| 22 |
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| 24 |
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| 25 |
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| 25 |
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| 26 |
|
\bibpunct{[}{]}{,}{s}{}{;} |
| 27 |
|
\bibliographystyle{aip} |
| 28 |
|
|
| 44 |
|
\begin{doublespace} |
| 45 |
|
|
| 46 |
|
\begin{abstract} |
| 47 |
< |
The abstract |
| 47 |
> |
|
| 48 |
> |
We have developed a Non-Isotropic Velocity Scaling algorithm for |
| 49 |
> |
setting up and maintaining stable thermal gradients in non-equilibrium |
| 50 |
> |
molecular dynamics simulations. This approach effectively imposes |
| 51 |
> |
unphysical thermal flux even between particles of different |
| 52 |
> |
identities, conserves linear momentum and kinetic energy, and |
| 53 |
> |
minimally perturbs the velocity profile of a system when compared with |
| 54 |
> |
previous RNEMD methods. We have used this method to simulate thermal |
| 55 |
> |
conductance at metal / organic solvent interfaces both with and |
| 56 |
> |
without the presence of thiol-based capping agents. We obtained |
| 57 |
> |
values comparable with experimental values, and observed significant |
| 58 |
> |
conductance enhancement with the presence of capping agents. Computed |
| 59 |
> |
power spectra indicate the acoustic impedance mismatch between metal |
| 60 |
> |
and liquid phase is greatly reduced by the capping agents and thus |
| 61 |
> |
leads to higher interfacial thermal transfer efficiency. |
| 62 |
> |
|
| 63 |
|
\end{abstract} |
| 64 |
|
|
| 65 |
|
\newpage |
| 71 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 72 |
|
|
| 73 |
|
\section{Introduction} |
| 74 |
+ |
[BACKGROUND FOR INTERFACIAL THERMAL CONDUCTANCE PROBLEM] |
| 75 |
+ |
Interfacial thermal conductance is extensively studied both |
| 76 |
+ |
experimentally and computationally, and systems with interfaces |
| 77 |
+ |
present are generally heterogeneous. Although interfaces are commonly |
| 78 |
+ |
barriers to heat transfer, it has been |
| 79 |
+ |
reported\cite{doi:10.1021/la904855s} that under specific circustances, |
| 80 |
+ |
e.g. with certain capping agents present on the surface, interfacial |
| 81 |
+ |
conductance can be significantly enhanced. However, heat conductance |
| 82 |
+ |
of molecular and nano-scale interfaces will be affected by the |
| 83 |
+ |
chemical details of the surface and is challenging to |
| 84 |
+ |
experimentalist. The lower thermal flux through interfaces is even |
| 85 |
+ |
more difficult to measure with EMD and forward NEMD simulation |
| 86 |
+ |
methods. Therefore, developing good simulation methods will be |
| 87 |
+ |
desirable in order to investigate thermal transport across interfaces. |
| 88 |
|
|
| 89 |
< |
The intro. |
| 89 |
> |
Recently, we have developed the Non-Isotropic Velocity Scaling (NIVS) |
| 90 |
> |
algorithm for RNEMD simulations\cite{kuang:164101}. This algorithm |
| 91 |
> |
retains the desirable features of RNEMD (conservation of linear |
| 92 |
> |
momentum and total energy, compatibility with periodic boundary |
| 93 |
> |
conditions) while establishing true thermal distributions in each of |
| 94 |
> |
the two slabs. Furthermore, it allows more effective thermal exchange |
| 95 |
> |
between particles of different identities, and thus enables extensive |
| 96 |
> |
study of interfacial conductance. |
| 97 |
> |
|
| 98 |
> |
\section{Methodology} |
| 99 |
> |
\subsection{Algorithm} |
| 100 |
> |
[BACKGROUND FOR MD METHODS] |
| 101 |
> |
There have been many algorithms for computing thermal conductivity |
| 102 |
> |
using molecular dynamics simulations. However, interfacial conductance |
| 103 |
> |
is at least an order of magnitude smaller. This would make the |
| 104 |
> |
calculation even more difficult for those slowly-converging |
| 105 |
> |
equilibrium methods. Imposed-flux non-equilibrium |
| 106 |
> |
methods\cite{MullerPlathe:1997xw} have the flux set {\it a priori} and |
| 107 |
> |
the response of temperature or momentum gradients are easier to |
| 108 |
> |
measure than the flux, if unknown, and thus, is a preferable way to |
| 109 |
> |
the forward NEMD methods. Although the momentum swapping approach for |
| 110 |
> |
flux-imposing can be used for exchanging energy between particles of |
| 111 |
> |
different identity, the kinetic energy transfer efficiency is affected |
| 112 |
> |
by the mass difference between the particles, which limits its |
| 113 |
> |
application on heterogeneous interfacial systems. |
| 114 |
> |
|
| 115 |
> |
The non-isotropic velocity scaling (NIVS)\cite{kuang:164101} approach in |
| 116 |
> |
non-equilibrium MD simulations is able to impose relatively large |
| 117 |
> |
kinetic energy flux without obvious perturbation to the velocity |
| 118 |
> |
distribution of the simulated systems. Furthermore, this approach has |
| 119 |
> |
the advantage in heterogeneous interfaces in that kinetic energy flux |
| 120 |
> |
can be applied between regions of particles of arbitary identity, and |
| 121 |
> |
the flux quantity is not restricted by particle mass difference. |
| 122 |
> |
|
| 123 |
> |
The NIVS algorithm scales the velocity vectors in two separate regions |
| 124 |
> |
of a simulation system with respective diagonal scaling matricies. To |
| 125 |
> |
determine these scaling factors in the matricies, a set of equations |
| 126 |
> |
including linear momentum conservation and kinetic energy conservation |
| 127 |
> |
constraints and target momentum/energy flux satisfaction is |
| 128 |
> |
solved. With the scaling operation applied to the system in a set |
| 129 |
> |
frequency, corresponding momentum/temperature gradients can be built, |
| 130 |
> |
which can be used for computing transportation properties and other |
| 131 |
> |
applications related to momentum/temperature gradients. The NIVS |
| 132 |
> |
algorithm conserves momenta and energy and does not depend on an |
| 133 |
> |
external thermostat. |
| 134 |
> |
|
| 135 |
> |
\subsection{Defining Interfacial Thermal Conductivity $G$} |
| 136 |
> |
For interfaces with a relatively low interfacial conductance, the bulk |
| 137 |
> |
regions on either side of an interface rapidly come to a state in |
| 138 |
> |
which the two phases have relatively homogeneous (but distinct) |
| 139 |
> |
temperatures. The interfacial thermal conductivity $G$ can therefore |
| 140 |
> |
be approximated as: |
| 141 |
> |
\begin{equation} |
| 142 |
> |
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle - |
| 143 |
> |
\langle T_\mathrm{cold}\rangle \right)} |
| 144 |
> |
\label{lowG} |
| 145 |
> |
\end{equation} |
| 146 |
> |
where ${E_{total}}$ is the imposed non-physical kinetic energy |
| 147 |
> |
transfer and ${\langle T_\mathrm{hot}\rangle}$ and ${\langle |
| 148 |
> |
T_\mathrm{cold}\rangle}$ are the average observed temperature of the |
| 149 |
> |
two separated phases. |
| 150 |
> |
|
| 151 |
> |
When the interfacial conductance is {\it not} small, two ways can be |
| 152 |
> |
used to define $G$. |
| 153 |
> |
|
| 154 |
> |
One way is to assume the temperature is discretely different on two |
| 155 |
> |
sides of the interface, $G$ can be calculated with the thermal flux |
| 156 |
> |
applied $J$ and the maximum temperature difference measured along the |
| 157 |
> |
thermal gradient max($\Delta T$), which occurs at the interface, as: |
| 158 |
> |
\begin{equation} |
| 159 |
> |
G=\frac{J}{\Delta T} |
| 160 |
> |
\label{discreteG} |
| 161 |
> |
\end{equation} |
| 162 |
> |
|
| 163 |
> |
The other approach is to assume a continuous temperature profile along |
| 164 |
> |
the thermal gradient axis (e.g. $z$) and define $G$ at the point where |
| 165 |
> |
the magnitude of thermal conductivity $\lambda$ change reach its |
| 166 |
> |
maximum, given that $\lambda$ is well-defined throughout the space: |
| 167 |
> |
\begin{equation} |
| 168 |
> |
G^\prime = \Big|\frac{\partial\lambda}{\partial z}\Big| |
| 169 |
> |
= \Big|\frac{\partial}{\partial z}\left(-J_z\Big/ |
| 170 |
> |
\left(\frac{\partial T}{\partial z}\right)\right)\Big| |
| 171 |
> |
= |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
| 172 |
> |
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
| 173 |
> |
\label{derivativeG} |
| 174 |
> |
\end{equation} |
| 175 |
> |
|
| 176 |
> |
With the temperature profile obtained from simulations, one is able to |
| 177 |
> |
approximate the first and second derivatives of $T$ with finite |
| 178 |
> |
difference method and thus calculate $G^\prime$. |
| 179 |
> |
|
| 180 |
> |
In what follows, both definitions are used for calculation and comparison. |
| 181 |
> |
|
| 182 |
> |
[IMPOSE G DEFINITION INTO OUR SYSTEMS] |
| 183 |
> |
To facilitate the use of the above definitions in calculating $G$ and |
| 184 |
> |
$G^\prime$, we have a metal slab with its (111) surfaces perpendicular |
| 185 |
> |
to the $z$-axis of our simulation cells. With or withour capping |
| 186 |
> |
agents on the surfaces, the metal slab is solvated with organic |
| 187 |
> |
solvents, as illustrated in Figure \ref{demoPic}. |
| 188 |
> |
|
| 189 |
> |
\begin{figure} |
| 190 |
> |
\includegraphics[width=\linewidth]{demoPic} |
| 191 |
> |
\caption{A sample showing how a metal slab has its (111) surface |
| 192 |
> |
covered by capping agent molecules and solvated by hexane.} |
| 193 |
> |
\label{demoPic} |
| 194 |
> |
\end{figure} |
| 195 |
> |
|
| 196 |
> |
With a simulation cell setup following the above manner, one is able |
| 197 |
> |
to equilibrate the system and impose an unphysical thermal flux |
| 198 |
> |
between the liquid and the metal phase with the NIVS algorithm. Under |
| 199 |
> |
a stablized thermal gradient induced by periodically applying the |
| 200 |
> |
unphysical flux, one is able to obtain a temperature profile and the |
| 201 |
> |
physical thermal flux corresponding to it, which equals to the |
| 202 |
> |
unphysical flux applied by NIVS. These data enables the evaluation of |
| 203 |
> |
the interfacial thermal conductance of a surface. Figure \ref{gradT} |
| 204 |
> |
is an example how those stablized thermal gradient can be used to |
| 205 |
> |
obtain the 1st and 2nd derivatives of the temperature profile. |
| 206 |
> |
|
| 207 |
> |
\begin{figure} |
| 208 |
> |
\includegraphics[width=\linewidth]{gradT} |
| 209 |
> |
\caption{The 1st and 2nd derivatives of temperature profile can be |
| 210 |
> |
obtained with finite difference approximation.} |
| 211 |
> |
\label{gradT} |
| 212 |
> |
\end{figure} |
| 213 |
> |
|
| 214 |
> |
[MAY INCLUDE POWER SPECTRUM PROTOCOL] |
| 215 |
> |
|
| 216 |
> |
\section{Computational Details} |
| 217 |
> |
\subsection{Simulation Protocol} |
| 218 |
> |
In our simulations, Au is used to construct a metal slab with bare |
| 219 |
> |
(111) surface perpendicular to the $z$-axis. Different slab thickness |
| 220 |
> |
(layer numbers of Au) are simulated. This metal slab is first |
| 221 |
> |
equilibrated under normal pressure (1 atm) and a desired |
| 222 |
> |
temperature. After equilibration, butanethiol is used as the capping |
| 223 |
> |
agent molecule to cover the bare Au (111) surfaces evenly. The sulfur |
| 224 |
> |
atoms in the butanethiol molecules would occupy the three-fold sites |
| 225 |
> |
of the surfaces, and the maximal butanethiol capacity on Au surface is |
| 226 |
> |
$1/3$ of the total number of surface Au atoms[CITATION]. A series of |
| 227 |
> |
different coverage surfaces is investigated in order to study the |
| 228 |
> |
relation between coverage and conductance. |
| 229 |
> |
|
| 230 |
> |
[COVERAGE DISCRIPTION] However, since the interactions between surface |
| 231 |
> |
Au and butanethiol is non-bonded, the capping agent molecules are |
| 232 |
> |
allowed to migrate to an empty neighbor three-fold site during a |
| 233 |
> |
simulation. Therefore, the initial configuration would not severely |
| 234 |
> |
affect the sampling of a variety of configurations of the same |
| 235 |
> |
coverage, and the final conductance measurement would be an average |
| 236 |
> |
effect of these configurations explored in the simulations. [MAY NEED FIGURES] |
| 237 |
> |
|
| 238 |
> |
After the modified Au-butanethiol surface systems are equilibrated |
| 239 |
> |
under canonical ensemble, Packmol\cite{packmol} is used to pack |
| 240 |
> |
organic solvent molecules in the previously vacuum part of the |
| 241 |
> |
simulation cells, which guarantees that short range repulsive |
| 242 |
> |
interactions do not disrupt the simulations. Two solvents are |
| 243 |
> |
investigated, one which has little vibrational overlap with the |
| 244 |
> |
alkanethiol and plane-like shape (toluene), and one which has similar |
| 245 |
> |
vibrational frequencies and chain-like shape ({\it n}-hexane). [MAY |
| 246 |
> |
EXPLAIN WHY WE CHOOSE THEM] |
| 247 |
> |
|
| 248 |
> |
The spacing filled by solvent molecules, i.e. the gap between |
| 249 |
> |
periodically repeated Au-butanethiol surfaces should be carefully |
| 250 |
> |
chosen. A very long length scale for the thermal gradient axis ($z$) |
| 251 |
> |
may cause excessively hot or cold temperatures in the middle of the |
| 252 |
> |
solvent region and lead to undesired phenomena such as solvent boiling |
| 253 |
> |
or freezing when a thermal flux is applied. Conversely, too few |
| 254 |
> |
solvent molecules would change the normal behavior of the liquid |
| 255 |
> |
phase. Therefore, our $N_{solvent}$ values were chosen to ensure that |
| 256 |
> |
these extreme cases did not happen to our simulations. And the |
| 257 |
> |
corresponding spacing is usually $35 \sim 60$\AA. |
| 258 |
> |
|
| 259 |
> |
The initial configurations generated by Packmol are further |
| 260 |
> |
equilibrated with the $x$ and $y$ dimensions fixed, only allowing |
| 261 |
> |
length scale change in $z$ dimension. This is to ensure that the |
| 262 |
> |
equilibration of liquid phase does not affect the metal crystal |
| 263 |
> |
structure in $x$ and $y$ dimensions. Further equilibration are run |
| 264 |
> |
under NVT and then NVE ensembles. |
| 265 |
> |
|
| 266 |
> |
After the systems reach equilibrium, NIVS is implemented to impose a |
| 267 |
> |
periodic unphysical thermal flux between the metal and the liquid |
| 268 |
> |
phase. Most of our simulations are under an average temperature of |
| 269 |
> |
$\sim$200K. Therefore, this flux usually comes from the metal to the |
| 270 |
> |
liquid so that the liquid has a higher temperature and would not |
| 271 |
> |
freeze due to excessively low temperature. This induced temperature |
| 272 |
> |
gradient is stablized and the simulation cell is devided evenly into |
| 273 |
> |
N slabs along the $z$-axis and the temperatures of each slab are |
| 274 |
> |
recorded. When the slab width $d$ of each slab is the same, the |
| 275 |
> |
derivatives of $T$ with respect to slab number $n$ can be directly |
| 276 |
> |
used for $G^\prime$ calculations: |
| 277 |
> |
\begin{equation} |
| 278 |
> |
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
| 279 |
> |
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
| 280 |
> |
= |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big| |
| 281 |
> |
\Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2 |
| 282 |
> |
= |J_z|\Big|\frac{\partial^2 T}{\partial n^2}\Big| |
| 283 |
> |
\Big/\left(\frac{\partial T}{\partial n}\right)^2 |
| 284 |
> |
\label{derivativeG2} |
| 285 |
> |
\end{equation} |
| 286 |
> |
|
| 287 |
> |
\subsection{Force Field Parameters} |
| 288 |
> |
Our simulations include various components. Therefore, force field |
| 289 |
> |
parameter descriptions are needed for interactions both between the |
| 290 |
> |
same type of particles and between particles of different species. |
| 291 |
> |
|
| 292 |
> |
The Au-Au interactions in metal lattice slab is described by the |
| 293 |
> |
quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
| 294 |
> |
potentials include zero-point quantum corrections and are |
| 295 |
> |
reparametrized for accurate surface energies compared to the |
| 296 |
> |
Sutton-Chen potentials\cite{Chen90}. |
| 297 |
> |
|
| 298 |
> |
Figure [REF] demonstrates how we name our pseudo-atoms of the |
| 299 |
> |
molecules in our simulations. |
| 300 |
> |
[FIGURE FOR MOLECULE NOMENCLATURE] |
| 301 |
> |
|
| 302 |
> |
For both solvent molecules, straight chain {\it n}-hexane and aromatic |
| 303 |
> |
toluene, United-Atom (UA) and All-Atom (AA) models are used |
| 304 |
> |
respectively. The TraPPE-UA |
| 305 |
> |
parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
| 306 |
> |
for our UA solvent molecules. In these models, pseudo-atoms are |
| 307 |
> |
located at the carbon centers for alkyl groups. By eliminating |
| 308 |
> |
explicit hydrogen atoms, these models are simple and computationally |
| 309 |
> |
efficient, while maintains good accuracy. However, the TraPPE-UA for |
| 310 |
> |
alkanes is known to predict a lower boiling point than experimental |
| 311 |
> |
values. Considering that after an unphysical thermal flux is applied |
| 312 |
> |
to a system, the temperature of ``hot'' area in the liquid phase would be |
| 313 |
> |
significantly higher than the average, to prevent over heating and |
| 314 |
> |
boiling of the liquid phase, the average temperature in our |
| 315 |
> |
simulations should be much lower than the liquid boiling point. [MORE DISCUSSION] |
| 316 |
> |
For UA-toluene model, rigid body constraints are applied, so that the |
| 317 |
> |
benzene ring and the methyl-CRar bond are kept rigid. This would save |
| 318 |
> |
computational time.[MORE DETAILS] |
| 319 |
> |
|
| 320 |
> |
Besides the TraPPE-UA models, AA models for both organic solvents are |
| 321 |
> |
included in our studies as well. For hexane, the OPLS-AA\cite{OPLSAA} |
| 322 |
> |
force field is used. [MORE DETAILS] |
| 323 |
> |
For toluene, the United Force Field developed by Rapp\'{e} {\it et |
| 324 |
> |
al.}\cite{doi:10.1021/ja00051a040} is adopted.[MORE DETAILS] |
| 325 |
> |
|
| 326 |
> |
The capping agent in our simulations, the butanethiol molecules can |
| 327 |
> |
either use UA or AA model. The TraPPE-UA force fields includes |
| 328 |
> |
parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for |
| 329 |
> |
UA butanethiol model in our simulations. The OPLS-AA also provides |
| 330 |
> |
parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) |
| 331 |
> |
surfaces do not have the hydrogen atom bonded to sulfur. To adapt this |
| 332 |
> |
change and derive suitable parameters for butanethiol adsorbed on |
| 333 |
> |
Au(111) surfaces, we adopt the S parameters from [CITATION CF VLUGT] |
| 334 |
> |
and modify parameters for its neighbor C atom for charge balance in |
| 335 |
> |
the molecule. Note that the model choice (UA or AA) of capping agent |
| 336 |
> |
can be different from the solvent. Regardless of model choice, the |
| 337 |
> |
force field parameters for interactions between capping agent and |
| 338 |
> |
solvent can be derived using Lorentz-Berthelot Mixing Rule: |
| 339 |
> |
|
| 340 |
> |
|
| 341 |
> |
To describe the interactions between metal Au and non-metal capping |
| 342 |
> |
agent and solvent particles, we refer to an adsorption study of alkyl |
| 343 |
> |
thiols on gold surfaces by Vlugt {\it et |
| 344 |
> |
al.}\cite{vlugt:cpc2007154} They fitted an effective Lennard-Jones |
| 345 |
> |
form of potential parameters for the interaction between Au and |
| 346 |
> |
pseudo-atoms CH$_x$ and S based on a well-established and widely-used |
| 347 |
> |
effective potential of Hautman and Klein[CITATION] for the Au(111) |
| 348 |
> |
surface. As our simulations require the gold lattice slab to be |
| 349 |
> |
non-rigid so that it could accommodate kinetic energy for thermal |
| 350 |
> |
transport study purpose, the pair-wise form of potentials is |
| 351 |
> |
preferred. |
| 352 |
> |
|
| 353 |
> |
Besides, the potentials developed from {\it ab initio} calculations by |
| 354 |
> |
Leng {\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
| 355 |
> |
interactions between Au and aromatic C/H atoms in toluene.[MORE DETAILS] |
| 356 |
> |
|
| 357 |
> |
However, the Lennard-Jones parameters between Au and other types of |
| 358 |
> |
particles in our simulations are not yet well-established. For these |
| 359 |
> |
interactions, we attempt to derive their parameters using the Mixing |
| 360 |
> |
Rule. To do this, the ``Metal-non-Metal'' (MnM) interaction parameters |
| 361 |
> |
for Au is first extracted from the Au-CH$_x$ parameters by applying |
| 362 |
> |
the Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM'' |
| 363 |
> |
parameters in our simulations. |
| 364 |
> |
|
| 365 |
> |
\begin{table*} |
| 366 |
> |
\begin{minipage}{\linewidth} |
| 367 |
> |
\begin{center} |
| 368 |
> |
\caption{Lennard-Jones parameters for Au-non-Metal |
| 369 |
> |
interactions in our simulations.} |
| 370 |
> |
|
| 371 |
> |
\begin{tabular}{ccc} |
| 372 |
> |
\hline\hline |
| 373 |
> |
Non-metal & $\sigma$/\AA & $\epsilon$/kcal/mol \\ |
| 374 |
> |
\hline |
| 375 |
> |
S & 2.40 & 8.465 \\ |
| 376 |
> |
CH3 & 3.54 & 0.2146 \\ |
| 377 |
> |
CH2 & 3.54 & 0.1749 \\ |
| 378 |
> |
CT3 & 3.365 & 0.1373 \\ |
| 379 |
> |
CT2 & 3.365 & 0.1373 \\ |
| 380 |
> |
CTT & 3.365 & 0.1373 \\ |
| 381 |
> |
HC & 2.865 & 0.09256 \\ |
| 382 |
> |
CHar & 3.4625 & 0.1680 \\ |
| 383 |
> |
CRar & 3.555 & 0.1604 \\ |
| 384 |
> |
CA & 3.173 & 0.0640 \\ |
| 385 |
> |
HA & 2.746 & 0.0414 \\ |
| 386 |
> |
\hline\hline |
| 387 |
> |
\end{tabular} |
| 388 |
> |
\label{MnM} |
| 389 |
> |
\end{center} |
| 390 |
> |
\end{minipage} |
| 391 |
> |
\end{table*} |
| 392 |
|
|
| 393 |
+ |
|
| 394 |
+ |
\section{Results and Discussions} |
| 395 |
+ |
[MAY HAVE A BRIEF SUMMARY] |
| 396 |
+ |
\subsection{How Simulation Parameters Affects $G$} |
| 397 |
+ |
[MAY NOT PUT AT FIRST] |
| 398 |
+ |
We have varied our protocol or other parameters of the simulations in |
| 399 |
+ |
order to investigate how these factors would affect the measurement of |
| 400 |
+ |
$G$'s. It turned out that while some of these parameters would not |
| 401 |
+ |
affect the results substantially, some other changes to the |
| 402 |
+ |
simulations would have a significant impact on the measurement |
| 403 |
+ |
results. |
| 404 |
+ |
|
| 405 |
+ |
In some of our simulations, we allowed $L_x$ and $L_y$ to change |
| 406 |
+ |
during equilibrating the liquid phase. Due to the stiffness of the Au |
| 407 |
+ |
slab, $L_x$ and $L_y$ would not change noticeably after |
| 408 |
+ |
equilibration. Although $L_z$ could fluctuate $\sim$1\% after a system |
| 409 |
+ |
is fully equilibrated in the NPT ensemble, this fluctuation, as well |
| 410 |
+ |
as those comparably smaller to $L_x$ and $L_y$, would not be magnified |
| 411 |
+ |
on the calculated $G$'s, as shown in Table \ref{AuThiolHexaneUA}. This |
| 412 |
+ |
insensivity to $L_i$ fluctuations allows reliable measurement of $G$'s |
| 413 |
+ |
without the necessity of extremely cautious equilibration process. |
| 414 |
+ |
|
| 415 |
+ |
As stated in our computational details, the spacing filled with |
| 416 |
+ |
solvent molecules can be chosen within a range. This allows some |
| 417 |
+ |
change of solvent molecule numbers for the same Au-butanethiol |
| 418 |
+ |
surfaces. We did this study on our Au-butanethiol/hexane |
| 419 |
+ |
simulations. Nevertheless, the results obtained from systems of |
| 420 |
+ |
different $N_{hexane}$ did not indicate that the measurement of $G$ is |
| 421 |
+ |
susceptible to this parameter. For computational efficiency concern, |
| 422 |
+ |
smaller system size would be preferable, given that the liquid phase |
| 423 |
+ |
structure is not affected. |
| 424 |
+ |
|
| 425 |
+ |
Our NIVS algorithm allows change of unphysical thermal flux both in |
| 426 |
+ |
direction and in quantity. This feature extends our investigation of |
| 427 |
+ |
interfacial thermal conductance. However, the magnitude of this |
| 428 |
+ |
thermal flux is not arbitary if one aims to obtain a stable and |
| 429 |
+ |
reliable thermal gradient. A temperature profile would be |
| 430 |
+ |
substantially affected by noise when $|J_z|$ has a much too low |
| 431 |
+ |
magnitude; while an excessively large $|J_z|$ that overwhelms the |
| 432 |
+ |
conductance capacity of the interface would prevent a thermal gradient |
| 433 |
+ |
to reach a stablized steady state. NIVS has the advantage of allowing |
| 434 |
+ |
$J$ to vary in a wide range such that the optimal flux range for $G$ |
| 435 |
+ |
measurement can generally be simulated by the algorithm. Within the |
| 436 |
+ |
optimal range, we were able to study how $G$ would change according to |
| 437 |
+ |
the thermal flux across the interface. For our simulations, we denote |
| 438 |
+ |
$J_z$ to be positive when the physical thermal flux is from the liquid |
| 439 |
+ |
to metal, and negative vice versa. The $G$'s measured under different |
| 440 |
+ |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and [REF]. These |
| 441 |
+ |
results do not suggest that $G$ is dependent on $J_z$ within this flux |
| 442 |
+ |
range. The linear response of flux to thermal gradient simplifies our |
| 443 |
+ |
investigations in that we can rely on $G$ measurement with only a |
| 444 |
+ |
couple $J_z$'s and do not need to test a large series of fluxes. |
| 445 |
+ |
|
| 446 |
+ |
%ADD MORE TO TABLE |
| 447 |
+ |
\begin{table*} |
| 448 |
+ |
\begin{minipage}{\linewidth} |
| 449 |
+ |
\begin{center} |
| 450 |
+ |
\caption{Computed interfacial thermal conductivity ($G$ and |
| 451 |
+ |
$G^\prime$) values for the Au/butanethiol/hexane interface |
| 452 |
+ |
with united-atom model and different capping agent coverage |
| 453 |
+ |
and solvent molecule numbers at different temperatures using a |
| 454 |
+ |
range of energy fluxes.} |
| 455 |
+ |
|
| 456 |
+ |
\begin{tabular}{cccccc} |
| 457 |
+ |
\hline\hline |
| 458 |
+ |
Thiol & $\langle T\rangle$ & & $J_z$ & $G$ & $G^\prime$ \\ |
| 459 |
+ |
coverage (\%) & (K) & $N_{hexane}$ & (GW/m$^2$) & |
| 460 |
+ |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 461 |
+ |
\hline |
| 462 |
+ |
0.0 & 200 & 200 & 0.96 & 43.3 & 42.7 \\ |
| 463 |
+ |
& & & 1.91 & 45.7 & 42.9 \\ |
| 464 |
+ |
& & 166 & 0.96 & 43.1 & 53.4 \\ |
| 465 |
+ |
88.9 & 200 & 166 & 1.94 & 172 & 108 \\ |
| 466 |
+ |
100.0 & 250 & 200 & 0.96 & 81.8 & 67.0 \\ |
| 467 |
+ |
& & 166 & 0.98 & 79.0 & 62.9 \\ |
| 468 |
+ |
& & & 1.44 & 76.2 & 64.8 \\ |
| 469 |
+ |
& 200 & 200 & 1.92 & 129 & 87.3 \\ |
| 470 |
+ |
& & & 1.93 & 131 & 77.5 \\ |
| 471 |
+ |
& & 166 & 0.97 & 115 & 69.3 \\ |
| 472 |
+ |
& & & 1.94 & 125 & 87.1 \\ |
| 473 |
+ |
\hline\hline |
| 474 |
+ |
\end{tabular} |
| 475 |
+ |
\label{AuThiolHexaneUA} |
| 476 |
+ |
\end{center} |
| 477 |
+ |
\end{minipage} |
| 478 |
+ |
\end{table*} |
| 479 |
+ |
|
| 480 |
+ |
Furthermore, we also attempted to increase system average temperatures |
| 481 |
+ |
to above 200K. These simulations are first equilibrated in the NPT |
| 482 |
+ |
ensemble under normal pressure. As stated above, the TraPPE-UA model |
| 483 |
+ |
for hexane tends to predict a lower boiling point. In our simulations, |
| 484 |
+ |
hexane had diffculty to remain in liquid phase when NPT equilibration |
| 485 |
+ |
temperature is higher than 250K. Additionally, the equilibrated liquid |
| 486 |
+ |
hexane density under 250K becomes lower than experimental value. This |
| 487 |
+ |
expanded liquid phase leads to lower contact between hexane and |
| 488 |
+ |
butanethiol as well.[MAY NEED FIGURE] And this reduced contact would |
| 489 |
+ |
probably be accountable for a lower interfacial thermal conductance, |
| 490 |
+ |
as shown in Table \ref{AuThiolHexaneUA}. |
| 491 |
+ |
|
| 492 |
+ |
A similar study for TraPPE-UA toluene agrees with the above result as |
| 493 |
+ |
well. Having a higher boiling point, toluene tends to remain liquid in |
| 494 |
+ |
our simulations even equilibrated under 300K in NPT |
| 495 |
+ |
ensembles. Furthermore, the expansion of the toluene liquid phase is |
| 496 |
+ |
not as significant as that of the hexane. This prevents severe |
| 497 |
+ |
decrease of liquid-capping agent contact and the results (Table |
| 498 |
+ |
\ref{AuThiolToluene}) show only a slightly decreased interface |
| 499 |
+ |
conductance. Therefore, solvent-capping agent contact should play an |
| 500 |
+ |
important role in the thermal transport process across the interface |
| 501 |
+ |
in that higher degree of contact could yield increased conductance. |
| 502 |
+ |
|
| 503 |
+ |
[ADD SIGNS AND ERROR ESTIMATE TO TABLE] |
| 504 |
+ |
\begin{table*} |
| 505 |
+ |
\begin{minipage}{\linewidth} |
| 506 |
+ |
\begin{center} |
| 507 |
+ |
\caption{Computed interfacial thermal conductivity ($G$ and |
| 508 |
+ |
$G^\prime$) values for the Au/butanethiol/toluene interface at |
| 509 |
+ |
different temperatures using a range of energy fluxes.} |
| 510 |
+ |
|
| 511 |
+ |
\begin{tabular}{cccc} |
| 512 |
+ |
\hline\hline |
| 513 |
+ |
$\langle T\rangle$ & $J_z$ & $G$ & $G^\prime$ \\ |
| 514 |
+ |
(K) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 515 |
+ |
\hline |
| 516 |
+ |
200 & 1.86 & 180 & 135 \\ |
| 517 |
+ |
& 2.15 & 204 & 113 \\ |
| 518 |
+ |
& 3.93 & 175 & 114 \\ |
| 519 |
+ |
300 & 1.91 & 143 & 125 \\ |
| 520 |
+ |
& 4.19 & 134 & 113 \\ |
| 521 |
+ |
\hline\hline |
| 522 |
+ |
\end{tabular} |
| 523 |
+ |
\label{AuThiolToluene} |
| 524 |
+ |
\end{center} |
| 525 |
+ |
\end{minipage} |
| 526 |
+ |
\end{table*} |
| 527 |
+ |
|
| 528 |
+ |
Besides lower interfacial thermal conductance, surfaces in relatively |
| 529 |
+ |
high temperatures are susceptible to reconstructions, when |
| 530 |
+ |
butanethiols have a full coverage on the Au(111) surface. These |
| 531 |
+ |
reconstructions include surface Au atoms migrated outward to the S |
| 532 |
+ |
atom layer, and butanethiol molecules embedded into the original |
| 533 |
+ |
surface Au layer. The driving force for this behavior is the strong |
| 534 |
+ |
Au-S interactions in our simulations. And these reconstructions lead |
| 535 |
+ |
to higher ratio of Au-S attraction and thus is energetically |
| 536 |
+ |
favorable. Furthermore, this phenomenon agrees with experimental |
| 537 |
+ |
results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt |
| 538 |
+ |
{\it et al.} had kept their Au(111) slab rigid so that their |
| 539 |
+ |
simulations can reach 300K without surface reconstructions. Without |
| 540 |
+ |
this practice, simulating 100\% thiol covered interfaces under higher |
| 541 |
+ |
temperatures could hardly avoid surface reconstructions. However, our |
| 542 |
+ |
measurement is based on assuming homogeneity on $x$ and $y$ dimensions |
| 543 |
+ |
so that measurement of $T$ at particular $z$ would be an effective |
| 544 |
+ |
average of the particles of the same type. Since surface |
| 545 |
+ |
reconstructions could eliminate the original $x$ and $y$ dimensional |
| 546 |
+ |
homogeneity, measurement of $G$ is more difficult to conduct under |
| 547 |
+ |
higher temperatures. Therefore, most of our measurements are |
| 548 |
+ |
undertaken at $<T>\sim$200K. |
| 549 |
+ |
|
| 550 |
+ |
However, when the surface is not completely covered by butanethiols, |
| 551 |
+ |
the simulated system is more resistent to the reconstruction |
| 552 |
+ |
above. Our Au-butanethiol/toluene system did not see this phenomena |
| 553 |
+ |
even at $<T>\sim$300K. The Au(111) surfaces have a 90\% coverage of |
| 554 |
+ |
butanethiols and have empty three-fold sites. These empty sites could |
| 555 |
+ |
help prevent surface reconstruction in that they provide other means |
| 556 |
+ |
of capping agent relaxation. It is observed that butanethiols can |
| 557 |
+ |
migrate to their neighbor empty sites during a simulation. Therefore, |
| 558 |
+ |
we were able to obtain $G$'s for these interfaces even at a relatively |
| 559 |
+ |
high temperature without being affected by surface reconstructions. |
| 560 |
+ |
|
| 561 |
+ |
\subsection{Influence of Capping Agent Coverage on $G$} |
| 562 |
+ |
To investigate the influence of butanethiol coverage on interfacial |
| 563 |
+ |
thermal conductance, a series of different coverage Au-butanethiol |
| 564 |
+ |
surfaces is prepared and solvated with various organic |
| 565 |
+ |
molecules. These systems are then equilibrated and their interfacial |
| 566 |
+ |
thermal conductivity are measured with our NIVS algorithm. Table |
| 567 |
+ |
\ref{tlnUhxnUhxnD} lists these results for direct comparison between |
| 568 |
+ |
different coverages of butanethiol. |
| 569 |
+ |
|
| 570 |
+ |
With high coverage of butanethiol on the gold surface, |
| 571 |
+ |
the interfacial thermal conductance is enhanced |
| 572 |
+ |
significantly. Interestingly, a slightly lower butanethiol coverage |
| 573 |
+ |
leads to a moderately higher conductivity. This is probably due to |
| 574 |
+ |
more solvent/capping agent contact when butanethiol molecules are |
| 575 |
+ |
not densely packed, which enhances the interactions between the two |
| 576 |
+ |
phases and lowers the thermal transfer barrier of this interface. |
| 577 |
+ |
[COMPARE TO AU/WATER IN PAPER] |
| 578 |
+ |
|
| 579 |
+ |
|
| 580 |
+ |
significant conductance enhancement compared to the gold/water |
| 581 |
+ |
interface without capping agent and agree with available experimental |
| 582 |
+ |
data. This indicates that the metal-metal potential, though not |
| 583 |
+ |
predicting an accurate bulk metal thermal conductivity, does not |
| 584 |
+ |
greatly interfere with the simulation of the thermal conductance |
| 585 |
+ |
behavior across a non-metal interface. |
| 586 |
+ |
The results show that the two definitions used for $G$ yield |
| 587 |
+ |
comparable values, though $G^\prime$ tends to be smaller. |
| 588 |
+ |
|
| 589 |
+ |
|
| 590 |
+ |
\begin{table*} |
| 591 |
+ |
\begin{minipage}{\linewidth} |
| 592 |
+ |
\begin{center} |
| 593 |
+ |
\caption{Computed interfacial thermal conductivity ($G$ and |
| 594 |
+ |
$G^\prime$) values for the Au/butanethiol/hexane interface |
| 595 |
+ |
with united-atom model and different capping agent coverage |
| 596 |
+ |
and solvent molecule numbers at different temperatures using a |
| 597 |
+ |
range of energy fluxes.} |
| 598 |
+ |
|
| 599 |
+ |
\begin{tabular}{cccccc} |
| 600 |
+ |
\hline\hline |
| 601 |
+ |
Thiol & $\langle T\rangle$ & & $J_z$ & $G$ & $G^\prime$ \\ |
| 602 |
+ |
coverage (\%) & (K) & $N_{hexane}$ & (GW/m$^2$) & |
| 603 |
+ |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 604 |
+ |
\hline |
| 605 |
+ |
0.0 & 200 & 200 & 0.96 & 43.3 & 42.7 \\ |
| 606 |
+ |
& & & 1.91 & 45.7 & 42.9 \\ |
| 607 |
+ |
& & 166 & 0.96 & 43.1 & 53.4 \\ |
| 608 |
+ |
88.9 & 200 & 166 & 1.94 & 172 & 108 \\ |
| 609 |
+ |
100.0 & 250 & 200 & 0.96 & 81.8 & 67.0 \\ |
| 610 |
+ |
& & 166 & 0.98 & 79.0 & 62.9 \\ |
| 611 |
+ |
& & & 1.44 & 76.2 & 64.8 \\ |
| 612 |
+ |
& 200 & 200 & 1.92 & 129 & 87.3 \\ |
| 613 |
+ |
& & & 1.93 & 131 & 77.5 \\ |
| 614 |
+ |
& & 166 & 0.97 & 115 & 69.3 \\ |
| 615 |
+ |
& & & 1.94 & 125 & 87.1 \\ |
| 616 |
+ |
\hline\hline |
| 617 |
+ |
\end{tabular} |
| 618 |
+ |
\label{tlnUhxnUhxnD} |
| 619 |
+ |
\end{center} |
| 620 |
+ |
\end{minipage} |
| 621 |
+ |
\end{table*} |
| 622 |
+ |
|
| 623 |
+ |
\subsection{Influence of Chosen Molecule Model on $G$} |
| 624 |
+ |
[MAY COMBINE W MECHANISM STUDY] |
| 625 |
+ |
|
| 626 |
+ |
For the all-atom model, the liquid hexane phase was not stable under NPT |
| 627 |
+ |
conditions. Therefore, the simulation length scale parameters are |
| 628 |
+ |
adopted from previous equilibration results of the united-atom model |
| 629 |
+ |
at 200K. Table \ref{AuThiolHexaneAA} shows the results of these |
| 630 |
+ |
simulations. The conductivity values calculated with full capping |
| 631 |
+ |
agent coverage are substantially larger than observed in the |
| 632 |
+ |
united-atom model, and is even higher than predicted by |
| 633 |
+ |
experiments. It is possible that our parameters for metal-non-metal |
| 634 |
+ |
particle interactions lead to an overestimate of the interfacial |
| 635 |
+ |
thermal conductivity, although the active C-H vibrations in the |
| 636 |
+ |
all-atom model (which should not be appreciably populated at normal |
| 637 |
+ |
temperatures) could also account for this high conductivity. The major |
| 638 |
+ |
thermal transfer barrier of Au/butanethiol/hexane interface is between |
| 639 |
+ |
the liquid phase and the capping agent, so extra degrees of freedom |
| 640 |
+ |
such as the C-H vibrations could enhance heat exchange between these |
| 641 |
+ |
two phases and result in a much higher conductivity. |
| 642 |
+ |
|
| 643 |
+ |
\begin{table*} |
| 644 |
+ |
\begin{minipage}{\linewidth} |
| 645 |
+ |
\begin{center} |
| 646 |
+ |
|
| 647 |
+ |
\caption{Computed interfacial thermal conductivity ($G$ and |
| 648 |
+ |
$G^\prime$) values for the Au/butanethiol/hexane interface |
| 649 |
+ |
with all-atom model and different capping agent coverage at |
| 650 |
+ |
200K using a range of energy fluxes.} |
| 651 |
+ |
|
| 652 |
+ |
\begin{tabular}{cccc} |
| 653 |
+ |
\hline\hline |
| 654 |
+ |
Thiol & $J_z$ & $G$ & $G^\prime$ \\ |
| 655 |
+ |
coverage (\%) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 656 |
+ |
\hline |
| 657 |
+ |
0.0 & 0.95 & 28.5 & 27.2 \\ |
| 658 |
+ |
& 1.88 & 30.3 & 28.9 \\ |
| 659 |
+ |
100.0 & 2.87 & 551 & 294 \\ |
| 660 |
+ |
& 3.81 & 494 & 193 \\ |
| 661 |
+ |
\hline\hline |
| 662 |
+ |
\end{tabular} |
| 663 |
+ |
\label{AuThiolHexaneAA} |
| 664 |
+ |
\end{center} |
| 665 |
+ |
\end{minipage} |
| 666 |
+ |
\end{table*} |
| 667 |
+ |
|
| 668 |
+ |
|
| 669 |
+ |
\subsection{Mechanism of Interfacial Thermal Conductance Enhancement |
| 670 |
+ |
by Capping Agent} |
| 671 |
+ |
[MAY INTRODUCE PROTOCOL IN METHODOLOGY/COMPUTATIONAL DETAIL] |
| 672 |
+ |
|
| 673 |
+ |
|
| 674 |
+ |
%subsubsection{Vibrational spectrum study on conductance mechanism} |
| 675 |
+ |
To investigate the mechanism of this interfacial thermal conductance, |
| 676 |
+ |
the vibrational spectra of various gold systems were obtained and are |
| 677 |
+ |
shown as in the upper panel of Fig. \ref{vibration}. To obtain these |
| 678 |
+ |
spectra, one first runs a simulation in the NVE ensemble and collects |
| 679 |
+ |
snapshots of configurations; these configurations are used to compute |
| 680 |
+ |
the velocity auto-correlation functions, which is used to construct a |
| 681 |
+ |
power spectrum via a Fourier transform. The gold surfaces covered by |
| 682 |
+ |
butanethiol molecules exhibit an additional peak observed at a |
| 683 |
+ |
frequency of $\sim$170cm$^{-1}$, which is attributed to the vibration |
| 684 |
+ |
of the S-Au bond. This vibration enables efficient thermal transport |
| 685 |
+ |
from surface Au atoms to the capping agents. Simultaneously, as shown |
| 686 |
+ |
in the lower panel of Fig. \ref{vibration}, the large overlap of the |
| 687 |
+ |
vibration spectra of butanethiol and hexane in the all-atom model, |
| 688 |
+ |
including the C-H vibration, also suggests high thermal exchange |
| 689 |
+ |
efficiency. The combination of these two effects produces the drastic |
| 690 |
+ |
interfacial thermal conductance enhancement in the all-atom model. |
| 691 |
+ |
|
| 692 |
+ |
\begin{figure} |
| 693 |
+ |
\includegraphics[width=\linewidth]{vibration} |
| 694 |
+ |
\caption{Vibrational spectra obtained for gold in different |
| 695 |
+ |
environments (upper panel) and for Au/thiol/hexane simulation in |
| 696 |
+ |
all-atom model (lower panel).} |
| 697 |
+ |
\label{vibration} |
| 698 |
+ |
\end{figure} |
| 699 |
+ |
% MAY NEED TO CONVERT TO JPEG |
| 700 |
+ |
|
| 701 |
+ |
\section{Conclusions} |
| 702 |
+ |
|
| 703 |
+ |
|
| 704 |
+ |
[NECESSITY TO STUDY THERMAL CONDUCTANCE IN NANOCRYSTAL STRUCTURE]\cite{vlugt:cpc2007154} |
| 705 |
+ |
|
| 706 |
|
\section{Acknowledgments} |
| 707 |
|
Support for this project was provided by the National Science |
| 708 |
|
Foundation under grant CHE-0848243. Computational time was provided by |
| 709 |
|
the Center for Research Computing (CRC) at the University of Notre |
| 710 |
< |
Dame. \newpage |
| 710 |
> |
Dame. \newpage |
| 711 |
|
|
| 712 |
|
\bibliography{interfacial} |
| 713 |
|
|
