--- interfacial/interfacial.tex 2011/01/27 16:29:20 3717 +++ interfacial/interfacial.tex 2011/09/23 17:31:50 3761 @@ -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 \begin{doublespace} \begin{abstract} -The abstract + With the Non-Isotropic Velocity Scaling (NIVS) approach to Reverse + Non-Equilibrium Molecular Dynamics (RNEMD) it is possible to impose + an unphysical thermal flux between different regions of + inhomogeneous systems such as solid / liquid interfaces. We have + applied NIVS to compute the interfacial thermal conductance at a + metal / organic solvent interface that has been chemically capped by + butanethiol molecules. Our calculations suggest that the acoustic + impedance mismatch between the metal and liquid phases is + effectively reduced by the capping agents, leading to a greatly + enhanced conductivity at the interface. Specifically, the chemical + bond between the metal and the capping agent introduces a + vibrational overlap that is not present without the capping agent, + and the overlap between the vibrational spectra (metal to cap, cap + to solvent) provides a mechanism for rapid thermal transport across + the interface. Our calculations also suggest that this is a + non-monotonic function of the fractional coverage of the surface, as + moderate coverages allow convective heat transport of solvent + molecules that have been in close contact with the capping agent. \end{abstract} \newpage @@ -56,14 +73,945 @@ The abstract %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} +Due to the importance of heat flow (and heat removal) in +nanotechnology, interfacial thermal conductance has been studied +extensively both experimentally and computationally.\cite{cahill:793} +Nanoscale materials have a significant fraction of their atoms at +interfaces, and the chemical details of these interfaces govern the +thermal transport properties. Furthermore, the interfaces are often +heterogeneous (e.g. solid - liquid), which provides a challenge to +computational methods which have been developed for homogeneous or +bulk systems. -The intro. +Experimentally, the thermal properties of a number of interfaces have +been investigated. Cahill and coworkers studied nanoscale thermal +transport from metal nanoparticle/fluid interfaces, to epitaxial +TiN/single crystal oxides interfaces, and hydrophilic and hydrophobic +interfaces between water and solids with different self-assembled +monolayers.\cite{Wilson:2002uq,PhysRevB.67.054302,doi:10.1021/jp048375k,PhysRevLett.96.186101} +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 +cetyltrimethylammonium bromide (CTAB) on the 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 resistance 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\cite{doi:10.1080/0026897031000068578} or forward NEMD simulation +methods. Therefore, the Reverse NEMD (RNEMD) +methods\cite{MullerPlathe:1997xw,kuang:164101} would be advantageous +in that they {\it apply} the difficult to measure quantity (flux), +while {\it measuring} the easily-computed quantity (the thermal +gradient). This is particularly true for inhomogeneous interfaces +where it would not be clear how to apply a gradient {\it a priori}. +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 chemical details of a +number of 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{Imposed-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 arbitrary 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 matrices. To +determine these scaling factors in the matrices, 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. For an applied flux $J_z$ +operating over a simulation time $t$ on a periodically-replicated slab +of dimensions $L_x \times L_y$, $E_{total} = J_z *(t)*(2 L_x L_y)$. + +When the interfacial conductance is {\it not} small, there are two +ways to define $G$. One common 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 dividing surface (Figure \ref{demoPic}). This is +known as the Kapitza conductance, which is the inverse of the Kapitza +resistance. +\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 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 (111) / butanethiol / hexane interfacial + system with the temperature profile after a kinetic energy flux has + been imposed. Note that the largest temperature jump in the thermal + profile (corresponding to the lowest interfacial conductance) is at + the interface between the butanethiol molecules (blue) and the + solvent (grey). First and second derivatives of the temperature + profile are obtained using a 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 $45 \sim 75$\AA in +our simulations. + +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$200K. Therefore, thermal flux usually came from the metal to the +liquid so that the liquid has a higher temperature and would not +freeze due to lowered temperatures. After this induced temperature +gradient had stabilized, the temperature profile of the simulation cell +was recorded. To do this, the simulation cell is divided evenly into +$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 References + \protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes,TraPPE-UA.thiols} + (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 slightly +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. The OPLS-AA\cite{OPLSAA} force fields +were used. For hexane, 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, 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*} + + +\section{Results} +There are many factors contributing to the measured interfacial +conductance; some of these factors are physically motivated +(e.g. coverage of the surface by the capping agent coverage and +solvent identity), while some are governed by parameters of the +methodology (e.g. applied flux and the formulas used to obtain the +conductance). In this section we discuss the major physical and +calculational effects on the computed conductivity. + +\subsection{Effects due to capping agent coverage} + +A series of different initial conditions with a range of surface +coverages was prepared and solvated with various with both of the +solvent molecules. These systems were then equilibrated and their +interfacial thermal conductivity was measured with the NIVS +algorithm. Figure \ref{coverage} demonstrates the trend of conductance +with respect to surface coverage. + +\begin{figure} +\includegraphics[width=\linewidth]{coverage} +\caption{The interfacial thermal conductivity ($G$) has a + non-monotonic dependence on the degree of surface capping. This + data is for the Au(111) / butanethiol / solvent interface with + various UA force fields at $\langle T\rangle \sim $200K.} +\label{coverage} +\end{figure} + +In partially covered surfaces, the derivative definition for +$G^\prime$ (Eq. \ref{derivativeG}) becomes difficult to apply, as the +location of maximum change of $\lambda$ becomes washed out. The +discrete definition (Eq. \ref{discreteG}) is easier to apply, as the +Gibbs dividing surface is still well-defined. Therefore, $G$ (not +$G^\prime$) was used in this section. + +From Figure \ref{coverage}, one can see the significance of the +presence of capping agents. When even a small fraction of the Au(111) +surface sites are covered with butanethiols, the conductivity exhibits +an enhancement by at least a factor of 3. Capping agents are clearly +playing a major role in thermal transport at metal / organic solvent +surfaces. +We note a non-monotonic behavior in the interfacial conductance as a +function of surface coverage. The maximum conductance (largest $G$) +happens when the surfaces are about 75\% covered with butanethiol +caps. The reason for this behavior is not entirely clear. One +explanation is that incomplete butanethiol coverage allows small gaps +between butanethiols to form. These gaps can be filled by transient +solvent molecules. These solvent molecules couple very strongly with +the hot capping agent molecules near the surface, and can then carry +away (diffusively) the excess thermal energy from the surface. + +There appears to be a competition between the conduction of the +thermal energy away from the surface by the capping agents (enhanced +by greater coverage) and the coupling of the capping agents with the +solvent (enhanced by interdigitation at lower coverages). This +competition would lead to the non-monotonic coverage behavior observed +here. + +Results for rigid body toluene solvent, as well as the UA hexane, are +within the ranges expected from prior experimental +work.\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406} This suggests +that explicit hydrogen atoms might not be required for modeling +thermal transport in these systems. C-H vibrational modes do not see +significant excited state population at low temperatures, and are not +likely to carry lower frequency excitations from the solid layer into +the bulk liquid. + +The toluene solvent does not exhibit the same behavior as hexane in +that $G$ remains at approximately the same magnitude when the capping +coverage increases from 25\% to 75\%. Toluene, as a rigid planar +molecule, cannot occupy the relatively small gaps between the capping +agents as easily as the chain-like {\it n}-hexane. The effect of +solvent coupling to the capping agent is therefore weaker in toluene +except at the very lowest coverage levels. This effect counters the +coverage-dependent conduction of heat away from the metal surface, +leading to a much flatter $G$ vs. coverage trend than is observed in +{\it n}-hexane. + +\subsection{Effects due to Solvent \& Solvent Models} +In addition to UA solvent and capping agent models, AA models have +also been included in our simulations. In most of this work, the same +(UA or AA) model for solvent and capping agent was used, but it is +also possible to utilize different models for different components. +We have also included isotopic substitutions (Hydrogen to Deuterium) +to decrease the explicit vibrational overlap between solvent and +capping agent. Table \ref{modelTest} summarizes the results of these +studies. + +\begin{table*} + \begin{minipage}{\linewidth} + \begin{center} + + \caption{Computed interfacial thermal conductance ($G$ and + $G^\prime$) values for interfaces using various models for + solvent and capping agent (or without capping agent) at + $\langle T\rangle\sim$200K. Here ``D'' stands for deuterated + solvent or capping agent molecules; ``Avg.'' denotes results + that are averages of simulations under different applied + thermal flux $(J_z)$ values. Error estimates are indicated in + parentheses.} + + \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 between force fields, systems with the +same capping agent and solvent were prepared with the same length +scales for the simulation cells. + +On bare metal / solvent surfaces, different force field models for +hexane yield similar results for both $G$ and $G^\prime$, and these +two definitions agree with each other very well. This is primarily an +indicator of weak interactions between the metal and the solvent, and +is a typical case for acoustic impedance mismatch between these two +phases. + +For the fully-covered surfaces, the choice of force field for the +capping agent and solvent has a large impact on the calculated values +of $G$ and $G^\prime$. The AA thiol to AA solvent conductivities are +much larger than their UA to UA counterparts, and these values exceed +the experimental estimates by a large measure. The AA force field +allows significant energy to go into C-H (or C-D) stretching modes, +and since these modes are high frequency, this non-quantum behavior is +likely responsible for the overestimate of the conductivity. Compared +to the AA model, the UA model yields more reasonable conductivity +values with much higher computational efficiency. + +\subsubsection{Are electronic excitations in the metal important?} +Because they lack electronic excitations, the QSC and related embedded +atom method (EAM) models for gold are known to predict unreasonably +low values for bulk conductivity +($\lambda$).\cite{kuang:164101,ISI:000207079300006,Clancy:1992} If the +conductance between the phases ($G$) is governed primarily by phonon +excitation (and not electronic degrees of freedom), one would expect a +classical model to capture most of the interfacial thermal +conductance. Our results for $G$ and $G^\prime$ indicate that this is +indeed the case, and suggest that the modeling of interfacial thermal +transport depends primarily on the description of the interactions +between the various components at the interface. When the metal is +chemically capped, the primary barrier to thermal conductivity appears +to be the interface between the capping agent and the surrounding +solvent, so the excitations in the metal have little impact on the +value of $G$. + +\subsection{Effects due to methodology and simulation parameters} + +We have varied the parameters of the simulations in order to +investigate how these factors would affect the computation of $G$. Of +particular interest are: 1) the length scale for the applied thermal +gradient (modified by increasing the amount of solvent in the system), +2) the sign and magnitude of the applied thermal flux, 3) the average +temperature of the simulation (which alters the solvent density during +equilibration), and 4) the definition of the interfacial conductance +(Eqs. (\ref{discreteG}) or (\ref{derivativeG})) used in the +calculation. + +Systems of different lengths were prepared by altering the number of +solvent molecules and extending the length of the box along the $z$ +axis to accomodate the extra solvent. Equilibration at the same +temperature and pressure conditions led to nearly identical surface +areas ($L_x$ and $L_y$) available to the metal and capping agent, +while the extra solvent served mainly to lengthen the axis that was +used to apply the thermal flux. For a given value of the applied +flux, the different $z$ length scale has only a weak effect on the +computed conductivities (Table \ref{AuThiolHexaneUA}). + +\subsubsection{Effects of applied flux} +The NIVS algorithm allows changes in both the sign and magnitude of +the applied flux. It is possible to reverse the direction of heat +flow simply by changing the sign of the flux, and thermal gradients +which would be difficult to obtain experimentally ($5$ K/\AA) can be +easily simulated. However, the magnitude of the applied flux is not +arbitrary if one aims to obtain a stable and reliable thermal gradient. +A temperature gradient can be lost in the noise if $|J_z|$ is too +small, and excessive $|J_z|$ values can cause phase transitions if the +extremes of the simulation cell become widely separated in +temperature. Also, if $|J_z|$ is too large for the bulk conductivity +of the materials, the thermal gradient will never reach a stable +state. + +Within a reasonable range of $J_z$ values, we were able to study how +$G$ changes as a function of this flux. In what follows, we use +positive $J_z$ values to denote the case where energy is being +transferred by the method from the metal phase and into the liquid. +The resulting gradient therefore has a higher temperature in the +liquid phase. Negative flux values reverse this transfer, and result +in higher temperature metal phases. The conductance measured under +different applied $J_z$ values is listed in Tables +\ref{AuThiolHexaneUA} and \ref{AuThiolToluene}. These results do not +indicate that $G$ depends strongly 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 +small number $J_z$ values. + +\begin{table*} + \begin{minipage}{\linewidth} + \begin{center} + \caption{In the hexane-solvated interfaces, the system size has + little effect on the calculated values for interfacial + conductance ($G$ and $G^\prime$), but the direction of heat + flow (i.e. the sign of $J_z$) can alter the average + temperature of the liquid phase and this can alter the + computed conductivity.} + + \begin{tabular}{ccccccc} + \hline\hline + $\langle T\rangle$ & $N_{hexane}$ & $\rho_{hexane}$ & + $J_z$ & $G$ & $G^\prime$ \\ + (K) & & (g/cm$^3$) & (GW/m$^2$) & + \multicolumn{2}{c}{(MW/m$^2$/K)} \\ + \hline + 200 & 266 & 0.672 & -0.96 & 102(3) & 80.0(0.8) \\ + & 200 & 0.688 & 0.96 & 125(16) & 90.2(15) \\ + & & & 1.91 & 139(10) & 101(10) \\ + & & & 2.83 & 141(6) & 89.9(9.8) \\ + & 166 & 0.681 & 0.97 & 141(30) & 78(22) \\ + & & & 1.92 & 138(4) & 98.9(9.5) \\ + \hline + 250 & 200 & 0.560 & 0.96 & 75(10) & 61.8(7.3) \\ + & & & -0.95 & 49.4(0.3) & 45.7(2.1) \\ + & 166 & 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*} + +The sign of $J_z$ is a different matter, however, as this can alter +the temperature on the two sides of the interface. The average +temperature values reported are for the entire system, and not for the +liquid phase, so at a given $\langle T \rangle$, the system with +positive $J_z$ has a warmer liquid phase. This means that if the +liquid carries thermal energy via convective transport, {\it positive} +$J_z$ values will result in increased molecular motion on the liquid +side of the interface, and this will increase the measured +conductivity. + +\subsubsection{Effects due to average temperature} + +We also studied the effect of average system temperature on the +interfacial conductance. The simulations are first equilibrated in +the NPT ensemble at 1 atm. The TraPPE-UA model for hexane tends to +predict a lower boiling point (and liquid state density) than +experiments. This lower-density liquid phase leads to reduced contact +between the hexane and butanethiol, and this accounts for our +observation of lower conductance at higher temperatures as shown in +Table \ref{AuThiolHexaneUA}. In raising the average temperature from +200K to 250K, the density drop of $\sim$20\% in the solvent phase +leads to a $\sim$40\% drop in the conductance. + +Similar behavior is observed in the TraPPE-UA model for toluene, +although this model has better agreement with the experimental +densities of toluene. The expansion of the toluene liquid phase is +not as significant as that of the hexane (8.3\% over 100K), and this +limits the effect to $\sim$20\% drop in thermal conductivity (Table +\ref{AuThiolToluene}). + +Although we have not mapped out the behavior at a large number of +temperatures, is clear that there will be a strong temperature +dependence in the interfacial conductance when the physical properties +of one side of the interface (notably the density) change rapidly as a +function of temperature. + +\begin{table*} + \begin{minipage}{\linewidth} + \begin{center} + \caption{When toluene is the solvent, the interfacial thermal + conductivity is less sensitive to temperature, but again, the + direction of the heat flow can alter the solvent temperature + and can change the computed conductance values.} + + \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 the lower interfacial thermal conductance, surfaces at +relatively high temperatures are susceptible to reconstructions, +particularly when butanethiols fully cover the Au(111) surface. These +reconstructions include surface Au atoms which migrate outward to the +S atom layer, and butanethiol molecules which embed into the surface +Au layer. The driving force for this behavior is the strong Au-S +interactions which are modeled here with a deep Lennard-Jones +potential. This phenomenon agrees with reconstructions that have been +experimentally +observed.\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt +{\it et al.} kept their Au(111) slab rigid so that their simulations +could reach 300K without surface +reconstructions.\cite{vlugt:cpc2007154} Since surface reconstructions +blur the interface, the measurement of $G$ becomes more difficult to +conduct at higher temperatures. For this reason, most of our +measurements are undertaken at $\langle T\rangle\sim$200K where +reconstruction is minimized. + +However, when the surface is not completely covered by butanethiols, +the simulated system appears to be more resistent to the +reconstruction. 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. That said, we did +observe butanethiols migrating to neighboring three-fold sites during +a simulation. Since the interface persisted in these simulations, +were able to obtain $G$'s for these interfaces even at a relatively +high temperature without being affected by surface reconstructions. + +\section{Discussion} + +The primary result of this work is that the capping agent acts as an +efficient thermal coupler between solid and solvent phases. One of +the ways the capping agent can carry out this role is to down-shift +between the phonon vibrations in the solid (which carry the heat from +the gold) and the molecular vibrations in the liquid (which carry some +of the heat in the solvent). + +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 +microcanonical (NVE) ensemble and without a thermal +gradient. Snapshots of configurations were collected at a frequency +that is higher than that of the fastest vibrations occurring 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} + +\subsection{The role of specific vibrations} +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 which are covered by butanethiol molecules +exhibit an additional peak observed at a frequency of +$\sim$165cm$^{-1}$. We attribute this peak to the S-Au bonding +vibration. This vibration enables efficient thermal coupling of the +surface Au layer to the capping agents. Therefore, in our simulations, +the Au / S interfaces do not appear to be the primary barrier to +thermal transport when compared with the butanethiol / solvent +interfaces. + +\begin{figure} +\includegraphics[width=\linewidth]{vibration} +\caption{The vibrational power spectrum for thiol-capped gold has an + additional vibrational peak at $\sim $165cm$^{-1}$. Bare gold + surfaces (both with and without a solvent over-layer) are missing + this peak. A similar peak at $\sim $165cm$^{-1}$ also appears in + the vibrational power spectrum for the butanethiol capping agents.} +\label{specAu} +\end{figure} + +Also in this figure, we show the vibrational power spectrum for the +bound butanethiol molecules, which also exhibits the same +$\sim$165cm$^{-1}$ peak. + +\subsection{Overlap of power spectra} +A comparison of the results obtained from the two different organic +solvents can also provide useful information of the interfacial +thermal transport process. In particular, the vibrational overlap +between the butanethiol and the organic solvents suggests a highly +efficient thermal exchange between these components. Very high +thermal conductivity was observed when AA models were used and C-H +vibrations were treated classically. The presence of extra degrees of +freedom in the AA force field yields higher heat exchange rates +between the two phases and results in a much higher conductivity than +in the UA force field. + +The similarity in the vibrational modes available to solvent and +capping agent can be reduced by deuterating one of the two components +(Fig. \ref{aahxntln}). Once either the hexanes or the butanethiols +are deuterated, one can observe a significantly lower $G$ and +$G^\prime$ values (Table \ref{modelTest}). + +\begin{figure} +\includegraphics[width=\linewidth]{aahxntln} +\caption{Spectra obtained for all-atom (AA) Au / butanethiol / solvent + systems. When butanethiol is deuterated (lower left), its + vibrational overlap with hexane decreases significantly. Since + aromatic molecules and the butanethiol are vibrationally dissimilar, + the change is not as dramatic when toluene is the solvent (right).} +\label{aahxntln} +\end{figure} + +For the Au / butanethiol / toluene interfaces, having the AA +butanethiol deuterated did not yield a significant change in the +measured conductance. Compared to the C-H vibrational overlap between +hexane and butanethiol, both of which have alkyl chains, the overlap +between toluene and butanethiol is not as significant and thus does +not contribute as much to the heat exchange process. + +Previous observations of Zhang {\it et al.}\cite{hase:2010} indicate +that the {\it intra}molecular heat transport due to alkylthiols is +highly efficient. Combining our observations with those of Zhang {\it + et al.}, it appears that butanethiol acts as a channel to expedite +heat flow from the gold surface and into the alkyl chain. The +acoustic impedance mismatch between the metal and the liquid phase can +therefore be effectively reduced with the presence of suitable capping +agents. + +Deuterated models in the UA force field did not decouple the thermal +transport as well as in the AA force field. The UA models, even +though they have eliminated the high frequency C-H vibrational +overlap, still have significant overlap in the lower-frequency +portions of the infrared spectra (Figure \ref{uahxnua}). Deuterating +the UA models did not decouple the low frequency region enough to +produce an observable difference for the results of $G$ (Table +\ref{modelTest}). + +\begin{figure} +\includegraphics[width=\linewidth]{uahxnua} +\caption{Vibrational power spectra for UA models for the butanethiol + and hexane solvent (upper panel) show the high degree of overlap + between these two molecules, particularly at lower frequencies. + Deuterating a UA model for the solvent (lower panel) does not + decouple the two spectra to the same degree as in the AA force + field (see Fig \ref{aahxntln}).} +\label{uahxnua} +\end{figure} + +\section{Conclusions} +The NIVS algorithm has been applied to simulations of +butanethiol-capped Au(111) surfaces in the presence of organic +solvents. This algorithm allows the application of unphysical thermal +flux to transfer heat between the metal and the liquid phase. With the +flux applied, we were able to measure the corresponding thermal +gradients and to obtain interfacial thermal conductivities. Under +steady states, 2-3 ns trajectory simulations are sufficient for +computation of this quantity. + +Our simulations have seen significant conductance enhancement in the +presence of capping agent, compared with the bare gold / liquid +interfaces. The acoustic impedance mismatch between the metal and the +liquid phase is effectively eliminated by a chemically-bonded capping +agent. Furthermore, the coverage percentage of the capping agent plays +an important role in the interfacial thermal transport +process. Moderately low coverages allow higher contact between capping +agent and solvent, and thus could further enhance the heat transfer +process, giving a non-monotonic behavior of conductance with +increasing coverage. + +Our results, particularly using the UA models, agree well with +available experimental data. The AA models tend to overestimate the +interfacial thermal conductance in that the classically treated C-H +vibrations become too easily populated. Compared to the AA models, the +UA models have higher computational efficiency with satisfactory +accuracy, and thus are preferable in modeling interfacial thermal +transport. + +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 +required for the use of a derivative form is limited by the number of +bins and the sampling required to obtain thermal gradient information. + +Vlugt {\it et al.} have investigated the surface thiol structures for +nanocrystalline gold and pointed out that they differ from those of +the Au(111) surface.\cite{landman:1998,vlugt:cpc2007154} This +difference could also cause differences in the interfacial thermal +transport behavior. To investigate this problem, one would need an +effective method for applying thermal gradients in non-planar +(i.e. spherical) geometries. + \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}