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{Simulating interfacial thermal conductance at metal-solvent |
32 |
interfaces: the role of chemical capping agents} |
33 |
|
34 |
\author{Shenyu Kuang and J. Daniel |
35 |
Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\ |
36 |
Department of Chemistry and Biochemistry,\\ |
37 |
University of Notre Dame\\ |
38 |
Notre Dame, Indiana 46556} |
39 |
|
40 |
\date{\today} |
41 |
|
42 |
\maketitle |
43 |
|
44 |
\begin{doublespace} |
45 |
|
46 |
\begin{abstract} |
47 |
|
48 |
With the Non-Isotropic Velocity Scaling algorithm (NIVS) we have |
49 |
developed, an unphysical thermal flux can be effectively set up even |
50 |
for non-homogeneous systems like interfaces in non-equilibrium |
51 |
molecular dynamics simulations. In this work, this algorithm is |
52 |
applied for simulating thermal conductance at metal / organic solvent |
53 |
interfaces with various coverages of butanethiol capping |
54 |
agents. Different solvents and force field models were tested. Our |
55 |
results suggest that the United-Atom models are able to provide an |
56 |
estimate of the interfacial thermal conductivity comparable to |
57 |
experiments in our simulations with satisfactory computational |
58 |
efficiency. From our results, the acoustic impedance mismatch between |
59 |
metal and liquid phase is effectively reduced by the capping |
60 |
agents, and thus leads to interfacial thermal conductance |
61 |
enhancement. Furthermore, this effect is closely related to the |
62 |
capping agent coverage on the metal surfaces and the type of solvent |
63 |
molecules, and is affected by the models used in the simulations. |
64 |
|
65 |
\end{abstract} |
66 |
|
67 |
\newpage |
68 |
|
69 |
%\narrowtext |
70 |
|
71 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
72 |
% BODY OF TEXT |
73 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
74 |
|
75 |
\section{Introduction} |
76 |
Due to the importance of heat flow in nanotechnology, interfacial |
77 |
thermal conductance has been studied extensively both experimentally |
78 |
and computationally.\cite{cahill:793} Unlike bulk materials, nanoscale |
79 |
materials have a significant fraction of their atoms at interfaces, |
80 |
and the chemical details of these interfaces govern the heat transfer |
81 |
behavior. Furthermore, the interfaces are |
82 |
heterogeneous (e.g. solid - liquid), which provides a challenge to |
83 |
traditional methods developed for homogeneous systems. |
84 |
|
85 |
Experimentally, various interfaces have been investigated for their |
86 |
thermal conductance. Wang {\it et al.} studied heat transport through |
87 |
long-chain hydrocarbon monolayers on gold substrate at individual |
88 |
molecular level,\cite{Wang10082007} Schmidt {\it et al.} studied the |
89 |
role of CTAB on thermal transport between gold nanorods and |
90 |
solvent,\cite{doi:10.1021/jp8051888} and Juv\'e {\it et al.} studied |
91 |
the cooling dynamics, which is controlled by thermal interface |
92 |
resistence of glass-embedded metal |
93 |
nanoparticles.\cite{PhysRevB.80.195406} Although interfaces are |
94 |
normally considered barriers for heat transport, Alper {\it et al.} |
95 |
suggested that specific ligands (capping agents) could completely |
96 |
eliminate this barrier |
97 |
($G\rightarrow\infty$).\cite{doi:10.1021/la904855s} |
98 |
|
99 |
Theoretical and computational models have also been used to study the |
100 |
interfacial thermal transport in order to gain an understanding of |
101 |
this phenomena at the molecular level. Recently, Hase and coworkers |
102 |
employed Non-Equilibrium Molecular Dynamics (NEMD) simulations to |
103 |
study thermal transport from hot Au(111) substrate to a self-assembled |
104 |
monolayer of alkylthiol with relatively long chain (8-20 carbon |
105 |
atoms).\cite{hase:2010,hase:2011} However, ensemble averaged |
106 |
measurements for heat conductance of interfaces between the capping |
107 |
monolayer on Au and a solvent phase have yet to be studied with their |
108 |
approach. The comparatively low thermal flux through interfaces is |
109 |
difficult to measure with Equilibrium MD or forward NEMD simulation |
110 |
methods. Therefore, the Reverse NEMD (RNEMD) |
111 |
methods\cite{MullerPlathe:1997xw,kuang:164101} would have the |
112 |
advantage of applying this difficult to measure flux (while measuring |
113 |
the resulting gradient), given that the simulation methods being able |
114 |
to effectively apply an unphysical flux in non-homogeneous systems. |
115 |
Garde and coworkers\cite{garde:nl2005,garde:PhysRevLett2009} applied |
116 |
this approach to various liquid interfaces and studied how thermal |
117 |
conductance (or resistance) is dependent on chemistry details of |
118 |
interfaces, e.g. hydrophobic and hydrophilic aqueous interfaces. |
119 |
|
120 |
Recently, we have developed a Non-Isotropic Velocity Scaling (NIVS) |
121 |
algorithm for RNEMD simulations\cite{kuang:164101}. This algorithm |
122 |
retains the desirable features of RNEMD (conservation of linear |
123 |
momentum and total energy, compatibility with periodic boundary |
124 |
conditions) while establishing true thermal distributions in each of |
125 |
the two slabs. Furthermore, it allows effective thermal exchange |
126 |
between particles of different identities, and thus makes the study of |
127 |
interfacial conductance much simpler. |
128 |
|
129 |
The work presented here deals with the Au(111) surface covered to |
130 |
varying degrees by butanethiol, a capping agent with short carbon |
131 |
chain, and solvated with organic solvents of different molecular |
132 |
properties. Different models were used for both the capping agent and |
133 |
the solvent force field parameters. Using the NIVS algorithm, the |
134 |
thermal transport across these interfaces was studied and the |
135 |
underlying mechanism for the phenomena was investigated. |
136 |
|
137 |
\section{Methodology} |
138 |
\subsection{Imposd-Flux Methods in MD Simulations} |
139 |
Steady state MD simulations have an advantage in that not many |
140 |
trajectories are needed to study the relationship between thermal flux |
141 |
and thermal gradients. For systems with low interfacial conductance, |
142 |
one must have a method capable of generating or measuring relatively |
143 |
small fluxes, compared to those required for bulk conductivity. This |
144 |
requirement makes the calculation even more difficult for |
145 |
slowly-converging equilibrium methods.\cite{Viscardy:2007lq} Forward |
146 |
NEMD methods impose a gradient (and measure a flux), but at interfaces |
147 |
it is not clear what behavior should be imposed at the boundaries |
148 |
between materials. Imposed-flux reverse non-equilibrium |
149 |
methods\cite{MullerPlathe:1997xw} set the flux {\it a priori} and |
150 |
the thermal response becomes an easy-to-measure quantity. Although |
151 |
M\"{u}ller-Plathe's original momentum swapping approach can be used |
152 |
for exchanging energy between particles of different identity, the |
153 |
kinetic energy transfer efficiency is affected by the mass difference |
154 |
between the particles, which limits its application on heterogeneous |
155 |
interfacial systems. |
156 |
|
157 |
The non-isotropic velocity scaling (NIVS) \cite{kuang:164101} approach |
158 |
to non-equilibrium MD simulations is able to impose a wide range of |
159 |
kinetic energy fluxes without obvious perturbation to the velocity |
160 |
distributions of the simulated systems. Furthermore, this approach has |
161 |
the advantage in heterogeneous interfaces in that kinetic energy flux |
162 |
can be applied between regions of particles of arbitary identity, and |
163 |
the flux will not be restricted by difference in particle mass. |
164 |
|
165 |
The NIVS algorithm scales the velocity vectors in two separate regions |
166 |
of a simulation system with respective diagonal scaling matricies. To |
167 |
determine these scaling factors in the matricies, a set of equations |
168 |
including linear momentum conservation and kinetic energy conservation |
169 |
constraints and target energy flux satisfaction is solved. With the |
170 |
scaling operation applied to the system in a set frequency, bulk |
171 |
temperature gradients can be easily established, and these can be used |
172 |
for computing thermal conductivities. The NIVS algorithm conserves |
173 |
momenta and energy and does not depend on an external thermostat. |
174 |
|
175 |
\subsection{Defining Interfacial Thermal Conductivity ($G$)} |
176 |
|
177 |
For an interface with relatively low interfacial conductance, and a |
178 |
thermal flux between two distinct bulk regions, the regions on either |
179 |
side of the interface rapidly come to a state in which the two phases |
180 |
have relatively homogeneous (but distinct) temperatures. The |
181 |
interfacial thermal conductivity $G$ can therefore be approximated as: |
182 |
\begin{equation} |
183 |
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle - |
184 |
\langle T_\mathrm{cold}\rangle \right)} |
185 |
\label{lowG} |
186 |
\end{equation} |
187 |
where ${E_{total}}$ is the total imposed non-physical kinetic energy |
188 |
transfer during the simulation and ${\langle T_\mathrm{hot}\rangle}$ |
189 |
and ${\langle T_\mathrm{cold}\rangle}$ are the average observed |
190 |
temperature of the two separated phases. |
191 |
|
192 |
When the interfacial conductance is {\it not} small, there are two |
193 |
ways to define $G$. One way is to assume the temperature is discrete |
194 |
on the two sides of the interface. $G$ can be calculated using the |
195 |
applied thermal flux $J$ and the maximum temperature difference |
196 |
measured along the thermal gradient max($\Delta T$), which occurs at |
197 |
the Gibbs deviding surface (Figure \ref{demoPic}): \begin{equation} |
198 |
G=\frac{J}{\Delta T} \label{discreteG} \end{equation} |
199 |
|
200 |
\begin{figure} |
201 |
\includegraphics[width=\linewidth]{method} |
202 |
\caption{Interfacial conductance can be calculated by applying an |
203 |
(unphysical) kinetic energy flux between two slabs, one located |
204 |
within the metal and another on the edge of the periodic box. The |
205 |
system responds by forming a thermal response or a gradient. In |
206 |
bulk liquids, this gradient typically has a single slope, but in |
207 |
interfacial systems, there are distinct thermal conductivity |
208 |
domains. The interfacial conductance, $G$ is found by measuring the |
209 |
temperature gap at the Gibbs dividing surface, or by using second |
210 |
derivatives of the thermal profile.} |
211 |
\label{demoPic} |
212 |
\end{figure} |
213 |
|
214 |
The other approach is to assume a continuous temperature profile along |
215 |
the thermal gradient axis (e.g. $z$) and define $G$ at the point where |
216 |
the magnitude of thermal conductivity ($\lambda$) change reaches its |
217 |
maximum, given that $\lambda$ is well-defined throughout the space: |
218 |
\begin{equation} |
219 |
G^\prime = \Big|\frac{\partial\lambda}{\partial z}\Big| |
220 |
= \Big|\frac{\partial}{\partial z}\left(-J_z\Big/ |
221 |
\left(\frac{\partial T}{\partial z}\right)\right)\Big| |
222 |
= |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
223 |
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
224 |
\label{derivativeG} |
225 |
\end{equation} |
226 |
|
227 |
With temperature profiles obtained from simulation, one is able to |
228 |
approximate the first and second derivatives of $T$ with finite |
229 |
difference methods and calculate $G^\prime$. In what follows, both |
230 |
definitions have been used, and are compared in the results. |
231 |
|
232 |
To investigate the interfacial conductivity at metal / solvent |
233 |
interfaces, we have modeled a metal slab with its (111) surfaces |
234 |
perpendicular to the $z$-axis of our simulation cells. The metal slab |
235 |
has been prepared both with and without capping agents on the exposed |
236 |
surface, and has been solvated with simple organic solvents, as |
237 |
illustrated in Figure \ref{gradT}. |
238 |
|
239 |
With the simulation cell described above, we are able to equilibrate |
240 |
the system and impose an unphysical thermal flux between the liquid |
241 |
and the metal phase using the NIVS algorithm. By periodically applying |
242 |
the unphysical flux, we obtained a temperature profile and its spatial |
243 |
derivatives. Figure \ref{gradT} shows how an applied thermal flux can |
244 |
be used to obtain the 1st and 2nd derivatives of the temperature |
245 |
profile. |
246 |
|
247 |
\begin{figure} |
248 |
\includegraphics[width=\linewidth]{gradT} |
249 |
\caption{A sample of Au-butanethiol/hexane interfacial system and the |
250 |
temperature profile after a kinetic energy flux is imposed to |
251 |
it. The 1st and 2nd derivatives of the temperature profile can be |
252 |
obtained with finite difference approximation (lower panel).} |
253 |
\label{gradT} |
254 |
\end{figure} |
255 |
|
256 |
\section{Computational Details} |
257 |
\subsection{Simulation Protocol} |
258 |
The NIVS algorithm has been implemented in our MD simulation code, |
259 |
OpenMD\cite{Meineke:2005gd,openmd}, and was used for our simulations. |
260 |
Metal slabs of 6 or 11 layers of Au atoms were first equilibrated |
261 |
under atmospheric pressure (1 atm) and 200K. After equilibration, |
262 |
butanethiol capping agents were placed at three-fold hollow sites on |
263 |
the Au(111) surfaces. These sites are either {\it fcc} or {\it |
264 |
hcp} sites, although Hase {\it et al.} found that they are |
265 |
equivalent in a heat transfer process,\cite{hase:2010} so we did not |
266 |
distinguish between these sites in our study. The maximum butanethiol |
267 |
capacity on Au surface is $1/3$ of the total number of surface Au |
268 |
atoms, and the packing forms a $(\sqrt{3}\times\sqrt{3})R30^\circ$ |
269 |
structure\cite{doi:10.1021/ja00008a001,doi:10.1021/cr9801317}. A |
270 |
series of lower coverages was also prepared by eliminating |
271 |
butanethiols from the higher coverage surface in a regular manner. The |
272 |
lower coverages were prepared in order to study the relation between |
273 |
coverage and interfacial conductance. |
274 |
|
275 |
The capping agent molecules were allowed to migrate during the |
276 |
simulations. They distributed themselves uniformly and sampled a |
277 |
number of three-fold sites throughout out study. Therefore, the |
278 |
initial configuration does not noticeably affect the sampling of a |
279 |
variety of configurations of the same coverage, and the final |
280 |
conductance measurement would be an average effect of these |
281 |
configurations explored in the simulations. |
282 |
|
283 |
After the modified Au-butanethiol surface systems were equilibrated in |
284 |
the canonical (NVT) ensemble, organic solvent molecules were packed in |
285 |
the previously empty part of the simulation cells.\cite{packmol} Two |
286 |
solvents were investigated, one which has little vibrational overlap |
287 |
with the alkanethiol and which has a planar shape (toluene), and one |
288 |
which has similar vibrational frequencies to the capping agent and |
289 |
chain-like shape ({\it n}-hexane). |
290 |
|
291 |
The simulation cells were not particularly extensive along the |
292 |
$z$-axis, as a very long length scale for the thermal gradient may |
293 |
cause excessively hot or cold temperatures in the middle of the |
294 |
solvent region and lead to undesired phenomena such as solvent boiling |
295 |
or freezing when a thermal flux is applied. Conversely, too few |
296 |
solvent molecules would change the normal behavior of the liquid |
297 |
phase. Therefore, our $N_{solvent}$ values were chosen to ensure that |
298 |
these extreme cases did not happen to our simulations. The spacing |
299 |
between periodic images of the gold interfaces is $35 \sim 75$\AA. |
300 |
|
301 |
The initial configurations generated are further equilibrated with the |
302 |
$x$ and $y$ dimensions fixed, only allowing the $z$-length scale to |
303 |
change. This is to ensure that the equilibration of liquid phase does |
304 |
not affect the metal's crystalline structure. Comparisons were made |
305 |
with simulations that allowed changes of $L_x$ and $L_y$ during NPT |
306 |
equilibration. No substantial changes in the box geometry were noticed |
307 |
in these simulations. After ensuring the liquid phase reaches |
308 |
equilibrium at atmospheric pressure (1 atm), further equilibration was |
309 |
carried out under canonical (NVT) and microcanonical (NVE) ensembles. |
310 |
|
311 |
After the systems reach equilibrium, NIVS was used to impose an |
312 |
unphysical thermal flux between the metal and the liquid phases. Most |
313 |
of our simulations were done under an average temperature of |
314 |
$\sim$200K. Therefore, thermal flux usually came from the metal to the |
315 |
liquid so that the liquid has a higher temperature and would not |
316 |
freeze due to lowered temperatures. After this induced temperature |
317 |
gradient had stablized, the temperature profile of the simulation cell |
318 |
was recorded. To do this, the simulation cell is devided evenly into |
319 |
$N$ slabs along the $z$-axis. The average temperatures of each slab |
320 |
are recorded for 1$\sim$2 ns. When the slab width $d$ of each slab is |
321 |
the same, the derivatives of $T$ with respect to slab number $n$ can |
322 |
be directly used for $G^\prime$ calculations: \begin{equation} |
323 |
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
324 |
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
325 |
= |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big| |
326 |
\Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2 |
327 |
= |J_z|\Big|\frac{\partial^2 T}{\partial n^2}\Big| |
328 |
\Big/\left(\frac{\partial T}{\partial n}\right)^2 |
329 |
\label{derivativeG2} |
330 |
\end{equation} |
331 |
|
332 |
All of the above simulation procedures use a time step of 1 fs. Each |
333 |
equilibration stage took a minimum of 100 ps, although in some cases, |
334 |
longer equilibration stages were utilized. |
335 |
|
336 |
\subsection{Force Field Parameters} |
337 |
Our simulations include a number of chemically distinct components. |
338 |
Figure \ref{demoMol} demonstrates the sites defined for both |
339 |
United-Atom and All-Atom models of the organic solvent and capping |
340 |
agents in our simulations. Force field parameters are needed for |
341 |
interactions both between the same type of particles and between |
342 |
particles of different species. |
343 |
|
344 |
\begin{figure} |
345 |
\includegraphics[width=\linewidth]{structures} |
346 |
\caption{Structures of the capping agent and solvents utilized in |
347 |
these simulations. The chemically-distinct sites (a-e) are expanded |
348 |
in terms of constituent atoms for both United Atom (UA) and All Atom |
349 |
(AA) force fields. Most parameters are from |
350 |
Refs. \protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} (UA) and |
351 |
\protect\cite{OPLSAA} (AA). Cross-interactions with the Au atoms are given |
352 |
in Table \ref{MnM}.} |
353 |
\label{demoMol} |
354 |
\end{figure} |
355 |
|
356 |
The Au-Au interactions in metal lattice slab is described by the |
357 |
quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
358 |
potentials include zero-point quantum corrections and are |
359 |
reparametrized for accurate surface energies compared to the |
360 |
Sutton-Chen potentials.\cite{Chen90} |
361 |
|
362 |
For the two solvent molecules, {\it n}-hexane and toluene, two |
363 |
different atomistic models were utilized. Both solvents were modeled |
364 |
using United-Atom (UA) and All-Atom (AA) models. The TraPPE-UA |
365 |
parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
366 |
for our UA solvent molecules. In these models, sites are located at |
367 |
the carbon centers for alkyl groups. Bonding interactions, including |
368 |
bond stretches and bends and torsions, were used for intra-molecular |
369 |
sites closer than 3 bonds. For non-bonded interactions, Lennard-Jones |
370 |
potentials are used. |
371 |
|
372 |
By eliminating explicit hydrogen atoms, the TraPPE-UA models are |
373 |
simple and computationally efficient, while maintaining good accuracy. |
374 |
However, the TraPPE-UA model for alkanes is known to predict a slighly |
375 |
lower boiling point than experimental values. This is one of the |
376 |
reasons we used a lower average temperature (200K) for our |
377 |
simulations. If heat is transferred to the liquid phase during the |
378 |
NIVS simulation, the liquid in the hot slab can actually be |
379 |
substantially warmer than the mean temperature in the simulation. The |
380 |
lower mean temperatures therefore prevent solvent boiling. |
381 |
|
382 |
For UA-toluene, the non-bonded potentials between intermolecular sites |
383 |
have a similar Lennard-Jones formulation. The toluene molecules were |
384 |
treated as a single rigid body, so there was no need for |
385 |
intramolecular interactions (including bonds, bends, or torsions) in |
386 |
this solvent model. |
387 |
|
388 |
Besides the TraPPE-UA models, AA models for both organic solvents are |
389 |
included in our studies as well. For hexane, the OPLS-AA\cite{OPLSAA} |
390 |
force field is used, and additional explicit hydrogen sites were |
391 |
included. Besides bonding and non-bonded site-site interactions, |
392 |
partial charges and the electrostatic interactions were added to each |
393 |
CT and HC site. For toluene, the United Force Field developed by |
394 |
Rapp\'{e} {\it et al.}\cite{doi:10.1021/ja00051a040} was adopted, and |
395 |
a flexible model for the toluene molecule was utilized which included |
396 |
bond, bend, torsion, and inversion potentials to enforce ring |
397 |
planarity. |
398 |
|
399 |
The butanethiol capping agent in our simulations, were also modeled |
400 |
with both UA and AA model. The TraPPE-UA force field includes |
401 |
parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for |
402 |
UA butanethiol model in our simulations. The OPLS-AA also provides |
403 |
parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) |
404 |
surfaces do not have the hydrogen atom bonded to sulfur. To derive |
405 |
suitable parameters for butanethiol adsorbed on Au(111) surfaces, we |
406 |
adopt the S parameters from Luedtke and Landman\cite{landman:1998} and |
407 |
modify the parameters for the CTS atom to maintain charge neutrality |
408 |
in the molecule. Note that the model choice (UA or AA) for the capping |
409 |
agent can be different from the solvent. Regardless of model choice, |
410 |
the force field parameters for interactions between capping agent and |
411 |
solvent can be derived using Lorentz-Berthelot Mixing Rule: |
412 |
\begin{eqnarray} |
413 |
\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\ |
414 |
\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}} |
415 |
\end{eqnarray} |
416 |
|
417 |
To describe the interactions between metal (Au) and non-metal atoms, |
418 |
we refer to an adsorption study of alkyl thiols on gold surfaces by |
419 |
Vlugt {\it et al.}\cite{vlugt:cpc2007154} They fitted an effective |
420 |
Lennard-Jones form of potential parameters for the interaction between |
421 |
Au and pseudo-atoms CH$_x$ and S based on a well-established and |
422 |
widely-used effective potential of Hautman and Klein for the Au(111) |
423 |
surface.\cite{hautman:4994} As our simulations require the gold slab |
424 |
to be flexible to accommodate thermal excitation, the pair-wise form |
425 |
of potentials they developed was used for our study. |
426 |
|
427 |
The potentials developed from {\it ab initio} calculations by Leng |
428 |
{\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
429 |
interactions between Au and aromatic C/H atoms in toluene. However, |
430 |
the Lennard-Jones parameters between Au and other types of particles, |
431 |
(e.g. AA alkanes) have not yet been established. For these |
432 |
interactions, the Lorentz-Berthelot mixing rule can be used to derive |
433 |
effective single-atom LJ parameters for the metal using the fit values |
434 |
for toluene. These are then used to construct reasonable mixing |
435 |
parameters for the interactions between the gold and other atoms. |
436 |
Table \ref{MnM} summarizes the ``metal/non-metal'' parameters used in |
437 |
our simulations. |
438 |
|
439 |
\begin{table*} |
440 |
\begin{minipage}{\linewidth} |
441 |
\begin{center} |
442 |
\caption{Non-bonded interaction parameters (including cross |
443 |
interactions with Au atoms) for both force fields used in this |
444 |
work.} |
445 |
\begin{tabular}{lllllll} |
446 |
\hline\hline |
447 |
& Site & $\sigma_{ii}$ & $\epsilon_{ii}$ & $q_i$ & |
448 |
$\sigma_{Au-i}$ & $\epsilon_{Au-i}$ \\ |
449 |
& & (\AA) & (kcal/mol) & ($e$) & (\AA) & (kcal/mol) \\ |
450 |
\hline |
451 |
United Atom (UA) |
452 |
&CH3 & 3.75 & 0.1947 & - & 3.54 & 0.2146 \\ |
453 |
&CH2 & 3.95 & 0.0914 & - & 3.54 & 0.1749 \\ |
454 |
&CHar & 3.695 & 0.1003 & - & 3.4625 & 0.1680 \\ |
455 |
&CRar & 3.88 & 0.04173 & - & 3.555 & 0.1604 \\ |
456 |
\hline |
457 |
All Atom (AA) |
458 |
&CT3 & 3.50 & 0.066 & -0.18 & 3.365 & 0.1373 \\ |
459 |
&CT2 & 3.50 & 0.066 & -0.12 & 3.365 & 0.1373 \\ |
460 |
&CTT & 3.50 & 0.066 & -0.065 & 3.365 & 0.1373 \\ |
461 |
&HC & 2.50 & 0.030 & 0.06 & 2.865 & 0.09256 \\ |
462 |
&CA & 3.55 & 0.070 & -0.115 & 3.173 & 0.0640 \\ |
463 |
&HA & 2.42 & 0.030 & 0.115 & 2.746 & 0.0414 \\ |
464 |
\hline |
465 |
Both UA and AA |
466 |
& S & 4.45 & 0.25 & - & 2.40 & 8.465 \\ |
467 |
\hline\hline |
468 |
\end{tabular} |
469 |
\label{MnM} |
470 |
\end{center} |
471 |
\end{minipage} |
472 |
\end{table*} |
473 |
|
474 |
\subsection{Vibrational Power Spectrum} |
475 |
|
476 |
To investigate the mechanism of interfacial thermal conductance, the |
477 |
vibrational power spectrum was computed. Power spectra were taken for |
478 |
individual components in different simulations. To obtain these |
479 |
spectra, simulations were run after equilibration, in the NVE |
480 |
ensemble, and without a thermal gradient. Snapshots of configurations |
481 |
were collected at a frequency that is higher than that of the fastest |
482 |
vibrations occuring in the simulations. With these configurations, the |
483 |
velocity auto-correlation functions can be computed: |
484 |
\begin{equation} |
485 |
C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle |
486 |
\label{vCorr} |
487 |
\end{equation} |
488 |
The power spectrum is constructed via a Fourier transform of the |
489 |
symmetrized velocity autocorrelation function, |
490 |
\begin{equation} |
491 |
\hat{f}(\omega) = |
492 |
\int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt |
493 |
\label{fourier} |
494 |
\end{equation} |
495 |
|
496 |
\section{Results and Discussions} |
497 |
In what follows, how the parameters and protocol of simulations would |
498 |
affect the measurement of $G$'s is first discussed. With a reliable |
499 |
protocol and set of parameters, the influence of capping agent |
500 |
coverage on thermal conductance is investigated. Besides, different |
501 |
force field models for both solvents and selected deuterated models |
502 |
were tested and compared. Finally, a summary of the role of capping |
503 |
agent in the interfacial thermal transport process is given. |
504 |
|
505 |
\subsection{How Simulation Parameters Affects $G$} |
506 |
We have varied our protocol or other parameters of the simulations in |
507 |
order to investigate how these factors would affect the measurement of |
508 |
$G$'s. It turned out that while some of these parameters would not |
509 |
affect the results substantially, some other changes to the |
510 |
simulations would have a significant impact on the measurement |
511 |
results. |
512 |
|
513 |
In some of our simulations, we allowed $L_x$ and $L_y$ to change |
514 |
during equilibrating the liquid phase. Due to the stiffness of the |
515 |
crystalline Au structure, $L_x$ and $L_y$ would not change noticeably |
516 |
after equilibration. Although $L_z$ could fluctuate $\sim$1\% after a |
517 |
system is fully equilibrated in the NPT ensemble, this fluctuation, as |
518 |
well as those of $L_x$ and $L_y$ (which is significantly smaller), |
519 |
would not be magnified on the calculated $G$'s, as shown in Table |
520 |
\ref{AuThiolHexaneUA}. This insensivity to $L_i$ fluctuations allows |
521 |
reliable measurement of $G$'s without the necessity of extremely |
522 |
cautious equilibration process. |
523 |
|
524 |
As stated in our computational details, the spacing filled with |
525 |
solvent molecules can be chosen within a range. This allows some |
526 |
change of solvent molecule numbers for the same Au-butanethiol |
527 |
surfaces. We did this study on our Au-butanethiol/hexane |
528 |
simulations. Nevertheless, the results obtained from systems of |
529 |
different $N_{hexane}$ did not indicate that the measurement of $G$ is |
530 |
susceptible to this parameter. For computational efficiency concern, |
531 |
smaller system size would be preferable, given that the liquid phase |
532 |
structure is not affected. |
533 |
|
534 |
Our NIVS algorithm allows change of unphysical thermal flux both in |
535 |
direction and in quantity. This feature extends our investigation of |
536 |
interfacial thermal conductance. However, the magnitude of this |
537 |
thermal flux is not arbitary if one aims to obtain a stable and |
538 |
reliable thermal gradient. A temperature profile would be |
539 |
substantially affected by noise when $|J_z|$ has a much too low |
540 |
magnitude; while an excessively large $|J_z|$ that overwhelms the |
541 |
conductance capacity of the interface would prevent a thermal gradient |
542 |
to reach a stablized steady state. NIVS has the advantage of allowing |
543 |
$J$ to vary in a wide range such that the optimal flux range for $G$ |
544 |
measurement can generally be simulated by the algorithm. Within the |
545 |
optimal range, we were able to study how $G$ would change according to |
546 |
the thermal flux across the interface. For our simulations, we denote |
547 |
$J_z$ to be positive when the physical thermal flux is from the liquid |
548 |
to metal, and negative vice versa. The $G$'s measured under different |
549 |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and |
550 |
\ref{AuThiolToluene}. These results do not suggest that $G$ is |
551 |
dependent on $J_z$ within this flux range. The linear response of flux |
552 |
to thermal gradient simplifies our investigations in that we can rely |
553 |
on $G$ measurement with only a couple $J_z$'s and do not need to test |
554 |
a large series of fluxes. |
555 |
|
556 |
\begin{table*} |
557 |
\begin{minipage}{\linewidth} |
558 |
\begin{center} |
559 |
\caption{Computed interfacial thermal conductivity ($G$ and |
560 |
$G^\prime$) values for the 100\% covered Au-butanethiol/hexane |
561 |
interfaces with UA model and different hexane molecule numbers |
562 |
at different temperatures using a range of energy |
563 |
fluxes. Error estimates indicated in parenthesis.} |
564 |
|
565 |
\begin{tabular}{ccccccc} |
566 |
\hline\hline |
567 |
$\langle T\rangle$ & $N_{hexane}$ & Fixed & $\rho_{hexane}$ & |
568 |
$J_z$ & $G$ & $G^\prime$ \\ |
569 |
(K) & & $L_x$ \& $L_y$? & (g/cm$^3$) & (GW/m$^2$) & |
570 |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
571 |
\hline |
572 |
200 & 266 & No & 0.672 & -0.96 & 102(3) & 80.0(0.8) \\ |
573 |
& 200 & Yes & 0.694 & 1.92 & 129(11) & 87.3(0.3) \\ |
574 |
& & Yes & 0.672 & 1.93 & 131(16) & 78(13) \\ |
575 |
& & No & 0.688 & 0.96 & 125(16) & 90.2(15) \\ |
576 |
& & & & 1.91 & 139(10) & 101(10) \\ |
577 |
& & & & 2.83 & 141(6) & 89.9(9.8) \\ |
578 |
& 166 & Yes & 0.679 & 0.97 & 115(19) & 69(18) \\ |
579 |
& & & & 1.94 & 125(9) & 87.1(0.2) \\ |
580 |
& & No & 0.681 & 0.97 & 141(30) & 78(22) \\ |
581 |
& & & & 1.92 & 138(4) & 98.9(9.5) \\ |
582 |
\hline |
583 |
250 & 200 & No & 0.560 & 0.96 & 75(10) & 61.8(7.3) \\ |
584 |
& & & & -0.95 & 49.4(0.3) & 45.7(2.1) \\ |
585 |
& 166 & Yes & 0.570 & 0.98 & 79.0(3.5) & 62.9(3.0) \\ |
586 |
& & No & 0.569 & 0.97 & 80.3(0.6) & 67(11) \\ |
587 |
& & & & 1.44 & 76.2(5.0) & 64.8(3.8) \\ |
588 |
& & & & -0.95 & 56.4(2.5) & 54.4(1.1) \\ |
589 |
& & & & -1.85 & 47.8(1.1) & 53.5(1.5) \\ |
590 |
\hline\hline |
591 |
\end{tabular} |
592 |
\label{AuThiolHexaneUA} |
593 |
\end{center} |
594 |
\end{minipage} |
595 |
\end{table*} |
596 |
|
597 |
Furthermore, we also attempted to increase system average temperatures |
598 |
to above 200K. These simulations are first equilibrated in the NPT |
599 |
ensemble under normal pressure. As stated above, the TraPPE-UA model |
600 |
for hexane tends to predict a lower boiling point. In our simulations, |
601 |
hexane had diffculty to remain in liquid phase when NPT equilibration |
602 |
temperature is higher than 250K. Additionally, the equilibrated liquid |
603 |
hexane density under 250K becomes lower than experimental value. This |
604 |
expanded liquid phase leads to lower contact between hexane and |
605 |
butanethiol as well.[MAY NEED SLAB DENSITY FIGURE] |
606 |
And this reduced contact would |
607 |
probably be accountable for a lower interfacial thermal conductance, |
608 |
as shown in Table \ref{AuThiolHexaneUA}. |
609 |
|
610 |
A similar study for TraPPE-UA toluene agrees with the above result as |
611 |
well. Having a higher boiling point, toluene tends to remain liquid in |
612 |
our simulations even equilibrated under 300K in NPT |
613 |
ensembles. Furthermore, the expansion of the toluene liquid phase is |
614 |
not as significant as that of the hexane. This prevents severe |
615 |
decrease of liquid-capping agent contact and the results (Table |
616 |
\ref{AuThiolToluene}) show only a slightly decreased interface |
617 |
conductance. Therefore, solvent-capping agent contact should play an |
618 |
important role in the thermal transport process across the interface |
619 |
in that higher degree of contact could yield increased conductance. |
620 |
|
621 |
\begin{table*} |
622 |
\begin{minipage}{\linewidth} |
623 |
\begin{center} |
624 |
\caption{Computed interfacial thermal conductivity ($G$ and |
625 |
$G^\prime$) values for a 90\% coverage Au-butanethiol/toluene |
626 |
interface at different temperatures using a range of energy |
627 |
fluxes. Error estimates indicated in parenthesis.} |
628 |
|
629 |
\begin{tabular}{ccccc} |
630 |
\hline\hline |
631 |
$\langle T\rangle$ & $\rho_{toluene}$ & $J_z$ & $G$ & $G^\prime$ \\ |
632 |
(K) & (g/cm$^3$) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
633 |
\hline |
634 |
200 & 0.933 & 2.15 & 204(12) & 113(12) \\ |
635 |
& & -1.86 & 180(3) & 135(21) \\ |
636 |
& & -3.93 & 176(5) & 113(12) \\ |
637 |
\hline |
638 |
300 & 0.855 & -1.91 & 143(5) & 125(2) \\ |
639 |
& & -4.19 & 135(9) & 113(12) \\ |
640 |
\hline\hline |
641 |
\end{tabular} |
642 |
\label{AuThiolToluene} |
643 |
\end{center} |
644 |
\end{minipage} |
645 |
\end{table*} |
646 |
|
647 |
Besides lower interfacial thermal conductance, surfaces in relatively |
648 |
high temperatures are susceptible to reconstructions, when |
649 |
butanethiols have a full coverage on the Au(111) surface. These |
650 |
reconstructions include surface Au atoms migrated outward to the S |
651 |
atom layer, and butanethiol molecules embedded into the original |
652 |
surface Au layer. The driving force for this behavior is the strong |
653 |
Au-S interactions in our simulations. And these reconstructions lead |
654 |
to higher ratio of Au-S attraction and thus is energetically |
655 |
favorable. Furthermore, this phenomenon agrees with experimental |
656 |
results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt |
657 |
{\it et al.} had kept their Au(111) slab rigid so that their |
658 |
simulations can reach 300K without surface reconstructions. Without |
659 |
this practice, simulating 100\% thiol covered interfaces under higher |
660 |
temperatures could hardly avoid surface reconstructions. However, our |
661 |
measurement is based on assuming homogeneity on $x$ and $y$ dimensions |
662 |
so that measurement of $T$ at particular $z$ would be an effective |
663 |
average of the particles of the same type. Since surface |
664 |
reconstructions could eliminate the original $x$ and $y$ dimensional |
665 |
homogeneity, measurement of $G$ is more difficult to conduct under |
666 |
higher temperatures. Therefore, most of our measurements are |
667 |
undertaken at $\langle T\rangle\sim$200K. |
668 |
|
669 |
However, when the surface is not completely covered by butanethiols, |
670 |
the simulated system is more resistent to the reconstruction |
671 |
above. Our Au-butanethiol/toluene system had the Au(111) surfaces 90\% |
672 |
covered by butanethiols, but did not see this above phenomena even at |
673 |
$\langle T\rangle\sim$300K. The empty three-fold sites not occupied by |
674 |
capping agents could help prevent surface reconstruction in that they |
675 |
provide other means of capping agent relaxation. It is observed that |
676 |
butanethiols can migrate to their neighbor empty sites during a |
677 |
simulation. Therefore, we were able to obtain $G$'s for these |
678 |
interfaces even at a relatively high temperature without being |
679 |
affected by surface reconstructions. |
680 |
|
681 |
\subsection{Influence of Capping Agent Coverage on $G$} |
682 |
To investigate the influence of butanethiol coverage on interfacial |
683 |
thermal conductance, a series of different coverage Au-butanethiol |
684 |
surfaces is prepared and solvated with various organic |
685 |
molecules. These systems are then equilibrated and their interfacial |
686 |
thermal conductivity are measured with our NIVS algorithm. Figure |
687 |
\ref{coverage} demonstrates the trend of conductance change with |
688 |
respect to different coverages of butanethiol. To study the isotope |
689 |
effect in interfacial thermal conductance, deuterated UA-hexane is |
690 |
included as well. |
691 |
|
692 |
\begin{figure} |
693 |
\includegraphics[width=\linewidth]{coverage} |
694 |
\caption{Comparison of interfacial thermal conductivity ($G$) values |
695 |
for the Au-butanethiol/solvent interface with various UA models and |
696 |
different capping agent coverages at $\langle T\rangle\sim$200K |
697 |
using certain energy flux respectively.} |
698 |
\label{coverage} |
699 |
\end{figure} |
700 |
|
701 |
It turned out that with partial covered butanethiol on the Au(111) |
702 |
surface, the derivative definition for $G^\prime$ |
703 |
(Eq. \ref{derivativeG}) was difficult to apply, due to the difficulty |
704 |
in locating the maximum of change of $\lambda$. Instead, the discrete |
705 |
definition (Eq. \ref{discreteG}) is easier to apply, as the Gibbs |
706 |
deviding surface can still be well-defined. Therefore, $G$ (not |
707 |
$G^\prime$) was used for this section. |
708 |
|
709 |
From Figure \ref{coverage}, one can see the significance of the |
710 |
presence of capping agents. Even when a fraction of the Au(111) |
711 |
surface sites are covered with butanethiols, the conductivity would |
712 |
see an enhancement by at least a factor of 3. This indicates the |
713 |
important role cappping agent is playing for thermal transport |
714 |
phenomena on metal / organic solvent surfaces. |
715 |
|
716 |
Interestingly, as one could observe from our results, the maximum |
717 |
conductance enhancement (largest $G$) happens while the surfaces are |
718 |
about 75\% covered with butanethiols. This again indicates that |
719 |
solvent-capping agent contact has an important role of the thermal |
720 |
transport process. Slightly lower butanethiol coverage allows small |
721 |
gaps between butanethiols to form. And these gaps could be filled with |
722 |
solvent molecules, which acts like ``heat conductors'' on the |
723 |
surface. The higher degree of interaction between these solvent |
724 |
molecules and capping agents increases the enhancement effect and thus |
725 |
produces a higher $G$ than densely packed butanethiol arrays. However, |
726 |
once this maximum conductance enhancement is reached, $G$ decreases |
727 |
when butanethiol coverage continues to decrease. Each capping agent |
728 |
molecule reaches its maximum capacity for thermal |
729 |
conductance. Therefore, even higher solvent-capping agent contact |
730 |
would not offset this effect. Eventually, when butanethiol coverage |
731 |
continues to decrease, solvent-capping agent contact actually |
732 |
decreases with the disappearing of butanethiol molecules. In this |
733 |
case, $G$ decrease could not be offset but instead accelerated. [NEED |
734 |
SNAPSHOT SHOWING THE PHENOMENA / SLAB DENSITY ANALYSIS] |
735 |
|
736 |
A comparison of the results obtained from differenet organic solvents |
737 |
can also provide useful information of the interfacial thermal |
738 |
transport process. The deuterated hexane (UA) results do not appear to |
739 |
be much different from those of normal hexane (UA), given that |
740 |
butanethiol (UA) is non-deuterated for both solvents. These UA model |
741 |
studies, even though eliminating C-H vibration samplings, still have |
742 |
C-C vibrational frequencies different from each other. However, these |
743 |
differences in the infrared range do not seem to produce an observable |
744 |
difference for the results of $G$ (Figure \ref{uahxnua}). |
745 |
|
746 |
\begin{figure} |
747 |
\includegraphics[width=\linewidth]{uahxnua} |
748 |
\caption{Vibrational spectra obtained for normal (upper) and |
749 |
deuterated (lower) hexane in Au-butanethiol/hexane |
750 |
systems. Butanethiol spectra are shown as reference. Both hexane and |
751 |
butanethiol were using United-Atom models.} |
752 |
\label{uahxnua} |
753 |
\end{figure} |
754 |
|
755 |
Furthermore, results for rigid body toluene solvent, as well as other |
756 |
UA-hexane solvents, are reasonable within the general experimental |
757 |
ranges\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406}. This |
758 |
suggests that explicit hydrogen might not be a required factor for |
759 |
modeling thermal transport phenomena of systems such as |
760 |
Au-thiol/organic solvent. |
761 |
|
762 |
However, results for Au-butanethiol/toluene do not show an identical |
763 |
trend with those for Au-butanethiol/hexane in that $G$ remains at |
764 |
approximately the same magnitue when butanethiol coverage differs from |
765 |
25\% to 75\%. This might be rooted in the molecule shape difference |
766 |
for planar toluene and chain-like {\it n}-hexane. Due to this |
767 |
difference, toluene molecules have more difficulty in occupying |
768 |
relatively small gaps among capping agents when their coverage is not |
769 |
too low. Therefore, the solvent-capping agent contact may keep |
770 |
increasing until the capping agent coverage reaches a relatively low |
771 |
level. This becomes an offset for decreasing butanethiol molecules on |
772 |
its effect to the process of interfacial thermal transport. Thus, one |
773 |
can see a plateau of $G$ vs. butanethiol coverage in our results. |
774 |
|
775 |
\subsection{Influence of Chosen Molecule Model on $G$} |
776 |
In addition to UA solvent/capping agent models, AA models are included |
777 |
in our simulations as well. Besides simulations of the same (UA or AA) |
778 |
model for solvent and capping agent, different models can be applied |
779 |
to different components. Furthermore, regardless of models chosen, |
780 |
either the solvent or the capping agent can be deuterated, similar to |
781 |
the previous section. Table \ref{modelTest} summarizes the results of |
782 |
these studies. |
783 |
|
784 |
\begin{table*} |
785 |
\begin{minipage}{\linewidth} |
786 |
\begin{center} |
787 |
|
788 |
\caption{Computed interfacial thermal conductivity ($G$ and |
789 |
$G^\prime$) values for interfaces using various models for |
790 |
solvent and capping agent (or without capping agent) at |
791 |
$\langle T\rangle\sim$200K. (D stands for deuterated solvent |
792 |
or capping agent molecules; ``Avg.'' denotes results that are |
793 |
averages of simulations under different $J_z$'s. Error |
794 |
estimates indicated in parenthesis.)} |
795 |
|
796 |
\begin{tabular}{llccc} |
797 |
\hline\hline |
798 |
Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ |
799 |
(or bare surface) & model & (GW/m$^2$) & |
800 |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
801 |
\hline |
802 |
UA & UA hexane & Avg. & 131(9) & 87(10) \\ |
803 |
& UA hexane(D) & 1.95 & 153(5) & 136(13) \\ |
804 |
& AA hexane & Avg. & 131(6) & 122(10) \\ |
805 |
& UA toluene & 1.96 & 187(16) & 151(11) \\ |
806 |
& AA toluene & 1.89 & 200(36) & 149(53) \\ |
807 |
\hline |
808 |
AA & UA hexane & 1.94 & 116(9) & 129(8) \\ |
809 |
& AA hexane & Avg. & 442(14) & 356(31) \\ |
810 |
& AA hexane(D) & 1.93 & 222(12) & 234(54) \\ |
811 |
& UA toluene & 1.98 & 125(25) & 97(60) \\ |
812 |
& AA toluene & 3.79 & 487(56) & 290(42) \\ |
813 |
\hline |
814 |
AA(D) & UA hexane & 1.94 & 158(25) & 172(4) \\ |
815 |
& AA hexane & 1.92 & 243(29) & 191(11) \\ |
816 |
& AA toluene & 1.93 & 364(36) & 322(67) \\ |
817 |
\hline |
818 |
bare & UA hexane & Avg. & 46.5(3.2) & 49.4(4.5) \\ |
819 |
& UA hexane(D) & 0.98 & 43.9(4.6) & 43.0(2.0) \\ |
820 |
& AA hexane & 0.96 & 31.0(1.4) & 29.4(1.3) \\ |
821 |
& UA toluene & 1.99 & 70.1(1.3) & 65.8(0.5) \\ |
822 |
\hline\hline |
823 |
\end{tabular} |
824 |
\label{modelTest} |
825 |
\end{center} |
826 |
\end{minipage} |
827 |
\end{table*} |
828 |
|
829 |
To facilitate direct comparison, the same system with differnt models |
830 |
for different components uses the same length scale for their |
831 |
simulation cells. Without the presence of capping agent, using |
832 |
different models for hexane yields similar results for both $G$ and |
833 |
$G^\prime$, and these two definitions agree with eath other very |
834 |
well. This indicates very weak interaction between the metal and the |
835 |
solvent, and is a typical case for acoustic impedance mismatch between |
836 |
these two phases. |
837 |
|
838 |
As for Au(111) surfaces completely covered by butanethiols, the choice |
839 |
of models for capping agent and solvent could impact the measurement |
840 |
of $G$ and $G^\prime$ quite significantly. For Au-butanethiol/hexane |
841 |
interfaces, using AA model for both butanethiol and hexane yields |
842 |
substantially higher conductivity values than using UA model for at |
843 |
least one component of the solvent and capping agent, which exceeds |
844 |
the general range of experimental measurement results. This is |
845 |
probably due to the classically treated C-H vibrations in the AA |
846 |
model, which should not be appreciably populated at normal |
847 |
temperatures. In comparison, once either the hexanes or the |
848 |
butanethiols are deuterated, one can see a significantly lower $G$ and |
849 |
$G^\prime$. In either of these cases, the C-H(D) vibrational overlap |
850 |
between the solvent and the capping agent is removed (Figure |
851 |
\ref{aahxntln}). Conclusively, the improperly treated C-H vibration in |
852 |
the AA model produced over-predicted results accordingly. Compared to |
853 |
the AA model, the UA model yields more reasonable results with higher |
854 |
computational efficiency. |
855 |
|
856 |
\begin{figure} |
857 |
\includegraphics[width=\linewidth]{aahxntln} |
858 |
\caption{Spectra obtained for All-Atom model Au-butanethil/solvent |
859 |
systems. When butanethiol is deuterated (lower left), its |
860 |
vibrational overlap with hexane would decrease significantly, |
861 |
compared with normal butanethiol (upper left). However, this |
862 |
dramatic change does not apply to toluene as much (right).} |
863 |
\label{aahxntln} |
864 |
\end{figure} |
865 |
|
866 |
However, for Au-butanethiol/toluene interfaces, having the AA |
867 |
butanethiol deuterated did not yield a significant change in the |
868 |
measurement results. Compared to the C-H vibrational overlap between |
869 |
hexane and butanethiol, both of which have alkyl chains, that overlap |
870 |
between toluene and butanethiol is not so significant and thus does |
871 |
not have as much contribution to the heat exchange |
872 |
process. Conversely, extra degrees of freedom such as the C-H |
873 |
vibrations could yield higher heat exchange rate between these two |
874 |
phases and result in a much higher conductivity. |
875 |
|
876 |
Although the QSC model for Au is known to predict an overly low value |
877 |
for bulk metal gold conductivity\cite{kuang:164101}, our computational |
878 |
results for $G$ and $G^\prime$ do not seem to be affected by this |
879 |
drawback of the model for metal. Instead, our results suggest that the |
880 |
modeling of interfacial thermal transport behavior relies mainly on |
881 |
the accuracy of the interaction descriptions between components |
882 |
occupying the interfaces. |
883 |
|
884 |
\subsection{Role of Capping Agent in Interfacial Thermal Conductance} |
885 |
The vibrational spectra for gold slabs in different environments are |
886 |
shown as in Figure \ref{specAu}. Regardless of the presence of |
887 |
solvent, the gold surfaces covered by butanethiol molecules, compared |
888 |
to bare gold surfaces, exhibit an additional peak observed at the |
889 |
frequency of $\sim$170cm$^{-1}$, which is attributed to the S-Au |
890 |
bonding vibration. This vibration enables efficient thermal transport |
891 |
from surface Au layer to the capping agents. Therefore, in our |
892 |
simulations, the Au/S interfaces do not appear major heat barriers |
893 |
compared to the butanethiol / solvent interfaces. |
894 |
|
895 |
Simultaneously, the vibrational overlap between butanethiol and |
896 |
organic solvents suggests higher thermal exchange efficiency between |
897 |
these two components. Even exessively high heat transport was observed |
898 |
when All-Atom models were used and C-H vibrations were treated |
899 |
classically. Compared to metal and organic liquid phase, the heat |
900 |
transfer efficiency between butanethiol and organic solvents is closer |
901 |
to that within bulk liquid phase. |
902 |
|
903 |
Furthermore, our observation validated previous |
904 |
results\cite{hase:2010} that the intramolecular heat transport of |
905 |
alkylthiols is highly effecient. As a combinational effects of these |
906 |
phenomena, butanethiol acts as a channel to expedite thermal transport |
907 |
process. The acoustic impedance mismatch between the metal and the |
908 |
liquid phase can be effectively reduced with the presence of suitable |
909 |
capping agents. |
910 |
|
911 |
\begin{figure} |
912 |
\includegraphics[width=\linewidth]{vibration} |
913 |
\caption{Vibrational spectra obtained for gold in different |
914 |
environments.} |
915 |
\label{specAu} |
916 |
\end{figure} |
917 |
|
918 |
[MAY ADD COMPARISON OF AU SLAB WIDTHS] |
919 |
|
920 |
\section{Conclusions} |
921 |
The NIVS algorithm we developed has been applied to simulations of |
922 |
Au-butanethiol surfaces with organic solvents. This algorithm allows |
923 |
effective unphysical thermal flux transferred between the metal and |
924 |
the liquid phase. With the flux applied, we were able to measure the |
925 |
corresponding thermal gradient and to obtain interfacial thermal |
926 |
conductivities. Under steady states, single trajectory simulation |
927 |
would be enough for accurate measurement. This would be advantageous |
928 |
compared to transient state simulations, which need multiple |
929 |
trajectories to produce reliable average results. |
930 |
|
931 |
Our simulations have seen significant conductance enhancement with the |
932 |
presence of capping agent, compared to the bare gold / liquid |
933 |
interfaces. The acoustic impedance mismatch between the metal and the |
934 |
liquid phase is effectively eliminated by proper capping |
935 |
agent. Furthermore, the coverage precentage of the capping agent plays |
936 |
an important role in the interfacial thermal transport |
937 |
process. Moderately lower coverages allow higher contact between |
938 |
capping agent and solvent, and thus could further enhance the heat |
939 |
transfer process. |
940 |
|
941 |
Our measurement results, particularly of the UA models, agree with |
942 |
available experimental data. This indicates that our force field |
943 |
parameters have a nice description of the interactions between the |
944 |
particles at the interfaces. AA models tend to overestimate the |
945 |
interfacial thermal conductance in that the classically treated C-H |
946 |
vibration would be overly sampled. Compared to the AA models, the UA |
947 |
models have higher computational efficiency with satisfactory |
948 |
accuracy, and thus are preferable in interfacial thermal transport |
949 |
modelings. Of the two definitions for $G$, the discrete form |
950 |
(Eq. \ref{discreteG}) was easier to use and gives out relatively |
951 |
consistent results, while the derivative form (Eq. \ref{derivativeG}) |
952 |
is not as versatile. Although $G^\prime$ gives out comparable results |
953 |
and follows similar trend with $G$ when measuring close to fully |
954 |
covered or bare surfaces, the spatial resolution of $T$ profile is |
955 |
limited for accurate computation of derivatives data. |
956 |
|
957 |
Vlugt {\it et al.} has investigated the surface thiol structures for |
958 |
nanocrystal gold and pointed out that they differs from those of the |
959 |
Au(111) surface\cite{vlugt:cpc2007154}. This difference might lead to |
960 |
change of interfacial thermal transport behavior as well. To |
961 |
investigate this problem, an effective means to introduce thermal flux |
962 |
and measure the corresponding thermal gradient is desirable for |
963 |
simulating structures with spherical symmetry. |
964 |
|
965 |
\section{Acknowledgments} |
966 |
Support for this project was provided by the National Science |
967 |
Foundation under grant CHE-0848243. Computational time was provided by |
968 |
the Center for Research Computing (CRC) at the University of Notre |
969 |
Dame. \newpage |
970 |
|
971 |
\bibliography{interfacial} |
972 |
|
973 |
\end{doublespace} |
974 |
\end{document} |
975 |
|