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Thermal conductivity at the nanoscale: A molecular dynamics study

Dissertation
Author: IV John W. Lyver
Abstract:
With the growing use of nanotechnology and nanodevices in many fields of engineering and science, a need for understanding the thermal properties of such devices has increased. The ability for nanomaterials to conduct heat is highly dependent on the purity of the material, internal boundaries due to material changes and the structure of the material itself. Experimentally measuring the heat transport at the nanoscale is extremely difficult and can only be done as a macro output from the device. Computational methods such as various Monte Carlo (MC) and molecular dynamics (MD) techniques for studying the contribution of atomic vibrations associated with heat transport properties are very useful. The Green-Kubo method in conjunction with Fourier's law for calculating the thermal conductivity, κ, has been used in this study and has shown promise as one approach well adapted for understanding nanosystems. Investigations were made of the thermal conductivity using noble gases, modeled with Lennard-Jones (LJ) interactions, in solid face-centered cubic (FCC) structures. MC and MD simulations were done to study homogeneous monatomic and binary materials as well as slabs of these materials possessing internal boundaries. Additionally, MD simulations were done on silicon carbide nanowires, nanotubes, and nanofilaments using a potential containing two-body and three-body terms. The results of the MC and MD simulations were matched against available experimental and other simulations and showed that both methods can accurately simulate real materials in a fraction of the time and effort. The results of the study show that in compositionally disordered materials the selection of atomic components by their mass, hard-core atomic diameter, well depth, and relative concentration can change the κ by as much as an order of magnitude. It was found that a 60% increase in mass produces a 25% decrease in κ. A 50% increase in interatomic strength produces a 25% increase in κ, while as little as a 10% change in the hard core radius can almost totally suppress a materials ability to conduct heat. Additionally, for two LJ materials sharing an interface, the atomic vibrations altering the heat energy depend on the type of internal boundary in the material. Mass increases across the interfacial boundary enhance excitation of the very low frequency (ballistic) vibrational modes, while the opposite effect is seen as increases in hard core radius and interatomic strength enhance excitation of higher frequency vibrational modes. Additionally, it was found that this effect was diminished for higher temperatures around half the Debye temperatures. In nanodevices and nanomachines, there is an additional factor that degrades heat transport at the boundary. In fact, the interface induces a temperature jump consistent with a thermal resistance created by the boundary. It was found that the temperature jump, which is due to a boundary resistance, was significant in boundaries involving small mass changes, lesser in changes in hard core radii changes and even lesser for interatomic strength changes. The study of SiC nanowires and nanotubes showed that the structural changes produced vastly different κ. The κ in closely packed structures like nanowires and nanofilaments approximated that of the bulk SiC, yet were less sensitive to temperature than the 1/T relationship traditionally found in bulk systems. The more open nanostructures, like nanotubes, had vastly lower κ values and are almost totally insensitive to temperature variation. The results of this study can be used in the design of nano-machines where heat generation and transport is a concern. Additionally, the design of nano-machines which transport heat like nano-refrigerators or nano-heaters may be possible due to a better selection of materials with the understanding of how the materials affect their nanothermal properties at the nano scale.

TABLE OF CONTENTS

Page

CHAPTER 1   INTRODUCTION 1   CHAPTER 2   THEORY AND METHODS 7   2.1 Modeling 7   2.1.1 Background on Statistical Mechanics Simulations 7   2.1.2 Gibbs Ensembles of Statistical Mechanics 9   2.2 Interatomic Potentials 15   2.2.1 Lennard-Jones Potential 15   2.2.2 Silicon Carbide Model Potential 17   2.3 Units 19   2.4 Simulation Details 21   2.4.1 Boundary Conditions 21   2.4.2 Cubic Computational Box 22   2.4.3 Slab Computational Box 22   2.4.4 Nanostructures Computational Box 24   2.5 Structural and Thermodynamic Quantities 25   2.5.1 Pressure 26   2.5.2 Positional Order Parameter 26   2.5.3 Boltzmann’s H-function 27   2.5.4 Mean Square Displacement (MSD) 28   2.5.5 Temperature Stabilization 29   2.5.6 Pair Distribution Function 29   2.6 Dynamical properties 31   2.6.1 Time Dependent Correlation Function and Green-Kubo Formalism 31   2.6.2 Heat Flux 32  

vi 2.6.3 Vibrational Spectrum from a Classical Approach 33   2.7 Non-Equlibrium MD Simulations 35   CHAPTER 3   THERMAL CONDUCTIVITY STUDY OF BINARY LENNARD- JONES SYSTEMS 38   3.1 Introduction 38   3.1.1 Setup 39   3.1.2 Determining Thermal Conductivity 43   3.1.3 Computational Disorder 44   3.2 Determination of the Lattice Thermal Conductivity 46   3.2.1 Calibration of Results 46   3.2.2 Thermal conductivity of binary mixtures as a function of the compositional disorder 47   3.2.3 Temperature Effects on the Thermal Conductivity 50   3.2.4 Concentration Effects 52   3.2.5 Compositional Disorder Effect on the Heat Current Autocorrelation Function Time 54   3.3 Summary and conclusions 54   CHAPTER 4   DETERMINATION OF INTERFACE EFFECTS ON THERMAL CONDUCTIVITY 57   4.1 Background 57   4.2 Modeling 59   4.2.1 Setup 59   4.2.2 Model Validation 61   4.2.3 Effect of the elongated computational box on the lattice vibrations 63   4.3 Effects on Thermal Conductivity 64   4.3.1 Vibrational Modes of Monatomic Systems 64   4.3.2 Vibrational Modes of Binary Systems across the Interface of Two LJ Systems in Equilibrium 65   4.3.3 Vibrational Mode Changes between Equilibrium and non-Equilibrium Systems with a Boundary Interface 67   4.4 Interface Effects on Thermal Conductivity 70   4.4.1 Thermal Boundary Resistance 70  

vii 4.4.2 Interface Width 74   4.4.3 Crystal Orientation at the Interface 74   4.5 Interface Boundary Effects On Thermal Conductivity Conclusions 75   CHAPTER 5   DETERMINATION OF THERMAL CONDUCTIVITY IN NANOWIRES, NANOTUBES, AND NANOFILAMENTS 77   5.1 Background 77   5.2 Modeling 79   5.2.1 Setup 79   5.2.2 Model Validation 82   5.3 Thermal Conductivity in Nanostructures 83   5.3.1 Nanowires 83   5.3.2 Armchair Nanotubes 85   5.3.3 Zigzag Nanotubes 86   5.4 Nanostructure Thermal Conductivity Conclusions 90   CHAPTER 6   COMPUTATIONAL CHALLENGES 91   6.1 Computational Effects 91   6.1.1 Counteracting PBC effects 91   6.1.2 Software Error Handling 93   6.1.3 Energy Conservation Test 94   6.2 Software Used 95   6.2.1 Computational Software Environments 95   6.2.2 Fortran Code 96   6.2.3 Graphics Software 105   6.2.4 Supporting Software 105   CHAPTER 7   CONCLUSIONS 106   APPENDIX A   PUBLISHED PAPERS 109   A.1   Published Paper: Acta Materialia, vol 54, pp4633-4639, 24 August 2006 [9] 110   A.2   Published Paper: J. Phys: Condens. Matter vol21, pp345402-345409, 5 August 2009 [10] 117  

viii A.3   Prepublished Paper [11] (Submitted: June 2010 to Computational and Theoretical Nanoscience) 125   APPENDIX B   DEVELOPED COMPUTER CODES 134   B.1   Lennard-Jones Potential 134   B.1.1 Annotated Code 134   B.1.2 Parameters 135   B.2   SiC Two-body Potential 135   B.2.1 Annotated Code 135   B.2.2 Parameters 137   B.3   SiC Three-body Potential 137   B.3.1 Annotated Code 137   B.3.2 Parameters 140   B.4   Determination of Heat Current: 140   B.4.1 Annotated Code 140   B.4.2 Parameters 141   B.5   Calculation of Autocorrelation and Density of States 142   B.5.1 Annotated Code 142   B.5.2 Parameters 145   B.6   Maintain Constant Temperature in Thermal Baths 145   B.6.1 Annotated Code 145   B.6.2 Parameters 146   BIBLIOGRAPHY 147   CURRICULUM VITAE 155  

ix a

LIST OF TABLES

Page

Table 1: Summary of reduced units for the LJ potential. Here k B is the Boltzmann constant. 20   Table 2: Frequency of the peaks in the vibrational spectrum for argon 35   Table 3: Lattice thermal conductivity for mixtures with various relative concentrations at temperatures 0.042, 0.083, 0.167, 0.333, and 0.5 in reduced units 53  

x

LIST OF FIGURES

Page

Figure 1: LJ Potential Model 16   Figure 2: Schematic view of the computational box used in the NEMD studies of Chapter 4 23   Figure 3: Expanded view of simulated system of slices for NEMD work with slabs 24   Figure 4: Expanded view of simulated system of slices for NEMD work with nanostructures described in Chapter 5 25   Figure 5: MSD as a function of time for a LJ fluid 28   Figure 6: Pair correlation function as a function of distance for a binary LJ system at T=36K, ρ=0.382. Solid line is for A-A atoms, short dashed line is for B-B atoms and long dashed line is for A-B atoms. 30   Figure 7: Typical DOS of vibrational states for a 50:50 solid mixture of LJ atoms at T=0.167 35   Figure 8: Schematic view of a compositionally disordered binary LJ solid with a 50:50 mixture of green and red atoms 40   Figure 9: Density as a function of temperature for the binary LJ system with 50:50 relative concentration. Quantities are in reduced units. 42   Figure 10: Calculated time dependent autocorrelation function of the heat current operator as a function of time lag 44   Figure 11: Pair distribution function of a 50:50 mixture at T=0.167: (top) g(r) vs. interatomic distance, (bottom) same results as in (a) as function of scaled distances by 13. Solid lines are g AA , dashed lines are g BB , and dotted lines are g AB 45   Figure 12: Thermal conductivity as a function of temperature ( ) compared to other theoretical and experimental works for pure Ar at zero pressure. Inset is expansion of region near (0,0) {Ref. [3] (◊), Ref. [28] (○), Ref. [25] (∆), Ref. [36] (+), Ref. [37] (□)} 47   Figure 13: Finite size effects due to computational box size for Ar systems at T=0.167 48  

xi Figure 14: Thermal conductivity as a function of parameter ratios for a sample at T=0.167 and ρ=1.035; (a) 1 , 1.0 ,1.25 ,1.5 , (b) 1, 1.0 ,1.6 ◊,2 .1 ,3.3 ; and (c) 1, 1.0 ,1.1 ◊,1.25 ,2.0 49   Figure 15: Thermal conductivity as a function of temperature for parameter ratios: (a) 1.0 , 1.0 ,1.25 ,1.5 , and (b) 1.1 (solid) 1.25 for 1.0 ,1.25,1.5 51   Figure 16: Density as a function of temperature for 50:50 relative concentrations in the mixture. Filled symbols for = 1.1 and open symbols for = 1.25. The circle, triangle and square are for = 1.0, 1.25 and 1.5, respectively. = = =1 is shown as crosses. 52   Figure 17: Relaxation time as a function of parameter ratios for the 50:50 sample for T=0.167, ρ=1.035, for 1.0 ,1.25 ,1.5 on each plot 55   Figure 18: Schematic representation of the system used to simulate the interface between two LJ solids 60   Figure 19: Thermal conductivity as a function of temperature. (a) = 0.7 (●), 0.8 (▲), 0.9 (■), 1.25 (□), 1.5 (∆), 2.0 (○), Stars and standard deviations are from [9]. Solid diamonds pertain to the reference LJ system and dotted line (in a,b,c) is the best fit to these values. Crosses and standard deviation are from [9].); (b): = 0.7 (●), 0.8 (▲), 0.9 (■), 1.1 (□), 1.2 (∆), 1.25 (○); (c) = 0.3 (●), 0.5 (▲), 0.7 (■), 1.6 (□), 2.1 (∆), 3.3 (○). 62   Figure 20: Normalized density of states (DOS) of one-component systems in thermal equilibrium for various LJ parameters. (a-c) are for T = 0.12 and (d-f) are for T = 0.33. The solid line is for 1 in all plots. In (a) and (d) the DOS for = 1.25 (●), 1.5 (□), and 2.0 (∆) with = = 1. In (b) and (e) the DOS for = 1.1 (●) and 1.2 (□) with and = = 1. In (c) and (f) show the DOS for = 1.6 (●), 2.1 (□), and 3.3 (∆) with = = 1. 65   Figure 21: Normalized density of states (DOS) of binary systems in thermal equilibrium relative to the DOS of the one-component reference system for various LJ parameters. (a-c) are for T = 0.12 and (d-f) are for T = 0.33. In (a) and (d) the DOS for = 1.25 (●), 1.5 (□), and 2.0 (∆) with = = 1. In (b) and (e) the DOS for = 1.1 (●) and 1.2 (□) with and = = 1. In (c) and (f) show the DOS for = 1.6 (●), 2.1 (□), and 3.3 (∆) with = = 1. 66  

xii Figure 22: Normalized vibrational modes DOS for non-equilibrium (dashed line) and equilibrium (solid line) monatomic system at (a) ρ = 1.07, T = 0.12, and (b) ρ = 1.04, T = 0.33. Error bars identify the average SD of the correlation functions. 68   Figure 23: Non-equilibrium DOS of binary systems relative to their DOS in thermal equilibrium for various LJ parameters. (a-c) are for T = 0.12 and (d-f) are for T = 0.33. In (a) and (d) the DOS for = 1.25 (●), 1.5 (□), and 2.0 (∆) with = = 1. In (b) and (e) the DOS for = 1.1 (●) and 1.2 (□) with = = 1. In (c) and (f) show the DOS for = 1.6 (●), 2.1 (□), and 3.3 (∆) with = = 1. 69   Figure 24: Temperature profile of the LJ system across the interface at T = 0.3 with = 3.3, = = 1 and ρ = 1.04. Horizontal lines show the Kapitza length calculated for hot side, interface, and cold side (top, middle, bottom), respectively). Temperature standard deviation is shown as vertical error bars. 71   Figure 25: Temperature drop ratio at the interface (left scale) and Kapitza resistance (right scale) as a function system parameters at various temperatures of : 0.10 (♦), 0.125 (■), 0.165 (▲), 0.33 (□), 0.425 (◊), and 0.5 (○). 73   Figure 26: Nanotube computational setup 80   Figure 27: Temperature and heat current profile across a nanowire in the steady state at 300K: (a) temperature profile (♦) with SD for temperature and position, (b) heat current profile (○) 81   Figure 28: SiC Nanowire cross sections 83 Figure 29: Thermal conductivity as a function of temperature for nanowires: 3C [-100] (♦), 3C [-111] (▲), 2H [001] (○), and 2H [110] (□) 84   Figure 30: Armchair nanotubes and nanofilament configurations 86   Figure 31: Thermal conductivity as a function of temperature for armchair nanotubes and nanofilaments: (2,2) (♦), (5,5) (▲) nanotubes, and (3,3) (□), (4,4) (○)nanofilaments 87   Figure 32: Zigzag nanotubes and nanofilament configurations 88   Figure 33: Thermal conductivity as a function of temperature for zigzag nanotubes and nanofilaments: nanotube (4,0) (♦), and nanofilaments (6,0) (∆), (8,0) (○) and (10,0) (□) 89   Figure 34: Square root of the standard deviation as a function of time step size 95   Figure 35: Processing time required for various system sizes 97   Figure 36: MC programmatic flow diagram 98  

xiii Figure 37: MD programmatic flow diagram 101   Figure 38: Comparison of MD and NEMD Flow 104  

xiv

LIST OF ACRONYMS AND ABBREVIATIONS

AMM Acoustic mismatch model Ar Argon C Carbon CPU Computer processing unit CSP Certified Safety Professional DMM Diffuse mismatch model DOS Density of states E Total energy FCC Face-centered cubic g(r) Radial distribution function GK Green–Kubo K Kelvin LJ Lennard-Jones MATLAB® MathWorks, Inc, product MC Monte Carlo MD Molecular dynamics MSD Mean square displacement N Number of particles NE Non-equlibrium NEMD Non-equilibrium molecular dynamics NPT Constant pressure, temperature and number of atoms ensemble

xv ODE Ordinary Differential Equation P Pressure PBC Periodic Boundary Condition PGHPF® Portland Group High Performance Fortran PGI® Portland Group, Inc Si Silicon T Temperature T cold Temperature of the cold bath T hot Temperature of the hot bath V Volume W Watt

Specific heat at constant volume

Correlation Function eV Electron volt

Total energy of each atom

Force for each atomic pair

Boltzmann’s constant

Thermal flux

Thermal flux due to lattice contribution ℓ Mean free path

Kapitza length

Local linear momentum r

Cutoff radius

Electric conductivity

Ratio of mass between the reference atom and an atom of a different type

Ratio of Lennard Jones interatomic strength parameters

xvi

Ratio of Lennard Jones hard core parameters

Interatomic vectors for each atomic pair

Energy of the computational box N Particle velocity N

Velocity of each atom ∆t Time step ∆

Temperature jump at the interface Å Angström Interatomic interaction strength Lattice thermal conductivity

Electrical thermal conductivity Positional disorder factor Chemical potential Grand canonical ensemble Density Hard core radius

Time for spatial translations to become periodic

Time correlation time

Frequency

Characteristic frequency Ω

Thermal boundary resistance

T Temperature gradient

ABSTRACT

THERMAL CONDUCTIVITY OF NANOMATERIALS: A MOLECULAR DYNAMICS STUDY John W. Lyver, IV, C.S.P., PhD George Mason University, 2010 Dissertation Director: Dr. Estela Blaisten-Barojas, Ph.D.

With the growing use of nanotechnology and nanodevices in many fields of engineering and science, a need for understanding the thermal properties of such devices has increased. The ability for nanomaterials to conduct heat is highly dependent on the purity of the material, internal boundaries due to material changes and the structure of the material itself. Experimentally measuring the heat transport at the nanoscale is extremely difficult and can only be done as a macro output from the device. Computational methods such as various Monte Carlo (MC) and molecular dynamics (MD) techniques for studying the contribution of atomic vibrations associated with heat transport properties are very useful. The Green–Kubo method in conjunction with Fourier’s law for calculating the thermal conductivity, κ , has been used in this study and has shown promise as one approach well adapted for understanding nanosystems. Investigations were made of the thermal conductivity using noble gases, modeled with

Lennard-Jones (LJ) interactions, in solid face-centered cubic (FCC) structures. MC and MD simulations were done to study homogeneous monatomic and binary materials as well as slabs of these materials possessing internal boundaries. Additionally, MD simulations were done on silicon carbide nanowires, nanotubes, and nanofilaments using a potential containing two-body and three-body terms. The results of the MC and MD simulations were matched against available experimental and other simulations and showed that both methods can accurately simulate real materials in a fraction of the time and effort. The results of the study show that in compositionally disordered materials the selection of atomic components by their mass, hard-core atomic diameter, well depth, and relative concentration can change the κ by as much as an order of magnitude. It was found that a 60% increase in mass produces a 25% decrease in κ. A 50% increase in interatomic strength produces a 25% increase in κ , while as little as a 10% change in the hard core radius can almost totally suppress a materials ability to conduct heat. Additionally, for two LJ materials sharing an interface, the atomic vibrations altering the heat energy depend on the type of internal boundary in the material. Mass increases across the interfacial boundary enhance excitation of the very low frequency (ballistic) vibrational modes, while the opposite effect is seen as increases in hard core radius and interatomic strength enhance excitation of higher frequency vibrational modes. Additionally, it was found that this effect was diminished for higher temperatures around half the Debye temperatures. In nanodevices and nanomachines, there is an additional factor that

degrades heat transport at the boundary. In fact, the interface induces a temperature jump consistent with a thermal resistance created by the boundary. It was found that the temperature jump, which is due to a boundary resistance, was significant in boundaries involving small mass changes, lesser in changes in hard core radii changes and even lesser for interatomic strength changes. The study of SiC nanowires and nanotubes showed that the structural changes produced vastly different κ . The κ in closely packed structures like nanowires and nanofilaments approximated that of the bulk SiC, yet were less sensitive to temperature than the 1/T relationship traditionally found in bulk systems. The more open nanostructures, like nanotubes, had vastly lower κ values and are almost totally insensitive to temperature variation. The results of this study can be used in the design of nano-machines where heat generation and transport is a concern. Additionally, the design of nano-machines which transport heat like nano-refrigerators or nano-heaters may be possible due to a better selection of materials with the understanding of how the materials affect their nanothermal properties at the nano scale.

1 CHAPTER 1 INTRODUCTION It is known that heat is transported better through solid materials that are pure and crystalline. Any type of impurity, defect, doping, void, or internal boundary within the material increases the resistance to heat transport, and thus, reduces the ability to transport thermal energy. With the growing interest in nanotechnology, the study of thermal conduction properties of systems with reduced dimensions, thin films, nanotubes, nanowires, and super lattices has increased. In nanomaterials and nanostructures, phenomena are highly dependent on the length scale where vibrations between nearest- neighbor atoms occur. The use of molecular dynamics (MD) and the Green–Kubo (GK) methods for calculating the thermal conductivity, κ , have shown promise as atomistic approaches for understanding nanosystems at the nanometer scale. For example, there are several recent calculations on pure noble gases with Lennard-Jones (LJ) interactions in which MD was the method of choice [1-5]. For binary crystals, the literature is not so abundant. There are over 6000 bi-atomic combinations of elements of which only a few hundred have been tested for their thermal conductivity. For some phenomena, such as thermoelectricity, to decrease the lattice thermal conductivity may increase the performance efficiency of the device by a factor of two or three. In a crystal, and in nanostructures, the thermal conductivity is composed of two additive contributions: lattice,

, and electronic,

. The lattice contribution captures

2 phenomena associated with lattice vibrations and phonon scattering and is dominated by the structural characteristics of the crystal or the nanostructure. The electronic contribution is proportional to the electric conductivity through the Wiedemann–Franz law [6-7]. The composition of a crystal affects the lattice symmetry characteristics and, consequently, the lattice vibrations. Therefore, the lattice contribution to the thermal conductivity in a crystal should reflect changes according to its composition. In contrast,

is a function of the conduction properties and these are expected to remain almost constant for families of solids with similar compositional components. A phenomenon that reduces

produces an overall reduction of the thermal conductivity if the electric conductivity is not affected. In dielectrics, and the noble gases specifically, changes in

do not simultaneously affect the electronic conductivity. The computational approach taken in this research is through atomic-level computer simulations using several simple models of binary LJ solids employing a variety of methodologies, including different types of MD and different types of Monte Carlo (MC) techniques. The simulations were expanded to MD investigation of silicon-carbide (SiC) nanostructures with a classical potential proposed by Vashishta [8]. The approach underlying this calculation, from an atomistic perspective, is linear response theory of many body systems. Under this approach, the lattice thermal conductivity is the “response” of a material to a time dependent perturbation, which is the thermal gradient established through it.

3 In this dissertation, research is presented which identifies various lattice changes and their effects on the thermal conductivity in binary LJ crystals. The study in this dissertation spans from changes in the lattice thermal conductivity due to atomic vibrations for binary crystals with compositional disorder, to size effects in SiC nanowires and nanotubes. This research work identified ranges of combinations of binary materials, disorder conditions, and nanodevice shapes and sizes which reduce the thermal conductivity of the simulated materials and may warrant further experimental work. This dissertation is organized as follows: Chapter 2 presents an overview of the theory and methods. The chapter begins with an overview of the modeling techniques employed with the MC and MD simulations. Several of the statistical mechanics ensembles proposed by Gibbs are presented as to their applicability to the simulations addressed in this research. A discussion of the two model potentials of interaction employed in the study is included in this chapter. The work presented in chapters 3 and 4 uses the LJ potential (two-body forces), and the work in chapter 5 uses a model potential proposed by Vastisha [8] containing Coulomb two- body interactions and three-body terms. The next section of this chapter discusses the methods used to analyze the atomic configurations for parameters, including: computational box size in simulations, shape of the computational box, structural and thermodynamic quantities, pressure, order parameters, velocity distribution, mean square displacement, radial distribution functions, and the dynamical properties studied (heat flux, frequency, density of states, and thermal conductivity). The chapter continues with

4 a discussion of how a thermal gradient was produced across the computational box and finally presents a few conclusions on the methods used. Chapter 3 presents the results of thermal conductivity as a function of temperature of homogeneous compositionally disordered binary crystals with atoms interacting through LJ potentials. The two species in the crystal differ in mass, hard-core atomic diameter, well depth and relative concentration. The isobaric MC was used to find the equilibration density of the samples at near-zero pressure and various temperatures. The isoenergy MD simulations combined with the Green–Kubo approach were taken to calculate the heat current time-dependent autocorrelation function and determine the lattice thermal conductivity of the sample. The chapter provides a step-by-step discussion of the process used in this phase of the research: (1) how the equilibrium crystal density was obtained for a near zero pressure simulation, (2) how the thermal conductivity was obtained, (3) how compositional disorder was modeled, and (4) how the models and methods were validated against experimental and other computational results. Next the chapter presents discussions of how each physical parameter effects the thermal conductivity (computational box size, changes in mass, hard core radius, and interatomic strengths, temperature, and density). The chapter concludes with a set of observations. This chapter is a synopsis of the work which was published in reference [9]. Chapter 4 contains results on the effects of internal boundaries on the thermal conductivity. The implementation of non-equilibrium molecular dynamics (NEMD) simulations is discussed in the context of determining the thermal conductivity effects of various monatomic and binary materials with internal boundaries. With this

5 computational strategy, a thermal bath was simulated on each side of the computational box and the Fourier law is used to determine κ . A thorough analysis of the heat contribution on the vibrations of atoms is discussed. Effects on the density of vibrational states due to the interface created between two types of solid LJ systems was investigated as a function of the atomic masses and model potential parameters. The chapter opens with a discussion of the setup and model validation, then proceeds with discussions of the effects of the interface between two LJ solids on the lattice vibrations due to material property changes (similar to those discussed in Chapter 3), as well as mutual orientation of the solid lattices. Further discussions present analyses of the thermal resistance as a function of temperature. The chapter concludes with a summary of conclusions obtained from the research. The results presented in Chapter 4 have been published in [10]. Chapter 5 contains a study of the thermal conductivity in SiC nanostructures. Within the past few years, extensive work has begun on the use of nanodevices in science. The nanodevices can consist of wires, pipes, storage tanks, motors, and pumps, just to name a few. Most of the initial work has been on the use of pure carbon nanowires and nanotubes. Recently, SiC highly ordered structures have begun to be synthesized. The research presented in this chapter is an expansion of the work presented in chapters 3 and 4, now specifically tailored to study SiC in various sizes of nanowires and nanotubes in both their armchair and zigzag chiral configurations. The thermal conductivity of these SiC nanostructures has not been experimentally tested as of today. Therefore, this work makes predictions for SiC nanostructures that will aid the laboratory researchers in their future measurements. This chapter discusses the processes used to determine the thermal

6 conductivity and determine if the nanostructure configuration was stable. The chapter concludes with a presentation of the results and conclusions. The results of this work are currently under review for publishing in [11]. Chapter 6 presents discussion of three computational challenges which include: (1) the counteracting of the effects of finite computational box sizes with the use of periodic boundary conditions (PBC), (2) software error handling used in the research, and (3) a discussion of the use of an energy balance test using the statistical deviations of the average atomic energies. The chapter also includes a brief overview of the software used in the computational and data analysis portions of the research. Chapter 7 presents a summary of the results and conclusions from the methods, processes and results in this dissertation. Additionally, a discussion of how this research has contributed to the general body of knowledge within the study of thermodynamic properties within materials and nanodevices is presented. The dissertation is supplemented by Appendix A, containing a copy of the published papers and Appendix B, containing sample original code developed along with the work. An extensive bibliography is presented after Appendix B.

7 CHAPTER 2 THEORY AND METHODS 2.1 MODELING 2.1.1 Background on Statistical Mechanics Simulations

Two simulation approaches are used in this research: Metropolis Monte Carlo (MC) and Molecular Dynamics (MD). The MC methods simulate how atoms will act/react as they seek thermal equilibrium using stochastically selected discrete changes of atomic positions for each atom in the computational box. MC methods in statistical physics model equilibrium and nonequilibrium thermodynamic systems by stochastic computer simulations. Starting from a description of the desired physical system in terms of modeling how atoms interact among themselves, pseudo-random numbers are used to construct the appropriate probability with which the various generated states of the system have to be weighted. For equilibrium systems, the probability is defined according to either the microcanonical, canonical, iso-pressure-iso-temperature, or grand canonical ensembles. The purpose of MC simulations is to obtain numerically the ensemble averages of desired system properties. In practice, the implementation is simple and can be seen as a sequence of simulation steps that build a Markov chain. At each MC step, a single atom is selected and then relocated to a new position. The new position is determined by

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Abstract: With the growing use of nanotechnology and nanodevices in many fields of engineering and science, a need for understanding the thermal properties of such devices has increased. The ability for nanomaterials to conduct heat is highly dependent on the purity of the material, internal boundaries due to material changes and the structure of the material itself. Experimentally measuring the heat transport at the nanoscale is extremely difficult and can only be done as a macro output from the device. Computational methods such as various Monte Carlo (MC) and molecular dynamics (MD) techniques for studying the contribution of atomic vibrations associated with heat transport properties are very useful. The Green-Kubo method in conjunction with Fourier's law for calculating the thermal conductivity, κ, has been used in this study and has shown promise as one approach well adapted for understanding nanosystems. Investigations were made of the thermal conductivity using noble gases, modeled with Lennard-Jones (LJ) interactions, in solid face-centered cubic (FCC) structures. MC and MD simulations were done to study homogeneous monatomic and binary materials as well as slabs of these materials possessing internal boundaries. Additionally, MD simulations were done on silicon carbide nanowires, nanotubes, and nanofilaments using a potential containing two-body and three-body terms. The results of the MC and MD simulations were matched against available experimental and other simulations and showed that both methods can accurately simulate real materials in a fraction of the time and effort. The results of the study show that in compositionally disordered materials the selection of atomic components by their mass, hard-core atomic diameter, well depth, and relative concentration can change the κ by as much as an order of magnitude. It was found that a 60% increase in mass produces a 25% decrease in κ. A 50% increase in interatomic strength produces a 25% increase in κ, while as little as a 10% change in the hard core radius can almost totally suppress a materials ability to conduct heat. Additionally, for two LJ materials sharing an interface, the atomic vibrations altering the heat energy depend on the type of internal boundary in the material. Mass increases across the interfacial boundary enhance excitation of the very low frequency (ballistic) vibrational modes, while the opposite effect is seen as increases in hard core radius and interatomic strength enhance excitation of higher frequency vibrational modes. Additionally, it was found that this effect was diminished for higher temperatures around half the Debye temperatures. In nanodevices and nanomachines, there is an additional factor that degrades heat transport at the boundary. In fact, the interface induces a temperature jump consistent with a thermal resistance created by the boundary. It was found that the temperature jump, which is due to a boundary resistance, was significant in boundaries involving small mass changes, lesser in changes in hard core radii changes and even lesser for interatomic strength changes. The study of SiC nanowires and nanotubes showed that the structural changes produced vastly different κ. The κ in closely packed structures like nanowires and nanofilaments approximated that of the bulk SiC, yet were less sensitive to temperature than the 1/T relationship traditionally found in bulk systems. The more open nanostructures, like nanotubes, had vastly lower κ values and are almost totally insensitive to temperature variation. The results of this study can be used in the design of nano-machines where heat generation and transport is a concern. Additionally, the design of nano-machines which transport heat like nano-refrigerators or nano-heaters may be possible due to a better selection of materials with the understanding of how the materials affect their nanothermal properties at the nano scale.