1 |
gezelter |
3717 |
\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 |
skuang |
3727 |
%\renewcommand\citemid{\ } % no comma in optional reference note |
26 |
gezelter |
3717 |
\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 |
skuang |
3725 |
|
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 |
gezelter |
3717 |
\end{abstract} |
64 |
|
|
|
65 |
|
|
\newpage |
66 |
|
|
|
67 |
|
|
%\narrowtext |
68 |
|
|
|
69 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
70 |
|
|
% BODY OF TEXT |
71 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
72 |
|
|
|
73 |
|
|
\section{Introduction} |
74 |
skuang |
3727 |
[BACKGROUND FOR INTERFACIAL THERMAL CONDUCTANCE PROBLEM] |
75 |
skuang |
3725 |
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 |
gezelter |
3717 |
|
89 |
skuang |
3725 |
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 |
skuang |
3721 |
\section{Methodology} |
99 |
|
|
\subsection{Algorithm} |
100 |
skuang |
3727 |
[BACKGROUND FOR MD METHODS] |
101 |
skuang |
3721 |
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 |
skuang |
3727 |
\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 |
skuang |
3721 |
|
151 |
skuang |
3727 |
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 |
|
|
\section{Computational Details} |
215 |
|
|
\subsection{System Geometry} |
216 |
|
|
In our simulations, Au is used to construct a metal slab with bare |
217 |
|
|
(111) surface perpendicular to the $z$-axis. Different slab thickness |
218 |
|
|
(layer numbers of Au) are simulated. This metal slab is first |
219 |
|
|
equilibrated under normal pressure (1 atm) and a desired |
220 |
|
|
temperature. After equilibration, butanethiol is used as the capping |
221 |
|
|
agent molecule to cover the bare Au (111) surfaces evenly. The sulfur |
222 |
|
|
atoms in the butanethiol molecules would occupy the three-fold sites |
223 |
|
|
of the surfaces, and the maximal butanethiol capacity on Au surface is |
224 |
|
|
$1/3$ of the total number of surface Au atoms[CITATION]. A series of |
225 |
|
|
different coverage surfaces is investigated in order to study the |
226 |
|
|
relation between coverage and conductance. |
227 |
|
|
|
228 |
|
|
[COVERAGE DISCRIPTION] However, since the interactions between surface |
229 |
|
|
Au and butanethiol is non-bonded, the capping agent molecules are |
230 |
|
|
allowed to migrate to an empty neighbor three-fold site during a |
231 |
|
|
simulation. Therefore, the initial configuration would not severely |
232 |
|
|
affect the sampling of a variety of configurations of the same |
233 |
|
|
coverage, and the final conductance measurement would be an average |
234 |
|
|
effect of these configurations explored in the simulations. [MAY NEED FIGURES] |
235 |
|
|
|
236 |
|
|
After the modified Au-butanethiol surface systems are equilibrated |
237 |
|
|
under canonical ensemble, Packmol\cite{packmol} is used to pack |
238 |
|
|
organic solvent molecules in the previously vacuum part of the |
239 |
|
|
simulation cells, which guarantees that short range repulsive |
240 |
|
|
interactions do not disrupt the simulations. Two solvents are |
241 |
|
|
investigated, one which has little vibrational overlap with the |
242 |
|
|
alkanethiol and plane-like shape (toluene), and one which has similar |
243 |
|
|
vibrational frequencies and chain-like shape ({\it n}-hexane). The |
244 |
skuang |
3728 |
spacing filled by solvent molecules, i.e. the gap between periodically |
245 |
|
|
repeated Au-butanethiol surfaces should be carefully chosen so that it |
246 |
|
|
would not be too short to affect the liquid phase structure, nor too |
247 |
|
|
long, leading to over cooling (freezing) or heating (boiling) when a |
248 |
|
|
thermal flux is applied. In our simulations, this spacing is usually |
249 |
|
|
$35 \sim 60$\AA. |
250 |
skuang |
3727 |
|
251 |
skuang |
3728 |
The initial configurations generated by Packmol are further |
252 |
|
|
equilibrated with the $x$ and $y$ dimensions fixed, only allowing |
253 |
|
|
length scale change in $z$ dimension. This is to ensure that the |
254 |
|
|
equilibration of liquid phase does not affect the metal crystal |
255 |
|
|
structure in $x$ and $y$ dimensions. Further equilibration are run |
256 |
|
|
under NVT and then NVE ensembles. |
257 |
|
|
|
258 |
skuang |
3727 |
After the systems reach equilibrium, NIVS is implemented to impose a |
259 |
|
|
periodic unphysical thermal flux between the metal and the liquid |
260 |
skuang |
3728 |
phase. Most of our simulations are under an average temperature of |
261 |
|
|
$\sim$200K. Therefore, this flux usually comes from the metal to the |
262 |
skuang |
3727 |
liquid so that the liquid has a higher temperature and would not |
263 |
|
|
freeze due to excessively low temperature. This induced temperature |
264 |
|
|
gradient is stablized and the simulation cell is devided evenly into |
265 |
|
|
N slabs along the $z$-axis and the temperatures of each slab are |
266 |
|
|
recorded. When the slab width $d$ of each slab is the same, the |
267 |
|
|
derivatives of $T$ with respect to slab number $n$ can be directly |
268 |
|
|
used for $G^\prime$ calculations: |
269 |
|
|
\begin{equation} |
270 |
|
|
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
271 |
|
|
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
272 |
|
|
= |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big| |
273 |
|
|
\Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2 |
274 |
|
|
= |J_z|\Big|\frac{\partial^2 T}{\partial n^2}\Big| |
275 |
|
|
\Big/\left(\frac{\partial T}{\partial n}\right)^2 |
276 |
|
|
\label{derivativeG2} |
277 |
|
|
\end{equation} |
278 |
|
|
|
279 |
skuang |
3725 |
\subsection{Force Field Parameters} |
280 |
skuang |
3728 |
Our simulations include various components. Therefore, force field |
281 |
|
|
parameter descriptions are needed for interactions both between the |
282 |
|
|
same type of particles and between particles of different species. |
283 |
skuang |
3721 |
|
284 |
|
|
The Au-Au interactions in metal lattice slab is described by the |
285 |
|
|
quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
286 |
|
|
potentials include zero-point quantum corrections and are |
287 |
|
|
reparametrized for accurate surface energies compared to the |
288 |
|
|
Sutton-Chen potentials\cite{Chen90}. |
289 |
|
|
|
290 |
skuang |
3728 |
For both solvent molecules, straight chain {\it n}-hexane and aromatic |
291 |
|
|
toluene, United-Atom (UA) and All-Atom (AA) models are used |
292 |
|
|
respectively. The TraPPE-UA |
293 |
|
|
parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
294 |
|
|
for our UA solvent molecules. In these models, pseudo-atoms are |
295 |
|
|
located at the carbon centers for alkyl groups. By eliminating |
296 |
|
|
explicit hydrogen atoms, these models are simple and computationally |
297 |
skuang |
3729 |
efficient, while maintains good accuracy. However, the TraPPE-UA for |
298 |
|
|
alkanes is known to predict a lower boiling point than experimental |
299 |
|
|
values. Considering that after an unphysical thermal flux is applied |
300 |
|
|
to a system, the temperature of ``hot'' area in the liquid phase would be |
301 |
|
|
significantly higher than the average, to prevent over heating and |
302 |
|
|
boiling of the liquid phase, the average temperature in our |
303 |
|
|
simulations should be much lower than the liquid boiling point. [NEED MORE DISCUSSION] |
304 |
|
|
For UA-toluene model, rigid body constraints are applied, so that the |
305 |
|
|
benzene ring and the methyl-C(aromatic) bond are kept rigid. This |
306 |
|
|
would save computational time.[MORE DETAILS NEEDED] |
307 |
skuang |
3721 |
|
308 |
skuang |
3729 |
Besides the TraPPE-UA models, AA models for both organic solvents are |
309 |
|
|
included in our studies as well. For hexane, the OPLS |
310 |
|
|
all-atom\cite{OPLSAA} force field is used. [MORE DETAILS] |
311 |
|
|
For toluene, the United Force Field developed by Rapp\'{e} {\it et |
312 |
|
|
al.}\cite{doi:10.1021/ja00051a040} is adopted.[MORE DETAILS] |
313 |
skuang |
3728 |
|
314 |
skuang |
3729 |
The capping agent in our simulations, the butanethiol molecules can |
315 |
|
|
either use UA or AA model. The TraPPE-UA force fields includes |
316 |
|
|
parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used in |
317 |
|
|
our simulations corresponding to our TraPPE-UA models for solvent. |
318 |
|
|
and All-Atom models [NEED CITATIONS] |
319 |
|
|
However, the model choice (UA or AA) of capping agent can be different |
320 |
|
|
from the solvent. Regardless of model choice, the force field |
321 |
|
|
parameters for interactions between capping agent and solvent can be |
322 |
|
|
derived using Lorentz-Berthelot Mixing Rule. |
323 |
skuang |
3721 |
|
324 |
|
|
To describe the interactions between metal Au and non-metal capping |
325 |
|
|
agent and solvent, we refer to Vlugt\cite{vlugt:cpc2007154} and derive |
326 |
skuang |
3729 |
other interactions which are not yet finely parametrized. [can add |
327 |
skuang |
3721 |
hautman and klein's paper here and more discussion; need to put |
328 |
skuang |
3729 |
aromatic-metal interaction approximation here]\cite{doi:10.1021/jp034405s} |
329 |
skuang |
3721 |
|
330 |
skuang |
3725 |
[TABULATED FORCE FIELD PARAMETERS NEEDED] |
331 |
|
|
|
332 |
skuang |
3729 |
|
333 |
|
|
[SURFACE RECONSTRUCTION PREVENTS SIMULATION TEMP TO GO HIGHER] |
334 |
|
|
|
335 |
|
|
|
336 |
skuang |
3725 |
\section{Results} |
337 |
skuang |
3729 |
[REARRANGEMENT NEEDED] |
338 |
skuang |
3725 |
\subsection{Toluene Solvent} |
339 |
|
|
|
340 |
skuang |
3727 |
The results (Table \ref{AuThiolToluene}) show a |
341 |
skuang |
3725 |
significant conductance enhancement compared to the gold/water |
342 |
|
|
interface without capping agent and agree with available experimental |
343 |
|
|
data. This indicates that the metal-metal potential, though not |
344 |
|
|
predicting an accurate bulk metal thermal conductivity, does not |
345 |
|
|
greatly interfere with the simulation of the thermal conductance |
346 |
|
|
behavior across a non-metal interface. The solvent model is not |
347 |
|
|
particularly volatile, so the simulation cell does not expand |
348 |
|
|
significantly under higher temperature. We did not observe a |
349 |
|
|
significant conductance decrease when the temperature was increased to |
350 |
|
|
300K. The results show that the two definitions used for $G$ yield |
351 |
|
|
comparable values, though $G^\prime$ tends to be smaller. |
352 |
|
|
|
353 |
|
|
\begin{table*} |
354 |
|
|
\begin{minipage}{\linewidth} |
355 |
|
|
\begin{center} |
356 |
|
|
\caption{Computed interfacial thermal conductivity ($G$ and |
357 |
|
|
$G^\prime$) values for the Au/butanethiol/toluene interface at |
358 |
|
|
different temperatures using a range of energy fluxes.} |
359 |
|
|
|
360 |
|
|
\begin{tabular}{cccc} |
361 |
|
|
\hline\hline |
362 |
|
|
$\langle T\rangle$ & $J_z$ & $G$ & $G^\prime$ \\ |
363 |
|
|
(K) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
364 |
|
|
\hline |
365 |
|
|
200 & 1.86 & 180 & 135 \\ |
366 |
|
|
& 2.15 & 204 & 113 \\ |
367 |
|
|
& 3.93 & 175 & 114 \\ |
368 |
|
|
300 & 1.91 & 143 & 125 \\ |
369 |
|
|
& 4.19 & 134 & 113 \\ |
370 |
|
|
\hline\hline |
371 |
|
|
\end{tabular} |
372 |
|
|
\label{AuThiolToluene} |
373 |
|
|
\end{center} |
374 |
|
|
\end{minipage} |
375 |
|
|
\end{table*} |
376 |
|
|
|
377 |
|
|
\subsection{Hexane Solvent} |
378 |
|
|
|
379 |
|
|
Using the united-atom model, different coverages of capping agent, |
380 |
|
|
temperatures of simulations and numbers of solvent molecules were all |
381 |
|
|
investigated and Table \ref{AuThiolHexaneUA} shows the results of |
382 |
|
|
these computations. The number of hexane molecules in our simulations |
383 |
|
|
does not affect the calculations significantly. However, a very long |
384 |
|
|
length scale for the thermal gradient axis ($z$) may cause excessively |
385 |
|
|
hot or cold temperatures in the middle of the solvent region and lead |
386 |
|
|
to undesired phenomena such as solvent boiling or freezing, while too |
387 |
|
|
few solvent molecules would change the normal behavior of the liquid |
388 |
|
|
phase. Our $N_{hexane}$ values were chosen to ensure that these |
389 |
|
|
extreme cases did not happen to our simulations. |
390 |
|
|
|
391 |
|
|
Table \ref{AuThiolHexaneUA} enables direct comparison between |
392 |
|
|
different coverages of capping agent, when other system parameters are |
393 |
|
|
held constant. With high coverage of butanethiol on the gold surface, |
394 |
|
|
the interfacial thermal conductance is enhanced |
395 |
|
|
significantly. Interestingly, a slightly lower butanethiol coverage |
396 |
|
|
leads to a moderately higher conductivity. This is probably due to |
397 |
|
|
more solvent/capping agent contact when butanethiol molecules are |
398 |
|
|
not densely packed, which enhances the interactions between the two |
399 |
|
|
phases and lowers the thermal transfer barrier of this interface. |
400 |
|
|
% [COMPARE TO AU/WATER IN PAPER] |
401 |
|
|
|
402 |
|
|
It is also noted that the overall simulation temperature is another |
403 |
|
|
factor that affects the interfacial thermal conductance. One |
404 |
|
|
possibility of this effect may be rooted in the decrease in density of |
405 |
|
|
the liquid phase. We observed that when the average temperature |
406 |
|
|
increases from 200K to 250K, the bulk hexane density becomes lower |
407 |
|
|
than experimental value, as the system is equilibrated under NPT |
408 |
|
|
ensemble. This leads to lower contact between solvent and capping |
409 |
|
|
agent, and thus lower conductivity. |
410 |
|
|
|
411 |
|
|
Conductivity values are more difficult to obtain under higher |
412 |
|
|
temperatures. This is because the Au surface tends to undergo |
413 |
|
|
reconstructions in relatively high temperatures. Surface Au atoms can |
414 |
|
|
migrate outward to reach higher Au-S contact; and capping agent |
415 |
|
|
molecules can be embedded into the surface Au layer due to the same |
416 |
|
|
driving force. This phenomenon agrees with experimental |
417 |
|
|
results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. A surface |
418 |
|
|
fully covered in capping agent is more susceptible to reconstruction, |
419 |
|
|
possibly because fully coverage prevents other means of capping agent |
420 |
|
|
relaxation, such as migration to an empty neighbor three-fold site. |
421 |
|
|
|
422 |
|
|
%MAY ADD MORE DATA TO TABLE |
423 |
|
|
\begin{table*} |
424 |
|
|
\begin{minipage}{\linewidth} |
425 |
|
|
\begin{center} |
426 |
|
|
\caption{Computed interfacial thermal conductivity ($G$ and |
427 |
|
|
$G^\prime$) values for the Au/butanethiol/hexane interface |
428 |
|
|
with united-atom model and different capping agent coverage |
429 |
|
|
and solvent molecule numbers at different temperatures using a |
430 |
|
|
range of energy fluxes.} |
431 |
|
|
|
432 |
|
|
\begin{tabular}{cccccc} |
433 |
|
|
\hline\hline |
434 |
|
|
Thiol & $\langle T\rangle$ & & $J_z$ & $G$ & $G^\prime$ \\ |
435 |
|
|
coverage (\%) & (K) & $N_{hexane}$ & (GW/m$^2$) & |
436 |
|
|
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
437 |
|
|
\hline |
438 |
|
|
0.0 & 200 & 200 & 0.96 & 43.3 & 42.7 \\ |
439 |
|
|
& & & 1.91 & 45.7 & 42.9 \\ |
440 |
|
|
& & 166 & 0.96 & 43.1 & 53.4 \\ |
441 |
|
|
88.9 & 200 & 166 & 1.94 & 172 & 108 \\ |
442 |
|
|
100.0 & 250 & 200 & 0.96 & 81.8 & 67.0 \\ |
443 |
|
|
& & 166 & 0.98 & 79.0 & 62.9 \\ |
444 |
|
|
& & & 1.44 & 76.2 & 64.8 \\ |
445 |
|
|
& 200 & 200 & 1.92 & 129 & 87.3 \\ |
446 |
|
|
& & & 1.93 & 131 & 77.5 \\ |
447 |
|
|
& & 166 & 0.97 & 115 & 69.3 \\ |
448 |
|
|
& & & 1.94 & 125 & 87.1 \\ |
449 |
|
|
\hline\hline |
450 |
|
|
\end{tabular} |
451 |
|
|
\label{AuThiolHexaneUA} |
452 |
|
|
\end{center} |
453 |
|
|
\end{minipage} |
454 |
|
|
\end{table*} |
455 |
|
|
|
456 |
|
|
For the all-atom model, the liquid hexane phase was not stable under NPT |
457 |
|
|
conditions. Therefore, the simulation length scale parameters are |
458 |
|
|
adopted from previous equilibration results of the united-atom model |
459 |
|
|
at 200K. Table \ref{AuThiolHexaneAA} shows the results of these |
460 |
|
|
simulations. The conductivity values calculated with full capping |
461 |
|
|
agent coverage are substantially larger than observed in the |
462 |
|
|
united-atom model, and is even higher than predicted by |
463 |
|
|
experiments. It is possible that our parameters for metal-non-metal |
464 |
|
|
particle interactions lead to an overestimate of the interfacial |
465 |
|
|
thermal conductivity, although the active C-H vibrations in the |
466 |
|
|
all-atom model (which should not be appreciably populated at normal |
467 |
|
|
temperatures) could also account for this high conductivity. The major |
468 |
|
|
thermal transfer barrier of Au/butanethiol/hexane interface is between |
469 |
|
|
the liquid phase and the capping agent, so extra degrees of freedom |
470 |
|
|
such as the C-H vibrations could enhance heat exchange between these |
471 |
|
|
two phases and result in a much higher conductivity. |
472 |
|
|
|
473 |
|
|
\begin{table*} |
474 |
|
|
\begin{minipage}{\linewidth} |
475 |
|
|
\begin{center} |
476 |
|
|
|
477 |
|
|
\caption{Computed interfacial thermal conductivity ($G$ and |
478 |
|
|
$G^\prime$) values for the Au/butanethiol/hexane interface |
479 |
|
|
with all-atom model and different capping agent coverage at |
480 |
|
|
200K using a range of energy fluxes.} |
481 |
|
|
|
482 |
|
|
\begin{tabular}{cccc} |
483 |
|
|
\hline\hline |
484 |
|
|
Thiol & $J_z$ & $G$ & $G^\prime$ \\ |
485 |
|
|
coverage (\%) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
486 |
|
|
\hline |
487 |
|
|
0.0 & 0.95 & 28.5 & 27.2 \\ |
488 |
|
|
& 1.88 & 30.3 & 28.9 \\ |
489 |
|
|
100.0 & 2.87 & 551 & 294 \\ |
490 |
|
|
& 3.81 & 494 & 193 \\ |
491 |
|
|
\hline\hline |
492 |
|
|
\end{tabular} |
493 |
|
|
\label{AuThiolHexaneAA} |
494 |
|
|
\end{center} |
495 |
|
|
\end{minipage} |
496 |
|
|
\end{table*} |
497 |
|
|
|
498 |
|
|
%subsubsection{Vibrational spectrum study on conductance mechanism} |
499 |
|
|
To investigate the mechanism of this interfacial thermal conductance, |
500 |
|
|
the vibrational spectra of various gold systems were obtained and are |
501 |
|
|
shown as in the upper panel of Fig. \ref{vibration}. To obtain these |
502 |
|
|
spectra, one first runs a simulation in the NVE ensemble and collects |
503 |
|
|
snapshots of configurations; these configurations are used to compute |
504 |
|
|
the velocity auto-correlation functions, which is used to construct a |
505 |
|
|
power spectrum via a Fourier transform. The gold surfaces covered by |
506 |
|
|
butanethiol molecules exhibit an additional peak observed at a |
507 |
|
|
frequency of $\sim$170cm$^{-1}$, which is attributed to the vibration |
508 |
|
|
of the S-Au bond. This vibration enables efficient thermal transport |
509 |
|
|
from surface Au atoms to the capping agents. Simultaneously, as shown |
510 |
|
|
in the lower panel of Fig. \ref{vibration}, the large overlap of the |
511 |
|
|
vibration spectra of butanethiol and hexane in the all-atom model, |
512 |
|
|
including the C-H vibration, also suggests high thermal exchange |
513 |
|
|
efficiency. The combination of these two effects produces the drastic |
514 |
|
|
interfacial thermal conductance enhancement in the all-atom model. |
515 |
|
|
|
516 |
|
|
\begin{figure} |
517 |
|
|
\includegraphics[width=\linewidth]{vibration} |
518 |
|
|
\caption{Vibrational spectra obtained for gold in different |
519 |
|
|
environments (upper panel) and for Au/thiol/hexane simulation in |
520 |
|
|
all-atom model (lower panel).} |
521 |
|
|
\label{vibration} |
522 |
|
|
\end{figure} |
523 |
|
|
% 600dpi, letter size. too large? |
524 |
|
|
|
525 |
|
|
|
526 |
gezelter |
3717 |
\section{Acknowledgments} |
527 |
|
|
Support for this project was provided by the National Science |
528 |
|
|
Foundation under grant CHE-0848243. Computational time was provided by |
529 |
|
|
the Center for Research Computing (CRC) at the University of Notre |
530 |
|
|
Dame. \newpage |
531 |
|
|
|
532 |
|
|
\bibliography{interfacial} |
533 |
|
|
|
534 |
|
|
\end{doublespace} |
535 |
|
|
\end{document} |
536 |
|
|
|