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\begin{document} |
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|
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\title{Simulating interfacial thermal conductance at metal-solvent |
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interfaces: the role of chemical capping agents} |
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\title{Simulating Interfacial Thermal Conductance at Metal-Solvent |
32 |
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Interfaces: the Role of Chemical Capping Agents} |
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|
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\author{Shenyu Kuang and J. Daniel |
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Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\ |
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\begin{doublespace} |
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|
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\begin{abstract} |
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|
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We have developed a Non-Isotropic Velocity Scaling algorithm for |
49 |
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setting up and maintaining stable thermal gradients in non-equilibrium |
50 |
< |
molecular dynamics simulations. This approach effectively imposes |
51 |
< |
unphysical thermal flux even between particles of different |
52 |
< |
identities, conserves linear momentum and kinetic energy, and |
53 |
< |
minimally perturbs the velocity profile of a system when compared with |
54 |
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previous RNEMD methods. We have used this method to simulate thermal |
55 |
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conductance at metal / organic solvent interfaces both with and |
56 |
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without the presence of thiol-based capping agents. We obtained |
57 |
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values comparable with experimental values, and observed significant |
58 |
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conductance enhancement with the presence of capping agents. Computed |
59 |
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power spectra indicate the acoustic impedance mismatch between metal |
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and liquid phase is greatly reduced by the capping agents and thus |
61 |
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leads to higher interfacial thermal transfer efficiency. |
62 |
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|
47 |
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With the Non-Isotropic Velocity Scaling (NIVS) approach to Reverse |
48 |
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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 the acoustic |
54 |
> |
impedance mismatch between the metal and liquid phases is |
55 |
> |
effectively reduced by the capping agents, leading to a greatly |
56 |
> |
enhanced conductivity at the interface. Specifically, the chemical |
57 |
> |
bond between the metal and the capping agent introduces a |
58 |
> |
vibrational overlap that is not present without the capping agent, |
59 |
> |
and the overlap between the vibrational spectra (metal to cap, cap |
60 |
> |
to solvent) provides a mechanism for rapid thermal transport across |
61 |
> |
the interface. Our calculations also suggest that this is a |
62 |
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non-monotonic function of the fractional coverage of the surface, as |
63 |
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moderate coverages allow {\bf vibrational heat diffusion} of solvent |
64 |
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molecules that have been in close contact with the capping agent. |
65 |
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\end{abstract} |
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|
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\newpage |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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|
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\section{Introduction} |
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[BACKGROUND FOR INTERFACIAL THERMAL CONDUCTANCE PROBLEM] |
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Interfacial thermal conductance is extensively studied both |
78 |
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experimentally and computationally, and systems with interfaces |
79 |
< |
present are generally heterogeneous. Although interfaces are commonly |
80 |
< |
barriers to heat transfer, it has been |
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reported\cite{doi:10.1021/la904855s} that under specific circustances, |
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e.g. with certain capping agents present on the surface, interfacial |
83 |
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conductance can be significantly enhanced. However, heat conductance |
84 |
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of molecular and nano-scale interfaces will be affected by the |
83 |
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chemical details of the surface and is challenging to |
84 |
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experimentalist. The lower thermal flux through interfaces is even |
85 |
< |
more difficult to measure with EMD and forward NEMD simulation |
86 |
< |
methods. Therefore, developing good simulation methods will be |
87 |
< |
desirable in order to investigate thermal transport across interfaces. |
76 |
> |
Due to the importance of heat flow (and heat removal) in |
77 |
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nanotechnology, interfacial thermal conductance has been studied |
78 |
> |
extensively both experimentally and computationally.\cite{cahill:793} |
79 |
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Nanoscale materials have a significant fraction of their atoms at |
80 |
> |
interfaces, and the chemical details of these interfaces govern the |
81 |
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thermal transport properties. Furthermore, the interfaces are often |
82 |
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heterogeneous (e.g. solid - liquid), which provides a challenge to |
83 |
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computational methods which have been developed for homogeneous or |
84 |
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bulk systems. |
85 |
|
|
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Recently, we have developed the Non-Isotropic Velocity Scaling (NIVS) |
86 |
> |
Experimentally, the thermal properties of a number of interfaces have |
87 |
> |
been investigated. Cahill and coworkers studied nanoscale thermal |
88 |
> |
transport from metal nanoparticle/fluid interfaces, to epitaxial |
89 |
> |
TiN/single crystal oxides interfaces, and hydrophilic and hydrophobic |
90 |
> |
interfaces between water and solids with different self-assembled |
91 |
> |
monolayers.\cite{Wilson:2002uq,PhysRevB.67.054302,doi:10.1021/jp048375k,PhysRevLett.96.186101} |
92 |
> |
Wang {\it et al.} studied heat transport through long-chain |
93 |
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hydrocarbon monolayers on gold substrate at individual molecular |
94 |
> |
level,\cite{Wang10082007} Schmidt {\it et al.} studied the role of |
95 |
> |
cetyltrimethylammonium bromide (CTAB) on the thermal transport between |
96 |
> |
gold nanorods and solvent,\cite{doi:10.1021/jp8051888} and Juv\'e {\it |
97 |
> |
et al.} studied the cooling dynamics, which is controlled by thermal |
98 |
> |
interface resistance of glass-embedded metal |
99 |
> |
nanoparticles.\cite{PhysRevB.80.195406} Although interfaces are |
100 |
> |
normally considered barriers for heat transport, Alper {\it et al.} |
101 |
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suggested that specific ligands (capping agents) could completely |
102 |
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eliminate this barrier |
103 |
> |
($G\rightarrow\infty$).\cite{doi:10.1021/la904855s} |
104 |
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|
105 |
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Theoretical and computational models have also been used to study the |
106 |
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interfacial thermal transport in order to gain an understanding of |
107 |
> |
this phenomena at the molecular level. Recently, Hase and coworkers |
108 |
> |
employed Non-Equilibrium Molecular Dynamics (NEMD) simulations to |
109 |
> |
study thermal transport from hot Au(111) substrate to a self-assembled |
110 |
> |
monolayer of alkylthiol with relatively long chain (8-20 carbon |
111 |
> |
atoms).\cite{hase:2010,hase:2011} However, ensemble averaged |
112 |
> |
measurements for heat conductance of interfaces between the capping |
113 |
> |
monolayer on Au and a solvent phase have yet to be studied with their |
114 |
> |
approach. The comparatively low thermal flux through interfaces is |
115 |
> |
difficult to measure with Equilibrium |
116 |
> |
MD\cite{doi:10.1080/0026897031000068578} or forward NEMD simulation |
117 |
> |
methods. Therefore, the Reverse NEMD (RNEMD) |
118 |
> |
methods\cite{MullerPlathe:1997xw,kuang:164101} would be advantageous |
119 |
> |
in that they {\it apply} the difficult to measure quantity (flux), |
120 |
> |
while {\it measuring} the easily-computed quantity (the thermal |
121 |
> |
gradient). This is particularly true for inhomogeneous interfaces |
122 |
> |
where it would not be clear how to apply a gradient {\it a priori}. |
123 |
> |
Garde and coworkers\cite{garde:nl2005,garde:PhysRevLett2009} applied |
124 |
> |
this approach to various liquid interfaces and studied how thermal |
125 |
> |
conductance (or resistance) is dependent on chemical details of a |
126 |
> |
number of hydrophobic and hydrophilic aqueous interfaces. {\bf And |
127 |
> |
Luo {\it et al.} studied the thermal conductance of Au-SAM-Au |
128 |
> |
junctions using the same approach, with comparison to a constant |
129 |
> |
temperature difference method\cite{Luo20101}. While this latter |
130 |
> |
approach establishes more thermal distributions compared to the |
131 |
> |
former RNEMD methods, it does not guarantee momentum or kinetic |
132 |
> |
energy conservations.} |
133 |
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|
134 |
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Recently, we have developed a Non-Isotropic Velocity Scaling (NIVS) |
135 |
|
algorithm for RNEMD simulations\cite{kuang:164101}. This algorithm |
136 |
|
retains the desirable features of RNEMD (conservation of linear |
137 |
|
momentum and total energy, compatibility with periodic boundary |
138 |
|
conditions) while establishing true thermal distributions in each of |
139 |
< |
the two slabs. Furthermore, it allows more effective thermal exchange |
140 |
< |
between particles of different identities, and thus enables extensive |
141 |
< |
study of interfacial conductance. |
139 |
> |
the two slabs. Furthermore, it allows effective thermal exchange |
140 |
> |
between particles of different identities, and thus makes the study of |
141 |
> |
interfacial conductance much simpler. |
142 |
|
|
143 |
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The work presented here deals with the Au(111) surface covered to |
144 |
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varying degrees by butanethiol, a capping agent with short carbon |
145 |
+ |
chain, and solvated with organic solvents of different molecular |
146 |
+ |
properties. {\bf To our knowledge, few previous MD inverstigations |
147 |
+ |
have been found to address to these systems yet.} Different models |
148 |
+ |
were used for both the capping agent and the solvent force field |
149 |
+ |
parameters. Using the NIVS algorithm, the thermal transport across |
150 |
+ |
these interfaces was studied and the underlying mechanism for the |
151 |
+ |
phenomena was investigated. |
152 |
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|
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|
\section{Methodology} |
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\subsection{Algorithm} |
155 |
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[BACKGROUND FOR MD METHODS] |
156 |
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There have been many algorithms for computing thermal conductivity |
157 |
< |
using molecular dynamics simulations. However, interfacial conductance |
158 |
< |
is at least an order of magnitude smaller. This would make the |
159 |
< |
calculation even more difficult for those slowly-converging |
160 |
< |
equilibrium methods. Imposed-flux non-equilibrium |
161 |
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methods\cite{MullerPlathe:1997xw} have the flux set {\it a priori} and |
162 |
< |
the response of temperature or momentum gradients are easier to |
163 |
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measure than the flux, if unknown, and thus, is a preferable way to |
164 |
< |
the forward NEMD methods. Although the momentum swapping approach for |
165 |
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flux-imposing can be used for exchanging energy between particles of |
166 |
< |
different identity, the kinetic energy transfer efficiency is affected |
167 |
< |
by the mass difference between the particles, which limits its |
168 |
< |
application on heterogeneous interfacial systems. |
154 |
> |
\subsection{Imposed-Flux Methods in MD Simulations} |
155 |
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Steady state MD simulations have an advantage in that not many |
156 |
> |
trajectories are needed to study the relationship between thermal flux |
157 |
> |
and thermal gradients. For systems with low interfacial conductance, |
158 |
> |
one must have a method capable of generating or measuring relatively |
159 |
> |
small fluxes, compared to those required for bulk conductivity. This |
160 |
> |
requirement makes the calculation even more difficult for |
161 |
> |
slowly-converging equilibrium methods.\cite{Viscardy:2007lq} Forward |
162 |
> |
NEMD methods impose a gradient (and measure a flux), but at interfaces |
163 |
> |
it is not clear what behavior should be imposed at the boundaries |
164 |
> |
between materials. Imposed-flux reverse non-equilibrium |
165 |
> |
methods\cite{MullerPlathe:1997xw} set the flux {\it a priori} and |
166 |
> |
the thermal response becomes an easy-to-measure quantity. Although |
167 |
> |
M\"{u}ller-Plathe's original momentum swapping approach can be used |
168 |
> |
for exchanging energy between particles of different identity, the |
169 |
> |
kinetic energy transfer efficiency is affected by the mass difference |
170 |
> |
between the particles, which limits its application on heterogeneous |
171 |
> |
interfacial systems. |
172 |
|
|
173 |
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The non-isotropic velocity scaling (NIVS)\cite{kuang:164101} approach in |
174 |
< |
non-equilibrium MD simulations is able to impose relatively large |
175 |
< |
kinetic energy flux without obvious perturbation to the velocity |
176 |
< |
distribution of the simulated systems. Furthermore, this approach has |
177 |
< |
the advantage in heterogeneous interfaces in that kinetic energy flux |
178 |
< |
can be applied between regions of particles of arbitary identity, and |
179 |
< |
the flux quantity is not restricted by particle mass difference. |
173 |
> |
The non-isotropic velocity scaling (NIVS) \cite{kuang:164101} approach |
174 |
> |
to non-equilibrium MD simulations is able to impose a wide range of |
175 |
> |
kinetic energy fluxes without obvious perturbation to the velocity |
176 |
> |
distributions of the simulated systems. Furthermore, this approach has |
177 |
> |
the advantage in heterogeneous interfaces in that kinetic energy flux |
178 |
> |
can be applied between regions of particles of arbitrary identity, and |
179 |
> |
the flux will not be restricted by difference in particle mass. |
180 |
|
|
181 |
|
The NIVS algorithm scales the velocity vectors in two separate regions |
182 |
< |
of a simulation system with respective diagonal scaling matricies. To |
183 |
< |
determine these scaling factors in the matricies, a set of equations |
182 |
> |
of a simulation system with respective diagonal scaling matrices. To |
183 |
> |
determine these scaling factors in the matrices, a set of equations |
184 |
|
including linear momentum conservation and kinetic energy conservation |
185 |
< |
constraints and target momentum/energy flux satisfaction is |
186 |
< |
solved. With the scaling operation applied to the system in a set |
187 |
< |
frequency, corresponding momentum/temperature gradients can be built, |
188 |
< |
which can be used for computing transportation properties and other |
189 |
< |
applications related to momentum/temperature gradients. The NIVS |
132 |
< |
algorithm conserves momenta and energy and does not depend on an |
133 |
< |
external thermostat. |
185 |
> |
constraints and target energy flux satisfaction is solved. With the |
186 |
> |
scaling operation applied to the system in a set frequency, bulk |
187 |
> |
temperature gradients can be easily established, and these can be used |
188 |
> |
for computing thermal conductivities. The NIVS algorithm conserves |
189 |
> |
momenta and energy and does not depend on an external thermostat. |
190 |
|
|
191 |
< |
\subsection{Defining Interfacial Thermal Conductivity $G$} |
192 |
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For interfaces with a relatively low interfacial conductance, the bulk |
193 |
< |
regions on either side of an interface rapidly come to a state in |
194 |
< |
which the two phases have relatively homogeneous (but distinct) |
195 |
< |
temperatures. The interfacial thermal conductivity $G$ can therefore |
196 |
< |
be approximated as: |
191 |
> |
\subsection{Defining Interfacial Thermal Conductivity ($G$)} |
192 |
> |
|
193 |
> |
For an interface with relatively low interfacial conductance, and a |
194 |
> |
thermal flux between two distinct bulk regions, the regions on either |
195 |
> |
side of the interface rapidly come to a state in which the two phases |
196 |
> |
have relatively homogeneous (but distinct) temperatures. The |
197 |
> |
interfacial thermal conductivity $G$ can therefore be approximated as: |
198 |
|
\begin{equation} |
199 |
< |
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle - |
199 |
> |
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle - |
200 |
|
\langle T_\mathrm{cold}\rangle \right)} |
201 |
|
\label{lowG} |
202 |
|
\end{equation} |
203 |
< |
where ${E_{total}}$ is the imposed non-physical kinetic energy |
204 |
< |
transfer and ${\langle T_\mathrm{hot}\rangle}$ and ${\langle |
205 |
< |
T_\mathrm{cold}\rangle}$ are the average observed temperature of the |
206 |
< |
two separated phases. |
203 |
> |
where ${E_{total}}$ is the total imposed non-physical kinetic energy |
204 |
> |
transfer during the simulation and ${\langle T_\mathrm{hot}\rangle}$ |
205 |
> |
and ${\langle T_\mathrm{cold}\rangle}$ are the average observed |
206 |
> |
temperature of the two separated phases. For an applied flux $J_z$ |
207 |
> |
operating over a simulation time $t$ on a periodically-replicated slab |
208 |
> |
of dimensions $L_x \times L_y$, $E_{total} = J_z *(t)*(2 L_x L_y)$. |
209 |
|
|
210 |
< |
When the interfacial conductance is {\it not} small, two ways can be |
211 |
< |
used to define $G$. |
212 |
< |
|
213 |
< |
One way is to assume the temperature is discretely different on two |
214 |
< |
sides of the interface, $G$ can be calculated with the thermal flux |
215 |
< |
applied $J$ and the maximum temperature difference measured along the |
216 |
< |
thermal gradient max($\Delta T$), which occurs at the interface, as: |
210 |
> |
When the interfacial conductance is {\it not} small, there are two |
211 |
> |
ways to define $G$. One common way is to assume the temperature is |
212 |
> |
discrete on the two sides of the interface. $G$ can be calculated |
213 |
> |
using the applied thermal flux $J$ and the maximum temperature |
214 |
> |
difference measured along the thermal gradient max($\Delta T$), which |
215 |
> |
occurs at the Gibbs dividing surface (Figure \ref{demoPic}). This is |
216 |
> |
known as the Kapitza conductance, which is the inverse of the Kapitza |
217 |
> |
resistance. |
218 |
|
\begin{equation} |
219 |
< |
G=\frac{J}{\Delta T} |
219 |
> |
G=\frac{J}{\Delta T} |
220 |
|
\label{discreteG} |
221 |
|
\end{equation} |
222 |
|
|
223 |
+ |
\begin{figure} |
224 |
+ |
\includegraphics[width=\linewidth]{method} |
225 |
+ |
\caption{Interfacial conductance can be calculated by applying an |
226 |
+ |
(unphysical) kinetic energy flux between two slabs, one located |
227 |
+ |
within the metal and another on the edge of the periodic box. The |
228 |
+ |
system responds by forming a thermal gradient. In bulk liquids, |
229 |
+ |
this gradient typically has a single slope, but in interfacial |
230 |
+ |
systems, there are distinct thermal conductivity domains. The |
231 |
+ |
interfacial conductance, $G$ is found by measuring the temperature |
232 |
+ |
gap at the Gibbs dividing surface, or by using second derivatives of |
233 |
+ |
the thermal profile.} |
234 |
+ |
\label{demoPic} |
235 |
+ |
\end{figure} |
236 |
+ |
|
237 |
+ |
{\bf We attempt another approach by assuming that temperature is |
238 |
+ |
continuous and differentiable throughout the space. Given that |
239 |
+ |
$\lambda$ is also differentiable, $G$ can be defined as its |
240 |
+ |
gradient. This quantity has the same unit as the commonly known $G$, |
241 |
+ |
and the maximum of its magnitude denotes where thermal conductivity |
242 |
+ |
has the largest change, i.e. the interface. And vector |
243 |
+ |
$\nabla\lambda$ is normal to the interface. In a simplified |
244 |
+ |
condition here, we have both $\vec{J}$ and the thermal gradient |
245 |
+ |
paralell to the $z$ axis and yield the formula used in our |
246 |
+ |
computations.} |
247 |
+ |
(original text) |
248 |
|
The other approach is to assume a continuous temperature profile along |
249 |
|
the thermal gradient axis (e.g. $z$) and define $G$ at the point where |
250 |
< |
the magnitude of thermal conductivity $\lambda$ change reach its |
250 |
> |
the magnitude of thermal conductivity ($\lambda$) change reaches its |
251 |
|
maximum, given that $\lambda$ is well-defined throughout the space: |
252 |
|
\begin{equation} |
253 |
|
G^\prime = \Big|\frac{\partial\lambda}{\partial z}\Big| |
258 |
|
\label{derivativeG} |
259 |
|
\end{equation} |
260 |
|
|
261 |
< |
With the temperature profile obtained from simulations, one is able to |
261 |
> |
With temperature profiles obtained from simulation, one is able to |
262 |
|
approximate the first and second derivatives of $T$ with finite |
263 |
< |
difference method and thus calculate $G^\prime$. |
263 |
> |
difference methods and calculate $G^\prime$. In what follows, both |
264 |
> |
definitions have been used, and are compared in the results. |
265 |
|
|
266 |
< |
In what follows, both definitions are used for calculation and comparison. |
266 |
> |
To investigate the interfacial conductivity at metal / solvent |
267 |
> |
interfaces, we have modeled a metal slab with its (111) surfaces |
268 |
> |
perpendicular to the $z$-axis of our simulation cells. The metal slab |
269 |
> |
has been prepared both with and without capping agents on the exposed |
270 |
> |
surface, and has been solvated with simple organic solvents, as |
271 |
> |
illustrated in Figure \ref{gradT}. |
272 |
|
|
273 |
< |
[IMPOSE G DEFINITION INTO OUR SYSTEMS] |
274 |
< |
To facilitate the use of the above definitions in calculating $G$ and |
275 |
< |
$G^\prime$, we have a metal slab with its (111) surfaces perpendicular |
276 |
< |
to the $z$-axis of our simulation cells. With or withour capping |
277 |
< |
agents on the surfaces, the metal slab is solvated with organic |
278 |
< |
solvents, as illustrated in Figure \ref{demoPic}. |
273 |
> |
With the simulation cell described above, we are able to equilibrate |
274 |
> |
the system and impose an unphysical thermal flux between the liquid |
275 |
> |
and the metal phase using the NIVS algorithm. By periodically applying |
276 |
> |
the unphysical flux, we obtained a temperature profile and its spatial |
277 |
> |
derivatives. Figure \ref{gradT} shows how an applied thermal flux can |
278 |
> |
be used to obtain the 1st and 2nd derivatives of the temperature |
279 |
> |
profile. |
280 |
|
|
281 |
|
\begin{figure} |
190 |
– |
\includegraphics[width=\linewidth]{demoPic} |
191 |
– |
\caption{A sample showing how a metal slab has its (111) surface |
192 |
– |
covered by capping agent molecules and solvated by hexane.} |
193 |
– |
\label{demoPic} |
194 |
– |
\end{figure} |
195 |
– |
|
196 |
– |
With a simulation cell setup following the above manner, one is able |
197 |
– |
to equilibrate the system and impose an unphysical thermal flux |
198 |
– |
between the liquid and the metal phase with the NIVS algorithm. Under |
199 |
– |
a stablized thermal gradient induced by periodically applying the |
200 |
– |
unphysical flux, one is able to obtain a temperature profile and the |
201 |
– |
physical thermal flux corresponding to it, which equals to the |
202 |
– |
unphysical flux applied by NIVS. These data enables the evaluation of |
203 |
– |
the interfacial thermal conductance of a surface. Figure \ref{gradT} |
204 |
– |
is an example how those stablized thermal gradient can be used to |
205 |
– |
obtain the 1st and 2nd derivatives of the temperature profile. |
206 |
– |
|
207 |
– |
\begin{figure} |
282 |
|
\includegraphics[width=\linewidth]{gradT} |
283 |
< |
\caption{The 1st and 2nd derivatives of temperature profile can be |
284 |
< |
obtained with finite difference approximation.} |
283 |
> |
\caption{A sample of Au (111) / butanethiol / hexane interfacial |
284 |
> |
system with the temperature profile after a kinetic energy flux has |
285 |
> |
been imposed. Note that the largest temperature jump in the thermal |
286 |
> |
profile (corresponding to the lowest interfacial conductance) is at |
287 |
> |
the interface between the butanethiol molecules (blue) and the |
288 |
> |
solvent (grey). First and second derivatives of the temperature |
289 |
> |
profile are obtained using a finite difference approximation (lower |
290 |
> |
panel).} |
291 |
|
\label{gradT} |
292 |
|
\end{figure} |
293 |
|
|
294 |
|
\section{Computational Details} |
295 |
< |
\subsection{System Geometry} |
296 |
< |
In our simulations, Au is used to construct a metal slab with bare |
297 |
< |
(111) surface perpendicular to the $z$-axis. Different slab thickness |
298 |
< |
(layer numbers of Au) are simulated. This metal slab is first |
299 |
< |
equilibrated under normal pressure (1 atm) and a desired |
300 |
< |
temperature. After equilibration, butanethiol is used as the capping |
301 |
< |
agent molecule to cover the bare Au (111) surfaces evenly. The sulfur |
302 |
< |
atoms in the butanethiol molecules would occupy the three-fold sites |
303 |
< |
of the surfaces, and the maximal butanethiol capacity on Au surface is |
304 |
< |
$1/3$ of the total number of surface Au atoms[CITATION]. A series of |
305 |
< |
different coverage surfaces is investigated in order to study the |
306 |
< |
relation between coverage and conductance. |
295 |
> |
\subsection{Simulation Protocol} |
296 |
> |
The NIVS algorithm has been implemented in our MD simulation code, |
297 |
> |
OpenMD\cite{Meineke:2005gd,openmd}, and was used for our simulations. |
298 |
> |
Metal slabs of 6 or 11 layers of Au atoms were first equilibrated |
299 |
> |
under atmospheric pressure (1 atm) and 200K. After equilibration, |
300 |
> |
butanethiol capping agents were placed at three-fold hollow sites on |
301 |
> |
the Au(111) surfaces. These sites are either {\it fcc} or {\it |
302 |
> |
hcp} sites, although Hase {\it et al.} found that they are |
303 |
> |
equivalent in a heat transfer process,\cite{hase:2010} so we did not |
304 |
> |
distinguish between these sites in our study. The maximum butanethiol |
305 |
> |
capacity on Au surface is $1/3$ of the total number of surface Au |
306 |
> |
atoms, and the packing forms a $(\sqrt{3}\times\sqrt{3})R30^\circ$ |
307 |
> |
structure\cite{doi:10.1021/ja00008a001,doi:10.1021/cr9801317}. A |
308 |
> |
series of lower coverages was also prepared by eliminating |
309 |
> |
butanethiols from the higher coverage surface in a regular manner. The |
310 |
> |
lower coverages were prepared in order to study the relation between |
311 |
> |
coverage and interfacial conductance. |
312 |
|
|
313 |
< |
[COVERAGE DISCRIPTION] However, since the interactions between surface |
314 |
< |
Au and butanethiol is non-bonded, the capping agent molecules are |
315 |
< |
allowed to migrate to an empty neighbor three-fold site during a |
316 |
< |
simulation. Therefore, the initial configuration would not severely |
317 |
< |
affect the sampling of a variety of configurations of the same |
318 |
< |
coverage, and the final conductance measurement would be an average |
319 |
< |
effect of these configurations explored in the simulations. [MAY NEED FIGURES] |
313 |
> |
The capping agent molecules were allowed to migrate during the |
314 |
> |
simulations. They distributed themselves uniformly and sampled a |
315 |
> |
number of three-fold sites throughout out study. Therefore, the |
316 |
> |
initial configuration does not noticeably affect the sampling of a |
317 |
> |
variety of configurations of the same coverage, and the final |
318 |
> |
conductance measurement would be an average effect of these |
319 |
> |
configurations explored in the simulations. |
320 |
|
|
321 |
< |
After the modified Au-butanethiol surface systems are equilibrated |
322 |
< |
under canonical ensemble, Packmol\cite{packmol} is used to pack |
323 |
< |
organic solvent molecules in the previously vacuum part of the |
324 |
< |
simulation cells, which guarantees that short range repulsive |
325 |
< |
interactions do not disrupt the simulations. Two solvents are |
326 |
< |
investigated, one which has little vibrational overlap with the |
327 |
< |
alkanethiol and plane-like shape (toluene), and one which has similar |
243 |
< |
vibrational frequencies and chain-like shape ({\it n}-hexane). The |
244 |
< |
spacing filled by solvent molecules, i.e. the gap between periodically |
245 |
< |
repeated Au-butanethiol surfaces should be carefully chosen so that it |
246 |
< |
would not be too short to affect the liquid phase structure, nor too |
247 |
< |
long, leading to over cooling (freezing) or heating (boiling) when a |
248 |
< |
thermal flux is applied. In our simulations, this spacing is usually |
249 |
< |
$35 \sim 60$\AA. |
321 |
> |
After the modified Au-butanethiol surface systems were equilibrated in |
322 |
> |
the canonical (NVT) ensemble, organic solvent molecules were packed in |
323 |
> |
the previously empty part of the simulation cells.\cite{packmol} Two |
324 |
> |
solvents were investigated, one which has little vibrational overlap |
325 |
> |
with the alkanethiol and which has a planar shape (toluene), and one |
326 |
> |
which has similar vibrational frequencies to the capping agent and |
327 |
> |
chain-like shape ({\it n}-hexane). |
328 |
|
|
329 |
< |
The initial configurations generated by Packmol are further |
330 |
< |
equilibrated with the $x$ and $y$ dimensions fixed, only allowing |
331 |
< |
length scale change in $z$ dimension. This is to ensure that the |
332 |
< |
equilibration of liquid phase does not affect the metal crystal |
333 |
< |
structure in $x$ and $y$ dimensions. Further equilibration are run |
334 |
< |
under NVT and then NVE ensembles. |
329 |
> |
The simulation cells were not particularly extensive along the |
330 |
> |
$z$-axis, as a very long length scale for the thermal gradient may |
331 |
> |
cause excessively hot or cold temperatures in the middle of the |
332 |
> |
solvent region and lead to undesired phenomena such as solvent boiling |
333 |
> |
or freezing when a thermal flux is applied. Conversely, too few |
334 |
> |
solvent molecules would change the normal behavior of the liquid |
335 |
> |
phase. Therefore, our $N_{solvent}$ values were chosen to ensure that |
336 |
> |
these extreme cases did not happen to our simulations. The spacing |
337 |
> |
between periodic images of the gold interfaces is $45 \sim 75$\AA in |
338 |
> |
our simulations. |
339 |
|
|
340 |
< |
After the systems reach equilibrium, NIVS is implemented to impose a |
341 |
< |
periodic unphysical thermal flux between the metal and the liquid |
342 |
< |
phase. Most of our simulations are under an average temperature of |
343 |
< |
$\sim$200K. Therefore, this flux usually comes from the metal to the |
340 |
> |
The initial configurations generated are further equilibrated with the |
341 |
> |
$x$ and $y$ dimensions fixed, only allowing the $z$-length scale to |
342 |
> |
change. This is to ensure that the equilibration of liquid phase does |
343 |
> |
not affect the metal's crystalline structure. Comparisons were made |
344 |
> |
with simulations that allowed changes of $L_x$ and $L_y$ during NPT |
345 |
> |
equilibration. No substantial changes in the box geometry were noticed |
346 |
> |
in these simulations. After ensuring the liquid phase reaches |
347 |
> |
equilibrium at atmospheric pressure (1 atm), further equilibration was |
348 |
> |
carried out under canonical (NVT) and microcanonical (NVE) ensembles. |
349 |
> |
|
350 |
> |
After the systems reach equilibrium, NIVS was used to impose an |
351 |
> |
unphysical thermal flux between the metal and the liquid phases. Most |
352 |
> |
of our simulations were done under an average temperature of |
353 |
> |
$\sim$200K. Therefore, thermal flux usually came from the metal to the |
354 |
|
liquid so that the liquid has a higher temperature and would not |
355 |
< |
freeze due to excessively low temperature. This induced temperature |
356 |
< |
gradient is stablized and the simulation cell is devided evenly into |
357 |
< |
N slabs along the $z$-axis and the temperatures of each slab are |
358 |
< |
recorded. When the slab width $d$ of each slab is the same, the |
359 |
< |
derivatives of $T$ with respect to slab number $n$ can be directly |
360 |
< |
used for $G^\prime$ calculations: |
361 |
< |
\begin{equation} |
362 |
< |
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
355 |
> |
freeze due to lowered temperatures. After this induced temperature |
356 |
> |
gradient had stabilized, the temperature profile of the simulation cell |
357 |
> |
was recorded. To do this, the simulation cell is divided evenly into |
358 |
> |
$N$ slabs along the $z$-axis. The average temperatures of each slab |
359 |
> |
are recorded for 1$\sim$2 ns. When the slab width $d$ of each slab is |
360 |
> |
the same, the derivatives of $T$ with respect to slab number $n$ can |
361 |
> |
be directly used for $G^\prime$ calculations: \begin{equation} |
362 |
> |
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
363 |
|
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
364 |
|
= |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big| |
365 |
|
\Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2 |
368 |
|
\label{derivativeG2} |
369 |
|
\end{equation} |
370 |
|
|
371 |
+ |
All of the above simulation procedures use a time step of 1 fs. Each |
372 |
+ |
equilibration stage took a minimum of 100 ps, although in some cases, |
373 |
+ |
longer equilibration stages were utilized. |
374 |
+ |
|
375 |
|
\subsection{Force Field Parameters} |
376 |
< |
Our simulations include various components. Therefore, force field |
377 |
< |
parameter descriptions are needed for interactions both between the |
378 |
< |
same type of particles and between particles of different species. |
376 |
> |
Our simulations include a number of chemically distinct components. |
377 |
> |
Figure \ref{demoMol} demonstrates the sites defined for both |
378 |
> |
United-Atom and All-Atom models of the organic solvent and capping |
379 |
> |
agents in our simulations. Force field parameters are needed for |
380 |
> |
interactions both between the same type of particles and between |
381 |
> |
particles of different species. |
382 |
|
|
383 |
+ |
\begin{figure} |
384 |
+ |
\includegraphics[width=\linewidth]{structures} |
385 |
+ |
\caption{Structures of the capping agent and solvents utilized in |
386 |
+ |
these simulations. The chemically-distinct sites (a-e) are expanded |
387 |
+ |
in terms of constituent atoms for both United Atom (UA) and All Atom |
388 |
+ |
(AA) force fields. Most parameters are from References |
389 |
+ |
\protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes,TraPPE-UA.thiols} |
390 |
+ |
(UA) and \protect\cite{OPLSAA} (AA). Cross-interactions with the Au |
391 |
+ |
atoms are given in Table \ref{MnM}.} |
392 |
+ |
\label{demoMol} |
393 |
+ |
\end{figure} |
394 |
+ |
|
395 |
|
The Au-Au interactions in metal lattice slab is described by the |
396 |
|
quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
397 |
|
potentials include zero-point quantum corrections and are |
398 |
|
reparametrized for accurate surface energies compared to the |
399 |
< |
Sutton-Chen potentials\cite{Chen90}. |
399 |
> |
Sutton-Chen potentials.\cite{Chen90} |
400 |
|
|
401 |
< |
For both solvent molecules, straight chain {\it n}-hexane and aromatic |
402 |
< |
toluene, United-Atom (UA) and All-Atom (AA) models are used |
403 |
< |
respectively. The TraPPE-UA |
401 |
> |
For the two solvent molecules, {\it n}-hexane and toluene, two |
402 |
> |
different atomistic models were utilized. Both solvents were modeled |
403 |
> |
using United-Atom (UA) and All-Atom (AA) models. The TraPPE-UA |
404 |
|
parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
405 |
< |
for our UA solvent molecules. In these models, pseudo-atoms are |
406 |
< |
located at the carbon centers for alkyl groups. By eliminating |
407 |
< |
explicit hydrogen atoms, these models are simple and computationally |
408 |
< |
efficient, while maintains good accuracy. [LOW BOILING POINT IS A |
409 |
< |
KNOWN PROBLEM FOR TRAPPE-UA ALKANES, NEED MORE DISCUSSION] |
299 |
< |
for |
300 |
< |
toluene, force fields are |
301 |
< |
used with rigid body constraints applied.[MORE DETAILS NEEDED] |
405 |
> |
for our UA solvent molecules. In these models, sites are located at |
406 |
> |
the carbon centers for alkyl groups. Bonding interactions, including |
407 |
> |
bond stretches and bends and torsions, were used for intra-molecular |
408 |
> |
sites closer than 3 bonds. For non-bonded interactions, Lennard-Jones |
409 |
> |
potentials are used. |
410 |
|
|
411 |
< |
Besides the TraPPE-UA models, AA models are included in our studies as |
412 |
< |
well. For hexane, the OPLS all-atom\cite{OPLSAA} force field is |
413 |
< |
used. [MORE DETAILS] |
414 |
< |
For toluene, |
411 |
> |
By eliminating explicit hydrogen atoms, the TraPPE-UA models are |
412 |
> |
simple and computationally efficient, while maintaining good accuracy. |
413 |
> |
However, the TraPPE-UA model for alkanes is known to predict a slightly |
414 |
> |
lower boiling point than experimental values. This is one of the |
415 |
> |
reasons we used a lower average temperature (200K) for our |
416 |
> |
simulations. If heat is transferred to the liquid phase during the |
417 |
> |
NIVS simulation, the liquid in the hot slab can actually be |
418 |
> |
substantially warmer than the mean temperature in the simulation. The |
419 |
> |
lower mean temperatures therefore prevent solvent boiling. |
420 |
|
|
421 |
< |
Buatnethiol molecules are used as capping agent for some of our |
422 |
< |
simulations. United-Atom\cite{TraPPE-UA.thiols} and All-Atom models |
423 |
< |
are respectively used corresponding to the force field type of |
424 |
< |
solvent. |
421 |
> |
For UA-toluene, the non-bonded potentials between intermolecular sites |
422 |
> |
have a similar Lennard-Jones formulation. The toluene molecules were |
423 |
> |
treated as a single rigid body, so there was no need for |
424 |
> |
intramolecular interactions (including bonds, bends, or torsions) in |
425 |
> |
this solvent model. |
426 |
|
|
427 |
< |
To describe the interactions between metal Au and non-metal capping |
428 |
< |
agent and solvent, we refer to Vlugt\cite{vlugt:cpc2007154} and derive |
429 |
< |
other interactions which are not parametrized in their work. (can add |
430 |
< |
hautman and klein's paper here and more discussion; need to put |
431 |
< |
aromatic-metal interaction approximation here) |
427 |
> |
Besides the TraPPE-UA models, AA models for both organic solvents are |
428 |
> |
included in our studies as well. The OPLS-AA\cite{OPLSAA} force fields |
429 |
> |
were used. For hexane, additional explicit hydrogen sites were |
430 |
> |
included. Besides bonding and non-bonded site-site interactions, |
431 |
> |
partial charges and the electrostatic interactions were added to each |
432 |
> |
CT and HC site. For toluene, a flexible model for the toluene molecule |
433 |
> |
was utilized which included bond, bend, torsion, and inversion |
434 |
> |
potentials to enforce ring planarity. |
435 |
|
|
436 |
< |
[TABULATED FORCE FIELD PARAMETERS NEEDED] |
436 |
> |
The butanethiol capping agent in our simulations, were also modeled |
437 |
> |
with both UA and AA model. The TraPPE-UA force field includes |
438 |
> |
parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for |
439 |
> |
UA butanethiol model in our simulations. The OPLS-AA also provides |
440 |
> |
parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) |
441 |
> |
surfaces do not have the hydrogen atom bonded to sulfur. To derive |
442 |
> |
suitable parameters for butanethiol adsorbed on Au(111) surfaces, we |
443 |
> |
adopt the S parameters from Luedtke and Landman\cite{landman:1998} and |
444 |
> |
modify the parameters for the CTS atom to maintain charge neutrality |
445 |
> |
in the molecule. Note that the model choice (UA or AA) for the capping |
446 |
> |
agent can be different from the solvent. Regardless of model choice, |
447 |
> |
the force field parameters for interactions between capping agent and |
448 |
> |
solvent can be derived using Lorentz-Berthelot Mixing Rule: |
449 |
> |
\begin{eqnarray} |
450 |
> |
\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\ |
451 |
> |
\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}} |
452 |
> |
\end{eqnarray} |
453 |
|
|
454 |
< |
\section{Results} |
455 |
< |
\subsection{Toluene Solvent} |
454 |
> |
To describe the interactions between metal (Au) and non-metal atoms, |
455 |
> |
we refer to an adsorption study of alkyl thiols on gold surfaces by |
456 |
> |
Vlugt {\it et al.}\cite{vlugt:cpc2007154} They fitted an effective |
457 |
> |
Lennard-Jones form of potential parameters for the interaction between |
458 |
> |
Au and pseudo-atoms CH$_x$ and S based on a well-established and |
459 |
> |
widely-used effective potential of Hautman and Klein for the Au(111) |
460 |
> |
surface.\cite{hautman:4994} As our simulations require the gold slab |
461 |
> |
to be flexible to accommodate thermal excitation, the pair-wise form |
462 |
> |
of potentials they developed was used for our study. |
463 |
|
|
464 |
< |
The results (Table \ref{AuThiolToluene}) show a |
465 |
< |
significant conductance enhancement compared to the gold/water |
466 |
< |
interface without capping agent and agree with available experimental |
467 |
< |
data. This indicates that the metal-metal potential, though not |
468 |
< |
predicting an accurate bulk metal thermal conductivity, does not |
469 |
< |
greatly interfere with the simulation of the thermal conductance |
470 |
< |
behavior across a non-metal interface. The solvent model is not |
471 |
< |
particularly volatile, so the simulation cell does not expand |
472 |
< |
significantly under higher temperature. We did not observe a |
473 |
< |
significant conductance decrease when the temperature was increased to |
474 |
< |
300K. The results show that the two definitions used for $G$ yield |
335 |
< |
comparable values, though $G^\prime$ tends to be smaller. |
464 |
> |
The potentials developed from {\it ab initio} calculations by Leng |
465 |
> |
{\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
466 |
> |
interactions between Au and aromatic C/H atoms in toluene. However, |
467 |
> |
the Lennard-Jones parameters between Au and other types of particles, |
468 |
> |
(e.g. AA alkanes) have not yet been established. For these |
469 |
> |
interactions, the Lorentz-Berthelot mixing rule can be used to derive |
470 |
> |
effective single-atom LJ parameters for the metal using the fit values |
471 |
> |
for toluene. These are then used to construct reasonable mixing |
472 |
> |
parameters for the interactions between the gold and other atoms. |
473 |
> |
Table \ref{MnM} summarizes the ``metal/non-metal'' parameters used in |
474 |
> |
our simulations. |
475 |
|
|
476 |
|
\begin{table*} |
477 |
|
\begin{minipage}{\linewidth} |
478 |
|
\begin{center} |
479 |
< |
\caption{Computed interfacial thermal conductivity ($G$ and |
480 |
< |
$G^\prime$) values for the Au/butanethiol/toluene interface at |
481 |
< |
different temperatures using a range of energy fluxes.} |
482 |
< |
|
344 |
< |
\begin{tabular}{cccc} |
479 |
> |
\caption{Non-bonded interaction parameters (including cross |
480 |
> |
interactions with Au atoms) for both force fields used in this |
481 |
> |
work.} |
482 |
> |
\begin{tabular}{lllllll} |
483 |
|
\hline\hline |
484 |
< |
$\langle T\rangle$ & $J_z$ & $G$ & $G^\prime$ \\ |
485 |
< |
(K) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
484 |
> |
& Site & $\sigma_{ii}$ & $\epsilon_{ii}$ & $q_i$ & |
485 |
> |
$\sigma_{Au-i}$ & $\epsilon_{Au-i}$ \\ |
486 |
> |
& & (\AA) & (kcal/mol) & ($e$) & (\AA) & (kcal/mol) \\ |
487 |
|
\hline |
488 |
< |
200 & 1.86 & 180 & 135 \\ |
489 |
< |
& 2.15 & 204 & 113 \\ |
490 |
< |
& 3.93 & 175 & 114 \\ |
491 |
< |
300 & 1.91 & 143 & 125 \\ |
492 |
< |
& 4.19 & 134 & 113 \\ |
488 |
> |
United Atom (UA) |
489 |
> |
&CH3 & 3.75 & 0.1947 & - & 3.54 & 0.2146 \\ |
490 |
> |
&CH2 & 3.95 & 0.0914 & - & 3.54 & 0.1749 \\ |
491 |
> |
&CHar & 3.695 & 0.1003 & - & 3.4625 & 0.1680 \\ |
492 |
> |
&CRar & 3.88 & 0.04173 & - & 3.555 & 0.1604 \\ |
493 |
> |
\hline |
494 |
> |
All Atom (AA) |
495 |
> |
&CT3 & 3.50 & 0.066 & -0.18 & 3.365 & 0.1373 \\ |
496 |
> |
&CT2 & 3.50 & 0.066 & -0.12 & 3.365 & 0.1373 \\ |
497 |
> |
&CTT & 3.50 & 0.066 & -0.065 & 3.365 & 0.1373 \\ |
498 |
> |
&HC & 2.50 & 0.030 & 0.06 & 2.865 & 0.09256 \\ |
499 |
> |
&CA & 3.55 & 0.070 & -0.115 & 3.173 & 0.0640 \\ |
500 |
> |
&HA & 2.42 & 0.030 & 0.115 & 2.746 & 0.0414 \\ |
501 |
> |
\hline |
502 |
> |
Both UA and AA |
503 |
> |
& S & 4.45 & 0.25 & - & 2.40 & 8.465 \\ |
504 |
|
\hline\hline |
505 |
|
\end{tabular} |
506 |
< |
\label{AuThiolToluene} |
506 |
> |
\label{MnM} |
507 |
|
\end{center} |
508 |
|
\end{minipage} |
509 |
|
\end{table*} |
510 |
|
|
361 |
– |
\subsection{Hexane Solvent} |
511 |
|
|
512 |
< |
Using the united-atom model, different coverages of capping agent, |
513 |
< |
temperatures of simulations and numbers of solvent molecules were all |
514 |
< |
investigated and Table \ref{AuThiolHexaneUA} shows the results of |
515 |
< |
these computations. The number of hexane molecules in our simulations |
516 |
< |
does not affect the calculations significantly. However, a very long |
517 |
< |
length scale for the thermal gradient axis ($z$) may cause excessively |
518 |
< |
hot or cold temperatures in the middle of the solvent region and lead |
519 |
< |
to undesired phenomena such as solvent boiling or freezing, while too |
371 |
< |
few solvent molecules would change the normal behavior of the liquid |
372 |
< |
phase. Our $N_{hexane}$ values were chosen to ensure that these |
373 |
< |
extreme cases did not happen to our simulations. |
512 |
> |
\section{Results} |
513 |
> |
There are many factors contributing to the measured interfacial |
514 |
> |
conductance; some of these factors are physically motivated |
515 |
> |
(e.g. coverage of the surface by the capping agent coverage and |
516 |
> |
solvent identity), while some are governed by parameters of the |
517 |
> |
methodology (e.g. applied flux and the formulas used to obtain the |
518 |
> |
conductance). In this section we discuss the major physical and |
519 |
> |
calculational effects on the computed conductivity. |
520 |
|
|
521 |
< |
Table \ref{AuThiolHexaneUA} enables direct comparison between |
376 |
< |
different coverages of capping agent, when other system parameters are |
377 |
< |
held constant. With high coverage of butanethiol on the gold surface, |
378 |
< |
the interfacial thermal conductance is enhanced |
379 |
< |
significantly. Interestingly, a slightly lower butanethiol coverage |
380 |
< |
leads to a moderately higher conductivity. This is probably due to |
381 |
< |
more solvent/capping agent contact when butanethiol molecules are |
382 |
< |
not densely packed, which enhances the interactions between the two |
383 |
< |
phases and lowers the thermal transfer barrier of this interface. |
384 |
< |
% [COMPARE TO AU/WATER IN PAPER] |
521 |
> |
\subsection{Effects due to capping agent coverage} |
522 |
|
|
523 |
< |
It is also noted that the overall simulation temperature is another |
524 |
< |
factor that affects the interfacial thermal conductance. One |
525 |
< |
possibility of this effect may be rooted in the decrease in density of |
526 |
< |
the liquid phase. We observed that when the average temperature |
527 |
< |
increases from 200K to 250K, the bulk hexane density becomes lower |
528 |
< |
than experimental value, as the system is equilibrated under NPT |
392 |
< |
ensemble. This leads to lower contact between solvent and capping |
393 |
< |
agent, and thus lower conductivity. |
523 |
> |
A series of different initial conditions with a range of surface |
524 |
> |
coverages was prepared and solvated with various with both of the |
525 |
> |
solvent molecules. These systems were then equilibrated and their |
526 |
> |
interfacial thermal conductivity was measured with the NIVS |
527 |
> |
algorithm. Figure \ref{coverage} demonstrates the trend of conductance |
528 |
> |
with respect to surface coverage. |
529 |
|
|
530 |
< |
Conductivity values are more difficult to obtain under higher |
531 |
< |
temperatures. This is because the Au surface tends to undergo |
532 |
< |
reconstructions in relatively high temperatures. Surface Au atoms can |
533 |
< |
migrate outward to reach higher Au-S contact; and capping agent |
534 |
< |
molecules can be embedded into the surface Au layer due to the same |
535 |
< |
driving force. This phenomenon agrees with experimental |
536 |
< |
results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. A surface |
537 |
< |
fully covered in capping agent is more susceptible to reconstruction, |
403 |
< |
possibly because fully coverage prevents other means of capping agent |
404 |
< |
relaxation, such as migration to an empty neighbor three-fold site. |
530 |
> |
\begin{figure} |
531 |
> |
\includegraphics[width=\linewidth]{coverage} |
532 |
> |
\caption{The interfacial thermal conductivity ($G$) has a |
533 |
> |
non-monotonic dependence on the degree of surface capping. This |
534 |
> |
data is for the Au(111) / butanethiol / solvent interface with |
535 |
> |
various UA force fields at $\langle T\rangle \sim $200K.} |
536 |
> |
\label{coverage} |
537 |
> |
\end{figure} |
538 |
|
|
539 |
< |
%MAY ADD MORE DATA TO TABLE |
539 |
> |
In partially covered surfaces, the derivative definition for |
540 |
> |
$G^\prime$ (Eq. \ref{derivativeG}) becomes difficult to apply, as the |
541 |
> |
location of maximum change of $\lambda$ becomes washed out. The |
542 |
> |
discrete definition (Eq. \ref{discreteG}) is easier to apply, as the |
543 |
> |
Gibbs dividing surface is still well-defined. Therefore, $G$ (not |
544 |
> |
$G^\prime$) was used in this section. |
545 |
> |
|
546 |
> |
From Figure \ref{coverage}, one can see the significance of the |
547 |
> |
presence of capping agents. When even a small fraction of the Au(111) |
548 |
> |
surface sites are covered with butanethiols, the conductivity exhibits |
549 |
> |
an enhancement by at least a factor of 3. Capping agents are clearly |
550 |
> |
playing a major role in thermal transport at metal / organic solvent |
551 |
> |
surfaces. |
552 |
> |
|
553 |
> |
We note a non-monotonic behavior in the interfacial conductance as a |
554 |
> |
function of surface coverage. The maximum conductance (largest $G$) |
555 |
> |
happens when the surfaces are about 75\% covered with butanethiol |
556 |
> |
caps. The reason for this behavior is not entirely clear. One |
557 |
> |
explanation is that incomplete butanethiol coverage allows small gaps |
558 |
> |
between butanethiols to form. These gaps can be filled by transient |
559 |
> |
solvent molecules. These solvent molecules couple very strongly with |
560 |
> |
the hot capping agent molecules near the surface, and can then carry |
561 |
> |
away (diffusively) the excess thermal energy from the surface. |
562 |
> |
|
563 |
> |
There appears to be a competition between the conduction of the |
564 |
> |
thermal energy away from the surface by the capping agents (enhanced |
565 |
> |
by greater coverage) and the coupling of the capping agents with the |
566 |
> |
solvent (enhanced by interdigitation at lower coverages). This |
567 |
> |
competition would lead to the non-monotonic coverage behavior observed |
568 |
> |
here. |
569 |
> |
|
570 |
> |
Results for rigid body toluene solvent, as well as the UA hexane, are |
571 |
> |
within the ranges expected from prior experimental |
572 |
> |
work.\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406} This suggests |
573 |
> |
that explicit hydrogen atoms might not be required for modeling |
574 |
> |
thermal transport in these systems. C-H vibrational modes do not see |
575 |
> |
significant excited state population at low temperatures, and are not |
576 |
> |
likely to carry lower frequency excitations from the solid layer into |
577 |
> |
the bulk liquid. |
578 |
> |
|
579 |
> |
The toluene solvent does not exhibit the same behavior as hexane in |
580 |
> |
that $G$ remains at approximately the same magnitude when the capping |
581 |
> |
coverage increases from 25\% to 75\%. Toluene, as a rigid planar |
582 |
> |
molecule, cannot occupy the relatively small gaps between the capping |
583 |
> |
agents as easily as the chain-like {\it n}-hexane. The effect of |
584 |
> |
solvent coupling to the capping agent is therefore weaker in toluene |
585 |
> |
except at the very lowest coverage levels. This effect counters the |
586 |
> |
coverage-dependent conduction of heat away from the metal surface, |
587 |
> |
leading to a much flatter $G$ vs. coverage trend than is observed in |
588 |
> |
{\it n}-hexane. |
589 |
> |
|
590 |
> |
\subsection{Effects due to Solvent \& Solvent Models} |
591 |
> |
In addition to UA solvent and capping agent models, AA models have |
592 |
> |
also been included in our simulations. In most of this work, the same |
593 |
> |
(UA or AA) model for solvent and capping agent was used, but it is |
594 |
> |
also possible to utilize different models for different components. |
595 |
> |
We have also included isotopic substitutions (Hydrogen to Deuterium) |
596 |
> |
to decrease the explicit vibrational overlap between solvent and |
597 |
> |
capping agent. Table \ref{modelTest} summarizes the results of these |
598 |
> |
studies. |
599 |
> |
|
600 |
> |
{\bf MAY NOT NEED $J_z$ IN TABLE} |
601 |
|
\begin{table*} |
602 |
|
\begin{minipage}{\linewidth} |
603 |
|
\begin{center} |
410 |
– |
\caption{Computed interfacial thermal conductivity ($G$ and |
411 |
– |
$G^\prime$) values for the Au/butanethiol/hexane interface |
412 |
– |
with united-atom model and different capping agent coverage |
413 |
– |
and solvent molecule numbers at different temperatures using a |
414 |
– |
range of energy fluxes.} |
604 |
|
|
605 |
< |
\begin{tabular}{cccccc} |
605 |
> |
\caption{Computed interfacial thermal conductance ($G$ and |
606 |
> |
$G^\prime$) values for interfaces using various models for |
607 |
> |
solvent and capping agent (or without capping agent) at |
608 |
> |
$\langle T\rangle\sim$200K. Here ``D'' stands for deuterated |
609 |
> |
solvent or capping agent molecules; ``Avg.'' denotes results |
610 |
> |
that are averages of simulations under different applied |
611 |
> |
thermal flux $(J_z)$ values. Error estimates are indicated in |
612 |
> |
parentheses.} |
613 |
> |
|
614 |
> |
\begin{tabular}{llccc} |
615 |
|
\hline\hline |
616 |
< |
Thiol & $\langle T\rangle$ & & $J_z$ & $G$ & $G^\prime$ \\ |
617 |
< |
coverage (\%) & (K) & $N_{hexane}$ & (GW/m$^2$) & |
616 |
> |
Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ |
617 |
> |
(or bare surface) & model & (GW/m$^2$) & |
618 |
|
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
619 |
|
\hline |
620 |
< |
0.0 & 200 & 200 & 0.96 & 43.3 & 42.7 \\ |
621 |
< |
& & & 1.91 & 45.7 & 42.9 \\ |
622 |
< |
& & 166 & 0.96 & 43.1 & 53.4 \\ |
623 |
< |
88.9 & 200 & 166 & 1.94 & 172 & 108 \\ |
624 |
< |
100.0 & 250 & 200 & 0.96 & 81.8 & 67.0 \\ |
625 |
< |
& & 166 & 0.98 & 79.0 & 62.9 \\ |
626 |
< |
& & & 1.44 & 76.2 & 64.8 \\ |
627 |
< |
& 200 & 200 & 1.92 & 129 & 87.3 \\ |
628 |
< |
& & & 1.93 & 131 & 77.5 \\ |
629 |
< |
& & 166 & 0.97 & 115 & 69.3 \\ |
630 |
< |
& & & 1.94 & 125 & 87.1 \\ |
620 |
> |
UA & UA hexane & Avg. & 131(9) & 87(10) \\ |
621 |
> |
& UA hexane(D) & 1.95 & 153(5) & 136(13) \\ |
622 |
> |
& AA hexane & Avg. & 131(6) & 122(10) \\ |
623 |
> |
& UA toluene & 1.96 & 187(16) & 151(11) \\ |
624 |
> |
& AA toluene & 1.89 & 200(36) & 149(53) \\ |
625 |
> |
\hline |
626 |
> |
AA & UA hexane & 1.94 & 116(9) & 129(8) \\ |
627 |
> |
& AA hexane & Avg. & 442(14) & 356(31) \\ |
628 |
> |
& AA hexane(D) & 1.93 & 222(12) & 234(54) \\ |
629 |
> |
& UA toluene & 1.98 & 125(25) & 97(60) \\ |
630 |
> |
& AA toluene & 3.79 & 487(56) & 290(42) \\ |
631 |
> |
\hline |
632 |
> |
AA(D) & UA hexane & 1.94 & 158(25) & 172(4) \\ |
633 |
> |
& AA hexane & 1.92 & 243(29) & 191(11) \\ |
634 |
> |
& AA toluene & 1.93 & 364(36) & 322(67) \\ |
635 |
> |
\hline |
636 |
> |
bare & UA hexane & Avg. & 46.5(3.2) & 49.4(4.5) \\ |
637 |
> |
& UA hexane(D) & 0.98 & 43.9(4.6) & 43.0(2.0) \\ |
638 |
> |
& AA hexane & 0.96 & 31.0(1.4) & 29.4(1.3) \\ |
639 |
> |
& UA toluene & 1.99 & 70.1(1.3) & 65.8(0.5) \\ |
640 |
|
\hline\hline |
641 |
|
\end{tabular} |
642 |
< |
\label{AuThiolHexaneUA} |
642 |
> |
\label{modelTest} |
643 |
|
\end{center} |
644 |
|
\end{minipage} |
645 |
|
\end{table*} |
646 |
|
|
647 |
< |
For the all-atom model, the liquid hexane phase was not stable under NPT |
648 |
< |
conditions. Therefore, the simulation length scale parameters are |
649 |
< |
adopted from previous equilibration results of the united-atom model |
443 |
< |
at 200K. Table \ref{AuThiolHexaneAA} shows the results of these |
444 |
< |
simulations. The conductivity values calculated with full capping |
445 |
< |
agent coverage are substantially larger than observed in the |
446 |
< |
united-atom model, and is even higher than predicted by |
447 |
< |
experiments. It is possible that our parameters for metal-non-metal |
448 |
< |
particle interactions lead to an overestimate of the interfacial |
449 |
< |
thermal conductivity, although the active C-H vibrations in the |
450 |
< |
all-atom model (which should not be appreciably populated at normal |
451 |
< |
temperatures) could also account for this high conductivity. The major |
452 |
< |
thermal transfer barrier of Au/butanethiol/hexane interface is between |
453 |
< |
the liquid phase and the capping agent, so extra degrees of freedom |
454 |
< |
such as the C-H vibrations could enhance heat exchange between these |
455 |
< |
two phases and result in a much higher conductivity. |
647 |
> |
To facilitate direct comparison between force fields, systems with the |
648 |
> |
same capping agent and solvent were prepared with the same length |
649 |
> |
scales for the simulation cells. |
650 |
|
|
651 |
+ |
On bare metal / solvent surfaces, different force field models for |
652 |
+ |
hexane yield similar results for both $G$ and $G^\prime$, and these |
653 |
+ |
two definitions agree with each other very well. This is primarily an |
654 |
+ |
indicator of weak interactions between the metal and the solvent, and |
655 |
+ |
is a typical case for acoustic impedance mismatch between these two |
656 |
+ |
phases. |
657 |
+ |
|
658 |
+ |
For the fully-covered surfaces, the choice of force field for the |
659 |
+ |
capping agent and solvent has a large impact on the calculated values |
660 |
+ |
of $G$ and $G^\prime$. The AA thiol to AA solvent conductivities are |
661 |
+ |
much larger than their UA to UA counterparts, and these values exceed |
662 |
+ |
the experimental estimates by a large measure. The AA force field |
663 |
+ |
allows significant energy to go into C-H (or C-D) stretching modes, |
664 |
+ |
and since these modes are high frequency, this non-quantum behavior is |
665 |
+ |
likely responsible for the overestimate of the conductivity. Compared |
666 |
+ |
to the AA model, the UA model yields more reasonable conductivity |
667 |
+ |
values with much higher computational efficiency. |
668 |
+ |
|
669 |
+ |
\subsubsection{Are electronic excitations in the metal important?} |
670 |
+ |
Because they lack electronic excitations, the QSC and related embedded |
671 |
+ |
atom method (EAM) models for gold are known to predict unreasonably |
672 |
+ |
low values for bulk conductivity |
673 |
+ |
($\lambda$).\cite{kuang:164101,ISI:000207079300006,Clancy:1992} If the |
674 |
+ |
conductance between the phases ($G$) is governed primarily by phonon |
675 |
+ |
excitation (and not electronic degrees of freedom), one would expect a |
676 |
+ |
classical model to capture most of the interfacial thermal |
677 |
+ |
conductance. Our results for $G$ and $G^\prime$ indicate that this is |
678 |
+ |
indeed the case, and suggest that the modeling of interfacial thermal |
679 |
+ |
transport depends primarily on the description of the interactions |
680 |
+ |
between the various components at the interface. When the metal is |
681 |
+ |
chemically capped, the primary barrier to thermal conductivity appears |
682 |
+ |
to be the interface between the capping agent and the surrounding |
683 |
+ |
solvent, so the excitations in the metal have little impact on the |
684 |
+ |
value of $G$. |
685 |
+ |
|
686 |
+ |
\subsection{Effects due to methodology and simulation parameters} |
687 |
+ |
|
688 |
+ |
We have varied the parameters of the simulations in order to |
689 |
+ |
investigate how these factors would affect the computation of $G$. Of |
690 |
+ |
particular interest are: 1) the length scale for the applied thermal |
691 |
+ |
gradient (modified by increasing the amount of solvent in the system), |
692 |
+ |
2) the sign and magnitude of the applied thermal flux, 3) the average |
693 |
+ |
temperature of the simulation (which alters the solvent density during |
694 |
+ |
equilibration), and 4) the definition of the interfacial conductance |
695 |
+ |
(Eqs. (\ref{discreteG}) or (\ref{derivativeG})) used in the |
696 |
+ |
calculation. |
697 |
+ |
|
698 |
+ |
Systems of different lengths were prepared by altering the number of |
699 |
+ |
solvent molecules and extending the length of the box along the $z$ |
700 |
+ |
axis to accomodate the extra solvent. Equilibration at the same |
701 |
+ |
temperature and pressure conditions led to nearly identical surface |
702 |
+ |
areas ($L_x$ and $L_y$) available to the metal and capping agent, |
703 |
+ |
while the extra solvent served mainly to lengthen the axis that was |
704 |
+ |
used to apply the thermal flux. For a given value of the applied |
705 |
+ |
flux, the different $z$ length scale has only a weak effect on the |
706 |
+ |
computed conductivities (Table \ref{AuThiolHexaneUA}). |
707 |
+ |
|
708 |
+ |
\subsubsection{Effects of applied flux} |
709 |
+ |
The NIVS algorithm allows changes in both the sign and magnitude of |
710 |
+ |
the applied flux. It is possible to reverse the direction of heat |
711 |
+ |
flow simply by changing the sign of the flux, and thermal gradients |
712 |
+ |
which would be difficult to obtain experimentally ($5$ K/\AA) can be |
713 |
+ |
easily simulated. However, the magnitude of the applied flux is not |
714 |
+ |
arbitrary if one aims to obtain a stable and reliable thermal gradient. |
715 |
+ |
A temperature gradient can be lost in the noise if $|J_z|$ is too |
716 |
+ |
small, and excessive $|J_z|$ values can cause phase transitions if the |
717 |
+ |
extremes of the simulation cell become widely separated in |
718 |
+ |
temperature. Also, if $|J_z|$ is too large for the bulk conductivity |
719 |
+ |
of the materials, the thermal gradient will never reach a stable |
720 |
+ |
state. |
721 |
+ |
|
722 |
+ |
Within a reasonable range of $J_z$ values, we were able to study how |
723 |
+ |
$G$ changes as a function of this flux. In what follows, we use |
724 |
+ |
positive $J_z$ values to denote the case where energy is being |
725 |
+ |
transferred by the method from the metal phase and into the liquid. |
726 |
+ |
The resulting gradient therefore has a higher temperature in the |
727 |
+ |
liquid phase. Negative flux values reverse this transfer, and result |
728 |
+ |
in higher temperature metal phases. The conductance measured under |
729 |
+ |
different applied $J_z$ values is listed in Tables |
730 |
+ |
\ref{AuThiolHexaneUA} and \ref{AuThiolToluene}. These results do not |
731 |
+ |
indicate that $G$ depends strongly on $J_z$ within this flux |
732 |
+ |
range. The linear response of flux to thermal gradient simplifies our |
733 |
+ |
investigations in that we can rely on $G$ measurement with only a |
734 |
+ |
small number $J_z$ values. |
735 |
+ |
|
736 |
+ |
{\bf MAY MOVE TO SUPPORT INFO} |
737 |
|
\begin{table*} |
738 |
|
\begin{minipage}{\linewidth} |
739 |
|
\begin{center} |
740 |
+ |
\caption{In the hexane-solvated interfaces, the system size has |
741 |
+ |
little effect on the calculated values for interfacial |
742 |
+ |
conductance ($G$ and $G^\prime$), but the direction of heat |
743 |
+ |
flow (i.e. the sign of $J_z$) can alter the average |
744 |
+ |
temperature of the liquid phase and this can alter the |
745 |
+ |
computed conductivity.} |
746 |
|
|
747 |
< |
\caption{Computed interfacial thermal conductivity ($G$ and |
462 |
< |
$G^\prime$) values for the Au/butanethiol/hexane interface |
463 |
< |
with all-atom model and different capping agent coverage at |
464 |
< |
200K using a range of energy fluxes.} |
465 |
< |
|
466 |
< |
\begin{tabular}{cccc} |
747 |
> |
\begin{tabular}{ccccccc} |
748 |
|
\hline\hline |
749 |
< |
Thiol & $J_z$ & $G$ & $G^\prime$ \\ |
750 |
< |
coverage (\%) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
749 |
> |
$\langle T\rangle$ & $N_{hexane}$ & $\rho_{hexane}$ & |
750 |
> |
$J_z$ & $G$ & $G^\prime$ \\ |
751 |
> |
(K) & & (g/cm$^3$) & (GW/m$^2$) & |
752 |
> |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
753 |
|
\hline |
754 |
< |
0.0 & 0.95 & 28.5 & 27.2 \\ |
755 |
< |
& 1.88 & 30.3 & 28.9 \\ |
756 |
< |
100.0 & 2.87 & 551 & 294 \\ |
757 |
< |
& 3.81 & 494 & 193 \\ |
754 |
> |
200 & 266 & 0.672 & -0.96 & 102(3) & 80.0(0.8) \\ |
755 |
> |
& 200 & 0.688 & 0.96 & 125(16) & 90.2(15) \\ |
756 |
> |
& & & 1.91 & 139(10) & 101(10) \\ |
757 |
> |
& & & 2.83 & 141(6) & 89.9(9.8) \\ |
758 |
> |
& 166 & 0.681 & 0.97 & 141(30) & 78(22) \\ |
759 |
> |
& & & 1.92 & 138(4) & 98.9(9.5) \\ |
760 |
> |
\hline |
761 |
> |
250 & 200 & 0.560 & 0.96 & 75(10) & 61.8(7.3) \\ |
762 |
> |
& & & -0.95 & 49.4(0.3) & 45.7(2.1) \\ |
763 |
> |
& 166 & 0.569 & 0.97 & 80.3(0.6) & 67(11) \\ |
764 |
> |
& & & 1.44 & 76.2(5.0) & 64.8(3.8) \\ |
765 |
> |
& & & -0.95 & 56.4(2.5) & 54.4(1.1) \\ |
766 |
> |
& & & -1.85 & 47.8(1.1) & 53.5(1.5) \\ |
767 |
|
\hline\hline |
768 |
|
\end{tabular} |
769 |
< |
\label{AuThiolHexaneAA} |
769 |
> |
\label{AuThiolHexaneUA} |
770 |
|
\end{center} |
771 |
|
\end{minipage} |
772 |
|
\end{table*} |
773 |
|
|
774 |
< |
%subsubsection{Vibrational spectrum study on conductance mechanism} |
775 |
< |
To investigate the mechanism of this interfacial thermal conductance, |
776 |
< |
the vibrational spectra of various gold systems were obtained and are |
777 |
< |
shown as in the upper panel of Fig. \ref{vibration}. To obtain these |
778 |
< |
spectra, one first runs a simulation in the NVE ensemble and collects |
779 |
< |
snapshots of configurations; these configurations are used to compute |
780 |
< |
the velocity auto-correlation functions, which is used to construct a |
781 |
< |
power spectrum via a Fourier transform. The gold surfaces covered by |
782 |
< |
butanethiol molecules exhibit an additional peak observed at a |
491 |
< |
frequency of $\sim$170cm$^{-1}$, which is attributed to the vibration |
492 |
< |
of the S-Au bond. This vibration enables efficient thermal transport |
493 |
< |
from surface Au atoms to the capping agents. Simultaneously, as shown |
494 |
< |
in the lower panel of Fig. \ref{vibration}, the large overlap of the |
495 |
< |
vibration spectra of butanethiol and hexane in the all-atom model, |
496 |
< |
including the C-H vibration, also suggests high thermal exchange |
497 |
< |
efficiency. The combination of these two effects produces the drastic |
498 |
< |
interfacial thermal conductance enhancement in the all-atom model. |
774 |
> |
The sign of $J_z$ is a different matter, however, as this can alter |
775 |
> |
the temperature on the two sides of the interface. The average |
776 |
> |
temperature values reported are for the entire system, and not for the |
777 |
> |
liquid phase, so at a given $\langle T \rangle$, the system with |
778 |
> |
positive $J_z$ has a warmer liquid phase. This means that if the |
779 |
> |
liquid carries thermal energy via diffusive transport, {\it positive} |
780 |
> |
$J_z$ values will result in increased molecular motion on the liquid |
781 |
> |
side of the interface, and this will increase the measured |
782 |
> |
conductivity. |
783 |
|
|
784 |
+ |
\subsubsection{Effects due to average temperature} |
785 |
+ |
|
786 |
+ |
We also studied the effect of average system temperature on the |
787 |
+ |
interfacial conductance. The simulations are first equilibrated in |
788 |
+ |
the NPT ensemble at 1 atm. The TraPPE-UA model for hexane tends to |
789 |
+ |
predict a lower boiling point (and liquid state density) than |
790 |
+ |
experiments. This lower-density liquid phase leads to reduced contact |
791 |
+ |
between the hexane and butanethiol, and this accounts for our |
792 |
+ |
observation of lower conductance at higher temperatures as shown in |
793 |
+ |
Table \ref{AuThiolHexaneUA}. In raising the average temperature from |
794 |
+ |
200K to 250K, the density drop of $\sim$20\% in the solvent phase |
795 |
+ |
leads to a $\sim$40\% drop in the conductance. |
796 |
+ |
|
797 |
+ |
Similar behavior is observed in the TraPPE-UA model for toluene, |
798 |
+ |
although this model has better agreement with the experimental |
799 |
+ |
densities of toluene. The expansion of the toluene liquid phase is |
800 |
+ |
not as significant as that of the hexane (8.3\% over 100K), and this |
801 |
+ |
limits the effect to $\sim$20\% drop in thermal conductivity (Table |
802 |
+ |
\ref{AuThiolToluene}). |
803 |
+ |
|
804 |
+ |
Although we have not mapped out the behavior at a large number of |
805 |
+ |
temperatures, is clear that there will be a strong temperature |
806 |
+ |
dependence in the interfacial conductance when the physical properties |
807 |
+ |
of one side of the interface (notably the density) change rapidly as a |
808 |
+ |
function of temperature. |
809 |
+ |
|
810 |
+ |
{\bf MAY MOVE TO SUPPORT INFO} |
811 |
+ |
\begin{table*} |
812 |
+ |
\begin{minipage}{\linewidth} |
813 |
+ |
\begin{center} |
814 |
+ |
\caption{When toluene is the solvent, the interfacial thermal |
815 |
+ |
conductivity is less sensitive to temperature, but again, the |
816 |
+ |
direction of the heat flow can alter the solvent temperature |
817 |
+ |
and can change the computed conductance values.} |
818 |
+ |
|
819 |
+ |
\begin{tabular}{ccccc} |
820 |
+ |
\hline\hline |
821 |
+ |
$\langle T\rangle$ & $\rho_{toluene}$ & $J_z$ & $G$ & $G^\prime$ \\ |
822 |
+ |
(K) & (g/cm$^3$) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
823 |
+ |
\hline |
824 |
+ |
200 & 0.933 & 2.15 & 204(12) & 113(12) \\ |
825 |
+ |
& & -1.86 & 180(3) & 135(21) \\ |
826 |
+ |
& & -3.93 & 176(5) & 113(12) \\ |
827 |
+ |
\hline |
828 |
+ |
300 & 0.855 & -1.91 & 143(5) & 125(2) \\ |
829 |
+ |
& & -4.19 & 135(9) & 113(12) \\ |
830 |
+ |
\hline\hline |
831 |
+ |
\end{tabular} |
832 |
+ |
\label{AuThiolToluene} |
833 |
+ |
\end{center} |
834 |
+ |
\end{minipage} |
835 |
+ |
\end{table*} |
836 |
+ |
|
837 |
+ |
Besides the lower interfacial thermal conductance, surfaces at |
838 |
+ |
relatively high temperatures are susceptible to reconstructions, |
839 |
+ |
particularly when butanethiols fully cover the Au(111) surface. These |
840 |
+ |
reconstructions include surface Au atoms which migrate outward to the |
841 |
+ |
S atom layer, and butanethiol molecules which embed into the surface |
842 |
+ |
Au layer. The driving force for this behavior is the strong Au-S |
843 |
+ |
interactions which are modeled here with a deep Lennard-Jones |
844 |
+ |
potential. This phenomenon agrees with reconstructions that have been |
845 |
+ |
experimentally |
846 |
+ |
observed.\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt |
847 |
+ |
{\it et al.} kept their Au(111) slab rigid so that their simulations |
848 |
+ |
could reach 300K without surface |
849 |
+ |
reconstructions.\cite{vlugt:cpc2007154} Since surface reconstructions |
850 |
+ |
blur the interface, the measurement of $G$ becomes more difficult to |
851 |
+ |
conduct at higher temperatures. For this reason, most of our |
852 |
+ |
measurements are undertaken at $\langle T\rangle\sim$200K where |
853 |
+ |
reconstruction is minimized. |
854 |
+ |
|
855 |
+ |
However, when the surface is not completely covered by butanethiols, |
856 |
+ |
the simulated system appears to be more resistent to the |
857 |
+ |
reconstruction. Our Au / butanethiol / toluene system had the Au(111) |
858 |
+ |
surfaces 90\% covered by butanethiols, but did not see this above |
859 |
+ |
phenomena even at $\langle T\rangle\sim$300K. That said, we did |
860 |
+ |
observe butanethiols migrating to neighboring three-fold sites during |
861 |
+ |
a simulation. Since the interface persisted in these simulations, we |
862 |
+ |
were able to obtain $G$'s for these interfaces even at a relatively |
863 |
+ |
high temperature without being affected by surface reconstructions. |
864 |
+ |
|
865 |
+ |
\section{Discussion} |
866 |
+ |
|
867 |
+ |
The primary result of this work is that the capping agent acts as an |
868 |
+ |
efficient thermal coupler between solid and solvent phases. One of |
869 |
+ |
the ways the capping agent can carry out this role is to down-shift |
870 |
+ |
between the phonon vibrations in the solid (which carry the heat from |
871 |
+ |
the gold) and the molecular vibrations in the liquid (which carry some |
872 |
+ |
of the heat in the solvent). |
873 |
+ |
|
874 |
+ |
To investigate the mechanism of interfacial thermal conductance, the |
875 |
+ |
vibrational power spectrum was computed. Power spectra were taken for |
876 |
+ |
individual components in different simulations. To obtain these |
877 |
+ |
spectra, simulations were run after equilibration in the |
878 |
+ |
microcanonical (NVE) ensemble and without a thermal |
879 |
+ |
gradient. Snapshots of configurations were collected at a frequency |
880 |
+ |
that is higher than that of the fastest vibrations occurring in the |
881 |
+ |
simulations. With these configurations, the velocity auto-correlation |
882 |
+ |
functions can be computed: |
883 |
+ |
\begin{equation} |
884 |
+ |
C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle |
885 |
+ |
\label{vCorr} |
886 |
+ |
\end{equation} |
887 |
+ |
The power spectrum is constructed via a Fourier transform of the |
888 |
+ |
symmetrized velocity autocorrelation function, |
889 |
+ |
\begin{equation} |
890 |
+ |
\hat{f}(\omega) = |
891 |
+ |
\int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt |
892 |
+ |
\label{fourier} |
893 |
+ |
\end{equation} |
894 |
+ |
|
895 |
+ |
\subsection{The role of specific vibrations} |
896 |
+ |
The vibrational spectra for gold slabs in different environments are |
897 |
+ |
shown as in Figure \ref{specAu}. Regardless of the presence of |
898 |
+ |
solvent, the gold surfaces which are covered by butanethiol molecules |
899 |
+ |
exhibit an additional peak observed at a frequency of |
900 |
+ |
$\sim$165cm$^{-1}$. We attribute this peak to the S-Au bonding |
901 |
+ |
vibration. This vibration enables efficient thermal coupling of the |
902 |
+ |
surface Au layer to the capping agents. Therefore, in our simulations, |
903 |
+ |
the Au / S interfaces do not appear to be the primary barrier to |
904 |
+ |
thermal transport when compared with the butanethiol / solvent |
905 |
+ |
interfaces. {\bf This confirms the results from Luo {\it et |
906 |
+ |
al.}\cite{Luo20101}, which reported $G$ for Au-SAM junctions |
907 |
+ |
generally twice larger than what we have computed for the |
908 |
+ |
thiol-liquid interfaces.} |
909 |
+ |
|
910 |
|
\begin{figure} |
911 |
|
\includegraphics[width=\linewidth]{vibration} |
912 |
< |
\caption{Vibrational spectra obtained for gold in different |
913 |
< |
environments (upper panel) and for Au/thiol/hexane simulation in |
914 |
< |
all-atom model (lower panel).} |
915 |
< |
\label{vibration} |
912 |
> |
\caption{The vibrational power spectrum for thiol-capped gold has an |
913 |
> |
additional vibrational peak at $\sim $165cm$^{-1}$. Bare gold |
914 |
> |
surfaces (both with and without a solvent over-layer) are missing |
915 |
> |
this peak. A similar peak at $\sim $165cm$^{-1}$ also appears in |
916 |
> |
the vibrational power spectrum for the butanethiol capping agents.} |
917 |
> |
\label{specAu} |
918 |
|
\end{figure} |
507 |
– |
% 600dpi, letter size. too large? |
919 |
|
|
920 |
+ |
Also in this figure, we show the vibrational power spectrum for the |
921 |
+ |
bound butanethiol molecules, which also exhibits the same |
922 |
+ |
$\sim$165cm$^{-1}$ peak. |
923 |
|
|
924 |
+ |
\subsection{Overlap of power spectra} |
925 |
+ |
A comparison of the results obtained from the two different organic |
926 |
+ |
solvents can also provide useful information of the interfacial |
927 |
+ |
thermal transport process. In particular, the vibrational overlap |
928 |
+ |
between the butanethiol and the organic solvents suggests a highly |
929 |
+ |
efficient thermal exchange between these components. Very high |
930 |
+ |
thermal conductivity was observed when AA models were used and C-H |
931 |
+ |
vibrations were treated classically. The presence of extra degrees of |
932 |
+ |
freedom in the AA force field yields higher heat exchange rates |
933 |
+ |
between the two phases and results in a much higher conductivity than |
934 |
+ |
in the UA force field. {\bf Due to the classical models used, this |
935 |
+ |
even includes those high frequency modes which should be unpopulated |
936 |
+ |
at our relatively low temperatures. This artifact causes high |
937 |
+ |
frequency vibrations accountable for thermal transport in classical |
938 |
+ |
MD simulations.} |
939 |
+ |
|
940 |
+ |
The similarity in the vibrational modes available to solvent and |
941 |
+ |
capping agent can be reduced by deuterating one of the two components |
942 |
+ |
(Fig. \ref{aahxntln}). Once either the hexanes or the butanethiols |
943 |
+ |
are deuterated, one can observe a significantly lower $G$ and |
944 |
+ |
$G^\prime$ values (Table \ref{modelTest}). |
945 |
+ |
|
946 |
+ |
\begin{figure} |
947 |
+ |
\includegraphics[width=\linewidth]{aahxntln} |
948 |
+ |
\caption{Spectra obtained for all-atom (AA) Au / butanethiol / solvent |
949 |
+ |
systems. When butanethiol is deuterated (lower left), its |
950 |
+ |
vibrational overlap with hexane decreases significantly. Since |
951 |
+ |
aromatic molecules and the butanethiol are vibrationally dissimilar, |
952 |
+ |
the change is not as dramatic when toluene is the solvent (right).} |
953 |
+ |
\label{aahxntln} |
954 |
+ |
\end{figure} |
955 |
+ |
|
956 |
+ |
For the Au / butanethiol / toluene interfaces, having the AA |
957 |
+ |
butanethiol deuterated did not yield a significant change in the |
958 |
+ |
measured conductance. Compared to the C-H vibrational overlap between |
959 |
+ |
hexane and butanethiol, both of which have alkyl chains, the overlap |
960 |
+ |
between toluene and butanethiol is not as significant and thus does |
961 |
+ |
not contribute as much to the heat exchange process. |
962 |
+ |
|
963 |
+ |
Previous observations of Zhang {\it et al.}\cite{hase:2010} indicate |
964 |
+ |
that the {\it intra}molecular heat transport due to alkylthiols is |
965 |
+ |
highly efficient. Combining our observations with those of Zhang {\it |
966 |
+ |
et al.}, it appears that butanethiol acts as a channel to expedite |
967 |
+ |
heat flow from the gold surface and into the alkyl chain. The |
968 |
+ |
acoustic impedance mismatch between the metal and the liquid phase can |
969 |
+ |
therefore be effectively reduced with the presence of suitable capping |
970 |
+ |
agents. |
971 |
+ |
|
972 |
+ |
Deuterated models in the UA force field did not decouple the thermal |
973 |
+ |
transport as well as in the AA force field. The UA models, even |
974 |
+ |
though they have eliminated the high frequency C-H vibrational |
975 |
+ |
overlap, still have significant overlap in the lower-frequency |
976 |
+ |
portions of the infrared spectra (Figure \ref{uahxnua}). Deuterating |
977 |
+ |
the UA models did not decouple the low frequency region enough to |
978 |
+ |
produce an observable difference for the results of $G$ (Table |
979 |
+ |
\ref{modelTest}). |
980 |
+ |
|
981 |
+ |
\begin{figure} |
982 |
+ |
\includegraphics[width=\linewidth]{uahxnua} |
983 |
+ |
\caption{Vibrational power spectra for UA models for the butanethiol |
984 |
+ |
and hexane solvent (upper panel) show the high degree of overlap |
985 |
+ |
between these two molecules, particularly at lower frequencies. |
986 |
+ |
Deuterating a UA model for the solvent (lower panel) does not |
987 |
+ |
decouple the two spectra to the same degree as in the AA force |
988 |
+ |
field (see Fig \ref{aahxntln}).} |
989 |
+ |
\label{uahxnua} |
990 |
+ |
\end{figure} |
991 |
+ |
|
992 |
+ |
\section{Conclusions} |
993 |
+ |
The NIVS algorithm has been applied to simulations of |
994 |
+ |
butanethiol-capped Au(111) surfaces in the presence of organic |
995 |
+ |
solvents. This algorithm allows the application of unphysical thermal |
996 |
+ |
flux to transfer heat between the metal and the liquid phase. With the |
997 |
+ |
flux applied, we were able to measure the corresponding thermal |
998 |
+ |
gradients and to obtain interfacial thermal conductivities. Under |
999 |
+ |
steady states, 2-3 ns trajectory simulations are sufficient for |
1000 |
+ |
computation of this quantity. |
1001 |
+ |
|
1002 |
+ |
Our simulations have seen significant conductance enhancement in the |
1003 |
+ |
presence of capping agent, compared with the bare gold / liquid |
1004 |
+ |
interfaces. The acoustic impedance mismatch between the metal and the |
1005 |
+ |
liquid phase is effectively eliminated by a chemically-bonded capping |
1006 |
+ |
agent. Furthermore, the coverage percentage of the capping agent plays |
1007 |
+ |
an important role in the interfacial thermal transport |
1008 |
+ |
process. Moderately low coverages allow higher contact between capping |
1009 |
+ |
agent and solvent, and thus could further enhance the heat transfer |
1010 |
+ |
process, giving a non-monotonic behavior of conductance with |
1011 |
+ |
increasing coverage. |
1012 |
+ |
|
1013 |
+ |
Our results, particularly using the UA models, agree well with |
1014 |
+ |
available experimental data. The AA models tend to overestimate the |
1015 |
+ |
interfacial thermal conductance in that the classically treated C-H |
1016 |
+ |
vibrations become too easily populated. Compared to the AA models, the |
1017 |
+ |
UA models have higher computational efficiency with satisfactory |
1018 |
+ |
accuracy, and thus are preferable in modeling interfacial thermal |
1019 |
+ |
transport. |
1020 |
+ |
|
1021 |
+ |
Of the two definitions for $G$, the discrete form |
1022 |
+ |
(Eq. \ref{discreteG}) was easier to use and gives out relatively |
1023 |
+ |
consistent results, while the derivative form (Eq. \ref{derivativeG}) |
1024 |
+ |
is not as versatile. Although $G^\prime$ gives out comparable results |
1025 |
+ |
and follows similar trend with $G$ when measuring close to fully |
1026 |
+ |
covered or bare surfaces, the spatial resolution of $T$ profile |
1027 |
+ |
required for the use of a derivative form is limited by the number of |
1028 |
+ |
bins and the sampling required to obtain thermal gradient information. |
1029 |
+ |
|
1030 |
+ |
Vlugt {\it et al.} have investigated the surface thiol structures for |
1031 |
+ |
nanocrystalline gold and pointed out that they differ from those of |
1032 |
+ |
the Au(111) surface.\cite{landman:1998,vlugt:cpc2007154} This |
1033 |
+ |
difference could also cause differences in the interfacial thermal |
1034 |
+ |
transport behavior. To investigate this problem, one would need an |
1035 |
+ |
effective method for applying thermal gradients in non-planar |
1036 |
+ |
(i.e. spherical) geometries. |
1037 |
+ |
|
1038 |
|
\section{Acknowledgments} |
1039 |
|
Support for this project was provided by the National Science |
1040 |
|
Foundation under grant CHE-0848243. Computational time was provided by |
1041 |
|
the Center for Research Computing (CRC) at the University of Notre |
1042 |
< |
Dame. \newpage |
1042 |
> |
Dame. |
1043 |
|
|
1044 |
+ |
\section{Supporting Information} |
1045 |
+ |
This information is available free of charge via the Internet at |
1046 |
+ |
http://pubs.acs.org. |
1047 |
+ |
|
1048 |
+ |
\newpage |
1049 |
+ |
|
1050 |
|
\bibliography{interfacial} |
1051 |
|
|
1052 |
|
\end{doublespace} |