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\begin{document} |
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\title{Simulating interfacial thermal conductance at metal-solvent |
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interfaces: the role of chemical capping agents} |
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|
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\author{Shenyu Kuang and J. Daniel |
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Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\ |
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Department of Chemistry and Biochemistry,\\ |
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University of Notre Dame\\ |
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Notre Dame, Indiana 46556} |
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\date{\today} |
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\maketitle |
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\begin{doublespace} |
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|
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\begin{abstract} |
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|
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With the Non-Isotropic Velocity Scaling algorithm (NIVS) we have |
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developed, an unphysical thermal flux can be effectively set up even |
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for non-homogeneous systems like interfaces in non-equilibrium |
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molecular dynamics simulations. In this work, this algorithm is |
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applied for simulating thermal conductance at metal / organic solvent |
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interfaces with various coverages of butanethiol capping |
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agents. Different solvents and force field models were tested. Our |
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results suggest that the United-Atom models are able to provide an |
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estimate of the interfacial thermal conductivity comparable to |
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experiments in our simulations with satisfactory computational |
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efficiency. From our results, the acoustic impedance mismatch between |
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metal and liquid phase is effectively reduced by the capping |
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agents, and thus leads to interfacial thermal conductance |
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enhancement. Furthermore, this effect is closely related to the |
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capping agent coverage on the metal surfaces and the type of solvent |
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molecules, and is affected by the models used in the simulations. |
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|
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\end{abstract} |
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|
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\newpage |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% BODY OF TEXT |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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|
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\section{Introduction} |
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Due to the importance of heat flow in nanotechnology, interfacial |
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thermal conductance has been studied extensively both experimentally |
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and computationally.\cite{cahill:793} Unlike bulk materials, nanoscale |
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materials have a significant fraction of their atoms at interfaces, |
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and the chemical details of these interfaces govern the heat transfer |
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behavior. Furthermore, the interfaces are |
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heterogeneous (e.g. solid - liquid), which provides a challenge to |
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traditional methods developed for homogeneous systems. |
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|
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Experimentally, various interfaces have been investigated for their |
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thermal conductance. Wang {\it et al.} studied heat transport through |
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long-chain hydrocarbon monolayers on gold substrate at individual |
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molecular level,\cite{Wang10082007} Schmidt {\it et al.} studied the |
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role of CTAB on thermal transport between gold nanorods and |
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solvent,\cite{doi:10.1021/jp8051888} and Juv\'e {\it et al.} studied |
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the cooling dynamics, which is controlled by thermal interface |
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resistence of glass-embedded metal |
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nanoparticles.\cite{PhysRevB.80.195406} Although interfaces are |
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normally considered barriers for heat transport, Alper {\it et al.} |
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suggested that specific ligands (capping agents) could completely |
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eliminate this barrier |
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($G\rightarrow\infty$).\cite{doi:10.1021/la904855s} |
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|
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Theoretical and computational models have also been used to study the |
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interfacial thermal transport in order to gain an understanding of |
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this phenomena at the molecular level. Recently, Hase and coworkers |
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employed Non-Equilibrium Molecular Dynamics (NEMD) simulations to |
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study thermal transport from hot Au(111) substrate to a self-assembled |
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monolayer of alkylthiol with relatively long chain (8-20 carbon |
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atoms).\cite{hase:2010,hase:2011} However, ensemble averaged |
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measurements for heat conductance of interfaces between the capping |
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monolayer on Au and a solvent phase have yet to be studied with their |
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approach. The comparatively low thermal flux through interfaces is |
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difficult to measure with Equilibrium MD or forward NEMD simulation |
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methods. Therefore, the Reverse NEMD (RNEMD) |
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methods\cite{MullerPlathe:1997xw,kuang:164101} would have the |
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advantage of applying this difficult to measure flux (while measuring |
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the resulting gradient), given that the simulation methods being able |
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to effectively apply an unphysical flux in non-homogeneous systems. |
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Garde and coworkers\cite{garde:nl2005,garde:PhysRevLett2009} applied |
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this approach to various liquid interfaces and studied how thermal |
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conductance (or resistance) is dependent on chemistry details of |
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interfaces, e.g. hydrophobic and hydrophilic aqueous interfaces. |
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|
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Recently, we have developed a Non-Isotropic Velocity Scaling (NIVS) |
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algorithm for RNEMD simulations\cite{kuang:164101}. This algorithm |
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retains the desirable features of RNEMD (conservation of linear |
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momentum and total energy, compatibility with periodic boundary |
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conditions) while establishing true thermal distributions in each of |
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the two slabs. Furthermore, it allows effective thermal exchange |
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between particles of different identities, and thus makes the study of |
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interfacial conductance much simpler. |
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|
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The work presented here deals with the Au(111) surface covered to |
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varying degrees by butanethiol, a capping agent with short carbon |
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chain, and solvated with organic solvents of different molecular |
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properties. Different models were used for both the capping agent and |
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the solvent force field parameters. Using the NIVS algorithm, the |
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thermal transport across these interfaces was studied and the |
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underlying mechanism for the phenomena was investigated. |
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|
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\section{Methodology} |
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\subsection{Imposd-Flux Methods in MD Simulations} |
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Steady state MD simulations have an advantage in that not many |
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trajectories are needed to study the relationship between thermal flux |
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and thermal gradients. For systems with low interfacial conductance, |
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one must have a method capable of generating or measuring relatively |
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small fluxes, compared to those required for bulk conductivity. This |
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requirement makes the calculation even more difficult for |
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slowly-converging equilibrium methods.\cite{Viscardy:2007lq} Forward |
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NEMD methods impose a gradient (and measure a flux), but at interfaces |
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it is not clear what behavior should be imposed at the boundaries |
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between materials. Imposed-flux reverse non-equilibrium |
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methods\cite{MullerPlathe:1997xw} set the flux {\it a priori} and |
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the thermal response becomes an easy-to-measure quantity. Although |
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M\"{u}ller-Plathe's original momentum swapping approach can be used |
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for exchanging energy between particles of different identity, the |
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kinetic energy transfer efficiency is affected by the mass difference |
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between the particles, which limits its application on heterogeneous |
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interfacial systems. |
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|
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The non-isotropic velocity scaling (NIVS) \cite{kuang:164101} approach |
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to non-equilibrium MD simulations is able to impose a wide range of |
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kinetic energy fluxes without obvious perturbation to the velocity |
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distributions of the simulated systems. Furthermore, this approach has |
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the advantage in heterogeneous interfaces in that kinetic energy flux |
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can be applied between regions of particles of arbitary identity, and |
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the flux will not be restricted by difference in particle mass. |
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|
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The NIVS algorithm scales the velocity vectors in two separate regions |
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of a simulation system with respective diagonal scaling matricies. To |
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determine these scaling factors in the matricies, a set of equations |
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including linear momentum conservation and kinetic energy conservation |
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constraints and target energy flux satisfaction is solved. With the |
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scaling operation applied to the system in a set frequency, bulk |
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temperature gradients can be easily established, and these can be used |
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for computing thermal conductivities. The NIVS algorithm conserves |
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momenta and energy and does not depend on an external thermostat. |
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|
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\subsection{Defining Interfacial Thermal Conductivity ($G$)} |
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|
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For an interface with relatively low interfacial conductance, and a |
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thermal flux between two distinct bulk regions, the regions on either |
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side of the interface rapidly come to a state in which the two phases |
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have relatively homogeneous (but distinct) temperatures. The |
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interfacial thermal conductivity $G$ can therefore be approximated as: |
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\begin{equation} |
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G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle - |
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\langle T_\mathrm{cold}\rangle \right)} |
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\label{lowG} |
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\end{equation} |
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where ${E_{total}}$ is the total imposed non-physical kinetic energy |
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transfer during the simulation and ${\langle T_\mathrm{hot}\rangle}$ |
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and ${\langle T_\mathrm{cold}\rangle}$ are the average observed |
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temperature of the two separated phases. |
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|
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When the interfacial conductance is {\it not} small, there are two |
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ways to define $G$. One common way is to assume the temperature is |
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discrete on the two sides of the interface. $G$ can be calculated |
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using the applied thermal flux $J$ and the maximum temperature |
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difference measured along the thermal gradient max($\Delta T$), which |
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occurs at the Gibbs deviding surface (Figure \ref{demoPic}): |
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\begin{equation} |
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G=\frac{J}{\Delta T} |
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\label{discreteG} |
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\end{equation} |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{method} |
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\caption{Interfacial conductance can be calculated by applying an |
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(unphysical) kinetic energy flux between two slabs, one located |
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within the metal and another on the edge of the periodic box. The |
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system responds by forming a thermal response or a gradient. In |
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bulk liquids, this gradient typically has a single slope, but in |
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interfacial systems, there are distinct thermal conductivity |
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domains. The interfacial conductance, $G$ is found by measuring the |
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temperature gap at the Gibbs dividing surface, or by using second |
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derivatives of the thermal profile.} |
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\label{demoPic} |
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\end{figure} |
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|
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The other approach is to assume a continuous temperature profile along |
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the thermal gradient axis (e.g. $z$) and define $G$ at the point where |
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the magnitude of thermal conductivity ($\lambda$) change reaches its |
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maximum, given that $\lambda$ is well-defined throughout the space: |
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\begin{equation} |
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G^\prime = \Big|\frac{\partial\lambda}{\partial z}\Big| |
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= \Big|\frac{\partial}{\partial z}\left(-J_z\Big/ |
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\left(\frac{\partial T}{\partial z}\right)\right)\Big| |
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= |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
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\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
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\label{derivativeG} |
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\end{equation} |
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|
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With temperature profiles obtained from simulation, one is able to |
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approximate the first and second derivatives of $T$ with finite |
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difference methods and calculate $G^\prime$. In what follows, both |
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definitions have been used, and are compared in the results. |
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|
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To investigate the interfacial conductivity at metal / solvent |
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interfaces, we have modeled a metal slab with its (111) surfaces |
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perpendicular to the $z$-axis of our simulation cells. The metal slab |
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has been prepared both with and without capping agents on the exposed |
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surface, and has been solvated with simple organic solvents, as |
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illustrated in Figure \ref{gradT}. |
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|
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With the simulation cell described above, we are able to equilibrate |
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the system and impose an unphysical thermal flux between the liquid |
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and the metal phase using the NIVS algorithm. By periodically applying |
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the unphysical flux, we obtained a temperature profile and its spatial |
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derivatives. Figure \ref{gradT} shows how an applied thermal flux can |
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be used to obtain the 1st and 2nd derivatives of the temperature |
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profile. |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{gradT} |
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\caption{A sample of Au-butanethiol/hexane interfacial system and the |
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temperature profile after a kinetic energy flux is imposed to |
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it. The 1st and 2nd derivatives of the temperature profile can be |
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obtained with finite difference approximation (lower panel).} |
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\label{gradT} |
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\end{figure} |
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|
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\section{Computational Details} |
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\subsection{Simulation Protocol} |
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The NIVS algorithm has been implemented in our MD simulation code, |
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OpenMD\cite{Meineke:2005gd,openmd}, and was used for our simulations. |
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Metal slabs of 6 or 11 layers of Au atoms were first equilibrated |
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under atmospheric pressure (1 atm) and 200K. After equilibration, |
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butanethiol capping agents were placed at three-fold hollow sites on |
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the Au(111) surfaces. These sites are either {\it fcc} or {\it |
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hcp} sites, although Hase {\it et al.} found that they are |
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equivalent in a heat transfer process,\cite{hase:2010} so we did not |
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distinguish between these sites in our study. The maximum butanethiol |
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capacity on Au surface is $1/3$ of the total number of surface Au |
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atoms, and the packing forms a $(\sqrt{3}\times\sqrt{3})R30^\circ$ |
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structure\cite{doi:10.1021/ja00008a001,doi:10.1021/cr9801317}. A |
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series of lower coverages was also prepared by eliminating |
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butanethiols from the higher coverage surface in a regular manner. The |
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lower coverages were prepared in order to study the relation between |
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coverage and interfacial conductance. |
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|
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The capping agent molecules were allowed to migrate during the |
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simulations. They distributed themselves uniformly and sampled a |
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number of three-fold sites throughout out study. Therefore, the |
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initial configuration does not noticeably affect the sampling of a |
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variety of configurations of the same coverage, and the final |
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conductance measurement would be an average effect of these |
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configurations explored in the simulations. |
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|
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After the modified Au-butanethiol surface systems were equilibrated in |
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the canonical (NVT) ensemble, organic solvent molecules were packed in |
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the previously empty part of the simulation cells.\cite{packmol} Two |
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solvents were investigated, one which has little vibrational overlap |
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with the alkanethiol and which has a planar shape (toluene), and one |
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which has similar vibrational frequencies to the capping agent and |
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chain-like shape ({\it n}-hexane). |
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|
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The simulation cells were not particularly extensive along the |
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$z$-axis, as a very long length scale for the thermal gradient may |
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cause excessively hot or cold temperatures in the middle of the |
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solvent region and lead to undesired phenomena such as solvent boiling |
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or freezing when a thermal flux is applied. Conversely, too few |
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solvent molecules would change the normal behavior of the liquid |
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phase. Therefore, our $N_{solvent}$ values were chosen to ensure that |
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these extreme cases did not happen to our simulations. The spacing |
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between periodic images of the gold interfaces is $45 \sim 75$\AA. |
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|
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The initial configurations generated are further equilibrated with the |
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$x$ and $y$ dimensions fixed, only allowing the $z$-length scale to |
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change. This is to ensure that the equilibration of liquid phase does |
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not affect the metal's crystalline structure. Comparisons were made |
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with simulations that allowed changes of $L_x$ and $L_y$ during NPT |
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equilibration. No substantial changes in the box geometry were noticed |
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in these simulations. After ensuring the liquid phase reaches |
311 |
equilibrium at atmospheric pressure (1 atm), further equilibration was |
312 |
carried out under canonical (NVT) and microcanonical (NVE) ensembles. |
313 |
|
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After the systems reach equilibrium, NIVS was used to impose an |
315 |
unphysical thermal flux between the metal and the liquid phases. Most |
316 |
of our simulations were done under an average temperature of |
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$\sim$200K. Therefore, thermal flux usually came from the metal to the |
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liquid so that the liquid has a higher temperature and would not |
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freeze due to lowered temperatures. After this induced temperature |
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gradient had stablized, the temperature profile of the simulation cell |
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was recorded. To do this, the simulation cell is devided evenly into |
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$N$ slabs along the $z$-axis. The average temperatures of each slab |
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are recorded for 1$\sim$2 ns. When the slab width $d$ of each slab is |
324 |
the same, the derivatives of $T$ with respect to slab number $n$ can |
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be directly used for $G^\prime$ calculations: \begin{equation} |
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G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
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\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
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= |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big| |
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\Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2 |
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= |J_z|\Big|\frac{\partial^2 T}{\partial n^2}\Big| |
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\Big/\left(\frac{\partial T}{\partial n}\right)^2 |
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\label{derivativeG2} |
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\end{equation} |
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|
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All of the above simulation procedures use a time step of 1 fs. Each |
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equilibration stage took a minimum of 100 ps, although in some cases, |
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longer equilibration stages were utilized. |
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|
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\subsection{Force Field Parameters} |
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Our simulations include a number of chemically distinct components. |
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Figure \ref{demoMol} demonstrates the sites defined for both |
342 |
United-Atom and All-Atom models of the organic solvent and capping |
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agents in our simulations. Force field parameters are needed for |
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interactions both between the same type of particles and between |
345 |
particles of different species. |
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|
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\begin{figure} |
348 |
\includegraphics[width=\linewidth]{structures} |
349 |
\caption{Structures of the capping agent and solvents utilized in |
350 |
these simulations. The chemically-distinct sites (a-e) are expanded |
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in terms of constituent atoms for both United Atom (UA) and All Atom |
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(AA) force fields. Most parameters are from |
353 |
Refs. \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}.} |
354 |
\label{demoMol} |
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\end{figure} |
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|
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The Au-Au interactions in metal lattice slab is described by the |
358 |
quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
359 |
potentials include zero-point quantum corrections and are |
360 |
reparametrized for accurate surface energies compared to the |
361 |
Sutton-Chen potentials.\cite{Chen90} |
362 |
|
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For the two solvent molecules, {\it n}-hexane and toluene, two |
364 |
different atomistic models were utilized. Both solvents were modeled |
365 |
using United-Atom (UA) and All-Atom (AA) models. The TraPPE-UA |
366 |
parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
367 |
for our UA solvent molecules. In these models, sites are located at |
368 |
the carbon centers for alkyl groups. Bonding interactions, including |
369 |
bond stretches and bends and torsions, were used for intra-molecular |
370 |
sites closer than 3 bonds. For non-bonded interactions, Lennard-Jones |
371 |
potentials are used. |
372 |
|
373 |
By eliminating explicit hydrogen atoms, the TraPPE-UA models are |
374 |
simple and computationally efficient, while maintaining good accuracy. |
375 |
However, the TraPPE-UA model for alkanes is known to predict a slighly |
376 |
lower boiling point than experimental values. This is one of the |
377 |
reasons we used a lower average temperature (200K) for our |
378 |
simulations. If heat is transferred to the liquid phase during the |
379 |
NIVS simulation, the liquid in the hot slab can actually be |
380 |
substantially warmer than the mean temperature in the simulation. The |
381 |
lower mean temperatures therefore prevent solvent boiling. |
382 |
|
383 |
For UA-toluene, the non-bonded potentials between intermolecular sites |
384 |
have a similar Lennard-Jones formulation. The toluene molecules were |
385 |
treated as a single rigid body, so there was no need for |
386 |
intramolecular interactions (including bonds, bends, or torsions) in |
387 |
this solvent model. |
388 |
|
389 |
Besides the TraPPE-UA models, AA models for both organic solvents are |
390 |
included in our studies as well. The OPLS-AA\cite{OPLSAA} force fields |
391 |
were used. For hexane, additional explicit hydrogen sites were |
392 |
included. Besides bonding and non-bonded site-site interactions, |
393 |
partial charges and the electrostatic interactions were added to each |
394 |
CT and HC site. For toluene, a flexible model for the toluene molecule |
395 |
was utilized which included bond, bend, torsion, and inversion |
396 |
potentials to enforce ring planarity. |
397 |
|
398 |
The butanethiol capping agent in our simulations, were also modeled |
399 |
with both UA and AA model. The TraPPE-UA force field includes |
400 |
parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for |
401 |
UA butanethiol model in our simulations. The OPLS-AA also provides |
402 |
parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) |
403 |
surfaces do not have the hydrogen atom bonded to sulfur. To derive |
404 |
suitable parameters for butanethiol adsorbed on Au(111) surfaces, we |
405 |
adopt the S parameters from Luedtke and Landman\cite{landman:1998} and |
406 |
modify the parameters for the CTS atom to maintain charge neutrality |
407 |
in the molecule. Note that the model choice (UA or AA) for the capping |
408 |
agent can be different from the solvent. Regardless of model choice, |
409 |
the force field parameters for interactions between capping agent and |
410 |
solvent can be derived using Lorentz-Berthelot Mixing Rule: |
411 |
\begin{eqnarray} |
412 |
\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\ |
413 |
\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}} |
414 |
\end{eqnarray} |
415 |
|
416 |
To describe the interactions between metal (Au) and non-metal atoms, |
417 |
we refer to an adsorption study of alkyl thiols on gold surfaces by |
418 |
Vlugt {\it et al.}\cite{vlugt:cpc2007154} They fitted an effective |
419 |
Lennard-Jones form of potential parameters for the interaction between |
420 |
Au and pseudo-atoms CH$_x$ and S based on a well-established and |
421 |
widely-used effective potential of Hautman and Klein for the Au(111) |
422 |
surface.\cite{hautman:4994} As our simulations require the gold slab |
423 |
to be flexible to accommodate thermal excitation, the pair-wise form |
424 |
of potentials they developed was used for our study. |
425 |
|
426 |
The potentials developed from {\it ab initio} calculations by Leng |
427 |
{\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
428 |
interactions between Au and aromatic C/H atoms in toluene. However, |
429 |
the Lennard-Jones parameters between Au and other types of particles, |
430 |
(e.g. AA alkanes) have not yet been established. For these |
431 |
interactions, the Lorentz-Berthelot mixing rule can be used to derive |
432 |
effective single-atom LJ parameters for the metal using the fit values |
433 |
for toluene. These are then used to construct reasonable mixing |
434 |
parameters for the interactions between the gold and other atoms. |
435 |
Table \ref{MnM} summarizes the ``metal/non-metal'' parameters used in |
436 |
our simulations. |
437 |
|
438 |
\begin{table*} |
439 |
\begin{minipage}{\linewidth} |
440 |
\begin{center} |
441 |
\caption{Non-bonded interaction parameters (including cross |
442 |
interactions with Au atoms) for both force fields used in this |
443 |
work.} |
444 |
\begin{tabular}{lllllll} |
445 |
\hline\hline |
446 |
& Site & $\sigma_{ii}$ & $\epsilon_{ii}$ & $q_i$ & |
447 |
$\sigma_{Au-i}$ & $\epsilon_{Au-i}$ \\ |
448 |
& & (\AA) & (kcal/mol) & ($e$) & (\AA) & (kcal/mol) \\ |
449 |
\hline |
450 |
United Atom (UA) |
451 |
&CH3 & 3.75 & 0.1947 & - & 3.54 & 0.2146 \\ |
452 |
&CH2 & 3.95 & 0.0914 & - & 3.54 & 0.1749 \\ |
453 |
&CHar & 3.695 & 0.1003 & - & 3.4625 & 0.1680 \\ |
454 |
&CRar & 3.88 & 0.04173 & - & 3.555 & 0.1604 \\ |
455 |
\hline |
456 |
All Atom (AA) |
457 |
&CT3 & 3.50 & 0.066 & -0.18 & 3.365 & 0.1373 \\ |
458 |
&CT2 & 3.50 & 0.066 & -0.12 & 3.365 & 0.1373 \\ |
459 |
&CTT & 3.50 & 0.066 & -0.065 & 3.365 & 0.1373 \\ |
460 |
&HC & 2.50 & 0.030 & 0.06 & 2.865 & 0.09256 \\ |
461 |
&CA & 3.55 & 0.070 & -0.115 & 3.173 & 0.0640 \\ |
462 |
&HA & 2.42 & 0.030 & 0.115 & 2.746 & 0.0414 \\ |
463 |
\hline |
464 |
Both UA and AA |
465 |
& S & 4.45 & 0.25 & - & 2.40 & 8.465 \\ |
466 |
\hline\hline |
467 |
\end{tabular} |
468 |
\label{MnM} |
469 |
\end{center} |
470 |
\end{minipage} |
471 |
\end{table*} |
472 |
|
473 |
\subsection{Vibrational Power Spectrum} |
474 |
|
475 |
To investigate the mechanism of interfacial thermal conductance, the |
476 |
vibrational power spectrum was computed. Power spectra were taken for |
477 |
individual components in different simulations. To obtain these |
478 |
spectra, simulations were run after equilibration, in the NVE |
479 |
ensemble, and without a thermal gradient. Snapshots of configurations |
480 |
were collected at a frequency that is higher than that of the fastest |
481 |
vibrations occuring in the simulations. With these configurations, the |
482 |
velocity auto-correlation functions can be computed: |
483 |
\begin{equation} |
484 |
C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle |
485 |
\label{vCorr} |
486 |
\end{equation} |
487 |
The power spectrum is constructed via a Fourier transform of the |
488 |
symmetrized velocity autocorrelation function, |
489 |
\begin{equation} |
490 |
\hat{f}(\omega) = |
491 |
\int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt |
492 |
\label{fourier} |
493 |
\end{equation} |
494 |
|
495 |
\section{Results and Discussions} |
496 |
In what follows, how the parameters and protocol of simulations would |
497 |
affect the measurement of $G$'s is first discussed. With a reliable |
498 |
protocol and set of parameters, the influence of capping agent |
499 |
coverage on thermal conductance is investigated. Besides, different |
500 |
force field models for both solvents and selected deuterated models |
501 |
were tested and compared. Finally, a summary of the role of capping |
502 |
agent in the interfacial thermal transport process is given. |
503 |
|
504 |
\subsection{How Simulation Parameters Affects $G$} |
505 |
We have varied our protocol or other parameters of the simulations in |
506 |
order to investigate how these factors would affect the measurement of |
507 |
$G$'s. It turned out that while some of these parameters would not |
508 |
affect the results substantially, some other changes to the |
509 |
simulations would have a significant impact on the measurement |
510 |
results. |
511 |
|
512 |
In some of our simulations, we allowed $L_x$ and $L_y$ to change |
513 |
during equilibrating the liquid phase. Due to the stiffness of the |
514 |
crystalline Au structure, $L_x$ and $L_y$ would not change noticeably |
515 |
after equilibration. Although $L_z$ could fluctuate $\sim$1\% after a |
516 |
system is fully equilibrated in the NPT ensemble, this fluctuation, as |
517 |
well as those of $L_x$ and $L_y$ (which is significantly smaller), |
518 |
would not be magnified on the calculated $G$'s, as shown in Table |
519 |
\ref{AuThiolHexaneUA}. This insensivity to $L_i$ fluctuations allows |
520 |
reliable measurement of $G$'s without the necessity of extremely |
521 |
cautious equilibration process. |
522 |
|
523 |
As stated in our computational details, the spacing filled with |
524 |
solvent molecules can be chosen within a range. This allows some |
525 |
change of solvent molecule numbers for the same Au-butanethiol |
526 |
surfaces. We did this study on our Au-butanethiol/hexane |
527 |
simulations. Nevertheless, the results obtained from systems of |
528 |
different $N_{hexane}$ did not indicate that the measurement of $G$ is |
529 |
susceptible to this parameter. For computational efficiency concern, |
530 |
smaller system size would be preferable, given that the liquid phase |
531 |
structure is not affected. |
532 |
|
533 |
Our NIVS algorithm allows change of unphysical thermal flux both in |
534 |
direction and in quantity. This feature extends our investigation of |
535 |
interfacial thermal conductance. However, the magnitude of this |
536 |
thermal flux is not arbitary if one aims to obtain a stable and |
537 |
reliable thermal gradient. A temperature profile would be |
538 |
substantially affected by noise when $|J_z|$ has a much too low |
539 |
magnitude; while an excessively large $|J_z|$ that overwhelms the |
540 |
conductance capacity of the interface would prevent a thermal gradient |
541 |
to reach a stablized steady state. NIVS has the advantage of allowing |
542 |
$J$ to vary in a wide range such that the optimal flux range for $G$ |
543 |
measurement can generally be simulated by the algorithm. Within the |
544 |
optimal range, we were able to study how $G$ would change according to |
545 |
the thermal flux across the interface. For our simulations, we denote |
546 |
$J_z$ to be positive when the physical thermal flux is from the liquid |
547 |
to metal, and negative vice versa. The $G$'s measured under different |
548 |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and |
549 |
\ref{AuThiolToluene}. These results do not suggest that $G$ is |
550 |
dependent on $J_z$ within this flux range. The linear response of flux |
551 |
to thermal gradient simplifies our investigations in that we can rely |
552 |
on $G$ measurement with only a couple $J_z$'s and do not need to test |
553 |
a large series of fluxes. |
554 |
|
555 |
\begin{table*} |
556 |
\begin{minipage}{\linewidth} |
557 |
\begin{center} |
558 |
\caption{Computed interfacial thermal conductivity ($G$ and |
559 |
$G^\prime$) values for the 100\% covered Au-butanethiol/hexane |
560 |
interfaces with UA model and different hexane molecule numbers |
561 |
at different temperatures using a range of energy |
562 |
fluxes. Error estimates indicated in parenthesis.} |
563 |
|
564 |
\begin{tabular}{ccccccc} |
565 |
\hline\hline |
566 |
$\langle T\rangle$ & $N_{hexane}$ & Fixed & $\rho_{hexane}$ & |
567 |
$J_z$ & $G$ & $G^\prime$ \\ |
568 |
(K) & & $L_x$ \& $L_y$? & (g/cm$^3$) & (GW/m$^2$) & |
569 |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
570 |
\hline |
571 |
200 & 266 & No & 0.672 & -0.96 & 102(3) & 80.0(0.8) \\ |
572 |
& 200 & Yes & 0.694 & 1.92 & 129(11) & 87.3(0.3) \\ |
573 |
& & Yes & 0.672 & 1.93 & 131(16) & 78(13) \\ |
574 |
& & No & 0.688 & 0.96 & 125(16) & 90.2(15) \\ |
575 |
& & & & 1.91 & 139(10) & 101(10) \\ |
576 |
& & & & 2.83 & 141(6) & 89.9(9.8) \\ |
577 |
& 166 & Yes & 0.679 & 0.97 & 115(19) & 69(18) \\ |
578 |
& & & & 1.94 & 125(9) & 87.1(0.2) \\ |
579 |
& & No & 0.681 & 0.97 & 141(30) & 78(22) \\ |
580 |
& & & & 1.92 & 138(4) & 98.9(9.5) \\ |
581 |
\hline |
582 |
250 & 200 & No & 0.560 & 0.96 & 75(10) & 61.8(7.3) \\ |
583 |
& & & & -0.95 & 49.4(0.3) & 45.7(2.1) \\ |
584 |
& 166 & Yes & 0.570 & 0.98 & 79.0(3.5) & 62.9(3.0) \\ |
585 |
& & No & 0.569 & 0.97 & 80.3(0.6) & 67(11) \\ |
586 |
& & & & 1.44 & 76.2(5.0) & 64.8(3.8) \\ |
587 |
& & & & -0.95 & 56.4(2.5) & 54.4(1.1) \\ |
588 |
& & & & -1.85 & 47.8(1.1) & 53.5(1.5) \\ |
589 |
\hline\hline |
590 |
\end{tabular} |
591 |
\label{AuThiolHexaneUA} |
592 |
\end{center} |
593 |
\end{minipage} |
594 |
\end{table*} |
595 |
|
596 |
Furthermore, we also attempted to increase system average temperatures |
597 |
to above 200K. These simulations are first equilibrated in the NPT |
598 |
ensemble under normal pressure. As stated above, the TraPPE-UA model |
599 |
for hexane tends to predict a lower boiling point. In our simulations, |
600 |
hexane had diffculty to remain in liquid phase when NPT equilibration |
601 |
temperature is higher than 250K. Additionally, the equilibrated liquid |
602 |
hexane density under 250K becomes lower than experimental value. This |
603 |
expanded liquid phase leads to lower contact between hexane and |
604 |
butanethiol as well.[MAY NEED SLAB DENSITY FIGURE] |
605 |
And this reduced contact would |
606 |
probably be accountable for a lower interfacial thermal conductance, |
607 |
as shown in Table \ref{AuThiolHexaneUA}. |
608 |
|
609 |
A similar study for TraPPE-UA toluene agrees with the above result as |
610 |
well. Having a higher boiling point, toluene tends to remain liquid in |
611 |
our simulations even equilibrated under 300K in NPT |
612 |
ensembles. Furthermore, the expansion of the toluene liquid phase is |
613 |
not as significant as that of the hexane. This prevents severe |
614 |
decrease of liquid-capping agent contact and the results (Table |
615 |
\ref{AuThiolToluene}) show only a slightly decreased interface |
616 |
conductance. Therefore, solvent-capping agent contact should play an |
617 |
important role in the thermal transport process across the interface |
618 |
in that higher degree of contact could yield increased conductance. |
619 |
|
620 |
\begin{table*} |
621 |
\begin{minipage}{\linewidth} |
622 |
\begin{center} |
623 |
\caption{Computed interfacial thermal conductivity ($G$ and |
624 |
$G^\prime$) values for a 90\% coverage Au-butanethiol/toluene |
625 |
interface at different temperatures using a range of energy |
626 |
fluxes. Error estimates indicated in parenthesis.} |
627 |
|
628 |
\begin{tabular}{ccccc} |
629 |
\hline\hline |
630 |
$\langle T\rangle$ & $\rho_{toluene}$ & $J_z$ & $G$ & $G^\prime$ \\ |
631 |
(K) & (g/cm$^3$) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
632 |
\hline |
633 |
200 & 0.933 & 2.15 & 204(12) & 113(12) \\ |
634 |
& & -1.86 & 180(3) & 135(21) \\ |
635 |
& & -3.93 & 176(5) & 113(12) \\ |
636 |
\hline |
637 |
300 & 0.855 & -1.91 & 143(5) & 125(2) \\ |
638 |
& & -4.19 & 135(9) & 113(12) \\ |
639 |
\hline\hline |
640 |
\end{tabular} |
641 |
\label{AuThiolToluene} |
642 |
\end{center} |
643 |
\end{minipage} |
644 |
\end{table*} |
645 |
|
646 |
Besides lower interfacial thermal conductance, surfaces in relatively |
647 |
high temperatures are susceptible to reconstructions, when |
648 |
butanethiols have a full coverage on the Au(111) surface. These |
649 |
reconstructions include surface Au atoms migrated outward to the S |
650 |
atom layer, and butanethiol molecules embedded into the original |
651 |
surface Au layer. The driving force for this behavior is the strong |
652 |
Au-S interactions in our simulations. And these reconstructions lead |
653 |
to higher ratio of Au-S attraction and thus is energetically |
654 |
favorable. Furthermore, this phenomenon agrees with experimental |
655 |
results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt |
656 |
{\it et al.} had kept their Au(111) slab rigid so that their |
657 |
simulations can reach 300K without surface reconstructions. Without |
658 |
this practice, simulating 100\% thiol covered interfaces under higher |
659 |
temperatures could hardly avoid surface reconstructions. However, our |
660 |
measurement is based on assuming homogeneity on $x$ and $y$ dimensions |
661 |
so that measurement of $T$ at particular $z$ would be an effective |
662 |
average of the particles of the same type. Since surface |
663 |
reconstructions could eliminate the original $x$ and $y$ dimensional |
664 |
homogeneity, measurement of $G$ is more difficult to conduct under |
665 |
higher temperatures. Therefore, most of our measurements are |
666 |
undertaken at $\langle T\rangle\sim$200K. |
667 |
|
668 |
However, when the surface is not completely covered by butanethiols, |
669 |
the simulated system is more resistent to the reconstruction |
670 |
above. Our Au-butanethiol/toluene system had the Au(111) surfaces 90\% |
671 |
covered by butanethiols, but did not see this above phenomena even at |
672 |
$\langle T\rangle\sim$300K. The empty three-fold sites not occupied by |
673 |
capping agents could help prevent surface reconstruction in that they |
674 |
provide other means of capping agent relaxation. It is observed that |
675 |
butanethiols can migrate to their neighbor empty sites during a |
676 |
simulation. Therefore, we were able to obtain $G$'s for these |
677 |
interfaces even at a relatively high temperature without being |
678 |
affected by surface reconstructions. |
679 |
|
680 |
\subsection{Influence of Capping Agent Coverage on $G$} |
681 |
To investigate the influence of butanethiol coverage on interfacial |
682 |
thermal conductance, a series of different coverage Au-butanethiol |
683 |
surfaces is prepared and solvated with various organic |
684 |
molecules. These systems are then equilibrated and their interfacial |
685 |
thermal conductivity are measured with our NIVS algorithm. Figure |
686 |
\ref{coverage} demonstrates the trend of conductance change with |
687 |
respect to different coverages of butanethiol. To study the isotope |
688 |
effect in interfacial thermal conductance, deuterated UA-hexane is |
689 |
included as well. |
690 |
|
691 |
\begin{figure} |
692 |
\includegraphics[width=\linewidth]{coverage} |
693 |
\caption{Comparison of interfacial thermal conductivity ($G$) values |
694 |
for the Au-butanethiol/solvent interface with various UA models and |
695 |
different capping agent coverages at $\langle T\rangle\sim$200K |
696 |
using certain energy flux respectively.} |
697 |
\label{coverage} |
698 |
\end{figure} |
699 |
|
700 |
It turned out that with partial covered butanethiol on the Au(111) |
701 |
surface, the derivative definition for $G^\prime$ |
702 |
(Eq. \ref{derivativeG}) was difficult to apply, due to the difficulty |
703 |
in locating the maximum of change of $\lambda$. Instead, the discrete |
704 |
definition (Eq. \ref{discreteG}) is easier to apply, as the Gibbs |
705 |
deviding surface can still be well-defined. Therefore, $G$ (not |
706 |
$G^\prime$) was used for this section. |
707 |
|
708 |
From Figure \ref{coverage}, one can see the significance of the |
709 |
presence of capping agents. Even when a fraction of the Au(111) |
710 |
surface sites are covered with butanethiols, the conductivity would |
711 |
see an enhancement by at least a factor of 3. This indicates the |
712 |
important role cappping agent is playing for thermal transport |
713 |
phenomena on metal / organic solvent surfaces. |
714 |
|
715 |
Interestingly, as one could observe from our results, the maximum |
716 |
conductance enhancement (largest $G$) happens while the surfaces are |
717 |
about 75\% covered with butanethiols. This again indicates that |
718 |
solvent-capping agent contact has an important role of the thermal |
719 |
transport process. Slightly lower butanethiol coverage allows small |
720 |
gaps between butanethiols to form. And these gaps could be filled with |
721 |
solvent molecules, which acts like ``heat conductors'' on the |
722 |
surface. The higher degree of interaction between these solvent |
723 |
molecules and capping agents increases the enhancement effect and thus |
724 |
produces a higher $G$ than densely packed butanethiol arrays. However, |
725 |
once this maximum conductance enhancement is reached, $G$ decreases |
726 |
when butanethiol coverage continues to decrease. Each capping agent |
727 |
molecule reaches its maximum capacity for thermal |
728 |
conductance. Therefore, even higher solvent-capping agent contact |
729 |
would not offset this effect. Eventually, when butanethiol coverage |
730 |
continues to decrease, solvent-capping agent contact actually |
731 |
decreases with the disappearing of butanethiol molecules. In this |
732 |
case, $G$ decrease could not be offset but instead accelerated. [MAY NEED |
733 |
SNAPSHOT SHOWING THE PHENOMENA / SLAB DENSITY ANALYSIS] |
734 |
|
735 |
A comparison of the results obtained from differenet organic solvents |
736 |
can also provide useful information of the interfacial thermal |
737 |
transport process. The deuterated hexane (UA) results do not appear to |
738 |
be much different from those of normal hexane (UA), given that |
739 |
butanethiol (UA) is non-deuterated for both solvents. These UA model |
740 |
studies, even though eliminating C-H vibration samplings, still have |
741 |
C-C vibrational frequencies different from each other. However, these |
742 |
differences in the infrared range do not seem to produce an observable |
743 |
difference for the results of $G$ (Figure \ref{uahxnua}). |
744 |
|
745 |
\begin{figure} |
746 |
\includegraphics[width=\linewidth]{uahxnua} |
747 |
\caption{Vibrational spectra obtained for normal (upper) and |
748 |
deuterated (lower) hexane in Au-butanethiol/hexane |
749 |
systems. Butanethiol spectra are shown as reference. Both hexane and |
750 |
butanethiol were using United-Atom models.} |
751 |
\label{uahxnua} |
752 |
\end{figure} |
753 |
|
754 |
Furthermore, results for rigid body toluene solvent, as well as other |
755 |
UA-hexane solvents, are reasonable within the general experimental |
756 |
ranges\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406}. This |
757 |
suggests that explicit hydrogen might not be a required factor for |
758 |
modeling thermal transport phenomena of systems such as |
759 |
Au-thiol/organic solvent. |
760 |
|
761 |
However, results for Au-butanethiol/toluene do not show an identical |
762 |
trend with those for Au-butanethiol/hexane in that $G$ remains at |
763 |
approximately the same magnitue when butanethiol coverage differs from |
764 |
25\% to 75\%. This might be rooted in the molecule shape difference |
765 |
for planar toluene and chain-like {\it n}-hexane. Due to this |
766 |
difference, toluene molecules have more difficulty in occupying |
767 |
relatively small gaps among capping agents when their coverage is not |
768 |
too low. Therefore, the solvent-capping agent contact may keep |
769 |
increasing until the capping agent coverage reaches a relatively low |
770 |
level. This becomes an offset for decreasing butanethiol molecules on |
771 |
its effect to the process of interfacial thermal transport. Thus, one |
772 |
can see a plateau of $G$ vs. butanethiol coverage in our results. |
773 |
|
774 |
\subsection{Influence of Chosen Molecule Model on $G$} |
775 |
In addition to UA solvent/capping agent models, AA models are included |
776 |
in our simulations as well. Besides simulations of the same (UA or AA) |
777 |
model for solvent and capping agent, different models can be applied |
778 |
to different components. Furthermore, regardless of models chosen, |
779 |
either the solvent or the capping agent can be deuterated, similar to |
780 |
the previous section. Table \ref{modelTest} summarizes the results of |
781 |
these studies. |
782 |
|
783 |
\begin{table*} |
784 |
\begin{minipage}{\linewidth} |
785 |
\begin{center} |
786 |
|
787 |
\caption{Computed interfacial thermal conductivity ($G$ and |
788 |
$G^\prime$) values for interfaces using various models for |
789 |
solvent and capping agent (or without capping agent) at |
790 |
$\langle T\rangle\sim$200K. (D stands for deuterated solvent |
791 |
or capping agent molecules; ``Avg.'' denotes results that are |
792 |
averages of simulations under different $J_z$'s. Error |
793 |
estimates indicated in parenthesis.)} |
794 |
|
795 |
\begin{tabular}{llccc} |
796 |
\hline\hline |
797 |
Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ |
798 |
(or bare surface) & model & (GW/m$^2$) & |
799 |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
800 |
\hline |
801 |
UA & UA hexane & Avg. & 131(9) & 87(10) \\ |
802 |
& UA hexane(D) & 1.95 & 153(5) & 136(13) \\ |
803 |
& AA hexane & Avg. & 131(6) & 122(10) \\ |
804 |
& UA toluene & 1.96 & 187(16) & 151(11) \\ |
805 |
& AA toluene & 1.89 & 200(36) & 149(53) \\ |
806 |
\hline |
807 |
AA & UA hexane & 1.94 & 116(9) & 129(8) \\ |
808 |
& AA hexane & Avg. & 442(14) & 356(31) \\ |
809 |
& AA hexane(D) & 1.93 & 222(12) & 234(54) \\ |
810 |
& UA toluene & 1.98 & 125(25) & 97(60) \\ |
811 |
& AA toluene & 3.79 & 487(56) & 290(42) \\ |
812 |
\hline |
813 |
AA(D) & UA hexane & 1.94 & 158(25) & 172(4) \\ |
814 |
& AA hexane & 1.92 & 243(29) & 191(11) \\ |
815 |
& AA toluene & 1.93 & 364(36) & 322(67) \\ |
816 |
\hline |
817 |
bare & UA hexane & Avg. & 46.5(3.2) & 49.4(4.5) \\ |
818 |
& UA hexane(D) & 0.98 & 43.9(4.6) & 43.0(2.0) \\ |
819 |
& AA hexane & 0.96 & 31.0(1.4) & 29.4(1.3) \\ |
820 |
& UA toluene & 1.99 & 70.1(1.3) & 65.8(0.5) \\ |
821 |
\hline\hline |
822 |
\end{tabular} |
823 |
\label{modelTest} |
824 |
\end{center} |
825 |
\end{minipage} |
826 |
\end{table*} |
827 |
|
828 |
To facilitate direct comparison, the same system with differnt models |
829 |
for different components uses the same length scale for their |
830 |
simulation cells. Without the presence of capping agent, using |
831 |
different models for hexane yields similar results for both $G$ and |
832 |
$G^\prime$, and these two definitions agree with eath other very |
833 |
well. This indicates very weak interaction between the metal and the |
834 |
solvent, and is a typical case for acoustic impedance mismatch between |
835 |
these two phases. |
836 |
|
837 |
As for Au(111) surfaces completely covered by butanethiols, the choice |
838 |
of models for capping agent and solvent could impact the measurement |
839 |
of $G$ and $G^\prime$ quite significantly. For Au-butanethiol/hexane |
840 |
interfaces, using AA model for both butanethiol and hexane yields |
841 |
substantially higher conductivity values than using UA model for at |
842 |
least one component of the solvent and capping agent, which exceeds |
843 |
the general range of experimental measurement results. This is |
844 |
probably due to the classically treated C-H vibrations in the AA |
845 |
model, which should not be appreciably populated at normal |
846 |
temperatures. In comparison, once either the hexanes or the |
847 |
butanethiols are deuterated, one can see a significantly lower $G$ and |
848 |
$G^\prime$. In either of these cases, the C-H(D) vibrational overlap |
849 |
between the solvent and the capping agent is removed (Figure |
850 |
\ref{aahxntln}). Conclusively, the improperly treated C-H vibration in |
851 |
the AA model produced over-predicted results accordingly. Compared to |
852 |
the AA model, the UA model yields more reasonable results with higher |
853 |
computational efficiency. |
854 |
|
855 |
\begin{figure} |
856 |
\includegraphics[width=\linewidth]{aahxntln} |
857 |
\caption{Spectra obtained for All-Atom model Au-butanethil/solvent |
858 |
systems. When butanethiol is deuterated (lower left), its |
859 |
vibrational overlap with hexane would decrease significantly, |
860 |
compared with normal butanethiol (upper left). However, this |
861 |
dramatic change does not apply to toluene as much (right).} |
862 |
\label{aahxntln} |
863 |
\end{figure} |
864 |
|
865 |
However, for Au-butanethiol/toluene interfaces, having the AA |
866 |
butanethiol deuterated did not yield a significant change in the |
867 |
measurement results. Compared to the C-H vibrational overlap between |
868 |
hexane and butanethiol, both of which have alkyl chains, that overlap |
869 |
between toluene and butanethiol is not so significant and thus does |
870 |
not have as much contribution to the heat exchange |
871 |
process. Conversely, extra degrees of freedom such as the C-H |
872 |
vibrations could yield higher heat exchange rate between these two |
873 |
phases and result in a much higher conductivity. |
874 |
|
875 |
Although the QSC model for Au is known to predict an overly low value |
876 |
for bulk metal gold conductivity\cite{kuang:164101}, our computational |
877 |
results for $G$ and $G^\prime$ do not seem to be affected by this |
878 |
drawback of the model for metal. Instead, our results suggest that the |
879 |
modeling of interfacial thermal transport behavior relies mainly on |
880 |
the accuracy of the interaction descriptions between components |
881 |
occupying the interfaces. |
882 |
|
883 |
\subsection{Role of Capping Agent in Interfacial Thermal Conductance} |
884 |
The vibrational spectra for gold slabs in different environments are |
885 |
shown as in Figure \ref{specAu}. Regardless of the presence of |
886 |
solvent, the gold surfaces covered by butanethiol molecules, compared |
887 |
to bare gold surfaces, exhibit an additional peak observed at the |
888 |
frequency of $\sim$170cm$^{-1}$, which is attributed to the S-Au |
889 |
bonding vibration. This vibration enables efficient thermal transport |
890 |
from surface Au layer to the capping agents. Therefore, in our |
891 |
simulations, the Au/S interfaces do not appear major heat barriers |
892 |
compared to the butanethiol / solvent interfaces. |
893 |
|
894 |
Simultaneously, the vibrational overlap between butanethiol and |
895 |
organic solvents suggests higher thermal exchange efficiency between |
896 |
these two components. Even exessively high heat transport was observed |
897 |
when All-Atom models were used and C-H vibrations were treated |
898 |
classically. Compared to metal and organic liquid phase, the heat |
899 |
transfer efficiency between butanethiol and organic solvents is closer |
900 |
to that within bulk liquid phase. |
901 |
|
902 |
Furthermore, our observation validated previous |
903 |
results\cite{hase:2010} that the intramolecular heat transport of |
904 |
alkylthiols is highly effecient. As a combinational effects of these |
905 |
phenomena, butanethiol acts as a channel to expedite thermal transport |
906 |
process. The acoustic impedance mismatch between the metal and the |
907 |
liquid phase can be effectively reduced with the presence of suitable |
908 |
capping agents. |
909 |
|
910 |
\begin{figure} |
911 |
\includegraphics[width=\linewidth]{vibration} |
912 |
\caption{Vibrational spectra obtained for gold in different |
913 |
environments.} |
914 |
\label{specAu} |
915 |
\end{figure} |
916 |
|
917 |
[MAY ADD COMPARISON OF AU SLAB WIDTHS] |
918 |
|
919 |
\section{Conclusions} |
920 |
The NIVS algorithm we developed has been applied to simulations of |
921 |
Au-butanethiol surfaces with organic solvents. This algorithm allows |
922 |
effective unphysical thermal flux transferred between the metal and |
923 |
the liquid phase. With the flux applied, we were able to measure the |
924 |
corresponding thermal gradient and to obtain interfacial thermal |
925 |
conductivities. Under steady states, single trajectory simulation |
926 |
would be enough for accurate measurement. This would be advantageous |
927 |
compared to transient state simulations, which need multiple |
928 |
trajectories to produce reliable average results. |
929 |
|
930 |
Our simulations have seen significant conductance enhancement with the |
931 |
presence of capping agent, compared to the bare gold / liquid |
932 |
interfaces. The acoustic impedance mismatch between the metal and the |
933 |
liquid phase is effectively eliminated by proper capping |
934 |
agent. Furthermore, the coverage precentage of the capping agent plays |
935 |
an important role in the interfacial thermal transport |
936 |
process. Moderately lower coverages allow higher contact between |
937 |
capping agent and solvent, and thus could further enhance the heat |
938 |
transfer process. |
939 |
|
940 |
Our measurement results, particularly of the UA models, agree with |
941 |
available experimental data. This indicates that our force field |
942 |
parameters have a nice description of the interactions between the |
943 |
particles at the interfaces. AA models tend to overestimate the |
944 |
interfacial thermal conductance in that the classically treated C-H |
945 |
vibration would be overly sampled. Compared to the AA models, the UA |
946 |
models have higher computational efficiency with satisfactory |
947 |
accuracy, and thus are preferable in interfacial thermal transport |
948 |
modelings. Of the two definitions for $G$, the discrete form |
949 |
(Eq. \ref{discreteG}) was easier to use and gives out relatively |
950 |
consistent results, while the derivative form (Eq. \ref{derivativeG}) |
951 |
is not as versatile. Although $G^\prime$ gives out comparable results |
952 |
and follows similar trend with $G$ when measuring close to fully |
953 |
covered or bare surfaces, the spatial resolution of $T$ profile is |
954 |
limited for accurate computation of derivatives data. |
955 |
|
956 |
Vlugt {\it et al.} has investigated the surface thiol structures for |
957 |
nanocrystal gold and pointed out that they differs from those of the |
958 |
Au(111) surface\cite{landman:1998,vlugt:cpc2007154}. This difference |
959 |
might lead to change of interfacial thermal transport behavior as |
960 |
well. To investigate this problem, an effective means to introduce |
961 |
thermal flux and measure the corresponding thermal gradient is |
962 |
desirable for simulating structures with spherical symmetry. |
963 |
|
964 |
\section{Acknowledgments} |
965 |
Support for this project was provided by the National Science |
966 |
Foundation under grant CHE-0848243. Computational time was provided by |
967 |
the Center for Research Computing (CRC) at the University of Notre |
968 |
Dame. \newpage |
969 |
|
970 |
\bibliography{interfacial} |
971 |
|
972 |
\end{doublespace} |
973 |
\end{document} |
974 |
|