1 |
\documentclass[11pt]{article} |
2 |
\usepackage{amsmath} |
3 |
\usepackage{amssymb} |
4 |
\usepackage{setspace} |
5 |
\usepackage{endfloat} |
6 |
\usepackage{caption} |
7 |
%\usepackage{tabularx} |
8 |
\usepackage{graphicx} |
9 |
\usepackage{multirow} |
10 |
%\usepackage{booktabs} |
11 |
%\usepackage{bibentry} |
12 |
%\usepackage{mathrsfs} |
13 |
%\usepackage[ref]{overcite} |
14 |
\usepackage[square, comma, sort&compress]{natbib} |
15 |
\usepackage{url} |
16 |
\pagestyle{plain} \pagenumbering{arabic} \oddsidemargin 0.0cm |
17 |
\evensidemargin 0.0cm \topmargin -21pt \headsep 10pt \textheight |
18 |
9.0in \textwidth 6.5in \brokenpenalty=10000 |
19 |
|
20 |
% double space list of tables and figures |
21 |
\AtBeginDelayedFloats{\renewcommand{\baselinestretch}{1.66}} |
22 |
\setlength{\abovecaptionskip}{20 pt} |
23 |
\setlength{\belowcaptionskip}{30 pt} |
24 |
|
25 |
%\renewcommand\citemid{\ } % no comma in optional reference note |
26 |
\bibpunct{[}{]}{,}{s}{}{;} |
27 |
\bibliographystyle{aip} |
28 |
|
29 |
\begin{document} |
30 |
|
31 |
\title{Simulating interfacial thermal conductance at metal-solvent |
32 |
interfaces: the role of chemical capping agents} |
33 |
|
34 |
\author{Shenyu Kuang and J. Daniel |
35 |
Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\ |
36 |
Department of Chemistry and Biochemistry,\\ |
37 |
University of Notre Dame\\ |
38 |
Notre Dame, Indiana 46556} |
39 |
|
40 |
\date{\today} |
41 |
|
42 |
\maketitle |
43 |
|
44 |
\begin{doublespace} |
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. |
62 |
|
63 |
\end{abstract} |
64 |
|
65 |
\newpage |
66 |
|
67 |
%\narrowtext |
68 |
|
69 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
70 |
% BODY OF TEXT |
71 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
72 |
|
73 |
\section{Introduction} |
74 |
[BACKGROUND FOR INTERFACIAL THERMAL CONDUCTANCE PROBLEM] |
75 |
Interfacial thermal conductance is extensively studied both |
76 |
experimentally and computationally, and systems with interfaces |
77 |
present are generally heterogeneous. Although interfaces are commonly |
78 |
barriers to heat transfer, it has been |
79 |
reported\cite{doi:10.1021/la904855s} that under specific circustances, |
80 |
e.g. with certain capping agents present on the surface, interfacial |
81 |
conductance can be significantly enhanced. However, heat conductance |
82 |
of molecular and nano-scale interfaces will be affected by the |
83 |
chemical details of the surface and is challenging to |
84 |
experimentalist. The lower thermal flux through interfaces is even |
85 |
more difficult to measure with EMD and forward NEMD simulation |
86 |
methods. Therefore, developing good simulation methods will be |
87 |
desirable in order to investigate thermal transport across interfaces. |
88 |
|
89 |
Recently, we have developed the Non-Isotropic Velocity Scaling (NIVS) |
90 |
algorithm for RNEMD simulations\cite{kuang:164101}. This algorithm |
91 |
retains the desirable features of RNEMD (conservation of linear |
92 |
momentum and total energy, compatibility with periodic boundary |
93 |
conditions) while establishing true thermal distributions in each of |
94 |
the two slabs. Furthermore, it allows more effective thermal exchange |
95 |
between particles of different identities, and thus enables extensive |
96 |
study of interfacial conductance. |
97 |
|
98 |
\section{Methodology} |
99 |
\subsection{Algorithm} |
100 |
[BACKGROUND FOR MD METHODS] |
101 |
There have been many algorithms for computing thermal conductivity |
102 |
using molecular dynamics simulations. However, interfacial conductance |
103 |
is at least an order of magnitude smaller. This would make the |
104 |
calculation even more difficult for those slowly-converging |
105 |
equilibrium methods. Imposed-flux non-equilibrium |
106 |
methods\cite{MullerPlathe:1997xw} have the flux set {\it a priori} and |
107 |
the response of temperature or momentum gradients are easier to |
108 |
measure than the flux, if unknown, and thus, is a preferable way to |
109 |
the forward NEMD methods. Although the momentum swapping approach for |
110 |
flux-imposing can be used for exchanging energy between particles of |
111 |
different identity, the kinetic energy transfer efficiency is affected |
112 |
by the mass difference between the particles, which limits its |
113 |
application on heterogeneous interfacial systems. |
114 |
|
115 |
The non-isotropic velocity scaling (NIVS)\cite{kuang:164101} approach in |
116 |
non-equilibrium MD simulations is able to impose relatively large |
117 |
kinetic energy flux without obvious perturbation to the velocity |
118 |
distribution of the simulated systems. Furthermore, this approach has |
119 |
the advantage in heterogeneous interfaces in that kinetic energy flux |
120 |
can be applied between regions of particles of arbitary identity, and |
121 |
the flux quantity is not restricted by particle mass difference. |
122 |
|
123 |
The NIVS algorithm scales the velocity vectors in two separate regions |
124 |
of a simulation system with respective diagonal scaling matricies. To |
125 |
determine these scaling factors in the matricies, a set of equations |
126 |
including linear momentum conservation and kinetic energy conservation |
127 |
constraints and target momentum/energy flux satisfaction is |
128 |
solved. With the scaling operation applied to the system in a set |
129 |
frequency, corresponding momentum/temperature gradients can be built, |
130 |
which can be used for computing transportation properties and other |
131 |
applications related to momentum/temperature gradients. The NIVS |
132 |
algorithm conserves momenta and energy and does not depend on an |
133 |
external thermostat. |
134 |
|
135 |
\subsection{Defining Interfacial Thermal Conductivity $G$} |
136 |
For interfaces with a relatively low interfacial conductance, the bulk |
137 |
regions on either side of an interface rapidly come to a state in |
138 |
which the two phases have relatively homogeneous (but distinct) |
139 |
temperatures. The interfacial thermal conductivity $G$ can therefore |
140 |
be approximated as: |
141 |
\begin{equation} |
142 |
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle - |
143 |
\langle T_\mathrm{cold}\rangle \right)} |
144 |
\label{lowG} |
145 |
\end{equation} |
146 |
where ${E_{total}}$ is the imposed non-physical kinetic energy |
147 |
transfer and ${\langle T_\mathrm{hot}\rangle}$ and ${\langle |
148 |
T_\mathrm{cold}\rangle}$ are the average observed temperature of the |
149 |
two separated phases. |
150 |
|
151 |
When the interfacial conductance is {\it not} small, two ways can be |
152 |
used to define $G$. |
153 |
|
154 |
One way is to assume the temperature is discretely different on two |
155 |
sides of the interface, $G$ can be calculated with the thermal flux |
156 |
applied $J$ and the maximum temperature difference measured along the |
157 |
thermal gradient max($\Delta T$), which occurs at the interface, as: |
158 |
\begin{equation} |
159 |
G=\frac{J}{\Delta T} |
160 |
\label{discreteG} |
161 |
\end{equation} |
162 |
|
163 |
The other approach is to assume a continuous temperature profile along |
164 |
the thermal gradient axis (e.g. $z$) and define $G$ at the point where |
165 |
the magnitude of thermal conductivity $\lambda$ change reach its |
166 |
maximum, given that $\lambda$ is well-defined throughout the space: |
167 |
\begin{equation} |
168 |
G^\prime = \Big|\frac{\partial\lambda}{\partial z}\Big| |
169 |
= \Big|\frac{\partial}{\partial z}\left(-J_z\Big/ |
170 |
\left(\frac{\partial T}{\partial z}\right)\right)\Big| |
171 |
= |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
172 |
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
173 |
\label{derivativeG} |
174 |
\end{equation} |
175 |
|
176 |
With the temperature profile obtained from simulations, one is able to |
177 |
approximate the first and second derivatives of $T$ with finite |
178 |
difference method and thus calculate $G^\prime$. |
179 |
|
180 |
In what follows, both definitions are used for calculation and comparison. |
181 |
|
182 |
[IMPOSE G DEFINITION INTO OUR SYSTEMS] |
183 |
To facilitate the use of the above definitions in calculating $G$ and |
184 |
$G^\prime$, we have a metal slab with its (111) surfaces perpendicular |
185 |
to the $z$-axis of our simulation cells. With or withour capping |
186 |
agents on the surfaces, the metal slab is solvated with organic |
187 |
solvents, as illustrated in Figure \ref{demoPic}. |
188 |
|
189 |
\begin{figure} |
190 |
\includegraphics[width=\linewidth]{demoPic} |
191 |
\caption{A sample showing how a metal slab has its (111) surface |
192 |
covered by capping agent molecules and solvated by hexane.} |
193 |
\label{demoPic} |
194 |
\end{figure} |
195 |
|
196 |
With a simulation cell setup following the above manner, one is able |
197 |
to equilibrate the system and impose an unphysical thermal flux |
198 |
between the liquid and the metal phase with the NIVS algorithm. Under |
199 |
a stablized thermal gradient induced by periodically applying the |
200 |
unphysical flux, one is able to obtain a temperature profile and the |
201 |
physical thermal flux corresponding to it, which equals to the |
202 |
unphysical flux applied by NIVS. These data enables the evaluation of |
203 |
the interfacial thermal conductance of a surface. Figure \ref{gradT} |
204 |
is an example how those stablized thermal gradient can be used to |
205 |
obtain the 1st and 2nd derivatives of the temperature profile. |
206 |
|
207 |
\begin{figure} |
208 |
\includegraphics[width=\linewidth]{gradT} |
209 |
\caption{The 1st and 2nd derivatives of temperature profile can be |
210 |
obtained with finite difference approximation.} |
211 |
\label{gradT} |
212 |
\end{figure} |
213 |
|
214 |
[MAY INCLUDE POWER SPECTRUM PROTOCOL] |
215 |
|
216 |
\section{Computational Details} |
217 |
\subsection{Simulation Protocol} |
218 |
In our simulations, Au is used to construct a metal slab with bare |
219 |
(111) surface perpendicular to the $z$-axis. Different slab thickness |
220 |
(layer numbers of Au) are simulated. This metal slab is first |
221 |
equilibrated under normal pressure (1 atm) and a desired |
222 |
temperature. After equilibration, butanethiol is used as the capping |
223 |
agent molecule to cover the bare Au (111) surfaces evenly. The sulfur |
224 |
atoms in the butanethiol molecules would occupy the three-fold sites |
225 |
of the surfaces, and the maximal butanethiol capacity on Au surface is |
226 |
$1/3$ of the total number of surface Au atoms[CITATION]. A series of |
227 |
different coverage surfaces is investigated in order to study the |
228 |
relation between coverage and conductance. |
229 |
|
230 |
[COVERAGE DISCRIPTION] However, since the interactions between surface |
231 |
Au and butanethiol is non-bonded, the capping agent molecules are |
232 |
allowed to migrate to an empty neighbor three-fold site during a |
233 |
simulation. Therefore, the initial configuration would not severely |
234 |
affect the sampling of a variety of configurations of the same |
235 |
coverage, and the final conductance measurement would be an average |
236 |
effect of these configurations explored in the simulations. [MAY NEED FIGURES] |
237 |
|
238 |
After the modified Au-butanethiol surface systems are equilibrated |
239 |
under canonical ensemble, Packmol\cite{packmol} is used to pack |
240 |
organic solvent molecules in the previously vacuum part of the |
241 |
simulation cells, which guarantees that short range repulsive |
242 |
interactions do not disrupt the simulations. Two solvents are |
243 |
investigated, one which has little vibrational overlap with the |
244 |
alkanethiol and plane-like shape (toluene), and one which has similar |
245 |
vibrational frequencies and chain-like shape ({\it n}-hexane). [MAY |
246 |
EXPLAIN WHY WE CHOOSE THEM] |
247 |
|
248 |
The spacing filled by solvent molecules, i.e. the gap between |
249 |
periodically repeated Au-butanethiol surfaces should be carefully |
250 |
chosen. A very long length scale for the thermal gradient axis ($z$) |
251 |
may cause excessively hot or cold temperatures in the middle of the |
252 |
solvent region and lead to undesired phenomena such as solvent boiling |
253 |
or freezing when a thermal flux is applied. Conversely, too few |
254 |
solvent molecules would change the normal behavior of the liquid |
255 |
phase. Therefore, our $N_{solvent}$ values were chosen to ensure that |
256 |
these extreme cases did not happen to our simulations. And the |
257 |
corresponding spacing is usually $35 \sim 60$\AA. |
258 |
|
259 |
The initial configurations generated by Packmol are further |
260 |
equilibrated with the $x$ and $y$ dimensions fixed, only allowing |
261 |
length scale change in $z$ dimension. This is to ensure that the |
262 |
equilibration of liquid phase does not affect the metal crystal |
263 |
structure in $x$ and $y$ dimensions. Further equilibration are run |
264 |
under NVT and then NVE ensembles. |
265 |
|
266 |
After the systems reach equilibrium, NIVS is implemented to impose a |
267 |
periodic unphysical thermal flux between the metal and the liquid |
268 |
phase. Most of our simulations are under an average temperature of |
269 |
$\sim$200K. Therefore, this flux usually comes from the metal to the |
270 |
liquid so that the liquid has a higher temperature and would not |
271 |
freeze due to excessively low temperature. This induced temperature |
272 |
gradient is stablized and the simulation cell is devided evenly into |
273 |
N slabs along the $z$-axis and the temperatures of each slab are |
274 |
recorded. When the slab width $d$ of each slab is the same, the |
275 |
derivatives of $T$ with respect to slab number $n$ can be directly |
276 |
used for $G^\prime$ calculations: |
277 |
\begin{equation} |
278 |
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
279 |
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
280 |
= |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big| |
281 |
\Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2 |
282 |
= |J_z|\Big|\frac{\partial^2 T}{\partial n^2}\Big| |
283 |
\Big/\left(\frac{\partial T}{\partial n}\right)^2 |
284 |
\label{derivativeG2} |
285 |
\end{equation} |
286 |
|
287 |
\subsection{Force Field Parameters} |
288 |
Our simulations include various components. Therefore, force field |
289 |
parameter descriptions are needed for interactions both between the |
290 |
same type of particles and between particles of different species. |
291 |
|
292 |
The Au-Au interactions in metal lattice slab is described by the |
293 |
quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
294 |
potentials include zero-point quantum corrections and are |
295 |
reparametrized for accurate surface energies compared to the |
296 |
Sutton-Chen potentials\cite{Chen90}. |
297 |
|
298 |
Figure [REF] demonstrates how we name our pseudo-atoms of the |
299 |
molecules in our simulations. |
300 |
[FIGURE FOR MOLECULE NOMENCLATURE] |
301 |
|
302 |
For both solvent molecules, straight chain {\it n}-hexane and aromatic |
303 |
toluene, United-Atom (UA) and All-Atom (AA) models are used |
304 |
respectively. The TraPPE-UA |
305 |
parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
306 |
for our UA solvent molecules. In these models, pseudo-atoms are |
307 |
located at the carbon centers for alkyl groups. By eliminating |
308 |
explicit hydrogen atoms, these models are simple and computationally |
309 |
efficient, while maintains good accuracy. However, the TraPPE-UA for |
310 |
alkanes is known to predict a lower boiling point than experimental |
311 |
values. Considering that after an unphysical thermal flux is applied |
312 |
to a system, the temperature of ``hot'' area in the liquid phase would be |
313 |
significantly higher than the average, to prevent over heating and |
314 |
boiling of the liquid phase, the average temperature in our |
315 |
simulations should be much lower than the liquid boiling point. [MORE DISCUSSION] |
316 |
For UA-toluene model, rigid body constraints are applied, so that the |
317 |
benzene ring and the methyl-CRar bond are kept rigid. This would save |
318 |
computational time.[MORE DETAILS] |
319 |
|
320 |
Besides the TraPPE-UA models, AA models for both organic solvents are |
321 |
included in our studies as well. For hexane, the OPLS-AA\cite{OPLSAA} |
322 |
force field is used. [MORE DETAILS] |
323 |
For toluene, the United Force Field developed by Rapp\'{e} {\it et |
324 |
al.}\cite{doi:10.1021/ja00051a040} is adopted.[MORE DETAILS] |
325 |
|
326 |
The capping agent in our simulations, the butanethiol molecules can |
327 |
either use UA or AA model. The TraPPE-UA force fields includes |
328 |
parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for |
329 |
UA butanethiol model in our simulations. The OPLS-AA also provides |
330 |
parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) |
331 |
surfaces do not have the hydrogen atom bonded to sulfur. To adapt this |
332 |
change and derive suitable parameters for butanethiol adsorbed on |
333 |
Au(111) surfaces, we adopt the S parameters from [CITATION CF VLUGT] |
334 |
and modify parameters for its neighbor C atom for charge balance in |
335 |
the molecule. Note that the model choice (UA or AA) of capping agent |
336 |
can be different from the solvent. Regardless of model choice, the |
337 |
force field parameters for interactions between capping agent and |
338 |
solvent can be derived using Lorentz-Berthelot Mixing Rule: |
339 |
|
340 |
|
341 |
To describe the interactions between metal Au and non-metal capping |
342 |
agent and solvent particles, we refer to an adsorption study of alkyl |
343 |
thiols on gold surfaces by Vlugt {\it et |
344 |
al.}\cite{vlugt:cpc2007154} They fitted an effective Lennard-Jones |
345 |
form of potential parameters for the interaction between Au and |
346 |
pseudo-atoms CH$_x$ and S based on a well-established and widely-used |
347 |
effective potential of Hautman and Klein[CITATION] for the Au(111) |
348 |
surface. As our simulations require the gold lattice slab to be |
349 |
non-rigid so that it could accommodate kinetic energy for thermal |
350 |
transport study purpose, the pair-wise form of potentials is |
351 |
preferred. |
352 |
|
353 |
Besides, the potentials developed from {\it ab initio} calculations by |
354 |
Leng {\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
355 |
interactions between Au and aromatic C/H atoms in toluene.[MORE DETAILS] |
356 |
|
357 |
However, the Lennard-Jones parameters between Au and other types of |
358 |
particles in our simulations are not yet well-established. For these |
359 |
interactions, we attempt to derive their parameters using the Mixing |
360 |
Rule. To do this, the ``Metal-non-Metal'' (MnM) interaction parameters |
361 |
for Au is first extracted from the Au-CH$_x$ parameters by applying |
362 |
the Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM'' |
363 |
parameters in our simulations. |
364 |
|
365 |
\begin{table*} |
366 |
\begin{minipage}{\linewidth} |
367 |
\begin{center} |
368 |
\caption{Lennard-Jones parameters for Au-non-Metal |
369 |
interactions in our simulations.} |
370 |
|
371 |
\begin{tabular}{ccc} |
372 |
\hline\hline |
373 |
Non-metal & $\sigma$/\AA & $\epsilon$/kcal/mol \\ |
374 |
\hline |
375 |
S & 2.40 & 8.465 \\ |
376 |
CH3 & 3.54 & 0.2146 \\ |
377 |
CH2 & 3.54 & 0.1749 \\ |
378 |
CT3 & 3.365 & 0.1373 \\ |
379 |
CT2 & 3.365 & 0.1373 \\ |
380 |
CTT & 3.365 & 0.1373 \\ |
381 |
HC & 2.865 & 0.09256 \\ |
382 |
CHar & 3.4625 & 0.1680 \\ |
383 |
CRar & 3.555 & 0.1604 \\ |
384 |
CA & 3.173 & 0.0640 \\ |
385 |
HA & 2.746 & 0.0414 \\ |
386 |
\hline\hline |
387 |
\end{tabular} |
388 |
\label{MnM} |
389 |
\end{center} |
390 |
\end{minipage} |
391 |
\end{table*} |
392 |
|
393 |
|
394 |
\section{Results and Discussions} |
395 |
[MAY HAVE A BRIEF SUMMARY] |
396 |
\subsection{How Simulation Parameters Affects $G$} |
397 |
[MAY NOT PUT AT FIRST] |
398 |
We have varied our protocol or other parameters of the simulations in |
399 |
order to investigate how these factors would affect the measurement of |
400 |
$G$'s. It turned out that while some of these parameters would not |
401 |
affect the results substantially, some other changes to the |
402 |
simulations would have a significant impact on the measurement |
403 |
results. |
404 |
|
405 |
In some of our simulations, we allowed $L_x$ and $L_y$ to change |
406 |
during equilibrating the liquid phase. Due to the stiffness of the Au |
407 |
slab, $L_x$ and $L_y$ would not change noticeably after |
408 |
equilibration. Although $L_z$ could fluctuate $\sim$1\% after a system |
409 |
is fully equilibrated in the NPT ensemble, this fluctuation, as well |
410 |
as those comparably smaller to $L_x$ and $L_y$, would not be magnified |
411 |
on the calculated $G$'s, as shown in Table \ref{AuThiolHexaneUA}. This |
412 |
insensivity to $L_i$ fluctuations allows reliable measurement of $G$'s |
413 |
without the necessity of extremely cautious equilibration process. |
414 |
|
415 |
As stated in our computational details, the spacing filled with |
416 |
solvent molecules can be chosen within a range. This allows some |
417 |
change of solvent molecule numbers for the same Au-butanethiol |
418 |
surfaces. We did this study on our Au-butanethiol/hexane |
419 |
simulations. Nevertheless, the results obtained from systems of |
420 |
different $N_{hexane}$ did not indicate that the measurement of $G$ is |
421 |
susceptible to this parameter. For computational efficiency concern, |
422 |
smaller system size would be preferable, given that the liquid phase |
423 |
structure is not affected. |
424 |
|
425 |
Our NIVS algorithm allows change of unphysical thermal flux both in |
426 |
direction and in quantity. This feature extends our investigation of |
427 |
interfacial thermal conductance. However, the magnitude of this |
428 |
thermal flux is not arbitary if one aims to obtain a stable and |
429 |
reliable thermal gradient. A temperature profile would be |
430 |
substantially affected by noise when $|J_z|$ has a much too low |
431 |
magnitude; while an excessively large $|J_z|$ that overwhelms the |
432 |
conductance capacity of the interface would prevent a thermal gradient |
433 |
to reach a stablized steady state. NIVS has the advantage of allowing |
434 |
$J$ to vary in a wide range such that the optimal flux range for $G$ |
435 |
measurement can generally be simulated by the algorithm. Within the |
436 |
optimal range, we were able to study how $G$ would change according to |
437 |
the thermal flux across the interface. For our simulations, we denote |
438 |
$J_z$ to be positive when the physical thermal flux is from the liquid |
439 |
to metal, and negative vice versa. The $G$'s measured under different |
440 |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and [REF]. These |
441 |
results do not suggest that $G$ is dependent on $J_z$ within this flux |
442 |
range. The linear response of flux to thermal gradient simplifies our |
443 |
investigations in that we can rely on $G$ measurement with only a |
444 |
couple $J_z$'s and do not need to test a large series of fluxes. |
445 |
|
446 |
%ADD MORE TO TABLE |
447 |
\begin{table*} |
448 |
\begin{minipage}{\linewidth} |
449 |
\begin{center} |
450 |
\caption{Computed interfacial thermal conductivity ($G$ and |
451 |
$G^\prime$) values for the Au/butanethiol/hexane interface |
452 |
with united-atom model and different capping agent coverage |
453 |
and solvent molecule numbers at different temperatures using a |
454 |
range of energy fluxes.} |
455 |
|
456 |
\begin{tabular}{cccccc} |
457 |
\hline\hline |
458 |
Thiol & $\langle T\rangle$ & & $J_z$ & $G$ & $G^\prime$ \\ |
459 |
coverage (\%) & (K) & $N_{hexane}$ & (GW/m$^2$) & |
460 |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
461 |
\hline |
462 |
0.0 & 200 & 200 & 0.96 & 43.3 & 42.7 \\ |
463 |
& & & 1.91 & 45.7 & 42.9 \\ |
464 |
& & 166 & 0.96 & 43.1 & 53.4 \\ |
465 |
88.9 & 200 & 166 & 1.94 & 172 & 108 \\ |
466 |
100.0 & 250 & 200 & 0.96 & 81.8 & 67.0 \\ |
467 |
& & 166 & 0.98 & 79.0 & 62.9 \\ |
468 |
& & & 1.44 & 76.2 & 64.8 \\ |
469 |
& 200 & 200 & 1.92 & 129 & 87.3 \\ |
470 |
& & & 1.93 & 131 & 77.5 \\ |
471 |
& & 166 & 0.97 & 115 & 69.3 \\ |
472 |
& & & 1.94 & 125 & 87.1 \\ |
473 |
\hline\hline |
474 |
\end{tabular} |
475 |
\label{AuThiolHexaneUA} |
476 |
\end{center} |
477 |
\end{minipage} |
478 |
\end{table*} |
479 |
|
480 |
Furthermore, we also attempted to increase system average temperatures |
481 |
to above 200K. These simulations are first equilibrated in the NPT |
482 |
ensemble under normal pressure. As stated above, the TraPPE-UA model |
483 |
for hexane tends to predict a lower boiling point. In our simulations, |
484 |
hexane had diffculty to remain in liquid phase when NPT equilibration |
485 |
temperature is higher than 250K. Additionally, the equilibrated liquid |
486 |
hexane density under 250K becomes lower than experimental value. This |
487 |
expanded liquid phase leads to lower contact between hexane and |
488 |
butanethiol as well.[MAY NEED FIGURE] And this reduced contact would |
489 |
probably be accountable for a lower interfacial thermal conductance, |
490 |
as shown in Table \ref{AuThiolHexaneUA}. |
491 |
|
492 |
A similar study for TraPPE-UA toluene agrees with the above result as |
493 |
well. Having a higher boiling point, toluene tends to remain liquid in |
494 |
our simulations even equilibrated under 300K in NPT |
495 |
ensembles. Furthermore, the expansion of the toluene liquid phase is |
496 |
not as significant as that of the hexane. This prevents severe |
497 |
decrease of liquid-capping agent contact and the results (Table |
498 |
\ref{AuThiolToluene}) show only a slightly decreased interface |
499 |
conductance. Therefore, solvent-capping agent contact should play an |
500 |
important role in the thermal transport process across the interface |
501 |
in that higher degree of contact could yield increased conductance. |
502 |
|
503 |
[ADD SIGNS AND ERROR ESTIMATE TO TABLE] |
504 |
\begin{table*} |
505 |
\begin{minipage}{\linewidth} |
506 |
\begin{center} |
507 |
\caption{Computed interfacial thermal conductivity ($G$ and |
508 |
$G^\prime$) values for the Au/butanethiol/toluene interface at |
509 |
different temperatures using a range of energy fluxes.} |
510 |
|
511 |
\begin{tabular}{cccc} |
512 |
\hline\hline |
513 |
$\langle T\rangle$ & $J_z$ & $G$ & $G^\prime$ \\ |
514 |
(K) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
515 |
\hline |
516 |
200 & 1.86 & 180 & 135 \\ |
517 |
& 2.15 & 204 & 113 \\ |
518 |
& 3.93 & 175 & 114 \\ |
519 |
300 & 1.91 & 143 & 125 \\ |
520 |
& 4.19 & 134 & 113 \\ |
521 |
\hline\hline |
522 |
\end{tabular} |
523 |
\label{AuThiolToluene} |
524 |
\end{center} |
525 |
\end{minipage} |
526 |
\end{table*} |
527 |
|
528 |
Besides lower interfacial thermal conductance, surfaces in relatively |
529 |
high temperatures are susceptible to reconstructions, when |
530 |
butanethiols have a full coverage on the Au(111) surface. These |
531 |
reconstructions include surface Au atoms migrated outward to the S |
532 |
atom layer, and butanethiol molecules embedded into the original |
533 |
surface Au layer. The driving force for this behavior is the strong |
534 |
Au-S interactions in our simulations. And these reconstructions lead |
535 |
to higher ratio of Au-S attraction and thus is energetically |
536 |
favorable. Furthermore, this phenomenon agrees with experimental |
537 |
results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt |
538 |
{\it et al.} had kept their Au(111) slab rigid so that their |
539 |
simulations can reach 300K without surface reconstructions. Without |
540 |
this practice, simulating 100\% thiol covered interfaces under higher |
541 |
temperatures could hardly avoid surface reconstructions. However, our |
542 |
measurement is based on assuming homogeneity on $x$ and $y$ dimensions |
543 |
so that measurement of $T$ at particular $z$ would be an effective |
544 |
average of the particles of the same type. Since surface |
545 |
reconstructions could eliminate the original $x$ and $y$ dimensional |
546 |
homogeneity, measurement of $G$ is more difficult to conduct under |
547 |
higher temperatures. Therefore, most of our measurements are |
548 |
undertaken at $<T>\sim$200K. |
549 |
|
550 |
However, when the surface is not completely covered by butanethiols, |
551 |
the simulated system is more resistent to the reconstruction |
552 |
above. Our Au-butanethiol/toluene system did not see this phenomena |
553 |
even at $<T>\sim$300K. The Au(111) surfaces have a 90\% coverage of |
554 |
butanethiols and have empty three-fold sites. These empty sites could |
555 |
help prevent surface reconstruction in that they provide other means |
556 |
of capping agent relaxation. It is observed that butanethiols can |
557 |
migrate to their neighbor empty sites during a simulation. Therefore, |
558 |
we were able to obtain $G$'s for these interfaces even at a relatively |
559 |
high temperature without being affected by surface reconstructions. |
560 |
|
561 |
\subsection{Influence of Capping Agent Coverage on $G$} |
562 |
To investigate the influence of butanethiol coverage on interfacial |
563 |
thermal conductance, a series of different coverage Au-butanethiol |
564 |
surfaces is prepared and solvated with various organic |
565 |
molecules. These systems are then equilibrated and their interfacial |
566 |
thermal conductivity are measured with our NIVS algorithm. Table |
567 |
\ref{tlnUhxnUhxnD} lists these results for direct comparison between |
568 |
different coverages of butanethiol. |
569 |
|
570 |
With high coverage of butanethiol on the gold surface, |
571 |
the interfacial thermal conductance is enhanced |
572 |
significantly. Interestingly, a slightly lower butanethiol coverage |
573 |
leads to a moderately higher conductivity. This is probably due to |
574 |
more solvent/capping agent contact when butanethiol molecules are |
575 |
not densely packed, which enhances the interactions between the two |
576 |
phases and lowers the thermal transfer barrier of this interface. |
577 |
[COMPARE TO AU/WATER IN PAPER] |
578 |
|
579 |
|
580 |
significant conductance enhancement compared to the gold/water |
581 |
interface without capping agent and agree with available experimental |
582 |
data. This indicates that the metal-metal potential, though not |
583 |
predicting an accurate bulk metal thermal conductivity, does not |
584 |
greatly interfere with the simulation of the thermal conductance |
585 |
behavior across a non-metal interface. |
586 |
The results show that the two definitions used for $G$ yield |
587 |
comparable values, though $G^\prime$ tends to be smaller. |
588 |
|
589 |
|
590 |
\begin{table*} |
591 |
\begin{minipage}{\linewidth} |
592 |
\begin{center} |
593 |
\caption{Computed interfacial thermal conductivity ($G$ and |
594 |
$G^\prime$) values for the Au/butanethiol/hexane interface |
595 |
with united-atom model and different capping agent coverage |
596 |
and solvent molecule numbers at different temperatures using a |
597 |
range of energy fluxes.} |
598 |
|
599 |
\begin{tabular}{cccccc} |
600 |
\hline\hline |
601 |
Thiol & $\langle T\rangle$ & & $J_z$ & $G$ & $G^\prime$ \\ |
602 |
coverage (\%) & (K) & $N_{hexane}$ & (GW/m$^2$) & |
603 |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
604 |
\hline |
605 |
0.0 & 200 & 200 & 0.96 & 43.3 & 42.7 \\ |
606 |
& & & 1.91 & 45.7 & 42.9 \\ |
607 |
& & 166 & 0.96 & 43.1 & 53.4 \\ |
608 |
88.9 & 200 & 166 & 1.94 & 172 & 108 \\ |
609 |
100.0 & 250 & 200 & 0.96 & 81.8 & 67.0 \\ |
610 |
& & 166 & 0.98 & 79.0 & 62.9 \\ |
611 |
& & & 1.44 & 76.2 & 64.8 \\ |
612 |
& 200 & 200 & 1.92 & 129 & 87.3 \\ |
613 |
& & & 1.93 & 131 & 77.5 \\ |
614 |
& & 166 & 0.97 & 115 & 69.3 \\ |
615 |
& & & 1.94 & 125 & 87.1 \\ |
616 |
\hline\hline |
617 |
\end{tabular} |
618 |
\label{tlnUhxnUhxnD} |
619 |
\end{center} |
620 |
\end{minipage} |
621 |
\end{table*} |
622 |
|
623 |
\subsection{Influence of Chosen Molecule Model on $G$} |
624 |
[MAY COMBINE W MECHANISM STUDY] |
625 |
|
626 |
For the all-atom model, the liquid hexane phase was not stable under NPT |
627 |
conditions. Therefore, the simulation length scale parameters are |
628 |
adopted from previous equilibration results of the united-atom model |
629 |
at 200K. Table \ref{AuThiolHexaneAA} shows the results of these |
630 |
simulations. The conductivity values calculated with full capping |
631 |
agent coverage are substantially larger than observed in the |
632 |
united-atom model, and is even higher than predicted by |
633 |
experiments. It is possible that our parameters for metal-non-metal |
634 |
particle interactions lead to an overestimate of the interfacial |
635 |
thermal conductivity, although the active C-H vibrations in the |
636 |
all-atom model (which should not be appreciably populated at normal |
637 |
temperatures) could also account for this high conductivity. The major |
638 |
thermal transfer barrier of Au/butanethiol/hexane interface is between |
639 |
the liquid phase and the capping agent, so extra degrees of freedom |
640 |
such as the C-H vibrations could enhance heat exchange between these |
641 |
two phases and result in a much higher conductivity. |
642 |
|
643 |
\begin{table*} |
644 |
\begin{minipage}{\linewidth} |
645 |
\begin{center} |
646 |
|
647 |
\caption{Computed interfacial thermal conductivity ($G$ and |
648 |
$G^\prime$) values for the Au/butanethiol/hexane interface |
649 |
with all-atom model and different capping agent coverage at |
650 |
200K using a range of energy fluxes.} |
651 |
|
652 |
\begin{tabular}{cccc} |
653 |
\hline\hline |
654 |
Thiol & $J_z$ & $G$ & $G^\prime$ \\ |
655 |
coverage (\%) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
656 |
\hline |
657 |
0.0 & 0.95 & 28.5 & 27.2 \\ |
658 |
& 1.88 & 30.3 & 28.9 \\ |
659 |
100.0 & 2.87 & 551 & 294 \\ |
660 |
& 3.81 & 494 & 193 \\ |
661 |
\hline\hline |
662 |
\end{tabular} |
663 |
\label{AuThiolHexaneAA} |
664 |
\end{center} |
665 |
\end{minipage} |
666 |
\end{table*} |
667 |
|
668 |
|
669 |
\subsection{Mechanism of Interfacial Thermal Conductance Enhancement |
670 |
by Capping Agent} |
671 |
[MAY INTRODUCE PROTOCOL IN METHODOLOGY/COMPUTATIONAL DETAIL] |
672 |
|
673 |
|
674 |
%subsubsection{Vibrational spectrum study on conductance mechanism} |
675 |
To investigate the mechanism of this interfacial thermal conductance, |
676 |
the vibrational spectra of various gold systems were obtained and are |
677 |
shown as in the upper panel of Fig. \ref{vibration}. To obtain these |
678 |
spectra, one first runs a simulation in the NVE ensemble and collects |
679 |
snapshots of configurations; these configurations are used to compute |
680 |
the velocity auto-correlation functions, which is used to construct a |
681 |
power spectrum via a Fourier transform. The gold surfaces covered by |
682 |
butanethiol molecules exhibit an additional peak observed at a |
683 |
frequency of $\sim$170cm$^{-1}$, which is attributed to the vibration |
684 |
of the S-Au bond. This vibration enables efficient thermal transport |
685 |
from surface Au atoms to the capping agents. Simultaneously, as shown |
686 |
in the lower panel of Fig. \ref{vibration}, the large overlap of the |
687 |
vibration spectra of butanethiol and hexane in the all-atom model, |
688 |
including the C-H vibration, also suggests high thermal exchange |
689 |
efficiency. The combination of these two effects produces the drastic |
690 |
interfacial thermal conductance enhancement in the all-atom model. |
691 |
|
692 |
\begin{figure} |
693 |
\includegraphics[width=\linewidth]{vibration} |
694 |
\caption{Vibrational spectra obtained for gold in different |
695 |
environments (upper panel) and for Au/thiol/hexane simulation in |
696 |
all-atom model (lower panel).} |
697 |
\label{vibration} |
698 |
\end{figure} |
699 |
% MAY NEED TO CONVERT TO JPEG |
700 |
|
701 |
\section{Conclusions} |
702 |
|
703 |
|
704 |
[NECESSITY TO STUDY THERMAL CONDUCTANCE IN NANOCRYSTAL STRUCTURE]\cite{vlugt:cpc2007154} |
705 |
|
706 |
\section{Acknowledgments} |
707 |
Support for this project was provided by the National Science |
708 |
Foundation under grant CHE-0848243. Computational time was provided by |
709 |
the Center for Research Computing (CRC) at the University of Notre |
710 |
Dame. \newpage |
711 |
|
712 |
\bibliography{interfacial} |
713 |
|
714 |
\end{doublespace} |
715 |
\end{document} |
716 |
|