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\begin{document} |
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|
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this phenomena at the molecular level. Recently, Hase and coworkers |
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employed Non-Equilibrium Molecular Dynamics (NEMD) simulations to |
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study thermal transport from hot Au(111) substrate to a self-assembled |
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monolayer of alkylthiolate with relatively long chain (8-20 carbon |
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monolayer of alkylthiol with relatively long chain (8-20 carbon |
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|
atoms)\cite{hase:2010,hase:2011}. However, ensemble averaged |
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measurements for heat conductance of interfaces between the capping |
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monolayer on Au and a solvent phase has yet to be studied. |
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The relatively low thermal flux through interfaces is |
111 |
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The comparatively low thermal flux through interfaces is |
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difficult to measure with Equilibrium MD or forward NEMD simulation |
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methods. Therefore, the Reverse NEMD (RNEMD) methods would have the |
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advantage of having this difficult to measure flux known when studying |
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One way is to assume the temperature is discrete on the two sides of |
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the interface. $G$ can be calculated using the applied thermal flux |
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$J$ and the maximum temperature difference measured along the thermal |
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gradient max($\Delta T$), which occurs at the Gibbs deviding surface, |
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as: |
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gradient max($\Delta T$), which occurs at the Gibbs deviding surface |
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(Figure \ref{demoPic}): |
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\begin{equation} |
199 |
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G=\frac{J}{\Delta T} |
200 |
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\label{discreteG} |
201 |
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\end{equation} |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{method} |
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\caption{Interfacial conductance can be calculated by applying an |
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(unphysical) kinetic energy flux between two slabs, one located |
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within the metal and another on the edge of the periodic box. The |
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system responds by forming a thermal response or a gradient. In |
209 |
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bulk liquids, this gradient typically has a single slope, but in |
210 |
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interfacial systems, there are distinct thermal conductivity |
211 |
+ |
domains. The interfacial conductance, $G$ is found by measuring the |
212 |
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temperature gap at the Gibbs dividing surface, or by using second |
213 |
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derivatives of the thermal profile.} |
214 |
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\label{demoPic} |
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\end{figure} |
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|
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The other approach is to assume a continuous temperature profile along |
218 |
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the thermal gradient axis (e.g. $z$) and define $G$ at the point where |
219 |
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the magnitude of thermal conductivity $\lambda$ change reach its |
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|
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To compare the above definitions ($G$ and $G^\prime$), we have modeled |
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a metal slab with its (111) surfaces perpendicular to the $z$-axis of |
239 |
< |
our simulation cells. Both with and withour capping agents on the |
239 |
> |
our simulation cells. Both with and without capping agents on the |
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|
surfaces, the metal slab is solvated with simple organic solvents, as |
241 |
|
illustrated in Figure \ref{demoPic}. |
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– |
|
229 |
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\begin{figure} |
230 |
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\includegraphics[width=\linewidth]{demoPic} |
231 |
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\caption{A sample showing how a metal slab has its (111) surface |
232 |
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covered by capping agent molecules and solvated by hexane.} |
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\label{demoPic} |
234 |
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\end{figure} |
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|
|
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With the simulation cell described above, we are able to equilibrate |
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the system and impose an unphysical thermal flux between the liquid |
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|
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\begin{figure} |
253 |
|
\includegraphics[width=\linewidth]{gradT} |
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\caption{The 1st and 2nd derivatives of temperature profile can be |
255 |
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obtained with finite difference approximation.} |
254 |
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\caption{A sample of Au-butanethiol/hexane interfacial system and the |
255 |
> |
temperature profile after a kinetic energy flux is imposed to |
256 |
> |
it. The 1st and 2nd derivatives of the temperature profile can be |
257 |
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obtained with finite difference approximation (lower panel).} |
258 |
|
\label{gradT} |
259 |
|
\end{figure} |
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|
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solvent molecules would change the normal behavior of the liquid |
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phase. Therefore, our $N_{solvent}$ values were chosen to ensure that |
298 |
|
these extreme cases did not happen to our simulations. And the |
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corresponding spacing is usually $35 \sim 60$\AA. |
299 |
> |
corresponding spacing is usually $35 \sim 75$\AA. |
300 |
|
|
301 |
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The initial configurations generated by Packmol are further |
302 |
|
equilibrated with the $x$ and $y$ dimensions fixed, only allowing |
327 |
|
\end{equation} |
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|
329 |
|
\subsection{Force Field Parameters} |
330 |
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Our simulations include various components. Therefore, force field |
331 |
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parameter descriptions are needed for interactions both between the |
332 |
< |
same type of particles and between particles of different species. |
330 |
> |
Our simulations include various components. Figure \ref{demoMol} |
331 |
> |
demonstrates the sites defined for both United-Atom and All-Atom |
332 |
> |
models of the organic solvent and capping agent molecules in our |
333 |
> |
simulations. Force field parameter descriptions are needed for |
334 |
> |
interactions both between the same type of particles and between |
335 |
> |
particles of different species. |
336 |
|
|
337 |
+ |
\begin{figure} |
338 |
+ |
\includegraphics[width=\linewidth]{structures} |
339 |
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\caption{Structures of the capping agent and solvents utilized in |
340 |
+ |
these simulations. The chemically-distinct sites (a-e) are expanded |
341 |
+ |
in terms of constituent atoms for both United Atom (UA) and All Atom |
342 |
+ |
(AA) force fields. Most parameters are from |
343 |
+ |
Refs. \protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} (UA) and |
344 |
+ |
\protect\cite{OPLSAA} (AA). Cross-interactions with the Au atoms are given |
345 |
+ |
in Table \ref{MnM}.} |
346 |
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\label{demoMol} |
347 |
+ |
\end{figure} |
348 |
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|
349 |
|
The Au-Au interactions in metal lattice slab is described by the |
350 |
|
quantum Sutton-Chen (QSC) formulation\cite{PhysRevB.59.3527}. The QSC |
351 |
|
potentials include zero-point quantum corrections and are |
352 |
|
reparametrized for accurate surface energies compared to the |
353 |
|
Sutton-Chen potentials\cite{Chen90}. |
354 |
|
|
331 |
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Figure \ref{demoMol} demonstrates how we name our pseudo-atoms of the |
332 |
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organic solvent molecules in our simulations. |
333 |
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|
334 |
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\begin{figure} |
335 |
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\includegraphics[width=\linewidth]{demoMol} |
336 |
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\caption{Denomination of atoms or pseudo-atoms in our simulations: a) |
337 |
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UA-hexane; b) AA-hexane; c) UA-toluene; d) AA-toluene.} |
338 |
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\label{demoMol} |
339 |
– |
\end{figure} |
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|
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For both solvent molecules, straight chain {\it n}-hexane and aromatic |
356 |
|
toluene, United-Atom (UA) and All-Atom (AA) models are used |
357 |
|
respectively. The TraPPE-UA |
358 |
|
parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
359 |
< |
for our UA solvent molecules. In these models, pseudo-atoms are |
360 |
< |
located at the carbon centers for alkyl groups. By eliminating |
361 |
< |
explicit hydrogen atoms, these models are simple and computationally |
362 |
< |
efficient, while maintains good accuracy. However, the TraPPE-UA for |
363 |
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alkanes is known to predict a lower boiling point than experimental |
350 |
< |
values. Considering that after an unphysical thermal flux is applied |
351 |
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to a system, the temperature of ``hot'' area in the liquid phase would be |
352 |
< |
significantly higher than the average, to prevent over heating and |
353 |
< |
boiling of the liquid phase, the average temperature in our |
354 |
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simulations should be much lower than the liquid boiling point. [MORE DISCUSSION] |
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For UA-toluene model, rigid body constraints are applied, so that the |
356 |
< |
benzene ring and the methyl-CRar bond are kept rigid. This would save |
357 |
< |
computational time.[MORE DETAILS] |
359 |
> |
for our UA solvent molecules. In these models, sites are located at |
360 |
> |
the carbon centers for alkyl groups. Bonding interactions, including |
361 |
> |
bond stretches and bends and torsions, were used for intra-molecular |
362 |
> |
sites not separated by more than 3 bonds. Otherwise, for non-bonded |
363 |
> |
interactions, Lennard-Jones potentials are used. [MORE CITATION?] |
364 |
|
|
365 |
+ |
By eliminating explicit hydrogen atoms, these models are simple and |
366 |
+ |
computationally efficient, while maintains good accuracy. However, the |
367 |
+ |
TraPPE-UA for alkanes is known to predict a lower boiling point than |
368 |
+ |
experimental values. Considering that after an unphysical thermal flux |
369 |
+ |
is applied to a system, the temperature of ``hot'' area in the liquid |
370 |
+ |
phase would be significantly higher than the average, to prevent over |
371 |
+ |
heating and boiling of the liquid phase, the average temperature in |
372 |
+ |
our simulations should be much lower than the liquid boiling point. |
373 |
+ |
|
374 |
+ |
For UA-toluene model, the non-bonded potentials between |
375 |
+ |
inter-molecular sites have a similar Lennard-Jones formulation. For |
376 |
+ |
intra-molecular interactions, considering the stiffness of the benzene |
377 |
+ |
ring, rigid body constraints are applied for further computational |
378 |
+ |
efficiency. All bonds in the benzene ring and between the ring and the |
379 |
+ |
methyl group remain rigid during the progress of simulations. |
380 |
+ |
|
381 |
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Besides the TraPPE-UA models, AA models for both organic solvents are |
382 |
|
included in our studies as well. For hexane, the OPLS-AA\cite{OPLSAA} |
383 |
< |
force field is used. [MORE DETAILS] |
384 |
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For toluene, the United Force Field developed by Rapp\'{e} {\it et |
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< |
al.}\cite{doi:10.1021/ja00051a040} is adopted.[MORE DETAILS] |
383 |
> |
force field is used. Additional explicit hydrogen sites were |
384 |
> |
included. Besides bonding and non-bonded site-site interactions, |
385 |
> |
partial charges and the electrostatic interactions were added to each |
386 |
> |
CT and HC site. For toluene, the United Force Field developed by |
387 |
> |
Rapp\'{e} {\it et al.}\cite{doi:10.1021/ja00051a040} is |
388 |
> |
adopted. Without the rigid body constraints, bonding interactions were |
389 |
> |
included. For the aromatic ring, improper torsions (inversions) were |
390 |
> |
added as an extra potential for maintaining the planar shape. |
391 |
> |
[MORE CITATION?] |
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|
|
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The capping agent in our simulations, the butanethiol molecules can |
394 |
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either use UA or AA model. The TraPPE-UA force fields includes |
403 |
|
(UA or AA) of capping agent can be different from the |
404 |
|
solvent. Regardless of model choice, the force field parameters for |
405 |
|
interactions between capping agent and solvent can be derived using |
406 |
< |
Lorentz-Berthelot Mixing Rule:[EQN'S] |
406 |
> |
Lorentz-Berthelot Mixing Rule: |
407 |
> |
\begin{eqnarray} |
408 |
> |
\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\ |
409 |
> |
\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}} |
410 |
> |
\end{eqnarray} |
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|
|
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|
412 |
|
To describe the interactions between metal Au and non-metal capping |
413 |
|
agent and solvent particles, we refer to an adsorption study of alkyl |
414 |
|
thiols on gold surfaces by Vlugt {\it et |
423 |
|
|
424 |
|
Besides, the potentials developed from {\it ab initio} calculations by |
425 |
|
Leng {\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
426 |
< |
interactions between Au and aromatic C/H atoms in toluene.[MORE DETAILS] |
426 |
> |
interactions between Au and aromatic C/H atoms in toluene. A set of |
427 |
> |
pseudo Lennard-Jones parameters were provided for Au in their force |
428 |
> |
fields. By using the Mixing Rule, this can be used to derive pair-wise |
429 |
> |
potentials for non-bonded interactions between Au and non-metal sites. |
430 |
|
|
431 |
|
However, the Lennard-Jones parameters between Au and other types of |
432 |
< |
particles in our simulations are not yet well-established. For these |
433 |
< |
interactions, we attempt to derive their parameters using the Mixing |
434 |
< |
Rule. To do this, the ``Metal-non-Metal'' (MnM) interaction parameters |
435 |
< |
for Au is first extracted from the Au-CH$_x$ parameters by applying |
436 |
< |
the Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM'' |
432 |
> |
particles, such as All-Atom normal alkanes in our simulations are not |
433 |
> |
yet well-established. For these interactions, we attempt to derive |
434 |
> |
their parameters using the Mixing Rule. To do this, Au pseudo |
435 |
> |
Lennard-Jones parameters for ``Metal-non-Metal'' (MnM) interactions |
436 |
> |
were first extracted from the Au-CH$_x$ parameters by applying the |
437 |
> |
Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM'' |
438 |
|
parameters in our simulations. |
439 |
|
|
440 |
|
\begin{table*} |
441 |
|
\begin{minipage}{\linewidth} |
442 |
|
\begin{center} |
443 |
< |
\caption{Lennard-Jones parameters for Au-non-Metal |
444 |
< |
interactions in our simulations.} |
445 |
< |
|
446 |
< |
\begin{tabular}{ccc} |
443 |
> |
\caption{Non-bonded interaction parameters (including cross |
444 |
> |
interactions with Au atoms) for both force fields used in this |
445 |
> |
work.} |
446 |
> |
\begin{tabular}{lllllll} |
447 |
|
\hline\hline |
448 |
< |
Non-metal atom & $\sigma$ & $\epsilon$ \\ |
449 |
< |
(or pseudo-atom) & \AA & kcal/mol \\ |
448 |
> |
& Site & $\sigma_{ii}$ & $\epsilon_{ii}$ & $q_i$ & |
449 |
> |
$\sigma_{Au-i}$ & $\epsilon_{Au-i}$ \\ |
450 |
> |
& & (\AA) & (kcal/mol) & ($e$) & (\AA) & (kcal/mol) \\ |
451 |
|
\hline |
452 |
< |
S & 2.40 & 8.465 \\ |
453 |
< |
CH3 & 3.54 & 0.2146 \\ |
454 |
< |
CH2 & 3.54 & 0.1749 \\ |
455 |
< |
CT3 & 3.365 & 0.1373 \\ |
456 |
< |
CT2 & 3.365 & 0.1373 \\ |
457 |
< |
CTT & 3.365 & 0.1373 \\ |
458 |
< |
HC & 2.865 & 0.09256 \\ |
459 |
< |
CHar & 3.4625 & 0.1680 \\ |
460 |
< |
CRar & 3.555 & 0.1604 \\ |
461 |
< |
CA & 3.173 & 0.0640 \\ |
462 |
< |
HA & 2.746 & 0.0414 \\ |
452 |
> |
United Atom (UA) |
453 |
> |
&CH3 & 3.75 & 0.1947 & - & 3.54 & 0.2146 \\ |
454 |
> |
&CH2 & 3.95 & 0.0914 & - & 3.54 & 0.1749 \\ |
455 |
> |
&CHar & 3.695 & 0.1003 & - & 3.4625 & 0.1680 \\ |
456 |
> |
&CRar & 3.88 & 0.04173 & - & 3.555 & 0.1604 \\ |
457 |
> |
\hline |
458 |
> |
All Atom (AA) |
459 |
> |
&CT3 & 3.50 & 0.066 & -0.18 & 3.365 & 0.1373 \\ |
460 |
> |
&CT2 & 3.50 & 0.066 & -0.12 & 3.365 & 0.1373 \\ |
461 |
> |
&CTT & 3.50 & 0.066 & -0.065 & 3.365 & 0.1373 \\ |
462 |
> |
&HC & 2.50 & 0.030 & 0.06 & 2.865 & 0.09256 \\ |
463 |
> |
&CA & 3.55 & 0.070 & -0.115 & 3.173 & 0.0640 \\ |
464 |
> |
&HA & 2.42 & 0.030 & 0.115 & 2.746 & 0.0414 \\ |
465 |
> |
\hline |
466 |
> |
Both UA and AA |
467 |
> |
& S & 4.45 & 0.25 & - & 2.40 & 8.465 \\ |
468 |
|
\hline\hline |
469 |
|
\end{tabular} |
470 |
|
\label{MnM} |
485 |
|
results. |
486 |
|
|
487 |
|
In some of our simulations, we allowed $L_x$ and $L_y$ to change |
488 |
< |
during equilibrating the liquid phase. Due to the stiffness of the Au |
489 |
< |
slab, $L_x$ and $L_y$ would not change noticeably after |
490 |
< |
equilibration. Although $L_z$ could fluctuate $\sim$1\% after a system |
491 |
< |
is fully equilibrated in the NPT ensemble, this fluctuation, as well |
492 |
< |
as those comparably smaller to $L_x$ and $L_y$, would not be magnified |
493 |
< |
on the calculated $G$'s, as shown in Table \ref{AuThiolHexaneUA}. This |
494 |
< |
insensivity to $L_i$ fluctuations allows reliable measurement of $G$'s |
495 |
< |
without the necessity of extremely cautious equilibration process. |
488 |
> |
during equilibrating the liquid phase. Due to the stiffness of the |
489 |
> |
crystalline Au structure, $L_x$ and $L_y$ would not change noticeably |
490 |
> |
after equilibration. Although $L_z$ could fluctuate $\sim$1\% after a |
491 |
> |
system is fully equilibrated in the NPT ensemble, this fluctuation, as |
492 |
> |
well as those of $L_x$ and $L_y$ (which is significantly smaller), |
493 |
> |
would not be magnified on the calculated $G$'s, as shown in Table |
494 |
> |
\ref{AuThiolHexaneUA}. This insensivity to $L_i$ fluctuations allows |
495 |
> |
reliable measurement of $G$'s without the necessity of extremely |
496 |
> |
cautious equilibration process. |
497 |
|
|
498 |
|
As stated in our computational details, the spacing filled with |
499 |
|
solvent molecules can be chosen within a range. This allows some |
520 |
|
the thermal flux across the interface. For our simulations, we denote |
521 |
|
$J_z$ to be positive when the physical thermal flux is from the liquid |
522 |
|
to metal, and negative vice versa. The $G$'s measured under different |
523 |
< |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and [REF]. These |
524 |
< |
results do not suggest that $G$ is dependent on $J_z$ within this flux |
525 |
< |
range. The linear response of flux to thermal gradient simplifies our |
526 |
< |
investigations in that we can rely on $G$ measurement with only a |
527 |
< |
couple $J_z$'s and do not need to test a large series of fluxes. |
523 |
> |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and |
524 |
> |
\ref{AuThiolToluene}. These results do not suggest that $G$ is |
525 |
> |
dependent on $J_z$ within this flux range. The linear response of flux |
526 |
> |
to thermal gradient simplifies our investigations in that we can rely |
527 |
> |
on $G$ measurement with only a couple $J_z$'s and do not need to test |
528 |
> |
a large series of fluxes. |
529 |
|
|
487 |
– |
%ADD MORE TO TABLE |
530 |
|
\begin{table*} |
531 |
|
\begin{minipage}{\linewidth} |
532 |
|
\begin{center} |
533 |
|
\caption{Computed interfacial thermal conductivity ($G$ and |
534 |
|
$G^\prime$) values for the 100\% covered Au-butanethiol/hexane |
535 |
|
interfaces with UA model and different hexane molecule numbers |
536 |
< |
at different temperatures using a range of energy fluxes.} |
536 |
> |
at different temperatures using a range of energy |
537 |
> |
fluxes. Error estimates indicated in parenthesis.} |
538 |
|
|
539 |
< |
\begin{tabular}{cccccccc} |
539 |
> |
\begin{tabular}{ccccccc} |
540 |
|
\hline\hline |
541 |
< |
$\langle T\rangle$ & & $L_x$ & $L_y$ & $L_z$ & $J_z$ & |
542 |
< |
$G$ & $G^\prime$ \\ |
543 |
< |
(K) & $N_{hexane}$ & \multicolumn{3}{c}{(\AA)} & (GW/m$^2$) & |
541 |
> |
$\langle T\rangle$ & $N_{hexane}$ & Fixed & $\rho_{hexane}$ & |
542 |
> |
$J_z$ & $G$ & $G^\prime$ \\ |
543 |
> |
(K) & & $L_x$ \& $L_y$? & (g/cm$^3$) & (GW/m$^2$) & |
544 |
|
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
545 |
|
\hline |
546 |
< |
200 & 266 & 29.86 & 25.80 & 113.1 & -0.96 & |
547 |
< |
102() & 80.0() \\ |
548 |
< |
& 200 & 29.84 & 25.81 & 93.9 & 1.92 & |
549 |
< |
129() & 87.3() \\ |
550 |
< |
& & 29.84 & 25.81 & 95.3 & 1.93 & |
551 |
< |
131() & 77.5() \\ |
552 |
< |
& 166 & 29.84 & 25.81 & 85.7 & 0.97 & |
553 |
< |
115() & 69.3() \\ |
554 |
< |
& & & & & 1.94 & |
555 |
< |
125() & 87.1() \\ |
556 |
< |
250 & 200 & 29.84 & 25.87 & 106.8 & 0.96 & |
557 |
< |
81.8() & 67.0() \\ |
558 |
< |
& 166 & 29.87 & 25.84 & 94.8 & 0.98 & |
559 |
< |
79.0() & 62.9() \\ |
560 |
< |
& & 29.84 & 25.85 & 95.0 & 1.44 & |
561 |
< |
76.2() & 64.8() \\ |
546 |
> |
200 & 266 & No & 0.672 & -0.96 & 102(3) & 80.0(0.8) \\ |
547 |
> |
& 200 & Yes & 0.694 & 1.92 & 129(11) & 87.3(0.3) \\ |
548 |
> |
& & Yes & 0.672 & 1.93 & 131(16) & 78(13) \\ |
549 |
> |
& & No & 0.688 & 0.96 & 125(16) & 90.2(15) \\ |
550 |
> |
& & & & 1.91 & 139(10) & 101(10) \\ |
551 |
> |
& & & & 2.83 & 141(6) & 89.9(9.8) \\ |
552 |
> |
& 166 & Yes & 0.679 & 0.97 & 115(19) & 69(18) \\ |
553 |
> |
& & & & 1.94 & 125(9) & 87.1(0.2) \\ |
554 |
> |
& & No & 0.681 & 0.97 & 141(30) & 78(22) \\ |
555 |
> |
& & & & 1.92 & 138(4) & 98.9(9.5) \\ |
556 |
> |
\hline |
557 |
> |
250 & 200 & No & 0.560 & 0.96 & 75(10) & 61.8(7.3) \\ |
558 |
> |
& & & & -0.95 & 49.4(0.3) & 45.7(2.1) \\ |
559 |
> |
& 166 & Yes & 0.570 & 0.98 & 79.0(3.5) & 62.9(3.0) \\ |
560 |
> |
& & No & 0.569 & 0.97 & 80.3(0.6) & 67(11) \\ |
561 |
> |
& & & & 1.44 & 76.2(5.0) & 64.8(3.8) \\ |
562 |
> |
& & & & -0.95 & 56.4(2.5) & 54.4(1.1) \\ |
563 |
> |
& & & & -1.85 & 47.8(1.1) & 53.5(1.5) \\ |
564 |
|
\hline\hline |
565 |
|
\end{tabular} |
566 |
|
\label{AuThiolHexaneUA} |
576 |
|
temperature is higher than 250K. Additionally, the equilibrated liquid |
577 |
|
hexane density under 250K becomes lower than experimental value. This |
578 |
|
expanded liquid phase leads to lower contact between hexane and |
579 |
< |
butanethiol as well.[MAY NEED FIGURE] And this reduced contact would |
579 |
> |
butanethiol as well.[MAY NEED SLAB DENSITY FIGURE] |
580 |
> |
And this reduced contact would |
581 |
|
probably be accountable for a lower interfacial thermal conductance, |
582 |
|
as shown in Table \ref{AuThiolHexaneUA}. |
583 |
|
|
592 |
|
important role in the thermal transport process across the interface |
593 |
|
in that higher degree of contact could yield increased conductance. |
594 |
|
|
549 |
– |
[ADD Lxyz AND ERROR ESTIMATE TO TABLE] |
595 |
|
\begin{table*} |
596 |
|
\begin{minipage}{\linewidth} |
597 |
|
\begin{center} |
598 |
|
\caption{Computed interfacial thermal conductivity ($G$ and |
599 |
|
$G^\prime$) values for a 90\% coverage Au-butanethiol/toluene |
600 |
|
interface at different temperatures using a range of energy |
601 |
< |
fluxes.} |
601 |
> |
fluxes. Error estimates indicated in parenthesis.} |
602 |
|
|
603 |
< |
\begin{tabular}{cccc} |
603 |
> |
\begin{tabular}{ccccc} |
604 |
|
\hline\hline |
605 |
< |
$\langle T\rangle$ & $J_z$ & $G$ & $G^\prime$ \\ |
606 |
< |
(K) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
605 |
> |
$\langle T\rangle$ & $\rho_{toluene}$ & $J_z$ & $G$ & $G^\prime$ \\ |
606 |
> |
(K) & (g/cm$^3$) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
607 |
|
\hline |
608 |
< |
200 & -1.86 & 180() & 135() \\ |
609 |
< |
& 2.15 & 204() & 113() \\ |
610 |
< |
& -3.93 & 175() & 114() \\ |
611 |
< |
300 & -1.91 & 143() & 125() \\ |
612 |
< |
& -4.19 & 134() & 113() \\ |
608 |
> |
200 & 0.933 & 2.15 & 204(12) & 113(12) \\ |
609 |
> |
& & -1.86 & 180(3) & 135(21) \\ |
610 |
> |
& & -3.93 & 176(5) & 113(12) \\ |
611 |
> |
\hline |
612 |
> |
300 & 0.855 & -1.91 & 143(5) & 125(2) \\ |
613 |
> |
& & -4.19 & 135(9) & 113(12) \\ |
614 |
|
\hline\hline |
615 |
|
\end{tabular} |
616 |
|
\label{AuThiolToluene} |
642 |
|
|
643 |
|
However, when the surface is not completely covered by butanethiols, |
644 |
|
the simulated system is more resistent to the reconstruction |
645 |
< |
above. Our Au-butanethiol/toluene system did not see this phenomena |
646 |
< |
even at $\langle T\rangle\sim$300K. The Au(111) surfaces have a 90\% coverage of |
647 |
< |
butanethiols and have empty three-fold sites. These empty sites could |
648 |
< |
help prevent surface reconstruction in that they provide other means |
649 |
< |
of capping agent relaxation. It is observed that butanethiols can |
650 |
< |
migrate to their neighbor empty sites during a simulation. Therefore, |
651 |
< |
we were able to obtain $G$'s for these interfaces even at a relatively |
652 |
< |
high temperature without being affected by surface reconstructions. |
645 |
> |
above. Our Au-butanethiol/toluene system had the Au(111) surfaces 90\% |
646 |
> |
covered by butanethiols, but did not see this above phenomena even at |
647 |
> |
$\langle T\rangle\sim$300K. The empty three-fold sites not occupied by |
648 |
> |
capping agents could help prevent surface reconstruction in that they |
649 |
> |
provide other means of capping agent relaxation. It is observed that |
650 |
> |
butanethiols can migrate to their neighbor empty sites during a |
651 |
> |
simulation. Therefore, we were able to obtain $G$'s for these |
652 |
> |
interfaces even at a relatively high temperature without being |
653 |
> |
affected by surface reconstructions. |
654 |
|
|
655 |
|
\subsection{Influence of Capping Agent Coverage on $G$} |
656 |
|
To investigate the influence of butanethiol coverage on interfacial |
657 |
|
thermal conductance, a series of different coverage Au-butanethiol |
658 |
|
surfaces is prepared and solvated with various organic |
659 |
|
molecules. These systems are then equilibrated and their interfacial |
660 |
< |
thermal conductivity are measured with our NIVS algorithm. Table |
661 |
< |
\ref{tlnUhxnUhxnD} lists these results for direct comparison between |
662 |
< |
different coverages of butanethiol. To study the isotope effect in |
663 |
< |
interfacial thermal conductance, deuterated UA-hexane is included as |
664 |
< |
well. |
660 |
> |
thermal conductivity are measured with our NIVS algorithm. Figure |
661 |
> |
\ref{coverage} demonstrates the trend of conductance change with |
662 |
> |
respect to different coverages of butanethiol. To study the isotope |
663 |
> |
effect in interfacial thermal conductance, deuterated UA-hexane is |
664 |
> |
included as well. |
665 |
|
|
666 |
|
It turned out that with partial covered butanethiol on the Au(111) |
667 |
< |
surface, the derivative definition for $G$ (Eq. \ref{derivativeG}) has |
668 |
< |
difficulty to apply, due to the difficulty in locating the maximum of |
669 |
< |
change of $\lambda$. Instead, the discrete definition |
670 |
< |
(Eq. \ref{discreteG}) is easier to apply, as max($\Delta T$) can still |
671 |
< |
be well-defined. Therefore, $G$'s (not $G^\prime$) are used for this |
672 |
< |
section. |
667 |
> |
surface, the derivative definition for $G^\prime$ |
668 |
> |
(Eq. \ref{derivativeG}) was difficult to apply, due to the difficulty |
669 |
> |
in locating the maximum of change of $\lambda$. Instead, the discrete |
670 |
> |
definition (Eq. \ref{discreteG}) is easier to apply, as the Gibbs |
671 |
> |
deviding surface can still be well-defined. Therefore, $G$ (not |
672 |
> |
$G^\prime$) was used for this section. |
673 |
|
|
674 |
< |
From Table \ref{tlnUhxnUhxnD}, one can see the significance of the |
674 |
> |
From Figure \ref{coverage}, one can see the significance of the |
675 |
|
presence of capping agents. Even when a fraction of the Au(111) |
676 |
|
surface sites are covered with butanethiols, the conductivity would |
677 |
|
see an enhancement by at least a factor of 3. This indicates the |
678 |
|
important role cappping agent is playing for thermal transport |
679 |
< |
phenomena on metal/organic solvent surfaces. |
679 |
> |
phenomena on metal / organic solvent surfaces. |
680 |
|
|
681 |
|
Interestingly, as one could observe from our results, the maximum |
682 |
|
conductance enhancement (largest $G$) happens while the surfaces are |
695 |
|
would not offset this effect. Eventually, when butanethiol coverage |
696 |
|
continues to decrease, solvent-capping agent contact actually |
697 |
|
decreases with the disappearing of butanethiol molecules. In this |
698 |
< |
case, $G$ decrease could not be offset but instead accelerated. |
698 |
> |
case, $G$ decrease could not be offset but instead accelerated. [NEED |
699 |
> |
SNAPSHOT SHOWING THE PHENOMENA] |
700 |
|
|
701 |
|
A comparison of the results obtained from differenet organic solvents |
702 |
|
can also provide useful information of the interfacial thermal |
706 |
|
studies, even though eliminating C-H vibration samplings, still have |
707 |
|
C-C vibrational frequencies different from each other. However, these |
708 |
|
differences in the infrared range do not seem to produce an observable |
709 |
< |
difference for the results of $G$. [MAY NEED FIGURE] |
709 |
> |
difference for the results of $G$. [MAY NEED SPECTRA FIGURE] |
710 |
|
|
711 |
|
Furthermore, results for rigid body toluene solvent, as well as other |
712 |
|
UA-hexane solvents, are reasonable within the general experimental |
715 |
|
such as Au-thiol/organic solvent. |
716 |
|
|
717 |
|
However, results for Au-butanethiol/toluene do not show an identical |
718 |
< |
trend with those for Au-butanethiol/hexane in that $G$'s remain at |
718 |
> |
trend with those for Au-butanethiol/hexane in that $G$ remains at |
719 |
|
approximately the same magnitue when butanethiol coverage differs from |
720 |
|
25\% to 75\%. This might be rooted in the molecule shape difference |
721 |
< |
for plane-like toluene and chain-like {\it n}-hexane. Due to this |
721 |
> |
for planar toluene and chain-like {\it n}-hexane. Due to this |
722 |
|
difference, toluene molecules have more difficulty in occupying |
723 |
|
relatively small gaps among capping agents when their coverage is not |
724 |
|
too low. Therefore, the solvent-capping agent contact may keep |
727 |
|
its effect to the process of interfacial thermal transport. Thus, one |
728 |
|
can see a plateau of $G$ vs. butanethiol coverage in our results. |
729 |
|
|
730 |
< |
[NEED ERROR ESTIMATE, MAY ALSO PUT J HERE] |
731 |
< |
\begin{table*} |
732 |
< |
\begin{minipage}{\linewidth} |
733 |
< |
\begin{center} |
734 |
< |
\caption{Computed interfacial thermal conductivity ($G$) values |
735 |
< |
for the Au-butanethiol/solvent interface with various UA |
736 |
< |
models and different capping agent coverages at $\langle |
737 |
< |
T\rangle\sim$200K using certain energy flux respectively.} |
690 |
< |
|
691 |
< |
\begin{tabular}{cccc} |
692 |
< |
\hline\hline |
693 |
< |
Thiol & \multicolumn{3}{c}{$G$(MW/m$^2$/K)} \\ |
694 |
< |
coverage (\%) & hexane & hexane(D) & toluene \\ |
695 |
< |
\hline |
696 |
< |
0.0 & 46.5() & 43.9() & 70.1() \\ |
697 |
< |
25.0 & 151() & 153() & 249() \\ |
698 |
< |
50.0 & 172() & 182() & 214() \\ |
699 |
< |
75.0 & 242() & 229() & 244() \\ |
700 |
< |
88.9 & 178() & - & - \\ |
701 |
< |
100.0 & 137() & 153() & 187() \\ |
702 |
< |
\hline\hline |
703 |
< |
\end{tabular} |
704 |
< |
\label{tlnUhxnUhxnD} |
705 |
< |
\end{center} |
706 |
< |
\end{minipage} |
707 |
< |
\end{table*} |
730 |
> |
\begin{figure} |
731 |
> |
\includegraphics[width=\linewidth]{coverage} |
732 |
> |
\caption{Comparison of interfacial thermal conductivity ($G$) values |
733 |
> |
for the Au-butanethiol/solvent interface with various UA models and |
734 |
> |
different capping agent coverages at $\langle T\rangle\sim$200K |
735 |
> |
using certain energy flux respectively.} |
736 |
> |
\label{coverage} |
737 |
> |
\end{figure} |
738 |
|
|
739 |
|
\subsection{Influence of Chosen Molecule Model on $G$} |
740 |
|
[MAY COMBINE W MECHANISM STUDY] |
747 |
|
the previous section. Table \ref{modelTest} summarizes the results of |
748 |
|
these studies. |
749 |
|
|
720 |
– |
[MORE DATA; ERROR ESTIMATE] |
750 |
|
\begin{table*} |
751 |
|
\begin{minipage}{\linewidth} |
752 |
|
\begin{center} |
754 |
|
\caption{Computed interfacial thermal conductivity ($G$ and |
755 |
|
$G^\prime$) values for interfaces using various models for |
756 |
|
solvent and capping agent (or without capping agent) at |
757 |
< |
$\langle T\rangle\sim$200K.} |
757 |
> |
$\langle T\rangle\sim$200K. (D stands for deuterated solvent |
758 |
> |
or capping agent molecules; ``Avg.'' denotes results that are |
759 |
> |
averages of simulations under different $J_z$'s. Error |
760 |
> |
estimates indicated in parenthesis.)} |
761 |
|
|
762 |
< |
\begin{tabular}{ccccc} |
762 |
> |
\begin{tabular}{llccc} |
763 |
|
\hline\hline |
764 |
|
Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ |
765 |
|
(or bare surface) & model & (GW/m$^2$) & |
766 |
|
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
767 |
|
\hline |
768 |
< |
UA & AA hexane & 1.94 & 135() & 129() \\ |
769 |
< |
& & 2.86 & 126() & 115() \\ |
770 |
< |
& AA toluene & 1.89 & 200() & 149() \\ |
771 |
< |
AA & UA hexane & 1.94 & 116() & 129() \\ |
772 |
< |
& AA hexane & 3.76 & 451() & 378() \\ |
773 |
< |
& & 4.71 & 432() & 334() \\ |
774 |
< |
& AA toluene & 3.79 & 487() & 290() \\ |
775 |
< |
AA(D) & UA hexane & 1.94 & 158() & 172() \\ |
776 |
< |
bare & AA hexane & 0.96 & 31.0() & 29.4() \\ |
768 |
> |
UA & UA hexane & Avg. & 131(9) & 87(10) \\ |
769 |
> |
& UA hexane(D) & 1.95 & 153(5) & 136(13) \\ |
770 |
> |
& AA hexane & Avg. & 131(6) & 122(10) \\ |
771 |
> |
& UA toluene & 1.96 & 187(16) & 151(11) \\ |
772 |
> |
& AA toluene & 1.89 & 200(36) & 149(53) \\ |
773 |
> |
\hline |
774 |
> |
AA & UA hexane & 1.94 & 116(9) & 129(8) \\ |
775 |
> |
& AA hexane & Avg. & 442(14) & 356(31) \\ |
776 |
> |
& AA hexane(D) & 1.93 & 222(12) & 234(54) \\ |
777 |
> |
& UA toluene & 1.98 & 125(25) & 97(60) \\ |
778 |
> |
& AA toluene & 3.79 & 487(56) & 290(42) \\ |
779 |
> |
\hline |
780 |
> |
AA(D) & UA hexane & 1.94 & 158(25) & 172(4) \\ |
781 |
> |
& AA hexane & 1.92 & 243(29) & 191(11) \\ |
782 |
> |
& AA toluene & 1.93 & 364(36) & 322(67) \\ |
783 |
> |
\hline |
784 |
> |
bare & UA hexane & Avg. & 46.5(3.2) & 49.4(4.5) \\ |
785 |
> |
& UA hexane(D) & 0.98 & 43.9(4.6) & 43.0(2.0) \\ |
786 |
> |
& AA hexane & 0.96 & 31.0(1.4) & 29.4(1.3) \\ |
787 |
> |
& UA toluene & 1.99 & 70.1(1.3) & 65.8(0.5) \\ |
788 |
|
\hline\hline |
789 |
|
\end{tabular} |
790 |
|
\label{modelTest} |
807 |
|
interfaces, using AA model for both butanethiol and hexane yields |
808 |
|
substantially higher conductivity values than using UA model for at |
809 |
|
least one component of the solvent and capping agent, which exceeds |
810 |
< |
the upper bond of experimental value range. This is probably due to |
811 |
< |
the classically treated C-H vibrations in the AA model, which should |
812 |
< |
not be appreciably populated at normal temperatures. In comparison, |
813 |
< |
once either the hexanes or the butanethiols are deuterated, one can |
814 |
< |
see a significantly lower $G$ and $G^\prime$. In either of these |
815 |
< |
cases, the C-H(D) vibrational overlap between the solvent and the |
816 |
< |
capping agent is removed. [MAY NEED FIGURE] Conclusively, the |
810 |
> |
the general range of experimental measurement results. This is |
811 |
> |
probably due to the classically treated C-H vibrations in the AA |
812 |
> |
model, which should not be appreciably populated at normal |
813 |
> |
temperatures. In comparison, once either the hexanes or the |
814 |
> |
butanethiols are deuterated, one can see a significantly lower $G$ and |
815 |
> |
$G^\prime$. In either of these cases, the C-H(D) vibrational overlap |
816 |
> |
between the solvent and the capping agent is removed. |
817 |
> |
[MAY NEED SPECTRA FIGURE] Conclusively, the |
818 |
|
improperly treated C-H vibration in the AA model produced |
819 |
|
over-predicted results accordingly. Compared to the AA model, the UA |
820 |
|
model yields more reasonable results with higher computational |
822 |
|
|
823 |
|
However, for Au-butanethiol/toluene interfaces, having the AA |
824 |
|
butanethiol deuterated did not yield a significant change in the |
825 |
< |
measurement results. |
826 |
< |
. , so extra degrees of freedom |
827 |
< |
such as the C-H vibrations could enhance heat exchange between these |
828 |
< |
two phases and result in a much higher conductivity. |
825 |
> |
measurement results. Compared to the C-H vibrational overlap between |
826 |
> |
hexane and butanethiol, both of which have alkyl chains, that overlap |
827 |
> |
between toluene and butanethiol is not so significant and thus does |
828 |
> |
not have as much contribution to the ``Intramolecular Vibration |
829 |
> |
Redistribution''[CITE HASE]. Conversely, extra degrees of freedom such |
830 |
> |
as the C-H vibrations could yield higher heat exchange rate between |
831 |
> |
these two phases and result in a much higher conductivity. |
832 |
|
|
786 |
– |
|
833 |
|
Although the QSC model for Au is known to predict an overly low value |
834 |
< |
for bulk metal gold conductivity[CITE NIVSRNEMD], our computational |
834 |
> |
for bulk metal gold conductivity\cite{kuang:164101}, our computational |
835 |
|
results for $G$ and $G^\prime$ do not seem to be affected by this |
836 |
< |
drawback of the model for metal. Instead, the modeling of interfacial |
837 |
< |
thermal transport behavior relies mainly on an accurate description of |
838 |
< |
the interactions between components occupying the interfaces. |
836 |
> |
drawback of the model for metal. Instead, our results suggest that the |
837 |
> |
modeling of interfacial thermal transport behavior relies mainly on |
838 |
> |
the accuracy of the interaction descriptions between components |
839 |
> |
occupying the interfaces. |
840 |
|
|
841 |
|
\subsection{Mechanism of Interfacial Thermal Conductance Enhancement |
842 |
|
by Capping Agent} |
843 |
< |
%OR\subsection{Vibrational spectrum study on conductance mechanism} |
843 |
> |
[OR: Vibrational Spectrum Study on Conductance Mechanism] |
844 |
|
|
845 |
|
[MAY INTRODUCE PROTOCOL IN METHODOLOGY/COMPUTATIONAL DETAIL, EQN'S] |
846 |
|
|
852 |
|
the velocity auto-correlation functions, which is used to construct a |
853 |
|
power spectrum via a Fourier transform. |
854 |
|
|
855 |
< |
The gold surfaces covered by |
856 |
< |
butanethiol molecules, compared to bare gold surfaces, exhibit an |
857 |
< |
additional peak observed at a frequency of $\sim$170cm$^{-1}$, which |
858 |
< |
is attributed to the vibration of the S-Au bond. This vibration |
859 |
< |
enables efficient thermal transport from surface Au atoms to the |
860 |
< |
capping agents. Simultaneously, as shown in the lower panel of |
861 |
< |
Fig. \ref{vibration}, the large overlap of the vibration spectra of |
862 |
< |
butanethiol and hexane in the all-atom model, including the C-H |
863 |
< |
vibration, also suggests high thermal exchange efficiency. The |
864 |
< |
combination of these two effects produces the drastic interfacial |
865 |
< |
thermal conductance enhancement in the all-atom model. |
855 |
> |
[MAY RELATE TO HASE'S] |
856 |
> |
The gold surfaces covered by butanethiol molecules, compared to bare |
857 |
> |
gold surfaces, exhibit an additional peak observed at the frequency of |
858 |
> |
$\sim$170cm$^{-1}$, which is attributed to the S-Au bonding |
859 |
> |
vibration. This vibration enables efficient thermal transport from |
860 |
> |
surface Au layer to the capping agents. |
861 |
> |
[MAY PUT IN OTHER SECTION] Simultaneously, as shown in |
862 |
> |
the lower panel of Fig. \ref{vibration}, the large overlap of the |
863 |
> |
vibration spectra of butanethiol and hexane in the All-Atom model, |
864 |
> |
including the C-H vibration, also suggests high thermal exchange |
865 |
> |
efficiency. The combination of these two effects produces the drastic |
866 |
> |
interfacial thermal conductance enhancement in the All-Atom model. |
867 |
|
|
868 |
< |
[MAY NEED TO CONVERT TO JPEG] |
868 |
> |
[NEED SEPARATE FIGURE. MAY NEED TO CONVERT TO JPEG] |
869 |
|
\begin{figure} |
870 |
|
\includegraphics[width=\linewidth]{vibration} |
871 |
|
\caption{Vibrational spectra obtained for gold in different |
872 |
< |
environments (upper panel) and for Au/thiol/hexane simulation in |
825 |
< |
all-atom model (lower panel).} |
872 |
> |
environments.} |
873 |
|
\label{vibration} |
874 |
|
\end{figure} |
875 |
|
|
876 |
< |
[COMPARISON OF TWO G'S; AU SLAB WIDTHS; ETC] |
876 |
> |
[MAY ADD COMPARISON OF G AND G', AU SLAB WIDTHS, ETC] |
877 |
|
% The results show that the two definitions used for $G$ yield |
878 |
|
% comparable values, though $G^\prime$ tends to be smaller. |
879 |
|
|
885 |
|
corresponding thermal gradient and to obtain interfacial thermal |
886 |
|
conductivities. Our simulations have seen significant conductance |
887 |
|
enhancement with the presence of capping agent, compared to the bare |
888 |
< |
gold/liquid interfaces. The acoustic impedance mismatch between the |
888 |
> |
gold / liquid interfaces. The acoustic impedance mismatch between the |
889 |
|
metal and the liquid phase is effectively eliminated by proper capping |
890 |
|
agent. Furthermore, the coverage precentage of the capping agent plays |
891 |
|
an important role in the interfacial thermal transport process. |