--- interfacial/interfacial.tex 2011/01/27 16:30:51 3718 +++ interfacial/interfacial.tex 2011/07/26 19:43:10 3751 @@ -22,9 +22,9 @@ \setlength{\abovecaptionskip}{20 pt} \setlength{\belowcaptionskip}{30 pt} -%\renewcommand\citemid{\ } % no comma in optional referenc note -\bibpunct{[}{]}{,}{s}{}{;} -\bibliographystyle{aip} +%\renewcommand\citemid{\ } % no comma in optional reference note +\bibpunct{[}{]}{,}{n}{}{;} +\bibliographystyle{achemso} \begin{document} @@ -44,7 +44,24 @@ The abstract version 2 \begin{doublespace} \begin{abstract} -The abstract version 2 + +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} \newpage @@ -56,14 +73,900 @@ The abstract version 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} +Due to the importance of heat flow in nanotechnology, interfacial +thermal conductance has been studied extensively both experimentally +and computationally.\cite{cahill:793} Unlike bulk materials, nanoscale +materials have a significant fraction of their atoms at interfaces, +and the chemical details of these interfaces govern the heat transfer +behavior. Furthermore, the interfaces are +heterogeneous (e.g. solid - liquid), which provides a challenge to +traditional methods developed for homogeneous systems. -The intro. +Experimentally, various interfaces have been investigated for their +thermal conductance. 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} and 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 +normally considered 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 have yet to be studied with their +approach. 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\cite{MullerPlathe:1997xw,kuang:164101} would have the +advantage of applying this difficult to measure flux (while measuring +the resulting gradient), given that the simulation methods being able +to effectively apply an unphysical flux in non-homogeneous systems. +Garde and coworkers\cite{garde:nl2005,garde:PhysRevLett2009} applied +this approach to various liquid interfaces and studied how thermal +conductance (or resistance) is dependent on chemistry details of +interfaces, e.g. hydrophobic and hydrophilic aqueous interfaces. + +Recently, we have developed a 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 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 the phenomena was investigated. + +\section{Methodology} +\subsection{Imposd-Flux Methods in MD Simulations} +Steady state MD simulations have an advantage in that not many +trajectories are needed to study the relationship between thermal flux +and thermal gradients. For systems with low interfacial conductance, +one must have a method capable of generating or measuring relatively +small fluxes, compared to those required for bulk conductivity. This +requirement makes the calculation even more difficult for +slowly-converging equilibrium methods.\cite{Viscardy:2007lq} Forward +NEMD methods impose a gradient (and measure a flux), but at interfaces +it is not clear what behavior should be imposed at the boundaries +between materials. Imposed-flux reverse non-equilibrium +methods\cite{MullerPlathe:1997xw} set the flux {\it a priori} and +the thermal response becomes an easy-to-measure quantity. 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 +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 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 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 an interface with relatively low interfacial conductance, and a +thermal flux between two distinct bulk regions, the regions on either +side of the interface rapidly come to a state in which the two phases +have relatively homogeneous (but distinct) temperatures. The +interfacial thermal conductivity $G$ can therefore be approximated as: +\begin{equation} + G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle - + \langle T_\mathrm{cold}\rangle \right)} +\label{lowG} +\end{equation} +where ${E_{total}}$ is the total imposed non-physical kinetic energy +transfer during the simulation and ${\langle T_\mathrm{hot}\rangle}$ +and ${\langle T_\mathrm{cold}\rangle}$ are the average observed +temperature of the two separated phases. + +When the interfacial conductance is {\it not} small, there are two +ways to define $G$. 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 (Figure \ref{demoPic}): \begin{equation} + G=\frac{J}{\Delta T} \label{discreteG} \end{equation} + +\begin{figure} +\includegraphics[width=\linewidth]{method} +\caption{Interfacial conductance can be calculated by applying an + (unphysical) kinetic energy flux between two slabs, one located + within the metal and another on the edge of the periodic box. The + system responds by forming a thermal response or a gradient. In + bulk liquids, this gradient typically has a single slope, but in + interfacial systems, there are distinct thermal conductivity + domains. The interfacial conductance, $G$ is found by measuring the + temperature gap at the Gibbs dividing surface, or by using second + derivatives of the thermal profile.} +\label{demoPic} +\end{figure} + +The other approach is to assume a continuous temperature profile along +the thermal gradient axis (e.g. $z$) and define $G$ at the point where +the magnitude of thermal conductivity ($\lambda$) change reaches its +maximum, given that $\lambda$ is well-defined throughout the space: +\begin{equation} +G^\prime = \Big|\frac{\partial\lambda}{\partial z}\Big| + = \Big|\frac{\partial}{\partial z}\left(-J_z\Big/ + \left(\frac{\partial T}{\partial z}\right)\right)\Big| + = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| + \Big/\left(\frac{\partial T}{\partial z}\right)^2 +\label{derivativeG} +\end{equation} + +With temperature profiles obtained from simulation, one is able to +approximate the first and second derivatives of $T$ with finite +difference methods and calculate $G^\prime$. In what follows, both +definitions have been used, and are compared in the results. + +To investigate the interfacial conductivity at metal / solvent +interfaces, we have modeled a metal slab with its (111) surfaces +perpendicular to the $z$-axis of our simulation cells. The metal slab +has been prepared both with and without capping agents on the exposed +surface, and has been solvated with simple organic solvents, as +illustrated in Figure \ref{gradT}. + +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 obtained a temperature profile and its spatial +derivatives. Figure \ref{gradT} shows how an applied thermal flux can +be used to obtain the 1st and 2nd derivatives of the temperature +profile. + +\begin{figure} +\includegraphics[width=\linewidth]{gradT} +\caption{A sample of Au-butanethiol/hexane interfacial system and the + temperature profile after a kinetic energy flux is imposed to + it. The 1st and 2nd derivatives of the temperature profile can be + obtained with finite difference approximation (lower panel).} +\label{gradT} +\end{figure} + +\section{Computational Details} +\subsection{Simulation Protocol} +The NIVS algorithm has been implemented in our MD simulation code, +OpenMD\cite{Meineke:2005gd,openmd}, and was used for our simulations. +Metal slabs of 6 or 11 layers of Au atoms were first equilibrated +under atmospheric pressure (1 atm) and 200K. After equilibration, +butanethiol capping agents were placed at three-fold hollow sites on +the Au(111) surfaces. These sites are either {\it fcc} or {\it + hcp} sites, although Hase {\it et al.} found that they are +equivalent in a heat transfer process,\cite{hase:2010} so we did not +distinguish between these sites in our study. The maximum butanethiol +capacity on Au surface is $1/3$ of the total number of surface Au +atoms, and the packing forms a $(\sqrt{3}\times\sqrt{3})R30^\circ$ +structure\cite{doi:10.1021/ja00008a001,doi:10.1021/cr9801317}. A +series of lower coverages was also prepared by eliminating +butanethiols from the higher coverage surface in a regular manner. The +lower coverages were prepared in order to study the relation between +coverage and interfacial conductance. + +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 does 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. + +After the modified Au-butanethiol surface systems were equilibrated in +the canonical (NVT) 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 which has a planar shape (toluene), and one +which has similar vibrational frequencies to the capping agent and +chain-like shape ({\it n}-hexane). + +The simulation cells were not particularly extensive along the +$z$-axis, as a very long length scale for the thermal gradient may +cause excessively hot or cold temperatures in the middle of the +solvent region and lead to undesired phenomena such as solvent boiling +or freezing when a thermal flux is applied. Conversely, too few +solvent molecules would change the normal behavior of the liquid +phase. Therefore, our $N_{solvent}$ values were chosen to ensure that +these extreme cases did not happen to our simulations. The spacing +between periodic images of the gold interfaces is $35 \sim 75$\AA. + +The initial configurations generated are further equilibrated with the +$x$ and $y$ dimensions fixed, only allowing the $z$-length scale to +change. This is to ensure that the equilibration of liquid phase does +not affect the metal's crystalline structure. Comparisons were made +with simulations that allowed changes of $L_x$ and $L_y$ during NPT +equilibration. No substantial changes in the box geometry were noticed +in these simulations. After ensuring the liquid phase reaches +equilibrium at atmospheric pressure (1 atm), further equilibration was +carried out under canonical (NVT) and microcanonical (NVE) ensembles. + +After the systems reach equilibrium, NIVS was used to impose an +unphysical thermal flux between the metal and the liquid phases. Most +of our simulations were done under an average temperature of +$\sim$200K. Therefore, thermal flux usually came from the metal to the +liquid so that the liquid has a higher temperature and would not +freeze due to lowered temperatures. After this induced temperature +gradient had stablized, the temperature profile of the simulation cell +was recorded. To do this, the simulation cell is devided evenly into +$N$ slabs along the $z$-axis. The average temperatures of each slab +are recorded for 1$\sim$2 ns. When the slab width $d$ of each slab is +the same, the derivatives of $T$ with respect to slab number $n$ can +be directly used for $G^\prime$ calculations: \begin{equation} + G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| + \Big/\left(\frac{\partial T}{\partial z}\right)^2 + = |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big| + \Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2 + = |J_z|\Big|\frac{\partial^2 T}{\partial n^2}\Big| + \Big/\left(\frac{\partial T}{\partial n}\right)^2 +\label{derivativeG2} +\end{equation} + +All of the above simulation procedures use a time step of 1 fs. Each +equilibration stage took a minimum of 100 ps, although in some cases, +longer equilibration stages were utilized. + +\subsection{Force Field Parameters} +Our simulations include a number of chemically distinct components. +Figure \ref{demoMol} demonstrates the sites defined for both +United-Atom and All-Atom models of the organic solvent and capping +agents in our simulations. Force field parameters are needed for +interactions both between the same type of particles and between +particles of different species. + +\begin{figure} +\includegraphics[width=\linewidth]{structures} +\caption{Structures of the capping agent and solvents utilized in + these simulations. The chemically-distinct sites (a-e) are expanded + in terms of constituent atoms for both United Atom (UA) and All Atom + (AA) force fields. Most parameters are from + Refs. \protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} (UA) and + \protect\cite{OPLSAA} (AA). Cross-interactions with the Au atoms are given + in Table \ref{MnM}.} +\label{demoMol} +\end{figure} + +The Au-Au interactions in metal lattice slab is described by the +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} + +For the two solvent molecules, {\it n}-hexane and toluene, two +different atomistic models were utilized. Both solvents were modeled +using United-Atom (UA) and All-Atom (AA) models. The TraPPE-UA +parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used +for our UA solvent molecules. In these models, sites are located at +the carbon centers for alkyl groups. Bonding interactions, including +bond stretches and bends and torsions, were used for intra-molecular +sites closer than 3 bonds. For non-bonded interactions, Lennard-Jones +potentials are used. + +By eliminating explicit hydrogen atoms, the TraPPE-UA models are +simple and computationally efficient, while maintaining good accuracy. +However, the TraPPE-UA model for alkanes is known to predict a slighly +lower boiling point than experimental values. This is one of the +reasons we used a lower average temperature (200K) for our +simulations. If heat is transferred to the liquid phase during the +NIVS simulation, the liquid in the hot slab can actually be +substantially warmer than the mean temperature in the simulation. The +lower mean temperatures therefore prevent solvent boiling. + +For UA-toluene, the non-bonded potentials between intermolecular sites +have a similar Lennard-Jones formulation. The toluene molecules were +treated as a single rigid body, so there was no need for +intramolecular interactions (including bonds, bends, or torsions) in +this solvent model. + +Besides the TraPPE-UA models, AA models for both organic solvents are +included in our studies as well. For hexane, the OPLS-AA\cite{OPLSAA} +force field is used, and additional explicit hydrogen sites were +included. Besides bonding and non-bonded site-site interactions, +partial charges and the electrostatic interactions were added to each +CT and HC site. For toluene, the United Force Field developed by +Rapp\'{e} {\it et al.}\cite{doi:10.1021/ja00051a040} was adopted, and +a flexible model for the toluene molecule was utilized which included +bond, bend, torsion, and inversion potentials to enforce ring +planarity. + +The butanethiol capping agent in our simulations, were also modeled +with both UA and AA model. The TraPPE-UA force field includes +parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for +UA butanethiol model in our simulations. The OPLS-AA also provides +parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) +surfaces do not have the hydrogen atom bonded to sulfur. To derive +suitable parameters for butanethiol adsorbed on Au(111) surfaces, we +adopt the S parameters from Luedtke and Landman\cite{landman:1998} and +modify the parameters for the CTS atom to maintain charge neutrality +in the molecule. Note that the model choice (UA or AA) for the 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 atoms, +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 for the Au(111) +surface.\cite{hautman:4994} As our simulations require the gold slab +to be flexible to accommodate thermal excitation, the pair-wise form +of potentials they developed was used for our study. + +The potentials developed from {\it ab initio} calculations by Leng +{\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the +interactions between Au and aromatic C/H atoms in toluene. However, +the Lennard-Jones parameters between Au and other types of particles, +(e.g. AA alkanes) have not yet been established. For these +interactions, the Lorentz-Berthelot mixing rule can be used to derive +effective single-atom LJ parameters for the metal using the fit values +for toluene. These are then used to construct reasonable mixing +parameters for the interactions between the gold and other atoms. +Table \ref{MnM} summarizes the ``metal/non-metal'' parameters used in +our simulations. + +\begin{table*} + \begin{minipage}{\linewidth} + \begin{center} + \caption{Non-bonded interaction parameters (including cross + interactions with Au atoms) for both force fields used in this + work.} + \begin{tabular}{lllllll} + \hline\hline + & Site & $\sigma_{ii}$ & $\epsilon_{ii}$ & $q_i$ & + $\sigma_{Au-i}$ & $\epsilon_{Au-i}$ \\ + & & (\AA) & (kcal/mol) & ($e$) & (\AA) & (kcal/mol) \\ + \hline + United Atom (UA) + &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 \\ + \hline + All Atom (AA) + &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 + Both UA and AA + & S & 4.45 & 0.25 & - & 2.40 & 8.465 \\ + \hline\hline + \end{tabular} + \label{MnM} + \end{center} + \end{minipage} +\end{table*} + +\subsection{Vibrational Power Spectrum} + +To investigate the mechanism of interfacial thermal conductance, the +vibrational power spectrum was computed. Power spectra were taken for +individual components in different simulations. To obtain these +spectra, simulations were run after equilibration, in the NVE +ensemble, and without a thermal gradient. Snapshots of configurations +were collected at a frequency that is higher than that of the fastest +vibrations occuring in the simulations. With these configurations, the +velocity auto-correlation functions can be computed: +\begin{equation} +C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle +\label{vCorr} +\end{equation} +The power spectrum is constructed via a Fourier transform of the +symmetrized velocity autocorrelation function, +\begin{equation} + \hat{f}(\omega) = + \int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt +\label{fourier} +\end{equation} + +\section{Results and Discussions} +In what follows, how the parameters and protocol of simulations would +affect the measurement of $G$'s is first discussed. With a reliable +protocol and set of parameters, the influence of capping agent +coverage on thermal conductance is investigated. Besides, different +force field models for both solvents and selected deuterated models +were tested and compared. Finally, a summary of the role of capping +agent in the interfacial thermal transport process is given. + +\subsection{How Simulation Parameters Affects $G$} +We have varied our protocol or other parameters of the simulations in +order to investigate how these factors would affect the measurement of +$G$'s. It turned out that while some of these parameters would not +affect the results substantially, some other changes to the +simulations would have a significant impact on the measurement +results. + +In some of our simulations, we allowed $L_x$ and $L_y$ to change +during equilibrating the liquid phase. Due to the stiffness of the +crystalline Au structure, $L_x$ and $L_y$ would not change noticeably +after equilibration. Although $L_z$ could fluctuate $\sim$1\% after a +system is fully equilibrated in the NPT ensemble, this fluctuation, as +well as those of $L_x$ and $L_y$ (which is significantly smaller), +would not be magnified on the calculated $G$'s, as shown in Table +\ref{AuThiolHexaneUA}. This insensivity to $L_i$ fluctuations allows +reliable measurement of $G$'s without the necessity of extremely +cautious equilibration process. + +As stated in our computational details, the spacing filled with +solvent molecules can be chosen within a range. This allows some +change of solvent molecule numbers for the same Au-butanethiol +surfaces. We did this study on our Au-butanethiol/hexane +simulations. Nevertheless, the results obtained from systems of +different $N_{hexane}$ did not indicate that the measurement of $G$ is +susceptible to this parameter. For computational efficiency concern, +smaller system size would be preferable, given that the liquid phase +structure is not affected. + +Our NIVS algorithm allows change of unphysical thermal flux both in +direction and in quantity. This feature extends our investigation of +interfacial thermal conductance. However, the magnitude of this +thermal flux is not arbitary if one aims to obtain a stable and +reliable thermal gradient. A temperature profile would be +substantially affected by noise when $|J_z|$ has a much too low +magnitude; while an excessively large $|J_z|$ that overwhelms the +conductance capacity of the interface would prevent a thermal gradient +to reach a stablized steady state. NIVS has the advantage of allowing +$J$ to vary in a wide range such that the optimal flux range for $G$ +measurement can generally be simulated by the algorithm. Within the +optimal range, we were able to study how $G$ would change according to +the thermal flux across the interface. For our simulations, we denote +$J_z$ to be positive when the physical thermal flux is from the liquid +to metal, and negative vice versa. The $G$'s measured under different +$J_z$ is listed in Table \ref{AuThiolHexaneUA} and +\ref{AuThiolToluene}. These results do not suggest that $G$ is +dependent on $J_z$ within this flux range. The linear response of flux +to thermal gradient simplifies our investigations in that we can rely +on $G$ measurement with only a couple $J_z$'s and do not need to test +a large series of fluxes. + +\begin{table*} + \begin{minipage}{\linewidth} + \begin{center} + \caption{Computed interfacial thermal conductivity ($G$ and + $G^\prime$) 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. Error estimates indicated in parenthesis.} + + \begin{tabular}{ccccccc} + \hline\hline + $\langle T\rangle$ & $N_{hexane}$ & 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 + 200 & 266 & No & 0.672 & -0.96 & 102(3) & 80.0(0.8) \\ + & 200 & Yes & 0.694 & 1.92 & 129(11) & 87.3(0.3) \\ + & & Yes & 0.672 & 1.93 & 131(16) & 78(13) \\ + & & No & 0.688 & 0.96 & 125(16) & 90.2(15) \\ + & & & & 1.91 & 139(10) & 101(10) \\ + & & & & 2.83 & 141(6) & 89.9(9.8) \\ + & 166 & Yes & 0.679 & 0.97 & 115(19) & 69(18) \\ + & & & & 1.94 & 125(9) & 87.1(0.2) \\ + & & No & 0.681 & 0.97 & 141(30) & 78(22) \\ + & & & & 1.92 & 138(4) & 98.9(9.5) \\ + \hline + 250 & 200 & No & 0.560 & 0.96 & 75(10) & 61.8(7.3) \\ + & & & & -0.95 & 49.4(0.3) & 45.7(2.1) \\ + & 166 & Yes & 0.570 & 0.98 & 79.0(3.5) & 62.9(3.0) \\ + & & No & 0.569 & 0.97 & 80.3(0.6) & 67(11) \\ + & & & & 1.44 & 76.2(5.0) & 64.8(3.8) \\ + & & & & -0.95 & 56.4(2.5) & 54.4(1.1) \\ + & & & & -1.85 & 47.8(1.1) & 53.5(1.5) \\ + \hline\hline + \end{tabular} + \label{AuThiolHexaneUA} + \end{center} + \end{minipage} +\end{table*} + +Furthermore, we also attempted to increase system average temperatures +to above 200K. These simulations are first equilibrated in the NPT +ensemble under normal pressure. As stated above, the TraPPE-UA model +for hexane tends to predict a lower boiling point. In our simulations, +hexane had diffculty to remain in liquid phase when NPT equilibration +temperature is higher than 250K. Additionally, the equilibrated liquid +hexane density under 250K becomes lower than experimental value. This +expanded liquid phase leads to lower contact between hexane and +butanethiol as well.[MAY NEED SLAB DENSITY FIGURE] +And this reduced contact would +probably be accountable for a lower interfacial thermal conductance, +as shown in Table \ref{AuThiolHexaneUA}. + +A similar study for TraPPE-UA toluene agrees with the above result as +well. Having a higher boiling point, toluene tends to remain liquid in +our simulations even equilibrated under 300K in NPT +ensembles. Furthermore, the expansion of the toluene liquid phase is +not as significant as that of the hexane. This prevents severe +decrease of liquid-capping agent contact and the results (Table +\ref{AuThiolToluene}) show only a slightly decreased interface +conductance. Therefore, solvent-capping agent contact should play an +important role in the thermal transport process across the interface +in that higher degree of contact could yield increased conductance. + +\begin{table*} + \begin{minipage}{\linewidth} + \begin{center} + \caption{Computed interfacial thermal conductivity ($G$ and + $G^\prime$) values for a 90\% coverage Au-butanethiol/toluene + interface at different temperatures using a range of energy + fluxes. Error estimates indicated in parenthesis.} + + \begin{tabular}{ccccc} + \hline\hline + $\langle T\rangle$ & $\rho_{toluene}$ & $J_z$ & $G$ & $G^\prime$ \\ + (K) & (g/cm$^3$) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ + \hline + 200 & 0.933 & 2.15 & 204(12) & 113(12) \\ + & & -1.86 & 180(3) & 135(21) \\ + & & -3.93 & 176(5) & 113(12) \\ + \hline + 300 & 0.855 & -1.91 & 143(5) & 125(2) \\ + & & -4.19 & 135(9) & 113(12) \\ + \hline\hline + \end{tabular} + \label{AuThiolToluene} + \end{center} + \end{minipage} +\end{table*} + +Besides lower interfacial thermal conductance, surfaces in relatively +high temperatures are susceptible to reconstructions, when +butanethiols have a full coverage on the Au(111) surface. These +reconstructions include surface Au atoms migrated outward to the S +atom layer, and butanethiol molecules embedded into the original +surface Au layer. The driving force for this behavior is the strong +Au-S interactions in our simulations. And these reconstructions lead +to higher ratio of Au-S attraction and thus is energetically +favorable. Furthermore, this phenomenon agrees with experimental +results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt +{\it et al.} had kept their Au(111) slab rigid so that their +simulations can reach 300K without surface reconstructions. Without +this practice, simulating 100\% thiol covered interfaces under higher +temperatures could hardly avoid surface reconstructions. However, our +measurement is based on assuming homogeneity on $x$ and $y$ dimensions +so that measurement of $T$ at particular $z$ would be an effective +average of the particles of the same type. Since surface +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 $\langle T\rangle\sim$200K. + +However, when the surface is not completely covered by butanethiols, +the simulated system is more resistent to the reconstruction +above. Our Au-butanethiol/toluene system had the Au(111) surfaces 90\% +covered by butanethiols, but did not see this above phenomena even at +$\langle T\rangle\sim$300K. The empty three-fold sites not occupied by +capping agents 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 +thermal conductance, a series of different coverage Au-butanethiol +surfaces is prepared and solvated with various organic +molecules. These systems are then equilibrated and their interfacial +thermal conductivity are measured with our NIVS algorithm. Figure +\ref{coverage} demonstrates the trend of conductance change with +respect to different coverages of butanethiol. To study the isotope +effect in interfacial thermal conductance, deuterated UA-hexane is +included as well. + +\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} + +It turned out that with partial covered butanethiol on the Au(111) +surface, the derivative definition for $G^\prime$ +(Eq. \ref{derivativeG}) was difficult 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 the Gibbs +deviding surface can still be well-defined. Therefore, $G$ (not +$G^\prime$) was used for this section. + +From Figure \ref{coverage}, 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. + +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. [NEED +SNAPSHOT SHOWING THE PHENOMENA / SLAB DENSITY ANALYSIS] + +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 $G$ (Figure \ref{uahxnua}). + +\begin{figure} +\includegraphics[width=\linewidth]{uahxnua} +\caption{Vibrational spectra obtained for normal (upper) and + deuterated (lower) hexane in Au-butanethiol/hexane + systems. Butanethiol spectra are shown as reference. Both hexane and + butanethiol were using United-Atom models.} +\label{uahxnua} +\end{figure} + +Furthermore, results for rigid body toluene solvent, as well as other +UA-hexane solvents, are reasonable within the general experimental +ranges\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406}. 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 $G$ remains at +approximately the same magnitue when butanethiol coverage differs from +25\% to 75\%. This might be rooted in the molecule shape difference +for planar 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. + +\subsection{Influence of Chosen Molecule Model on $G$} +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. + +\begin{table*} + \begin{minipage}{\linewidth} + \begin{center} + + \caption{Computed interfacial thermal conductivity ($G$ and + $G^\prime$) 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 simulations under different $J_z$'s. Error + estimates indicated in parenthesis.)} + + \begin{tabular}{llccc} + \hline\hline + Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ + (or bare surface) & model & (GW/m$^2$) & + \multicolumn{2}{c}{(MW/m$^2$/K)} \\ + \hline + UA & UA hexane & Avg. & 131(9) & 87(10) \\ + & UA hexane(D) & 1.95 & 153(5) & 136(13) \\ + & AA hexane & Avg. & 131(6) & 122(10) \\ + & UA toluene & 1.96 & 187(16) & 151(11) \\ + & AA toluene & 1.89 & 200(36) & 149(53) \\ + \hline + AA & UA hexane & 1.94 & 116(9) & 129(8) \\ + & AA hexane & Avg. & 442(14) & 356(31) \\ + & AA hexane(D) & 1.93 & 222(12) & 234(54) \\ + & UA toluene & 1.98 & 125(25) & 97(60) \\ + & AA toluene & 3.79 & 487(56) & 290(42) \\ + \hline + AA(D) & UA hexane & 1.94 & 158(25) & 172(4) \\ + & AA hexane & 1.92 & 243(29) & 191(11) \\ + & AA toluene & 1.93 & 364(36) & 322(67) \\ + \hline + bare & UA hexane & Avg. & 46.5(3.2) & 49.4(4.5) \\ + & UA hexane(D) & 0.98 & 43.9(4.6) & 43.0(2.0) \\ + & AA hexane & 0.96 & 31.0(1.4) & 29.4(1.3) \\ + & UA toluene & 1.99 & 70.1(1.3) & 65.8(0.5) \\ + \hline\hline + \end{tabular} + \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 general range of experimental measurement results. 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 (Figure +\ref{aahxntln}). 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. + +\begin{figure} +\includegraphics[width=\linewidth]{aahxntln} +\caption{Spectra obtained for All-Atom model Au-butanethil/solvent + systems. When butanethiol is deuterated (lower left), its + vibrational overlap with hexane would decrease significantly, + compared with normal butanethiol (upper left). However, this + dramatic change does not apply to toluene as much (right).} +\label{aahxntln} +\end{figure} + +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 heat exchange +process. 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{Role of Capping Agent in Interfacial Thermal Conductance} +The vibrational spectra for gold slabs in different environments are +shown as in Figure \ref{specAu}. Regardless of the presence of +solvent, the gold surfaces covered by butanethiol molecules, compared +to bare gold surfaces, exhibit an additional peak observed at the +frequency of $\sim$170cm$^{-1}$, which is attributed to the S-Au +bonding vibration. This vibration enables efficient thermal transport +from surface Au layer to the capping agents. Therefore, in our +simulations, the Au/S interfaces do not appear major heat barriers +compared to the butanethiol / solvent interfaces. + +Simultaneously, the vibrational overlap between butanethiol and +organic solvents suggests higher thermal exchange efficiency between +these two components. Even exessively high heat transport was observed +when All-Atom models were used and C-H vibrations were treated +classically. Compared to metal and organic liquid phase, the heat +transfer efficiency between butanethiol and organic solvents is closer +to that within bulk liquid phase. + +Furthermore, our observation validated previous +results\cite{hase:2010} that the intramolecular heat transport of +alkylthiols is highly effecient. As a combinational effects of these +phenomena, butanethiol acts as a channel to expedite thermal transport +process. The acoustic impedance mismatch between the metal and the +liquid phase can be effectively reduced with the presence of suitable +capping agents. + +\begin{figure} +\includegraphics[width=\linewidth]{vibration} +\caption{Vibrational spectra obtained for gold in different + environments.} +\label{specAu} +\end{figure} + +[MAY ADD COMPARISON OF AU SLAB WIDTHS] + +\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. Under steady states, single trajectory simulation +would be enough for accurate measurement. This would be advantageous +compared to transient state simulations, which need multiple +trajectories to produce reliable average results. + +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. Moderately lower coverages allow higher contact between +capping agent and solvent, and thus could further enhance the heat +transfer 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. Of the two definitions for $G$, the discrete form +(Eq. \ref{discreteG}) was easier to use and gives out relatively +consistent results, while the derivative form (Eq. \ref{derivativeG}) +is not as versatile. Although $G^\prime$ gives out comparable results +and follows similar trend with $G$ when measuring close to fully +covered or bare surfaces, the spatial resolution of $T$ profile is +limited for accurate computation of derivatives data. + +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 the Center for Research Computing (CRC) at the University of Notre -Dame. \newpage +Dame. \newpage \bibliography{interfacial}