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{[}{]}{,}{n}{}{;} |
27 |
\bibliographystyle{achemso} |
28 |
|
29 |
\begin{document} |
30 |
|
31 |
\title{ENTER TITLE HERE} |
32 |
|
33 |
\author{Shenyu Kuang and J. Daniel |
34 |
Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\ |
35 |
Department of Chemistry and Biochemistry,\\ |
36 |
University of Notre Dame\\ |
37 |
Notre Dame, Indiana 46556} |
38 |
|
39 |
\date{\today} |
40 |
|
41 |
\maketitle |
42 |
|
43 |
\begin{doublespace} |
44 |
|
45 |
\begin{abstract} |
46 |
REPLACE ABSTRACT HERE |
47 |
With the Non-Isotropic Velocity Scaling (NIVS) approach to Reverse |
48 |
Non-Equilibrium Molecular Dynamics (RNEMD) it is possible to impose |
49 |
an unphysical thermal flux between different regions of |
50 |
inhomogeneous systems such as solid / liquid interfaces. We have |
51 |
applied NIVS to compute the interfacial thermal conductance at a |
52 |
metal / organic solvent interface that has been chemically capped by |
53 |
butanethiol molecules. Our calculations suggest that coupling |
54 |
between the metal and liquid phases is enhanced by the capping |
55 |
agents, leading to a greatly enhanced conductivity at the interface. |
56 |
Specifically, the chemical bond between the metal and the capping |
57 |
agent introduces a vibrational overlap that is not present without |
58 |
the capping agent, and the overlap between the vibrational spectra |
59 |
(metal to cap, cap to solvent) provides a mechanism for rapid |
60 |
thermal transport across the interface. Our calculations also |
61 |
suggest that this is a non-monotonic function of the fractional |
62 |
coverage of the surface, as moderate coverages allow diffusive heat |
63 |
transport of solvent molecules that have been in close contact with |
64 |
the capping agent. |
65 |
|
66 |
\end{abstract} |
67 |
|
68 |
\newpage |
69 |
|
70 |
%\narrowtext |
71 |
|
72 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
73 |
% BODY OF TEXT |
74 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
75 |
|
76 |
\section{Introduction} |
77 |
[REFINE LATER, ADD MORE REF.S] |
78 |
Imposed-flux methods in Molecular Dynamics (MD) |
79 |
simulations\cite{MullerPlathe:1997xw} can establish steady state |
80 |
systems with a set applied flux vs a corresponding gradient that can |
81 |
be measured. These methods does not need many trajectories to provide |
82 |
information of transport properties of a given system. Thus, they are |
83 |
utilized in computing thermal and mechanical transfer of homogeneous |
84 |
or bulk systems as well as heterogeneous systems such as liquid-solid |
85 |
interfaces.\cite{kuang:AuThl} |
86 |
|
87 |
The Reverse Non-Equilibrium MD (RNEMD) methods adopt constraints that |
88 |
satisfy linear momentum and total energy conservation of a system when |
89 |
imposing fluxes in a simulation. Thus they are compatible with various |
90 |
ensembles, including the micro-canonical (NVE) ensemble, without the |
91 |
need of an external thermostat. The original approaches by |
92 |
M\"{u}ller-Plathe {\it et |
93 |
al.}\cite{MullerPlathe:1997xw,ISI:000080382700030} utilize simple |
94 |
momentum swapping for generating energy/momentum fluxes, which is also |
95 |
compatible with particles of different identities. Although simple to |
96 |
implement in a simulation, this approach can create nonthermal |
97 |
velocity distributions, as discovered by Tenney and |
98 |
Maginn\cite{Maginn:2010}. Furthermore, this approach to kinetic energy |
99 |
transfer between particles of different identities is less efficient |
100 |
when the mass difference between the particles becomes significant, |
101 |
which also limits its application on heterogeneous interfacial |
102 |
systems. |
103 |
|
104 |
Recently, we developed a different approach, using Non-Isotropic |
105 |
Velocity Scaling (NIVS) \cite{kuang:164101} algorithm to impose |
106 |
fluxes. Compared to the momentum swapping move, it scales the velocity |
107 |
vectors in two separate regions of a simulated system with respective |
108 |
diagonal scaling matrices. These matrices are determined by solving a |
109 |
set of equations including linear momentum and kinetic energy |
110 |
conservation constraints and target flux satisfaction. This method is |
111 |
able to effectively impose a wide range of kinetic energy fluxes |
112 |
without obvious perturbation to the velocity distributions of the |
113 |
simulated systems, regardless of the presence of heterogeneous |
114 |
interfaces. We have successfully applied this approach in studying the |
115 |
interfacial thermal conductance at metal-solvent |
116 |
interfaces.\cite{kuang:AuThl} |
117 |
|
118 |
However, the NIVS approach limits its application in imposing momentum |
119 |
fluxes. Temperature anisotropy can happen under high momentum fluxes, |
120 |
due to the nature of the algorithm. Thus, combining thermal and |
121 |
momentum flux is also difficult to implement with this |
122 |
approach. However, such combination may provide a means to simulate |
123 |
thermal/momentum gradient coupled processes such as freeze |
124 |
desalination. Therefore, developing novel approaches to extend the |
125 |
application of imposed-flux method is desired. |
126 |
|
127 |
In this paper, we improve the NIVS method and propose a novel approach |
128 |
to impose fluxes. This approach separate the means of applying |
129 |
momentum and thermal flux with operations in one time step and thus is |
130 |
able to simutaneously impose thermal and momentum flux. Furthermore, |
131 |
the approach retains desirable features of previous RNEMD approaches |
132 |
and is simpler to implement compared to the NIVS method. In what |
133 |
follows, we first present the method to implement the method in a |
134 |
simulation. Then we compare the method on bulk fluids to previous |
135 |
methods. Also, interfacial frictions are computed for a series of |
136 |
interfaces. |
137 |
|
138 |
\section{Methodology} |
139 |
Similar to the NIVS methodology,\cite{kuang:164101} we consider a |
140 |
periodic system divided into a series of slabs along a certain axis |
141 |
(e.g. $z$). The unphysical thermal and/or momentum flux is designated |
142 |
from the center slab to one of the end slabs, and thus the center slab |
143 |
would have a lower temperature than the end slab (unless the thermal |
144 |
flux is negative). Therefore, the center slab is denoted as ``$c$'' |
145 |
while the end slab as ``$h$''. |
146 |
|
147 |
To impose these fluxes, we periodically apply separate operations to |
148 |
velocities of particles {$i$} within the center slab and of particles |
149 |
{$j$} within the end slab: |
150 |
\begin{eqnarray} |
151 |
\vec{v}_i & \leftarrow & c\cdot\left(\vec{v}_i - \langle\vec{v}_c |
152 |
\rangle\right) + \left(\langle\vec{v}_c\rangle + \vec{a}_c\right) \\ |
153 |
\vec{v}_j & \leftarrow & h\cdot\left(\vec{v}_j - \langle\vec{v}_h |
154 |
\rangle\right) + \left(\langle\vec{v}_h\rangle + \vec{a}_h\right) |
155 |
\end{eqnarray} |
156 |
where $\langle\vec{v}_c\rangle$ and $\langle\vec{v}_h\rangle$ denotes |
157 |
the instantaneous bulk velocity of slabs $c$ and $h$ respectively |
158 |
before an operation occurs. When a momentum flux $\vec{j}_z(\vec{p})$ |
159 |
presents, these bulk velocities would have a corresponding change |
160 |
($\vec{a}_c$ and $\vec{a}_h$ respectively) according to Newton's |
161 |
second law: |
162 |
\begin{eqnarray} |
163 |
M_c \vec{a}_c & = & -\vec{j}_z(\vec{p}) \Delta t \\ |
164 |
M_h \vec{a}_h & = & \vec{j}_z(\vec{p}) \Delta t |
165 |
\end{eqnarray} |
166 |
where |
167 |
\begin{eqnarray} |
168 |
M_c & = & \sum_{i = 1}^{N_c} m_i \\ |
169 |
M_h & = & \sum_{j = 1}^{N_h} m_j |
170 |
\end{eqnarray} |
171 |
and $\Delta t$ is the interval between two operations. |
172 |
|
173 |
The above operations conserve the linear momentum of a periodic |
174 |
system. To satisfy total energy conservation as well as to impose a |
175 |
thermal flux $J_z$, one would have |
176 |
[SUPPORT INFO MIGHT BE NECESSARY TO PUT EXTRA MATH IN] |
177 |
\begin{eqnarray} |
178 |
K_c - J_z\Delta t & = & c^2 (K_c - \frac{1}{2}M_c \langle\vec{v}_c |
179 |
\rangle^2) + \frac{1}{2}M_c (\langle \vec{v}_c \rangle + \vec{a}_c)^2 \\ |
180 |
K_h + J_z\Delta t & = & h^2 (K_h - \frac{1}{2}M_h \langle\vec{v}_h |
181 |
\rangle^2) + \frac{1}{2}M_h (\langle \vec{v}_h \rangle + \vec{a}_h)^2 |
182 |
\end{eqnarray} |
183 |
where $K_c$ and $K_h$ denotes translational kinetic energy of slabs |
184 |
$c$ and $h$ respectively before an operation occurs. These |
185 |
translational kinetic energy conservation equations are sufficient to |
186 |
ensure total energy conservation, as the operations applied do not |
187 |
change the potential energy of a system, given that the potential |
188 |
energy does not depend on particle velocity. |
189 |
|
190 |
The above sets of equations are sufficient to determine the velocity |
191 |
scaling coefficients ($c$ and $h$) as well as $\vec{a}_c$ and |
192 |
$\vec{a}_h$. Note that two roots of $c$ and $h$ exist |
193 |
respectively. However, to avoid dramatic perturbations to a system, |
194 |
the positive roots (which are closer to 1) are chosen. Figure |
195 |
\ref{method} illustrates the implementation of this algorithm in an |
196 |
individual step. |
197 |
|
198 |
\begin{figure} |
199 |
\includegraphics[width=\linewidth]{method} |
200 |
\caption{Illustration of the implementation of the algorithm in a |
201 |
single step. Starting from an ideal velocity distribution, the |
202 |
transformation is used to apply both thermal and momentum flux from |
203 |
the ``c'' slab to the ``h'' slab. As the figure shows, the thermal |
204 |
distributions preserve after this operation.} |
205 |
\label{method} |
206 |
\end{figure} |
207 |
|
208 |
By implementing these operations at a certain frequency, a steady |
209 |
thermal and/or momentum flux can be applied and the corresponding |
210 |
temperature and/or momentum gradients can be established. |
211 |
|
212 |
This approach is more computationaly efficient compared to the |
213 |
previous NIVS method, in that only quadratic equations are involved, |
214 |
while the NIVS method needs to solve a quartic equations. Furthermore, |
215 |
the method implements isotropic scaling of velocities in respective |
216 |
slabs, unlike the NIVS, where an extra criteria function is necessary |
217 |
to choose a set of coefficients that performs the most isotropic |
218 |
scaling. More importantly, separating the momentum flux imposing from |
219 |
velocity scaling avoids the underlying cause that NIVS produced |
220 |
thermal anisotropy when applying a momentum flux. |
221 |
%NEW METHOD DOESN'T CAUSE UNDESIRED CONCOMITENT MOMENTUM FLUX WHEN |
222 |
%IMPOSING A THERMAL FLUX |
223 |
|
224 |
The advantages of the approach over the original momentum swapping |
225 |
approach lies in its nature to preserve a Gaussian |
226 |
distribution. Because the momentum swapping tends to render a |
227 |
nonthermal distribution, when the imposed flux is relatively large, |
228 |
diffusion of the neighboring slabs could no longer remedy this effect, |
229 |
and nonthermal distributions would be observed. Results in later |
230 |
section will illustrate this effect. |
231 |
%NEW METHOD (AND NIVS) HAVE LESS PERTURBATION THAN MOMENTUM SWAPPING |
232 |
|
233 |
\section{Computational Details} |
234 |
The algorithm has been implemented in our MD simulation code, |
235 |
OpenMD\cite{Meineke:2005gd,openmd}. We compare the method with |
236 |
previous RNEMD methods or equilibrium MD methods in homogeneous fluids |
237 |
(Lennard-Jones and SPC/E water). And taking advantage of the method, |
238 |
we simulate the interfacial friction of different heterogeneous |
239 |
interfaces (gold-organic solvent and gold-SPC/E water and ice-liquid |
240 |
water). |
241 |
|
242 |
\subsection{Simulation Protocols} |
243 |
The systems to be investigated are set up in a orthorhombic simulation |
244 |
cell with periodic boundary conditions in all three dimensions. The |
245 |
$z$ axis of these cells were longer and was set as the gradient axis |
246 |
of temperature and/or momentum. Thus the cells were divided into $N$ |
247 |
slabs along this axis, with various $N$ depending on individual |
248 |
system. The $x$ and $y$ axis were usually of the same length in |
249 |
homogeneous systems or close to each other where interfaces |
250 |
presents. In all cases, before introducing a nonequilibrium method to |
251 |
establish steady thermal and/or momentum gradients for further |
252 |
measurements and calculations, canonical ensemble with a Nos\'e-Hoover |
253 |
thermostat\cite{hoover85} and microcanonical ensemble equilibrations |
254 |
were used to prepare systems ready for data |
255 |
collections. Isobaric-isothermal equilibrations are performed before |
256 |
this for SPC/E water systems to reach normal pressure (1 bar), while |
257 |
similar equilibrations are used for interfacial systems to relax the |
258 |
surface tensions. |
259 |
|
260 |
While homogeneous fluid systems can be set up with random |
261 |
configurations, our interfacial systems needs extra steps to ensure |
262 |
the interfaces be established properly for computations. The |
263 |
preparation and equilibration of butanethiol covered gold (111) |
264 |
surface and further solvation and equilibration process is described |
265 |
as in reference \ref{kuang:AuThl}. |
266 |
|
267 |
As for the ice/liquid water interfaces, the basal surface of ice |
268 |
lattice was first constructed. [REF JPCB PAPER] explored the energeics |
269 |
of ice lattices with different proton orders. We refer to their |
270 |
results and choose the configuration of the lowest energy after |
271 |
relaxations as the unit cells of our ice lattices. Although |
272 |
experimental solid/liquid coexistant temperature near normal pressure |
273 |
is 273K, [REF HAYMET] simulations of ice/liquid water interfaces with |
274 |
different models suggest that for SPC/E, stable interfaces could only |
275 |
exist no higher than 225K. Therefore, all our ice/liquid water |
276 |
simulations were carried out under 225K. To have extra protection of |
277 |
the ice lattice during initial equilibration (when the randomly |
278 |
generated liquid configuration could release large amount of energy in |
279 |
relaxation), a constraint method (REF?) was adopted until the high |
280 |
energy configuration was relaxed. |
281 |
|
282 |
\subsection{Force Field Parameters} |
283 |
For comparison of our new method with previous work, we retain our |
284 |
force field parameters consistent with the results we will compare |
285 |
with. The Lennard-Jones Fluid used here is argon, and reduced unit |
286 |
results are reported when it is favorable for comparison purpose. |
287 |
|
288 |
As for our water simulations, SPC/E model is used throughout this work |
289 |
for consistency. Previous work for transport properties of SPC/E water |
290 |
model is available so that unnecessary repetition of previous methods |
291 |
can be avoided. |
292 |
|
293 |
The Au-Au interaction parameters in all simulations are described by |
294 |
the quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The |
295 |
QSC potentials include zero-point quantum corrections and are |
296 |
reparametrized for accurate surface energies compared to the |
297 |
Sutton-Chen potentials.\cite{Chen90} When Au-H$_2$O interactions are |
298 |
involved, the Spohr potential was adopted.[CITE NIVS REF.46] |
299 |
|
300 |
The small organic molecules included in our simulations are the Au |
301 |
surface capping agent butanethiol and liquid hexane and toluene. The |
302 |
United-Atom |
303 |
models\cite{TraPPE-UA.thiols,TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} |
304 |
for these components were used in this work for better computational |
305 |
efficiency, while maintaining good accuracy. We refer readers to our |
306 |
previous work\cite{kuang:AuThl} for further details of these models, |
307 |
as well as the interactions between Au and the above organic molecule |
308 |
components. |
309 |
|
310 |
\subsection{Thermal conductivities} |
311 |
\subsection{Shear viscosities} |
312 |
\subsection{Interfacial friction and Slip length} |
313 |
|
314 |
|
315 |
\section{Results} |
316 |
[L-J COMPARED TO RNEMD NIVS; WATER COMPARED TO RNEMD NIVS AND EMD; |
317 |
SLIP BOUNDARY VS STICK BOUNDARY; ICE-WATER INTERFACES] |
318 |
|
319 |
There are many factors contributing to the measured interfacial |
320 |
conductance; some of these factors are physically motivated |
321 |
(e.g. coverage of the surface by the capping agent coverage and |
322 |
solvent identity), while some are governed by parameters of the |
323 |
methodology (e.g. applied flux and the formulas used to obtain the |
324 |
conductance). In this section we discuss the major physical and |
325 |
calculational effects on the computed conductivity. |
326 |
|
327 |
\subsection{Effects due to capping agent coverage} |
328 |
|
329 |
A series of different initial conditions with a range of surface |
330 |
coverages was prepared and solvated with various with both of the |
331 |
solvent molecules. These systems were then equilibrated and their |
332 |
interfacial thermal conductivity was measured with the NIVS |
333 |
algorithm. Figure \ref{coverage} demonstrates the trend of conductance |
334 |
with respect to surface coverage. |
335 |
|
336 |
\begin{figure} |
337 |
\includegraphics[width=\linewidth]{coverage} |
338 |
\caption{The interfacial thermal conductivity ($G$) has a |
339 |
non-monotonic dependence on the degree of surface capping. This |
340 |
data is for the Au(111) / butanethiol / solvent interface with |
341 |
various UA force fields at $\langle T\rangle \sim $200K.} |
342 |
\label{coverage} |
343 |
\end{figure} |
344 |
|
345 |
In partially covered surfaces, the derivative definition for |
346 |
$G^\prime$ (Eq. \ref{derivativeG}) becomes difficult to apply, as the |
347 |
location of maximum change of $\lambda$ becomes washed out. The |
348 |
discrete definition (Eq. \ref{discreteG}) is easier to apply, as the |
349 |
Gibbs dividing surface is still well-defined. Therefore, $G$ (not |
350 |
$G^\prime$) was used in this section. |
351 |
|
352 |
From Figure \ref{coverage}, one can see the significance of the |
353 |
presence of capping agents. When even a small fraction of the Au(111) |
354 |
surface sites are covered with butanethiols, the conductivity exhibits |
355 |
an enhancement by at least a factor of 3. Capping agents are clearly |
356 |
playing a major role in thermal transport at metal / organic solvent |
357 |
surfaces. |
358 |
|
359 |
We note a non-monotonic behavior in the interfacial conductance as a |
360 |
function of surface coverage. The maximum conductance (largest $G$) |
361 |
happens when the surfaces are about 75\% covered with butanethiol |
362 |
caps. The reason for this behavior is not entirely clear. One |
363 |
explanation is that incomplete butanethiol coverage allows small gaps |
364 |
between butanethiols to form. These gaps can be filled by transient |
365 |
solvent molecules. These solvent molecules couple very strongly with |
366 |
the hot capping agent molecules near the surface, and can then carry |
367 |
away (diffusively) the excess thermal energy from the surface. |
368 |
|
369 |
There appears to be a competition between the conduction of the |
370 |
thermal energy away from the surface by the capping agents (enhanced |
371 |
by greater coverage) and the coupling of the capping agents with the |
372 |
solvent (enhanced by interdigitation at lower coverages). This |
373 |
competition would lead to the non-monotonic coverage behavior observed |
374 |
here. |
375 |
|
376 |
Results for rigid body toluene solvent, as well as the UA hexane, are |
377 |
within the ranges expected from prior experimental |
378 |
work.\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406} This suggests |
379 |
that explicit hydrogen atoms might not be required for modeling |
380 |
thermal transport in these systems. C-H vibrational modes do not see |
381 |
significant excited state population at low temperatures, and are not |
382 |
likely to carry lower frequency excitations from the solid layer into |
383 |
the bulk liquid. |
384 |
|
385 |
The toluene solvent does not exhibit the same behavior as hexane in |
386 |
that $G$ remains at approximately the same magnitude when the capping |
387 |
coverage increases from 25\% to 75\%. Toluene, as a rigid planar |
388 |
molecule, cannot occupy the relatively small gaps between the capping |
389 |
agents as easily as the chain-like {\it n}-hexane. The effect of |
390 |
solvent coupling to the capping agent is therefore weaker in toluene |
391 |
except at the very lowest coverage levels. This effect counters the |
392 |
coverage-dependent conduction of heat away from the metal surface, |
393 |
leading to a much flatter $G$ vs. coverage trend than is observed in |
394 |
{\it n}-hexane. |
395 |
|
396 |
\subsection{Effects due to Solvent \& Solvent Models} |
397 |
In addition to UA solvent and capping agent models, AA models have |
398 |
also been included in our simulations. In most of this work, the same |
399 |
(UA or AA) model for solvent and capping agent was used, but it is |
400 |
also possible to utilize different models for different components. |
401 |
We have also included isotopic substitutions (Hydrogen to Deuterium) |
402 |
to decrease the explicit vibrational overlap between solvent and |
403 |
capping agent. Table \ref{modelTest} summarizes the results of these |
404 |
studies. |
405 |
|
406 |
\begin{table*} |
407 |
\begin{minipage}{\linewidth} |
408 |
\begin{center} |
409 |
|
410 |
\caption{Computed interfacial thermal conductance ($G$ and |
411 |
$G^\prime$) values for interfaces using various models for |
412 |
solvent and capping agent (or without capping agent) at |
413 |
$\langle T\rangle\sim$200K. Here ``D'' stands for deuterated |
414 |
solvent or capping agent molecules. Error estimates are |
415 |
indicated in parentheses.} |
416 |
|
417 |
\begin{tabular}{llccc} |
418 |
\hline\hline |
419 |
Butanethiol model & Solvent & $G$ & $G^\prime$ \\ |
420 |
(or bare surface) & model & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
421 |
\hline |
422 |
UA & UA hexane & 131(9) & 87(10) \\ |
423 |
& UA hexane(D) & 153(5) & 136(13) \\ |
424 |
& AA hexane & 131(6) & 122(10) \\ |
425 |
& UA toluene & 187(16) & 151(11) \\ |
426 |
& AA toluene & 200(36) & 149(53) \\ |
427 |
\hline |
428 |
AA & UA hexane & 116(9) & 129(8) \\ |
429 |
& AA hexane & 442(14) & 356(31) \\ |
430 |
& AA hexane(D) & 222(12) & 234(54) \\ |
431 |
& UA toluene & 125(25) & 97(60) \\ |
432 |
& AA toluene & 487(56) & 290(42) \\ |
433 |
\hline |
434 |
AA(D) & UA hexane & 158(25) & 172(4) \\ |
435 |
& AA hexane & 243(29) & 191(11) \\ |
436 |
& AA toluene & 364(36) & 322(67) \\ |
437 |
\hline |
438 |
bare & UA hexane & 46.5(3.2) & 49.4(4.5) \\ |
439 |
& UA hexane(D) & 43.9(4.6) & 43.0(2.0) \\ |
440 |
& AA hexane & 31.0(1.4) & 29.4(1.3) \\ |
441 |
& UA toluene & 70.1(1.3) & 65.8(0.5) \\ |
442 |
\hline\hline |
443 |
\end{tabular} |
444 |
\label{modelTest} |
445 |
\end{center} |
446 |
\end{minipage} |
447 |
\end{table*} |
448 |
|
449 |
To facilitate direct comparison between force fields, systems with the |
450 |
same capping agent and solvent were prepared with the same length |
451 |
scales for the simulation cells. |
452 |
|
453 |
On bare metal / solvent surfaces, different force field models for |
454 |
hexane yield similar results for both $G$ and $G^\prime$, and these |
455 |
two definitions agree with each other very well. This is primarily an |
456 |
indicator of weak interactions between the metal and the solvent. |
457 |
|
458 |
For the fully-covered surfaces, the choice of force field for the |
459 |
capping agent and solvent has a large impact on the calculated values |
460 |
of $G$ and $G^\prime$. The AA thiol to AA solvent conductivities are |
461 |
much larger than their UA to UA counterparts, and these values exceed |
462 |
the experimental estimates by a large measure. The AA force field |
463 |
allows significant energy to go into C-H (or C-D) stretching modes, |
464 |
and since these modes are high frequency, this non-quantum behavior is |
465 |
likely responsible for the overestimate of the conductivity. Compared |
466 |
to the AA model, the UA model yields more reasonable conductivity |
467 |
values with much higher computational efficiency. |
468 |
|
469 |
\subsubsection{Are electronic excitations in the metal important?} |
470 |
Because they lack electronic excitations, the QSC and related embedded |
471 |
atom method (EAM) models for gold are known to predict unreasonably |
472 |
low values for bulk conductivity |
473 |
($\lambda$).\cite{kuang:164101,ISI:000207079300006,Clancy:1992} If the |
474 |
conductance between the phases ($G$) is governed primarily by phonon |
475 |
excitation (and not electronic degrees of freedom), one would expect a |
476 |
classical model to capture most of the interfacial thermal |
477 |
conductance. Our results for $G$ and $G^\prime$ indicate that this is |
478 |
indeed the case, and suggest that the modeling of interfacial thermal |
479 |
transport depends primarily on the description of the interactions |
480 |
between the various components at the interface. When the metal is |
481 |
chemically capped, the primary barrier to thermal conductivity appears |
482 |
to be the interface between the capping agent and the surrounding |
483 |
solvent, so the excitations in the metal have little impact on the |
484 |
value of $G$. |
485 |
|
486 |
\subsection{Effects due to methodology and simulation parameters} |
487 |
|
488 |
We have varied the parameters of the simulations in order to |
489 |
investigate how these factors would affect the computation of $G$. Of |
490 |
particular interest are: 1) the length scale for the applied thermal |
491 |
gradient (modified by increasing the amount of solvent in the system), |
492 |
2) the sign and magnitude of the applied thermal flux, 3) the average |
493 |
temperature of the simulation (which alters the solvent density during |
494 |
equilibration), and 4) the definition of the interfacial conductance |
495 |
(Eqs. (\ref{discreteG}) or (\ref{derivativeG})) used in the |
496 |
calculation. |
497 |
|
498 |
Systems of different lengths were prepared by altering the number of |
499 |
solvent molecules and extending the length of the box along the $z$ |
500 |
axis to accomodate the extra solvent. Equilibration at the same |
501 |
temperature and pressure conditions led to nearly identical surface |
502 |
areas ($L_x$ and $L_y$) available to the metal and capping agent, |
503 |
while the extra solvent served mainly to lengthen the axis that was |
504 |
used to apply the thermal flux. For a given value of the applied |
505 |
flux, the different $z$ length scale has only a weak effect on the |
506 |
computed conductivities. |
507 |
|
508 |
\subsubsection{Effects of applied flux} |
509 |
The NIVS algorithm allows changes in both the sign and magnitude of |
510 |
the applied flux. It is possible to reverse the direction of heat |
511 |
flow simply by changing the sign of the flux, and thermal gradients |
512 |
which would be difficult to obtain experimentally ($5$ K/\AA) can be |
513 |
easily simulated. However, the magnitude of the applied flux is not |
514 |
arbitrary if one aims to obtain a stable and reliable thermal gradient. |
515 |
A temperature gradient can be lost in the noise if $|J_z|$ is too |
516 |
small, and excessive $|J_z|$ values can cause phase transitions if the |
517 |
extremes of the simulation cell become widely separated in |
518 |
temperature. Also, if $|J_z|$ is too large for the bulk conductivity |
519 |
of the materials, the thermal gradient will never reach a stable |
520 |
state. |
521 |
|
522 |
Within a reasonable range of $J_z$ values, we were able to study how |
523 |
$G$ changes as a function of this flux. In what follows, we use |
524 |
positive $J_z$ values to denote the case where energy is being |
525 |
transferred by the method from the metal phase and into the liquid. |
526 |
The resulting gradient therefore has a higher temperature in the |
527 |
liquid phase. Negative flux values reverse this transfer, and result |
528 |
in higher temperature metal phases. The conductance measured under |
529 |
different applied $J_z$ values is listed in Tables 2 and 3 in the |
530 |
supporting information. These results do not indicate that $G$ depends |
531 |
strongly on $J_z$ within this flux range. The linear response of flux |
532 |
to thermal gradient simplifies our investigations in that we can rely |
533 |
on $G$ measurement with only a small number $J_z$ values. |
534 |
|
535 |
The sign of $J_z$ is a different matter, however, as this can alter |
536 |
the temperature on the two sides of the interface. The average |
537 |
temperature values reported are for the entire system, and not for the |
538 |
liquid phase, so at a given $\langle T \rangle$, the system with |
539 |
positive $J_z$ has a warmer liquid phase. This means that if the |
540 |
liquid carries thermal energy via diffusive transport, {\it positive} |
541 |
$J_z$ values will result in increased molecular motion on the liquid |
542 |
side of the interface, and this will increase the measured |
543 |
conductivity. |
544 |
|
545 |
\subsubsection{Effects due to average temperature} |
546 |
|
547 |
We also studied the effect of average system temperature on the |
548 |
interfacial conductance. The simulations are first equilibrated in |
549 |
the NPT ensemble at 1 atm. The TraPPE-UA model for hexane tends to |
550 |
predict a lower boiling point (and liquid state density) than |
551 |
experiments. This lower-density liquid phase leads to reduced contact |
552 |
between the hexane and butanethiol, and this accounts for our |
553 |
observation of lower conductance at higher temperatures. In raising |
554 |
the average temperature from 200K to 250K, the density drop of |
555 |
$\sim$20\% in the solvent phase leads to a $\sim$40\% drop in the |
556 |
conductance. |
557 |
|
558 |
Similar behavior is observed in the TraPPE-UA model for toluene, |
559 |
although this model has better agreement with the experimental |
560 |
densities of toluene. The expansion of the toluene liquid phase is |
561 |
not as significant as that of the hexane (8.3\% over 100K), and this |
562 |
limits the effect to $\sim$20\% drop in thermal conductivity. |
563 |
|
564 |
Although we have not mapped out the behavior at a large number of |
565 |
temperatures, is clear that there will be a strong temperature |
566 |
dependence in the interfacial conductance when the physical properties |
567 |
of one side of the interface (notably the density) change rapidly as a |
568 |
function of temperature. |
569 |
|
570 |
Besides the lower interfacial thermal conductance, surfaces at |
571 |
relatively high temperatures are susceptible to reconstructions, |
572 |
particularly when butanethiols fully cover the Au(111) surface. These |
573 |
reconstructions include surface Au atoms which migrate outward to the |
574 |
S atom layer, and butanethiol molecules which embed into the surface |
575 |
Au layer. The driving force for this behavior is the strong Au-S |
576 |
interactions which are modeled here with a deep Lennard-Jones |
577 |
potential. This phenomenon agrees with reconstructions that have been |
578 |
experimentally |
579 |
observed.\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt |
580 |
{\it et al.} kept their Au(111) slab rigid so that their simulations |
581 |
could reach 300K without surface |
582 |
reconstructions.\cite{vlugt:cpc2007154} Since surface reconstructions |
583 |
blur the interface, the measurement of $G$ becomes more difficult to |
584 |
conduct at higher temperatures. For this reason, most of our |
585 |
measurements are undertaken at $\langle T\rangle\sim$200K where |
586 |
reconstruction is minimized. |
587 |
|
588 |
However, when the surface is not completely covered by butanethiols, |
589 |
the simulated system appears to be more resistent to the |
590 |
reconstruction. Our Au / butanethiol / toluene system had the Au(111) |
591 |
surfaces 90\% covered by butanethiols, but did not see this above |
592 |
phenomena even at $\langle T\rangle\sim$300K. That said, we did |
593 |
observe butanethiols migrating to neighboring three-fold sites during |
594 |
a simulation. Since the interface persisted in these simulations, we |
595 |
were able to obtain $G$'s for these interfaces even at a relatively |
596 |
high temperature without being affected by surface reconstructions. |
597 |
|
598 |
\section{Discussion} |
599 |
[COMBINE W. RESULTS] |
600 |
The primary result of this work is that the capping agent acts as an |
601 |
efficient thermal coupler between solid and solvent phases. One of |
602 |
the ways the capping agent can carry out this role is to down-shift |
603 |
between the phonon vibrations in the solid (which carry the heat from |
604 |
the gold) and the molecular vibrations in the liquid (which carry some |
605 |
of the heat in the solvent). |
606 |
|
607 |
To investigate the mechanism of interfacial thermal conductance, the |
608 |
vibrational power spectrum was computed. Power spectra were taken for |
609 |
individual components in different simulations. To obtain these |
610 |
spectra, simulations were run after equilibration in the |
611 |
microcanonical (NVE) ensemble and without a thermal |
612 |
gradient. Snapshots of configurations were collected at a frequency |
613 |
that is higher than that of the fastest vibrations occurring in the |
614 |
simulations. With these configurations, the velocity auto-correlation |
615 |
functions can be computed: |
616 |
\begin{equation} |
617 |
C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle |
618 |
\label{vCorr} |
619 |
\end{equation} |
620 |
The power spectrum is constructed via a Fourier transform of the |
621 |
symmetrized velocity autocorrelation function, |
622 |
\begin{equation} |
623 |
\hat{f}(\omega) = |
624 |
\int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt |
625 |
\label{fourier} |
626 |
\end{equation} |
627 |
|
628 |
\subsection{The role of specific vibrations} |
629 |
The vibrational spectra for gold slabs in different environments are |
630 |
shown as in Figure \ref{specAu}. Regardless of the presence of |
631 |
solvent, the gold surfaces which are covered by butanethiol molecules |
632 |
exhibit an additional peak observed at a frequency of |
633 |
$\sim$165cm$^{-1}$. We attribute this peak to the S-Au bonding |
634 |
vibration. This vibration enables efficient thermal coupling of the |
635 |
surface Au layer to the capping agents. Therefore, in our simulations, |
636 |
the Au / S interfaces do not appear to be the primary barrier to |
637 |
thermal transport when compared with the butanethiol / solvent |
638 |
interfaces. This supports the results of Luo {\it et |
639 |
al.}\cite{Luo20101}, who reported $G$ for Au-SAM junctions roughly |
640 |
twice as large as what we have computed for the thiol-liquid |
641 |
interfaces. |
642 |
|
643 |
\begin{figure} |
644 |
\includegraphics[width=\linewidth]{vibration} |
645 |
\caption{The vibrational power spectrum for thiol-capped gold has an |
646 |
additional vibrational peak at $\sim $165cm$^{-1}$. Bare gold |
647 |
surfaces (both with and without a solvent over-layer) are missing |
648 |
this peak. A similar peak at $\sim $165cm$^{-1}$ also appears in |
649 |
the vibrational power spectrum for the butanethiol capping agents.} |
650 |
\label{specAu} |
651 |
\end{figure} |
652 |
|
653 |
Also in this figure, we show the vibrational power spectrum for the |
654 |
bound butanethiol molecules, which also exhibits the same |
655 |
$\sim$165cm$^{-1}$ peak. |
656 |
|
657 |
\subsection{Overlap of power spectra} |
658 |
A comparison of the results obtained from the two different organic |
659 |
solvents can also provide useful information of the interfacial |
660 |
thermal transport process. In particular, the vibrational overlap |
661 |
between the butanethiol and the organic solvents suggests a highly |
662 |
efficient thermal exchange between these components. Very high |
663 |
thermal conductivity was observed when AA models were used and C-H |
664 |
vibrations were treated classically. The presence of extra degrees of |
665 |
freedom in the AA force field yields higher heat exchange rates |
666 |
between the two phases and results in a much higher conductivity than |
667 |
in the UA force field. The all-atom classical models include high |
668 |
frequency modes which should be unpopulated at our relatively low |
669 |
temperatures. This artifact is likely the cause of the high thermal |
670 |
conductance in all-atom MD simulations. |
671 |
|
672 |
The similarity in the vibrational modes available to solvent and |
673 |
capping agent can be reduced by deuterating one of the two components |
674 |
(Fig. \ref{aahxntln}). Once either the hexanes or the butanethiols |
675 |
are deuterated, one can observe a significantly lower $G$ and |
676 |
$G^\prime$ values (Table \ref{modelTest}). |
677 |
|
678 |
\begin{figure} |
679 |
\includegraphics[width=\linewidth]{aahxntln} |
680 |
\caption{Spectra obtained for all-atom (AA) Au / butanethiol / solvent |
681 |
systems. When butanethiol is deuterated (lower left), its |
682 |
vibrational overlap with hexane decreases significantly. Since |
683 |
aromatic molecules and the butanethiol are vibrationally dissimilar, |
684 |
the change is not as dramatic when toluene is the solvent (right).} |
685 |
\label{aahxntln} |
686 |
\end{figure} |
687 |
|
688 |
For the Au / butanethiol / toluene interfaces, having the AA |
689 |
butanethiol deuterated did not yield a significant change in the |
690 |
measured conductance. Compared to the C-H vibrational overlap between |
691 |
hexane and butanethiol, both of which have alkyl chains, the overlap |
692 |
between toluene and butanethiol is not as significant and thus does |
693 |
not contribute as much to the heat exchange process. |
694 |
|
695 |
Previous observations of Zhang {\it et al.}\cite{hase:2010} indicate |
696 |
that the {\it intra}molecular heat transport due to alkylthiols is |
697 |
highly efficient. Combining our observations with those of Zhang {\it |
698 |
et al.}, it appears that butanethiol acts as a channel to expedite |
699 |
heat flow from the gold surface and into the alkyl chain. The |
700 |
vibrational coupling between the metal and the liquid phase can |
701 |
therefore be enhanced with the presence of suitable capping agents. |
702 |
|
703 |
Deuterated models in the UA force field did not decouple the thermal |
704 |
transport as well as in the AA force field. The UA models, even |
705 |
though they have eliminated the high frequency C-H vibrational |
706 |
overlap, still have significant overlap in the lower-frequency |
707 |
portions of the infrared spectra (Figure \ref{uahxnua}). Deuterating |
708 |
the UA models did not decouple the low frequency region enough to |
709 |
produce an observable difference for the results of $G$ (Table |
710 |
\ref{modelTest}). |
711 |
|
712 |
\begin{figure} |
713 |
\includegraphics[width=\linewidth]{uahxnua} |
714 |
\caption{Vibrational power spectra for UA models for the butanethiol |
715 |
and hexane solvent (upper panel) show the high degree of overlap |
716 |
between these two molecules, particularly at lower frequencies. |
717 |
Deuterating a UA model for the solvent (lower panel) does not |
718 |
decouple the two spectra to the same degree as in the AA force |
719 |
field (see Fig \ref{aahxntln}).} |
720 |
\label{uahxnua} |
721 |
\end{figure} |
722 |
|
723 |
\section{Conclusions} |
724 |
The NIVS algorithm has been applied to simulations of |
725 |
butanethiol-capped Au(111) surfaces in the presence of organic |
726 |
solvents. This algorithm allows the application of unphysical thermal |
727 |
flux to transfer heat between the metal and the liquid phase. With the |
728 |
flux applied, we were able to measure the corresponding thermal |
729 |
gradients and to obtain interfacial thermal conductivities. Under |
730 |
steady states, 2-3 ns trajectory simulations are sufficient for |
731 |
computation of this quantity. |
732 |
|
733 |
Our simulations have seen significant conductance enhancement in the |
734 |
presence of capping agent, compared with the bare gold / liquid |
735 |
interfaces. The vibrational coupling between the metal and the liquid |
736 |
phase is enhanced by a chemically-bonded capping agent. Furthermore, |
737 |
the coverage percentage of the capping agent plays an important role |
738 |
in the interfacial thermal transport process. Moderately low coverages |
739 |
allow higher contact between capping agent and solvent, and thus could |
740 |
further enhance the heat transfer process, giving a non-monotonic |
741 |
behavior of conductance with increasing coverage. |
742 |
|
743 |
Our results, particularly using the UA models, agree well with |
744 |
available experimental data. The AA models tend to overestimate the |
745 |
interfacial thermal conductance in that the classically treated C-H |
746 |
vibrations become too easily populated. Compared to the AA models, the |
747 |
UA models have higher computational efficiency with satisfactory |
748 |
accuracy, and thus are preferable in modeling interfacial thermal |
749 |
transport. |
750 |
|
751 |
Of the two definitions for $G$, the discrete form |
752 |
(Eq. \ref{discreteG}) was easier to use and gives out relatively |
753 |
consistent results, while the derivative form (Eq. \ref{derivativeG}) |
754 |
is not as versatile. Although $G^\prime$ gives out comparable results |
755 |
and follows similar trend with $G$ when measuring close to fully |
756 |
covered or bare surfaces, the spatial resolution of $T$ profile |
757 |
required for the use of a derivative form is limited by the number of |
758 |
bins and the sampling required to obtain thermal gradient information. |
759 |
|
760 |
Vlugt {\it et al.} have investigated the surface thiol structures for |
761 |
nanocrystalline gold and pointed out that they differ from those of |
762 |
the Au(111) surface.\cite{landman:1998,vlugt:cpc2007154} This |
763 |
difference could also cause differences in the interfacial thermal |
764 |
transport behavior. To investigate this problem, one would need an |
765 |
effective method for applying thermal gradients in non-planar |
766 |
(i.e. spherical) geometries. |
767 |
|
768 |
\section{Acknowledgments} |
769 |
Support for this project was provided by the National Science |
770 |
Foundation under grant CHE-0848243. Computational time was provided by |
771 |
the Center for Research Computing (CRC) at the University of Notre |
772 |
Dame. |
773 |
|
774 |
\newpage |
775 |
|
776 |
\bibliography{stokes} |
777 |
|
778 |
\end{doublespace} |
779 |
\end{document} |
780 |
|