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\begin{doublespace} |
45 |
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|
46 |
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\begin{abstract} |
47 |
< |
The abstract version 2 |
47 |
> |
|
48 |
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We have developed a Non-Isotropic Velocity Scaling algorithm for |
49 |
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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 |
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leads to higher interfacial thermal transfer efficiency. |
62 |
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|
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\end{abstract} |
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|
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\newpage |
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|
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\section{Introduction} |
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|
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< |
The intro. |
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Interfacial thermal conductance is extensively studied both |
76 |
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experimentally and computationally, and systems with interfaces |
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> |
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 |
+ |
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 |
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|
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|
\section{Methodology} |
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|
\subsection{Algorithm} |
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|
There have been many algorithms for computing thermal conductivity |
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|
134 |
|
(wondering how much detail of algorithm should be put here...) |
135 |
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|
136 |
< |
\subsection{Simulation Parameters} |
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> |
\subsection{Force Field Parameters} |
137 |
|
Our simulation systems consists of metal gold lattice slab solvated by |
138 |
|
organic solvents. In order to study the role of capping agents in |
139 |
|
interfacial thermal conductance, butanethiol is chosen to cover gold |
163 |
|
hautman and klein's paper here and more discussion; need to put |
164 |
|
aromatic-metal interaction approximation here) |
165 |
|
|
166 |
+ |
[TABULATED FORCE FIELD PARAMETERS NEEDED] |
167 |
+ |
|
168 |
|
\section{Computational Details} |
169 |
+ |
\subsection{System Geometry} |
170 |
|
Our simulation systems consists of a lattice Au slab with the (111) |
171 |
|
surface perpendicular to the $z$-axis, and a solvent layer between the |
172 |
|
periodic Au slabs along the $z$-axis. To set up the interfacial |
180 |
|
|
181 |
|
\cite{packmol} |
182 |
|
|
183 |
+ |
\subsection{Simulation Parameters} |
184 |
+ |
|
185 |
+ |
When the interfacial conductance is {\it not} small, there are two |
186 |
+ |
ways to define $G$. If we assume the temperature is discretely |
187 |
+ |
different on two sides of the interface, $G$ can be calculated with |
188 |
+ |
the thermal flux applied $J$ and the temperature difference measured |
189 |
+ |
$\Delta T$ as: |
190 |
+ |
\begin{equation} |
191 |
+ |
G=\frac{J}{\Delta T} |
192 |
+ |
\label{discreteG} |
193 |
+ |
\end{equation} |
194 |
+ |
We can as well assume a continuous temperature profile along the |
195 |
+ |
thermal gradient axis $z$ and define $G$ as the change of bulk thermal |
196 |
+ |
conductivity $\lambda$ at a defined interfacial point: |
197 |
+ |
\begin{equation} |
198 |
+ |
G^\prime = \Big|\frac{\partial\lambda}{\partial z}\Big| |
199 |
+ |
= \Big|\frac{\partial}{\partial z}\left(-J_z\Big/ |
200 |
+ |
\left(\frac{\partial T}{\partial z}\right)\right)\Big| |
201 |
+ |
= J_z\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
202 |
+ |
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
203 |
+ |
\label{derivativeG} |
204 |
+ |
\end{equation} |
205 |
+ |
With the temperature profile obtained from simulations, one is able to |
206 |
+ |
approximate the first and second derivatives of $T$ with finite |
207 |
+ |
difference method and thus calculate $G^\prime$. |
208 |
+ |
|
209 |
+ |
In what follows, both definitions are used for calculation and comparison. |
210 |
+ |
|
211 |
+ |
\section{Results} |
212 |
+ |
\subsection{Toluene Solvent} |
213 |
+ |
|
214 |
+ |
The simulations follow a protocol similar to the previous gold/water |
215 |
+ |
interfacial systems. The results (Table \ref{AuThiolToluene}) show a |
216 |
+ |
significant conductance enhancement compared to the gold/water |
217 |
+ |
interface without capping agent and agree with available experimental |
218 |
+ |
data. This indicates that the metal-metal potential, though not |
219 |
+ |
predicting an accurate bulk metal thermal conductivity, does not |
220 |
+ |
greatly interfere with the simulation of the thermal conductance |
221 |
+ |
behavior across a non-metal interface. The solvent model is not |
222 |
+ |
particularly volatile, so the simulation cell does not expand |
223 |
+ |
significantly under higher temperature. We did not observe a |
224 |
+ |
significant conductance decrease when the temperature was increased to |
225 |
+ |
300K. The results show that the two definitions used for $G$ yield |
226 |
+ |
comparable values, though $G^\prime$ tends to be smaller. |
227 |
+ |
|
228 |
+ |
\begin{table*} |
229 |
+ |
\begin{minipage}{\linewidth} |
230 |
+ |
\begin{center} |
231 |
+ |
\caption{Computed interfacial thermal conductivity ($G$ and |
232 |
+ |
$G^\prime$) values for the Au/butanethiol/toluene interface at |
233 |
+ |
different temperatures using a range of energy fluxes.} |
234 |
+ |
|
235 |
+ |
\begin{tabular}{cccc} |
236 |
+ |
\hline\hline |
237 |
+ |
$\langle T\rangle$ & $J_z$ & $G$ & $G^\prime$ \\ |
238 |
+ |
(K) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
239 |
+ |
\hline |
240 |
+ |
200 & 1.86 & 180 & 135 \\ |
241 |
+ |
& 2.15 & 204 & 113 \\ |
242 |
+ |
& 3.93 & 175 & 114 \\ |
243 |
+ |
300 & 1.91 & 143 & 125 \\ |
244 |
+ |
& 4.19 & 134 & 113 \\ |
245 |
+ |
\hline\hline |
246 |
+ |
\end{tabular} |
247 |
+ |
\label{AuThiolToluene} |
248 |
+ |
\end{center} |
249 |
+ |
\end{minipage} |
250 |
+ |
\end{table*} |
251 |
+ |
|
252 |
+ |
\subsection{Hexane Solvent} |
253 |
+ |
|
254 |
+ |
Using the united-atom model, different coverages of capping agent, |
255 |
+ |
temperatures of simulations and numbers of solvent molecules were all |
256 |
+ |
investigated and Table \ref{AuThiolHexaneUA} shows the results of |
257 |
+ |
these computations. The number of hexane molecules in our simulations |
258 |
+ |
does not affect the calculations significantly. However, a very long |
259 |
+ |
length scale for the thermal gradient axis ($z$) may cause excessively |
260 |
+ |
hot or cold temperatures in the middle of the solvent region and lead |
261 |
+ |
to undesired phenomena such as solvent boiling or freezing, while too |
262 |
+ |
few solvent molecules would change the normal behavior of the liquid |
263 |
+ |
phase. Our $N_{hexane}$ values were chosen to ensure that these |
264 |
+ |
extreme cases did not happen to our simulations. |
265 |
+ |
|
266 |
+ |
Table \ref{AuThiolHexaneUA} enables direct comparison between |
267 |
+ |
different coverages of capping agent, when other system parameters are |
268 |
+ |
held constant. With high coverage of butanethiol on the gold surface, |
269 |
+ |
the interfacial thermal conductance is enhanced |
270 |
+ |
significantly. Interestingly, a slightly lower butanethiol coverage |
271 |
+ |
leads to a moderately higher conductivity. This is probably due to |
272 |
+ |
more solvent/capping agent contact when butanethiol molecules are |
273 |
+ |
not densely packed, which enhances the interactions between the two |
274 |
+ |
phases and lowers the thermal transfer barrier of this interface. |
275 |
+ |
% [COMPARE TO AU/WATER IN PAPER] |
276 |
+ |
|
277 |
+ |
It is also noted that the overall simulation temperature is another |
278 |
+ |
factor that affects the interfacial thermal conductance. One |
279 |
+ |
possibility of this effect may be rooted in the decrease in density of |
280 |
+ |
the liquid phase. We observed that when the average temperature |
281 |
+ |
increases from 200K to 250K, the bulk hexane density becomes lower |
282 |
+ |
than experimental value, as the system is equilibrated under NPT |
283 |
+ |
ensemble. This leads to lower contact between solvent and capping |
284 |
+ |
agent, and thus lower conductivity. |
285 |
+ |
|
286 |
+ |
Conductivity values are more difficult to obtain under higher |
287 |
+ |
temperatures. This is because the Au surface tends to undergo |
288 |
+ |
reconstructions in relatively high temperatures. Surface Au atoms can |
289 |
+ |
migrate outward to reach higher Au-S contact; and capping agent |
290 |
+ |
molecules can be embedded into the surface Au layer due to the same |
291 |
+ |
driving force. This phenomenon agrees with experimental |
292 |
+ |
results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. A surface |
293 |
+ |
fully covered in capping agent is more susceptible to reconstruction, |
294 |
+ |
possibly because fully coverage prevents other means of capping agent |
295 |
+ |
relaxation, such as migration to an empty neighbor three-fold site. |
296 |
+ |
|
297 |
+ |
%MAY ADD MORE DATA TO TABLE |
298 |
+ |
\begin{table*} |
299 |
+ |
\begin{minipage}{\linewidth} |
300 |
+ |
\begin{center} |
301 |
+ |
\caption{Computed interfacial thermal conductivity ($G$ and |
302 |
+ |
$G^\prime$) values for the Au/butanethiol/hexane interface |
303 |
+ |
with united-atom model and different capping agent coverage |
304 |
+ |
and solvent molecule numbers at different temperatures using a |
305 |
+ |
range of energy fluxes.} |
306 |
+ |
|
307 |
+ |
\begin{tabular}{cccccc} |
308 |
+ |
\hline\hline |
309 |
+ |
Thiol & $\langle T\rangle$ & & $J_z$ & $G$ & $G^\prime$ \\ |
310 |
+ |
coverage (\%) & (K) & $N_{hexane}$ & (GW/m$^2$) & |
311 |
+ |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
312 |
+ |
\hline |
313 |
+ |
0.0 & 200 & 200 & 0.96 & 43.3 & 42.7 \\ |
314 |
+ |
& & & 1.91 & 45.7 & 42.9 \\ |
315 |
+ |
& & 166 & 0.96 & 43.1 & 53.4 \\ |
316 |
+ |
88.9 & 200 & 166 & 1.94 & 172 & 108 \\ |
317 |
+ |
100.0 & 250 & 200 & 0.96 & 81.8 & 67.0 \\ |
318 |
+ |
& & 166 & 0.98 & 79.0 & 62.9 \\ |
319 |
+ |
& & & 1.44 & 76.2 & 64.8 \\ |
320 |
+ |
& 200 & 200 & 1.92 & 129 & 87.3 \\ |
321 |
+ |
& & & 1.93 & 131 & 77.5 \\ |
322 |
+ |
& & 166 & 0.97 & 115 & 69.3 \\ |
323 |
+ |
& & & 1.94 & 125 & 87.1 \\ |
324 |
+ |
\hline\hline |
325 |
+ |
\end{tabular} |
326 |
+ |
\label{AuThiolHexaneUA} |
327 |
+ |
\end{center} |
328 |
+ |
\end{minipage} |
329 |
+ |
\end{table*} |
330 |
+ |
|
331 |
+ |
For the all-atom model, the liquid hexane phase was not stable under NPT |
332 |
+ |
conditions. Therefore, the simulation length scale parameters are |
333 |
+ |
adopted from previous equilibration results of the united-atom model |
334 |
+ |
at 200K. Table \ref{AuThiolHexaneAA} shows the results of these |
335 |
+ |
simulations. The conductivity values calculated with full capping |
336 |
+ |
agent coverage are substantially larger than observed in the |
337 |
+ |
united-atom model, and is even higher than predicted by |
338 |
+ |
experiments. It is possible that our parameters for metal-non-metal |
339 |
+ |
particle interactions lead to an overestimate of the interfacial |
340 |
+ |
thermal conductivity, although the active C-H vibrations in the |
341 |
+ |
all-atom model (which should not be appreciably populated at normal |
342 |
+ |
temperatures) could also account for this high conductivity. The major |
343 |
+ |
thermal transfer barrier of Au/butanethiol/hexane interface is between |
344 |
+ |
the liquid phase and the capping agent, so extra degrees of freedom |
345 |
+ |
such as the C-H vibrations could enhance heat exchange between these |
346 |
+ |
two phases and result in a much higher conductivity. |
347 |
+ |
|
348 |
+ |
\begin{table*} |
349 |
+ |
\begin{minipage}{\linewidth} |
350 |
+ |
\begin{center} |
351 |
+ |
|
352 |
+ |
\caption{Computed interfacial thermal conductivity ($G$ and |
353 |
+ |
$G^\prime$) values for the Au/butanethiol/hexane interface |
354 |
+ |
with all-atom model and different capping agent coverage at |
355 |
+ |
200K using a range of energy fluxes.} |
356 |
+ |
|
357 |
+ |
\begin{tabular}{cccc} |
358 |
+ |
\hline\hline |
359 |
+ |
Thiol & $J_z$ & $G$ & $G^\prime$ \\ |
360 |
+ |
coverage (\%) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
361 |
+ |
\hline |
362 |
+ |
0.0 & 0.95 & 28.5 & 27.2 \\ |
363 |
+ |
& 1.88 & 30.3 & 28.9 \\ |
364 |
+ |
100.0 & 2.87 & 551 & 294 \\ |
365 |
+ |
& 3.81 & 494 & 193 \\ |
366 |
+ |
\hline\hline |
367 |
+ |
\end{tabular} |
368 |
+ |
\label{AuThiolHexaneAA} |
369 |
+ |
\end{center} |
370 |
+ |
\end{minipage} |
371 |
+ |
\end{table*} |
372 |
+ |
|
373 |
+ |
%subsubsection{Vibrational spectrum study on conductance mechanism} |
374 |
+ |
To investigate the mechanism of this interfacial thermal conductance, |
375 |
+ |
the vibrational spectra of various gold systems were obtained and are |
376 |
+ |
shown as in the upper panel of Fig. \ref{vibration}. To obtain these |
377 |
+ |
spectra, one first runs a simulation in the NVE ensemble and collects |
378 |
+ |
snapshots of configurations; these configurations are used to compute |
379 |
+ |
the velocity auto-correlation functions, which is used to construct a |
380 |
+ |
power spectrum via a Fourier transform. The gold surfaces covered by |
381 |
+ |
butanethiol molecules exhibit an additional peak observed at a |
382 |
+ |
frequency of $\sim$170cm$^{-1}$, which is attributed to the vibration |
383 |
+ |
of the S-Au bond. This vibration enables efficient thermal transport |
384 |
+ |
from surface Au atoms to the capping agents. Simultaneously, as shown |
385 |
+ |
in the lower panel of Fig. \ref{vibration}, the large overlap of the |
386 |
+ |
vibration spectra of butanethiol and hexane in the all-atom model, |
387 |
+ |
including the C-H vibration, also suggests high thermal exchange |
388 |
+ |
efficiency. The combination of these two effects produces the drastic |
389 |
+ |
interfacial thermal conductance enhancement in the all-atom model. |
390 |
+ |
|
391 |
+ |
\begin{figure} |
392 |
+ |
\includegraphics[width=\linewidth]{vibration} |
393 |
+ |
\caption{Vibrational spectra obtained for gold in different |
394 |
+ |
environments (upper panel) and for Au/thiol/hexane simulation in |
395 |
+ |
all-atom model (lower panel).} |
396 |
+ |
\label{vibration} |
397 |
+ |
\end{figure} |
398 |
+ |
% 600dpi, letter size. too large? |
399 |
+ |
|
400 |
+ |
|
401 |
|
\section{Acknowledgments} |
402 |
|
Support for this project was provided by the National Science |
403 |
|
Foundation under grant CHE-0848243. Computational time was provided by |