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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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|
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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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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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[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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[MAY INCLUDE POWER SPECTRUM PROTOCOL] |
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|
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\section{Computational Details} |
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\subsection{Simulation Protocol} |
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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). [MAY |
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EXPLAIN WHY WE CHOOSE THEM] |
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|
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The spacing filled by solvent molecules, i.e. the gap between |
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periodically repeated Au-butanethiol surfaces should be carefully |
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chosen. A very long length scale for the thermal gradient axis ($z$) |
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may 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. And the |
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corresponding spacing is usually $35 \sim 60$\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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Figure [REF] demonstrates how we name our pseudo-atoms of the |
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molecules in our simulations. |
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[FIGURE FOR MOLECULE NOMENCLATURE] |
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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. However, the TraPPE-UA for |
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alkanes is known to predict a lower boiling point than experimental |
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values. Considering that after an unphysical thermal flux is applied |
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to a system, the temperature of ``hot'' area in the liquid phase would be |
315 |
significantly higher than the average, to prevent over heating and |
316 |
boiling of the liquid phase, the average temperature in our |
317 |
simulations should be much lower than the liquid boiling point. [MORE DISCUSSION] |
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For UA-toluene model, rigid body constraints are applied, so that the |
319 |
benzene ring and the methyl-CRar bond are kept rigid. This would save |
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computational time.[MORE DETAILS] |
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|
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Besides the TraPPE-UA models, AA models for both organic solvents are |
323 |
included in our studies as well. For hexane, the OPLS-AA\cite{OPLSAA} |
324 |
force field is used. [MORE DETAILS] |
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For toluene, the United Force Field developed by Rapp\'{e} {\it et |
326 |
al.}\cite{doi:10.1021/ja00051a040} is adopted.[MORE DETAILS] |
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|
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The capping agent in our simulations, the butanethiol molecules can |
329 |
either use UA or AA model. The TraPPE-UA force fields includes |
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parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for |
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UA butanethiol model in our simulations. The OPLS-AA also provides |
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parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) |
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surfaces do not have the hydrogen atom bonded to sulfur. To adapt this |
334 |
change and derive suitable parameters for butanethiol adsorbed on |
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Au(111) surfaces, we adopt the S parameters from [CITATION CF VLUGT] |
336 |
and modify parameters for its neighbor C atom for charge balance in |
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the molecule. Note that the model choice (UA or AA) of capping agent |
338 |
can be different from the solvent. Regardless of model choice, the |
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force field parameters for interactions between capping agent and |
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solvent can be derived using Lorentz-Berthelot Mixing Rule: |
341 |
|
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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 particles, we refer to an adsorption study of alkyl |
345 |
thiols on gold surfaces by Vlugt {\it et |
346 |
al.}\cite{vlugt:cpc2007154} They fitted an effective Lennard-Jones |
347 |
form of potential parameters for the interaction between Au and |
348 |
pseudo-atoms CH$_x$ and S based on a well-established and widely-used |
349 |
effective potential of Hautman and Klein[CITATION] for the Au(111) |
350 |
surface. As our simulations require the gold lattice slab to be |
351 |
non-rigid so that it could accommodate kinetic energy for thermal |
352 |
transport study purpose, the pair-wise form of potentials is |
353 |
preferred. |
354 |
|
355 |
Besides, the potentials developed from {\it ab initio} calculations by |
356 |
Leng {\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
357 |
interactions between Au and aromatic C/H atoms in toluene.[MORE DETAILS] |
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|
359 |
However, the Lennard-Jones parameters between Au and other types of |
360 |
particles in our simulations are not yet well-established. For these |
361 |
interactions, we attempt to derive their parameters using the Mixing |
362 |
Rule. To do this, the ``Metal-non-Metal'' (MnM) interaction parameters |
363 |
for Au is first extracted from the Au-CH$_x$ parameters by applying |
364 |
the Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM'' |
365 |
parameters in our simulations. |
366 |
|
367 |
\begin{table*} |
368 |
\begin{minipage}{\linewidth} |
369 |
\begin{center} |
370 |
\caption{Lennard-Jones parameters for Au-non-Metal |
371 |
interactions in our simulations.} |
372 |
|
373 |
\begin{tabular}{ccc} |
374 |
\hline\hline |
375 |
Non-metal atom & $\sigma$ & $\epsilon$ \\ |
376 |
(or pseudo-atom) & \AA & kcal/mol \\ |
377 |
\hline |
378 |
S & 2.40 & 8.465 \\ |
379 |
CH3 & 3.54 & 0.2146 \\ |
380 |
CH2 & 3.54 & 0.1749 \\ |
381 |
CT3 & 3.365 & 0.1373 \\ |
382 |
CT2 & 3.365 & 0.1373 \\ |
383 |
CTT & 3.365 & 0.1373 \\ |
384 |
HC & 2.865 & 0.09256 \\ |
385 |
CHar & 3.4625 & 0.1680 \\ |
386 |
CRar & 3.555 & 0.1604 \\ |
387 |
CA & 3.173 & 0.0640 \\ |
388 |
HA & 2.746 & 0.0414 \\ |
389 |
\hline\hline |
390 |
\end{tabular} |
391 |
\label{MnM} |
392 |
\end{center} |
393 |
\end{minipage} |
394 |
\end{table*} |
395 |
|
396 |
|
397 |
\section{Results and Discussions} |
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[MAY HAVE A BRIEF SUMMARY] |
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\subsection{How Simulation Parameters Affects $G$} |
400 |
[MAY NOT PUT AT FIRST] |
401 |
We have varied our protocol or other parameters of the simulations in |
402 |
order to investigate how these factors would affect the measurement of |
403 |
$G$'s. It turned out that while some of these parameters would not |
404 |
affect the results substantially, some other changes to the |
405 |
simulations would have a significant impact on the measurement |
406 |
results. |
407 |
|
408 |
In some of our simulations, we allowed $L_x$ and $L_y$ to change |
409 |
during equilibrating the liquid phase. Due to the stiffness of the Au |
410 |
slab, $L_x$ and $L_y$ would not change noticeably after |
411 |
equilibration. Although $L_z$ could fluctuate $\sim$1\% after a system |
412 |
is fully equilibrated in the NPT ensemble, this fluctuation, as well |
413 |
as those comparably smaller to $L_x$ and $L_y$, would not be magnified |
414 |
on the calculated $G$'s, as shown in Table \ref{AuThiolHexaneUA}. This |
415 |
insensivity to $L_i$ fluctuations allows reliable measurement of $G$'s |
416 |
without the necessity of extremely cautious equilibration process. |
417 |
|
418 |
As stated in our computational details, the spacing filled with |
419 |
solvent molecules can be chosen within a range. This allows some |
420 |
change of solvent molecule numbers for the same Au-butanethiol |
421 |
surfaces. We did this study on our Au-butanethiol/hexane |
422 |
simulations. Nevertheless, the results obtained from systems of |
423 |
different $N_{hexane}$ did not indicate that the measurement of $G$ is |
424 |
susceptible to this parameter. For computational efficiency concern, |
425 |
smaller system size would be preferable, given that the liquid phase |
426 |
structure is not affected. |
427 |
|
428 |
Our NIVS algorithm allows change of unphysical thermal flux both in |
429 |
direction and in quantity. This feature extends our investigation of |
430 |
interfacial thermal conductance. However, the magnitude of this |
431 |
thermal flux is not arbitary if one aims to obtain a stable and |
432 |
reliable thermal gradient. A temperature profile would be |
433 |
substantially affected by noise when $|J_z|$ has a much too low |
434 |
magnitude; while an excessively large $|J_z|$ that overwhelms the |
435 |
conductance capacity of the interface would prevent a thermal gradient |
436 |
to reach a stablized steady state. NIVS has the advantage of allowing |
437 |
$J$ to vary in a wide range such that the optimal flux range for $G$ |
438 |
measurement can generally be simulated by the algorithm. Within the |
439 |
optimal range, we were able to study how $G$ would change according to |
440 |
the thermal flux across the interface. For our simulations, we denote |
441 |
$J_z$ to be positive when the physical thermal flux is from the liquid |
442 |
to metal, and negative vice versa. The $G$'s measured under different |
443 |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and [REF]. These |
444 |
results do not suggest that $G$ is dependent on $J_z$ within this flux |
445 |
range. The linear response of flux to thermal gradient simplifies our |
446 |
investigations in that we can rely on $G$ measurement with only a |
447 |
couple $J_z$'s and do not need to test a large series of fluxes. |
448 |
|
449 |
%ADD MORE TO TABLE |
450 |
\begin{table*} |
451 |
\begin{minipage}{\linewidth} |
452 |
\begin{center} |
453 |
\caption{Computed interfacial thermal conductivity ($G$ and |
454 |
$G^\prime$) values for the 100\% covered Au-butanethiol/hexane |
455 |
interfaces with UA model and different hexane molecule numbers |
456 |
at different temperatures using a range of energy fluxes.} |
457 |
|
458 |
\begin{tabular}{cccccccc} |
459 |
\hline\hline |
460 |
$\langle T\rangle$ & & $L_x$ & $L_y$ & $L_z$ & $J_z$ & |
461 |
$G$ & $G^\prime$ \\ |
462 |
(K) & $N_{hexane}$ & \multicolumn{3}{c}{(\AA)} & (GW/m$^2$) & |
463 |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
464 |
\hline |
465 |
200 & 266 & 29.86 & 25.80 & 113.1 & -0.96 & |
466 |
102() & 80.0() \\ |
467 |
& 200 & 29.84 & 25.81 & 93.9 & 1.92 & |
468 |
129() & 87.3() \\ |
469 |
& & 29.84 & 25.81 & 95.3 & 1.93 & |
470 |
131() & 77.5() \\ |
471 |
& 166 & 29.84 & 25.81 & 85.7 & 0.97 & |
472 |
115() & 69.3() \\ |
473 |
& & & & & 1.94 & |
474 |
125() & 87.1() \\ |
475 |
250 & 200 & 29.84 & 25.87 & 106.8 & 0.96 & |
476 |
81.8() & 67.0() \\ |
477 |
& 166 & 29.87 & 25.84 & 94.8 & 0.98 & |
478 |
79.0() & 62.9() \\ |
479 |
& & 29.84 & 25.85 & 95.0 & 1.44 & |
480 |
76.2() & 64.8() \\ |
481 |
\hline\hline |
482 |
\end{tabular} |
483 |
\label{AuThiolHexaneUA} |
484 |
\end{center} |
485 |
\end{minipage} |
486 |
\end{table*} |
487 |
|
488 |
Furthermore, we also attempted to increase system average temperatures |
489 |
to above 200K. These simulations are first equilibrated in the NPT |
490 |
ensemble under normal pressure. As stated above, the TraPPE-UA model |
491 |
for hexane tends to predict a lower boiling point. In our simulations, |
492 |
hexane had diffculty to remain in liquid phase when NPT equilibration |
493 |
temperature is higher than 250K. Additionally, the equilibrated liquid |
494 |
hexane density under 250K becomes lower than experimental value. This |
495 |
expanded liquid phase leads to lower contact between hexane and |
496 |
butanethiol as well.[MAY NEED FIGURE] And this reduced contact would |
497 |
probably be accountable for a lower interfacial thermal conductance, |
498 |
as shown in Table \ref{AuThiolHexaneUA}. |
499 |
|
500 |
A similar study for TraPPE-UA toluene agrees with the above result as |
501 |
well. Having a higher boiling point, toluene tends to remain liquid in |
502 |
our simulations even equilibrated under 300K in NPT |
503 |
ensembles. Furthermore, the expansion of the toluene liquid phase is |
504 |
not as significant as that of the hexane. This prevents severe |
505 |
decrease of liquid-capping agent contact and the results (Table |
506 |
\ref{AuThiolToluene}) show only a slightly decreased interface |
507 |
conductance. Therefore, solvent-capping agent contact should play an |
508 |
important role in the thermal transport process across the interface |
509 |
in that higher degree of contact could yield increased conductance. |
510 |
|
511 |
[ADD Lxyz AND ERROR ESTIMATE TO TABLE] |
512 |
\begin{table*} |
513 |
\begin{minipage}{\linewidth} |
514 |
\begin{center} |
515 |
\caption{Computed interfacial thermal conductivity ($G$ and |
516 |
$G^\prime$) values for a 90\% coverage Au-butanethiol/toluene |
517 |
interface at different temperatures using a range of energy |
518 |
fluxes.} |
519 |
|
520 |
\begin{tabular}{cccc} |
521 |
\hline\hline |
522 |
$\langle T\rangle$ & $J_z$ & $G$ & $G^\prime$ \\ |
523 |
(K) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
524 |
\hline |
525 |
200 & -1.86 & 180() & 135() \\ |
526 |
& 2.15 & 204() & 113() \\ |
527 |
& -3.93 & 175() & 114() \\ |
528 |
300 & -1.91 & 143() & 125() \\ |
529 |
& -4.19 & 134() & 113() \\ |
530 |
\hline\hline |
531 |
\end{tabular} |
532 |
\label{AuThiolToluene} |
533 |
\end{center} |
534 |
\end{minipage} |
535 |
\end{table*} |
536 |
|
537 |
Besides lower interfacial thermal conductance, surfaces in relatively |
538 |
high temperatures are susceptible to reconstructions, when |
539 |
butanethiols have a full coverage on the Au(111) surface. These |
540 |
reconstructions include surface Au atoms migrated outward to the S |
541 |
atom layer, and butanethiol molecules embedded into the original |
542 |
surface Au layer. The driving force for this behavior is the strong |
543 |
Au-S interactions in our simulations. And these reconstructions lead |
544 |
to higher ratio of Au-S attraction and thus is energetically |
545 |
favorable. Furthermore, this phenomenon agrees with experimental |
546 |
results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt |
547 |
{\it et al.} had kept their Au(111) slab rigid so that their |
548 |
simulations can reach 300K without surface reconstructions. Without |
549 |
this practice, simulating 100\% thiol covered interfaces under higher |
550 |
temperatures could hardly avoid surface reconstructions. However, our |
551 |
measurement is based on assuming homogeneity on $x$ and $y$ dimensions |
552 |
so that measurement of $T$ at particular $z$ would be an effective |
553 |
average of the particles of the same type. Since surface |
554 |
reconstructions could eliminate the original $x$ and $y$ dimensional |
555 |
homogeneity, measurement of $G$ is more difficult to conduct under |
556 |
higher temperatures. Therefore, most of our measurements are |
557 |
undertaken at $\langle T\rangle\sim$200K. |
558 |
|
559 |
However, when the surface is not completely covered by butanethiols, |
560 |
the simulated system is more resistent to the reconstruction |
561 |
above. Our Au-butanethiol/toluene system did not see this phenomena |
562 |
even at $<T>\sim$300K. The Au(111) surfaces have a 90\% coverage of |
563 |
butanethiols and have empty three-fold sites. These empty sites could |
564 |
help prevent surface reconstruction in that they provide other means |
565 |
of capping agent relaxation. It is observed that butanethiols can |
566 |
migrate to their neighbor empty sites during a simulation. Therefore, |
567 |
we were able to obtain $G$'s for these interfaces even at a relatively |
568 |
high temperature without being affected by surface reconstructions. |
569 |
|
570 |
\subsection{Influence of Capping Agent Coverage on $G$} |
571 |
To investigate the influence of butanethiol coverage on interfacial |
572 |
thermal conductance, a series of different coverage Au-butanethiol |
573 |
surfaces is prepared and solvated with various organic |
574 |
molecules. These systems are then equilibrated and their interfacial |
575 |
thermal conductivity are measured with our NIVS algorithm. Table |
576 |
\ref{tlnUhxnUhxnD} lists these results for direct comparison between |
577 |
different coverages of butanethiol. To study the isotope effect in |
578 |
interfacial thermal conductance, deuterated UA-hexane is included as |
579 |
well. |
580 |
|
581 |
It turned out that with partial covered butanethiol on the Au(111) |
582 |
surface, the derivative definition for $G$ (Eq. \ref{derivativeG}) has |
583 |
difficulty to apply, due to the difficulty in locating the maximum of |
584 |
change of $\lambda$. Instead, the discrete definition |
585 |
(Eq. \ref{discreteG}) is easier to apply, as max($\Delta T$) can still |
586 |
be well-defined. Therefore, $G$'s (not $G^\prime$) are used for this |
587 |
section. |
588 |
|
589 |
From Table \ref{tlnUhxnUhxnD}, one can see the significance of the |
590 |
presence of capping agents. Even when a fraction of the Au(111) |
591 |
surface sites are covered with butanethiols, the conductivity would |
592 |
see an enhancement by at least a factor of 3. This indicates the |
593 |
important role cappping agent is playing for thermal transport |
594 |
phenomena on metal/organic solvent surfaces. |
595 |
|
596 |
Interestingly, as one could observe from our results, the maximum |
597 |
conductance enhancement (largest $G$) happens while the surfaces are |
598 |
about 75\% covered with butanethiols. This again indicates that |
599 |
solvent-capping agent contact has an important role of the thermal |
600 |
transport process. Slightly lower butanethiol coverage allows small |
601 |
gaps between butanethiols to form. And these gaps could be filled with |
602 |
solvent molecules, which acts like ``heat conductors'' on the |
603 |
surface. The higher degree of interaction between these solvent |
604 |
molecules and capping agents increases the enhancement effect and thus |
605 |
produces a higher $G$ than densely packed butanethiol arrays. However, |
606 |
once this maximum conductance enhancement is reached, $G$ decreases |
607 |
when butanethiol coverage continues to decrease. Each capping agent |
608 |
molecule reaches its maximum capacity for thermal |
609 |
conductance. Therefore, even higher solvent-capping agent contact |
610 |
would not offset this effect. Eventually, when butanethiol coverage |
611 |
continues to decrease, solvent-capping agent contact actually |
612 |
decreases with the disappearing of butanethiol molecules. In this |
613 |
case, $G$ decrease could not be offset but instead accelerated. |
614 |
|
615 |
A comparison of the results obtained from differenet organic solvents |
616 |
can also provide useful information of the interfacial thermal |
617 |
transport process. The deuterated hexane (UA) results do not appear to |
618 |
be much different from those of normal hexane (UA), given that |
619 |
butanethiol (UA) is non-deuterated for both solvents. These UA model |
620 |
studies, even though eliminating C-H vibration samplings, still have |
621 |
C-C vibrational frequencies different from each other. However, these |
622 |
differences in the infrared range do not seem to produce an observable |
623 |
difference for the results of $G$. [MAY NEED FIGURE] |
624 |
|
625 |
Furthermore, results for rigid body toluene solvent, as well as other |
626 |
UA-hexane solvents, are reasonable within the general experimental |
627 |
ranges[CITATIONS]. This suggests that explicit hydrogen might not be a |
628 |
required factor for modeling thermal transport phenomena of systems |
629 |
such as Au-thiol/organic solvent. |
630 |
|
631 |
However, results for Au-butanethiol/toluene do not show an identical |
632 |
trend with those for Au-butanethiol/hexane in that $G$'s remain at |
633 |
approximately the same magnitue when butanethiol coverage differs from |
634 |
25\% to 75\%. This might be rooted in the molecule shape difference |
635 |
for plane-like toluene and chain-like {\it n}-hexane. Due to this |
636 |
difference, toluene molecules have more difficulty in occupying |
637 |
relatively small gaps among capping agents when their coverage is not |
638 |
too low. Therefore, the solvent-capping agent contact may keep |
639 |
increasing until the capping agent coverage reaches a relatively low |
640 |
level. This becomes an offset for decreasing butanethiol molecules on |
641 |
its effect to the process of interfacial thermal transport. Thus, one |
642 |
can see a plateau of $G$ vs. butanethiol coverage in our results. |
643 |
|
644 |
[NEED ERROR ESTIMATE, MAY ALSO PUT J HERE] |
645 |
\begin{table*} |
646 |
\begin{minipage}{\linewidth} |
647 |
\begin{center} |
648 |
\caption{Computed interfacial thermal conductivity ($G$) values |
649 |
for the Au-butanethiol/solvent interface with various UA |
650 |
models and different capping agent coverages at $\langle |
651 |
T\rangle\sim$200K using certain energy flux respectively.} |
652 |
|
653 |
\begin{tabular}{cccc} |
654 |
\hline\hline |
655 |
Thiol & \multicolumn{3}{c}{$G$(MW/m$^2$/K)} \\ |
656 |
coverage (\%) & hexane & hexane(D) & toluene \\ |
657 |
\hline |
658 |
0.0 & 46.5() & 43.9() & 70.1() \\ |
659 |
25.0 & 151() & 153() & 249() \\ |
660 |
50.0 & 172() & 182() & 214() \\ |
661 |
75.0 & 242() & 229() & 244() \\ |
662 |
88.9 & 178() & - & - \\ |
663 |
100.0 & 137() & 153() & 187() \\ |
664 |
\hline\hline |
665 |
\end{tabular} |
666 |
\label{tlnUhxnUhxnD} |
667 |
\end{center} |
668 |
\end{minipage} |
669 |
\end{table*} |
670 |
|
671 |
\subsection{Influence of Chosen Molecule Model on $G$} |
672 |
[MAY COMBINE W MECHANISM STUDY] |
673 |
|
674 |
In addition to UA solvent/capping agent models, AA models are included |
675 |
in our simulations as well. Besides simulations of the same (UA or AA) |
676 |
model for solvent and capping agent, different models can be applied |
677 |
to different components. Furthermore, regardless of models chosen, |
678 |
either the solvent or the capping agent can be deuterated, similar to |
679 |
the previous section. Table \ref{modelTest} summarizes the results of |
680 |
these studies. |
681 |
|
682 |
[MORE DATA; ERROR ESTIMATE] |
683 |
\begin{table*} |
684 |
\begin{minipage}{\linewidth} |
685 |
\begin{center} |
686 |
|
687 |
\caption{Computed interfacial thermal conductivity ($G$ and |
688 |
$G^\prime$) values for interfaces using various models for |
689 |
solvent and capping agent (or without capping agent) at |
690 |
$\langle T\rangle\sim$200K.} |
691 |
|
692 |
\begin{tabular}{ccccc} |
693 |
\hline\hline |
694 |
Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ |
695 |
(or bare surface) & model & (GW/m$^2$) & |
696 |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
697 |
\hline |
698 |
UA & AA hexane & 1.94 & 135() & 129() \\ |
699 |
& & 2.86 & 126() & 115() \\ |
700 |
& AA toluene & 1.89 & 200() & 149() \\ |
701 |
AA & UA hexane & 1.94 & 116() & 129() \\ |
702 |
& AA hexane & 3.76 & 451() & 378() \\ |
703 |
& & 4.71 & 432() & 334() \\ |
704 |
& AA toluene & 3.79 & 487() & 290() \\ |
705 |
AA(D) & UA hexane & 1.94 & 158() & 172() \\ |
706 |
bare & AA hexane & 0.96 & 31.0() & 29.4() \\ |
707 |
\hline\hline |
708 |
\end{tabular} |
709 |
\label{modelTest} |
710 |
\end{center} |
711 |
\end{minipage} |
712 |
\end{table*} |
713 |
|
714 |
To facilitate direct comparison, the same system with differnt models |
715 |
for different components uses the same length scale for their |
716 |
simulation cells. Without the presence of capping agent, using |
717 |
different models for hexane yields similar results for both $G$ and |
718 |
$G^\prime$, and these two definitions agree with eath other very |
719 |
well. This indicates very weak interaction between the metal and the |
720 |
solvent, and is a typical case for acoustic impedance mismatch between |
721 |
these two phases. |
722 |
|
723 |
As for Au(111) surfaces completely covered by butanethiols, the choice |
724 |
of models for capping agent and solvent could impact the measurement |
725 |
of $G$ and $G^\prime$ quite significantly. For Au-butanethiol/hexane |
726 |
interfaces, using AA model for both butanethiol and hexane yields |
727 |
substantially higher conductivity values than using UA model for at |
728 |
least one component of the solvent and capping agent, which exceeds |
729 |
the upper bond of experimental value range. This is probably due to |
730 |
the classically treated C-H vibrations in the AA model, which should |
731 |
not be appreciably populated at normal temperatures. In comparison, |
732 |
once either the hexanes or the butanethiols are deuterated, one can |
733 |
see a significantly lower $G$ and $G^\prime$. In either of these |
734 |
cases, the C-H(D) vibrational overlap between the solvent and the |
735 |
capping agent is removed. [MAY NEED FIGURE] Conclusively, the |
736 |
improperly treated C-H vibration in the AA model produced |
737 |
over-predicted results accordingly. Compared to the AA model, the UA |
738 |
model yields more reasonable results with higher computational |
739 |
efficiency. |
740 |
|
741 |
However, for Au-butanethiol/toluene interfaces, having the AA |
742 |
butanethiol deuterated did not yield a significant change in the |
743 |
measurement results. |
744 |
. , so extra degrees of freedom |
745 |
such as the C-H vibrations could enhance heat exchange between these |
746 |
two phases and result in a much higher conductivity. |
747 |
|
748 |
|
749 |
Although the QSC model for Au is known to predict an overly low value |
750 |
for bulk metal gold conductivity[CITE NIVSRNEMD], our computational |
751 |
results for $G$ and $G^\prime$ do not seem to be affected by this |
752 |
drawback of the model for metal. Instead, the modeling of interfacial |
753 |
thermal transport behavior relies mainly on an accurate description of |
754 |
the interactions between components occupying the interfaces. |
755 |
|
756 |
\subsection{Mechanism of Interfacial Thermal Conductance Enhancement |
757 |
by Capping Agent} |
758 |
%OR\subsection{Vibrational spectrum study on conductance mechanism} |
759 |
|
760 |
[MAY INTRODUCE PROTOCOL IN METHODOLOGY/COMPUTATIONAL DETAIL, EQN'S] |
761 |
|
762 |
To investigate the mechanism of this interfacial thermal conductance, |
763 |
the vibrational spectra of various gold systems were obtained and are |
764 |
shown as in the upper panel of Fig. \ref{vibration}. To obtain these |
765 |
spectra, one first runs a simulation in the NVE ensemble and collects |
766 |
snapshots of configurations; these configurations are used to compute |
767 |
the velocity auto-correlation functions, which is used to construct a |
768 |
power spectrum via a Fourier transform. |
769 |
|
770 |
The gold surfaces covered by |
771 |
butanethiol molecules, compared to bare gold surfaces, exhibit an |
772 |
additional peak observed at a frequency of $\sim$170cm$^{-1}$, which |
773 |
is attributed to the vibration of the S-Au bond. This vibration |
774 |
enables efficient thermal transport from surface Au atoms to the |
775 |
capping agents. Simultaneously, as shown in the lower panel of |
776 |
Fig. \ref{vibration}, the large overlap of the vibration spectra of |
777 |
butanethiol and hexane in the all-atom model, including the C-H |
778 |
vibration, also suggests high thermal exchange efficiency. The |
779 |
combination of these two effects produces the drastic interfacial |
780 |
thermal conductance enhancement in the all-atom model. |
781 |
|
782 |
[MAY NEED TO CONVERT TO JPEG] |
783 |
\begin{figure} |
784 |
\includegraphics[width=\linewidth]{vibration} |
785 |
\caption{Vibrational spectra obtained for gold in different |
786 |
environments (upper panel) and for Au/thiol/hexane simulation in |
787 |
all-atom model (lower panel).} |
788 |
\label{vibration} |
789 |
\end{figure} |
790 |
|
791 |
[COMPARISON OF TWO G'S; AU SLAB WIDTHS; ETC] |
792 |
% The results show that the two definitions used for $G$ yield |
793 |
% comparable values, though $G^\prime$ tends to be smaller. |
794 |
|
795 |
\section{Conclusions} |
796 |
The NIVS algorithm we developed has been applied to simulations of |
797 |
Au-butanethiol surfaces with organic solvents. This algorithm allows |
798 |
effective unphysical thermal flux transferred between the metal and |
799 |
the liquid phase. With the flux applied, we were able to measure the |
800 |
corresponding thermal gradient and to obtain interfacial thermal |
801 |
conductivities. Our simulations have seen significant conductance |
802 |
enhancement with the presence of capping agent, compared to the bare |
803 |
gold/liquid interfaces. The acoustic impedance mismatch between the |
804 |
metal and the liquid phase is effectively eliminated by proper capping |
805 |
agent. Furthermore, the coverage precentage of the capping agent plays |
806 |
an important role in the interfacial thermal transport process. |
807 |
|
808 |
Our measurement results, particularly of the UA models, agree with |
809 |
available experimental data. This indicates that our force field |
810 |
parameters have a nice description of the interactions between the |
811 |
particles at the interfaces. AA models tend to overestimate the |
812 |
interfacial thermal conductance in that the classically treated C-H |
813 |
vibration would be overly sampled. Compared to the AA models, the UA |
814 |
models have higher computational efficiency with satisfactory |
815 |
accuracy, and thus are preferable in interfacial thermal transport |
816 |
modelings. |
817 |
|
818 |
Vlugt {\it et al.} has investigated the surface thiol structures for |
819 |
nanocrystal gold and pointed out that they differs from those of the |
820 |
Au(111) surface\cite{vlugt:cpc2007154}. This difference might lead to |
821 |
change of interfacial thermal transport behavior as well. To |
822 |
investigate this problem, an effective means to introduce thermal flux |
823 |
and measure the corresponding thermal gradient is desirable for |
824 |
simulating structures with spherical symmetry. |
825 |
|
826 |
|
827 |
\section{Acknowledgments} |
828 |
Support for this project was provided by the National Science |
829 |
Foundation under grant CHE-0848243. Computational time was provided by |
830 |
the Center for Research Computing (CRC) at the University of Notre |
831 |
Dame. \newpage |
832 |
|
833 |
\bibliography{interfacial} |
834 |
|
835 |
\end{doublespace} |
836 |
\end{document} |
837 |
|