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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 |
< |
importance in nanoscale science and technology. Reliability of |
| 79 |
< |
nanoscale devices depends on their thermal transport |
| 80 |
< |
properties. Unlike bulk homogeneous materials, nanoscale materials |
| 81 |
< |
features significant presence of interfaces, and these interfaces |
| 82 |
< |
could dominate the heat transfer behavior of these |
| 83 |
< |
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 |
| 77 |
> |
thermal conductance has been studied extensively both experimentally |
| 78 |
> |
and computationally.\cite{cahill:793} Unlike bulk materials, nanoscale |
| 79 |
> |
materials have a significant fraction of their atoms at interfaces, |
| 80 |
> |
and the chemical details of these interfaces govern the heat transfer |
| 81 |
> |
behavior. Furthermore, the interfaces are |
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> |
heterogeneous (e.g. solid - liquid), which provides a challenge to |
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traditional methods developed for homogeneous systems. |
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|
|
| 85 |
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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.} |
| 91 |
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studied the role of CTAB on thermal transport between gold nanorods |
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and solvent\cite{doi:10.1021/jp8051888}; Juv\'e {\it et al.} studied |
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> |
Experimentally, various interfaces have been investigated for their |
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> |
thermal conductance. Cahill and coworkers studied nanoscale thermal |
| 87 |
> |
transport from metal nanoparticle/fluid interfaces, to epitaxial |
| 88 |
> |
TiN/single crystal oxides interfaces, to hydrophilic and hydrophobic |
| 89 |
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interfaces between water and solids with different self-assembled |
| 90 |
> |
monolayers.\cite{Wilson:2002uq,PhysRevB.67.054302,doi:10.1021/jp048375k,PhysRevLett.96.186101} |
| 91 |
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Wang {\it et al.} studied heat transport through |
| 92 |
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long-chain hydrocarbon monolayers on gold substrate at individual |
| 93 |
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molecular level,\cite{Wang10082007} Schmidt {\it et al.} studied the |
| 94 |
> |
role of CTAB on thermal transport between gold nanorods and |
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> |
solvent,\cite{doi:10.1021/jp8051888} and Juv\'e {\it et al.} studied |
| 96 |
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the cooling dynamics, which is controlled by thermal interface |
| 97 |
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resistence of glass-embedded metal |
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nanoparticles\cite{PhysRevB.80.195406}. Although interfaces are |
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commonly barriers for heat transport, Alper {\it et al.} suggested |
| 100 |
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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}. |
| 98 |
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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.} |
| 100 |
> |
suggested that specific ligands (capping agents) could completely |
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eliminate this barrier |
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($G\rightarrow\infty$).\cite{doi:10.1021/la904855s} |
| 103 |
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|
| 104 |
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Theoretical and computational models have also been used to study the |
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interfacial thermal transport in order to gain an understanding of |
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employed Non-Equilibrium Molecular Dynamics (NEMD) simulations to |
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study thermal transport from hot Au(111) substrate to a self-assembled |
| 109 |
|
monolayer of alkylthiol with relatively long chain (8-20 carbon |
| 110 |
< |
atoms)\cite{hase:2010,hase:2011}. However, ensemble averaged |
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> |
atoms).\cite{hase:2010,hase:2011} However, ensemble averaged |
| 111 |
|
measurements for heat conductance of interfaces between the capping |
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monolayer on Au and a solvent phase has yet to be studied. |
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The comparatively low thermal flux through interfaces is |
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difficult to measure with Equilibrium MD or forward NEMD simulation |
| 115 |
< |
methods. Therefore, the Reverse NEMD (RNEMD) methods would have the |
| 116 |
< |
advantage of having this difficult to measure flux known when studying |
| 117 |
< |
the thermal transport across interfaces, given that the simulation |
| 118 |
< |
methods being able to effectively apply an unphysical flux in |
| 119 |
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non-homogeneous systems. |
| 112 |
> |
monolayer on Au and a solvent phase have yet to be studied with their |
| 113 |
> |
approach. The comparatively low thermal flux through interfaces is |
| 114 |
> |
difficult to measure with Equilibrium |
| 115 |
> |
MD\cite{doi:10.1080/0026897031000068578} or forward NEMD simulation |
| 116 |
> |
methods. Therefore, the Reverse NEMD (RNEMD) |
| 117 |
> |
methods\cite{MullerPlathe:1997xw,kuang:164101} would have the |
| 118 |
> |
advantage of applying this difficult to measure flux (while measuring |
| 119 |
> |
the resulting gradient), given that the simulation methods being able |
| 120 |
> |
to effectively apply an unphysical flux in non-homogeneous systems. |
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> |
Garde and coworkers\cite{garde:nl2005,garde:PhysRevLett2009} applied |
| 122 |
> |
this approach to various liquid interfaces and studied how thermal |
| 123 |
> |
conductance (or resistance) is dependent on chemistry details of |
| 124 |
> |
interfaces, e.g. hydrophobic and hydrophilic aqueous interfaces. |
| 125 |
|
|
| 126 |
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Recently, we have developed the Non-Isotropic Velocity Scaling (NIVS) |
| 126 |
> |
Recently, we have developed a Non-Isotropic Velocity Scaling (NIVS) |
| 127 |
|
algorithm for RNEMD simulations\cite{kuang:164101}. This algorithm |
| 128 |
|
retains the desirable features of RNEMD (conservation of linear |
| 129 |
|
momentum and total energy, compatibility with periodic boundary |
| 140 |
|
thermal transport across these interfaces was studied and the |
| 141 |
|
underlying mechanism for the phenomena was investigated. |
| 142 |
|
|
| 136 |
– |
[MAY ADD WHY STUDY AU-THIOL SURFACE; CITE SHAOYI JIANG] |
| 137 |
– |
|
| 143 |
|
\section{Methodology} |
| 144 |
|
\subsection{Imposd-Flux Methods in MD Simulations} |
| 145 |
< |
Steady state MD simulations has the advantage that not many |
| 145 |
> |
Steady state MD simulations have an advantage in that not many |
| 146 |
|
trajectories are needed to study the relationship between thermal flux |
| 147 |
< |
and thermal gradients. For systems including low conductance |
| 148 |
< |
interfaces one must have a method capable of generating or measuring |
| 149 |
< |
relatively small fluxes, compared to those required for bulk |
| 150 |
< |
conductivity. This requirement makes the calculation even more |
| 151 |
< |
difficult for those slowly-converging equilibrium |
| 152 |
< |
methods\cite{Viscardy:2007lq}. Forward methods may impose gradient, |
| 153 |
< |
but in interfacial conditions it is not clear what behavior to impose |
| 154 |
< |
at the interfacial boundaries. Imposed-flux reverse non-equilibrium |
| 155 |
< |
methods\cite{MullerPlathe:1997xw} have the flux set {\it a priori} and |
| 156 |
< |
the thermal response becomes easier to measure than the flux. Although |
| 147 |
> |
and thermal gradients. For systems with low interfacial conductance, |
| 148 |
> |
one must have a method capable of generating or measuring relatively |
| 149 |
> |
small fluxes, compared to those required for bulk conductivity. This |
| 150 |
> |
requirement makes the calculation even more difficult for |
| 151 |
> |
slowly-converging equilibrium methods.\cite{Viscardy:2007lq} Forward |
| 152 |
> |
NEMD methods impose a gradient (and measure a flux), but at interfaces |
| 153 |
> |
it is not clear what behavior should be imposed at the boundaries |
| 154 |
> |
between materials. Imposed-flux reverse non-equilibrium |
| 155 |
> |
methods\cite{MullerPlathe:1997xw} set the flux {\it a priori} and |
| 156 |
> |
the thermal response becomes an easy-to-measure quantity. Although |
| 157 |
|
M\"{u}ller-Plathe's original momentum swapping approach can be used |
| 158 |
|
for exchanging energy between particles of different identity, the |
| 159 |
|
kinetic energy transfer efficiency is affected by the mass difference |
| 160 |
|
between the particles, which limits its application on heterogeneous |
| 161 |
|
interfacial systems. |
| 162 |
|
|
| 163 |
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The non-isotropic velocity scaling (NIVS)\cite{kuang:164101} approach to |
| 164 |
< |
non-equilibrium MD simulations is able to impose a wide range of |
| 163 |
> |
The non-isotropic velocity scaling (NIVS) \cite{kuang:164101} approach |
| 164 |
> |
to non-equilibrium MD simulations is able to impose a wide range of |
| 165 |
|
kinetic energy fluxes without obvious perturbation to the velocity |
| 166 |
|
distributions of the simulated systems. Furthermore, this approach has |
| 167 |
|
the advantage in heterogeneous interfaces in that kinetic energy flux |
| 178 |
|
for computing thermal conductivities. The NIVS algorithm conserves |
| 179 |
|
momenta and energy and does not depend on an external thermostat. |
| 180 |
|
|
| 181 |
< |
\subsection{Defining Interfacial Thermal Conductivity $G$} |
| 182 |
< |
Given a system with thermal gradients and the corresponding thermal |
| 183 |
< |
flux, for interfaces with a relatively low interfacial conductance, |
| 184 |
< |
the bulk regions on either side of an interface rapidly come to a |
| 185 |
< |
state in which the two phases have relatively homogeneous (but |
| 186 |
< |
distinct) temperatures. The interfacial thermal conductivity $G$ can |
| 187 |
< |
therefore be approximated as: |
| 181 |
> |
\subsection{Defining Interfacial Thermal Conductivity ($G$)} |
| 182 |
> |
|
| 183 |
> |
For an interface with relatively low interfacial conductance, and a |
| 184 |
> |
thermal flux between two distinct bulk regions, the regions on either |
| 185 |
> |
side of the interface rapidly come to a state in which the two phases |
| 186 |
> |
have relatively homogeneous (but distinct) temperatures. The |
| 187 |
> |
interfacial thermal conductivity $G$ can therefore be approximated as: |
| 188 |
|
\begin{equation} |
| 189 |
< |
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle - |
| 189 |
> |
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle - |
| 190 |
|
\langle T_\mathrm{cold}\rangle \right)} |
| 191 |
|
\label{lowG} |
| 192 |
|
\end{equation} |
| 193 |
< |
where ${E_{total}}$ is the imposed non-physical kinetic energy |
| 194 |
< |
transfer and ${\langle T_\mathrm{hot}\rangle}$ and ${\langle |
| 195 |
< |
T_\mathrm{cold}\rangle}$ are the average observed temperature of the |
| 196 |
< |
two separated phases. |
| 193 |
> |
where ${E_{total}}$ is the total imposed non-physical kinetic energy |
| 194 |
> |
transfer during the simulation and ${\langle T_\mathrm{hot}\rangle}$ |
| 195 |
> |
and ${\langle T_\mathrm{cold}\rangle}$ are the average observed |
| 196 |
> |
temperature of the two separated phases. |
| 197 |
|
|
| 198 |
|
When the interfacial conductance is {\it not} small, there are two |
| 199 |
< |
ways to define $G$. |
| 200 |
< |
|
| 201 |
< |
One way is to assume the temperature is discrete on the two sides of |
| 202 |
< |
the interface. $G$ can be calculated using the applied thermal flux |
| 203 |
< |
$J$ and the maximum temperature difference measured along the thermal |
| 204 |
< |
gradient max($\Delta T$), which occurs at the Gibbs deviding surface |
| 205 |
< |
(Figure \ref{demoPic}): |
| 199 |
> |
ways to define $G$. One common way is to assume the temperature is |
| 200 |
> |
discrete on the two sides of the interface. $G$ can be calculated |
| 201 |
> |
using the applied thermal flux $J$ and the maximum temperature |
| 202 |
> |
difference measured along the thermal gradient max($\Delta T$), which |
| 203 |
> |
occurs at the Gibbs deviding surface (Figure \ref{demoPic}). This is |
| 204 |
> |
known as the Kapitza conductance, which is the inverse of the Kapitza |
| 205 |
> |
resistance. |
| 206 |
|
\begin{equation} |
| 207 |
< |
G=\frac{J}{\Delta T} |
| 207 |
> |
G=\frac{J}{\Delta T} |
| 208 |
|
\label{discreteG} |
| 209 |
|
\end{equation} |
| 210 |
|
|
| 224 |
|
|
| 225 |
|
The other approach is to assume a continuous temperature profile along |
| 226 |
|
the thermal gradient axis (e.g. $z$) and define $G$ at the point where |
| 227 |
< |
the magnitude of thermal conductivity $\lambda$ change reach its |
| 227 |
> |
the magnitude of thermal conductivity ($\lambda$) change reaches its |
| 228 |
|
maximum, given that $\lambda$ is well-defined throughout the space: |
| 229 |
|
\begin{equation} |
| 230 |
|
G^\prime = \Big|\frac{\partial\lambda}{\partial z}\Big| |
| 235 |
|
\label{derivativeG} |
| 236 |
|
\end{equation} |
| 237 |
|
|
| 238 |
< |
With the temperature profile obtained from simulations, one is able to |
| 238 |
> |
With temperature profiles obtained from simulation, one is able to |
| 239 |
|
approximate the first and second derivatives of $T$ with finite |
| 240 |
< |
difference methods and thus calculate $G^\prime$. |
| 240 |
> |
difference methods and calculate $G^\prime$. In what follows, both |
| 241 |
> |
definitions have been used, and are compared in the results. |
| 242 |
|
|
| 243 |
< |
In what follows, both definitions have been used for calculation and |
| 244 |
< |
are compared in the results. |
| 245 |
< |
|
| 246 |
< |
To compare the above definitions ($G$ and $G^\prime$), we have modeled |
| 247 |
< |
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 |
| 243 |
> |
To investigate the interfacial conductivity at metal / solvent |
| 244 |
> |
interfaces, we have modeled a metal slab with its (111) surfaces |
| 245 |
> |
perpendicular to the $z$-axis of our simulation cells. The metal slab |
| 246 |
> |
has been prepared both with and without capping agents on the exposed |
| 247 |
> |
surface, and has been solvated with simple organic solvents, as |
| 248 |
|
illustrated in Figure \ref{gradT}. |
| 249 |
|
|
| 250 |
|
With the simulation cell described above, we are able to equilibrate |
| 251 |
|
the system and impose an unphysical thermal flux between the liquid |
| 252 |
|
and the metal phase using the NIVS algorithm. By periodically applying |
| 253 |
< |
the unphysical flux, we are able to obtain a temperature profile and |
| 254 |
< |
its spatial derivatives. These quantities enable the evaluation of the |
| 255 |
< |
interfacial thermal conductance of a surface. Figure \ref{gradT} is an |
| 256 |
< |
example of how an applied thermal flux can be used to obtain the 1st |
| 253 |
< |
and 2nd derivatives of the temperature profile. |
| 253 |
> |
the unphysical flux, we obtained a temperature profile and its spatial |
| 254 |
> |
derivatives. Figure \ref{gradT} shows how an applied thermal flux can |
| 255 |
> |
be used to obtain the 1st and 2nd derivatives of the temperature |
| 256 |
> |
profile. |
| 257 |
|
|
| 258 |
|
\begin{figure} |
| 259 |
|
\includegraphics[width=\linewidth]{gradT} |
| 267 |
|
\section{Computational Details} |
| 268 |
|
\subsection{Simulation Protocol} |
| 269 |
|
The NIVS algorithm has been implemented in our MD simulation code, |
| 270 |
< |
OpenMD\cite{Meineke:2005gd,openmd}, and was used for our |
| 271 |
< |
simulations. Different metal slab thickness (layer numbers of Au) was |
| 272 |
< |
simulated. Metal slabs were first equilibrated under atmospheric |
| 273 |
< |
pressure (1 atm) and a desired temperature (e.g. 200K). After |
| 274 |
< |
equilibration, butanethiol capping agents were placed at three-fold |
| 275 |
< |
hollow sites on the Au(111) surfaces. These sites could be either a |
| 276 |
< |
{\it fcc} or {\it hcp} site. However, Hase {\it et al.} found that |
| 277 |
< |
they are equivalent in a heat transfer process\cite{hase:2010}, so |
| 275 |
< |
they are not distinguished in our study. The maximum butanethiol |
| 270 |
> |
OpenMD\cite{Meineke:2005gd,openmd}, and was used for our simulations. |
| 271 |
> |
Metal slabs of 6 or 11 layers of Au atoms were first equilibrated |
| 272 |
> |
under atmospheric pressure (1 atm) and 200K. After equilibration, |
| 273 |
> |
butanethiol capping agents were placed at three-fold hollow sites on |
| 274 |
> |
the Au(111) surfaces. These sites are either {\it fcc} or {\it |
| 275 |
> |
hcp} sites, although Hase {\it et al.} found that they are |
| 276 |
> |
equivalent in a heat transfer process,\cite{hase:2010} so we did not |
| 277 |
> |
distinguish between these sites in our study. The maximum butanethiol |
| 278 |
|
capacity on Au surface is $1/3$ of the total number of surface Au |
| 279 |
|
atoms, and the packing forms a $(\sqrt{3}\times\sqrt{3})R30^\circ$ |
| 280 |
|
structure\cite{doi:10.1021/ja00008a001,doi:10.1021/cr9801317}. A |
| 281 |
< |
series of different coverages was derived by evenly eliminating |
| 282 |
< |
butanethiols on the surfaces, and was investigated in order to study |
| 283 |
< |
the relation between coverage and interfacial conductance. |
| 281 |
> |
series of lower coverages was also prepared by eliminating |
| 282 |
> |
butanethiols from the higher coverage surface in a regular manner. The |
| 283 |
> |
lower coverages were prepared in order to study the relation between |
| 284 |
> |
coverage and interfacial conductance. |
| 285 |
|
|
| 286 |
|
The capping agent molecules were allowed to migrate during the |
| 287 |
|
simulations. They distributed themselves uniformly and sampled a |
| 288 |
|
number of three-fold sites throughout out study. Therefore, the |
| 289 |
< |
initial configuration would not noticeably affect the sampling of a |
| 289 |
> |
initial configuration does not noticeably affect the sampling of a |
| 290 |
|
variety of configurations of the same coverage, and the final |
| 291 |
|
conductance measurement would be an average effect of these |
| 292 |
< |
configurations explored in the simulations. [MAY NEED SNAPSHOTS] |
| 292 |
> |
configurations explored in the simulations. |
| 293 |
|
|
| 294 |
< |
After the modified Au-butanethiol surface systems were equilibrated |
| 295 |
< |
under canonical ensemble, organic solvent molecules were packed in the |
| 296 |
< |
previously empty part of the simulation cells\cite{packmol}. Two |
| 294 |
> |
After the modified Au-butanethiol surface systems were equilibrated in |
| 295 |
> |
the canonical (NVT) ensemble, organic solvent molecules were packed in |
| 296 |
> |
the previously empty part of the simulation cells.\cite{packmol} Two |
| 297 |
|
solvents were investigated, one which has little vibrational overlap |
| 298 |
< |
with the alkanethiol and a planar shape (toluene), and one which has |
| 299 |
< |
similar vibrational frequencies and chain-like shape ({\it n}-hexane). |
| 298 |
> |
with the alkanethiol and which has a planar shape (toluene), and one |
| 299 |
> |
which has similar vibrational frequencies to the capping agent and |
| 300 |
> |
chain-like shape ({\it n}-hexane). |
| 301 |
|
|
| 302 |
< |
The space filled by solvent molecules, i.e. the gap between |
| 303 |
< |
periodically repeated Au-butanethiol surfaces should be carefully |
| 304 |
< |
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 |
| 302 |
> |
The simulation cells were not particularly extensive along the |
| 303 |
> |
$z$-axis, as a very long length scale for the thermal gradient may |
| 304 |
> |
cause excessively hot or cold temperatures in the middle of the |
| 305 |
|
solvent region and lead to undesired phenomena such as solvent boiling |
| 306 |
|
or freezing when a thermal flux is applied. Conversely, too few |
| 307 |
|
solvent molecules would change the normal behavior of the liquid |
| 308 |
|
phase. Therefore, our $N_{solvent}$ values were chosen to ensure that |
| 309 |
< |
these extreme cases did not happen to our simulations. And the |
| 310 |
< |
corresponding spacing is usually $35 \sim 75$\AA. |
| 309 |
> |
these extreme cases did not happen to our simulations. The spacing |
| 310 |
> |
between periodic images of the gold interfaces is $45 \sim 75$\AA. |
| 311 |
|
|
| 312 |
|
The initial configurations generated are further equilibrated with the |
| 313 |
< |
$x$ and $y$ dimensions fixed, only allowing length scale change in $z$ |
| 314 |
< |
dimension. This is to ensure that the equilibration of liquid phase |
| 315 |
< |
does not affect the metal crystal structure in $x$ and $y$ dimensions. |
| 316 |
< |
To investigate this effect, comparisons were made with simulations |
| 317 |
< |
that allow changes of $L_x$ and $L_y$ during NPT equilibration, and |
| 318 |
< |
the results are shown in later sections. After ensuring the liquid |
| 319 |
< |
phase reaches equilibrium at atmospheric pressure (1 atm), further |
| 320 |
< |
equilibration are followed under NVT and then NVE ensembles. |
| 313 |
> |
$x$ and $y$ dimensions fixed, only allowing the $z$-length scale to |
| 314 |
> |
change. This is to ensure that the equilibration of liquid phase does |
| 315 |
> |
not affect the metal's crystalline structure. Comparisons were made |
| 316 |
> |
with simulations that allowed changes of $L_x$ and $L_y$ during NPT |
| 317 |
> |
equilibration. No substantial changes in the box geometry were noticed |
| 318 |
> |
in these simulations. After ensuring the liquid phase reaches |
| 319 |
> |
equilibrium at atmospheric pressure (1 atm), further equilibration was |
| 320 |
> |
carried out under canonical (NVT) and microcanonical (NVE) ensembles. |
| 321 |
|
|
| 322 |
< |
After the systems reach equilibrium, NIVS is implemented to impose a |
| 323 |
< |
periodic unphysical thermal flux between the metal and the liquid |
| 324 |
< |
phase. Most of our simulations are under an average temperature of |
| 325 |
< |
$\sim$200K. Therefore, this flux usually comes from the metal to the |
| 322 |
> |
After the systems reach equilibrium, NIVS was used to impose an |
| 323 |
> |
unphysical thermal flux between the metal and the liquid phases. Most |
| 324 |
> |
of our simulations were done under an average temperature of |
| 325 |
> |
$\sim$200K. Therefore, thermal flux usually came from the metal to the |
| 326 |
|
liquid so that the liquid has a higher temperature and would not |
| 327 |
< |
freeze due to excessively low temperature. After this induced |
| 328 |
< |
temperature gradient is stablized, the temperature profile of the |
| 329 |
< |
simulation cell is recorded. To do this, the simulation cell is |
| 330 |
< |
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 |
| 327 |
> |
freeze due to lowered temperatures. After this induced temperature |
| 328 |
> |
gradient had stablized, the temperature profile of the simulation cell |
| 329 |
> |
was recorded. To do this, the simulation cell is devided evenly into |
| 330 |
> |
$N$ slabs along the $z$-axis. The average temperatures of each slab |
| 331 |
|
are recorded for 1$\sim$2 ns. When the slab width $d$ of each slab is |
| 332 |
|
the same, the derivatives of $T$ with respect to slab number $n$ can |
| 333 |
< |
be directly used for $G^\prime$ calculations: |
| 334 |
< |
\begin{equation} |
| 334 |
< |
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
| 333 |
> |
be directly used for $G^\prime$ calculations: \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 |
| 336 |
|
= |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big| |
| 337 |
|
\Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\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. |
| 343 |
> |
All of the above simulation procedures use a time step of 1 fs. Each |
| 344 |
> |
equilibration stage took a minimum of 100 ps, although in some cases, |
| 345 |
> |
longer equilibration stages were utilized. |
| 346 |
|
|
| 347 |
|
\subsection{Force Field Parameters} |
| 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 |
| 348 |
> |
Our simulations include a number of chemically distinct components. |
| 349 |
> |
Figure \ref{demoMol} demonstrates the sites defined for both |
| 350 |
> |
United-Atom and All-Atom models of the organic solvent and capping |
| 351 |
> |
agents in our simulations. Force field parameters are needed for |
| 352 |
|
interactions both between the same type of particles and between |
| 353 |
|
particles of different species. |
| 354 |
|
|
| 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}.} |
| 361 |
> |
Refs. \protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes,TraPPE-UA.thiols} |
| 362 |
> |
(UA) and \protect\cite{OPLSAA} (AA). Cross-interactions with the Au |
| 363 |
> |
atoms are given 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}. |
| 371 |
> |
Sutton-Chen potentials.\cite{Chen90} |
| 372 |
|
|
| 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 |
| 373 |
> |
For the two solvent molecules, {\it n}-hexane and toluene, two |
| 374 |
> |
different atomistic models were utilized. Both solvents were modeled |
| 375 |
> |
using United-Atom (UA) and All-Atom (AA) models. The TraPPE-UA |
| 376 |
|
parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
| 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] |
| 380 |
> |
sites closer than 3 bonds. For non-bonded interactions, Lennard-Jones |
| 381 |
> |
potentials are used. |
| 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. |
| 383 |
> |
By eliminating explicit hydrogen atoms, the TraPPE-UA models are |
| 384 |
> |
simple and computationally efficient, while maintaining good accuracy. |
| 385 |
> |
However, the TraPPE-UA model for alkanes is known to predict a slighly |
| 386 |
> |
lower boiling point than experimental values. This is one of the |
| 387 |
> |
reasons we used a lower average temperature (200K) for our |
| 388 |
> |
simulations. If heat is transferred to the liquid phase during the |
| 389 |
> |
NIVS simulation, the liquid in the hot slab can actually be |
| 390 |
> |
substantially warmer than the mean temperature in the simulation. The |
| 391 |
> |
lower mean temperatures therefore prevent solvent boiling. |
| 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. |
| 393 |
> |
For UA-toluene, the non-bonded potentials between intermolecular sites |
| 394 |
> |
have a similar Lennard-Jones formulation. The toluene molecules were |
| 395 |
> |
treated as a single rigid body, so there was no need for |
| 396 |
> |
intramolecular interactions (including bonds, bends, or torsions) in |
| 397 |
> |
this solvent model. |
| 398 |
|
|
| 399 |
|
Besides the TraPPE-UA models, AA models for both organic solvents are |
| 400 |
< |
included in our studies as well. For hexane, the OPLS-AA\cite{OPLSAA} |
| 401 |
< |
force field is used. Additional explicit hydrogen sites were |
| 400 |
> |
included in our studies as well. The OPLS-AA\cite{OPLSAA} force fields |
| 401 |
> |
were used. For hexane, additional explicit hydrogen sites were |
| 402 |
|
included. Besides bonding and non-bonded site-site interactions, |
| 403 |
|
partial charges and the electrostatic interactions were added to each |
| 404 |
< |
CT and HC site. For toluene, the United Force Field developed by |
| 405 |
< |
Rapp\'{e} {\it et al.}\cite{doi:10.1021/ja00051a040} is |
| 406 |
< |
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] |
| 404 |
> |
CT and HC site. For toluene, a flexible model for the toluene molecule |
| 405 |
> |
was utilized which included bond, bend, torsion, and inversion |
| 406 |
> |
potentials to enforce ring planarity. |
| 407 |
|
|
| 408 |
< |
The capping agent in our simulations, the butanethiol molecules can |
| 409 |
< |
either use UA or AA model. The TraPPE-UA force fields includes |
| 408 |
> |
The butanethiol capping agent in our simulations, were also modeled |
| 409 |
> |
with both UA and AA model. The TraPPE-UA force field includes |
| 410 |
|
parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for |
| 411 |
|
UA butanethiol model in our simulations. The OPLS-AA also provides |
| 412 |
|
parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) |
| 413 |
< |
surfaces do not have the hydrogen atom bonded to sulfur. To adapt this |
| 414 |
< |
change and derive suitable parameters for butanethiol adsorbed on |
| 415 |
< |
Au(111) surfaces, we adopt the S parameters from Luedtke and |
| 416 |
< |
Landman\cite{landman:1998}[CHECK CITATION] |
| 417 |
< |
and modify parameters for its neighbor C |
| 418 |
< |
atom for charge balance in the molecule. Note that the model choice |
| 419 |
< |
(UA or AA) of capping agent can be different from the |
| 420 |
< |
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: |
| 413 |
> |
surfaces do not have the hydrogen atom bonded to sulfur. To derive |
| 414 |
> |
suitable parameters for butanethiol adsorbed on Au(111) surfaces, we |
| 415 |
> |
adopt the S parameters from Luedtke and Landman\cite{landman:1998} and |
| 416 |
> |
modify the parameters for the CTS atom to maintain charge neutrality |
| 417 |
> |
in the molecule. Note that the model choice (UA or AA) for the capping |
| 418 |
> |
agent can be different from the solvent. Regardless of model choice, |
| 419 |
> |
the force field parameters for interactions between capping agent and |
| 420 |
> |
solvent can be derived using Lorentz-Berthelot Mixing Rule: |
| 421 |
|
\begin{eqnarray} |
| 422 |
< |
\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\ |
| 423 |
< |
\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}} |
| 422 |
> |
\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\ |
| 423 |
> |
\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}} |
| 424 |
|
\end{eqnarray} |
| 425 |
|
|
| 426 |
< |
To describe the interactions between metal Au and non-metal capping |
| 427 |
< |
agent and solvent particles, we refer to an adsorption study of alkyl |
| 428 |
< |
thiols on gold surfaces by Vlugt {\it et |
| 429 |
< |
al.}\cite{vlugt:cpc2007154} They fitted an effective Lennard-Jones |
| 430 |
< |
form of potential parameters for the interaction between Au and |
| 431 |
< |
pseudo-atoms CH$_x$ and S based on a well-established and widely-used |
| 432 |
< |
effective potential of Hautman and Klein\cite{hautman:4994} for the |
| 433 |
< |
Au(111) surface. As our simulations require the gold lattice slab to |
| 434 |
< |
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. |
| 426 |
> |
To describe the interactions between metal (Au) and non-metal atoms, |
| 427 |
> |
we refer to an adsorption study of alkyl thiols on gold surfaces by |
| 428 |
> |
Vlugt {\it et al.}\cite{vlugt:cpc2007154} They fitted an effective |
| 429 |
> |
Lennard-Jones form of potential parameters for the interaction between |
| 430 |
> |
Au and pseudo-atoms CH$_x$ and S based on a well-established and |
| 431 |
> |
widely-used effective potential of Hautman and Klein for the Au(111) |
| 432 |
> |
surface.\cite{hautman:4994} As our simulations require the gold slab |
| 433 |
> |
to be flexible to accommodate thermal excitation, the pair-wise form |
| 434 |
> |
of potentials they developed was used for our study. |
| 435 |
|
|
| 436 |
< |
Besides, the potentials developed from {\it ab initio} calculations by |
| 437 |
< |
Leng {\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
| 438 |
< |
interactions between Au and aromatic C/H atoms in toluene. A set of |
| 439 |
< |
pseudo Lennard-Jones parameters were provided for Au in their force |
| 440 |
< |
fields. By using the Mixing Rule, this can be used to derive pair-wise |
| 441 |
< |
potentials for non-bonded interactions between Au and non-metal sites. |
| 436 |
> |
The potentials developed from {\it ab initio} calculations by Leng |
| 437 |
> |
{\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
| 438 |
> |
interactions between Au and aromatic C/H atoms in toluene. However, |
| 439 |
> |
the Lennard-Jones parameters between Au and other types of particles, |
| 440 |
> |
(e.g. AA alkanes) have not yet been established. For these |
| 441 |
> |
interactions, the Lorentz-Berthelot mixing rule can be used to derive |
| 442 |
> |
effective single-atom LJ parameters for the metal using the fit values |
| 443 |
> |
for toluene. These are then used to construct reasonable mixing |
| 444 |
> |
parameters for the interactions between the gold and other atoms. |
| 445 |
> |
Table \ref{MnM} summarizes the ``metal/non-metal'' parameters used in |
| 446 |
> |
our simulations. |
| 447 |
|
|
| 451 |
– |
However, the Lennard-Jones parameters between Au and other types of |
| 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 |
– |
|
| 448 |
|
\begin{table*} |
| 449 |
|
\begin{minipage}{\linewidth} |
| 450 |
|
\begin{center} |
| 480 |
|
\end{minipage} |
| 481 |
|
\end{table*} |
| 482 |
|
|
| 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} |
| 483 |
|
|
| 484 |
< |
\section{Results and Discussions} |
| 485 |
< |
In what follows, how the parameters and protocol of simulations would |
| 486 |
< |
affect the measurement of $G$'s is first discussed. With a reliable |
| 487 |
< |
protocol and set of parameters, the influence of capping agent |
| 488 |
< |
coverage on thermal conductance is investigated. Besides, different |
| 489 |
< |
force field models for both solvents and selected deuterated models |
| 490 |
< |
were tested and compared. Finally, a summary of the role of capping |
| 491 |
< |
agent in the interfacial thermal transport process is given. |
| 484 |
> |
\section{Results} |
| 485 |
> |
There are many factors contributing to the measured interfacial |
| 486 |
> |
conductance; some of these factors are physically motivated |
| 487 |
> |
(e.g. coverage of the surface by the capping agent coverage and |
| 488 |
> |
solvent identity), while some are governed by parameters of the |
| 489 |
> |
methodology (e.g. applied flux and the formulas used to obtain the |
| 490 |
> |
conductance). In this section we discuss the major physical and |
| 491 |
> |
calculational effects on the computed conductivity. |
| 492 |
|
|
| 493 |
< |
\subsection{How Simulation Parameters Affects $G$} |
| 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 |
| 527 |
< |
affect the results substantially, some other changes to the |
| 528 |
< |
simulations would have a significant impact on the measurement |
| 529 |
< |
results. |
| 493 |
> |
\subsection{Effects due to capping agent coverage} |
| 494 |
|
|
| 495 |
< |
In some of our simulations, we allowed $L_x$ and $L_y$ to change |
| 496 |
< |
during equilibrating the liquid phase. Due to the stiffness of the |
| 497 |
< |
crystalline Au structure, $L_x$ and $L_y$ would not change noticeably |
| 498 |
< |
after equilibration. Although $L_z$ could fluctuate $\sim$1\% after a |
| 499 |
< |
system is fully equilibrated in the NPT ensemble, this fluctuation, as |
| 500 |
< |
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. |
| 495 |
> |
A series of different initial conditions with a range of surface |
| 496 |
> |
coverages was prepared and solvated with various with both of the |
| 497 |
> |
solvent molecules. These systems were then equilibrated and their |
| 498 |
> |
interfacial thermal conductivity was measured with the NIVS |
| 499 |
> |
algorithm. Figure \ref{coverage} demonstrates the trend of conductance |
| 500 |
> |
with respect to surface coverage. |
| 501 |
|
|
| 502 |
+ |
\begin{figure} |
| 503 |
+ |
\includegraphics[width=\linewidth]{coverage} |
| 504 |
+ |
\caption{Comparison of interfacial thermal conductivity ($G$) values |
| 505 |
+ |
for the Au-butanethiol/solvent interface with various UA models and |
| 506 |
+ |
different capping agent coverages at $\langle T\rangle\sim$200K.} |
| 507 |
+ |
\label{coverage} |
| 508 |
+ |
\end{figure} |
| 509 |
+ |
|
| 510 |
+ |
In partially covered surfaces, the derivative definition for $G^\prime$ (Eq. \ref{derivativeG}) becomes difficult to apply, as the location of maximum change of $\lambda$ becomes washed out. The discrete definition (Eq. \ref{discreteG}) is easier to apply, as the Gibbs dividing surface is still well-defined. Therefore, $G$ (not $G^\prime$) was used in this section. |
| 511 |
+ |
|
| 512 |
+ |
From Figure \ref{coverage}, one can see the significance of the presence of capping agents. When even a small fraction of the Au(111) surface sites are covered with butanethiols, the conductivity exhibits an enhancement by at least a factor of 3. Cappping agents are clearly playing a major role in thermal transport at metal / organic solvent surfaces. |
| 513 |
+ |
|
| 514 |
+ |
We note a non-monotonic behavior in the interfacial conductance as a function of surface coverage. The maximum conductance (largest $G$) happens when the surfaces are about 75\% covered with butanethiol caps. The reason for this behavior is not entirely clear. One explanation is that incomplete butanethiol coverage allows small gaps between butanethiols to form. These gaps can be filled by transient solvent molecules. These solvent molecules couple very strongly with the hot capping agent molecules near the surface, and can then carry away (diffusively) the excess thermal energy from the surface. |
| 515 |
+ |
|
| 516 |
+ |
There appears to be a competition between the conduction of the thermal energy away from the surface by the capping agents (enhanced by greater coverage) and the coupling of the capping agents with the solvent (enhanced by interdigitation at lower coverages). This competition would lead to the non-monotonic coverage behavior observed here. |
| 517 |
+ |
|
| 518 |
+ |
Results for rigid body toluene solvent, as well as the UA hexane, are within the ranges expected from prior experimental work.\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406} This suggests that explicit hydrogen atoms might not be required for modeling thermal transport in these systems. C-H vibrational modes do not see significant excited state population at low temperatures, and are not likely to carry lower frequency excitations from the solid layer into the bulk liquid. |
| 519 |
+ |
|
| 520 |
+ |
The toluene solvent does not exhibit the same behavior as hexane in that $G$ remains at approximately the same magnitude when the capping coverage increases from 25\% to 75\%. Toluene, as a rigid planar molecule, cannot occupy the relatively small gaps between the capping agents as easily as the chain-like {\it n}-hexane. The effect of solvent coupling to the capping agent is therefore weaker in toluene except at the very lowest coverage levels. This effect counters the coverage-dependent conduction of heat away from the metal surface, leading to a much flatter $G$ vs. coverage trend than is observed in {\it n}-hexane. |
| 521 |
+ |
|
| 522 |
+ |
\subsection{Effects due to Solvent \& Solvent Models} |
| 523 |
+ |
In addition to UA solvent and capping agent models, AA models have also been included in our simulations. In most of this work, the same (UA or AA) model for solvent and capping agent was used, but it is also possible to utilize different models for different components. We have also included isotopic substitutions (Hydrogen to Deuterium) to decrease the explicit vibrational overlap between solvent and capping agent. Table \ref{modelTest} summarizes the results of these studies. |
| 524 |
+ |
|
| 525 |
+ |
\begin{table*} |
| 526 |
+ |
\begin{minipage}{\linewidth} |
| 527 |
+ |
\begin{center} |
| 528 |
+ |
|
| 529 |
+ |
\caption{Computed interfacial thermal conductance ($G$ and |
| 530 |
+ |
$G^\prime$) values for interfaces using various models for |
| 531 |
+ |
solvent and capping agent (or without capping agent) at |
| 532 |
+ |
$\langle T\rangle\sim$200K. (D stands for deuterated solvent |
| 533 |
+ |
or capping agent molecules; ``Avg.'' denotes results that are |
| 534 |
+ |
averages of simulations under different applied thermal flux values $(J_z)$. Error |
| 535 |
+ |
estimates are indicated in parentheses.)} |
| 536 |
+ |
|
| 537 |
+ |
\begin{tabular}{llccc} |
| 538 |
+ |
\hline\hline |
| 539 |
+ |
Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ |
| 540 |
+ |
(or bare surface) & model & (GW/m$^2$) & |
| 541 |
+ |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 542 |
+ |
\hline |
| 543 |
+ |
UA & UA hexane & Avg. & 131(9) & 87(10) \\ |
| 544 |
+ |
& UA hexane(D) & 1.95 & 153(5) & 136(13) \\ |
| 545 |
+ |
& AA hexane & Avg. & 131(6) & 122(10) \\ |
| 546 |
+ |
& UA toluene & 1.96 & 187(16) & 151(11) \\ |
| 547 |
+ |
& AA toluene & 1.89 & 200(36) & 149(53) \\ |
| 548 |
+ |
\hline |
| 549 |
+ |
AA & UA hexane & 1.94 & 116(9) & 129(8) \\ |
| 550 |
+ |
& AA hexane & Avg. & 442(14) & 356(31) \\ |
| 551 |
+ |
& AA hexane(D) & 1.93 & 222(12) & 234(54) \\ |
| 552 |
+ |
& UA toluene & 1.98 & 125(25) & 97(60) \\ |
| 553 |
+ |
& AA toluene & 3.79 & 487(56) & 290(42) \\ |
| 554 |
+ |
\hline |
| 555 |
+ |
AA(D) & UA hexane & 1.94 & 158(25) & 172(4) \\ |
| 556 |
+ |
& AA hexane & 1.92 & 243(29) & 191(11) \\ |
| 557 |
+ |
& AA toluene & 1.93 & 364(36) & 322(67) \\ |
| 558 |
+ |
\hline |
| 559 |
+ |
bare & UA hexane & Avg. & 46.5(3.2) & 49.4(4.5) \\ |
| 560 |
+ |
& UA hexane(D) & 0.98 & 43.9(4.6) & 43.0(2.0) \\ |
| 561 |
+ |
& AA hexane & 0.96 & 31.0(1.4) & 29.4(1.3) \\ |
| 562 |
+ |
& UA toluene & 1.99 & 70.1(1.3) & 65.8(0.5) \\ |
| 563 |
+ |
\hline\hline |
| 564 |
+ |
\end{tabular} |
| 565 |
+ |
\label{modelTest} |
| 566 |
+ |
\end{center} |
| 567 |
+ |
\end{minipage} |
| 568 |
+ |
\end{table*} |
| 569 |
+ |
|
| 570 |
+ |
To facilitate direct comparison between force fields, systems with the same capping agent and solvent were prepared with the same length scales for the simulation cells. |
| 571 |
+ |
|
| 572 |
+ |
On bare metal / solvent surfaces, different force field models for hexane yield similar results for both $G$ and $G^\prime$, and these two definitions agree with each other very well. This is primarily an indicator of weak interactions between the metal and the solvent, and is a typical case for acoustic impedance mismatch between these two phases. |
| 573 |
+ |
|
| 574 |
+ |
For the fully-covered surfaces, the choice of force field for the capping agent and solvent has a large impact on the calulated values of $G$ and $G^\prime$. The AA thiol to AA solvent conductivities are much larger than their UA to UA counterparts, and these values exceed the experimental estimates by a large measure. The AA force field allows significant energy to go into C-H (or C-D) stretching modes, and since these modes are high frequency, this non-quantum behavior is likely responsible for the overestimate of the conductivity. |
| 575 |
+ |
|
| 576 |
+ |
The similarity in the vibrational modes available to solvent and capping agent can be reduced by deuterating one of the two components. Once either the hexanes or the butanethiols are deuterated, one can see a significantly lower $G$ and $G^\prime$ (Figure \ref{aahxntln}). Compared to the AA model, the UA model yields more reasonable conductivity values with much higher computational efficiency. |
| 577 |
+ |
|
| 578 |
+ |
\begin{figure} |
| 579 |
+ |
\includegraphics[width=\linewidth]{aahxntln} |
| 580 |
+ |
\caption{Spectra obtained for all-atom (AA) Au / butanethiol / solvent |
| 581 |
+ |
systems. When butanethiol is deuterated (lower left), its |
| 582 |
+ |
vibrational overlap with hexane decreases significantly. Since aromatic molecules and the butanethiol are vibrationally dissimilar, the change is not as dramatic when toluene is the solvent (right).} |
| 583 |
+ |
\label{aahxntln} |
| 584 |
+ |
\end{figure} |
| 585 |
+ |
|
| 586 |
+ |
For the Au / butanethiol / toluene interfaces, having the AA butanethiol deuterated did not yield a significant change in the measured conductance. Compared to the C-H vibrational overlap between hexane and butanethiol, both of which have alkyl chains, the overlap between toluene and butanethiol is not as significant and thus does not contribute as much to the heat exchange process. The presence of extra degrees of freedom in the AA force field for toluene yields higher heat exchange rates between the two phases and results in a much higher conductivity than in the UA force field. |
| 587 |
+ |
|
| 588 |
+ |
\subsubsection{Are electronic excitations in the metal important?} |
| 589 |
+ |
Because they lack electronic excitations, the QSC and related embedded atom method (EAM) models for gold are known to predict unreasonably low values for bulk conductivity ($\lambda$).\cite{kuang:164101,ISI:000207079300006,Clancy:1992} If the conductance between the phases ($G$) is governed primarily by phonon excitation (and not electronic degrees of freedom), one would expect a classical model to capture most of the interfacial thermal conductance. Our results for $G$ and $G^\prime$ indicate that this is indeed the case, and suggest that the modeling of interfacial thermal transport depends primarily on the description of the interactions between the various components at the interface. When the metal is chemically capped, the primary barrier to thermal conductivity appears to be the interface between the capping agent and the surrounding solvent, so the excitations in the metal have little impact on the value of $G$. |
| 590 |
+ |
|
| 591 |
+ |
\subsection{Effects due to methodology and simulation parameters} |
| 592 |
+ |
|
| 593 |
+ |
START HERE |
| 594 |
+ |
|
| 595 |
+ |
We have varied our protocol or other parameters of the simulations in order to investigate how these factors would affect the computation of $G$. |
| 596 |
+ |
|
| 597 |
+ |
We allowed $L_x$ and $L_y$ to change during equilibrating the liquid phase. Due to the stiffness of the crystalline Au structure, $L_x$ and $L_y$ would not change noticeably after equilibration. Although $L_z$ could fluctuate $\sim$1\% after a system is fully equilibrated in the NPT ensemble, this fluctuation, as well as those of $L_x$ and $L_y$ (which is significantly smaller), would not be magnified on the calculated $G$'s, as shown in Table \ref{AuThiolHexaneUA}. This insensivity to $L_i$ fluctuations allows reliable measurement of $G$'s without the necessity of extremely cautious equilibration process. |
| 598 |
+ |
|
| 599 |
|
As stated in our computational details, the spacing filled with |
| 600 |
|
solvent molecules can be chosen within a range. This allows some |
| 601 |
|
change of solvent molecule numbers for the same Au-butanethiol |
| 606 |
|
smaller system size would be preferable, given that the liquid phase |
| 607 |
|
structure is not affected. |
| 608 |
|
|
| 609 |
+ |
\subsubsection{Effects of applied flux} |
| 610 |
|
Our NIVS algorithm allows change of unphysical thermal flux both in |
| 611 |
|
direction and in quantity. This feature extends our investigation of |
| 612 |
|
interfacial thermal conductance. However, the magnitude of this |
| 670 |
|
\end{minipage} |
| 671 |
|
\end{table*} |
| 672 |
|
|
| 673 |
+ |
\subsubsection{Effects due to average temperature} |
| 674 |
+ |
|
| 675 |
|
Furthermore, we also attempted to increase system average temperatures |
| 676 |
|
to above 200K. These simulations are first equilibrated in the NPT |
| 677 |
|
ensemble under normal pressure. As stated above, the TraPPE-UA model |
| 756 |
|
interfaces even at a relatively high temperature without being |
| 757 |
|
affected by surface reconstructions. |
| 758 |
|
|
| 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. 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. |
| 759 |
|
|
| 760 |
< |
\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} |
| 760 |
> |
\section{Discussion} |
| 761 |
|
|
| 762 |
< |
It turned out that with partial covered butanethiol on the Au(111) |
| 763 |
< |
surface, the derivative definition for $G^\prime$ |
| 764 |
< |
(Eq. \ref{derivativeG}) was difficult to apply, due to the difficulty |
| 765 |
< |
in locating the maximum of change of $\lambda$. Instead, the discrete |
| 766 |
< |
definition (Eq. \ref{discreteG}) is easier to apply, as the Gibbs |
| 767 |
< |
deviding surface can still be well-defined. Therefore, $G$ (not |
| 768 |
< |
$G^\prime$) was used for this section. |
| 762 |
> |
\subsection{Capping agent acts as a vibrational coupler between solid |
| 763 |
> |
and solvent phases} |
| 764 |
> |
To investigate the mechanism of interfacial thermal conductance, the |
| 765 |
> |
vibrational power spectrum was computed. Power spectra were taken for |
| 766 |
> |
individual components in different simulations. To obtain these |
| 767 |
> |
spectra, simulations were run after equilibration, in the NVE |
| 768 |
> |
ensemble, and without a thermal gradient. Snapshots of configurations |
| 769 |
> |
were collected at a frequency that is higher than that of the fastest |
| 770 |
> |
vibrations occuring in the simulations. With these configurations, the |
| 771 |
> |
velocity auto-correlation functions can be computed: |
| 772 |
> |
\begin{equation} |
| 773 |
> |
C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle |
| 774 |
> |
\label{vCorr} |
| 775 |
> |
\end{equation} |
| 776 |
> |
The power spectrum is constructed via a Fourier transform of the |
| 777 |
> |
symmetrized velocity autocorrelation function, |
| 778 |
> |
\begin{equation} |
| 779 |
> |
\hat{f}(\omega) = |
| 780 |
> |
\int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt |
| 781 |
> |
\label{fourier} |
| 782 |
> |
\end{equation} |
| 783 |
|
|
| 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. |
| 784 |
|
|
| 785 |
< |
Interestingly, as one could observe from our results, the maximum |
| 735 |
< |
conductance enhancement (largest $G$) happens while the surfaces are |
| 736 |
< |
about 75\% covered with butanethiols. This again indicates that |
| 737 |
< |
solvent-capping agent contact has an important role of the thermal |
| 738 |
< |
transport process. Slightly lower butanethiol coverage allows small |
| 739 |
< |
gaps between butanethiols to form. And these gaps could be filled with |
| 740 |
< |
solvent molecules, which acts like ``heat conductors'' on the |
| 741 |
< |
surface. The higher degree of interaction between these solvent |
| 742 |
< |
molecules and capping agents increases the enhancement effect and thus |
| 743 |
< |
produces a higher $G$ than densely packed butanethiol arrays. However, |
| 744 |
< |
once this maximum conductance enhancement is reached, $G$ decreases |
| 745 |
< |
when butanethiol coverage continues to decrease. Each capping agent |
| 746 |
< |
molecule reaches its maximum capacity for thermal |
| 747 |
< |
conductance. Therefore, even higher solvent-capping agent contact |
| 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. [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 |
| 756 |
< |
transport process. The deuterated hexane (UA) results do not appear to |
| 757 |
< |
be much different from those of normal hexane (UA), given that |
| 758 |
< |
butanethiol (UA) is non-deuterated for both solvents. These UA model |
| 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 $G$ (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\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 $G$ 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 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 |
| 788 |
< |
increasing until the capping agent coverage reaches a relatively low |
| 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. |
| 792 |
< |
|
| 793 |
< |
\subsection{Influence of Chosen Molecule Model on $G$} |
| 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 |
| 797 |
< |
to different components. Furthermore, regardless of models chosen, |
| 798 |
< |
either the solvent or the capping agent can be deuterated, similar to |
| 799 |
< |
the previous section. Table \ref{modelTest} summarizes the results of |
| 800 |
< |
these studies. |
| 801 |
< |
|
| 802 |
< |
\begin{table*} |
| 803 |
< |
\begin{minipage}{\linewidth} |
| 804 |
< |
\begin{center} |
| 805 |
< |
|
| 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. (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}{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 & 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} |
| 843 |
< |
\end{center} |
| 844 |
< |
\end{minipage} |
| 845 |
< |
\end{table*} |
| 846 |
< |
|
| 847 |
< |
To facilitate direct comparison, the same system with differnt models |
| 848 |
< |
for different components uses the same length scale for their |
| 849 |
< |
simulation cells. Without the presence of capping agent, using |
| 850 |
< |
different models for hexane yields similar results for both $G$ and |
| 851 |
< |
$G^\prime$, and these two definitions agree with eath other very |
| 852 |
< |
well. This indicates very weak interaction between the metal and the |
| 853 |
< |
solvent, and is a typical case for acoustic impedance mismatch between |
| 854 |
< |
these two phases. |
| 855 |
< |
|
| 856 |
< |
As for Au(111) surfaces completely covered by butanethiols, the choice |
| 857 |
< |
of models for capping agent and solvent could impact the measurement |
| 858 |
< |
of $G$ and $G^\prime$ quite significantly. For Au-butanethiol/hexane |
| 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 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. 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 |
< |
|
| 894 |
< |
Although the QSC model for Au is known to predict an overly low value |
| 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, 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{Role of Capping Agent in Interfacial Thermal Conductance} |
| 785 |
> |
\subsubsection{The role of specific vibrations} |
| 786 |
|
The vibrational spectra for gold slabs in different environments are |
| 787 |
|
shown as in Figure \ref{specAu}. Regardless of the presence of |
| 788 |
|
solvent, the gold surfaces covered by butanethiol molecules, compared |
| 793 |
|
simulations, the Au/S interfaces do not appear major heat barriers |
| 794 |
|
compared to the butanethiol / solvent interfaces. |
| 795 |
|
|
| 796 |
+ |
\subsubsection{Overlap of power spectrum} |
| 797 |
|
Simultaneously, the vibrational overlap between butanethiol and |
| 798 |
|
organic solvents suggests higher thermal exchange efficiency between |
| 799 |
|
these two components. Even exessively high heat transport was observed |
| 817 |
|
\label{specAu} |
| 818 |
|
\end{figure} |
| 819 |
|
|
| 820 |
< |
[MAY ADD COMPARISON OF AU SLAB WIDTHS] |
| 820 |
> |
\subsubsection{Isotopic substitution and vibrational overlap} |
| 821 |
> |
A comparison of the results obtained from the two different organic |
| 822 |
> |
solvents can also provide useful information of the interfacial |
| 823 |
> |
thermal transport process. The deuterated hexane (UA) results do not |
| 824 |
> |
appear to be substantially different from those of normal hexane (UA), |
| 825 |
> |
given that butanethiol (UA) is non-deuterated for both solvents. The |
| 826 |
> |
UA models, even though they have eliminated C-H vibrational overlap, |
| 827 |
> |
still have significant overlap in the infrared spectra. Because |
| 828 |
> |
differences in the infrared range do not seem to produce an observable |
| 829 |
> |
difference for the results of $G$ (Figure \ref{uahxnua}). |
| 830 |
|
|
| 831 |
+ |
\begin{figure} |
| 832 |
+ |
\includegraphics[width=\linewidth]{uahxnua} |
| 833 |
+ |
\caption{Vibrational spectra obtained for normal (upper) and |
| 834 |
+ |
deuterated (lower) hexane in Au-butanethiol/hexane |
| 835 |
+ |
systems. Butanethiol spectra are shown as reference. Both hexane and |
| 836 |
+ |
butanethiol were using United-Atom models.} |
| 837 |
+ |
\label{uahxnua} |
| 838 |
+ |
\end{figure} |
| 839 |
+ |
|
| 840 |
|
\section{Conclusions} |
| 841 |
|
The NIVS algorithm we developed has been applied to simulations of |
| 842 |
|
Au-butanethiol surfaces with organic solvents. This algorithm allows |
| 876 |
|
|
| 877 |
|
Vlugt {\it et al.} has investigated the surface thiol structures for |
| 878 |
|
nanocrystal gold and pointed out that they differs from those of the |
| 879 |
< |
Au(111) surface\cite{vlugt:cpc2007154}. This difference might lead to |
| 880 |
< |
change of interfacial thermal transport behavior as well. To |
| 881 |
< |
investigate this problem, an effective means to introduce thermal flux |
| 882 |
< |
and measure the corresponding thermal gradient is desirable for |
| 883 |
< |
simulating structures with spherical symmetry. |
| 879 |
> |
Au(111) surface\cite{landman:1998,vlugt:cpc2007154}. This difference |
| 880 |
> |
might lead to change of interfacial thermal transport behavior as |
| 881 |
> |
well. To investigate this problem, an effective means to introduce |
| 882 |
> |
thermal flux and measure the corresponding thermal gradient is |
| 883 |
> |
desirable for simulating structures with spherical symmetry. |
| 884 |
|
|
| 885 |
|
\section{Acknowledgments} |
| 886 |
|
Support for this project was provided by the National Science |
| 887 |
|
Foundation under grant CHE-0848243. Computational time was provided by |
| 888 |
|
the Center for Research Computing (CRC) at the University of Notre |
| 889 |
< |
Dame. \newpage |
| 889 |
> |
Dame. |
| 890 |
> |
\newpage |
| 891 |
|
|
| 892 |
|
\bibliography{interfacial} |
| 893 |
|
|
