Topics in Mechanics of Two Dimensional Crystalline Materials by Akbar Bagri B.Sc., Mechanical Engineering, MUT, Iran, 2000 M.Sc., Mechanical Engineering, AUT, Iran, 2003 M.A., Chemistry, Brown University, USA, 2011 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 May, 2013 i   © Copyright 2013 by Akbar Bagri ii   This dissertation by Akbar Bagri is accepted in its present form by the School of Engineering as satisfying the dissertation requirement for the degree of Doctor of Philosophy. Date________________ ________________________________ Vivek B. Shenoy, Advisor Recommended to the Graduate Council Date________________ ________________________________ Pradeep R. Guduru, Reader Date________________ ________________________________ Brian W. Sheldon, Reader Approved by the Graduate Council Date________________ ________________________________ Peter Weber, Dean of the Graduate School iii   Vita Akbar Bagri was born in Andimeshk, Khouzestan, Iran on August 23, 1976. He received his B.Sc. degree from MUT, in 2000, and his M.Sc. degree from AUT, in 2003 both in Iran. He entered the solid mechanics group at Brown University for pursuing his Ph.D. degree in Fall 2008. En route to his Ph.D., he received a Masters in Chemistry from Brown University in 2011. iv   Acknowledgements I would like to extend my deepest gratitude and appreciation to my advisor, Professor Vivek Shenoy. During my graduate studies, I have been privileged to work under his guidance and he has been very patient and encouraging with me throughout the course of the work. I would also like to express my gratitude to the members of the thesis committee, Professor Sheldon and Professor Guduru for taking time out of their schedules to review and critique my work. The fruitful collaboration with Prof. Rodney Ruoff, Prof. Manish Chhowalla, Dr. Cecilia Mattevi, Dr. Muge Acik, and Prof. Yves Chabal, is gratefully acknowledged. I would like to thank all the current and former members of the research group of Prof. Shenoy. My thanks are due to my colleagues and friends who have helped me during the course of the work. Especially, I would like to convey my thanks to Nikhil Medhekar, Sang-Pil Kim, Rassin Grantab, Ivan Milas, Priya Johari, Hamed Haftbaradaran and Maria Stournara for their wonderful company. I gratefully acknowledge support from the NSF and NRI through the Brown University MRSEC program and the NSF through grants CMMI-0825771 and CMMI-0855853, and the support of the Army Research Office through Contract W911NF-11-1-0171. I wish to express my sincere thanks to Patricia Capece, Peggy Mercurio, Stephanie Gesualdi, Jim Scheuerman, for their help over the years. Finally, I would like to thank my family - my parents and my siblings who have been a source of constant support and encouragement. v   Contents Signature page ………………………………………………………………………... iii Vita …………………………………………………………………………………… iv Acknowledgements …………………………………………………………………... v Table of Contents …………………………………………………………………...... vi List of Tables ….……………………………………………………………………... viii List of Figures .……………………………………………………………………….. ix 1 Introduction and overview …………………………………………………………. 1 1.1 History of graphene ………………………………………………………. 1 1.2 Synthesis of graphene .…………………………………………………… 3 1.2.1 Reduction of graphene oxide ………………………………….. 3 1.2.2 Epitaxial growth on metal substrates ………………………….. 3 1.3 Atomistic simulation methods …………………………………………… 4 1.3.1 Molecular dynamics method ………………………………….. 5 1.3.1.1 Velocity-Verlet algorithm ………………………….. 7 1.3.1.2 Potential energy function .………………………….. 7 1.3.2 Density functional theory calculation …………………………. 11 1.4 Organization of this thesis ……………………………………………….. 14 2 Structural evolution of graphene oxide during thermal reduction process ………… 15 2.1 Introduction ………………………………………………………………. 15 2.2 Computational and experimental methods ………………………………. 17 2.3 Results and Discussion …………………………………………………... 21 vi   2.4 Summary of the chapter ………………………………………………….. 54 3 Thermal transport across twin grain boundaries in polycrystalline graphene ……… 56 3.1 Introduction ………………………………………………………………. 56 3.2 Computational method …………………………………………………… 58 3.3 Results and Discussion …………………………………………………... 62 3.4 Summary of the chapter ………………………………………………….. 68 4 Concluding remarks ………………………………………………………………... 70 Bibliography ………………………………………………………………………….. 73 vii   List of Tables 2.1 Simulation results of oxygen concentration after heating (1000K, 1500K) of graphene oxide with different initial oxygen concentrations. The values shown in black indicate percentage of remaining oxygen after heating (1000K, 1500K) while the values shown in red are after introducing H2 and reheating the structure up to thermal reduction temperature (1000K, 1500K). The ratio of hydroxyls/epoxides are reported in parentheses for each initial oxygen concentration. ………………………………………………………... 25 3.1 Temperature jumps at the grain boundaries and heat flux (given in parentheses) for different grain sizes and angles. Units for the given temperature and heat flux values are Kelvin and Watt per square meter, respectively. ………………………………………………………………….. 66 viii   List of Figures 1.1 Atomic structure of graphene ………………………………………………... 1 2.1 (a) Initial configuration of hydroxyl and epoxy groups used in the MD calculations based on the observations of Cai et al. [86] who found that hydroxyl and epoxy groups are bonded to neighboring carbon atoms. The hydroxyl group is shown to be on the same side of the basal plane as the basal plane in (i) while it is on the opposite side in (ii). Similarly the epoxy group can be on either side of the basal plane (not shown). Simulated GO structures for different initial oxygen contents having hydroxyl to epoxy ratio of 3:2 at 300 K. Structures for 20% and 33% oxygen contents before (b and c, respectively) and after (d-e, respectively) annealing at 1500 K are shown. Carbon, oxygen and hydrogen atoms are color-coded as gray, red and white, respectively. ……………………………………………………........... 22 2.2 Oxygen functional groups and carbon arrangements formed after annealing: (a) a pair of carbonyls, (b) carbon chain, (c) pyran, (d) furan, (e) pyrone, (f) 1,2-quinone, (g) 1,4-quinone, (h) carbon pentagon, (i) carbon triangle, (j) phenol. Carbon, oxygen and hydrogen atoms are color-coded as gray, red and white, respectively. ……………………………………………………… 23 2.3 Side views of GO sheets with hydroxyl to epoxy ratio of 3:2 at oxygen concentrations of 20% (a,b) and 25% (c,d). (a) and (c) correspond to the configurations before reduction, while (b) and (d) are the configurations after reduction at 1500K. …………………………………………………………... 24 ix   2.4 Concentrations of oxygen functional group remaining in GO after heating to (a) 1500K (b) 1000K versus the initial oxygen concentration. The different functional groups are denoted by the color: black for carbonyls, blue for hydroxyls, green for epoxides and red for furan and pyran. The filled dots indicate initial hydroxyl to epoxy ratio of 2:3 whereas the empty dots refer to ratio of 3:2. The lines are guides for the eye. ……………………................... 27 2.5 The atomic structure of relaxed GO sheets with an epoxy pair (a, d), a hole decorated by a carbonyl pair (b, e), and a hole decorated by a carbonyl and a hydroxyl group (c, f). The configurations (a-c) are obtained using the ReaxFF potential, while (d-f) are obtained by first principles methods. In each case, the strain (in %) in the C-C bonds near the functional groups is marked in the plan view (top), while the C-O bond lengths (in Å) are shown in the side view (bottom). The lengths of the remaining C-C bonds can be determined using symmetry considerations. In all the calculations, periodic boundary conditions are enforced along both coordinate directions. ............... 29 2.6 The potential energy map – as predicted using ReaxFF – for the relaxed GO sheet with pairs of epoxy and carbonyl groups arranged in different configurations. For each configuration, the absolute potential energy of the system in eV is given, while the energy relative to state (g), the lowest energy configuration, is noted in parentheses. ……………………………….. 31 2.7 The potential energy map for graphene oxide with a row of ethers and an epoxy group (a,b), and a row of ether and a carbonyl pair (c), as predicted using the ReaxFF potential. For each case, the absolute potential energy of x   the system in eV is given, while the energy relative to state (c), the lowest energy configuration, is noted in parentheses. The relative energies of these configurations are in qualitative agreement with an earlier first principles study [93]. ……………………………………………………………………. 32 2.8 Hole formation in GO due to the interaction of epoxy and hydroxyl groups during thermal reduction at 2000 K. The snapshots present a chronological evolution of the structure at 3.5 ps, 3.6 ps, 4.0 ps, and 4.25 ps. Carbon, oxygen and hydrogen atoms are color-coded as gray, red and white, respectively. ………………………………………………………………….. 35 2.9 Variations in the bond lengths and bond orders of the most relevant bonds during hole formation in GO due to the interaction of epoxy and hydroxyl groups given in Figure 2.8. C1, C2, C3, and O4 correspond to the carbon and oxygen atoms marked in Figure 2.8a. ………………………………………... 36 2.10 Hole formation in GO due to the interaction of hydroxyl groups during thermal reduction at 2000K. The snapshots present a chronological evolution of the structure at 0.75 ps, 1.0 ps, 1.1 ps, and 3.5 ps. Carbon, oxygen and hydrogen atoms are color-coded as gray, red and white, respectively. ……… 38 2.11 Variations in the bond lengths and bond orders of the most relevant bonds during hole formation in GO due to the interaction of hydroxyl groups given in Figure 2.10. C1, C2, C3, and O4, O5 and H6 refer to the carbon, oxygen and hydrogen atoms marked in Figure 2.10a. ………………………………... 39 2.12 Hole formation in GO due to the interaction of epoxy groups during thermal reduction at 2000 K. The snapshots present a chronological evolution of the xi   structure at 0.4 ps, 0.55 ps, 0.7 ps, 1.12 ps, 1.2 ps and 5.0 ps. Carbon and oxygen atoms are color-coded as gray and red, respectively. ……………….. 41 2.13 Variations in the bond lengths and bond orders of the most relevant bonds during hole formation in GO due to the interaction of epoxy groups given in Figure 2.12. C1, C2, C3, C4 and O5 and O6 refer to the carbon and oxygen atoms marked in Figure 2.12a. ……………………………………………….. 42 2.14 Initial configuration of hydroxyl and epoxy groups (top) leading to formation of carbonyl and hydroxyl decorated holes (bottom). In (a) transfer of hydrogen between neighboring hydroxyl groups (indicated by arrows) leads to the formation of a carbonyl pair. In (b-d) strain created by epoxy groups leads to the creation of carbonyl and phenol groups that relax the strain the neighboring carbon atoms. The energy difference (Ei – Ef in eV) between the initial configurations (top) and the final configurations (bottom) in each case obtained using DFT calculations (refer to methods section for details). Note that formation of these holes is energetically favorable in all cases. Carbon, oxygen and hydrogen atoms are color-coded as gray, red and white, respectively. ………………………………………………………………….. 44 2.15 Energy barriers and transition states for the formation of (a) a carbonyl pair from a pair of epoxies and (b) a phenol-carbonyl pair from a hydroxyl and epoxy groups. The transition states in both cases are characterized by “atop” oxygen atoms on the basal plane. ……………………………………………. 45 2.16 (a) Initial configuration with hydroxyl and epoxies that leads to the incorporation of an oxygen atoms in the basal plane (f). The ephemeral xii   intermediate configurations seen in the MD simulations are shown in panels (b-e). The sequence of events resulting in the incorporation of the oxygen molecule in the basal plane are: 1) Formation of a carbonyl –phenol hole (b), 2) Transfer of hydrogen via hydrogen bonding (b-c), 3) Formation of carboxyl group (d) Hydrogen migration and release of CO2 (e-f). …………... 47 2.17 (a) Initial configuration with epoxies only that leads to the incorporation of oxygen atom in the basal plane (f). A CO molecule is produced as a byproduct of this reaction (refer panel (e)). The ephemeral intermediate configurations that facilitate the formation this molecule and the pyran configuration are indicated in panels (b-e). The sequence of events shown in this figure are: 1) breaking the C-O bonds shown arrows in (b) leading to the creation of a hole decorated by carbonyl pair shown in (c). 2) formation of an atop C-atom leading to C-C bond breaking and creation of an out-of-plane C=O configuration as shown in (d). 3) connection of the oxygen which is part of carbonyl to the under-coordinated carbon atom in (e) resulting in the formation of a CO molecule as shown in (f). ………………………………... 48 2.18 In-situ transmission infrared and XPS spectra of GO: (a) (i) Absorbance spectrum of single-layer GO after annealing at 423 K, referenced to the bare oxidized silicon substrate, showing hydroxyls (broad peak at 3050-3800 cm- 1 ), carbonyls (1750-1850 cm-1), carboxyls (1650-1750 cm-1), C=C (1500- 1600 cm-1), and epoxides and/or ethers (1280–1000cm-1). (ii) Differential spectrum after annealing to 448 K referenced to spectrum (i), indicating epoxide loss, carbonyl formation and hydroxyl desorption. (iii) Differential xiii   spectrum after annealing to 1023 K referenced to spectrum at 448 K anneal, showing ether formation (1000-1060 cm-1 and 1080-1240 cm-1) and hydroxyl desorption. (iv) Absorbance spectrum (referenced to H-terminated Si), showing the full SiO2 absorption of the oxide (750-1500 cm-1 range), characterized by longitudinal optical (LO, 1250 cm-1 ) and transverse optical (TO, 1060 cm-1 ) vibrational modes. (b) O1s XPS spectra collected on one layered GO thin film deposited on Au (10nm)/SiO2(300nm)/Si and annealed in UHV at the indicated temperatures for 15 min. By deconvolving, the O 1s peaks collected after heating at 573 K and 723 K, two component have been identified as C-O bonds (533 eV) and C=O (531.2 eV) [74, 99] bonds. It has been found that 50% of the oxygen atoms are in C=O configurations. ……………………………………………………………….. 50 2.19 Evolution of residual oxygen groups during annealing of reduced GO in hydrogen atmosphere. (i) The epoxy group labeled 1 in (a) evolves to a hydroxyl group (1’) in (b). (ii) The hydroxyl group labeled 2 leaves the basal plane in the form of a water molecule (2’). (iii) The carbonyl labeled 3 in (a) is converted to a hydroxyl (3’). (iv) The carbonyl labeled (4) leaves the basal plane in the form of a water molecule (4’) leading of the healing of the hole and the restoration of sp2 bonding in the basal plane. ……………………….. 52 2.20 Improvement in reduction efficiency upon annealing of reduced GO in a hydrogen atmosphere. (a) Initial configuration before with hydroxyl to epoxy ratio of 3:2 at 300 K (b) structure after annealing at 1500 K (c) structure after annealing the reduced configuration from (b) in hydrogen. …………………. 53 xiv   3.1 Geometry of the RNEMD simulation box. The cold slabs are placed at the ends of the simulation cell, while the hot slab is located in the middle of the cell. …………………………………………………………………………… 58 3.2 Structure of tilt grain boundaries with misorientation angles of (a) 5.5 (b) 13.2 and (c) 21.7 degree. ……………………………………………………... 61 3.3 Typical temperature profile through the geometry of (a) graphene and (b) graphene with grain boundaries obtained by using the RNEMD method. …... 62 3.4 Inverse of thermal conductivity for zigzag-oriented graphene as a function of grain orientation versus the inverse of grain size (lg). The intercepts for the case of zig-zag graphene, 5.5, 13.2, and 21.7 degree oriented grains are 37x10-5, 45 x10-5, 42 x10-5 and 42 x10-5, respectively. ……………………… 64 3.5 Thermal conductivity of graphene grains of different sizes (25, 50, and 125 nm) versus the orientation of the grain. ……………………………………… 65 3.6 Boundary conductance of grain boundaries as a function of orientation. The curves are labeled by the size of the grains used to compute the boundary conductance. …………………………………………………………………. 68 xv   1   Chapter 1 Introduction and overview 1.1 History of graphene Following from the fullerenes of the 1980s and the nanotubes of the 1990s, graphene– also known as graphite layer– is one of the allotrope of carbon whose structure is a two- dimensional atomic layer of sp2-bonded carbon atoms in a honeycomb crystal lattice [1- 3]. Like diamonds and coal, graphene is made up entirely of carbon. But unlike those materials, two-dimensional arrangement of carbon atoms in graphene makes it incredibly strong and flexible. At the molecular level, it looks like chickenwire, see Figure 1.1. Figure 1.1: Atomic structure of graphene 2   Acheson [4] had developed exfoliation methods to synthesize graphite layers and these exfoliated graphite flakes have been extensively used in electronics devices as a conducting point to produce conducting surfaces in vacuum tubes [5] . In 1859, Brodie used nitric acid and potassium chromate to produce graphitic oxide, which consists graphene layers with oxygen functional groups connected to the carbon atoms [6]. Then by immersing graphite oxide into water the individual graphene oxide layers dispersed and separate layers were obtained. In 1962, using hydrazine, Boehm reduced monolayers of graphene oxide to graphene and for the first time coined term “graphene” as “ene” ending refers to hydrogen terminated edges of macromolecules of graphene [7] . Despite a long history of discovery of graphene, it took a century that graphene became important for its extraordinary electronic properties. But after work of Novoselov and Geim in 2004 [1], several research activities have been devoted to exploring the exceptional properties of graphene, including high carrier mobilities [8, 9], mechanical strength [10], optical transparency [11], and thermal conductivity [12]. Due to its extraordinary transport properties [3], it has also emerged as a promising candidate for a variety of novel applications that include nanoelectronics [13, 14], spintronic devices[15, 16], electromechanical resonators [17], impermeable atomic membranes [18], and quantum computing [19, 20]. For the realization of many of these device architectures, large sheets of defect-free graphene must be synthesized via a controlled and repeatable process. Several methods of production of grapehene have been proposed: Mechanical exfoliation, epitaxial growth on silicon carbide, epitaxial growth on metal substrates, and reduction of graphene oxide sheet. In order to guide the experimental researches, it is crucial to have a 3   thorough understanding of the nature of materials and their properties. In the following we briefly introduce two of these experimental methods and present the atomistic simulation methods used in present study to shed light on details of physical/chemical phenomena which are happening during the experiments. 1.2 Synthesis of graphene 1.2.1 Reduction of graphene oxide Historically, graphene oxide (GO) reduction was probably the first method of graphene synthesis. Among the several methods that are currently being pursued, one attractive route for large-scale production and uniform deposition of graphene is solution exfoliation of graphite into individual layers [21-24]. It has been demonstrated that efficient exfoliation of graphite can be achieved via chemical or thermal oxidation to graphite oxide [25, 26]. Due to its hydrophilic nature, graphite oxide readily disperses into individual atomically thin sheets in aqueous media. Recently it has been shown that graphite oxide can be deposited in the form of thin films from solution with controllable thicknesses [23, 24, 27]. Once deposited, these films can in principle be converted to graphene by removing oxygen-containing function groups via chemical [23, 24, 27-29] or thermal reduction [30, 31]. Thus, graphite oxide suspensions provide a solution-based process for uniform deposition of graphene over large areas on a variety of substrates. 1.2.2 Epitaxial growth on metal substrates Among the methods of production of graphene, the epitaxial growth on transition metal substrates (chemical vapor deposition method) seems to be most promising method for synthesizing large area graphene with high quality at relatively low cost. Chemical vapor 4   deposition (CVD) technique has been devised that exploits the low solubility of carbon in metals such as nickel [32, 33] and copper [34, 35] in order to grow graphene on metal foils. Similar to the synthesis of carbon nanotubes (CNT), the CVD growth mechanism can be understood by a vapor-liquid-solid or vapor-solid-solid model. CVD is a process whereby gaseous mixtures react, usually by an endothermic reaction to form a deposit on a heated substrate. The process generally involves the following steps: (1) initial stage (2) nucleation stage (3) growth stage until covering the whole area of metal substrate. A consequence of this technique is that the large-area graphene sheets typically contain grain boundaries, because each grain in the metallic foil serves as a nucleation site for individual grains of graphene [34]. 1.3 Atomistic simulation methods Atomic and molecular simulations can provide insights into the structural, electronic, physical and chemical properties of nanoscale structures. In recent years, with technological advancements these computational tools have been widely used to predict the properties of complex and new materials. The ability to manipulate the material at the atomic scale can lead to produce devices of unprecedented speed and efficiency. The outcome of such efforts is to revolutionize the technologies and materials in ways that will enable us to manipulate even individual atoms toward desire properties and features. The limitation of atomistic methods to simulating systems containing a small number of particles is a pathological problem in spite of continuous progress in pushing the limit toward systems of ever increasing sizes. System size is a critical issue when one desires a high degree of accuracy in modeling the inter-atomic forces between the atoms 5   constituting the system with quantum calculation approaches. While small size simulation is an issue for prediction of properties of bulk materials, it provides the opportunity for understanding the physical phenomena or material properties of the structures at micro/nano scales. The last decade have witnessed important theoretical and algorithmic advances in material science. Among the computational tools in materials science, Monte Carlo (MC) method, molecular dynamics (MD) and density functional theory (DFT) calculations have led to great strides in the description of materials properties. In the following, a brief review of MD method and DFT calculations, which have been used in this dissertation, is presented. 1.3.1 Molecular dynamics method One of the promising tools in the theoretical study of materials science is the molecular dynamics method. This computational method calculates the time dependent behavior of a molecular system. MD simulations have provided detailed information on the physical/ chemical properties of structures and systems. These methods are now routinely used to investigate the structure, dynamics and thermodynamics of materials/molecules and their complexes. Alder and Wainwright first introduced the MD method in the late 1950's [36, 37] to study the interactions of hard spheres. Many important insights concerning the behavior of simple liquids emerged from their studies. Later, Rahman carried out the first simulation using a realistic potential for liquid argon [38] and the first MD simulation of a realistic system was done by Rahman and Stillinger in their simulation of liquid water [39]. The goal of this section is to provide an overview of the theoretical foundations of 6   classical molecular dynamics simulations which has been used in the rest of this dissertation. The general steps in MD simulations are 1) Introducing initial positions and velocities of the atoms 2) Defining the boundary conditions 3) Selection of potential energy function 4) Solving the equations of motion during an MD time span 5) Determining the thermodynamic material properties The molecular dynamics simulation consists of the numerical, step-by-step, solution of the equations of motion, which for a simple atomic system may be written as Fi = mi˙r˙i # (1.1) Fi =" V #ri ! where Fi is force exerted on atom i, mi is mass, ri is the position of the atom and V is the inter-atomic potential. By knowing the force on each atom and solving the equations of motions the acceleration of each atom in the system can be obtained and then using integration schemes one can determine the positions and velocities of the particles at next time step. Once the positions and velocities of each atom are known, the state of the system can be predicted at any time in the future or the past and properties of the system can be determined. Thus, in molecular dynamics simulation we need to define inter- atomic potential, which is a function of position of atoms and determines the way that particles interact with each other, and use an integration algorithm to derive the velocity and position at each time step. First, we introduce velocity-Verlet integration algorithm 7   used in our molecular dynamics simulations and then a brief review of different types of inter-atomic potentials will be presented. 1.3.1.1 Velocity-Verlet algorithm In three-dimensional (3D) systems with N atoms, the potential energy is a function of the atomic positions with 3N variables. Due to the presence of such a complicated function in equations of motion, it would be very difficult to solve them analytically and; therefore, they are solved using numerical methods. Among the numerical algorithms developed for integrating the equations of motion, velocity-Verlet is fast and reliable and has been used in many MD simulations. The velocity-Verlet method requires current forces be calculated before the first time-step, so these routines compute forces due to all atomic interactions. Thus, the initial positions and velocities of the atoms must be defined before starting the simulations. The equations used in velocity-Verlet algorithm to compute the positions and velocities of atoms at each time step are Fi (t) 2 ri (t + "t) = ri (t) + v i (t)"t + "t 2mi (1.2) F (t) + Fi (t + "t) v i (t + "t) = v i (t) + i "t 2mi ! 1.3.1.2 Potential energy function Selection of the potential function plays an important role in MD simulations. This selection depends on the type of materials in the system. On the other hand, potential energy used for the simulations determines the accuracy and computational time of simulations. 8   Pair-wise potentials The simplest way to introduce the interaction between the particles (atoms) is to use the total potential energy of a system as a sum of energy contributions between pairs of atoms. These potential functions take the van der Waals forces into account and represent the non-bonded interactions between the atoms. Thus, they are mostly useful for rough estimation of the behavior of the systems or calculation of properties of materials where there is no bonding or weak bonds are present. Two examples of such potential are Lenard-Jones (also known as the 6–12 potential) and Morse potentials. Many-body potentials The development of microscopic models beyond pair potentials made it possible to describe more realistic systems. In many-body potentials, the potential energy is not defined by a simple sum over pairs of atoms interactions, but is calculated explicitly as a combination of higher-order terms. For example, embedded-atom method (EAM) [40] and tight-binding [41] potentials consider the electron density of states around each atom from sum of contributions of the atoms in the neighborhood, and then calculate the potential energy as a function of this sum. EAM potentials have been successfully used for modeling the structures, properties and defects of metals and its general form is 1 V = #" i (n i ) + # $ (r ) (1.3) i 2 i% j ij ij ! where φij is a two-body central potential between atoms i and j, rij is the distance between atoms i and j, εi is the embedding energy, and n i is the electronic density of atom i due to the surrounding atoms and is defined as ! 9   n i = # n i (rij ) (1.4) i" j ! in which ni is the contribution of electronic density of atom i due to atom j at distance rij. Another example of many body potentials is Tersoff potential [42, 43], which has been used for carbon, silicon, germanium and some other materials. This potential involves a sum over groups of three atoms, and considers the angles between the atoms as an important factor. A particularly successful version of Tersoffian potential is reactive empirical bond-order (REBO) developed by Brenner [44, 45] (also known as Brenner potential), which has been widely used to model the hydrocarbons. The Tersoff potential for a system is 1 V= [ # # f (r ) f (r ) + bij f A (rij ) 2 i j "i C ij R ij ] & 1 r , indicate the average of the quantities over time as well as over the particles in the simulation cell. The above approach for computing the thermal conductivity of a homogenous system can also be generalized to the case of a system with defects. In the case of grain boundaries we consider here, imposing a heat flux leads to a “jump” in temperature across the boundary. The jump gives a measure of the boundary conductance (Kapitza conductance) G of the grain boundary through the relation [123]   60   (3.3) Furthermore, the temperature profiles that develop across the grains can be analyzed to obtain their thermal conductivity as a function of their orientation. The structures of tilt grain boundaries in zigzag-oriented graphene are shown in Figure 3.2 for different grain boundary angles. The grain boundaries consist of repeating five- and seven-membered ring pairs (5-7 pairs) that are separated by several hexagonal rings (hex rings). As the grain boundary angle increases, the number of hex rings separating the 5-7 defects decreases, with the ultimate limit occurring at 21.7° when only a single hex ring separates the periodic 5-7 defects. Therefore, more severe grain boundary angles are composed of higher defect densities. The repeating defect pairs can also be thought of as an array of edge dislocations with horizontal Burgers vectors where the five-membered rings represent the extra plane of atoms. In our simulations, periodic boundary conditions are employed both along the direction of heat flow (x) and perpendicular to the direction of heat flow (y). The atomic interactions are defined by a modified version of the Tersoff potential [124] which has been recently shown to yield values of the acoustic- phonon velocities that are in excellent agreement with measured data. The potential also provides lattice thermal conductivity values in single-walled carbon nanotubes and graphene that are considerably improved compared to those obtained from the original parameter sets [125, 126]. The atomic coordinates and the overall periodic dimensions of the simulation cell is first optimized using the gradient-based minimization method implemented in the Large-Scale Atomic/Molecular Massively Parallel Simulation (LAMMPS) molecular dynamics package [127] in a microcanonical NVE ensemble until   61   the forces on atoms are less than 10-8 eV/Å. RNEMD simulations are then carried out on the relaxed structure at room temperature with a time step of 0.5 fs. Before the temperature profiles are computed to infer thermal conductivity, the system is allowed to evolve for 44,000,000 MD steps during which the velocities of the atoms in the hot and cold region are exchanged every 100 MD time steps. After reaching the steady state regime, the temperature gradient through the structure is obtained by averaging over 8,000,000 MD steps. The temperature profiles are determined by dividing the structure into slabs that are approximately 10Å wide. Figure 3.2: Structure of tilt grain boundaries with misorientation angles of (a) 5.5 (b) 13.2 and (c) 21.7 degree.   62   Figure 3.3: Typical temperature profile through the geometry of (a) graphene and (b) graphene with grain boundaries obtained by using the RNEMD method. 3.3 Results and discussion First, we validate our approach by computing the thermal conductivity of defect-free graphene. The temperature variation obtained in our calculations in this case is shown in Figure 3.3a. The temperature profile is nonlinear near the hot and cold ends due to finite size effects as noted in previous work [128] and the high thermal conductivity of graphene. Similar temperature profiles have also been noted in carbon nanotubes (CNTs) [129]. Care must be taken in the extraction of thermal conductivity from this nonlinear profile, which indicates that thermal transport is not fully diffusive. To obtain the correct diffusive thermal conductivity [129] we take the temperature gradient of the middle portion between the thermostats to avoid edge effects. Even with this correction, the thermal conductivity inferred from NEMD calculations depends on the size of the system. We find that the conductivity of cells with periodic lengths of 50nm, 100nm, and   63   250nm, are 532, 898, and 1460 W/mK, respectively. To calculate the thermal conductivity of 2-D graphene sheets, the cross-sectional area is defined as Ayz=wd, where w is the width of the sheet and d is the thickness (chosen as the interplanar distance in graphite =3.35 Å). The dependence of the thermal conductivity on the length of the simulation cell can be understood by noting that the mean free path of phonons in graphene is of the order of 775nm [130], which is bigger than the size of our simulation cells. Therefore, in addition to phonon-phonon scattering, scattering at the heat baths (or boundaries) of the system must be considered. Based on the kinetic theory of phonon transport [131], the thermal conductivity is proportional to the mean free path for phonon scattering. In the case where phonons scatter at the heat reservoir, the effective mean free path is given by (3.4) where lph-ph denotes phonon-phonon scattering length and lg is scattering length due to the boundaries in a finite system and can be approximated to be the length of the simulation box. Based on this relation, the thermal conductivity satisfies the relation (3.5) which implies that a plot of the inverse of thermal conductivity, k, versus the inverse of the system size lg, should be a linear curve, the intercept being the thermal conductivity of the infinitely large system. Indeed, the plot of the inverse of the thermal conductivity as a function of the size of the system in Figure 3.4 confirms this scaling relation. The scaling approach we use here has also been used in earlier work on 3D systems [128,   64   132]. From the intercepts of the plot in Figure 3.4, we find the thermal conductivity of defect-free graphene along the zig-zag direction to be 2650 W/mK. This result is in excellent agreement with the result (2600 W/mK) obtained using the same potential we use here, but a different approach, namely the non-equilibrium molecular dynamics (NEMD) method [126]. Furthermore, our results agree with the theoretical [133, 134] and experimental [12, 135] values, which lie in the range 2000-5000 W/mK. Our results for the thermal conductivity of finite-sized graphene also agree with the results reported in Ref. ([136]), where NEMD has been used and the thermal conductivity of graphene of length 29.5nm was found to be 256 W/mK. Figure 3.4: Inverse of thermal conductivity for zigzag-oriented graphene as a function of grain orientation versus the inverse of grain size (lg). The intercepts for the case of zig-zag graphene, 5.5, 13.2, and 21.7 degree oriented grains are 37x10-5, 45 x10-5, 42 x10-5 and 42 x10-5, respectively. Next, we consider the temperature profile in the case of graphene with tilt grain boundaries (Figure 3.3b). We find a nearly linear temperature profile in the grains, but observe a jump in the temperature at the grain boundaries. A plot of the thermal   65   conductivity of grains (inferred from the slope of the linear part of the temperature profile) of different orientations as a function of size is plotted in Figure 3.5. As in the case of zigzag oriented grains, the inverse of the thermal conductivity decreases linearly with the size of the simulation cell. The intercepts of the curves in Figure 3.4 show that thermal conductivity is anisotropic, but only weakly depends on the orientation of the grains (k= 2220, 2380, 2380 W/mK for 5.5, 13.2, and 21.7 degree grains, respectively). Figure 3.5: Thermal conductivity of graphene grains of different sizes (25, 50, and 125 nm) versus the orientation of the grain. The jump in the temperature across the grain boundary can be used in Equation 3.3 to obtain the boundary conductance of the grain boundaries. A summary of these temperature jumps as a function of the grain sizes and angles is given in Table 3.1. Using the measured jumps, we find that the boundary conductance for the grain boundaries we have considered fall in the range 1.5x1010-4.5x1010 W/m2K (refer to Figure 3.6). These values are 6-to-12, 10-to-50, and 6-to-30 times larger than the boundary conductance’s reported for grain boundaries in ultra-nano-crystalline diamond thin films with grain   66   boundaries on the (001) plane [137], silicon-silicon (001) ∑29 grain boundaries [123, 138], and the Si-Ge interface with the <100> orientation [139], respectively. Note that the boundary conductance decreases with increasing the misorientation angle of the grain boundaries. This can be qualitatively understood by considering the defect density as a function of grain boundary angle – the higher the misorientation angle the larger is the density of 5-7 defect pairs per unit length of the boundary, which can lead to increase of scattering of phonons and hence a drop in the boundary conductance. We also observe that computed conductance depends slightly on the size of the grains. A similar dependence of the boundary conductance on size has also been reported in Refs. ([123]) and ([139]) for Si-Ge and silicon grain boundaries. This has been attributed to scattering of long wavelength phonons at the heat reservoirs and at the boundaries [131, 140], but a scaling of the conductance with length has not been provided. Table 3.1: Temperature jumps at the grain boundaries and heat flux (given in parentheses) for different grain sizes and angles. Units for the given temperature and heat flux values are Kelvin and Watt per square meter, respectively. Grain orientation Grain size (nm) angle (degree) 25 50 125 5.5 8.13 (1.95x1011) 6.18 (1.89x1011) 4.40 (1.89x1011) 13.2 12.17 (2.15x1011) 8.97 (2.13x1011) 7.36 (2.19x1011) 21.7 12.98 (2.12x1011) 10.24 (2.21x1011) 7.63 (2.13x1011) Note that when grains are very large in size, the scattering of phonons within the grains will primarily determine the thermal conductivity of the polycrystalline graphene, but with decreasing grain size the contribution to thermal conductivity due to scattering from   67   grain boundaries will become more significant. For the polycrystalline graphene sheet in Figure 3.2 with grain spacing lg , the thermal conductivity k p can be written as "1 ( ) k p"1 = k g"1 + lg G , (3.6) ! ! ! where kg is the thermal conductivity of the grain. Using this expression and the computed values of boundary conductance for the tilt boundaries, we can now estimate the critical ! size of grains below which the contribution from the grain boundaries becomes comparable to the scattering from the grains. This length scale is simply the ratio of the thermal conductivity to the boundary conductance. For the tilt boundaries considered here, this length scale is of the order of 0.1 microns. While a systematic study of the scaling of the thermal conductivity of polycrystalline graphene as a function of grain size has not been reported, the thermal conductivity of exfoliated graphene [141] is generally observed to be higher than the thermal conductivity of CVD-graphene [12]. Experiments on graphene with well controlled grain sizes and orientations can help verify the predictions of this study, namely, the scaling of the boundary conductance (Equation 3.6) and the computed value of boundary conductance as a function of the grain boundary angle. The former can be studied by measuring the overall thermal conductivity of polycrystalline graphene for different grain sizes, while the latter will involve measurement of temperature drop across individual boundaries (once they have been identified using appropriate microscopy techniques).   68   Figure 3.6: Boundary conductance of grain boundaries as a function of orientation. The curves are labeled by the size of the grains used to compute the boundary conductance. 2.4 Summary of the chapter In summary we have studied thermal transport across tilt grain boundaries in polycrystalline graphene. As in the case of interfaces in the dissimilar materials, we find a jump in temperature at grain boundary when a constant heat flux is applied. We have used this information to extract the boundary conductance, which lies in the range of 1.5x1010-4.5x1010 W/m2K in the case of tilt grain boundaries. Based on this information, we have identified a critical grain size of about 0.1 microns below which the contribution of tilt boundaries becomes comparable to that of the contribution from the grains themselves. We also note that here we have considered the most common type of tilt boundaries reported in the literature [112-115]; future work on thermal conductance of other types of grain boundaries can shed further light on the thermal transport properties of polycrystalline graphene. Recent experiments have shown that defects such as vacancies and voids tend to segregate at grain boundaries [116, 117]. These defects can   69   be expected to further lower the thermal conductance of the boundaries. We hope to consider the effect of these defects in forthcoming publications.   70   Chapter 4 Concluding remarks Having deep knowledge of nature of materials and their properties is crucial to guide the experiments and make new devices. In this regard and to reduce the costs of experiments, researchers have tried to use analytical and numerical tools in modeling and simulation of the processes and calculation of the material properties. In this thesis, the combinations of atomistic simulations and experimental measurements have been used to elucidate the atomic evolution of graphene oxide structure during thermal reduction process. We also have used molecular dynamics simulations to calculate the thermal conductance across boundaries of polycrystalline graphene and analyzed the effect of size and orientation of the grains. The main conclusions of this work are as follows Structural evolution of graphene oxide during thermal reduction process Using MD and DFT calculations in conjunction with experimental measurements the following results have been revealed about the atomic details of reduced graphene oxide. 1) Transformation of the initial hydroxyl and epoxy groups during thermal annealing can lead to formation of carbonyl and ether groups.   71   2) Hydroxyl groups desorb at low temperatures without altering the graphene basal plane. 3) Isolated epoxy groups are relatively more stable but substantially distort the graphene lattice upon desorption. 4) The simulations indicate that thermal reduction may result in removal of carbon from the graphene plane when the initial hydroxyl and epoxy groups are in close proximity to each other. Our simulations suggest that careful reduction processes must be designed if highly ordered graphene is to be achieved from graphene oxide. To this end, one may consider using chemical reactants other than hydrogen gas to remove further functional groups. Also, the effects of remaining functional groups on optoelectronic and thermomechanical properties of graphene oxide must be investigated. Thermal conductance across grain boundaries of polycrystalline graphene We have simulated polycrystalline graphene with different size and orientation of grains and performed MD calculations to determine the thermal conductance across grain boundaries. The summary of our results is 1) We find a jump in temperature at grain boundary when a constant heat flux is applied. 2) The boundary conductance extracted from these temperature jumps lies in the range of 1.5x1010-4.5x1010 W/m2K in the case of tilt grain boundaries.   72   3) We have identified a critical grain size of about 0.1 microns below which the contribution of tilt boundaries becomes comparable to that of the contribution from the grains themselves 4) Results show that when the size of structure is much less than phonon mean free path both thermal conductivity and boundary thermal conductance of polycrystalline are size dependent properties. 5) MD simulations revealed that thermal conductivity of graphene is anisotropic, but only weakly depends on the orientation of the grains. While we considered the most common type of tilt boundaries reported in the literature, future works on thermal conductance of other types of grain boundaries can shed further light on the thermal transport properties of polycrystalline graphene. Also, other type of defects such as vacancies and voids, which tend to segregate at grain boundaries, may have significant effect on thermal conductance of graphene and may be considered as future works. 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