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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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|
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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 |
| 78 |
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importance in nanoscale science and technology. Reliability of |
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nanoscale devices depends on their thermal transport |
| 80 |
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properties. Unlike bulk homogeneous materials, nanoscale materials |
| 81 |
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features significant presence of interfaces, and these interfaces |
| 82 |
< |
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 |
| 78 |
> |
and computationally.\cite{cahill:793} Unlike bulk materials, nanoscale |
| 79 |
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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 |
| 83 |
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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 |
| 89 |
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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.} |
| 93 |
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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 |
| 85 |
> |
Experimentally, various interfaces have been investigated for their |
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> |
thermal conductance. Wang {\it et al.} studied heat transport through |
| 87 |
> |
long-chain hydrocarbon monolayers on gold substrate at individual |
| 88 |
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molecular level,\cite{Wang10082007} Schmidt {\it et al.} studied the |
| 89 |
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role of CTAB on thermal transport between gold nanorods and |
| 90 |
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solvent,\cite{doi:10.1021/jp8051888} and Juv\'e {\it et al.} studied |
| 91 |
|
the cooling dynamics, which is controlled by thermal interface |
| 92 |
|
resistence of glass-embedded metal |
| 93 |
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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 |
| 95 |
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that specific ligands (capping agents) could completely eliminate this |
| 96 |
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barrier ($G\rightarrow\infty$)\cite{doi:10.1021/la904855s}. |
| 93 |
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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.} |
| 95 |
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suggested that specific ligands (capping agents) could completely |
| 96 |
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eliminate this barrier |
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> |
($G\rightarrow\infty$).\cite{doi:10.1021/la904855s} |
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|
| 99 |
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Theoretical and computational models have also been used to study the |
| 100 |
|
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 |
| 104 |
|
monolayer of alkylthiol with relatively long chain (8-20 carbon |
| 105 |
< |
atoms)\cite{hase:2010,hase:2011}. However, ensemble averaged |
| 105 |
> |
atoms).\cite{hase:2010,hase:2011} However, ensemble averaged |
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|
measurements for heat conductance of interfaces between the capping |
| 107 |
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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 |
| 108 |
> |
approach. The comparatively low thermal flux through interfaces is |
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|
difficult to measure with Equilibrium MD or forward NEMD simulation |
| 110 |
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methods. Therefore, the Reverse NEMD (RNEMD) methods would have the |
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< |
advantage of having this difficult to measure flux known when studying |
| 112 |
< |
the thermal transport across interfaces, given that the simulation |
| 113 |
< |
methods being able to effectively apply an unphysical flux in |
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non-homogeneous systems. |
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> |
methods. Therefore, the Reverse NEMD (RNEMD) |
| 111 |
> |
methods\cite{MullerPlathe:1997xw,kuang:164101} would have the |
| 112 |
> |
advantage of applying this difficult to measure flux (while measuring |
| 113 |
> |
the resulting gradient), given that the simulation methods being able |
| 114 |
> |
to effectively apply an unphysical flux in non-homogeneous systems. |
| 115 |
> |
Garde and coworkers\cite{garde:nl2005,garde:PhysRevLett2009} applied |
| 116 |
> |
this approach to various liquid interfaces and studied how thermal |
| 117 |
> |
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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|
|
| 120 |
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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 |
| 122 |
|
retains the desirable features of RNEMD (conservation of linear |
| 123 |
|
momentum and total energy, compatibility with periodic boundary |
| 132 |
|
properties. Different models were used for both the capping agent and |
| 133 |
|
the solvent force field parameters. Using the NIVS algorithm, the |
| 134 |
|
thermal transport across these interfaces was studied and the |
| 135 |
< |
underlying mechanism for this phenomena was investigated. |
| 135 |
> |
underlying mechanism for the phenomena was investigated. |
| 136 |
|
|
| 136 |
– |
[MAY ADD WHY STUDY AU-THIOL SURFACE; CITE SHAOYI JIANG] |
| 137 |
– |
|
| 137 |
|
\section{Methodology} |
| 138 |
|
\subsection{Imposd-Flux Methods in MD Simulations} |
| 139 |
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For systems with low interfacial conductivity one must have a method |
| 140 |
< |
capable of generating relatively small fluxes, compared to those |
| 141 |
< |
required for bulk conductivity. This requirement makes the calculation |
| 142 |
< |
even more difficult for those slowly-converging equilibrium |
| 143 |
< |
methods\cite{Viscardy:2007lq}. |
| 144 |
< |
Forward methods impose gradient, but in interfacail conditions it is |
| 145 |
< |
not clear what behavior to impose at the boundary... |
| 146 |
< |
Imposed-flux reverse non-equilibrium |
| 147 |
< |
methods\cite{MullerPlathe:1997xw} have the flux set {\it a priori} and |
| 148 |
< |
the thermal response becomes easier to |
| 149 |
< |
measure than the flux. Although M\"{u}ller-Plathe's original momentum |
| 150 |
< |
swapping approach can be used for exchanging energy between particles |
| 151 |
< |
of different identity, the kinetic energy transfer efficiency is |
| 152 |
< |
affected by the mass difference between the particles, which limits |
| 153 |
< |
its application on heterogeneous interfacial systems. |
| 139 |
> |
Steady state MD simulations have an advantage in that not many |
| 140 |
> |
trajectories are needed to study the relationship between thermal flux |
| 141 |
> |
and thermal gradients. For systems with low interfacial conductance, |
| 142 |
> |
one must have a method capable of generating or measuring relatively |
| 143 |
> |
small fluxes, compared to those required for bulk conductivity. This |
| 144 |
> |
requirement makes the calculation even more difficult for |
| 145 |
> |
slowly-converging equilibrium methods.\cite{Viscardy:2007lq} Forward |
| 146 |
> |
NEMD methods impose a gradient (and measure a flux), but at interfaces |
| 147 |
> |
it is not clear what behavior should be imposed at the boundaries |
| 148 |
> |
between materials. Imposed-flux reverse non-equilibrium |
| 149 |
> |
methods\cite{MullerPlathe:1997xw} set the flux {\it a priori} and |
| 150 |
> |
the thermal response becomes an easy-to-measure quantity. Although |
| 151 |
> |
M\"{u}ller-Plathe's original momentum swapping approach can be used |
| 152 |
> |
for exchanging energy between particles of different identity, the |
| 153 |
> |
kinetic energy transfer efficiency is affected by the mass difference |
| 154 |
> |
between the particles, which limits its application on heterogeneous |
| 155 |
> |
interfacial systems. |
| 156 |
|
|
| 157 |
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The non-isotropic velocity scaling (NIVS)\cite{kuang:164101} approach to |
| 158 |
< |
non-equilibrium MD simulations is able to impose a wide range of |
| 157 |
> |
The non-isotropic velocity scaling (NIVS) \cite{kuang:164101} approach |
| 158 |
> |
to non-equilibrium MD simulations is able to impose a wide range of |
| 159 |
|
kinetic energy fluxes without obvious perturbation to the velocity |
| 160 |
|
distributions of the simulated systems. Furthermore, this approach has |
| 161 |
|
the advantage in heterogeneous interfaces in that kinetic energy flux |
| 172 |
|
for computing thermal conductivities. The NIVS algorithm conserves |
| 173 |
|
momenta and energy and does not depend on an external thermostat. |
| 174 |
|
|
| 175 |
< |
\subsection{Defining Interfacial Thermal Conductivity $G$} |
| 176 |
< |
For interfaces with a relatively low interfacial conductance, the bulk |
| 177 |
< |
regions on either side of an interface rapidly come to a state in |
| 178 |
< |
which the two phases have relatively homogeneous (but distinct) |
| 179 |
< |
temperatures. The interfacial thermal conductivity $G$ can therefore |
| 180 |
< |
be approximated as: |
| 175 |
> |
\subsection{Defining Interfacial Thermal Conductivity ($G$)} |
| 176 |
> |
|
| 177 |
> |
For an interface with relatively low interfacial conductance, and a |
| 178 |
> |
thermal flux between two distinct bulk regions, the regions on either |
| 179 |
> |
side of the interface rapidly come to a state in which the two phases |
| 180 |
> |
have relatively homogeneous (but distinct) temperatures. The |
| 181 |
> |
interfacial thermal conductivity $G$ can therefore be approximated as: |
| 182 |
|
\begin{equation} |
| 183 |
< |
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)} |
| 185 |
|
\label{lowG} |
| 186 |
|
\end{equation} |
| 187 |
< |
where ${E_{total}}$ is the imposed non-physical kinetic energy |
| 188 |
< |
transfer and ${\langle T_\mathrm{hot}\rangle}$ and ${\langle |
| 189 |
< |
T_\mathrm{cold}\rangle}$ are the average observed temperature of the |
| 190 |
< |
two separated phases. |
| 187 |
> |
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 |
|
|
| 192 |
|
When the interfacial conductance is {\it not} small, there are two |
| 193 |
< |
ways to define $G$. |
| 194 |
< |
|
| 195 |
< |
One way is to assume the temperature is discrete on the two sides of |
| 196 |
< |
the interface. $G$ can be calculated using the applied thermal flux |
| 197 |
< |
$J$ and the maximum temperature difference measured along the thermal |
| 196 |
< |
gradient max($\Delta T$), which occurs at the Gibbs deviding surface, |
| 197 |
< |
as: |
| 193 |
> |
ways to define $G$. One common way is to assume the temperature is |
| 194 |
> |
discrete on the two sides of the interface. $G$ can be calculated |
| 195 |
> |
using the applied thermal flux $J$ and the maximum temperature |
| 196 |
> |
difference measured along the thermal gradient max($\Delta T$), which |
| 197 |
> |
occurs at the Gibbs deviding surface (Figure \ref{demoPic}): |
| 198 |
|
\begin{equation} |
| 199 |
< |
G=\frac{J}{\Delta T} |
| 199 |
> |
G=\frac{J}{\Delta T} |
| 200 |
|
\label{discreteG} |
| 201 |
|
\end{equation} |
| 202 |
|
|
| 203 |
– |
The other approach is to assume a continuous temperature profile along |
| 204 |
– |
the thermal gradient axis (e.g. $z$) and define $G$ at the point where |
| 205 |
– |
the magnitude of thermal conductivity $\lambda$ change reach its |
| 206 |
– |
maximum, given that $\lambda$ is well-defined throughout the space: |
| 207 |
– |
\begin{equation} |
| 208 |
– |
G^\prime = \Big|\frac{\partial\lambda}{\partial z}\Big| |
| 209 |
– |
= \Big|\frac{\partial}{\partial z}\left(-J_z\Big/ |
| 210 |
– |
\left(\frac{\partial T}{\partial z}\right)\right)\Big| |
| 211 |
– |
= |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
| 212 |
– |
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
| 213 |
– |
\label{derivativeG} |
| 214 |
– |
\end{equation} |
| 215 |
– |
|
| 216 |
– |
With the temperature profile obtained from simulations, one is able to |
| 217 |
– |
approximate the first and second derivatives of $T$ with finite |
| 218 |
– |
difference methods and thus calculate $G^\prime$. |
| 219 |
– |
|
| 220 |
– |
In what follows, both definitions have been used for calculation and |
| 221 |
– |
are compared in the results. |
| 222 |
– |
|
| 223 |
– |
To compare the above definitions ($G$ and $G^\prime$), we have modeled |
| 224 |
– |
a metal slab with its (111) surfaces perpendicular to the $z$-axis of |
| 225 |
– |
our simulation cells. Both with and withour capping agents on the |
| 226 |
– |
surfaces, the metal slab is solvated with simple organic solvents, as |
| 227 |
– |
illustrated in Figure \ref{demoPic}. |
| 228 |
– |
|
| 203 |
|
\begin{figure} |
| 204 |
|
\includegraphics[width=\linewidth]{method} |
| 205 |
|
\caption{Interfacial conductance can be calculated by applying an |
| 214 |
|
\label{demoPic} |
| 215 |
|
\end{figure} |
| 216 |
|
|
| 217 |
+ |
The other approach is to assume a continuous temperature profile along |
| 218 |
+ |
the thermal gradient axis (e.g. $z$) and define $G$ at the point where |
| 219 |
+ |
the magnitude of thermal conductivity ($\lambda$) change reaches its |
| 220 |
+ |
maximum, given that $\lambda$ is well-defined throughout the space: |
| 221 |
+ |
\begin{equation} |
| 222 |
+ |
G^\prime = \Big|\frac{\partial\lambda}{\partial z}\Big| |
| 223 |
+ |
= \Big|\frac{\partial}{\partial z}\left(-J_z\Big/ |
| 224 |
+ |
\left(\frac{\partial T}{\partial z}\right)\right)\Big| |
| 225 |
+ |
= |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
| 226 |
+ |
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
| 227 |
+ |
\label{derivativeG} |
| 228 |
+ |
\end{equation} |
| 229 |
+ |
|
| 230 |
+ |
With temperature profiles obtained from simulation, one is able to |
| 231 |
+ |
approximate the first and second derivatives of $T$ with finite |
| 232 |
+ |
difference methods and calculate $G^\prime$. In what follows, both |
| 233 |
+ |
definitions have been used, and are compared in the results. |
| 234 |
+ |
|
| 235 |
+ |
To investigate the interfacial conductivity at metal / solvent |
| 236 |
+ |
interfaces, we have modeled a metal slab with its (111) surfaces |
| 237 |
+ |
perpendicular to the $z$-axis of our simulation cells. The metal slab |
| 238 |
+ |
has been prepared both with and without capping agents on the exposed |
| 239 |
+ |
surface, and has been solvated with simple organic solvents, as |
| 240 |
+ |
illustrated in Figure \ref{gradT}. |
| 241 |
+ |
|
| 242 |
|
With the simulation cell described above, we are able to equilibrate |
| 243 |
|
the system and impose an unphysical thermal flux between the liquid |
| 244 |
|
and the metal phase using the NIVS algorithm. By periodically applying |
| 245 |
< |
the unphysical flux, we are able to obtain a temperature profile and |
| 246 |
< |
its spatial derivatives. These quantities enable the evaluation of the |
| 247 |
< |
interfacial thermal conductance of a surface. Figure \ref{gradT} is an |
| 248 |
< |
example how those applied thermal fluxes can be used to obtain the 1st |
| 250 |
< |
and 2nd derivatives of the temperature profile. |
| 245 |
> |
the unphysical flux, we obtained a temperature profile and its spatial |
| 246 |
> |
derivatives. Figure \ref{gradT} shows how an applied thermal flux can |
| 247 |
> |
be used to obtain the 1st and 2nd derivatives of the temperature |
| 248 |
> |
profile. |
| 249 |
|
|
| 250 |
|
\begin{figure} |
| 251 |
|
\includegraphics[width=\linewidth]{gradT} |
| 252 |
< |
\caption{The 1st and 2nd derivatives of temperature profile can be |
| 253 |
< |
obtained with finite difference approximation.} |
| 252 |
> |
\caption{A sample of Au-butanethiol/hexane interfacial system and the |
| 253 |
> |
temperature profile after a kinetic energy flux is imposed to |
| 254 |
> |
it. The 1st and 2nd derivatives of the temperature profile can be |
| 255 |
> |
obtained with finite difference approximation (lower panel).} |
| 256 |
|
\label{gradT} |
| 257 |
|
\end{figure} |
| 258 |
|
|
| 259 |
|
\section{Computational Details} |
| 260 |
|
\subsection{Simulation Protocol} |
| 261 |
|
The NIVS algorithm has been implemented in our MD simulation code, |
| 262 |
< |
OpenMD\cite{Meineke:2005gd,openmd}, and was used for our |
| 263 |
< |
simulations. Different slab thickness (layer numbers of Au) were |
| 264 |
< |
simulated. Metal slabs were first equilibrated under atmospheric |
| 265 |
< |
pressure (1 atm) and a desired temperature (e.g. 200K). After |
| 266 |
< |
equilibration, butanethiol capping agents were placed at three-fold |
| 267 |
< |
sites on the Au(111) surfaces. The maximum butanethiol capacity on Au |
| 268 |
< |
surface is $1/3$ of the total number of surface Au |
| 269 |
< |
atoms\cite{vlugt:cpc2007154}. A series of different coverages was |
| 270 |
< |
investigated in order to study the relation between coverage and |
| 271 |
< |
interfacial conductance. |
| 262 |
> |
OpenMD\cite{Meineke:2005gd,openmd}, and was used for our simulations. |
| 263 |
> |
Metal slabs of 6 or 11 layers of Au atoms were first equilibrated |
| 264 |
> |
under atmospheric pressure (1 atm) and 200K. After equilibration, |
| 265 |
> |
butanethiol capping agents were placed at three-fold hollow sites on |
| 266 |
> |
the Au(111) surfaces. These sites are either {\it fcc} or {\it |
| 267 |
> |
hcp} sites, although Hase {\it et al.} found that they are |
| 268 |
> |
equivalent in a heat transfer process,\cite{hase:2010} so we did not |
| 269 |
> |
distinguish between these sites in our study. The maximum butanethiol |
| 270 |
> |
capacity on Au surface is $1/3$ of the total number of surface Au |
| 271 |
> |
atoms, and the packing forms a $(\sqrt{3}\times\sqrt{3})R30^\circ$ |
| 272 |
> |
structure\cite{doi:10.1021/ja00008a001,doi:10.1021/cr9801317}. A |
| 273 |
> |
series of lower coverages was also prepared by eliminating |
| 274 |
> |
butanethiols from the higher coverage surface in a regular manner. The |
| 275 |
> |
lower coverages were prepared in order to study the relation between |
| 276 |
> |
coverage and interfacial conductance. |
| 277 |
|
|
| 278 |
|
The capping agent molecules were allowed to migrate during the |
| 279 |
|
simulations. They distributed themselves uniformly and sampled a |
| 280 |
|
number of three-fold sites throughout out study. Therefore, the |
| 281 |
< |
initial configuration would not noticeably affect the sampling of a |
| 281 |
> |
initial configuration does not noticeably affect the sampling of a |
| 282 |
|
variety of configurations of the same coverage, and the final |
| 283 |
|
conductance measurement would be an average effect of these |
| 284 |
< |
configurations explored in the simulations. [MAY NEED FIGURES] |
| 284 |
> |
configurations explored in the simulations. |
| 285 |
|
|
| 286 |
< |
After the modified Au-butanethiol surface systems were equilibrated |
| 287 |
< |
under canonical ensemble, organic solvent molecules were packed in the |
| 288 |
< |
previously empty part of the simulation cells\cite{packmol}. Two |
| 286 |
> |
After the modified Au-butanethiol surface systems were equilibrated in |
| 287 |
> |
the canonical (NVT) ensemble, organic solvent molecules were packed in |
| 288 |
> |
the previously empty part of the simulation cells.\cite{packmol} Two |
| 289 |
|
solvents were investigated, one which has little vibrational overlap |
| 290 |
< |
with the alkanethiol and a planar shape (toluene), and one which has |
| 291 |
< |
similar vibrational frequencies and chain-like shape ({\it n}-hexane). |
| 290 |
> |
with the alkanethiol and which has a planar shape (toluene), and one |
| 291 |
> |
which has similar vibrational frequencies to the capping agent and |
| 292 |
> |
chain-like shape ({\it n}-hexane). |
| 293 |
|
|
| 294 |
< |
The space filled by solvent molecules, i.e. the gap between |
| 295 |
< |
periodically repeated Au-butanethiol surfaces should be carefully |
| 296 |
< |
chosen. A very long length scale for the thermal gradient axis ($z$) |
| 291 |
< |
may cause excessively hot or cold temperatures in the middle of the |
| 294 |
> |
The simulation cells were not particularly extensive along the |
| 295 |
> |
$z$-axis, as a very long length scale for the thermal gradient may |
| 296 |
> |
cause excessively hot or cold temperatures in the middle of the |
| 297 |
|
solvent region and lead to undesired phenomena such as solvent boiling |
| 298 |
|
or freezing when a thermal flux is applied. Conversely, too few |
| 299 |
|
solvent molecules would change the normal behavior of the liquid |
| 300 |
|
phase. Therefore, our $N_{solvent}$ values were chosen to ensure that |
| 301 |
< |
these extreme cases did not happen to our simulations. And the |
| 302 |
< |
corresponding spacing is usually $35 \sim 60$\AA. |
| 301 |
> |
these extreme cases did not happen to our simulations. The spacing |
| 302 |
> |
between periodic images of the gold interfaces is $45 \sim 75$\AA. |
| 303 |
|
|
| 304 |
< |
The initial configurations generated by Packmol are further |
| 305 |
< |
equilibrated with the $x$ and $y$ dimensions fixed, only allowing |
| 306 |
< |
length scale change in $z$ dimension. This is to ensure that the |
| 307 |
< |
equilibration of liquid phase does not affect the metal crystal |
| 308 |
< |
structure in $x$ and $y$ dimensions. Further equilibration are run |
| 309 |
< |
under NVT and then NVE ensembles. |
| 304 |
> |
The initial configurations generated are further equilibrated with the |
| 305 |
> |
$x$ and $y$ dimensions fixed, only allowing the $z$-length scale to |
| 306 |
> |
change. This is to ensure that the equilibration of liquid phase does |
| 307 |
> |
not affect the metal's crystalline structure. Comparisons were made |
| 308 |
> |
with simulations that allowed changes of $L_x$ and $L_y$ during NPT |
| 309 |
> |
equilibration. No substantial changes in the box geometry were noticed |
| 310 |
> |
in these simulations. After ensuring the liquid phase reaches |
| 311 |
> |
equilibrium at atmospheric pressure (1 atm), further equilibration was |
| 312 |
> |
carried out under canonical (NVT) and microcanonical (NVE) ensembles. |
| 313 |
|
|
| 314 |
< |
After the systems reach equilibrium, NIVS is implemented to impose a |
| 315 |
< |
periodic unphysical thermal flux between the metal and the liquid |
| 316 |
< |
phase. Most of our simulations are under an average temperature of |
| 317 |
< |
$\sim$200K. Therefore, this flux usually comes from the metal to the |
| 314 |
> |
After the systems reach equilibrium, NIVS was used to impose an |
| 315 |
> |
unphysical thermal flux between the metal and the liquid phases. Most |
| 316 |
> |
of our simulations were done under an average temperature of |
| 317 |
> |
$\sim$200K. Therefore, thermal flux usually came from the metal to the |
| 318 |
|
liquid so that the liquid has a higher temperature and would not |
| 319 |
< |
freeze due to excessively low temperature. This induced temperature |
| 320 |
< |
gradient is stablized and the simulation cell is devided evenly into |
| 321 |
< |
N slabs along the $z$-axis and the temperatures of each slab are |
| 322 |
< |
recorded. When the slab width $d$ of each slab is the same, the |
| 323 |
< |
derivatives of $T$ with respect to slab number $n$ can be directly |
| 324 |
< |
used for $G^\prime$ calculations: |
| 325 |
< |
\begin{equation} |
| 326 |
< |
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
| 319 |
> |
freeze due to lowered temperatures. After this induced temperature |
| 320 |
> |
gradient had stablized, the temperature profile of the simulation cell |
| 321 |
> |
was recorded. To do this, the simulation cell is devided evenly into |
| 322 |
> |
$N$ slabs along the $z$-axis. The average temperatures of each slab |
| 323 |
> |
are recorded for 1$\sim$2 ns. When the slab width $d$ of each slab is |
| 324 |
> |
the same, the derivatives of $T$ with respect to slab number $n$ can |
| 325 |
> |
be directly used for $G^\prime$ calculations: \begin{equation} |
| 326 |
> |
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
| 327 |
|
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
| 328 |
|
= |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big| |
| 329 |
|
\Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2 |
| 332 |
|
\label{derivativeG2} |
| 333 |
|
\end{equation} |
| 334 |
|
|
| 335 |
+ |
All of the above simulation procedures use a time step of 1 fs. Each |
| 336 |
+ |
equilibration stage took a minimum of 100 ps, although in some cases, |
| 337 |
+ |
longer equilibration stages were utilized. |
| 338 |
+ |
|
| 339 |
|
\subsection{Force Field Parameters} |
| 340 |
< |
Our simulations include various components. Therefore, force field |
| 341 |
< |
parameter descriptions are needed for interactions both between the |
| 342 |
< |
same type of particles and between particles of different species. |
| 340 |
> |
Our simulations include a number of chemically distinct components. |
| 341 |
> |
Figure \ref{demoMol} demonstrates the sites defined for both |
| 342 |
> |
United-Atom and All-Atom models of the organic solvent and capping |
| 343 |
> |
agents in our simulations. Force field parameters are needed for |
| 344 |
> |
interactions both between the same type of particles and between |
| 345 |
> |
particles of different species. |
| 346 |
|
|
| 332 |
– |
The Au-Au interactions in metal lattice slab is described by the |
| 333 |
– |
quantum Sutton-Chen (QSC) formulation\cite{PhysRevB.59.3527}. The QSC |
| 334 |
– |
potentials include zero-point quantum corrections and are |
| 335 |
– |
reparametrized for accurate surface energies compared to the |
| 336 |
– |
Sutton-Chen potentials\cite{Chen90}. |
| 337 |
– |
|
| 338 |
– |
Figure \ref{demoMol} demonstrates how we name our pseudo-atoms of the |
| 339 |
– |
organic solvent molecules in our simulations. |
| 340 |
– |
|
| 347 |
|
\begin{figure} |
| 348 |
|
\includegraphics[width=\linewidth]{structures} |
| 349 |
|
\caption{Structures of the capping agent and solvents utilized in |
| 350 |
|
these simulations. The chemically-distinct sites (a-e) are expanded |
| 351 |
|
in terms of constituent atoms for both United Atom (UA) and All Atom |
| 352 |
|
(AA) force fields. Most parameters are from |
| 353 |
< |
Refs. \protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} (UA) and |
| 348 |
< |
\protect\cite{OPLSAA} (AA). Cross-interactions with the Au atoms are given |
| 349 |
< |
in Table \ref{MnM}.} |
| 353 |
> |
Refs. \protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes,TraPPE-UA.thiols} (UA) and \protect\cite{OPLSAA} (AA). Cross-interactions with the Au atoms are given in Table \ref{MnM}.} |
| 354 |
|
\label{demoMol} |
| 355 |
|
\end{figure} |
| 356 |
|
|
| 357 |
< |
For both solvent molecules, straight chain {\it n}-hexane and aromatic |
| 358 |
< |
toluene, United-Atom (UA) and All-Atom (AA) models are used |
| 359 |
< |
respectively. The TraPPE-UA |
| 357 |
> |
The Au-Au interactions in metal lattice slab is described by the |
| 358 |
> |
quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
| 359 |
> |
potentials include zero-point quantum corrections and are |
| 360 |
> |
reparametrized for accurate surface energies compared to the |
| 361 |
> |
Sutton-Chen potentials.\cite{Chen90} |
| 362 |
> |
|
| 363 |
> |
For the two solvent molecules, {\it n}-hexane and toluene, two |
| 364 |
> |
different atomistic models were utilized. Both solvents were modeled |
| 365 |
> |
using United-Atom (UA) and All-Atom (AA) models. The TraPPE-UA |
| 366 |
|
parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
| 367 |
< |
for our UA solvent molecules. In these models, pseudo-atoms are |
| 368 |
< |
located at the carbon centers for alkyl groups. By eliminating |
| 369 |
< |
explicit hydrogen atoms, these models are simple and computationally |
| 370 |
< |
efficient, while maintains good accuracy. However, the TraPPE-UA for |
| 371 |
< |
alkanes is known to predict a lower boiling point than experimental |
| 362 |
< |
values. Considering that after an unphysical thermal flux is applied |
| 363 |
< |
to a system, the temperature of ``hot'' area in the liquid phase would be |
| 364 |
< |
significantly higher than the average, to prevent over heating and |
| 365 |
< |
boiling of the liquid phase, the average temperature in our |
| 366 |
< |
simulations should be much lower than the liquid boiling point. [MORE DISCUSSION] |
| 367 |
< |
For UA-toluene model, rigid body constraints are applied, so that the |
| 368 |
< |
benzene ring and the methyl-CRar bond are kept rigid. This would save |
| 369 |
< |
computational time.[MORE DETAILS] |
| 367 |
> |
for our UA solvent molecules. In these models, sites are located at |
| 368 |
> |
the carbon centers for alkyl groups. Bonding interactions, including |
| 369 |
> |
bond stretches and bends and torsions, were used for intra-molecular |
| 370 |
> |
sites closer than 3 bonds. For non-bonded interactions, Lennard-Jones |
| 371 |
> |
potentials are used. |
| 372 |
|
|
| 373 |
+ |
By eliminating explicit hydrogen atoms, the TraPPE-UA models are |
| 374 |
+ |
simple and computationally efficient, while maintaining good accuracy. |
| 375 |
+ |
However, the TraPPE-UA model for alkanes is known to predict a slighly |
| 376 |
+ |
lower boiling point than experimental values. This is one of the |
| 377 |
+ |
reasons we used a lower average temperature (200K) for our |
| 378 |
+ |
simulations. If heat is transferred to the liquid phase during the |
| 379 |
+ |
NIVS simulation, the liquid in the hot slab can actually be |
| 380 |
+ |
substantially warmer than the mean temperature in the simulation. The |
| 381 |
+ |
lower mean temperatures therefore prevent solvent boiling. |
| 382 |
+ |
|
| 383 |
+ |
For UA-toluene, the non-bonded potentials between intermolecular sites |
| 384 |
+ |
have a similar Lennard-Jones formulation. The toluene molecules were |
| 385 |
+ |
treated as a single rigid body, so there was no need for |
| 386 |
+ |
intramolecular interactions (including bonds, bends, or torsions) in |
| 387 |
+ |
this solvent model. |
| 388 |
+ |
|
| 389 |
|
Besides the TraPPE-UA models, AA models for both organic solvents are |
| 390 |
< |
included in our studies as well. For hexane, the OPLS-AA\cite{OPLSAA} |
| 391 |
< |
force field is used. [MORE DETAILS] |
| 392 |
< |
For toluene, the United Force Field developed by Rapp\'{e} {\it et |
| 393 |
< |
al.}\cite{doi:10.1021/ja00051a040} is adopted.[MORE DETAILS] |
| 390 |
> |
included in our studies as well. The OPLS-AA\cite{OPLSAA} force fields |
| 391 |
> |
were used. For hexane, additional explicit hydrogen sites were |
| 392 |
> |
included. Besides bonding and non-bonded site-site interactions, |
| 393 |
> |
partial charges and the electrostatic interactions were added to each |
| 394 |
> |
CT and HC site. For toluene, a flexible model for the toluene molecule |
| 395 |
> |
was utilized which included bond, bend, torsion, and inversion |
| 396 |
> |
potentials to enforce ring planarity. |
| 397 |
|
|
| 398 |
< |
The capping agent in our simulations, the butanethiol molecules can |
| 399 |
< |
either use UA or AA model. The TraPPE-UA force fields includes |
| 398 |
> |
The butanethiol capping agent in our simulations, were also modeled |
| 399 |
> |
with both UA and AA model. The TraPPE-UA force field includes |
| 400 |
|
parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for |
| 401 |
|
UA butanethiol model in our simulations. The OPLS-AA also provides |
| 402 |
|
parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) |
| 403 |
< |
surfaces do not have the hydrogen atom bonded to sulfur. To adapt this |
| 404 |
< |
change and derive suitable parameters for butanethiol adsorbed on |
| 405 |
< |
Au(111) surfaces, we adopt the S parameters from Luedtke and |
| 406 |
< |
Landman\cite{landman:1998} and modify parameters for its neighbor C |
| 407 |
< |
atom for charge balance in the molecule. Note that the model choice |
| 408 |
< |
(UA or AA) of capping agent can be different from the |
| 409 |
< |
solvent. Regardless of model choice, the force field parameters for |
| 410 |
< |
interactions between capping agent and solvent can be derived using |
| 390 |
< |
Lorentz-Berthelot Mixing Rule: |
| 403 |
> |
surfaces do not have the hydrogen atom bonded to sulfur. To derive |
| 404 |
> |
suitable parameters for butanethiol adsorbed on Au(111) surfaces, we |
| 405 |
> |
adopt the S parameters from Luedtke and Landman\cite{landman:1998} and |
| 406 |
> |
modify the parameters for the CTS atom to maintain charge neutrality |
| 407 |
> |
in the molecule. Note that the model choice (UA or AA) for the capping |
| 408 |
> |
agent can be different from the solvent. Regardless of model choice, |
| 409 |
> |
the force field parameters for interactions between capping agent and |
| 410 |
> |
solvent can be derived using Lorentz-Berthelot Mixing Rule: |
| 411 |
|
\begin{eqnarray} |
| 412 |
< |
\sigma_{IJ} & = & \frac{1}{2} \left(\sigma_{II} + \sigma_{JJ}\right) \\ |
| 413 |
< |
\epsilon_{IJ} & = & \sqrt{\epsilon_{II}\epsilon_{JJ}} |
| 412 |
> |
\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\ |
| 413 |
> |
\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}} |
| 414 |
|
\end{eqnarray} |
| 415 |
|
|
| 416 |
< |
To describe the interactions between metal Au and non-metal capping |
| 417 |
< |
agent and solvent particles, we refer to an adsorption study of alkyl |
| 418 |
< |
thiols on gold surfaces by Vlugt {\it et |
| 419 |
< |
al.}\cite{vlugt:cpc2007154} They fitted an effective Lennard-Jones |
| 420 |
< |
form of potential parameters for the interaction between Au and |
| 421 |
< |
pseudo-atoms CH$_x$ and S based on a well-established and widely-used |
| 422 |
< |
effective potential of Hautman and Klein\cite{hautman:4994} for the |
| 423 |
< |
Au(111) surface. As our simulations require the gold lattice slab to |
| 424 |
< |
be non-rigid so that it could accommodate kinetic energy for thermal |
| 405 |
< |
transport study purpose, the pair-wise form of potentials is |
| 406 |
< |
preferred. |
| 416 |
> |
To describe the interactions between metal (Au) and non-metal atoms, |
| 417 |
> |
we refer to an adsorption study of alkyl thiols on gold surfaces by |
| 418 |
> |
Vlugt {\it et al.}\cite{vlugt:cpc2007154} They fitted an effective |
| 419 |
> |
Lennard-Jones form of potential parameters for the interaction between |
| 420 |
> |
Au and pseudo-atoms CH$_x$ and S based on a well-established and |
| 421 |
> |
widely-used effective potential of Hautman and Klein for the Au(111) |
| 422 |
> |
surface.\cite{hautman:4994} As our simulations require the gold slab |
| 423 |
> |
to be flexible to accommodate thermal excitation, the pair-wise form |
| 424 |
> |
of potentials they developed was used for our study. |
| 425 |
|
|
| 426 |
< |
Besides, the potentials developed from {\it ab initio} calculations by |
| 427 |
< |
Leng {\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
| 428 |
< |
interactions between Au and aromatic C/H atoms in toluene.[MORE DETAILS] |
| 426 |
> |
The potentials developed from {\it ab initio} calculations by Leng |
| 427 |
> |
{\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
| 428 |
> |
interactions between Au and aromatic C/H atoms in toluene. However, |
| 429 |
> |
the Lennard-Jones parameters between Au and other types of particles, |
| 430 |
> |
(e.g. AA alkanes) have not yet been established. For these |
| 431 |
> |
interactions, the Lorentz-Berthelot mixing rule can be used to derive |
| 432 |
> |
effective single-atom LJ parameters for the metal using the fit values |
| 433 |
> |
for toluene. These are then used to construct reasonable mixing |
| 434 |
> |
parameters for the interactions between the gold and other atoms. |
| 435 |
> |
Table \ref{MnM} summarizes the ``metal/non-metal'' parameters used in |
| 436 |
> |
our simulations. |
| 437 |
|
|
| 412 |
– |
However, the Lennard-Jones parameters between Au and other types of |
| 413 |
– |
particles in our simulations are not yet well-established. For these |
| 414 |
– |
interactions, we attempt to derive their parameters using the Mixing |
| 415 |
– |
Rule. To do this, the ``Metal-non-Metal'' (MnM) interaction parameters |
| 416 |
– |
for Au is first extracted from the Au-CH$_x$ parameters by applying |
| 417 |
– |
the Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM'' |
| 418 |
– |
parameters in our simulations. |
| 419 |
– |
|
| 438 |
|
\begin{table*} |
| 439 |
|
\begin{minipage}{\linewidth} |
| 440 |
|
\begin{center} |
| 441 |
< |
\caption{Non-bonded interaction paramters for non-metal |
| 442 |
< |
particles and metal-non-metal interactions in our |
| 443 |
< |
simulations.} |
| 444 |
< |
|
| 427 |
< |
\begin{tabular}{cccccc} |
| 441 |
> |
\caption{Non-bonded interaction parameters (including cross |
| 442 |
> |
interactions with Au atoms) for both force fields used in this |
| 443 |
> |
work.} |
| 444 |
> |
\begin{tabular}{lllllll} |
| 445 |
|
\hline\hline |
| 446 |
< |
Non-metal atom $I$ & $\sigma_{II}$ & $\epsilon_{II}$ & $q_I$ & |
| 447 |
< |
$\sigma_{AuI}$ & $\epsilon_{AuI}$ \\ |
| 448 |
< |
(or pseudo-atom) & \AA & kcal/mol & & \AA & kcal/mol \\ |
| 446 |
> |
& Site & $\sigma_{ii}$ & $\epsilon_{ii}$ & $q_i$ & |
| 447 |
> |
$\sigma_{Au-i}$ & $\epsilon_{Au-i}$ \\ |
| 448 |
> |
& & (\AA) & (kcal/mol) & ($e$) & (\AA) & (kcal/mol) \\ |
| 449 |
|
\hline |
| 450 |
< |
CH3 & 3.75 & 0.1947 & - & 3.54 & 0.2146 \\ |
| 451 |
< |
CH2 & 3.95 & 0.0914 & - & 3.54 & 0.1749 \\ |
| 452 |
< |
CHar & 3.695 & 0.1003 & - & 3.4625 & 0.1680 \\ |
| 453 |
< |
CRar & 3.88 & 0.04173 & - & 3.555 & 0.1604 \\ |
| 454 |
< |
S & 4.45 & 0.25 & - & 2.40 & 8.465 \\ |
| 455 |
< |
CT3 & 3.50 & 0.066 & -0.18 & 3.365 & 0.1373 \\ |
| 456 |
< |
CT2 & 3.50 & 0.066 & -0.12 & 3.365 & 0.1373 \\ |
| 457 |
< |
CTT & 3.50 & 0.066 & -0.065 & 3.365 & 0.1373 \\ |
| 458 |
< |
HC & 2.50 & 0.030 & 0.06 & 2.865 & 0.09256 \\ |
| 459 |
< |
CA & 3.55 & 0.070 & -0.115 & 3.173 & 0.0640 \\ |
| 460 |
< |
HA & 2.42 & 0.030 & 0.115 & 2.746 & 0.0414 \\ |
| 450 |
> |
United Atom (UA) |
| 451 |
> |
&CH3 & 3.75 & 0.1947 & - & 3.54 & 0.2146 \\ |
| 452 |
> |
&CH2 & 3.95 & 0.0914 & - & 3.54 & 0.1749 \\ |
| 453 |
> |
&CHar & 3.695 & 0.1003 & - & 3.4625 & 0.1680 \\ |
| 454 |
> |
&CRar & 3.88 & 0.04173 & - & 3.555 & 0.1604 \\ |
| 455 |
> |
\hline |
| 456 |
> |
All Atom (AA) |
| 457 |
> |
&CT3 & 3.50 & 0.066 & -0.18 & 3.365 & 0.1373 \\ |
| 458 |
> |
&CT2 & 3.50 & 0.066 & -0.12 & 3.365 & 0.1373 \\ |
| 459 |
> |
&CTT & 3.50 & 0.066 & -0.065 & 3.365 & 0.1373 \\ |
| 460 |
> |
&HC & 2.50 & 0.030 & 0.06 & 2.865 & 0.09256 \\ |
| 461 |
> |
&CA & 3.55 & 0.070 & -0.115 & 3.173 & 0.0640 \\ |
| 462 |
> |
&HA & 2.42 & 0.030 & 0.115 & 2.746 & 0.0414 \\ |
| 463 |
> |
\hline |
| 464 |
> |
Both UA and AA |
| 465 |
> |
& S & 4.45 & 0.25 & - & 2.40 & 8.465 \\ |
| 466 |
|
\hline\hline |
| 467 |
|
\end{tabular} |
| 468 |
|
\label{MnM} |
| 471 |
|
\end{table*} |
| 472 |
|
|
| 473 |
|
|
| 474 |
< |
\section{Results and Discussions} |
| 475 |
< |
[MAY HAVE A BRIEF SUMMARY] |
| 476 |
< |
\subsection{How Simulation Parameters Affects $G$} |
| 477 |
< |
[MAY NOT PUT AT FIRST] |
| 474 |
> |
\section{Results} |
| 475 |
> |
There are many factors contributing to the measured interfacial |
| 476 |
> |
conductance; some of these factors are physically motivated |
| 477 |
> |
(e.g. coverage of the surface by the capping agent coverage and |
| 478 |
> |
solvent identity), while some are governed by parameters of the |
| 479 |
> |
methodology (e.g. applied flux and the formulas used to obtain the |
| 480 |
> |
conductance). In this section we discuss the major physical and |
| 481 |
> |
calculational effects on the computed conductivity. |
| 482 |
> |
|
| 483 |
> |
\subsection{Effects due to capping agent coverage} |
| 484 |
> |
|
| 485 |
> |
A series of different initial conditions with a range of surface |
| 486 |
> |
coverages was prepared and solvated with various with both of the |
| 487 |
> |
solvent molecules. These systems were then equilibrated and their |
| 488 |
> |
interfacial thermal conductivity was measured with our NIVS |
| 489 |
> |
algorithm. Figure \ref{coverage} demonstrates the trend of conductance |
| 490 |
> |
with respect to surface coverage. |
| 491 |
> |
|
| 492 |
> |
\begin{figure} |
| 493 |
> |
\includegraphics[width=\linewidth]{coverage} |
| 494 |
> |
\caption{Comparison of interfacial thermal conductivity ($G$) values |
| 495 |
> |
for the Au-butanethiol/solvent interface with various UA models and |
| 496 |
> |
different capping agent coverages at $\langle T\rangle\sim$200K.} |
| 497 |
> |
\label{coverage} |
| 498 |
> |
\end{figure} |
| 499 |
> |
|
| 500 |
> |
|
| 501 |
> |
In partially covered butanethiol on the Au(111) surface, the |
| 502 |
> |
derivative definition for $G^\prime$ (Eq. \ref{derivativeG}) becomes |
| 503 |
> |
difficult to apply, as the location of maximum change of $\lambda$ |
| 504 |
> |
becomes washed out. The discrete definition (Eq. \ref{discreteG}) is |
| 505 |
> |
easier to apply, as the Gibbs dividing surface is still |
| 506 |
> |
well-defined. Therefore, $G$ (not $G^\prime$) was used in this |
| 507 |
> |
section. |
| 508 |
> |
|
| 509 |
> |
From Figure \ref{coverage}, one can see the significance of the |
| 510 |
> |
presence of capping agents. When even a small fraction of the Au(111) |
| 511 |
> |
surface sites are covered with butanethiols, the conductivity exhibits |
| 512 |
> |
an enhancement by at least a factor of 3. This indicates the important |
| 513 |
> |
role cappping agents are playing for thermal transport at metal / |
| 514 |
> |
organic solvent surfaces. |
| 515 |
> |
|
| 516 |
> |
We note a non-monotonic behavior in the interfacial conductance as a |
| 517 |
> |
function of surface coverage. The maximum conductance (largest $G$) |
| 518 |
> |
happens when the surfaces are about 75\% covered with butanethiol |
| 519 |
> |
caps. The reason for this behavior is not entirely clear. One |
| 520 |
> |
explanation is that incomplete butanethiol coverage allows small gaps |
| 521 |
> |
between butanethiols to form. These gaps can be filled by transient |
| 522 |
> |
solvent molecules. These solvent molecules couple very strongly with |
| 523 |
> |
the hot capping agent molecules near the surface, and can then carry |
| 524 |
> |
(diffusively) the excess thermal energy away from the surface. |
| 525 |
> |
|
| 526 |
> |
There appears to be a competition between the conduction of the |
| 527 |
> |
thermal energy away from the surface by the capping agents (enhanced |
| 528 |
> |
by greater coverage) and the coupling of the capping agents with the |
| 529 |
> |
solvent (enhanced by physical contact at lower coverages). This |
| 530 |
> |
competition would lead to the non-monotonic coverage behavior observed |
| 531 |
> |
here. |
| 532 |
> |
|
| 533 |
> |
A comparison of the results obtained from the two different organic |
| 534 |
> |
solvents can also provide useful information of the interfacial |
| 535 |
> |
thermal transport process. The deuterated hexane (UA) results do not |
| 536 |
> |
appear to be substantially different from those of normal hexane (UA), |
| 537 |
> |
given that butanethiol (UA) is non-deuterated for both solvents. The |
| 538 |
> |
UA models, even though they have eliminated C-H vibrational overlap, |
| 539 |
> |
still have significant overlap in the infrared spectra. Because |
| 540 |
> |
differences in the infrared range do not seem to produce an observable |
| 541 |
> |
difference for the results of $G$ (Figure \ref{uahxnua}). |
| 542 |
> |
|
| 543 |
> |
\begin{figure} |
| 544 |
> |
\includegraphics[width=\linewidth]{uahxnua} |
| 545 |
> |
\caption{Vibrational spectra obtained for normal (upper) and |
| 546 |
> |
deuterated (lower) hexane in Au-butanethiol/hexane |
| 547 |
> |
systems. Butanethiol spectra are shown as reference. Both hexane and |
| 548 |
> |
butanethiol were using United-Atom models.} |
| 549 |
> |
\label{uahxnua} |
| 550 |
> |
\end{figure} |
| 551 |
> |
|
| 552 |
> |
Furthermore, results for rigid body toluene solvent, as well as other |
| 553 |
> |
UA-hexane solvents, are reasonable within the general experimental |
| 554 |
> |
ranges\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406}. This |
| 555 |
> |
suggests that explicit hydrogen might not be a required factor for |
| 556 |
> |
modeling thermal transport phenomena of systems such as |
| 557 |
> |
Au-thiol/organic solvent. |
| 558 |
> |
|
| 559 |
> |
However, results for Au-butanethiol/toluene do not show an identical |
| 560 |
> |
trend with those for Au-butanethiol/hexane in that $G$ remains at |
| 561 |
> |
approximately the same magnitue when butanethiol coverage differs from |
| 562 |
> |
25\% to 75\%. This might be rooted in the molecule shape difference |
| 563 |
> |
for planar toluene and chain-like {\it n}-hexane. Due to this |
| 564 |
> |
difference, toluene molecules have more difficulty in occupying |
| 565 |
> |
relatively small gaps among capping agents when their coverage is not |
| 566 |
> |
too low. Therefore, the solvent-capping agent contact may keep |
| 567 |
> |
increasing until the capping agent coverage reaches a relatively low |
| 568 |
> |
level. This becomes an offset for decreasing butanethiol molecules on |
| 569 |
> |
its effect to the process of interfacial thermal transport. Thus, one |
| 570 |
> |
can see a plateau of $G$ vs. butanethiol coverage in our results. |
| 571 |
> |
|
| 572 |
> |
\subsection{Effects due to Solvent \& Solvent Models} |
| 573 |
> |
In addition to UA solvent/capping agent models, AA models are included |
| 574 |
> |
in our simulations as well. Besides simulations of the same (UA or AA) |
| 575 |
> |
model for solvent and capping agent, different models can be applied |
| 576 |
> |
to different components. Furthermore, regardless of models chosen, |
| 577 |
> |
either the solvent or the capping agent can be deuterated, similar to |
| 578 |
> |
the previous section. Table \ref{modelTest} summarizes the results of |
| 579 |
> |
these studies. |
| 580 |
> |
|
| 581 |
> |
\begin{table*} |
| 582 |
> |
\begin{minipage}{\linewidth} |
| 583 |
> |
\begin{center} |
| 584 |
> |
|
| 585 |
> |
\caption{Computed interfacial thermal conductivity ($G$ and |
| 586 |
> |
$G^\prime$) values for interfaces using various models for |
| 587 |
> |
solvent and capping agent (or without capping agent) at |
| 588 |
> |
$\langle T\rangle\sim$200K. (D stands for deuterated solvent |
| 589 |
> |
or capping agent molecules; ``Avg.'' denotes results that are |
| 590 |
> |
averages of simulations under different $J_z$'s. Error |
| 591 |
> |
estimates indicated in parenthesis.)} |
| 592 |
> |
|
| 593 |
> |
\begin{tabular}{llccc} |
| 594 |
> |
\hline\hline |
| 595 |
> |
Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ |
| 596 |
> |
(or bare surface) & model & (GW/m$^2$) & |
| 597 |
> |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 598 |
> |
\hline |
| 599 |
> |
UA & UA hexane & Avg. & 131(9) & 87(10) \\ |
| 600 |
> |
& UA hexane(D) & 1.95 & 153(5) & 136(13) \\ |
| 601 |
> |
& AA hexane & Avg. & 131(6) & 122(10) \\ |
| 602 |
> |
& UA toluene & 1.96 & 187(16) & 151(11) \\ |
| 603 |
> |
& AA toluene & 1.89 & 200(36) & 149(53) \\ |
| 604 |
> |
\hline |
| 605 |
> |
AA & UA hexane & 1.94 & 116(9) & 129(8) \\ |
| 606 |
> |
& AA hexane & Avg. & 442(14) & 356(31) \\ |
| 607 |
> |
& AA hexane(D) & 1.93 & 222(12) & 234(54) \\ |
| 608 |
> |
& UA toluene & 1.98 & 125(25) & 97(60) \\ |
| 609 |
> |
& AA toluene & 3.79 & 487(56) & 290(42) \\ |
| 610 |
> |
\hline |
| 611 |
> |
AA(D) & UA hexane & 1.94 & 158(25) & 172(4) \\ |
| 612 |
> |
& AA hexane & 1.92 & 243(29) & 191(11) \\ |
| 613 |
> |
& AA toluene & 1.93 & 364(36) & 322(67) \\ |
| 614 |
> |
\hline |
| 615 |
> |
bare & UA hexane & Avg. & 46.5(3.2) & 49.4(4.5) \\ |
| 616 |
> |
& UA hexane(D) & 0.98 & 43.9(4.6) & 43.0(2.0) \\ |
| 617 |
> |
& AA hexane & 0.96 & 31.0(1.4) & 29.4(1.3) \\ |
| 618 |
> |
& UA toluene & 1.99 & 70.1(1.3) & 65.8(0.5) \\ |
| 619 |
> |
\hline\hline |
| 620 |
> |
\end{tabular} |
| 621 |
> |
\label{modelTest} |
| 622 |
> |
\end{center} |
| 623 |
> |
\end{minipage} |
| 624 |
> |
\end{table*} |
| 625 |
> |
|
| 626 |
> |
To facilitate direct comparison, the same system with differnt models |
| 627 |
> |
for different components uses the same length scale for their |
| 628 |
> |
simulation cells. Without the presence of capping agent, using |
| 629 |
> |
different models for hexane yields similar results for both $G$ and |
| 630 |
> |
$G^\prime$, and these two definitions agree with eath other very |
| 631 |
> |
well. This indicates very weak interaction between the metal and the |
| 632 |
> |
solvent, and is a typical case for acoustic impedance mismatch between |
| 633 |
> |
these two phases. |
| 634 |
> |
|
| 635 |
> |
As for Au(111) surfaces completely covered by butanethiols, the choice |
| 636 |
> |
of models for capping agent and solvent could impact the measurement |
| 637 |
> |
of $G$ and $G^\prime$ quite significantly. For Au-butanethiol/hexane |
| 638 |
> |
interfaces, using AA model for both butanethiol and hexane yields |
| 639 |
> |
substantially higher conductivity values than using UA model for at |
| 640 |
> |
least one component of the solvent and capping agent, which exceeds |
| 641 |
> |
the general range of experimental measurement results. This is |
| 642 |
> |
probably due to the classically treated C-H vibrations in the AA |
| 643 |
> |
model, which should not be appreciably populated at normal |
| 644 |
> |
temperatures. In comparison, once either the hexanes or the |
| 645 |
> |
butanethiols are deuterated, one can see a significantly lower $G$ and |
| 646 |
> |
$G^\prime$. In either of these cases, the C-H(D) vibrational overlap |
| 647 |
> |
between the solvent and the capping agent is removed (Figure |
| 648 |
> |
\ref{aahxntln}). Conclusively, the improperly treated C-H vibration in |
| 649 |
> |
the AA model produced over-predicted results accordingly. Compared to |
| 650 |
> |
the AA model, the UA model yields more reasonable results with higher |
| 651 |
> |
computational efficiency. |
| 652 |
> |
|
| 653 |
> |
\begin{figure} |
| 654 |
> |
\includegraphics[width=\linewidth]{aahxntln} |
| 655 |
> |
\caption{Spectra obtained for All-Atom model Au-butanethil/solvent |
| 656 |
> |
systems. When butanethiol is deuterated (lower left), its |
| 657 |
> |
vibrational overlap with hexane would decrease significantly, |
| 658 |
> |
compared with normal butanethiol (upper left). However, this |
| 659 |
> |
dramatic change does not apply to toluene as much (right).} |
| 660 |
> |
\label{aahxntln} |
| 661 |
> |
\end{figure} |
| 662 |
> |
|
| 663 |
> |
However, for Au-butanethiol/toluene interfaces, having the AA |
| 664 |
> |
butanethiol deuterated did not yield a significant change in the |
| 665 |
> |
measurement results. Compared to the C-H vibrational overlap between |
| 666 |
> |
hexane and butanethiol, both of which have alkyl chains, that overlap |
| 667 |
> |
between toluene and butanethiol is not so significant and thus does |
| 668 |
> |
not have as much contribution to the heat exchange |
| 669 |
> |
process. Conversely, extra degrees of freedom such as the C-H |
| 670 |
> |
vibrations could yield higher heat exchange rate between these two |
| 671 |
> |
phases and result in a much higher conductivity. |
| 672 |
> |
|
| 673 |
> |
Although the QSC model for Au is known to predict an overly low value |
| 674 |
> |
for bulk metal gold conductivity\cite{kuang:164101}, our computational |
| 675 |
> |
results for $G$ and $G^\prime$ do not seem to be affected by this |
| 676 |
> |
drawback of the model for metal. Instead, our results suggest that the |
| 677 |
> |
modeling of interfacial thermal transport behavior relies mainly on |
| 678 |
> |
the accuracy of the interaction descriptions between components |
| 679 |
> |
occupying the interfaces. |
| 680 |
> |
|
| 681 |
> |
\subsection{Effects due to methodology and simulation parameters} |
| 682 |
> |
|
| 683 |
|
We have varied our protocol or other parameters of the simulations in |
| 684 |
|
order to investigate how these factors would affect the measurement of |
| 685 |
|
$G$'s. It turned out that while some of these parameters would not |
| 688 |
|
results. |
| 689 |
|
|
| 690 |
|
In some of our simulations, we allowed $L_x$ and $L_y$ to change |
| 691 |
< |
during equilibrating the liquid phase. Due to the stiffness of the Au |
| 692 |
< |
slab, $L_x$ and $L_y$ would not change noticeably after |
| 693 |
< |
equilibration. Although $L_z$ could fluctuate $\sim$1\% after a system |
| 694 |
< |
is fully equilibrated in the NPT ensemble, this fluctuation, as well |
| 695 |
< |
as those comparably smaller to $L_x$ and $L_y$, would not be magnified |
| 696 |
< |
on the calculated $G$'s, as shown in Table \ref{AuThiolHexaneUA}. This |
| 697 |
< |
insensivity to $L_i$ fluctuations allows reliable measurement of $G$'s |
| 698 |
< |
without the necessity of extremely cautious equilibration process. |
| 691 |
> |
during equilibrating the liquid phase. Due to the stiffness of the |
| 692 |
> |
crystalline Au structure, $L_x$ and $L_y$ would not change noticeably |
| 693 |
> |
after equilibration. Although $L_z$ could fluctuate $\sim$1\% after a |
| 694 |
> |
system is fully equilibrated in the NPT ensemble, this fluctuation, as |
| 695 |
> |
well as those of $L_x$ and $L_y$ (which is significantly smaller), |
| 696 |
> |
would not be magnified on the calculated $G$'s, as shown in Table |
| 697 |
> |
\ref{AuThiolHexaneUA}. This insensivity to $L_i$ fluctuations allows |
| 698 |
> |
reliable measurement of $G$'s without the necessity of extremely |
| 699 |
> |
cautious equilibration process. |
| 700 |
|
|
| 701 |
|
As stated in our computational details, the spacing filled with |
| 702 |
|
solvent molecules can be chosen within a range. This allows some |
| 708 |
|
smaller system size would be preferable, given that the liquid phase |
| 709 |
|
structure is not affected. |
| 710 |
|
|
| 711 |
+ |
\subsubsection{Effects of applied flux} |
| 712 |
|
Our NIVS algorithm allows change of unphysical thermal flux both in |
| 713 |
|
direction and in quantity. This feature extends our investigation of |
| 714 |
|
interfacial thermal conductance. However, the magnitude of this |
| 724 |
|
the thermal flux across the interface. For our simulations, we denote |
| 725 |
|
$J_z$ to be positive when the physical thermal flux is from the liquid |
| 726 |
|
to metal, and negative vice versa. The $G$'s measured under different |
| 727 |
< |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and [REF]. These |
| 728 |
< |
results do not suggest that $G$ is dependent on $J_z$ within this flux |
| 729 |
< |
range. The linear response of flux to thermal gradient simplifies our |
| 730 |
< |
investigations in that we can rely on $G$ measurement with only a |
| 731 |
< |
couple $J_z$'s and do not need to test a large series of fluxes. |
| 727 |
> |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and |
| 728 |
> |
\ref{AuThiolToluene}. These results do not suggest that $G$ is |
| 729 |
> |
dependent on $J_z$ within this flux range. The linear response of flux |
| 730 |
> |
to thermal gradient simplifies our investigations in that we can rely |
| 731 |
> |
on $G$ measurement with only a couple $J_z$'s and do not need to test |
| 732 |
> |
a large series of fluxes. |
| 733 |
|
|
| 504 |
– |
%ADD MORE TO TABLE |
| 734 |
|
\begin{table*} |
| 735 |
|
\begin{minipage}{\linewidth} |
| 736 |
|
\begin{center} |
| 737 |
|
\caption{Computed interfacial thermal conductivity ($G$ and |
| 738 |
|
$G^\prime$) values for the 100\% covered Au-butanethiol/hexane |
| 739 |
|
interfaces with UA model and different hexane molecule numbers |
| 740 |
< |
at different temperatures using a range of energy fluxes.} |
| 740 |
> |
at different temperatures using a range of energy |
| 741 |
> |
fluxes. Error estimates indicated in parenthesis.} |
| 742 |
|
|
| 743 |
|
\begin{tabular}{ccccccc} |
| 744 |
|
\hline\hline |
| 747 |
|
(K) & & $L_x$ \& $L_y$? & (g/cm$^3$) & (GW/m$^2$) & |
| 748 |
|
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 749 |
|
\hline |
| 750 |
< |
200 & 266 & No & 0.672 & -0.96 & 102() & 80.0() \\ |
| 751 |
< |
& 200 & Yes & 0.694 & 1.92 & 129() & 87.3() \\ |
| 752 |
< |
& & Yes & 0.672 & 1.93 & 131() & 77.5() \\ |
| 753 |
< |
& & No & 0.688 & 0.96 & 125() & 90.2() \\ |
| 754 |
< |
& & & & 1.91 & 139() & 101() \\ |
| 755 |
< |
& & & & 2.83 & 141() & 89.9() \\ |
| 756 |
< |
& 166 & Yes & 0.679 & 0.97 & 115() & 69.3() \\ |
| 757 |
< |
& & & & 1.94 & 125() & 87.1() \\ |
| 758 |
< |
& & No & 0.681 & 0.97 & 141() & 77.7() \\ |
| 759 |
< |
& & & & 1.92 & 138() & 98.9() \\ |
| 750 |
> |
200 & 266 & No & 0.672 & -0.96 & 102(3) & 80.0(0.8) \\ |
| 751 |
> |
& 200 & Yes & 0.694 & 1.92 & 129(11) & 87.3(0.3) \\ |
| 752 |
> |
& & Yes & 0.672 & 1.93 & 131(16) & 78(13) \\ |
| 753 |
> |
& & No & 0.688 & 0.96 & 125(16) & 90.2(15) \\ |
| 754 |
> |
& & & & 1.91 & 139(10) & 101(10) \\ |
| 755 |
> |
& & & & 2.83 & 141(6) & 89.9(9.8) \\ |
| 756 |
> |
& 166 & Yes & 0.679 & 0.97 & 115(19) & 69(18) \\ |
| 757 |
> |
& & & & 1.94 & 125(9) & 87.1(0.2) \\ |
| 758 |
> |
& & No & 0.681 & 0.97 & 141(30) & 78(22) \\ |
| 759 |
> |
& & & & 1.92 & 138(4) & 98.9(9.5) \\ |
| 760 |
|
\hline |
| 761 |
< |
250 & 200 & No & 0.560 & 0.96 & 74.8() & 61.8() \\ |
| 762 |
< |
& & & & -0.95 & 49.4() & 45.7() \\ |
| 763 |
< |
& 166 & Yes & 0.570 & 0.98 & 79.0() & 62.9() \\ |
| 764 |
< |
& & No & 0.569 & 0.97 & 80.3() & 67.1() \\ |
| 765 |
< |
& & & & 1.44 & 76.2() & 64.8() \\ |
| 766 |
< |
& & & & -0.95 & 56.4() & 54.4() \\ |
| 767 |
< |
& & & & -1.85 & 47.8() & 53.5() \\ |
| 761 |
> |
250 & 200 & No & 0.560 & 0.96 & 75(10) & 61.8(7.3) \\ |
| 762 |
> |
& & & & -0.95 & 49.4(0.3) & 45.7(2.1) \\ |
| 763 |
> |
& 166 & Yes & 0.570 & 0.98 & 79.0(3.5) & 62.9(3.0) \\ |
| 764 |
> |
& & No & 0.569 & 0.97 & 80.3(0.6) & 67(11) \\ |
| 765 |
> |
& & & & 1.44 & 76.2(5.0) & 64.8(3.8) \\ |
| 766 |
> |
& & & & -0.95 & 56.4(2.5) & 54.4(1.1) \\ |
| 767 |
> |
& & & & -1.85 & 47.8(1.1) & 53.5(1.5) \\ |
| 768 |
|
\hline\hline |
| 769 |
|
\end{tabular} |
| 770 |
|
\label{AuThiolHexaneUA} |
| 772 |
|
\end{minipage} |
| 773 |
|
\end{table*} |
| 774 |
|
|
| 775 |
+ |
\subsubsection{Effects due to average temperature} |
| 776 |
+ |
|
| 777 |
|
Furthermore, we also attempted to increase system average temperatures |
| 778 |
|
to above 200K. These simulations are first equilibrated in the NPT |
| 779 |
|
ensemble under normal pressure. As stated above, the TraPPE-UA model |
| 782 |
|
temperature is higher than 250K. Additionally, the equilibrated liquid |
| 783 |
|
hexane density under 250K becomes lower than experimental value. This |
| 784 |
|
expanded liquid phase leads to lower contact between hexane and |
| 785 |
< |
butanethiol as well.[MAY NEED FIGURE] And this reduced contact would |
| 785 |
> |
butanethiol as well.[MAY NEED SLAB DENSITY FIGURE] |
| 786 |
> |
And this reduced contact would |
| 787 |
|
probably be accountable for a lower interfacial thermal conductance, |
| 788 |
|
as shown in Table \ref{AuThiolHexaneUA}. |
| 789 |
|
|
| 798 |
|
important role in the thermal transport process across the interface |
| 799 |
|
in that higher degree of contact could yield increased conductance. |
| 800 |
|
|
| 568 |
– |
[ADD ERROR ESTIMATE TO TABLE] |
| 801 |
|
\begin{table*} |
| 802 |
|
\begin{minipage}{\linewidth} |
| 803 |
|
\begin{center} |
| 804 |
|
\caption{Computed interfacial thermal conductivity ($G$ and |
| 805 |
|
$G^\prime$) values for a 90\% coverage Au-butanethiol/toluene |
| 806 |
|
interface at different temperatures using a range of energy |
| 807 |
< |
fluxes.} |
| 807 |
> |
fluxes. Error estimates indicated in parenthesis.} |
| 808 |
|
|
| 809 |
|
\begin{tabular}{ccccc} |
| 810 |
|
\hline\hline |
| 811 |
|
$\langle T\rangle$ & $\rho_{toluene}$ & $J_z$ & $G$ & $G^\prime$ \\ |
| 812 |
|
(K) & (g/cm$^3$) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 813 |
|
\hline |
| 814 |
< |
200 & 0.933 & -1.86 & 180() & 135() \\ |
| 815 |
< |
& & 2.15 & 204() & 113() \\ |
| 816 |
< |
& & -3.93 & 175() & 114() \\ |
| 814 |
> |
200 & 0.933 & 2.15 & 204(12) & 113(12) \\ |
| 815 |
> |
& & -1.86 & 180(3) & 135(21) \\ |
| 816 |
> |
& & -3.93 & 176(5) & 113(12) \\ |
| 817 |
|
\hline |
| 818 |
< |
300 & 0.855 & -1.91 & 143() & 125() \\ |
| 819 |
< |
& & -4.19 & 134() & 113() \\ |
| 818 |
> |
300 & 0.855 & -1.91 & 143(5) & 125(2) \\ |
| 819 |
> |
& & -4.19 & 135(9) & 113(12) \\ |
| 820 |
|
\hline\hline |
| 821 |
|
\end{tabular} |
| 822 |
|
\label{AuThiolToluene} |
| 848 |
|
|
| 849 |
|
However, when the surface is not completely covered by butanethiols, |
| 850 |
|
the simulated system is more resistent to the reconstruction |
| 851 |
< |
above. Our Au-butanethiol/toluene system did not see this phenomena |
| 852 |
< |
even at $\langle T\rangle\sim$300K. The Au(111) surfaces have a 90\% |
| 853 |
< |
coverage of butanethiols and have empty three-fold sites. These empty |
| 854 |
< |
sites could help prevent surface reconstruction in that they provide |
| 855 |
< |
other means of capping agent relaxation. It is observed that |
| 851 |
> |
above. Our Au-butanethiol/toluene system had the Au(111) surfaces 90\% |
| 852 |
> |
covered by butanethiols, but did not see this above phenomena even at |
| 853 |
> |
$\langle T\rangle\sim$300K. The empty three-fold sites not occupied by |
| 854 |
> |
capping agents could help prevent surface reconstruction in that they |
| 855 |
> |
provide other means of capping agent relaxation. It is observed that |
| 856 |
|
butanethiols can migrate to their neighbor empty sites during a |
| 857 |
|
simulation. Therefore, we were able to obtain $G$'s for these |
| 858 |
|
interfaces even at a relatively high temperature without being |
| 859 |
|
affected by surface reconstructions. |
| 860 |
|
|
| 629 |
– |
\subsection{Influence of Capping Agent Coverage on $G$} |
| 630 |
– |
To investigate the influence of butanethiol coverage on interfacial |
| 631 |
– |
thermal conductance, a series of different coverage Au-butanethiol |
| 632 |
– |
surfaces is prepared and solvated with various organic |
| 633 |
– |
molecules. These systems are then equilibrated and their interfacial |
| 634 |
– |
thermal conductivity are measured with our NIVS algorithm. Table |
| 635 |
– |
\ref{tlnUhxnUhxnD} lists these results for direct comparison between |
| 636 |
– |
different coverages of butanethiol. To study the isotope effect in |
| 637 |
– |
interfacial thermal conductance, deuterated UA-hexane is included as |
| 638 |
– |
well. |
| 861 |
|
|
| 862 |
< |
It turned out that with partial covered butanethiol on the Au(111) |
| 641 |
< |
surface, the derivative definition for $G$ (Eq. \ref{derivativeG}) has |
| 642 |
< |
difficulty to apply, due to the difficulty in locating the maximum of |
| 643 |
< |
change of $\lambda$. Instead, the discrete definition |
| 644 |
< |
(Eq. \ref{discreteG}) is easier to apply, as max($\Delta T$) can still |
| 645 |
< |
be well-defined. Therefore, $G$'s (not $G^\prime$) are used for this |
| 646 |
< |
section. |
| 862 |
> |
\section{Discussion} |
| 863 |
|
|
| 864 |
< |
From Table \ref{tlnUhxnUhxnD}, one can see the significance of the |
| 864 |
> |
\subsection{Capping agent acts as a vibrational coupler between solid |
| 865 |
> |
and solvent phases} |
| 866 |
> |
To investigate the mechanism of interfacial thermal conductance, the |
| 867 |
> |
vibrational power spectrum was computed. Power spectra were taken for |
| 868 |
> |
individual components in different simulations. To obtain these |
| 869 |
> |
spectra, simulations were run after equilibration, in the NVE |
| 870 |
> |
ensemble, and without a thermal gradient. Snapshots of configurations |
| 871 |
> |
were collected at a frequency that is higher than that of the fastest |
| 872 |
> |
vibrations occuring in the simulations. With these configurations, the |
| 873 |
> |
velocity auto-correlation functions can be computed: |
| 874 |
> |
\begin{equation} |
| 875 |
> |
C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle |
| 876 |
> |
\label{vCorr} |
| 877 |
> |
\end{equation} |
| 878 |
> |
The power spectrum is constructed via a Fourier transform of the |
| 879 |
> |
symmetrized velocity autocorrelation function, |
| 880 |
> |
\begin{equation} |
| 881 |
> |
\hat{f}(\omega) = |
| 882 |
> |
\int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt |
| 883 |
> |
\label{fourier} |
| 884 |
> |
\end{equation} |
| 885 |
> |
|
| 886 |
> |
From Figure \ref{coverage}, one can see the significance of the |
| 887 |
|
presence of capping agents. Even when a fraction of the Au(111) |
| 888 |
|
surface sites are covered with butanethiols, the conductivity would |
| 889 |
|
see an enhancement by at least a factor of 3. This indicates the |
| 890 |
|
important role cappping agent is playing for thermal transport |
| 891 |
< |
phenomena on metal/organic solvent surfaces. |
| 891 |
> |
phenomena on metal / organic solvent surfaces. |
| 892 |
|
|
| 893 |
|
Interestingly, as one could observe from our results, the maximum |
| 894 |
|
conductance enhancement (largest $G$) happens while the surfaces are |
| 907 |
|
would not offset this effect. Eventually, when butanethiol coverage |
| 908 |
|
continues to decrease, solvent-capping agent contact actually |
| 909 |
|
decreases with the disappearing of butanethiol molecules. In this |
| 910 |
< |
case, $G$ decrease could not be offset but instead accelerated. |
| 910 |
> |
case, $G$ decrease could not be offset but instead accelerated. [MAY NEED |
| 911 |
> |
SNAPSHOT SHOWING THE PHENOMENA / SLAB DENSITY ANALYSIS] |
| 912 |
|
|
| 913 |
|
A comparison of the results obtained from differenet organic solvents |
| 914 |
|
can also provide useful information of the interfacial thermal |
| 918 |
|
studies, even though eliminating C-H vibration samplings, still have |
| 919 |
|
C-C vibrational frequencies different from each other. However, these |
| 920 |
|
differences in the infrared range do not seem to produce an observable |
| 921 |
< |
difference for the results of $G$. [MAY NEED FIGURE] |
| 921 |
> |
difference for the results of $G$ (Figure \ref{uahxnua}). |
| 922 |
|
|
| 923 |
+ |
\begin{figure} |
| 924 |
+ |
\includegraphics[width=\linewidth]{uahxnua} |
| 925 |
+ |
\caption{Vibrational spectra obtained for normal (upper) and |
| 926 |
+ |
deuterated (lower) hexane in Au-butanethiol/hexane |
| 927 |
+ |
systems. Butanethiol spectra are shown as reference. Both hexane and |
| 928 |
+ |
butanethiol were using United-Atom models.} |
| 929 |
+ |
\label{uahxnua} |
| 930 |
+ |
\end{figure} |
| 931 |
+ |
|
| 932 |
|
Furthermore, results for rigid body toluene solvent, as well as other |
| 933 |
|
UA-hexane solvents, are reasonable within the general experimental |
| 934 |
< |
ranges[CITATIONS]. This suggests that explicit hydrogen might not be a |
| 935 |
< |
required factor for modeling thermal transport phenomena of systems |
| 936 |
< |
such as Au-thiol/organic solvent. |
| 934 |
> |
ranges\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406}. This |
| 935 |
> |
suggests that explicit hydrogen might not be a required factor for |
| 936 |
> |
modeling thermal transport phenomena of systems such as |
| 937 |
> |
Au-thiol/organic solvent. |
| 938 |
|
|
| 939 |
|
However, results for Au-butanethiol/toluene do not show an identical |
| 940 |
< |
trend with those for Au-butanethiol/hexane in that $G$'s remain at |
| 940 |
> |
trend with those for Au-butanethiol/hexane in that $G$ remains at |
| 941 |
|
approximately the same magnitue when butanethiol coverage differs from |
| 942 |
|
25\% to 75\%. This might be rooted in the molecule shape difference |
| 943 |
< |
for plane-like toluene and chain-like {\it n}-hexane. Due to this |
| 943 |
> |
for planar toluene and chain-like {\it n}-hexane. Due to this |
| 944 |
|
difference, toluene molecules have more difficulty in occupying |
| 945 |
|
relatively small gaps among capping agents when their coverage is not |
| 946 |
|
too low. Therefore, the solvent-capping agent contact may keep |
| 949 |
|
its effect to the process of interfacial thermal transport. Thus, one |
| 950 |
|
can see a plateau of $G$ vs. butanethiol coverage in our results. |
| 951 |
|
|
| 703 |
– |
[NEED ERROR ESTIMATE] |
| 704 |
– |
\begin{figure} |
| 705 |
– |
\includegraphics[width=\linewidth]{coverage} |
| 706 |
– |
\caption{Comparison of interfacial thermal conductivity ($G$) values |
| 707 |
– |
for the Au-butanethiol/solvent interface with various UA models and |
| 708 |
– |
different capping agent coverages at $\langle T\rangle\sim$200K |
| 709 |
– |
using certain energy flux respectively.} |
| 710 |
– |
\label{coverage} |
| 711 |
– |
\end{figure} |
| 712 |
– |
|
| 952 |
|
\subsection{Influence of Chosen Molecule Model on $G$} |
| 714 |
– |
[MAY COMBINE W MECHANISM STUDY] |
| 715 |
– |
|
| 953 |
|
In addition to UA solvent/capping agent models, AA models are included |
| 954 |
|
in our simulations as well. Besides simulations of the same (UA or AA) |
| 955 |
|
model for solvent and capping agent, different models can be applied |
| 958 |
|
the previous section. Table \ref{modelTest} summarizes the results of |
| 959 |
|
these studies. |
| 960 |
|
|
| 724 |
– |
[MORE DATA; ERROR ESTIMATE] |
| 961 |
|
\begin{table*} |
| 962 |
|
\begin{minipage}{\linewidth} |
| 963 |
|
\begin{center} |
| 967 |
|
solvent and capping agent (or without capping agent) at |
| 968 |
|
$\langle T\rangle\sim$200K. (D stands for deuterated solvent |
| 969 |
|
or capping agent molecules; ``Avg.'' denotes results that are |
| 970 |
< |
averages of several simulations.)} |
| 970 |
> |
averages of simulations under different $J_z$'s. Error |
| 971 |
> |
estimates indicated in parenthesis.)} |
| 972 |
|
|
| 973 |
< |
\begin{tabular}{ccccc} |
| 973 |
> |
\begin{tabular}{llccc} |
| 974 |
|
\hline\hline |
| 975 |
|
Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ |
| 976 |
|
(or bare surface) & model & (GW/m$^2$) & |
| 977 |
|
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 978 |
|
\hline |
| 979 |
< |
UA & UA hexane & Avg. & 131() & 86.5() \\ |
| 980 |
< |
& UA hexane(D) & 1.95 & 153() & 136() \\ |
| 981 |
< |
& AA hexane & 1.94 & 135() & 129() \\ |
| 982 |
< |
& & 2.86 & 126() & 115() \\ |
| 983 |
< |
& UA toluene & 1.96 & 187() & 151() \\ |
| 747 |
< |
& AA toluene & 1.89 & 200() & 149() \\ |
| 979 |
> |
UA & UA hexane & Avg. & 131(9) & 87(10) \\ |
| 980 |
> |
& UA hexane(D) & 1.95 & 153(5) & 136(13) \\ |
| 981 |
> |
& AA hexane & Avg. & 131(6) & 122(10) \\ |
| 982 |
> |
& UA toluene & 1.96 & 187(16) & 151(11) \\ |
| 983 |
> |
& AA toluene & 1.89 & 200(36) & 149(53) \\ |
| 984 |
|
\hline |
| 985 |
< |
AA & UA hexane & 1.94 & 116() & 129() \\ |
| 986 |
< |
& AA hexane & Avg. & 442() & 356() \\ |
| 987 |
< |
& AA hexane(D) & 1.93 & 222() & 234() \\ |
| 988 |
< |
& UA toluene & 1.98 & 125() & 96.5() \\ |
| 989 |
< |
& AA toluene & 3.79 & 487() & 290() \\ |
| 985 |
> |
AA & UA hexane & 1.94 & 116(9) & 129(8) \\ |
| 986 |
> |
& AA hexane & Avg. & 442(14) & 356(31) \\ |
| 987 |
> |
& AA hexane(D) & 1.93 & 222(12) & 234(54) \\ |
| 988 |
> |
& UA toluene & 1.98 & 125(25) & 97(60) \\ |
| 989 |
> |
& AA toluene & 3.79 & 487(56) & 290(42) \\ |
| 990 |
|
\hline |
| 991 |
< |
AA(D) & UA hexane & 1.94 & 158() & 172() \\ |
| 992 |
< |
& AA hexane & 1.92 & 243() & 191() \\ |
| 993 |
< |
& AA toluene & 1.93 & 364() & 322() \\ |
| 991 |
> |
AA(D) & UA hexane & 1.94 & 158(25) & 172(4) \\ |
| 992 |
> |
& AA hexane & 1.92 & 243(29) & 191(11) \\ |
| 993 |
> |
& AA toluene & 1.93 & 364(36) & 322(67) \\ |
| 994 |
|
\hline |
| 995 |
< |
bare & UA hexane & Avg. & 46.5() & 49.4() \\ |
| 996 |
< |
& UA hexane(D) & 0.98 & 43.9() & 43.0() \\ |
| 997 |
< |
& AA hexane & 0.96 & 31.0() & 29.4() \\ |
| 998 |
< |
& UA toluene & 1.99 & 70.1() & 65.8() \\ |
| 995 |
> |
bare & UA hexane & Avg. & 46.5(3.2) & 49.4(4.5) \\ |
| 996 |
> |
& UA hexane(D) & 0.98 & 43.9(4.6) & 43.0(2.0) \\ |
| 997 |
> |
& AA hexane & 0.96 & 31.0(1.4) & 29.4(1.3) \\ |
| 998 |
> |
& UA toluene & 1.99 & 70.1(1.3) & 65.8(0.5) \\ |
| 999 |
|
\hline\hline |
| 1000 |
|
\end{tabular} |
| 1001 |
|
\label{modelTest} |
| 1018 |
|
interfaces, using AA model for both butanethiol and hexane yields |
| 1019 |
|
substantially higher conductivity values than using UA model for at |
| 1020 |
|
least one component of the solvent and capping agent, which exceeds |
| 1021 |
< |
the upper bond of experimental value range. This is probably due to |
| 1022 |
< |
the classically treated C-H vibrations in the AA model, which should |
| 1023 |
< |
not be appreciably populated at normal temperatures. In comparison, |
| 1024 |
< |
once either the hexanes or the butanethiols are deuterated, one can |
| 1025 |
< |
see a significantly lower $G$ and $G^\prime$. In either of these |
| 1026 |
< |
cases, the C-H(D) vibrational overlap between the solvent and the |
| 1027 |
< |
capping agent is removed. [MAY NEED FIGURE] Conclusively, the |
| 1028 |
< |
improperly treated C-H vibration in the AA model produced |
| 1029 |
< |
over-predicted results accordingly. Compared to the AA model, the UA |
| 1030 |
< |
model yields more reasonable results with higher computational |
| 1031 |
< |
efficiency. |
| 1021 |
> |
the general range of experimental measurement results. This is |
| 1022 |
> |
probably due to the classically treated C-H vibrations in the AA |
| 1023 |
> |
model, which should not be appreciably populated at normal |
| 1024 |
> |
temperatures. In comparison, once either the hexanes or the |
| 1025 |
> |
butanethiols are deuterated, one can see a significantly lower $G$ and |
| 1026 |
> |
$G^\prime$. In either of these cases, the C-H(D) vibrational overlap |
| 1027 |
> |
between the solvent and the capping agent is removed (Figure |
| 1028 |
> |
\ref{aahxntln}). Conclusively, the improperly treated C-H vibration in |
| 1029 |
> |
the AA model produced over-predicted results accordingly. Compared to |
| 1030 |
> |
the AA model, the UA model yields more reasonable results with higher |
| 1031 |
> |
computational efficiency. |
| 1032 |
|
|
| 1033 |
+ |
\begin{figure} |
| 1034 |
+ |
\includegraphics[width=\linewidth]{aahxntln} |
| 1035 |
+ |
\caption{Spectra obtained for All-Atom model Au-butanethil/solvent |
| 1036 |
+ |
systems. When butanethiol is deuterated (lower left), its |
| 1037 |
+ |
vibrational overlap with hexane would decrease significantly, |
| 1038 |
+ |
compared with normal butanethiol (upper left). However, this |
| 1039 |
+ |
dramatic change does not apply to toluene as much (right).} |
| 1040 |
+ |
\label{aahxntln} |
| 1041 |
+ |
\end{figure} |
| 1042 |
+ |
|
| 1043 |
|
However, for Au-butanethiol/toluene interfaces, having the AA |
| 1044 |
|
butanethiol deuterated did not yield a significant change in the |
| 1045 |
|
measurement results. Compared to the C-H vibrational overlap between |
| 1046 |
|
hexane and butanethiol, both of which have alkyl chains, that overlap |
| 1047 |
|
between toluene and butanethiol is not so significant and thus does |
| 1048 |
< |
not have as much contribution to the ``Intramolecular Vibration |
| 1049 |
< |
Redistribution''[CITE HASE]. Conversely, extra degrees of freedom such |
| 1050 |
< |
as the C-H vibrations could yield higher heat exchange rate between |
| 1051 |
< |
these two phases and result in a much higher conductivity. |
| 1048 |
> |
not have as much contribution to the heat exchange |
| 1049 |
> |
process. Conversely, extra degrees of freedom such as the C-H |
| 1050 |
> |
vibrations could yield higher heat exchange rate between these two |
| 1051 |
> |
phases and result in a much higher conductivity. |
| 1052 |
|
|
| 1053 |
|
Although the QSC model for Au is known to predict an overly low value |
| 1054 |
|
for bulk metal gold conductivity\cite{kuang:164101}, our computational |
| 1058 |
|
the accuracy of the interaction descriptions between components |
| 1059 |
|
occupying the interfaces. |
| 1060 |
|
|
| 1061 |
< |
\subsection{Mechanism of Interfacial Thermal Conductance Enhancement |
| 1062 |
< |
by Capping Agent} |
| 1063 |
< |
%OR\subsection{Vibrational spectrum study on conductance mechanism} |
| 1061 |
> |
\subsection{Role of Capping Agent in Interfacial Thermal Conductance} |
| 1062 |
> |
The vibrational spectra for gold slabs in different environments are |
| 1063 |
> |
shown as in Figure \ref{specAu}. Regardless of the presence of |
| 1064 |
> |
solvent, the gold surfaces covered by butanethiol molecules, compared |
| 1065 |
> |
to bare gold surfaces, exhibit an additional peak observed at the |
| 1066 |
> |
frequency of $\sim$170cm$^{-1}$, which is attributed to the S-Au |
| 1067 |
> |
bonding vibration. This vibration enables efficient thermal transport |
| 1068 |
> |
from surface Au layer to the capping agents. Therefore, in our |
| 1069 |
> |
simulations, the Au/S interfaces do not appear major heat barriers |
| 1070 |
> |
compared to the butanethiol / solvent interfaces. |
| 1071 |
|
|
| 1072 |
< |
[MAY INTRODUCE PROTOCOL IN METHODOLOGY/COMPUTATIONAL DETAIL, EQN'S] |
| 1072 |
> |
\subsubsection{Overlap of power spectrum} |
| 1073 |
> |
Simultaneously, the vibrational overlap between butanethiol and |
| 1074 |
> |
organic solvents suggests higher thermal exchange efficiency between |
| 1075 |
> |
these two components. Even exessively high heat transport was observed |
| 1076 |
> |
when All-Atom models were used and C-H vibrations were treated |
| 1077 |
> |
classically. Compared to metal and organic liquid phase, the heat |
| 1078 |
> |
transfer efficiency between butanethiol and organic solvents is closer |
| 1079 |
> |
to that within bulk liquid phase. |
| 1080 |
|
|
| 1081 |
< |
To investigate the mechanism of this interfacial thermal conductance, |
| 1082 |
< |
the vibrational spectra of various gold systems were obtained and are |
| 1083 |
< |
shown as in the upper panel of Fig. \ref{vibration}. To obtain these |
| 1084 |
< |
spectra, one first runs a simulation in the NVE ensemble and collects |
| 1085 |
< |
snapshots of configurations; these configurations are used to compute |
| 1086 |
< |
the velocity auto-correlation functions, which is used to construct a |
| 1087 |
< |
power spectrum via a Fourier transform. |
| 1081 |
> |
Furthermore, our observation validated previous |
| 1082 |
> |
results\cite{hase:2010} that the intramolecular heat transport of |
| 1083 |
> |
alkylthiols is highly effecient. As a combinational effects of these |
| 1084 |
> |
phenomena, butanethiol acts as a channel to expedite thermal transport |
| 1085 |
> |
process. The acoustic impedance mismatch between the metal and the |
| 1086 |
> |
liquid phase can be effectively reduced with the presence of suitable |
| 1087 |
> |
capping agents. |
| 1088 |
|
|
| 829 |
– |
[MAY RELATE TO HASE'S] |
| 830 |
– |
The gold surfaces covered by |
| 831 |
– |
butanethiol molecules, compared to bare gold surfaces, exhibit an |
| 832 |
– |
additional peak observed at a frequency of $\sim$170cm$^{-1}$, which |
| 833 |
– |
is attributed to the vibration of the S-Au bond. This vibration |
| 834 |
– |
enables efficient thermal transport from surface Au atoms to the |
| 835 |
– |
capping agents. Simultaneously, as shown in the lower panel of |
| 836 |
– |
Fig. \ref{vibration}, the large overlap of the vibration spectra of |
| 837 |
– |
butanethiol and hexane in the all-atom model, including the C-H |
| 838 |
– |
vibration, also suggests high thermal exchange efficiency. The |
| 839 |
– |
combination of these two effects produces the drastic interfacial |
| 840 |
– |
thermal conductance enhancement in the all-atom model. |
| 841 |
– |
|
| 842 |
– |
[REDO. MAY NEED TO CONVERT TO JPEG] |
| 1089 |
|
\begin{figure} |
| 1090 |
|
\includegraphics[width=\linewidth]{vibration} |
| 1091 |
|
\caption{Vibrational spectra obtained for gold in different |
| 1092 |
< |
environments (upper panel) and for Au/thiol/hexane simulation in |
| 1093 |
< |
all-atom model (lower panel).} |
| 848 |
< |
\label{vibration} |
| 1092 |
> |
environments.} |
| 1093 |
> |
\label{specAu} |
| 1094 |
|
\end{figure} |
| 1095 |
|
|
| 1096 |
< |
[COMPARISON OF TWO G'S; AU SLAB WIDTHS; ETC] |
| 852 |
< |
% The results show that the two definitions used for $G$ yield |
| 853 |
< |
% comparable values, though $G^\prime$ tends to be smaller. |
| 1096 |
> |
[MAY ADD COMPARISON OF AU SLAB WIDTHS BUT NOT MUCH TO TALK ABOUT...] |
| 1097 |
|
|
| 1098 |
|
\section{Conclusions} |
| 1099 |
|
The NIVS algorithm we developed has been applied to simulations of |
| 1101 |
|
effective unphysical thermal flux transferred between the metal and |
| 1102 |
|
the liquid phase. With the flux applied, we were able to measure the |
| 1103 |
|
corresponding thermal gradient and to obtain interfacial thermal |
| 1104 |
< |
conductivities. Our simulations have seen significant conductance |
| 1105 |
< |
enhancement with the presence of capping agent, compared to the bare |
| 1106 |
< |
gold/liquid interfaces. The acoustic impedance mismatch between the |
| 1107 |
< |
metal and the liquid phase is effectively eliminated by proper capping |
| 1104 |
> |
conductivities. Under steady states, single trajectory simulation |
| 1105 |
> |
would be enough for accurate measurement. This would be advantageous |
| 1106 |
> |
compared to transient state simulations, which need multiple |
| 1107 |
> |
trajectories to produce reliable average results. |
| 1108 |
> |
|
| 1109 |
> |
Our simulations have seen significant conductance enhancement with the |
| 1110 |
> |
presence of capping agent, compared to the bare gold / liquid |
| 1111 |
> |
interfaces. The acoustic impedance mismatch between the metal and the |
| 1112 |
> |
liquid phase is effectively eliminated by proper capping |
| 1113 |
|
agent. Furthermore, the coverage precentage of the capping agent plays |
| 1114 |
< |
an important role in the interfacial thermal transport process. |
| 1114 |
> |
an important role in the interfacial thermal transport |
| 1115 |
> |
process. Moderately lower coverages allow higher contact between |
| 1116 |
> |
capping agent and solvent, and thus could further enhance the heat |
| 1117 |
> |
transfer process. |
| 1118 |
|
|
| 1119 |
|
Our measurement results, particularly of the UA models, agree with |
| 1120 |
|
available experimental data. This indicates that our force field |
| 1124 |
|
vibration would be overly sampled. Compared to the AA models, the UA |
| 1125 |
|
models have higher computational efficiency with satisfactory |
| 1126 |
|
accuracy, and thus are preferable in interfacial thermal transport |
| 1127 |
< |
modelings. |
| 1127 |
> |
modelings. Of the two definitions for $G$, the discrete form |
| 1128 |
> |
(Eq. \ref{discreteG}) was easier to use and gives out relatively |
| 1129 |
> |
consistent results, while the derivative form (Eq. \ref{derivativeG}) |
| 1130 |
> |
is not as versatile. Although $G^\prime$ gives out comparable results |
| 1131 |
> |
and follows similar trend with $G$ when measuring close to fully |
| 1132 |
> |
covered or bare surfaces, the spatial resolution of $T$ profile is |
| 1133 |
> |
limited for accurate computation of derivatives data. |
| 1134 |
|
|
| 1135 |
|
Vlugt {\it et al.} has investigated the surface thiol structures for |
| 1136 |
|
nanocrystal gold and pointed out that they differs from those of the |
| 1137 |
< |
Au(111) surface\cite{vlugt:cpc2007154}. This difference might lead to |
| 1138 |
< |
change of interfacial thermal transport behavior as well. To |
| 1139 |
< |
investigate this problem, an effective means to introduce thermal flux |
| 1140 |
< |
and measure the corresponding thermal gradient is desirable for |
| 1141 |
< |
simulating structures with spherical symmetry. |
| 1137 |
> |
Au(111) surface\cite{landman:1998,vlugt:cpc2007154}. This difference |
| 1138 |
> |
might lead to change of interfacial thermal transport behavior as |
| 1139 |
> |
well. To investigate this problem, an effective means to introduce |
| 1140 |
> |
thermal flux and measure the corresponding thermal gradient is |
| 1141 |
> |
desirable for simulating structures with spherical symmetry. |
| 1142 |
|
|
| 886 |
– |
|
| 1143 |
|
\section{Acknowledgments} |
| 1144 |
|
Support for this project was provided by the National Science |
| 1145 |
|
Foundation under grant CHE-0848243. Computational time was provided by |
| 1146 |
|
the Center for Research Computing (CRC) at the University of Notre |
| 1147 |
< |
Dame. \newpage |
| 1147 |
> |
Dame. |
| 1148 |
> |
\newpage |
| 1149 |
|
|
| 1150 |
|
\bibliography{interfacial} |
| 1151 |
|
|
