--- interfacial/interfacial.tex 2011/01/27 16:30:51 3718 +++ interfacial/interfacial.tex 2011/07/25 03:28:20 3748 @@ -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 +73,914 @@ The abstract version 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} +Interfacial thermal conductance is extensively studied both +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. -The intro. +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 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. + +[MAY ADD WHY STUDY AU-THIOL SURFACE; CITE SHAOYI JIANG] + +\section{Methodology} +\subsection{Imposd-Flux Methods in MD Simulations} +[CF. CAHILL] +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 interfacial 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 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 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 $G$} +Given a system with thermal gradients and the corresponding thermal +flux, for interfaces with a relatively low interfacial conductance, +the bulk regions on either side of an 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 imposed non-physical kinetic energy +transfer 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 reach 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 the temperature profile obtained from simulations, one is able to +approximate the first and second derivatives of $T$ with finite +difference methods and thus calculate $G^\prime$. + +In what follows, both definitions have been used for calculation and +are compared in the results. + +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 without capping agents on the +surfaces, the metal slab is 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 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 of 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. Different metal slab thickness (layer numbers of Au) was +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 +hollow sites on the Au(111) surfaces. These sites could be either a +{\it fcc} or {\it hcp} site. However, Hase {\it et al.} found that +they are equivalent in a heat transfer process\cite{hase:2010}, so +they are not distinguished 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 PORTER]. + A series of different coverages was derived by evenly eliminating + butanethiols on the surfaces, and was investigated 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 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 SNAPSHOTS] +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 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 +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. And the +corresponding spacing is usually $35[DOUBLE CHECK] \sim 75$\AA. + +The initial configurations generated are further equilibrated with the +$x$ and $y$ dimensions fixed, only allowing length scale change in $z$ +dimension. This is to ensure that the equilibration of liquid phase +does not affect the metal crystal structure in $x$ and $y$ dimensions. +To investigate this effect, comparisons were made with simulations +that allow changes of $L_x$ and $L_y$ during NPT equilibration, and +the results are shown in later sections. After ensuring the liquid +phase reaches equilibrium at atmospheric pressure (1 atm), further +equilibration are followed under NVT and then NVE ensembles. + +After the systems reach equilibrium, NIVS is implemented to impose a +periodic unphysical thermal flux between the metal and the liquid +phase. Most of our simulations are under an average temperature of +$\sim$200K. Therefore, this flux usually comes from the metal to the +liquid so that the liquid has a higher temperature and would not +freeze due to excessively low temperature. After this induced +temperature gradient is stablized, the temperature profile of the +simulation cell is recorded. To do this, the simulation cell is +devided evenly into $N$ slabs along the $z$-axis and $N$ is maximized +for highest possible spatial resolution but not too many to have some +slabs empty most of the time. 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. And +each equilibration / stabilization step usually takes 100 ps, or +longer, if necessary. + +\subsection{Force Field Parameters} +Our simulations include various components. Figure \ref{demoMol} +demonstrates the sites defined for both United-Atom and All-Atom +models of the organic solvent and capping agent molecules in our +simulations. Force field parameter descriptions 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 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 +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 not separated by more than 3 bonds. Otherwise, for non-bonded +interactions, Lennard-Jones potentials are used. [CHECK CITATION] + +By eliminating explicit hydrogen atoms, these models are simple and +computationally efficient, while maintains good accuracy. However, the +TraPPE-UA for alkanes is known to predict a lower boiling point than +experimental values. Considering that after an unphysical thermal flux +is applied to a system, the temperature of ``hot'' area in the liquid +phase would be significantly higher than the average of the system, to +prevent over heating and boiling of the liquid phase, the average +temperature in our simulations should be much lower than the liquid +boiling point. + +For UA-toluene model, the non-bonded potentials between +inter-molecular sites have a similar Lennard-Jones formulation. For +intra-molecular interactions, considering the stiffness of the benzene +ring, rigid body constraints are applied for further computational +efficiency. All bonds in the benzene ring and between the ring and the +methyl group remain rigid during the progress of simulations. + +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. 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} is +adopted. Without the rigid body constraints, bonding interactions were +included. For the aromatic ring, improper torsions (inversions) were +added as an extra potential for maintaining the planar shape. +[CHECK CITATION] + +The capping agent in our simulations, the butanethiol molecules can +either use UA or AA model. The TraPPE-UA force fields 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 adapt this +change and derive suitable parameters for butanethiol adsorbed on +Au(111) surfaces, we adopt the S parameters from Luedtke and +Landman\cite{landman:1998}[CHECK CITATION] + 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\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. + +Besides, 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. A set of +pseudo Lennard-Jones parameters were provided for Au in their force +fields. By using the Mixing Rule, this can be used to derive pair-wise +potentials for non-bonded interactions between Au and non-metal sites. + +However, the Lennard-Jones parameters between Au and other types of +particles, such as All-Atom normal alkanes in our simulations are not +yet well-established. For these interactions, we attempt to derive +their parameters using the Mixing Rule. To do this, Au pseudo +Lennard-Jones parameters for ``Metal-non-Metal'' (MnM) interactions +were first extracted from the Au-CH$_x$ parameters by applying the +Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM'' +parameters 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 Spectrum} +To investigate the mechanism of interfacial thermal conductance, the +vibrational spectrum is utilized as a complementary tool. Vibrational +spectra were taken for individual components in different +simulations. To obtain these spectra, simulations were run after +equilibration, in the NVE ensemble. 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} + +Followed by Fourier transforms, the power spectrum can be constructed: +\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[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 $G$ 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 ``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{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. + +As a combinational effects of the above two, 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}