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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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|
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\begin{doublespace} |
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|
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\begin{abstract} |
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|
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We have developed a Non-Isotropic Velocity Scaling algorithm for |
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setting up and maintaining stable thermal gradients in non-equilibrium |
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molecular dynamics simulations. This approach effectively imposes |
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unphysical thermal flux even between particles of different |
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identities, conserves linear momentum and kinetic energy, and |
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minimally perturbs the velocity profile of a system when compared with |
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previous RNEMD methods. We have used this method to simulate thermal |
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conductance at metal / organic solvent interfaces both with and |
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without the presence of thiol-based capping agents. We obtained |
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values comparable with experimental values, and observed significant |
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conductance enhancement with the presence of capping agents. Computed |
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power spectra indicate the acoustic impedance mismatch between metal |
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and liquid phase is greatly reduced by the capping agents and thus |
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leads to higher interfacial thermal transfer efficiency. |
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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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[BACKGROUND FOR INTERFACIAL THERMAL CONDUCTANCE PROBLEM] |
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Interfacial thermal conductance is extensively studied both |
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experimentally and computationally, and systems with interfaces |
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present are generally heterogeneous. Although interfaces are commonly |
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barriers to heat transfer, it has been |
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reported\cite{doi:10.1021/la904855s} that under specific circustances, |
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e.g. with certain capping agents present on the surface, interfacial |
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conductance can be significantly enhanced. However, heat conductance |
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of molecular and nano-scale interfaces will be affected by the |
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chemical details of the surface and is challenging to |
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experimentalist. The lower thermal flux through interfaces is even |
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more difficult to measure with EMD and forward NEMD simulation |
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methods. Therefore, developing good simulation methods will be |
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desirable in order to investigate thermal transport across interfaces. |
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|
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Recently, we have developed the 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 more effective thermal exchange |
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between particles of different identities, and thus enables extensive |
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study of interfacial conductance. |
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|
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\section{Methodology} |
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\subsection{Algorithm} |
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[BACKGROUND FOR MD METHODS] |
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There have been many algorithms for computing thermal conductivity |
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using molecular dynamics simulations. However, interfacial conductance |
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is at least an order of magnitude smaller. This would make the |
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calculation even more difficult for those slowly-converging |
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equilibrium methods. Imposed-flux non-equilibrium |
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methods\cite{MullerPlathe:1997xw} have the flux set {\it a priori} and |
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the response of temperature or momentum gradients are easier to |
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measure than the flux, if unknown, and thus, is a preferable way to |
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the forward NEMD methods. Although the momentum swapping approach for |
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flux-imposing can be used for exchanging energy between particles of |
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different identity, the kinetic energy transfer efficiency is affected |
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by the mass difference between the particles, which limits its |
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application on heterogeneous interfacial systems. |
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|
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The non-isotropic velocity scaling (NIVS)\cite{kuang:164101} approach in |
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non-equilibrium MD simulations is able to impose relatively large |
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kinetic energy flux without obvious perturbation to the velocity |
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distribution 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 quantity is not restricted by particle mass difference. |
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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 momentum/energy flux satisfaction is |
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solved. With the scaling operation applied to the system in a set |
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frequency, corresponding momentum/temperature gradients can be built, |
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which can be used for computing transportation properties and other |
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applications related to momentum/temperature gradients. The NIVS |
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algorithm conserves momenta and energy and does not depend on an |
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external thermostat. |
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|
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\subsection{Defining Interfacial Thermal Conductivity $G$} |
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For interfaces with a relatively low interfacial conductance, the bulk |
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regions on either side of an interface rapidly come to a state in |
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which the two phases have relatively homogeneous (but distinct) |
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temperatures. The interfacial thermal conductivity $G$ can therefore |
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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 imposed non-physical kinetic energy |
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transfer and ${\langle T_\mathrm{hot}\rangle}$ and ${\langle |
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T_\mathrm{cold}\rangle}$ are the average observed temperature of the |
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two separated phases. |
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|
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When the interfacial conductance is {\it not} small, two ways can be |
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used to define $G$. |
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|
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One way is to assume the temperature is discretely different on two |
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sides of the interface, $G$ can be calculated with the thermal flux |
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applied $J$ and the maximum temperature difference measured along the |
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thermal gradient max($\Delta T$), which occurs at the interface, as: |
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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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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 reach 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 the temperature profile obtained from simulations, one is able to |
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approximate the first and second derivatives of $T$ with finite |
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difference method and thus calculate $G^\prime$. |
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|
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In what follows, both definitions are used for calculation and comparison. |
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|
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[IMPOSE G DEFINITION INTO OUR SYSTEMS] |
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To facilitate the use of the above definitions in calculating $G$ and |
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$G^\prime$, we have a metal slab with its (111) surfaces perpendicular |
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to the $z$-axis of our simulation cells. With or withour capping |
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agents on the surfaces, the metal slab is solvated with organic |
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solvents, as illustrated in Figure \ref{demoPic}. |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{demoPic} |
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\caption{A sample showing how a metal slab has its (111) surface |
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covered by capping agent molecules and solvated by hexane.} |
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\label{demoPic} |
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\end{figure} |
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|
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With a simulation cell setup following the above manner, one is able |
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to equilibrate the system and impose an unphysical thermal flux |
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between the liquid and the metal phase with the NIVS algorithm. Under |
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a stablized thermal gradient induced by periodically applying the |
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unphysical flux, one is able to obtain a temperature profile and the |
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physical thermal flux corresponding to it, which equals to the |
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unphysical flux applied by NIVS. These data enables the evaluation of |
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the interfacial thermal conductance of a surface. Figure \ref{gradT} |
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is an example how those stablized thermal gradient can be used to |
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obtain the 1st and 2nd derivatives of the temperature profile. |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{gradT} |
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\caption{The 1st and 2nd derivatives of temperature profile can be |
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obtained with finite difference approximation.} |
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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{System Geometry} |
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In our simulations, Au is used to construct a metal slab with bare |
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(111) surface perpendicular to the $z$-axis. Different slab thickness |
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(layer numbers of Au) are simulated. This metal slab is first |
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equilibrated under normal pressure (1 atm) and a desired |
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temperature. After equilibration, butanethiol is used as the capping |
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agent molecule to cover the bare Au (111) surfaces evenly. The sulfur |
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atoms in the butanethiol molecules would occupy the three-fold sites |
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of the surfaces, and the maximal butanethiol capacity on Au surface is |
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$1/3$ of the total number of surface Au atoms[CITATION]. A series of |
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different coverage surfaces is investigated in order to study the |
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relation between coverage and conductance. |
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|
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[COVERAGE DISCRIPTION] However, since the interactions between surface |
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Au and butanethiol is non-bonded, the capping agent molecules are |
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allowed to migrate to an empty neighbor three-fold site during a |
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simulation. Therefore, the initial configuration would not severely |
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affect the sampling of a variety of configurations of the same |
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coverage, and the final conductance measurement would be an average |
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effect of these configurations explored in the simulations. [MAY NEED FIGURES] |
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|
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After the modified Au-butanethiol surface systems are equilibrated |
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under canonical ensemble, Packmol\cite{packmol} is used to pack |
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organic solvent molecules in the previously vacuum part of the |
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simulation cells, which guarantees that short range repulsive |
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interactions do not disrupt the simulations. Two solvents are |
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investigated, one which has little vibrational overlap with the |
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alkanethiol and plane-like shape (toluene), and one which has similar |
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vibrational frequencies and chain-like shape ({\it n}-hexane). The |
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spacing filled by solvent molecules, i.e. the gap between periodically |
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repeated Au-butanethiol surfaces should be carefully chosen so that it |
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would not be too short to affect the liquid phase structure, nor too |
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long, leading to over cooling (freezing) or heating (boiling) when a |
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thermal flux is applied. In our simulations, this spacing is usually |
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$35 \sim 60$\AA. |
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|
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The initial configurations generated by Packmol are further |
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equilibrated with the $x$ and $y$ dimensions fixed, only allowing |
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length scale change in $z$ dimension. This is to ensure that the |
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equilibration of liquid phase does not affect the metal crystal |
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structure in $x$ and $y$ dimensions. Further equilibration are run |
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under NVT and then NVE ensembles. |
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|
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After the systems reach equilibrium, NIVS is implemented to impose a |
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periodic unphysical thermal flux between the metal and the liquid |
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phase. Most of our simulations are under an average temperature of |
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$\sim$200K. Therefore, this flux usually comes 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 excessively low temperature. This induced temperature |
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gradient is stablized and the simulation cell is devided evenly into |
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N slabs along the $z$-axis and the temperatures of each slab are |
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recorded. When the slab width $d$ of each slab is the same, the |
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derivatives of $T$ with respect to slab number $n$ can be directly |
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used for $G^\prime$ calculations: |
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\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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\subsection{Force Field Parameters} |
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Our simulations include various components. Therefore, force field |
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parameter descriptions are needed for interactions both between the |
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same type of particles and between particles of different species. |
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|
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The Au-Au interactions in metal lattice slab is described by the |
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quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
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potentials include zero-point quantum corrections and are |
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reparametrized for accurate surface energies compared to the |
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Sutton-Chen potentials\cite{Chen90}. |
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|
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For both solvent molecules, straight chain {\it n}-hexane and aromatic |
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toluene, United-Atom (UA) and All-Atom (AA) models are used |
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respectively. The TraPPE-UA |
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parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
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for our UA solvent molecules. In these models, pseudo-atoms are |
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located at the carbon centers for alkyl groups. By eliminating |
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explicit hydrogen atoms, these models are simple and computationally |
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efficient, while maintains good accuracy. [LOW BOILING POINT IS A |
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KNOWN PROBLEM FOR TRAPPE-UA ALKANES, NEED MORE DISCUSSION] |
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for |
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toluene, force fields are |
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used with rigid body constraints applied.[MORE DETAILS NEEDED] |
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|
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Besides the TraPPE-UA models, AA models are included in our studies as |
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well. For hexane, the OPLS all-atom\cite{OPLSAA} force field is |
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used. [MORE DETAILS] |
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For toluene, |
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|
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Buatnethiol molecules are used as capping agent for some of our |
309 |
simulations. United-Atom\cite{TraPPE-UA.thiols} and All-Atom models |
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are respectively used corresponding to the force field type of |
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solvent. |
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|
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To describe the interactions between metal Au and non-metal capping |
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agent and solvent, we refer to Vlugt\cite{vlugt:cpc2007154} and derive |
315 |
other interactions which are not parametrized in their work. (can add |
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hautman and klein's paper here and more discussion; need to put |
317 |
aromatic-metal interaction approximation here) |
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|
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[TABULATED FORCE FIELD PARAMETERS NEEDED] |
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|
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\section{Results} |
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\subsection{Toluene Solvent} |
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|
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The results (Table \ref{AuThiolToluene}) show a |
325 |
significant conductance enhancement compared to the gold/water |
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interface without capping agent and agree with available experimental |
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data. This indicates that the metal-metal potential, though not |
328 |
predicting an accurate bulk metal thermal conductivity, does not |
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greatly interfere with the simulation of the thermal conductance |
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behavior across a non-metal interface. The solvent model is not |
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particularly volatile, so the simulation cell does not expand |
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significantly under higher temperature. We did not observe a |
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significant conductance decrease when the temperature was increased to |
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300K. The results show that the two definitions used for $G$ yield |
335 |
comparable values, though $G^\prime$ tends to be smaller. |
336 |
|
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\begin{table*} |
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\begin{minipage}{\linewidth} |
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\begin{center} |
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\caption{Computed interfacial thermal conductivity ($G$ and |
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$G^\prime$) values for the Au/butanethiol/toluene interface at |
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different temperatures using a range of energy fluxes.} |
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|
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\begin{tabular}{cccc} |
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\hline\hline |
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$\langle T\rangle$ & $J_z$ & $G$ & $G^\prime$ \\ |
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(K) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
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\hline |
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200 & 1.86 & 180 & 135 \\ |
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& 2.15 & 204 & 113 \\ |
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& 3.93 & 175 & 114 \\ |
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300 & 1.91 & 143 & 125 \\ |
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& 4.19 & 134 & 113 \\ |
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\hline\hline |
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\end{tabular} |
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\label{AuThiolToluene} |
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\end{center} |
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\end{minipage} |
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\end{table*} |
360 |
|
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\subsection{Hexane Solvent} |
362 |
|
363 |
Using the united-atom model, different coverages of capping agent, |
364 |
temperatures of simulations and numbers of solvent molecules were all |
365 |
investigated and Table \ref{AuThiolHexaneUA} shows the results of |
366 |
these computations. The number of hexane molecules in our simulations |
367 |
does not affect the calculations significantly. However, a very long |
368 |
length scale for the thermal gradient axis ($z$) may cause excessively |
369 |
hot or cold temperatures in the middle of the solvent region and lead |
370 |
to undesired phenomena such as solvent boiling or freezing, while too |
371 |
few solvent molecules would change the normal behavior of the liquid |
372 |
phase. Our $N_{hexane}$ values were chosen to ensure that these |
373 |
extreme cases did not happen to our simulations. |
374 |
|
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Table \ref{AuThiolHexaneUA} enables direct comparison between |
376 |
different coverages of capping agent, when other system parameters are |
377 |
held constant. With high coverage of butanethiol on the gold surface, |
378 |
the interfacial thermal conductance is enhanced |
379 |
significantly. Interestingly, a slightly lower butanethiol coverage |
380 |
leads to a moderately higher conductivity. This is probably due to |
381 |
more solvent/capping agent contact when butanethiol molecules are |
382 |
not densely packed, which enhances the interactions between the two |
383 |
phases and lowers the thermal transfer barrier of this interface. |
384 |
% [COMPARE TO AU/WATER IN PAPER] |
385 |
|
386 |
It is also noted that the overall simulation temperature is another |
387 |
factor that affects the interfacial thermal conductance. One |
388 |
possibility of this effect may be rooted in the decrease in density of |
389 |
the liquid phase. We observed that when the average temperature |
390 |
increases from 200K to 250K, the bulk hexane density becomes lower |
391 |
than experimental value, as the system is equilibrated under NPT |
392 |
ensemble. This leads to lower contact between solvent and capping |
393 |
agent, and thus lower conductivity. |
394 |
|
395 |
Conductivity values are more difficult to obtain under higher |
396 |
temperatures. This is because the Au surface tends to undergo |
397 |
reconstructions in relatively high temperatures. Surface Au atoms can |
398 |
migrate outward to reach higher Au-S contact; and capping agent |
399 |
molecules can be embedded into the surface Au layer due to the same |
400 |
driving force. This phenomenon agrees with experimental |
401 |
results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. A surface |
402 |
fully covered in capping agent is more susceptible to reconstruction, |
403 |
possibly because fully coverage prevents other means of capping agent |
404 |
relaxation, such as migration to an empty neighbor three-fold site. |
405 |
|
406 |
%MAY ADD MORE DATA TO TABLE |
407 |
\begin{table*} |
408 |
\begin{minipage}{\linewidth} |
409 |
\begin{center} |
410 |
\caption{Computed interfacial thermal conductivity ($G$ and |
411 |
$G^\prime$) values for the Au/butanethiol/hexane interface |
412 |
with united-atom model and different capping agent coverage |
413 |
and solvent molecule numbers at different temperatures using a |
414 |
range of energy fluxes.} |
415 |
|
416 |
\begin{tabular}{cccccc} |
417 |
\hline\hline |
418 |
Thiol & $\langle T\rangle$ & & $J_z$ & $G$ & $G^\prime$ \\ |
419 |
coverage (\%) & (K) & $N_{hexane}$ & (GW/m$^2$) & |
420 |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
421 |
\hline |
422 |
0.0 & 200 & 200 & 0.96 & 43.3 & 42.7 \\ |
423 |
& & & 1.91 & 45.7 & 42.9 \\ |
424 |
& & 166 & 0.96 & 43.1 & 53.4 \\ |
425 |
88.9 & 200 & 166 & 1.94 & 172 & 108 \\ |
426 |
100.0 & 250 & 200 & 0.96 & 81.8 & 67.0 \\ |
427 |
& & 166 & 0.98 & 79.0 & 62.9 \\ |
428 |
& & & 1.44 & 76.2 & 64.8 \\ |
429 |
& 200 & 200 & 1.92 & 129 & 87.3 \\ |
430 |
& & & 1.93 & 131 & 77.5 \\ |
431 |
& & 166 & 0.97 & 115 & 69.3 \\ |
432 |
& & & 1.94 & 125 & 87.1 \\ |
433 |
\hline\hline |
434 |
\end{tabular} |
435 |
\label{AuThiolHexaneUA} |
436 |
\end{center} |
437 |
\end{minipage} |
438 |
\end{table*} |
439 |
|
440 |
For the all-atom model, the liquid hexane phase was not stable under NPT |
441 |
conditions. Therefore, the simulation length scale parameters are |
442 |
adopted from previous equilibration results of the united-atom model |
443 |
at 200K. Table \ref{AuThiolHexaneAA} shows the results of these |
444 |
simulations. The conductivity values calculated with full capping |
445 |
agent coverage are substantially larger than observed in the |
446 |
united-atom model, and is even higher than predicted by |
447 |
experiments. It is possible that our parameters for metal-non-metal |
448 |
particle interactions lead to an overestimate of the interfacial |
449 |
thermal conductivity, although the active C-H vibrations in the |
450 |
all-atom model (which should not be appreciably populated at normal |
451 |
temperatures) could also account for this high conductivity. The major |
452 |
thermal transfer barrier of Au/butanethiol/hexane interface is between |
453 |
the liquid phase and the capping agent, so extra degrees of freedom |
454 |
such as the C-H vibrations could enhance heat exchange between these |
455 |
two phases and result in a much higher conductivity. |
456 |
|
457 |
\begin{table*} |
458 |
\begin{minipage}{\linewidth} |
459 |
\begin{center} |
460 |
|
461 |
\caption{Computed interfacial thermal conductivity ($G$ and |
462 |
$G^\prime$) values for the Au/butanethiol/hexane interface |
463 |
with all-atom model and different capping agent coverage at |
464 |
200K using a range of energy fluxes.} |
465 |
|
466 |
\begin{tabular}{cccc} |
467 |
\hline\hline |
468 |
Thiol & $J_z$ & $G$ & $G^\prime$ \\ |
469 |
coverage (\%) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
470 |
\hline |
471 |
0.0 & 0.95 & 28.5 & 27.2 \\ |
472 |
& 1.88 & 30.3 & 28.9 \\ |
473 |
100.0 & 2.87 & 551 & 294 \\ |
474 |
& 3.81 & 494 & 193 \\ |
475 |
\hline\hline |
476 |
\end{tabular} |
477 |
\label{AuThiolHexaneAA} |
478 |
\end{center} |
479 |
\end{minipage} |
480 |
\end{table*} |
481 |
|
482 |
%subsubsection{Vibrational spectrum study on conductance mechanism} |
483 |
To investigate the mechanism of this interfacial thermal conductance, |
484 |
the vibrational spectra of various gold systems were obtained and are |
485 |
shown as in the upper panel of Fig. \ref{vibration}. To obtain these |
486 |
spectra, one first runs a simulation in the NVE ensemble and collects |
487 |
snapshots of configurations; these configurations are used to compute |
488 |
the velocity auto-correlation functions, which is used to construct a |
489 |
power spectrum via a Fourier transform. The gold surfaces covered by |
490 |
butanethiol molecules exhibit an additional peak observed at a |
491 |
frequency of $\sim$170cm$^{-1}$, which is attributed to the vibration |
492 |
of the S-Au bond. This vibration enables efficient thermal transport |
493 |
from surface Au atoms to the capping agents. Simultaneously, as shown |
494 |
in the lower panel of Fig. \ref{vibration}, the large overlap of the |
495 |
vibration spectra of butanethiol and hexane in the all-atom model, |
496 |
including the C-H vibration, also suggests high thermal exchange |
497 |
efficiency. The combination of these two effects produces the drastic |
498 |
interfacial thermal conductance enhancement in the all-atom model. |
499 |
|
500 |
\begin{figure} |
501 |
\includegraphics[width=\linewidth]{vibration} |
502 |
\caption{Vibrational spectra obtained for gold in different |
503 |
environments (upper panel) and for Au/thiol/hexane simulation in |
504 |
all-atom model (lower panel).} |
505 |
\label{vibration} |
506 |
\end{figure} |
507 |
% 600dpi, letter size. too large? |
508 |
|
509 |
|
510 |
\section{Acknowledgments} |
511 |
Support for this project was provided by the National Science |
512 |
Foundation under grant CHE-0848243. Computational time was provided by |
513 |
the Center for Research Computing (CRC) at the University of Notre |
514 |
Dame. \newpage |
515 |
|
516 |
\bibliography{interfacial} |
517 |
|
518 |
\end{doublespace} |
519 |
\end{document} |
520 |
|