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\section{Introduction} |
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Interfacial thermal conductance is extensively studied both |
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experimentally and computationally\cite{cahill:793}, due to its |
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importance in nanoscale science and technology. Reliability of |
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nanoscale devices depends on their thermal transport |
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properties. Unlike bulk homogeneous materials, nanoscale materials |
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features significant presence of interfaces, and these interfaces |
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could dominate the heat transfer behavior of these |
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materials. Furthermore, these materials are generally heterogeneous, |
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which challenges traditional research methods for homogeneous |
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systems. |
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Due to the importance of heat flow in nanotechnology, interfacial |
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thermal conductance has been studied extensively both experimentally |
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and computationally.\cite{cahill:793} Unlike bulk materials, nanoscale |
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materials have a significant fraction of their atoms at interfaces, |
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and the chemical details of these interfaces govern the heat transfer |
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behavior. Furthermore, the interfaces are |
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heterogeneous (e.g. solid - liquid), which provides a challenge to |
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traditional methods developed for homogeneous systems. |
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|
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Heat conductance of molecular and nano-scale interfaces will be |
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affected by the chemical details of the surface. Experimentally, |
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various interfaces have been investigated for their thermal |
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conductance properties. Wang {\it et al.} studied heat transport |
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through long-chain hydrocarbon monolayers on gold substrate at |
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individual molecular level\cite{Wang10082007}; Schmidt {\it et al.} |
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studied the role of CTAB on thermal transport between gold nanorods |
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and solvent\cite{doi:10.1021/jp8051888}; Juv\'e {\it et al.} studied |
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Experimentally, various interfaces have been investigated for their |
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thermal conductance. Wang {\it et al.} studied heat transport through |
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long-chain hydrocarbon monolayers on gold substrate at individual |
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molecular level,\cite{Wang10082007} Schmidt {\it et al.} studied the |
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role of CTAB on thermal transport between gold nanorods and |
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solvent,\cite{doi:10.1021/jp8051888} and Juv\'e {\it et al.} studied |
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the cooling dynamics, which is controlled by thermal interface |
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resistence of glass-embedded metal |
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nanoparticles\cite{PhysRevB.80.195406}. Although interfaces are |
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commonly barriers for heat transport, Alper {\it et al.} suggested |
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that specific ligands (capping agents) could completely eliminate this |
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barrier ($G\rightarrow\infty$)\cite{doi:10.1021/la904855s}. |
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nanoparticles.\cite{PhysRevB.80.195406} Although interfaces are |
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normally considered barriers for heat transport, Alper {\it et al.} |
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suggested that specific ligands (capping agents) could completely |
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eliminate this barrier |
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($G\rightarrow\infty$).\cite{doi:10.1021/la904855s} |
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|
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Theoretical and computational models have also been used to study the |
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interfacial thermal transport in order to gain an understanding of |
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employed Non-Equilibrium Molecular Dynamics (NEMD) simulations to |
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study thermal transport from hot Au(111) substrate to a self-assembled |
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monolayer of alkylthiol with relatively long chain (8-20 carbon |
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atoms)\cite{hase:2010,hase:2011}. However, ensemble averaged |
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atoms).\cite{hase:2010,hase:2011} However, ensemble averaged |
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|
measurements for heat conductance of interfaces between the capping |
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monolayer on Au and a solvent phase has yet to be studied. |
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The comparatively low thermal flux through interfaces is |
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monolayer on Au and a solvent phase have yet to be studied with their |
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approach. The comparatively low thermal flux through interfaces is |
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difficult to measure with Equilibrium MD or forward NEMD simulation |
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methods. Therefore, the Reverse NEMD (RNEMD) |
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methods\cite{MullerPlathe:1997xw} would have the advantage of having |
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this difficult to measure flux known when studying the thermal |
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transport across interfaces, given that the simulation methods being |
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able to effectively apply an unphysical flux in non-homogeneous |
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systems. Garde and coworkers\cite{garde:nl2005,garde:PhysRevLett2009} |
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applied this approach to various liquid interfaces and studied how |
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thermal conductance (or resistance) is dependent on chemistry details |
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of interfaces, e.g. hydrophobic and hydrophilic aqueous interfaces. |
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methods\cite{MullerPlathe:1997xw,kuang:164101} would have the |
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advantage of applying this difficult to measure flux (while measuring |
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the resulting gradient), given that the simulation methods being able |
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to effectively apply an unphysical flux in non-homogeneous systems. |
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Garde and coworkers\cite{garde:nl2005,garde:PhysRevLett2009} applied |
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this approach to various liquid interfaces and studied how thermal |
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conductance (or resistance) is dependent on chemistry details of |
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interfaces, e.g. hydrophobic and hydrophilic aqueous interfaces. |
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|
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Recently, we have developed the Non-Isotropic Velocity Scaling (NIVS) |
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Recently, we have developed a Non-Isotropic Velocity Scaling (NIVS) |
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algorithm for RNEMD simulations\cite{kuang:164101}. This algorithm |
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retains the desirable features of RNEMD (conservation of linear |
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momentum and total energy, compatibility with periodic boundary |
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thermal transport across these interfaces was studied and the |
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underlying mechanism for the phenomena was investigated. |
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|
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[MAY ADD WHY STUDY AU-THIOL SURFACE; CITE SHAOYI JIANG] |
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|
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\section{Methodology} |
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\subsection{Imposd-Flux Methods in MD Simulations} |
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Steady state MD simulations has the advantage that not many |
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Steady state MD simulations have an advantage in that not many |
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|
trajectories are needed to study the relationship between thermal flux |
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and thermal gradients. For systems including low conductance |
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interfaces one must have a method capable of generating or measuring |
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relatively small fluxes, compared to those required for bulk |
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conductivity. This requirement makes the calculation even more |
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difficult for those slowly-converging equilibrium |
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methods\cite{Viscardy:2007lq}. Forward methods may impose gradient, |
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but in interfacial conditions it is not clear what behavior to impose |
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at the interfacial boundaries. Imposed-flux reverse non-equilibrium |
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methods\cite{MullerPlathe:1997xw} have the flux set {\it a priori} and |
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the thermal response becomes easier to measure than the flux. Although |
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and thermal gradients. For systems with low interfacial conductance, |
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one must have a method capable of generating or measuring relatively |
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small fluxes, compared to those required for bulk conductivity. This |
144 |
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requirement makes the calculation even more difficult for |
145 |
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slowly-converging equilibrium methods.\cite{Viscardy:2007lq} Forward |
146 |
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NEMD methods impose a gradient (and measure a flux), but at interfaces |
147 |
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it is not clear what behavior should be imposed at the boundaries |
148 |
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between materials. Imposed-flux reverse non-equilibrium |
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methods\cite{MullerPlathe:1997xw} set the flux {\it a priori} and |
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the thermal response becomes an easy-to-measure quantity. Although |
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M\"{u}ller-Plathe's original momentum swapping approach can be used |
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for exchanging energy between particles of different identity, the |
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kinetic energy transfer efficiency is affected by the mass difference |
154 |
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between the particles, which limits its application on heterogeneous |
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interfacial systems. |
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|
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The non-isotropic velocity scaling (NIVS)\cite{kuang:164101} approach to |
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non-equilibrium MD simulations is able to impose a wide range of |
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The non-isotropic velocity scaling (NIVS) \cite{kuang:164101} approach |
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to non-equilibrium MD simulations is able to impose a wide range of |
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kinetic energy fluxes without obvious perturbation to the velocity |
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distributions of the simulated systems. Furthermore, this approach has |
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the advantage in heterogeneous interfaces in that kinetic energy flux |
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for computing thermal conductivities. The NIVS algorithm conserves |
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momenta and energy and does not depend on an external thermostat. |
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|
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\subsection{Defining Interfacial Thermal Conductivity $G$} |
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Given a system with thermal gradients and the corresponding thermal |
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flux, for interfaces with a relatively low interfacial conductance, |
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the bulk regions on either side of an interface rapidly come to a |
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state in which the two phases have relatively homogeneous (but |
180 |
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distinct) temperatures. The interfacial thermal conductivity $G$ can |
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therefore be approximated as: |
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\subsection{Defining Interfacial Thermal Conductivity ($G$)} |
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|
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For an interface with relatively low interfacial conductance, and a |
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thermal flux between two distinct bulk regions, the regions on either |
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side of the interface rapidly come to a state in which the two phases |
180 |
> |
have relatively homogeneous (but distinct) temperatures. The |
181 |
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interfacial thermal conductivity $G$ can therefore be approximated as: |
182 |
|
\begin{equation} |
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G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle - |
183 |
> |
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle - |
184 |
|
\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 |
188 |
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transfer and ${\langle T_\mathrm{hot}\rangle}$ and ${\langle |
189 |
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T_\mathrm{cold}\rangle}$ are the average observed temperature of the |
190 |
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two separated phases. |
187 |
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where ${E_{total}}$ is the total imposed non-physical kinetic energy |
188 |
> |
transfer during the simulation and ${\langle T_\mathrm{hot}\rangle}$ |
189 |
> |
and ${\langle T_\mathrm{cold}\rangle}$ are the average observed |
190 |
> |
temperature of the two separated phases. |
191 |
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|
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|
When the interfacial conductance is {\it not} small, there are two |
193 |
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ways to define $G$. |
193 |
> |
ways to define $G$. One way is to assume the temperature is discrete |
194 |
> |
on the two sides of the interface. $G$ can be calculated using the |
195 |
> |
applied thermal flux $J$ and the maximum temperature difference |
196 |
> |
measured along the thermal gradient max($\Delta T$), which occurs at |
197 |
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the Gibbs deviding surface (Figure \ref{demoPic}): \begin{equation} |
198 |
> |
G=\frac{J}{\Delta T} \label{discreteG} \end{equation} |
199 |
|
|
200 |
– |
One way is to assume the temperature is discrete on the two sides of |
201 |
– |
the interface. $G$ can be calculated using the applied thermal flux |
202 |
– |
$J$ and the maximum temperature difference measured along the thermal |
203 |
– |
gradient max($\Delta T$), which occurs at the Gibbs deviding surface |
204 |
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(Figure \ref{demoPic}): |
205 |
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\begin{equation} |
206 |
– |
G=\frac{J}{\Delta T} |
207 |
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\label{discreteG} |
208 |
– |
\end{equation} |
209 |
– |
|
200 |
|
\begin{figure} |
201 |
|
\includegraphics[width=\linewidth]{method} |
202 |
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\caption{Interfacial conductance can be calculated by applying an |
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|
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The other approach is to assume a continuous temperature profile along |
215 |
|
the thermal gradient axis (e.g. $z$) and define $G$ at the point where |
216 |
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the magnitude of thermal conductivity $\lambda$ change reach its |
216 |
> |
the magnitude of thermal conductivity ($\lambda$) change reaches its |
217 |
|
maximum, given that $\lambda$ is well-defined throughout the space: |
218 |
|
\begin{equation} |
219 |
|
G^\prime = \Big|\frac{\partial\lambda}{\partial z}\Big| |
224 |
|
\label{derivativeG} |
225 |
|
\end{equation} |
226 |
|
|
227 |
< |
With the temperature profile obtained from simulations, one is able to |
227 |
> |
With temperature profiles obtained from simulation, one is able to |
228 |
|
approximate the first and second derivatives of $T$ with finite |
229 |
< |
difference methods and thus calculate $G^\prime$. |
229 |
> |
difference methods and calculate $G^\prime$. In what follows, both |
230 |
> |
definitions have been used, and are compared in the results. |
231 |
|
|
232 |
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In what follows, both definitions have been used for calculation and |
233 |
< |
are compared in the results. |
234 |
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|
235 |
< |
To compare the above definitions ($G$ and $G^\prime$), we have modeled |
236 |
< |
a metal slab with its (111) surfaces perpendicular to the $z$-axis of |
246 |
< |
our simulation cells. Both with and without capping agents on the |
247 |
< |
surfaces, the metal slab is solvated with simple organic solvents, as |
232 |
> |
To investigate the interfacial conductivity at metal / solvent |
233 |
> |
interfaces, we have modeled a metal slab with its (111) surfaces |
234 |
> |
perpendicular to the $z$-axis of our simulation cells. The metal slab |
235 |
> |
has been prepared both with and without capping agents on the exposed |
236 |
> |
surface, and has been solvated with simple organic solvents, as |
237 |
|
illustrated in Figure \ref{gradT}. |
238 |
|
|
239 |
|
With the simulation cell described above, we are able to equilibrate |
240 |
|
the system and impose an unphysical thermal flux between the liquid |
241 |
|
and the metal phase using the NIVS algorithm. By periodically applying |
242 |
< |
the unphysical flux, we are able to obtain a temperature profile and |
243 |
< |
its spatial derivatives. These quantities enable the evaluation of the |
244 |
< |
interfacial thermal conductance of a surface. Figure \ref{gradT} is an |
245 |
< |
example of how an applied thermal flux can be used to obtain the 1st |
257 |
< |
and 2nd derivatives of the temperature profile. |
242 |
> |
the unphysical flux, we obtained a temperature profile and its spatial |
243 |
> |
derivatives. Figure \ref{gradT} shows how an applied thermal flux can |
244 |
> |
be used to obtain the 1st and 2nd derivatives of the temperature |
245 |
> |
profile. |
246 |
|
|
247 |
|
\begin{figure} |
248 |
|
\includegraphics[width=\linewidth]{gradT} |
256 |
|
\section{Computational Details} |
257 |
|
\subsection{Simulation Protocol} |
258 |
|
The NIVS algorithm has been implemented in our MD simulation code, |
259 |
< |
OpenMD\cite{Meineke:2005gd,openmd}, and was used for our |
260 |
< |
simulations. Different metal slab thickness (layer numbers of Au) was |
261 |
< |
simulated. Metal slabs were first equilibrated under atmospheric |
262 |
< |
pressure (1 atm) and a desired temperature (e.g. 200K). After |
263 |
< |
equilibration, butanethiol capping agents were placed at three-fold |
264 |
< |
hollow sites on the Au(111) surfaces. These sites could be either a |
265 |
< |
{\it fcc} or {\it hcp} site. However, Hase {\it et al.} found that |
266 |
< |
they are equivalent in a heat transfer process\cite{hase:2010}, so |
279 |
< |
they are not distinguished in our study. The maximum butanethiol |
259 |
> |
OpenMD\cite{Meineke:2005gd,openmd}, and was used for our simulations. |
260 |
> |
Metal slabs of 6 or 11 layers of Au atoms were first equilibrated |
261 |
> |
under atmospheric pressure (1 atm) and 200K. After equilibration, |
262 |
> |
butanethiol capping agents were placed at three-fold hollow sites on |
263 |
> |
the Au(111) surfaces. These sites are either {\it fcc} or {\it |
264 |
> |
hcp} sites, although Hase {\it et al.} found that they are |
265 |
> |
equivalent in a heat transfer process,\cite{hase:2010} so we did not |
266 |
> |
distinguish between these sites in our study. The maximum butanethiol |
267 |
|
capacity on Au surface is $1/3$ of the total number of surface Au |
268 |
|
atoms, and the packing forms a $(\sqrt{3}\times\sqrt{3})R30^\circ$ |
269 |
|
structure\cite{doi:10.1021/ja00008a001,doi:10.1021/cr9801317}. A |
270 |
< |
series of different coverages was derived by evenly eliminating |
271 |
< |
butanethiols on the surfaces, and was investigated in order to study |
272 |
< |
the relation between coverage and interfacial conductance. |
270 |
> |
series of lower coverages was also prepared by eliminating |
271 |
> |
butanethiols from the higher coverage surface in a regular manner. The |
272 |
> |
lower coverages were prepared in order to study the relation between |
273 |
> |
coverage and interfacial conductance. |
274 |
|
|
275 |
|
The capping agent molecules were allowed to migrate during the |
276 |
|
simulations. They distributed themselves uniformly and sampled a |
277 |
|
number of three-fold sites throughout out study. Therefore, the |
278 |
< |
initial configuration would not noticeably affect the sampling of a |
278 |
> |
initial configuration does not noticeably affect the sampling of a |
279 |
|
variety of configurations of the same coverage, and the final |
280 |
|
conductance measurement would be an average effect of these |
281 |
< |
configurations explored in the simulations. [MAY NEED SNAPSHOTS] |
281 |
> |
configurations explored in the simulations. |
282 |
|
|
283 |
< |
After the modified Au-butanethiol surface systems were equilibrated |
284 |
< |
under canonical ensemble, organic solvent molecules were packed in the |
285 |
< |
previously empty part of the simulation cells\cite{packmol}. Two |
283 |
> |
After the modified Au-butanethiol surface systems were equilibrated in |
284 |
> |
the canonical (NVT) ensemble, organic solvent molecules were packed in |
285 |
> |
the previously empty part of the simulation cells.\cite{packmol} Two |
286 |
|
solvents were investigated, one which has little vibrational overlap |
287 |
< |
with the alkanethiol and a planar shape (toluene), and one which has |
288 |
< |
similar vibrational frequencies and chain-like shape ({\it n}-hexane). |
287 |
> |
with the alkanethiol and which has a planar shape (toluene), and one |
288 |
> |
which has similar vibrational frequencies to the capping agent and |
289 |
> |
chain-like shape ({\it n}-hexane). |
290 |
|
|
291 |
< |
The space filled by solvent molecules, i.e. the gap between |
292 |
< |
periodically repeated Au-butanethiol surfaces should be carefully |
293 |
< |
chosen. A very long length scale for the thermal gradient axis ($z$) |
305 |
< |
may cause excessively hot or cold temperatures in the middle of the |
291 |
> |
The simulation cells were not particularly extensive along the |
292 |
> |
$z$-axis, as a very long length scale for the thermal gradient may |
293 |
> |
cause excessively hot or cold temperatures in the middle of the |
294 |
|
solvent region and lead to undesired phenomena such as solvent boiling |
295 |
|
or freezing when a thermal flux is applied. Conversely, too few |
296 |
|
solvent molecules would change the normal behavior of the liquid |
297 |
|
phase. Therefore, our $N_{solvent}$ values were chosen to ensure that |
298 |
< |
these extreme cases did not happen to our simulations. And the |
299 |
< |
corresponding spacing is usually $35 \sim 75$\AA. |
298 |
> |
these extreme cases did not happen to our simulations. The spacing |
299 |
> |
between periodic images of the gold interfaces is $35 \sim 75$\AA. |
300 |
|
|
301 |
|
The initial configurations generated are further equilibrated with the |
302 |
< |
$x$ and $y$ dimensions fixed, only allowing length scale change in $z$ |
303 |
< |
dimension. This is to ensure that the equilibration of liquid phase |
304 |
< |
does not affect the metal crystal structure in $x$ and $y$ dimensions. |
305 |
< |
To investigate this effect, comparisons were made with simulations |
306 |
< |
that allow changes of $L_x$ and $L_y$ during NPT equilibration, and |
307 |
< |
the results are shown in later sections. After ensuring the liquid |
308 |
< |
phase reaches equilibrium at atmospheric pressure (1 atm), further |
309 |
< |
equilibration are followed under NVT and then NVE ensembles. |
302 |
> |
$x$ and $y$ dimensions fixed, only allowing the $z$-length scale to |
303 |
> |
change. This is to ensure that the equilibration of liquid phase does |
304 |
> |
not affect the metal's crystalline structure. Comparisons were made |
305 |
> |
with simulations that allowed changes of $L_x$ and $L_y$ during NPT |
306 |
> |
equilibration. No substantial changes in the box geometry were noticed |
307 |
> |
in these simulations. After ensuring the liquid phase reaches |
308 |
> |
equilibrium at atmospheric pressure (1 atm), further equilibration was |
309 |
> |
carried out under canonical (NVT) and microcanonical (NVE) ensembles. |
310 |
|
|
311 |
< |
After the systems reach equilibrium, NIVS is implemented to impose a |
312 |
< |
periodic unphysical thermal flux between the metal and the liquid |
313 |
< |
phase. Most of our simulations are under an average temperature of |
314 |
< |
$\sim$200K. Therefore, this flux usually comes from the metal to the |
311 |
> |
After the systems reach equilibrium, NIVS was used to impose an |
312 |
> |
unphysical thermal flux between the metal and the liquid phases. Most |
313 |
> |
of our simulations were done under an average temperature of |
314 |
> |
$\sim$200K. Therefore, thermal flux usually came from the metal to the |
315 |
|
liquid so that the liquid has a higher temperature and would not |
316 |
< |
freeze due to excessively low temperature. After this induced |
317 |
< |
temperature gradient is stablized, the temperature profile of the |
318 |
< |
simulation cell is recorded. To do this, the simulation cell is |
319 |
< |
devided evenly into $N$ slabs along the $z$-axis and $N$ is maximized |
332 |
< |
for highest possible spatial resolution but not too many to have some |
333 |
< |
slabs empty most of the time. The average temperatures of each slab |
316 |
> |
freeze due to lowered temperatures. After this induced temperature |
317 |
> |
gradient had stablized, the temperature profile of the simulation cell |
318 |
> |
was recorded. To do this, the simulation cell is devided evenly into |
319 |
> |
$N$ slabs along the $z$-axis. The average temperatures of each slab |
320 |
|
are recorded for 1$\sim$2 ns. When the slab width $d$ of each slab is |
321 |
|
the same, the derivatives of $T$ with respect to slab number $n$ can |
322 |
< |
be directly used for $G^\prime$ calculations: |
323 |
< |
\begin{equation} |
338 |
< |
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
322 |
> |
be directly used for $G^\prime$ calculations: \begin{equation} |
323 |
> |
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
324 |
|
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
325 |
|
= |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big| |
326 |
|
\Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2 |
329 |
|
\label{derivativeG2} |
330 |
|
\end{equation} |
331 |
|
|
332 |
< |
All of the above simulation procedures use a time step of 1 fs. And |
333 |
< |
each equilibration / stabilization step usually takes 100 ps, or |
334 |
< |
longer, if necessary. |
332 |
> |
All of the above simulation procedures use a time step of 1 fs. Each |
333 |
> |
equilibration stage took a minimum of 100 ps, although in some cases, |
334 |
> |
longer equilibration stages were utilized. |
335 |
|
|
336 |
|
\subsection{Force Field Parameters} |
337 |
< |
Our simulations include various components. Figure \ref{demoMol} |
338 |
< |
demonstrates the sites defined for both United-Atom and All-Atom |
339 |
< |
models of the organic solvent and capping agent molecules in our |
340 |
< |
simulations. Force field parameter descriptions are needed for |
337 |
> |
Our simulations include a number of chemically distinct components. |
338 |
> |
Figure \ref{demoMol} demonstrates the sites defined for both |
339 |
> |
United-Atom and All-Atom models of the organic solvent and capping |
340 |
> |
agents in our simulations. Force field parameters are needed for |
341 |
|
interactions both between the same type of particles and between |
342 |
|
particles of different species. |
343 |
|
|
354 |
|
\end{figure} |
355 |
|
|
356 |
|
The Au-Au interactions in metal lattice slab is described by the |
357 |
< |
quantum Sutton-Chen (QSC) formulation\cite{PhysRevB.59.3527}. The QSC |
357 |
> |
quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
358 |
|
potentials include zero-point quantum corrections and are |
359 |
|
reparametrized for accurate surface energies compared to the |
360 |
< |
Sutton-Chen potentials\cite{Chen90}. |
360 |
> |
Sutton-Chen potentials.\cite{Chen90} |
361 |
|
|
362 |
< |
For both solvent molecules, straight chain {\it n}-hexane and aromatic |
363 |
< |
toluene, United-Atom (UA) and All-Atom (AA) models are used |
364 |
< |
respectively. The TraPPE-UA |
362 |
> |
For the two solvent molecules, {\it n}-hexane and toluene, two |
363 |
> |
different atomistic models were utilized. Both solvents were modeled |
364 |
> |
using United-Atom (UA) and All-Atom (AA) models. The TraPPE-UA |
365 |
|
parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
366 |
|
for our UA solvent molecules. In these models, sites are located at |
367 |
|
the carbon centers for alkyl groups. Bonding interactions, including |
368 |
|
bond stretches and bends and torsions, were used for intra-molecular |
369 |
< |
sites not separated by more than 3 bonds. Otherwise, for non-bonded |
370 |
< |
interactions, Lennard-Jones potentials are used. [CHECK CITATION] |
369 |
> |
sites closer than 3 bonds. For non-bonded interactions, Lennard-Jones |
370 |
> |
potentials are used. |
371 |
|
|
372 |
< |
By eliminating explicit hydrogen atoms, these models are simple and |
373 |
< |
computationally efficient, while maintains good accuracy. However, the |
374 |
< |
TraPPE-UA for alkanes is known to predict a lower boiling point than |
375 |
< |
experimental values. Considering that after an unphysical thermal flux |
376 |
< |
is applied to a system, the temperature of ``hot'' area in the liquid |
377 |
< |
phase would be significantly higher than the average of the system, to |
378 |
< |
prevent over heating and boiling of the liquid phase, the average |
379 |
< |
temperature in our simulations should be much lower than the liquid |
380 |
< |
boiling point. |
372 |
> |
By eliminating explicit hydrogen atoms, the TraPPE-UA models are |
373 |
> |
simple and computationally efficient, while maintaining good accuracy. |
374 |
> |
However, the TraPPE-UA model for alkanes is known to predict a slighly |
375 |
> |
lower boiling point than experimental values. This is one of the |
376 |
> |
reasons we used a lower average temperature (200K) for our |
377 |
> |
simulations. If heat is transferred to the liquid phase during the |
378 |
> |
NIVS simulation, the liquid in the hot slab can actually be |
379 |
> |
substantially warmer than the mean temperature in the simulation. The |
380 |
> |
lower mean temperatures therefore prevent solvent boiling. |
381 |
|
|
382 |
< |
For UA-toluene model, the non-bonded potentials between |
383 |
< |
inter-molecular sites have a similar Lennard-Jones formulation. For |
384 |
< |
intra-molecular interactions, considering the stiffness of the benzene |
385 |
< |
ring, rigid body constraints are applied for further computational |
386 |
< |
efficiency. All bonds in the benzene ring and between the ring and the |
402 |
< |
methyl group remain rigid during the progress of simulations. |
382 |
> |
For UA-toluene, the non-bonded potentials between intermolecular sites |
383 |
> |
have a similar Lennard-Jones formulation. The toluene molecules were |
384 |
> |
treated as a single rigid body, so there was no need for |
385 |
> |
intramolecular interactions (including bonds, bends, or torsions) in |
386 |
> |
this solvent model. |
387 |
|
|
388 |
|
Besides the TraPPE-UA models, AA models for both organic solvents are |
389 |
|
included in our studies as well. For hexane, the OPLS-AA\cite{OPLSAA} |
390 |
< |
force field is used. Additional explicit hydrogen sites were |
390 |
> |
force field is used, and additional explicit hydrogen sites were |
391 |
|
included. Besides bonding and non-bonded site-site interactions, |
392 |
|
partial charges and the electrostatic interactions were added to each |
393 |
|
CT and HC site. For toluene, the United Force Field developed by |
394 |
< |
Rapp\'{e} {\it et al.}\cite{doi:10.1021/ja00051a040} is |
395 |
< |
adopted. Without the rigid body constraints, bonding interactions were |
396 |
< |
included. For the aromatic ring, improper torsions (inversions) were |
397 |
< |
added as an extra potential for maintaining the planar shape. |
414 |
< |
[CHECK CITATION] |
394 |
> |
Rapp\'{e} {\it et al.}\cite{doi:10.1021/ja00051a040} was adopted, and |
395 |
> |
a flexible model for the toluene molecule was utilized which included |
396 |
> |
bond, bend, torsion, and inversion potentials to enforce ring |
397 |
> |
planarity. |
398 |
|
|
399 |
< |
The capping agent in our simulations, the butanethiol molecules can |
400 |
< |
either use UA or AA model. The TraPPE-UA force fields includes |
399 |
> |
The butanethiol capping agent in our simulations, were also modeled |
400 |
> |
with both UA and AA model. The TraPPE-UA force field includes |
401 |
|
parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for |
402 |
|
UA butanethiol model in our simulations. The OPLS-AA also provides |
403 |
|
parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) |
404 |
< |
surfaces do not have the hydrogen atom bonded to sulfur. To adapt this |
405 |
< |
change and derive suitable parameters for butanethiol adsorbed on |
406 |
< |
Au(111) surfaces, we adopt the S parameters from Luedtke and |
407 |
< |
Landman\cite{landman:1998}[CHECK CITATION] |
408 |
< |
and modify parameters for its neighbor C |
409 |
< |
atom for charge balance in the molecule. Note that the model choice |
410 |
< |
(UA or AA) of capping agent can be different from the |
411 |
< |
solvent. Regardless of model choice, the force field parameters for |
429 |
< |
interactions between capping agent and solvent can be derived using |
430 |
< |
Lorentz-Berthelot Mixing Rule: |
404 |
> |
surfaces do not have the hydrogen atom bonded to sulfur. To derive |
405 |
> |
suitable parameters for butanethiol adsorbed on Au(111) surfaces, we |
406 |
> |
adopt the S parameters from Luedtke and Landman\cite{landman:1998} and |
407 |
> |
modify the parameters for the CTS atom to maintain charge neutrality |
408 |
> |
in the molecule. Note that the model choice (UA or AA) for the capping |
409 |
> |
agent can be different from the solvent. Regardless of model choice, |
410 |
> |
the force field parameters for interactions between capping agent and |
411 |
> |
solvent can be derived using Lorentz-Berthelot Mixing Rule: |
412 |
|
\begin{eqnarray} |
413 |
< |
\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\ |
414 |
< |
\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}} |
413 |
> |
\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\ |
414 |
> |
\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}} |
415 |
|
\end{eqnarray} |
416 |
|
|
417 |
< |
To describe the interactions between metal Au and non-metal capping |
418 |
< |
agent and solvent particles, we refer to an adsorption study of alkyl |
419 |
< |
thiols on gold surfaces by Vlugt {\it et |
420 |
< |
al.}\cite{vlugt:cpc2007154} They fitted an effective Lennard-Jones |
421 |
< |
form of potential parameters for the interaction between Au and |
422 |
< |
pseudo-atoms CH$_x$ and S based on a well-established and widely-used |
423 |
< |
effective potential of Hautman and Klein\cite{hautman:4994} for the |
424 |
< |
Au(111) surface. As our simulations require the gold lattice slab to |
425 |
< |
be non-rigid so that it could accommodate kinetic energy for thermal |
445 |
< |
transport study purpose, the pair-wise form of potentials is |
446 |
< |
preferred. |
417 |
> |
To describe the interactions between metal (Au) and non-metal atoms, |
418 |
> |
we refer to an adsorption study of alkyl thiols on gold surfaces by |
419 |
> |
Vlugt {\it et al.}\cite{vlugt:cpc2007154} They fitted an effective |
420 |
> |
Lennard-Jones form of potential parameters for the interaction between |
421 |
> |
Au and pseudo-atoms CH$_x$ and S based on a well-established and |
422 |
> |
widely-used effective potential of Hautman and Klein for the Au(111) |
423 |
> |
surface.\cite{hautman:4994} As our simulations require the gold slab |
424 |
> |
to be flexible to accommodate thermal excitation, the pair-wise form |
425 |
> |
of potentials they developed was used for our study. |
426 |
|
|
427 |
< |
Besides, the potentials developed from {\it ab initio} calculations by |
428 |
< |
Leng {\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
429 |
< |
interactions between Au and aromatic C/H atoms in toluene. A set of |
430 |
< |
pseudo Lennard-Jones parameters were provided for Au in their force |
431 |
< |
fields. By using the Mixing Rule, this can be used to derive pair-wise |
432 |
< |
potentials for non-bonded interactions between Au and non-metal sites. |
433 |
< |
|
434 |
< |
However, the Lennard-Jones parameters between Au and other types of |
435 |
< |
particles, such as All-Atom normal alkanes in our simulations are not |
436 |
< |
yet well-established. For these interactions, we attempt to derive |
437 |
< |
their parameters using the Mixing Rule. To do this, Au pseudo |
459 |
< |
Lennard-Jones parameters for ``Metal-non-Metal'' (MnM) interactions |
460 |
< |
were first extracted from the Au-CH$_x$ parameters by applying the |
461 |
< |
Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM'' |
462 |
< |
parameters in our simulations. |
427 |
> |
The potentials developed from {\it ab initio} calculations by Leng |
428 |
> |
{\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
429 |
> |
interactions between Au and aromatic C/H atoms in toluene. However, |
430 |
> |
the Lennard-Jones parameters between Au and other types of particles, |
431 |
> |
(e.g. AA alkanes) have not yet been established. For these |
432 |
> |
interactions, the Lorentz-Berthelot mixing rule can be used to derive |
433 |
> |
effective single-atom LJ parameters for the metal using the fit values |
434 |
> |
for toluene. These are then used to construct reasonable mixing |
435 |
> |
parameters for the interactions between the gold and other atoms. |
436 |
> |
Table \ref{MnM} summarizes the ``metal/non-metal'' parameters used in |
437 |
> |
our simulations. |
438 |
|
|
439 |
|
\begin{table*} |
440 |
|
\begin{minipage}{\linewidth} |
471 |
|
\end{minipage} |
472 |
|
\end{table*} |
473 |
|
|
474 |
< |
\subsection{Vibrational Spectrum} |
474 |
> |
\subsection{Vibrational Power Spectrum} |
475 |
> |
|
476 |
|
To investigate the mechanism of interfacial thermal conductance, the |
477 |
< |
vibrational spectrum is utilized as a complementary tool. Vibrational |
478 |
< |
spectra were taken for individual components in different |
479 |
< |
simulations. To obtain these spectra, simulations were run after |
480 |
< |
equilibration, in the NVE ensemble. Snapshots of configurations were |
481 |
< |
collected at a frequency that is higher than that of the fastest |
482 |
< |
vibrations occuring in the simulations. With these configurations, the |
483 |
< |
velocity auto-correlation functions can be computed: |
477 |
> |
vibrational power spectrum was computed. Power spectra were taken for |
478 |
> |
individual components in different simulations. To obtain these |
479 |
> |
spectra, simulations were run after equilibration, in the NVE |
480 |
> |
ensemble, and without a thermal gradient. Snapshots of configurations |
481 |
> |
were collected at a frequency that is higher than that of the fastest |
482 |
> |
vibrations occuring in the simulations. With these configurations, the |
483 |
> |
velocity auto-correlation functions can be computed: |
484 |
|
\begin{equation} |
485 |
|
C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle |
486 |
< |
\label{vCorr} |
487 |
< |
\end{equation} |
488 |
< |
Followed by Fourier transforms, the power spectrum can be constructed: |
486 |
> |
\label{vCorr} |
487 |
> |
\end{equation} |
488 |
> |
The power spectrum is constructed via a Fourier transform of the |
489 |
> |
symmetrized velocity autocorrelation function, |
490 |
|
\begin{equation} |
491 |
< |
\hat{f}(\omega) = \int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt |
492 |
< |
\label{fourier} |
491 |
> |
\hat{f}(\omega) = |
492 |
> |
\int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt |
493 |
> |
\label{fourier} |
494 |
|
\end{equation} |
495 |
|
|
496 |
|
\section{Results and Discussions} |