in nanomedicine are now making it possible to deliver drugs and therapeutic proteins to local areas of disease or tumors to maximize clinical benefit while limiting unwanted side effects. 136 • The complexity of cancer and the vast amount of experimental data avail- able have made computer-aided approaches necessary. Biomolecular mod- elling techniques are becoming increasingly easier to use, whereas hardware and software are becoming better and cheaper. 4. Nanocomposites • More investigation is required about the interfacial properties of polymer and different nanomaterials 1 . • In recent years, two-dimensional polymers (2DPs) have drawn great atten- tions as these are expected to be superb membrane materials because of their defined pore sizes. 11.2.2 B. Energy Research Battery research is very much interdisciplinary integrating ideas from different disciplines. Among many aspects, battery research links mechanics and chemistry i.e. essential ingredients of solid mechanics e.g. plasticity, elasticity, mass transport, fracture are linked to chemistry (chemomechanics). Here are some of the intriguing problems worth studying : • High-Capacity Li-ion Battery: Our recent paper in Nature Communication proposed graphene with point defects (Stone-Wales and Vacancy) as potential high capacity anode materials. Supported by experimental results, we showed that for the maximum possible divacancy defect densities, Li storage capacities of up to ∼ 1675 mAh/g can be achieved. While for Stone-Wales defects, we find that a maximum capacity of up to ∼ 1100 mAh/g is possible. So as a possible 1 Some preliminary results are reported in Appendix G 137 high capacity anode materials for Lithium Ion Batteries (LIBs), it is recom- mended to continue investigating different nanomaterials: graphene allotropes (graphdiyne, graphyne, biphenylene), MXene (Mn+1 AXn , where n = 1,2, or 3) element, and X is C and/or N, from laminated structures with anisotropic prop- erties), Molybdenum Disulfide (MoS2), Germanene etc. In addition, it will be interesting to investigate the effect of different point defect, surface termination, and doping on battery capacity. • High-Capacity Na-ion and other ion Batteries : LIBs are ubiquitous in terms of energy storage application. However, among light metals, Li is a very rare element (concentration 35 ppm). Our paper on Na- and Ca- ion battery, published in ACS Applied Materials and Interfaces, has already 15 citations in last 10 months. Among the non-LIBs, Na-ion battery (NIBs) is getting a lot of attention because of its abundance (10320 ppm in seawater and 28300 ppm in the lithosphere). It will be interesting to investigate above-mentioned nanomaterials for high capacity NIBs and other ion-batteries e.g. Ca-, K-, and Mg. • Phase Boundary and Solid-Electrolyte Interface (SEI) : In our re- cent paper in JPCC, we have analyzed the formation and propagation of phase boundary during lithiation of Silicon. Also, there have been studies on SEI for- mation of LIBs. It will be interesting to study more aspects of these interface structures like stress field, atomic structure etc. • Segregation of Silicon/Copper Interface : Recently there has been tremendous interest in interfacial properties of Si anode and underlying metal substrate. Upon lithiation, there is segregation and consequent reduction of adhesion strength leading to debonding, fracture, and several other problems. 138 It will be interesting to study this for different Si orientations and also other ion batteries (Na,Ca,K,Mg etc). • Interface of Si/C Core - Shell Particles : Recently, a lot of attention got into the study of Si/C hollow core-shell nanoparticles - a promising anode archi- tecture, which can successfully sustain thousands of cycles with high Coulombic efficiency. Besides LIBs, it will be interesting to study these structures for other ion batteries. There are many issues to be solved: Constitutive modeling, stress field development during ionation in Si and C, capacity-voltage dependence etc. • Mechanical Properties of Electrodes : Despite intensive work on battery, there is no study on the systematic computation of different properties of elec- trodes e.g. yield strength, young modulus, fracture energy etc. • Study of Other Alloy Anodes : Until now, most of the study on alloy anode is Silicon. It will be interesting for different ion batteries to study insertion in other alloy anodes e.g. Al, Ag, Sn etc. • Fatigue in the Cyclic Life of Electrodes : Theory has been developed to describe cyclic plasticity of electrodes, but it will be interesting to quantify the contribution of lithiation-induced plasticity to the fatigue of electrodes. This study can be extended for other ion batteries. 11.2.3 C. Emerging Nanomaterials • Besides Germanene and MoS2 , recently there have been several emerging nano- materials reported e.g. Borophene 2 , Phosphorene etc. There new materials have already got tremendous attentions. It will be interesting to study different properties of these nanomaterials. 2 It was infact invented at Chemistry Department of Brown University Appendix A Mechanical Strength of Hydrogen Functionalized Graphene Allotropes A.1 Introduction Over the last few decades, tremendous research attention has been devoted to the extraordinary electrical, mechanical and thermal properties of fullerene, nanotubes, and graphene. Recent advancement in the synthesis and assembly process has led to the development of many new carbon materials. Particularly, the two dimen- sional structures of carbon network with the same symmetry as graphene, such as carbine, graphane and graphyne, have been extensively investigated experimentally and theoretically due to their promising electrical and optical mechanical properties. These two-dimensional graphene allotropes (GAs) can serve as precursors to build various nanotubes, fullerenes, nanoribbons, and other low-dimensional nanomateri- als. Depending on their structural composition, the GAs can also be functionalized 139 140 via chemical addition reaction in which the carbon atoms are converted from sp2 to sp3 hybrids to bond with the added chemical groups. Hydrogen adsorption on GAs has been acknowledged as an efficient way to modify their properties, such as band gap, ferromagnetism and thermal conductivity. The enhanced properties provided by hydrogenation are also tunable by changing the hydrogen coverage. To efficiently utilize such chemical functionalization with hydrogen atoms, it is necessary to understand the mechanical properties of the hydrogenated structures. Hydrogenation process causes membrane shrinkage and extensive membrane corru- gation. This can lead to the deterioration of graphene mechanical properties, such as Young’s modulus, shear modulus and wrinkling properties. Many studies have been devoted to evaluate the mechanical properties of GAs using molecular dynamics (MD) simulations. Pei et al. (2010) studied the effect of the degree of hydrogen cov- erage on mechanical properties of hydrogenated graphene and found that the Young’s modulus, tensile strength, and fracture strain deteriorate drastically with increasing H-coverage. Their results suggest that the coverage-dependent deterioration of the mechanical properties must be taken into account when analyzing the performance characteristics of nanodevices fabricated from hydrogenated graphene allotrope sheets (GA sheets). However, no investigation about the properties of other hydrogenated GAs has been reported. Further research to date about the influence of hydrogen coverage on the mechanical properties of GAs is in great need for design and practi- cal application. In this chapter, we report the hydrogen coverage-dependent mechanical properties for graphyne and three new stable GAs. Young’s modulus and intrinsic strength of the chosen GAs are evaluated with varying H-coverage in the range of 0-100%. Moreover, 141 the failure processes of some new GAs, such as Biphenylene, are reported for the first time. The mechanical properties of the investigated GAs deteriorate with increasing H coverage, and show different sensitivity to the functionalization. Our results suggest that novel failure mechanics is unique to a functionalized two-dimensional system. A.2 Models and Method The atomic structures of the examined carbon networks are depicted in Fig. A.1. The simulated allotropes have periodic boundary conditions with a lateral size of ap- proximately 7.5 nm. We arranged the carbon sheets by orienting armchair and zigzag edges along the X and Y axes, respectively. We used the LAMMPS package for the MD simulations with an Adaptive Intermolecular Reactive Bond Order (AIREBO) potential with an interaction cut-off parameter of 1.92 ˚ A. Prior to loading, the initial configurations were first relaxed to reach equilibrium. Tensile loading with a strain rate of 0.0005/ps was applied by displacing the simulation box followed by a relax- ation for 10,000 MD steps. The time step of our simulations was 1 fs. This procedure of relaxation and stretching was repeated for all the allotropes to evaluate their me- chanical properties. All sets of the simulation were performed at room temperature under NVT ensemble. We first generated models of fully hydrogen functionalized GA sheets (H-100%) by bonding hydrogen atoms on one side of the carbon structure. Further hydrogenated GA sheets with certain H-coverage were achieved by randomly removing H atoms from fully hydrogenated hydrocarbon models. The stressstrain curves during deformation can be obtained by following the study of Pei et al. (2010) on the mechanical properties of hydrogen functionalized graphene. The atomic volume is taken from the initial (relaxed) sheet with the thickness of 3.4 ˚ A. The stress of the carbon sheet is computed by averaging over all the carbon atoms 142 in the sheet. From the simulated stressstrain curves, the Young’s modulus E, ultimate strength σ can be obtained. The Young’s modulus is calculated as the initial slope of the stress strain curve and the strength is defined at the point where the peak stress is reached. A.3 Results and Discussion The described MD approach is first verified by studying the mechanical properties of pristine graphene and hydrogen-functionalized graphene. The simulated Young’s modulus and tensile strength of graphene are around 0.86 TPa and 121 GPa, which are in good agreement with experimental results of 1.0 TPa and 123.5 GPa, respec- tively. For the mechanical properties of graphene with different hydrogen coverage, our calculations show the same deterioration as reported by Pei et al. (2010). We also simulated the mechanical properties for GAs nanoribbons with a width of 7.5 nm to test size effect on the simulated results. By comparing the results of nanoribbons with the case of the periodic boundary conditions, we find a very small change in the Young’s modulus and tensile strength. We now proceed to study how mechanical properties are altered as GA sheets are functionalized with hydrogen. As shown in Fig. A.1, we choose four different allotropes, which have been predicted by first-principles total energy calculations, in- cluding graphyne (benzene rings linked by diacetylene) as well as three stable 2D car- bon supra crystals (biphenylene, cyclic and octagonal graphene). The chosen allotrope sheets are mixture of sp- and sp2 - hybridized carbon atoms network with the area of unit lattice being considerably larger than that of graphene. Compared to graphene, their larger surface areas allow a variety of potential applications for energy storage, such as hydrogen storage and lithium-ion batteries. Typical stressstrain curves of the 143 functionalized allotropes with varying H-coverage are calculated. Fig. A.2 shows the stressstrain curves of hydrogen functionalized Graphyne, Cyclic Graphene, Octago- nal Graphene, and Biphenylene for H-coverage of 10%, 50% and 100%, together with that of the pristine allotrope sheet. The tensile strength of the investigated pristine GA sheets is much higher than pristine graphene implying their potential for wider applications. It can be seen that functionalized sheets fail at much lower stress and the corresponding fracture strain is also lower compared to that of the pristine sheet. The Young’s modulus, tensile strength and fracture strain of the pristine GAs ob- tained from the stressstrain curves in Fig. A.2. The mechanical properties of the functionalized sheets obtained from extensive molecular dynamics simulations for different H-coverage are shown in Fig. A.3. The strength and Young’s modulus deteriorate because of the formation of weaker sp3 bonds after H-functionalization. Local stress rearrangement induced by the conver- sion of local carbon bonding from sp2 to sp3 hybridization also contributes to the decay. For a given coverage, the hydrogen atoms considered in this paper are ran- domly distributed, and the error bars in the curves of Fig. A.3 are obtained from simulations of statistically independent realizations of functionalized. As Fig. A.3a shows, the Young’s modulus of these four allotropes exhibits sharp decreases with the increase of hydrogen coverage from 0% to 50% with different sensitivities. The Young’s modulus of cyclic graphene shows noticeable decay as H- coverage increases from 0% to 60%, beyond which the decay is not obvious. For graphyne, it reduces almost linearly to 30% of that of pristine as H-coverage increase from 0% to 100%. In case of cyclic graphene, there is sharp decay (almost 50%) until 30% H-coverage followed by about 15% more decay until 60% coverage. Beyond this 144 percentage, however, there is no change in modulus. For octagonal graphene, two lin- ear decay regimes can be identified in the range of 0-50% (50% decay) and 50-100% (10% more decay). The Young’s moduli are less sensitive to the increase of hydrogen coverage after 90%. Fully hydrogenated biphenylene shows the largest decay (similar pattern like octagonal graphene) among the investigated allotropes. The tensile strength of these four allotropes shows coverage sensitive and insen- sitive regimes. It can be seen that the sensitive regime for cyclic graphene and gra- phyne is 0-40%. For octagonal graphene and biphenylene, it is 0-60%. In the sensitive regime, strength reduces by 65-70% for the first two allotropes while for the latter two cases, strength reduces by around 80%. In the coverage sensitive regime, the drop in strength is faster than the drop in Young’s modulus. When H-coverage increases from 0% to 20%, the cyclic graphene shows lowest decay in tensile strength while graphyne shows fastest decay. It is interesting to notice that for the four allotropes considered, the order of decay speed in strength is opposite to that of the Young’s modulus. Fig. A.4 shows a typical bond breaking, crack nucleation, and growth scenario in biphenylene functionalized with 40% H-coverage. The stress distribution on car- bon atoms in the sheet before bond breaking is shown in Fig. A.4b. It can be seen that the lattices deformed under external tension and the structure of benzene rings (hexagons C6), which are interconnected by CC linkages, sustain the largest stress. It is interesting to note that the bond breaking always initiates at the sp3 bonds. Subsequently bonds outside the hydrogenated regions break leading to the crack for- mation (Fig. A.4c), nucleation (Fig. A.4d), propagation (Fig. A.4e), and finally tearing of the sheet (Fig. A.4f). We have also noted in all other allotropes that sp3 145 bonds always break before the sp2 bonds even when the latter bonds are subjected to larger stresses. This clearly shows that the sp2 to sp3 bonding transition results in the drop of mechanical strength of hydrogenated GAs. The failure occurs in the limit of large strain and is primarily controlled by breaking of weakest bonds, which can be triggered by even small H-coverage. This conclusion explains the noticeable drop of tensile strength when the hydrogen coverage increases from 0% to 40%. Since the Young’s modulus measures an average deformation of the system in a small am- plitude regime, its deterioration is less sensitive to hydrogenation than the decay of tensile strength, which has been demonstrated in Fig. A.3. As the hydrogen coverage increases above the sensitive threshold, the stronger sp2 bond network finally begins to disrupt. But weaker sp3 bonds that make the mechanical properties insensitive to the H-coverage govern the failure behavior of the functionalized allotropes. Pei et al. (2010) reported the unconstrained rotation of sp3 bonds caused by the stretching of the two dimensional graphene sheets. This unique phenomenon is also applicable for the investigated allotropes. The formation of unsupported sp3 bonds leads to the reduction in the elastic modulus and the strength of graphene. A.4 Conclusions In conclusion, we have carried out systematic MD simulations to study the me- chanical properties of H-functionalized GAs with coverage spanning the entire range from 0% to 100%. Here we have found that the tensile strength is more sensitive to the increase of H-coverage than Young’s modulus for all the allotropes, and it drops sharply even at small coverage. Among the chosen GAs, graphyne shows sharpest decay in tensile strength and lowest deterioration in Young’s modulus as H-coverage increases from 0% to 20% because of its lowest surface density. Different allotropes 146 exhibit different deterioration pattern and sensitive regimes, whichcan provide guid- ance for the potential application for hydrogen storage. Our simulations also show that bond breaking always initiates at sp3 bonds even with the presence of hybrid sp2 bonds, and therefore the sp3 bonding transition contributes to the loss of strength of functionalized GAs. Our results suggest that the coverage-dependence of the me- chanical properties should be taken into account in analyzing the performance char- acteristics of mass sensors, nanoresonators, and impermeable membrane structures fabricated from functionalized GA sheets. In this paper, we only focused on the decay trend and mechanism induced by hydrogen functionalization. Recent experiments re- ported the possibility to form nanopatterns on graphene via tip-induced desorption of hydrogen, and simulation also verified the influence of hydrogen distribution. Fur- ther analysis is expected into the dependence of mechanical and other properties on stretch directions and optimized hydrogen pattern on the surface of carbon sheets. The experimental verifications of our results are also expected. 147 Figure A.1: Optimized pristine carbon atomic structures for the examined graphene allotropes. 148 Figure A.2: Stressstrain curves of hydrogen functionalized (a) graphyne (b) cyclic graphene (c) octagonal graphene and (d) biphenylene for H-coverage of 0%, 10%, 50% and 100%. Figure A.3: Deterioration of (a) Young’s modulus and (b) tensile strength for the investigated GAs for different H-coverage. 149 Figure A.4: Fracture in biphenylene sheet functionalized with 40% H-coverage: (a) configuration at initial stage (b) onset of bond breaking (c) crack formation, (d) nucleation, (e) propagation, and (f) tearing of the sheet. Appendix B Plastic Fracture of Silicon Nanowire B.1 Introduction A nanowire is a nanostructure, with the diameter of the order of a nanometer (10−9 meters). Alternatively, nanowires can be defined as structures that have a thickness or diameter constrained to tens of nanometers or less and an unconstrained length. At these scales, quantum mechanical effects are important which coined the term “quantum wires”. The nanowires could be used, in the near future, to link tiny components into extremely small circuits. Using nanotechnology, such compo- nents could be created out of chemical compounds. Nanowires are being studied for use as photon ballistic waveguides as interconnects in quantum dot/quantum affect well photon logic arrays. Photons travel inside the tube, electrons travel on the out- side shell. When two nanowires acting as photon waveguides cross each other the juncture acts as a quantum dot. Conducting nanowires over the possibility of con- necting molecular-scale entities in a molecular computer. Dispersions of conducting 150 151 nanowires in different polymers are being investigated for use as transparent elec- trodes for flexible flat-screen displays. Because of their high Young’s moduli, their use in mechanically enhancing composites is being investigated. Because nanowires appear in bundles, they may be used as tribological additives to improve friction characteristics and reliability of electronic transducers and actuators. Because of their high aspect ratio, nanowires are also uniquely suited to dielectrophoretic ma- nipulation. Many researchers characterizing the electrical, mechanical, and thermal properties of various metallic nanowires have performed experimental work. With the advancement of computing power, it is now possible to simulate the behavior of nanowire under various conditions. The objective of this work is to study the yield and fracture properties of gold nanowires are determined by varying the loading rate and cross-sectional areas of the wires. B.2 Simulation Methods B.2.1 MD Potential and Procedure The MD simulations with LAMMPS package were performed here uses Embedded Atom Method (EAM)[]. Periodic BCs (PBC) is enforced in all three directions. Around 7−8 ˚ A of lengths on both sides are kept fixed (no movement of atoms in these regions). Simulation is performed at room temperature (300K). For 5 picoseconds, the simulation is performed to equilibrate the system and the energy of the system is minimized. For different strain rate, the velocity linearly applied first for all the atoms in the wire. In next step, fixed velocity is applied at the two ends the of bar. Time step considered is 0.001 ps. The samples considered this study are described in Table B.1. 152 Width Fixed Lengths at Ends Length of Wire Ratio (l/w) 10.1970 8.0000, 7.8518 80.0000 7.8455 18.3546 7.0000, 7.1130 147.0000 8.0090 30.5910 7.0000, 7.0040 245.0000 8.0090 38.7486 7.0000, 7.2650 310.0000 8.0003 50.9850 7.0000, 7.1560 408.0000 8.0024 Table B.1: Geometry of the Sample (All data are in Angstrom) B.3 Results and Discussion The results of the nanowire mechanical properties such as yield strain, yield stress, and fracture strain as functions of the wire cross-sectional area and different strain rate are shown. Strain is defined as  = (l − l0 )/l0 , where l current wire length and l0 is the wire length after the energy minimization. Yield strain is defined as the strain at which the maximum tensile stress occurs, while fracture strain is defined as the strain at which the wire breaks into two distinct entities. The stress reported here is Virial stress. The results of the nanowire mechanical properties such as yield strain, yield stress, and fracture strain as functions of the wire cross-sectional area and loading rate are shown. Figure B.1 shows stress vs strain curve for wire width of 5.09850 nm for two dif- ferent strain rate of 108 /s and 109 /s. The initial linear part is the elastic region where stress and strain has linear relationship. The slope of this portion is the elastic modulus if the gold nanowire. For strain rate of 109 /s, yielding starts at 2.75 GPa and yielding strain is around 0.062. Fracture strain is around 0.36. We can observe that for lower strain rate of 108 /s, yielding strain and corresponding stress is lower as compared to higher case. However, initial elastic response for all cases is same be- cause the slope of this curve is the elastic modulus, which is a property of the material. 153 Figure B.2 shows stress/strain curve for strain rate of 109 /s for different cross- sectional area. However, in all cases l/w ratio is 8. Yield Stress, strain and Fracture strain is less for the case of lower cross-sectional area. However, for very low cross- sectional case, the stress/strain behavior is not properly observable because in this case, the averaging is done in small cross-sectional area. This induces more noise. For large cross-sectional area, noise is reduced. Yield strain increases with decreasing wire cross-sectional area for all applied strain rates. We can conclude that the larger wire is more prone to slip and dislocation initiation from free surface due to the overall surface area. In general, the fracture strain increases with wire cross-sectional area for a given strain rate. Furthermore, the fracture strain increases with strain rate for all wire cross sectional areas considered, indicating the existence of a strain rate hardening mechanism in the wires. Figure B.3 shows the typical deformation process for a nanowire. The nanowires experience clearly defined stages of deformation, including yield and the onset of necking. The ATC within the wire neck gradually elongates un- der continued tensile loading until fracture occurs. We do not observe any twinning in FCC structure in this length scale. In general, of the three common crystalline struc- tures BCC, FCC, and HCP, the HCP structure is the most likely to form deformation twins when strained, mainly due to the lack of slip systems in this structure. B.4 Conclusions We have studied mechanical behavior of gold nanowire of different ratio of l/w. For l/w = 8.0024, for strain rate of 109 /s, yielding starts at 2.75 GPa with a yielding strain of 0.062. Fracture strain is around 0.36. For lower strain rate (e.g. 108 /s), yielding strain and corresponding stress is lower. However, the initial elastic response 154 for all cases is same as it corresponds to the elastic modulus of the wire. For lower cross-sectional area, yield stress, strain, and fracture strain is less. Larger wire is more prone to slip and dislocation initiation from free surface due to overall surface area. In general, fracture strain increases with wire cross-sectional area for a given strain rate. This study will help to understand fracture behavior of gold nanowire. Figure B.1: Nanowire Stress/Strain Curve for width = 5.098 nm. Two different strain rates (109 /s and 108 /s) are considered. 155 Figure B.2: Stress-strain curves for nanowires loaded at a strain rate of ( 109 /s for different cross sectional area. Length to Width ratio is same for all cases. Figure B.3: Failure Mechanism of Gold Nanowire Appendix C Viscoelastic Fracture of Silly-Putty C.1 What is Silly Putty Figure C.1: Silly Putty subjected to (a) low and (b) high strain rate. Silly putty 1 has unusual rheological or flow properties, it is a non-Newtonian fluid with dilatent behaviour. The deformation of Silly Putty is significantly affected by strain rate (Rheopecty). When Silly Putty is deformed slowly (low strain rate) as shown in Figure C.1a, its polymeric chains are allowed to uncoil, detangle, and move 1 Discovered in 1943 by a Scottish engineer James Wright 156 157 relative to one another, and we observe ductile fracture with great plastic deformation. When Silly Putty is deformed quickly (high strain rate) as shown in Figure C.1b , the polymeric chains are not able to move relative to each other, bonds are broken, and the Silly Putty exhibits brittle fracture. Unusual flow characteristics are due to the ingredient polydimethylsiloxane (PDMS), a viscoelastic liquid. C.2 Modeling Breaking of Silly Putty Figure C.2: Modeling breaking of Silly-Putty by Morse Bond. • At t = 0, the length of the breakable bond : Xbond and total system length : Xtot . • At t = tbreak , Xbreak break bond = Xbond + dXbond ; Xtot = Xtot + dXtot . • dXbond is same irrespective of external applied strain rate. • For Silly Putty behaviour, there will be variation in dXtot vsdXbond plot for different applied strain rate i.e. dXtot (lower strain rate)  dXtot (higher strain rate). 158 • For NO Silly Putty behaviour, there will be data collapse for different applied strain rate i.e. dXtot (lower strain rate)  dXtot (higher strain rate). Morse Bond to Model Failure : The Morse Potential is given as : E = D0 e−2α(r−r0 ) − 2e−α(r−r0 )   (C.1) Where r < rc (the cut-off distance). Where r0 is the equilibrium bond distance, α is a stiffness parameter, and D is the depth of the potential well. The force between the beads is given by : dE = 2αD0 e−2α(r−r0 ) − e−α(r−r0 )   F =− (C.2) dr dF The force between the beads will be maximum at a distance when dr = 0. Using Equation C.1 , we find Fmax that will occur at a distance given by : 1 rbreak = r0 + ln(2) (C.3) α The force causing the breaking is : 1 Fmax = − αD0 (C.4) 2 Modeling in LAMMPS : • Apply thermostat to bead 1 & 2 only. Bead 3 is subjected to applied velocity. So 159 we do not apply thermostat for this bead following the Equipartition Theorem. Z ∞ 1 1 2 3 < Hkin >=< mv 2 >= mv f (v)dv = kB T (C.5) 2 0 2 2 • We compute velocity of bead 2 & 3 at every stages and compute the differ- ence. Multiplication of velocity difference by damping coefficient of dashpot (we assume this value) gives the damping force. • This damping force is the additional force the bead 2 & 3 are experiencing. We use the command addforce in LAMMPS to apply this additional force. Figure C.3: Morse Bond : Variation of Energy (eV) and Force (eV ˚ A) w.r.t. to ˚ distance (A) between connecting atoms. 160 Figure C.4: Data collapse : No addition of Dashpot will results in data collapse and can’t capture Silly-Putty behavior. 161 Figure C.5: Capturing silly-putty behavior with the inclusion of dash-pot. Appendix D Atomic Stress Computation D.1 Stress at Atomic Scale Figure D.1: Atomistic System and Internal Force Between Atoms From the Principle of Virtual Work, we can derive that the volume average stress < σij > as : Z Z 1 1 < σij >= xi tj dS + ρxi (bj − aj ) dV (D.1) V ∂V V V 162 163 Figure D.2: Discrete Molecular System For a discrete atomistic system as shown in Figure D.1, we introduce : k(ext) δ2D xi − xki Select Boundary for xki ∈    Traction := tj = fj / ∂V (D.2) k(ext) δ3D xi − xki for xki ∈ V  Body Force := ρbj = fj (D.3) Density Acceleration := ρaj = mk akj δ3D xi − xki for xki ∈ V  (D.4) Using Equations D.2, D.3, and D.4 in Equation D.1, we get : k 1 X  k(ext)  < σij >= fj − mk akj (D.5) V i k(int) k(ext) Describing fj as internal force and fj as external force on the k particle and 164 using Equation ??, we will get : M (n) k(int) k(ext) k(int) X fj + fj = mk akj & fj = f nl(int) (D.6) l=1 N N N 1 X k k(int) 1 XX j  kl(int) < σij >= − xi f j = xi − xki fj (D.7) V k=1 2V k=1 l=k Referring to Figure D.2, we introduce the mass center position as: P  N mk xki k=1 xˆ = P  (D.8) N k k=1 m We assume that : " # N d 1 X k   x − xˆki x˙ ki − xˆ˙ ki  =0 (D.9) dt V k=1 i From Equation D.17, we get : N N X   X    mk xki − xˆki ¨ k k xˆj − x¨j = k k ˙ m x˙ i − xˆik k ˙ x˙ j − xˆjk (D.10) k=1 k=1  Here xki − xˆki is the position vector of the k th particle w.r.t. the mass center position. Referring to Figure D.2, we can select the reference frame in a way to coincide point O and M. Thus we can write as xki as the position vector we desire: N X  N  X    mk xki ¨ k k xˆj − x¨j = mk x˙ ki − xˆ˙ ki x˙ kj − xˆ˙ kj (D.11) k=1 k=1 Equating Equation D.5 and Equation D.7, we get : N N N 1 XX j k  kl(int) 1 X k  k(ext) k k  x − xi f j = x f − m aj (D.12) 2V k=1 l=k i V k=1 i j 165 From R.H.S. of Equation D.12, we can write : N N N 1 X k  k(ext)  1 X k 1 X k k k xi f j − mk akj = xi − x m x¨j (D.13) V k=1 V k=1 V k=1 i | {z } From Equation D.11, considering the L.H.S., we can write N X N X N X    − mk xki x¨kj = − mk xki x¨ˆj + mk x˙ ki − xˆ˙ i x˙ kj − xˆ˙ j (D.14) k=1 k=1 k=1 Substituting it in Equation D.12, we get: N N N N 1 XX j  kl(int) 1 X  ˙  ˙  1 X k  k(ext)  k xi − xi f j − k k k m x˙ i − xˆi x˙ j − xˆj = xi f j − mk x¨ˆj 2V k=1 l=k V k=1 V k=1 (D.15) Hence we have derived the Virial Stress as : N N N 1 XX l  kl(int) 1 X k  k(ext)  < σijV >= k xi − xi f j − x i fj − mk x¨ˆj (D.16) 2V k=1 l6=k V k=1 We rewrite Equation D.17 as : " N # d 1 X k k   m xi − xˆki x˙ kj − xˆ˙ kj  =0 (D.17) dt V k=1   The term L = mk xki − xˆki x˙ kj − xˆ˙ kj is defined as the Moment of the Momentum. 1 dL Hence V dt (considering volume to be constant) implies that the rate of change of moment of momentum is zero or moment of momentum of the system is constant. Appendix E Supporting Figures of Complex Fracture of Graphene Figure E.1: GNR (with crack of length 2a) is subjected to loading (marked in red) in both (a) zigzag and (b) armchair direction 166 167 Figure E.2: Crack surface does not matter in Classical Mechanics at continuum scale as it matters in atomic scale 168 Figure E.3: Stress Intensity Factors (SIF) for cracks with armchair (a-c) and zigzag (d-f) edges for different a/b 169 Figure E.4: Variation of maximum normal stress for different loading angle Figure E.5: Stress along the direction perpendicular to the crack in the graphene sheet pulled in armchair direction 170 Figure E.6: Variation of Potential Energy for crack with armchair and zigzag edges for the case of pure tension Figure E.7: Surface energy plays a critical role in Atomistic System Figure E.8: Potential energy for different slit angle for GNR pulled in zigzag direction Appendix F Supporting Information : Friction between Bilayer of Nanomaterials F.1 Introduction Figure F.1: Computation of equilibrium lattice constant of graphene. 171 172 Figure F.2: Increase in friction between graphene/graphene bilayer due to the pres- ence of stone-wales defect. 173 Figure F.3: Computation of equilibrium distance between h-BN/h-BN bilayer. Figure F.4: Computation of equilibrium distance between h-BN/graphene bilayer. Appendix G Graphene-Polystyrene Nanocomposites Figure G.1: Motivation of Nanocomposites. Source: Nanowerk 174 175 Figure G.2: Structure of Polystyrene Polymer. 176 Figure G.3: Computation fo Density of Polystyrene: (a) Different number of Polystyrene monomer chains (20, 60, 100, 150). (b) Volume and (c) Density at different MD steps (d) NVT and NPT Stages : Analogous to Squeezing and Relaxing a paper (e) Temperature fluctuation at different MD stages. 177 Figure G.4: Adhesion between polystyrene and graphene nanocomposites. Figure G.5: Shear between polystyrene and graphene nanocomposites. (a)-(b) Trans- verse and Side View (c) Effect of hydrogenation (d) Shear between polystyrene and pristine graphene. Appendix H Supporting Figures : Enhanced Lithiation in Defective Graphene H.1 Introduction In this study, we have considered triclinic system instead of square sheet. Since we are considering periodic systems i.e. there is no edge stress, overall conclusions for triclinic and square sheet will be same. To generate triclinic super cell, we used unit cell consisting of 2 atoms. For square sheet, we need unit cell consisting at least 4 atoms. For DFT computation, we are using VASP package, which cannot handle too many atoms. For computational efficiency, it is desirable to have less number of atoms. Therefore, it is better to consider unit cell with less number of atoms. 178 179 Figure H.1: Li adsorption on DV Hex (a-c) and Top (d-f) site at (a,d) over the defect (O position) (b,e) neighborhood of the defect (N position) and (c,f) away from the defect (N position). 180 Figure H.2: Li adsorption on SW Hex (a-c) and Top (d-f) site at (a,d) over the defect (O position) (b,e) neighborhood of the defect (N position) and (c,f) away from the defect (N position). 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