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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 |
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importance in nanoscale science and technology. Reliability of |
| 79 |
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nanoscale devices depends on their thermal transport |
| 80 |
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properties. Unlike bulk homogeneous materials, nanoscale materials |
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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 |
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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, |
| 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 |
| 83 |
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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 |
| 86 |
< |
affected by the chemical details of the surface. Experimentally, |
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various interfaces have been investigated for their thermal |
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conductance properties. Wang {\it et al.} studied heat transport |
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through long-chain hydrocarbon monolayers on gold substrate at |
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individual molecular level\cite{Wang10082007}; Schmidt {\it et al.} |
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studied the role of CTAB on thermal transport between gold nanorods |
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and solvent\cite{doi:10.1021/jp8051888}; Juv\'e {\it et al.} studied |
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Experimentally, various interfaces have been investigated for their |
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thermal conductance. Cahill and coworkers studied nanoscale thermal |
| 87 |
> |
transport from metal nanoparticle/fluid interfaces, to epitaxial |
| 88 |
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TiN/single crystal oxides interfaces, to hydrophilic and hydrophobic |
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interfaces between water and solids with different self-assembled |
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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 |
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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 |
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role of CTAB on thermal transport between gold nanorods and |
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solvent,\cite{doi:10.1021/jp8051888} and Juv\'e {\it et al.} studied |
| 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 |
| 99 |
> |
normally considered barriers for heat transport, Alper {\it et al.} |
| 100 |
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suggested that specific ligands (capping agents) could completely |
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eliminate this barrier |
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($G\rightarrow\infty$).\cite{doi:10.1021/la904855s} |
| 103 |
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|
| 104 |
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Theoretical and computational models have also been used to study the |
| 105 |
|
interfacial thermal transport in order to gain an understanding of |
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employed Non-Equilibrium Molecular Dynamics (NEMD) simulations to |
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study thermal transport from hot Au(111) substrate to a self-assembled |
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monolayer of alkylthiol with relatively long chain (8-20 carbon |
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< |
atoms)\cite{hase:2010,hase:2011}. However, ensemble averaged |
| 110 |
> |
atoms).\cite{hase:2010,hase:2011} However, ensemble averaged |
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|
measurements for heat conductance of interfaces between the capping |
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monolayer on Au and a solvent phase has yet to be studied. |
| 113 |
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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 |
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methods. Therefore, the Reverse NEMD (RNEMD) methods would have the |
| 116 |
< |
advantage of having this difficult to measure flux known when studying |
| 117 |
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the thermal transport across interfaces, given that the simulation |
| 118 |
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methods being able to effectively apply an unphysical flux in |
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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 |
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difficult to measure with Equilibrium |
| 115 |
> |
MD\cite{doi:10.1080/0026897031000068578} or forward NEMD simulation |
| 116 |
> |
methods. Therefore, the Reverse NEMD (RNEMD) |
| 117 |
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methods\cite{MullerPlathe:1997xw,kuang:164101} would have the |
| 118 |
> |
advantage of applying this difficult to measure flux (while measuring |
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> |
the resulting gradient), given that the simulation methods being able |
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to effectively apply an unphysical flux in non-homogeneous systems. |
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Garde and coworkers\cite{garde:nl2005,garde:PhysRevLett2009} applied |
| 122 |
> |
this approach to various liquid interfaces and studied how thermal |
| 123 |
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conductance (or resistance) is dependent on chemistry details of |
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> |
interfaces, e.g. hydrophobic and hydrophilic aqueous interfaces. |
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|
|
| 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 |
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|
retains the desirable features of RNEMD (conservation of linear |
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|
momentum and total energy, compatibility with periodic boundary |
| 138 |
|
properties. Different models were used for both the capping agent and |
| 139 |
|
the solvent force field parameters. Using the NIVS algorithm, the |
| 140 |
|
thermal transport across these interfaces was studied and the |
| 141 |
< |
underlying mechanism for this phenomena was investigated. |
| 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 |
< |
For systems with low interfacial conductivity one must have a method |
| 146 |
< |
capable of generating relatively small fluxes, compared to those |
| 147 |
< |
required for bulk conductivity. This requirement makes the calculation |
| 148 |
< |
even more difficult for those slowly-converging equilibrium |
| 149 |
< |
methods\cite{Viscardy:2007lq}. |
| 150 |
< |
Forward methods impose gradient, but in interfacial conditions it is |
| 151 |
< |
not clear what behavior to impose at the boundary... |
| 152 |
< |
Imposed-flux reverse non-equilibrium |
| 153 |
< |
methods\cite{MullerPlathe:1997xw} have the flux set {\it a priori} and |
| 154 |
< |
the thermal response becomes easier to |
| 155 |
< |
measure than the flux. Although M\"{u}ller-Plathe's original momentum |
| 156 |
< |
swapping approach can be used for exchanging energy between particles |
| 157 |
< |
of different identity, the kinetic energy transfer efficiency is |
| 158 |
< |
affected by the mass difference between the particles, which limits |
| 159 |
< |
its application on heterogeneous interfacial systems. |
| 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 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 |
< |
For interfaces with a relatively low interfacial conductance, the bulk |
| 183 |
< |
regions on either side of an interface rapidly come to a state in |
| 184 |
< |
which the two phases have relatively homogeneous (but distinct) |
| 185 |
< |
temperatures. The interfacial thermal conductivity $G$ can therefore |
| 186 |
< |
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. For an applied flux $J_z$ |
| 197 |
> |
operating over a simulation time $t$ on a periodically-replicated slab |
| 198 |
> |
of dimensions $L_x \times L_y$, $E_{total} = J_z *(t)*(2 L_x L_y)$. |
| 199 |
|
|
| 200 |
|
When the interfacial conductance is {\it not} small, there are two |
| 201 |
< |
ways to define $G$. |
| 202 |
< |
|
| 203 |
< |
One way is to assume the temperature is discrete on the two sides of |
| 204 |
< |
the interface. $G$ can be calculated using the applied thermal flux |
| 205 |
< |
$J$ and the maximum temperature difference measured along the thermal |
| 206 |
< |
gradient max($\Delta T$), which occurs at the Gibbs deviding surface |
| 207 |
< |
(Figure \ref{demoPic}): |
| 201 |
> |
ways to define $G$. One common way is to assume the temperature is |
| 202 |
> |
discrete on the two sides of the interface. $G$ can be calculated |
| 203 |
> |
using the applied thermal flux $J$ and the maximum temperature |
| 204 |
> |
difference measured along the thermal gradient max($\Delta T$), which |
| 205 |
> |
occurs at the Gibbs deviding surface (Figure \ref{demoPic}). This is |
| 206 |
> |
known as the Kapitza conductance, which is the inverse of the Kapitza |
| 207 |
> |
resistance. |
| 208 |
|
\begin{equation} |
| 209 |
< |
G=\frac{J}{\Delta T} |
| 209 |
> |
G=\frac{J}{\Delta T} |
| 210 |
|
\label{discreteG} |
| 211 |
|
\end{equation} |
| 212 |
|
|
| 226 |
|
|
| 227 |
|
The other approach is to assume a continuous temperature profile along |
| 228 |
|
the thermal gradient axis (e.g. $z$) and define $G$ at the point where |
| 229 |
< |
the magnitude of thermal conductivity $\lambda$ change reach its |
| 229 |
> |
the magnitude of thermal conductivity ($\lambda$) change reaches its |
| 230 |
|
maximum, given that $\lambda$ is well-defined throughout the space: |
| 231 |
|
\begin{equation} |
| 232 |
|
G^\prime = \Big|\frac{\partial\lambda}{\partial z}\Big| |
| 237 |
|
\label{derivativeG} |
| 238 |
|
\end{equation} |
| 239 |
|
|
| 240 |
< |
With the temperature profile obtained from simulations, one is able to |
| 240 |
> |
With temperature profiles obtained from simulation, one is able to |
| 241 |
|
approximate the first and second derivatives of $T$ with finite |
| 242 |
< |
difference methods and thus calculate $G^\prime$. |
| 242 |
> |
difference methods and calculate $G^\prime$. In what follows, both |
| 243 |
> |
definitions have been used, and are compared in the results. |
| 244 |
|
|
| 245 |
< |
In what follows, both definitions have been used for calculation and |
| 246 |
< |
are compared in the results. |
| 247 |
< |
|
| 248 |
< |
To compare the above definitions ($G$ and $G^\prime$), we have modeled |
| 249 |
< |
a metal slab with its (111) surfaces perpendicular to the $z$-axis of |
| 239 |
< |
our simulation cells. Both with and without capping agents on the |
| 240 |
< |
surfaces, the metal slab is solvated with simple organic solvents, as |
| 245 |
> |
To investigate the interfacial conductivity at metal / solvent |
| 246 |
> |
interfaces, we have modeled a metal slab with its (111) surfaces |
| 247 |
> |
perpendicular to the $z$-axis of our simulation cells. The metal slab |
| 248 |
> |
has been prepared both with and without capping agents on the exposed |
| 249 |
> |
surface, and has been solvated with simple organic solvents, as |
| 250 |
|
illustrated in Figure \ref{gradT}. |
| 251 |
|
|
| 252 |
|
With the simulation cell described above, we are able to equilibrate |
| 253 |
|
the system and impose an unphysical thermal flux between the liquid |
| 254 |
|
and the metal phase using the NIVS algorithm. By periodically applying |
| 255 |
< |
the unphysical flux, we are able to obtain a temperature profile and |
| 256 |
< |
its spatial derivatives. These quantities enable the evaluation of the |
| 257 |
< |
interfacial thermal conductance of a surface. Figure \ref{gradT} is an |
| 258 |
< |
example how those applied thermal fluxes can be used to obtain the 1st |
| 250 |
< |
and 2nd derivatives of the temperature profile. |
| 255 |
> |
the unphysical flux, we obtained a temperature profile and its spatial |
| 256 |
> |
derivatives. Figure \ref{gradT} shows how an applied thermal flux can |
| 257 |
> |
be used to obtain the 1st and 2nd derivatives of the temperature |
| 258 |
> |
profile. |
| 259 |
|
|
| 260 |
|
\begin{figure} |
| 261 |
|
\includegraphics[width=\linewidth]{gradT} |
| 269 |
|
\section{Computational Details} |
| 270 |
|
\subsection{Simulation Protocol} |
| 271 |
|
The NIVS algorithm has been implemented in our MD simulation code, |
| 272 |
< |
OpenMD\cite{Meineke:2005gd,openmd}, and was used for our |
| 273 |
< |
simulations. Different metal slab thickness (layer numbers of Au) were |
| 274 |
< |
simulated. Metal slabs were first equilibrated under atmospheric |
| 275 |
< |
pressure (1 atm) and a desired temperature (e.g. 200K). After |
| 276 |
< |
equilibration, butanethiol capping agents were placed at three-fold |
| 277 |
< |
sites on the Au(111) surfaces. The maximum butanethiol capacity on Au |
| 278 |
< |
surface is $1/3$ of the total number of surface Au |
| 279 |
< |
atoms\cite{vlugt:cpc2007154}[CITE CHEM REV]. |
| 280 |
< |
A series of different coverages was |
| 281 |
< |
investigated in order to study the relation between coverage and |
| 282 |
< |
interfacial conductance. |
| 272 |
> |
OpenMD\cite{Meineke:2005gd,openmd}, and was used for our simulations. |
| 273 |
> |
Metal slabs of 6 or 11 layers of Au atoms were first equilibrated |
| 274 |
> |
under atmospheric pressure (1 atm) and 200K. After equilibration, |
| 275 |
> |
butanethiol capping agents were placed at three-fold hollow sites on |
| 276 |
> |
the Au(111) surfaces. These sites are either {\it fcc} or {\it |
| 277 |
> |
hcp} sites, although Hase {\it et al.} found that they are |
| 278 |
> |
equivalent in a heat transfer process,\cite{hase:2010} so we did not |
| 279 |
> |
distinguish between these sites in our study. The maximum butanethiol |
| 280 |
> |
capacity on Au surface is $1/3$ of the total number of surface Au |
| 281 |
> |
atoms, and the packing forms a $(\sqrt{3}\times\sqrt{3})R30^\circ$ |
| 282 |
> |
structure\cite{doi:10.1021/ja00008a001,doi:10.1021/cr9801317}. A |
| 283 |
> |
series of lower coverages was also prepared by eliminating |
| 284 |
> |
butanethiols from the higher coverage surface in a regular manner. The |
| 285 |
> |
lower coverages were prepared in order to study the relation between |
| 286 |
> |
coverage and interfacial conductance. |
| 287 |
|
|
| 288 |
|
The capping agent molecules were allowed to migrate during the |
| 289 |
|
simulations. They distributed themselves uniformly and sampled a |
| 290 |
|
number of three-fold sites throughout out study. Therefore, the |
| 291 |
< |
initial configuration would not noticeably affect the sampling of a |
| 291 |
> |
initial configuration does not noticeably affect the sampling of a |
| 292 |
|
variety of configurations of the same coverage, and the final |
| 293 |
|
conductance measurement would be an average effect of these |
| 294 |
< |
configurations explored in the simulations. [MAY NEED SNAPSHOTS] |
| 294 |
> |
configurations explored in the simulations. |
| 295 |
|
|
| 296 |
< |
After the modified Au-butanethiol surface systems were equilibrated |
| 297 |
< |
under canonical ensemble, organic solvent molecules were packed in the |
| 298 |
< |
previously empty part of the simulation cells\cite{packmol}. Two |
| 296 |
> |
After the modified Au-butanethiol surface systems were equilibrated in |
| 297 |
> |
the canonical (NVT) ensemble, organic solvent molecules were packed in |
| 298 |
> |
the previously empty part of the simulation cells.\cite{packmol} Two |
| 299 |
|
solvents were investigated, one which has little vibrational overlap |
| 300 |
< |
with the alkanethiol and a planar shape (toluene), and one which has |
| 301 |
< |
similar vibrational frequencies and chain-like shape ({\it n}-hexane). |
| 300 |
> |
with the alkanethiol and which has a planar shape (toluene), and one |
| 301 |
> |
which has similar vibrational frequencies to the capping agent and |
| 302 |
> |
chain-like shape ({\it n}-hexane). |
| 303 |
|
|
| 304 |
< |
The space filled by solvent molecules, i.e. the gap between |
| 305 |
< |
periodically repeated Au-butanethiol surfaces should be carefully |
| 306 |
< |
chosen. A very long length scale for the thermal gradient axis ($z$) |
| 294 |
< |
may cause excessively hot or cold temperatures in the middle of the |
| 304 |
> |
The simulation cells were not particularly extensive along the |
| 305 |
> |
$z$-axis, as a very long length scale for the thermal gradient may |
| 306 |
> |
cause excessively hot or cold temperatures in the middle of the |
| 307 |
|
solvent region and lead to undesired phenomena such as solvent boiling |
| 308 |
|
or freezing when a thermal flux is applied. Conversely, too few |
| 309 |
|
solvent molecules would change the normal behavior of the liquid |
| 310 |
|
phase. Therefore, our $N_{solvent}$ values were chosen to ensure that |
| 311 |
< |
these extreme cases did not happen to our simulations. And the |
| 312 |
< |
corresponding spacing is usually $35[?] \sim 75$\AA. |
| 311 |
> |
these extreme cases did not happen to our simulations. The spacing |
| 312 |
> |
between periodic images of the gold interfaces is $45 \sim 75$\AA. |
| 313 |
|
|
| 314 |
|
The initial configurations generated are further equilibrated with the |
| 315 |
< |
$x$ and $y$ dimensions fixed, only allowing length scale change in $z$ |
| 316 |
< |
dimension. This is to ensure that the equilibration of liquid phase |
| 317 |
< |
does not affect the metal crystal structure in $x$ and $y$ dimensions. |
| 318 |
< |
To investigate this effect, comparisons were made with simulations |
| 319 |
< |
allowed to change $L_x$ and $L_y$ during NPT equilibration, and the |
| 320 |
< |
results are shown in later sections. After ensuring the liquid phase |
| 321 |
< |
reaches equilibrium at atmospheric pressure (1 atm), further |
| 322 |
< |
equilibration are followed under NVT and then NVE ensembles. |
| 315 |
> |
$x$ and $y$ dimensions fixed, only allowing the $z$-length scale to |
| 316 |
> |
change. This is to ensure that the equilibration of liquid phase does |
| 317 |
> |
not affect the metal's crystalline structure. Comparisons were made |
| 318 |
> |
with simulations that allowed changes of $L_x$ and $L_y$ during NPT |
| 319 |
> |
equilibration. No substantial changes in the box geometry were noticed |
| 320 |
> |
in these simulations. After ensuring the liquid phase reaches |
| 321 |
> |
equilibrium at atmospheric pressure (1 atm), further equilibration was |
| 322 |
> |
carried out under canonical (NVT) and microcanonical (NVE) ensembles. |
| 323 |
|
|
| 324 |
< |
After the systems reach equilibrium, NIVS is implemented to impose a |
| 325 |
< |
periodic unphysical thermal flux between the metal and the liquid |
| 326 |
< |
phase. Most of our simulations are under an average temperature of |
| 327 |
< |
$\sim$200K. Therefore, this flux usually comes from the metal to the |
| 324 |
> |
After the systems reach equilibrium, NIVS was used to impose an |
| 325 |
> |
unphysical thermal flux between the metal and the liquid phases. Most |
| 326 |
> |
of our simulations were done under an average temperature of |
| 327 |
> |
$\sim$200K. Therefore, thermal flux usually came from the metal to the |
| 328 |
|
liquid so that the liquid has a higher temperature and would not |
| 329 |
< |
freeze due to excessively low temperature. This induced temperature |
| 330 |
< |
gradient is stablized and the simulation cell is devided evenly into |
| 331 |
< |
N slabs along the $z$-axis and the temperatures of each slab are |
| 332 |
< |
recorded. When the slab width $d$ of each slab is the same, the |
| 333 |
< |
derivatives of $T$ with respect to slab number $n$ can be directly |
| 334 |
< |
used for $G^\prime$ calculations: |
| 335 |
< |
\begin{equation} |
| 336 |
< |
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
| 329 |
> |
freeze due to lowered temperatures. After this induced temperature |
| 330 |
> |
gradient had stablized, the temperature profile of the simulation cell |
| 331 |
> |
was recorded. To do this, the simulation cell is devided evenly into |
| 332 |
> |
$N$ slabs along the $z$-axis. The average temperatures of each slab |
| 333 |
> |
are recorded for 1$\sim$2 ns. When the slab width $d$ of each slab is |
| 334 |
> |
the same, the derivatives of $T$ with respect to slab number $n$ can |
| 335 |
> |
be directly used for $G^\prime$ calculations: \begin{equation} |
| 336 |
> |
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
| 337 |
|
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
| 338 |
|
= |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big| |
| 339 |
|
\Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2 |
| 342 |
|
\label{derivativeG2} |
| 343 |
|
\end{equation} |
| 344 |
|
|
| 345 |
+ |
All of the above simulation procedures use a time step of 1 fs. Each |
| 346 |
+ |
equilibration stage took a minimum of 100 ps, although in some cases, |
| 347 |
+ |
longer equilibration stages were utilized. |
| 348 |
+ |
|
| 349 |
|
\subsection{Force Field Parameters} |
| 350 |
< |
Our simulations include various components. Figure \ref{demoMol} |
| 351 |
< |
demonstrates the sites defined for both United-Atom and All-Atom |
| 352 |
< |
models of the organic solvent and capping agent molecules in our |
| 353 |
< |
simulations. Force field parameter descriptions are needed for |
| 350 |
> |
Our simulations include a number of chemically distinct components. |
| 351 |
> |
Figure \ref{demoMol} demonstrates the sites defined for both |
| 352 |
> |
United-Atom and All-Atom models of the organic solvent and capping |
| 353 |
> |
agents in our simulations. Force field parameters are needed for |
| 354 |
|
interactions both between the same type of particles and between |
| 355 |
|
particles of different species. |
| 356 |
|
|
| 360 |
|
these simulations. The chemically-distinct sites (a-e) are expanded |
| 361 |
|
in terms of constituent atoms for both United Atom (UA) and All Atom |
| 362 |
|
(AA) force fields. Most parameters are from |
| 363 |
< |
Refs. \protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} (UA) and |
| 364 |
< |
\protect\cite{OPLSAA} (AA). Cross-interactions with the Au atoms are given |
| 365 |
< |
in Table \ref{MnM}.} |
| 363 |
> |
Refs. \protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes,TraPPE-UA.thiols} |
| 364 |
> |
(UA) and \protect\cite{OPLSAA} (AA). Cross-interactions with the Au |
| 365 |
> |
atoms are given in Table \ref{MnM}.} |
| 366 |
|
\label{demoMol} |
| 367 |
|
\end{figure} |
| 368 |
|
|
| 369 |
|
The Au-Au interactions in metal lattice slab is described by the |
| 370 |
< |
quantum Sutton-Chen (QSC) formulation\cite{PhysRevB.59.3527}. The QSC |
| 370 |
> |
quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
| 371 |
|
potentials include zero-point quantum corrections and are |
| 372 |
|
reparametrized for accurate surface energies compared to the |
| 373 |
< |
Sutton-Chen potentials\cite{Chen90}. |
| 373 |
> |
Sutton-Chen potentials.\cite{Chen90} |
| 374 |
|
|
| 375 |
< |
For both solvent molecules, straight chain {\it n}-hexane and aromatic |
| 376 |
< |
toluene, United-Atom (UA) and All-Atom (AA) models are used |
| 377 |
< |
respectively. The TraPPE-UA |
| 375 |
> |
For the two solvent molecules, {\it n}-hexane and toluene, two |
| 376 |
> |
different atomistic models were utilized. Both solvents were modeled |
| 377 |
> |
using United-Atom (UA) and All-Atom (AA) models. The TraPPE-UA |
| 378 |
|
parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
| 379 |
|
for our UA solvent molecules. In these models, sites are located at |
| 380 |
|
the carbon centers for alkyl groups. Bonding interactions, including |
| 381 |
|
bond stretches and bends and torsions, were used for intra-molecular |
| 382 |
< |
sites not separated by more than 3 bonds. Otherwise, for non-bonded |
| 383 |
< |
interactions, Lennard-Jones potentials are used. [MORE CITATION?] |
| 382 |
> |
sites closer than 3 bonds. For non-bonded interactions, Lennard-Jones |
| 383 |
> |
potentials are used. |
| 384 |
|
|
| 385 |
< |
By eliminating explicit hydrogen atoms, these models are simple and |
| 386 |
< |
computationally efficient, while maintains good accuracy. However, the |
| 387 |
< |
TraPPE-UA for alkanes is known to predict a lower boiling point than |
| 388 |
< |
experimental values. Considering that after an unphysical thermal flux |
| 389 |
< |
is applied to a system, the temperature of ``hot'' area in the liquid |
| 390 |
< |
phase would be significantly higher than the average, to prevent over |
| 391 |
< |
heating and boiling of the liquid phase, the average temperature in |
| 392 |
< |
our simulations should be much lower than the liquid boiling point. |
| 385 |
> |
By eliminating explicit hydrogen atoms, the TraPPE-UA models are |
| 386 |
> |
simple and computationally efficient, while maintaining good accuracy. |
| 387 |
> |
However, the TraPPE-UA model for alkanes is known to predict a slighly |
| 388 |
> |
lower boiling point than experimental values. This is one of the |
| 389 |
> |
reasons we used a lower average temperature (200K) for our |
| 390 |
> |
simulations. If heat is transferred to the liquid phase during the |
| 391 |
> |
NIVS simulation, the liquid in the hot slab can actually be |
| 392 |
> |
substantially warmer than the mean temperature in the simulation. The |
| 393 |
> |
lower mean temperatures therefore prevent solvent boiling. |
| 394 |
|
|
| 395 |
< |
For UA-toluene model, the non-bonded potentials between |
| 396 |
< |
inter-molecular sites have a similar Lennard-Jones formulation. For |
| 397 |
< |
intra-molecular interactions, considering the stiffness of the benzene |
| 398 |
< |
ring, rigid body constraints are applied for further computational |
| 399 |
< |
efficiency. All bonds in the benzene ring and between the ring and the |
| 383 |
< |
methyl group remain rigid during the progress of simulations. |
| 395 |
> |
For UA-toluene, the non-bonded potentials between intermolecular sites |
| 396 |
> |
have a similar Lennard-Jones formulation. The toluene molecules were |
| 397 |
> |
treated as a single rigid body, so there was no need for |
| 398 |
> |
intramolecular interactions (including bonds, bends, or torsions) in |
| 399 |
> |
this solvent model. |
| 400 |
|
|
| 401 |
|
Besides the TraPPE-UA models, AA models for both organic solvents are |
| 402 |
< |
included in our studies as well. For hexane, the OPLS-AA\cite{OPLSAA} |
| 403 |
< |
force field is used. Additional explicit hydrogen sites were |
| 402 |
> |
included in our studies as well. The OPLS-AA\cite{OPLSAA} force fields |
| 403 |
> |
were used. For hexane, additional explicit hydrogen sites were |
| 404 |
|
included. Besides bonding and non-bonded site-site interactions, |
| 405 |
|
partial charges and the electrostatic interactions were added to each |
| 406 |
< |
CT and HC site. For toluene, the United Force Field developed by |
| 407 |
< |
Rapp\'{e} {\it et al.}\cite{doi:10.1021/ja00051a040} is |
| 408 |
< |
adopted. Without the rigid body constraints, bonding interactions were |
| 393 |
< |
included. For the aromatic ring, improper torsions (inversions) were |
| 394 |
< |
added as an extra potential for maintaining the planar shape. |
| 395 |
< |
[MORE CITATION?] |
| 406 |
> |
CT and HC site. For toluene, a flexible model for the toluene molecule |
| 407 |
> |
was utilized which included bond, bend, torsion, and inversion |
| 408 |
> |
potentials to enforce ring planarity. |
| 409 |
|
|
| 410 |
< |
The capping agent in our simulations, the butanethiol molecules can |
| 411 |
< |
either use UA or AA model. The TraPPE-UA force fields includes |
| 410 |
> |
The butanethiol capping agent in our simulations, were also modeled |
| 411 |
> |
with both UA and AA model. The TraPPE-UA force field includes |
| 412 |
|
parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for |
| 413 |
|
UA butanethiol model in our simulations. The OPLS-AA also provides |
| 414 |
|
parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) |
| 415 |
< |
surfaces do not have the hydrogen atom bonded to sulfur. To adapt this |
| 416 |
< |
change and derive suitable parameters for butanethiol adsorbed on |
| 417 |
< |
Au(111) surfaces, we adopt the S parameters from Luedtke and |
| 418 |
< |
Landman\cite{landman:1998} and modify parameters for its neighbor C |
| 419 |
< |
atom for charge balance in the molecule. Note that the model choice |
| 420 |
< |
(UA or AA) of capping agent can be different from the |
| 421 |
< |
solvent. Regardless of model choice, the force field parameters for |
| 422 |
< |
interactions between capping agent and solvent can be derived using |
| 410 |
< |
Lorentz-Berthelot Mixing Rule: |
| 415 |
> |
surfaces do not have the hydrogen atom bonded to sulfur. To derive |
| 416 |
> |
suitable parameters for butanethiol adsorbed on Au(111) surfaces, we |
| 417 |
> |
adopt the S parameters from Luedtke and Landman\cite{landman:1998} and |
| 418 |
> |
modify the parameters for the CTS atom to maintain charge neutrality |
| 419 |
> |
in the molecule. Note that the model choice (UA or AA) for the capping |
| 420 |
> |
agent can be different from the solvent. Regardless of model choice, |
| 421 |
> |
the force field parameters for interactions between capping agent and |
| 422 |
> |
solvent can be derived using Lorentz-Berthelot Mixing Rule: |
| 423 |
|
\begin{eqnarray} |
| 424 |
< |
\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\ |
| 425 |
< |
\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}} |
| 424 |
> |
\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\ |
| 425 |
> |
\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}} |
| 426 |
|
\end{eqnarray} |
| 427 |
|
|
| 428 |
< |
To describe the interactions between metal Au and non-metal capping |
| 429 |
< |
agent and solvent particles, we refer to an adsorption study of alkyl |
| 430 |
< |
thiols on gold surfaces by Vlugt {\it et |
| 431 |
< |
al.}\cite{vlugt:cpc2007154} They fitted an effective Lennard-Jones |
| 432 |
< |
form of potential parameters for the interaction between Au and |
| 433 |
< |
pseudo-atoms CH$_x$ and S based on a well-established and widely-used |
| 434 |
< |
effective potential of Hautman and Klein\cite{hautman:4994} for the |
| 435 |
< |
Au(111) surface. As our simulations require the gold lattice slab to |
| 436 |
< |
be non-rigid so that it could accommodate kinetic energy for thermal |
| 425 |
< |
transport study purpose, the pair-wise form of potentials is |
| 426 |
< |
preferred. |
| 428 |
> |
To describe the interactions between metal (Au) and non-metal atoms, |
| 429 |
> |
we refer to an adsorption study of alkyl thiols on gold surfaces by |
| 430 |
> |
Vlugt {\it et al.}\cite{vlugt:cpc2007154} They fitted an effective |
| 431 |
> |
Lennard-Jones form of potential parameters for the interaction between |
| 432 |
> |
Au and pseudo-atoms CH$_x$ and S based on a well-established and |
| 433 |
> |
widely-used effective potential of Hautman and Klein for the Au(111) |
| 434 |
> |
surface.\cite{hautman:4994} As our simulations require the gold slab |
| 435 |
> |
to be flexible to accommodate thermal excitation, the pair-wise form |
| 436 |
> |
of potentials they developed was used for our study. |
| 437 |
|
|
| 438 |
< |
Besides, the potentials developed from {\it ab initio} calculations by |
| 439 |
< |
Leng {\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
| 440 |
< |
interactions between Au and aromatic C/H atoms in toluene. A set of |
| 441 |
< |
pseudo Lennard-Jones parameters were provided for Au in their force |
| 442 |
< |
fields. By using the Mixing Rule, this can be used to derive pair-wise |
| 443 |
< |
potentials for non-bonded interactions between Au and non-metal sites. |
| 438 |
> |
The potentials developed from {\it ab initio} calculations by Leng |
| 439 |
> |
{\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
| 440 |
> |
interactions between Au and aromatic C/H atoms in toluene. However, |
| 441 |
> |
the Lennard-Jones parameters between Au and other types of particles, |
| 442 |
> |
(e.g. AA alkanes) have not yet been established. For these |
| 443 |
> |
interactions, the Lorentz-Berthelot mixing rule can be used to derive |
| 444 |
> |
effective single-atom LJ parameters for the metal using the fit values |
| 445 |
> |
for toluene. These are then used to construct reasonable mixing |
| 446 |
> |
parameters for the interactions between the gold and other atoms. |
| 447 |
> |
Table \ref{MnM} summarizes the ``metal/non-metal'' parameters used in |
| 448 |
> |
our simulations. |
| 449 |
|
|
| 435 |
– |
However, the Lennard-Jones parameters between Au and other types of |
| 436 |
– |
particles, such as All-Atom normal alkanes in our simulations are not |
| 437 |
– |
yet well-established. For these interactions, we attempt to derive |
| 438 |
– |
their parameters using the Mixing Rule. To do this, Au pseudo |
| 439 |
– |
Lennard-Jones parameters for ``Metal-non-Metal'' (MnM) interactions |
| 440 |
– |
were first extracted from the Au-CH$_x$ parameters by applying the |
| 441 |
– |
Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM'' |
| 442 |
– |
parameters in our simulations. |
| 443 |
– |
|
| 450 |
|
\begin{table*} |
| 451 |
|
\begin{minipage}{\linewidth} |
| 452 |
|
\begin{center} |
| 482 |
|
\end{minipage} |
| 483 |
|
\end{table*} |
| 484 |
|
|
| 479 |
– |
\subsection{Vibrational Spectrum} |
| 485 |
|
|
| 486 |
< |
[MAY ADD EQN'S] |
| 487 |
< |
To obtain these |
| 488 |
< |
spectra, one first runs a simulation in the NVE ensemble and collects |
| 489 |
< |
snapshots of configurations; these configurations are used to compute |
| 490 |
< |
the velocity auto-correlation functions, which is used to construct a |
| 491 |
< |
power spectrum via a Fourier transform. |
| 486 |
> |
\section{Results} |
| 487 |
> |
There are many factors contributing to the measured interfacial |
| 488 |
> |
conductance; some of these factors are physically motivated |
| 489 |
> |
(e.g. coverage of the surface by the capping agent coverage and |
| 490 |
> |
solvent identity), while some are governed by parameters of the |
| 491 |
> |
methodology (e.g. applied flux and the formulas used to obtain the |
| 492 |
> |
conductance). In this section we discuss the major physical and |
| 493 |
> |
calculational effects on the computed conductivity. |
| 494 |
|
|
| 495 |
< |
\section{Results and Discussions} |
| 489 |
< |
[MAY HAVE A BRIEF SUMMARY] |
| 490 |
< |
\subsection{How Simulation Parameters Affects $G$} |
| 491 |
< |
[MAY NOT PUT AT FIRST] |
| 492 |
< |
We have varied our protocol or other parameters of the simulations in |
| 493 |
< |
order to investigate how these factors would affect the measurement of |
| 494 |
< |
$G$'s. It turned out that while some of these parameters would not |
| 495 |
< |
affect the results substantially, some other changes to the |
| 496 |
< |
simulations would have a significant impact on the measurement |
| 497 |
< |
results. |
| 495 |
> |
\subsection{Effects due to capping agent coverage} |
| 496 |
|
|
| 497 |
< |
In some of our simulations, we allowed $L_x$ and $L_y$ to change |
| 498 |
< |
during equilibrating the liquid phase. Due to the stiffness of the |
| 499 |
< |
crystalline Au structure, $L_x$ and $L_y$ would not change noticeably |
| 500 |
< |
after equilibration. Although $L_z$ could fluctuate $\sim$1\% after a |
| 501 |
< |
system is fully equilibrated in the NPT ensemble, this fluctuation, as |
| 502 |
< |
well as those of $L_x$ and $L_y$ (which is significantly smaller), |
| 505 |
< |
would not be magnified on the calculated $G$'s, as shown in Table |
| 506 |
< |
\ref{AuThiolHexaneUA}. This insensivity to $L_i$ fluctuations allows |
| 507 |
< |
reliable measurement of $G$'s without the necessity of extremely |
| 508 |
< |
cautious equilibration process. |
| 497 |
> |
A series of different initial conditions with a range of surface |
| 498 |
> |
coverages was prepared and solvated with various with both of the |
| 499 |
> |
solvent molecules. These systems were then equilibrated and their |
| 500 |
> |
interfacial thermal conductivity was measured with the NIVS |
| 501 |
> |
algorithm. Figure \ref{coverage} demonstrates the trend of conductance |
| 502 |
> |
with respect to surface coverage. |
| 503 |
|
|
| 504 |
< |
As stated in our computational details, the spacing filled with |
| 505 |
< |
solvent molecules can be chosen within a range. This allows some |
| 506 |
< |
change of solvent molecule numbers for the same Au-butanethiol |
| 507 |
< |
surfaces. We did this study on our Au-butanethiol/hexane |
| 508 |
< |
simulations. Nevertheless, the results obtained from systems of |
| 509 |
< |
different $N_{hexane}$ did not indicate that the measurement of $G$ is |
| 510 |
< |
susceptible to this parameter. For computational efficiency concern, |
| 517 |
< |
smaller system size would be preferable, given that the liquid phase |
| 518 |
< |
structure is not affected. |
| 504 |
> |
\begin{figure} |
| 505 |
> |
\includegraphics[width=\linewidth]{coverage} |
| 506 |
> |
\caption{Comparison of interfacial thermal conductivity ($G$) values |
| 507 |
> |
for the Au-butanethiol/solvent interface with various UA models and |
| 508 |
> |
different capping agent coverages at $\langle T\rangle\sim$200K.} |
| 509 |
> |
\label{coverage} |
| 510 |
> |
\end{figure} |
| 511 |
|
|
| 512 |
< |
Our NIVS algorithm allows change of unphysical thermal flux both in |
| 513 |
< |
direction and in quantity. This feature extends our investigation of |
| 514 |
< |
interfacial thermal conductance. However, the magnitude of this |
| 515 |
< |
thermal flux is not arbitary if one aims to obtain a stable and |
| 516 |
< |
reliable thermal gradient. A temperature profile would be |
| 517 |
< |
substantially affected by noise when $|J_z|$ has a much too low |
| 518 |
< |
magnitude; while an excessively large $|J_z|$ that overwhelms the |
| 519 |
< |
conductance capacity of the interface would prevent a thermal gradient |
| 520 |
< |
to reach a stablized steady state. NIVS has the advantage of allowing |
| 521 |
< |
$J$ to vary in a wide range such that the optimal flux range for $G$ |
| 522 |
< |
measurement can generally be simulated by the algorithm. Within the |
| 523 |
< |
optimal range, we were able to study how $G$ would change according to |
| 524 |
< |
the thermal flux across the interface. For our simulations, we denote |
| 525 |
< |
$J_z$ to be positive when the physical thermal flux is from the liquid |
| 526 |
< |
to metal, and negative vice versa. The $G$'s measured under different |
| 527 |
< |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and |
| 528 |
< |
\ref{AuThiolToluene}. These results do not suggest that $G$ is |
| 529 |
< |
dependent on $J_z$ within this flux range. The linear response of flux |
| 530 |
< |
to thermal gradient simplifies our investigations in that we can rely |
| 531 |
< |
on $G$ measurement with only a couple $J_z$'s and do not need to test |
| 532 |
< |
a large series of fluxes. |
| 512 |
> |
In partially covered surfaces, the derivative definition for |
| 513 |
> |
$G^\prime$ (Eq. \ref{derivativeG}) becomes difficult to apply, as the |
| 514 |
> |
location of maximum change of $\lambda$ becomes washed out. The |
| 515 |
> |
discrete definition (Eq. \ref{discreteG}) is easier to apply, as the |
| 516 |
> |
Gibbs dividing surface is still well-defined. Therefore, $G$ (not |
| 517 |
> |
$G^\prime$) was used in this section. |
| 518 |
> |
|
| 519 |
> |
From Figure \ref{coverage}, one can see the significance of the |
| 520 |
> |
presence of capping agents. When even a small fraction of the Au(111) |
| 521 |
> |
surface sites are covered with butanethiols, the conductivity exhibits |
| 522 |
> |
an enhancement by at least a factor of 3. Cappping agents are clearly |
| 523 |
> |
playing a major role in thermal transport at metal / organic solvent |
| 524 |
> |
surfaces. |
| 525 |
> |
|
| 526 |
> |
We note a non-monotonic behavior in the interfacial conductance as a |
| 527 |
> |
function of surface coverage. The maximum conductance (largest $G$) |
| 528 |
> |
happens when the surfaces are about 75\% covered with butanethiol |
| 529 |
> |
caps. The reason for this behavior is not entirely clear. One |
| 530 |
> |
explanation is that incomplete butanethiol coverage allows small gaps |
| 531 |
> |
between butanethiols to form. These gaps can be filled by transient |
| 532 |
> |
solvent molecules. These solvent molecules couple very strongly with |
| 533 |
> |
the hot capping agent molecules near the surface, and can then carry |
| 534 |
> |
away (diffusively) the excess thermal energy from the surface. |
| 535 |
> |
|
| 536 |
> |
There appears to be a competition between the conduction of the |
| 537 |
> |
thermal energy away from the surface by the capping agents (enhanced |
| 538 |
> |
by greater coverage) and the coupling of the capping agents with the |
| 539 |
> |
solvent (enhanced by interdigitation at lower coverages). This |
| 540 |
> |
competition would lead to the non-monotonic coverage behavior observed |
| 541 |
> |
here. |
| 542 |
> |
|
| 543 |
> |
Results for rigid body toluene solvent, as well as the UA hexane, are |
| 544 |
> |
within the ranges expected from prior experimental |
| 545 |
> |
work.\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406} This suggests |
| 546 |
> |
that explicit hydrogen atoms might not be required for modeling |
| 547 |
> |
thermal transport in these systems. C-H vibrational modes do not see |
| 548 |
> |
significant excited state population at low temperatures, and are not |
| 549 |
> |
likely to carry lower frequency excitations from the solid layer into |
| 550 |
> |
the bulk liquid. |
| 551 |
> |
|
| 552 |
> |
The toluene solvent does not exhibit the same behavior as hexane in |
| 553 |
> |
that $G$ remains at approximately the same magnitude when the capping |
| 554 |
> |
coverage increases from 25\% to 75\%. Toluene, as a rigid planar |
| 555 |
> |
molecule, cannot occupy the relatively small gaps between the capping |
| 556 |
> |
agents as easily as the chain-like {\it n}-hexane. The effect of |
| 557 |
> |
solvent coupling to the capping agent is therefore weaker in toluene |
| 558 |
> |
except at the very lowest coverage levels. This effect counters the |
| 559 |
> |
coverage-dependent conduction of heat away from the metal surface, |
| 560 |
> |
leading to a much flatter $G$ vs. coverage trend than is observed in |
| 561 |
> |
{\it n}-hexane. |
| 562 |
> |
|
| 563 |
> |
\subsection{Effects due to Solvent \& Solvent Models} |
| 564 |
> |
In addition to UA solvent and capping agent models, AA models have |
| 565 |
> |
also been included in our simulations. In most of this work, the same |
| 566 |
> |
(UA or AA) model for solvent and capping agent was used, but it is |
| 567 |
> |
also possible to utilize different models for different components. |
| 568 |
> |
We have also included isotopic substitutions (Hydrogen to Deuterium) |
| 569 |
> |
to decrease the explicit vibrational overlap between solvent and |
| 570 |
> |
capping agent. Table \ref{modelTest} summarizes the results of these |
| 571 |
> |
studies. |
| 572 |
> |
|
| 573 |
> |
\begin{table*} |
| 574 |
> |
\begin{minipage}{\linewidth} |
| 575 |
> |
\begin{center} |
| 576 |
> |
|
| 577 |
> |
\caption{Computed interfacial thermal conductance ($G$ and |
| 578 |
> |
$G^\prime$) values for interfaces using various models for |
| 579 |
> |
solvent and capping agent (or without capping agent) at |
| 580 |
> |
$\langle T\rangle\sim$200K. (D stands for deuterated solvent |
| 581 |
> |
or capping agent molecules; ``Avg.'' denotes results that are |
| 582 |
> |
averages of simulations under different applied thermal flux |
| 583 |
> |
values $(J_z)$. Error estimates are indicated in |
| 584 |
> |
parentheses.)} |
| 585 |
> |
|
| 586 |
> |
\begin{tabular}{llccc} |
| 587 |
> |
\hline\hline |
| 588 |
> |
Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ |
| 589 |
> |
(or bare surface) & model & (GW/m$^2$) & |
| 590 |
> |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 591 |
> |
\hline |
| 592 |
> |
UA & UA hexane & Avg. & 131(9) & 87(10) \\ |
| 593 |
> |
& UA hexane(D) & 1.95 & 153(5) & 136(13) \\ |
| 594 |
> |
& AA hexane & Avg. & 131(6) & 122(10) \\ |
| 595 |
> |
& UA toluene & 1.96 & 187(16) & 151(11) \\ |
| 596 |
> |
& AA toluene & 1.89 & 200(36) & 149(53) \\ |
| 597 |
> |
\hline |
| 598 |
> |
AA & UA hexane & 1.94 & 116(9) & 129(8) \\ |
| 599 |
> |
& AA hexane & Avg. & 442(14) & 356(31) \\ |
| 600 |
> |
& AA hexane(D) & 1.93 & 222(12) & 234(54) \\ |
| 601 |
> |
& UA toluene & 1.98 & 125(25) & 97(60) \\ |
| 602 |
> |
& AA toluene & 3.79 & 487(56) & 290(42) \\ |
| 603 |
> |
\hline |
| 604 |
> |
AA(D) & UA hexane & 1.94 & 158(25) & 172(4) \\ |
| 605 |
> |
& AA hexane & 1.92 & 243(29) & 191(11) \\ |
| 606 |
> |
& AA toluene & 1.93 & 364(36) & 322(67) \\ |
| 607 |
> |
\hline |
| 608 |
> |
bare & UA hexane & Avg. & 46.5(3.2) & 49.4(4.5) \\ |
| 609 |
> |
& UA hexane(D) & 0.98 & 43.9(4.6) & 43.0(2.0) \\ |
| 610 |
> |
& AA hexane & 0.96 & 31.0(1.4) & 29.4(1.3) \\ |
| 611 |
> |
& UA toluene & 1.99 & 70.1(1.3) & 65.8(0.5) \\ |
| 612 |
> |
\hline\hline |
| 613 |
> |
\end{tabular} |
| 614 |
> |
\label{modelTest} |
| 615 |
> |
\end{center} |
| 616 |
> |
\end{minipage} |
| 617 |
> |
\end{table*} |
| 618 |
> |
|
| 619 |
> |
To facilitate direct comparison between force fields, systems with the |
| 620 |
> |
same capping agent and solvent were prepared with the same length |
| 621 |
> |
scales for the simulation cells. |
| 622 |
> |
|
| 623 |
> |
On bare metal / solvent surfaces, different force field models for |
| 624 |
> |
hexane yield similar results for both $G$ and $G^\prime$, and these |
| 625 |
> |
two definitions agree with each other very well. This is primarily an |
| 626 |
> |
indicator of weak interactions between the metal and the solvent, and |
| 627 |
> |
is a typical case for acoustic impedance mismatch between these two |
| 628 |
> |
phases. |
| 629 |
> |
|
| 630 |
> |
For the fully-covered surfaces, the choice of force field for the |
| 631 |
> |
capping agent and solvent has a large impact on the calulated values |
| 632 |
> |
of $G$ and $G^\prime$. The AA thiol to AA solvent conductivities are |
| 633 |
> |
much larger than their UA to UA counterparts, and these values exceed |
| 634 |
> |
the experimental estimates by a large measure. The AA force field |
| 635 |
> |
allows significant energy to go into C-H (or C-D) stretching modes, |
| 636 |
> |
and since these modes are high frequency, this non-quantum behavior is |
| 637 |
> |
likely responsible for the overestimate of the conductivity. Compared |
| 638 |
> |
to the AA model, the UA model yields more reasonable conductivity |
| 639 |
> |
values with much higher computational efficiency. |
| 640 |
> |
|
| 641 |
> |
\subsubsection{Are electronic excitations in the metal important?} |
| 642 |
> |
Because they lack electronic excitations, the QSC and related embedded |
| 643 |
> |
atom method (EAM) models for gold are known to predict unreasonably |
| 644 |
> |
low values for bulk conductivity |
| 645 |
> |
($\lambda$).\cite{kuang:164101,ISI:000207079300006,Clancy:1992} If the |
| 646 |
> |
conductance between the phases ($G$) is governed primarily by phonon |
| 647 |
> |
excitation (and not electronic degrees of freedom), one would expect a |
| 648 |
> |
classical model to capture most of the interfacial thermal |
| 649 |
> |
conductance. Our results for $G$ and $G^\prime$ indicate that this is |
| 650 |
> |
indeed the case, and suggest that the modeling of interfacial thermal |
| 651 |
> |
transport depends primarily on the description of the interactions |
| 652 |
> |
between the various components at the interface. When the metal is |
| 653 |
> |
chemically capped, the primary barrier to thermal conductivity appears |
| 654 |
> |
to be the interface between the capping agent and the surrounding |
| 655 |
> |
solvent, so the excitations in the metal have little impact on the |
| 656 |
> |
value of $G$. |
| 657 |
|
|
| 658 |
+ |
\subsection{Effects due to methodology and simulation parameters} |
| 659 |
+ |
|
| 660 |
+ |
We have varied the parameters of the simulations in order to |
| 661 |
+ |
investigate how these factors would affect the computation of $G$. Of |
| 662 |
+ |
particular interest are: 1) the length scale for the applied thermal |
| 663 |
+ |
gradient (modified by increasing the amount of solvent in the system), |
| 664 |
+ |
2) the sign and magnitude of the applied thermal flux, 3) the average |
| 665 |
+ |
temperature of the simulation (which alters the solvent density during |
| 666 |
+ |
equilibration), and 4) the definition of the interfacial conductance |
| 667 |
+ |
(Eqs. (\ref{discreteG}) or (\ref{derivativeG})) used in the |
| 668 |
+ |
calculation. |
| 669 |
+ |
|
| 670 |
+ |
Systems of different lengths were prepared by altering the number of |
| 671 |
+ |
solvent molecules and extending the length of the box along the $z$ |
| 672 |
+ |
axis to accomodate the extra solvent. Equilibration at the same |
| 673 |
+ |
temperature and pressure conditions led to nearly identical surface |
| 674 |
+ |
areas ($L_x$ and $L_y$) available to the metal and capping agent, |
| 675 |
+ |
while the extra solvent served mainly to lengthen the axis that was |
| 676 |
+ |
used to apply the thermal flux. For a given value of the applied |
| 677 |
+ |
flux, the different $z$ length scale has only a weak effect on the |
| 678 |
+ |
computed conductivities (Table \ref{AuThiolHexaneUA}). |
| 679 |
+ |
|
| 680 |
+ |
\subsubsection{Effects of applied flux} |
| 681 |
+ |
The NIVS algorithm allows changes in both the sign and magnitude of |
| 682 |
+ |
the applied flux. It is possible to reverse the direction of heat |
| 683 |
+ |
flow simply by changing the sign of the flux, and thermal gradients |
| 684 |
+ |
which would be difficult to obtain experimentally ($5$ K/\AA) can be |
| 685 |
+ |
easily simulated. However, the magnitude of the applied flux is not |
| 686 |
+ |
arbitary if one aims to obtain a stable and reliable thermal gradient. |
| 687 |
+ |
A temperature gradient can be lost in the noise if $|J_z|$ is too |
| 688 |
+ |
small, and excessive $|J_z|$ values can cause phase transitions if the |
| 689 |
+ |
extremes of the simulation cell become widely separated in |
| 690 |
+ |
temperature. Also, if $|J_z|$ is too large for the bulk conductivity |
| 691 |
+ |
of the materials, the thermal gradient will never reach a stable |
| 692 |
+ |
state. |
| 693 |
+ |
|
| 694 |
+ |
Within a reasonable range of $J_z$ values, we were able to study how |
| 695 |
+ |
$G$ changes as a function of this flux. In what follows, we use |
| 696 |
+ |
positive $J_z$ values to denote the case where energy is being |
| 697 |
+ |
transferred by the method from the metal phase and into the liquid. |
| 698 |
+ |
The resulting gradient therefore has a higher temperature in the |
| 699 |
+ |
liquid phase. Negative flux values reverse this transfer, and result |
| 700 |
+ |
in higher temperature metal phases. The conductance measured under |
| 701 |
+ |
different applied $J_z$ values is listed in Tables |
| 702 |
+ |
\ref{AuThiolHexaneUA} and \ref{AuThiolToluene}. These results do not |
| 703 |
+ |
indicate that $G$ depends strongly on $J_z$ within this flux |
| 704 |
+ |
range. The linear response of flux to thermal gradient simplifies our |
| 705 |
+ |
investigations in that we can rely on $G$ measurement with only a |
| 706 |
+ |
small number $J_z$ values. |
| 707 |
+ |
|
| 708 |
|
\begin{table*} |
| 709 |
|
\begin{minipage}{\linewidth} |
| 710 |
|
\begin{center} |
| 716 |
|
|
| 717 |
|
\begin{tabular}{ccccccc} |
| 718 |
|
\hline\hline |
| 719 |
< |
$\langle T\rangle$ & $N_{hexane}$ & Fixed & $\rho_{hexane}$ & |
| 719 |
> |
$\langle T\rangle$ & $N_{hexane}$ & $\rho_{hexane}$ & |
| 720 |
|
$J_z$ & $G$ & $G^\prime$ \\ |
| 721 |
< |
(K) & & $L_x$ \& $L_y$? & (g/cm$^3$) & (GW/m$^2$) & |
| 721 |
> |
(K) & & (g/cm$^3$) & (GW/m$^2$) & |
| 722 |
|
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 723 |
|
\hline |
| 724 |
< |
200 & 266 & No & 0.672 & -0.96 & 102(3) & 80.0(0.8) \\ |
| 725 |
< |
& 200 & Yes & 0.694 & 1.92 & 129(11) & 87.3(0.3) \\ |
| 726 |
< |
& & Yes & 0.672 & 1.93 & 131(16) & 78(13) \\ |
| 727 |
< |
& & No & 0.688 & 0.96 & 125(16) & 90.2(15) \\ |
| 728 |
< |
& & & & 1.91 & 139(10) & 101(10) \\ |
| 729 |
< |
& & & & 2.83 & 141(6) & 89.9(9.8) \\ |
| 564 |
< |
& 166 & Yes & 0.679 & 0.97 & 115(19) & 69(18) \\ |
| 565 |
< |
& & & & 1.94 & 125(9) & 87.1(0.2) \\ |
| 566 |
< |
& & No & 0.681 & 0.97 & 141(30) & 78(22) \\ |
| 567 |
< |
& & & & 1.92 & 138(4) & 98.9(9.5) \\ |
| 724 |
> |
200 & 266 & 0.672 & -0.96 & 102(3) & 80.0(0.8) \\ |
| 725 |
> |
& 200 & 0.688 & 0.96 & 125(16) & 90.2(15) \\ |
| 726 |
> |
& & & 1.91 & 139(10) & 101(10) \\ |
| 727 |
> |
& & & 2.83 & 141(6) & 89.9(9.8) \\ |
| 728 |
> |
& 166 & 0.681 & 0.97 & 141(30) & 78(22) \\ |
| 729 |
> |
& & & 1.92 & 138(4) & 98.9(9.5) \\ |
| 730 |
|
\hline |
| 731 |
< |
250 & 200 & No & 0.560 & 0.96 & 75(10) & 61.8(7.3) \\ |
| 732 |
< |
& & & & -0.95 & 49.4(0.3) & 45.7(2.1) \\ |
| 733 |
< |
& 166 & Yes & 0.570 & 0.98 & 79.0(3.5) & 62.9(3.0) \\ |
| 734 |
< |
& & No & 0.569 & 0.97 & 80.3(0.6) & 67(11) \\ |
| 735 |
< |
& & & & 1.44 & 76.2(5.0) & 64.8(3.8) \\ |
| 736 |
< |
& & & & -0.95 & 56.4(2.5) & 54.4(1.1) \\ |
| 575 |
< |
& & & & -1.85 & 47.8(1.1) & 53.5(1.5) \\ |
| 731 |
> |
250 & 200 & 0.560 & 0.96 & 75(10) & 61.8(7.3) \\ |
| 732 |
> |
& & & -0.95 & 49.4(0.3) & 45.7(2.1) \\ |
| 733 |
> |
& 166 & 0.569 & 0.97 & 80.3(0.6) & 67(11) \\ |
| 734 |
> |
& & & 1.44 & 76.2(5.0) & 64.8(3.8) \\ |
| 735 |
> |
& & & -0.95 & 56.4(2.5) & 54.4(1.1) \\ |
| 736 |
> |
& & & -1.85 & 47.8(1.1) & 53.5(1.5) \\ |
| 737 |
|
\hline\hline |
| 738 |
|
\end{tabular} |
| 739 |
|
\label{AuThiolHexaneUA} |
| 741 |
|
\end{minipage} |
| 742 |
|
\end{table*} |
| 743 |
|
|
| 744 |
< |
Furthermore, we also attempted to increase system average temperatures |
| 745 |
< |
to above 200K. These simulations are first equilibrated in the NPT |
| 746 |
< |
ensemble under normal pressure. As stated above, the TraPPE-UA model |
| 747 |
< |
for hexane tends to predict a lower boiling point. In our simulations, |
| 748 |
< |
hexane had diffculty to remain in liquid phase when NPT equilibration |
| 749 |
< |
temperature is higher than 250K. Additionally, the equilibrated liquid |
| 750 |
< |
hexane density under 250K becomes lower than experimental value. This |
| 751 |
< |
expanded liquid phase leads to lower contact between hexane and |
| 752 |
< |
butanethiol as well.[MAY NEED SLAB DENSITY FIGURE] |
| 592 |
< |
And this reduced contact would |
| 593 |
< |
probably be accountable for a lower interfacial thermal conductance, |
| 594 |
< |
as shown in Table \ref{AuThiolHexaneUA}. |
| 744 |
> |
The sign of $J_z$ is a different matter, however, as this can alter |
| 745 |
> |
the temperature on the two sides of the interface. The average |
| 746 |
> |
temperature values reported are for the entire system, and not for the |
| 747 |
> |
liquid phase, so at a given $\langle T \rangle$, the system with |
| 748 |
> |
positive $J_z$ has a warmer liquid phase. This means that if the |
| 749 |
> |
liquid carries thermal energy via convective transport, {\it positive} |
| 750 |
> |
$J_z$ values will result in increased molecular motion on the liquid |
| 751 |
> |
side of the interface, and this will increase the measured |
| 752 |
> |
conductivity. |
| 753 |
|
|
| 754 |
< |
A similar study for TraPPE-UA toluene agrees with the above result as |
| 597 |
< |
well. Having a higher boiling point, toluene tends to remain liquid in |
| 598 |
< |
our simulations even equilibrated under 300K in NPT |
| 599 |
< |
ensembles. Furthermore, the expansion of the toluene liquid phase is |
| 600 |
< |
not as significant as that of the hexane. This prevents severe |
| 601 |
< |
decrease of liquid-capping agent contact and the results (Table |
| 602 |
< |
\ref{AuThiolToluene}) show only a slightly decreased interface |
| 603 |
< |
conductance. Therefore, solvent-capping agent contact should play an |
| 604 |
< |
important role in the thermal transport process across the interface |
| 605 |
< |
in that higher degree of contact could yield increased conductance. |
| 754 |
> |
\subsubsection{Effects due to average temperature} |
| 755 |
|
|
| 756 |
+ |
We also studied the effect of average system temperature on the |
| 757 |
+ |
interfacial conductance. The simulations are first equilibrated in |
| 758 |
+ |
the NPT ensemble at 1 atm. The TraPPE-UA model for hexane tends to |
| 759 |
+ |
predict a lower boiling point (and liquid state density) than |
| 760 |
+ |
experiments. This lower-density liquid phase leads to reduced contact |
| 761 |
+ |
between the hexane and butanethiol, and this accounts for our |
| 762 |
+ |
observation of lower conductance at higher temperatures as shown in |
| 763 |
+ |
Table \ref{AuThiolHexaneUA}. In raising the average temperature from |
| 764 |
+ |
200K to 250K, the density drop of ~20\% in the solvent phase leads to |
| 765 |
+ |
a ~65\% drop in the conductance. |
| 766 |
+ |
|
| 767 |
+ |
Similar behavior is observed in the TraPPE-UA model for toluene, |
| 768 |
+ |
although this model has better agreement with the experimental |
| 769 |
+ |
densities of toluene. The expansion of the toluene liquid phase is |
| 770 |
+ |
not as significant as that of the hexane (8.3\% over 100K), and this |
| 771 |
+ |
limits the effect to ~20\% drop in thermal conductivity (Table |
| 772 |
+ |
\ref{AuThiolToluene}). |
| 773 |
+ |
|
| 774 |
+ |
Although we have not mapped out the behavior at a large number of |
| 775 |
+ |
temperatures, is clear that there will be a strong temperature |
| 776 |
+ |
dependence in the interfacial conductance when the physical properties |
| 777 |
+ |
of one side of the interface (notably the density) change rapidly as a |
| 778 |
+ |
function of temperature. |
| 779 |
+ |
|
| 780 |
|
\begin{table*} |
| 781 |
|
\begin{minipage}{\linewidth} |
| 782 |
|
\begin{center} |
| 803 |
|
\end{minipage} |
| 804 |
|
\end{table*} |
| 805 |
|
|
| 806 |
< |
Besides lower interfacial thermal conductance, surfaces in relatively |
| 807 |
< |
high temperatures are susceptible to reconstructions, when |
| 808 |
< |
butanethiols have a full coverage on the Au(111) surface. These |
| 809 |
< |
reconstructions include surface Au atoms migrated outward to the S |
| 810 |
< |
atom layer, and butanethiol molecules embedded into the original |
| 811 |
< |
surface Au layer. The driving force for this behavior is the strong |
| 812 |
< |
Au-S interactions in our simulations. And these reconstructions lead |
| 813 |
< |
to higher ratio of Au-S attraction and thus is energetically |
| 814 |
< |
favorable. Furthermore, this phenomenon agrees with experimental |
| 815 |
< |
results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt |
| 816 |
< |
{\it et al.} had kept their Au(111) slab rigid so that their |
| 817 |
< |
simulations can reach 300K without surface reconstructions. Without |
| 818 |
< |
this practice, simulating 100\% thiol covered interfaces under higher |
| 819 |
< |
temperatures could hardly avoid surface reconstructions. However, our |
| 820 |
< |
measurement is based on assuming homogeneity on $x$ and $y$ dimensions |
| 821 |
< |
so that measurement of $T$ at particular $z$ would be an effective |
| 822 |
< |
average of the particles of the same type. Since surface |
| 650 |
< |
reconstructions could eliminate the original $x$ and $y$ dimensional |
| 651 |
< |
homogeneity, measurement of $G$ is more difficult to conduct under |
| 652 |
< |
higher temperatures. Therefore, most of our measurements are |
| 653 |
< |
undertaken at $\langle T\rangle\sim$200K. |
| 806 |
> |
Besides the lower interfacial thermal conductance, surfaces at |
| 807 |
> |
relatively high temperatures are susceptible to reconstructions, |
| 808 |
> |
particularly when butanethiols fully cover the Au(111) surface. These |
| 809 |
> |
reconstructions include surface Au atoms which migrate outward to the |
| 810 |
> |
S atom layer, and butanethiol molecules which embed into the surface |
| 811 |
> |
Au layer. The driving force for this behavior is the strong Au-S |
| 812 |
> |
interactions which are modeled here with a deep Lennard-Jones |
| 813 |
> |
potential. This phenomenon agrees with reconstructions that have beeen |
| 814 |
> |
experimentally |
| 815 |
> |
observed.\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt |
| 816 |
> |
{\it et al.} kept their Au(111) slab rigid so that their simulations |
| 817 |
> |
could reach 300K without surface |
| 818 |
> |
reconstructions.\cite{vlugt:cpc2007154} Since surface reconstructions |
| 819 |
> |
blur the interface, the measurement of $G$ becomes more difficult to |
| 820 |
> |
conduct at higher temperatures. For this reason, most of our |
| 821 |
> |
measurements are undertaken at $\langle T\rangle\sim$200K where |
| 822 |
> |
reconstruction is minimized. |
| 823 |
|
|
| 824 |
|
However, when the surface is not completely covered by butanethiols, |
| 825 |
< |
the simulated system is more resistent to the reconstruction |
| 826 |
< |
above. Our Au-butanethiol/toluene system had the Au(111) surfaces 90\% |
| 827 |
< |
covered by butanethiols, but did not see this above phenomena even at |
| 828 |
< |
$\langle T\rangle\sim$300K. The empty three-fold sites not occupied by |
| 829 |
< |
capping agents could help prevent surface reconstruction in that they |
| 830 |
< |
provide other means of capping agent relaxation. It is observed that |
| 831 |
< |
butanethiols can migrate to their neighbor empty sites during a |
| 832 |
< |
simulation. Therefore, we were able to obtain $G$'s for these |
| 664 |
< |
interfaces even at a relatively high temperature without being |
| 665 |
< |
affected by surface reconstructions. |
| 825 |
> |
the simulated system appears to be more resistent to the |
| 826 |
> |
reconstruction. O ur Au / butanethiol / toluene system had the Au(111) |
| 827 |
> |
surfaces 90\% covered by butanethiols, but did not see this above |
| 828 |
> |
phenomena even at $\langle T\rangle\sim$300K. That said, we did |
| 829 |
> |
observe butanethiols migrating to neighboring three-fold sites during |
| 830 |
> |
a simulation. Since the interface persisted in these simulations, |
| 831 |
> |
were able to obtain $G$'s for these interfaces even at a relatively |
| 832 |
> |
high temperature without being affected by surface reconstructions. |
| 833 |
|
|
| 834 |
< |
\subsection{Influence of Capping Agent Coverage on $G$} |
| 668 |
< |
To investigate the influence of butanethiol coverage on interfacial |
| 669 |
< |
thermal conductance, a series of different coverage Au-butanethiol |
| 670 |
< |
surfaces is prepared and solvated with various organic |
| 671 |
< |
molecules. These systems are then equilibrated and their interfacial |
| 672 |
< |
thermal conductivity are measured with our NIVS algorithm. Figure |
| 673 |
< |
\ref{coverage} demonstrates the trend of conductance change with |
| 674 |
< |
respect to different coverages of butanethiol. To study the isotope |
| 675 |
< |
effect in interfacial thermal conductance, deuterated UA-hexane is |
| 676 |
< |
included as well. |
| 834 |
> |
\section{Discussion} |
| 835 |
|
|
| 836 |
< |
It turned out that with partial covered butanethiol on the Au(111) |
| 837 |
< |
surface, the derivative definition for $G^\prime$ |
| 838 |
< |
(Eq. \ref{derivativeG}) was difficult to apply, due to the difficulty |
| 839 |
< |
in locating the maximum of change of $\lambda$. Instead, the discrete |
| 840 |
< |
definition (Eq. \ref{discreteG}) is easier to apply, as the Gibbs |
| 841 |
< |
deviding surface can still be well-defined. Therefore, $G$ (not |
| 684 |
< |
$G^\prime$) was used for this section. |
| 836 |
> |
The primary result of this work is that the capping agent acts as an |
| 837 |
> |
efficient thermal coupler between solid and solvent phases. One of |
| 838 |
> |
the ways the capping agent can carry out this role is to down-shift |
| 839 |
> |
between the phonon vibrations in the solid (which carry the heat from |
| 840 |
> |
the gold) and the molecular vibrations in the liquid (which carry some |
| 841 |
> |
of the heat in the solvent). |
| 842 |
|
|
| 843 |
< |
From Figure \ref{coverage}, one can see the significance of the |
| 844 |
< |
presence of capping agents. Even when a fraction of the Au(111) |
| 845 |
< |
surface sites are covered with butanethiols, the conductivity would |
| 846 |
< |
see an enhancement by at least a factor of 3. This indicates the |
| 847 |
< |
important role cappping agent is playing for thermal transport |
| 848 |
< |
phenomena on metal / organic solvent surfaces. |
| 843 |
> |
To investigate the mechanism of interfacial thermal conductance, the |
| 844 |
> |
vibrational power spectrum was computed. Power spectra were taken for |
| 845 |
> |
individual components in different simulations. To obtain these |
| 846 |
> |
spectra, simulations were run after equilibration in the |
| 847 |
> |
microcanonical (NVE) ensemble and without a thermal |
| 848 |
> |
gradient. Snapshots of configurations were collected at a frequency |
| 849 |
> |
that is higher than that of the fastest vibrations occuring in the |
| 850 |
> |
simulations. With these configurations, the velocity auto-correlation |
| 851 |
> |
functions can be computed: |
| 852 |
> |
\begin{equation} |
| 853 |
> |
C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle |
| 854 |
> |
\label{vCorr} |
| 855 |
> |
\end{equation} |
| 856 |
> |
The power spectrum is constructed via a Fourier transform of the |
| 857 |
> |
symmetrized velocity autocorrelation function, |
| 858 |
> |
\begin{equation} |
| 859 |
> |
\hat{f}(\omega) = |
| 860 |
> |
\int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt |
| 861 |
> |
\label{fourier} |
| 862 |
> |
\end{equation} |
| 863 |
|
|
| 864 |
< |
Interestingly, as one could observe from our results, the maximum |
| 865 |
< |
conductance enhancement (largest $G$) happens while the surfaces are |
| 866 |
< |
about 75\% covered with butanethiols. This again indicates that |
| 867 |
< |
solvent-capping agent contact has an important role of the thermal |
| 868 |
< |
transport process. Slightly lower butanethiol coverage allows small |
| 869 |
< |
gaps between butanethiols to form. And these gaps could be filled with |
| 870 |
< |
solvent molecules, which acts like ``heat conductors'' on the |
| 871 |
< |
surface. The higher degree of interaction between these solvent |
| 872 |
< |
molecules and capping agents increases the enhancement effect and thus |
| 873 |
< |
produces a higher $G$ than densely packed butanethiol arrays. However, |
| 874 |
< |
once this maximum conductance enhancement is reached, $G$ decreases |
| 704 |
< |
when butanethiol coverage continues to decrease. Each capping agent |
| 705 |
< |
molecule reaches its maximum capacity for thermal |
| 706 |
< |
conductance. Therefore, even higher solvent-capping agent contact |
| 707 |
< |
would not offset this effect. Eventually, when butanethiol coverage |
| 708 |
< |
continues to decrease, solvent-capping agent contact actually |
| 709 |
< |
decreases with the disappearing of butanethiol molecules. In this |
| 710 |
< |
case, $G$ decrease could not be offset but instead accelerated. [NEED |
| 711 |
< |
SNAPSHOT SHOWING THE PHENOMENA / SLAB DENSITY ANALYSIS] |
| 864 |
> |
\subsection{The role of specific vibrations} |
| 865 |
> |
The vibrational spectra for gold slabs in different environments are |
| 866 |
> |
shown as in Figure \ref{specAu}. Regardless of the presence of |
| 867 |
> |
solvent, the gold surfaces which are covered by butanethiol molecules |
| 868 |
> |
exhibit an additional peak observed at a frequency of |
| 869 |
> |
$\sim$165cm$^{-1}$. We attribute this peak to the S-Au bonding |
| 870 |
> |
vibration. This vibration enables efficient thermal coupling of the |
| 871 |
> |
surface Au layer to the capping agents. Therefore, in our simulations, |
| 872 |
> |
the Au / S interfaces do not appear to be the primary barrier to |
| 873 |
> |
thermal transport when compared with the butanethiol / solvent |
| 874 |
> |
interfaces. |
| 875 |
|
|
| 876 |
< |
A comparison of the results obtained from differenet organic solvents |
| 877 |
< |
can also provide useful information of the interfacial thermal |
| 878 |
< |
transport process. The deuterated hexane (UA) results do not appear to |
| 879 |
< |
be much different from those of normal hexane (UA), given that |
| 880 |
< |
butanethiol (UA) is non-deuterated for both solvents. These UA model |
| 881 |
< |
studies, even though eliminating C-H vibration samplings, still have |
| 882 |
< |
C-C vibrational frequencies different from each other. However, these |
| 883 |
< |
differences in the infrared range do not seem to produce an observable |
| 721 |
< |
difference for the results of $G$. [MAY NEED SPECTRA FIGURE] |
| 876 |
> |
\begin{figure} |
| 877 |
> |
\includegraphics[width=\linewidth]{vibration} |
| 878 |
> |
\caption{Vibrational power spectra for gold in different solvent |
| 879 |
> |
environments. The presence of the butanethiol capping molecules |
| 880 |
> |
adds a vibrational peak at $\sim$165cm$^{-1}$. The butanethiol |
| 881 |
> |
spectra exhibit a corresponding peak.} |
| 882 |
> |
\label{specAu} |
| 883 |
> |
\end{figure} |
| 884 |
|
|
| 885 |
< |
Furthermore, results for rigid body toluene solvent, as well as other |
| 886 |
< |
UA-hexane solvents, are reasonable within the general experimental |
| 887 |
< |
ranges[CITATIONS]. This suggests that explicit hydrogen might not be a |
| 726 |
< |
required factor for modeling thermal transport phenomena of systems |
| 727 |
< |
such as Au-thiol/organic solvent. |
| 885 |
> |
Also in this figure, we show the vibrational power spectrum for the |
| 886 |
> |
bound butanethiol molecules, which also exhibits the same |
| 887 |
> |
$\sim$165cm$^{-1}$ peak. |
| 888 |
|
|
| 889 |
< |
However, results for Au-butanethiol/toluene do not show an identical |
| 890 |
< |
trend with those for Au-butanethiol/hexane in that $G$ remains at |
| 891 |
< |
approximately the same magnitue when butanethiol coverage differs from |
| 892 |
< |
25\% to 75\%. This might be rooted in the molecule shape difference |
| 893 |
< |
for planar toluene and chain-like {\it n}-hexane. Due to this |
| 894 |
< |
difference, toluene molecules have more difficulty in occupying |
| 895 |
< |
relatively small gaps among capping agents when their coverage is not |
| 896 |
< |
too low. Therefore, the solvent-capping agent contact may keep |
| 897 |
< |
increasing until the capping agent coverage reaches a relatively low |
| 898 |
< |
level. This becomes an offset for decreasing butanethiol molecules on |
| 899 |
< |
its effect to the process of interfacial thermal transport. Thus, one |
| 740 |
< |
can see a plateau of $G$ vs. butanethiol coverage in our results. |
| 889 |
> |
\subsection{Overlap of power spectra} |
| 890 |
> |
A comparison of the results obtained from the two different organic |
| 891 |
> |
solvents can also provide useful information of the interfacial |
| 892 |
> |
thermal transport process. In particular, the vibrational overlap |
| 893 |
> |
between the butanethiol and the organic solvents suggests a highly |
| 894 |
> |
efficient thermal exchange between these components. Very high |
| 895 |
> |
thermal conductivity was observed when AA models were used and C-H |
| 896 |
> |
vibrations were treated classically. The presence of extra degrees of |
| 897 |
> |
freedom in the AA force field yields higher heat exchange rates |
| 898 |
> |
between the two phases and results in a much higher conductivity than |
| 899 |
> |
in the UA force field. |
| 900 |
|
|
| 901 |
+ |
The similarity in the vibrational modes available to solvent and |
| 902 |
+ |
capping agent can be reduced by deuterating one of the two components |
| 903 |
+ |
(Fig. \ref{aahxntln}). Once either the hexanes or the butanethiols |
| 904 |
+ |
are deuterated, one can observe a significantly lower $G$ and |
| 905 |
+ |
$G^\prime$ values (Table \ref{modelTest}). |
| 906 |
+ |
|
| 907 |
|
\begin{figure} |
| 908 |
< |
\includegraphics[width=\linewidth]{coverage} |
| 909 |
< |
\caption{Comparison of interfacial thermal conductivity ($G$) values |
| 910 |
< |
for the Au-butanethiol/solvent interface with various UA models and |
| 911 |
< |
different capping agent coverages at $\langle T\rangle\sim$200K |
| 912 |
< |
using certain energy flux respectively.} |
| 913 |
< |
\label{coverage} |
| 908 |
> |
\includegraphics[width=\linewidth]{aahxntln} |
| 909 |
> |
\caption{Spectra obtained for all-atom (AA) Au / butanethiol / solvent |
| 910 |
> |
systems. When butanethiol is deuterated (lower left), its |
| 911 |
> |
vibrational overlap with hexane decreases significantly. Since |
| 912 |
> |
aromatic molecules and the butanethiol are vibrationally dissimilar, |
| 913 |
> |
the change is not as dramatic when toluene is the solvent (right).} |
| 914 |
> |
\label{aahxntln} |
| 915 |
|
\end{figure} |
| 916 |
|
|
| 917 |
< |
\subsection{Influence of Chosen Molecule Model on $G$} |
| 752 |
< |
In addition to UA solvent/capping agent models, AA models are included |
| 753 |
< |
in our simulations as well. Besides simulations of the same (UA or AA) |
| 754 |
< |
model for solvent and capping agent, different models can be applied |
| 755 |
< |
to different components. Furthermore, regardless of models chosen, |
| 756 |
< |
either the solvent or the capping agent can be deuterated, similar to |
| 757 |
< |
the previous section. Table \ref{modelTest} summarizes the results of |
| 758 |
< |
these studies. |
| 759 |
< |
|
| 760 |
< |
\begin{table*} |
| 761 |
< |
\begin{minipage}{\linewidth} |
| 762 |
< |
\begin{center} |
| 763 |
< |
|
| 764 |
< |
\caption{Computed interfacial thermal conductivity ($G$ and |
| 765 |
< |
$G^\prime$) values for interfaces using various models for |
| 766 |
< |
solvent and capping agent (or without capping agent) at |
| 767 |
< |
$\langle T\rangle\sim$200K. (D stands for deuterated solvent |
| 768 |
< |
or capping agent molecules; ``Avg.'' denotes results that are |
| 769 |
< |
averages of simulations under different $J_z$'s. Error |
| 770 |
< |
estimates indicated in parenthesis.)} |
| 771 |
< |
|
| 772 |
< |
\begin{tabular}{llccc} |
| 773 |
< |
\hline\hline |
| 774 |
< |
Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ |
| 775 |
< |
(or bare surface) & model & (GW/m$^2$) & |
| 776 |
< |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 777 |
< |
\hline |
| 778 |
< |
UA & UA hexane & Avg. & 131(9) & 87(10) \\ |
| 779 |
< |
& UA hexane(D) & 1.95 & 153(5) & 136(13) \\ |
| 780 |
< |
& AA hexane & Avg. & 131(6) & 122(10) \\ |
| 781 |
< |
& UA toluene & 1.96 & 187(16) & 151(11) \\ |
| 782 |
< |
& AA toluene & 1.89 & 200(36) & 149(53) \\ |
| 783 |
< |
\hline |
| 784 |
< |
AA & UA hexane & 1.94 & 116(9) & 129(8) \\ |
| 785 |
< |
& AA hexane & Avg. & 442(14) & 356(31) \\ |
| 786 |
< |
& AA hexane(D) & 1.93 & 222(12) & 234(54) \\ |
| 787 |
< |
& UA toluene & 1.98 & 125(25) & 97(60) \\ |
| 788 |
< |
& AA toluene & 3.79 & 487(56) & 290(42) \\ |
| 789 |
< |
\hline |
| 790 |
< |
AA(D) & UA hexane & 1.94 & 158(25) & 172(4) \\ |
| 791 |
< |
& AA hexane & 1.92 & 243(29) & 191(11) \\ |
| 792 |
< |
& AA toluene & 1.93 & 364(36) & 322(67) \\ |
| 793 |
< |
\hline |
| 794 |
< |
bare & UA hexane & Avg. & 46.5(3.2) & 49.4(4.5) \\ |
| 795 |
< |
& UA hexane(D) & 0.98 & 43.9(4.6) & 43.0(2.0) \\ |
| 796 |
< |
& AA hexane & 0.96 & 31.0(1.4) & 29.4(1.3) \\ |
| 797 |
< |
& UA toluene & 1.99 & 70.1(1.3) & 65.8(0.5) \\ |
| 798 |
< |
\hline\hline |
| 799 |
< |
\end{tabular} |
| 800 |
< |
\label{modelTest} |
| 801 |
< |
\end{center} |
| 802 |
< |
\end{minipage} |
| 803 |
< |
\end{table*} |
| 804 |
< |
|
| 805 |
< |
To facilitate direct comparison, the same system with differnt models |
| 806 |
< |
for different components uses the same length scale for their |
| 807 |
< |
simulation cells. Without the presence of capping agent, using |
| 808 |
< |
different models for hexane yields similar results for both $G$ and |
| 809 |
< |
$G^\prime$, and these two definitions agree with eath other very |
| 810 |
< |
well. This indicates very weak interaction between the metal and the |
| 811 |
< |
solvent, and is a typical case for acoustic impedance mismatch between |
| 812 |
< |
these two phases. |
| 813 |
< |
|
| 814 |
< |
As for Au(111) surfaces completely covered by butanethiols, the choice |
| 815 |
< |
of models for capping agent and solvent could impact the measurement |
| 816 |
< |
of $G$ and $G^\prime$ quite significantly. For Au-butanethiol/hexane |
| 817 |
< |
interfaces, using AA model for both butanethiol and hexane yields |
| 818 |
< |
substantially higher conductivity values than using UA model for at |
| 819 |
< |
least one component of the solvent and capping agent, which exceeds |
| 820 |
< |
the general range of experimental measurement results. This is |
| 821 |
< |
probably due to the classically treated C-H vibrations in the AA |
| 822 |
< |
model, which should not be appreciably populated at normal |
| 823 |
< |
temperatures. In comparison, once either the hexanes or the |
| 824 |
< |
butanethiols are deuterated, one can see a significantly lower $G$ and |
| 825 |
< |
$G^\prime$. In either of these cases, the C-H(D) vibrational overlap |
| 826 |
< |
between the solvent and the capping agent is removed. |
| 827 |
< |
[MAY NEED SPECTRA FIGURE] Conclusively, the |
| 828 |
< |
improperly treated C-H vibration in the AA model produced |
| 829 |
< |
over-predicted results accordingly. Compared to the AA model, the UA |
| 830 |
< |
model yields more reasonable results with higher computational |
| 831 |
< |
efficiency. |
| 832 |
< |
|
| 833 |
< |
However, for Au-butanethiol/toluene interfaces, having the AA |
| 917 |
> |
For the Au / butanethiol / toluene interfaces, having the AA |
| 918 |
|
butanethiol deuterated did not yield a significant change in the |
| 919 |
< |
measurement results. Compared to the C-H vibrational overlap between |
| 920 |
< |
hexane and butanethiol, both of which have alkyl chains, that overlap |
| 921 |
< |
between toluene and butanethiol is not so significant and thus does |
| 922 |
< |
not have as much contribution to the ``Intramolecular Vibration |
| 839 |
< |
Redistribution''[CITE HASE]. Conversely, extra degrees of freedom such |
| 840 |
< |
as the C-H vibrations could yield higher heat exchange rate between |
| 841 |
< |
these two phases and result in a much higher conductivity. |
| 919 |
> |
measured conductance. Compared to the C-H vibrational overlap between |
| 920 |
> |
hexane and butanethiol, both of which have alkyl chains, the overlap |
| 921 |
> |
between toluene and butanethiol is not as significant and thus does |
| 922 |
> |
not contribute as much to the heat exchange process. |
| 923 |
|
|
| 924 |
< |
Although the QSC model for Au is known to predict an overly low value |
| 925 |
< |
for bulk metal gold conductivity\cite{kuang:164101}, our computational |
| 926 |
< |
results for $G$ and $G^\prime$ do not seem to be affected by this |
| 927 |
< |
drawback of the model for metal. Instead, our results suggest that the |
| 928 |
< |
modeling of interfacial thermal transport behavior relies mainly on |
| 929 |
< |
the accuracy of the interaction descriptions between components |
| 930 |
< |
occupying the interfaces. |
| 924 |
> |
Previous observations of Zhang {\it et al.}\cite{hase:2010} indicate |
| 925 |
> |
that the {\it intra}molecular heat transport due to alkylthiols is |
| 926 |
> |
highly efficient. Combining our observations with those of Zhang {\it |
| 927 |
> |
et al.}, it appears that butanethiol acts as a channel to expedite |
| 928 |
> |
heat flow from the gold surface and into the alkyl chain. The |
| 929 |
> |
acoustic impedance mismatch between the metal and the liquid phase can |
| 930 |
> |
therefore be effectively reduced with the presence of suitable capping |
| 931 |
> |
agents. |
| 932 |
|
|
| 933 |
< |
\subsection{Role of Capping Agent in Interfacial Thermal Conductance} |
| 934 |
< |
To investigate the mechanism of this interfacial thermal conductance, |
| 935 |
< |
the vibrational spectra of various gold systems were obtained and are |
| 936 |
< |
shown as in Fig. \ref{vibration}. |
| 937 |
< |
[MAY RELATE TO HASE'S] |
| 938 |
< |
The gold surfaces covered by butanethiol molecules, compared to bare |
| 939 |
< |
gold surfaces, exhibit an additional peak observed at the frequency of |
| 940 |
< |
$\sim$170cm$^{-1}$, which is attributed to the S-Au bonding |
| 859 |
< |
vibration. This vibration enables efficient thermal transport from |
| 860 |
< |
surface Au layer to the capping agents. |
| 861 |
< |
[MAY PUT IN OTHER SECTION] Simultaneously, as shown in |
| 862 |
< |
the lower panel of Fig. \ref{vibration}, the large overlap of the |
| 863 |
< |
vibration spectra of butanethiol and hexane in the All-Atom model, |
| 864 |
< |
including the C-H vibration, also suggests high thermal exchange |
| 865 |
< |
efficiency. The combination of these two effects produces the drastic |
| 866 |
< |
interfacial thermal conductance enhancement in the All-Atom model. |
| 933 |
> |
Deuterated models in the UA force field did not decouple the thermal |
| 934 |
> |
transport as well as in the AA force field. The UA models, even |
| 935 |
> |
though they have eliminated the high frequency C-H vibrational |
| 936 |
> |
overlap, still have significant overlap in the lower-frequency |
| 937 |
> |
portions of the infrared spectra (Figure \ref{uahxnua}). Deuterating |
| 938 |
> |
the UA models did not decouple the low frequency region enough to |
| 939 |
> |
produce an observable difference for the results of $G$ (Table |
| 940 |
> |
\ref{modelTest}). |
| 941 |
|
|
| 868 |
– |
[NEED SEPARATE FIGURE. MAY NEED TO CONVERT TO JPEG] |
| 942 |
|
\begin{figure} |
| 943 |
< |
\includegraphics[width=\linewidth]{vibration} |
| 944 |
< |
\caption{Vibrational spectra obtained for gold in different |
| 945 |
< |
environments.} |
| 946 |
< |
\label{vibration} |
| 943 |
> |
\includegraphics[width=\linewidth]{uahxnua} |
| 944 |
> |
\caption{Vibrational spectra obtained for normal (upper) and |
| 945 |
> |
deuterated (lower) hexane in Au-butanethiol/hexane |
| 946 |
> |
systems. Butanethiol spectra are shown as reference. Both hexane and |
| 947 |
> |
butanethiol were using United-Atom models.} |
| 948 |
> |
\label{uahxnua} |
| 949 |
|
\end{figure} |
| 950 |
|
|
| 876 |
– |
[MAY ADD COMPARISON OF G AND G', AU SLAB WIDTHS, ETC] |
| 877 |
– |
% The results show that the two definitions used for $G$ yield |
| 878 |
– |
% comparable values, though $G^\prime$ tends to be smaller. |
| 879 |
– |
|
| 951 |
|
\section{Conclusions} |
| 952 |
< |
The NIVS algorithm we developed has been applied to simulations of |
| 953 |
< |
Au-butanethiol surfaces with organic solvents. This algorithm allows |
| 954 |
< |
effective unphysical thermal flux transferred between the metal and |
| 955 |
< |
the liquid phase. With the flux applied, we were able to measure the |
| 956 |
< |
corresponding thermal gradient and to obtain interfacial thermal |
| 957 |
< |
conductivities. Our simulations have seen significant conductance |
| 958 |
< |
enhancement with the presence of capping agent, compared to the bare |
| 959 |
< |
gold / liquid interfaces. The acoustic impedance mismatch between the |
| 960 |
< |
metal and the liquid phase is effectively eliminated by proper capping |
| 952 |
> |
The NIVS algorithm has been applied to simulations of |
| 953 |
> |
butanethiol-capped Au(111) surfaces in the presence of organic |
| 954 |
> |
solvents. This algorithm allows the application of unphysical thermal |
| 955 |
> |
flux to transfer heat between the metal and the liquid phase. With the |
| 956 |
> |
flux applied, we were able to measure the corresponding thermal |
| 957 |
> |
gradients and to obtain interfacial thermal conductivities. Under |
| 958 |
> |
steady states, 2-3 ns trajectory simulations are sufficient for |
| 959 |
> |
computation of this quantity. |
| 960 |
> |
|
| 961 |
> |
Our simulations have seen significant conductance enhancement in the |
| 962 |
> |
presence of capping agent, compared with the bare gold / liquid |
| 963 |
> |
interfaces. The acoustic impedance mismatch between the metal and the |
| 964 |
> |
liquid phase is effectively eliminated by a chemically-bonded capping |
| 965 |
|
agent. Furthermore, the coverage precentage of the capping agent plays |
| 966 |
< |
an important role in the interfacial thermal transport process. |
| 966 |
> |
an important role in the interfacial thermal transport |
| 967 |
> |
process. Moderately low coverages allow higher contact between capping |
| 968 |
> |
agent and solvent, and thus could further enhance the heat transfer |
| 969 |
> |
process, giving a non-monotonic behavior of conductance with |
| 970 |
> |
increasing coverage. |
| 971 |
|
|
| 972 |
< |
Our measurement results, particularly of the UA models, agree with |
| 973 |
< |
available experimental data. This indicates that our force field |
| 895 |
< |
parameters have a nice description of the interactions between the |
| 896 |
< |
particles at the interfaces. AA models tend to overestimate the |
| 972 |
> |
Our results, particularly using the UA models, agree well with |
| 973 |
> |
available experimental data. The AA models tend to overestimate the |
| 974 |
|
interfacial thermal conductance in that the classically treated C-H |
| 975 |
< |
vibration would be overly sampled. Compared to the AA models, the UA |
| 976 |
< |
models have higher computational efficiency with satisfactory |
| 977 |
< |
accuracy, and thus are preferable in interfacial thermal transport |
| 978 |
< |
modelings. |
| 975 |
> |
vibrations become too easily populated. Compared to the AA models, the |
| 976 |
> |
UA models have higher computational efficiency with satisfactory |
| 977 |
> |
accuracy, and thus are preferable in modeling interfacial thermal |
| 978 |
> |
transport. |
| 979 |
|
|
| 980 |
< |
Vlugt {\it et al.} has investigated the surface thiol structures for |
| 981 |
< |
nanocrystal gold and pointed out that they differs from those of the |
| 982 |
< |
Au(111) surface\cite{vlugt:cpc2007154}. This difference might lead to |
| 983 |
< |
change of interfacial thermal transport behavior as well. To |
| 984 |
< |
investigate this problem, an effective means to introduce thermal flux |
| 985 |
< |
and measure the corresponding thermal gradient is desirable for |
| 986 |
< |
simulating structures with spherical symmetry. |
| 980 |
> |
Of the two definitions for $G$, the discrete form |
| 981 |
> |
(Eq. \ref{discreteG}) was easier to use and gives out relatively |
| 982 |
> |
consistent results, while the derivative form (Eq. \ref{derivativeG}) |
| 983 |
> |
is not as versatile. Although $G^\prime$ gives out comparable results |
| 984 |
> |
and follows similar trend with $G$ when measuring close to fully |
| 985 |
> |
covered or bare surfaces, the spatial resolution of $T$ profile |
| 986 |
> |
required for the use of a derivative form is limited by the number of |
| 987 |
> |
bins and the sampling required to obtain thermal gradient information. |
| 988 |
|
|
| 989 |
+ |
Vlugt {\it et al.} have investigated the surface thiol structures for |
| 990 |
+ |
nanocrystalline gold and pointed out that they differ from those of |
| 991 |
+ |
the Au(111) surface.\cite{landman:1998,vlugt:cpc2007154} This |
| 992 |
+ |
difference could also cause differences in the interfacial thermal |
| 993 |
+ |
transport behavior. To investigate this problem, one would need an |
| 994 |
+ |
effective method for applying thermal gradients in non-planar |
| 995 |
+ |
(i.e. spherical) geometries. |
| 996 |
|
|
| 997 |
|
\section{Acknowledgments} |
| 998 |
|
Support for this project was provided by the National Science |
| 999 |
|
Foundation under grant CHE-0848243. Computational time was provided by |
| 1000 |
|
the Center for Research Computing (CRC) at the University of Notre |
| 1001 |
< |
Dame. \newpage |
| 1001 |
> |
Dame. |
| 1002 |
> |
\newpage |
| 1003 |
|
|
| 1004 |
|
\bibliography{interfacial} |
| 1005 |
|
|
