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\begin{document} |
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\section{Introduction} |
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Interfacial thermal conductance is extensively studied both |
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experimentally and computationally, due to its importance in nanoscale |
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science and technology. Reliability of nanoscale devices depends on |
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their thermal transport properties. Unlike bulk homogeneous materials, |
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nanoscale materials features significant presence of interfaces, and |
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these interfaces could dominate the heat transfer behavior of these |
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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 systems. |
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which challenges traditional research methods for homogeneous |
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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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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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|
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Theoretical and computational studies were also engaged in the |
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interfacial thermal transport research in order to gain an |
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understanding of this phenomena at the molecular level. Hase and |
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coworkers employed Non-Equilibrium Molecular Dynamics (NEMD) |
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simulations to study thermal transport from hot Au(111) substrate to a |
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self-assembled monolayer of alkylthiolate with relatively long chain |
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(8-20 carbon atoms)[CITE TWO PAPERS]. However, emsemble average measurements for heat |
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conductance of interfaces between the capping monolayer on Au and a |
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solvent phase has yet to be studied. The relatively low thermal flux |
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through interfaces is difficult to measure with Equilibrium MD or |
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forward NEMD simulation methods. Therefore, the Reverse NEMD (RNEMD) |
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methods would have the advantage of having this difficult to measure |
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flux known when studying the thermal transport |
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across interfaces, given that the simulation |
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Theoretical and computational models have also been used to study the |
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interfacial thermal transport in order to gain an understanding of |
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this phenomena at the molecular level. Recently, Hase and coworkers |
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employed Non-Equilibrium Molecular Dynamics (NEMD) simulations to |
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study thermal transport from hot Au(111) substrate to a self-assembled |
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monolayer of alkylthiol with relatively long chain (8-20 carbon |
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atoms)\cite{hase:2010,hase:2011}. However, ensemble averaged |
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measurements for heat conductance of interfaces between the capping |
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monolayer on Au and a solvent phase has yet to be studied. |
111 |
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The comparatively low thermal flux through interfaces is |
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difficult to measure with Equilibrium MD or forward NEMD simulation |
113 |
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methods. Therefore, the Reverse NEMD (RNEMD) methods would have the |
114 |
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advantage of having this difficult to measure flux known when studying |
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the thermal transport across interfaces, given that the simulation |
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methods being able to effectively apply an unphysical flux in |
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non-homogeneous systems. |
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|
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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 under steady states. |
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the two slabs. Furthermore, it allows effective thermal exchange |
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between particles of different identities, and thus makes the study of |
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interfacial conductance much simpler. |
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|
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Our work presented here investigated the Au(111) surface with various |
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coverage of butanethiol, a capping agent with shorter carbon chain, |
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solvated with organic solvents of different molecular shapes. And |
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different models were used for both the capping agent and the solvent |
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force field parameters. With the NIVS algorithm applied, the thermal |
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transport through these interfacial systems was studied and the |
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underlying mechanism for this phenomena was investigated. |
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The work presented here deals with the Au(111) surface covered to |
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varying degrees by butanethiol, a capping agent with short carbon |
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chain, and solvated with organic solvents of different molecular |
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properties. Different models were used for both the capping agent and |
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the solvent force field parameters. Using the NIVS algorithm, the |
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thermal transport across these interfaces was studied and the |
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underlying mechanism for the phenomena was investigated. |
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|
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[WHY STUDY AU-THIOL SURFACE; MAY CITE SHAOYI JIANG] |
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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{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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\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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trajectories are needed to study the relationship between thermal flux |
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and thermal gradients. For systems including low conductance |
143 |
> |
interfaces one must have a method capable of generating or measuring |
144 |
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relatively small fluxes, compared to those required for bulk |
145 |
> |
conductivity. This requirement makes the calculation even more |
146 |
> |
difficult for those slowly-converging equilibrium |
147 |
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methods\cite{Viscardy:2007lq}. Forward methods may impose gradient, |
148 |
> |
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 |
150 |
|
methods\cite{MullerPlathe:1997xw} have the flux set {\it a priori} and |
151 |
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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 |
153 |
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the forward NEMD methods. Although the momentum swapping approach for |
154 |
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flux-imposing can be used for exchanging energy between particles of |
155 |
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different identity, the kinetic energy transfer efficiency is affected |
156 |
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by the mass difference between the particles, which limits its |
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application on heterogeneous interfacial systems. |
151 |
> |
the thermal response becomes easier to measure than the flux. Although |
152 |
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M\"{u}ller-Plathe's original momentum swapping approach can be used |
153 |
> |
for exchanging energy between particles of different identity, the |
154 |
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kinetic energy transfer efficiency is affected by the mass difference |
155 |
> |
between the particles, which limits its application on heterogeneous |
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interfacial systems. |
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|
158 |
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The non-isotropic velocity scaling (NIVS)\cite{kuang:164101} approach in |
159 |
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non-equilibrium MD simulations is able to impose relatively large |
160 |
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kinetic energy flux without obvious perturbation to the velocity |
161 |
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distribution of the simulated systems. Furthermore, this approach has |
158 |
> |
The non-isotropic velocity scaling (NIVS)\cite{kuang:164101} approach to |
159 |
> |
non-equilibrium MD simulations is able to impose a wide range of |
160 |
> |
kinetic energy fluxes without obvious perturbation to the velocity |
161 |
> |
distributions of the simulated systems. Furthermore, this approach has |
162 |
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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 |
164 |
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the flux quantity is not restricted by particle mass difference. |
164 |
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the flux will not be restricted by difference in particle mass. |
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|
166 |
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The NIVS algorithm scales the velocity vectors in two separate regions |
167 |
|
of a simulation system with respective diagonal scaling matricies. To |
168 |
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determine these scaling factors in the matricies, a set of equations |
169 |
|
including linear momentum conservation and kinetic energy conservation |
170 |
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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 |
172 |
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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 |
171 |
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external thermostat. |
170 |
> |
constraints and target energy flux satisfaction is solved. With the |
171 |
> |
scaling operation applied to the system in a set frequency, bulk |
172 |
> |
temperature gradients can be easily established, and these can be used |
173 |
> |
for computing thermal conductivities. The NIVS algorithm conserves |
174 |
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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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For interfaces with a relatively low interfacial conductance, the bulk |
178 |
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regions on either side of an interface rapidly come to a state in |
179 |
< |
which the two phases have relatively homogeneous (but distinct) |
180 |
< |
temperatures. The interfacial thermal conductivity $G$ can therefore |
181 |
< |
be approximated as: |
177 |
> |
Given a system with thermal gradients and the corresponding thermal |
178 |
> |
flux, for interfaces with a relatively low interfacial conductance, |
179 |
> |
the bulk regions on either side of an interface rapidly come to a |
180 |
> |
state in which the two phases have relatively homogeneous (but |
181 |
> |
distinct) temperatures. The interfacial thermal conductivity $G$ can |
182 |
> |
therefore be approximated as: |
183 |
|
\begin{equation} |
184 |
|
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle - |
185 |
|
\langle T_\mathrm{cold}\rangle \right)} |
190 |
|
T_\mathrm{cold}\rangle}$ are the average observed temperature of the |
191 |
|
two separated phases. |
192 |
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|
193 |
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When the interfacial conductance is {\it not} small, two ways can be |
194 |
< |
used to define $G$. |
193 |
> |
When the interfacial conductance is {\it not} small, there are two |
194 |
> |
ways to define $G$. |
195 |
|
|
196 |
< |
One way is to assume the temperature is discretely different on two |
197 |
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sides of the interface, $G$ can be calculated with the thermal flux |
198 |
< |
applied $J$ and the maximum temperature difference measured along the |
199 |
< |
thermal gradient max($\Delta T$), which occurs at the interface, as: |
196 |
> |
One way is to assume the temperature is discrete on the two sides of |
197 |
> |
the interface. $G$ can be calculated using the applied thermal flux |
198 |
> |
$J$ and the maximum temperature difference measured along the thermal |
199 |
> |
gradient max($\Delta T$), which occurs at the Gibbs deviding surface |
200 |
> |
(Figure \ref{demoPic}): |
201 |
|
\begin{equation} |
202 |
|
G=\frac{J}{\Delta T} |
203 |
|
\label{discreteG} |
204 |
|
\end{equation} |
205 |
|
|
206 |
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\begin{figure} |
207 |
+ |
\includegraphics[width=\linewidth]{method} |
208 |
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\caption{Interfacial conductance can be calculated by applying an |
209 |
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(unphysical) kinetic energy flux between two slabs, one located |
210 |
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within the metal and another on the edge of the periodic box. The |
211 |
+ |
system responds by forming a thermal response or a gradient. In |
212 |
+ |
bulk liquids, this gradient typically has a single slope, but in |
213 |
+ |
interfacial systems, there are distinct thermal conductivity |
214 |
+ |
domains. The interfacial conductance, $G$ is found by measuring the |
215 |
+ |
temperature gap at the Gibbs dividing surface, or by using second |
216 |
+ |
derivatives of the thermal profile.} |
217 |
+ |
\label{demoPic} |
218 |
+ |
\end{figure} |
219 |
+ |
|
220 |
|
The other approach is to assume a continuous temperature profile along |
221 |
|
the thermal gradient axis (e.g. $z$) and define $G$ at the point where |
222 |
|
the magnitude of thermal conductivity $\lambda$ change reach its |
232 |
|
|
233 |
|
With the temperature profile obtained from simulations, one is able to |
234 |
|
approximate the first and second derivatives of $T$ with finite |
235 |
< |
difference method and thus calculate $G^\prime$. |
235 |
> |
difference methods and thus calculate $G^\prime$. |
236 |
|
|
237 |
< |
In what follows, both definitions are used for calculation and comparison. |
237 |
> |
In what follows, both definitions have been used for calculation and |
238 |
> |
are compared in the results. |
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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 |
242 |
< |
$G^\prime$, we have a metal slab with its (111) surfaces perpendicular |
243 |
< |
to the $z$-axis of our simulation cells. With or withour capping |
244 |
< |
agents on the surfaces, the metal slab is solvated with organic |
225 |
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solvents, as illustrated in Figure \ref{demoPic}. |
240 |
> |
To compare the above definitions ($G$ and $G^\prime$), we have modeled |
241 |
> |
a metal slab with its (111) surfaces perpendicular to the $z$-axis of |
242 |
> |
our simulation cells. Both with and without capping agents on the |
243 |
> |
surfaces, the metal slab is solvated with simple organic solvents, as |
244 |
> |
illustrated in Figure \ref{gradT}. |
245 |
|
|
246 |
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\begin{figure} |
247 |
< |
\includegraphics[width=\linewidth]{demoPic} |
248 |
< |
\caption{A sample showing how a metal slab has its (111) surface |
249 |
< |
covered by capping agent molecules and solvated by hexane.} |
250 |
< |
\label{demoPic} |
251 |
< |
\end{figure} |
246 |
> |
With the simulation cell described above, we are able to equilibrate |
247 |
> |
the system and impose an unphysical thermal flux between the liquid |
248 |
> |
and the metal phase using the NIVS algorithm. By periodically applying |
249 |
> |
the unphysical flux, we are able to obtain a temperature profile and |
250 |
> |
its spatial derivatives. These quantities enable the evaluation of the |
251 |
> |
interfacial thermal conductance of a surface. Figure \ref{gradT} is an |
252 |
> |
example of how an applied thermal flux can be used to obtain the 1st |
253 |
> |
and 2nd derivatives of the temperature profile. |
254 |
|
|
234 |
– |
With a simulation cell setup following the above manner, one is able |
235 |
– |
to equilibrate the system and impose an unphysical thermal flux |
236 |
– |
between the liquid and the metal phase with the NIVS algorithm. Under |
237 |
– |
a stablized thermal gradient induced by periodically applying the |
238 |
– |
unphysical flux, one is able to obtain a temperature profile and the |
239 |
– |
physical thermal flux corresponding to it, which equals to the |
240 |
– |
unphysical flux applied by NIVS. These data enables the evaluation of |
241 |
– |
the interfacial thermal conductance of a surface. Figure \ref{gradT} |
242 |
– |
is an example how those stablized thermal gradient can be used to |
243 |
– |
obtain the 1st and 2nd derivatives of the temperature profile. |
244 |
– |
|
255 |
|
\begin{figure} |
256 |
|
\includegraphics[width=\linewidth]{gradT} |
257 |
< |
\caption{The 1st and 2nd derivatives of temperature profile can be |
258 |
< |
obtained with finite difference approximation.} |
257 |
> |
\caption{A sample of Au-butanethiol/hexane interfacial system and the |
258 |
> |
temperature profile after a kinetic energy flux is imposed to |
259 |
> |
it. The 1st and 2nd derivatives of the temperature profile can be |
260 |
> |
obtained with finite difference approximation (lower panel).} |
261 |
|
\label{gradT} |
262 |
|
\end{figure} |
263 |
|
|
252 |
– |
[MAY INCLUDE POWER SPECTRUM PROTOCOL] |
253 |
– |
|
264 |
|
\section{Computational Details} |
265 |
|
\subsection{Simulation Protocol} |
266 |
< |
In our simulations, Au is used to construct a metal slab with bare |
267 |
< |
(111) surface perpendicular to the $z$-axis. Different slab thickness |
268 |
< |
(layer numbers of Au) are simulated. This metal slab is first |
269 |
< |
equilibrated under normal pressure (1 atm) and a desired |
270 |
< |
temperature. After equilibration, butanethiol is used as the capping |
271 |
< |
agent molecule to cover the bare Au (111) surfaces evenly. The sulfur |
272 |
< |
atoms in the butanethiol molecules would occupy the three-fold sites |
273 |
< |
of the surfaces, and the maximal butanethiol capacity on Au surface is |
274 |
< |
$1/3$ of the total number of surface Au atoms[CITATION]. A series of |
275 |
< |
different coverage surfaces is investigated in order to study the |
276 |
< |
relation between coverage and conductance. |
277 |
< |
|
278 |
< |
[COVERAGE DISCRIPTION] However, since the interactions between surface |
279 |
< |
Au and butanethiol is non-bonded, the capping agent molecules are |
280 |
< |
allowed to migrate to an empty neighbor three-fold site during a |
281 |
< |
simulation. Therefore, the initial configuration would not severely |
272 |
< |
affect the sampling of a variety of configurations of the same |
273 |
< |
coverage, and the final conductance measurement would be an average |
274 |
< |
effect of these configurations explored in the simulations. [MAY NEED FIGURES] |
266 |
> |
The NIVS algorithm has been implemented in our MD simulation code, |
267 |
> |
OpenMD\cite{Meineke:2005gd,openmd}, and was used for our |
268 |
> |
simulations. Different metal slab thickness (layer numbers of Au) was |
269 |
> |
simulated. Metal slabs were first equilibrated under atmospheric |
270 |
> |
pressure (1 atm) and a desired temperature (e.g. 200K). After |
271 |
> |
equilibration, butanethiol capping agents were placed at three-fold |
272 |
> |
hollow sites on the Au(111) surfaces. These sites could be either a |
273 |
> |
{\it fcc} or {\it hcp} site. However, Hase {\it et al.} found that |
274 |
> |
they are equivalent in a heat transfer process\cite{hase:2010}, so |
275 |
> |
they are not distinguished in our study. The maximum butanethiol |
276 |
> |
capacity on Au surface is $1/3$ of the total number of surface Au |
277 |
> |
atoms, and the packing forms a $(\sqrt{3}\times\sqrt{3})R30^\circ$ |
278 |
> |
structure\cite{doi:10.1021/ja00008a001,doi:10.1021/cr9801317}. A |
279 |
> |
series of different coverages was derived by evenly eliminating |
280 |
> |
butanethiols on the surfaces, and was investigated in order to study |
281 |
> |
the relation between coverage and interfacial conductance. |
282 |
|
|
283 |
< |
After the modified Au-butanethiol surface systems are equilibrated |
284 |
< |
under canonical ensemble, Packmol\cite{packmol} is used to pack |
285 |
< |
organic solvent molecules in the previously vacuum part of the |
286 |
< |
simulation cells, which guarantees that short range repulsive |
287 |
< |
interactions do not disrupt the simulations. Two solvents are |
288 |
< |
investigated, one which has little vibrational overlap with the |
289 |
< |
alkanethiol and plane-like shape (toluene), and one which has similar |
283 |
< |
vibrational frequencies and chain-like shape ({\it n}-hexane). [MAY |
284 |
< |
EXPLAIN WHY WE CHOOSE THEM] |
283 |
> |
The capping agent molecules were allowed to migrate during the |
284 |
> |
simulations. They distributed themselves uniformly and sampled a |
285 |
> |
number of three-fold sites throughout out study. Therefore, the |
286 |
> |
initial configuration would not noticeably affect the sampling of a |
287 |
> |
variety of configurations of the same coverage, and the final |
288 |
> |
conductance measurement would be an average effect of these |
289 |
> |
configurations explored in the simulations. [MAY NEED SNAPSHOTS] |
290 |
|
|
291 |
< |
The spacing filled by solvent molecules, i.e. the gap between |
291 |
> |
After the modified Au-butanethiol surface systems were equilibrated |
292 |
> |
under canonical ensemble, organic solvent molecules were packed in the |
293 |
> |
previously empty part of the simulation cells\cite{packmol}. Two |
294 |
> |
solvents were investigated, one which has little vibrational overlap |
295 |
> |
with the alkanethiol and a planar shape (toluene), and one which has |
296 |
> |
similar vibrational frequencies and chain-like shape ({\it n}-hexane). |
297 |
> |
|
298 |
> |
The space filled by solvent molecules, i.e. the gap between |
299 |
|
periodically repeated Au-butanethiol surfaces should be carefully |
300 |
|
chosen. A very long length scale for the thermal gradient axis ($z$) |
301 |
|
may cause excessively hot or cold temperatures in the middle of the |
304 |
|
solvent molecules would change the normal behavior of the liquid |
305 |
|
phase. Therefore, our $N_{solvent}$ values were chosen to ensure that |
306 |
|
these extreme cases did not happen to our simulations. And the |
307 |
< |
corresponding spacing is usually $35 \sim 60$\AA. |
307 |
> |
corresponding spacing is usually $35 \sim 75$\AA. |
308 |
|
|
309 |
< |
The initial configurations generated by Packmol are further |
310 |
< |
equilibrated with the $x$ and $y$ dimensions fixed, only allowing |
311 |
< |
length scale change in $z$ dimension. This is to ensure that the |
312 |
< |
equilibration of liquid phase does not affect the metal crystal |
313 |
< |
structure in $x$ and $y$ dimensions. Further equilibration are run |
314 |
< |
under NVT and then NVE ensembles. |
309 |
> |
The initial configurations generated are further equilibrated with the |
310 |
> |
$x$ and $y$ dimensions fixed, only allowing length scale change in $z$ |
311 |
> |
dimension. This is to ensure that the equilibration of liquid phase |
312 |
> |
does not affect the metal crystal structure in $x$ and $y$ dimensions. |
313 |
> |
To investigate this effect, comparisons were made with simulations |
314 |
> |
that allow changes of $L_x$ and $L_y$ during NPT equilibration, and |
315 |
> |
the results are shown in later sections. After ensuring the liquid |
316 |
> |
phase reaches equilibrium at atmospheric pressure (1 atm), further |
317 |
> |
equilibration are followed under NVT and then NVE ensembles. |
318 |
|
|
319 |
|
After the systems reach equilibrium, NIVS is implemented to impose a |
320 |
|
periodic unphysical thermal flux between the metal and the liquid |
321 |
|
phase. Most of our simulations are under an average temperature of |
322 |
|
$\sim$200K. Therefore, this flux usually comes from the metal to the |
323 |
|
liquid so that the liquid has a higher temperature and would not |
324 |
< |
freeze due to excessively low temperature. This induced temperature |
325 |
< |
gradient is stablized and the simulation cell is devided evenly into |
326 |
< |
N slabs along the $z$-axis and the temperatures of each slab are |
327 |
< |
recorded. When the slab width $d$ of each slab is the same, the |
328 |
< |
derivatives of $T$ with respect to slab number $n$ can be directly |
329 |
< |
used for $G^\prime$ calculations: |
324 |
> |
freeze due to excessively low temperature. After this induced |
325 |
> |
temperature gradient is stablized, the temperature profile of the |
326 |
> |
simulation cell is recorded. To do this, the simulation cell is |
327 |
> |
devided evenly into $N$ slabs along the $z$-axis and $N$ is maximized |
328 |
> |
for highest possible spatial resolution but not too many to have some |
329 |
> |
slabs empty most of the time. The average temperatures of each slab |
330 |
> |
are recorded for 1$\sim$2 ns. When the slab width $d$ of each slab is |
331 |
> |
the same, the derivatives of $T$ with respect to slab number $n$ can |
332 |
> |
be directly used for $G^\prime$ calculations: |
333 |
|
\begin{equation} |
334 |
|
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
335 |
|
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
340 |
|
\label{derivativeG2} |
341 |
|
\end{equation} |
342 |
|
|
343 |
+ |
All of the above simulation procedures use a time step of 1 fs. And |
344 |
+ |
each equilibration / stabilization step usually takes 100 ps, or |
345 |
+ |
longer, if necessary. |
346 |
+ |
|
347 |
|
\subsection{Force Field Parameters} |
348 |
< |
Our simulations include various components. Therefore, force field |
349 |
< |
parameter descriptions are needed for interactions both between the |
350 |
< |
same type of particles and between particles of different species. |
348 |
> |
Our simulations include various components. Figure \ref{demoMol} |
349 |
> |
demonstrates the sites defined for both United-Atom and All-Atom |
350 |
> |
models of the organic solvent and capping agent molecules in our |
351 |
> |
simulations. Force field parameter descriptions are needed for |
352 |
> |
interactions both between the same type of particles and between |
353 |
> |
particles of different species. |
354 |
|
|
355 |
+ |
\begin{figure} |
356 |
+ |
\includegraphics[width=\linewidth]{structures} |
357 |
+ |
\caption{Structures of the capping agent and solvents utilized in |
358 |
+ |
these simulations. The chemically-distinct sites (a-e) are expanded |
359 |
+ |
in terms of constituent atoms for both United Atom (UA) and All Atom |
360 |
+ |
(AA) force fields. Most parameters are from |
361 |
+ |
Refs. \protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} (UA) and |
362 |
+ |
\protect\cite{OPLSAA} (AA). Cross-interactions with the Au atoms are given |
363 |
+ |
in Table \ref{MnM}.} |
364 |
+ |
\label{demoMol} |
365 |
+ |
\end{figure} |
366 |
+ |
|
367 |
|
The Au-Au interactions in metal lattice slab is described by the |
368 |
< |
quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
368 |
> |
quantum Sutton-Chen (QSC) formulation\cite{PhysRevB.59.3527}. The QSC |
369 |
|
potentials include zero-point quantum corrections and are |
370 |
|
reparametrized for accurate surface energies compared to the |
371 |
|
Sutton-Chen potentials\cite{Chen90}. |
372 |
|
|
336 |
– |
Figure [REF] demonstrates how we name our pseudo-atoms of the |
337 |
– |
molecules in our simulations. |
338 |
– |
[FIGURE FOR MOLECULE NOMENCLATURE] |
339 |
– |
|
373 |
|
For both solvent molecules, straight chain {\it n}-hexane and aromatic |
374 |
|
toluene, United-Atom (UA) and All-Atom (AA) models are used |
375 |
|
respectively. The TraPPE-UA |
376 |
|
parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
377 |
< |
for our UA solvent molecules. In these models, pseudo-atoms are |
378 |
< |
located at the carbon centers for alkyl groups. By eliminating |
379 |
< |
explicit hydrogen atoms, these models are simple and computationally |
380 |
< |
efficient, while maintains good accuracy. However, the TraPPE-UA for |
381 |
< |
alkanes is known to predict a lower boiling point than experimental |
349 |
< |
values. Considering that after an unphysical thermal flux is applied |
350 |
< |
to a system, the temperature of ``hot'' area in the liquid phase would be |
351 |
< |
significantly higher than the average, to prevent over heating and |
352 |
< |
boiling of the liquid phase, the average temperature in our |
353 |
< |
simulations should be much lower than the liquid boiling point. [MORE DISCUSSION] |
354 |
< |
For UA-toluene model, rigid body constraints are applied, so that the |
355 |
< |
benzene ring and the methyl-CRar bond are kept rigid. This would save |
356 |
< |
computational time.[MORE DETAILS] |
377 |
> |
for our UA solvent molecules. In these models, sites are located at |
378 |
> |
the carbon centers for alkyl groups. Bonding interactions, including |
379 |
> |
bond stretches and bends and torsions, were used for intra-molecular |
380 |
> |
sites not separated by more than 3 bonds. Otherwise, for non-bonded |
381 |
> |
interactions, Lennard-Jones potentials are used. [CHECK CITATION] |
382 |
|
|
383 |
+ |
By eliminating explicit hydrogen atoms, these models are simple and |
384 |
+ |
computationally efficient, while maintains good accuracy. However, the |
385 |
+ |
TraPPE-UA for alkanes is known to predict a lower boiling point than |
386 |
+ |
experimental values. Considering that after an unphysical thermal flux |
387 |
+ |
is applied to a system, the temperature of ``hot'' area in the liquid |
388 |
+ |
phase would be significantly higher than the average of the system, to |
389 |
+ |
prevent over heating and boiling of the liquid phase, the average |
390 |
+ |
temperature in our simulations should be much lower than the liquid |
391 |
+ |
boiling point. |
392 |
+ |
|
393 |
+ |
For UA-toluene model, the non-bonded potentials between |
394 |
+ |
inter-molecular sites have a similar Lennard-Jones formulation. For |
395 |
+ |
intra-molecular interactions, considering the stiffness of the benzene |
396 |
+ |
ring, rigid body constraints are applied for further computational |
397 |
+ |
efficiency. All bonds in the benzene ring and between the ring and the |
398 |
+ |
methyl group remain rigid during the progress of simulations. |
399 |
+ |
|
400 |
|
Besides the TraPPE-UA models, AA models for both organic solvents are |
401 |
|
included in our studies as well. For hexane, the OPLS-AA\cite{OPLSAA} |
402 |
< |
force field is used. [MORE DETAILS] |
403 |
< |
For toluene, the United Force Field developed by Rapp\'{e} {\it et |
404 |
< |
al.}\cite{doi:10.1021/ja00051a040} is adopted.[MORE DETAILS] |
402 |
> |
force field is used. Additional explicit hydrogen sites were |
403 |
> |
included. Besides bonding and non-bonded site-site interactions, |
404 |
> |
partial charges and the electrostatic interactions were added to each |
405 |
> |
CT and HC site. For toluene, the United Force Field developed by |
406 |
> |
Rapp\'{e} {\it et al.}\cite{doi:10.1021/ja00051a040} is |
407 |
> |
adopted. Without the rigid body constraints, bonding interactions were |
408 |
> |
included. For the aromatic ring, improper torsions (inversions) were |
409 |
> |
added as an extra potential for maintaining the planar shape. |
410 |
> |
[CHECK CITATION] |
411 |
|
|
412 |
|
The capping agent in our simulations, the butanethiol molecules can |
413 |
|
either use UA or AA model. The TraPPE-UA force fields includes |
416 |
|
parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) |
417 |
|
surfaces do not have the hydrogen atom bonded to sulfur. To adapt this |
418 |
|
change and derive suitable parameters for butanethiol adsorbed on |
419 |
< |
Au(111) surfaces, we adopt the S parameters from [CITATION CF VLUGT] |
420 |
< |
and modify parameters for its neighbor C atom for charge balance in |
421 |
< |
the molecule. Note that the model choice (UA or AA) of capping agent |
422 |
< |
can be different from the solvent. Regardless of model choice, the |
423 |
< |
force field parameters for interactions between capping agent and |
424 |
< |
solvent can be derived using Lorentz-Berthelot Mixing Rule: |
419 |
> |
Au(111) surfaces, we adopt the S parameters from Luedtke and |
420 |
> |
Landman\cite{landman:1998}[CHECK CITATION] |
421 |
> |
and modify parameters for its neighbor C |
422 |
> |
atom for charge balance in the molecule. Note that the model choice |
423 |
> |
(UA or AA) of capping agent can be different from the |
424 |
> |
solvent. Regardless of model choice, the force field parameters for |
425 |
> |
interactions between capping agent and solvent can be derived using |
426 |
> |
Lorentz-Berthelot Mixing Rule: |
427 |
> |
\begin{eqnarray} |
428 |
> |
\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\ |
429 |
> |
\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}} |
430 |
> |
\end{eqnarray} |
431 |
|
|
378 |
– |
|
432 |
|
To describe the interactions between metal Au and non-metal capping |
433 |
|
agent and solvent particles, we refer to an adsorption study of alkyl |
434 |
|
thiols on gold surfaces by Vlugt {\it et |
435 |
|
al.}\cite{vlugt:cpc2007154} They fitted an effective Lennard-Jones |
436 |
|
form of potential parameters for the interaction between Au and |
437 |
|
pseudo-atoms CH$_x$ and S based on a well-established and widely-used |
438 |
< |
effective potential of Hautman and Klein[CITATION] for the Au(111) |
439 |
< |
surface. As our simulations require the gold lattice slab to be |
440 |
< |
non-rigid so that it could accommodate kinetic energy for thermal |
438 |
> |
effective potential of Hautman and Klein\cite{hautman:4994} for the |
439 |
> |
Au(111) surface. As our simulations require the gold lattice slab to |
440 |
> |
be non-rigid so that it could accommodate kinetic energy for thermal |
441 |
|
transport study purpose, the pair-wise form of potentials is |
442 |
|
preferred. |
443 |
|
|
444 |
|
Besides, the potentials developed from {\it ab initio} calculations by |
445 |
|
Leng {\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
446 |
< |
interactions between Au and aromatic C/H atoms in toluene.[MORE DETAILS] |
446 |
> |
interactions between Au and aromatic C/H atoms in toluene. A set of |
447 |
> |
pseudo Lennard-Jones parameters were provided for Au in their force |
448 |
> |
fields. By using the Mixing Rule, this can be used to derive pair-wise |
449 |
> |
potentials for non-bonded interactions between Au and non-metal sites. |
450 |
|
|
451 |
|
However, the Lennard-Jones parameters between Au and other types of |
452 |
< |
particles in our simulations are not yet well-established. For these |
453 |
< |
interactions, we attempt to derive their parameters using the Mixing |
454 |
< |
Rule. To do this, the ``Metal-non-Metal'' (MnM) interaction parameters |
455 |
< |
for Au is first extracted from the Au-CH$_x$ parameters by applying |
456 |
< |
the Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM'' |
452 |
> |
particles, such as All-Atom normal alkanes in our simulations are not |
453 |
> |
yet well-established. For these interactions, we attempt to derive |
454 |
> |
their parameters using the Mixing Rule. To do this, Au pseudo |
455 |
> |
Lennard-Jones parameters for ``Metal-non-Metal'' (MnM) interactions |
456 |
> |
were first extracted from the Au-CH$_x$ parameters by applying the |
457 |
> |
Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM'' |
458 |
|
parameters in our simulations. |
459 |
|
|
460 |
|
\begin{table*} |
461 |
|
\begin{minipage}{\linewidth} |
462 |
|
\begin{center} |
463 |
< |
\caption{Lennard-Jones parameters for Au-non-Metal |
464 |
< |
interactions in our simulations.} |
465 |
< |
|
466 |
< |
\begin{tabular}{ccc} |
463 |
> |
\caption{Non-bonded interaction parameters (including cross |
464 |
> |
interactions with Au atoms) for both force fields used in this |
465 |
> |
work.} |
466 |
> |
\begin{tabular}{lllllll} |
467 |
|
\hline\hline |
468 |
< |
Non-metal atom & $\sigma$ & $\epsilon$ \\ |
469 |
< |
(or pseudo-atom) & \AA & kcal/mol \\ |
468 |
> |
& Site & $\sigma_{ii}$ & $\epsilon_{ii}$ & $q_i$ & |
469 |
> |
$\sigma_{Au-i}$ & $\epsilon_{Au-i}$ \\ |
470 |
> |
& & (\AA) & (kcal/mol) & ($e$) & (\AA) & (kcal/mol) \\ |
471 |
|
\hline |
472 |
< |
S & 2.40 & 8.465 \\ |
473 |
< |
CH3 & 3.54 & 0.2146 \\ |
474 |
< |
CH2 & 3.54 & 0.1749 \\ |
475 |
< |
CT3 & 3.365 & 0.1373 \\ |
476 |
< |
CT2 & 3.365 & 0.1373 \\ |
477 |
< |
CTT & 3.365 & 0.1373 \\ |
478 |
< |
HC & 2.865 & 0.09256 \\ |
479 |
< |
CHar & 3.4625 & 0.1680 \\ |
480 |
< |
CRar & 3.555 & 0.1604 \\ |
481 |
< |
CA & 3.173 & 0.0640 \\ |
482 |
< |
HA & 2.746 & 0.0414 \\ |
472 |
> |
United Atom (UA) |
473 |
> |
&CH3 & 3.75 & 0.1947 & - & 3.54 & 0.2146 \\ |
474 |
> |
&CH2 & 3.95 & 0.0914 & - & 3.54 & 0.1749 \\ |
475 |
> |
&CHar & 3.695 & 0.1003 & - & 3.4625 & 0.1680 \\ |
476 |
> |
&CRar & 3.88 & 0.04173 & - & 3.555 & 0.1604 \\ |
477 |
> |
\hline |
478 |
> |
All Atom (AA) |
479 |
> |
&CT3 & 3.50 & 0.066 & -0.18 & 3.365 & 0.1373 \\ |
480 |
> |
&CT2 & 3.50 & 0.066 & -0.12 & 3.365 & 0.1373 \\ |
481 |
> |
&CTT & 3.50 & 0.066 & -0.065 & 3.365 & 0.1373 \\ |
482 |
> |
&HC & 2.50 & 0.030 & 0.06 & 2.865 & 0.09256 \\ |
483 |
> |
&CA & 3.55 & 0.070 & -0.115 & 3.173 & 0.0640 \\ |
484 |
> |
&HA & 2.42 & 0.030 & 0.115 & 2.746 & 0.0414 \\ |
485 |
> |
\hline |
486 |
> |
Both UA and AA |
487 |
> |
& S & 4.45 & 0.25 & - & 2.40 & 8.465 \\ |
488 |
|
\hline\hline |
489 |
|
\end{tabular} |
490 |
|
\label{MnM} |
492 |
|
\end{minipage} |
493 |
|
\end{table*} |
494 |
|
|
495 |
+ |
\subsection{Vibrational Spectrum} |
496 |
+ |
To investigate the mechanism of interfacial thermal conductance, the |
497 |
+ |
vibrational spectrum is utilized as a complementary tool. Vibrational |
498 |
+ |
spectra were taken for individual components in different |
499 |
+ |
simulations. To obtain these spectra, simulations were run after |
500 |
+ |
equilibration, in the NVE ensemble. Snapshots of configurations were |
501 |
+ |
collected at a frequency that is higher than that of the fastest |
502 |
+ |
vibrations occuring in the simulations. With these configurations, the |
503 |
+ |
velocity auto-correlation functions can be computed: |
504 |
+ |
\begin{equation} |
505 |
+ |
C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle |
506 |
+ |
\label{vCorr} |
507 |
+ |
\end{equation} |
508 |
+ |
Followed by Fourier transforms, the power spectrum can be constructed: |
509 |
+ |
\begin{equation} |
510 |
+ |
\hat{f}(\omega) = \int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt |
511 |
+ |
\label{fourier} |
512 |
+ |
\end{equation} |
513 |
|
|
514 |
|
\section{Results and Discussions} |
515 |
< |
[MAY HAVE A BRIEF SUMMARY] |
515 |
> |
In what follows, how the parameters and protocol of simulations would |
516 |
> |
affect the measurement of $G$'s is first discussed. With a reliable |
517 |
> |
protocol and set of parameters, the influence of capping agent |
518 |
> |
coverage on thermal conductance is investigated. Besides, different |
519 |
> |
force field models for both solvents and selected deuterated models |
520 |
> |
were tested and compared. Finally, a summary of the role of capping |
521 |
> |
agent in the interfacial thermal transport process is given. |
522 |
> |
|
523 |
|
\subsection{How Simulation Parameters Affects $G$} |
436 |
– |
[MAY NOT PUT AT FIRST] |
524 |
|
We have varied our protocol or other parameters of the simulations in |
525 |
|
order to investigate how these factors would affect the measurement of |
526 |
|
$G$'s. It turned out that while some of these parameters would not |
529 |
|
results. |
530 |
|
|
531 |
|
In some of our simulations, we allowed $L_x$ and $L_y$ to change |
532 |
< |
during equilibrating the liquid phase. Due to the stiffness of the Au |
533 |
< |
slab, $L_x$ and $L_y$ would not change noticeably after |
534 |
< |
equilibration. Although $L_z$ could fluctuate $\sim$1\% after a system |
535 |
< |
is fully equilibrated in the NPT ensemble, this fluctuation, as well |
536 |
< |
as those comparably smaller to $L_x$ and $L_y$, would not be magnified |
537 |
< |
on the calculated $G$'s, as shown in Table \ref{AuThiolHexaneUA}. This |
538 |
< |
insensivity to $L_i$ fluctuations allows reliable measurement of $G$'s |
539 |
< |
without the necessity of extremely cautious equilibration process. |
532 |
> |
during equilibrating the liquid phase. Due to the stiffness of the |
533 |
> |
crystalline Au structure, $L_x$ and $L_y$ would not change noticeably |
534 |
> |
after equilibration. Although $L_z$ could fluctuate $\sim$1\% after a |
535 |
> |
system is fully equilibrated in the NPT ensemble, this fluctuation, as |
536 |
> |
well as those of $L_x$ and $L_y$ (which is significantly smaller), |
537 |
> |
would not be magnified on the calculated $G$'s, as shown in Table |
538 |
> |
\ref{AuThiolHexaneUA}. This insensivity to $L_i$ fluctuations allows |
539 |
> |
reliable measurement of $G$'s without the necessity of extremely |
540 |
> |
cautious equilibration process. |
541 |
|
|
542 |
|
As stated in our computational details, the spacing filled with |
543 |
|
solvent molecules can be chosen within a range. This allows some |
564 |
|
the thermal flux across the interface. For our simulations, we denote |
565 |
|
$J_z$ to be positive when the physical thermal flux is from the liquid |
566 |
|
to metal, and negative vice versa. The $G$'s measured under different |
567 |
< |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and [REF]. These |
568 |
< |
results do not suggest that $G$ is dependent on $J_z$ within this flux |
569 |
< |
range. The linear response of flux to thermal gradient simplifies our |
570 |
< |
investigations in that we can rely on $G$ measurement with only a |
571 |
< |
couple $J_z$'s and do not need to test a large series of fluxes. |
567 |
> |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and |
568 |
> |
\ref{AuThiolToluene}. These results do not suggest that $G$ is |
569 |
> |
dependent on $J_z$ within this flux range. The linear response of flux |
570 |
> |
to thermal gradient simplifies our investigations in that we can rely |
571 |
> |
on $G$ measurement with only a couple $J_z$'s and do not need to test |
572 |
> |
a large series of fluxes. |
573 |
|
|
485 |
– |
%ADD MORE TO TABLE |
574 |
|
\begin{table*} |
575 |
|
\begin{minipage}{\linewidth} |
576 |
|
\begin{center} |
577 |
|
\caption{Computed interfacial thermal conductivity ($G$ and |
578 |
|
$G^\prime$) values for the 100\% covered Au-butanethiol/hexane |
579 |
|
interfaces with UA model and different hexane molecule numbers |
580 |
< |
at different temperatures using a range of energy fluxes.} |
580 |
> |
at different temperatures using a range of energy |
581 |
> |
fluxes. Error estimates indicated in parenthesis.} |
582 |
|
|
583 |
< |
\begin{tabular}{cccccccc} |
583 |
> |
\begin{tabular}{ccccccc} |
584 |
|
\hline\hline |
585 |
< |
$\langle T\rangle$ & & $L_x$ & $L_y$ & $L_z$ & $J_z$ & |
586 |
< |
$G$ & $G^\prime$ \\ |
587 |
< |
(K) & $N_{hexane}$ & \multicolumn{3}{c}{(\AA)} & (GW/m$^2$) & |
585 |
> |
$\langle T\rangle$ & $N_{hexane}$ & Fixed & $\rho_{hexane}$ & |
586 |
> |
$J_z$ & $G$ & $G^\prime$ \\ |
587 |
> |
(K) & & $L_x$ \& $L_y$? & (g/cm$^3$) & (GW/m$^2$) & |
588 |
|
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
589 |
|
\hline |
590 |
< |
200 & 266 & 29.86 & 25.80 & 113.1 & -0.96 & |
591 |
< |
102() & 80.0() \\ |
592 |
< |
& 200 & 29.84 & 25.81 & 93.9 & 1.92 & |
593 |
< |
129() & 87.3() \\ |
594 |
< |
& & 29.84 & 25.81 & 95.3 & 1.93 & |
595 |
< |
131() & 77.5() \\ |
596 |
< |
& 166 & 29.84 & 25.81 & 85.7 & 0.97 & |
597 |
< |
115() & 69.3() \\ |
598 |
< |
& & & & & 1.94 & |
599 |
< |
125() & 87.1() \\ |
600 |
< |
250 & 200 & 29.84 & 25.87 & 106.8 & 0.96 & |
601 |
< |
81.8() & 67.0() \\ |
602 |
< |
& 166 & 29.87 & 25.84 & 94.8 & 0.98 & |
603 |
< |
79.0() & 62.9() \\ |
604 |
< |
& & 29.84 & 25.85 & 95.0 & 1.44 & |
605 |
< |
76.2() & 64.8() \\ |
590 |
> |
200 & 266 & No & 0.672 & -0.96 & 102(3) & 80.0(0.8) \\ |
591 |
> |
& 200 & Yes & 0.694 & 1.92 & 129(11) & 87.3(0.3) \\ |
592 |
> |
& & Yes & 0.672 & 1.93 & 131(16) & 78(13) \\ |
593 |
> |
& & No & 0.688 & 0.96 & 125(16) & 90.2(15) \\ |
594 |
> |
& & & & 1.91 & 139(10) & 101(10) \\ |
595 |
> |
& & & & 2.83 & 141(6) & 89.9(9.8) \\ |
596 |
> |
& 166 & Yes & 0.679 & 0.97 & 115(19) & 69(18) \\ |
597 |
> |
& & & & 1.94 & 125(9) & 87.1(0.2) \\ |
598 |
> |
& & No & 0.681 & 0.97 & 141(30) & 78(22) \\ |
599 |
> |
& & & & 1.92 & 138(4) & 98.9(9.5) \\ |
600 |
> |
\hline |
601 |
> |
250 & 200 & No & 0.560 & 0.96 & 75(10) & 61.8(7.3) \\ |
602 |
> |
& & & & -0.95 & 49.4(0.3) & 45.7(2.1) \\ |
603 |
> |
& 166 & Yes & 0.570 & 0.98 & 79.0(3.5) & 62.9(3.0) \\ |
604 |
> |
& & No & 0.569 & 0.97 & 80.3(0.6) & 67(11) \\ |
605 |
> |
& & & & 1.44 & 76.2(5.0) & 64.8(3.8) \\ |
606 |
> |
& & & & -0.95 & 56.4(2.5) & 54.4(1.1) \\ |
607 |
> |
& & & & -1.85 & 47.8(1.1) & 53.5(1.5) \\ |
608 |
|
\hline\hline |
609 |
|
\end{tabular} |
610 |
|
\label{AuThiolHexaneUA} |
620 |
|
temperature is higher than 250K. Additionally, the equilibrated liquid |
621 |
|
hexane density under 250K becomes lower than experimental value. This |
622 |
|
expanded liquid phase leads to lower contact between hexane and |
623 |
< |
butanethiol as well.[MAY NEED FIGURE] And this reduced contact would |
623 |
> |
butanethiol as well.[MAY NEED SLAB DENSITY FIGURE] |
624 |
> |
And this reduced contact would |
625 |
|
probably be accountable for a lower interfacial thermal conductance, |
626 |
|
as shown in Table \ref{AuThiolHexaneUA}. |
627 |
|
|
636 |
|
important role in the thermal transport process across the interface |
637 |
|
in that higher degree of contact could yield increased conductance. |
638 |
|
|
547 |
– |
[ADD Lxyz AND ERROR ESTIMATE TO TABLE] |
639 |
|
\begin{table*} |
640 |
|
\begin{minipage}{\linewidth} |
641 |
|
\begin{center} |
642 |
|
\caption{Computed interfacial thermal conductivity ($G$ and |
643 |
|
$G^\prime$) values for a 90\% coverage Au-butanethiol/toluene |
644 |
|
interface at different temperatures using a range of energy |
645 |
< |
fluxes.} |
645 |
> |
fluxes. Error estimates indicated in parenthesis.} |
646 |
|
|
647 |
< |
\begin{tabular}{cccc} |
647 |
> |
\begin{tabular}{ccccc} |
648 |
|
\hline\hline |
649 |
< |
$\langle T\rangle$ & $J_z$ & $G$ & $G^\prime$ \\ |
650 |
< |
(K) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
649 |
> |
$\langle T\rangle$ & $\rho_{toluene}$ & $J_z$ & $G$ & $G^\prime$ \\ |
650 |
> |
(K) & (g/cm$^3$) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
651 |
|
\hline |
652 |
< |
200 & -1.86 & 180() & 135() \\ |
653 |
< |
& 2.15 & 204() & 113() \\ |
654 |
< |
& -3.93 & 175() & 114() \\ |
655 |
< |
300 & -1.91 & 143() & 125() \\ |
656 |
< |
& -4.19 & 134() & 113() \\ |
652 |
> |
200 & 0.933 & 2.15 & 204(12) & 113(12) \\ |
653 |
> |
& & -1.86 & 180(3) & 135(21) \\ |
654 |
> |
& & -3.93 & 176(5) & 113(12) \\ |
655 |
> |
\hline |
656 |
> |
300 & 0.855 & -1.91 & 143(5) & 125(2) \\ |
657 |
> |
& & -4.19 & 135(9) & 113(12) \\ |
658 |
|
\hline\hline |
659 |
|
\end{tabular} |
660 |
|
\label{AuThiolToluene} |
686 |
|
|
687 |
|
However, when the surface is not completely covered by butanethiols, |
688 |
|
the simulated system is more resistent to the reconstruction |
689 |
< |
above. Our Au-butanethiol/toluene system did not see this phenomena |
690 |
< |
even at $\langle T\rangle\sim$300K. The Au(111) surfaces have a 90\% coverage of |
691 |
< |
butanethiols and have empty three-fold sites. These empty sites could |
692 |
< |
help prevent surface reconstruction in that they provide other means |
693 |
< |
of capping agent relaxation. It is observed that butanethiols can |
694 |
< |
migrate to their neighbor empty sites during a simulation. Therefore, |
695 |
< |
we were able to obtain $G$'s for these interfaces even at a relatively |
696 |
< |
high temperature without being affected by surface reconstructions. |
689 |
> |
above. Our Au-butanethiol/toluene system had the Au(111) surfaces 90\% |
690 |
> |
covered by butanethiols, but did not see this above phenomena even at |
691 |
> |
$\langle T\rangle\sim$300K. The empty three-fold sites not occupied by |
692 |
> |
capping agents could help prevent surface reconstruction in that they |
693 |
> |
provide other means of capping agent relaxation. It is observed that |
694 |
> |
butanethiols can migrate to their neighbor empty sites during a |
695 |
> |
simulation. Therefore, we were able to obtain $G$'s for these |
696 |
> |
interfaces even at a relatively high temperature without being |
697 |
> |
affected by surface reconstructions. |
698 |
|
|
699 |
|
\subsection{Influence of Capping Agent Coverage on $G$} |
700 |
|
To investigate the influence of butanethiol coverage on interfacial |
701 |
|
thermal conductance, a series of different coverage Au-butanethiol |
702 |
|
surfaces is prepared and solvated with various organic |
703 |
|
molecules. These systems are then equilibrated and their interfacial |
704 |
< |
thermal conductivity are measured with our NIVS algorithm. Table |
705 |
< |
\ref{tlnUhxnUhxnD} lists these results for direct comparison between |
706 |
< |
different coverages of butanethiol. To study the isotope effect in |
707 |
< |
interfacial thermal conductance, deuterated UA-hexane is included as |
708 |
< |
well. |
704 |
> |
thermal conductivity are measured with our NIVS algorithm. Figure |
705 |
> |
\ref{coverage} demonstrates the trend of conductance change with |
706 |
> |
respect to different coverages of butanethiol. To study the isotope |
707 |
> |
effect in interfacial thermal conductance, deuterated UA-hexane is |
708 |
> |
included as well. |
709 |
|
|
710 |
+ |
\begin{figure} |
711 |
+ |
\includegraphics[width=\linewidth]{coverage} |
712 |
+ |
\caption{Comparison of interfacial thermal conductivity ($G$) values |
713 |
+ |
for the Au-butanethiol/solvent interface with various UA models and |
714 |
+ |
different capping agent coverages at $\langle T\rangle\sim$200K |
715 |
+ |
using certain energy flux respectively.} |
716 |
+ |
\label{coverage} |
717 |
+ |
\end{figure} |
718 |
+ |
|
719 |
|
It turned out that with partial covered butanethiol on the Au(111) |
720 |
< |
surface, the derivative definition for $G$ (Eq. \ref{derivativeG}) has |
721 |
< |
difficulty to apply, due to the difficulty in locating the maximum of |
722 |
< |
change of $\lambda$. Instead, the discrete definition |
723 |
< |
(Eq. \ref{discreteG}) is easier to apply, as max($\Delta T$) can still |
724 |
< |
be well-defined. Therefore, $G$'s (not $G^\prime$) are used for this |
725 |
< |
section. |
720 |
> |
surface, the derivative definition for $G^\prime$ |
721 |
> |
(Eq. \ref{derivativeG}) was difficult to apply, due to the difficulty |
722 |
> |
in locating the maximum of change of $\lambda$. Instead, the discrete |
723 |
> |
definition (Eq. \ref{discreteG}) is easier to apply, as the Gibbs |
724 |
> |
deviding surface can still be well-defined. Therefore, $G$ (not |
725 |
> |
$G^\prime$) was used for this section. |
726 |
|
|
727 |
< |
From Table \ref{tlnUhxnUhxnD}, one can see the significance of the |
727 |
> |
From Figure \ref{coverage}, one can see the significance of the |
728 |
|
presence of capping agents. Even when a fraction of the Au(111) |
729 |
|
surface sites are covered with butanethiols, the conductivity would |
730 |
|
see an enhancement by at least a factor of 3. This indicates the |
731 |
|
important role cappping agent is playing for thermal transport |
732 |
< |
phenomena on metal/organic solvent surfaces. |
732 |
> |
phenomena on metal / organic solvent surfaces. |
733 |
|
|
734 |
|
Interestingly, as one could observe from our results, the maximum |
735 |
|
conductance enhancement (largest $G$) happens while the surfaces are |
748 |
|
would not offset this effect. Eventually, when butanethiol coverage |
749 |
|
continues to decrease, solvent-capping agent contact actually |
750 |
|
decreases with the disappearing of butanethiol molecules. In this |
751 |
< |
case, $G$ decrease could not be offset but instead accelerated. |
751 |
> |
case, $G$ decrease could not be offset but instead accelerated. [NEED |
752 |
> |
SNAPSHOT SHOWING THE PHENOMENA / SLAB DENSITY ANALYSIS] |
753 |
|
|
754 |
|
A comparison of the results obtained from differenet organic solvents |
755 |
|
can also provide useful information of the interfacial thermal |
759 |
|
studies, even though eliminating C-H vibration samplings, still have |
760 |
|
C-C vibrational frequencies different from each other. However, these |
761 |
|
differences in the infrared range do not seem to produce an observable |
762 |
< |
difference for the results of $G$. [MAY NEED FIGURE] |
762 |
> |
difference for the results of $G$ (Figure \ref{uahxnua}). |
763 |
|
|
764 |
+ |
\begin{figure} |
765 |
+ |
\includegraphics[width=\linewidth]{uahxnua} |
766 |
+ |
\caption{Vibrational spectra obtained for normal (upper) and |
767 |
+ |
deuterated (lower) hexane in Au-butanethiol/hexane |
768 |
+ |
systems. Butanethiol spectra are shown as reference. Both hexane and |
769 |
+ |
butanethiol were using United-Atom models.} |
770 |
+ |
\label{uahxnua} |
771 |
+ |
\end{figure} |
772 |
+ |
|
773 |
|
Furthermore, results for rigid body toluene solvent, as well as other |
774 |
|
UA-hexane solvents, are reasonable within the general experimental |
775 |
< |
ranges[CITATIONS]. This suggests that explicit hydrogen might not be a |
776 |
< |
required factor for modeling thermal transport phenomena of systems |
777 |
< |
such as Au-thiol/organic solvent. |
775 |
> |
ranges\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406}. This |
776 |
> |
suggests that explicit hydrogen might not be a required factor for |
777 |
> |
modeling thermal transport phenomena of systems such as |
778 |
> |
Au-thiol/organic solvent. |
779 |
|
|
780 |
|
However, results for Au-butanethiol/toluene do not show an identical |
781 |
< |
trend with those for Au-butanethiol/hexane in that $G$'s remain at |
781 |
> |
trend with those for Au-butanethiol/hexane in that $G$ remains at |
782 |
|
approximately the same magnitue when butanethiol coverage differs from |
783 |
|
25\% to 75\%. This might be rooted in the molecule shape difference |
784 |
< |
for plane-like toluene and chain-like {\it n}-hexane. Due to this |
784 |
> |
for planar toluene and chain-like {\it n}-hexane. Due to this |
785 |
|
difference, toluene molecules have more difficulty in occupying |
786 |
|
relatively small gaps among capping agents when their coverage is not |
787 |
|
too low. Therefore, the solvent-capping agent contact may keep |
789 |
|
level. This becomes an offset for decreasing butanethiol molecules on |
790 |
|
its effect to the process of interfacial thermal transport. Thus, one |
791 |
|
can see a plateau of $G$ vs. butanethiol coverage in our results. |
679 |
– |
|
680 |
– |
[NEED ERROR ESTIMATE, MAY ALSO PUT J HERE] |
681 |
– |
\begin{table*} |
682 |
– |
\begin{minipage}{\linewidth} |
683 |
– |
\begin{center} |
684 |
– |
\caption{Computed interfacial thermal conductivity ($G$) values |
685 |
– |
for the Au-butanethiol/solvent interface with various UA |
686 |
– |
models and different capping agent coverages at $\langle |
687 |
– |
T\rangle\sim$200K using certain energy flux respectively.} |
688 |
– |
|
689 |
– |
\begin{tabular}{cccc} |
690 |
– |
\hline\hline |
691 |
– |
Thiol & \multicolumn{3}{c}{$G$(MW/m$^2$/K)} \\ |
692 |
– |
coverage (\%) & hexane & hexane(D) & toluene \\ |
693 |
– |
\hline |
694 |
– |
0.0 & 46.5() & 43.9() & 70.1() \\ |
695 |
– |
25.0 & 151() & 153() & 249() \\ |
696 |
– |
50.0 & 172() & 182() & 214() \\ |
697 |
– |
75.0 & 242() & 229() & 244() \\ |
698 |
– |
88.9 & 178() & - & - \\ |
699 |
– |
100.0 & 137() & 153() & 187() \\ |
700 |
– |
\hline\hline |
701 |
– |
\end{tabular} |
702 |
– |
\label{tlnUhxnUhxnD} |
703 |
– |
\end{center} |
704 |
– |
\end{minipage} |
705 |
– |
\end{table*} |
792 |
|
|
793 |
|
\subsection{Influence of Chosen Molecule Model on $G$} |
708 |
– |
[MAY COMBINE W MECHANISM STUDY] |
709 |
– |
|
794 |
|
In addition to UA solvent/capping agent models, AA models are included |
795 |
|
in our simulations as well. Besides simulations of the same (UA or AA) |
796 |
|
model for solvent and capping agent, different models can be applied |
799 |
|
the previous section. Table \ref{modelTest} summarizes the results of |
800 |
|
these studies. |
801 |
|
|
718 |
– |
[MORE DATA; ERROR ESTIMATE] |
802 |
|
\begin{table*} |
803 |
|
\begin{minipage}{\linewidth} |
804 |
|
\begin{center} |
806 |
|
\caption{Computed interfacial thermal conductivity ($G$ and |
807 |
|
$G^\prime$) values for interfaces using various models for |
808 |
|
solvent and capping agent (or without capping agent) at |
809 |
< |
$\langle T\rangle\sim$200K.} |
809 |
> |
$\langle T\rangle\sim$200K. (D stands for deuterated solvent |
810 |
> |
or capping agent molecules; ``Avg.'' denotes results that are |
811 |
> |
averages of simulations under different $J_z$'s. Error |
812 |
> |
estimates indicated in parenthesis.)} |
813 |
|
|
814 |
< |
\begin{tabular}{ccccc} |
814 |
> |
\begin{tabular}{llccc} |
815 |
|
\hline\hline |
816 |
|
Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ |
817 |
|
(or bare surface) & model & (GW/m$^2$) & |
818 |
|
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
819 |
|
\hline |
820 |
< |
UA & AA hexane & 1.94 & 135() & 129() \\ |
821 |
< |
& & 2.86 & 126() & 115() \\ |
822 |
< |
& AA toluene & 1.89 & 200() & 149() \\ |
823 |
< |
AA & UA hexane & 1.94 & 116() & 129() \\ |
824 |
< |
& AA hexane & 3.76 & 451() & 378() \\ |
825 |
< |
& & 4.71 & 432() & 334() \\ |
826 |
< |
& AA toluene & 3.79 & 487() & 290() \\ |
827 |
< |
AA(D) & UA hexane & 1.94 & 158() & 172() \\ |
828 |
< |
bare & AA hexane & 0.96 & 31.0() & 29.4() \\ |
820 |
> |
UA & UA hexane & Avg. & 131(9) & 87(10) \\ |
821 |
> |
& UA hexane(D) & 1.95 & 153(5) & 136(13) \\ |
822 |
> |
& AA hexane & Avg. & 131(6) & 122(10) \\ |
823 |
> |
& UA toluene & 1.96 & 187(16) & 151(11) \\ |
824 |
> |
& AA toluene & 1.89 & 200(36) & 149(53) \\ |
825 |
> |
\hline |
826 |
> |
AA & UA hexane & 1.94 & 116(9) & 129(8) \\ |
827 |
> |
& AA hexane & Avg. & 442(14) & 356(31) \\ |
828 |
> |
& AA hexane(D) & 1.93 & 222(12) & 234(54) \\ |
829 |
> |
& UA toluene & 1.98 & 125(25) & 97(60) \\ |
830 |
> |
& AA toluene & 3.79 & 487(56) & 290(42) \\ |
831 |
> |
\hline |
832 |
> |
AA(D) & UA hexane & 1.94 & 158(25) & 172(4) \\ |
833 |
> |
& AA hexane & 1.92 & 243(29) & 191(11) \\ |
834 |
> |
& AA toluene & 1.93 & 364(36) & 322(67) \\ |
835 |
> |
\hline |
836 |
> |
bare & UA hexane & Avg. & 46.5(3.2) & 49.4(4.5) \\ |
837 |
> |
& UA hexane(D) & 0.98 & 43.9(4.6) & 43.0(2.0) \\ |
838 |
> |
& AA hexane & 0.96 & 31.0(1.4) & 29.4(1.3) \\ |
839 |
> |
& UA toluene & 1.99 & 70.1(1.3) & 65.8(0.5) \\ |
840 |
|
\hline\hline |
841 |
|
\end{tabular} |
842 |
|
\label{modelTest} |
859 |
|
interfaces, using AA model for both butanethiol and hexane yields |
860 |
|
substantially higher conductivity values than using UA model for at |
861 |
|
least one component of the solvent and capping agent, which exceeds |
862 |
< |
the upper bond of experimental value range. This is probably due to |
863 |
< |
the classically treated C-H vibrations in the AA model, which should |
864 |
< |
not be appreciably populated at normal temperatures. In comparison, |
865 |
< |
once either the hexanes or the butanethiols are deuterated, one can |
866 |
< |
see a significantly lower $G$ and $G^\prime$. In either of these |
867 |
< |
cases, the C-H(D) vibrational overlap between the solvent and the |
868 |
< |
capping agent is removed. [MAY NEED FIGURE] Conclusively, the |
869 |
< |
improperly treated C-H vibration in the AA model produced |
870 |
< |
over-predicted results accordingly. Compared to the AA model, the UA |
871 |
< |
model yields more reasonable results with higher computational |
872 |
< |
efficiency. |
862 |
> |
the general range of experimental measurement results. This is |
863 |
> |
probably due to the classically treated C-H vibrations in the AA |
864 |
> |
model, which should not be appreciably populated at normal |
865 |
> |
temperatures. In comparison, once either the hexanes or the |
866 |
> |
butanethiols are deuterated, one can see a significantly lower $G$ and |
867 |
> |
$G^\prime$. In either of these cases, the C-H(D) vibrational overlap |
868 |
> |
between the solvent and the capping agent is removed (Figure |
869 |
> |
\ref{aahxntln}). Conclusively, the improperly treated C-H vibration in |
870 |
> |
the AA model produced over-predicted results accordingly. Compared to |
871 |
> |
the AA model, the UA model yields more reasonable results with higher |
872 |
> |
computational efficiency. |
873 |
|
|
874 |
+ |
\begin{figure} |
875 |
+ |
\includegraphics[width=\linewidth]{aahxntln} |
876 |
+ |
\caption{Spectra obtained for All-Atom model Au-butanethil/solvent |
877 |
+ |
systems. When butanethiol is deuterated (lower left), its |
878 |
+ |
vibrational overlap with hexane would decrease significantly, |
879 |
+ |
compared with normal butanethiol (upper left). However, this |
880 |
+ |
dramatic change does not apply to toluene as much (right).} |
881 |
+ |
\label{aahxntln} |
882 |
+ |
\end{figure} |
883 |
+ |
|
884 |
|
However, for Au-butanethiol/toluene interfaces, having the AA |
885 |
|
butanethiol deuterated did not yield a significant change in the |
886 |
< |
measurement results. |
887 |
< |
. , so extra degrees of freedom |
888 |
< |
such as the C-H vibrations could enhance heat exchange between these |
889 |
< |
two phases and result in a much higher conductivity. |
886 |
> |
measurement results. Compared to the C-H vibrational overlap between |
887 |
> |
hexane and butanethiol, both of which have alkyl chains, that overlap |
888 |
> |
between toluene and butanethiol is not so significant and thus does |
889 |
> |
not have as much contribution to the heat exchange |
890 |
> |
process. Conversely, extra degrees of freedom such as the C-H |
891 |
> |
vibrations could yield higher heat exchange rate between these two |
892 |
> |
phases and result in a much higher conductivity. |
893 |
|
|
784 |
– |
|
894 |
|
Although the QSC model for Au is known to predict an overly low value |
895 |
< |
for bulk metal gold conductivity[CITE NIVSRNEMD], our computational |
895 |
> |
for bulk metal gold conductivity\cite{kuang:164101}, our computational |
896 |
|
results for $G$ and $G^\prime$ do not seem to be affected by this |
897 |
< |
drawback of the model for metal. Instead, the modeling of interfacial |
898 |
< |
thermal transport behavior relies mainly on an accurate description of |
899 |
< |
the interactions between components occupying the interfaces. |
897 |
> |
drawback of the model for metal. Instead, our results suggest that the |
898 |
> |
modeling of interfacial thermal transport behavior relies mainly on |
899 |
> |
the accuracy of the interaction descriptions between components |
900 |
> |
occupying the interfaces. |
901 |
|
|
902 |
< |
\subsection{Mechanism of Interfacial Thermal Conductance Enhancement |
903 |
< |
by Capping Agent} |
904 |
< |
%OR\subsection{Vibrational spectrum study on conductance mechanism} |
902 |
> |
\subsection{Role of Capping Agent in Interfacial Thermal Conductance} |
903 |
> |
The vibrational spectra for gold slabs in different environments are |
904 |
> |
shown as in Figure \ref{specAu}. Regardless of the presence of |
905 |
> |
solvent, the gold surfaces covered by butanethiol molecules, compared |
906 |
> |
to bare gold surfaces, exhibit an additional peak observed at the |
907 |
> |
frequency of $\sim$170cm$^{-1}$, which is attributed to the S-Au |
908 |
> |
bonding vibration. This vibration enables efficient thermal transport |
909 |
> |
from surface Au layer to the capping agents. Therefore, in our |
910 |
> |
simulations, the Au/S interfaces do not appear major heat barriers |
911 |
> |
compared to the butanethiol / solvent interfaces. |
912 |
|
|
913 |
< |
[MAY INTRODUCE PROTOCOL IN METHODOLOGY/COMPUTATIONAL DETAIL, EQN'S] |
913 |
> |
Simultaneously, the vibrational overlap between butanethiol and |
914 |
> |
organic solvents suggests higher thermal exchange efficiency between |
915 |
> |
these two components. Even exessively high heat transport was observed |
916 |
> |
when All-Atom models were used and C-H vibrations were treated |
917 |
> |
classically. Compared to metal and organic liquid phase, the heat |
918 |
> |
transfer efficiency between butanethiol and organic solvents is closer |
919 |
> |
to that within bulk liquid phase. |
920 |
|
|
921 |
< |
To investigate the mechanism of this interfacial thermal conductance, |
922 |
< |
the vibrational spectra of various gold systems were obtained and are |
923 |
< |
shown as in the upper panel of Fig. \ref{vibration}. To obtain these |
924 |
< |
spectra, one first runs a simulation in the NVE ensemble and collects |
925 |
< |
snapshots of configurations; these configurations are used to compute |
926 |
< |
the velocity auto-correlation functions, which is used to construct a |
927 |
< |
power spectrum via a Fourier transform. |
921 |
> |
Furthermore, our observation validated previous |
922 |
> |
results\cite{hase:2010} that the intramolecular heat transport of |
923 |
> |
alkylthiols is highly effecient. As a combinational effects of these |
924 |
> |
phenomena, butanethiol acts as a channel to expedite thermal transport |
925 |
> |
process. The acoustic impedance mismatch between the metal and the |
926 |
> |
liquid phase can be effectively reduced with the presence of suitable |
927 |
> |
capping agents. |
928 |
|
|
806 |
– |
The gold surfaces covered by |
807 |
– |
butanethiol molecules, compared to bare gold surfaces, exhibit an |
808 |
– |
additional peak observed at a frequency of $\sim$170cm$^{-1}$, which |
809 |
– |
is attributed to the vibration of the S-Au bond. This vibration |
810 |
– |
enables efficient thermal transport from surface Au atoms to the |
811 |
– |
capping agents. Simultaneously, as shown in the lower panel of |
812 |
– |
Fig. \ref{vibration}, the large overlap of the vibration spectra of |
813 |
– |
butanethiol and hexane in the all-atom model, including the C-H |
814 |
– |
vibration, also suggests high thermal exchange efficiency. The |
815 |
– |
combination of these two effects produces the drastic interfacial |
816 |
– |
thermal conductance enhancement in the all-atom model. |
817 |
– |
|
818 |
– |
[MAY NEED TO CONVERT TO JPEG] |
929 |
|
\begin{figure} |
930 |
|
\includegraphics[width=\linewidth]{vibration} |
931 |
|
\caption{Vibrational spectra obtained for gold in different |
932 |
< |
environments (upper panel) and for Au/thiol/hexane simulation in |
933 |
< |
all-atom model (lower panel).} |
824 |
< |
\label{vibration} |
932 |
> |
environments.} |
933 |
> |
\label{specAu} |
934 |
|
\end{figure} |
935 |
|
|
936 |
< |
[COMPARISON OF TWO G'S; AU SLAB WIDTHS; ETC] |
828 |
< |
% The results show that the two definitions used for $G$ yield |
829 |
< |
% comparable values, though $G^\prime$ tends to be smaller. |
936 |
> |
[MAY ADD COMPARISON OF AU SLAB WIDTHS] |
937 |
|
|
938 |
|
\section{Conclusions} |
939 |
|
The NIVS algorithm we developed has been applied to simulations of |
941 |
|
effective unphysical thermal flux transferred between the metal and |
942 |
|
the liquid phase. With the flux applied, we were able to measure the |
943 |
|
corresponding thermal gradient and to obtain interfacial thermal |
944 |
< |
conductivities. Our simulations have seen significant conductance |
945 |
< |
enhancement with the presence of capping agent, compared to the bare |
946 |
< |
gold/liquid interfaces. The acoustic impedance mismatch between the |
947 |
< |
metal and the liquid phase is effectively eliminated by proper capping |
944 |
> |
conductivities. Under steady states, single trajectory simulation |
945 |
> |
would be enough for accurate measurement. This would be advantageous |
946 |
> |
compared to transient state simulations, which need multiple |
947 |
> |
trajectories to produce reliable average results. |
948 |
> |
|
949 |
> |
Our simulations have seen significant conductance enhancement with the |
950 |
> |
presence of capping agent, compared to the bare gold / liquid |
951 |
> |
interfaces. The acoustic impedance mismatch between the metal and the |
952 |
> |
liquid phase is effectively eliminated by proper capping |
953 |
|
agent. Furthermore, the coverage precentage of the capping agent plays |
954 |
< |
an important role in the interfacial thermal transport process. |
954 |
> |
an important role in the interfacial thermal transport |
955 |
> |
process. Moderately lower coverages allow higher contact between |
956 |
> |
capping agent and solvent, and thus could further enhance the heat |
957 |
> |
transfer process. |
958 |
|
|
959 |
|
Our measurement results, particularly of the UA models, agree with |
960 |
|
available experimental data. This indicates that our force field |
964 |
|
vibration would be overly sampled. Compared to the AA models, the UA |
965 |
|
models have higher computational efficiency with satisfactory |
966 |
|
accuracy, and thus are preferable in interfacial thermal transport |
967 |
< |
modelings. |
967 |
> |
modelings. Of the two definitions for $G$, the discrete form |
968 |
> |
(Eq. \ref{discreteG}) was easier to use and gives out relatively |
969 |
> |
consistent results, while the derivative form (Eq. \ref{derivativeG}) |
970 |
> |
is not as versatile. Although $G^\prime$ gives out comparable results |
971 |
> |
and follows similar trend with $G$ when measuring close to fully |
972 |
> |
covered or bare surfaces, the spatial resolution of $T$ profile is |
973 |
> |
limited for accurate computation of derivatives data. |
974 |
|
|
975 |
|
Vlugt {\it et al.} has investigated the surface thiol structures for |
976 |
|
nanocrystal gold and pointed out that they differs from those of the |
980 |
|
and measure the corresponding thermal gradient is desirable for |
981 |
|
simulating structures with spherical symmetry. |
982 |
|
|
862 |
– |
|
983 |
|
\section{Acknowledgments} |
984 |
|
Support for this project was provided by the National Science |
985 |
|
Foundation under grant CHE-0848243. Computational time was provided by |