45 |
|
|
46 |
|
\begin{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. |
48 |
> |
With the Non-Isotropic Velocity Scaling algorithm (NIVS) we have |
49 |
> |
developed, an unphysical thermal flux can be effectively set up even |
50 |
> |
for non-homogeneous systems like interfaces in non-equilibrium |
51 |
> |
molecular dynamics simulations. In this work, this algorithm is |
52 |
> |
applied for simulating thermal conductance at metal / organic solvent |
53 |
> |
interfaces with various coverages of butanethiol capping |
54 |
> |
agents. Different solvents and force field models were tested. Our |
55 |
> |
results suggest that the United-Atom models are able to provide an |
56 |
> |
estimate of the interfacial thermal conductivity comparable to |
57 |
> |
experiments in our simulations with satisfactory computational |
58 |
> |
efficiency. From our results, the acoustic impedance mismatch between |
59 |
> |
metal and liquid phase is effectively reduced by the capping |
60 |
> |
agents, and thus leads to interfacial thermal conductance |
61 |
> |
enhancement. Furthermore, this effect is closely related to the |
62 |
> |
capping agent coverage on the metal surfaces and the type of solvent |
63 |
> |
molecules, and is affected by the models used in the simulations. |
64 |
|
|
65 |
|
\end{abstract} |
66 |
|
|
213 |
|
\label{gradT} |
214 |
|
\end{figure} |
215 |
|
|
216 |
+ |
[MAY INCLUDE POWER SPECTRUM PROTOCOL] |
217 |
+ |
|
218 |
|
\section{Computational Details} |
219 |
< |
\subsection{System Geometry} |
219 |
> |
\subsection{Simulation Protocol} |
220 |
|
In our simulations, Au is used to construct a metal slab with bare |
221 |
|
(111) surface perpendicular to the $z$-axis. Different slab thickness |
222 |
|
(layer numbers of Au) are simulated. This metal slab is first |
244 |
|
interactions do not disrupt the simulations. Two solvents are |
245 |
|
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). The |
248 |
< |
spacing filled by solvent molecules, i.e. the gap between periodically |
249 |
< |
repeated Au-butanethiol surfaces should be carefully chosen so that it |
250 |
< |
would not be too short to affect the liquid phase structure, nor too |
251 |
< |
long, leading to over cooling (freezing) or heating (boiling) when a |
252 |
< |
thermal flux is applied. In our simulations, this spacing is usually |
253 |
< |
$35 \sim 60$\AA. |
247 |
> |
vibrational frequencies and chain-like shape ({\it n}-hexane). [MAY |
248 |
> |
EXPLAIN WHY WE CHOOSE THEM] |
249 |
> |
|
250 |
> |
The spacing filled by solvent molecules, i.e. the gap between |
251 |
> |
periodically repeated Au-butanethiol surfaces should be carefully |
252 |
> |
chosen. A very long length scale for the thermal gradient axis ($z$) |
253 |
> |
may cause excessively hot or cold temperatures in the middle of the |
254 |
> |
solvent region and lead to undesired phenomena such as solvent boiling |
255 |
> |
or freezing when a thermal flux is applied. Conversely, too few |
256 |
> |
solvent molecules would change the normal behavior of the liquid |
257 |
> |
phase. Therefore, our $N_{solvent}$ values were chosen to ensure that |
258 |
> |
these extreme cases did not happen to our simulations. And the |
259 |
> |
corresponding spacing is usually $35 \sim 60$\AA. |
260 |
|
|
261 |
|
The initial configurations generated by Packmol are further |
262 |
|
equilibrated with the $x$ and $y$ dimensions fixed, only allowing |
297 |
|
reparametrized for accurate surface energies compared to the |
298 |
|
Sutton-Chen potentials\cite{Chen90}. |
299 |
|
|
300 |
+ |
Figure [REF] demonstrates how we name our pseudo-atoms of the |
301 |
+ |
molecules in our simulations. |
302 |
+ |
[FIGURE FOR MOLECULE NOMENCLATURE] |
303 |
+ |
|
304 |
|
For both solvent molecules, straight chain {\it n}-hexane and aromatic |
305 |
|
toluene, United-Atom (UA) and All-Atom (AA) models are used |
306 |
|
respectively. The TraPPE-UA |
308 |
|
for our UA solvent molecules. In these models, pseudo-atoms are |
309 |
|
located at the carbon centers for alkyl groups. By eliminating |
310 |
|
explicit hydrogen atoms, these models are simple and computationally |
311 |
< |
efficient, while maintains good accuracy. [LOW BOILING POINT IS A |
312 |
< |
KNOWN PROBLEM FOR TRAPPE-UA ALKANES, NEED MORE DISCUSSION] |
313 |
< |
for |
314 |
< |
toluene, force fields are |
315 |
< |
used with rigid body constraints applied.[MORE DETAILS NEEDED] |
311 |
> |
efficient, while maintains good accuracy. However, the TraPPE-UA for |
312 |
> |
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] |
321 |
|
|
322 |
< |
Besides the TraPPE-UA models, AA models are included in our studies as |
323 |
< |
well. For hexane, the OPLS all-atom\cite{OPLSAA} force field is |
324 |
< |
used. [MORE DETAILS] |
325 |
< |
For toluene, |
322 |
> |
Besides the TraPPE-UA models, AA models for both organic solvents are |
323 |
> |
included in our studies as well. For hexane, the OPLS-AA\cite{OPLSAA} |
324 |
> |
force field is used. [MORE DETAILS] |
325 |
> |
For toluene, the United Force Field developed by Rapp\'{e} {\it et |
326 |
> |
al.}\cite{doi:10.1021/ja00051a040} is adopted.[MORE DETAILS] |
327 |
|
|
328 |
< |
Buatnethiol molecules are used as capping agent for some of our |
329 |
< |
simulations. United-Atom\cite{TraPPE-UA.thiols} and All-Atom models |
330 |
< |
are respectively used corresponding to the force field type of |
331 |
< |
solvent. |
328 |
> |
The capping agent in our simulations, the butanethiol molecules can |
329 |
> |
either use UA or AA model. The TraPPE-UA force fields includes |
330 |
> |
parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for |
331 |
> |
UA butanethiol model in our simulations. The OPLS-AA also provides |
332 |
> |
parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) |
333 |
> |
surfaces do not have the hydrogen atom bonded to sulfur. To adapt this |
334 |
> |
change and derive suitable parameters for butanethiol adsorbed on |
335 |
> |
Au(111) surfaces, we adopt the S parameters from [CITATION CF VLUGT] |
336 |
> |
and modify parameters for its neighbor C atom for charge balance in |
337 |
> |
the molecule. Note that the model choice (UA or AA) of capping agent |
338 |
> |
can be different from the solvent. Regardless of model choice, the |
339 |
> |
force field parameters for interactions between capping agent and |
340 |
> |
solvent can be derived using Lorentz-Berthelot Mixing Rule: |
341 |
|
|
342 |
+ |
|
343 |
|
To describe the interactions between metal Au and non-metal capping |
344 |
< |
agent and solvent, we refer to Vlugt\cite{vlugt:cpc2007154} and derive |
345 |
< |
other interactions which are not parametrized in their work. (can add |
346 |
< |
hautman and klein's paper here and more discussion; need to put |
347 |
< |
aromatic-metal interaction approximation here) |
344 |
> |
agent and solvent particles, we refer to an adsorption study of alkyl |
345 |
> |
thiols on gold surfaces by Vlugt {\it et |
346 |
> |
al.}\cite{vlugt:cpc2007154} They fitted an effective Lennard-Jones |
347 |
> |
form of potential parameters for the interaction between Au and |
348 |
> |
pseudo-atoms CH$_x$ and S based on a well-established and widely-used |
349 |
> |
effective potential of Hautman and Klein[CITATION] for the Au(111) |
350 |
> |
surface. As our simulations require the gold lattice slab to be |
351 |
> |
non-rigid so that it could accommodate kinetic energy for thermal |
352 |
> |
transport study purpose, the pair-wise form of potentials is |
353 |
> |
preferred. |
354 |
|
|
355 |
< |
[TABULATED FORCE FIELD PARAMETERS NEEDED] |
355 |
> |
Besides, the potentials developed from {\it ab initio} calculations by |
356 |
> |
Leng {\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
357 |
> |
interactions between Au and aromatic C/H atoms in toluene.[MORE DETAILS] |
358 |
|
|
359 |
< |
\section{Results} |
360 |
< |
\subsection{Toluene Solvent} |
359 |
> |
However, the Lennard-Jones parameters between Au and other types of |
360 |
> |
particles in our simulations are not yet well-established. For these |
361 |
> |
interactions, we attempt to derive their parameters using the Mixing |
362 |
> |
Rule. To do this, the ``Metal-non-Metal'' (MnM) interaction parameters |
363 |
> |
for Au is first extracted from the Au-CH$_x$ parameters by applying |
364 |
> |
the Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM'' |
365 |
> |
parameters in our simulations. |
366 |
|
|
324 |
– |
The results (Table \ref{AuThiolToluene}) show a |
325 |
– |
significant conductance enhancement compared to the gold/water |
326 |
– |
interface without capping agent and agree with available experimental |
327 |
– |
data. This indicates that the metal-metal potential, though not |
328 |
– |
predicting an accurate bulk metal thermal conductivity, does not |
329 |
– |
greatly interfere with the simulation of the thermal conductance |
330 |
– |
behavior across a non-metal interface. The solvent model is not |
331 |
– |
particularly volatile, so the simulation cell does not expand |
332 |
– |
significantly under higher temperature. We did not observe a |
333 |
– |
significant conductance decrease when the temperature was increased to |
334 |
– |
300K. The results show that the two definitions used for $G$ yield |
335 |
– |
comparable values, though $G^\prime$ tends to be smaller. |
336 |
– |
|
367 |
|
\begin{table*} |
368 |
|
\begin{minipage}{\linewidth} |
369 |
|
\begin{center} |
370 |
< |
\caption{Computed interfacial thermal conductivity ($G$ and |
371 |
< |
$G^\prime$) values for the Au/butanethiol/toluene interface at |
342 |
< |
different temperatures using a range of energy fluxes.} |
370 |
> |
\caption{Lennard-Jones parameters for Au-non-Metal |
371 |
> |
interactions in our simulations.} |
372 |
|
|
373 |
< |
\begin{tabular}{cccc} |
373 |
> |
\begin{tabular}{ccc} |
374 |
|
\hline\hline |
375 |
< |
$\langle T\rangle$ & $J_z$ & $G$ & $G^\prime$ \\ |
376 |
< |
(K) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
375 |
> |
Non-metal atom & $\sigma$ & $\epsilon$ \\ |
376 |
> |
(or pseudo-atom) & \AA & kcal/mol \\ |
377 |
|
\hline |
378 |
< |
200 & 1.86 & 180 & 135 \\ |
379 |
< |
& 2.15 & 204 & 113 \\ |
380 |
< |
& 3.93 & 175 & 114 \\ |
381 |
< |
300 & 1.91 & 143 & 125 \\ |
382 |
< |
& 4.19 & 134 & 113 \\ |
378 |
> |
S & 2.40 & 8.465 \\ |
379 |
> |
CH3 & 3.54 & 0.2146 \\ |
380 |
> |
CH2 & 3.54 & 0.1749 \\ |
381 |
> |
CT3 & 3.365 & 0.1373 \\ |
382 |
> |
CT2 & 3.365 & 0.1373 \\ |
383 |
> |
CTT & 3.365 & 0.1373 \\ |
384 |
> |
HC & 2.865 & 0.09256 \\ |
385 |
> |
CHar & 3.4625 & 0.1680 \\ |
386 |
> |
CRar & 3.555 & 0.1604 \\ |
387 |
> |
CA & 3.173 & 0.0640 \\ |
388 |
> |
HA & 2.746 & 0.0414 \\ |
389 |
|
\hline\hline |
390 |
|
\end{tabular} |
391 |
< |
\label{AuThiolToluene} |
391 |
> |
\label{MnM} |
392 |
|
\end{center} |
393 |
|
\end{minipage} |
394 |
|
\end{table*} |
395 |
|
|
361 |
– |
\subsection{Hexane Solvent} |
396 |
|
|
397 |
< |
Using the united-atom model, different coverages of capping agent, |
398 |
< |
temperatures of simulations and numbers of solvent molecules were all |
399 |
< |
investigated and Table \ref{AuThiolHexaneUA} shows the results of |
400 |
< |
these computations. The number of hexane molecules in our simulations |
401 |
< |
does not affect the calculations significantly. However, a very long |
402 |
< |
length scale for the thermal gradient axis ($z$) may cause excessively |
403 |
< |
hot or cold temperatures in the middle of the solvent region and lead |
404 |
< |
to undesired phenomena such as solvent boiling or freezing, while too |
405 |
< |
few solvent molecules would change the normal behavior of the liquid |
406 |
< |
phase. Our $N_{hexane}$ values were chosen to ensure that these |
373 |
< |
extreme cases did not happen to our simulations. |
397 |
> |
\section{Results and Discussions} |
398 |
> |
[MAY HAVE A BRIEF SUMMARY] |
399 |
> |
\subsection{How Simulation Parameters Affects $G$} |
400 |
> |
[MAY NOT PUT AT FIRST] |
401 |
> |
We have varied our protocol or other parameters of the simulations in |
402 |
> |
order to investigate how these factors would affect the measurement of |
403 |
> |
$G$'s. It turned out that while some of these parameters would not |
404 |
> |
affect the results substantially, some other changes to the |
405 |
> |
simulations would have a significant impact on the measurement |
406 |
> |
results. |
407 |
|
|
408 |
< |
Table \ref{AuThiolHexaneUA} enables direct comparison between |
409 |
< |
different coverages of capping agent, when other system parameters are |
410 |
< |
held constant. With high coverage of butanethiol on the gold surface, |
411 |
< |
the interfacial thermal conductance is enhanced |
412 |
< |
significantly. Interestingly, a slightly lower butanethiol coverage |
413 |
< |
leads to a moderately higher conductivity. This is probably due to |
414 |
< |
more solvent/capping agent contact when butanethiol molecules are |
415 |
< |
not densely packed, which enhances the interactions between the two |
416 |
< |
phases and lowers the thermal transfer barrier of this interface. |
384 |
< |
% [COMPARE TO AU/WATER IN PAPER] |
408 |
> |
In some of our simulations, we allowed $L_x$ and $L_y$ to change |
409 |
> |
during equilibrating the liquid phase. Due to the stiffness of the Au |
410 |
> |
slab, $L_x$ and $L_y$ would not change noticeably after |
411 |
> |
equilibration. Although $L_z$ could fluctuate $\sim$1\% after a system |
412 |
> |
is fully equilibrated in the NPT ensemble, this fluctuation, as well |
413 |
> |
as those comparably smaller to $L_x$ and $L_y$, would not be magnified |
414 |
> |
on the calculated $G$'s, as shown in Table \ref{AuThiolHexaneUA}. This |
415 |
> |
insensivity to $L_i$ fluctuations allows reliable measurement of $G$'s |
416 |
> |
without the necessity of extremely cautious equilibration process. |
417 |
|
|
418 |
< |
It is also noted that the overall simulation temperature is another |
419 |
< |
factor that affects the interfacial thermal conductance. One |
420 |
< |
possibility of this effect may be rooted in the decrease in density of |
421 |
< |
the liquid phase. We observed that when the average temperature |
422 |
< |
increases from 200K to 250K, the bulk hexane density becomes lower |
423 |
< |
than experimental value, as the system is equilibrated under NPT |
424 |
< |
ensemble. This leads to lower contact between solvent and capping |
425 |
< |
agent, and thus lower conductivity. |
418 |
> |
As stated in our computational details, the spacing filled with |
419 |
> |
solvent molecules can be chosen within a range. This allows some |
420 |
> |
change of solvent molecule numbers for the same Au-butanethiol |
421 |
> |
surfaces. We did this study on our Au-butanethiol/hexane |
422 |
> |
simulations. Nevertheless, the results obtained from systems of |
423 |
> |
different $N_{hexane}$ did not indicate that the measurement of $G$ is |
424 |
> |
susceptible to this parameter. For computational efficiency concern, |
425 |
> |
smaller system size would be preferable, given that the liquid phase |
426 |
> |
structure is not affected. |
427 |
|
|
428 |
< |
Conductivity values are more difficult to obtain under higher |
429 |
< |
temperatures. This is because the Au surface tends to undergo |
430 |
< |
reconstructions in relatively high temperatures. Surface Au atoms can |
431 |
< |
migrate outward to reach higher Au-S contact; and capping agent |
432 |
< |
molecules can be embedded into the surface Au layer due to the same |
433 |
< |
driving force. This phenomenon agrees with experimental |
434 |
< |
results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. A surface |
435 |
< |
fully covered in capping agent is more susceptible to reconstruction, |
436 |
< |
possibly because fully coverage prevents other means of capping agent |
437 |
< |
relaxation, such as migration to an empty neighbor three-fold site. |
428 |
> |
Our NIVS algorithm allows change of unphysical thermal flux both in |
429 |
> |
direction and in quantity. This feature extends our investigation of |
430 |
> |
interfacial thermal conductance. However, the magnitude of this |
431 |
> |
thermal flux is not arbitary if one aims to obtain a stable and |
432 |
> |
reliable thermal gradient. A temperature profile would be |
433 |
> |
substantially affected by noise when $|J_z|$ has a much too low |
434 |
> |
magnitude; while an excessively large $|J_z|$ that overwhelms the |
435 |
> |
conductance capacity of the interface would prevent a thermal gradient |
436 |
> |
to reach a stablized steady state. NIVS has the advantage of allowing |
437 |
> |
$J$ to vary in a wide range such that the optimal flux range for $G$ |
438 |
> |
measurement can generally be simulated by the algorithm. Within the |
439 |
> |
optimal range, we were able to study how $G$ would change according to |
440 |
> |
the thermal flux across the interface. For our simulations, we denote |
441 |
> |
$J_z$ to be positive when the physical thermal flux is from the liquid |
442 |
> |
to metal, and negative vice versa. The $G$'s measured under different |
443 |
> |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and [REF]. These |
444 |
> |
results do not suggest that $G$ is dependent on $J_z$ within this flux |
445 |
> |
range. The linear response of flux to thermal gradient simplifies our |
446 |
> |
investigations in that we can rely on $G$ measurement with only a |
447 |
> |
couple $J_z$'s and do not need to test a large series of fluxes. |
448 |
|
|
449 |
< |
%MAY ADD MORE DATA TO TABLE |
449 |
> |
%ADD MORE TO TABLE |
450 |
|
\begin{table*} |
451 |
|
\begin{minipage}{\linewidth} |
452 |
|
\begin{center} |
453 |
|
\caption{Computed interfacial thermal conductivity ($G$ and |
454 |
< |
$G^\prime$) values for the Au/butanethiol/hexane interface |
455 |
< |
with united-atom model and different capping agent coverage |
456 |
< |
and solvent molecule numbers at different temperatures using a |
414 |
< |
range of energy fluxes.} |
454 |
> |
$G^\prime$) values for the 100\% covered Au-butanethiol/hexane |
455 |
> |
interfaces with UA model and different hexane molecule numbers |
456 |
> |
at different temperatures using a range of energy fluxes.} |
457 |
|
|
458 |
< |
\begin{tabular}{cccccc} |
458 |
> |
\begin{tabular}{cccccccc} |
459 |
|
\hline\hline |
460 |
< |
Thiol & $\langle T\rangle$ & & $J_z$ & $G$ & $G^\prime$ \\ |
461 |
< |
coverage (\%) & (K) & $N_{hexane}$ & (GW/m$^2$) & |
460 |
> |
$\langle T\rangle$ & & $L_x$ & $L_y$ & $L_z$ & $J_z$ & |
461 |
> |
$G$ & $G^\prime$ \\ |
462 |
> |
(K) & $N_{hexane}$ & \multicolumn{3}{c}{(\AA)} & (GW/m$^2$) & |
463 |
|
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
464 |
|
\hline |
465 |
< |
0.0 & 200 & 200 & 0.96 & 43.3 & 42.7 \\ |
466 |
< |
& & & 1.91 & 45.7 & 42.9 \\ |
467 |
< |
& & 166 & 0.96 & 43.1 & 53.4 \\ |
468 |
< |
88.9 & 200 & 166 & 1.94 & 172 & 108 \\ |
469 |
< |
100.0 & 250 & 200 & 0.96 & 81.8 & 67.0 \\ |
470 |
< |
& & 166 & 0.98 & 79.0 & 62.9 \\ |
471 |
< |
& & & 1.44 & 76.2 & 64.8 \\ |
472 |
< |
& 200 & 200 & 1.92 & 129 & 87.3 \\ |
473 |
< |
& & & 1.93 & 131 & 77.5 \\ |
474 |
< |
& & 166 & 0.97 & 115 & 69.3 \\ |
475 |
< |
& & & 1.94 & 125 & 87.1 \\ |
465 |
> |
200 & 266 & 29.86 & 25.80 & 113.1 & -0.96 & |
466 |
> |
102() & 80.0() \\ |
467 |
> |
& 200 & 29.84 & 25.81 & 93.9 & 1.92 & |
468 |
> |
129() & 87.3() \\ |
469 |
> |
& & 29.84 & 25.81 & 95.3 & 1.93 & |
470 |
> |
131() & 77.5() \\ |
471 |
> |
& 166 & 29.84 & 25.81 & 85.7 & 0.97 & |
472 |
> |
115() & 69.3() \\ |
473 |
> |
& & & & & 1.94 & |
474 |
> |
125() & 87.1() \\ |
475 |
> |
250 & 200 & 29.84 & 25.87 & 106.8 & 0.96 & |
476 |
> |
81.8() & 67.0() \\ |
477 |
> |
& 166 & 29.87 & 25.84 & 94.8 & 0.98 & |
478 |
> |
79.0() & 62.9() \\ |
479 |
> |
& & 29.84 & 25.85 & 95.0 & 1.44 & |
480 |
> |
76.2() & 64.8() \\ |
481 |
|
\hline\hline |
482 |
|
\end{tabular} |
483 |
|
\label{AuThiolHexaneUA} |
485 |
|
\end{minipage} |
486 |
|
\end{table*} |
487 |
|
|
488 |
< |
For the all-atom model, the liquid hexane phase was not stable under NPT |
489 |
< |
conditions. Therefore, the simulation length scale parameters are |
490 |
< |
adopted from previous equilibration results of the united-atom model |
491 |
< |
at 200K. Table \ref{AuThiolHexaneAA} shows the results of these |
492 |
< |
simulations. The conductivity values calculated with full capping |
493 |
< |
agent coverage are substantially larger than observed in the |
494 |
< |
united-atom model, and is even higher than predicted by |
495 |
< |
experiments. It is possible that our parameters for metal-non-metal |
496 |
< |
particle interactions lead to an overestimate of the interfacial |
497 |
< |
thermal conductivity, although the active C-H vibrations in the |
498 |
< |
all-atom model (which should not be appreciably populated at normal |
451 |
< |
temperatures) could also account for this high conductivity. The major |
452 |
< |
thermal transfer barrier of Au/butanethiol/hexane interface is between |
453 |
< |
the liquid phase and the capping agent, so extra degrees of freedom |
454 |
< |
such as the C-H vibrations could enhance heat exchange between these |
455 |
< |
two phases and result in a much higher conductivity. |
488 |
> |
Furthermore, we also attempted to increase system average temperatures |
489 |
> |
to above 200K. These simulations are first equilibrated in the NPT |
490 |
> |
ensemble under normal pressure. As stated above, the TraPPE-UA model |
491 |
> |
for hexane tends to predict a lower boiling point. In our simulations, |
492 |
> |
hexane had diffculty to remain in liquid phase when NPT equilibration |
493 |
> |
temperature is higher than 250K. Additionally, the equilibrated liquid |
494 |
> |
hexane density under 250K becomes lower than experimental value. This |
495 |
> |
expanded liquid phase leads to lower contact between hexane and |
496 |
> |
butanethiol as well.[MAY NEED FIGURE] And this reduced contact would |
497 |
> |
probably be accountable for a lower interfacial thermal conductance, |
498 |
> |
as shown in Table \ref{AuThiolHexaneUA}. |
499 |
|
|
500 |
+ |
A similar study for TraPPE-UA toluene agrees with the above result as |
501 |
+ |
well. Having a higher boiling point, toluene tends to remain liquid in |
502 |
+ |
our simulations even equilibrated under 300K in NPT |
503 |
+ |
ensembles. Furthermore, the expansion of the toluene liquid phase is |
504 |
+ |
not as significant as that of the hexane. This prevents severe |
505 |
+ |
decrease of liquid-capping agent contact and the results (Table |
506 |
+ |
\ref{AuThiolToluene}) show only a slightly decreased interface |
507 |
+ |
conductance. Therefore, solvent-capping agent contact should play an |
508 |
+ |
important role in the thermal transport process across the interface |
509 |
+ |
in that higher degree of contact could yield increased conductance. |
510 |
+ |
|
511 |
+ |
[ADD Lxyz AND ERROR ESTIMATE TO TABLE] |
512 |
|
\begin{table*} |
513 |
|
\begin{minipage}{\linewidth} |
514 |
|
\begin{center} |
460 |
– |
|
515 |
|
\caption{Computed interfacial thermal conductivity ($G$ and |
516 |
< |
$G^\prime$) values for the Au/butanethiol/hexane interface |
517 |
< |
with all-atom model and different capping agent coverage at |
518 |
< |
200K using a range of energy fluxes.} |
516 |
> |
$G^\prime$) values for a 90\% coverage Au-butanethiol/toluene |
517 |
> |
interface at different temperatures using a range of energy |
518 |
> |
fluxes.} |
519 |
|
|
520 |
|
\begin{tabular}{cccc} |
521 |
|
\hline\hline |
522 |
< |
Thiol & $J_z$ & $G$ & $G^\prime$ \\ |
523 |
< |
coverage (\%) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
522 |
> |
$\langle T\rangle$ & $J_z$ & $G$ & $G^\prime$ \\ |
523 |
> |
(K) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
524 |
|
\hline |
525 |
< |
0.0 & 0.95 & 28.5 & 27.2 \\ |
526 |
< |
& 1.88 & 30.3 & 28.9 \\ |
527 |
< |
100.0 & 2.87 & 551 & 294 \\ |
528 |
< |
& 3.81 & 494 & 193 \\ |
525 |
> |
200 & -1.86 & 180() & 135() \\ |
526 |
> |
& 2.15 & 204() & 113() \\ |
527 |
> |
& -3.93 & 175() & 114() \\ |
528 |
> |
300 & -1.91 & 143() & 125() \\ |
529 |
> |
& -4.19 & 134() & 113() \\ |
530 |
|
\hline\hline |
531 |
|
\end{tabular} |
532 |
< |
\label{AuThiolHexaneAA} |
532 |
> |
\label{AuThiolToluene} |
533 |
|
\end{center} |
534 |
|
\end{minipage} |
535 |
|
\end{table*} |
536 |
|
|
537 |
< |
%subsubsection{Vibrational spectrum study on conductance mechanism} |
537 |
> |
Besides lower interfacial thermal conductance, surfaces in relatively |
538 |
> |
high temperatures are susceptible to reconstructions, when |
539 |
> |
butanethiols have a full coverage on the Au(111) surface. These |
540 |
> |
reconstructions include surface Au atoms migrated outward to the S |
541 |
> |
atom layer, and butanethiol molecules embedded into the original |
542 |
> |
surface Au layer. The driving force for this behavior is the strong |
543 |
> |
Au-S interactions in our simulations. And these reconstructions lead |
544 |
> |
to higher ratio of Au-S attraction and thus is energetically |
545 |
> |
favorable. Furthermore, this phenomenon agrees with experimental |
546 |
> |
results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt |
547 |
> |
{\it et al.} had kept their Au(111) slab rigid so that their |
548 |
> |
simulations can reach 300K without surface reconstructions. Without |
549 |
> |
this practice, simulating 100\% thiol covered interfaces under higher |
550 |
> |
temperatures could hardly avoid surface reconstructions. However, our |
551 |
> |
measurement is based on assuming homogeneity on $x$ and $y$ dimensions |
552 |
> |
so that measurement of $T$ at particular $z$ would be an effective |
553 |
> |
average of the particles of the same type. Since surface |
554 |
> |
reconstructions could eliminate the original $x$ and $y$ dimensional |
555 |
> |
homogeneity, measurement of $G$ is more difficult to conduct under |
556 |
> |
higher temperatures. Therefore, most of our measurements are |
557 |
> |
undertaken at $\langle T\rangle\sim$200K. |
558 |
> |
|
559 |
> |
However, when the surface is not completely covered by butanethiols, |
560 |
> |
the simulated system is more resistent to the reconstruction |
561 |
> |
above. Our Au-butanethiol/toluene system did not see this phenomena |
562 |
> |
even at $<T>\sim$300K. The Au(111) surfaces have a 90\% coverage of |
563 |
> |
butanethiols and have empty three-fold sites. These empty sites could |
564 |
> |
help prevent surface reconstruction in that they provide other means |
565 |
> |
of capping agent relaxation. It is observed that butanethiols can |
566 |
> |
migrate to their neighbor empty sites during a simulation. Therefore, |
567 |
> |
we were able to obtain $G$'s for these interfaces even at a relatively |
568 |
> |
high temperature without being affected by surface reconstructions. |
569 |
> |
|
570 |
> |
\subsection{Influence of Capping Agent Coverage on $G$} |
571 |
> |
To investigate the influence of butanethiol coverage on interfacial |
572 |
> |
thermal conductance, a series of different coverage Au-butanethiol |
573 |
> |
surfaces is prepared and solvated with various organic |
574 |
> |
molecules. These systems are then equilibrated and their interfacial |
575 |
> |
thermal conductivity are measured with our NIVS algorithm. Table |
576 |
> |
\ref{tlnUhxnUhxnD} lists these results for direct comparison between |
577 |
> |
different coverages of butanethiol. To study the isotope effect in |
578 |
> |
interfacial thermal conductance, deuterated UA-hexane is included as |
579 |
> |
well. |
580 |
> |
|
581 |
> |
It turned out that with partial covered butanethiol on the Au(111) |
582 |
> |
surface, the derivative definition for $G$ (Eq. \ref{derivativeG}) has |
583 |
> |
difficulty to apply, due to the difficulty in locating the maximum of |
584 |
> |
change of $\lambda$. Instead, the discrete definition |
585 |
> |
(Eq. \ref{discreteG}) is easier to apply, as max($\Delta T$) can still |
586 |
> |
be well-defined. Therefore, $G$'s (not $G^\prime$) are used for this |
587 |
> |
section. |
588 |
> |
|
589 |
> |
From Table \ref{tlnUhxnUhxnD}, one can see the significance of the |
590 |
> |
presence of capping agents. Even when a fraction of the Au(111) |
591 |
> |
surface sites are covered with butanethiols, the conductivity would |
592 |
> |
see an enhancement by at least a factor of 3. This indicates the |
593 |
> |
important role cappping agent is playing for thermal transport |
594 |
> |
phenomena on metal/organic solvent surfaces. |
595 |
> |
|
596 |
> |
Interestingly, as one could observe from our results, the maximum |
597 |
> |
conductance enhancement (largest $G$) happens while the surfaces are |
598 |
> |
about 75\% covered with butanethiols. This again indicates that |
599 |
> |
solvent-capping agent contact has an important role of the thermal |
600 |
> |
transport process. Slightly lower butanethiol coverage allows small |
601 |
> |
gaps between butanethiols to form. And these gaps could be filled with |
602 |
> |
solvent molecules, which acts like ``heat conductors'' on the |
603 |
> |
surface. The higher degree of interaction between these solvent |
604 |
> |
molecules and capping agents increases the enhancement effect and thus |
605 |
> |
produces a higher $G$ than densely packed butanethiol arrays. However, |
606 |
> |
once this maximum conductance enhancement is reached, $G$ decreases |
607 |
> |
when butanethiol coverage continues to decrease. Each capping agent |
608 |
> |
molecule reaches its maximum capacity for thermal |
609 |
> |
conductance. Therefore, even higher solvent-capping agent contact |
610 |
> |
would not offset this effect. Eventually, when butanethiol coverage |
611 |
> |
continues to decrease, solvent-capping agent contact actually |
612 |
> |
decreases with the disappearing of butanethiol molecules. In this |
613 |
> |
case, $G$ decrease could not be offset but instead accelerated. |
614 |
> |
|
615 |
> |
A comparison of the results obtained from differenet organic solvents |
616 |
> |
can also provide useful information of the interfacial thermal |
617 |
> |
transport process. The deuterated hexane (UA) results do not appear to |
618 |
> |
be much different from those of normal hexane (UA), given that |
619 |
> |
butanethiol (UA) is non-deuterated for both solvents. These UA model |
620 |
> |
studies, even though eliminating C-H vibration samplings, still have |
621 |
> |
C-C vibrational frequencies different from each other. However, these |
622 |
> |
differences in the infrared range do not seem to produce an observable |
623 |
> |
difference for the results of $G$. [MAY NEED FIGURE] |
624 |
> |
|
625 |
> |
Furthermore, results for rigid body toluene solvent, as well as other |
626 |
> |
UA-hexane solvents, are reasonable within the general experimental |
627 |
> |
ranges[CITATIONS]. This suggests that explicit hydrogen might not be a |
628 |
> |
required factor for modeling thermal transport phenomena of systems |
629 |
> |
such as Au-thiol/organic solvent. |
630 |
> |
|
631 |
> |
However, results for Au-butanethiol/toluene do not show an identical |
632 |
> |
trend with those for Au-butanethiol/hexane in that $G$'s remain at |
633 |
> |
approximately the same magnitue when butanethiol coverage differs from |
634 |
> |
25\% to 75\%. This might be rooted in the molecule shape difference |
635 |
> |
for plane-like toluene and chain-like {\it n}-hexane. Due to this |
636 |
> |
difference, toluene molecules have more difficulty in occupying |
637 |
> |
relatively small gaps among capping agents when their coverage is not |
638 |
> |
too low. Therefore, the solvent-capping agent contact may keep |
639 |
> |
increasing until the capping agent coverage reaches a relatively low |
640 |
> |
level. This becomes an offset for decreasing butanethiol molecules on |
641 |
> |
its effect to the process of interfacial thermal transport. Thus, one |
642 |
> |
can see a plateau of $G$ vs. butanethiol coverage in our results. |
643 |
> |
|
644 |
> |
[NEED ERROR ESTIMATE, MAY ALSO PUT J HERE] |
645 |
> |
\begin{table*} |
646 |
> |
\begin{minipage}{\linewidth} |
647 |
> |
\begin{center} |
648 |
> |
\caption{Computed interfacial thermal conductivity ($G$) values |
649 |
> |
for the Au-butanethiol/solvent interface with various UA |
650 |
> |
models and different capping agent coverages at $\langle |
651 |
> |
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*} |
670 |
> |
|
671 |
> |
\subsection{Influence of Chosen Molecule Model on $G$} |
672 |
> |
[MAY COMBINE W MECHANISM STUDY] |
673 |
> |
|
674 |
> |
In addition to UA solvent/capping agent models, AA models are included |
675 |
> |
in our simulations as well. Besides simulations of the same (UA or AA) |
676 |
> |
model for solvent and capping agent, different models can be applied |
677 |
> |
to different components. Furthermore, regardless of models chosen, |
678 |
> |
either the solvent or the capping agent can be deuterated, similar to |
679 |
> |
the previous section. Table \ref{modelTest} summarizes the results of |
680 |
> |
these studies. |
681 |
> |
|
682 |
> |
[MORE DATA; ERROR ESTIMATE] |
683 |
> |
\begin{table*} |
684 |
> |
\begin{minipage}{\linewidth} |
685 |
> |
\begin{center} |
686 |
> |
|
687 |
> |
\caption{Computed interfacial thermal conductivity ($G$ and |
688 |
> |
$G^\prime$) values for interfaces using various models for |
689 |
> |
solvent and capping agent (or without capping agent) at |
690 |
> |
$\langle T\rangle\sim$200K.} |
691 |
> |
|
692 |
> |
\begin{tabular}{ccccc} |
693 |
> |
\hline\hline |
694 |
> |
Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ |
695 |
> |
(or bare surface) & model & (GW/m$^2$) & |
696 |
> |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
697 |
> |
\hline |
698 |
> |
UA & AA hexane & 1.94 & 135() & 129() \\ |
699 |
> |
& & 2.86 & 126() & 115() \\ |
700 |
> |
& AA toluene & 1.89 & 200() & 149() \\ |
701 |
> |
AA & UA hexane & 1.94 & 116() & 129() \\ |
702 |
> |
& AA hexane & 3.76 & 451() & 378() \\ |
703 |
> |
& & 4.71 & 432() & 334() \\ |
704 |
> |
& AA toluene & 3.79 & 487() & 290() \\ |
705 |
> |
AA(D) & UA hexane & 1.94 & 158() & 172() \\ |
706 |
> |
bare & AA hexane & 0.96 & 31.0() & 29.4() \\ |
707 |
> |
\hline\hline |
708 |
> |
\end{tabular} |
709 |
> |
\label{modelTest} |
710 |
> |
\end{center} |
711 |
> |
\end{minipage} |
712 |
> |
\end{table*} |
713 |
> |
|
714 |
> |
To facilitate direct comparison, the same system with differnt models |
715 |
> |
for different components uses the same length scale for their |
716 |
> |
simulation cells. Without the presence of capping agent, using |
717 |
> |
different models for hexane yields similar results for both $G$ and |
718 |
> |
$G^\prime$, and these two definitions agree with eath other very |
719 |
> |
well. This indicates very weak interaction between the metal and the |
720 |
> |
solvent, and is a typical case for acoustic impedance mismatch between |
721 |
> |
these two phases. |
722 |
> |
|
723 |
> |
As for Au(111) surfaces completely covered by butanethiols, the choice |
724 |
> |
of models for capping agent and solvent could impact the measurement |
725 |
> |
of $G$ and $G^\prime$ quite significantly. For Au-butanethiol/hexane |
726 |
> |
interfaces, using AA model for both butanethiol and hexane yields |
727 |
> |
substantially higher conductivity values than using UA model for at |
728 |
> |
least one component of the solvent and capping agent, which exceeds |
729 |
> |
the upper bond of experimental value range. This is probably due to |
730 |
> |
the classically treated C-H vibrations in the AA model, which should |
731 |
> |
not be appreciably populated at normal temperatures. In comparison, |
732 |
> |
once either the hexanes or the butanethiols are deuterated, one can |
733 |
> |
see a significantly lower $G$ and $G^\prime$. In either of these |
734 |
> |
cases, the C-H(D) vibrational overlap between the solvent and the |
735 |
> |
capping agent is removed. [MAY NEED FIGURE] Conclusively, the |
736 |
> |
improperly treated C-H vibration in the AA model produced |
737 |
> |
over-predicted results accordingly. Compared to the AA model, the UA |
738 |
> |
model yields more reasonable results with higher computational |
739 |
> |
efficiency. |
740 |
> |
|
741 |
> |
However, for Au-butanethiol/toluene interfaces, having the AA |
742 |
> |
butanethiol deuterated did not yield a significant change in the |
743 |
> |
measurement results. |
744 |
> |
. , so extra degrees of freedom |
745 |
> |
such as the C-H vibrations could enhance heat exchange between these |
746 |
> |
two phases and result in a much higher conductivity. |
747 |
> |
|
748 |
> |
|
749 |
> |
Although the QSC model for Au is known to predict an overly low value |
750 |
> |
for bulk metal gold conductivity[CITE NIVSRNEMD], our computational |
751 |
> |
results for $G$ and $G^\prime$ do not seem to be affected by this |
752 |
> |
drawback of the model for metal. Instead, the modeling of interfacial |
753 |
> |
thermal transport behavior relies mainly on an accurate description of |
754 |
> |
the interactions between components occupying the interfaces. |
755 |
> |
|
756 |
> |
\subsection{Mechanism of Interfacial Thermal Conductance Enhancement |
757 |
> |
by Capping Agent} |
758 |
> |
%OR\subsection{Vibrational spectrum study on conductance mechanism} |
759 |
> |
|
760 |
> |
[MAY INTRODUCE PROTOCOL IN METHODOLOGY/COMPUTATIONAL DETAIL, EQN'S] |
761 |
> |
|
762 |
|
To investigate the mechanism of this interfacial thermal conductance, |
763 |
|
the vibrational spectra of various gold systems were obtained and are |
764 |
|
shown as in the upper panel of Fig. \ref{vibration}. To obtain these |
765 |
|
spectra, one first runs a simulation in the NVE ensemble and collects |
766 |
|
snapshots of configurations; these configurations are used to compute |
767 |
|
the velocity auto-correlation functions, which is used to construct a |
768 |
< |
power spectrum via a Fourier transform. The gold surfaces covered by |
490 |
< |
butanethiol molecules exhibit an additional peak observed at a |
491 |
< |
frequency of $\sim$170cm$^{-1}$, which is attributed to the vibration |
492 |
< |
of the S-Au bond. This vibration enables efficient thermal transport |
493 |
< |
from surface Au atoms to the capping agents. Simultaneously, as shown |
494 |
< |
in the lower panel of Fig. \ref{vibration}, the large overlap of the |
495 |
< |
vibration spectra of butanethiol and hexane in the all-atom model, |
496 |
< |
including the C-H vibration, also suggests high thermal exchange |
497 |
< |
efficiency. The combination of these two effects produces the drastic |
498 |
< |
interfacial thermal conductance enhancement in the all-atom model. |
768 |
> |
power spectrum via a Fourier transform. |
769 |
|
|
770 |
+ |
The gold surfaces covered by |
771 |
+ |
butanethiol molecules, compared to bare gold surfaces, exhibit an |
772 |
+ |
additional peak observed at a frequency of $\sim$170cm$^{-1}$, which |
773 |
+ |
is attributed to the vibration of the S-Au bond. This vibration |
774 |
+ |
enables efficient thermal transport from surface Au atoms to the |
775 |
+ |
capping agents. Simultaneously, as shown in the lower panel of |
776 |
+ |
Fig. \ref{vibration}, the large overlap of the vibration spectra of |
777 |
+ |
butanethiol and hexane in the all-atom model, including the C-H |
778 |
+ |
vibration, also suggests high thermal exchange efficiency. The |
779 |
+ |
combination of these two effects produces the drastic interfacial |
780 |
+ |
thermal conductance enhancement in the all-atom model. |
781 |
+ |
|
782 |
+ |
[MAY NEED TO CONVERT TO JPEG] |
783 |
|
\begin{figure} |
784 |
|
\includegraphics[width=\linewidth]{vibration} |
785 |
|
\caption{Vibrational spectra obtained for gold in different |
787 |
|
all-atom model (lower panel).} |
788 |
|
\label{vibration} |
789 |
|
\end{figure} |
507 |
– |
% 600dpi, letter size. too large? |
790 |
|
|
791 |
+ |
[COMPARISON OF TWO G'S; AU SLAB WIDTHS; ETC] |
792 |
+ |
% The results show that the two definitions used for $G$ yield |
793 |
+ |
% comparable values, though $G^\prime$ tends to be smaller. |
794 |
|
|
795 |
+ |
\section{Conclusions} |
796 |
+ |
The NIVS algorithm we developed has been applied to simulations of |
797 |
+ |
Au-butanethiol surfaces with organic solvents. This algorithm allows |
798 |
+ |
effective unphysical thermal flux transferred between the metal and |
799 |
+ |
the liquid phase. With the flux applied, we were able to measure the |
800 |
+ |
corresponding thermal gradient and to obtain interfacial thermal |
801 |
+ |
conductivities. Our simulations have seen significant conductance |
802 |
+ |
enhancement with the presence of capping agent, compared to the bare |
803 |
+ |
gold/liquid interfaces. The acoustic impedance mismatch between the |
804 |
+ |
metal and the liquid phase is effectively eliminated by proper capping |
805 |
+ |
agent. Furthermore, the coverage precentage of the capping agent plays |
806 |
+ |
an important role in the interfacial thermal transport process. |
807 |
+ |
|
808 |
+ |
Our measurement results, particularly of the UA models, agree with |
809 |
+ |
available experimental data. This indicates that our force field |
810 |
+ |
parameters have a nice description of the interactions between the |
811 |
+ |
particles at the interfaces. AA models tend to overestimate the |
812 |
+ |
interfacial thermal conductance in that the classically treated C-H |
813 |
+ |
vibration would be overly sampled. Compared to the AA models, the UA |
814 |
+ |
models have higher computational efficiency with satisfactory |
815 |
+ |
accuracy, and thus are preferable in interfacial thermal transport |
816 |
+ |
modelings. |
817 |
+ |
|
818 |
+ |
Vlugt {\it et al.} has investigated the surface thiol structures for |
819 |
+ |
nanocrystal gold and pointed out that they differs from those of the |
820 |
+ |
Au(111) surface\cite{vlugt:cpc2007154}. This difference might lead to |
821 |
+ |
change of interfacial thermal transport behavior as well. To |
822 |
+ |
investigate this problem, an effective means to introduce thermal flux |
823 |
+ |
and measure the corresponding thermal gradient is desirable for |
824 |
+ |
simulating structures with spherical symmetry. |
825 |
+ |
|
826 |
+ |
|
827 |
|
\section{Acknowledgments} |
828 |
|
Support for this project was provided by the National Science |
829 |
|
Foundation under grant CHE-0848243. Computational time was provided by |
830 |
|
the Center for Research Computing (CRC) at the University of Notre |
831 |
< |
Dame. \newpage |
831 |
> |
Dame. \newpage |
832 |
|
|
833 |
|
\bibliography{interfacial} |
834 |
|
|