UNIVERSITY OF CALGARY

Molecular dynamics study of nanoscale heat transfer

at liquid-solid interfaces (LSIs)

by

Khaled M. Issa

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILISOPHY

DEPARTMENT OF MECHANICAL AND MANUFACTURING ENGINEERING

CALGARY, ALBERTA

September, 2012

c© Khaled M. Issa 2012

Abstract

Phonons, or lattice vibrations, are the primary thermal energy carriers in dielectric solids. At

liquid-solid interfaces (LSIs), there is a considerable mismatch in the vibrational properties

between the two phases. Consequently, during heat transfer across LSIs, the transmission

of phonons is impeded. This gives rise to a temperature drop that is proportional to what

is termed as the Kapitza resistance (RLSI). The present work examines various important

aspects of nanoscale heat transfer across LSIs. This is accomplished computationally using

the Molecular Dynamics (MD) method, in which the two phases are treated at the atomic

scale. First, the effect of the solid surface geometry is investigated in light of advancements in

nanopatterning. It is found that the surface atomic structure can be tailored to significantly

reduce RLSI, by simultaneously influencing two key factors: (i) the interaction energy between

liquid and solid atoms at the LSI, and (ii) the vibrational characteristics of the nanopatterned

surfaces. The second study focuses on the effects of system pressure on RLSI for wetting (W),

and nonwetting (NW) surfaces, respectively. It is demonstrated that, in contrast to the W

surfaces, the system pressure has a strong effect on lowering RLSI for the NW surfaces. For

the pressure range considered, it is concluded that the central cause of this behavior is the

relative increase in adsorbed liquid density accompanying the pressure increase. In the third

part of the thesis, the aim is shifted towards carbon nanotubes (CNTs). Using a novel

technique, a spectral analysis for the frequency dependence of thermal energy exchange at

CNT LSIs is conducted. The results confirm the notion of thermal coupling between a CNT

and its surrounding liquid being limited to the low frequency range. The CNT inner high-to-

low frequency heat transfer is a limiting factor that results in a high RLSI. More importantly,

the findings provide evidence of the origin of a heat transfer ’bottle-neck’ within the CNT.

This could provide new avenues for improving RLSI in CNTs.

ii

Acknowledgements

My deepest gratitude has to go to my PhD supervisor, Professor Abdulmajeed Mohamad,

who gave me this incredible opportunity. Throughout my PhD studies he has been a constant

source of advice, patience, and encouragement.

I would like to thank the supervisory committee: Dr. Oleg Vinogradov, and Dr. Les

Sudak, for their time and feedback on the thesis.

Special thanks go to Dr. Doug Phillips from Information Technologies, and the staff of

WestGrid. I have never come across a more informative, and reliable, group of professionals.

They were crucial in limiting interruptions to the computational work carried out during my

PhD.

I also want to acknowledge the financial support from the Natural Sciences and Engi-

neering Research Council of Canada (NSERC).

I would not have come this far without the love and support of those closest to me. I

feel blessed to have Betty El-Chouli, and Mohamed Issa as my parents. To my life partner

Kate, I can find no words to express my indebtedness for your continuing support, and

encouragement throughout this journey. It is only fair that I dedicate this thesis to you.

iii

Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Importance of nanoscale heat transfer at LSIs . . . . . . . . . . . . . . . . . 11.2 Heat conduction at the nanoscale . . . . . . . . . . . . . . . . . . . . . . . . 22 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Molecular dynamics: An introduction . . . . . . . . . . . . . . . . . . . . . . 163.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Equations of motion for MD systems . . . . . . . . . . . . . . . . . . . . . . 173.3 Thermodynamic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4 Simulation strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.5 Thermostatting, heating, and cooling . . . . . . . . . . . . . . . . . . . . . . 223.6 LAMMPS, WestGrid, VMD, and Octave . . . . . . . . . . . . . . . . . . . . 254 Lowering LSI thermal resistance with nanopatterned surfaces . . . . . . . . . 264.1 Introduction & Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2 Simulation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.3 Results & Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Pressure effects on liquid-solid interfacial thermal resistance . . . . . . . . . 385.1 Introduction & Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.2 Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.3 Results & Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 Spectral analysis of LSI heat transfer in carbon nanotubes (CNTs) . . . . . . 466.1 Introduction & Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.3 Results & Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 Contributions and future research direction . . . . . . . . . . . . . . . . . . . 577.1 Summary of contribution and findings . . . . . . . . . . . . . . . . . . . . . 57Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60A Velocity-Verlet algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74B MD potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76B.1 Lennard-Jones (LJ) potential . . . . . . . . . . . . . . . . . . . . . . . . . . 76B.2 Morse bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78B.3 Reactive Empirical Bond Order potential (REBO) potential . . . . . . . . . 79C Phonon density of states - Lattice dynamics approach . . . . . . . . . . . . . 81D Velocity-autocorrelation function . . . . . . . . . . . . . . . . . . . . . . . . 86

iv

E Modeling Logistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88E.1 LAMMPS input script for Nanopatterning study (Chapter 4) . . . . . . . . . 88E.2 Velocity-autocorrelation function Fortran code . . . . . . . . . . . . . . . . . 91

v

List of Tables

4.1 Simulation results. The LSI thermal resistanceRLSI has units of (×10−9Km2/W).The % difference in interfacial resistance is based on S1. . . . . . . . . . . . 30

vi

List of Figures and Illustrations

1.1 LSIs encountered in various nanotechnologies. . . . . . . . . . . . . . . . . . 2

3.1 Separation distance and force vectors between atoms i and j. . . . . . . . . . 173.2 Velocity-Verlet algorithm steps. Cells with darker shades indicate current

variable time step. Solid arrows are for updates, and broken arrows indicatethe variables used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 MD system with periodic boundary conditions, and minimum imaging, shownhere in two dimensions. Simulation cells B-I are images of cell A. Each cellis of size L× L. Cutoff circles of radius rc are shown by darker shades, withthe one at the H-G border representing minimum image convention. Arrowsshow atoms crossing cell boundaries. . . . . . . . . . . . . . . . . . . . . . . 21

3.4 General setup for the direct method. Arrows indicate the direction of heat flowout of the heat source (red) and into the heat sink (blue). Periodic boundaryconditions are imposed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.1 Simulation cell for S1. The heat source is placed at the solid film mid-plane,with the heat sink at the outermost liquid layers. . . . . . . . . . . . . . . . 28

4.2 Top and side views of the solid surface nanopatterning examined: (a) S2, (b)S3, (c) S4, and (d) S5. The patterned unit cell layer is shown in dark gray. . 30

4.3 Temperature profile for S4, the temperatures used in calculating RLSI areshown with darker shades. The liquid temperatures represent those of the IAand EA liquid. In the absence of nanopatterning (S1), only the EA liquid andthe outermost solid film layer contribute to ∆Tav. Inset shows temperatureprofile over entire domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.4 Liquid density profile for S1, and corresponding surface minimum energy contour 314.5 Liquid density profile for S2, and corresponding surface minimum energy contour 324.6 Liquid density profile for S3, and corresponding surface minimum energy contour 324.7 Liquid density profile for S4, and corresponding surface minimum energy contour 324.8 Liquid density profile for S5, and corresponding surface minimum energy contour 334.9 Vibrational density of states (VDOS): (a) S1, (b) S2, (c) S3, (d) S4, and (e)

S5. SM: solid film mid-plane, SO: solid film patterned layers, IA: internallyadsorbed liquid, EA: externally adsorbed liquid. . . . . . . . . . . . . . . . . 36

4.10 Correlation between normalized RLSI and CVDOS for surfaces S1-S5. . . . . . 37

5.1 Pressure effect MD simulation cell. . . . . . . . . . . . . . . . . . . . . . . . 395.2 Temperature profile for case C4-NW. . . . . . . . . . . . . . . . . . . . . . . 405.3 Pressure effect on RLSI for wetting (W), and nonwetting (NW) surfaces. C1-

C4 in order of increasing pressure. . . . . . . . . . . . . . . . . . . . . . . . . 425.4 Liquid density profiles: (a) W surface, (b) NW surface. Insets show the %

relative increase in adsorbed liquid density peak based on C1 . . . . . . . . . 435.5 Averaged VDOS of LSI atoms for cases C1-C4: (a) W, and (b) NW. SM:

solid mid-plane, SO: solid outer surface, AL: adsorbed liquid. Insets show theVDOS of the adsorbed liquid, and the solid surface superimposed for C1-C4. 44

vii

5.6 Correlation between relative increase in adsorbed liquid density for the NWsurface, and the corresponding relative drop in RLSI. All computations arebased on C1 for the NW surface. . . . . . . . . . . . . . . . . . . . . . . . . 45

6.1 Honeycomb structure of a graphite sheet. Chiral vectors Cnm, and transla-tional vectors Tnm are shown for armchair (5, 5), and zig-zag (8, 0) CNTs. . . 47

6.2 Simulation cell for (5,5) CNT immersed in liquid. Darker region in the liquiddenotes the thermostatted region. The simulation cell has a depth of Lz = 250A. 49

6.3 Temperature history of the CNT (TCNT), and surrounding liquid (TL) in theNVE ensemble following equilibration at 120K. . . . . . . . . . . . . . . . . . 51

6.4 (a) Temperature profile of CNT cooling in liquid after sudden heating to 300K,(b) exponential fit of ∆T (t). . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.5 (a) Close correspondence between total liquid and adsorbed liquid temper-ature (TAds) (b) Radial density distribution starting from the CNT surface.The shaded region denotes where TAds was measured. . . . . . . . . . . . . . 51

6.6 Equilibrium VDOS of CNT (green) and adsorbed liquid (blue) at 120K. Insetis a zoom on 0− 15THz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.7 Time history of CNT temperature (TCNT). The temperature of the adsorbedliquid (TAds) is unaffected by the low TCNT as a result of the high RLSI. . . . 53

6.8 VDOS plots and their corresponding averaging periods following the CNTquench to 1K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.9 Projection of CNT VDOSNE onto VDOSEQ. Also shown is the overlap withthe liquid VDOS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

A.1 Time series of atomic trajectories. . . . . . . . . . . . . . . . . . . . . . . . . 75

B.1 LJ potential and force magnitude in reduced units. . . . . . . . . . . . . . . 77B.2 Morse potential and force magnitude in reduced units. Harmonic potential

shown for comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78B.3 CNT atom and angle labeling for REBO potential . . . . . . . . . . . . . . . 80

C.1 Monatomic linear chain of atoms. . . . . . . . . . . . . . . . . . . . . . . . . 82C.2 Dispersion curve for monatomic chain (N = 100) . . . . . . . . . . . . . . . 84C.3 Density of states of monatomic chain (N = 100). The values for VDOS(ω)

are normalized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

D.1 Normalized VACF for adsorbed liquid, and solid, atoms. . . . . . . . . . . . 87

viii

List of Symbols, Abbreviations and Nomenclature

Symbol Definition

ao unit cell length

a acceleration vector

A area

c speed of sound

C specific heat

C chirality vector

D dissociation energy

E total energy

F force vector

k,k thermal conductivity vector, thermal conductivity scalar

kb Boltzmann constant

K kinetic energy

L length

m mass

N number of atoms

O correlation parameter

p periodicity

P pressure

q,q heat flux vector, heat flux scalar

rc cutoff radius

ro nearest neighbor distance

r, rij position vector, separation distance vector

R Kapitza resistance

ix

t, δt time, time step

T ,∆T temperature, temperature drop

T translational vector

u potential energy

U total potential energy

v velocity vector

V volume

Greek symbols:

ε Lennard-Jones energy parameter

θ bond angle

κ wave vector

λ wavelength

µ distribution function

ρ density

σ Lennard-Jones size parameter

τ relaxation time

φ angle

Φ dihedral

χ velocity scaling factor

ω frequency

Subscripts:

eq equilibrium

l liquid

ll liquid-liquid

x

ls liquid-solid

LSI liquid-solid interface

ne nonequilibrium

R relative

s solid

sp spectral

ss solid-solid

Prefixes:

f femto (×10−15)

n nano (×10−9)

p pico (×10−12)

T tera (×1012)

Abbreviations:

AL adsorbed liquid

CNT carbon nanotube

EA externally adsorbed

IA internally adsorbed

NVE microcanonical ensemble

NVT canonical ensemble

NW nonwetting

SM solid mid-plane

SO solid outer plane

VDOS vibrational density of states

W wetting

xi

Operators:

∇ gradient

xii

Chapter 1

Introduction

1.1 Importance of nanoscale heat transfer at LSIs

With the emergence of Nanotechnology, production and processing of structures that vary

over length scales of several nanometers, has been made standard by recent improvements

in manufacturing, and microanalysis [1]. These nanostructures may interact with a liquid

medium as part of their operation. In many instances, the nature of this interaction is ther-

mal, and dominated by heat transfer between the two phases. Examples of such occurrences

are wide ranging, as depicted in Fig. 1.1. Thermal management is perhaps one of the fields

most influenced by heat transfer at liquid-solid interfaces (LSIs), specifically in relation to

cooling future generation electronic devices. The projected miniaturization of these devices,

coupled with a demand for higher performance [2], renders gas cooling techniques insufficient.

A high heat flux must be dissipated efficiently to protect the reliability, integrity, and per-

formance of the nanoscale device. Liquid-based cooling methods are a promising technique

in dissipating excess heat in high power devices [3–5]. In some liquid-based cooling systems,

the liquid is confined in nanoscale channels that are attached to the heat source, giving rise

to heat transfer across LSIs. Thermal energy transport at LSIs is also a key factor in un-

derstanding heat transfer in nanoparticle suspensions, also termed nanofluids [6]. While the

issue of thermal conductivity improvements in such suspensions remains a contested subject,

a more definitive conclusion will require a broad understanding of LSI heat transfer. Another

field that is affected by LSI heat transfer is the thermal therapy of cancers, based on the

selective heating of magnetic nanoparticles [7]. Hence, the subject of nanoscale heat transfer

at LSIs is of both: theoretical, and industrial significance.

1

Figure 1.1: LSIs encountered in various nanotechnologies.

1.2 Heat conduction at the nanoscale

Heat transfer by conduction is the flow of energy within a medium as a result of a temperature

difference. How well a material allows this form of energy flow is dictated by its thermal

conductivity (k). From Fourier’s law, k can be seen as a constant of proportionality between

the temperature gradient (∇T ) within the material, and the corresponding heat flux (q):

q = −k∇T (1.1)

By combining Eq. 1.1 with an energy balance on a material control volume, the heat diffusion

equation can be derived [8]. In an isotropic material, where the thermal conductivity reduces

to a scalar (k→ k), the heat diffusion equation becomes:

k∇2T = ρC∂T

∂t(1.2)

where ρ is the mass density (kg/m3), C is the specific heat (J/kgK), and t is time. With

suitable initial and boundary conditions, the heat diffusion equation can be solved for the

spatiotemporal temperature profile. In the presence of a LSI, conservation of energy neces-

sitates that the heat flux be continuous at the boundary. Such an assumption is also often

imposed on the temperature. This, however, has no concrete physical basis, especially fol-

lowing the pioneering work by Kapitza [9] on the thermal resistance at liquid helium-metal

2

interfaces, termed the Kapitza resistance (RLSI). The Kapitza resistance can be estimated

using:

RLSI =∆TLSIq

(1.3)

where ∆TLSI is the temperature drop at the interface. When expressed in terms of the

Kapitza length (LLSI = kRLSI), it provides a measure of the bulk material over which an

equivalent temperature drop can be attained. From the macroscopic perspective, the tem-

perature drop at the LSI can be negligible, with LLSI ranging from an order of a molecular

size (for a wetting surface), to tens of molecular sizes (non-wetting surface) [10]. However,

for structures at the nanoscale, this can have a significant impact on the thermal energy

transport across the LSI.

One of the major limitations in applying Eq. (1.2) to nanoscale heat transfer at the LSI,

comes from the assumption that k is independent of the length, and timescales, involved. In

other words, the thermal conductivity of the material is unaffected by the size of the domain,

or the rate at which heat is added. At the nanoscale, this becomes problematic from both

the solid’s, and the liquid’s perspective. In crystalline solids that are dielectric, and when

photon’s contribution to heat transfer are negligible [11], thermal transport is dominated

by crystal lattice vibrations, or phonons. The highly efficient heat transfer in such solids is

brought about by the ability of phonons to travel ballistically over relatively long distances

(phonon mean free path), before being scattered by other phonons, structural defects, or

boundaries. The mean free path of phonons in solids can cover a range of 10− 100nm [12].

The length scales associated with a nanostructure that is present at the LSI, are therefore

comparable to these thermal energy carriers. In such scenarios, heat conduction cannot be

described solely by the heat diffusion-based Eq. 1.2. Furthermore, the phonon spectrum

is sensitive to the local temperature, and can be affected during a nanoscale heat transfer

process. Consequently, the assumption of constant thermal conductivity, k, can be argued

at the nanoscale.

3

When in contact with a solid surface, the structure of a liquid can exhibit a higher degree

of order in comparison to the liquid bulk. The density distribution is also affected, with a

profile that displays an oscillatory behavior at the LSI. This occurs as a result of the strong

interaction between the liquid and solid atoms. The oscillation in density, or layering, can

extend over several atomic distances, depending on the interaction strength with the solid

atoms [13, 14]. The structural variations in the liquid at the LSI have important effects

on properties such as: liquid-solid phase transition [15–17], flow and tribology [18–21],and

viscosity increase due to confinement and solidification [22]. The lack of a highly ordered

atomic structure in the bulk of confined liquids prevents the superior form of thermal energy

transport witnessed in crystalline solids, and results in an effective mean free path of the

order of an atom size [23]. The ordering of the liquid at the LSI, on the other hand, is

believed to facilitate ballistic heat transport, enhancing by that its thermal conductivity

[24]. This is thought to be one of the factors for the improved thermal conductivity in

nanofluids [25]. These effects on the physical properties of liquids at the nanoscale LSI are

ignored in continuum based modeling. The mentioned modifications to the liquid structure

and distribution at a nanoscale LSI, are affected not only by the strength of interaction

between the liquid and solid on an atomic level, but also by the atomic surface structure of

the solid itself. In continuum based approaches, the lack of a proper treatment of the above

factors in modeling nanoscale heat transfer across a LSI, severely limits its applicability.

Theoretical models based on phonon physics have been proposed to attempt alleviating

the deficiencies in continuum modeling. Such models include the Acoustic Mismatch Model

(AMM), and the Diffusive Mismatch Model (DMM) [26,27]. In the AMM, the transmission

and reflection of thermal phonons is governed by the acoustic impedance (which is the

product of density and speed of sound ρc) of the materials at the interface. This is usually

many orders of magnitude larger in a solid than in a liquid. As a result of this acoustic

mismatch, a sizable fraction of the phonons impinging upon a LSI from both sides is impeded,

4

resulting in a temperature discontinuity at the LSI. The DMM, is stochastic in nature, and

assumes all phonons to be diffusively scattered with a probability that is dependent on the

bulk density mismatch between the solid and the liquid. The AMM is known to over predict

the LSI thermal resistance, and thus, provide an upper bound on RLSI. The DMM, on the

other hand, provides a lower bound on the LSI thermal resistance. Their lack of accuracy

stems mainly from the assumption that the bulk material properties are applicable at the

nanoscale LSI, in addition to neglecting the atomic details of the liquid and solid structures.

While these models suffer from some of the disadvantages of continuum modeling, they still

provide better insight into the parameters that can be controlled to manipulate the LSI

thermal resistance.

For a proper modeling of thermal energy transport at a nanoscale LSI, an atomic resolu-

tion is necessary. This is more the case, considering the fact that experimental work at these

scales is still in its infancy [1]. Thus, computational modeling is a vital tool for the proper

understanding of this phenomenon, and the overall advancement in the field of nanoscale

heat transfer. One such powerful computational tool is Molecular Dynamics (MD), which

has been invaluable in shedding light on the physics behind nanoscale heat transfer. This

is the primary tool used in the thesis to examine various aspects of LSI heat transfer at

the nanoscale. In Chapter 2, a literature review of MD modeling of LSI heat transfer is

summarized, based on which the objectives of the thesis are laid out. An introduction to

MD is given in Chapter 3, along with the simulation strategies employed throughout the

thesis. Chapters 4-5 contain the first two studies in the thesis which examine the effects of

nanopatterning, and system pressure on RLSI, respectively. In Chapter 6, a frequency based

analysis of RLSI in Carbon Nanotubes (CNTs) is investigated. Important derivations, and

supporting documents, can be found in the Appendices.

5

Chapter 2

Literature Review

In one of the earliest studies on nanoscale LSI heat transfer [28], MD simulations were carried

out to measure the LSI thermal resistance. The system consisted of a saturated vapor region

sandwiched between two liquid regions that were in contact with the solid walls. Fluid atoms

interacted via the Lennard-Jones (LJ) potential with the parameters of Argon (Sec. B.1).

The solid walls were modeled using three layers of a face-centered-cubic (fcc) (111) surface

of harmonic particles. The solid atomic mass, spacing, and spring constant, corresponded

to those of Platinum (Pt), which in this case was assumed to be electrically insulating.

The temperature of the solid surfaces was controlled through the use of external phantom

molecules [29]. The interaction strength parameter between the liquid and solid atoms was

varied to study the effects of the degree of surface wettability. The wall temperatures were

maintained at 100K and 120K, respectively. The importance of this study, is that in almost

all of the following work in the literature, a similar computational setup was used. The study

reported a temperature jump at the LSI, which decreased with increasing surface wettability.

Consequently, LSI thermal resistance was found to drastically drop with stronger liquid-solid

interaction. A similar model was later incorporated [30], excluding the vapor region. The

study attempted to explain the temperature jump in terms of an intermolecular energy

transfer model that they developed. It was proposed that the temperature jump was related

to an energy exchange at the LSI that was opposing the ’macroscopic’ heat flux direction.

Through experiments, and MD simulations, the work in [31] investigated the interfacial

thermal resistance of individual Carbon Nanotubes (CNTs) immersed in a hydrocarbon

liquid. They discovered that despite the high thermal conductivity of CNTs, the interfacial

thermal resistance was remarkably excessive with most of the temperature drop occurring

6

at the interface between the heated CNT and the liquid. They attributed this trend to the

coupling of only a small number of low frequency vibrational modes between the CNT and

the surrounding liquid. This in turn required that the high frequency vibrational modes

of the CNT be first transferred to the low frequency modes, via phonon-phonon coupling,

before being exchanged with the liquid matrix. The proposed explanation was later further

supported by MD, in which shorter CNTs displayed higher interfacial resistance. This was

attributed to shorter CNTs possessing less of the low frequency modes, resulting in a weaker

coupling to the liquid [32].

Two different regimes in LSI Kapitza resistance were reported in [33], based on the

wetting/nonwetting characteristics of the solid surface. Liquid and solid atoms were modeled

using the LJ potential, but with the solid-solid interaction being ten times larger than that

within the liquid. The liquid-solid interaction strength was varied to study its effect on

the Kapitza resistance. In addition to confirming the sizable drop in the Kapitza resistance

with increased wettability, the study pointed out a switch in the functional dependence of

the Kapitza resistance on the liquid-solid interaction strength. In the wetting case, the LSI

thermal resistance was inversely proportional to the interaction strength parameter. Whereas

in the non-wetting case, there was an exponential drop in the LSI thermal resistance as the

interaction strength was increased. One of the interesting findings of the study was that

the liquid layering at the solid surface was found to exhibit no noticeable effect on the

thermal transport properties. This aspect was later investigated [23], and it was concluded

that the liquid layering at the LSI had a surprisingly small effect on the thermal transport

characteristics of the liquid, with a thermal conductivity that was indistinguishable from

that of the bulk. This was found to be true regardless of whether the solid surface was

wetting or non-wetting, and for both cases where the heat traveled normal, and parallel

to the layering. To explain the findings, three main factors were suggested as follows: (i)

although the liquid at the LSI was more ordered relative to the bulk, it still lacked the degree

7

of order witnessed in solids, (ii) the ordering extended only over several atomic distances in

the liquid, and (iii) the liquid atoms interaction with solid atoms can result in the scattering

of collective motions of the layered liquid. Thus, it was concluded that while layering can

significantly affect the liquid mechanical properties, that was not the case when it comes to

thermal characteristics. Contrary to a gas-solid interface, for which the degree of layering at

the solid surface enhances thermal transport [34]. The study did, however, acknowledge that

the findings can be different for more complex liquids, or for other computational setups.

The effect of nanostructural patterning on LSI heat transfer was first reported in [35], in

which the effect of the solid wall structure on the transfer of momentum and thermal energy

at the LSI was investigated. Three different crystal planes of an fcc structure were used in

shearing a confined liquid (both atomic and linear molecular). Two types of energy transfers

were studied: parallel, and normal to the wall velocity. For the parallel component, the

dictating factor was the atomic roughness of the solid wall (or the unevenness of the potential

surface). As for the normal component, which was responsible for the heat transport at the

LSI, the key factor was the number density of the solid surface plane interacting with the

liquid at the LSI.

An investigation of the relationship between velocity slip and thermal slip in a nanochan-

nel flow was carried out in [10]. Two types of liquids were tested, simple monatomic and

polymeric liquids. For both cases, all atomic interactions were governed by the LJ poten-

tial. In the case of the polymeric liquid, FENE springs [36] were used to connect adjacent

atoms. The liquids were heated in two forms: (i) by shearing, and (ii) by adding thermal

energy to the liquid in the absence of shear. The interaction between the liquid and solid

surfaces was found to strongly influence both: the velocity, and thermal discontinuities. The

thermal discontinuity was found to exist even in the absence of a velocity slip at the LSI.

The polymeric liquid witnessed a higher velocity slip which was coupled by an increase in

the Kapitza length by up to a factor of two. The main conclusion was that the LSI heat

8

transfer was strongly coupled to LSI momentum transfer.

The effect that different nanometer structures had on the energy exchange at the LSI was

investigated in [37]. Three different rectangular structures were patterned on the wall, with

the same number of atoms each. Atoms for the liquid and solid were modeled using the LJ

potential. A direct correlation between the density of particles that are adsorbed along the

nanostructures, and the effectiveness of energy exchange was concluded. The performance

of the various nanostructures was, however, also dependent on other factors such as the fluid

density, and the temperature gradient.

While all previous studies focused on systems that involved periodic boundary conditions,

the work in [38] explored the LSI thermal transport in a triangular heat pipe by extending

the system in [28] to include side walls. It was reported that for a weak interaction strength

between the liquid and the solid atoms, the lateral walls played a minor role, and the LSI

thermal resistance was dominated by the evaporation and condensation zones at the lower

and upper walls, respectively. The further increase in the interaction potential resulted

in the increase of the LSI thermal resistance along the lateral walls. This also acted to

transfer the liquid from the condensed region to the evaporation region. At its initiation,

this phenomenon was found to provide an optimal LSI thermal transport, since cooler liquid

was reversely transported to the evaporative region.

The work reported in [39] proposed a solid wall model that is more efficient computation-

ally for the purpose of LSI thermal transport studies. This was accomplished by excluding

interatomic interactions within the solid surface layers. Instead, each wall atom was tethered

to an equilibrium position by harmonic springs. The temperature control of the solid surface

was maintained by scaling the atomic velocities on a per layer basis. The effectiveness of the

wall model was demonstrated by simulating the effects of liquid-solid interaction strength

on the Kapitza resistance. Also, the effect of the solid wall stiffness on the temperature

jump at the LSI was shown, with the stiffer walls resulting in a higher temperature jump.

9

These effects were later investigated by the authors in more detail, in an attempt to extract

empirical models [40].

The effect of the nanostructural clearances along a nanochannel wall on the LSI thermal

resistance was studied in [41]. The SPC/E water model [42] was used for the liquid part,

while the solid was modeled using the LJ potential, with parameters representative of iron.

It was reported that the LSI thermal resistance passed through a minimum value at which

the residence time of the water molecules at the LSI was found to be maximum. Thus,

facilitating by that optimum thermal energy transport.

Building on the concept of the acoustic mismatch between the solid and the liquid at the

LSI, the study in [43] demonstrated that it was possible to reduce the LSI thermal resistance

by: increasing the fluid pressure (through a temperature increase in a fixed volume), and

enabling additional liquid layering through increased surface hydrophilicity. The SPC/E

water model was used for the liquid, while for the solid wall, silicon atoms were utilized that

were tethered, and interacted via an LJ potential.

The transient behavior of nanoscale LSI heat transfer was reported in [44], in which

an argon-iron system was modeled. The LJ potential was used for the liquid argon, and

the embedded atom model (EAM) [45] for the solid iron. It was demonstrated that the

LSI thermal resistance is time dependent, increasing as the liquid equilibrated towards the

increased solid temperature. The system was also found to equilibrate at a much slower rate

compared to analytical predictions based on continuum modeling of heat conduction.

The LSI thermal transport in a water-silica (SiO2) system was modeled in [46]. The

water molecules, in addition to forming an adsorbed layer at the solid surface, also pene-

trated the silica cells (modeled using an LJ type potential) for a depth of approximately

two molecular diameters. This resulted in significantly lowering the LSI thermal resistance,

relative to silicon membranes where no such water penetration is present. The successive

absorption/adsorption water layers was believed to have promoted the ballistic transport of

10

phonons across the LSI.

The thermal boundary resistance of single-walled carbon nanotubes (CNTs) in both

solid, and liquid argon matrices, was investigated in [47]. The aim was also to extend the

previous work in [31] to longer CNTs. It was shown that the interfacial thermal resistance

remained constant for nanotube lengths over a 20 − 500A(1A = 10−10m) range. The LSI

thermal resistance was higher for the liquid matrix compared to the solid one. In support of

the findings in [31], resonant coupling between the low frequency CNT phonon modes and

the matrices was observed. However, these low frequency modes carried only very little of

the total energy. The CNT heat dissipation was, therefore, strongly retarded by the slow

intermode energy transfer from higher to lower frequencies within the CNT.

The mechanism of friction induced fluid heating in a nanoscale Poiseuille flow was exam-

ined in [48]. Two liquids, argon and helium, and two different solid wall materials, silver and

aluminum, were tested. The heat generation in the liquid originated from friction both: at

the LSI, and within the liquid itself. The latter was found to be the dominating contributor

at high flow driving forces. At a low driving force, the stick condition applied at the wall and

the liquid heating was caused by the friction within the liquid. The LSI thermal resistance

affected the steady state liquid temperature, and consequently, the temperature jump at the

solid wall. In the cases where different materials were used for the top and bottom walls, it

was shown that the liquid temperature profile lost its symmetry about the channel center.

Additionally, the wall temperature itself was found to affect the source of friction induced

heating. When the liquid-solid binding energy was lower than the wall temperature, slip

occurred at relatively smaller driving forces, causing the heating to be influenced more by

friction at the LSI.

With previous LSI heat transfer studies focused on controlling the solid temperature, the

work in [49] demonstrated the existence of a temperature jump within the solid surface that

is adjacent to the temperature controlled liquid. The heating and cooling was carried out

11

through liquid water surrounding two silicon walls. This ’solid-side’ temperature jump, in

conjunction with the Kapitza resistance, contributed to the overall hindrance to nanoscale

LSI thermal transport. In contrast to the case of the Kapitza resistance, increasing the

temperature of the interfacial fluid resulted in a higher solid-side thermal resistance.

The study in [50] reported LSI thermal conductance results in functionalized hydrophobic

and hydrophilic silica-water interfaces. It showed an increase in the hydrophilic surface

conductance with temperature. More importantly, a rectifying effect was demonstrated

when reversing the direction of heat flow in the silica slab. This rectification was further

studied with the use of self-assembled monolayers (SAMs) at the silica surface [51]. It was

attributed to the hydrogen bonding in water being highly sensitive to temperature. When

the heat flows from the water side towards the SAMs, the higher temperature results in a

break in the hydrogen bonding, affecting by that the local hydrogen bond network, and the

vibrational characteristics of water. Consequently, causing a relatively lower conductance

of heat in this direction. A more recent study [52] showed the possibility of reducing the

Kapitza resistance across graphene sheet-oil interfaces by functionalizing the graphene sheet

with alkanes that possessed vibrational modes, which coupled well with the surrounding oil

matrix.

In [53], MD simulations were performed for the LSI thermal conductance across water-

SAMs interfaces with varying chemistries. The results were found to be in excellent agree-

ment with experimentally obtained data [54]. The interfaces ranged from the highly hy-

drophobic to the highly hydrophilic. The accuracy of the MD simulations suggested the

possibility of utilizing LSI thermal transport measurements in predicting LSI environments,

and bonding strength.

The work in [55] focused on the effects of water nanoconfinement between two quartz

surfaces on the LSI thermal conductance. An optimum thickness for the water layer was ob-

served, which resulted in the freezing of the water molecules at such an extreme confinement.

12

This was coupled with a good match in vibrational states between the water molecules and

the quartz hydrophilic surface.

The effect of argon liquid layer thickness on the LSI thermal conductivity was investigates

in [56]. The study showed that the thermal conductivity of nanoconfined liquid argon was

higher for a larger film thickness, and a stronger liquid-solid interaction strength. In contrast

to the findings in [23], the liquid layering at the LSI was found to affect the thermal conduc-

tivity. The degree of layering was dependent on the temperature in addition to liquid-solid

interaction strength. Interestingly, the study reported a drop in the thermal conductivity

of the film for stronger liquid-solid interactions due to higher depletion in the second liquid

layer. This was argued to affect the frequency of migration of higher temperature liquid

atoms.

The study in [57] investigated the effect of the magnitude of the driving force on the LSI

thermal resistance in nanochannel driven flows. Liquid argon flows in both copper, and silver

solid channels were tested. It was shown that the LSI thermal resistance had an optimum

value as a function of the driving force. The initial increase in the LSI thermal resistance

was attributed to reduction in the vibrational mobility of the liquid atoms due to a stick

boundary condition, and a stiffening effect from the applied force. Once the force is high

enough to induce slip at the LSI, the friction heating of the liquid atoms at the LSI enhanced

their ability to transfer energy to the walls, resulting in a drop in the LSI thermal resistance.

Thus, the liquid-solid interaction strength was concluded to be an important factor for this

phenomenon to exist.

The thermal conductivity of adsorbed liquid argon layers was compared to that of the

bulk liquid in [24]. An improvement of up to a factor of 2 was reported. This was attributed

mostly to the relatively higher density of the liquid at the LSI, with a less pronounced

influence of the increased structural order of the liquid. The same group later examined

the effects of thickness of the thermal conductivity of a liquid confined between two solid

13

slabs [58]. It was shown that the thermal conductivity improved as the thickness was reduced,

with a maximum enhancement in the case of a liquid monolayer. This was demonstrated to

be caused by a relatively better vibrational matching between the liquid and solid atoms at

the LSI.

The work in [59] examined the effects of surface functionalization on the LSI thermal

resistance of water-silica interfaces. First, they demonstrated a significant improvement in

the LSI thermal conductance with increased surface hydrophilicity, via (-OH) terminated

headgroups. For the second part, geometric functionalization was explored, in which silica

nanopillars were added to the surface. This was also found to improve the LSI thermal con-

ductance at the base surface. This rise in conductance plateaued as the nanopillar heights

were increased. This behavior was attributed to the variations in vibrational coupling be-

tween the two phases at the LSI.

A similar, but more detailed, analysis was conducted in [60] for water-SAM interfaces.

The study demonstrated a nonlinear drop in LSI thermal resistance when hydrophobic (-

CF3) headgroups were replaced by hydrophilic ones (-OH). Surface roughness was controlled

by using SAMs with varying chain lengths, and was shown to improve the LSI thermal

conductance. Furthermore, a thermal rectification effect was reported in which the LSI

thermal resistance was lower when heat flow was directed from the SAMs to liquid water.

Over the past years, a substantial amount of knowledge has been accumulated on the

subject of LSI thermal resistance, using MD simulations. The work in this thesis attempts

to contribute to these findings, by exploring new areas in this exciting field. This is broken

down into three main parts, as follows:

1. The issue of surface geometric functionalization is considered via nanopat-

terning in Chapter 4. In this regard, the studies summarized above lacked a

systematic approach in analyzing the important factors contributing to ther-

mal energy exchange between liquid and solid atoms at the LSI. Therefore,

14

this part of the thesis examines how the level of nanopatterning affects both:

the interaction potential between the liquid and solid atoms, in addition to

the solid surface vibrational characteristics and its coupling to that of the

adsorbed liquid.

2. The effects of pressure on LSI thermal resistance have not being explored yet.

For nanoscale heat transfer at solid-solid interfaces, pressure has been shown

to have a contrasting effect, depending on the strength of bonding between

the two solids [61]. In Chapter 5, this issue is examined for both: wetting,

and nonwetting surfaces. Apart from it being a possible avenue for tuning the

thermal resistance at LSIs, a good understanding of pressure effects on LSI

thermal resistance is important for reliable MD modeling. This is given that

pressure levels can vary during a MD simulation as a result of energy addition,

or extraction, in a constant volume system.

3. There is strong evidence that the vibrational coupling between a solid and

liquid at the LSI is limited to a relatively low frequency range. Under such

a scenario of frequency dependent thermal energy exchange, the efficiency of

heat conduction between low, and high frequency modes within the solid, be-

comes a crucial factor. Such is the case in CNTs, where it has been shown

that the high LSI thermal resistance is not merely a result of weak interaction

strength with the liquid, but rather as a result of the impeded inner low-to-

high frequency energy exchange within the CNT itself. In Chapter 6, a novel

technique is devised to provide a clearer view for the frequency dependence of

energy exchange between a CNT, and its surrounding liquid. This is carried

out with the aim of locating the origin of the inadequate internal energy ex-

change in CNTs, with potentially significant implications on their LSI thermal

resistance.

15

Chapter 3

Molecular dynamics: An introduction

3.1 Overview

Molecular dynamics (MD) is a computational tool that can predict the behavior of materials

at the atomic level. It is a discrete method in which the MD particles represent atoms

constituting the examined material. The two main components in MD modeling are the

atomic structure, and the interatomic potential function (u). The functional form of u can

be fitted to match experimental data, or quantum mechanics calculations [62]. Ideally a

potential energy function must combine simplicity, accuracy, efficiency, and transferability.

Generally, the total potential energy (U) of a system of MD particles can be expressed as a

sum of terms involving single, pairs, triplets, and so forth, of atoms [63]:

U =∑i

u1(ri) +∑i

∑j>i

u2(ri, rj) +∑i

∑j>i

∑k>j>i

u3(ri, rj, rk) + ... (3.1)

The summation limits prevent the inclusion of the interaction potential between a set of

atoms more than once. The first term in the potential energy function, u1(ri), can be used

to incorporate the effect of an external field acting on the system. All the remaining terms

represent particle-particle interactions. The inclusion of more than two-body interactions is

computationally demanding, and when possible is either ignored, or its effects are incorpo-

rated into u2. In that case, and in the absence of an external potential field, the potential

function reduces to a two-body, or pair potential. The total pair potential depends only on

the distance of separation between two particles (rij), and U is now1:

U =∑i

∑j>i

u2(ri, rj) =∑i

∑j>i

u(|ri − rj|) =∑i

∑j>i

u(rij) (3.2)

1The 2 subscript in u2 is dropped for clarity.

16

The motion of each atom in MD is governed by Newton’s second law, which is given by:

d2ridt2

=Fi

mi

(3.3)

where t is time, mi is the mass of atom i, ri is its position vector, and Fi is the total force

vector acting on the atom. In the case of the simple pair potential, Fi is calculated as follows:

Fi = −∑i6=j

Fij = −∑i6=j

∂u(rij)

∂rij= −

∑i6=j

∂u(rij)

∂rij

∂rij∂rij

= −∑i6=j

∂u(rij)

∂rijrij (3.4)

where Fij is the force acting on atom i by atom j, rij = ri − rj is the separation distance

vector, with rij as its unit vector. A depiction of these vectors is shown in Fig. 3.1.

Figure 3.1: Separation distance and force vectors between atoms i and j.

3.2 Equations of motion for MD systems

Following the calculation of forces, Eq. 3.3 must be solved to advance the trajectory of the

system in time. One of the most commonly used methods in solving the atomic equations

of motion is the finite-difference Verlet algorithm [64]:

rt+δt = 2rt − rt−δt + δt2at (3.5)

vt+δt = 12δt [rt+δt − rt−δt] (3.6)

17

where δt is the time step, v, and a = F/m, are the velocity and acceleration vectors,

respectively. One drawback of the Verlet algorithm is that velocities do not appear in Eq.

3.5, and are only calculated following the atomic positions update. To add, or remove, energy

(kinetic) from a MD system, it is important that velocities appear in Eq. 3.5. This issue

is discussed further in Sec. 3.5. Furthermore, Eq. 3.6 has errors of O(δt2), compared to

O(δt4) for Eq. 3.5. To resolve the above shortcomings, the velocity-Verlet algorithm was

proposed [65], which is carried out in the following order:

rt+δt = rt + δt vt + 12δt2at (3.7)

vt+ 12δt = vt + 1

2δt at (3.8)

vt+δt = vt+ 12δt + 1

2δt at+δt (3.9)

The first step Eq. 3.7 updates the positions of the atoms to t + δt using the positions,

velocities, and accelerations at t. The velocity update takes place in two steps, and starts

with calculating the velocity at t+ 12δt using the acceleration at t. Then there is a ’hidden’

step between Eq. 3.8 and Eq. 3.9 in which the forces at t + δt are calculated based on the

updated positions. These forces are then used to complete the velocity update to t+δt. The

steps for the velocity-verlet algorithm are depicted in Fig. 3.2. The derivation of Eqs. 3.7-3.9

is given in Appendix A. The velocity-Verlet algorithm is mathematically equivalent to the

original Verlet algorithm, and is the algorithm choice in this reported work. The choice for

a time step δt is problem dependent. A larger time step would require less computational

time, but it should always be limited to ensure that the system dynamics with the highest

frequencies are sufficiently captured. An additional good measure of the suitability of the

time step is how strongly the total energy of the system is conserved during a run.

18

Figure 3.2: Velocity-Verlet algorithm steps. Cells with darker shades indicate current vari-able time step. Solid arrows are for updates, and broken arrows indicate the variables used.

3.3 Thermodynamic properties

In MD simulations, thermodynamic properties are calculated from the dynamics of atoms

in the system, in what could be viewed as a ’bottom up’ approach. The total energy of the

system, E, is the sum of the total potential energy U , and kinetic energy K:

〈E〉 = 〈U〉+ 〈K〉 = 〈N−1∑i

N∑j>i

u(rij)〉+ 12〈N∑i

miv2i 〉 (3.10)

where the brackets 〈...〉 denote ensemble averaging, N is the total number of atoms in the

system, and the pair potential assumption is implied (Eq. 3.2). In heat transfer studies,

two important thermodynamic parameters are the temperature (T ), and pressure (P ), of

the system. For an atomic system these are calculated using:

T =2〈K〉3Nkb

=〈∑N

i miv2i 〉

3Nkb(3.11)

P =NkbT

V+

1

3V〈N−1∑i

N∑j>i

rij · Fij〉 (3.12)

where kb is the Boltzmann constant. Both expressions can be derived from the generalized

equipartition theorem [63]. A quick check of Eq. 3.11 shows that 〈K〉 = (3N)12kbT , i.e.

each degree of freedom contributes on average an equal amount of 12kbT , in accordance with

the equipartition principle. The two terms in the pressure expression Eq. 3.12 represent

19

contributions from the system kinetic and potential energies, respectively. For an ideal gas

the second term, which is referred to as the virial, vanishes. An extensive derivation of Eq.

3.12 can be found in [66].

3.4 Simulation strategies

The proper initialization of a MD simulation is essential in avoiding instabilities. This is best

accomplished by ensuring that atoms start at their equilibrium solid state positions, where

they witness no forces. Liquids are initialized in a similar manner, but with the initial solid

structures melting into a fluid phase. To set the simulation in motion, atoms are assigned

random velocities from the equilibrium Maxwell-Boltzmann distribution given by:

µ(vα,i) =

(mi

2πkBT

) 12

exp

[−1

2miv

2α,i

kBT

](3.13)

where α = x, y, z. This velocity assignment is done under the condition that there is zero

net momentum:∑N

i=1mivα,i = 0, in all three dimensions.

The most computationally demanding element of any MD simulation is the calculation

of the atomic forces, as given for example in Eq. 3.4 for a pair potential. When the potential

function is short-ranged, as is the case for the Lennard-Jones (LJ) potential (see Appendix

B), the majority of the contribution to the potential energy, and forces, comes from nearby

atoms. If only atoms within a cutoff sphere of radius rc are considered (see Fig. 3.3), the

computations can be reduced by a factor of 43πr3c/V , where V is the system volume. In fact,

at rc = 2.5σ (σ is the LJ size parameter) , the potential is only 1.63% of the well depth. It

can also be shown that for this choice of rc, the total potential from rc to∞ is only ≈ 2% of

the total potential from σ to rc. Hence, given the computational savings, this is the common

practice in MD modeling. To avoid the discontinuity in the truncated potential at r = rc,

the potential can be adjusted by removing its constant term at that separation distance:

20

u(rij) = u(rij)− u(rc), for rij ≤ rc.

The size of systems accessible to conventional MD simulations are (for the time being)

limited to a few hundred nanometers. This limitation can be overcome by employing periodic

boundary conditions as depicted in Fig. 3.3. The central cell A is infinitely replicated to

produce an unbound two-dimensional space. Each atom and its infinite images have exactly

the same dynamics. When an atom crosses a boundary, as shown at the top of cell A, its

image from cell H renters cell A, which ensures the conservation of the number of atoms.

Atoms at boundaries interact with images of the atoms in the adjacent cells, which is known

as the minimum image convention. An example of that is shown by the atom at the left

boundary of cell H.

Figure 3.3: MD system with periodic boundary conditions, and minimum imaging, shownhere in two dimensions. Simulation cells B-I are images of cell A. Each cell is of size L× L.Cutoff circles of radius rc are shown by darker shades, with the one at the H-G borderrepresenting minimum image convention. Arrows show atoms crossing cell boundaries.

21

3.5 Thermostatting, heating, and cooling

MD simulations in the microcanonical ensemble (NV E) involve a fixed volume (V ), and a

constant number of atoms (N), and total energy (E). An alternative to the NV E ensemble

is to control the system temperature by adding/removing energy from the system, in what

is termed the canonical (NV T ) ensemble. The temperature control can be performed by

adjusting the system kinetic energy via velocity scaling. If the current ’instantaneous’ tem-

perature (as opposed to the actual ensemble averaged temperature) of the system is T1, and

the aim is to adjust the system temperature to T2, then from Eq. 3.11, the temperature

ratio can be expressed as:

T2T1

=

∑Ni (χvxi)

2 + (χvyi)2 + (χv2zi)∑N

i v2xi + v2yi + v2zi

orT2T1

= χ2 (3.14)

Therefore, the scaling factor is χ =√T2/T1. Although this form of direct velocity scaling

does achieve the required temperature T2, it lacks physical soundness since it prevents fluc-

tuations in the system temperature, that are present in the NVT ensemble. The Berendsen

thermostat [67] is an alternative method that allows such temperature fluctuations. This

is done by relaxing the system temperature towards the target temperature, To over a pe-

riod of time, τ , according to: T2 = T1 + (δt/τ)(To − T1), where T2 is now an intermediate

temperature. Using the result from Eq. 3.14:

χ2T1 = T1 +δt

τ(To − T1) or χ =

√1 +

δt

τ

[ToT1− 1

](3.15)

Thus, τ can be seen as a coupling parameter to a heat bath, the value of which should be

chosen to allow adequate temperature control without overly suppressing the fluctuations.

The Berendsen thermostat does not sample the NVT ensemble, but rather (depending on the

choice of τ) an ensemble that lies between those of the NVE and the NVT [68]. Nevertheless,

22

because of its simple approach and effectiveness, the Berendsen thermostat is frequently used

to initialize the system temperature. This is the thermostat of choice in this reported work.

In MD modeling of thermal conductivity, one of two methods can be employed: equilib-

rium MD, as in the Green-Kubo method, or nonequilibrium MD, as in the direct method.

Each of these methods have their advantages and drawbacks [69]. In the Green-Kubo

method, the thermal conductivity is deduced from integrating the autocorrelation of the

thermal current, for a system in equilibrium. The direct method, on the other hand, mim-

ics an experimental setup numerically by imposing a temperature gradient (or a heat flux)

across the system. The thermal conductivity is then computed using the resulting heat flux

(or temperature gradient) in Fourier’s law. Relative to the direct method, the Green-Kubo

method requires longer simulation times. Although the Green-Kubo method can be applied

at the LSI [70], the smaller number of atoms present at the LSI region only exacerbates the

computational requirements. This renders the direct method a more convenient approach

towards MD studies of LSI thermal conductance, as has been the case in the overwhelming

majority of reported studies.

In this work, the direct method is used in Chapters 4, and 5, to model the LSI Kapitza

resistance. This is accomplished by applying a heat flux (q) to the system, then using the

temperature drop across the interface (∆TLSI) to estimate the Kapitza resistance: RLSI =

∆TLSI/q. There are advantages to applying a heat flux as opposed to imposing a tempera-

ture gradient via temperature control. The latter does not conserve the initial total energy

of the equilibrated system, and is more susceptible to momentum drifts. Additionally, the

calculation of heat flux is far more demanding compared to temperature, and longer simu-

lation times would be needed to ensure sufficient convergence in the measured flux. On the

other hand, the application of a heat flux can be carried out in the NVE ensemble, where

total energy is strongly conserved. This is accomplished by moving a predetermined amount

of kinetic energy (−∆K) from atoms in the heat sink, and delivering that same energy to

23

atoms in the heat source (+∆K). When repeated every time step, this induces a heat flux

in the system, as is depicted in Fig. 3.4 for a solid film. In this scenario, the heat flux is

given by: q = ∆K/(2δtA), where A is the cross-sectional area normal to the flux, and the

division by 2 is to account for the fact that the heat flux travels along ±x.

Figure 3.4: General setup for the direct method. Arrows indicate the direction of heat flowout of the heat source (red) and into the heat sink (blue). Periodic boundary conditions areimposed.

The heat addition/subtraction can be carried out without affecting the net linear mo-

mentum of the atoms in the heat source/sink as follows [71]:

KR =1

2

N∑i

miv2i −

1

2

N∑i

miv2g (3.16)

χ =

√KR ±∆K

KR

(3.17)

vi,new = vg + χ (vi − vg) (3.18)

where vg =∑N

i mivi is the net momentum of the atoms in the heat source/sink, and KR

is the relative kinetic energy, from which the energy associated with the net momentum is

deducted. In Eq. 3.18, vg is added back after scaling is applied to the net velocities.

A different technique is used in Chapter 6, which is based on the ’relaxation’ method.

This involves applying a heat pulse to the solid by direct velocity scaling. The solid is then

allowed to equilibrate thermally with the surrounding liquid, without any further tempera-

ture controls. Such a method is suitable for LSI heat transfer studies that involve nanofibers,

24

and/or nanoparticles, in liquid suspensions. This is in order to avoid the excessive perturba-

tions to the dynamics of the system, when a heat flux or a temperature control is constantly

imposed on the entire solid.

3.6 LAMMPS, WestGrid, VMD, and Octave

Prior to utilizing any MD softwares, codes were developed personally, compiled in Fortran,

and validated from literature. This was carried out for LJ systems [63], SPC/E water [42],

in addition to carbon nanotubes, and graphene sheets [72]. The aim of this process was to

experience firsthand the various computational aspects of MD. One of these aspects is the

statistical nature of the measured thermodynamic variables. By opting for larger systems,

variable statistics can be considerably improved. This was the motivation to take advan-

tage of the parallel computing resources, provided by the WestGrid clusters. LAMMPS [73]

(Large-scale Atomic Molecular Massively Parallel Simulator) is one of the MD softwares

installed on WestGrid. It is extensively used in research involving nanoscale heat transfer,

as it is well suited for these types of MD simulations. For Chapters 4-5, the initial atomic

structures were created using personally developed codes. In Chapter 6, the CNT structure

was created through the Nanotube Builder module in VMD [74] (Visual Molecular Dynam-

ics). In both cases, after importing the structures into VMD, atomic structural data files

appropriate for LAMMPS were produced using the Tk Console in VMD.

The calculation of the VDOS from atomic velocities was carried out in a post-processing

fashion. A program was written that extracts velocities from a LAMMPS output file, and

simultaneously computes the velocity-autocorrelation function. This is then imported into

Octave, which contains a biult-in FFT routine, out of which the VDOS plots are produced.

An example of the LAMMPS input script used in Chapter 4 is given in Appendix E, along

with a copy of the velocity-autocorrelation function program.

25

Chapter 4

Lowering LSI thermal resistance with nanopatterned surfaces

4.1 Introduction & Objectives

The ability to fabricate surfaces with nanoscale features, facilitates a means of controlling

the behavior of a liquid in contact with the solid surface [75]. For instance, the wetting

characteristics of a surface can be lowered to enable self-cleaning [76], and reduce friction

drag [77]. Or, on the other hand, the wetting can be enhanced to improve multiphase

flow [78]. Investigations of the effect of surface nanopatterning on wetting [79], and flow

behavior [80] have been reported in recent years. When it comes to analyzing the effects of

surface nanopatterning on LSI heat transfer, factors other than the wetting properties of the

surface need to be examined. Specifically, it is important to understand how modifications

to the surface topology affect its vibrational properties, i.e. its vibrational density of states

(VDOS) (Appendix C). In that regard, previous studies have controlled the surface topology

via self-assembled monolayers (SAMs) [60], or by the addition of nanopillars of the same

material to the surface [37, 41, 59]. While all these studies displayed that surface topology

can affect the LSI thermal conductance, with the exception of [59], none of them examined

the vibrational behavior of the surface. In [59], both the height, and cross-sectional size of the

nanopillars were varied. The improvements in RLSI were attributed to a better vibrational

matching between the two phases, in addition to liquid confinement effects.

In this study, nanopatterning is introduced by periodically removing atoms from the solid

film surface. This is at a relatively smaller scale than has been previously reported. Such

atomic scale surface manipulations can be accomplished via techniques such as scanning

probe lithography (SPL) [81]. With this form of patterning, the connectivity of the modified

atomic layers, apart from the removed portions, is retained. The aim is to examine how

26

nanopatterning at this level can be utilized to lower RLSI. The results for the nanopattered

surfaces are compared to a crystalline smooth surface. The cases that are studied allow a

systematic investigation of how the degree of patterning affects liquid adsorption, and the

vibrational properties of the solid film surface. The combined effect of these two factors

directly influences the liquid-solid vibrational coupling at the LSI, and consequentially, RLSI.

4.2 Simulation details

MD simulations are conducted to model the effects of solid surface nanopatterning on the

LSI thermal resistance. Liquid-liquid, and liquid-solid, interactions are governed by the

Lennard-Jones (LJ) potential (see Appendix B). For the liquid, the interaction parameters

of argon are used with: σll = 3.40 A, and εll = 0.998 kJ/mol. The solid atoms interact via

the Morse potential, with D = 32.0kJ/mol, α = 1.37A−1

, and a nearest neighbor distance of

ro = 2.89A. The interaction between the solid film atoms is limited to the bonded nearest

neighbors in order to prevent the disintegration of the patterned structures. The mass of

the solid atoms is set to ms = 1.6ml, with ml = 0.04kg/mol. The liquid-solid interaction

strength is taken equal to that within the liquid, εls = εll, which results in a wetting surface. A

relatively strong εls is chosen to prevent a wetting-dewetting transition that can be induced

by changes to the surface structure [82]. A size parameter of σls = 0.8σll is used. The

chosen solid film properties are representative of cubic metals [83], and produce vibrational

characteristics that are in good agreement with fcc metals such as Cu.

The simulation cell consists of a solid film surrounded on both sides by liquid, as shown

in Fig. 4.1. The solid film is constructed from a face-centered cubic (fcc) lattice, with a unit

cell length of ao = 4.09A. The simulation cell is 54ao along the z-axis, with two-thirds of the

domain occupied by the liquid. The cross sectional area is 16ao × 16ao in the x-y plane. An

additional half unit cell layer of atoms is added to one of the sides of the solid film to ensure

symmetry of the solid atomic structure that is in contact with the liquid. In all cases there

27

are 12,584 liquid atoms with a bulk density of ≈ 1330kg/m3, which ensures a liquid state at

the simulated temperature, and pressure. Five cases are examined in total (S1-S5), which

includes the case without nanopatterning (S1). In the remaining four cases (S2-S5), the solid

surface at the LSI is patterned by periodically removing atoms in unit cell increments. The

patterning periodicity (pi) is two-dimensional, uniform (px = py), and is equivalent in the

number of unit cells extracted in the x-y plane. Thus, resulting in a checkerboard structure

as shown in Fig. 4.2, where a total of half the atoms in the outermost unit cell are extracted.

The nanopatterning periodicities in unit cells are: px × py =8 × 8 (S2), 4 × 4 (S3), 2 × 2

(S4), and 1 × 1 (S5). Consequent to this choice of patterning, there are a total of 17,920

and 18,944 atoms for the cases with, and without patterning, respectively. The resulting

surface cavities allow a relatively equivalent number of interstitial liquid atoms, which aids

in maintaining the bulk liquid density, and pressure, at comparable levels.

Figure 4.1: Simulation cell for S1. The heat source is placed at the solid film mid-plane,with the heat sink at the outermost liquid layers.

Periodic boundary conditions are used in all directions throughout the simulation cell.

The equations of motion are integrated using the velocity Verlet algorithm [63], with a

timestep of δt = 1.0fs, which was found to provide sufficient energy conservation. A cutoff

radius of rc = 2.5σll is employed for the LJ interactions. The system temperature is first

equilibrated at To = 140K for a period of 1ns using the Berendsen thermostat [67], with

28

a time constant of 0.5ps. Following the equilibration period, at each time step an assigned

amount of energy, ∆Ek = 0.01kJ/mol, is transferred from the liquid atoms in the heat sink

to the solid film mid-plane, as depicted in Fig. 4.1. This is accomplished using an algorithm

that conserves the linear momentum of the system [71]. Therefore, the calculation of the

heat flux along the z-axis is given by: q = ∆Ek/(2δtA) = 207MW/m2, where A is the cross

sectional area in the x-y plane. The simulation is then run without any thermostatting for

10ns, with the last 5ns used for data gathering. This ensures a sufficient convergence of the

temperature profiles in the system.

4.3 Results & Discussion

Temperature data is recorded over bins of size ao/2 ≈ 2.05A along the z-axis. In the cases

with nanopatterned surfaces (S2-S5), the adsorbed liquid is analyzed in two parts: the inter-

nally adsorbed (IA) liquid which resides within the formed surface cavities, and the externally

adsorbed (EA) liquid which adheres to the outermost solid surface. From the final tempera-

ture profiles, the temperature drop across the LSI is used to estimate RLSI = ∆Tav/q, where

∆Tav is taken as the difference between the average temperature of the adsorbed liquid, and

that of the exposed solid film planes (Fig. 4.3). The simulation results for the LSI thermal

resistances are given in Table 4.1. There is a steady drop in RLSI that is associated with

the decrease in the periodicity of nanopatterning, pi. To help explain the trends in RLSI,

the liquid density, and the liquid-solid interaction potential energy distributions, are first

examined. Furthermore, the vibrational properties of the solid film surface, and their level

of matching with the IA and EA liquid, are investigated for all the surfaces. This part of

the study is carried out using separate simulations in the NVE ensemble, at a temperature

of 140K.

29

Figure 4.2: Top and side views of the solid surface nanopatterning examined: (a) S2, (b) S3,(c) S4, and (d) S5. The patterned unit cell layer is shown in dark gray.

Table 4.1: Simulation results. The LSI thermal resistance RLSI has units of (×10−9Km2/W).The % difference in interfacial resistance is based on S1.

Case ∆Tav(K) RLSI (% Diff)S1 3.8 18.4S2 3.0 14.5 (-21.1)S3 2.4 11.6 (-36.8)S4 2.1 10.1 (-44.7)S5 1.7 8.20 (-55.3)

30

Figure 4.3: Temperature profile for S4, the temperatures used in calculating RLSI are shownwith darker shades. The liquid temperatures represent those of the IA and EA liquid. Inthe absence of nanopatterning (S1), only the EA liquid and the outermost solid film layercontribute to ∆Tav. Inset shows temperature profile over entire domain.

Figure 4.4: Liquid density profile for S1, and corresponding surface minimum energy contour.

31

Figure 4.5: Liquid density profile for S2, and corresponding surface minimum energy contour.

Figure 4.6: Liquid density profile for S3, and corresponding surface minimum energy contour.

Figure 4.7: Liquid density profile for S4, and corresponding surface minimum energy contour.

32

Figure 4.8: Liquid density profile for S5, and corresponding surface minimum energy contour.

The liquid density profiles are shown in Figs. 4.4-4.8. A higher resolution of 1A is

employed for the liquid density bins in order to accurately capture the locations of the

density peaks for the IA, and EA liquid. The nanopatterning employed exposes half the

surface area of the lower planes in the solid film, which is consistent with the reduction in

the density of the IA liquid by a factor of ≈ 0.5, relative to the EA liquid density in S1.

The EA liquid in S2-S5 includes the liquid adsorbed to the outer solid film surface, and

the liquid that resides in that same bin over the cavities. This explains the slight reduction

in the EA liquid density from S2-S5, as the area of the cavities is reduced. Although the

density of the IA liquid is comparable among S2-S5, the distribution of the liquid atoms

within the formed cavities is strongly influenced by the nature of patterning. The accessible

area of these cavities in the x-y plane, dictates how the IA liquid atoms are distributed.

For example, in S5, this accessible area is ≈ 3.30A × 3.30A, or ≈ 1.2σls × 1.2σls. This in

turn permits the access of a single liquid atom per cavity at a given time. In addition to

affecting the distribution of the liquid atoms, the nanopatterning influences the adsorption

characteristics of the surfaces. To further illustrate this point, the surface minimum potential

energy contours are also presented in Figs. 4.4-4.8. The contours are constructed over an

area of 8ao × 8ao about the center of the surfaces, using a 250 × 250 grid, which gives a

resolution of ≈ 0.13A in the x-y plane. At each point, the scanning begins at the solid film

33

surface using a liquid atom, and moves along the z-axis in 0.05A increments until the edge of

the liquid-solid interaction zone (at rc from the solid surface). The minimum potential energy

value is then recorded and projected onto the corresponding x-y coordinates. As expected,

for S1 the contour is uniform. When nanopatterning is introduced to the surface, regions of

lowered potential minima appear in the cavities, with the highest intensity resulting from

S5. Hence, with nanopatterning the interaction of the IA liquid with the solid film surface

becomes more localized.

In order to examine the liquid-solid vibrational coupling, the vibrational density of states

(VDOS) are computed by taking the Fourier transform of the velocity autocorrelation func-

tion (Appendix D). For the solid film, this was conducted for all atoms in the middle (SM),

and patterned (SO), planes. Whereas for the liquid, the VDOS is calculated for the IA and

EA liquid layers. As can be seen in Figs. 4.4-4.8, the VDOS for SM (VDOSSM), which is

far removed from the surface, remains essentially unaffected for S1-S5. Compared to SM,

VDOSSO is shifted to lower frequencies. Such a shift is expected as a result of the lower

coordination number of the surface atoms [84, 85]. This effect is enhanced from S2 to S5,

with each patterned surface possessing a lower average coordination number for the surface

atoms. Hence, the most drastic modification for VDOSSO appears for S5, with a pronounced

peak in the lower frequency range (≈ 1.25THz). The more noticeable shift in VDOSSO for S5

is consistent with the vibrational properties of adatoms on metallic surfaces [86,87]. An im-

portant feature in VDOSIA, and VDOSEA, is the zero frequency value, which is proportional

to the liquid self-diffusion coefficient [88]. For VDOSEA, there is an upward shift in the zero

frequency value for S5, compared to S2-S4. This results from the reduction in the EA liquid

adsorption for this surface as indicated in the density profile plot of S5. More noticeable

differences appear in VDOSIA, with the value for S5 being significantly lower. This can be

interpreted using the minimum potential energy contours from Fig. 4.8, where the potential

minima are highly localized in S5. A similar trend has been reported for flows in rough

34

nanochannels [89], for which the residence time of the adsorbed liquid correlated with the

degree of solid wall roughness. There is also a slight shift in VDOSIA to higher frequencies,

when compared to VDOSEA, as a result of the stronger vibrational coupling to the solid film.

This is the most apparent in S5, which develops an additional peak that coincides closely

with that of VDOSSO.

There is a correlation between the trends in RLSI, and the degree of vibrational cou-

pling between the adsorbed liquid, and the solid film surface. To elucidate this relation-

ship, the following quantity is computed separately for the IA and EA liquid: OVDOS,i =∫ +∞−∞ [VDOSi (ω)] [VDOSSO (ω)] dω, where: i = IA, EA, and ω is the frequency. For each sur-

face, the average: 12

[OVDOS,IA +OVDOS,EA] is then normalized by the maximum value, which

belongs to S5. When plotted along with the normalized RLSI, a strong inverse relationship

exists between the two parameters, as shown in Fig. 4.10.

4.4 Conclusion

It is shown through MD simulations that solid surface nanopatterning can notably influence

the LSI thermal resistance. Through nanopatterning, the wetting characteristics of the

solid surface, and its vibrational properties can be tailored to provide a stronger vibrational

coupling with the internally adsorbed liquid. The VDOS of the solid film surface are shifted

to lower frequencies due to the drop in the surface atoms coordination number. When

combined with a highly localized interaction potential minima, as was the case for S5, a

clear drop in RLSI is witnessed. This drop correlates directly with the degree of vibrational

coupling that is computed from the VDOS profiles of the two phases at the LSI. It is hoped

that the findings in this study provide some insight for designing surface nanoscale geometries

with the goal of improved thermal management.

35

Figure 4.9: Vibrational density of states (VDOS): (a) S1, (b) S2, (c) S3, (d) S4, and (e) S5.SM: solid film mid-plane, SO: solid film patterned layers, IA: internally adsorbed liquid, EA:externally adsorbed liquid.

36

Figure 4.10: Correlation between normalized RLSI and CVDOS for surfaces S1-S5.

37

Chapter 5

Pressure effects on liquid-solid interfacial thermal resistance

5.1 Introduction & Objectives

The transmission of thermal energy carriers, or phonons, across interfaces is impeded due

to the vibrational mismatch between the materials in contact. In the case of LSIs, there

is a considerable disparity in the phonon spectrum between the two phases, resulting in

a relatively high RLSI. Recently, the possibility of pressure tuning the interfacial thermal

resistance at solid-solid interfaces has been investigated experimentally [90], and computa-

tionally [61], through MD simulations. It was found that the interfacial thermal resistance

dropped with increasing pressure when the bonding between the two solids was weaker than

that in the bulk. This was attributed to a rise in the interfacial bonding stiffness, resulting

in an enhanced phonon transmission at the interface [91]. In the case where interfacial bond-

ing is stronger, relative to the bulk, MD results indicated that the conductance is relatively

unchanged, with a slight drop at higher pressures. This was explained by the changes to the

VDOS of the two strongly bonded solids at high pressures, where the overlap was somewhat

reduced due to a disproportionate shift to higher frequencies among the two solids.

The effect of pressure on RLSI has not been fully investigated yet, with the focus mainly

on surface functionalization [59,60,92], and liquid confinement effects [58]. In this work, MD

simulations are conducted to examine the effects of pressure on the RLSI. Analogous to the

case of solid-solid interfaces, this is carried out for a wetting (W), and a nonwetting (NW)

solid surface. This is achieved by controlling the liquid-solid interaction strength.

38

5.2 Simulation Details

The simulation setup is similar to that from Sec. 4.2, and consists of a solid film sandwiched

between two liquid regions, as shown in Fig. 5.1. Here, all atoms interact via the Lennard-

Jones (LJ) potential. For all interactions, σss,ll,ls = 3.4A. The interaction strength for the

liquid atoms is taken to be that of argon εll = 0.998kJ/mol. A stronger interaction is chosen

for the solid atoms with εss = 10εll, which provides sufficient cohesiveness for the simulated

conditions. For the wetting case, εls = 1.0, whereas for the nonwetting case εls = 0.1. The

solid film is constructed from a face-centered cubic (fcc) lattice with a lattice constant of

ao = 1.56σ. The simulation cell has dimensions of Lx = Ly = 12ao, in the x-y plane. For

all cases studied, the solid film has a thickness of 12ao along the z-axis. The total Lz of

the simulation domain is adjusted, for a fixed number of liquid atoms, in order to vary the

system pressure. Four cases (C1-C4) are examined for the wetting (W), and nonwetting

(NW) surfaces, respectively. In the initial case (C1), Lz = 33ao, which is subsequently

reduced in increments of 2.526A to steadily increase the system pressure. There are a total

of 15,200 atoms in the system.

Figure 5.1: Pressure effect MD simulation cell.

Periodic boundary conditions are used in all three dimensions. The equations of motion

are integrated using the velocity Verlet algorithm, with a timestep of δt = 1fs. For computa-

tional efficiency, a cutoff radius of 2.5σ is used. The system is first equilibrated at 130K for

39

a period of 1.0ns using the Berendsen thermostat, with a time constant of 0.5ps. After the

equilibration period, a small amount of energy ∆E = 4.90× 10−3kJ/mol, is removed at each

time step from the liquid atoms in the heat sink, and added to the midplane of the solid film,

as depicted in Fig. 5.1. This process is equivalent to inducing a heat flux along the z-axis

with a magnitude of: q = ∆E/(2δtA) = 100MW/m2, where A is the cross-sectional area in

the x-y plane. The energy exchange is run for an additional 15ns, with the last 10ns used for

temperature averaging. This provides sufficient convergence in the developed temperature

profiles.

5.3 Results & Discussion

Data for the temperature profiles along the z-axis is recorded in bins of size ∆z = 0.5ao. The

temperature drop ∆T across the LSI is calculated as shown in Fig. 5.2, and is the average

temperature difference between the outermost solid film plane, and the adsorbed liquid. As is

expected, the temperature profile in the liquid region displays a higher temperature gradient

due to the lower thermal conductivity compared to the solid film.

Figure 5.2: Temperature profile for case C4-NW.

The results for RLSI are given in Fig. 5.3, for both the W and NW surfaces, as a function

40

of pressure. There is a stark difference in how pressure affects the heat transfer across the

LSI for a W, and a NW surface. The pressure effect is negligible for the W surface, with all

resistances ≈ 20 × 10−9m2K/W, as the pressure is increased from ≈ 35MPa to ≈ 60MPa.

For the NW case, RLSI is highly sensitive to the system pressure, and decreases by a factor of

≈ 2, as the pressure is raised from ≈ 34MPa to ≈ 64MPa. To explain the stronger response

to pressure for RLSI of the NW surface, two important factors are analyzed: (i) adsorbed

liquid density, and (ii) vibrational density of states (VDOS) of the atoms at the LSI. The

liquid density profiles are averaged over bins of size ∆z = 0.1ao which sufficiently captures

the liquid layering at the solid film surface. The VDOS is computed by taking the Fourier

transform of the velocity autocorrelation function. This was conducted for all the liquid and

solid atoms used in calculating ∆T , in addition to the solid mid-plane. The calculation of

the VDOS is carried out in the NVE ensemble at 130K.

The adsorbed liquid density profiles are given in Fig. 5.4. When compared to the NW

surface, the stronger interaction between solid and liquid atoms for the W surface results in

a considerably higher liquid density peak (at ≈ 35A) next to the solid film. The pressure

increase has a minor effect on the adsorbed liquid density peak for the W surface. In the

NW case, the increase in the pressure results in a hike in the density peak by up to ≈ 50%,

from C1 to C4. This in turn permits more liquid atoms at the LSI, which enhances the

thermal energy exchange between the liquid and solid atoms. Consequently, this causes a

more significant drop in RLSI for the NW surface as the pressure is increased.

The vibrational properties of the LSI atoms were found to be relatively unaffected by

the pressure increase (from C1-C4) for the W, and NW surfaces, respectively. The averaged

VDOS of the LSI atoms (AL: adsorbed liquid, SO: solid outer surface) are shown in Fig. 5.5,

along with that of the solid film mid-plane (SM). For both types of surfaces, the VDOS of

the SO and SM are similar, with the SO occupying a lower frequency range as a result of the

lower coordination number of the atoms at the film surface. The frequency range occupied

41

Figure 5.3: Pressure effect on RLSI for wetting (W), and nonwetting (NW) surfaces. C1-C4in order of increasing pressure.

by the solid atoms (up to 6.0THz) is, as anticipated, higher than that of the adsorbed liquid

(mostly below 2.0THz). The liquid adsorbed to the W surface vibrates at higher frequencies,

compared to the NW liquid. This is a consequence of the stronger interaction between the

liquid, and the W surface atoms. This improves the AL vibrational coupling with the SO

atoms, and combined with the relatively higher AL density for the W surface, results in a

much lower RLSI (see Fig. 5.3) when compared to the NW surface. An important aspect

of the VDOS in liquids is the zero frequency value, which is directly proportional to the

adsorbed liquids self-diffusion coefficient [88]. The strong interaction between the W surface

and the AL considerably lowers its self-diffusion coefficient [93] relative to the NW surface.

This can be deduced from the AL zero frequency values in Fig. 5.5 (a) and (b), respectively.

In the case of the W surface, within the pressure range studied here, there is little effect

on the adsorbed liquid density, or on the VDOS of the LSI atoms. Hence, RLSI appears

impervious to system pressure. Similarly for the NW surface, there are no noticeable shifts

in the VDOS as a result of system pressure increase. In that case, the significant reduction

in RLSI has to be related to the increase in the AL density. This is demonstrated in Fig. 5.6,

by plotting the relative increase in the adsorbed liquid density peak, along with the relative

42

Figure 5.4: Liquid density profiles: (a) W surface, (b) NW surface. Insets show the %relative increase in adsorbed liquid density peak based on C1

43

Figure 5.5: Averaged VDOS of LSI atoms for cases C1-C4: (a) W, and (b) NW. SM: solidmid-plane, SO: solid outer surface, AL: adsorbed liquid. Insets show the VDOS of theadsorbed liquid, and the solid surface superimposed for C1-C4.

44

Figure 5.6: Correlation between relative increase in adsorbed liquid density for the NWsurface, and the corresponding relative drop in RLSI. All computations are based on C1 forthe NW surface.

drop in RLSI, based on C1 values. There is a clear correlation between the two parameters.

A similar trend of the effect of adsorbed liquid density on RLSI was recently reported for a

nonwetting carbon nanotube [94].

5.4 Conclusion

Through MD simulations, it was shown that the interfacial thermal resistance across a LSI is

appreciably more sensitive to the system pressure when the solid surface is nonwetting. For

the W surface, the pressure appears to have a limited effect on the adsorbed liquid density,

and the vibrational characteristics of the atoms at the LSI. The latter also applies to the

NW surface. On the other hand, the adsorbed liquid density for the NW surface experiences

a significant relative increase with system pressure. This facilitates a noticeable reduction

in RLSI as more liquid atoms exchange thermal energy with the solid atoms at the LSI. The

results imply the possibility of tuning RLSI for NW surfaces through controlling the system

pressure. It also shows that when studying LSI heat transfer involving wetting surfaces,

pressure is not expected to be a key parameter.

45

Chapter 6

Spectral analysis of LSI heat transfer in carbon nanotubes (CNTs)

6.1 Introduction & Objectives

Since the discovery of carbon nanotubes (CNTs) [95], there has been a steady increase in their

projected microscale, and nanoscale applications [96]. While their discovery might be fairly

recent, the treatment of various aspects of CNTs in the literature is exhaustive [97,98]. This

is primarily driven by their superior mechanical, electrical, and thermal properties [97, 99].

A review of synthesis procedures can be found in [98]. Among the attractive applications of

CNTs is employing them as additives for enhancing the thermal conductivity of polymers

[100], organic liquids [101], and water [102], for thermal management applications. Thermal

conductivities of up to 3000W/mK have been measured [103] for CNTs. However, when

submerged in liquids, CNTs suffer from high LSI thermal resistance [31]. A comprehensive

understanding of thermal interactions of CNTs in liquid media is imperative for many of

their expected applications in thermal management.

On the theoretical front, CNTs provide an ideal model for MD studies of thermal energy

transport, with heat conduction in CNTs being primarily due to phonons [104]. In itself, a

CNT is a molecular scale carbon fiber that possesses a well defined vibrational spectrum.

Thus, enabling the mode coupling studies that explained the roots of their low LSI thermal

conductance. When visualizing a CNT, it can be thought of as a rolled up graphite sheet

with a honeycomb pattern, as can be seen in Figure 6.1. The chiral vector, C, defines the

roll up direction in terms of the basis vectors a1 and a2:

Cnm = na1 +ma2, n ≥ m (6.1)

When m = 0, the edges of the CNT take on a zig-zag form, whereas with m = n, the CNT

46

Figure 6.1: Honeycomb structure of a graphite sheet. Chiral vectors Cnm, and translationalvectors Tnm are shown for armchair (5, 5), and zig-zag (8, 0) CNTs.

edges resemble an armchair. For all other chiralities the CNT is termed simply as chiral.

The translational vector, Tnm points along the CNT axis, and is normal to Cnm. The length

of the chiral vector represents the circumferential length of the CNT, and the CNT radius

rCNT is given by:

rCNT =|Cnm|

2π=a√m2 +mn+ n2

2π(6.2)

where a = 1.42√

3A, is the lattice constant of the graphite sheet.

In addition to the vibrational mismatch between a CNT and its liquid surroundings,

there is evidence that the weak interfacial thermal conductance in CNTs is induced by

the the slow transfer of thermal energy between high, and low frequency modes within

the nanotube [32, 47]. This was primarily based on the spectral temperature analysis which

displays a disparity between the energy of modes having frequencies comparable to the liquid,

and those that are higher. The spectral temperature (Tsp) for a frequency range between ω1,

47

and ω2 is given by:

Tsp(t) = Teq

(1

ω2 − ω1

)∫ ω2

ω1

VDOSNE(ω, t)

VDOSEQ(ω)dω (6.3)

where VDOSEQ and VDOSNE are the vibrational density of states at equilibrium and during

thermal energy exchange, respectively. The temperature Teq is the average equilibration

temperature on which VDOSEQ is based. It is believed that the rate of thermal energy

exchange between low and high frequency modes is the limiting factor in LSI heat transfer

for CNTs. However, the causes of this type of obstruction to heat transfer in CNTs remains

unclear. Methods of improving RLSI in CNTs have primarily focused on utilizing chemical

functionalization [40, 105], which provides a vibrational bridge between the CNT and the

surrounding liquid. By understanding the nature of the inter-modal energy exchange in

CNTs, new avenues can be explored to improve RLSI .

6.2 Methodology

MD simulations are conducted to provide a detailed analysis of the frequency dependent

energy exchange in CNTs, that are interacting with a liquid medium. Interactions between

the carbon atoms in the CNT are modeled using the REBO potential [106] (see Appendix

B). The REBO potential is one of the most commonly used potentials in MD of CNTs, and

is capable of capturing important features in the phonon spectrum of the CNT [107]. Liquid,

and liquid-solid interactions are modeled with the LJ potential. Argon parameters are used

for the liquid with σll = 3.4A, and εll = 0.998kJ/mol. For the liquid-solid interactions:

σls = 3.38A, and εls = 0.48kJ/mol [47]. The simulation cell consists of an armchair CNT

with a chirality of (5,5), immersed in liquid, as shown in Fig. 6.2. The CNT has a radius

of ≈3.39A. The simulation cell has a cross section of Lx × Ly = 50A × 50A, and a depth

that is equal to the CNT length of 250A. The CNT contains 2040 carbon atoms, while the

liquid consists of 10,935 atoms which results in a bulk liquid density of 1200kg/m3. Periodic

48

boundary conditions are used in all dimensions, with a time step of 0.5fs. A cutoff radius of

3σll was utilized.

Figure 6.2: Simulation cell for (5,5) CNT immersed in liquid. Darker region in the liquiddenotes the thermostatted region. The simulation cell has a depth of Lz = 250A.

The simulation procedure is based on the thermal relaxation method [108]. Generally, this

involves applying a heat pulse to the solid, followed by an equilibration period during which

the solid is cooled by the liquid. Using the decay of the temperature difference, an exponential

fit can provide an estimate of RLSI based on the lumped capacitance assumption [8]:

∆T (t) = ∆Toe−[

1mlCl

+ 1msCs

]AsR

−1LSIt (6.4)

where ∆T is the time dependent temperature difference between the CNT and the liquid,

∆To is the initial temperature difference, ml and ms are the total mass of the liquid and

the solid, respectively, whereas Cl and Cs are their corresponding heat capacities. For the

system under consideration Cl = 312J/kgK, and Cs = 1000J/kgK [47]. The effective surface

area of the CNT is taken to be: As = πLCNT(dCNT + σls), where LCNT, and dCNT, are the

49

CNT length, and diameter, respectively. Using the relaxation time (τ) from the exponential

fit, the LSI thermal resistance is finally obtained through:

RLSI = τ

[1

mlCl

+1

msCs

]As (6.5)

6.3 Results & Discussion

In this study, the relaxation method is used first to validate the simulation model. The

system is first equilibrated at 120K using the Berendsen thermostat [67], with a time period

of 0.25ps. Due to the high RLSI in this system, separate thermostats are applied to the CNT,

and the liquid, during which momentum conservation is enforced. This circumvents the

need for unnecessarily long equilibration times, and was found to provide a well equilibrated

system, as is shown in Fig. 6.3. Following equilibration, the CNT temperature is suddenly

raised to 300K by velocity scaling, and is held at that value for 10ps. All temperature

controls are then turned off, and the CNT is allowed to cool as shown in Fig. 6.4(a). The

exponential fit to the temperature decay is given in Fig. 6.4(b), and it produced a time

constant of τ ≈ 417ps. This results in a Kapitza resistance of RLSI = 0.971× 10−6m2K/W,

or alternatively a conductance of 1.03MW/m2K, which is in good agreement with values

reported in the literature [47].

During the CNT cooling process, there is little variation in the temperature of the liquid

spatially. This is a consequence of the high RLSI for the CNT, relative to the thermal conduc-

tivity of liquid argon (≈ 0.1W/mK [109]), which was the basis for the lumped capacitance

approximation discussed earlier. To put this into perspective, the Kapitza length in this case

would be LLSI = 97.1nm, which exceeds the radial distance covered by the simulation cell.

To further illustrate this point, the temperature of the adsorbed liquid (TAds) within 4A of

the CNT surface was measured and plotted against the entire liquid temperature, as shown

in Fig. 6.5. There is almost no deviation between TAds, and TL, throughout the simulation

time, regardless of the fact that TCNT starts at 300K.

50

Figure 6.3: Temperature history of the CNT (TCNT), and surrounding liquid (TL) in theNVE ensemble following equilibration at 120K.

Figure 6.4: (a) Temperature profile of CNT cooling in liquid after sudden heating to 300K,(b) exponential fit of ∆T (t).

Figure 6.5: (a) Close correspondence between total liquid and adsorbed liquid temperature(TAds) (b) Radial density distribution starting from the CNT surface. The shaded regiondenotes where TAds was measured.

51

The spectral temperature analysis is primarily based on the disruptions caused to the

VDOS of a heated CNT, relative to a CNT in thermal equilibrium. An example of the VDOS

of the CNT, and the liquid, in an NVE ensemble at 120K is shown in Fig. 6.6. The inherent

CNT peak at a frequency ≈ 55THz [110] can be seen. Additionally, the liquid VDOS extends

to only ≈ 2THz, with a peak at 1THz [47]. If the liquid surrounding the CNT is linked to

an infinite heat bath by thermostatting an outer ring area (Fig. 6.2), then it would be

fair to assume that the adsorbed liquid temperature will remain relatively unchanged when

TCNT is increased, or decreased. To provide a clearer picture of the frequency dependence

of energy exchange at the LSI of the CNT, an alternative would be to quench the CNT

to a lower temperature. By that, the thermal noise in the CNT can be greatly reduced,

and the evolution of the VDOS would be statistically easier to track. A similar approach

has been previously utilized to study heat pulse propagation in CNTs [111]. To examine

this hypothesis, following the equilibration period described earlier, the temperature of the

CNT was quenched to TCNT = 1K, while the liquid outside a radius of 22A was constantly

thermostatted at 120K, via velocity scaling. As can be seen in Fig. 6.7, the adsorbed liquid

temperature is maintained at 120K throughout the CNT thermal equilibration process.

Following the 10ps quenching period, the velocities of the CNT atoms were recorded

every time step (0.5fs) for a total period of ≈ 5.0 × 105 time steps (≈ 250ps). From these

velocities, 10 sets of VDOS curves were computed by averaging the velocity autocorrelation

function over periods of 25ps. The VDOS plots and their corresponding time frames are

shown in Fig. 6.8. In the period following the quench (0 − 25ps), the density of states are

overwhelmingly confined in the low frequency range. This is further confirmation of the low

frequency coupling between the CNT and the liquid. With each time period, the VDOS

begin to extend to higher frequencies, and the initial confinement to the low frequency range

diminishes. This trend validates the view that thermal energy in the high frequency range

communicates with the liquid via the low frequency zone [32].

52

Figure 6.6: Equilibrium VDOS of CNT (green) and adsorbed liquid (blue) at 120K. Inset isa zoom on 0− 15THz

Figure 6.7: Time history of CNT temperature (TCNT). The temperature of the adsorbedliquid (TAds) is unaffected by the low TCNT as a result of the high RLSI.

53

Figure 6.8: VDOS plots and their corresponding averaging periods following the CNT quenchto 1K.

As the CNT is being heated by the liquid, its VDOS is attempting to regain its equilibrium

distribution prior to the quench, which is given in Fig. 6.6. To illustrate this point, the

CNT VDOS plots in Fig. 6.8 are projected onto the equilibrium VDOS, as shown in Fig.

6.9. To facilitate a clearer view, VDOSNE is set equal to VDOSEQ at all points where:

VDOSNE ≥ VDOSEQ. There is an interesting trend that signals the presence of a ’bottle-

neck’ type behavior during the evolution of VDOSNE. The recovery of VDOSNE is fully

completed in the range 0 − 15THz within 50ps of CNT heating by the liquid. Beyond this

frequency range, the response is significantly slower, with the frequency range of 20−30THz

still lagging up to 200ps later. It is important to note that the response in the vicinity of

≈ 10THz is not linked to the liquid, given that the VDOS overlap with the liquid region is

limited to only ≤ 2THz. Instead, this phenomenon can be explained by exploring the shape

of the VDOS, and consequently, the dispersion relation in the frequency range 10− 20THz.

As can be seen in Fig. 6.6, within this region, there is a sharp drop in the VDOS, which

54

signals a relatively depleted region in the phonon dispersion of the CNT [112]. In analogy to

phononic crystals [113], when thermal energy enters the CNT at relatively low frequencies,

this ’pseudo’ phonon band gap impedes the energy exchange between the low and high

frequency modes. Hence, limiting by that the ability of the CNT to exchange heat efficiently

with the surrounding liquid, and consequently resulting in a high RLSI. Moreover, in a recent

study [55], a steady-state spectral temperature profile for a heated graphene sheet displayed

a sharp drop in spectral temperature that was also in the frequency range 10− 20THz. This

is additional supporting evidence for the above hypothesis.

6.4 Conclusion

By quenching the CNT to a low temperature, a vivid picture emerged of the frequency de-

pendence of thermal energy exchange at the LSI. There is an obvious propagation of thermal

energy in frequency space. From the analysis of the dynamics of this energy propagation, an

energy exchange bottle-neck appears in the frequency range of 10− 20THz. As a result, the

low-to-high frequency heat conduction dominates the thermal interaction at the LSI of the

CNT. This frequency range coincides with a depleted region in the VDOS of the CNT, and

agrees well with recent findings in the literature. This can provide new avenues of improving

RLSI in CNTs via defect engineering [114] or isotope doping [115].

55

Figure 6.9: Projection of CNT VDOSNE onto VDOSEQ. Also shown is the overlap with theliquid VDOS.

56

Chapter 7

Contributions and future research direction

7.1 Summary of contribution and findings

Following the introduction in Chapter 1, a detailed literature review was provided in Chapter

2. At the end of Chapter 2, the objectives of the thesis were given, which aimed to: (i)

provide a systematic examination of the factors affecting the LSI thermal resistance in the

presence of surface nanopatterning, (ii) explore the potential for pressure-tuning the LSI

thermal resistance for wetting and nonwetting surfaces, and (iii) demonstrate the frequency

dependence of thermal energy exchange at LSIs involving CNTs. In Chapter 3, the molecular

dynamics method was introduced, along with the techniques relevant for studying nanoscale

heat transfer. In Chapters 4-6 the above listed objectives were carried out, respectively. In

this chapter, a summary of the contributions, and recommendations for future research are

given as follows:

• Surface nanopatterning and heat transfer at LSIs : It was demonstrated in

Chapter 4 that modifying the nanoscale topology of a solid surface can have

a significant effect on lowering the Kapitza resistance. This was attributed to

how surface nanopatterning affects the interaction strength with the adsorbed

liquid, as well as the vibratonal properties of the patterned surface. In this

study the nanopatterning was limited to unit cell depth, in order to focus

more on the role of vibrational coupling between the solid and liquid atoms

at the LSI. The depth of the patterning, which also translates to the height

of surface nanopillars, could directly affect the thermal conductivity of the

solid surface itself. Increasing the depth of patterning for case S5 would result

in a larger jump in the liquid-exposed surface area relative to S2. However,

57

the hindrance to phonon transport within the solid surface is expected to be

higher in S5, due to the increased nanoporosity of the surface [116, 117]. For

future work, the interplay between these factors can be further investigated.

Another potentially important factor is the geometry of the surface pillars.

Consideration has mostly been given to nanopillars of rectangular shape. The

effect of other geometries, such as pyramidal and hemispherical, would provide

more insight into the role that surface geometric functionalization plays in LSI

heat conduction.

• Pressure effects on LSI thermal resistance: In Chapter 5, the effects of system

pressure on the LSI thermal resistance were found to be significantly more

pronounced for nonwetting surfaces. For all the cases considered, the phonon

coupling at the interface was relatively unaffected by the increase in pressure.

The relative increase in adsorbed liquid density for the nonwetting surface was

shown to be the major factor in lowering the Kapitza resistance. The outcome

demonstrates the possibility for pressure tunable LSI thermal resistance for

nonwetting surfaces. A future extension of this study can examine this pressure

effect for surfaces of intermediate wetting properties. This would provide a

more detailed picture of how the wetting characteristics of the solid surface

affects the relative increase in adsorbed liquid density, and consequently the

range of tuning the LSI thermal resistance.

• Frequency dependence of heat conduction at LSIs involving CNTs : The find-

ings in Chapter 6 provided a strong confirmation of the low frequency coupling

between CNTs, and the surrounding liquid medium. The high Kapitza resis-

tance was also linked to the poor exchange of thermal energy between high,

and low, frequency modes withing the CNT. Hence, one way to improve the

LSI thermal conduction for CNTs is to promote the high-to-low frequency en-

58

ergy exchange. This is directly linked to altering the dispersion relations in

the CNT. A further investigation into this phenomena should focus on two

important avenues: (i) isotope doping, and (ii) CNT diameter. Both of these

parameters can influence the CNT dispersion relations, and thermal conductiv-

ity. For purposes of thermal management, a balance must be struck between

the improvement in LSI thermal conduction, and the possible reduction in

the CNT thermal conductivity as a result of modifications to the dispersion

relations.

The study of nanoscale heat transfer using molecular dynamics remains a highly active

field of research. It is hoped that the above findings contribute to our understanding of

thermal energy exchange at LSIs. The recommendations for future studies aim at extending

these findings to continue exploring this phenomena. This is crucial for the proper thermal

design of nanodevices, which continue to play an expanding role in various sectors and

industries.

59

Bibliography

[1] D.G. Cahill, W.K. Ford, K.E. Goodson, G.D. Mahan, A. Majumdar, H.J. Maris,

R. Merlin, and S.R. Phillpot. ”Nanoscale Thermal Transport”. Journal of Applied

Physics, 93(2):793–818, 2003.

[2] S. Krishnan, S.V. Garimella, G.M. Chrysler, and R.V. Mahajan. ”Towards a thermal

Moore’s law”. IEEE Transactions on Advanced Packaging, 30(3):462–474, 2007.

[3] R.B. Oueslati, D. Therriault, and S. Martel. ”PCB-Integrated Heat Exchanger for

Cooling Electronics Using Microchannels Fabricated With the Direct-Wire Method”.

IEEE Transactions on Components and Packaging Technologies, 31(4):869–874, 2008.

[4] E. Baird and K. Mohseni. ”Digitized Heat Transfer: A New Paradigm for Thermal

Management of Compact Micro Systems”. IEEE Transactions on Components and

Packaging Technologies, 13(46):143–151, 2008.

[5] L.L. Vasiliev. ”Heat Pipes in Moderm Heat Exchangers”. Applied Thermal Engineer-

ing, 25(1):1–19, 2005.

[6] P. Keblinski, S.R. Phillpot, S.U.S. Choi, and J.A. Eastman. ”Mechanisms of heat flow

in suspensions of nano-sized particles (nanofluids)”. International Journal of Heat and

Mass Transfer, 45:855–863, 2002.

[7] L.R. Hirsch, R.J. Stafford, J.A. Bankson, S.R. Sershen, B. Rivera, R.E¿ Price, J.D.

Hazle, N.J. Halas, and J.L. West. ”Nanoshell-mediated near-infrared thermal therapy

of tumors under magnetic resonance guidance”. Proceedings of the National Academy

of Sciences of the United States of America, 100(23):13549–13554, 2003.

[8] F.P. Incropera, D.P. DeWitt, T.L. Bergman, and A.S. Lavine. Fundamentals of Heat

and Mass Transfer. John Wiley & Sons, USA, 2007.

60

[9] P.L. Kapitza. ”The Study of Heat Transfer in Helium II”. Journal of Physics - USSR,

4(1-6):181–210, 1941.

[10] R. Khare, P. Keblinski, and A. Yethiraj. ”Molecular Dynamics Simulations of Heat

and Momentum Transfer At a Solid-Liquid Interface: Relationship Between Thermal

and Velocity Slip”. International Journal of Heat and Mass Transfer, 49:3401–3407,

2006.

[11] A.J.H. McGaughey and M. Kaviany. ”Phonon transport in molecular dynamics simu-

lations: formulation and thermal conductivity prediction”. Advances in Heat Transfer,

39:169–255, 2006.

[12] D.G. Cahill, K.E. Goodson, and A. Majumdar. ”Thermometry and Thermal Transport

in Micro/Nanoscale Solid-State Devices and Structures”. Journal of Heat Transfer-

Transactions of the ASMe, 124(2):223–241, 2002.

[13] J.R. Henderson and F. Vanswolf. ”On the Interface Between a Fluid and a Planar

Wall - Theory and Simulations of a Hard-Sphere Fluid At a Hard-Wall”. Molecular

Physics, 51:991–1010, 1984.

[14] C.J. Yu, A.G. Richter, A. Datta, M.K. Durbin, and P. Dutta. ”Molecular Layering in

a Liquid on a Solid Substrate: An X-ray Reflective Study”. Physica B, 283(1-3):27–31,

2000.

[15] P.A. Thompson, G.S. Grest, and M.O. Robbins. ”Phase-Transitions and Universal

Dynamics in Confined Films”. Physical Review Letters, 68(23):3448–3451, 1992.

[16] J. Klein and E. Kumacheva. ”Confinement-Induced Phase-Transitions in Simple Flu-

ids”. Science, 269(5225):816–819, 1995.

[17] A.L. Demirel and S. Granick. ”Glasslike Transition of a Confined Simple Fluid”.

Physical Review Letters, 77(11):2261–2264, 1996.

61

[18] P.A. Thompson and M.O. Robbins. ”Shear-Flow Near Solids - Epitaxial Order and

Flow Boundary-Conditions”. Physical Review A, 41(12):6830–6837, 1990.

[19] B. Bhushan, J.N. Israelachvili, and U. Landman. ”Nanotribology - Friction, Wear and

Lubrication at the Atomic-Scale”. Nature, 374(6523):607–616, 1995.

[20] S.T. Cui, P.T. Cummings, and H.D. Cochran. ”Molecular Dynamics Simulation of the

Rheological and Dynamical Properties of a Model Alkane Fluid Under Confinement”.

Journal of Chemical Physics, 111(3):1273–1280, 1999.

[21] J.P. Gao, W.D. Luedtke, and U. Landman. ”Structures, Solvation Forces and Shear of

Molecular Films in a Rough Nano-Confinement”. Tribology Letters, 9(1-2):3–13, 2000.

[22] Y. Zhu and S. Granick. ”Reassessment of Solidification in Fluids Confined Between

Mica Sheets”. Langmuir, 19(20):8148–8151, 2003.

[23] L. Xue, P. Keblinski, S.R. Phillpot, S.U.-S. Choi, and J.A. Eastman. Effect of Liquid

Layering at the Liquid-Solid Interface on Thermal Transport. International Journal

of Heat and Mass Transfer, 47:4277–4284, 2004.

[24] Z. Liang and H.-T. Tsai. ”Thermal conductivity of interfacial layers in nanofluids”.

Physical Review E, 83:0416021–0416028, 2011.

[25] S.K. Das, S.U.S. Choi, W. Yu, and T. Pradeep. Nanofluids. John Wiley & Sons, USA,

2008.

[26] E.T. Swartz and R.O. Pohl. ”Thermal Boundary Resistance”. Reviews of Modern

Physics, 61(3):605–668, 1989.

[27] G.L. Pollack. ”Kapitza Resistance”. Reviews of Modern Physics, 41(1):48–81, 1969.

[28] S. Maruyama and T. Kimura. ”A Study on Thermal Resistance Over a Solid-Liquid

Interface by the Molecular Dynamics Method”. Thermal Science and Engineering,

62

7(1):63–68, 1999.

[29] J.C. Tully. ”Dynamics of Gas-Surface Interactions: 3D Generalized Langevin Model

Applied to fcc and bcc Surfaces”. Journal of Chemical Physics, 73(4):1975–1985, 1980.

[30] T. Ohara and D. Suzuki. ”Intermolecular Energy Transfer at a Solid-Liquid Interface”.

Microscale Thermophysical Engineering, 4:189–196, 2000.

[31] S.T. Huxtable, D.G. Cahill, S. Shenogin, L. Xue, R. Ozisik, P. Barone, M. Usrey,

M. Strano, G. Siddons, M. Shim, and P. Keblinski. ”Interfacial Heat Flow in Carbon

Nanotube Suspensions”. Nature Materials, 2:731–734, 2003.

[32] S. Shenogin, L. Xue, R. Ozisik, P. Keblinski, and D. Cahill. Role of Thermal Boundary

Resistance on the Heat Flow in Carbon-Nanotube Composites. Journal of Applied

Physics, 95(12):8136–8144, 2004.

[33] L. Xue, P. Keblinski, S.R. Phillpot, S.U.-S. Choi, and J.A. Eastman. ”Two Regimes

of Thermal Resistance at a Liquid-Solid Interface”. Journal of Chemical Physics,

118(1):337–339, 2003.

[34] A.J. Markvoort and P.A. Hilbers. ”Molecular Dynamics Study of the Influence of

Wall-Gas Interactions on Heat Flow in Nanochannels”. Physical Review E, 71:066702–

1–066702–9, 2005.

[35] T. Ohara and D. Torii. ”Molecular Dynamics Study of Thermal Phenomena in an

Ultrathin Liquid Film Sheared Between Solid Surfaces: The Influence of the Crystal

Plane on Energy and Momentum Transfer at Solid-Liquid Interfaces”. The Journal of

Chemical Physics, 122:214717–1–214717–9, 2005.

[36] K. Kremer and G.S. Grest. ”Dynamics of Entangled Linear Polymer Melts - A

Molecular-Dynamics Simulation”. Journal of Chemical Physics, 92(8):5057–5086,

1990.

63

[37] M. Shibahara, M. Katsuki, T. Kunugi, and K. Muko. ”Effects of Nanometer Scale

Structure on Energy Transfer Between a Surface and Fluid Molecules”. Nanoscale and

Microscale Thermophysical Engineering, 10:197–204, 2006.

[38] C.S. Wang, J.S. Chen, J. Shiomi, and S. Maruyama. A Study on the Thermal Resis-

tance Over Solid-Liquid-Vapor Interfaces in a Finite-Space by a Molecular Dynamics

Method. International Journal of Thermal Sciences, 46:1203–1210, 2007.

[39] B.H. Kim, A. Beskok, and T. Cagin. Thermal Interactions in Nanoscale Fluid Flow:

Molecular Dynamics Simulations With Solid-Liquid Interfaces. Microfluidics and

Nanofluidics, 5(4):551–559, 2008.

[40] B.H. Kim, A. Beskok, and T. Cagin. Molecular Dynamics Simulations of Thermal Re-

sistance At the Liquid-Solid Interface. The Journal of Chemical Physics, 129:174701–

1–174701–9, 2008.

[41] M. Shibahara and K. Takeuchi. ”A Molecular Dynamics Study on the Effect of Nanos-

tructural Clearances on the Thermal Resistance at a Liquid Water-Solid Interface”.

Nanoscale and Microscale Thermophysical Engineering, 12:311–319, 2008.

[42] H.J.C. Berendsen, J.R. Grigera, and T.P. Straatsma. ”The Missing Term in Effective

Pair Potential”. Journal of Physical Chemistry, 91:6269–6271, 1987.

[43] S. Murad and I.K. Puri. ”Thermal Transport Across Nanoscale Solid-Fluid Interfaces”.

Applied Physics Letters, 92:133105–1–133105–3, 2008.

[44] G. Balasubramanian, S. Banerjee, and I.K. Puri. ”Unsteady Nanoscale Thermal

Transport Across A Solid-Fluid Interface”. Journal of Applied Physics, 104:064306–1–

064306–4, 2008.

[45] M.I. Mendelev, S. Han, D.J. Srolovitz, G.J. Ackland, D.Y. Sun, and M. Asta. ”Devel-

opment of New Interatomic Potentials Appropriate for Crystalline and Liquid Iron”.

64

Philosophical Magazine, 83(35):3977–3994, 2003.

[46] S. Murad and I.K. Puri. ”Molecular Simulation of Thermal Transport Across Hy-

drophilic Interfaces”. Chemical Physics Letters, 467:110–113, 2008.

[47] C.F. Carlborg, J. Shiomi, and S. Maruyama. ”Thermal Boundary Resistance Between

Single-Walled Carbon Nanotubes and Surrounding Matrices”. Physical Review B,

78:205406–1–205406–8, 2008.

[48] Z. Li. ”Surface Effects on Friction-Induced Fluid Heating in Nanochannel Flows”.

Physical Review E, 79:026312–1–026312–6, 2009.

[49] S. Murad and I.K. Puri. ”Thermal Transport Through a Fluid-Solid Interface”. Chem-

ical Physics Letters, 476:267–270, 2009.

[50] P.A.E Schoen, B. Michel, A. Curioni, and D. Poulikakos. ”Hydrogen-Bond Enhanced

Thermal Energy Transport At Functionalized, Hydrophobic and Hydrophilic Silica-

Water Interfaces”. Chemical Physics Letters, 476:271–276, 2009.

[51] M. Hu, J.V. Goicochea, B. Michel, and D. Poulikakos. ”Thermal Rectification At a Wa-

ter/Functionalized Silica Interfaces”. Applied Physics Letters, 95:151903–1–151903–3,

2009.

[52] D. Konatham and A. Striolo. ”Thermal Boundary Resistance at Graphene-Oil Inter-

faces”. Applied Physics Letters, 95:163105–1–163105–3, 2009.

[53] N. Shenogina, R. Godawat, P. Keblinski, and S. Garde. ”How Wetting and Adhe-

sion Affect Thermal Conductance of a Range of Hydrophobic to Hydrophilic Aqueous

Interfaces”. Physical Review Letters, 102:156101–1–156101–4, 2009.

[54] Z.B. Ge, D.G. Cahill, and P.V. Braun. ”Thermal Conductance of Hydrophilic and

Hydrophobic Interfaces”. Physical Review Letters, 96:186101–1–186101–4, 2006.

65

[55] M. Hu, J.V. Goicochea, B. Michel, and D. Poulikakos. ”Water Nanoconfinement

Induced Thermal Enhancement at Hydrophobic Quartz Interfaces”. Nano Letters,

10:279–285, 2010.

[56] Q.X. Liu, P.X. Jiang, and H. Xiang. ”Molecular Dynamics Simulation of Thermal

Conductivity of an Argon Liquid Layer Confined in Nanospace”. Molecular Simulation,

36(13):1080–1085, 2010.

[57] C. Liu, H.B. Fan, K. Zhang, M.M.F. Yuen, and Z. Li. ”Flow Dependence of Interfacial

Thermal Resistance in Nanochannels”. The Journal of Chemical Physics, 132:094703–

1–094703–5, 2010.

[58] Z. Liang and H.-L. Tsai. ”Effect of molecular film thickness on thermal conduction

across solid-film interfaces”. Physical Review E, 83:0616031–0616037, 2011.

[59] J.V. Goicochea, M. Hu, B. Michel, and D. Poulikakos. ”Surface functionalization

mechanisms of enhancing heat transfer at solid-liquid interfaces”. Journal of Heat

Transfer, 133:0824011–0824016, 2011.

[60] H. Acharya, N.J. Mozdzierz, P. Keblinski, and S. Garde. ”How chemistry, nanoscale

roughness, and the direction of heat flow affect thermal conductance of solid-water

interfaces”. Industrial & Engineering Chemistry Research, 51:1767–1773, 2012.

[61] M. Shen, W.J. Evans, D. Cahill, and P. Keblinski. ”Bonding and pressure-tunable

interfacial thermal conductance”. Physical Review B, 84:1954321–1954326, 2011.

[62] D. Poulikakos, S. Arcidiacono, and S. Maruyama. ”Molecular Dynamics Simulation in

Nanoscale Heat Transfer: A Review”. Microscale Thermophysical Engineering, 7:181–

206, 2003.

[63] M.P. Allen and D.J. Tildesley. Computer Simulation of Liqiuds. Clarendon Press,

Oxford, 1987.

66

[64] L. Verlet. ”Computer Experiments on Classical Fluids .I. Thermodynamical Properties

of Lennard Jones Molecules”. Physical Review, 159(1):98–103, 1967.

[65] W.C. Swope, H.C. Andersen, P.H. Berens, and K.R. Wilson. ”A Computer Simulation

Method for the Calculation of Equilibrium Constants for the Formation of Physical

Clusters of Molecules: Application to Small Water Clusters”. Journal of Chemical

Physics, 76(1):637–649, 1982.

[66] J.M. Haile. Molecular Dynamics Simulation - Elementary Methods. John Wiley &

Sons, Inc., address =.

[67] H.J.C. Berendsen, J.P.M. Postma, W.F. van Gunsteren, A. DiNola, and J.R. Haak.

”Molecular Dynamics with Coupling to an External Bath”. Journal of Chemical

Physics, 81(8):3684–3690, 1984.

[68] P.H. Hunenberger. ”Thermostat algorithms for molecular dynamics simulations”. Ad-

vances in Polymer Science, 173:105–149, 2005.

[69] P.K. Schelling, S.R. Phillpot, and P. Keblinski. ”Comparison of Atomic-Level Simula-

tion Methods for Computing Thermal Conductivity”. Physical Review B, 65:144306–

1–144306–12, 2002.

[70] J. Petravic and P. Harrowell. ”Equilibrium Calculations of Viscosity and Thermal Con-

ductivity Across a Solid-Liquid Interface Using Boundary Fluctuations”. The Journal

of Chemical Physics, 128:194710–1–194710–12, 2008.

[71] P. Jund and R. Jullien. ”Molecular dynamics calculation of the thermal conductivity

of vitreous silica”. Physical Review B, 59(2):13707–13711, 1999-I.

[72] J.H. Walther, R. Jaffe, T. Halicioglu, and P. Koumoutsakos. ”Carbon Nanotubes in

Water: Structural Characteristics and Energetics”. Journal of Physical Chemistry B,

105:9980–9987, 2001.

67

[73] S. Plimpton. ”Fast parallel algorithms for short-ranged molecular dynamics”. Journal

of Computational Physics, 117(1):1–19, 1995.

[74] W. Humphrey, A. Dalke, and K. Schulten. ”VMD: Visual molecular dynamics”. Jour-

nal of Molecular Graphics, 14(1):33–38, 1996.

[75] X. Feng and L. Jiang. ”Design and creation of superwetting/antiwetting surfaces”.

Advanced Materials, 18(23):3063–3078, 2006.

[76] R. Furstner, W. Barthlott, C. Neinhuis, and P. Walzel. ”Wetting and self-cleaning

properties of artificial superhydrophobic surfaces”. Langmuir, 21(3):956–961, 2005.

[77] Z. Yoshimitsu, A. Nakajima, T. Watanabe, and K. Hashimoto. ”Effects of surface

structure on the hydrophobicity and sliding behavior of water droplets”. Langmuir,

18(15):5818–5822, 2002.

[78] C.W. Extrand, S.I. Moon, P. Hall, and D. Schmidt. ”Superwetting of structured

surfaces”. Langmuir, 23(17):8882–8890, 2007.

[79] X. Yong and L.T. Zhang. ”Nanoscale wetting on groove-patterned surfaces”. Lang-

muir, 25(9):5045–5053, 2009.

[80] S.C. Yang. ”Effects of surface roughness and interface wettability on nanoscale flow in

a nanochannel”. Microfluid Nanofluid, 2:501–511, 2006.

[81] I.-W. Lyo and Phaedon Avouris. ”Field-induced nanometer- to atomic-scale manipu-

lation of silicon surfaces with STM”. Science, 253(5016):173–176, 1991.

[82] Y. Wang and P. Keblinski. ”Role of wetting and nanoscale roughness on thermal

conductance”. Applied Physics Letters, 99(7):0731121–0731123, 2011.

[83] L. Girifalco and V.G. Weizer. ”Application of the Morse potential function to cubic

metals”. Physcial Review, 114(3):687–690, 1959.

68

[84] A. Kara and T.S. Rahman. ”Vibrational properties of metallic nanocrystals”. Physical

Review Letters, 81(7):1453–1456, 1998.

[85] P.M. Derlet, R. Meyer, L.J. Lewis, U. Stuhr, and H. Van Swygenhoven. ”Low-

frequency vibrational properties of nanocrystalline materials”. Physical Review Letters,

87(20):2055011–2055014, 2001.

[86] N.I. Papanicolaou, I.E. Lagaris, and G.A. Evangelakis. ”Modification of phonon spec-

tral densities of the (001) copper surface due to copper adatoms by molecular dynam-

ics”. Surface Science, 337(1-2):L819–L824, 1995.

[87] E. Vamvakopoulos and G.A. Evangelakis. ”Dynamical behaviour and size dependence

of 2D copper islands on the Cu(111) surface: a molecular dynamics study”. Journal

of Physics: Condensed Matter, 13:10757–10766, 2001.

[88] S.-T. Lin, M. Blanco, and W.A. Goddard III. ”The two-phase model for calculating

thermodynamic properties of liquids from molecular dynamics: Validation for the phase

diagram of Lennard-Jones fluids”. Journal of Chemical Physics, 119(22):11792–11805,

2003.

[89] F.D. Sofos, T.E. Karakasidis, and A. Liakopoulos. ”Effect of wall roughness on flow

in nanochannels”. Physical Revew E, 79:0263051–0263057, 2009.

[90] W.-P. Hsieh, A.S. Lyons, E. Pop, P. Keblinski, and D.G. Cahill. ”Pressure tuning of

the thermal conductance of weak interfaces”. Physical Review B, 84:1841071–1841075,

2011.

[91] D.A. Young and H.J. Marris. ”Lattice-dynamical calculation of the Kapitza resistance

between fcc lattices”. Physical Review B, 40(6):3685–3693, 1989-II.

[92] K.M. Issa and A.A. Mohamad. ”Lowering liquid-solid interfacial thermal resistance

with nanopatterned surfaces”. Physical Review E, 85:0316021–0316025, 2012.

69

[93] J.A. Thomas and A.J.H. McGaughey. ”Effect of surface wettability on liquid den-

sity, structure, and diffusion near a solid surface”. The Journal of Chemical Physics,

126:0347071–0347079, 2007.

[94] J.H. Cha, S. Chiashi, J. Shiomi, and S. Maruyama. ”Generalized model of thermal

boundary conductance between SWNT and surrounding supercritical Lennard-Jones

fluid - derivation from molecular dynamics simulations”. International Journal of Heat

and Mass Transfer, 55:2008–2013, 2012.

[95] S. Iijima. ”Helical Microtubules of Graphitic Carbon”. Nature, 354(6348):56–58, 1991.

[96] S.V. Rotkin and S. Subramoney. Applied Physics of Carbon Nanotubes. Springer,

USA, 2005.

[97] R. Saito, G. Dresselhaus, and M.S. Dresselhaus. Physical Properties of Carbon Nan-

otubes. Imperial College Press, London, 1998.

[98] M.J. O’Connell. Carbon Nanotubes Properties and Applications. CRC Press, USA,

2006.

[99] A. Jorio, G. Dresselhaus, and M.S. Dresselhaus. Carbon Nanotubes: Advanced Topics

in the Synthesis, Structure, Properties and Applications. Springer-Verlag, Berlin, 2008.

[100] M.J. Biercuk, M.C. Llaguno, M. Radosavljevic, J.K. Hyun, A.T. Johnson, and J.E.

Fischer. ”Carbon Nanotube Composites for Thermal Management”. Applied Physics

Letters, 80(15):2767–2769, 2002.

[101] S.U.S. Choi, Z.G. Zhang, W. Yu, F.E. Lockwood, and E.A. Grulke. ”Anomalous Ther-

mal Conductivity Enhancement in Nanotube Suspensions”. Applied Physics Letters,

79(14):2252–2254, 2001.

70

[102] M.J. Assael, C.F. Chen, I. Metaxa, and W.A. Wakeham. ”Thermal Conductivity of

Suspensions of Carbon Nanotubes in Water”. International Journal of Thermophysics,

54(14):971–985, 2004.

[103] S. Berber, Y.K. Kwon, and D. Tomanek. ”Unusually High Thermal Conductivity of

Carbon Nanotubes”. Physical Review Letters, 84(20):4613–4616, 2000.

[104] J. Hone, M. Whitney, C. Piskoti, and A. Zettl. ”Thermal conductivity of single-walled

carbon nanotubes”. Physical Revew B, 59(4):R2514–R2516, 1999-II.

[105] S. Shenogin, P. Keblinski, D. Bedrov, and G. Smith. ”Thermal relaxation mechanism

and role of chemical functionalization in fullerene solutions”. The Journal of Chemical

Physics, 124(1):0147021–0147027, 2006.

[106] D.W. Brenner, O.A. Shenderova, J.A. Harrison, S.J. Stuart, B. Ni, and S.B. Sinnott.

”A Second-Generation Reactive Empirical Bond Order (REBO) Potential Energy Ex-

pression For Hydrocarbons”. Journal of Physics - Condensed Matter, 14(4):783–802,

2002.

[107] M.S. Dresselhaus and P.C. Eklund. ”Phonons in carbon nanotubes”. Advances in

Physics, 49(6):705–814, 2000.

[108] L. Hu, T. Desai, and P. Keblinski. ”Determination of interfacial thermal resistance at

the nanoscale”. Physical Review B, 83:1954231–1954235, 2011.

[109] B.A. Younglove. ”Thermophysical properties of fluids. I. Argon, ethylene, parahydro-

gen, nitrogen, nitrogen trifluoride, and oxygen”. Journal of Physical and Chemical

Reference Data, 11(1):1–353, 1982.

[110] J.R. Lukes and H. Zhong. ”Thermal conductivity of individual single-wall carbon

nanotubes”. Journal of Heat Transfer, 129:705–716, 2007.

71

[111] M.A. Osman and D. Srivastava. ”Molecular Dynamics Simulation of Heat Pulse Propa-

gation in Single-Wall Carbon Nanotubes”. Physical Review B, 72:125413–1–125413–7,

2005.

[112] S. Maruyama. ”A molecular dynamics simulation of heat conduction of a finite length

single-walled carbon nanotube”. Microscale Thermophysical Engineering, 7(1):41–50,

2003.

[113] J. Fornleitner, G. Kahl, and C.N. Likos. ”Tailoring the phonon band structure in

binary colloidal mixtures”. Physical Review E, 81:0604011–0604014, 2010.

[114] C. Sevik, H. Sevincli, G. Cuniberti, and T. Cagin. ”Phonon engineering in carbon

nanotubes by controlling defect concentration”. Nano Letters, 11:4971–4977, 2011.

[115] I. Savic, N. Mingo, and D.A. Stewart. ”Phonon transport in isotope-disordered carbon

and boron-nitride nanotubes: Is localization observable?”. Physical Review Letters,

101:1655021–1655024, 2008.

[116] Y. He, D. Donaido, J-H Lee, J. C. Grossman, and G. Galli. ”Thermal transport in

nanoporous silicon: interplay between disorder at mesoscopic and atomic scales”. ACS

Nano, 5(3):1839–1844, 2011.

[117] J. Fang and L. Pilon. ”Scaling laes for thermal conductivity of crystalline nanoporous

silicon based on molecular dynamics simulations”. Journal of Applied Physics,

110:0643051–06430510, 2011.

[118] C. Kittel. Introduction to Solid State Physics. John Wiley & Sons, Inc., Canada, 1996.

[119] D.J. Evans and G.P. Morriss. Statistical Mechanics of Nonequilibrium Liquids. ANU

E Press, Australia, 2007.

[120] J.N. Israelachvili. Intermolecular and Surface Forces. Academic Press, England, 1992.

72

[121] A.R. Leach. Molecular Modelling: Principles and Application. Prentice Hall, England,

2001.

[122] P. Yi, D. Poulikakos, J. Walther, and G. Yadigaroglu. ”Molecular Dynamics Simulation

of Vaporization of an Ultra-Thin Liquid Argon Layer on a Surface”. International

Journal of Heat and Mass Transfer, 45:2087–2100, 2002.

[123] A. Cheng and K.M Merz Jr. ”The pressure and pressure tensor for macromolecular

systems”. Journal of Physical Chemistry, 100:905–908, 1996.

[124] M.T. Dove. Introduction to Lattice Dynamics. Cambridge University Press, USA,

1993.

[125] J. Shiomi and S. Maruyama. ”Non-Fourier heat conduction in a single-walled carbon

nanotube: Classical molecular dynamics simulations”. Physical Revew B, 73:2054201–

2054207, 2006.

[126] J.M. Dickey and A. Paskin. ”Computer simulation of the Lattice Dynamics of Solids”.

Physical Review, 188(3):1407–1418, 1969.

73

Appendix A

Velocity-Verlet algorithm

It is convenient when deriving the expressions for the velocity-Verlet algorithm [65] to start

with the original Verlet algorithm [64]. First let the position, velocity, and acceleration

vectors of an atom at time step n be given by: rn, vn, and an, respectively. A Taylor

expansion for rn−1, and rn+1 about rn is given by:

rn−1 = rn − δtvn + (1/2) δt2an − .... (A.1)

rn+1 = rn + δtvn + (1/2) δt2an + .... (A.2)

where δt is the time step as shown in Fig. A.1. By adding Eqs. A.1- A.2, and neglecting

higher order terms, an expression for rn+1 can be obtained:

rn+1 = 2rn − rn−1 + δt2an (A.3)

What can be immediately noted about Eq. A.3 is that there are no velocity terms, given

that they canceled out in the above addition. The velocities can still be calculated, however,

this always comes following the position updates:

vn =1

2δt(rn+1 − rn−1) (A.4)

The premise behind the velocity-verlet algorithm is to reintroduce the velocity into Eq. A.3.

One way to accomplish that is by adding rn+1 to both sides of Eq. A.3, then dividing by 2

to obtain:

rn+1 = rn + (1/2) (rn+1 − rn−1) +(δt2/2

)an (A.5)

The second term on the r.h.s. of Eq. A.5 is nothing but: δtvn, given that: vn = (1/2δt)(rn+1−

rn−1). Thus, the position is now expressed in terms of the velocity, which forms the first step

74

Figure A.1: Time series of atomic trajectories.

in the velocity-Verlet algorithm in which the atomic positions at n+1 are updated using the

velocities and accelerations from the previous timestep:

rn+1 = rn + δtvn +(δt2/2

)an (A.6)

A velocity update to n + 1 is now required. An expression for vn+1 can be derived by

substituting rn+2 = rn+1 + δtvn+1 + (δt2/2)an+1 into vn+1 = (1/2δt)(rn+2− rn) which gives:

vn+1 = (1/2δt)[rn+1 + δtvn+1 + (δt2/2)an+1 − rn

]vn+1 = (1/2δt)(rn+1 − rn) + (1/2)vn+1 + (δt/4)an+1

(1/2)vn+1 = (1/2)vn+ 12

+ (δt/4)an+1

vn+1 = vn+ 12

+ (δt/2)an+1 (A.7)

The values for an+1 are obtained by evaluating the forces following the positions update

from Eq. A.6. However, Eq. A.7 also requires the half step velocity vn+ 12. This can be

achieved by a simple rearrangement of Eq. A.6 to give: (1/δt)(rn+1 − rn) = vn + (δt/2)an,

and therefore:

vn+ 12

= vn + (δt/2)an (A.8)

Which completes the requirements to advance the trajectories by δt in the order of Eq.

A.6→Eq. A.8→Eq. A.7.

75

Appendix B

MD potentials

B.1 Lennard-Jones (LJ) potential

The LJ interatomic potential is one of the most employed potentials in MD studies of LSI

heat transfer. It is an additive pair potential, that depends on the sepration distance between

two atoms i, and j (rij) and is of the form:

uLJ(rij) = 4ε

[(σ

rij

)12

−(σ

rij

)6]

(B.1)

The LJ potential is very effective in modeling inert gas solids (crystals) [118], and liquids

[119], in addition to the interaction between an adsorbed liquid and a solid surface [120].

The parameters controlling the LJ potential are: the collision diameter, σ, and the potential

well depth, ε. These parameters are determined from empirical data. For example, liquid

argon has σ ≈ 3.40A, and ε ≈ 0.998kJ/mol. The LJ force is derived from the relation:

FLJ(rij) = −∇rijuLJ(rij) , for which in any one dimension:

∂uLJ∂rxij

=∂uLJ∂rij

∂rij∂rxij

=∂uLJ∂rij

rxijrij

(B.2)

and:

∂uLJ∂rij

=24ε

rij

[2

(σ

rij

)12

−(σ

rij

)6]

(B.3)

therefore, the three-dimensional expression for the force is:

FLJ(rij) =24ε

r2ij

[2

(σ

rij

)12

−(σ

rij

)6]rij (B.4)

76

A plot of the LJ potential, and force, is given in Fig. B.1. The LJ potential and force

are zero when the separation between two atoms is at σ, and 1.12σ, respectively. Also at

rij = 1.12σ, the potential reaches its minimum value of −ε. Beyond this separation, the force

is attractive, which is responsible for cohesion in condensed phases. At separations below σ,

the potential and force become highly repulsive due to non-bonded overlap between electron

clouds.

Figure B.1: LJ potential and force magnitude in reduced units.

In cases where a mixture of different species (A and B) is simulated, the interaction between

them can be approximated using the Lorentz-Berthelot mixing rule [63]:

σAB =1

2[σAA + σBB] (B.5)

εAB =√εAAεBB (B.6)

77

B.2 Morse bonds

A plot of the Morse potential, and resulting force, is given in Fig. B.2. The Morse potential

has the following form:

uM(rij) = D[1− e−α(rij−ro)

]2(B.7)

Through a similar procedure to that in Sec. B.1, the resulting force between two bonded

atoms is expressed as:

FM(rij) =2Dαe−α(rij−ro)

rij

[1− e−α(rij−ro)

]rij (B.8)

where D is the well depth, or the bond dissociation energy, α is a parameter that controls the

width of the well, and ro is the equilibrium bond length. The form of the Morse potential

closely resembles that encountered in bond dynamics [121], for which the compression of

the bond results in a rapid increase in energy, as opposed to stretching with a slower hike

in energy, up to the dissociation value D. Previous MD studies on LSI heat transfer have

utilized a harmonic potential [28,122] to represent the solid, which is a good approximation

for small deviations from ro, as can be seen in Fig. B.2.

Figure B.2: Morse potential and force magnitude in reduced units. Harmonic potentialshown for comparison

78

B.3 Reactive Empirical Bond Order potential (REBO) potential

REBO is a many-body potential that is capable of describing hydrocarbon molecules, dia-

mond lattices, and graphite structures. The REBO potential predicts the phonon density of

states in carbon nanotubes to good accuracy, and for a pristine CNT can be expressed as:

uREBO =1

2

∑i

∑j

[uR(rij)− bijuA(rij)

](B.9)

where the repulsive (uR), and attractive (uA) components of the potential are given by:

uR(rij) =

(1 +

Q

rij

)Ae−αrij (B.10)

uA(rij) =3∑

n=1

Bne−βnrij (B.11)

The parameters Q, A, α, Bn(n = 1, 3), and βn(n = 1, 3), are fitting parameters determined

to reproduce carbon bonding characteristics in CNTs [106].The bond order variable in Eq.

B.9 is given by:

bij =1

2

[bσ−πij + bσ−πji

]+ bDHij (B.12)

bσ−πij = [1 +G(cos θijk1) +G(cos θijk2)]− 1

2 (B.13)

bσ−πji = [1 +G(cos θjil1) +G(cos θjil2)]− 1

2 (B.14)

bDHij = Tij∑k 6=i,j

∑l 6=i,j

(1− cos2 Φijkl

)(B.15)

where the terms bσ−πij and bσ−πji represent the contributions from bond angles, as shown in

Fig. B.3. The term bDHij accounts for the dihedral angle torsion. The functions G(cos θ),

and Tij are sixth order, and tricubic spline functions, respectively. The indices, and angles

appearing in Eqs. B.11-B.15, excluding the dihedral angle (Φijkl), are also shown in Fig.

B.3.

Expressions for the forces arising from the REBO potential can be found by the appro-

priate differentiation of Eq. B.9 with respect to rij. Given the additional dependencies on

79

Figure B.3: CNT atom and angle labeling for REBO potential

the bending angles, and the dihedrals, chain rules must be used of the form:

Fi = −∂uREBO

∂φ∇ri cosφ (B.16)

where φ is the angle under consideration. Force expressions arising from Eq. B.16 can be

quite tedious. An excellent presentation of these expressions can be found in [123], with

universal applicability.

80

Appendix C

Phonon density of states - Lattice dynamics approach

Lattice dynamics can help provide a better view of how the vibrational (phonon) density of

states (VDOS) of a material relates directly to the atomic vibrations. For simplicity, the

analysis is limited to a monatomic linear chain of N atoms, with an equilibrium separation

distance that is equal to the unit cell length a, as shown in Fig. C.1. The atoms interact via

the potential u, which depends only on the separation distance between the atoms. Periodic

boundary conditions are assumed for which atoms 1 and N interact. At the equilibrium

separation distance, where all the displacements qn = 0 (n = 1, 2, 3, ...N), the total energy of

the chain is given by: E = Nu(a). If the atoms are displaced by a small quantity, such that

qn � a, then the distance between atoms n and n+ 1 can be expressed as: a+ (qn − qn+1).

Using a Taylor series expansion, the contribution to the energy of the chain by atom n is

given by:

En = u(a) +∂u(a)

∂q[a+ (qn − qn+1)− a] +

1

2!

∂2u(a)

∂2q[a+ (qn − qn+1)− a]2 +

1

3!

∂3u(a)

∂3q[a+ (qn − qn+1)− a]3 + .... (C.1)

with a canceling out in Eq. C.1, and by summing over all the atoms in the chain, the total

energy can be written as:

E = Nu(a) +∞∑s=1

1

s!

∂su(a)

∂sq

N∑n=1

(qn − qn+1)s (C.2)

The atoms do not experience any forces at the equilibrium separation, and therefore: ∂u(a)/∂q =

0. Based on the assumption that qi � a, Eq. C.2 can be further simplified by ignoring the

terms of order s ≥ 3 in what is known as the harmonic approximation. The total energy is

81

now approximated by:

E = Nu(a) +1

2J

N∑n=1

(qn − qn+1)2; J =

∂2u(a)

∂2q(C.3)

Figure C.1: Monatomic linear chain of atoms.

Newton’s equation of motion for an atom n becomes:

m∂2qn∂2t

= − ∂E∂qn

(C.4)

where m is the mass. Note that the term u(a) in Eq. C.3 disappears again as a result of

the ∂/∂qn term in Eq. C.4. The only terms in the total energy expression of Eq. C.3 that

involve qn are: E(qn) = (qn−1 − qn)2 + (qn − qn+1)2. Hence, Eq. C.4 can be expressed as:

m∂2qn∂2t

= −J(2un − un+1 − un−1) (C.5)

The solution to the harmonic equation Eq. C.5 is a sinusoidal wave. The motion of the atoms

in the chain as a whole, is equivalent to a set of traveling waves. Therefore, the motion of

an atom n is expressed as the superposition of each of the traveling waves:

qn(t) =∑κ

qκei(κx−ωκt) (C.6)

where qκ, and ωκ are the wave amplitude and angular frequency, respectively, both of which

correspond to a wave vector κ. The wave vector is given by κ = 2π/λ, with λ as the

wavelength. Given that the chain is discrete, x can only take on values that are a multiple

of the unit cell length, i.e. xn = na. The relationship between ωκ, and κ can be inferred

by substituting Eq. C.6 into Eq. C.5. Considering each wave vector separately, and noting

82

that: ∂2qn/∂t2 = −ω2

κqκei(κna−ωκt), Eq. C.5 becomes:

−mω2κqκe

i(κna−ωκt) = −Jqκei(κna−ωκt)[2− eiκa − e−iκa

]= −2qκJe

i(κna−ωκt)[1− eiκa − e−iκa

2

](C.7)

Using the identity: cos(x) = 12(eix + e−ix):

mω2κ = 2J [1− cos(κa)]

= 4J[sin2(κa

2)]

(C.8)

where use was made of the identity: cos(2x) = 1− 2 sin2(x). Solving for ωκ, and taking only

the positive root for the frequency:

ωκ =

[4J

m

] 12

| sin(κa2

)| (C.9)

Eq. C.9 is known as the dispersion relation, and a plot of the dispersion curve is given in Fig.

C.2. With only the positive values considered for ωκ in Eq. C.9, the dispersion curve has a

periodicity of π, and hence, all the useful information can be obtained from the highlighted

region in Fig. C.2, which extends over: −πa> κ ≤ π

a, and is known as the first Brillouin

zone. In a discrete system such as the linear atomic chain, κ can only take on a discrete set

of values. These values can be inferred from the periodic boundary condition imposed on

the chain for which the atom at x = Na, is equivalent to that at x = 0, therefore:

eiκNa = e0 = 1

cos(κNa) + i sin(κNa) = 1

κNa = 2πs→ κ = 2πs/Na (C.10)

where s = 1, 2, 3, ....N , given that κ has a range of 2π/a. The minimum and maximum

allowable values for the wavelength are λmin = 2a, and λmax = 2L. This in turn results in

κmin = π/L, and κmax = π/a = Nπ/L. Hence, there is one mode in every dκ/dN = π/L,

where L = Na, is the length of the monatomic chain. The vibrational density of states

83

Figure C.2: Dispersion curve for monatomic chain (N = 100)

VDOS(ω), is defined as the number of modes per unit frequency (dN/dω) per unit volume

in real space (L):

VDOS(ω) =1

L

dN

dω=

1

L

dN

dκ

dκ

dω=

1

π

1

dω/dκ(C.11)

where the denominator in the last term is known as the group velocity, The group velocity

(dω/dκ) is the transmission velocity of a wave packet (with mean frequency ω), and directly

relates to the propagation velocity of energy flow in a medium [118]. Using Eq. C.9, the

group velocity is found to be:

dω/dκ = a

[J

m

] 12

| cos(κa2

)| (C.12)

As can be seen from Eq. C.12, a value of κ = ±π/a results in a ’standing wave’ with a zero

group velocity. This also causes a singularity in the expression for the density of states. A

plot of VDOS(ω) for the same monatomic chain is shown in Fig. C.3. There is a peak around

a frequency value of 1.0 which indicates a larger number of modes around this frequency.

This can be verified by visual inspection of Fig. C.2, where the number of points per dω

increases around the peaks at frequency values in the vicinity of 1.0.

The above derivation provides a straightforward link between atomic vibrations and their

corresponding density of states. For more realistic three dimensional systems, this process

becomes exponentially more complicated when using lattice dynamics [124]. Apart from the

84

Figure C.3: Density of states of monatomic chain (N = 100). The values for VDOS(ω) arenormalized.

complexity issues, one of the key assumptions made in the above derivation is the harmonic

approximation in Eq. C.3, for which higher order terms were ignored. This might be accurate

at low temperatures where the atomic vibrations are limited, however, as the temperature

rises, the deviation in the atomic vibrations can no longer be ignored, and anharmonic effects

must be incorporated. With the current available computational resources, MD lends itself

to such problems as the anharmonic effects are naturally captured. In this reported work,

the focus is on the phonon density of states which are computed from the Fourier transform

of the velocity atuocorrelation function (Appendix D). The dispersion relations can also be

obtained computationally using MD via the the spectral energy density. This requires a

two-dimensional Fourier transform [125] in both: time, and space.

85

Appendix D

Velocity-autocorrelation function

The velocity-autocorrelation function (VACF) provides a measure of how the atomic veloci-

ties in a system evolve, relative to an initial starting point. The calculation of the normalized

VACF is given by:

VACF(∆t) =N∑i=1

〈vi(to) · vi(to + ∆t)〉〈vi(to) · vi(to)〉

(D.1)

where ∆t is the correlation period, following the starting time to. The above equation

is averaged over N atoms to improve the accuracy of the measure. An example of the

calculated VACF for the adsorbed liquid, and solid surface, atoms is shown in Fig. D.1.

This corresponds to case S1 from Chapter 4. An important feature of the normalized VACF

is the start value of 1.0. If there was no interaction between the atoms in the system, then

the initial atomic velocities will remain unchanged, which will result in a normalized VACF

of 1.0 for all the correlation periods. In the case of weak interactions, as in a gas, the atomic

velocities decorrelate at a slow exponential rate. As the density increases, atoms begin to

seek energetically stable positions where the repulsive, and attractive forces tend to cancel

on average. This phenomenon has two different outcomes in liquids and solids, as reflected

in the VACF in Fig. D.1. For a solid, the atoms are tethered to their equilibrium lattice

positions, and therefore, vibrate about these locations. In this process, the solid atoms

reverse their velocities at the end of each oscillation, which results in the initial dip in their

VACF to ≈ −1.0. This back and forth motion persists producing the oscillations seen in the

solid atoms VACF. Over time, however, these oscillations diminish, due to the random forces

from surrounding atoms, preventing an ideal oscillatory motion. A similar process exists in

86

Figure D.1: Normalized VACF for adsorbed liquid, and solid, atoms.

liquids, except for the fact that the adsorbed liquid experiences self-diffusion. Hence, the

dip in the liquid atoms VACF is not as pronounced, and is not sustainable, given that the

initial velocities are ’forgotten’ at a relatively much shorter time scale. By taking the Fourier

transform of the VACF, the various frequencies present in the VACF oscillations are brought

to light in the form of the VDOS. A proof that the Fourier transform of the VACF produces

the phonon density of states is given in [126] for a harmonic system.

87

Appendix E

Modeling Logistics

E.1 LAMMPS input script for Nanopatterning study (Chapter 4)

dimension 3

boundary p p p

un i t s r e a l

a tom sty l e molecu lar

ne ighbor 0 .3 bin

ne igh modi fy de lay 1

read data data . nanopatt

mass 1 63 .546

mass 2 39 .95

# Create groups by type :

group s o l type 1

group l i q type 2

# Def ine heat ing / coo l i n g r e g i on s

r eg i on qin block INF INF INF INF −1.0 1 .0 un i t s box

group qin r eg i on qin

r eg i on qout block INF INF INF INF −107.774822 107.774822 &

s i d e out un i t s box

# In t e r a c t i o n type :

p a i r s t y l e l j / cut 8 .5

pa i r mod i fy s h i f t yes

88

p a i r c o e f f 1 1 0 .0 0 .0

p a i r c o e f f 2 2 0.2385277 3 .40

p a i r c o e f f 1 2 0.2385277 2 .8385

bond s ty l e morse

bond coe f f 1 7 .663017 1 .369 2 .89

v e l o c i t y l i q c r e a t e 140 451216 un i t s box

v e l o c i t y s o l c r e a t e 140 453516 un i t s box

t imestep 1 .0

va r i ab l e etk equal 503.23082 #Temperature conver s i on f a c t o r

compute temper a l l ke/atom

va r i ab l e temps1 atom c temper /1 .5

va r i ab l e temps1k atom v temps1∗${ etk }

# Calcu la te l i q u i d p r e s su r e

compute s t en s l i q s t r e s s /atom

compute p1 l i q reduce sum c s t e n s [ 1 ]

compute p2 l i q reduce sum c s t e n s [ 2 ]

compute p3 l i q reduce sum c s t e n s [ 3 ]

v a r i ab l e v o l l i q equal 629188.8

va r i ab l e p r e s l equal −(c p1+c p2+c p3 )/(3∗ v v o l l i q )

v a r i ab l e p r e s l z equal −(c p3 )/ ( v v o l l i q )

# Compute tamperature p r o f i l e s in s o l i d and l i q u i d s epa r a t e l y

f i x tps s o l ave/ s p a t i a l 1 250000 250000 z 0 .0 2 .043539 &

v temps1k f i l e tmps . p r o f i l e un i t s box

89

f i x t p l l i q ave/ s p a t i a l 1 250000 250000 z 0 .0 2 .043539 &

v temps1k f i l e tmpl . p r o f i l e un i t s box

# Animation f i l e

dump anim a l l atom 5000 anim . atom

thermo sty l e custom step ke pe e t o t a l temp pre s s v p r e s l v p r e s l z

thermo modify f l u s h yes

thermo 1000

# Equ i l i b r a t i on with Berendsen thermostat

f i x 1 a l l nve

f i x 2 a l l temp/berendsen 140 140 500

f i x un i f q in r e c en t e r NULL NULL 0 .0 s h i f t a l l un i t s box

run 1000000

un f ix 2

f i x hot qin heat 1 0.00254427

f i x co ld l i q heat 1 −0.00254427 reg i on qout

run 10000000

90

E.2 Velocity-autocorrelation function Fortran code

program pdos sk ip

imp l i c i t none

i n t e g e r nwor

parameter ( nwor = 10000 )

i n t e g e r i , i i , trun , t tot , sk ip

i n t e g e r icheck , n1o , n2o , t s

i n t e g e r ido ( 10000 ) , id , check , n2 , check i ( nwor )

i n t e g e r tcur , s tep

r e a l ∗8 vcxo ( nwor ) , vcyo ( nwor ) , vczo ( nwor )

r e a l ∗8 vac fo ( 0 :20000 )

r e a l ∗8 vxw( nwor ) , vyw( nwor ) , vzw( nwor )

c ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗

c I n i t i a l i z e v a r i a b l e s :

vac fo = 0 .0

ido = 0

check = 0

c Open LAMMPS f i l e with atomic v e l o c i t i e s :

open ( 112 , f i l e =’dump . vdossm ’ )

n1o = 1

c Assign c o r r e l a t i o n per iod :

trun = 10000

t t o t = trun ∗ 100

91

sk ip = −100 ! This a l l ows a sk ip between c o r r e l a t i o n s o f l ength trun

do 1000 step = 1 , t t o t

check i = 1 ! Check to i gnore l i q u i d atom escape from adsorbed l ay e r

i f ( mod( step , trun ) . eq . 0 ) p r i n t ∗ , ’ s t ep = ’ , step , ’ out o f : ’ , t t o t

i f ( s tep . eq . 1 . or . mod( step , trun ) . eq . 0 ) then

sk ip = sk ip + 100

tcur = 0

ido = 0

c l o s e ( 112 )

open ( 112 , f i l e =’dump . vdossm ’ )

do 100 i i = 1 , sk ip

read ( 112 , ∗ )

read ( 112 , ∗ )

read ( 112 , ∗ )

read ( 112 , ∗ ) n2o

do 101 i = 1 , 5

read ( 112 , ∗ )

101 cont inue

do 102 i = 1 , n2o

read ( 112 , ∗ )

102 cont inue

100 cont inue

read ( 112 , ∗ )

read ( 112 , ∗ ) t s

read ( 112 , ∗ )

read ( 112 , ∗ ) n2o

do 103 i = 1 , 5

read ( 112 , ∗ )

103 cont inue

do 104 i = n1o , n2o

92

read ( 112 , ∗ ) ido ( i ) , vcxo ( ido ( i ) ) ,

& vcyo ( ido ( i ) ) , vczo ( ido ( i ) )

104 cont inue

open ( 1136 , f i l e =’vacfsm . dat ’ )

do 105 i = 0 , trun

wr i t e ( 1136 , 1004 ) vac fo ( i ) / vac fo ( 0 )

105 cont inue

c l o s e ( 1136 )

end i f

do 106 i = 1 , 3

read ( 112 , ∗ )

106 cont inue

read ( 112 , ∗ ) n2

do 107 i = 1 , 5

read ( 112 , ∗ )

107 cont inue

do 108 i = n1o , n2

read ( 112 , ∗ ) id , vxw( id ) , vyw( id ) , vzw( id )

do 109 i i = n1o , n2o

i f ( check i ( id ) . eq . 1 . and . id . eq . ido ( i i ) ) then

goto 555

end i f

109 cont inue

check i ( id ) = 0

goto 108

555 vac fo ( tcur ) = vac fo ( tcur ) + vxw( id ) ∗ vcxo ( id ) +

& vyw( id ) ∗ vcyo ( id ) +

& vzw( id ) ∗ vczo ( id )

108 cont inue

93

tcur = tcur + 1

1000 cont inue

1004 format (1(2 x , e14 . 8 ) )

stop

end

94