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\begin{document} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Introduction} |
| 76 |
– |
[BACKGROUND FOR INTERFACIAL THERMAL CONDUCTANCE PROBLEM] |
| 76 |
|
Interfacial thermal conductance is extensively studied both |
| 77 |
< |
experimentally and computationally, and systems with interfaces |
| 78 |
< |
present are generally heterogeneous. Although interfaces are commonly |
| 79 |
< |
barriers to heat transfer, it has been |
| 80 |
< |
reported\cite{doi:10.1021/la904855s} that under specific circustances, |
| 81 |
< |
e.g. with certain capping agents present on the surface, interfacial |
| 82 |
< |
conductance can be significantly enhanced. However, heat conductance |
| 83 |
< |
of molecular and nano-scale interfaces will be affected by the |
| 84 |
< |
chemical details of the surface and is challenging to |
| 85 |
< |
experimentalist. The lower thermal flux through interfaces is even |
| 87 |
< |
more difficult to measure with EMD and forward NEMD simulation |
| 88 |
< |
methods. Therefore, developing good simulation methods will be |
| 89 |
< |
desirable in order to investigate thermal transport across interfaces. |
| 77 |
> |
experimentally and computationally\cite{cahill:793}, due to its |
| 78 |
> |
importance in nanoscale science and technology. Reliability of |
| 79 |
> |
nanoscale devices depends on their thermal transport |
| 80 |
> |
properties. Unlike bulk homogeneous materials, nanoscale materials |
| 81 |
> |
features significant presence of interfaces, and these interfaces |
| 82 |
> |
could dominate the heat transfer behavior of these |
| 83 |
> |
materials. Furthermore, these materials are generally heterogeneous, |
| 84 |
> |
which challenges traditional research methods for homogeneous |
| 85 |
> |
systems. |
| 86 |
|
|
| 87 |
+ |
Heat conductance of molecular and nano-scale interfaces will be |
| 88 |
+ |
affected by the chemical details of the surface. Experimentally, |
| 89 |
+ |
various interfaces have been investigated for their thermal |
| 90 |
+ |
conductance properties. Wang {\it et al.} studied heat transport |
| 91 |
+ |
through long-chain hydrocarbon monolayers on gold substrate at |
| 92 |
+ |
individual molecular level\cite{Wang10082007}; Schmidt {\it et al.} |
| 93 |
+ |
studied the role of CTAB on thermal transport between gold nanorods |
| 94 |
+ |
and solvent\cite{doi:10.1021/jp8051888}; Juv\'e {\it et al.} studied |
| 95 |
+ |
the cooling dynamics, which is controlled by thermal interface |
| 96 |
+ |
resistence of glass-embedded metal |
| 97 |
+ |
nanoparticles\cite{PhysRevB.80.195406}. Although interfaces are |
| 98 |
+ |
commonly barriers for heat transport, Alper {\it et al.} suggested |
| 99 |
+ |
that specific ligands (capping agents) could completely eliminate this |
| 100 |
+ |
barrier ($G\rightarrow\infty$)\cite{doi:10.1021/la904855s}. |
| 101 |
+ |
|
| 102 |
+ |
Theoretical and computational models have also been used to study the |
| 103 |
+ |
interfacial thermal transport in order to gain an understanding of |
| 104 |
+ |
this phenomena at the molecular level. Recently, Hase and coworkers |
| 105 |
+ |
employed Non-Equilibrium Molecular Dynamics (NEMD) simulations to |
| 106 |
+ |
study thermal transport from hot Au(111) substrate to a self-assembled |
| 107 |
+ |
monolayer of alkylthiol with relatively long chain (8-20 carbon |
| 108 |
+ |
atoms)\cite{hase:2010,hase:2011}. However, ensemble averaged |
| 109 |
+ |
measurements for heat conductance of interfaces between the capping |
| 110 |
+ |
monolayer on Au and a solvent phase has yet to be studied. |
| 111 |
+ |
The comparatively low thermal flux through interfaces is |
| 112 |
+ |
difficult to measure with Equilibrium MD or forward NEMD simulation |
| 113 |
+ |
methods. Therefore, the Reverse NEMD (RNEMD) methods would have the |
| 114 |
+ |
advantage of having this difficult to measure flux known when studying |
| 115 |
+ |
the thermal transport across interfaces, given that the simulation |
| 116 |
+ |
methods being able to effectively apply an unphysical flux in |
| 117 |
+ |
non-homogeneous systems. |
| 118 |
+ |
|
| 119 |
|
Recently, we have developed the Non-Isotropic Velocity Scaling (NIVS) |
| 120 |
|
algorithm for RNEMD simulations\cite{kuang:164101}. This algorithm |
| 121 |
|
retains the desirable features of RNEMD (conservation of linear |
| 122 |
|
momentum and total energy, compatibility with periodic boundary |
| 123 |
|
conditions) while establishing true thermal distributions in each of |
| 124 |
< |
the two slabs. Furthermore, it allows more effective thermal exchange |
| 125 |
< |
between particles of different identities, and thus enables extensive |
| 126 |
< |
study of interfacial conductance. |
| 124 |
> |
the two slabs. Furthermore, it allows effective thermal exchange |
| 125 |
> |
between particles of different identities, and thus makes the study of |
| 126 |
> |
interfacial conductance much simpler. |
| 127 |
|
|
| 128 |
+ |
The work presented here deals with the Au(111) surface covered to |
| 129 |
+ |
varying degrees by butanethiol, a capping agent with short carbon |
| 130 |
+ |
chain, and solvated with organic solvents of different molecular |
| 131 |
+ |
properties. Different models were used for both the capping agent and |
| 132 |
+ |
the solvent force field parameters. Using the NIVS algorithm, the |
| 133 |
+ |
thermal transport across these interfaces was studied and the |
| 134 |
+ |
underlying mechanism for this phenomena was investigated. |
| 135 |
+ |
|
| 136 |
+ |
[MAY ADD WHY STUDY AU-THIOL SURFACE; CITE SHAOYI JIANG] |
| 137 |
+ |
|
| 138 |
|
\section{Methodology} |
| 139 |
< |
\subsection{Algorithm} |
| 140 |
< |
[BACKGROUND FOR MD METHODS] |
| 141 |
< |
There have been many algorithms for computing thermal conductivity |
| 142 |
< |
using molecular dynamics simulations. However, interfacial conductance |
| 143 |
< |
is at least an order of magnitude smaller. This would make the |
| 144 |
< |
calculation even more difficult for those slowly-converging |
| 145 |
< |
equilibrium methods. Imposed-flux non-equilibrium |
| 139 |
> |
\subsection{Imposd-Flux Methods in MD Simulations} |
| 140 |
> |
For systems with low interfacial conductivity one must have a method |
| 141 |
> |
capable of generating relatively small fluxes, compared to those |
| 142 |
> |
required for bulk conductivity. This requirement makes the calculation |
| 143 |
> |
even more difficult for those slowly-converging equilibrium |
| 144 |
> |
methods\cite{Viscardy:2007lq}. |
| 145 |
> |
Forward methods impose gradient, but in interfacail conditions it is |
| 146 |
> |
not clear what behavior to impose at the boundary... |
| 147 |
> |
Imposed-flux reverse non-equilibrium |
| 148 |
|
methods\cite{MullerPlathe:1997xw} have the flux set {\it a priori} and |
| 149 |
< |
the response of temperature or momentum gradients are easier to |
| 150 |
< |
measure than the flux, if unknown, and thus, is a preferable way to |
| 151 |
< |
the forward NEMD methods. Although the momentum swapping approach for |
| 152 |
< |
flux-imposing can be used for exchanging energy between particles of |
| 153 |
< |
different identity, the kinetic energy transfer efficiency is affected |
| 154 |
< |
by the mass difference between the particles, which limits its |
| 115 |
< |
application on heterogeneous interfacial systems. |
| 149 |
> |
the thermal response becomes easier to |
| 150 |
> |
measure than the flux. Although M\"{u}ller-Plathe's original momentum |
| 151 |
> |
swapping approach can be used for exchanging energy between particles |
| 152 |
> |
of different identity, the kinetic energy transfer efficiency is |
| 153 |
> |
affected by the mass difference between the particles, which limits |
| 154 |
> |
its application on heterogeneous interfacial systems. |
| 155 |
|
|
| 156 |
< |
The non-isotropic velocity scaling (NIVS)\cite{kuang:164101} approach in |
| 157 |
< |
non-equilibrium MD simulations is able to impose relatively large |
| 158 |
< |
kinetic energy flux without obvious perturbation to the velocity |
| 159 |
< |
distribution of the simulated systems. Furthermore, this approach has |
| 156 |
> |
The non-isotropic velocity scaling (NIVS)\cite{kuang:164101} approach to |
| 157 |
> |
non-equilibrium MD simulations is able to impose a wide range of |
| 158 |
> |
kinetic energy fluxes without obvious perturbation to the velocity |
| 159 |
> |
distributions of the simulated systems. Furthermore, this approach has |
| 160 |
|
the advantage in heterogeneous interfaces in that kinetic energy flux |
| 161 |
|
can be applied between regions of particles of arbitary identity, and |
| 162 |
< |
the flux quantity is not restricted by particle mass difference. |
| 162 |
> |
the flux will not be restricted by difference in particle mass. |
| 163 |
|
|
| 164 |
|
The NIVS algorithm scales the velocity vectors in two separate regions |
| 165 |
|
of a simulation system with respective diagonal scaling matricies. To |
| 166 |
|
determine these scaling factors in the matricies, a set of equations |
| 167 |
|
including linear momentum conservation and kinetic energy conservation |
| 168 |
< |
constraints and target momentum/energy flux satisfaction is |
| 169 |
< |
solved. With the scaling operation applied to the system in a set |
| 170 |
< |
frequency, corresponding momentum/temperature gradients can be built, |
| 171 |
< |
which can be used for computing transportation properties and other |
| 172 |
< |
applications related to momentum/temperature gradients. The NIVS |
| 134 |
< |
algorithm conserves momenta and energy and does not depend on an |
| 135 |
< |
external thermostat. |
| 168 |
> |
constraints and target energy flux satisfaction is solved. With the |
| 169 |
> |
scaling operation applied to the system in a set frequency, bulk |
| 170 |
> |
temperature gradients can be easily established, and these can be used |
| 171 |
> |
for computing thermal conductivities. The NIVS algorithm conserves |
| 172 |
> |
momenta and energy and does not depend on an external thermostat. |
| 173 |
|
|
| 174 |
|
\subsection{Defining Interfacial Thermal Conductivity $G$} |
| 175 |
|
For interfaces with a relatively low interfacial conductance, the bulk |
| 187 |
|
T_\mathrm{cold}\rangle}$ are the average observed temperature of the |
| 188 |
|
two separated phases. |
| 189 |
|
|
| 190 |
< |
When the interfacial conductance is {\it not} small, two ways can be |
| 191 |
< |
used to define $G$. |
| 190 |
> |
When the interfacial conductance is {\it not} small, there are two |
| 191 |
> |
ways to define $G$. |
| 192 |
|
|
| 193 |
< |
One way is to assume the temperature is discretely different on two |
| 194 |
< |
sides of the interface, $G$ can be calculated with the thermal flux |
| 195 |
< |
applied $J$ and the maximum temperature difference measured along the |
| 196 |
< |
thermal gradient max($\Delta T$), which occurs at the interface, as: |
| 193 |
> |
One way is to assume the temperature is discrete on the two sides of |
| 194 |
> |
the interface. $G$ can be calculated using the applied thermal flux |
| 195 |
> |
$J$ and the maximum temperature difference measured along the thermal |
| 196 |
> |
gradient max($\Delta T$), which occurs at the Gibbs deviding surface |
| 197 |
> |
(Figure \ref{demoPic}): |
| 198 |
|
\begin{equation} |
| 199 |
|
G=\frac{J}{\Delta T} |
| 200 |
|
\label{discreteG} |
| 201 |
|
\end{equation} |
| 202 |
|
|
| 203 |
+ |
\begin{figure} |
| 204 |
+ |
\includegraphics[width=\linewidth]{method} |
| 205 |
+ |
\caption{Interfacial conductance can be calculated by applying an |
| 206 |
+ |
(unphysical) kinetic energy flux between two slabs, one located |
| 207 |
+ |
within the metal and another on the edge of the periodic box. The |
| 208 |
+ |
system responds by forming a thermal response or a gradient. In |
| 209 |
+ |
bulk liquids, this gradient typically has a single slope, but in |
| 210 |
+ |
interfacial systems, there are distinct thermal conductivity |
| 211 |
+ |
domains. The interfacial conductance, $G$ is found by measuring the |
| 212 |
+ |
temperature gap at the Gibbs dividing surface, or by using second |
| 213 |
+ |
derivatives of the thermal profile.} |
| 214 |
+ |
\label{demoPic} |
| 215 |
+ |
\end{figure} |
| 216 |
+ |
|
| 217 |
|
The other approach is to assume a continuous temperature profile along |
| 218 |
|
the thermal gradient axis (e.g. $z$) and define $G$ at the point where |
| 219 |
|
the magnitude of thermal conductivity $\lambda$ change reach its |
| 229 |
|
|
| 230 |
|
With the temperature profile obtained from simulations, one is able to |
| 231 |
|
approximate the first and second derivatives of $T$ with finite |
| 232 |
< |
difference method and thus calculate $G^\prime$. |
| 232 |
> |
difference methods and thus calculate $G^\prime$. |
| 233 |
|
|
| 234 |
< |
In what follows, both definitions are used for calculation and comparison. |
| 234 |
> |
In what follows, both definitions have been used for calculation and |
| 235 |
> |
are compared in the results. |
| 236 |
|
|
| 237 |
< |
[IMPOSE G DEFINITION INTO OUR SYSTEMS] |
| 238 |
< |
To facilitate the use of the above definitions in calculating $G$ and |
| 239 |
< |
$G^\prime$, we have a metal slab with its (111) surfaces perpendicular |
| 240 |
< |
to the $z$-axis of our simulation cells. With or withour capping |
| 241 |
< |
agents on the surfaces, the metal slab is solvated with organic |
| 189 |
< |
solvents, as illustrated in Figure \ref{demoPic}. |
| 237 |
> |
To compare the above definitions ($G$ and $G^\prime$), we have modeled |
| 238 |
> |
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 |
| 241 |
> |
illustrated in Figure \ref{demoPic}. |
| 242 |
|
|
| 243 |
< |
\begin{figure} |
| 244 |
< |
\includegraphics[width=\linewidth]{demoPic} |
| 245 |
< |
\caption{A sample showing how a metal slab has its (111) surface |
| 246 |
< |
covered by capping agent molecules and solvated by hexane.} |
| 247 |
< |
\label{demoPic} |
| 248 |
< |
\end{figure} |
| 249 |
< |
|
| 250 |
< |
With a simulation cell setup following the above manner, one is able |
| 199 |
< |
to equilibrate the system and impose an unphysical thermal flux |
| 200 |
< |
between the liquid and the metal phase with the NIVS algorithm. Under |
| 201 |
< |
a stablized thermal gradient induced by periodically applying the |
| 202 |
< |
unphysical flux, one is able to obtain a temperature profile and the |
| 203 |
< |
physical thermal flux corresponding to it, which equals to the |
| 204 |
< |
unphysical flux applied by NIVS. These data enables the evaluation of |
| 205 |
< |
the interfacial thermal conductance of a surface. Figure \ref{gradT} |
| 206 |
< |
is an example how those stablized thermal gradient can be used to |
| 207 |
< |
obtain the 1st and 2nd derivatives of the temperature profile. |
| 243 |
> |
With the simulation cell described above, we are able to equilibrate |
| 244 |
> |
the system and impose an unphysical thermal flux between the liquid |
| 245 |
> |
and the metal phase using the NIVS algorithm. By periodically applying |
| 246 |
> |
the unphysical flux, we are able to obtain a temperature profile and |
| 247 |
> |
its spatial derivatives. These quantities enable the evaluation of the |
| 248 |
> |
interfacial thermal conductance of a surface. Figure \ref{gradT} is an |
| 249 |
> |
example how those applied thermal fluxes can be used to obtain the 1st |
| 250 |
> |
and 2nd derivatives of the temperature profile. |
| 251 |
|
|
| 252 |
|
\begin{figure} |
| 253 |
|
\includegraphics[width=\linewidth]{gradT} |
| 254 |
< |
\caption{The 1st and 2nd derivatives of temperature profile can be |
| 255 |
< |
obtained with finite difference approximation.} |
| 254 |
> |
\caption{A sample of Au-butanethiol/hexane interfacial system and the |
| 255 |
> |
temperature profile after a kinetic energy flux is imposed to |
| 256 |
> |
it. The 1st and 2nd derivatives of the temperature profile can be |
| 257 |
> |
obtained with finite difference approximation (lower panel).} |
| 258 |
|
\label{gradT} |
| 259 |
|
\end{figure} |
| 260 |
|
|
| 216 |
– |
[MAY INCLUDE POWER SPECTRUM PROTOCOL] |
| 217 |
– |
|
| 261 |
|
\section{Computational Details} |
| 262 |
|
\subsection{Simulation Protocol} |
| 263 |
< |
In our simulations, Au is used to construct a metal slab with bare |
| 264 |
< |
(111) surface perpendicular to the $z$-axis. Different slab thickness |
| 265 |
< |
(layer numbers of Au) are simulated. This metal slab is first |
| 266 |
< |
equilibrated under normal pressure (1 atm) and a desired |
| 267 |
< |
temperature. After equilibration, butanethiol is used as the capping |
| 268 |
< |
agent molecule to cover the bare Au (111) surfaces evenly. The sulfur |
| 269 |
< |
atoms in the butanethiol molecules would occupy the three-fold sites |
| 270 |
< |
of the surfaces, and the maximal butanethiol capacity on Au surface is |
| 271 |
< |
$1/3$ of the total number of surface Au atoms[CITATION]. A series of |
| 272 |
< |
different coverage surfaces is investigated in order to study the |
| 273 |
< |
relation between coverage and conductance. |
| 263 |
> |
The NIVS algorithm has been implemented in our MD simulation code, |
| 264 |
> |
OpenMD\cite{Meineke:2005gd,openmd}, and was used for our |
| 265 |
> |
simulations. Different slab thickness (layer numbers of Au) were |
| 266 |
> |
simulated. Metal slabs were first equilibrated under atmospheric |
| 267 |
> |
pressure (1 atm) and a desired temperature (e.g. 200K). After |
| 268 |
> |
equilibration, butanethiol capping agents were placed at three-fold |
| 269 |
> |
sites on the Au(111) surfaces. The maximum butanethiol capacity on Au |
| 270 |
> |
surface is $1/3$ of the total number of surface Au |
| 271 |
> |
atoms\cite{vlugt:cpc2007154}. A series of different coverages was |
| 272 |
> |
investigated in order to study the relation between coverage and |
| 273 |
> |
interfacial conductance. |
| 274 |
|
|
| 275 |
< |
[COVERAGE DISCRIPTION] However, since the interactions between surface |
| 276 |
< |
Au and butanethiol is non-bonded, the capping agent molecules are |
| 277 |
< |
allowed to migrate to an empty neighbor three-fold site during a |
| 278 |
< |
simulation. Therefore, the initial configuration would not severely |
| 279 |
< |
affect the sampling of a variety of configurations of the same |
| 280 |
< |
coverage, and the final conductance measurement would be an average |
| 281 |
< |
effect of these configurations explored in the simulations. [MAY NEED FIGURES] |
| 275 |
> |
The capping agent molecules were allowed to migrate during the |
| 276 |
> |
simulations. They distributed themselves uniformly and sampled a |
| 277 |
> |
number of three-fold sites throughout out study. Therefore, the |
| 278 |
> |
initial configuration would not noticeably affect the sampling of a |
| 279 |
> |
variety of configurations of the same coverage, and the final |
| 280 |
> |
conductance measurement would be an average effect of these |
| 281 |
> |
configurations explored in the simulations. [MAY NEED FIGURES] |
| 282 |
|
|
| 283 |
< |
After the modified Au-butanethiol surface systems are equilibrated |
| 284 |
< |
under canonical ensemble, Packmol\cite{packmol} is used to pack |
| 285 |
< |
organic solvent molecules in the previously vacuum part of the |
| 286 |
< |
simulation cells, which guarantees that short range repulsive |
| 287 |
< |
interactions do not disrupt the simulations. Two solvents are |
| 288 |
< |
investigated, one which has little vibrational overlap with the |
| 246 |
< |
alkanethiol and plane-like shape (toluene), and one which has similar |
| 247 |
< |
vibrational frequencies and chain-like shape ({\it n}-hexane). [MAY |
| 248 |
< |
EXPLAIN WHY WE CHOOSE THEM] |
| 283 |
> |
After the modified Au-butanethiol surface systems were equilibrated |
| 284 |
> |
under canonical ensemble, organic solvent molecules were packed in the |
| 285 |
> |
previously empty part of the simulation cells\cite{packmol}. Two |
| 286 |
> |
solvents were investigated, one which has little vibrational overlap |
| 287 |
> |
with the alkanethiol and a planar shape (toluene), and one which has |
| 288 |
> |
similar vibrational frequencies and chain-like shape ({\it n}-hexane). |
| 289 |
|
|
| 290 |
< |
The spacing filled by solvent molecules, i.e. the gap between |
| 290 |
> |
The space filled by solvent molecules, i.e. the gap between |
| 291 |
|
periodically repeated Au-butanethiol surfaces should be carefully |
| 292 |
|
chosen. A very long length scale for the thermal gradient axis ($z$) |
| 293 |
|
may cause excessively hot or cold temperatures in the middle of the |
| 296 |
|
solvent molecules would change the normal behavior of the liquid |
| 297 |
|
phase. Therefore, our $N_{solvent}$ values were chosen to ensure that |
| 298 |
|
these extreme cases did not happen to our simulations. And the |
| 299 |
< |
corresponding spacing is usually $35 \sim 60$\AA. |
| 299 |
> |
corresponding spacing is usually $35 \sim 75$\AA. |
| 300 |
|
|
| 301 |
|
The initial configurations generated by Packmol are further |
| 302 |
|
equilibrated with the $x$ and $y$ dimensions fixed, only allowing |
| 327 |
|
\end{equation} |
| 328 |
|
|
| 329 |
|
\subsection{Force Field Parameters} |
| 330 |
< |
Our simulations include various components. Therefore, force field |
| 331 |
< |
parameter descriptions are needed for interactions both between the |
| 332 |
< |
same type of particles and between particles of different species. |
| 330 |
> |
Our simulations include various components. Figure \ref{demoMol} |
| 331 |
> |
demonstrates the sites defined for both United-Atom and All-Atom |
| 332 |
> |
models of the organic solvent and capping agent molecules in our |
| 333 |
> |
simulations. Force field parameter descriptions are needed for |
| 334 |
> |
interactions both between the same type of particles and between |
| 335 |
> |
particles of different species. |
| 336 |
|
|
| 337 |
+ |
\begin{figure} |
| 338 |
+ |
\includegraphics[width=\linewidth]{structures} |
| 339 |
+ |
\caption{Structures of the capping agent and solvents utilized in |
| 340 |
+ |
these simulations. The chemically-distinct sites (a-e) are expanded |
| 341 |
+ |
in terms of constituent atoms for both United Atom (UA) and All Atom |
| 342 |
+ |
(AA) force fields. Most parameters are from |
| 343 |
+ |
Refs. \protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} (UA) and |
| 344 |
+ |
\protect\cite{OPLSAA} (AA). Cross-interactions with the Au atoms are given |
| 345 |
+ |
in Table \ref{MnM}.} |
| 346 |
+ |
\label{demoMol} |
| 347 |
+ |
\end{figure} |
| 348 |
+ |
|
| 349 |
|
The Au-Au interactions in metal lattice slab is described by the |
| 350 |
< |
quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
| 350 |
> |
quantum Sutton-Chen (QSC) formulation\cite{PhysRevB.59.3527}. The QSC |
| 351 |
|
potentials include zero-point quantum corrections and are |
| 352 |
|
reparametrized for accurate surface energies compared to the |
| 353 |
|
Sutton-Chen potentials\cite{Chen90}. |
| 354 |
|
|
| 300 |
– |
Figure [REF] demonstrates how we name our pseudo-atoms of the |
| 301 |
– |
molecules in our simulations. |
| 302 |
– |
[FIGURE FOR MOLECULE NOMENCLATURE] |
| 303 |
– |
|
| 355 |
|
For both solvent molecules, straight chain {\it n}-hexane and aromatic |
| 356 |
|
toluene, United-Atom (UA) and All-Atom (AA) models are used |
| 357 |
|
respectively. The TraPPE-UA |
| 358 |
|
parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
| 359 |
< |
for our UA solvent molecules. In these models, pseudo-atoms are |
| 360 |
< |
located at the carbon centers for alkyl groups. By eliminating |
| 361 |
< |
explicit hydrogen atoms, these models are simple and computationally |
| 362 |
< |
efficient, while maintains good accuracy. However, the TraPPE-UA for |
| 363 |
< |
alkanes is known to predict a lower boiling point than experimental |
| 313 |
< |
values. Considering that after an unphysical thermal flux is applied |
| 314 |
< |
to a system, the temperature of ``hot'' area in the liquid phase would be |
| 315 |
< |
significantly higher than the average, to prevent over heating and |
| 316 |
< |
boiling of the liquid phase, the average temperature in our |
| 317 |
< |
simulations should be much lower than the liquid boiling point. [MORE DISCUSSION] |
| 318 |
< |
For UA-toluene model, rigid body constraints are applied, so that the |
| 319 |
< |
benzene ring and the methyl-CRar bond are kept rigid. This would save |
| 320 |
< |
computational time.[MORE DETAILS] |
| 359 |
> |
for our UA solvent molecules. In these models, sites are located at |
| 360 |
> |
the carbon centers for alkyl groups. Bonding interactions, including |
| 361 |
> |
bond stretches and bends and torsions, were used for intra-molecular |
| 362 |
> |
sites not separated by more than 3 bonds. Otherwise, for non-bonded |
| 363 |
> |
interactions, Lennard-Jones potentials are used. [MORE CITATION?] |
| 364 |
|
|
| 365 |
+ |
By eliminating explicit hydrogen atoms, these models are simple and |
| 366 |
+ |
computationally efficient, while maintains good accuracy. However, the |
| 367 |
+ |
TraPPE-UA for alkanes is known to predict a lower boiling point than |
| 368 |
+ |
experimental values. Considering that after an unphysical thermal flux |
| 369 |
+ |
is applied to a system, the temperature of ``hot'' area in the liquid |
| 370 |
+ |
phase would be significantly higher than the average, to prevent over |
| 371 |
+ |
heating and boiling of the liquid phase, the average temperature in |
| 372 |
+ |
our simulations should be much lower than the liquid boiling point. |
| 373 |
+ |
|
| 374 |
+ |
For UA-toluene model, the non-bonded potentials between |
| 375 |
+ |
inter-molecular sites have a similar Lennard-Jones formulation. For |
| 376 |
+ |
intra-molecular interactions, considering the stiffness of the benzene |
| 377 |
+ |
ring, rigid body constraints are applied for further computational |
| 378 |
+ |
efficiency. All bonds in the benzene ring and between the ring and the |
| 379 |
+ |
methyl group remain rigid during the progress of simulations. |
| 380 |
+ |
|
| 381 |
|
Besides the TraPPE-UA models, AA models for both organic solvents are |
| 382 |
|
included in our studies as well. For hexane, the OPLS-AA\cite{OPLSAA} |
| 383 |
< |
force field is used. [MORE DETAILS] |
| 384 |
< |
For toluene, the United Force Field developed by Rapp\'{e} {\it et |
| 385 |
< |
al.}\cite{doi:10.1021/ja00051a040} is adopted.[MORE DETAILS] |
| 383 |
> |
force field is used. Additional explicit hydrogen sites were |
| 384 |
> |
included. Besides bonding and non-bonded site-site interactions, |
| 385 |
> |
partial charges and the electrostatic interactions were added to each |
| 386 |
> |
CT and HC site. For toluene, the United Force Field developed by |
| 387 |
> |
Rapp\'{e} {\it et al.}\cite{doi:10.1021/ja00051a040} is |
| 388 |
> |
adopted. Without the rigid body constraints, bonding interactions were |
| 389 |
> |
included. For the aromatic ring, improper torsions (inversions) were |
| 390 |
> |
added as an extra potential for maintaining the planar shape. |
| 391 |
> |
[MORE CITATION?] |
| 392 |
|
|
| 393 |
|
The capping agent in our simulations, the butanethiol molecules can |
| 394 |
|
either use UA or AA model. The TraPPE-UA force fields includes |
| 397 |
|
parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) |
| 398 |
|
surfaces do not have the hydrogen atom bonded to sulfur. To adapt this |
| 399 |
|
change and derive suitable parameters for butanethiol adsorbed on |
| 400 |
< |
Au(111) surfaces, we adopt the S parameters from [CITATION CF VLUGT] |
| 401 |
< |
and modify parameters for its neighbor C atom for charge balance in |
| 402 |
< |
the molecule. Note that the model choice (UA or AA) of capping agent |
| 403 |
< |
can be different from the solvent. Regardless of model choice, the |
| 404 |
< |
force field parameters for interactions between capping agent and |
| 405 |
< |
solvent can be derived using Lorentz-Berthelot Mixing Rule: |
| 400 |
> |
Au(111) surfaces, we adopt the S parameters from Luedtke and |
| 401 |
> |
Landman\cite{landman:1998} and modify parameters for its neighbor C |
| 402 |
> |
atom for charge balance in the molecule. Note that the model choice |
| 403 |
> |
(UA or AA) of capping agent can be different from the |
| 404 |
> |
solvent. Regardless of model choice, the force field parameters for |
| 405 |
> |
interactions between capping agent and solvent can be derived using |
| 406 |
> |
Lorentz-Berthelot Mixing Rule: |
| 407 |
> |
\begin{eqnarray} |
| 408 |
> |
\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\ |
| 409 |
> |
\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}} |
| 410 |
> |
\end{eqnarray} |
| 411 |
|
|
| 342 |
– |
|
| 412 |
|
To describe the interactions between metal Au and non-metal capping |
| 413 |
|
agent and solvent particles, we refer to an adsorption study of alkyl |
| 414 |
|
thiols on gold surfaces by Vlugt {\it et |
| 415 |
|
al.}\cite{vlugt:cpc2007154} They fitted an effective Lennard-Jones |
| 416 |
|
form of potential parameters for the interaction between Au and |
| 417 |
|
pseudo-atoms CH$_x$ and S based on a well-established and widely-used |
| 418 |
< |
effective potential of Hautman and Klein[CITATION] for the Au(111) |
| 419 |
< |
surface. As our simulations require the gold lattice slab to be |
| 420 |
< |
non-rigid so that it could accommodate kinetic energy for thermal |
| 418 |
> |
effective potential of Hautman and Klein\cite{hautman:4994} for the |
| 419 |
> |
Au(111) surface. As our simulations require the gold lattice slab to |
| 420 |
> |
be non-rigid so that it could accommodate kinetic energy for thermal |
| 421 |
|
transport study purpose, the pair-wise form of potentials is |
| 422 |
|
preferred. |
| 423 |
|
|
| 424 |
|
Besides, the potentials developed from {\it ab initio} calculations by |
| 425 |
|
Leng {\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
| 426 |
< |
interactions between Au and aromatic C/H atoms in toluene.[MORE DETAILS] |
| 426 |
> |
interactions between Au and aromatic C/H atoms in toluene. A set of |
| 427 |
> |
pseudo Lennard-Jones parameters were provided for Au in their force |
| 428 |
> |
fields. By using the Mixing Rule, this can be used to derive pair-wise |
| 429 |
> |
potentials for non-bonded interactions between Au and non-metal sites. |
| 430 |
|
|
| 431 |
|
However, the Lennard-Jones parameters between Au and other types of |
| 432 |
< |
particles in our simulations are not yet well-established. For these |
| 433 |
< |
interactions, we attempt to derive their parameters using the Mixing |
| 434 |
< |
Rule. To do this, the ``Metal-non-Metal'' (MnM) interaction parameters |
| 435 |
< |
for Au is first extracted from the Au-CH$_x$ parameters by applying |
| 436 |
< |
the Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM'' |
| 432 |
> |
particles, such as All-Atom normal alkanes in our simulations are not |
| 433 |
> |
yet well-established. For these interactions, we attempt to derive |
| 434 |
> |
their parameters using the Mixing Rule. To do this, Au pseudo |
| 435 |
> |
Lennard-Jones parameters for ``Metal-non-Metal'' (MnM) interactions |
| 436 |
> |
were first extracted from the Au-CH$_x$ parameters by applying the |
| 437 |
> |
Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM'' |
| 438 |
|
parameters in our simulations. |
| 439 |
|
|
| 440 |
|
\begin{table*} |
| 441 |
|
\begin{minipage}{\linewidth} |
| 442 |
|
\begin{center} |
| 443 |
< |
\caption{Lennard-Jones parameters for Au-non-Metal |
| 444 |
< |
interactions in our simulations.} |
| 445 |
< |
|
| 446 |
< |
\begin{tabular}{ccc} |
| 443 |
> |
\caption{Non-bonded interaction parameters (including cross |
| 444 |
> |
interactions with Au atoms) for both force fields used in this |
| 445 |
> |
work.} |
| 446 |
> |
\begin{tabular}{lllllll} |
| 447 |
|
\hline\hline |
| 448 |
< |
Non-metal atom & $\sigma$ & $\epsilon$ \\ |
| 449 |
< |
(or pseudo-atom) & \AA & kcal/mol \\ |
| 448 |
> |
& Site & $\sigma_{ii}$ & $\epsilon_{ii}$ & $q_i$ & |
| 449 |
> |
$\sigma_{Au-i}$ & $\epsilon_{Au-i}$ \\ |
| 450 |
> |
& & (\AA) & (kcal/mol) & ($e$) & (\AA) & (kcal/mol) \\ |
| 451 |
|
\hline |
| 452 |
< |
S & 2.40 & 8.465 \\ |
| 453 |
< |
CH3 & 3.54 & 0.2146 \\ |
| 454 |
< |
CH2 & 3.54 & 0.1749 \\ |
| 455 |
< |
CT3 & 3.365 & 0.1373 \\ |
| 456 |
< |
CT2 & 3.365 & 0.1373 \\ |
| 457 |
< |
CTT & 3.365 & 0.1373 \\ |
| 458 |
< |
HC & 2.865 & 0.09256 \\ |
| 459 |
< |
CHar & 3.4625 & 0.1680 \\ |
| 460 |
< |
CRar & 3.555 & 0.1604 \\ |
| 461 |
< |
CA & 3.173 & 0.0640 \\ |
| 462 |
< |
HA & 2.746 & 0.0414 \\ |
| 452 |
> |
United Atom (UA) |
| 453 |
> |
&CH3 & 3.75 & 0.1947 & - & 3.54 & 0.2146 \\ |
| 454 |
> |
&CH2 & 3.95 & 0.0914 & - & 3.54 & 0.1749 \\ |
| 455 |
> |
&CHar & 3.695 & 0.1003 & - & 3.4625 & 0.1680 \\ |
| 456 |
> |
&CRar & 3.88 & 0.04173 & - & 3.555 & 0.1604 \\ |
| 457 |
> |
\hline |
| 458 |
> |
All Atom (AA) |
| 459 |
> |
&CT3 & 3.50 & 0.066 & -0.18 & 3.365 & 0.1373 \\ |
| 460 |
> |
&CT2 & 3.50 & 0.066 & -0.12 & 3.365 & 0.1373 \\ |
| 461 |
> |
&CTT & 3.50 & 0.066 & -0.065 & 3.365 & 0.1373 \\ |
| 462 |
> |
&HC & 2.50 & 0.030 & 0.06 & 2.865 & 0.09256 \\ |
| 463 |
> |
&CA & 3.55 & 0.070 & -0.115 & 3.173 & 0.0640 \\ |
| 464 |
> |
&HA & 2.42 & 0.030 & 0.115 & 2.746 & 0.0414 \\ |
| 465 |
> |
\hline |
| 466 |
> |
Both UA and AA |
| 467 |
> |
& S & 4.45 & 0.25 & - & 2.40 & 8.465 \\ |
| 468 |
|
\hline\hline |
| 469 |
|
\end{tabular} |
| 470 |
|
\label{MnM} |
| 485 |
|
results. |
| 486 |
|
|
| 487 |
|
In some of our simulations, we allowed $L_x$ and $L_y$ to change |
| 488 |
< |
during equilibrating the liquid phase. Due to the stiffness of the Au |
| 489 |
< |
slab, $L_x$ and $L_y$ would not change noticeably after |
| 490 |
< |
equilibration. Although $L_z$ could fluctuate $\sim$1\% after a system |
| 491 |
< |
is fully equilibrated in the NPT ensemble, this fluctuation, as well |
| 492 |
< |
as those comparably smaller to $L_x$ and $L_y$, would not be magnified |
| 493 |
< |
on the calculated $G$'s, as shown in Table \ref{AuThiolHexaneUA}. This |
| 494 |
< |
insensivity to $L_i$ fluctuations allows reliable measurement of $G$'s |
| 495 |
< |
without the necessity of extremely cautious equilibration process. |
| 488 |
> |
during equilibrating the liquid phase. Due to the stiffness of the |
| 489 |
> |
crystalline Au structure, $L_x$ and $L_y$ would not change noticeably |
| 490 |
> |
after equilibration. Although $L_z$ could fluctuate $\sim$1\% after a |
| 491 |
> |
system is fully equilibrated in the NPT ensemble, this fluctuation, as |
| 492 |
> |
well as those of $L_x$ and $L_y$ (which is significantly smaller), |
| 493 |
> |
would not be magnified on the calculated $G$'s, as shown in Table |
| 494 |
> |
\ref{AuThiolHexaneUA}. This insensivity to $L_i$ fluctuations allows |
| 495 |
> |
reliable measurement of $G$'s without the necessity of extremely |
| 496 |
> |
cautious equilibration process. |
| 497 |
|
|
| 498 |
|
As stated in our computational details, the spacing filled with |
| 499 |
|
solvent molecules can be chosen within a range. This allows some |
| 520 |
|
the thermal flux across the interface. For our simulations, we denote |
| 521 |
|
$J_z$ to be positive when the physical thermal flux is from the liquid |
| 522 |
|
to metal, and negative vice versa. The $G$'s measured under different |
| 523 |
< |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and [REF]. These |
| 524 |
< |
results do not suggest that $G$ is dependent on $J_z$ within this flux |
| 525 |
< |
range. The linear response of flux to thermal gradient simplifies our |
| 526 |
< |
investigations in that we can rely on $G$ measurement with only a |
| 527 |
< |
couple $J_z$'s and do not need to test a large series of fluxes. |
| 523 |
> |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and |
| 524 |
> |
\ref{AuThiolToluene}. These results do not suggest that $G$ is |
| 525 |
> |
dependent on $J_z$ within this flux range. The linear response of flux |
| 526 |
> |
to thermal gradient simplifies our investigations in that we can rely |
| 527 |
> |
on $G$ measurement with only a couple $J_z$'s and do not need to test |
| 528 |
> |
a large series of fluxes. |
| 529 |
|
|
| 449 |
– |
%ADD MORE TO TABLE |
| 530 |
|
\begin{table*} |
| 531 |
|
\begin{minipage}{\linewidth} |
| 532 |
|
\begin{center} |
| 533 |
|
\caption{Computed interfacial thermal conductivity ($G$ and |
| 534 |
|
$G^\prime$) values for the 100\% covered Au-butanethiol/hexane |
| 535 |
|
interfaces with UA model and different hexane molecule numbers |
| 536 |
< |
at different temperatures using a range of energy fluxes.} |
| 536 |
> |
at different temperatures using a range of energy |
| 537 |
> |
fluxes. Error estimates indicated in parenthesis.} |
| 538 |
|
|
| 539 |
< |
\begin{tabular}{cccccccc} |
| 539 |
> |
\begin{tabular}{ccccccc} |
| 540 |
|
\hline\hline |
| 541 |
< |
$\langle T\rangle$ & & $L_x$ & $L_y$ & $L_z$ & $J_z$ & |
| 542 |
< |
$G$ & $G^\prime$ \\ |
| 543 |
< |
(K) & $N_{hexane}$ & \multicolumn{3}{c}{(\AA)} & (GW/m$^2$) & |
| 541 |
> |
$\langle T\rangle$ & $N_{hexane}$ & Fixed & $\rho_{hexane}$ & |
| 542 |
> |
$J_z$ & $G$ & $G^\prime$ \\ |
| 543 |
> |
(K) & & $L_x$ \& $L_y$? & (g/cm$^3$) & (GW/m$^2$) & |
| 544 |
|
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 545 |
|
\hline |
| 546 |
< |
200 & 266 & 29.86 & 25.80 & 113.1 & -0.96 & |
| 547 |
< |
102() & 80.0() \\ |
| 548 |
< |
& 200 & 29.84 & 25.81 & 93.9 & 1.92 & |
| 549 |
< |
129() & 87.3() \\ |
| 550 |
< |
& & 29.84 & 25.81 & 95.3 & 1.93 & |
| 551 |
< |
131() & 77.5() \\ |
| 552 |
< |
& 166 & 29.84 & 25.81 & 85.7 & 0.97 & |
| 553 |
< |
115() & 69.3() \\ |
| 554 |
< |
& & & & & 1.94 & |
| 555 |
< |
125() & 87.1() \\ |
| 556 |
< |
250 & 200 & 29.84 & 25.87 & 106.8 & 0.96 & |
| 557 |
< |
81.8() & 67.0() \\ |
| 558 |
< |
& 166 & 29.87 & 25.84 & 94.8 & 0.98 & |
| 559 |
< |
79.0() & 62.9() \\ |
| 560 |
< |
& & 29.84 & 25.85 & 95.0 & 1.44 & |
| 561 |
< |
76.2() & 64.8() \\ |
| 546 |
> |
200 & 266 & No & 0.672 & -0.96 & 102(3) & 80.0(0.8) \\ |
| 547 |
> |
& 200 & Yes & 0.694 & 1.92 & 129(11) & 87.3(0.3) \\ |
| 548 |
> |
& & Yes & 0.672 & 1.93 & 131(16) & 78(13) \\ |
| 549 |
> |
& & No & 0.688 & 0.96 & 125(16) & 90.2(15) \\ |
| 550 |
> |
& & & & 1.91 & 139(10) & 101(10) \\ |
| 551 |
> |
& & & & 2.83 & 141(6) & 89.9(9.8) \\ |
| 552 |
> |
& 166 & Yes & 0.679 & 0.97 & 115(19) & 69(18) \\ |
| 553 |
> |
& & & & 1.94 & 125(9) & 87.1(0.2) \\ |
| 554 |
> |
& & No & 0.681 & 0.97 & 141(30) & 78(22) \\ |
| 555 |
> |
& & & & 1.92 & 138(4) & 98.9(9.5) \\ |
| 556 |
> |
\hline |
| 557 |
> |
250 & 200 & No & 0.560 & 0.96 & 75(10) & 61.8(7.3) \\ |
| 558 |
> |
& & & & -0.95 & 49.4(0.3) & 45.7(2.1) \\ |
| 559 |
> |
& 166 & Yes & 0.570 & 0.98 & 79.0(3.5) & 62.9(3.0) \\ |
| 560 |
> |
& & No & 0.569 & 0.97 & 80.3(0.6) & 67(11) \\ |
| 561 |
> |
& & & & 1.44 & 76.2(5.0) & 64.8(3.8) \\ |
| 562 |
> |
& & & & -0.95 & 56.4(2.5) & 54.4(1.1) \\ |
| 563 |
> |
& & & & -1.85 & 47.8(1.1) & 53.5(1.5) \\ |
| 564 |
|
\hline\hline |
| 565 |
|
\end{tabular} |
| 566 |
|
\label{AuThiolHexaneUA} |
| 576 |
|
temperature is higher than 250K. Additionally, the equilibrated liquid |
| 577 |
|
hexane density under 250K becomes lower than experimental value. This |
| 578 |
|
expanded liquid phase leads to lower contact between hexane and |
| 579 |
< |
butanethiol as well.[MAY NEED FIGURE] And this reduced contact would |
| 579 |
> |
butanethiol as well.[MAY NEED SLAB DENSITY FIGURE] |
| 580 |
> |
And this reduced contact would |
| 581 |
|
probably be accountable for a lower interfacial thermal conductance, |
| 582 |
|
as shown in Table \ref{AuThiolHexaneUA}. |
| 583 |
|
|
| 592 |
|
important role in the thermal transport process across the interface |
| 593 |
|
in that higher degree of contact could yield increased conductance. |
| 594 |
|
|
| 511 |
– |
[ADD Lxyz AND ERROR ESTIMATE TO TABLE] |
| 595 |
|
\begin{table*} |
| 596 |
|
\begin{minipage}{\linewidth} |
| 597 |
|
\begin{center} |
| 598 |
|
\caption{Computed interfacial thermal conductivity ($G$ and |
| 599 |
|
$G^\prime$) values for a 90\% coverage Au-butanethiol/toluene |
| 600 |
|
interface at different temperatures using a range of energy |
| 601 |
< |
fluxes.} |
| 601 |
> |
fluxes. Error estimates indicated in parenthesis.} |
| 602 |
|
|
| 603 |
< |
\begin{tabular}{cccc} |
| 603 |
> |
\begin{tabular}{ccccc} |
| 604 |
|
\hline\hline |
| 605 |
< |
$\langle T\rangle$ & $J_z$ & $G$ & $G^\prime$ \\ |
| 606 |
< |
(K) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 605 |
> |
$\langle T\rangle$ & $\rho_{toluene}$ & $J_z$ & $G$ & $G^\prime$ \\ |
| 606 |
> |
(K) & (g/cm$^3$) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 607 |
|
\hline |
| 608 |
< |
200 & -1.86 & 180() & 135() \\ |
| 609 |
< |
& 2.15 & 204() & 113() \\ |
| 610 |
< |
& -3.93 & 175() & 114() \\ |
| 611 |
< |
300 & -1.91 & 143() & 125() \\ |
| 612 |
< |
& -4.19 & 134() & 113() \\ |
| 608 |
> |
200 & 0.933 & 2.15 & 204(12) & 113(12) \\ |
| 609 |
> |
& & -1.86 & 180(3) & 135(21) \\ |
| 610 |
> |
& & -3.93 & 176(5) & 113(12) \\ |
| 611 |
> |
\hline |
| 612 |
> |
300 & 0.855 & -1.91 & 143(5) & 125(2) \\ |
| 613 |
> |
& & -4.19 & 135(9) & 113(12) \\ |
| 614 |
|
\hline\hline |
| 615 |
|
\end{tabular} |
| 616 |
|
\label{AuThiolToluene} |
| 642 |
|
|
| 643 |
|
However, when the surface is not completely covered by butanethiols, |
| 644 |
|
the simulated system is more resistent to the reconstruction |
| 645 |
< |
above. Our Au-butanethiol/toluene system did not see this phenomena |
| 646 |
< |
even at $<T>\sim$300K. The Au(111) surfaces have a 90\% coverage of |
| 647 |
< |
butanethiols and have empty three-fold sites. These empty sites could |
| 648 |
< |
help prevent surface reconstruction in that they provide other means |
| 649 |
< |
of capping agent relaxation. It is observed that butanethiols can |
| 650 |
< |
migrate to their neighbor empty sites during a simulation. Therefore, |
| 651 |
< |
we were able to obtain $G$'s for these interfaces even at a relatively |
| 652 |
< |
high temperature without being affected by surface reconstructions. |
| 645 |
> |
above. Our Au-butanethiol/toluene system had the Au(111) surfaces 90\% |
| 646 |
> |
covered by butanethiols, but did not see this above phenomena even at |
| 647 |
> |
$\langle T\rangle\sim$300K. The empty three-fold sites not occupied by |
| 648 |
> |
capping agents could help prevent surface reconstruction in that they |
| 649 |
> |
provide other means of capping agent relaxation. It is observed that |
| 650 |
> |
butanethiols can migrate to their neighbor empty sites during a |
| 651 |
> |
simulation. Therefore, we were able to obtain $G$'s for these |
| 652 |
> |
interfaces even at a relatively high temperature without being |
| 653 |
> |
affected by surface reconstructions. |
| 654 |
|
|
| 655 |
|
\subsection{Influence of Capping Agent Coverage on $G$} |
| 656 |
|
To investigate the influence of butanethiol coverage on interfacial |
| 657 |
|
thermal conductance, a series of different coverage Au-butanethiol |
| 658 |
|
surfaces is prepared and solvated with various organic |
| 659 |
|
molecules. These systems are then equilibrated and their interfacial |
| 660 |
< |
thermal conductivity are measured with our NIVS algorithm. Table |
| 661 |
< |
\ref{tlnUhxnUhxnD} lists these results for direct comparison between |
| 662 |
< |
different coverages of butanethiol. To study the isotope effect in |
| 663 |
< |
interfacial thermal conductance, deuterated UA-hexane is included as |
| 664 |
< |
well. |
| 660 |
> |
thermal conductivity are measured with our NIVS algorithm. Figure |
| 661 |
> |
\ref{coverage} demonstrates the trend of conductance change with |
| 662 |
> |
respect to different coverages of butanethiol. To study the isotope |
| 663 |
> |
effect in interfacial thermal conductance, deuterated UA-hexane is |
| 664 |
> |
included as well. |
| 665 |
|
|
| 666 |
|
It turned out that with partial covered butanethiol on the Au(111) |
| 667 |
< |
surface, the derivative definition for $G$ (Eq. \ref{derivativeG}) has |
| 668 |
< |
difficulty to apply, due to the difficulty in locating the maximum of |
| 669 |
< |
change of $\lambda$. Instead, the discrete definition |
| 670 |
< |
(Eq. \ref{discreteG}) is easier to apply, as max($\Delta T$) can still |
| 671 |
< |
be well-defined. Therefore, $G$'s (not $G^\prime$) are used for this |
| 672 |
< |
section. |
| 667 |
> |
surface, the derivative definition for $G^\prime$ |
| 668 |
> |
(Eq. \ref{derivativeG}) was difficult to apply, due to the difficulty |
| 669 |
> |
in locating the maximum of change of $\lambda$. Instead, the discrete |
| 670 |
> |
definition (Eq. \ref{discreteG}) is easier to apply, as the Gibbs |
| 671 |
> |
deviding surface can still be well-defined. Therefore, $G$ (not |
| 672 |
> |
$G^\prime$) was used for this section. |
| 673 |
|
|
| 674 |
< |
From Table \ref{tlnUhxnUhxnD}, one can see the significance of the |
| 674 |
> |
From Figure \ref{coverage}, one can see the significance of the |
| 675 |
|
presence of capping agents. Even when a fraction of the Au(111) |
| 676 |
|
surface sites are covered with butanethiols, the conductivity would |
| 677 |
|
see an enhancement by at least a factor of 3. This indicates the |
| 678 |
|
important role cappping agent is playing for thermal transport |
| 679 |
< |
phenomena on metal/organic solvent surfaces. |
| 679 |
> |
phenomena on metal / organic solvent surfaces. |
| 680 |
|
|
| 681 |
|
Interestingly, as one could observe from our results, the maximum |
| 682 |
|
conductance enhancement (largest $G$) happens while the surfaces are |
| 695 |
|
would not offset this effect. Eventually, when butanethiol coverage |
| 696 |
|
continues to decrease, solvent-capping agent contact actually |
| 697 |
|
decreases with the disappearing of butanethiol molecules. In this |
| 698 |
< |
case, $G$ decrease could not be offset but instead accelerated. |
| 698 |
> |
case, $G$ decrease could not be offset but instead accelerated. [NEED |
| 699 |
> |
SNAPSHOT SHOWING THE PHENOMENA] |
| 700 |
|
|
| 701 |
|
A comparison of the results obtained from differenet organic solvents |
| 702 |
|
can also provide useful information of the interfacial thermal |
| 706 |
|
studies, even though eliminating C-H vibration samplings, still have |
| 707 |
|
C-C vibrational frequencies different from each other. However, these |
| 708 |
|
differences in the infrared range do not seem to produce an observable |
| 709 |
< |
difference for the results of $G$. [MAY NEED FIGURE] |
| 709 |
> |
difference for the results of $G$. [MAY NEED SPECTRA FIGURE] |
| 710 |
|
|
| 711 |
|
Furthermore, results for rigid body toluene solvent, as well as other |
| 712 |
|
UA-hexane solvents, are reasonable within the general experimental |
| 715 |
|
such as Au-thiol/organic solvent. |
| 716 |
|
|
| 717 |
|
However, results for Au-butanethiol/toluene do not show an identical |
| 718 |
< |
trend with those for Au-butanethiol/hexane in that $G$'s remain at |
| 718 |
> |
trend with those for Au-butanethiol/hexane in that $G$ remains at |
| 719 |
|
approximately the same magnitue when butanethiol coverage differs from |
| 720 |
|
25\% to 75\%. This might be rooted in the molecule shape difference |
| 721 |
< |
for plane-like toluene and chain-like {\it n}-hexane. Due to this |
| 721 |
> |
for planar toluene and chain-like {\it n}-hexane. Due to this |
| 722 |
|
difference, toluene molecules have more difficulty in occupying |
| 723 |
|
relatively small gaps among capping agents when their coverage is not |
| 724 |
|
too low. Therefore, the solvent-capping agent contact may keep |
| 727 |
|
its effect to the process of interfacial thermal transport. Thus, one |
| 728 |
|
can see a plateau of $G$ vs. butanethiol coverage in our results. |
| 729 |
|
|
| 730 |
< |
[NEED ERROR ESTIMATE, MAY ALSO PUT J HERE] |
| 731 |
< |
\begin{table*} |
| 732 |
< |
\begin{minipage}{\linewidth} |
| 733 |
< |
\begin{center} |
| 734 |
< |
\caption{Computed interfacial thermal conductivity ($G$) values |
| 735 |
< |
for the Au-butanethiol/solvent interface with various UA |
| 736 |
< |
models and different capping agent coverages at $\langle |
| 737 |
< |
T\rangle\sim$200K using certain energy flux respectively.} |
| 652 |
< |
|
| 653 |
< |
\begin{tabular}{cccc} |
| 654 |
< |
\hline\hline |
| 655 |
< |
Thiol & \multicolumn{3}{c}{$G$(MW/m$^2$/K)} \\ |
| 656 |
< |
coverage (\%) & hexane & hexane(D) & toluene \\ |
| 657 |
< |
\hline |
| 658 |
< |
0.0 & 46.5() & 43.9() & 70.1() \\ |
| 659 |
< |
25.0 & 151() & 153() & 249() \\ |
| 660 |
< |
50.0 & 172() & 182() & 214() \\ |
| 661 |
< |
75.0 & 242() & 229() & 244() \\ |
| 662 |
< |
88.9 & 178() & - & - \\ |
| 663 |
< |
100.0 & 137() & 153() & 187() \\ |
| 664 |
< |
\hline\hline |
| 665 |
< |
\end{tabular} |
| 666 |
< |
\label{tlnUhxnUhxnD} |
| 667 |
< |
\end{center} |
| 668 |
< |
\end{minipage} |
| 669 |
< |
\end{table*} |
| 730 |
> |
\begin{figure} |
| 731 |
> |
\includegraphics[width=\linewidth]{coverage} |
| 732 |
> |
\caption{Comparison of interfacial thermal conductivity ($G$) values |
| 733 |
> |
for the Au-butanethiol/solvent interface with various UA models and |
| 734 |
> |
different capping agent coverages at $\langle T\rangle\sim$200K |
| 735 |
> |
using certain energy flux respectively.} |
| 736 |
> |
\label{coverage} |
| 737 |
> |
\end{figure} |
| 738 |
|
|
| 739 |
|
\subsection{Influence of Chosen Molecule Model on $G$} |
| 740 |
|
[MAY COMBINE W MECHANISM STUDY] |
| 747 |
|
the previous section. Table \ref{modelTest} summarizes the results of |
| 748 |
|
these studies. |
| 749 |
|
|
| 682 |
– |
[MORE DATA; ERROR ESTIMATE] |
| 750 |
|
\begin{table*} |
| 751 |
|
\begin{minipage}{\linewidth} |
| 752 |
|
\begin{center} |
| 754 |
|
\caption{Computed interfacial thermal conductivity ($G$ and |
| 755 |
|
$G^\prime$) values for interfaces using various models for |
| 756 |
|
solvent and capping agent (or without capping agent) at |
| 757 |
< |
$\langle T\rangle\sim$200K.} |
| 757 |
> |
$\langle T\rangle\sim$200K. (D stands for deuterated solvent |
| 758 |
> |
or capping agent molecules; ``Avg.'' denotes results that are |
| 759 |
> |
averages of simulations under different $J_z$'s. Error |
| 760 |
> |
estimates indicated in parenthesis.)} |
| 761 |
|
|
| 762 |
< |
\begin{tabular}{ccccc} |
| 762 |
> |
\begin{tabular}{llccc} |
| 763 |
|
\hline\hline |
| 764 |
|
Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ |
| 765 |
|
(or bare surface) & model & (GW/m$^2$) & |
| 766 |
|
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 767 |
|
\hline |
| 768 |
< |
UA & AA hexane & 1.94 & 135() & 129() \\ |
| 769 |
< |
& & 2.86 & 126() & 115() \\ |
| 770 |
< |
& AA toluene & 1.89 & 200() & 149() \\ |
| 771 |
< |
AA & UA hexane & 1.94 & 116() & 129() \\ |
| 772 |
< |
& AA hexane & 3.76 & 451() & 378() \\ |
| 773 |
< |
& & 4.71 & 432() & 334() \\ |
| 774 |
< |
& AA toluene & 3.79 & 487() & 290() \\ |
| 775 |
< |
AA(D) & UA hexane & 1.94 & 158() & 172() \\ |
| 776 |
< |
bare & AA hexane & 0.96 & 31.0() & 29.4() \\ |
| 768 |
> |
UA & UA hexane & Avg. & 131(9) & 87(10) \\ |
| 769 |
> |
& UA hexane(D) & 1.95 & 153(5) & 136(13) \\ |
| 770 |
> |
& AA hexane & Avg. & 131(6) & 122(10) \\ |
| 771 |
> |
& UA toluene & 1.96 & 187(16) & 151(11) \\ |
| 772 |
> |
& AA toluene & 1.89 & 200(36) & 149(53) \\ |
| 773 |
> |
\hline |
| 774 |
> |
AA & UA hexane & 1.94 & 116(9) & 129(8) \\ |
| 775 |
> |
& AA hexane & Avg. & 442(14) & 356(31) \\ |
| 776 |
> |
& AA hexane(D) & 1.93 & 222(12) & 234(54) \\ |
| 777 |
> |
& UA toluene & 1.98 & 125(25) & 97(60) \\ |
| 778 |
> |
& AA toluene & 3.79 & 487(56) & 290(42) \\ |
| 779 |
> |
\hline |
| 780 |
> |
AA(D) & UA hexane & 1.94 & 158(25) & 172(4) \\ |
| 781 |
> |
& AA hexane & 1.92 & 243(29) & 191(11) \\ |
| 782 |
> |
& AA toluene & 1.93 & 364(36) & 322(67) \\ |
| 783 |
> |
\hline |
| 784 |
> |
bare & UA hexane & Avg. & 46.5(3.2) & 49.4(4.5) \\ |
| 785 |
> |
& UA hexane(D) & 0.98 & 43.9(4.6) & 43.0(2.0) \\ |
| 786 |
> |
& AA hexane & 0.96 & 31.0(1.4) & 29.4(1.3) \\ |
| 787 |
> |
& UA toluene & 1.99 & 70.1(1.3) & 65.8(0.5) \\ |
| 788 |
|
\hline\hline |
| 789 |
|
\end{tabular} |
| 790 |
|
\label{modelTest} |
| 807 |
|
interfaces, using AA model for both butanethiol and hexane yields |
| 808 |
|
substantially higher conductivity values than using UA model for at |
| 809 |
|
least one component of the solvent and capping agent, which exceeds |
| 810 |
< |
the upper bond of experimental value range. This is probably due to |
| 811 |
< |
the classically treated C-H vibrations in the AA model, which should |
| 812 |
< |
not be appreciably populated at normal temperatures. In comparison, |
| 813 |
< |
once either the hexanes or the butanethiols are deuterated, one can |
| 814 |
< |
see a significantly lower $G$ and $G^\prime$. In either of these |
| 815 |
< |
cases, the C-H(D) vibrational overlap between the solvent and the |
| 816 |
< |
capping agent is removed. [MAY NEED FIGURE] Conclusively, the |
| 810 |
> |
the general range of experimental measurement results. This is |
| 811 |
> |
probably due to the classically treated C-H vibrations in the AA |
| 812 |
> |
model, which should not be appreciably populated at normal |
| 813 |
> |
temperatures. In comparison, once either the hexanes or the |
| 814 |
> |
butanethiols are deuterated, one can see a significantly lower $G$ and |
| 815 |
> |
$G^\prime$. In either of these cases, the C-H(D) vibrational overlap |
| 816 |
> |
between the solvent and the capping agent is removed. |
| 817 |
> |
[MAY NEED SPECTRA FIGURE] Conclusively, the |
| 818 |
|
improperly treated C-H vibration in the AA model produced |
| 819 |
|
over-predicted results accordingly. Compared to the AA model, the UA |
| 820 |
|
model yields more reasonable results with higher computational |
| 822 |
|
|
| 823 |
|
However, for Au-butanethiol/toluene interfaces, having the AA |
| 824 |
|
butanethiol deuterated did not yield a significant change in the |
| 825 |
< |
measurement results. |
| 826 |
< |
. , so extra degrees of freedom |
| 827 |
< |
such as the C-H vibrations could enhance heat exchange between these |
| 828 |
< |
two phases and result in a much higher conductivity. |
| 825 |
> |
measurement results. Compared to the C-H vibrational overlap between |
| 826 |
> |
hexane and butanethiol, both of which have alkyl chains, that overlap |
| 827 |
> |
between toluene and butanethiol is not so significant and thus does |
| 828 |
> |
not have as much contribution to the ``Intramolecular Vibration |
| 829 |
> |
Redistribution''[CITE HASE]. Conversely, extra degrees of freedom such |
| 830 |
> |
as the C-H vibrations could yield higher heat exchange rate between |
| 831 |
> |
these two phases and result in a much higher conductivity. |
| 832 |
|
|
| 748 |
– |
|
| 833 |
|
Although the QSC model for Au is known to predict an overly low value |
| 834 |
< |
for bulk metal gold conductivity[CITE NIVSRNEMD], our computational |
| 834 |
> |
for bulk metal gold conductivity\cite{kuang:164101}, our computational |
| 835 |
|
results for $G$ and $G^\prime$ do not seem to be affected by this |
| 836 |
< |
drawback of the model for metal. Instead, the modeling of interfacial |
| 837 |
< |
thermal transport behavior relies mainly on an accurate description of |
| 838 |
< |
the interactions between components occupying the interfaces. |
| 836 |
> |
drawback of the model for metal. Instead, our results suggest that the |
| 837 |
> |
modeling of interfacial thermal transport behavior relies mainly on |
| 838 |
> |
the accuracy of the interaction descriptions between components |
| 839 |
> |
occupying the interfaces. |
| 840 |
|
|
| 841 |
|
\subsection{Mechanism of Interfacial Thermal Conductance Enhancement |
| 842 |
|
by Capping Agent} |
| 843 |
< |
%OR\subsection{Vibrational spectrum study on conductance mechanism} |
| 843 |
> |
[OR: Vibrational Spectrum Study on Conductance Mechanism] |
| 844 |
|
|
| 845 |
|
[MAY INTRODUCE PROTOCOL IN METHODOLOGY/COMPUTATIONAL DETAIL, EQN'S] |
| 846 |
|
|
| 852 |
|
the velocity auto-correlation functions, which is used to construct a |
| 853 |
|
power spectrum via a Fourier transform. |
| 854 |
|
|
| 855 |
< |
The gold surfaces covered by |
| 856 |
< |
butanethiol molecules, compared to bare gold surfaces, exhibit an |
| 857 |
< |
additional peak observed at a frequency of $\sim$170cm$^{-1}$, which |
| 858 |
< |
is attributed to the vibration of the S-Au bond. This vibration |
| 859 |
< |
enables efficient thermal transport from surface Au atoms to the |
| 860 |
< |
capping agents. Simultaneously, as shown in the lower panel of |
| 861 |
< |
Fig. \ref{vibration}, the large overlap of the vibration spectra of |
| 862 |
< |
butanethiol and hexane in the all-atom model, including the C-H |
| 863 |
< |
vibration, also suggests high thermal exchange efficiency. The |
| 864 |
< |
combination of these two effects produces the drastic interfacial |
| 865 |
< |
thermal conductance enhancement in the all-atom model. |
| 855 |
> |
[MAY RELATE TO HASE'S] |
| 856 |
> |
The gold surfaces covered by butanethiol molecules, compared to bare |
| 857 |
> |
gold surfaces, exhibit an additional peak observed at the frequency of |
| 858 |
> |
$\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. |
| 867 |
|
|
| 868 |
< |
[MAY NEED TO CONVERT TO JPEG] |
| 868 |
> |
[NEED SEPARATE FIGURE. MAY NEED TO CONVERT TO JPEG] |
| 869 |
|
\begin{figure} |
| 870 |
|
\includegraphics[width=\linewidth]{vibration} |
| 871 |
|
\caption{Vibrational spectra obtained for gold in different |
| 872 |
< |
environments (upper panel) and for Au/thiol/hexane simulation in |
| 787 |
< |
all-atom model (lower panel).} |
| 872 |
> |
environments.} |
| 873 |
|
\label{vibration} |
| 874 |
|
\end{figure} |
| 875 |
|
|
| 876 |
< |
[COMPARISON OF TWO G'S; AU SLAB WIDTHS; ETC] |
| 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 |
|
|
| 885 |
|
corresponding thermal gradient and to obtain interfacial thermal |
| 886 |
|
conductivities. Our simulations have seen significant conductance |
| 887 |
|
enhancement with the presence of capping agent, compared to the bare |
| 888 |
< |
gold/liquid interfaces. The acoustic impedance mismatch between the |
| 888 |
> |
gold / liquid interfaces. The acoustic impedance mismatch between the |
| 889 |
|
metal and the liquid phase is effectively eliminated by proper capping |
| 890 |
|
agent. Furthermore, the coverage precentage of the capping agent plays |
| 891 |
|
an important role in the interfacial thermal transport process. |
