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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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\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 |
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significantly higher than the average, to prevent over heating and |
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boiling of the liquid phase, the average temperature in our |
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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 |
317 |
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 |
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included in our studies as well. For hexane, the OPLS-AA\cite{OPLSAA} |
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force field is used. [MORE DETAILS] |
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For toluene, the United Force Field developed by Rapp\'{e} {\it et |
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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 |
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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 |
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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] |
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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 |
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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: |
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|
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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 |
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thiols on gold surfaces by Vlugt {\it et |
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al.}\cite{vlugt:cpc2007154} They fitted an effective Lennard-Jones |
345 |
form of potential parameters for the interaction between Au and |
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pseudo-atoms CH$_x$ and S based on a well-established and widely-used |
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effective potential of Hautman and Klein[CITATION] for the Au(111) |
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surface. As our simulations require the gold lattice slab to be |
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non-rigid so that it could accommodate kinetic energy for thermal |
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transport study purpose, the pair-wise form of potentials is |
351 |
preferred. |
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|
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Besides, the potentials developed from {\it ab initio} calculations by |
354 |
Leng {\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
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interactions between Au and aromatic C/H atoms in toluene.[MORE DETAILS] |
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|
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However, the Lennard-Jones parameters between Au and other types of |
358 |
particles in our simulations are not yet well-established. For these |
359 |
interactions, we attempt to derive their parameters using the Mixing |
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Rule. To do this, the ``Metal-non-Metal'' (MnM) interaction parameters |
361 |
for Au is first extracted from the Au-CH$_x$ parameters by applying |
362 |
the Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM'' |
363 |
parameters in our simulations. |
364 |
|
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\begin{table*} |
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\begin{minipage}{\linewidth} |
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\begin{center} |
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\caption{Lennard-Jones parameters for Au-non-Metal |
369 |
interactions in our simulations.} |
370 |
|
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\begin{tabular}{ccc} |
372 |
\hline\hline |
373 |
Non-metal & $\sigma$/\AA & $\epsilon$/kcal/mol \\ |
374 |
\hline |
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S & 2.40 & 8.465 \\ |
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CH3 & 3.54 & 0.2146 \\ |
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CH2 & 3.54 & 0.1749 \\ |
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CT3 & 3.365 & 0.1373 \\ |
379 |
CT2 & 3.365 & 0.1373 \\ |
380 |
CTT & 3.365 & 0.1373 \\ |
381 |
HC & 2.865 & 0.09256 \\ |
382 |
CHar & 3.4625 & 0.1680 \\ |
383 |
CRar & 3.555 & 0.1604 \\ |
384 |
CA & 3.173 & 0.0640 \\ |
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HA & 2.746 & 0.0414 \\ |
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\hline\hline |
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\end{tabular} |
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\label{MnM} |
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\end{center} |
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\end{minipage} |
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\end{table*} |
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|
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|
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\section{Results and Discussions} |
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[MAY HAVE A BRIEF SUMMARY] |
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\subsection{How Simulation Parameters Affects $G$} |
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[MAY NOT PUT AT FIRST] |
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We have varied our protocol or other parameters of the simulations in |
399 |
order to investigate how these factors would affect the measurement of |
400 |
$G$'s. It turned out that while some of these parameters would not |
401 |
affect the results substantially, some other changes to the |
402 |
simulations would have a significant impact on the measurement |
403 |
results. |
404 |
|
405 |
In some of our simulations, we allowed $L_x$ and $L_y$ to change |
406 |
during equilibrating the liquid phase. Due to the stiffness of the Au |
407 |
slab, $L_x$ and $L_y$ would not change noticeably after |
408 |
equilibration. Although $L_z$ could fluctuate $\sim$1\% after a system |
409 |
is fully equilibrated in the NPT ensemble, this fluctuation, as well |
410 |
as those comparably smaller to $L_x$ and $L_y$, would not be magnified |
411 |
on the calculated $G$'s, as shown in Table \ref{AuThiolHexaneUA}. This |
412 |
insensivity to $L_i$ fluctuations allows reliable measurement of $G$'s |
413 |
without the necessity of extremely cautious equilibration process. |
414 |
|
415 |
As stated in our computational details, the spacing filled with |
416 |
solvent molecules can be chosen within a range. This allows some |
417 |
change of solvent molecule numbers for the same Au-butanethiol |
418 |
surfaces. We did this study on our Au-butanethiol/hexane |
419 |
simulations. Nevertheless, the results obtained from systems of |
420 |
different $N_{hexane}$ did not indicate that the measurement of $G$ is |
421 |
susceptible to this parameter. For computational efficiency concern, |
422 |
smaller system size would be preferable, given that the liquid phase |
423 |
structure is not affected. |
424 |
|
425 |
Our NIVS algorithm allows change of unphysical thermal flux both in |
426 |
direction and in quantity. This feature extends our investigation of |
427 |
interfacial thermal conductance. However, the magnitude of this |
428 |
thermal flux is not arbitary if one aims to obtain a stable and |
429 |
reliable thermal gradient. A temperature profile would be |
430 |
substantially affected by noise when $|J_z|$ has a much too low |
431 |
magnitude; while an excessively large $|J_z|$ that overwhelms the |
432 |
conductance capacity of the interface would prevent a thermal gradient |
433 |
to reach a stablized steady state. NIVS has the advantage of allowing |
434 |
$J$ to vary in a wide range such that the optimal flux range for $G$ |
435 |
measurement can generally be simulated by the algorithm. Within the |
436 |
optimal range, we were able to study how $G$ would change according to |
437 |
the thermal flux across the interface. For our simulations, we denote |
438 |
$J_z$ to be positive when the physical thermal flux is from the liquid |
439 |
to metal, and negative vice versa. The $G$'s measured under different |
440 |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and [REF]. These |
441 |
results do not suggest that $G$ is dependent on $J_z$ within this flux |
442 |
range. The linear response of flux to thermal gradient simplifies our |
443 |
investigations in that we can rely on $G$ measurement with only a |
444 |
couple $J_z$'s and do not need to test a large series of fluxes. |
445 |
|
446 |
%ADD MORE TO TABLE |
447 |
\begin{table*} |
448 |
\begin{minipage}{\linewidth} |
449 |
\begin{center} |
450 |
\caption{Computed interfacial thermal conductivity ($G$ and |
451 |
$G^\prime$) values for the 100\% covered Au-butanethiol/hexane |
452 |
interfaces with UA model and different hexane molecule numbers |
453 |
at different temperatures using a range of energy fluxes.} |
454 |
|
455 |
\begin{tabular}{cccccccc} |
456 |
\hline\hline |
457 |
$\langle T\rangle$ & & $L_x$ & $L_y$ & $L_z$ & $J_z$ & |
458 |
$G$ & $G^\prime$ \\ |
459 |
(K) & $N_{hexane}$ & \multicolumn{3}{c}\AA & (GW/m$^2$) & |
460 |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
461 |
\hline |
462 |
200 & 266 & 29.86 & 25.80 & 113.1 & -0.96 & |
463 |
102() & 80.0() \\ |
464 |
& 200 & 29.84 & 25.81 & 93.9 & 1.92 & |
465 |
129() & 87.3() \\ |
466 |
& & 29.84 & 25.81 & 95.3 & 1.93 & |
467 |
131() & 77.5() \\ |
468 |
& 166 & 29.84 & 25.81 & 85.7 & 0.97 & |
469 |
115() & 69.3() \\ |
470 |
& & & & & 1.94 & |
471 |
125() & 87.1() \\ |
472 |
250 & 200 & 29.84 & 25.87 & 106.8 & 0.96 & |
473 |
81.8() & 67.0() \\ |
474 |
& 166 & 29.87 & 25.84 & 94.8 & 0.98 & |
475 |
79.0() & 62.9() \\ |
476 |
& & 29.84 & 25.85 & 95.0 & 1.44 & |
477 |
76.2() & 64.8() \\ |
478 |
\hline\hline |
479 |
\end{tabular} |
480 |
\label{AuThiolHexaneUA} |
481 |
\end{center} |
482 |
\end{minipage} |
483 |
\end{table*} |
484 |
|
485 |
Furthermore, we also attempted to increase system average temperatures |
486 |
to above 200K. These simulations are first equilibrated in the NPT |
487 |
ensemble under normal pressure. As stated above, the TraPPE-UA model |
488 |
for hexane tends to predict a lower boiling point. In our simulations, |
489 |
hexane had diffculty to remain in liquid phase when NPT equilibration |
490 |
temperature is higher than 250K. Additionally, the equilibrated liquid |
491 |
hexane density under 250K becomes lower than experimental value. This |
492 |
expanded liquid phase leads to lower contact between hexane and |
493 |
butanethiol as well.[MAY NEED FIGURE] And this reduced contact would |
494 |
probably be accountable for a lower interfacial thermal conductance, |
495 |
as shown in Table \ref{AuThiolHexaneUA}. |
496 |
|
497 |
A similar study for TraPPE-UA toluene agrees with the above result as |
498 |
well. Having a higher boiling point, toluene tends to remain liquid in |
499 |
our simulations even equilibrated under 300K in NPT |
500 |
ensembles. Furthermore, the expansion of the toluene liquid phase is |
501 |
not as significant as that of the hexane. This prevents severe |
502 |
decrease of liquid-capping agent contact and the results (Table |
503 |
\ref{AuThiolToluene}) show only a slightly decreased interface |
504 |
conductance. Therefore, solvent-capping agent contact should play an |
505 |
important role in the thermal transport process across the interface |
506 |
in that higher degree of contact could yield increased conductance. |
507 |
|
508 |
[ADD Lxyz AND ERROR ESTIMATE TO TABLE] |
509 |
\begin{table*} |
510 |
\begin{minipage}{\linewidth} |
511 |
\begin{center} |
512 |
\caption{Computed interfacial thermal conductivity ($G$ and |
513 |
$G^\prime$) values for a 90\% coverage Au-butanethiol/toluene |
514 |
interface at different temperatures using a range of energy |
515 |
fluxes.} |
516 |
|
517 |
\begin{tabular}{cccc} |
518 |
\hline\hline |
519 |
$\langle T\rangle$ & $J_z$ & $G$ & $G^\prime$ \\ |
520 |
(K) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
521 |
\hline |
522 |
200 & -1.86 & 180() & 135() \\ |
523 |
& 2.15 & 204() & 113() \\ |
524 |
& -3.93 & 175() & 114() \\ |
525 |
300 & -1.91 & 143() & 125() \\ |
526 |
& -4.19 & 134() & 113() \\ |
527 |
\hline\hline |
528 |
\end{tabular} |
529 |
\label{AuThiolToluene} |
530 |
\end{center} |
531 |
\end{minipage} |
532 |
\end{table*} |
533 |
|
534 |
Besides lower interfacial thermal conductance, surfaces in relatively |
535 |
high temperatures are susceptible to reconstructions, when |
536 |
butanethiols have a full coverage on the Au(111) surface. These |
537 |
reconstructions include surface Au atoms migrated outward to the S |
538 |
atom layer, and butanethiol molecules embedded into the original |
539 |
surface Au layer. The driving force for this behavior is the strong |
540 |
Au-S interactions in our simulations. And these reconstructions lead |
541 |
to higher ratio of Au-S attraction and thus is energetically |
542 |
favorable. Furthermore, this phenomenon agrees with experimental |
543 |
results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt |
544 |
{\it et al.} had kept their Au(111) slab rigid so that their |
545 |
simulations can reach 300K without surface reconstructions. Without |
546 |
this practice, simulating 100\% thiol covered interfaces under higher |
547 |
temperatures could hardly avoid surface reconstructions. However, our |
548 |
measurement is based on assuming homogeneity on $x$ and $y$ dimensions |
549 |
so that measurement of $T$ at particular $z$ would be an effective |
550 |
average of the particles of the same type. Since surface |
551 |
reconstructions could eliminate the original $x$ and $y$ dimensional |
552 |
homogeneity, measurement of $G$ is more difficult to conduct under |
553 |
higher temperatures. Therefore, most of our measurements are |
554 |
undertaken at $<T>\sim$200K. |
555 |
|
556 |
However, when the surface is not completely covered by butanethiols, |
557 |
the simulated system is more resistent to the reconstruction |
558 |
above. Our Au-butanethiol/toluene system did not see this phenomena |
559 |
even at $<T>\sim$300K. The Au(111) surfaces have a 90\% coverage of |
560 |
butanethiols and have empty three-fold sites. These empty sites could |
561 |
help prevent surface reconstruction in that they provide other means |
562 |
of capping agent relaxation. It is observed that butanethiols can |
563 |
migrate to their neighbor empty sites during a simulation. Therefore, |
564 |
we were able to obtain $G$'s for these interfaces even at a relatively |
565 |
high temperature without being affected by surface reconstructions. |
566 |
|
567 |
\subsection{Influence of Capping Agent Coverage on $G$} |
568 |
To investigate the influence of butanethiol coverage on interfacial |
569 |
thermal conductance, a series of different coverage Au-butanethiol |
570 |
surfaces is prepared and solvated with various organic |
571 |
molecules. These systems are then equilibrated and their interfacial |
572 |
thermal conductivity are measured with our NIVS algorithm. Table |
573 |
\ref{tlnUhxnUhxnD} lists these results for direct comparison between |
574 |
different coverages of butanethiol. To study the isotope effect in |
575 |
interfacial thermal conductance, deuterated UA-hexane is included as |
576 |
well. |
577 |
|
578 |
It turned out that with partial covered butanethiol on the Au(111) |
579 |
surface, the derivative definition for $G$ (Eq. \ref{derivativeG}) has |
580 |
difficulty to apply, due to the difficulty in locating the maximum of |
581 |
change of $\lambda$. Instead, the discrete definition |
582 |
(Eq. \ref{discreteG}) is easier to apply, as max($\Delta T$) can still |
583 |
be well-defined. Therefore, $G$'s (not $G^\prime$) are used for this |
584 |
section. |
585 |
|
586 |
From Table \ref{tlnUhxnUhxnD}, one can see the significance of the |
587 |
presence of capping agents. Even when a fraction of the Au(111) |
588 |
surface sites are covered with butanethiols, the conductivity would |
589 |
see an enhancement by at least a factor of 3. This indicates the |
590 |
important role cappping agent is playing for thermal transport |
591 |
phenomena on metal/organic solvent surfaces. |
592 |
|
593 |
Interestingly, as one could observe from our results, the maximum |
594 |
conductance enhancement (largest $G$) happens while the surfaces are |
595 |
about 75\% covered with butanethiols. This again indicates that |
596 |
solvent-capping agent contact has an important role of the thermal |
597 |
transport process. Slightly lower butanethiol coverage allows small |
598 |
gaps between butanethiols to form. And these gaps could be filled with |
599 |
solvent molecules, which acts like ``heat conductors'' on the |
600 |
surface. The higher degree of interaction between these solvent |
601 |
molecules and capping agents increases the enhancement effect and thus |
602 |
produces a higher $G$ than densely packed butanethiol arrays. However, |
603 |
once this maximum conductance enhancement is reached, $G$ decreases |
604 |
when butanethiol coverage continues to decrease. Each capping agent |
605 |
molecule reaches its maximum capacity for thermal |
606 |
conductance. Therefore, even higher solvent-capping agent contact |
607 |
would not offset this effect. Eventually, when butanethiol coverage |
608 |
continues to decrease, solvent-capping agent contact actually |
609 |
decreases with the disappearing of butanethiol molecules. In this |
610 |
case, $G$ decrease could not be offset but instead accelerated. |
611 |
|
612 |
A comparison of the results obtained from differenet organic solvents |
613 |
can also provide useful information of the interfacial thermal |
614 |
transport process. The deuterated hexane (UA) results do not appear to |
615 |
be much different from those of normal hexane (UA), given that |
616 |
butanethiol (UA) is non-deuterated for both solvents. These UA model |
617 |
studies, even though eliminating C-H vibration samplings, still have |
618 |
C-C vibrational frequencies different from each other. However, these |
619 |
differences in the IR range do not seem to produce an observable |
620 |
difference for the results of $G$. [MAY NEED FIGURE] |
621 |
|
622 |
Furthermore, results for rigid body toluene solvent, as well as other |
623 |
UA-hexane solvents, are reasonable within the general experimental |
624 |
ranges[CITATIONS]. This suggests that explicit hydrogen might not be a |
625 |
required factor for modeling thermal transport phenomena of systems |
626 |
such as Au-thiol/organic solvent. |
627 |
|
628 |
However, results for Au-butanethiol/toluene do not show an identical |
629 |
trend with those for Au-butanethiol/hexane in that $G$'s remain at |
630 |
approximately the same magnitue when butanethiol coverage differs from |
631 |
25\% to 75\%. This might be rooted in the molecule shape difference |
632 |
for plane-like toluene and chain-like {\it n}-hexane. Due to this |
633 |
difference, toluene molecules have more difficulty in occupying |
634 |
relatively small gaps among capping agents when their coverage is not |
635 |
too low. Therefore, the solvent-capping agent contact may keep |
636 |
increasing until the capping agent coverage reaches a relatively low |
637 |
level. This becomes an offset for decreasing butanethiol molecules on |
638 |
its effect to the process of interfacial thermal transport. Thus, one |
639 |
can see a plateau of $G$ vs. butanethiol coverage in our results. |
640 |
|
641 |
[NEED ERROR ESTIMATE, MAY ALSO PUT J HERE] |
642 |
\begin{table*} |
643 |
\begin{minipage}{\linewidth} |
644 |
\begin{center} |
645 |
\caption{Computed interfacial thermal conductivity ($G$ in |
646 |
MW/m$^2$/K) values for the Au-butanethiol/solvent interface |
647 |
with various UA models and different capping agent coverages |
648 |
at $<T>\sim$200K using certain energy flux respectively.} |
649 |
|
650 |
\begin{tabular}{cccc} |
651 |
\hline\hline |
652 |
Thiol & & & \\ |
653 |
coverage (\%) & hexane & hexane-D & toluene \\ |
654 |
\hline |
655 |
0.0 & 46.5 & 43.9 & 70.1 \\ |
656 |
25.0 & 151 & 153 & 249 \\ |
657 |
50.0 & 172 & 182 & 214 \\ |
658 |
75.0 & 242 & 229 & 244 \\ |
659 |
88.9 & 178 & - & - \\ |
660 |
100.0 & 137 & 153 & 187 \\ |
661 |
\hline\hline |
662 |
\end{tabular} |
663 |
\label{tlnUhxnUhxnD} |
664 |
\end{center} |
665 |
\end{minipage} |
666 |
\end{table*} |
667 |
|
668 |
\subsection{Influence of Chosen Molecule Model on $G$} |
669 |
[MAY COMBINE W MECHANISM STUDY] |
670 |
|
671 |
For the all-atom model, the liquid hexane phase was not stable under NPT |
672 |
conditions. Therefore, the simulation length scale parameters are |
673 |
adopted from previous equilibration results of the united-atom model |
674 |
at 200K. Table \ref{AuThiolHexaneAA} shows the results of these |
675 |
simulations. The conductivity values calculated with full capping |
676 |
agent coverage are substantially larger than observed in the |
677 |
united-atom model, and is even higher than predicted by |
678 |
experiments. It is possible that our parameters for metal-non-metal |
679 |
particle interactions lead to an overestimate of the interfacial |
680 |
thermal conductivity, although the active C-H vibrations in the |
681 |
all-atom model (which should not be appreciably populated at normal |
682 |
temperatures) could also account for this high conductivity. The major |
683 |
thermal transfer barrier of Au/butanethiol/hexane interface is between |
684 |
the liquid phase and the capping agent, so extra degrees of freedom |
685 |
such as the C-H vibrations could enhance heat exchange between these |
686 |
two phases and result in a much higher conductivity. |
687 |
|
688 |
\begin{table*} |
689 |
\begin{minipage}{\linewidth} |
690 |
\begin{center} |
691 |
|
692 |
\caption{Computed interfacial thermal conductivity ($G$ and |
693 |
$G^\prime$) values for the Au/butanethiol/hexane interface |
694 |
with all-atom model and different capping agent coverage at |
695 |
200K using a range of energy fluxes.} |
696 |
|
697 |
\begin{tabular}{cccc} |
698 |
\hline\hline |
699 |
Thiol & $J_z$ & $G$ & $G^\prime$ \\ |
700 |
coverage (\%) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
701 |
\hline |
702 |
0.0 & 0.95 & 28.5 & 27.2 \\ |
703 |
& 1.88 & 30.3 & 28.9 \\ |
704 |
100.0 & 2.87 & 551 & 294 \\ |
705 |
& 3.81 & 494 & 193 \\ |
706 |
\hline\hline |
707 |
\end{tabular} |
708 |
\label{AuThiolHexaneAA} |
709 |
\end{center} |
710 |
\end{minipage} |
711 |
\end{table*} |
712 |
|
713 |
|
714 |
significant conductance enhancement compared to the gold/water |
715 |
interface without capping agent and agree with available experimental |
716 |
data. This indicates that the metal-metal potential, though not |
717 |
predicting an accurate bulk metal thermal conductivity, does not |
718 |
greatly interfere with the simulation of the thermal conductance |
719 |
behavior across a non-metal interface. |
720 |
|
721 |
% The results show that the two definitions used for $G$ yield |
722 |
% comparable values, though $G^\prime$ tends to be smaller. |
723 |
|
724 |
\subsection{Mechanism of Interfacial Thermal Conductance Enhancement |
725 |
by Capping Agent} |
726 |
[MAY INTRODUCE PROTOCOL IN METHODOLOGY/COMPUTATIONAL DETAIL] |
727 |
|
728 |
|
729 |
%subsubsection{Vibrational spectrum study on conductance mechanism} |
730 |
To investigate the mechanism of this interfacial thermal conductance, |
731 |
the vibrational spectra of various gold systems were obtained and are |
732 |
shown as in the upper panel of Fig. \ref{vibration}. To obtain these |
733 |
spectra, one first runs a simulation in the NVE ensemble and collects |
734 |
snapshots of configurations; these configurations are used to compute |
735 |
the velocity auto-correlation functions, which is used to construct a |
736 |
power spectrum via a Fourier transform. The gold surfaces covered by |
737 |
butanethiol molecules exhibit an additional peak observed at a |
738 |
frequency of $\sim$170cm$^{-1}$, which is attributed to the vibration |
739 |
of the S-Au bond. This vibration enables efficient thermal transport |
740 |
from surface Au atoms to the capping agents. Simultaneously, as shown |
741 |
in the lower panel of Fig. \ref{vibration}, the large overlap of the |
742 |
vibration spectra of butanethiol and hexane in the all-atom model, |
743 |
including the C-H vibration, also suggests high thermal exchange |
744 |
efficiency. The combination of these two effects produces the drastic |
745 |
interfacial thermal conductance enhancement in the all-atom model. |
746 |
|
747 |
\begin{figure} |
748 |
\includegraphics[width=\linewidth]{vibration} |
749 |
\caption{Vibrational spectra obtained for gold in different |
750 |
environments (upper panel) and for Au/thiol/hexane simulation in |
751 |
all-atom model (lower panel).} |
752 |
\label{vibration} |
753 |
\end{figure} |
754 |
% MAY NEED TO CONVERT TO JPEG |
755 |
|
756 |
\section{Conclusions} |
757 |
|
758 |
|
759 |
[NECESSITY TO STUDY THERMAL CONDUCTANCE IN NANOCRYSTAL STRUCTURE]\cite{vlugt:cpc2007154} |
760 |
|
761 |
\section{Acknowledgments} |
762 |
Support for this project was provided by the National Science |
763 |
Foundation under grant CHE-0848243. Computational time was provided by |
764 |
the Center for Research Computing (CRC) at the University of Notre |
765 |
Dame. \newpage |
766 |
|
767 |
\bibliography{interfacial} |
768 |
|
769 |
\end{doublespace} |
770 |
\end{document} |
771 |
|