1 |
gezelter |
3769 |
\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 |
skuang |
3770 |
\title{ENTER TITLE HERE} |
32 |
gezelter |
3769 |
|
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 |
skuang |
3770 |
REPLACE ABSTRACT HERE |
47 |
gezelter |
3769 |
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 |
skuang |
3770 |
[DO THIS LATER] |
78 |
|
|
|
79 |
|
|
[IMPORTANCE OF NANOSCALE TRANSPORT PROPERTIES STUDIES] |
80 |
|
|
|
81 |
gezelter |
3769 |
Due to the importance of heat flow (and heat removal) in |
82 |
|
|
nanotechnology, interfacial thermal conductance has been studied |
83 |
|
|
extensively both experimentally and computationally.\cite{cahill:793} |
84 |
|
|
Nanoscale materials have a significant fraction of their atoms at |
85 |
|
|
interfaces, and the chemical details of these interfaces govern the |
86 |
|
|
thermal transport properties. Furthermore, the interfaces are often |
87 |
|
|
heterogeneous (e.g. solid - liquid), which provides a challenge to |
88 |
|
|
computational methods which have been developed for homogeneous or |
89 |
|
|
bulk systems. |
90 |
|
|
|
91 |
|
|
Experimentally, the thermal properties of a number of interfaces have |
92 |
|
|
been investigated. Cahill and coworkers studied nanoscale thermal |
93 |
|
|
transport from metal nanoparticle/fluid interfaces, to epitaxial |
94 |
|
|
TiN/single crystal oxides interfaces, and hydrophilic and hydrophobic |
95 |
|
|
interfaces between water and solids with different self-assembled |
96 |
|
|
monolayers.\cite{Wilson:2002uq,PhysRevB.67.054302,doi:10.1021/jp048375k,PhysRevLett.96.186101} |
97 |
|
|
Wang {\it et al.} studied heat transport through long-chain |
98 |
|
|
hydrocarbon monolayers on gold substrate at individual molecular |
99 |
|
|
level,\cite{Wang10082007} Schmidt {\it et al.} studied the role of |
100 |
|
|
cetyltrimethylammonium bromide (CTAB) on the thermal transport between |
101 |
|
|
gold nanorods and solvent,\cite{doi:10.1021/jp8051888} and Juv\'e {\it |
102 |
|
|
et al.} studied the cooling dynamics, which is controlled by thermal |
103 |
|
|
interface resistance of glass-embedded metal |
104 |
|
|
nanoparticles.\cite{PhysRevB.80.195406} Although interfaces are |
105 |
|
|
normally considered barriers for heat transport, Alper {\it et al.} |
106 |
|
|
suggested that specific ligands (capping agents) could completely |
107 |
|
|
eliminate this barrier |
108 |
|
|
($G\rightarrow\infty$).\cite{doi:10.1021/la904855s} |
109 |
|
|
|
110 |
|
|
The acoustic mismatch model for interfacial conductance utilizes the |
111 |
|
|
acoustic impedance ($Z_a = \rho_a v^s_a$) on both sides of the |
112 |
|
|
interface.\cite{swartz1989} Here, $\rho_a$ and $v^s_a$ are the density |
113 |
|
|
and speed of sound in material $a$. The phonon transmission |
114 |
|
|
probability at the $a-b$ interface is |
115 |
|
|
\begin{equation} |
116 |
|
|
t_{ab} = \frac{4 Z_a Z_b}{\left(Z_a + Z_b \right)^2}, |
117 |
|
|
\end{equation} |
118 |
|
|
and the interfacial conductance can then be approximated as |
119 |
|
|
\begin{equation} |
120 |
|
|
G_{ab} \approx \frac{1}{4} C_D v_D t_{ab} |
121 |
|
|
\end{equation} |
122 |
|
|
where $C_D$ is the Debye heat capacity of the hot side, and $v_D$ is |
123 |
|
|
the Debye phonon velocity ($1/v_D^3 = 1/3v_L^3 + 2/3 v_T^3$) where |
124 |
|
|
$v_L$ and $v_T$ are the longitudinal and transverse speeds of sound, |
125 |
|
|
respectively. For the Au/hexane and Au/toluene interfaces, the |
126 |
|
|
acoustic mismatch model predicts room-temperature $G \approx 87 \mbox{ |
127 |
|
|
and } 129$ MW m$^{-2}$ K$^{-1}$, respectively. However, it is not |
128 |
|
|
clear how to apply the acoustic mismatch model to a |
129 |
|
|
chemically-modified surface, particularly when the acoustic properties |
130 |
|
|
of a monolayer film may not be well characterized. |
131 |
|
|
|
132 |
skuang |
3770 |
[PREVIOUS METHODS INCLUDING NIVS AND THEIR LIMITATIONS] |
133 |
|
|
[DIFFICULTY TO GENERATE JZKE AND JZP SIMUTANEOUSLY] |
134 |
|
|
|
135 |
gezelter |
3769 |
More precise computational models have also been used to study the |
136 |
|
|
interfacial thermal transport in order to gain an understanding of |
137 |
|
|
this phenomena at the molecular level. Recently, Hase and coworkers |
138 |
|
|
employed Non-Equilibrium Molecular Dynamics (NEMD) simulations to |
139 |
|
|
study thermal transport from hot Au(111) substrate to a self-assembled |
140 |
|
|
monolayer of alkylthiol with relatively long chain (8-20 carbon |
141 |
|
|
atoms).\cite{hase:2010,hase:2011} However, ensemble averaged |
142 |
|
|
measurements for heat conductance of interfaces between the capping |
143 |
|
|
monolayer on Au and a solvent phase have yet to be studied with their |
144 |
|
|
approach. The comparatively low thermal flux through interfaces is |
145 |
|
|
difficult to measure with Equilibrium |
146 |
|
|
MD\cite{doi:10.1080/0026897031000068578} or forward NEMD simulation |
147 |
|
|
methods. Therefore, the Reverse NEMD (RNEMD) |
148 |
|
|
methods\cite{MullerPlathe:1997xw,kuang:164101} would be advantageous |
149 |
|
|
in that they {\it apply} the difficult to measure quantity (flux), |
150 |
|
|
while {\it measuring} the easily-computed quantity (the thermal |
151 |
|
|
gradient). This is particularly true for inhomogeneous interfaces |
152 |
|
|
where it would not be clear how to apply a gradient {\it a priori}. |
153 |
|
|
Garde and coworkers\cite{garde:nl2005,garde:PhysRevLett2009} applied |
154 |
|
|
this approach to various liquid interfaces and studied how thermal |
155 |
|
|
conductance (or resistance) is dependent on chemical details of a |
156 |
|
|
number of hydrophobic and hydrophilic aqueous interfaces. And |
157 |
|
|
recently, Luo {\it et al.} studied the thermal conductance of |
158 |
|
|
Au-SAM-Au junctions using the same approach, comparing to a constant |
159 |
|
|
temperature difference method.\cite{Luo20101} While this latter |
160 |
|
|
approach establishes more ideal Maxwell-Boltzmann distributions than |
161 |
|
|
previous RNEMD methods, it does not guarantee momentum or kinetic |
162 |
|
|
energy conservation. |
163 |
|
|
|
164 |
|
|
Recently, we have developed a Non-Isotropic Velocity Scaling (NIVS) |
165 |
|
|
algorithm for RNEMD simulations\cite{kuang:164101}. This algorithm |
166 |
|
|
retains the desirable features of RNEMD (conservation of linear |
167 |
|
|
momentum and total energy, compatibility with periodic boundary |
168 |
|
|
conditions) while establishing true thermal distributions in each of |
169 |
|
|
the two slabs. Furthermore, it allows effective thermal exchange |
170 |
|
|
between particles of different identities, and thus makes the study of |
171 |
|
|
interfacial conductance much simpler. |
172 |
|
|
|
173 |
skuang |
3770 |
[WHAT IS COVERED IN THIS MANUSCRIPT] |
174 |
|
|
[MAY PUT FIGURE 1 HERE] |
175 |
gezelter |
3769 |
The work presented here deals with the Au(111) surface covered to |
176 |
|
|
varying degrees by butanethiol, a capping agent with short carbon |
177 |
|
|
chain, and solvated with organic solvents of different molecular |
178 |
|
|
properties. Different models were used for both the capping agent and |
179 |
|
|
the solvent force field parameters. Using the NIVS algorithm, the |
180 |
|
|
thermal transport across these interfaces was studied and the |
181 |
|
|
underlying mechanism for the phenomena was investigated. |
182 |
|
|
|
183 |
|
|
\section{Methodology} |
184 |
skuang |
3770 |
Similar to the NIVS methodology,\cite{kuang:164101} we consider a |
185 |
|
|
periodic system divided into a series of slabs along a certain axis |
186 |
|
|
(e.g. $z$). The unphysical thermal and/or momentum flux is designated |
187 |
|
|
from the center slab to one of the end slabs, and thus the center slab |
188 |
|
|
would have a lower temperature than the end slab (unless the thermal |
189 |
|
|
flux is negative). Therefore, the center slab is denoted as ``$c$'' |
190 |
|
|
while the end slab as ``$h$''. |
191 |
|
|
|
192 |
|
|
To impose these fluxes, we periodically apply separate operations to |
193 |
|
|
velocities of particles {$i$} within the center slab and of particles |
194 |
|
|
{$j$} within the end slab: |
195 |
|
|
\begin{eqnarray} |
196 |
|
|
\vec{v}_i & \leftarrow & c\cdot\left(\vec{v}_i - \langle\vec{v}_c |
197 |
|
|
\rangle\right) + \left(\langle\vec{v}_c\rangle + \vec{a}_c\right) \\ |
198 |
|
|
\vec{v}_j & \leftarrow & h\cdot\left(\vec{v}_j - \langle\vec{v}_h |
199 |
|
|
\rangle\right) + \left(\langle\vec{v}_h\rangle + \vec{a}_h\right) |
200 |
|
|
\end{eqnarray} |
201 |
|
|
where $\langle\vec{v}_c\rangle$ and $\langle\vec{v}_h\rangle$ denotes |
202 |
|
|
the instantaneous bulk velocity of slabs $c$ and $h$ respectively |
203 |
|
|
before an operation occurs. When a momentum flux $\vec{j}_z(\vec{p})$ |
204 |
|
|
presents, these bulk velocities would have a corresponding change |
205 |
|
|
($\vec{a}_c$ and $\vec{a}_h$ respectively) according to Newton's |
206 |
|
|
second law: |
207 |
|
|
\begin{eqnarray} |
208 |
|
|
M_c \vec{a}_c & = & -\vec{j}_z(\vec{p}) \Delta t \\ |
209 |
|
|
M_h \vec{a}_h & = & \vec{j}_z(\vec{p}) \Delta t |
210 |
|
|
\end{eqnarray} |
211 |
|
|
where |
212 |
|
|
\begin{eqnarray} |
213 |
|
|
M_c & = & \sum_{i = 1}^{N_c} m_i \\ |
214 |
|
|
M_h & = & \sum_{j = 1}^{N_h} m_j |
215 |
|
|
\end{eqnarray} |
216 |
|
|
and $\Delta t$ is the interval between two operations. |
217 |
|
|
|
218 |
|
|
The above operations conserve the linear momentum of a periodic |
219 |
|
|
system. To satisfy total energy conservation as well as to impose a |
220 |
|
|
thermal flux $J_z$, one would have |
221 |
|
|
%SUPPORT INFO MIGHT BE NECESSARY TO PUT EXTRA MATH IN |
222 |
|
|
\begin{eqnarray} |
223 |
|
|
K_c - J_z\Delta t & = & c^2 (K_c - \frac{1}{2}M_c \langle\vec{v}_c |
224 |
|
|
\rangle^2) + \frac{1}{2}M_c (\langle \vec{v}_c \rangle + \vec{a}_c)^2 \\ |
225 |
|
|
K_h + J_z\Delta t & = & h^2 (K_h - \frac{1}{2}M_h \langle\vec{v}_h |
226 |
|
|
\rangle^2) + \frac{1}{2}M_h (\langle \vec{v}_h \rangle + \vec{a}_h)^2 |
227 |
|
|
\end{eqnarray} |
228 |
|
|
where $K_c$ and $K_h$ denotes translational kinetic energy of slabs |
229 |
|
|
$c$ and $h$ respectively before an operation occurs. These |
230 |
|
|
translational kinetic energy conservation equations are sufficient to |
231 |
|
|
ensure total energy conservation, as the operations applied do not |
232 |
|
|
change the potential energy of a system, given that the potential |
233 |
|
|
energy does not depend on particle velocity. |
234 |
|
|
|
235 |
|
|
The above sets of equations are sufficient to determine the velocity |
236 |
|
|
scaling coefficients ($c$ and $h$) as well as $\vec{a}_c$ and |
237 |
|
|
$\vec{a}_h$. Note that two roots of $c$ and $h$ exist |
238 |
|
|
respectively. However, to avoid dramatic perturbations to a system, |
239 |
|
|
the positive roots (which are closer to 1) are chosen. |
240 |
|
|
|
241 |
|
|
By implementing these operations at a certain frequency, a steady |
242 |
|
|
thermal and/or momentum flux can be applied and the corresponding |
243 |
|
|
temperature and/or momentum gradients can be established. |
244 |
|
|
[REFER TO NIVS PAPER] |
245 |
|
|
[ADVANTAGES] |
246 |
|
|
|
247 |
gezelter |
3769 |
Steady state MD simulations have an advantage in that not many |
248 |
|
|
trajectories are needed to study the relationship between thermal flux |
249 |
|
|
and thermal gradients. For systems with low interfacial conductance, |
250 |
|
|
one must have a method capable of generating or measuring relatively |
251 |
|
|
small fluxes, compared to those required for bulk conductivity. This |
252 |
|
|
requirement makes the calculation even more difficult for |
253 |
|
|
slowly-converging equilibrium methods.\cite{Viscardy:2007lq} Forward |
254 |
|
|
NEMD methods impose a gradient (and measure a flux), but at interfaces |
255 |
|
|
it is not clear what behavior should be imposed at the boundaries |
256 |
|
|
between materials. Imposed-flux reverse non-equilibrium |
257 |
|
|
methods\cite{MullerPlathe:1997xw} set the flux {\it a priori} and |
258 |
|
|
the thermal response becomes an easy-to-measure quantity. Although |
259 |
|
|
M\"{u}ller-Plathe's original momentum swapping approach can be used |
260 |
|
|
for exchanging energy between particles of different identity, the |
261 |
|
|
kinetic energy transfer efficiency is affected by the mass difference |
262 |
|
|
between the particles, which limits its application on heterogeneous |
263 |
|
|
interfacial systems. |
264 |
|
|
|
265 |
|
|
The non-isotropic velocity scaling (NIVS) \cite{kuang:164101} approach |
266 |
|
|
to non-equilibrium MD simulations is able to impose a wide range of |
267 |
|
|
kinetic energy fluxes without obvious perturbation to the velocity |
268 |
|
|
distributions of the simulated systems. Furthermore, this approach has |
269 |
|
|
the advantage in heterogeneous interfaces in that kinetic energy flux |
270 |
|
|
can be applied between regions of particles of arbitrary identity, and |
271 |
|
|
the flux will not be restricted by difference in particle mass. |
272 |
|
|
|
273 |
|
|
The NIVS algorithm scales the velocity vectors in two separate regions |
274 |
|
|
of a simulation system with respective diagonal scaling matrices. To |
275 |
|
|
determine these scaling factors in the matrices, a set of equations |
276 |
|
|
including linear momentum conservation and kinetic energy conservation |
277 |
|
|
constraints and target energy flux satisfaction is solved. With the |
278 |
|
|
scaling operation applied to the system in a set frequency, bulk |
279 |
|
|
temperature gradients can be easily established, and these can be used |
280 |
|
|
for computing thermal conductivities. The NIVS algorithm conserves |
281 |
|
|
momenta and energy and does not depend on an external thermostat. |
282 |
|
|
|
283 |
|
|
\subsection{Defining Interfacial Thermal Conductivity ($G$)} |
284 |
|
|
|
285 |
|
|
For an interface with relatively low interfacial conductance, and a |
286 |
|
|
thermal flux between two distinct bulk regions, the regions on either |
287 |
|
|
side of the interface rapidly come to a state in which the two phases |
288 |
|
|
have relatively homogeneous (but distinct) temperatures. The |
289 |
|
|
interfacial thermal conductivity $G$ can therefore be approximated as: |
290 |
|
|
\begin{equation} |
291 |
|
|
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle - |
292 |
|
|
\langle T_\mathrm{cold}\rangle \right)} |
293 |
|
|
\label{lowG} |
294 |
|
|
\end{equation} |
295 |
|
|
where ${E_{total}}$ is the total imposed non-physical kinetic energy |
296 |
|
|
transfer during the simulation and ${\langle T_\mathrm{hot}\rangle}$ |
297 |
|
|
and ${\langle T_\mathrm{cold}\rangle}$ are the average observed |
298 |
|
|
temperature of the two separated phases. For an applied flux $J_z$ |
299 |
|
|
operating over a simulation time $t$ on a periodically-replicated slab |
300 |
|
|
of dimensions $L_x \times L_y$, $E_{total} = 2 J_z t L_x L_y$. |
301 |
|
|
|
302 |
|
|
When the interfacial conductance is {\it not} small, there are two |
303 |
|
|
ways to define $G$. One common way is to assume the temperature is |
304 |
|
|
discrete on the two sides of the interface. $G$ can be calculated |
305 |
|
|
using the applied thermal flux $J$ and the maximum temperature |
306 |
|
|
difference measured along the thermal gradient max($\Delta T$), which |
307 |
|
|
occurs at the Gibbs dividing surface (Figure \ref{demoPic}). This is |
308 |
|
|
known as the Kapitza conductance, which is the inverse of the Kapitza |
309 |
|
|
resistance. |
310 |
|
|
\begin{equation} |
311 |
|
|
G=\frac{J}{\Delta T} |
312 |
|
|
\label{discreteG} |
313 |
|
|
\end{equation} |
314 |
|
|
|
315 |
|
|
\begin{figure} |
316 |
|
|
\includegraphics[width=\linewidth]{method} |
317 |
|
|
\caption{Interfacial conductance can be calculated by applying an |
318 |
|
|
(unphysical) kinetic energy flux between two slabs, one located |
319 |
|
|
within the metal and another on the edge of the periodic box. The |
320 |
|
|
system responds by forming a thermal gradient. In bulk liquids, |
321 |
|
|
this gradient typically has a single slope, but in interfacial |
322 |
|
|
systems, there are distinct thermal conductivity domains. The |
323 |
|
|
interfacial conductance, $G$ is found by measuring the temperature |
324 |
|
|
gap at the Gibbs dividing surface, or by using second derivatives of |
325 |
|
|
the thermal profile.} |
326 |
|
|
\label{demoPic} |
327 |
|
|
\end{figure} |
328 |
|
|
|
329 |
|
|
Another approach is to assume that the temperature is continuous and |
330 |
|
|
differentiable throughout the space. Given that $\lambda$ is also |
331 |
|
|
differentiable, $G$ can be defined as its gradient ($\nabla\lambda$) |
332 |
|
|
projected along a vector normal to the interface ($\mathbf{\hat{n}}$) |
333 |
|
|
and evaluated at the interface location ($z_0$). This quantity, |
334 |
|
|
\begin{align} |
335 |
|
|
G^\prime &= \left(\nabla\lambda \cdot \mathbf{\hat{n}}\right)_{z_0} \\ |
336 |
|
|
&= \frac{\partial}{\partial z}\left(-\frac{J_z}{ |
337 |
|
|
\left(\frac{\partial T}{\partial z}\right)}\right)_{z_0} \\ |
338 |
|
|
&= J_z\left(\frac{\partial^2 T}{\partial z^2}\right)_{z_0} \Big/ |
339 |
|
|
\left(\frac{\partial T}{\partial z}\right)_{z_0}^2 \label{derivativeG} |
340 |
|
|
\end{align} |
341 |
|
|
has the same units as the common definition for $G$, and the maximum |
342 |
|
|
of its magnitude denotes where thermal conductivity has the largest |
343 |
|
|
change, i.e. the interface. In the geometry used in this study, the |
344 |
|
|
vector normal to the interface points along the $z$ axis, as do |
345 |
|
|
$\vec{J}$ and the thermal gradient. This yields the simplified |
346 |
|
|
expressions in Eq. \ref{derivativeG}. |
347 |
|
|
|
348 |
|
|
With temperature profiles obtained from simulation, one is able to |
349 |
|
|
approximate the first and second derivatives of $T$ with finite |
350 |
|
|
difference methods and calculate $G^\prime$. In what follows, both |
351 |
|
|
definitions have been used, and are compared in the results. |
352 |
|
|
|
353 |
|
|
To investigate the interfacial conductivity at metal / solvent |
354 |
|
|
interfaces, we have modeled a metal slab with its (111) surfaces |
355 |
|
|
perpendicular to the $z$-axis of our simulation cells. The metal slab |
356 |
|
|
has been prepared both with and without capping agents on the exposed |
357 |
|
|
surface, and has been solvated with simple organic solvents, as |
358 |
|
|
illustrated in Figure \ref{gradT}. |
359 |
|
|
|
360 |
|
|
With the simulation cell described above, we are able to equilibrate |
361 |
|
|
the system and impose an unphysical thermal flux between the liquid |
362 |
|
|
and the metal phase using the NIVS algorithm. By periodically applying |
363 |
|
|
the unphysical flux, we obtained a temperature profile and its spatial |
364 |
|
|
derivatives. Figure \ref{gradT} shows how an applied thermal flux can |
365 |
|
|
be used to obtain the 1st and 2nd derivatives of the temperature |
366 |
|
|
profile. |
367 |
|
|
|
368 |
|
|
\begin{figure} |
369 |
|
|
\includegraphics[width=\linewidth]{gradT} |
370 |
|
|
\caption{A sample of Au (111) / butanethiol / hexane interfacial |
371 |
|
|
system with the temperature profile after a kinetic energy flux has |
372 |
|
|
been imposed. Note that the largest temperature jump in the thermal |
373 |
|
|
profile (corresponding to the lowest interfacial conductance) is at |
374 |
|
|
the interface between the butanethiol molecules (blue) and the |
375 |
|
|
solvent (grey). First and second derivatives of the temperature |
376 |
|
|
profile are obtained using a finite difference approximation (lower |
377 |
|
|
panel).} |
378 |
|
|
\label{gradT} |
379 |
|
|
\end{figure} |
380 |
|
|
|
381 |
|
|
\section{Computational Details} |
382 |
|
|
\subsection{Simulation Protocol} |
383 |
|
|
The NIVS algorithm has been implemented in our MD simulation code, |
384 |
|
|
OpenMD\cite{Meineke:2005gd,openmd}, and was used for our simulations. |
385 |
|
|
Metal slabs of 6 or 11 layers of Au atoms were first equilibrated |
386 |
|
|
under atmospheric pressure (1 atm) and 200K. After equilibration, |
387 |
|
|
butanethiol capping agents were placed at three-fold hollow sites on |
388 |
|
|
the Au(111) surfaces. These sites are either {\it fcc} or {\it |
389 |
|
|
hcp} sites, although Hase {\it et al.} found that they are |
390 |
|
|
equivalent in a heat transfer process,\cite{hase:2010} so we did not |
391 |
|
|
distinguish between these sites in our study. The maximum butanethiol |
392 |
|
|
capacity on Au surface is $1/3$ of the total number of surface Au |
393 |
|
|
atoms, and the packing forms a $(\sqrt{3}\times\sqrt{3})R30^\circ$ |
394 |
|
|
structure\cite{doi:10.1021/ja00008a001,doi:10.1021/cr9801317}. A |
395 |
|
|
series of lower coverages was also prepared by eliminating |
396 |
|
|
butanethiols from the higher coverage surface in a regular manner. The |
397 |
|
|
lower coverages were prepared in order to study the relation between |
398 |
|
|
coverage and interfacial conductance. |
399 |
|
|
|
400 |
|
|
The capping agent molecules were allowed to migrate during the |
401 |
|
|
simulations. They distributed themselves uniformly and sampled a |
402 |
|
|
number of three-fold sites throughout out study. Therefore, the |
403 |
|
|
initial configuration does not noticeably affect the sampling of a |
404 |
|
|
variety of configurations of the same coverage, and the final |
405 |
|
|
conductance measurement would be an average effect of these |
406 |
|
|
configurations explored in the simulations. |
407 |
|
|
|
408 |
|
|
After the modified Au-butanethiol surface systems were equilibrated in |
409 |
|
|
the canonical (NVT) ensemble, organic solvent molecules were packed in |
410 |
|
|
the previously empty part of the simulation cells.\cite{packmol} Two |
411 |
|
|
solvents were investigated, one which has little vibrational overlap |
412 |
|
|
with the alkanethiol and which has a planar shape (toluene), and one |
413 |
|
|
which has similar vibrational frequencies to the capping agent and |
414 |
|
|
chain-like shape ({\it n}-hexane). |
415 |
|
|
|
416 |
|
|
The simulation cells were not particularly extensive along the |
417 |
|
|
$z$-axis, as a very long length scale for the thermal gradient may |
418 |
|
|
cause excessively hot or cold temperatures in the middle of the |
419 |
|
|
solvent region and lead to undesired phenomena such as solvent boiling |
420 |
|
|
or freezing when a thermal flux is applied. Conversely, too few |
421 |
|
|
solvent molecules would change the normal behavior of the liquid |
422 |
|
|
phase. Therefore, our $N_{solvent}$ values were chosen to ensure that |
423 |
|
|
these extreme cases did not happen to our simulations. The spacing |
424 |
|
|
between periodic images of the gold interfaces is $45 \sim 75$\AA in |
425 |
|
|
our simulations. |
426 |
|
|
|
427 |
|
|
The initial configurations generated are further equilibrated with the |
428 |
|
|
$x$ and $y$ dimensions fixed, only allowing the $z$-length scale to |
429 |
|
|
change. This is to ensure that the equilibration of liquid phase does |
430 |
|
|
not affect the metal's crystalline structure. Comparisons were made |
431 |
|
|
with simulations that allowed changes of $L_x$ and $L_y$ during NPT |
432 |
|
|
equilibration. No substantial changes in the box geometry were noticed |
433 |
|
|
in these simulations. After ensuring the liquid phase reaches |
434 |
|
|
equilibrium at atmospheric pressure (1 atm), further equilibration was |
435 |
|
|
carried out under canonical (NVT) and microcanonical (NVE) ensembles. |
436 |
|
|
|
437 |
|
|
After the systems reach equilibrium, NIVS was used to impose an |
438 |
|
|
unphysical thermal flux between the metal and the liquid phases. Most |
439 |
|
|
of our simulations were done under an average temperature of |
440 |
|
|
$\sim$200K. Therefore, thermal flux usually came from the metal to the |
441 |
|
|
liquid so that the liquid has a higher temperature and would not |
442 |
|
|
freeze due to lowered temperatures. After this induced temperature |
443 |
|
|
gradient had stabilized, the temperature profile of the simulation cell |
444 |
|
|
was recorded. To do this, the simulation cell is divided evenly into |
445 |
|
|
$N$ slabs along the $z$-axis. The average temperatures of each slab |
446 |
|
|
are recorded for 1$\sim$2 ns. When the slab width $d$ of each slab is |
447 |
|
|
the same, the derivatives of $T$ with respect to slab number $n$ can |
448 |
|
|
be directly used for $G^\prime$ calculations: \begin{equation} |
449 |
|
|
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
450 |
|
|
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
451 |
|
|
= |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big| |
452 |
|
|
\Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2 |
453 |
|
|
= |J_z|\Big|\frac{\partial^2 T}{\partial n^2}\Big| |
454 |
|
|
\Big/\left(\frac{\partial T}{\partial n}\right)^2 |
455 |
|
|
\label{derivativeG2} |
456 |
|
|
\end{equation} |
457 |
|
|
The absolute values in Eq. \ref{derivativeG2} appear because the |
458 |
|
|
direction of the flux $\vec{J}$ is in an opposing direction on either |
459 |
|
|
side of the metal slab. |
460 |
|
|
|
461 |
|
|
All of the above simulation procedures use a time step of 1 fs. Each |
462 |
|
|
equilibration stage took a minimum of 100 ps, although in some cases, |
463 |
|
|
longer equilibration stages were utilized. |
464 |
|
|
|
465 |
|
|
\subsection{Force Field Parameters} |
466 |
|
|
Our simulations include a number of chemically distinct components. |
467 |
|
|
Figure \ref{demoMol} demonstrates the sites defined for both |
468 |
|
|
United-Atom and All-Atom models of the organic solvent and capping |
469 |
|
|
agents in our simulations. Force field parameters are needed for |
470 |
|
|
interactions both between the same type of particles and between |
471 |
|
|
particles of different species. |
472 |
|
|
|
473 |
|
|
\begin{figure} |
474 |
|
|
\includegraphics[width=\linewidth]{structures} |
475 |
|
|
\caption{Structures of the capping agent and solvents utilized in |
476 |
|
|
these simulations. The chemically-distinct sites (a-e) are expanded |
477 |
|
|
in terms of constituent atoms for both United Atom (UA) and All Atom |
478 |
|
|
(AA) force fields. Most parameters are from References |
479 |
|
|
\protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes,TraPPE-UA.thiols} |
480 |
|
|
(UA) and \protect\cite{OPLSAA} (AA). Cross-interactions with the Au |
481 |
|
|
atoms are given in Table 1 in the supporting information.} |
482 |
|
|
\label{demoMol} |
483 |
|
|
\end{figure} |
484 |
|
|
|
485 |
|
|
The Au-Au interactions in metal lattice slab is described by the |
486 |
|
|
quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
487 |
|
|
potentials include zero-point quantum corrections and are |
488 |
|
|
reparametrized for accurate surface energies compared to the |
489 |
|
|
Sutton-Chen potentials.\cite{Chen90} |
490 |
|
|
|
491 |
|
|
For the two solvent molecules, {\it n}-hexane and toluene, two |
492 |
|
|
different atomistic models were utilized. Both solvents were modeled |
493 |
|
|
using United-Atom (UA) and All-Atom (AA) models. The TraPPE-UA |
494 |
|
|
parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
495 |
|
|
for our UA solvent molecules. In these models, sites are located at |
496 |
|
|
the carbon centers for alkyl groups. Bonding interactions, including |
497 |
|
|
bond stretches and bends and torsions, were used for intra-molecular |
498 |
|
|
sites closer than 3 bonds. For non-bonded interactions, Lennard-Jones |
499 |
|
|
potentials are used. |
500 |
|
|
|
501 |
|
|
By eliminating explicit hydrogen atoms, the TraPPE-UA models are |
502 |
|
|
simple and computationally efficient, while maintaining good accuracy. |
503 |
|
|
However, the TraPPE-UA model for alkanes is known to predict a slightly |
504 |
|
|
lower boiling point than experimental values. This is one of the |
505 |
|
|
reasons we used a lower average temperature (200K) for our |
506 |
|
|
simulations. If heat is transferred to the liquid phase during the |
507 |
|
|
NIVS simulation, the liquid in the hot slab can actually be |
508 |
|
|
substantially warmer than the mean temperature in the simulation. The |
509 |
|
|
lower mean temperatures therefore prevent solvent boiling. |
510 |
|
|
|
511 |
|
|
For UA-toluene, the non-bonded potentials between intermolecular sites |
512 |
|
|
have a similar Lennard-Jones formulation. The toluene molecules were |
513 |
|
|
treated as a single rigid body, so there was no need for |
514 |
|
|
intramolecular interactions (including bonds, bends, or torsions) in |
515 |
|
|
this solvent model. |
516 |
|
|
|
517 |
|
|
Besides the TraPPE-UA models, AA models for both organic solvents are |
518 |
|
|
included in our studies as well. The OPLS-AA\cite{OPLSAA} force fields |
519 |
|
|
were used. For hexane, additional explicit hydrogen sites were |
520 |
|
|
included. Besides bonding and non-bonded site-site interactions, |
521 |
|
|
partial charges and the electrostatic interactions were added to each |
522 |
|
|
CT and HC site. For toluene, a flexible model for the toluene molecule |
523 |
|
|
was utilized which included bond, bend, torsion, and inversion |
524 |
|
|
potentials to enforce ring planarity. |
525 |
|
|
|
526 |
|
|
The butanethiol capping agent in our simulations, were also modeled |
527 |
|
|
with both UA and AA model. The TraPPE-UA force field includes |
528 |
|
|
parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for |
529 |
|
|
UA butanethiol model in our simulations. The OPLS-AA also provides |
530 |
|
|
parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) |
531 |
|
|
surfaces do not have the hydrogen atom bonded to sulfur. To derive |
532 |
|
|
suitable parameters for butanethiol adsorbed on Au(111) surfaces, we |
533 |
|
|
adopt the S parameters from Luedtke and Landman\cite{landman:1998} and |
534 |
|
|
modify the parameters for the CTS atom to maintain charge neutrality |
535 |
|
|
in the molecule. Note that the model choice (UA or AA) for the capping |
536 |
|
|
agent can be different from the solvent. Regardless of model choice, |
537 |
|
|
the force field parameters for interactions between capping agent and |
538 |
|
|
solvent can be derived using Lorentz-Berthelot Mixing Rule: |
539 |
|
|
\begin{eqnarray} |
540 |
|
|
\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\ |
541 |
|
|
\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}} |
542 |
|
|
\end{eqnarray} |
543 |
|
|
|
544 |
|
|
To describe the interactions between metal (Au) and non-metal atoms, |
545 |
|
|
we refer to an adsorption study of alkyl thiols on gold surfaces by |
546 |
|
|
Vlugt {\it et al.}\cite{vlugt:cpc2007154} They fitted an effective |
547 |
|
|
Lennard-Jones form of potential parameters for the interaction between |
548 |
|
|
Au and pseudo-atoms CH$_x$ and S based on a well-established and |
549 |
|
|
widely-used effective potential of Hautman and Klein for the Au(111) |
550 |
|
|
surface.\cite{hautman:4994} As our simulations require the gold slab |
551 |
|
|
to be flexible to accommodate thermal excitation, the pair-wise form |
552 |
|
|
of potentials they developed was used for our study. |
553 |
|
|
|
554 |
|
|
The potentials developed from {\it ab initio} calculations by Leng |
555 |
|
|
{\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
556 |
|
|
interactions between Au and aromatic C/H atoms in toluene. However, |
557 |
|
|
the Lennard-Jones parameters between Au and other types of particles, |
558 |
|
|
(e.g. AA alkanes) have not yet been established. For these |
559 |
|
|
interactions, the Lorentz-Berthelot mixing rule can be used to derive |
560 |
|
|
effective single-atom LJ parameters for the metal using the fit values |
561 |
|
|
for toluene. These are then used to construct reasonable mixing |
562 |
|
|
parameters for the interactions between the gold and other atoms. |
563 |
|
|
Table 1 in the supporting information summarizes the |
564 |
|
|
``metal/non-metal'' parameters utilized in our simulations. |
565 |
|
|
|
566 |
|
|
\section{Results} |
567 |
skuang |
3770 |
[L-J COMPARED TO RENMD NIVS; WATER COMPARED TO RNEMD NIVS; |
568 |
|
|
SLIP BOUNDARY VS STICK BOUNDARY; ICE-WATER INTERFACES] |
569 |
|
|
|
570 |
gezelter |
3769 |
There are many factors contributing to the measured interfacial |
571 |
|
|
conductance; some of these factors are physically motivated |
572 |
|
|
(e.g. coverage of the surface by the capping agent coverage and |
573 |
|
|
solvent identity), while some are governed by parameters of the |
574 |
|
|
methodology (e.g. applied flux and the formulas used to obtain the |
575 |
|
|
conductance). In this section we discuss the major physical and |
576 |
|
|
calculational effects on the computed conductivity. |
577 |
|
|
|
578 |
|
|
\subsection{Effects due to capping agent coverage} |
579 |
|
|
|
580 |
|
|
A series of different initial conditions with a range of surface |
581 |
|
|
coverages was prepared and solvated with various with both of the |
582 |
|
|
solvent molecules. These systems were then equilibrated and their |
583 |
|
|
interfacial thermal conductivity was measured with the NIVS |
584 |
|
|
algorithm. Figure \ref{coverage} demonstrates the trend of conductance |
585 |
|
|
with respect to surface coverage. |
586 |
|
|
|
587 |
|
|
\begin{figure} |
588 |
|
|
\includegraphics[width=\linewidth]{coverage} |
589 |
|
|
\caption{The interfacial thermal conductivity ($G$) has a |
590 |
|
|
non-monotonic dependence on the degree of surface capping. This |
591 |
|
|
data is for the Au(111) / butanethiol / solvent interface with |
592 |
|
|
various UA force fields at $\langle T\rangle \sim $200K.} |
593 |
|
|
\label{coverage} |
594 |
|
|
\end{figure} |
595 |
|
|
|
596 |
|
|
In partially covered surfaces, the derivative definition for |
597 |
|
|
$G^\prime$ (Eq. \ref{derivativeG}) becomes difficult to apply, as the |
598 |
|
|
location of maximum change of $\lambda$ becomes washed out. The |
599 |
|
|
discrete definition (Eq. \ref{discreteG}) is easier to apply, as the |
600 |
|
|
Gibbs dividing surface is still well-defined. Therefore, $G$ (not |
601 |
|
|
$G^\prime$) was used in this section. |
602 |
|
|
|
603 |
|
|
From Figure \ref{coverage}, one can see the significance of the |
604 |
|
|
presence of capping agents. When even a small fraction of the Au(111) |
605 |
|
|
surface sites are covered with butanethiols, the conductivity exhibits |
606 |
|
|
an enhancement by at least a factor of 3. Capping agents are clearly |
607 |
|
|
playing a major role in thermal transport at metal / organic solvent |
608 |
|
|
surfaces. |
609 |
|
|
|
610 |
|
|
We note a non-monotonic behavior in the interfacial conductance as a |
611 |
|
|
function of surface coverage. The maximum conductance (largest $G$) |
612 |
|
|
happens when the surfaces are about 75\% covered with butanethiol |
613 |
|
|
caps. The reason for this behavior is not entirely clear. One |
614 |
|
|
explanation is that incomplete butanethiol coverage allows small gaps |
615 |
|
|
between butanethiols to form. These gaps can be filled by transient |
616 |
|
|
solvent molecules. These solvent molecules couple very strongly with |
617 |
|
|
the hot capping agent molecules near the surface, and can then carry |
618 |
|
|
away (diffusively) the excess thermal energy from the surface. |
619 |
|
|
|
620 |
|
|
There appears to be a competition between the conduction of the |
621 |
|
|
thermal energy away from the surface by the capping agents (enhanced |
622 |
|
|
by greater coverage) and the coupling of the capping agents with the |
623 |
|
|
solvent (enhanced by interdigitation at lower coverages). This |
624 |
|
|
competition would lead to the non-monotonic coverage behavior observed |
625 |
|
|
here. |
626 |
|
|
|
627 |
|
|
Results for rigid body toluene solvent, as well as the UA hexane, are |
628 |
|
|
within the ranges expected from prior experimental |
629 |
|
|
work.\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406} This suggests |
630 |
|
|
that explicit hydrogen atoms might not be required for modeling |
631 |
|
|
thermal transport in these systems. C-H vibrational modes do not see |
632 |
|
|
significant excited state population at low temperatures, and are not |
633 |
|
|
likely to carry lower frequency excitations from the solid layer into |
634 |
|
|
the bulk liquid. |
635 |
|
|
|
636 |
|
|
The toluene solvent does not exhibit the same behavior as hexane in |
637 |
|
|
that $G$ remains at approximately the same magnitude when the capping |
638 |
|
|
coverage increases from 25\% to 75\%. Toluene, as a rigid planar |
639 |
|
|
molecule, cannot occupy the relatively small gaps between the capping |
640 |
|
|
agents as easily as the chain-like {\it n}-hexane. The effect of |
641 |
|
|
solvent coupling to the capping agent is therefore weaker in toluene |
642 |
|
|
except at the very lowest coverage levels. This effect counters the |
643 |
|
|
coverage-dependent conduction of heat away from the metal surface, |
644 |
|
|
leading to a much flatter $G$ vs. coverage trend than is observed in |
645 |
|
|
{\it n}-hexane. |
646 |
|
|
|
647 |
|
|
\subsection{Effects due to Solvent \& Solvent Models} |
648 |
|
|
In addition to UA solvent and capping agent models, AA models have |
649 |
|
|
also been included in our simulations. In most of this work, the same |
650 |
|
|
(UA or AA) model for solvent and capping agent was used, but it is |
651 |
|
|
also possible to utilize different models for different components. |
652 |
|
|
We have also included isotopic substitutions (Hydrogen to Deuterium) |
653 |
|
|
to decrease the explicit vibrational overlap between solvent and |
654 |
|
|
capping agent. Table \ref{modelTest} summarizes the results of these |
655 |
|
|
studies. |
656 |
|
|
|
657 |
|
|
\begin{table*} |
658 |
|
|
\begin{minipage}{\linewidth} |
659 |
|
|
\begin{center} |
660 |
|
|
|
661 |
|
|
\caption{Computed interfacial thermal conductance ($G$ and |
662 |
|
|
$G^\prime$) values for interfaces using various models for |
663 |
|
|
solvent and capping agent (or without capping agent) at |
664 |
|
|
$\langle T\rangle\sim$200K. Here ``D'' stands for deuterated |
665 |
|
|
solvent or capping agent molecules. Error estimates are |
666 |
|
|
indicated in parentheses.} |
667 |
|
|
|
668 |
|
|
\begin{tabular}{llccc} |
669 |
|
|
\hline\hline |
670 |
|
|
Butanethiol model & Solvent & $G$ & $G^\prime$ \\ |
671 |
|
|
(or bare surface) & model & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
672 |
|
|
\hline |
673 |
|
|
UA & UA hexane & 131(9) & 87(10) \\ |
674 |
|
|
& UA hexane(D) & 153(5) & 136(13) \\ |
675 |
|
|
& AA hexane & 131(6) & 122(10) \\ |
676 |
|
|
& UA toluene & 187(16) & 151(11) \\ |
677 |
|
|
& AA toluene & 200(36) & 149(53) \\ |
678 |
|
|
\hline |
679 |
|
|
AA & UA hexane & 116(9) & 129(8) \\ |
680 |
|
|
& AA hexane & 442(14) & 356(31) \\ |
681 |
|
|
& AA hexane(D) & 222(12) & 234(54) \\ |
682 |
|
|
& UA toluene & 125(25) & 97(60) \\ |
683 |
|
|
& AA toluene & 487(56) & 290(42) \\ |
684 |
|
|
\hline |
685 |
|
|
AA(D) & UA hexane & 158(25) & 172(4) \\ |
686 |
|
|
& AA hexane & 243(29) & 191(11) \\ |
687 |
|
|
& AA toluene & 364(36) & 322(67) \\ |
688 |
|
|
\hline |
689 |
|
|
bare & UA hexane & 46.5(3.2) & 49.4(4.5) \\ |
690 |
|
|
& UA hexane(D) & 43.9(4.6) & 43.0(2.0) \\ |
691 |
|
|
& AA hexane & 31.0(1.4) & 29.4(1.3) \\ |
692 |
|
|
& UA toluene & 70.1(1.3) & 65.8(0.5) \\ |
693 |
|
|
\hline\hline |
694 |
|
|
\end{tabular} |
695 |
|
|
\label{modelTest} |
696 |
|
|
\end{center} |
697 |
|
|
\end{minipage} |
698 |
|
|
\end{table*} |
699 |
|
|
|
700 |
|
|
To facilitate direct comparison between force fields, systems with the |
701 |
|
|
same capping agent and solvent were prepared with the same length |
702 |
|
|
scales for the simulation cells. |
703 |
|
|
|
704 |
|
|
On bare metal / solvent surfaces, different force field models for |
705 |
|
|
hexane yield similar results for both $G$ and $G^\prime$, and these |
706 |
|
|
two definitions agree with each other very well. This is primarily an |
707 |
|
|
indicator of weak interactions between the metal and the solvent. |
708 |
|
|
|
709 |
|
|
For the fully-covered surfaces, the choice of force field for the |
710 |
|
|
capping agent and solvent has a large impact on the calculated values |
711 |
|
|
of $G$ and $G^\prime$. The AA thiol to AA solvent conductivities are |
712 |
|
|
much larger than their UA to UA counterparts, and these values exceed |
713 |
|
|
the experimental estimates by a large measure. The AA force field |
714 |
|
|
allows significant energy to go into C-H (or C-D) stretching modes, |
715 |
|
|
and since these modes are high frequency, this non-quantum behavior is |
716 |
|
|
likely responsible for the overestimate of the conductivity. Compared |
717 |
|
|
to the AA model, the UA model yields more reasonable conductivity |
718 |
|
|
values with much higher computational efficiency. |
719 |
|
|
|
720 |
|
|
\subsubsection{Are electronic excitations in the metal important?} |
721 |
|
|
Because they lack electronic excitations, the QSC and related embedded |
722 |
|
|
atom method (EAM) models for gold are known to predict unreasonably |
723 |
|
|
low values for bulk conductivity |
724 |
|
|
($\lambda$).\cite{kuang:164101,ISI:000207079300006,Clancy:1992} If the |
725 |
|
|
conductance between the phases ($G$) is governed primarily by phonon |
726 |
|
|
excitation (and not electronic degrees of freedom), one would expect a |
727 |
|
|
classical model to capture most of the interfacial thermal |
728 |
|
|
conductance. Our results for $G$ and $G^\prime$ indicate that this is |
729 |
|
|
indeed the case, and suggest that the modeling of interfacial thermal |
730 |
|
|
transport depends primarily on the description of the interactions |
731 |
|
|
between the various components at the interface. When the metal is |
732 |
|
|
chemically capped, the primary barrier to thermal conductivity appears |
733 |
|
|
to be the interface between the capping agent and the surrounding |
734 |
|
|
solvent, so the excitations in the metal have little impact on the |
735 |
|
|
value of $G$. |
736 |
|
|
|
737 |
|
|
\subsection{Effects due to methodology and simulation parameters} |
738 |
|
|
|
739 |
|
|
We have varied the parameters of the simulations in order to |
740 |
|
|
investigate how these factors would affect the computation of $G$. Of |
741 |
|
|
particular interest are: 1) the length scale for the applied thermal |
742 |
|
|
gradient (modified by increasing the amount of solvent in the system), |
743 |
|
|
2) the sign and magnitude of the applied thermal flux, 3) the average |
744 |
|
|
temperature of the simulation (which alters the solvent density during |
745 |
|
|
equilibration), and 4) the definition of the interfacial conductance |
746 |
|
|
(Eqs. (\ref{discreteG}) or (\ref{derivativeG})) used in the |
747 |
|
|
calculation. |
748 |
|
|
|
749 |
|
|
Systems of different lengths were prepared by altering the number of |
750 |
|
|
solvent molecules and extending the length of the box along the $z$ |
751 |
|
|
axis to accomodate the extra solvent. Equilibration at the same |
752 |
|
|
temperature and pressure conditions led to nearly identical surface |
753 |
|
|
areas ($L_x$ and $L_y$) available to the metal and capping agent, |
754 |
|
|
while the extra solvent served mainly to lengthen the axis that was |
755 |
|
|
used to apply the thermal flux. For a given value of the applied |
756 |
|
|
flux, the different $z$ length scale has only a weak effect on the |
757 |
|
|
computed conductivities. |
758 |
|
|
|
759 |
|
|
\subsubsection{Effects of applied flux} |
760 |
|
|
The NIVS algorithm allows changes in both the sign and magnitude of |
761 |
|
|
the applied flux. It is possible to reverse the direction of heat |
762 |
|
|
flow simply by changing the sign of the flux, and thermal gradients |
763 |
|
|
which would be difficult to obtain experimentally ($5$ K/\AA) can be |
764 |
|
|
easily simulated. However, the magnitude of the applied flux is not |
765 |
|
|
arbitrary if one aims to obtain a stable and reliable thermal gradient. |
766 |
|
|
A temperature gradient can be lost in the noise if $|J_z|$ is too |
767 |
|
|
small, and excessive $|J_z|$ values can cause phase transitions if the |
768 |
|
|
extremes of the simulation cell become widely separated in |
769 |
|
|
temperature. Also, if $|J_z|$ is too large for the bulk conductivity |
770 |
|
|
of the materials, the thermal gradient will never reach a stable |
771 |
|
|
state. |
772 |
|
|
|
773 |
|
|
Within a reasonable range of $J_z$ values, we were able to study how |
774 |
|
|
$G$ changes as a function of this flux. In what follows, we use |
775 |
|
|
positive $J_z$ values to denote the case where energy is being |
776 |
|
|
transferred by the method from the metal phase and into the liquid. |
777 |
|
|
The resulting gradient therefore has a higher temperature in the |
778 |
|
|
liquid phase. Negative flux values reverse this transfer, and result |
779 |
|
|
in higher temperature metal phases. The conductance measured under |
780 |
|
|
different applied $J_z$ values is listed in Tables 2 and 3 in the |
781 |
|
|
supporting information. These results do not indicate that $G$ depends |
782 |
|
|
strongly on $J_z$ within this flux range. The linear response of flux |
783 |
|
|
to thermal gradient simplifies our investigations in that we can rely |
784 |
|
|
on $G$ measurement with only a small number $J_z$ values. |
785 |
|
|
|
786 |
|
|
The sign of $J_z$ is a different matter, however, as this can alter |
787 |
|
|
the temperature on the two sides of the interface. The average |
788 |
|
|
temperature values reported are for the entire system, and not for the |
789 |
|
|
liquid phase, so at a given $\langle T \rangle$, the system with |
790 |
|
|
positive $J_z$ has a warmer liquid phase. This means that if the |
791 |
|
|
liquid carries thermal energy via diffusive transport, {\it positive} |
792 |
|
|
$J_z$ values will result in increased molecular motion on the liquid |
793 |
|
|
side of the interface, and this will increase the measured |
794 |
|
|
conductivity. |
795 |
|
|
|
796 |
|
|
\subsubsection{Effects due to average temperature} |
797 |
|
|
|
798 |
|
|
We also studied the effect of average system temperature on the |
799 |
|
|
interfacial conductance. The simulations are first equilibrated in |
800 |
|
|
the NPT ensemble at 1 atm. The TraPPE-UA model for hexane tends to |
801 |
|
|
predict a lower boiling point (and liquid state density) than |
802 |
|
|
experiments. This lower-density liquid phase leads to reduced contact |
803 |
|
|
between the hexane and butanethiol, and this accounts for our |
804 |
|
|
observation of lower conductance at higher temperatures. In raising |
805 |
|
|
the average temperature from 200K to 250K, the density drop of |
806 |
|
|
$\sim$20\% in the solvent phase leads to a $\sim$40\% drop in the |
807 |
|
|
conductance. |
808 |
|
|
|
809 |
|
|
Similar behavior is observed in the TraPPE-UA model for toluene, |
810 |
|
|
although this model has better agreement with the experimental |
811 |
|
|
densities of toluene. The expansion of the toluene liquid phase is |
812 |
|
|
not as significant as that of the hexane (8.3\% over 100K), and this |
813 |
|
|
limits the effect to $\sim$20\% drop in thermal conductivity. |
814 |
|
|
|
815 |
|
|
Although we have not mapped out the behavior at a large number of |
816 |
|
|
temperatures, is clear that there will be a strong temperature |
817 |
|
|
dependence in the interfacial conductance when the physical properties |
818 |
|
|
of one side of the interface (notably the density) change rapidly as a |
819 |
|
|
function of temperature. |
820 |
|
|
|
821 |
|
|
Besides the lower interfacial thermal conductance, surfaces at |
822 |
|
|
relatively high temperatures are susceptible to reconstructions, |
823 |
|
|
particularly when butanethiols fully cover the Au(111) surface. These |
824 |
|
|
reconstructions include surface Au atoms which migrate outward to the |
825 |
|
|
S atom layer, and butanethiol molecules which embed into the surface |
826 |
|
|
Au layer. The driving force for this behavior is the strong Au-S |
827 |
|
|
interactions which are modeled here with a deep Lennard-Jones |
828 |
|
|
potential. This phenomenon agrees with reconstructions that have been |
829 |
|
|
experimentally |
830 |
|
|
observed.\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt |
831 |
|
|
{\it et al.} kept their Au(111) slab rigid so that their simulations |
832 |
|
|
could reach 300K without surface |
833 |
|
|
reconstructions.\cite{vlugt:cpc2007154} Since surface reconstructions |
834 |
|
|
blur the interface, the measurement of $G$ becomes more difficult to |
835 |
|
|
conduct at higher temperatures. For this reason, most of our |
836 |
|
|
measurements are undertaken at $\langle T\rangle\sim$200K where |
837 |
|
|
reconstruction is minimized. |
838 |
|
|
|
839 |
|
|
However, when the surface is not completely covered by butanethiols, |
840 |
|
|
the simulated system appears to be more resistent to the |
841 |
|
|
reconstruction. Our Au / butanethiol / toluene system had the Au(111) |
842 |
|
|
surfaces 90\% covered by butanethiols, but did not see this above |
843 |
|
|
phenomena even at $\langle T\rangle\sim$300K. That said, we did |
844 |
|
|
observe butanethiols migrating to neighboring three-fold sites during |
845 |
|
|
a simulation. Since the interface persisted in these simulations, we |
846 |
|
|
were able to obtain $G$'s for these interfaces even at a relatively |
847 |
|
|
high temperature without being affected by surface reconstructions. |
848 |
|
|
|
849 |
|
|
\section{Discussion} |
850 |
skuang |
3770 |
[COMBINE W. RESULTS] |
851 |
gezelter |
3769 |
The primary result of this work is that the capping agent acts as an |
852 |
|
|
efficient thermal coupler between solid and solvent phases. One of |
853 |
|
|
the ways the capping agent can carry out this role is to down-shift |
854 |
|
|
between the phonon vibrations in the solid (which carry the heat from |
855 |
|
|
the gold) and the molecular vibrations in the liquid (which carry some |
856 |
|
|
of the heat in the solvent). |
857 |
|
|
|
858 |
|
|
To investigate the mechanism of interfacial thermal conductance, the |
859 |
|
|
vibrational power spectrum was computed. Power spectra were taken for |
860 |
|
|
individual components in different simulations. To obtain these |
861 |
|
|
spectra, simulations were run after equilibration in the |
862 |
|
|
microcanonical (NVE) ensemble and without a thermal |
863 |
|
|
gradient. Snapshots of configurations were collected at a frequency |
864 |
|
|
that is higher than that of the fastest vibrations occurring in the |
865 |
|
|
simulations. With these configurations, the velocity auto-correlation |
866 |
|
|
functions can be computed: |
867 |
|
|
\begin{equation} |
868 |
|
|
C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle |
869 |
|
|
\label{vCorr} |
870 |
|
|
\end{equation} |
871 |
|
|
The power spectrum is constructed via a Fourier transform of the |
872 |
|
|
symmetrized velocity autocorrelation function, |
873 |
|
|
\begin{equation} |
874 |
|
|
\hat{f}(\omega) = |
875 |
|
|
\int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt |
876 |
|
|
\label{fourier} |
877 |
|
|
\end{equation} |
878 |
|
|
|
879 |
|
|
\subsection{The role of specific vibrations} |
880 |
|
|
The vibrational spectra for gold slabs in different environments are |
881 |
|
|
shown as in Figure \ref{specAu}. Regardless of the presence of |
882 |
|
|
solvent, the gold surfaces which are covered by butanethiol molecules |
883 |
|
|
exhibit an additional peak observed at a frequency of |
884 |
|
|
$\sim$165cm$^{-1}$. We attribute this peak to the S-Au bonding |
885 |
|
|
vibration. This vibration enables efficient thermal coupling of the |
886 |
|
|
surface Au layer to the capping agents. Therefore, in our simulations, |
887 |
|
|
the Au / S interfaces do not appear to be the primary barrier to |
888 |
|
|
thermal transport when compared with the butanethiol / solvent |
889 |
|
|
interfaces. This supports the results of Luo {\it et |
890 |
|
|
al.}\cite{Luo20101}, who reported $G$ for Au-SAM junctions roughly |
891 |
|
|
twice as large as what we have computed for the thiol-liquid |
892 |
|
|
interfaces. |
893 |
|
|
|
894 |
|
|
\begin{figure} |
895 |
|
|
\includegraphics[width=\linewidth]{vibration} |
896 |
|
|
\caption{The vibrational power spectrum for thiol-capped gold has an |
897 |
|
|
additional vibrational peak at $\sim $165cm$^{-1}$. Bare gold |
898 |
|
|
surfaces (both with and without a solvent over-layer) are missing |
899 |
|
|
this peak. A similar peak at $\sim $165cm$^{-1}$ also appears in |
900 |
|
|
the vibrational power spectrum for the butanethiol capping agents.} |
901 |
|
|
\label{specAu} |
902 |
|
|
\end{figure} |
903 |
|
|
|
904 |
|
|
Also in this figure, we show the vibrational power spectrum for the |
905 |
|
|
bound butanethiol molecules, which also exhibits the same |
906 |
|
|
$\sim$165cm$^{-1}$ peak. |
907 |
|
|
|
908 |
|
|
\subsection{Overlap of power spectra} |
909 |
|
|
A comparison of the results obtained from the two different organic |
910 |
|
|
solvents can also provide useful information of the interfacial |
911 |
|
|
thermal transport process. In particular, the vibrational overlap |
912 |
|
|
between the butanethiol and the organic solvents suggests a highly |
913 |
|
|
efficient thermal exchange between these components. Very high |
914 |
|
|
thermal conductivity was observed when AA models were used and C-H |
915 |
|
|
vibrations were treated classically. The presence of extra degrees of |
916 |
|
|
freedom in the AA force field yields higher heat exchange rates |
917 |
|
|
between the two phases and results in a much higher conductivity than |
918 |
|
|
in the UA force field. The all-atom classical models include high |
919 |
|
|
frequency modes which should be unpopulated at our relatively low |
920 |
|
|
temperatures. This artifact is likely the cause of the high thermal |
921 |
|
|
conductance in all-atom MD simulations. |
922 |
|
|
|
923 |
|
|
The similarity in the vibrational modes available to solvent and |
924 |
|
|
capping agent can be reduced by deuterating one of the two components |
925 |
|
|
(Fig. \ref{aahxntln}). Once either the hexanes or the butanethiols |
926 |
|
|
are deuterated, one can observe a significantly lower $G$ and |
927 |
|
|
$G^\prime$ values (Table \ref{modelTest}). |
928 |
|
|
|
929 |
|
|
\begin{figure} |
930 |
|
|
\includegraphics[width=\linewidth]{aahxntln} |
931 |
|
|
\caption{Spectra obtained for all-atom (AA) Au / butanethiol / solvent |
932 |
|
|
systems. When butanethiol is deuterated (lower left), its |
933 |
|
|
vibrational overlap with hexane decreases significantly. Since |
934 |
|
|
aromatic molecules and the butanethiol are vibrationally dissimilar, |
935 |
|
|
the change is not as dramatic when toluene is the solvent (right).} |
936 |
|
|
\label{aahxntln} |
937 |
|
|
\end{figure} |
938 |
|
|
|
939 |
|
|
For the Au / butanethiol / toluene interfaces, having the AA |
940 |
|
|
butanethiol deuterated did not yield a significant change in the |
941 |
|
|
measured conductance. Compared to the C-H vibrational overlap between |
942 |
|
|
hexane and butanethiol, both of which have alkyl chains, the overlap |
943 |
|
|
between toluene and butanethiol is not as significant and thus does |
944 |
|
|
not contribute as much to the heat exchange process. |
945 |
|
|
|
946 |
|
|
Previous observations of Zhang {\it et al.}\cite{hase:2010} indicate |
947 |
|
|
that the {\it intra}molecular heat transport due to alkylthiols is |
948 |
|
|
highly efficient. Combining our observations with those of Zhang {\it |
949 |
|
|
et al.}, it appears that butanethiol acts as a channel to expedite |
950 |
|
|
heat flow from the gold surface and into the alkyl chain. The |
951 |
|
|
vibrational coupling between the metal and the liquid phase can |
952 |
|
|
therefore be enhanced with the presence of suitable capping agents. |
953 |
|
|
|
954 |
|
|
Deuterated models in the UA force field did not decouple the thermal |
955 |
|
|
transport as well as in the AA force field. The UA models, even |
956 |
|
|
though they have eliminated the high frequency C-H vibrational |
957 |
|
|
overlap, still have significant overlap in the lower-frequency |
958 |
|
|
portions of the infrared spectra (Figure \ref{uahxnua}). Deuterating |
959 |
|
|
the UA models did not decouple the low frequency region enough to |
960 |
|
|
produce an observable difference for the results of $G$ (Table |
961 |
|
|
\ref{modelTest}). |
962 |
|
|
|
963 |
|
|
\begin{figure} |
964 |
|
|
\includegraphics[width=\linewidth]{uahxnua} |
965 |
|
|
\caption{Vibrational power spectra for UA models for the butanethiol |
966 |
|
|
and hexane solvent (upper panel) show the high degree of overlap |
967 |
|
|
between these two molecules, particularly at lower frequencies. |
968 |
|
|
Deuterating a UA model for the solvent (lower panel) does not |
969 |
|
|
decouple the two spectra to the same degree as in the AA force |
970 |
|
|
field (see Fig \ref{aahxntln}).} |
971 |
|
|
\label{uahxnua} |
972 |
|
|
\end{figure} |
973 |
|
|
|
974 |
|
|
\section{Conclusions} |
975 |
|
|
The NIVS algorithm has been applied to simulations of |
976 |
|
|
butanethiol-capped Au(111) surfaces in the presence of organic |
977 |
|
|
solvents. This algorithm allows the application of unphysical thermal |
978 |
|
|
flux to transfer heat between the metal and the liquid phase. With the |
979 |
|
|
flux applied, we were able to measure the corresponding thermal |
980 |
|
|
gradients and to obtain interfacial thermal conductivities. Under |
981 |
|
|
steady states, 2-3 ns trajectory simulations are sufficient for |
982 |
|
|
computation of this quantity. |
983 |
|
|
|
984 |
|
|
Our simulations have seen significant conductance enhancement in the |
985 |
|
|
presence of capping agent, compared with the bare gold / liquid |
986 |
|
|
interfaces. The vibrational coupling between the metal and the liquid |
987 |
|
|
phase is enhanced by a chemically-bonded capping agent. Furthermore, |
988 |
|
|
the coverage percentage of the capping agent plays an important role |
989 |
|
|
in the interfacial thermal transport process. Moderately low coverages |
990 |
|
|
allow higher contact between capping agent and solvent, and thus could |
991 |
|
|
further enhance the heat transfer process, giving a non-monotonic |
992 |
|
|
behavior of conductance with increasing coverage. |
993 |
|
|
|
994 |
|
|
Our results, particularly using the UA models, agree well with |
995 |
|
|
available experimental data. The AA models tend to overestimate the |
996 |
|
|
interfacial thermal conductance in that the classically treated C-H |
997 |
|
|
vibrations become too easily populated. Compared to the AA models, the |
998 |
|
|
UA models have higher computational efficiency with satisfactory |
999 |
|
|
accuracy, and thus are preferable in modeling interfacial thermal |
1000 |
|
|
transport. |
1001 |
|
|
|
1002 |
|
|
Of the two definitions for $G$, the discrete form |
1003 |
|
|
(Eq. \ref{discreteG}) was easier to use and gives out relatively |
1004 |
|
|
consistent results, while the derivative form (Eq. \ref{derivativeG}) |
1005 |
|
|
is not as versatile. Although $G^\prime$ gives out comparable results |
1006 |
|
|
and follows similar trend with $G$ when measuring close to fully |
1007 |
|
|
covered or bare surfaces, the spatial resolution of $T$ profile |
1008 |
|
|
required for the use of a derivative form is limited by the number of |
1009 |
|
|
bins and the sampling required to obtain thermal gradient information. |
1010 |
|
|
|
1011 |
|
|
Vlugt {\it et al.} have investigated the surface thiol structures for |
1012 |
|
|
nanocrystalline gold and pointed out that they differ from those of |
1013 |
|
|
the Au(111) surface.\cite{landman:1998,vlugt:cpc2007154} This |
1014 |
|
|
difference could also cause differences in the interfacial thermal |
1015 |
|
|
transport behavior. To investigate this problem, one would need an |
1016 |
|
|
effective method for applying thermal gradients in non-planar |
1017 |
|
|
(i.e. spherical) geometries. |
1018 |
|
|
|
1019 |
|
|
\section{Acknowledgments} |
1020 |
|
|
Support for this project was provided by the National Science |
1021 |
|
|
Foundation under grant CHE-0848243. Computational time was provided by |
1022 |
|
|
the Center for Research Computing (CRC) at the University of Notre |
1023 |
|
|
Dame. |
1024 |
|
|
|
1025 |
|
|
\newpage |
1026 |
|
|
|
1027 |
skuang |
3770 |
\bibliography{stokes} |
1028 |
gezelter |
3769 |
|
1029 |
|
|
\end{doublespace} |
1030 |
|
|
\end{document} |
1031 |
|
|
|