--- interfacial/interfacial.tex 2011/07/05 17:39:21 3730 +++ interfacial/interfacial.tex 2011/07/14 19:49:12 3739 @@ -45,20 +45,22 @@ We have developed a Non-Isotropic Velocity Scaling alg \begin{abstract} -We have developed a Non-Isotropic Velocity Scaling algorithm for -setting up and maintaining stable thermal gradients in non-equilibrium -molecular dynamics simulations. This approach effectively imposes -unphysical thermal flux even between particles of different -identities, conserves linear momentum and kinetic energy, and -minimally perturbs the velocity profile of a system when compared with -previous RNEMD methods. We have used this method to simulate thermal -conductance at metal / organic solvent interfaces both with and -without the presence of thiol-based capping agents. We obtained -values comparable with experimental values, and observed significant -conductance enhancement with the presence of capping agents. Computed -power spectra indicate the acoustic impedance mismatch between metal -and liquid phase is greatly reduced by the capping agents and thus -leads to higher interfacial thermal transfer efficiency. +With the Non-Isotropic Velocity Scaling algorithm (NIVS) we have +developed, an unphysical thermal flux can be effectively set up even +for non-homogeneous systems like interfaces in non-equilibrium +molecular dynamics simulations. In this work, this algorithm is +applied for simulating thermal conductance at metal / organic solvent +interfaces with various coverages of butanethiol capping +agents. Different solvents and force field models were tested. Our +results suggest that the United-Atom models are able to provide an +estimate of the interfacial thermal conductivity comparable to +experiments in our simulations with satisfactory computational +efficiency. From our results, the acoustic impedance mismatch between +metal and liquid phase is effectively reduced by the capping +agents, and thus leads to interfacial thermal conductance +enhancement. Furthermore, this effect is closely related to the +capping agent coverage on the metal surfaces and the type of solvent +molecules, and is affected by the models used in the simulations. \end{abstract} @@ -71,66 +73,103 @@ leads to higher interfacial thermal transfer efficienc %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} -[BACKGROUND FOR INTERFACIAL THERMAL CONDUCTANCE PROBLEM] Interfacial thermal conductance is extensively studied both -experimentally and computationally, and systems with interfaces -present are generally heterogeneous. Although interfaces are commonly -barriers to heat transfer, it has been -reported\cite{doi:10.1021/la904855s} that under specific circustances, -e.g. with certain capping agents present on the surface, interfacial -conductance can be significantly enhanced. However, heat conductance -of molecular and nano-scale interfaces will be affected by the -chemical details of the surface and is challenging to -experimentalist. The lower thermal flux through interfaces is even -more difficult to measure with EMD and forward NEMD simulation -methods. Therefore, developing good simulation methods will be -desirable in order to investigate thermal transport across interfaces. +experimentally and computationally\cite{cahill:793}, due to its +importance in nanoscale science and technology. Reliability of +nanoscale devices depends on their thermal transport +properties. Unlike bulk homogeneous materials, nanoscale materials +features significant presence of interfaces, and these interfaces +could dominate the heat transfer behavior of these +materials. Furthermore, these materials are generally heterogeneous, +which challenges traditional research methods for homogeneous +systems. +Heat conductance of molecular and nano-scale interfaces will be +affected by the chemical details of the surface. Experimentally, +various interfaces have been investigated for their thermal +conductance properties. Wang {\it et al.} studied heat transport +through long-chain hydrocarbon monolayers on gold substrate at +individual molecular level\cite{Wang10082007}; Schmidt {\it et al.} +studied the role of CTAB on thermal transport between gold nanorods +and solvent\cite{doi:10.1021/jp8051888}; Juv\'e {\it et al.} studied +the cooling dynamics, which is controlled by thermal interface +resistence of glass-embedded metal +nanoparticles\cite{PhysRevB.80.195406}. Although interfaces are +commonly barriers for heat transport, Alper {\it et al.} suggested +that specific ligands (capping agents) could completely eliminate this +barrier ($G\rightarrow\infty$)\cite{doi:10.1021/la904855s}. + +Theoretical and computational models have also been used to study the +interfacial thermal transport in order to gain an understanding of +this phenomena at the molecular level. Recently, Hase and coworkers +employed Non-Equilibrium Molecular Dynamics (NEMD) simulations to +study thermal transport from hot Au(111) substrate to a self-assembled +monolayer of alkylthiol with relatively long chain (8-20 carbon +atoms)\cite{hase:2010,hase:2011}. However, ensemble averaged +measurements for heat conductance of interfaces between the capping +monolayer on Au and a solvent phase has yet to be studied. +The comparatively low thermal flux through interfaces is +difficult to measure with Equilibrium MD or forward NEMD simulation +methods. Therefore, the Reverse NEMD (RNEMD) methods would have the +advantage of having this difficult to measure flux known when studying +the thermal transport across interfaces, given that the simulation +methods being able to effectively apply an unphysical flux in +non-homogeneous systems. + Recently, we have developed the Non-Isotropic Velocity Scaling (NIVS) algorithm for RNEMD simulations\cite{kuang:164101}. This algorithm retains the desirable features of RNEMD (conservation of linear momentum and total energy, compatibility with periodic boundary conditions) while establishing true thermal distributions in each of -the two slabs. Furthermore, it allows more effective thermal exchange -between particles of different identities, and thus enables extensive -study of interfacial conductance. +the two slabs. Furthermore, it allows effective thermal exchange +between particles of different identities, and thus makes the study of +interfacial conductance much simpler. +The work presented here deals with the Au(111) surface covered to +varying degrees by butanethiol, a capping agent with short carbon +chain, and solvated with organic solvents of different molecular +properties. Different models were used for both the capping agent and +the solvent force field parameters. Using the NIVS algorithm, the +thermal transport across these interfaces was studied and the +underlying mechanism for this phenomena was investigated. + +[MAY ADD WHY STUDY AU-THIOL SURFACE; CITE SHAOYI JIANG] + \section{Methodology} -\subsection{Algorithm} -[BACKGROUND FOR MD METHODS] -There have been many algorithms for computing thermal conductivity -using molecular dynamics simulations. However, interfacial conductance -is at least an order of magnitude smaller. This would make the -calculation even more difficult for those slowly-converging -equilibrium methods. Imposed-flux non-equilibrium +\subsection{Imposd-Flux Methods in MD Simulations} +For systems with low interfacial conductivity one must have a method +capable of generating relatively small fluxes, compared to those +required for bulk conductivity. This requirement makes the calculation +even more difficult for those slowly-converging equilibrium +methods\cite{Viscardy:2007lq}. +Forward methods impose gradient, but in interfacail conditions it is +not clear what behavior to impose at the boundary... + Imposed-flux reverse non-equilibrium methods\cite{MullerPlathe:1997xw} have the flux set {\it a priori} and -the response of temperature or momentum gradients are easier to -measure than the flux, if unknown, and thus, is a preferable way to -the forward NEMD methods. Although the momentum swapping approach for -flux-imposing can be used for exchanging energy between particles of -different identity, the kinetic energy transfer efficiency is affected -by the mass difference between the particles, which limits its -application on heterogeneous interfacial systems. +the thermal response becomes easier to +measure than the flux. Although M\"{u}ller-Plathe's original momentum +swapping approach can be used for exchanging energy between particles +of different identity, the kinetic energy transfer efficiency is +affected by the mass difference between the particles, which limits +its application on heterogeneous interfacial systems. -The non-isotropic velocity scaling (NIVS)\cite{kuang:164101} approach in -non-equilibrium MD simulations is able to impose relatively large -kinetic energy flux without obvious perturbation to the velocity -distribution of the simulated systems. Furthermore, this approach has +The non-isotropic velocity scaling (NIVS)\cite{kuang:164101} approach to +non-equilibrium MD simulations is able to impose a wide range of +kinetic energy fluxes without obvious perturbation to the velocity +distributions of the simulated systems. Furthermore, this approach has the advantage in heterogeneous interfaces in that kinetic energy flux can be applied between regions of particles of arbitary identity, and -the flux quantity is not restricted by particle mass difference. +the flux will not be restricted by difference in particle mass. The NIVS algorithm scales the velocity vectors in two separate regions of a simulation system with respective diagonal scaling matricies. To determine these scaling factors in the matricies, a set of equations including linear momentum conservation and kinetic energy conservation -constraints and target momentum/energy flux satisfaction is -solved. With the scaling operation applied to the system in a set -frequency, corresponding momentum/temperature gradients can be built, -which can be used for computing transportation properties and other -applications related to momentum/temperature gradients. The NIVS -algorithm conserves momenta and energy and does not depend on an -external thermostat. +constraints and target energy flux satisfaction is solved. With the +scaling operation applied to the system in a set frequency, bulk +temperature gradients can be easily established, and these can be used +for computing thermal conductivities. The NIVS algorithm conserves +momenta and energy and does not depend on an external thermostat. \subsection{Defining Interfacial Thermal Conductivity $G$} For interfaces with a relatively low interfacial conductance, the bulk @@ -148,13 +187,14 @@ When the interfacial conductance is {\it not} small, t T_\mathrm{cold}\rangle}$ are the average observed temperature of the two separated phases. -When the interfacial conductance is {\it not} small, two ways can be -used to define $G$. +When the interfacial conductance is {\it not} small, there are two +ways to define $G$. -One way is to assume the temperature is discretely different on two -sides of the interface, $G$ can be calculated with the thermal flux -applied $J$ and the maximum temperature difference measured along the -thermal gradient max($\Delta T$), which occurs at the interface, as: +One way is to assume the temperature is discrete on the two sides of +the interface. $G$ can be calculated using the applied thermal flux +$J$ and the maximum temperature difference measured along the thermal +gradient max($\Delta T$), which occurs at the Gibbs deviding surface, +as: \begin{equation} G=\frac{J}{\Delta T} \label{discreteG} @@ -175,16 +215,16 @@ difference method and thus calculate $G^\prime$. With the temperature profile obtained from simulations, one is able to approximate the first and second derivatives of $T$ with finite -difference method and thus calculate $G^\prime$. +difference methods and thus calculate $G^\prime$. -In what follows, both definitions are used for calculation and comparison. +In what follows, both definitions have been used for calculation and +are compared in the results. -[IMPOSE G DEFINITION INTO OUR SYSTEMS] -To facilitate the use of the above definitions in calculating $G$ and -$G^\prime$, we have a metal slab with its (111) surfaces perpendicular -to the $z$-axis of our simulation cells. With or withour capping -agents on the surfaces, the metal slab is solvated with organic -solvents, as illustrated in Figure \ref{demoPic}. +To compare the above definitions ($G$ and $G^\prime$), we have modeled +a metal slab with its (111) surfaces perpendicular to the $z$-axis of +our simulation cells. Both with and withour capping agents on the +surfaces, the metal slab is solvated with simple organic solvents, as +illustrated in Figure \ref{demoPic}. \begin{figure} \includegraphics[width=\linewidth]{demoPic} @@ -193,16 +233,14 @@ With a simulation cell setup following the above manne \label{demoPic} \end{figure} -With a simulation cell setup following the above manner, one is able -to equilibrate the system and impose an unphysical thermal flux -between the liquid and the metal phase with the NIVS algorithm. Under -a stablized thermal gradient induced by periodically applying the -unphysical flux, one is able to obtain a temperature profile and the -physical thermal flux corresponding to it, which equals to the -unphysical flux applied by NIVS. These data enables the evaluation of -the interfacial thermal conductance of a surface. Figure \ref{gradT} -is an example how those stablized thermal gradient can be used to -obtain the 1st and 2nd derivatives of the temperature profile. +With the simulation cell described above, we are able to equilibrate +the system and impose an unphysical thermal flux between the liquid +and the metal phase using the NIVS algorithm. By periodically applying +the unphysical flux, we are able to obtain a temperature profile and +its spatial derivatives. These quantities enable the evaluation of the +interfacial thermal conductance of a surface. Figure \ref{gradT} is an +example how those applied thermal fluxes can be used to obtain the 1st +and 2nd derivatives of the temperature profile. \begin{figure} \includegraphics[width=\linewidth]{gradT} @@ -211,41 +249,36 @@ obtain the 1st and 2nd derivatives of the temperature \label{gradT} \end{figure} -[MAY INCLUDE POWER SPECTRUM PROTOCOL] - \section{Computational Details} \subsection{Simulation Protocol} -In our simulations, Au is used to construct a metal slab with bare -(111) surface perpendicular to the $z$-axis. Different slab thickness -(layer numbers of Au) are simulated. This metal slab is first -equilibrated under normal pressure (1 atm) and a desired -temperature. After equilibration, butanethiol is used as the capping -agent molecule to cover the bare Au (111) surfaces evenly. The sulfur -atoms in the butanethiol molecules would occupy the three-fold sites -of the surfaces, and the maximal butanethiol capacity on Au surface is -$1/3$ of the total number of surface Au atoms[CITATION]. A series of -different coverage surfaces is investigated in order to study the -relation between coverage and conductance. +The NIVS algorithm has been implemented in our MD simulation code, +OpenMD\cite{Meineke:2005gd,openmd}, and was used for our +simulations. Different slab thickness (layer numbers of Au) were +simulated. Metal slabs were first equilibrated under atmospheric +pressure (1 atm) and a desired temperature (e.g. 200K). After +equilibration, butanethiol capping agents were placed at three-fold +sites on the Au(111) surfaces. The maximum butanethiol capacity on Au +surface is $1/3$ of the total number of surface Au +atoms\cite{vlugt:cpc2007154}. A series of different coverages was +investigated in order to study the relation between coverage and +interfacial conductance. -[COVERAGE DISCRIPTION] However, since the interactions between surface -Au and butanethiol is non-bonded, the capping agent molecules are -allowed to migrate to an empty neighbor three-fold site during a -simulation. Therefore, the initial configuration would not severely -affect the sampling of a variety of configurations of the same -coverage, and the final conductance measurement would be an average -effect of these configurations explored in the simulations. [MAY NEED FIGURES] +The capping agent molecules were allowed to migrate during the +simulations. They distributed themselves uniformly and sampled a +number of three-fold sites throughout out study. Therefore, the +initial configuration would not noticeably affect the sampling of a +variety of configurations of the same coverage, and the final +conductance measurement would be an average effect of these +configurations explored in the simulations. [MAY NEED FIGURES] -After the modified Au-butanethiol surface systems are equilibrated -under canonical ensemble, Packmol\cite{packmol} is used to pack -organic solvent molecules in the previously vacuum part of the -simulation cells, which guarantees that short range repulsive -interactions do not disrupt the simulations. Two solvents are -investigated, one which has little vibrational overlap with the -alkanethiol and plane-like shape (toluene), and one which has similar -vibrational frequencies and chain-like shape ({\it n}-hexane). [MAY -EXPLAIN WHY WE CHOOSE THEM] +After the modified Au-butanethiol surface systems were equilibrated +under canonical ensemble, organic solvent molecules were packed in the +previously empty part of the simulation cells\cite{packmol}. Two +solvents were investigated, one which has little vibrational overlap +with the alkanethiol and a planar shape (toluene), and one which has +similar vibrational frequencies and chain-like shape ({\it n}-hexane). -The spacing filled by solvent molecules, i.e. the gap between +The space filled by solvent molecules, i.e. the gap between periodically repeated Au-butanethiol surfaces should be carefully chosen. A very long length scale for the thermal gradient axis ($z$) may cause excessively hot or cold temperatures in the middle of the @@ -290,15 +323,21 @@ quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.5 same type of particles and between particles of different species. The Au-Au interactions in metal lattice slab is described by the -quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC +quantum Sutton-Chen (QSC) formulation\cite{PhysRevB.59.3527}. The QSC potentials include zero-point quantum corrections and are reparametrized for accurate surface energies compared to the Sutton-Chen potentials\cite{Chen90}. -Figure [REF] demonstrates how we name our pseudo-atoms of the -molecules in our simulations. -[FIGURE FOR MOLECULE NOMENCLATURE] +Figure \ref{demoMol} demonstrates how we name our pseudo-atoms of the +organic solvent molecules in our simulations. +\begin{figure} +\includegraphics[width=\linewidth]{demoMol} +\caption{Denomination of atoms or pseudo-atoms in our simulations: a) + UA-hexane; b) AA-hexane; c) UA-toluene; d) AA-toluene.} +\label{demoMol} +\end{figure} + For both solvent molecules, straight chain {\it n}-hexane and aromatic toluene, United-Atom (UA) and All-Atom (AA) models are used respectively. The TraPPE-UA @@ -330,23 +369,27 @@ Au(111) surfaces, we adopt the S parameters from [CITA parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) surfaces do not have the hydrogen atom bonded to sulfur. To adapt this change and derive suitable parameters for butanethiol adsorbed on -Au(111) surfaces, we adopt the S parameters from [CITATION CF VLUGT] -and modify parameters for its neighbor C atom for charge balance in -the molecule. Note that the model choice (UA or AA) of capping agent -can be different from the solvent. Regardless of model choice, the -force field parameters for interactions between capping agent and -solvent can be derived using Lorentz-Berthelot Mixing Rule: +Au(111) surfaces, we adopt the S parameters from Luedtke and +Landman\cite{landman:1998} and modify parameters for its neighbor C +atom for charge balance in the molecule. Note that the model choice +(UA or AA) of capping agent can be different from the +solvent. Regardless of model choice, the force field parameters for +interactions between capping agent and solvent can be derived using +Lorentz-Berthelot Mixing Rule: +\begin{eqnarray} +\sigma_{IJ} & = & \frac{1}{2} \left(\sigma_{II} + \sigma_{JJ}\right) \\ +\epsilon_{IJ} & = & \sqrt{\epsilon_{II}\epsilon_{JJ}} +\end{eqnarray} - To describe the interactions between metal Au and non-metal capping agent and solvent particles, we refer to an adsorption study of alkyl thiols on gold surfaces by Vlugt {\it et al.}\cite{vlugt:cpc2007154} They fitted an effective Lennard-Jones form of potential parameters for the interaction between Au and pseudo-atoms CH$_x$ and S based on a well-established and widely-used -effective potential of Hautman and Klein[CITATION] for the Au(111) -surface. As our simulations require the gold lattice slab to be -non-rigid so that it could accommodate kinetic energy for thermal +effective potential of Hautman and Klein\cite{hautman:4994} for the +Au(111) surface. As our simulations require the gold lattice slab to +be non-rigid so that it could accommodate kinetic energy for thermal transport study purpose, the pair-wise form of potentials is preferred. @@ -365,24 +408,27 @@ parameters in our simulations. \begin{table*} \begin{minipage}{\linewidth} \begin{center} - \caption{Lennard-Jones parameters for Au-non-Metal - interactions in our simulations.} + \caption{Non-bonded interaction paramters for non-metal + particles and metal-non-metal interactions in our + simulations.} - \begin{tabular}{ccc} + \begin{tabular}{cccccc} \hline\hline - Non-metal & $\sigma$/\AA & $\epsilon$/kcal/mol \\ + Non-metal atom $I$ & $\sigma_{II}$ & $\epsilon_{II}$ & $q_I$ & + $\sigma_{AuI}$ & $\epsilon_{AuI}$ \\ + (or pseudo-atom) & \AA & kcal/mol & & \AA & kcal/mol \\ \hline - S & 2.40 & 8.465 \\ - CH3 & 3.54 & 0.2146 \\ - CH2 & 3.54 & 0.1749 \\ - CT3 & 3.365 & 0.1373 \\ - CT2 & 3.365 & 0.1373 \\ - CTT & 3.365 & 0.1373 \\ - HC & 2.865 & 0.09256 \\ - CHar & 3.4625 & 0.1680 \\ - CRar & 3.555 & 0.1604 \\ - CA & 3.173 & 0.0640 \\ - HA & 2.746 & 0.0414 \\ + CH3 & 3.75 & 0.1947 & - & 3.54 & 0.2146 \\ + CH2 & 3.95 & 0.0914 & - & 3.54 & 0.1749 \\ + CHar & 3.695 & 0.1003 & - & 3.4625 & 0.1680 \\ + CRar & 3.88 & 0.04173 & - & 3.555 & 0.1604 \\ + S & 4.45 & 0.25 & - & 2.40 & 8.465 \\ + CT3 & 3.50 & 0.066 & -0.18 & 3.365 & 0.1373 \\ + CT2 & 3.50 & 0.066 & -0.12 & 3.365 & 0.1373 \\ + CTT & 3.50 & 0.066 & -0.065 & 3.365 & 0.1373 \\ + HC & 2.50 & 0.030 & 0.06 & 2.865 & 0.09256 \\ + CA & 3.55 & 0.070 & -0.115 & 3.173 & 0.0640 \\ + HA & 2.42 & 0.030 & 0.115 & 2.746 & 0.0414 \\ \hline\hline \end{tabular} \label{MnM} @@ -448,28 +494,35 @@ couple $J_z$'s and do not need to test a large series \begin{minipage}{\linewidth} \begin{center} \caption{Computed interfacial thermal conductivity ($G$ and - $G^\prime$) values for the Au/butanethiol/hexane interface - with united-atom model and different capping agent coverage - and solvent molecule numbers at different temperatures using a - range of energy fluxes.} + $G^\prime$) values for the 100\% covered Au-butanethiol/hexane + interfaces with UA model and different hexane molecule numbers + at different temperatures using a range of energy fluxes.} - \begin{tabular}{cccccc} + \begin{tabular}{ccccccc} \hline\hline - Thiol & $\langle T\rangle$ & & $J_z$ & $G$ & $G^\prime$ \\ - coverage (\%) & (K) & $N_{hexane}$ & (GW/m$^2$) & + $\langle T\rangle$ & $N_{hexane}$ & Fixed & $\rho_{hexane}$ & + $J_z$ & $G$ & $G^\prime$ \\ + (K) & & $L_x$ \& $L_y$? & (g/cm$^3$) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ \hline - 0.0 & 200 & 200 & 0.96 & 43.3 & 42.7 \\ - & & & 1.91 & 45.7 & 42.9 \\ - & & 166 & 0.96 & 43.1 & 53.4 \\ - 88.9 & 200 & 166 & 1.94 & 172 & 108 \\ - 100.0 & 250 & 200 & 0.96 & 81.8 & 67.0 \\ - & & 166 & 0.98 & 79.0 & 62.9 \\ - & & & 1.44 & 76.2 & 64.8 \\ - & 200 & 200 & 1.92 & 129 & 87.3 \\ - & & & 1.93 & 131 & 77.5 \\ - & & 166 & 0.97 & 115 & 69.3 \\ - & & & 1.94 & 125 & 87.1 \\ + 200 & 266 & No & 0.672 & -0.96 & 102() & 80.0() \\ + & 200 & Yes & 0.694 & 1.92 & 129() & 87.3() \\ + & & Yes & 0.672 & 1.93 & 131() & 77.5() \\ + & & No & 0.688 & 0.96 & 125() & 90.2() \\ + & & & & 1.91 & 139() & 101() \\ + & & & & 2.83 & 141() & 89.9() \\ + & 166 & Yes & 0.679 & 0.97 & 115() & 69.3() \\ + & & & & 1.94 & 125() & 87.1() \\ + & & No & 0.681 & 0.97 & 141() & 77.7() \\ + & & & & 1.92 & 138() & 98.9() \\ + \hline + 250 & 200 & No & 0.560 & 0.96 & 74.8() & 61.8() \\ + & & & & -0.95 & 49.4() & 45.7() \\ + & 166 & Yes & 0.570 & 0.98 & 79.0() & 62.9() \\ + & & No & 0.569 & 0.97 & 80.3() & 67.1() \\ + & & & & 1.44 & 76.2() & 64.8() \\ + & & & & -0.95 & 56.4() & 54.4() \\ + & & & & -1.85 & 47.8() & 53.5() \\ \hline\hline \end{tabular} \label{AuThiolHexaneUA} @@ -500,24 +553,26 @@ in that higher degree of contact could yield increased important role in the thermal transport process across the interface in that higher degree of contact could yield increased conductance. -[ADD SIGNS AND ERROR ESTIMATE TO TABLE] +[ADD ERROR ESTIMATE TO TABLE] \begin{table*} \begin{minipage}{\linewidth} \begin{center} \caption{Computed interfacial thermal conductivity ($G$ and - $G^\prime$) values for the Au/butanethiol/toluene interface at - different temperatures using a range of energy fluxes.} + $G^\prime$) values for a 90\% coverage Au-butanethiol/toluene + interface at different temperatures using a range of energy + fluxes.} - \begin{tabular}{cccc} + \begin{tabular}{ccccc} \hline\hline - $\langle T\rangle$ & $J_z$ & $G$ & $G^\prime$ \\ - (K) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ + $\langle T\rangle$ & $\rho_{toluene}$ & $J_z$ & $G$ & $G^\prime$ \\ + (K) & (g/cm$^3$) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ \hline - 200 & 1.86 & 180 & 135 \\ - & 2.15 & 204 & 113 \\ - & 3.93 & 175 & 114 \\ - 300 & 1.91 & 143 & 125 \\ - & 4.19 & 134 & 113 \\ + 200 & 0.933 & -1.86 & 180() & 135() \\ + & & 2.15 & 204() & 113() \\ + & & -3.93 & 175() & 114() \\ + \hline + 300 & 0.855 & -1.91 & 143() & 125() \\ + & & -4.19 & 134() & 113() \\ \hline\hline \end{tabular} \label{AuThiolToluene} @@ -545,18 +600,19 @@ undertaken at $\sim$200K. reconstructions could eliminate the original $x$ and $y$ dimensional homogeneity, measurement of $G$ is more difficult to conduct under higher temperatures. Therefore, most of our measurements are -undertaken at $\sim$200K. +undertaken at $\langle T\rangle\sim$200K. However, when the surface is not completely covered by butanethiols, the simulated system is more resistent to the reconstruction above. Our Au-butanethiol/toluene system did not see this phenomena -even at $\sim$300K. The Au(111) surfaces have a 90\% coverage of -butanethiols and have empty three-fold sites. These empty sites could -help prevent surface reconstruction in that they provide other means -of capping agent relaxation. It is observed that butanethiols can -migrate to their neighbor empty sites during a simulation. Therefore, -we were able to obtain $G$'s for these interfaces even at a relatively -high temperature without being affected by surface reconstructions. +even at $\langle T\rangle\sim$300K. The Au(111) surfaces have a 90\% +coverage of butanethiols and have empty three-fold sites. These empty +sites could help prevent surface reconstruction in that they provide +other means of capping agent relaxation. It is observed that +butanethiols can migrate to their neighbor empty sites during a +simulation. Therefore, we were able to obtain $G$'s for these +interfaces even at a relatively high temperature without being +affected by surface reconstructions. \subsection{Influence of Capping Agent Coverage on $G$} To investigate the influence of butanethiol coverage on interfacial @@ -565,130 +621,213 @@ different coverages of butanethiol. molecules. These systems are then equilibrated and their interfacial thermal conductivity are measured with our NIVS algorithm. Table \ref{tlnUhxnUhxnD} lists these results for direct comparison between -different coverages of butanethiol. +different coverages of butanethiol. To study the isotope effect in +interfacial thermal conductance, deuterated UA-hexane is included as +well. - With high coverage of butanethiol on the gold surface, -the interfacial thermal conductance is enhanced -significantly. Interestingly, a slightly lower butanethiol coverage -leads to a moderately higher conductivity. This is probably due to -more solvent/capping agent contact when butanethiol molecules are -not densely packed, which enhances the interactions between the two -phases and lowers the thermal transfer barrier of this interface. -[COMPARE TO AU/WATER IN PAPER] +It turned out that with partial covered butanethiol on the Au(111) +surface, the derivative definition for $G$ (Eq. \ref{derivativeG}) has +difficulty to apply, due to the difficulty in locating the maximum of +change of $\lambda$. Instead, the discrete definition +(Eq. \ref{discreteG}) is easier to apply, as max($\Delta T$) can still +be well-defined. Therefore, $G$'s (not $G^\prime$) are used for this +section. +From Table \ref{tlnUhxnUhxnD}, one can see the significance of the +presence of capping agents. Even when a fraction of the Au(111) +surface sites are covered with butanethiols, the conductivity would +see an enhancement by at least a factor of 3. This indicates the +important role cappping agent is playing for thermal transport +phenomena on metal/organic solvent surfaces. -significant conductance enhancement compared to the gold/water -interface without capping agent and agree with available experimental -data. This indicates that the metal-metal potential, though not -predicting an accurate bulk metal thermal conductivity, does not -greatly interfere with the simulation of the thermal conductance -behavior across a non-metal interface. - The results show that the two definitions used for $G$ yield -comparable values, though $G^\prime$ tends to be smaller. +Interestingly, as one could observe from our results, the maximum +conductance enhancement (largest $G$) happens while the surfaces are +about 75\% covered with butanethiols. This again indicates that +solvent-capping agent contact has an important role of the thermal +transport process. Slightly lower butanethiol coverage allows small +gaps between butanethiols to form. And these gaps could be filled with +solvent molecules, which acts like ``heat conductors'' on the +surface. The higher degree of interaction between these solvent +molecules and capping agents increases the enhancement effect and thus +produces a higher $G$ than densely packed butanethiol arrays. However, +once this maximum conductance enhancement is reached, $G$ decreases +when butanethiol coverage continues to decrease. Each capping agent +molecule reaches its maximum capacity for thermal +conductance. Therefore, even higher solvent-capping agent contact +would not offset this effect. Eventually, when butanethiol coverage +continues to decrease, solvent-capping agent contact actually +decreases with the disappearing of butanethiol molecules. In this +case, $G$ decrease could not be offset but instead accelerated. +A comparison of the results obtained from differenet organic solvents +can also provide useful information of the interfacial thermal +transport process. The deuterated hexane (UA) results do not appear to +be much different from those of normal hexane (UA), given that +butanethiol (UA) is non-deuterated for both solvents. These UA model +studies, even though eliminating C-H vibration samplings, still have +C-C vibrational frequencies different from each other. However, these +differences in the infrared range do not seem to produce an observable +difference for the results of $G$. [MAY NEED FIGURE] -\begin{table*} - \begin{minipage}{\linewidth} - \begin{center} - \caption{Computed interfacial thermal conductivity ($G$ and - $G^\prime$) values for the Au/butanethiol/hexane interface - with united-atom model and different capping agent coverage - and solvent molecule numbers at different temperatures using a - range of energy fluxes.} - - \begin{tabular}{cccccc} - \hline\hline - Thiol & $\langle T\rangle$ & & $J_z$ & $G$ & $G^\prime$ \\ - coverage (\%) & (K) & $N_{hexane}$ & (GW/m$^2$) & - \multicolumn{2}{c}{(MW/m$^2$/K)} \\ - \hline - 0.0 & 200 & 200 & 0.96 & 43.3 & 42.7 \\ - & & & 1.91 & 45.7 & 42.9 \\ - & & 166 & 0.96 & 43.1 & 53.4 \\ - 88.9 & 200 & 166 & 1.94 & 172 & 108 \\ - 100.0 & 250 & 200 & 0.96 & 81.8 & 67.0 \\ - & & 166 & 0.98 & 79.0 & 62.9 \\ - & & & 1.44 & 76.2 & 64.8 \\ - & 200 & 200 & 1.92 & 129 & 87.3 \\ - & & & 1.93 & 131 & 77.5 \\ - & & 166 & 0.97 & 115 & 69.3 \\ - & & & 1.94 & 125 & 87.1 \\ - \hline\hline - \end{tabular} - \label{tlnUhxnUhxnD} - \end{center} - \end{minipage} -\end{table*} +Furthermore, results for rigid body toluene solvent, as well as other +UA-hexane solvents, are reasonable within the general experimental +ranges[CITATIONS]. This suggests that explicit hydrogen might not be a +required factor for modeling thermal transport phenomena of systems +such as Au-thiol/organic solvent. +However, results for Au-butanethiol/toluene do not show an identical +trend with those for Au-butanethiol/hexane in that $G$'s remain at +approximately the same magnitue when butanethiol coverage differs from +25\% to 75\%. This might be rooted in the molecule shape difference +for plane-like toluene and chain-like {\it n}-hexane. Due to this +difference, toluene molecules have more difficulty in occupying +relatively small gaps among capping agents when their coverage is not +too low. Therefore, the solvent-capping agent contact may keep +increasing until the capping agent coverage reaches a relatively low +level. This becomes an offset for decreasing butanethiol molecules on +its effect to the process of interfacial thermal transport. Thus, one +can see a plateau of $G$ vs. butanethiol coverage in our results. + +[NEED ERROR ESTIMATE] +\begin{figure} +\includegraphics[width=\linewidth]{coverage} +\caption{Comparison of interfacial thermal conductivity ($G$) values + for the Au-butanethiol/solvent interface with various UA models and + different capping agent coverages at $\langle T\rangle\sim$200K + using certain energy flux respectively.} +\label{coverage} +\end{figure} + \subsection{Influence of Chosen Molecule Model on $G$} [MAY COMBINE W MECHANISM STUDY] -For the all-atom model, the liquid hexane phase was not stable under NPT -conditions. Therefore, the simulation length scale parameters are -adopted from previous equilibration results of the united-atom model -at 200K. Table \ref{AuThiolHexaneAA} shows the results of these -simulations. The conductivity values calculated with full capping -agent coverage are substantially larger than observed in the -united-atom model, and is even higher than predicted by -experiments. It is possible that our parameters for metal-non-metal -particle interactions lead to an overestimate of the interfacial -thermal conductivity, although the active C-H vibrations in the -all-atom model (which should not be appreciably populated at normal -temperatures) could also account for this high conductivity. The major -thermal transfer barrier of Au/butanethiol/hexane interface is between -the liquid phase and the capping agent, so extra degrees of freedom -such as the C-H vibrations could enhance heat exchange between these -two phases and result in a much higher conductivity. +In addition to UA solvent/capping agent models, AA models are included +in our simulations as well. Besides simulations of the same (UA or AA) +model for solvent and capping agent, different models can be applied +to different components. Furthermore, regardless of models chosen, +either the solvent or the capping agent can be deuterated, similar to +the previous section. Table \ref{modelTest} summarizes the results of +these studies. +[MORE DATA; ERROR ESTIMATE] \begin{table*} \begin{minipage}{\linewidth} \begin{center} \caption{Computed interfacial thermal conductivity ($G$ and - $G^\prime$) values for the Au/butanethiol/hexane interface - with all-atom model and different capping agent coverage at - 200K using a range of energy fluxes.} + $G^\prime$) values for interfaces using various models for + solvent and capping agent (or without capping agent) at + $\langle T\rangle\sim$200K. (D stands for deuterated solvent + or capping agent molecules; ``Avg.'' denotes results that are + averages of several simulations.)} - \begin{tabular}{cccc} + \begin{tabular}{ccccc} \hline\hline - Thiol & $J_z$ & $G$ & $G^\prime$ \\ - coverage (\%) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ + Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ + (or bare surface) & model & (GW/m$^2$) & + \multicolumn{2}{c}{(MW/m$^2$/K)} \\ \hline - 0.0 & 0.95 & 28.5 & 27.2 \\ - & 1.88 & 30.3 & 28.9 \\ - 100.0 & 2.87 & 551 & 294 \\ - & 3.81 & 494 & 193 \\ + UA & UA hexane & Avg. & 131() & 86.5() \\ + & UA hexane(D) & 1.95 & 153() & 136() \\ + & AA hexane & 1.94 & 135() & 129() \\ + & & 2.86 & 126() & 115() \\ + & UA toluene & 1.96 & 187() & 151() \\ + & AA toluene & 1.89 & 200() & 149() \\ + \hline + AA & UA hexane & 1.94 & 116() & 129() \\ + & AA hexane & Avg. & 442() & 356() \\ + & AA hexane(D) & 1.93 & 222() & 234() \\ + & UA toluene & 1.98 & 125() & 96.5() \\ + & AA toluene & 3.79 & 487() & 290() \\ + \hline + AA(D) & UA hexane & 1.94 & 158() & 172() \\ + & AA hexane & 1.92 & 243() & 191() \\ + & AA toluene & 1.93 & 364() & 322() \\ + \hline + bare & UA hexane & Avg. & 46.5() & 49.4() \\ + & UA hexane(D) & 0.98 & 43.9() & 43.0() \\ + & AA hexane & 0.96 & 31.0() & 29.4() \\ + & UA toluene & 1.99 & 70.1() & 65.8() \\ \hline\hline \end{tabular} - \label{AuThiolHexaneAA} + \label{modelTest} \end{center} \end{minipage} \end{table*} +To facilitate direct comparison, the same system with differnt models +for different components uses the same length scale for their +simulation cells. Without the presence of capping agent, using +different models for hexane yields similar results for both $G$ and +$G^\prime$, and these two definitions agree with eath other very +well. This indicates very weak interaction between the metal and the +solvent, and is a typical case for acoustic impedance mismatch between +these two phases. +As for Au(111) surfaces completely covered by butanethiols, the choice +of models for capping agent and solvent could impact the measurement +of $G$ and $G^\prime$ quite significantly. For Au-butanethiol/hexane +interfaces, using AA model for both butanethiol and hexane yields +substantially higher conductivity values than using UA model for at +least one component of the solvent and capping agent, which exceeds +the upper bond of experimental value range. This is probably due to +the classically treated C-H vibrations in the AA model, which should +not be appreciably populated at normal temperatures. In comparison, +once either the hexanes or the butanethiols are deuterated, one can +see a significantly lower $G$ and $G^\prime$. In either of these +cases, the C-H(D) vibrational overlap between the solvent and the +capping agent is removed. [MAY NEED FIGURE] Conclusively, the +improperly treated C-H vibration in the AA model produced +over-predicted results accordingly. Compared to the AA model, the UA +model yields more reasonable results with higher computational +efficiency. + +However, for Au-butanethiol/toluene interfaces, having the AA +butanethiol deuterated did not yield a significant change in the +measurement results. Compared to the C-H vibrational overlap between +hexane and butanethiol, both of which have alkyl chains, that overlap +between toluene and butanethiol is not so significant and thus does +not have as much contribution to the ``Intramolecular Vibration +Redistribution''[CITE HASE]. Conversely, extra degrees of freedom such +as the C-H vibrations could yield higher heat exchange rate between +these two phases and result in a much higher conductivity. + +Although the QSC model for Au is known to predict an overly low value +for bulk metal gold conductivity\cite{kuang:164101}, our computational +results for $G$ and $G^\prime$ do not seem to be affected by this +drawback of the model for metal. Instead, our results suggest that the +modeling of interfacial thermal transport behavior relies mainly on +the accuracy of the interaction descriptions between components +occupying the interfaces. + \subsection{Mechanism of Interfacial Thermal Conductance Enhancement by Capping Agent} -[MAY INTRODUCE PROTOCOL IN METHODOLOGY/COMPUTATIONAL DETAIL] +%OR\subsection{Vibrational spectrum study on conductance mechanism} +[MAY INTRODUCE PROTOCOL IN METHODOLOGY/COMPUTATIONAL DETAIL, EQN'S] -%subsubsection{Vibrational spectrum study on conductance mechanism} To investigate the mechanism of this interfacial thermal conductance, the vibrational spectra of various gold systems were obtained and are shown as in the upper panel of Fig. \ref{vibration}. To obtain these spectra, one first runs a simulation in the NVE ensemble and collects snapshots of configurations; these configurations are used to compute the velocity auto-correlation functions, which is used to construct a -power spectrum via a Fourier transform. The gold surfaces covered by -butanethiol molecules exhibit an additional peak observed at a -frequency of $\sim$170cm$^{-1}$, which is attributed to the vibration -of the S-Au bond. This vibration enables efficient thermal transport -from surface Au atoms to the capping agents. Simultaneously, as shown -in the lower panel of Fig. \ref{vibration}, the large overlap of the -vibration spectra of butanethiol and hexane in the all-atom model, -including the C-H vibration, also suggests high thermal exchange -efficiency. The combination of these two effects produces the drastic -interfacial thermal conductance enhancement in the all-atom model. +power spectrum via a Fourier transform. +[MAY RELATE TO HASE'S] + The gold surfaces covered by +butanethiol molecules, compared to bare gold surfaces, exhibit an +additional peak observed at a frequency of $\sim$170cm$^{-1}$, which +is attributed to the vibration of the S-Au bond. This vibration +enables efficient thermal transport from surface Au atoms to the +capping agents. Simultaneously, as shown in the lower panel of +Fig. \ref{vibration}, the large overlap of the vibration spectra of +butanethiol and hexane in the all-atom model, including the C-H +vibration, also suggests high thermal exchange efficiency. The +combination of these two effects produces the drastic interfacial +thermal conductance enhancement in the all-atom model. + +[REDO. MAY NEED TO CONVERT TO JPEG] \begin{figure} \includegraphics[width=\linewidth]{vibration} \caption{Vibrational spectra obtained for gold in different @@ -696,13 +835,43 @@ interfacial thermal conductance enhancement in the all all-atom model (lower panel).} \label{vibration} \end{figure} -% MAY NEED TO CONVERT TO JPEG +[COMPARISON OF TWO G'S; AU SLAB WIDTHS; ETC] +% The results show that the two definitions used for $G$ yield +% comparable values, though $G^\prime$ tends to be smaller. + \section{Conclusions} +The NIVS algorithm we developed has been applied to simulations of +Au-butanethiol surfaces with organic solvents. This algorithm allows +effective unphysical thermal flux transferred between the metal and +the liquid phase. With the flux applied, we were able to measure the +corresponding thermal gradient and to obtain interfacial thermal +conductivities. Our simulations have seen significant conductance +enhancement with the presence of capping agent, compared to the bare +gold/liquid interfaces. The acoustic impedance mismatch between the +metal and the liquid phase is effectively eliminated by proper capping +agent. Furthermore, the coverage precentage of the capping agent plays +an important role in the interfacial thermal transport process. +Our measurement results, particularly of the UA models, agree with +available experimental data. This indicates that our force field +parameters have a nice description of the interactions between the +particles at the interfaces. AA models tend to overestimate the +interfacial thermal conductance in that the classically treated C-H +vibration would be overly sampled. Compared to the AA models, the UA +models have higher computational efficiency with satisfactory +accuracy, and thus are preferable in interfacial thermal transport +modelings. -[NECESSITY TO STUDY THERMAL CONDUCTANCE IN NANOCRYSTAL STRUCTURE]\cite{vlugt:cpc2007154} +Vlugt {\it et al.} has investigated the surface thiol structures for +nanocrystal gold and pointed out that they differs from those of the +Au(111) surface\cite{vlugt:cpc2007154}. This difference might lead to +change of interfacial thermal transport behavior as well. To +investigate this problem, an effective means to introduce thermal flux +and measure the corresponding thermal gradient is desirable for +simulating structures with spherical symmetry. + \section{Acknowledgments} Support for this project was provided by the National Science Foundation under grant CHE-0848243. Computational time was provided by