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. |
195 |
|
196 |
By implementing these operations at a certain frequency, a steady |
197 |
thermal and/or momentum flux can be applied and the corresponding |
198 |
temperature and/or momentum gradients can be established. |
199 |
|
200 |
This approach is more computationaly efficient compared to the |
201 |
previous NIVS method, in that only quadratic equations are involved, |
202 |
while the NIVS method needs to solve a quartic equations. Furthermore, |
203 |
the method implements isotropic scaling of velocities in respective |
204 |
slabs, unlike the NIVS, where an extra criteria function is necessary |
205 |
to choose a set of coefficients that performs the most isotropic |
206 |
scaling. More importantly, separating the momentum flux imposing from |
207 |
velocity scaling avoids the underlying cause that NIVS produced |
208 |
thermal anisotropy when applying a momentum flux. |
209 |
%NEW METHOD DOESN'T CAUSE UNDESIRED CONCOMITENT MOMENTUM FLUX WHEN |
210 |
%IMPOSING A THERMAL FLUX |
211 |
|
212 |
Figure \ref{method} illustrates the advantages of the approach over |
213 |
the original momentum swapping approach. For simplification purpose, |
214 |
this figure only shows one dimension (e.g. $x$) of the $h$ and $c$ |
215 |
slabs. As one can see, the operations used in this approach have the |
216 |
nature to preserve a Gaussian distribution. However, the momentum |
217 |
swapping tends to render a nonthermal distribution. Therefore, when |
218 |
the imposed flux is relatively large, diffusion of the neighboring |
219 |
slabs could no longer remedy this effect, and nonthermal distributions |
220 |
would be observed. |
221 |
|
222 |
\begin{figure} |
223 |
\includegraphics[width=\linewidth]{method} |
224 |
\caption{.} |
225 |
\label{method} |
226 |
\end{figure} |
227 |
|
228 |
\section{Computational Details} |
229 |
\subsection{Simulation Protocol} |
230 |
The NIVS algorithm has been implemented in our MD simulation code, |
231 |
OpenMD\cite{Meineke:2005gd,openmd}, and was used for our simulations. |
232 |
Metal slabs of 6 or 11 layers of Au atoms were first equilibrated |
233 |
under atmospheric pressure (1 atm) and 200K. After equilibration, |
234 |
butanethiol capping agents were placed at three-fold hollow sites on |
235 |
the Au(111) surfaces. These sites are either {\it fcc} or {\it |
236 |
hcp} sites, although Hase {\it et al.} found that they are |
237 |
equivalent in a heat transfer process,\cite{hase:2010} so we did not |
238 |
distinguish between these sites in our study. The maximum butanethiol |
239 |
capacity on Au surface is $1/3$ of the total number of surface Au |
240 |
atoms, and the packing forms a $(\sqrt{3}\times\sqrt{3})R30^\circ$ |
241 |
structure\cite{doi:10.1021/ja00008a001,doi:10.1021/cr9801317}. A |
242 |
series of lower coverages was also prepared by eliminating |
243 |
butanethiols from the higher coverage surface in a regular manner. The |
244 |
lower coverages were prepared in order to study the relation between |
245 |
coverage and interfacial conductance. |
246 |
|
247 |
The capping agent molecules were allowed to migrate during the |
248 |
simulations. They distributed themselves uniformly and sampled a |
249 |
number of three-fold sites throughout out study. Therefore, the |
250 |
initial configuration does not noticeably affect the sampling of a |
251 |
variety of configurations of the same coverage, and the final |
252 |
conductance measurement would be an average effect of these |
253 |
configurations explored in the simulations. |
254 |
|
255 |
After the modified Au-butanethiol surface systems were equilibrated in |
256 |
the canonical (NVT) ensemble, organic solvent molecules were packed in |
257 |
the previously empty part of the simulation cells.\cite{packmol} Two |
258 |
solvents were investigated, one which has little vibrational overlap |
259 |
with the alkanethiol and which has a planar shape (toluene), and one |
260 |
which has similar vibrational frequencies to the capping agent and |
261 |
chain-like shape ({\it n}-hexane). |
262 |
|
263 |
The simulation cells were not particularly extensive along the |
264 |
$z$-axis, as a very long length scale for the thermal gradient may |
265 |
cause excessively hot or cold temperatures in the middle of the |
266 |
solvent region and lead to undesired phenomena such as solvent boiling |
267 |
or freezing when a thermal flux is applied. Conversely, too few |
268 |
solvent molecules would change the normal behavior of the liquid |
269 |
phase. Therefore, our $N_{solvent}$ values were chosen to ensure that |
270 |
these extreme cases did not happen to our simulations. The spacing |
271 |
between periodic images of the gold interfaces is $45 \sim 75$\AA in |
272 |
our simulations. |
273 |
|
274 |
The initial configurations generated are further equilibrated with the |
275 |
$x$ and $y$ dimensions fixed, only allowing the $z$-length scale to |
276 |
change. This is to ensure that the equilibration of liquid phase does |
277 |
not affect the metal's crystalline structure. Comparisons were made |
278 |
with simulations that allowed changes of $L_x$ and $L_y$ during NPT |
279 |
equilibration. No substantial changes in the box geometry were noticed |
280 |
in these simulations. After ensuring the liquid phase reaches |
281 |
equilibrium at atmospheric pressure (1 atm), further equilibration was |
282 |
carried out under canonical (NVT) and microcanonical (NVE) ensembles. |
283 |
|
284 |
After the systems reach equilibrium, NIVS was used to impose an |
285 |
unphysical thermal flux between the metal and the liquid phases. Most |
286 |
of our simulations were done under an average temperature of |
287 |
$\sim$200K. Therefore, thermal flux usually came from the metal to the |
288 |
liquid so that the liquid has a higher temperature and would not |
289 |
freeze due to lowered temperatures. After this induced temperature |
290 |
gradient had stabilized, the temperature profile of the simulation cell |
291 |
was recorded. To do this, the simulation cell is divided evenly into |
292 |
$N$ slabs along the $z$-axis. The average temperatures of each slab |
293 |
are recorded for 1$\sim$2 ns. When the slab width $d$ of each slab is |
294 |
the same, the derivatives of $T$ with respect to slab number $n$ can |
295 |
be directly used for $G^\prime$ calculations: \begin{equation} |
296 |
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
297 |
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
298 |
= |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big| |
299 |
\Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2 |
300 |
= |J_z|\Big|\frac{\partial^2 T}{\partial n^2}\Big| |
301 |
\Big/\left(\frac{\partial T}{\partial n}\right)^2 |
302 |
\label{derivativeG2} |
303 |
\end{equation} |
304 |
The absolute values in Eq. \ref{derivativeG2} appear because the |
305 |
direction of the flux $\vec{J}$ is in an opposing direction on either |
306 |
side of the metal slab. |
307 |
|
308 |
All of the above simulation procedures use a time step of 1 fs. Each |
309 |
equilibration stage took a minimum of 100 ps, although in some cases, |
310 |
longer equilibration stages were utilized. |
311 |
|
312 |
\subsection{Force Field Parameters} |
313 |
Our simulations include a number of chemically distinct components. |
314 |
Figure \ref{demoMol} demonstrates the sites defined for both |
315 |
United-Atom and All-Atom models of the organic solvent and capping |
316 |
agents in our simulations. Force field parameters are needed for |
317 |
interactions both between the same type of particles and between |
318 |
particles of different species. |
319 |
|
320 |
\begin{figure} |
321 |
\includegraphics[width=\linewidth]{structures} |
322 |
\caption{Structures of the capping agent and solvents utilized in |
323 |
these simulations. The chemically-distinct sites (a-e) are expanded |
324 |
in terms of constituent atoms for both United Atom (UA) and All Atom |
325 |
(AA) force fields. Most parameters are from References |
326 |
\protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes,TraPPE-UA.thiols} |
327 |
(UA) and \protect\cite{OPLSAA} (AA). Cross-interactions with the Au |
328 |
atoms are given in Table 1 in the supporting information.} |
329 |
\label{demoMol} |
330 |
\end{figure} |
331 |
|
332 |
The Au-Au interactions in metal lattice slab is described by the |
333 |
quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
334 |
potentials include zero-point quantum corrections and are |
335 |
reparametrized for accurate surface energies compared to the |
336 |
Sutton-Chen potentials.\cite{Chen90} |
337 |
|
338 |
For the two solvent molecules, {\it n}-hexane and toluene, two |
339 |
different atomistic models were utilized. Both solvents were modeled |
340 |
using United-Atom (UA) and All-Atom (AA) models. The TraPPE-UA |
341 |
parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
342 |
for our UA solvent molecules. In these models, sites are located at |
343 |
the carbon centers for alkyl groups. Bonding interactions, including |
344 |
bond stretches and bends and torsions, were used for intra-molecular |
345 |
sites closer than 3 bonds. For non-bonded interactions, Lennard-Jones |
346 |
potentials are used. |
347 |
|
348 |
By eliminating explicit hydrogen atoms, the TraPPE-UA models are |
349 |
simple and computationally efficient, while maintaining good accuracy. |
350 |
However, the TraPPE-UA model for alkanes is known to predict a slightly |
351 |
lower boiling point than experimental values. This is one of the |
352 |
reasons we used a lower average temperature (200K) for our |
353 |
simulations. If heat is transferred to the liquid phase during the |
354 |
NIVS simulation, the liquid in the hot slab can actually be |
355 |
substantially warmer than the mean temperature in the simulation. The |
356 |
lower mean temperatures therefore prevent solvent boiling. |
357 |
|
358 |
For UA-toluene, the non-bonded potentials between intermolecular sites |
359 |
have a similar Lennard-Jones formulation. The toluene molecules were |
360 |
treated as a single rigid body, so there was no need for |
361 |
intramolecular interactions (including bonds, bends, or torsions) in |
362 |
this solvent model. |
363 |
|
364 |
Besides the TraPPE-UA models, AA models for both organic solvents are |
365 |
included in our studies as well. The OPLS-AA\cite{OPLSAA} force fields |
366 |
were used. For hexane, additional explicit hydrogen sites were |
367 |
included. Besides bonding and non-bonded site-site interactions, |
368 |
partial charges and the electrostatic interactions were added to each |
369 |
CT and HC site. For toluene, a flexible model for the toluene molecule |
370 |
was utilized which included bond, bend, torsion, and inversion |
371 |
potentials to enforce ring planarity. |
372 |
|
373 |
The butanethiol capping agent in our simulations, were also modeled |
374 |
with both UA and AA model. The TraPPE-UA force field includes |
375 |
parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for |
376 |
UA butanethiol model in our simulations. The OPLS-AA also provides |
377 |
parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) |
378 |
surfaces do not have the hydrogen atom bonded to sulfur. To derive |
379 |
suitable parameters for butanethiol adsorbed on Au(111) surfaces, we |
380 |
adopt the S parameters from Luedtke and Landman\cite{landman:1998} and |
381 |
modify the parameters for the CTS atom to maintain charge neutrality |
382 |
in the molecule. Note that the model choice (UA or AA) for the capping |
383 |
agent can be different from the solvent. Regardless of model choice, |
384 |
the force field parameters for interactions between capping agent and |
385 |
solvent can be derived using Lorentz-Berthelot Mixing Rule: |
386 |
\begin{eqnarray} |
387 |
\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\ |
388 |
\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}} |
389 |
\end{eqnarray} |
390 |
|
391 |
To describe the interactions between metal (Au) and non-metal atoms, |
392 |
we refer to an adsorption study of alkyl thiols on gold surfaces by |
393 |
Vlugt {\it et al.}\cite{vlugt:cpc2007154} They fitted an effective |
394 |
Lennard-Jones form of potential parameters for the interaction between |
395 |
Au and pseudo-atoms CH$_x$ and S based on a well-established and |
396 |
widely-used effective potential of Hautman and Klein for the Au(111) |
397 |
surface.\cite{hautman:4994} As our simulations require the gold slab |
398 |
to be flexible to accommodate thermal excitation, the pair-wise form |
399 |
of potentials they developed was used for our study. |
400 |
|
401 |
The potentials developed from {\it ab initio} calculations by Leng |
402 |
{\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
403 |
interactions between Au and aromatic C/H atoms in toluene. However, |
404 |
the Lennard-Jones parameters between Au and other types of particles, |
405 |
(e.g. AA alkanes) have not yet been established. For these |
406 |
interactions, the Lorentz-Berthelot mixing rule can be used to derive |
407 |
effective single-atom LJ parameters for the metal using the fit values |
408 |
for toluene. These are then used to construct reasonable mixing |
409 |
parameters for the interactions between the gold and other atoms. |
410 |
Table 1 in the supporting information summarizes the |
411 |
``metal/non-metal'' parameters utilized in our simulations. |
412 |
|
413 |
\section{Results} |
414 |
[L-J COMPARED TO RENMD NIVS; WATER COMPARED TO RNEMD NIVS; |
415 |
SLIP BOUNDARY VS STICK BOUNDARY; ICE-WATER INTERFACES] |
416 |
|
417 |
There are many factors contributing to the measured interfacial |
418 |
conductance; some of these factors are physically motivated |
419 |
(e.g. coverage of the surface by the capping agent coverage and |
420 |
solvent identity), while some are governed by parameters of the |
421 |
methodology (e.g. applied flux and the formulas used to obtain the |
422 |
conductance). In this section we discuss the major physical and |
423 |
calculational effects on the computed conductivity. |
424 |
|
425 |
\subsection{Effects due to capping agent coverage} |
426 |
|
427 |
A series of different initial conditions with a range of surface |
428 |
coverages was prepared and solvated with various with both of the |
429 |
solvent molecules. These systems were then equilibrated and their |
430 |
interfacial thermal conductivity was measured with the NIVS |
431 |
algorithm. Figure \ref{coverage} demonstrates the trend of conductance |
432 |
with respect to surface coverage. |
433 |
|
434 |
\begin{figure} |
435 |
\includegraphics[width=\linewidth]{coverage} |
436 |
\caption{The interfacial thermal conductivity ($G$) has a |
437 |
non-monotonic dependence on the degree of surface capping. This |
438 |
data is for the Au(111) / butanethiol / solvent interface with |
439 |
various UA force fields at $\langle T\rangle \sim $200K.} |
440 |
\label{coverage} |
441 |
\end{figure} |
442 |
|
443 |
In partially covered surfaces, the derivative definition for |
444 |
$G^\prime$ (Eq. \ref{derivativeG}) becomes difficult to apply, as the |
445 |
location of maximum change of $\lambda$ becomes washed out. The |
446 |
discrete definition (Eq. \ref{discreteG}) is easier to apply, as the |
447 |
Gibbs dividing surface is still well-defined. Therefore, $G$ (not |
448 |
$G^\prime$) was used in this section. |
449 |
|
450 |
From Figure \ref{coverage}, one can see the significance of the |
451 |
presence of capping agents. When even a small fraction of the Au(111) |
452 |
surface sites are covered with butanethiols, the conductivity exhibits |
453 |
an enhancement by at least a factor of 3. Capping agents are clearly |
454 |
playing a major role in thermal transport at metal / organic solvent |
455 |
surfaces. |
456 |
|
457 |
We note a non-monotonic behavior in the interfacial conductance as a |
458 |
function of surface coverage. The maximum conductance (largest $G$) |
459 |
happens when the surfaces are about 75\% covered with butanethiol |
460 |
caps. The reason for this behavior is not entirely clear. One |
461 |
explanation is that incomplete butanethiol coverage allows small gaps |
462 |
between butanethiols to form. These gaps can be filled by transient |
463 |
solvent molecules. These solvent molecules couple very strongly with |
464 |
the hot capping agent molecules near the surface, and can then carry |
465 |
away (diffusively) the excess thermal energy from the surface. |
466 |
|
467 |
There appears to be a competition between the conduction of the |
468 |
thermal energy away from the surface by the capping agents (enhanced |
469 |
by greater coverage) and the coupling of the capping agents with the |
470 |
solvent (enhanced by interdigitation at lower coverages). This |
471 |
competition would lead to the non-monotonic coverage behavior observed |
472 |
here. |
473 |
|
474 |
Results for rigid body toluene solvent, as well as the UA hexane, are |
475 |
within the ranges expected from prior experimental |
476 |
work.\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406} This suggests |
477 |
that explicit hydrogen atoms might not be required for modeling |
478 |
thermal transport in these systems. C-H vibrational modes do not see |
479 |
significant excited state population at low temperatures, and are not |
480 |
likely to carry lower frequency excitations from the solid layer into |
481 |
the bulk liquid. |
482 |
|
483 |
The toluene solvent does not exhibit the same behavior as hexane in |
484 |
that $G$ remains at approximately the same magnitude when the capping |
485 |
coverage increases from 25\% to 75\%. Toluene, as a rigid planar |
486 |
molecule, cannot occupy the relatively small gaps between the capping |
487 |
agents as easily as the chain-like {\it n}-hexane. The effect of |
488 |
solvent coupling to the capping agent is therefore weaker in toluene |
489 |
except at the very lowest coverage levels. This effect counters the |
490 |
coverage-dependent conduction of heat away from the metal surface, |
491 |
leading to a much flatter $G$ vs. coverage trend than is observed in |
492 |
{\it n}-hexane. |
493 |
|
494 |
\subsection{Effects due to Solvent \& Solvent Models} |
495 |
In addition to UA solvent and capping agent models, AA models have |
496 |
also been included in our simulations. In most of this work, the same |
497 |
(UA or AA) model for solvent and capping agent was used, but it is |
498 |
also possible to utilize different models for different components. |
499 |
We have also included isotopic substitutions (Hydrogen to Deuterium) |
500 |
to decrease the explicit vibrational overlap between solvent and |
501 |
capping agent. Table \ref{modelTest} summarizes the results of these |
502 |
studies. |
503 |
|
504 |
\begin{table*} |
505 |
\begin{minipage}{\linewidth} |
506 |
\begin{center} |
507 |
|
508 |
\caption{Computed interfacial thermal conductance ($G$ and |
509 |
$G^\prime$) values for interfaces using various models for |
510 |
solvent and capping agent (or without capping agent) at |
511 |
$\langle T\rangle\sim$200K. Here ``D'' stands for deuterated |
512 |
solvent or capping agent molecules. Error estimates are |
513 |
indicated in parentheses.} |
514 |
|
515 |
\begin{tabular}{llccc} |
516 |
\hline\hline |
517 |
Butanethiol model & Solvent & $G$ & $G^\prime$ \\ |
518 |
(or bare surface) & model & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
519 |
\hline |
520 |
UA & UA hexane & 131(9) & 87(10) \\ |
521 |
& UA hexane(D) & 153(5) & 136(13) \\ |
522 |
& AA hexane & 131(6) & 122(10) \\ |
523 |
& UA toluene & 187(16) & 151(11) \\ |
524 |
& AA toluene & 200(36) & 149(53) \\ |
525 |
\hline |
526 |
AA & UA hexane & 116(9) & 129(8) \\ |
527 |
& AA hexane & 442(14) & 356(31) \\ |
528 |
& AA hexane(D) & 222(12) & 234(54) \\ |
529 |
& UA toluene & 125(25) & 97(60) \\ |
530 |
& AA toluene & 487(56) & 290(42) \\ |
531 |
\hline |
532 |
AA(D) & UA hexane & 158(25) & 172(4) \\ |
533 |
& AA hexane & 243(29) & 191(11) \\ |
534 |
& AA toluene & 364(36) & 322(67) \\ |
535 |
\hline |
536 |
bare & UA hexane & 46.5(3.2) & 49.4(4.5) \\ |
537 |
& UA hexane(D) & 43.9(4.6) & 43.0(2.0) \\ |
538 |
& AA hexane & 31.0(1.4) & 29.4(1.3) \\ |
539 |
& UA toluene & 70.1(1.3) & 65.8(0.5) \\ |
540 |
\hline\hline |
541 |
\end{tabular} |
542 |
\label{modelTest} |
543 |
\end{center} |
544 |
\end{minipage} |
545 |
\end{table*} |
546 |
|
547 |
To facilitate direct comparison between force fields, systems with the |
548 |
same capping agent and solvent were prepared with the same length |
549 |
scales for the simulation cells. |
550 |
|
551 |
On bare metal / solvent surfaces, different force field models for |
552 |
hexane yield similar results for both $G$ and $G^\prime$, and these |
553 |
two definitions agree with each other very well. This is primarily an |
554 |
indicator of weak interactions between the metal and the solvent. |
555 |
|
556 |
For the fully-covered surfaces, the choice of force field for the |
557 |
capping agent and solvent has a large impact on the calculated values |
558 |
of $G$ and $G^\prime$. The AA thiol to AA solvent conductivities are |
559 |
much larger than their UA to UA counterparts, and these values exceed |
560 |
the experimental estimates by a large measure. The AA force field |
561 |
allows significant energy to go into C-H (or C-D) stretching modes, |
562 |
and since these modes are high frequency, this non-quantum behavior is |
563 |
likely responsible for the overestimate of the conductivity. Compared |
564 |
to the AA model, the UA model yields more reasonable conductivity |
565 |
values with much higher computational efficiency. |
566 |
|
567 |
\subsubsection{Are electronic excitations in the metal important?} |
568 |
Because they lack electronic excitations, the QSC and related embedded |
569 |
atom method (EAM) models for gold are known to predict unreasonably |
570 |
low values for bulk conductivity |
571 |
($\lambda$).\cite{kuang:164101,ISI:000207079300006,Clancy:1992} If the |
572 |
conductance between the phases ($G$) is governed primarily by phonon |
573 |
excitation (and not electronic degrees of freedom), one would expect a |
574 |
classical model to capture most of the interfacial thermal |
575 |
conductance. Our results for $G$ and $G^\prime$ indicate that this is |
576 |
indeed the case, and suggest that the modeling of interfacial thermal |
577 |
transport depends primarily on the description of the interactions |
578 |
between the various components at the interface. When the metal is |
579 |
chemically capped, the primary barrier to thermal conductivity appears |
580 |
to be the interface between the capping agent and the surrounding |
581 |
solvent, so the excitations in the metal have little impact on the |
582 |
value of $G$. |
583 |
|
584 |
\subsection{Effects due to methodology and simulation parameters} |
585 |
|
586 |
We have varied the parameters of the simulations in order to |
587 |
investigate how these factors would affect the computation of $G$. Of |
588 |
particular interest are: 1) the length scale for the applied thermal |
589 |
gradient (modified by increasing the amount of solvent in the system), |
590 |
2) the sign and magnitude of the applied thermal flux, 3) the average |
591 |
temperature of the simulation (which alters the solvent density during |
592 |
equilibration), and 4) the definition of the interfacial conductance |
593 |
(Eqs. (\ref{discreteG}) or (\ref{derivativeG})) used in the |
594 |
calculation. |
595 |
|
596 |
Systems of different lengths were prepared by altering the number of |
597 |
solvent molecules and extending the length of the box along the $z$ |
598 |
axis to accomodate the extra solvent. Equilibration at the same |
599 |
temperature and pressure conditions led to nearly identical surface |
600 |
areas ($L_x$ and $L_y$) available to the metal and capping agent, |
601 |
while the extra solvent served mainly to lengthen the axis that was |
602 |
used to apply the thermal flux. For a given value of the applied |
603 |
flux, the different $z$ length scale has only a weak effect on the |
604 |
computed conductivities. |
605 |
|
606 |
\subsubsection{Effects of applied flux} |
607 |
The NIVS algorithm allows changes in both the sign and magnitude of |
608 |
the applied flux. It is possible to reverse the direction of heat |
609 |
flow simply by changing the sign of the flux, and thermal gradients |
610 |
which would be difficult to obtain experimentally ($5$ K/\AA) can be |
611 |
easily simulated. However, the magnitude of the applied flux is not |
612 |
arbitrary if one aims to obtain a stable and reliable thermal gradient. |
613 |
A temperature gradient can be lost in the noise if $|J_z|$ is too |
614 |
small, and excessive $|J_z|$ values can cause phase transitions if the |
615 |
extremes of the simulation cell become widely separated in |
616 |
temperature. Also, if $|J_z|$ is too large for the bulk conductivity |
617 |
of the materials, the thermal gradient will never reach a stable |
618 |
state. |
619 |
|
620 |
Within a reasonable range of $J_z$ values, we were able to study how |
621 |
$G$ changes as a function of this flux. In what follows, we use |
622 |
positive $J_z$ values to denote the case where energy is being |
623 |
transferred by the method from the metal phase and into the liquid. |
624 |
The resulting gradient therefore has a higher temperature in the |
625 |
liquid phase. Negative flux values reverse this transfer, and result |
626 |
in higher temperature metal phases. The conductance measured under |
627 |
different applied $J_z$ values is listed in Tables 2 and 3 in the |
628 |
supporting information. These results do not indicate that $G$ depends |
629 |
strongly on $J_z$ within this flux range. The linear response of flux |
630 |
to thermal gradient simplifies our investigations in that we can rely |
631 |
on $G$ measurement with only a small number $J_z$ values. |
632 |
|
633 |
The sign of $J_z$ is a different matter, however, as this can alter |
634 |
the temperature on the two sides of the interface. The average |
635 |
temperature values reported are for the entire system, and not for the |
636 |
liquid phase, so at a given $\langle T \rangle$, the system with |
637 |
positive $J_z$ has a warmer liquid phase. This means that if the |
638 |
liquid carries thermal energy via diffusive transport, {\it positive} |
639 |
$J_z$ values will result in increased molecular motion on the liquid |
640 |
side of the interface, and this will increase the measured |
641 |
conductivity. |
642 |
|
643 |
\subsubsection{Effects due to average temperature} |
644 |
|
645 |
We also studied the effect of average system temperature on the |
646 |
interfacial conductance. The simulations are first equilibrated in |
647 |
the NPT ensemble at 1 atm. The TraPPE-UA model for hexane tends to |
648 |
predict a lower boiling point (and liquid state density) than |
649 |
experiments. This lower-density liquid phase leads to reduced contact |
650 |
between the hexane and butanethiol, and this accounts for our |
651 |
observation of lower conductance at higher temperatures. In raising |
652 |
the average temperature from 200K to 250K, the density drop of |
653 |
$\sim$20\% in the solvent phase leads to a $\sim$40\% drop in the |
654 |
conductance. |
655 |
|
656 |
Similar behavior is observed in the TraPPE-UA model for toluene, |
657 |
although this model has better agreement with the experimental |
658 |
densities of toluene. The expansion of the toluene liquid phase is |
659 |
not as significant as that of the hexane (8.3\% over 100K), and this |
660 |
limits the effect to $\sim$20\% drop in thermal conductivity. |
661 |
|
662 |
Although we have not mapped out the behavior at a large number of |
663 |
temperatures, is clear that there will be a strong temperature |
664 |
dependence in the interfacial conductance when the physical properties |
665 |
of one side of the interface (notably the density) change rapidly as a |
666 |
function of temperature. |
667 |
|
668 |
Besides the lower interfacial thermal conductance, surfaces at |
669 |
relatively high temperatures are susceptible to reconstructions, |
670 |
particularly when butanethiols fully cover the Au(111) surface. These |
671 |
reconstructions include surface Au atoms which migrate outward to the |
672 |
S atom layer, and butanethiol molecules which embed into the surface |
673 |
Au layer. The driving force for this behavior is the strong Au-S |
674 |
interactions which are modeled here with a deep Lennard-Jones |
675 |
potential. This phenomenon agrees with reconstructions that have been |
676 |
experimentally |
677 |
observed.\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt |
678 |
{\it et al.} kept their Au(111) slab rigid so that their simulations |
679 |
could reach 300K without surface |
680 |
reconstructions.\cite{vlugt:cpc2007154} Since surface reconstructions |
681 |
blur the interface, the measurement of $G$ becomes more difficult to |
682 |
conduct at higher temperatures. For this reason, most of our |
683 |
measurements are undertaken at $\langle T\rangle\sim$200K where |
684 |
reconstruction is minimized. |
685 |
|
686 |
However, when the surface is not completely covered by butanethiols, |
687 |
the simulated system appears to be more resistent to the |
688 |
reconstruction. Our Au / butanethiol / toluene system had the Au(111) |
689 |
surfaces 90\% covered by butanethiols, but did not see this above |
690 |
phenomena even at $\langle T\rangle\sim$300K. That said, we did |
691 |
observe butanethiols migrating to neighboring three-fold sites during |
692 |
a simulation. Since the interface persisted in these simulations, we |
693 |
were able to obtain $G$'s for these interfaces even at a relatively |
694 |
high temperature without being affected by surface reconstructions. |
695 |
|
696 |
\section{Discussion} |
697 |
[COMBINE W. RESULTS] |
698 |
The primary result of this work is that the capping agent acts as an |
699 |
efficient thermal coupler between solid and solvent phases. One of |
700 |
the ways the capping agent can carry out this role is to down-shift |
701 |
between the phonon vibrations in the solid (which carry the heat from |
702 |
the gold) and the molecular vibrations in the liquid (which carry some |
703 |
of the heat in the solvent). |
704 |
|
705 |
To investigate the mechanism of interfacial thermal conductance, the |
706 |
vibrational power spectrum was computed. Power spectra were taken for |
707 |
individual components in different simulations. To obtain these |
708 |
spectra, simulations were run after equilibration in the |
709 |
microcanonical (NVE) ensemble and without a thermal |
710 |
gradient. Snapshots of configurations were collected at a frequency |
711 |
that is higher than that of the fastest vibrations occurring in the |
712 |
simulations. With these configurations, the velocity auto-correlation |
713 |
functions can be computed: |
714 |
\begin{equation} |
715 |
C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle |
716 |
\label{vCorr} |
717 |
\end{equation} |
718 |
The power spectrum is constructed via a Fourier transform of the |
719 |
symmetrized velocity autocorrelation function, |
720 |
\begin{equation} |
721 |
\hat{f}(\omega) = |
722 |
\int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt |
723 |
\label{fourier} |
724 |
\end{equation} |
725 |
|
726 |
\subsection{The role of specific vibrations} |
727 |
The vibrational spectra for gold slabs in different environments are |
728 |
shown as in Figure \ref{specAu}. Regardless of the presence of |
729 |
solvent, the gold surfaces which are covered by butanethiol molecules |
730 |
exhibit an additional peak observed at a frequency of |
731 |
$\sim$165cm$^{-1}$. We attribute this peak to the S-Au bonding |
732 |
vibration. This vibration enables efficient thermal coupling of the |
733 |
surface Au layer to the capping agents. Therefore, in our simulations, |
734 |
the Au / S interfaces do not appear to be the primary barrier to |
735 |
thermal transport when compared with the butanethiol / solvent |
736 |
interfaces. This supports the results of Luo {\it et |
737 |
al.}\cite{Luo20101}, who reported $G$ for Au-SAM junctions roughly |
738 |
twice as large as what we have computed for the thiol-liquid |
739 |
interfaces. |
740 |
|
741 |
\begin{figure} |
742 |
\includegraphics[width=\linewidth]{vibration} |
743 |
\caption{The vibrational power spectrum for thiol-capped gold has an |
744 |
additional vibrational peak at $\sim $165cm$^{-1}$. Bare gold |
745 |
surfaces (both with and without a solvent over-layer) are missing |
746 |
this peak. A similar peak at $\sim $165cm$^{-1}$ also appears in |
747 |
the vibrational power spectrum for the butanethiol capping agents.} |
748 |
\label{specAu} |
749 |
\end{figure} |
750 |
|
751 |
Also in this figure, we show the vibrational power spectrum for the |
752 |
bound butanethiol molecules, which also exhibits the same |
753 |
$\sim$165cm$^{-1}$ peak. |
754 |
|
755 |
\subsection{Overlap of power spectra} |
756 |
A comparison of the results obtained from the two different organic |
757 |
solvents can also provide useful information of the interfacial |
758 |
thermal transport process. In particular, the vibrational overlap |
759 |
between the butanethiol and the organic solvents suggests a highly |
760 |
efficient thermal exchange between these components. Very high |
761 |
thermal conductivity was observed when AA models were used and C-H |
762 |
vibrations were treated classically. The presence of extra degrees of |
763 |
freedom in the AA force field yields higher heat exchange rates |
764 |
between the two phases and results in a much higher conductivity than |
765 |
in the UA force field. The all-atom classical models include high |
766 |
frequency modes which should be unpopulated at our relatively low |
767 |
temperatures. This artifact is likely the cause of the high thermal |
768 |
conductance in all-atom MD simulations. |
769 |
|
770 |
The similarity in the vibrational modes available to solvent and |
771 |
capping agent can be reduced by deuterating one of the two components |
772 |
(Fig. \ref{aahxntln}). Once either the hexanes or the butanethiols |
773 |
are deuterated, one can observe a significantly lower $G$ and |
774 |
$G^\prime$ values (Table \ref{modelTest}). |
775 |
|
776 |
\begin{figure} |
777 |
\includegraphics[width=\linewidth]{aahxntln} |
778 |
\caption{Spectra obtained for all-atom (AA) Au / butanethiol / solvent |
779 |
systems. When butanethiol is deuterated (lower left), its |
780 |
vibrational overlap with hexane decreases significantly. Since |
781 |
aromatic molecules and the butanethiol are vibrationally dissimilar, |
782 |
the change is not as dramatic when toluene is the solvent (right).} |
783 |
\label{aahxntln} |
784 |
\end{figure} |
785 |
|
786 |
For the Au / butanethiol / toluene interfaces, having the AA |
787 |
butanethiol deuterated did not yield a significant change in the |
788 |
measured conductance. Compared to the C-H vibrational overlap between |
789 |
hexane and butanethiol, both of which have alkyl chains, the overlap |
790 |
between toluene and butanethiol is not as significant and thus does |
791 |
not contribute as much to the heat exchange process. |
792 |
|
793 |
Previous observations of Zhang {\it et al.}\cite{hase:2010} indicate |
794 |
that the {\it intra}molecular heat transport due to alkylthiols is |
795 |
highly efficient. Combining our observations with those of Zhang {\it |
796 |
et al.}, it appears that butanethiol acts as a channel to expedite |
797 |
heat flow from the gold surface and into the alkyl chain. The |
798 |
vibrational coupling between the metal and the liquid phase can |
799 |
therefore be enhanced with the presence of suitable capping agents. |
800 |
|
801 |
Deuterated models in the UA force field did not decouple the thermal |
802 |
transport as well as in the AA force field. The UA models, even |
803 |
though they have eliminated the high frequency C-H vibrational |
804 |
overlap, still have significant overlap in the lower-frequency |
805 |
portions of the infrared spectra (Figure \ref{uahxnua}). Deuterating |
806 |
the UA models did not decouple the low frequency region enough to |
807 |
produce an observable difference for the results of $G$ (Table |
808 |
\ref{modelTest}). |
809 |
|
810 |
\begin{figure} |
811 |
\includegraphics[width=\linewidth]{uahxnua} |
812 |
\caption{Vibrational power spectra for UA models for the butanethiol |
813 |
and hexane solvent (upper panel) show the high degree of overlap |
814 |
between these two molecules, particularly at lower frequencies. |
815 |
Deuterating a UA model for the solvent (lower panel) does not |
816 |
decouple the two spectra to the same degree as in the AA force |
817 |
field (see Fig \ref{aahxntln}).} |
818 |
\label{uahxnua} |
819 |
\end{figure} |
820 |
|
821 |
\section{Conclusions} |
822 |
The NIVS algorithm has been applied to simulations of |
823 |
butanethiol-capped Au(111) surfaces in the presence of organic |
824 |
solvents. This algorithm allows the application of unphysical thermal |
825 |
flux to transfer heat between the metal and the liquid phase. With the |
826 |
flux applied, we were able to measure the corresponding thermal |
827 |
gradients and to obtain interfacial thermal conductivities. Under |
828 |
steady states, 2-3 ns trajectory simulations are sufficient for |
829 |
computation of this quantity. |
830 |
|
831 |
Our simulations have seen significant conductance enhancement in the |
832 |
presence of capping agent, compared with the bare gold / liquid |
833 |
interfaces. The vibrational coupling between the metal and the liquid |
834 |
phase is enhanced by a chemically-bonded capping agent. Furthermore, |
835 |
the coverage percentage of the capping agent plays an important role |
836 |
in the interfacial thermal transport process. Moderately low coverages |
837 |
allow higher contact between capping agent and solvent, and thus could |
838 |
further enhance the heat transfer process, giving a non-monotonic |
839 |
behavior of conductance with increasing coverage. |
840 |
|
841 |
Our results, particularly using the UA models, agree well with |
842 |
available experimental data. The AA models tend to overestimate the |
843 |
interfacial thermal conductance in that the classically treated C-H |
844 |
vibrations become too easily populated. Compared to the AA models, the |
845 |
UA models have higher computational efficiency with satisfactory |
846 |
accuracy, and thus are preferable in modeling interfacial thermal |
847 |
transport. |
848 |
|
849 |
Of the two definitions for $G$, the discrete form |
850 |
(Eq. \ref{discreteG}) was easier to use and gives out relatively |
851 |
consistent results, while the derivative form (Eq. \ref{derivativeG}) |
852 |
is not as versatile. Although $G^\prime$ gives out comparable results |
853 |
and follows similar trend with $G$ when measuring close to fully |
854 |
covered or bare surfaces, the spatial resolution of $T$ profile |
855 |
required for the use of a derivative form is limited by the number of |
856 |
bins and the sampling required to obtain thermal gradient information. |
857 |
|
858 |
Vlugt {\it et al.} have investigated the surface thiol structures for |
859 |
nanocrystalline gold and pointed out that they differ from those of |
860 |
the Au(111) surface.\cite{landman:1998,vlugt:cpc2007154} This |
861 |
difference could also cause differences in the interfacial thermal |
862 |
transport behavior. To investigate this problem, one would need an |
863 |
effective method for applying thermal gradients in non-planar |
864 |
(i.e. spherical) geometries. |
865 |
|
866 |
\section{Acknowledgments} |
867 |
Support for this project was provided by the National Science |
868 |
Foundation under grant CHE-0848243. Computational time was provided by |
869 |
the Center for Research Computing (CRC) at the University of Notre |
870 |
Dame. |
871 |
|
872 |
\newpage |
873 |
|
874 |
\bibliography{stokes} |
875 |
|
876 |
\end{doublespace} |
877 |
\end{document} |
878 |
|