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\begin{document} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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|
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\section{Introduction} |
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Interfacial thermal conductance is extensively studied both |
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experimentally and computationally, due to its importance in nanoscale |
78 |
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science and technology. Reliability of nanoscale devices depends on |
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their thermal transport properties. Unlike bulk homogeneous materials, |
80 |
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nanoscale materials features significant presence of interfaces, and |
81 |
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these interfaces could dominate the heat transfer behavior of these |
82 |
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materials. Furthermore, these materials are generally heterogeneous, |
83 |
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which challenges traditional research methods for homogeneous systems. |
76 |
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Due to the importance of heat flow in nanotechnology, interfacial |
77 |
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thermal conductance has been studied extensively both experimentally |
78 |
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and computationally.\cite{cahill:793} Unlike bulk materials, nanoscale |
79 |
> |
materials have a significant fraction of their atoms at interfaces, |
80 |
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and the chemical details of these interfaces govern the heat transfer |
81 |
> |
behavior. Furthermore, the interfaces are |
82 |
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heterogeneous (e.g. solid - liquid), which provides a challenge to |
83 |
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traditional methods developed for homogeneous systems. |
84 |
|
|
85 |
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Heat conductance of molecular and nano-scale interfaces will be |
86 |
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affected by the chemical details of the surface. Experimentally, |
87 |
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various interfaces have been investigated for their thermal |
88 |
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conductance properties. Wang {\it et al.} studied heat transport |
89 |
< |
through long-chain hydrocarbon monolayers on gold substrate at |
90 |
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individual molecular level\cite{Wang10082007}; Schmidt {\it et al.} |
91 |
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studied the role of CTAB on thermal transport between gold nanorods |
92 |
< |
and solvent\cite{doi:10.1021/jp8051888}; Juv\'e {\it et al.} studied |
85 |
> |
Experimentally, various interfaces have been investigated for their |
86 |
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thermal conductance. Cahill and coworkers studied nanoscale thermal |
87 |
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transport from metal nanoparticle/fluid interfaces, to epitaxial |
88 |
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TiN/single crystal oxides interfaces, to hydrophilic and hydrophobic |
89 |
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interfaces between water and solids with different self-assembled |
90 |
> |
monolayers.\cite{Wilson:2002uq,PhysRevB.67.054302,doi:10.1021/jp048375k,PhysRevLett.96.186101} |
91 |
> |
Wang {\it et al.} studied heat transport through |
92 |
> |
long-chain hydrocarbon monolayers on gold substrate at individual |
93 |
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molecular level,\cite{Wang10082007} Schmidt {\it et al.} studied the |
94 |
> |
role of CTAB on thermal transport between gold nanorods and |
95 |
> |
solvent,\cite{doi:10.1021/jp8051888} and Juv\'e {\it et al.} studied |
96 |
|
the cooling dynamics, which is controlled by thermal interface |
97 |
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resistence of glass-embedded metal |
98 |
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nanoparticles\cite{PhysRevB.80.195406}. Although interfaces are |
99 |
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commonly barriers for heat transport, Alper {\it et al.} suggested |
100 |
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that specific ligands (capping agents) could completely eliminate this |
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barrier ($G\rightarrow\infty$)\cite{doi:10.1021/la904855s}. |
98 |
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nanoparticles.\cite{PhysRevB.80.195406} Although interfaces are |
99 |
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normally considered barriers for heat transport, Alper {\it et al.} |
100 |
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suggested that specific ligands (capping agents) could completely |
101 |
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eliminate this barrier |
102 |
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($G\rightarrow\infty$).\cite{doi:10.1021/la904855s} |
103 |
|
|
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Theoretical and computational studies were also engaged in the |
105 |
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interfacial thermal transport research in order to gain an |
106 |
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understanding of this phenomena at the molecular level. Hase and |
107 |
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coworkers employed Non-Equilibrium Molecular Dynamics (NEMD) |
108 |
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simulations to study thermal transport from hot Au(111) substrate to a |
109 |
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self-assembled monolayer of alkylthiolate with relatively long chain |
110 |
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(8-20 carbon atoms)\cite{hase:2010,hase:2011}. However, |
111 |
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emsemble average measurements for heat conductance of interfaces |
112 |
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between the capping monolayer on Au and a solvent phase has yet to be |
113 |
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studied. The relatively low thermal flux through interfaces is |
114 |
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difficult to measure with Equilibrium MD or forward NEMD simulation |
115 |
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methods. Therefore, the Reverse NEMD (RNEMD) methods would have the |
116 |
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advantage of having this difficult to measure flux known when studying |
117 |
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the thermal transport across interfaces, given that the simulation |
118 |
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methods being able to effectively apply an unphysical flux in |
119 |
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non-homogeneous systems. |
104 |
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Theoretical and computational models have also been used to study the |
105 |
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interfacial thermal transport in order to gain an understanding of |
106 |
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this phenomena at the molecular level. Recently, Hase and coworkers |
107 |
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employed Non-Equilibrium Molecular Dynamics (NEMD) simulations to |
108 |
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study thermal transport from hot Au(111) substrate to a self-assembled |
109 |
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monolayer of alkylthiol with relatively long chain (8-20 carbon |
110 |
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atoms).\cite{hase:2010,hase:2011} However, ensemble averaged |
111 |
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measurements for heat conductance of interfaces between the capping |
112 |
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monolayer on Au and a solvent phase have yet to be studied with their |
113 |
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approach. The comparatively low thermal flux through interfaces is |
114 |
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difficult to measure with Equilibrium |
115 |
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MD\cite{doi:10.1080/0026897031000068578} or forward NEMD simulation |
116 |
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methods. Therefore, the Reverse NEMD (RNEMD) |
117 |
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methods\cite{MullerPlathe:1997xw,kuang:164101} would have the |
118 |
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advantage of applying this difficult to measure flux (while measuring |
119 |
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the resulting gradient), given that the simulation methods being able |
120 |
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to effectively apply an unphysical flux in non-homogeneous systems. |
121 |
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Garde and coworkers\cite{garde:nl2005,garde:PhysRevLett2009} applied |
122 |
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this approach to various liquid interfaces and studied how thermal |
123 |
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conductance (or resistance) is dependent on chemistry details of |
124 |
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interfaces, e.g. hydrophobic and hydrophilic aqueous interfaces. |
125 |
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|
126 |
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Recently, we have developed the Non-Isotropic Velocity Scaling (NIVS) |
126 |
> |
Recently, we have developed a Non-Isotropic Velocity Scaling (NIVS) |
127 |
|
algorithm for RNEMD simulations\cite{kuang:164101}. This algorithm |
128 |
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retains the desirable features of RNEMD (conservation of linear |
129 |
|
momentum and total energy, compatibility with periodic boundary |
130 |
|
conditions) while establishing true thermal distributions in each of |
131 |
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the two slabs. Furthermore, it allows more effective thermal exchange |
132 |
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between particles of different identities, and thus enables extensive |
133 |
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study of interfacial conductance under steady states. |
131 |
> |
the two slabs. Furthermore, it allows effective thermal exchange |
132 |
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between particles of different identities, and thus makes the study of |
133 |
> |
interfacial conductance much simpler. |
134 |
|
|
135 |
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Our work presented here investigated the Au(111) surface with various |
136 |
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coverage of butanethiol, a capping agent with shorter carbon chain, |
137 |
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solvated with organic solvents of different molecular shapes. And |
138 |
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different models were used for both the capping agent and the solvent |
139 |
< |
force field parameters. With the NIVS algorithm applied, the thermal |
140 |
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transport through these interfacial systems was studied and the |
141 |
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underlying mechanism for this phenomena was investigated. |
135 |
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The work presented here deals with the Au(111) surface covered to |
136 |
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varying degrees by butanethiol, a capping agent with short carbon |
137 |
> |
chain, and solvated with organic solvents of different molecular |
138 |
> |
properties. Different models were used for both the capping agent and |
139 |
> |
the solvent force field parameters. Using the NIVS algorithm, the |
140 |
> |
thermal transport across these interfaces was studied and the |
141 |
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underlying mechanism for the phenomena was investigated. |
142 |
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|
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[WHY STUDY AU-THIOL SURFACE; MAY CITE SHAOYI JIANG] |
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|
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|
\section{Methodology} |
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\subsection{Algorithm} |
145 |
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[BACKGROUND FOR MD METHODS] |
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There have been many algorithms for computing thermal conductivity |
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using molecular dynamics simulations. However, interfacial conductance |
148 |
< |
is at least an order of magnitude smaller. This would make the |
149 |
< |
calculation even more difficult for those slowly-converging |
150 |
< |
equilibrium methods. Imposed-flux non-equilibrium |
151 |
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methods\cite{MullerPlathe:1997xw} have the flux set {\it a priori} and |
152 |
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the response of temperature or momentum gradients are easier to |
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measure than the flux, if unknown, and thus, is a preferable way to |
154 |
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the forward NEMD methods. Although the momentum swapping approach for |
155 |
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flux-imposing can be used for exchanging energy between particles of |
156 |
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different identity, the kinetic energy transfer efficiency is affected |
157 |
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by the mass difference between the particles, which limits its |
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application on heterogeneous interfacial systems. |
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\subsection{Imposd-Flux Methods in MD Simulations} |
145 |
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Steady state MD simulations have an advantage in that not many |
146 |
> |
trajectories are needed to study the relationship between thermal flux |
147 |
> |
and thermal gradients. For systems with low interfacial conductance, |
148 |
> |
one must have a method capable of generating or measuring relatively |
149 |
> |
small fluxes, compared to those required for bulk conductivity. This |
150 |
> |
requirement makes the calculation even more difficult for |
151 |
> |
slowly-converging equilibrium methods.\cite{Viscardy:2007lq} Forward |
152 |
> |
NEMD methods impose a gradient (and measure a flux), but at interfaces |
153 |
> |
it is not clear what behavior should be imposed at the boundaries |
154 |
> |
between materials. Imposed-flux reverse non-equilibrium |
155 |
> |
methods\cite{MullerPlathe:1997xw} set the flux {\it a priori} and |
156 |
> |
the thermal response becomes an easy-to-measure quantity. Although |
157 |
> |
M\"{u}ller-Plathe's original momentum swapping approach can be used |
158 |
> |
for exchanging energy between particles of different identity, the |
159 |
> |
kinetic energy transfer efficiency is affected by the mass difference |
160 |
> |
between the particles, which limits its application on heterogeneous |
161 |
> |
interfacial systems. |
162 |
|
|
163 |
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The non-isotropic velocity scaling (NIVS)\cite{kuang:164101} approach in |
164 |
< |
non-equilibrium MD simulations is able to impose relatively large |
165 |
< |
kinetic energy flux without obvious perturbation to the velocity |
166 |
< |
distribution of the simulated systems. Furthermore, this approach has |
163 |
> |
The non-isotropic velocity scaling (NIVS) \cite{kuang:164101} approach |
164 |
> |
to non-equilibrium MD simulations is able to impose a wide range of |
165 |
> |
kinetic energy fluxes without obvious perturbation to the velocity |
166 |
> |
distributions of the simulated systems. Furthermore, this approach has |
167 |
|
the advantage in heterogeneous interfaces in that kinetic energy flux |
168 |
|
can be applied between regions of particles of arbitary identity, and |
169 |
< |
the flux quantity is not restricted by particle mass difference. |
169 |
> |
the flux will not be restricted by difference in particle mass. |
170 |
|
|
171 |
|
The NIVS algorithm scales the velocity vectors in two separate regions |
172 |
|
of a simulation system with respective diagonal scaling matricies. To |
173 |
|
determine these scaling factors in the matricies, a set of equations |
174 |
|
including linear momentum conservation and kinetic energy conservation |
175 |
< |
constraints and target momentum / energy flux satisfaction is |
176 |
< |
solved. With the scaling operation applied to the system in a set |
177 |
< |
frequency, corresponding momentum / temperature gradients can be |
178 |
< |
built, which can be used for computing transport properties and other |
179 |
< |
applications related to momentum / temperature gradients. The NIVS |
170 |
< |
algorithm conserves momenta and energy and does not depend on an |
171 |
< |
external thermostat. |
175 |
> |
constraints and target energy flux satisfaction is solved. With the |
176 |
> |
scaling operation applied to the system in a set frequency, bulk |
177 |
> |
temperature gradients can be easily established, and these can be used |
178 |
> |
for computing thermal conductivities. The NIVS algorithm conserves |
179 |
> |
momenta and energy and does not depend on an external thermostat. |
180 |
|
|
181 |
< |
\subsection{Defining Interfacial Thermal Conductivity $G$} |
182 |
< |
For interfaces with a relatively low interfacial conductance, the bulk |
183 |
< |
regions on either side of an interface rapidly come to a state in |
184 |
< |
which the two phases have relatively homogeneous (but distinct) |
185 |
< |
temperatures. The interfacial thermal conductivity $G$ can therefore |
186 |
< |
be approximated as: |
181 |
> |
\subsection{Defining Interfacial Thermal Conductivity ($G$)} |
182 |
> |
|
183 |
> |
For an interface with relatively low interfacial conductance, and a |
184 |
> |
thermal flux between two distinct bulk regions, the regions on either |
185 |
> |
side of the interface rapidly come to a state in which the two phases |
186 |
> |
have relatively homogeneous (but distinct) temperatures. The |
187 |
> |
interfacial thermal conductivity $G$ can therefore be approximated as: |
188 |
|
\begin{equation} |
189 |
< |
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle - |
189 |
> |
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle - |
190 |
|
\langle T_\mathrm{cold}\rangle \right)} |
191 |
|
\label{lowG} |
192 |
|
\end{equation} |
193 |
< |
where ${E_{total}}$ is the imposed non-physical kinetic energy |
194 |
< |
transfer and ${\langle T_\mathrm{hot}\rangle}$ and ${\langle |
195 |
< |
T_\mathrm{cold}\rangle}$ are the average observed temperature of the |
196 |
< |
two separated phases. |
197 |
< |
|
198 |
< |
When the interfacial conductance is {\it not} small, two ways can be |
199 |
< |
used to define $G$. |
200 |
< |
|
201 |
< |
One way is to assume the temperature is discretely different on two |
202 |
< |
sides of the interface, $G$ can be calculated with the thermal flux |
203 |
< |
applied $J$ and the maximum temperature difference measured along the |
204 |
< |
thermal gradient max($\Delta T$), which occurs at the interface, as: |
193 |
> |
where ${E_{total}}$ is the total imposed non-physical kinetic energy |
194 |
> |
transfer during the simulation and ${\langle T_\mathrm{hot}\rangle}$ |
195 |
> |
and ${\langle T_\mathrm{cold}\rangle}$ are the average observed |
196 |
> |
temperature of the two separated phases. For an applied flux $J_z$ |
197 |
> |
operating over a simulation time $t$ on a periodically-replicated slab |
198 |
> |
of dimensions $L_x \times L_y$, $E_{total} = J_z *(t)*(2 L_x L_y)$. |
199 |
> |
|
200 |
> |
When the interfacial conductance is {\it not} small, there are two |
201 |
> |
ways to define $G$. One common way is to assume the temperature is |
202 |
> |
discrete on the two sides of the interface. $G$ can be calculated |
203 |
> |
using the applied thermal flux $J$ and the maximum temperature |
204 |
> |
difference measured along the thermal gradient max($\Delta T$), which |
205 |
> |
occurs at the Gibbs deviding surface (Figure \ref{demoPic}). This is |
206 |
> |
known as the Kapitza conductance, which is the inverse of the Kapitza |
207 |
> |
resistance. |
208 |
|
\begin{equation} |
209 |
< |
G=\frac{J}{\Delta T} |
209 |
> |
G=\frac{J}{\Delta T} |
210 |
|
\label{discreteG} |
211 |
|
\end{equation} |
212 |
|
|
213 |
+ |
\begin{figure} |
214 |
+ |
\includegraphics[width=\linewidth]{method} |
215 |
+ |
\caption{Interfacial conductance can be calculated by applying an |
216 |
+ |
(unphysical) kinetic energy flux between two slabs, one located |
217 |
+ |
within the metal and another on the edge of the periodic box. The |
218 |
+ |
system responds by forming a thermal response or a gradient. In |
219 |
+ |
bulk liquids, this gradient typically has a single slope, but in |
220 |
+ |
interfacial systems, there are distinct thermal conductivity |
221 |
+ |
domains. The interfacial conductance, $G$ is found by measuring the |
222 |
+ |
temperature gap at the Gibbs dividing surface, or by using second |
223 |
+ |
derivatives of the thermal profile.} |
224 |
+ |
\label{demoPic} |
225 |
+ |
\end{figure} |
226 |
+ |
|
227 |
|
The other approach is to assume a continuous temperature profile along |
228 |
|
the thermal gradient axis (e.g. $z$) and define $G$ at the point where |
229 |
< |
the magnitude of thermal conductivity $\lambda$ change reach its |
229 |
> |
the magnitude of thermal conductivity ($\lambda$) change reaches its |
230 |
|
maximum, given that $\lambda$ is well-defined throughout the space: |
231 |
|
\begin{equation} |
232 |
|
G^\prime = \Big|\frac{\partial\lambda}{\partial z}\Big| |
237 |
|
\label{derivativeG} |
238 |
|
\end{equation} |
239 |
|
|
240 |
< |
With the temperature profile obtained from simulations, one is able to |
240 |
> |
With temperature profiles obtained from simulation, one is able to |
241 |
|
approximate the first and second derivatives of $T$ with finite |
242 |
< |
difference method and thus calculate $G^\prime$. |
242 |
> |
difference methods and calculate $G^\prime$. In what follows, both |
243 |
> |
definitions have been used, and are compared in the results. |
244 |
|
|
245 |
< |
In what follows, both definitions are used for calculation and comparison. |
245 |
> |
To investigate the interfacial conductivity at metal / solvent |
246 |
> |
interfaces, we have modeled a metal slab with its (111) surfaces |
247 |
> |
perpendicular to the $z$-axis of our simulation cells. The metal slab |
248 |
> |
has been prepared both with and without capping agents on the exposed |
249 |
> |
surface, and has been solvated with simple organic solvents, as |
250 |
> |
illustrated in Figure \ref{gradT}. |
251 |
|
|
252 |
< |
[IMPOSE G DEFINITION INTO OUR SYSTEMS] |
253 |
< |
To facilitate the use of the above definitions in calculating $G$ and |
254 |
< |
$G^\prime$, we have a metal slab with its (111) surfaces perpendicular |
255 |
< |
to the $z$-axis of our simulation cells. With or withour capping |
256 |
< |
agents on the surfaces, the metal slab is solvated with organic |
257 |
< |
solvents, as illustrated in Figure \ref{demoPic}. |
252 |
> |
With the simulation cell described above, we are able to equilibrate |
253 |
> |
the system and impose an unphysical thermal flux between the liquid |
254 |
> |
and the metal phase using the NIVS algorithm. By periodically applying |
255 |
> |
the unphysical flux, we obtained a temperature profile and its spatial |
256 |
> |
derivatives. Figure \ref{gradT} shows how an applied thermal flux can |
257 |
> |
be used to obtain the 1st and 2nd derivatives of the temperature |
258 |
> |
profile. |
259 |
|
|
260 |
|
\begin{figure} |
228 |
– |
\includegraphics[width=\linewidth]{demoPic} |
229 |
– |
\caption{A sample showing how a metal slab has its (111) surface |
230 |
– |
covered by capping agent molecules and solvated by hexane.} |
231 |
– |
\label{demoPic} |
232 |
– |
\end{figure} |
233 |
– |
|
234 |
– |
With a simulation cell setup following the above manner, one is able |
235 |
– |
to equilibrate the system and impose an unphysical thermal flux |
236 |
– |
between the liquid and the metal phase with the NIVS algorithm. Under |
237 |
– |
a stablized thermal gradient induced by periodically applying the |
238 |
– |
unphysical flux, one is able to obtain a temperature profile and the |
239 |
– |
physical thermal flux corresponding to it, which equals to the |
240 |
– |
unphysical flux applied by NIVS. These data enables the evaluation of |
241 |
– |
the interfacial thermal conductance of a surface. Figure \ref{gradT} |
242 |
– |
is an example how those stablized thermal gradient can be used to |
243 |
– |
obtain the 1st and 2nd derivatives of the temperature profile. |
244 |
– |
|
245 |
– |
\begin{figure} |
261 |
|
\includegraphics[width=\linewidth]{gradT} |
262 |
< |
\caption{The 1st and 2nd derivatives of temperature profile can be |
263 |
< |
obtained with finite difference approximation.} |
262 |
> |
\caption{A sample of Au-butanethiol/hexane interfacial system and the |
263 |
> |
temperature profile after a kinetic energy flux is imposed to |
264 |
> |
it. The 1st and 2nd derivatives of the temperature profile can be |
265 |
> |
obtained with finite difference approximation (lower panel).} |
266 |
|
\label{gradT} |
267 |
|
\end{figure} |
268 |
|
|
252 |
– |
[MAY INCLUDE POWER SPECTRUM PROTOCOL] |
253 |
– |
|
269 |
|
\section{Computational Details} |
270 |
|
\subsection{Simulation Protocol} |
271 |
< |
Our MD simulation code, OpenMD\cite{Meineke:2005gd,openmd}, has the |
272 |
< |
NIVS algorithm integrated and was used for our simulations. In our |
273 |
< |
simulations, Au is used to construct a metal slab with bare (111) |
274 |
< |
surface perpendicular to the $z$-axis. Different slab thickness (layer |
275 |
< |
numbers of Au) are simulated. This metal slab is first equilibrated |
276 |
< |
under normal pressure (1 atm) and a desired temperature. After |
277 |
< |
equilibration, butanethiol is used as the capping agent molecule to |
278 |
< |
cover the bare Au (111) surfaces evenly. The sulfur atoms in the |
279 |
< |
butanethiol molecules would occupy the three-fold sites of the |
280 |
< |
surfaces, and the maximal butanethiol capacity on Au surface is $1/3$ |
281 |
< |
of the total number of surface Au atoms[CITATIONs]. A series of |
282 |
< |
different coverage surfaces is investigated in order to study the |
283 |
< |
relation between coverage and conductance. |
271 |
> |
The NIVS algorithm has been implemented in our MD simulation code, |
272 |
> |
OpenMD\cite{Meineke:2005gd,openmd}, and was used for our simulations. |
273 |
> |
Metal slabs of 6 or 11 layers of Au atoms were first equilibrated |
274 |
> |
under atmospheric pressure (1 atm) and 200K. After equilibration, |
275 |
> |
butanethiol capping agents were placed at three-fold hollow sites on |
276 |
> |
the Au(111) surfaces. These sites are either {\it fcc} or {\it |
277 |
> |
hcp} sites, although Hase {\it et al.} found that they are |
278 |
> |
equivalent in a heat transfer process,\cite{hase:2010} so we did not |
279 |
> |
distinguish between these sites in our study. The maximum butanethiol |
280 |
> |
capacity on Au surface is $1/3$ of the total number of surface Au |
281 |
> |
atoms, and the packing forms a $(\sqrt{3}\times\sqrt{3})R30^\circ$ |
282 |
> |
structure\cite{doi:10.1021/ja00008a001,doi:10.1021/cr9801317}. A |
283 |
> |
series of lower coverages was also prepared by eliminating |
284 |
> |
butanethiols from the higher coverage surface in a regular manner. The |
285 |
> |
lower coverages were prepared in order to study the relation between |
286 |
> |
coverage and interfacial conductance. |
287 |
|
|
288 |
< |
[COVERAGE DISCRIPTION] |
289 |
< |
In the initial configurations for each coverage precentage, |
290 |
< |
butanethiols were distributed evenly on the Au(111) surfaces. However, |
291 |
< |
since the interaction descriptions between surface Au and butanethiol |
292 |
< |
is non-bonded in our simulations, the capping agent molecules are |
293 |
< |
allowed to migrate to an empty neighbor three-fold site during a |
294 |
< |
simulation. Therefore, the initial configuration would not severely |
277 |
< |
affect the sampling of a variety of configurations of the same |
278 |
< |
coverage, and the final conductance measurement would be an average |
279 |
< |
effect of these configurations explored in the simulations. [MAY NEED FIGURES] |
288 |
> |
The capping agent molecules were allowed to migrate during the |
289 |
> |
simulations. They distributed themselves uniformly and sampled a |
290 |
> |
number of three-fold sites throughout out study. Therefore, the |
291 |
> |
initial configuration does not noticeably affect the sampling of a |
292 |
> |
variety of configurations of the same coverage, and the final |
293 |
> |
conductance measurement would be an average effect of these |
294 |
> |
configurations explored in the simulations. |
295 |
|
|
296 |
< |
After the modified Au-butanethiol surface systems are equilibrated |
297 |
< |
under canonical ensemble, organic solvent molecules are packed in the |
298 |
< |
previously vacuum part of the simulation cells and guarantees that |
299 |
< |
short range repulsive interactions do not disrupt the |
300 |
< |
simulations\cite{packmol}. Two solvents are investigated, one which |
301 |
< |
has little vibrational overlap with the alkanethiol and plane-like |
302 |
< |
shape (toluene), and one which has similar vibrational frequencies and |
288 |
< |
chain-like shape ({\it n}-hexane). [MAY EXPLAIN WHY WE CHOOSE THEM] |
296 |
> |
After the modified Au-butanethiol surface systems were equilibrated in |
297 |
> |
the canonical (NVT) ensemble, organic solvent molecules were packed in |
298 |
> |
the previously empty part of the simulation cells.\cite{packmol} Two |
299 |
> |
solvents were investigated, one which has little vibrational overlap |
300 |
> |
with the alkanethiol and which has a planar shape (toluene), and one |
301 |
> |
which has similar vibrational frequencies to the capping agent and |
302 |
> |
chain-like shape ({\it n}-hexane). |
303 |
|
|
304 |
< |
The spacing filled by solvent molecules, i.e. the gap between |
305 |
< |
periodically repeated Au-butanethiol surfaces should be carefully |
306 |
< |
chosen. A very long length scale for the thermal gradient axis ($z$) |
293 |
< |
may cause excessively hot or cold temperatures in the middle of the |
304 |
> |
The simulation cells were not particularly extensive along the |
305 |
> |
$z$-axis, as a very long length scale for the thermal gradient may |
306 |
> |
cause excessively hot or cold temperatures in the middle of the |
307 |
|
solvent region and lead to undesired phenomena such as solvent boiling |
308 |
|
or freezing when a thermal flux is applied. Conversely, too few |
309 |
|
solvent molecules would change the normal behavior of the liquid |
310 |
|
phase. Therefore, our $N_{solvent}$ values were chosen to ensure that |
311 |
< |
these extreme cases did not happen to our simulations. And the |
312 |
< |
corresponding spacing is usually $35 \sim 60$\AA. |
311 |
> |
these extreme cases did not happen to our simulations. The spacing |
312 |
> |
between periodic images of the gold interfaces is $45 \sim 75$\AA. |
313 |
|
|
314 |
< |
The initial configurations generated by Packmol are further |
315 |
< |
equilibrated with the $x$ and $y$ dimensions fixed, only allowing |
316 |
< |
length scale change in $z$ dimension. This is to ensure that the |
317 |
< |
equilibration of liquid phase does not affect the metal crystal |
318 |
< |
structure in $x$ and $y$ dimensions. Further equilibration are run |
319 |
< |
under NVT and then NVE ensembles. |
314 |
> |
The initial configurations generated are further equilibrated with the |
315 |
> |
$x$ and $y$ dimensions fixed, only allowing the $z$-length scale to |
316 |
> |
change. This is to ensure that the equilibration of liquid phase does |
317 |
> |
not affect the metal's crystalline structure. Comparisons were made |
318 |
> |
with simulations that allowed changes of $L_x$ and $L_y$ during NPT |
319 |
> |
equilibration. No substantial changes in the box geometry were noticed |
320 |
> |
in these simulations. After ensuring the liquid phase reaches |
321 |
> |
equilibrium at atmospheric pressure (1 atm), further equilibration was |
322 |
> |
carried out under canonical (NVT) and microcanonical (NVE) ensembles. |
323 |
|
|
324 |
< |
After the systems reach equilibrium, NIVS is implemented to impose a |
325 |
< |
periodic unphysical thermal flux between the metal and the liquid |
326 |
< |
phase. Most of our simulations are under an average temperature of |
327 |
< |
$\sim$200K. Therefore, this flux usually comes from the metal to the |
324 |
> |
After the systems reach equilibrium, NIVS was used to impose an |
325 |
> |
unphysical thermal flux between the metal and the liquid phases. Most |
326 |
> |
of our simulations were done under an average temperature of |
327 |
> |
$\sim$200K. Therefore, thermal flux usually came from the metal to the |
328 |
|
liquid so that the liquid has a higher temperature and would not |
329 |
< |
freeze due to excessively low temperature. This induced temperature |
330 |
< |
gradient is stablized and the simulation cell is devided evenly into |
331 |
< |
N slabs along the $z$-axis and the temperatures of each slab are |
332 |
< |
recorded. When the slab width $d$ of each slab is the same, the |
333 |
< |
derivatives of $T$ with respect to slab number $n$ can be directly |
334 |
< |
used for $G^\prime$ calculations: |
335 |
< |
\begin{equation} |
336 |
< |
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
329 |
> |
freeze due to lowered temperatures. After this induced temperature |
330 |
> |
gradient had stablized, the temperature profile of the simulation cell |
331 |
> |
was recorded. To do this, the simulation cell is devided evenly into |
332 |
> |
$N$ slabs along the $z$-axis. The average temperatures of each slab |
333 |
> |
are recorded for 1$\sim$2 ns. When the slab width $d$ of each slab is |
334 |
> |
the same, the derivatives of $T$ with respect to slab number $n$ can |
335 |
> |
be directly used for $G^\prime$ calculations: \begin{equation} |
336 |
> |
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
337 |
|
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
338 |
|
= |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big| |
339 |
|
\Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2 |
342 |
|
\label{derivativeG2} |
343 |
|
\end{equation} |
344 |
|
|
345 |
+ |
All of the above simulation procedures use a time step of 1 fs. Each |
346 |
+ |
equilibration stage took a minimum of 100 ps, although in some cases, |
347 |
+ |
longer equilibration stages were utilized. |
348 |
+ |
|
349 |
|
\subsection{Force Field Parameters} |
350 |
< |
Our simulations include various components. Therefore, force field |
351 |
< |
parameter descriptions are needed for interactions both between the |
352 |
< |
same type of particles and between particles of different species. |
350 |
> |
Our simulations include a number of chemically distinct components. |
351 |
> |
Figure \ref{demoMol} demonstrates the sites defined for both |
352 |
> |
United-Atom and All-Atom models of the organic solvent and capping |
353 |
> |
agents in our simulations. Force field parameters are needed for |
354 |
> |
interactions both between the same type of particles and between |
355 |
> |
particles of different species. |
356 |
|
|
357 |
+ |
\begin{figure} |
358 |
+ |
\includegraphics[width=\linewidth]{structures} |
359 |
+ |
\caption{Structures of the capping agent and solvents utilized in |
360 |
+ |
these simulations. The chemically-distinct sites (a-e) are expanded |
361 |
+ |
in terms of constituent atoms for both United Atom (UA) and All Atom |
362 |
+ |
(AA) force fields. Most parameters are from |
363 |
+ |
Refs. \protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes,TraPPE-UA.thiols} |
364 |
+ |
(UA) and \protect\cite{OPLSAA} (AA). Cross-interactions with the Au |
365 |
+ |
atoms are given in Table \ref{MnM}.} |
366 |
+ |
\label{demoMol} |
367 |
+ |
\end{figure} |
368 |
+ |
|
369 |
|
The Au-Au interactions in metal lattice slab is described by the |
370 |
< |
quantum Sutton-Chen (QSC) formulation\cite{PhysRevB.59.3527}. The QSC |
370 |
> |
quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
371 |
|
potentials include zero-point quantum corrections and are |
372 |
|
reparametrized for accurate surface energies compared to the |
373 |
< |
Sutton-Chen potentials\cite{Chen90}. |
373 |
> |
Sutton-Chen potentials.\cite{Chen90} |
374 |
|
|
375 |
< |
Figure \ref{demoMol} demonstrates how we name our pseudo-atoms of the |
376 |
< |
organic solvent molecules in our simulations. |
375 |
> |
For the two solvent molecules, {\it n}-hexane and toluene, two |
376 |
> |
different atomistic models were utilized. Both solvents were modeled |
377 |
> |
using United-Atom (UA) and All-Atom (AA) models. The TraPPE-UA |
378 |
> |
parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
379 |
> |
for our UA solvent molecules. In these models, sites are located at |
380 |
> |
the carbon centers for alkyl groups. Bonding interactions, including |
381 |
> |
bond stretches and bends and torsions, were used for intra-molecular |
382 |
> |
sites closer than 3 bonds. For non-bonded interactions, Lennard-Jones |
383 |
> |
potentials are used. |
384 |
|
|
385 |
< |
\begin{figure} |
386 |
< |
\includegraphics[width=\linewidth]{demoMol} |
387 |
< |
\caption{Denomination of atoms or pseudo-atoms in our simulations: a) |
388 |
< |
UA-hexane; b) AA-hexane; c) UA-toluene; d) AA-toluene.} |
389 |
< |
\label{demoMol} |
390 |
< |
\end{figure} |
385 |
> |
By eliminating explicit hydrogen atoms, the TraPPE-UA models are |
386 |
> |
simple and computationally efficient, while maintaining good accuracy. |
387 |
> |
However, the TraPPE-UA model for alkanes is known to predict a slighly |
388 |
> |
lower boiling point than experimental values. This is one of the |
389 |
> |
reasons we used a lower average temperature (200K) for our |
390 |
> |
simulations. If heat is transferred to the liquid phase during the |
391 |
> |
NIVS simulation, the liquid in the hot slab can actually be |
392 |
> |
substantially warmer than the mean temperature in the simulation. The |
393 |
> |
lower mean temperatures therefore prevent solvent boiling. |
394 |
|
|
395 |
< |
For both solvent molecules, straight chain {\it n}-hexane and aromatic |
396 |
< |
toluene, United-Atom (UA) and All-Atom (AA) models are used |
397 |
< |
respectively. The TraPPE-UA |
398 |
< |
parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
399 |
< |
for our UA solvent molecules. In these models, pseudo-atoms are |
355 |
< |
located at the carbon centers for alkyl groups. By eliminating |
356 |
< |
explicit hydrogen atoms, these models are simple and computationally |
357 |
< |
efficient, while maintains good accuracy. However, the TraPPE-UA for |
358 |
< |
alkanes is known to predict a lower boiling point than experimental |
359 |
< |
values. Considering that after an unphysical thermal flux is applied |
360 |
< |
to a system, the temperature of ``hot'' area in the liquid phase would be |
361 |
< |
significantly higher than the average, to prevent over heating and |
362 |
< |
boiling of the liquid phase, the average temperature in our |
363 |
< |
simulations should be much lower than the liquid boiling point. [MORE DISCUSSION] |
364 |
< |
For UA-toluene model, rigid body constraints are applied, so that the |
365 |
< |
benzene ring and the methyl-CRar bond are kept rigid. This would save |
366 |
< |
computational time.[MORE DETAILS] |
395 |
> |
For UA-toluene, the non-bonded potentials between intermolecular sites |
396 |
> |
have a similar Lennard-Jones formulation. The toluene molecules were |
397 |
> |
treated as a single rigid body, so there was no need for |
398 |
> |
intramolecular interactions (including bonds, bends, or torsions) in |
399 |
> |
this solvent model. |
400 |
|
|
401 |
|
Besides the TraPPE-UA models, AA models for both organic solvents are |
402 |
< |
included in our studies as well. For hexane, the OPLS-AA\cite{OPLSAA} |
403 |
< |
force field is used. [MORE DETAILS] |
404 |
< |
For toluene, the United Force Field developed by Rapp\'{e} {\it et |
405 |
< |
al.}\cite{doi:10.1021/ja00051a040} is adopted.[MORE DETAILS] |
402 |
> |
included in our studies as well. The OPLS-AA\cite{OPLSAA} force fields |
403 |
> |
were used. For hexane, additional explicit hydrogen sites were |
404 |
> |
included. Besides bonding and non-bonded site-site interactions, |
405 |
> |
partial charges and the electrostatic interactions were added to each |
406 |
> |
CT and HC site. For toluene, a flexible model for the toluene molecule |
407 |
> |
was utilized which included bond, bend, torsion, and inversion |
408 |
> |
potentials to enforce ring planarity. |
409 |
|
|
410 |
< |
The capping agent in our simulations, the butanethiol molecules can |
411 |
< |
either use UA or AA model. The TraPPE-UA force fields includes |
410 |
> |
The butanethiol capping agent in our simulations, were also modeled |
411 |
> |
with both UA and AA model. The TraPPE-UA force field includes |
412 |
|
parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for |
413 |
|
UA butanethiol model in our simulations. The OPLS-AA also provides |
414 |
|
parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) |
415 |
< |
surfaces do not have the hydrogen atom bonded to sulfur. To adapt this |
416 |
< |
change and derive suitable parameters for butanethiol adsorbed on |
417 |
< |
Au(111) surfaces, we adopt the S parameters from Luedtke and |
418 |
< |
Landman\cite{landman:1998} and modify parameters for its neighbor C |
419 |
< |
atom for charge balance in the molecule. Note that the model choice |
420 |
< |
(UA or AA) of capping agent can be different from the |
421 |
< |
solvent. Regardless of model choice, the force field parameters for |
422 |
< |
interactions between capping agent and solvent can be derived using |
423 |
< |
Lorentz-Berthelot Mixing Rule:[EQN'S] |
415 |
> |
surfaces do not have the hydrogen atom bonded to sulfur. To derive |
416 |
> |
suitable parameters for butanethiol adsorbed on Au(111) surfaces, we |
417 |
> |
adopt the S parameters from Luedtke and Landman\cite{landman:1998} and |
418 |
> |
modify the parameters for the CTS atom to maintain charge neutrality |
419 |
> |
in the molecule. Note that the model choice (UA or AA) for the capping |
420 |
> |
agent can be different from the solvent. Regardless of model choice, |
421 |
> |
the force field parameters for interactions between capping agent and |
422 |
> |
solvent can be derived using Lorentz-Berthelot Mixing Rule: |
423 |
> |
\begin{eqnarray} |
424 |
> |
\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\ |
425 |
> |
\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}} |
426 |
> |
\end{eqnarray} |
427 |
|
|
428 |
+ |
To describe the interactions between metal (Au) and non-metal atoms, |
429 |
+ |
we refer to an adsorption study of alkyl thiols on gold surfaces by |
430 |
+ |
Vlugt {\it et al.}\cite{vlugt:cpc2007154} They fitted an effective |
431 |
+ |
Lennard-Jones form of potential parameters for the interaction between |
432 |
+ |
Au and pseudo-atoms CH$_x$ and S based on a well-established and |
433 |
+ |
widely-used effective potential of Hautman and Klein for the Au(111) |
434 |
+ |
surface.\cite{hautman:4994} As our simulations require the gold slab |
435 |
+ |
to be flexible to accommodate thermal excitation, the pair-wise form |
436 |
+ |
of potentials they developed was used for our study. |
437 |
|
|
438 |
< |
To describe the interactions between metal Au and non-metal capping |
439 |
< |
agent and solvent particles, we refer to an adsorption study of alkyl |
440 |
< |
thiols on gold surfaces by Vlugt {\it et |
441 |
< |
al.}\cite{vlugt:cpc2007154} They fitted an effective Lennard-Jones |
442 |
< |
form of potential parameters for the interaction between Au and |
443 |
< |
pseudo-atoms CH$_x$ and S based on a well-established and widely-used |
444 |
< |
effective potential of Hautman and Klein\cite{hautman:4994} for the |
445 |
< |
Au(111) surface. As our simulations require the gold lattice slab to |
446 |
< |
be non-rigid so that it could accommodate kinetic energy for thermal |
447 |
< |
transport study purpose, the pair-wise form of potentials is |
448 |
< |
preferred. |
438 |
> |
The potentials developed from {\it ab initio} calculations by Leng |
439 |
> |
{\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
440 |
> |
interactions between Au and aromatic C/H atoms in toluene. However, |
441 |
> |
the Lennard-Jones parameters between Au and other types of particles, |
442 |
> |
(e.g. AA alkanes) have not yet been established. For these |
443 |
> |
interactions, the Lorentz-Berthelot mixing rule can be used to derive |
444 |
> |
effective single-atom LJ parameters for the metal using the fit values |
445 |
> |
for toluene. These are then used to construct reasonable mixing |
446 |
> |
parameters for the interactions between the gold and other atoms. |
447 |
> |
Table \ref{MnM} summarizes the ``metal/non-metal'' parameters used in |
448 |
> |
our simulations. |
449 |
|
|
402 |
– |
Besides, the potentials developed from {\it ab initio} calculations by |
403 |
– |
Leng {\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
404 |
– |
interactions between Au and aromatic C/H atoms in toluene.[MORE DETAILS] |
405 |
– |
|
406 |
– |
However, the Lennard-Jones parameters between Au and other types of |
407 |
– |
particles in our simulations are not yet well-established. For these |
408 |
– |
interactions, we attempt to derive their parameters using the Mixing |
409 |
– |
Rule. To do this, the ``Metal-non-Metal'' (MnM) interaction parameters |
410 |
– |
for Au is first extracted from the Au-CH$_x$ parameters by applying |
411 |
– |
the Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM'' |
412 |
– |
parameters in our simulations. |
413 |
– |
|
450 |
|
\begin{table*} |
451 |
|
\begin{minipage}{\linewidth} |
452 |
|
\begin{center} |
453 |
< |
\caption{Lennard-Jones parameters for Au-non-Metal |
454 |
< |
interactions in our simulations.} |
455 |
< |
|
456 |
< |
\begin{tabular}{ccc} |
453 |
> |
\caption{Non-bonded interaction parameters (including cross |
454 |
> |
interactions with Au atoms) for both force fields used in this |
455 |
> |
work.} |
456 |
> |
\begin{tabular}{lllllll} |
457 |
|
\hline\hline |
458 |
< |
Non-metal atom & $\sigma$ & $\epsilon$ \\ |
459 |
< |
(or pseudo-atom) & \AA & kcal/mol \\ |
458 |
> |
& Site & $\sigma_{ii}$ & $\epsilon_{ii}$ & $q_i$ & |
459 |
> |
$\sigma_{Au-i}$ & $\epsilon_{Au-i}$ \\ |
460 |
> |
& & (\AA) & (kcal/mol) & ($e$) & (\AA) & (kcal/mol) \\ |
461 |
|
\hline |
462 |
< |
S & 2.40 & 8.465 \\ |
463 |
< |
CH3 & 3.54 & 0.2146 \\ |
464 |
< |
CH2 & 3.54 & 0.1749 \\ |
465 |
< |
CT3 & 3.365 & 0.1373 \\ |
466 |
< |
CT2 & 3.365 & 0.1373 \\ |
467 |
< |
CTT & 3.365 & 0.1373 \\ |
468 |
< |
HC & 2.865 & 0.09256 \\ |
469 |
< |
CHar & 3.4625 & 0.1680 \\ |
470 |
< |
CRar & 3.555 & 0.1604 \\ |
471 |
< |
CA & 3.173 & 0.0640 \\ |
472 |
< |
HA & 2.746 & 0.0414 \\ |
462 |
> |
United Atom (UA) |
463 |
> |
&CH3 & 3.75 & 0.1947 & - & 3.54 & 0.2146 \\ |
464 |
> |
&CH2 & 3.95 & 0.0914 & - & 3.54 & 0.1749 \\ |
465 |
> |
&CHar & 3.695 & 0.1003 & - & 3.4625 & 0.1680 \\ |
466 |
> |
&CRar & 3.88 & 0.04173 & - & 3.555 & 0.1604 \\ |
467 |
> |
\hline |
468 |
> |
All Atom (AA) |
469 |
> |
&CT3 & 3.50 & 0.066 & -0.18 & 3.365 & 0.1373 \\ |
470 |
> |
&CT2 & 3.50 & 0.066 & -0.12 & 3.365 & 0.1373 \\ |
471 |
> |
&CTT & 3.50 & 0.066 & -0.065 & 3.365 & 0.1373 \\ |
472 |
> |
&HC & 2.50 & 0.030 & 0.06 & 2.865 & 0.09256 \\ |
473 |
> |
&CA & 3.55 & 0.070 & -0.115 & 3.173 & 0.0640 \\ |
474 |
> |
&HA & 2.42 & 0.030 & 0.115 & 2.746 & 0.0414 \\ |
475 |
> |
\hline |
476 |
> |
Both UA and AA |
477 |
> |
& S & 4.45 & 0.25 & - & 2.40 & 8.465 \\ |
478 |
|
\hline\hline |
479 |
|
\end{tabular} |
480 |
|
\label{MnM} |
483 |
|
\end{table*} |
484 |
|
|
485 |
|
|
486 |
< |
\section{Results and Discussions} |
487 |
< |
[MAY HAVE A BRIEF SUMMARY] |
488 |
< |
\subsection{How Simulation Parameters Affects $G$} |
489 |
< |
[MAY NOT PUT AT FIRST] |
490 |
< |
We have varied our protocol or other parameters of the simulations in |
491 |
< |
order to investigate how these factors would affect the measurement of |
492 |
< |
$G$'s. It turned out that while some of these parameters would not |
493 |
< |
affect the results substantially, some other changes to the |
452 |
< |
simulations would have a significant impact on the measurement |
453 |
< |
results. |
486 |
> |
\section{Results} |
487 |
> |
There are many factors contributing to the measured interfacial |
488 |
> |
conductance; some of these factors are physically motivated |
489 |
> |
(e.g. coverage of the surface by the capping agent coverage and |
490 |
> |
solvent identity), while some are governed by parameters of the |
491 |
> |
methodology (e.g. applied flux and the formulas used to obtain the |
492 |
> |
conductance). In this section we discuss the major physical and |
493 |
> |
calculational effects on the computed conductivity. |
494 |
|
|
495 |
< |
In some of our simulations, we allowed $L_x$ and $L_y$ to change |
456 |
< |
during equilibrating the liquid phase. Due to the stiffness of the Au |
457 |
< |
slab, $L_x$ and $L_y$ would not change noticeably after |
458 |
< |
equilibration. Although $L_z$ could fluctuate $\sim$1\% after a system |
459 |
< |
is fully equilibrated in the NPT ensemble, this fluctuation, as well |
460 |
< |
as those comparably smaller to $L_x$ and $L_y$, would not be magnified |
461 |
< |
on the calculated $G$'s, as shown in Table \ref{AuThiolHexaneUA}. This |
462 |
< |
insensivity to $L_i$ fluctuations allows reliable measurement of $G$'s |
463 |
< |
without the necessity of extremely cautious equilibration process. |
495 |
> |
\subsection{Effects due to capping agent coverage} |
496 |
|
|
497 |
< |
As stated in our computational details, the spacing filled with |
498 |
< |
solvent molecules can be chosen within a range. This allows some |
499 |
< |
change of solvent molecule numbers for the same Au-butanethiol |
500 |
< |
surfaces. We did this study on our Au-butanethiol/hexane |
501 |
< |
simulations. Nevertheless, the results obtained from systems of |
502 |
< |
different $N_{hexane}$ did not indicate that the measurement of $G$ is |
471 |
< |
susceptible to this parameter. For computational efficiency concern, |
472 |
< |
smaller system size would be preferable, given that the liquid phase |
473 |
< |
structure is not affected. |
497 |
> |
A series of different initial conditions with a range of surface |
498 |
> |
coverages was prepared and solvated with various with both of the |
499 |
> |
solvent molecules. These systems were then equilibrated and their |
500 |
> |
interfacial thermal conductivity was measured with the NIVS |
501 |
> |
algorithm. Figure \ref{coverage} demonstrates the trend of conductance |
502 |
> |
with respect to surface coverage. |
503 |
|
|
504 |
< |
Our NIVS algorithm allows change of unphysical thermal flux both in |
505 |
< |
direction and in quantity. This feature extends our investigation of |
506 |
< |
interfacial thermal conductance. However, the magnitude of this |
507 |
< |
thermal flux is not arbitary if one aims to obtain a stable and |
508 |
< |
reliable thermal gradient. A temperature profile would be |
509 |
< |
substantially affected by noise when $|J_z|$ has a much too low |
510 |
< |
magnitude; while an excessively large $|J_z|$ that overwhelms the |
482 |
< |
conductance capacity of the interface would prevent a thermal gradient |
483 |
< |
to reach a stablized steady state. NIVS has the advantage of allowing |
484 |
< |
$J$ to vary in a wide range such that the optimal flux range for $G$ |
485 |
< |
measurement can generally be simulated by the algorithm. Within the |
486 |
< |
optimal range, we were able to study how $G$ would change according to |
487 |
< |
the thermal flux across the interface. For our simulations, we denote |
488 |
< |
$J_z$ to be positive when the physical thermal flux is from the liquid |
489 |
< |
to metal, and negative vice versa. The $G$'s measured under different |
490 |
< |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and [REF]. These |
491 |
< |
results do not suggest that $G$ is dependent on $J_z$ within this flux |
492 |
< |
range. The linear response of flux to thermal gradient simplifies our |
493 |
< |
investigations in that we can rely on $G$ measurement with only a |
494 |
< |
couple $J_z$'s and do not need to test a large series of fluxes. |
504 |
> |
\begin{figure} |
505 |
> |
\includegraphics[width=\linewidth]{coverage} |
506 |
> |
\caption{Comparison of interfacial thermal conductivity ($G$) values |
507 |
> |
for the Au-butanethiol/solvent interface with various UA models and |
508 |
> |
different capping agent coverages at $\langle T\rangle\sim$200K.} |
509 |
> |
\label{coverage} |
510 |
> |
\end{figure} |
511 |
|
|
512 |
< |
%ADD MORE TO TABLE |
512 |
> |
In partially covered surfaces, the derivative definition for |
513 |
> |
$G^\prime$ (Eq. \ref{derivativeG}) becomes difficult to apply, as the |
514 |
> |
location of maximum change of $\lambda$ becomes washed out. The |
515 |
> |
discrete definition (Eq. \ref{discreteG}) is easier to apply, as the |
516 |
> |
Gibbs dividing surface is still well-defined. Therefore, $G$ (not |
517 |
> |
$G^\prime$) was used in this section. |
518 |
> |
|
519 |
> |
From Figure \ref{coverage}, one can see the significance of the |
520 |
> |
presence of capping agents. When even a small fraction of the Au(111) |
521 |
> |
surface sites are covered with butanethiols, the conductivity exhibits |
522 |
> |
an enhancement by at least a factor of 3. Cappping agents are clearly |
523 |
> |
playing a major role in thermal transport at metal / organic solvent |
524 |
> |
surfaces. |
525 |
> |
|
526 |
> |
We note a non-monotonic behavior in the interfacial conductance as a |
527 |
> |
function of surface coverage. The maximum conductance (largest $G$) |
528 |
> |
happens when the surfaces are about 75\% covered with butanethiol |
529 |
> |
caps. The reason for this behavior is not entirely clear. One |
530 |
> |
explanation is that incomplete butanethiol coverage allows small gaps |
531 |
> |
between butanethiols to form. These gaps can be filled by transient |
532 |
> |
solvent molecules. These solvent molecules couple very strongly with |
533 |
> |
the hot capping agent molecules near the surface, and can then carry |
534 |
> |
away (diffusively) the excess thermal energy from the surface. |
535 |
> |
|
536 |
> |
There appears to be a competition between the conduction of the |
537 |
> |
thermal energy away from the surface by the capping agents (enhanced |
538 |
> |
by greater coverage) and the coupling of the capping agents with the |
539 |
> |
solvent (enhanced by interdigitation at lower coverages). This |
540 |
> |
competition would lead to the non-monotonic coverage behavior observed |
541 |
> |
here. |
542 |
> |
|
543 |
> |
Results for rigid body toluene solvent, as well as the UA hexane, are |
544 |
> |
within the ranges expected from prior experimental |
545 |
> |
work.\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406} This suggests |
546 |
> |
that explicit hydrogen atoms might not be required for modeling |
547 |
> |
thermal transport in these systems. C-H vibrational modes do not see |
548 |
> |
significant excited state population at low temperatures, and are not |
549 |
> |
likely to carry lower frequency excitations from the solid layer into |
550 |
> |
the bulk liquid. |
551 |
> |
|
552 |
> |
The toluene solvent does not exhibit the same behavior as hexane in |
553 |
> |
that $G$ remains at approximately the same magnitude when the capping |
554 |
> |
coverage increases from 25\% to 75\%. Toluene, as a rigid planar |
555 |
> |
molecule, cannot occupy the relatively small gaps between the capping |
556 |
> |
agents as easily as the chain-like {\it n}-hexane. The effect of |
557 |
> |
solvent coupling to the capping agent is therefore weaker in toluene |
558 |
> |
except at the very lowest coverage levels. This effect counters the |
559 |
> |
coverage-dependent conduction of heat away from the metal surface, |
560 |
> |
leading to a much flatter $G$ vs. coverage trend than is observed in |
561 |
> |
{\it n}-hexane. |
562 |
> |
|
563 |
> |
\subsection{Effects due to Solvent \& Solvent Models} |
564 |
> |
In addition to UA solvent and capping agent models, AA models have |
565 |
> |
also been included in our simulations. In most of this work, the same |
566 |
> |
(UA or AA) model for solvent and capping agent was used, but it is |
567 |
> |
also possible to utilize different models for different components. |
568 |
> |
We have also included isotopic substitutions (Hydrogen to Deuterium) |
569 |
> |
to decrease the explicit vibrational overlap between solvent and |
570 |
> |
capping agent. Table \ref{modelTest} summarizes the results of these |
571 |
> |
studies. |
572 |
> |
|
573 |
|
\begin{table*} |
574 |
|
\begin{minipage}{\linewidth} |
575 |
|
\begin{center} |
500 |
– |
\caption{Computed interfacial thermal conductivity ($G$ and |
501 |
– |
$G^\prime$) values for the 100\% covered Au-butanethiol/hexane |
502 |
– |
interfaces with UA model and different hexane molecule numbers |
503 |
– |
at different temperatures using a range of energy fluxes.} |
576 |
|
|
577 |
< |
\begin{tabular}{cccccccc} |
577 |
> |
\caption{Computed interfacial thermal conductance ($G$ and |
578 |
> |
$G^\prime$) values for interfaces using various models for |
579 |
> |
solvent and capping agent (or without capping agent) at |
580 |
> |
$\langle T\rangle\sim$200K. (D stands for deuterated solvent |
581 |
> |
or capping agent molecules; ``Avg.'' denotes results that are |
582 |
> |
averages of simulations under different applied thermal flux values $(J_z)$. Error |
583 |
> |
estimates are indicated in parentheses.)} |
584 |
> |
|
585 |
> |
\begin{tabular}{llccc} |
586 |
|
\hline\hline |
587 |
< |
$\langle T\rangle$ & & $L_x$ & $L_y$ & $L_z$ & $J_z$ & |
588 |
< |
$G$ & $G^\prime$ \\ |
509 |
< |
(K) & $N_{hexane}$ & \multicolumn{3}{c}{(\AA)} & (GW/m$^2$) & |
587 |
> |
Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ |
588 |
> |
(or bare surface) & model & (GW/m$^2$) & |
589 |
|
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
590 |
|
\hline |
591 |
< |
200 & 266 & 29.86 & 25.80 & 113.1 & -0.96 & |
592 |
< |
102() & 80.0() \\ |
593 |
< |
& 200 & 29.84 & 25.81 & 93.9 & 1.92 & |
594 |
< |
129() & 87.3() \\ |
595 |
< |
& & 29.84 & 25.81 & 95.3 & 1.93 & |
517 |
< |
131() & 77.5() \\ |
518 |
< |
& 166 & 29.84 & 25.81 & 85.7 & 0.97 & |
519 |
< |
115() & 69.3() \\ |
520 |
< |
& & & & & 1.94 & |
521 |
< |
125() & 87.1() \\ |
522 |
< |
250 & 200 & 29.84 & 25.87 & 106.8 & 0.96 & |
523 |
< |
81.8() & 67.0() \\ |
524 |
< |
& 166 & 29.87 & 25.84 & 94.8 & 0.98 & |
525 |
< |
79.0() & 62.9() \\ |
526 |
< |
& & 29.84 & 25.85 & 95.0 & 1.44 & |
527 |
< |
76.2() & 64.8() \\ |
528 |
< |
\hline\hline |
529 |
< |
\end{tabular} |
530 |
< |
\label{AuThiolHexaneUA} |
531 |
< |
\end{center} |
532 |
< |
\end{minipage} |
533 |
< |
\end{table*} |
534 |
< |
|
535 |
< |
Furthermore, we also attempted to increase system average temperatures |
536 |
< |
to above 200K. These simulations are first equilibrated in the NPT |
537 |
< |
ensemble under normal pressure. As stated above, the TraPPE-UA model |
538 |
< |
for hexane tends to predict a lower boiling point. In our simulations, |
539 |
< |
hexane had diffculty to remain in liquid phase when NPT equilibration |
540 |
< |
temperature is higher than 250K. Additionally, the equilibrated liquid |
541 |
< |
hexane density under 250K becomes lower than experimental value. This |
542 |
< |
expanded liquid phase leads to lower contact between hexane and |
543 |
< |
butanethiol as well.[MAY NEED FIGURE] And this reduced contact would |
544 |
< |
probably be accountable for a lower interfacial thermal conductance, |
545 |
< |
as shown in Table \ref{AuThiolHexaneUA}. |
546 |
< |
|
547 |
< |
A similar study for TraPPE-UA toluene agrees with the above result as |
548 |
< |
well. Having a higher boiling point, toluene tends to remain liquid in |
549 |
< |
our simulations even equilibrated under 300K in NPT |
550 |
< |
ensembles. Furthermore, the expansion of the toluene liquid phase is |
551 |
< |
not as significant as that of the hexane. This prevents severe |
552 |
< |
decrease of liquid-capping agent contact and the results (Table |
553 |
< |
\ref{AuThiolToluene}) show only a slightly decreased interface |
554 |
< |
conductance. Therefore, solvent-capping agent contact should play an |
555 |
< |
important role in the thermal transport process across the interface |
556 |
< |
in that higher degree of contact could yield increased conductance. |
557 |
< |
|
558 |
< |
[ADD Lxyz AND ERROR ESTIMATE TO TABLE] |
559 |
< |
\begin{table*} |
560 |
< |
\begin{minipage}{\linewidth} |
561 |
< |
\begin{center} |
562 |
< |
\caption{Computed interfacial thermal conductivity ($G$ and |
563 |
< |
$G^\prime$) values for a 90\% coverage Au-butanethiol/toluene |
564 |
< |
interface at different temperatures using a range of energy |
565 |
< |
fluxes.} |
566 |
< |
|
567 |
< |
\begin{tabular}{cccc} |
568 |
< |
\hline\hline |
569 |
< |
$\langle T\rangle$ & $J_z$ & $G$ & $G^\prime$ \\ |
570 |
< |
(K) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
591 |
> |
UA & UA hexane & Avg. & 131(9) & 87(10) \\ |
592 |
> |
& UA hexane(D) & 1.95 & 153(5) & 136(13) \\ |
593 |
> |
& AA hexane & Avg. & 131(6) & 122(10) \\ |
594 |
> |
& UA toluene & 1.96 & 187(16) & 151(11) \\ |
595 |
> |
& AA toluene & 1.89 & 200(36) & 149(53) \\ |
596 |
|
\hline |
597 |
< |
200 & -1.86 & 180() & 135() \\ |
598 |
< |
& 2.15 & 204() & 113() \\ |
599 |
< |
& -3.93 & 175() & 114() \\ |
600 |
< |
300 & -1.91 & 143() & 125() \\ |
601 |
< |
& -4.19 & 134() & 113() \\ |
597 |
> |
AA & UA hexane & 1.94 & 116(9) & 129(8) \\ |
598 |
> |
& AA hexane & Avg. & 442(14) & 356(31) \\ |
599 |
> |
& AA hexane(D) & 1.93 & 222(12) & 234(54) \\ |
600 |
> |
& UA toluene & 1.98 & 125(25) & 97(60) \\ |
601 |
> |
& AA toluene & 3.79 & 487(56) & 290(42) \\ |
602 |
> |
\hline |
603 |
> |
AA(D) & UA hexane & 1.94 & 158(25) & 172(4) \\ |
604 |
> |
& AA hexane & 1.92 & 243(29) & 191(11) \\ |
605 |
> |
& AA toluene & 1.93 & 364(36) & 322(67) \\ |
606 |
> |
\hline |
607 |
> |
bare & UA hexane & Avg. & 46.5(3.2) & 49.4(4.5) \\ |
608 |
> |
& UA hexane(D) & 0.98 & 43.9(4.6) & 43.0(2.0) \\ |
609 |
> |
& AA hexane & 0.96 & 31.0(1.4) & 29.4(1.3) \\ |
610 |
> |
& UA toluene & 1.99 & 70.1(1.3) & 65.8(0.5) \\ |
611 |
|
\hline\hline |
612 |
|
\end{tabular} |
613 |
< |
\label{AuThiolToluene} |
613 |
> |
\label{modelTest} |
614 |
|
\end{center} |
615 |
|
\end{minipage} |
616 |
|
\end{table*} |
617 |
|
|
618 |
< |
Besides lower interfacial thermal conductance, surfaces in relatively |
619 |
< |
high temperatures are susceptible to reconstructions, when |
620 |
< |
butanethiols have a full coverage on the Au(111) surface. These |
587 |
< |
reconstructions include surface Au atoms migrated outward to the S |
588 |
< |
atom layer, and butanethiol molecules embedded into the original |
589 |
< |
surface Au layer. The driving force for this behavior is the strong |
590 |
< |
Au-S interactions in our simulations. And these reconstructions lead |
591 |
< |
to higher ratio of Au-S attraction and thus is energetically |
592 |
< |
favorable. Furthermore, this phenomenon agrees with experimental |
593 |
< |
results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt |
594 |
< |
{\it et al.} had kept their Au(111) slab rigid so that their |
595 |
< |
simulations can reach 300K without surface reconstructions. Without |
596 |
< |
this practice, simulating 100\% thiol covered interfaces under higher |
597 |
< |
temperatures could hardly avoid surface reconstructions. However, our |
598 |
< |
measurement is based on assuming homogeneity on $x$ and $y$ dimensions |
599 |
< |
so that measurement of $T$ at particular $z$ would be an effective |
600 |
< |
average of the particles of the same type. Since surface |
601 |
< |
reconstructions could eliminate the original $x$ and $y$ dimensional |
602 |
< |
homogeneity, measurement of $G$ is more difficult to conduct under |
603 |
< |
higher temperatures. Therefore, most of our measurements are |
604 |
< |
undertaken at $\langle T\rangle\sim$200K. |
618 |
> |
To facilitate direct comparison between force fields, systems with the |
619 |
> |
same capping agent and solvent were prepared with the same length |
620 |
> |
scales for the simulation cells. |
621 |
|
|
622 |
< |
However, when the surface is not completely covered by butanethiols, |
623 |
< |
the simulated system is more resistent to the reconstruction |
624 |
< |
above. Our Au-butanethiol/toluene system did not see this phenomena |
625 |
< |
even at $\langle T\rangle\sim$300K. The Au(111) surfaces have a 90\% coverage of |
626 |
< |
butanethiols and have empty three-fold sites. These empty sites could |
627 |
< |
help prevent surface reconstruction in that they provide other means |
612 |
< |
of capping agent relaxation. It is observed that butanethiols can |
613 |
< |
migrate to their neighbor empty sites during a simulation. Therefore, |
614 |
< |
we were able to obtain $G$'s for these interfaces even at a relatively |
615 |
< |
high temperature without being affected by surface reconstructions. |
622 |
> |
On bare metal / solvent surfaces, different force field models for |
623 |
> |
hexane yield similar results for both $G$ and $G^\prime$, and these |
624 |
> |
two definitions agree with each other very well. This is primarily an |
625 |
> |
indicator of weak interactions between the metal and the solvent, and |
626 |
> |
is a typical case for acoustic impedance mismatch between these two |
627 |
> |
phases. |
628 |
|
|
629 |
< |
\subsection{Influence of Capping Agent Coverage on $G$} |
630 |
< |
To investigate the influence of butanethiol coverage on interfacial |
631 |
< |
thermal conductance, a series of different coverage Au-butanethiol |
632 |
< |
surfaces is prepared and solvated with various organic |
633 |
< |
molecules. These systems are then equilibrated and their interfacial |
634 |
< |
thermal conductivity are measured with our NIVS algorithm. Table |
635 |
< |
\ref{tlnUhxnUhxnD} lists these results for direct comparison between |
636 |
< |
different coverages of butanethiol. To study the isotope effect in |
637 |
< |
interfacial thermal conductance, deuterated UA-hexane is included as |
638 |
< |
well. |
629 |
> |
For the fully-covered surfaces, the choice of force field for the |
630 |
> |
capping agent and solvent has a large impact on the calulated values |
631 |
> |
of $G$ and $G^\prime$. The AA thiol to AA solvent conductivities are |
632 |
> |
much larger than their UA to UA counterparts, and these values exceed |
633 |
> |
the experimental estimates by a large measure. The AA force field |
634 |
> |
allows significant energy to go into C-H (or C-D) stretching modes, |
635 |
> |
and since these modes are high frequency, this non-quantum behavior is |
636 |
> |
likely responsible for the overestimate of the conductivity. Compared |
637 |
> |
to the AA model, the UA model yields more reasonable conductivity |
638 |
> |
values with much higher computational efficiency. |
639 |
|
|
640 |
< |
It turned out that with partial covered butanethiol on the Au(111) |
641 |
< |
surface, the derivative definition for $G$ (Eq. \ref{derivativeG}) has |
642 |
< |
difficulty to apply, due to the difficulty in locating the maximum of |
643 |
< |
change of $\lambda$. Instead, the discrete definition |
644 |
< |
(Eq. \ref{discreteG}) is easier to apply, as max($\Delta T$) can still |
645 |
< |
be well-defined. Therefore, $G$'s (not $G^\prime$) are used for this |
646 |
< |
section. |
640 |
> |
\subsubsection{Are electronic excitations in the metal important?} |
641 |
> |
Because they lack electronic excitations, the QSC and related embedded |
642 |
> |
atom method (EAM) models for gold are known to predict unreasonably |
643 |
> |
low values for bulk conductivity |
644 |
> |
($\lambda$).\cite{kuang:164101,ISI:000207079300006,Clancy:1992} If the |
645 |
> |
conductance between the phases ($G$) is governed primarily by phonon |
646 |
> |
excitation (and not electronic degrees of freedom), one would expect a |
647 |
> |
classical model to capture most of the interfacial thermal |
648 |
> |
conductance. Our results for $G$ and $G^\prime$ indicate that this is |
649 |
> |
indeed the case, and suggest that the modeling of interfacial thermal |
650 |
> |
transport depends primarily on the description of the interactions |
651 |
> |
between the various components at the interface. When the metal is |
652 |
> |
chemically capped, the primary barrier to thermal conductivity appears |
653 |
> |
to be the interface between the capping agent and the surrounding |
654 |
> |
solvent, so the excitations in the metal have little impact on the |
655 |
> |
value of $G$. |
656 |
|
|
657 |
< |
From Table \ref{tlnUhxnUhxnD}, one can see the significance of the |
637 |
< |
presence of capping agents. Even when a fraction of the Au(111) |
638 |
< |
surface sites are covered with butanethiols, the conductivity would |
639 |
< |
see an enhancement by at least a factor of 3. This indicates the |
640 |
< |
important role cappping agent is playing for thermal transport |
641 |
< |
phenomena on metal/organic solvent surfaces. |
657 |
> |
\subsection{Effects due to methodology and simulation parameters} |
658 |
|
|
659 |
< |
Interestingly, as one could observe from our results, the maximum |
660 |
< |
conductance enhancement (largest $G$) happens while the surfaces are |
661 |
< |
about 75\% covered with butanethiols. This again indicates that |
662 |
< |
solvent-capping agent contact has an important role of the thermal |
663 |
< |
transport process. Slightly lower butanethiol coverage allows small |
664 |
< |
gaps between butanethiols to form. And these gaps could be filled with |
665 |
< |
solvent molecules, which acts like ``heat conductors'' on the |
666 |
< |
surface. The higher degree of interaction between these solvent |
667 |
< |
molecules and capping agents increases the enhancement effect and thus |
652 |
< |
produces a higher $G$ than densely packed butanethiol arrays. However, |
653 |
< |
once this maximum conductance enhancement is reached, $G$ decreases |
654 |
< |
when butanethiol coverage continues to decrease. Each capping agent |
655 |
< |
molecule reaches its maximum capacity for thermal |
656 |
< |
conductance. Therefore, even higher solvent-capping agent contact |
657 |
< |
would not offset this effect. Eventually, when butanethiol coverage |
658 |
< |
continues to decrease, solvent-capping agent contact actually |
659 |
< |
decreases with the disappearing of butanethiol molecules. In this |
660 |
< |
case, $G$ decrease could not be offset but instead accelerated. |
659 |
> |
We have varied the parameters of the simulations in order to |
660 |
> |
investigate how these factors would affect the computation of $G$. Of |
661 |
> |
particular interest are: 1) the length scale for the applied thermal |
662 |
> |
gradient (modified by increasing the amount of solvent in the system), |
663 |
> |
2) the sign and magnitude of the applied thermal flux, 3) the average |
664 |
> |
temperature of the simulation (which alters the solvent density during |
665 |
> |
equilibration), and 4) the definition of the interfacial conductance |
666 |
> |
(Eqs. (\ref{discreteG}) or (\ref{derivativeG})) used in the |
667 |
> |
calculation. |
668 |
|
|
669 |
< |
A comparison of the results obtained from differenet organic solvents |
670 |
< |
can also provide useful information of the interfacial thermal |
671 |
< |
transport process. The deuterated hexane (UA) results do not appear to |
672 |
< |
be much different from those of normal hexane (UA), given that |
673 |
< |
butanethiol (UA) is non-deuterated for both solvents. These UA model |
674 |
< |
studies, even though eliminating C-H vibration samplings, still have |
675 |
< |
C-C vibrational frequencies different from each other. However, these |
676 |
< |
differences in the infrared range do not seem to produce an observable |
677 |
< |
difference for the results of $G$. [MAY NEED FIGURE] |
669 |
> |
Systems of different lengths were prepared by altering the number of |
670 |
> |
solvent molecules and extending the length of the box along the $z$ |
671 |
> |
axis to accomodate the extra solvent. Equilibration at the same |
672 |
> |
temperature and pressure conditions led to nearly identical surface |
673 |
> |
areas ($L_x$ and $L_y$) available to the metal and capping agent, |
674 |
> |
while the extra solvent served mainly to lengthen the axis that was |
675 |
> |
used to apply the thermal flux. For a given value of the applied |
676 |
> |
flux, the different $z$ length scale has only a weak effect on the |
677 |
> |
computed conductivities (Table \ref{AuThiolHexaneUA}). |
678 |
|
|
679 |
< |
Furthermore, results for rigid body toluene solvent, as well as other |
680 |
< |
UA-hexane solvents, are reasonable within the general experimental |
681 |
< |
ranges[CITATIONS]. This suggests that explicit hydrogen might not be a |
682 |
< |
required factor for modeling thermal transport phenomena of systems |
683 |
< |
such as Au-thiol/organic solvent. |
679 |
> |
\subsubsection{Effects of applied flux} |
680 |
> |
The NIVS algorithm allows changes in both the sign and magnitude of |
681 |
> |
the applied flux. It is possible to reverse the direction of heat |
682 |
> |
flow simply by changing the sign of the flux, and thermal gradients |
683 |
> |
which would be difficult to obtain experimentally ($5$ K/\AA) can be |
684 |
> |
easily simulated. However, the magnitude of the applied flux is not |
685 |
> |
arbitary if one aims to obtain a stable and reliable thermal gradient. |
686 |
> |
A temperature gradient can be lost in the noise if $|J_z|$ is too |
687 |
> |
small, and excessive $|J_z|$ values can cause phase transitions if the |
688 |
> |
extremes of the simulation cell become widely separated in |
689 |
> |
temperature. Also, if $|J_z|$ is too large for the bulk conductivity |
690 |
> |
of the materials, the thermal gradient will never reach a stable |
691 |
> |
state. |
692 |
|
|
693 |
< |
However, results for Au-butanethiol/toluene do not show an identical |
694 |
< |
trend with those for Au-butanethiol/hexane in that $G$'s remain at |
695 |
< |
approximately the same magnitue when butanethiol coverage differs from |
696 |
< |
25\% to 75\%. This might be rooted in the molecule shape difference |
697 |
< |
for plane-like toluene and chain-like {\it n}-hexane. Due to this |
698 |
< |
difference, toluene molecules have more difficulty in occupying |
699 |
< |
relatively small gaps among capping agents when their coverage is not |
700 |
< |
too low. Therefore, the solvent-capping agent contact may keep |
701 |
< |
increasing until the capping agent coverage reaches a relatively low |
702 |
< |
level. This becomes an offset for decreasing butanethiol molecules on |
703 |
< |
its effect to the process of interfacial thermal transport. Thus, one |
704 |
< |
can see a plateau of $G$ vs. butanethiol coverage in our results. |
693 |
> |
Within a reasonable range of $J_z$ values, we were able to study how |
694 |
> |
$G$ changes as a function of this flux. In what follows, we use |
695 |
> |
positive $J_z$ values to denote the case where energy is being |
696 |
> |
transferred by the method from the metal phase and into the liquid. |
697 |
> |
The resulting gradient therefore has a higher temperature in the |
698 |
> |
liquid phase. Negative flux values reverse this transfer, and result |
699 |
> |
in higher temperature metal phases. The conductance measured under |
700 |
> |
different applied $J_z$ values is listed in Tables |
701 |
> |
\ref{AuThiolHexaneUA} and \ref{AuThiolToluene}. These results do not |
702 |
> |
indicate that $G$ depends strongly on $J_z$ within this flux |
703 |
> |
range. The linear response of flux to thermal gradient simplifies our |
704 |
> |
investigations in that we can rely on $G$ measurement with only a |
705 |
> |
small number $J_z$ values. |
706 |
|
|
691 |
– |
[NEED ERROR ESTIMATE, MAY ALSO PUT J HERE] |
707 |
|
\begin{table*} |
708 |
|
\begin{minipage}{\linewidth} |
709 |
|
\begin{center} |
710 |
< |
\caption{Computed interfacial thermal conductivity ($G$) values |
711 |
< |
for the Au-butanethiol/solvent interface with various UA |
712 |
< |
models and different capping agent coverages at $\langle |
713 |
< |
T\rangle\sim$200K using certain energy flux respectively.} |
710 |
> |
\caption{Computed interfacial thermal conductivity ($G$ and |
711 |
> |
$G^\prime$) values for the 100\% covered Au-butanethiol/hexane |
712 |
> |
interfaces with UA model and different hexane molecule numbers |
713 |
> |
at different temperatures using a range of energy |
714 |
> |
fluxes. Error estimates indicated in parenthesis.} |
715 |
|
|
716 |
< |
\begin{tabular}{cccc} |
716 |
> |
\begin{tabular}{ccccccc} |
717 |
|
\hline\hline |
718 |
< |
Thiol & \multicolumn{3}{c}{$G$(MW/m$^2$/K)} \\ |
719 |
< |
coverage (\%) & hexane & hexane(D) & toluene \\ |
718 |
> |
$\langle T\rangle$ & $N_{hexane}$ & $\rho_{hexane}$ & |
719 |
> |
$J_z$ & $G$ & $G^\prime$ \\ |
720 |
> |
(K) & & (g/cm$^3$) & (GW/m$^2$) & |
721 |
> |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
722 |
|
\hline |
723 |
< |
0.0 & 46.5() & 43.9() & 70.1() \\ |
724 |
< |
25.0 & 151() & 153() & 249() \\ |
725 |
< |
50.0 & 172() & 182() & 214() \\ |
726 |
< |
75.0 & 242() & 229() & 244() \\ |
727 |
< |
88.9 & 178() & - & - \\ |
728 |
< |
100.0 & 137() & 153() & 187() \\ |
723 |
> |
200 & 266 & 0.672 & -0.96 & 102(3) & 80.0(0.8) \\ |
724 |
> |
& 200 & 0.688 & 0.96 & 125(16) & 90.2(15) \\ |
725 |
> |
& & & 1.91 & 139(10) & 101(10) \\ |
726 |
> |
& & & 2.83 & 141(6) & 89.9(9.8) \\ |
727 |
> |
& 166 & 0.681 & 0.97 & 141(30) & 78(22) \\ |
728 |
> |
& & & 1.92 & 138(4) & 98.9(9.5) \\ |
729 |
> |
\hline |
730 |
> |
250 & 200 & 0.560 & 0.96 & 75(10) & 61.8(7.3) \\ |
731 |
> |
& & & -0.95 & 49.4(0.3) & 45.7(2.1) \\ |
732 |
> |
& 166 & 0.569 & 0.97 & 80.3(0.6) & 67(11) \\ |
733 |
> |
& & & 1.44 & 76.2(5.0) & 64.8(3.8) \\ |
734 |
> |
& & & -0.95 & 56.4(2.5) & 54.4(1.1) \\ |
735 |
> |
& & & -1.85 & 47.8(1.1) & 53.5(1.5) \\ |
736 |
|
\hline\hline |
737 |
|
\end{tabular} |
738 |
< |
\label{tlnUhxnUhxnD} |
738 |
> |
\label{AuThiolHexaneUA} |
739 |
|
\end{center} |
740 |
|
\end{minipage} |
741 |
|
\end{table*} |
742 |
|
|
743 |
< |
\subsection{Influence of Chosen Molecule Model on $G$} |
744 |
< |
[MAY COMBINE W MECHANISM STUDY] |
743 |
> |
The sign of $J_z$ is a different matter, however, as this can alter |
744 |
> |
the temperature on the two sides of the interface. The average |
745 |
> |
temperature values reported are for the entire system, and not for the |
746 |
> |
liquid phase, so at a given $\langle T \rangle$, the system with |
747 |
> |
positive $J_z$ has a warmer liquid phase. This means that if the |
748 |
> |
liquid carries thermal energy via convective transport, {\it positive} |
749 |
> |
$J_z$ values will result in increased molecular motion on the liquid |
750 |
> |
side of the interface, and this will increase the measured |
751 |
> |
conductivity. |
752 |
|
|
753 |
< |
In addition to UA solvent/capping agent models, AA models are included |
722 |
< |
in our simulations as well. Besides simulations of the same (UA or AA) |
723 |
< |
model for solvent and capping agent, different models can be applied |
724 |
< |
to different components. Furthermore, regardless of models chosen, |
725 |
< |
either the solvent or the capping agent can be deuterated, similar to |
726 |
< |
the previous section. Table \ref{modelTest} summarizes the results of |
727 |
< |
these studies. |
753 |
> |
\subsubsection{Effects due to average temperature} |
754 |
|
|
755 |
< |
[MORE DATA; ERROR ESTIMATE] |
755 |
> |
We also studied the effect of average system temperature on the |
756 |
> |
interfacial conductance. The simulations are first equilibrated in |
757 |
> |
the NPT ensemble at 1 atm. The TraPPE-UA model for hexane tends to |
758 |
> |
predict a lower boiling point (and liquid state density) than |
759 |
> |
experiments. This lower-density liquid phase leads to reduced contact |
760 |
> |
between the hexane and butanethiol, and this accounts for our |
761 |
> |
observation of lower conductance at higher temperatures as shown in |
762 |
> |
Table \ref{AuThiolHexaneUA}. In raising the average temperature from |
763 |
> |
200K to 250K, the density drop of ~20\% in the solvent phase leads to |
764 |
> |
a ~65\% drop in the conductance. |
765 |
> |
|
766 |
> |
Similar behavior is observed in the TraPPE-UA model for toluene, |
767 |
> |
although this model has better agreement with the experimental |
768 |
> |
densities of toluene. The expansion of the toluene liquid phase is |
769 |
> |
not as significant as that of the hexane (8.3\% over 100K), and this |
770 |
> |
limits the effect to ~20\% drop in thermal conductivity (Table |
771 |
> |
\ref{AuThiolToluene}). |
772 |
> |
|
773 |
> |
Although we have not mapped out the behavior at a large number of |
774 |
> |
temperatures, is clear that there will be a strong temperature |
775 |
> |
dependence in the interfacial conductance when the physical properties |
776 |
> |
of one side of the interface (notably the density) change rapidly as a |
777 |
> |
function of temperature. |
778 |
> |
|
779 |
|
\begin{table*} |
780 |
|
\begin{minipage}{\linewidth} |
781 |
|
\begin{center} |
733 |
– |
|
782 |
|
\caption{Computed interfacial thermal conductivity ($G$ and |
783 |
< |
$G^\prime$) values for interfaces using various models for |
784 |
< |
solvent and capping agent (or without capping agent) at |
785 |
< |
$\langle T\rangle\sim$200K.} |
783 |
> |
$G^\prime$) values for a 90\% coverage Au-butanethiol/toluene |
784 |
> |
interface at different temperatures using a range of energy |
785 |
> |
fluxes. Error estimates indicated in parenthesis.} |
786 |
|
|
787 |
|
\begin{tabular}{ccccc} |
788 |
|
\hline\hline |
789 |
< |
Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ |
790 |
< |
(or bare surface) & model & (GW/m$^2$) & |
743 |
< |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
789 |
> |
$\langle T\rangle$ & $\rho_{toluene}$ & $J_z$ & $G$ & $G^\prime$ \\ |
790 |
> |
(K) & (g/cm$^3$) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
791 |
|
\hline |
792 |
< |
UA & AA hexane & 1.94 & 135() & 129() \\ |
793 |
< |
& & 2.86 & 126() & 115() \\ |
794 |
< |
& AA toluene & 1.89 & 200() & 149() \\ |
795 |
< |
AA & UA hexane & 1.94 & 116() & 129() \\ |
796 |
< |
& AA hexane & 3.76 & 451() & 378() \\ |
797 |
< |
& & 4.71 & 432() & 334() \\ |
751 |
< |
& AA toluene & 3.79 & 487() & 290() \\ |
752 |
< |
AA(D) & UA hexane & 1.94 & 158() & 172() \\ |
753 |
< |
bare & AA hexane & 0.96 & 31.0() & 29.4() \\ |
792 |
> |
200 & 0.933 & 2.15 & 204(12) & 113(12) \\ |
793 |
> |
& & -1.86 & 180(3) & 135(21) \\ |
794 |
> |
& & -3.93 & 176(5) & 113(12) \\ |
795 |
> |
\hline |
796 |
> |
300 & 0.855 & -1.91 & 143(5) & 125(2) \\ |
797 |
> |
& & -4.19 & 135(9) & 113(12) \\ |
798 |
|
\hline\hline |
799 |
|
\end{tabular} |
800 |
< |
\label{modelTest} |
800 |
> |
\label{AuThiolToluene} |
801 |
|
\end{center} |
802 |
|
\end{minipage} |
803 |
|
\end{table*} |
804 |
|
|
805 |
< |
To facilitate direct comparison, the same system with differnt models |
806 |
< |
for different components uses the same length scale for their |
807 |
< |
simulation cells. Without the presence of capping agent, using |
808 |
< |
different models for hexane yields similar results for both $G$ and |
809 |
< |
$G^\prime$, and these two definitions agree with eath other very |
810 |
< |
well. This indicates very weak interaction between the metal and the |
811 |
< |
solvent, and is a typical case for acoustic impedance mismatch between |
812 |
< |
these two phases. |
805 |
> |
Besides the lower interfacial thermal conductance, surfaces at |
806 |
> |
relatively high temperatures are susceptible to reconstructions, |
807 |
> |
particularly when butanethiols fully cover the Au(111) surface. These |
808 |
> |
reconstructions include surface Au atoms which migrate outward to the |
809 |
> |
S atom layer, and butanethiol molecules which embed into the surface |
810 |
> |
Au layer. The driving force for this behavior is the strong Au-S |
811 |
> |
interactions which are modeled here with a deep Lennard-Jones |
812 |
> |
potential. This phenomenon agrees with reconstructions that have beeen |
813 |
> |
experimentally |
814 |
> |
observed.\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt |
815 |
> |
{\it et al.} kept their Au(111) slab rigid so that their simulations |
816 |
> |
could reach 300K without surface |
817 |
> |
reconstructions.\cite{vlugt:cpc2007154} Since surface reconstructions |
818 |
> |
blur the interface, the measurement of $G$ becomes more difficult to |
819 |
> |
conduct at higher temperatures. For this reason, most of our |
820 |
> |
measurements are undertaken at $\langle T\rangle\sim$200K where |
821 |
> |
reconstruction is minimized. |
822 |
|
|
823 |
< |
As for Au(111) surfaces completely covered by butanethiols, the choice |
824 |
< |
of models for capping agent and solvent could impact the measurement |
825 |
< |
of $G$ and $G^\prime$ quite significantly. For Au-butanethiol/hexane |
826 |
< |
interfaces, using AA model for both butanethiol and hexane yields |
827 |
< |
substantially higher conductivity values than using UA model for at |
828 |
< |
least one component of the solvent and capping agent, which exceeds |
829 |
< |
the upper bond of experimental value range. This is probably due to |
830 |
< |
the classically treated C-H vibrations in the AA model, which should |
831 |
< |
not be appreciably populated at normal temperatures. In comparison, |
779 |
< |
once either the hexanes or the butanethiols are deuterated, one can |
780 |
< |
see a significantly lower $G$ and $G^\prime$. In either of these |
781 |
< |
cases, the C-H(D) vibrational overlap between the solvent and the |
782 |
< |
capping agent is removed. [MAY NEED FIGURE] Conclusively, the |
783 |
< |
improperly treated C-H vibration in the AA model produced |
784 |
< |
over-predicted results accordingly. Compared to the AA model, the UA |
785 |
< |
model yields more reasonable results with higher computational |
786 |
< |
efficiency. |
823 |
> |
However, when the surface is not completely covered by butanethiols, |
824 |
> |
the simulated system appears to be more resistent to the |
825 |
> |
reconstruction. O ur Au / butanethiol / toluene system had the Au(111) |
826 |
> |
surfaces 90\% covered by butanethiols, but did not see this above |
827 |
> |
phenomena even at $\langle T\rangle\sim$300K. That said, we did |
828 |
> |
observe butanethiols migrating to neighboring three-fold sites during |
829 |
> |
a simulation. Since the interface persisted in these simulations, |
830 |
> |
were able to obtain $G$'s for these interfaces even at a relatively |
831 |
> |
high temperature without being affected by surface reconstructions. |
832 |
|
|
833 |
< |
However, for Au-butanethiol/toluene interfaces, having the AA |
789 |
< |
butanethiol deuterated did not yield a significant change in the |
790 |
< |
measurement results. |
791 |
< |
. , so extra degrees of freedom |
792 |
< |
such as the C-H vibrations could enhance heat exchange between these |
793 |
< |
two phases and result in a much higher conductivity. |
833 |
> |
\section{Discussion} |
834 |
|
|
835 |
+ |
The primary result of this work is that the capping agent acts as an |
836 |
+ |
efficient thermal coupler between solid and solvent phases. One of |
837 |
+ |
the ways the capping agent can carry out this role is to down-shift |
838 |
+ |
between the phonon vibrations in the solid (which carry the heat from |
839 |
+ |
the gold) and the molecular vibrations in the liquid (which carry some |
840 |
+ |
of the heat in the solvent). |
841 |
|
|
842 |
< |
Although the QSC model for Au is known to predict an overly low value |
843 |
< |
for bulk metal gold conductivity[CITE NIVSRNEMD], our computational |
844 |
< |
results for $G$ and $G^\prime$ do not seem to be affected by this |
845 |
< |
drawback of the model for metal. Instead, the modeling of interfacial |
846 |
< |
thermal transport behavior relies mainly on an accurate description of |
847 |
< |
the interactions between components occupying the interfaces. |
842 |
> |
To investigate the mechanism of interfacial thermal conductance, the |
843 |
> |
vibrational power spectrum was computed. Power spectra were taken for |
844 |
> |
individual components in different simulations. To obtain these |
845 |
> |
spectra, simulations were run after equilibration in the |
846 |
> |
microcanonical (NVE) ensemble and without a thermal |
847 |
> |
gradient. Snapshots of configurations were collected at a frequency |
848 |
> |
that is higher than that of the fastest vibrations occuring in the |
849 |
> |
simulations. With these configurations, the velocity auto-correlation |
850 |
> |
functions can be computed: |
851 |
> |
\begin{equation} |
852 |
> |
C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle |
853 |
> |
\label{vCorr} |
854 |
> |
\end{equation} |
855 |
> |
The power spectrum is constructed via a Fourier transform of the |
856 |
> |
symmetrized velocity autocorrelation function, |
857 |
> |
\begin{equation} |
858 |
> |
\hat{f}(\omega) = |
859 |
> |
\int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt |
860 |
> |
\label{fourier} |
861 |
> |
\end{equation} |
862 |
|
|
863 |
< |
\subsection{Mechanism of Interfacial Thermal Conductance Enhancement |
864 |
< |
by Capping Agent} |
865 |
< |
%OR\subsection{Vibrational spectrum study on conductance mechanism} |
863 |
> |
\subsection{The role of specific vibrations} |
864 |
> |
The vibrational spectra for gold slabs in different environments are |
865 |
> |
shown as in Figure \ref{specAu}. Regardless of the presence of |
866 |
> |
solvent, the gold surfaces which are covered by butanethiol molecules |
867 |
> |
exhibit an additional peak observed at a frequency of |
868 |
> |
$\sim$170cm$^{-1}$. We attribute this peak to the S-Au bonding |
869 |
> |
vibration. This vibration enables efficient thermal coupling of the |
870 |
> |
surface Au layer to the capping agents. Therefore, in our simulations, |
871 |
> |
the Au / S interfaces do not appear to be the primary barrier to |
872 |
> |
thermal transport when compared with the butanethiol / solvent |
873 |
> |
interfaces. |
874 |
|
|
875 |
< |
[MAY INTRODUCE PROTOCOL IN METHODOLOGY/COMPUTATIONAL DETAIL, EQN'S] |
875 |
> |
\begin{figure} |
876 |
> |
\includegraphics[width=\linewidth]{vibration} |
877 |
> |
\caption{Vibrational power spectra for gold in different solvent |
878 |
> |
environments. The presence of the butanethiol capping molecules |
879 |
> |
adds a vibrational peak at $\sim$170cm$^{-1}$.} |
880 |
> |
\label{specAu} |
881 |
> |
\end{figure} |
882 |
|
|
883 |
< |
To investigate the mechanism of this interfacial thermal conductance, |
884 |
< |
the vibrational spectra of various gold systems were obtained and are |
885 |
< |
shown as in the upper panel of Fig. \ref{vibration}. To obtain these |
812 |
< |
spectra, one first runs a simulation in the NVE ensemble and collects |
813 |
< |
snapshots of configurations; these configurations are used to compute |
814 |
< |
the velocity auto-correlation functions, which is used to construct a |
815 |
< |
power spectrum via a Fourier transform. |
883 |
> |
Also in this figure, we show the vibrational power spectrum for the |
884 |
> |
bound butanethiol molecules, which also exhibits the same |
885 |
> |
$\sim$170cm$^{-1}$ peak. |
886 |
|
|
887 |
< |
The gold surfaces covered by |
888 |
< |
butanethiol molecules, compared to bare gold surfaces, exhibit an |
889 |
< |
additional peak observed at a frequency of $\sim$170cm$^{-1}$, which |
890 |
< |
is attributed to the vibration of the S-Au bond. This vibration |
891 |
< |
enables efficient thermal transport from surface Au atoms to the |
892 |
< |
capping agents. Simultaneously, as shown in the lower panel of |
893 |
< |
Fig. \ref{vibration}, the large overlap of the vibration spectra of |
894 |
< |
butanethiol and hexane in the all-atom model, including the C-H |
895 |
< |
vibration, also suggests high thermal exchange efficiency. The |
896 |
< |
combination of these two effects produces the drastic interfacial |
897 |
< |
thermal conductance enhancement in the all-atom model. |
887 |
> |
\subsection{Overlap of power spectra} |
888 |
> |
A comparison of the results obtained from the two different organic |
889 |
> |
solvents can also provide useful information of the interfacial |
890 |
> |
thermal transport process. In particular, the vibrational overlap |
891 |
> |
between the butanethiol and the organic solvents suggests a highly |
892 |
> |
efficient thermal exchange between these components. Very high |
893 |
> |
thermal conductivity was observed when AA models were used and C-H |
894 |
> |
vibrations were treated classically. The presence of extra degrees of |
895 |
> |
freedom in the AA force field yields higher heat exchange rates |
896 |
> |
between the two phases and results in a much higher conductivity than |
897 |
> |
in the UA force field. |
898 |
|
|
899 |
< |
[MAY NEED TO CONVERT TO JPEG] |
899 |
> |
The similarity in the vibrational modes available to solvent and |
900 |
> |
capping agent can be reduced by deuterating one of the two components |
901 |
> |
(Fig. \ref{aahxntln}). Once either the hexanes or the butanethiols |
902 |
> |
are deuterated, one can observe a significantly lower $G$ and |
903 |
> |
$G^\prime$ values (Table \ref{modelTest}). |
904 |
> |
|
905 |
|
\begin{figure} |
906 |
< |
\includegraphics[width=\linewidth]{vibration} |
907 |
< |
\caption{Vibrational spectra obtained for gold in different |
908 |
< |
environments (upper panel) and for Au/thiol/hexane simulation in |
909 |
< |
all-atom model (lower panel).} |
910 |
< |
\label{vibration} |
906 |
> |
\includegraphics[width=\linewidth]{aahxntln} |
907 |
> |
\caption{Spectra obtained for all-atom (AA) Au / butanethiol / solvent |
908 |
> |
systems. When butanethiol is deuterated (lower left), its |
909 |
> |
vibrational overlap with hexane decreases significantly. Since |
910 |
> |
aromatic molecules and the butanethiol are vibrationally dissimilar, |
911 |
> |
the change is not as dramatic when toluene is the solvent (right).} |
912 |
> |
\label{aahxntln} |
913 |
|
\end{figure} |
914 |
|
|
915 |
< |
[COMPARISON OF TWO G'S; AU SLAB WIDTHS; ETC] |
916 |
< |
% The results show that the two definitions used for $G$ yield |
917 |
< |
% comparable values, though $G^\prime$ tends to be smaller. |
915 |
> |
For the Au / butanethiol / toluene interfaces, having the AA |
916 |
> |
butanethiol deuterated did not yield a significant change in the |
917 |
> |
measured conductance. Compared to the C-H vibrational overlap between |
918 |
> |
hexane and butanethiol, both of which have alkyl chains, the overlap |
919 |
> |
between toluene and butanethiol is not as significant and thus does |
920 |
> |
not contribute as much to the heat exchange process. |
921 |
|
|
922 |
+ |
Previous observations of Zhang {\it et al.}\cite{hase:2010} indicate |
923 |
+ |
that the {\it intra}molecular heat transport due to alkylthiols is |
924 |
+ |
highly efficient. Combining our observations with those of Zhang {\it |
925 |
+ |
et al.}, it appears that butanethiol acts as a channel to expedite |
926 |
+ |
heat flow from the gold surface and into the alkyl chain. The |
927 |
+ |
acoustic impedance mismatch between the metal and the liquid phase can |
928 |
+ |
therefore be effectively reduced with the presence of suitable capping |
929 |
+ |
agents. |
930 |
+ |
|
931 |
+ |
Deuterated models in the UA force field did not decouple the thermal |
932 |
+ |
transport as well as in the AA force field. The UA models, even |
933 |
+ |
though they have eliminated the high frequency C-H vibrational |
934 |
+ |
overlap, still have significant overlap in the lower-frequency |
935 |
+ |
portions of the infrared spectra (Figure \ref{uahxnua}). Deuterating |
936 |
+ |
the UA models did not decouple the low frequency region enough to |
937 |
+ |
produce an observable difference for the results of $G$ (Table |
938 |
+ |
\ref{modelTest}). |
939 |
+ |
|
940 |
+ |
\begin{figure} |
941 |
+ |
\includegraphics[width=\linewidth]{uahxnua} |
942 |
+ |
\caption{Vibrational spectra obtained for normal (upper) and |
943 |
+ |
deuterated (lower) hexane in Au-butanethiol/hexane |
944 |
+ |
systems. Butanethiol spectra are shown as reference. Both hexane and |
945 |
+ |
butanethiol were using United-Atom models.} |
946 |
+ |
\label{uahxnua} |
947 |
+ |
\end{figure} |
948 |
+ |
|
949 |
|
\section{Conclusions} |
950 |
< |
The NIVS algorithm we developed has been applied to simulations of |
951 |
< |
Au-butanethiol surfaces with organic solvents. This algorithm allows |
952 |
< |
effective unphysical thermal flux transferred between the metal and |
953 |
< |
the liquid phase. With the flux applied, we were able to measure the |
954 |
< |
corresponding thermal gradient and to obtain interfacial thermal |
955 |
< |
conductivities. Our simulations have seen significant conductance |
956 |
< |
enhancement with the presence of capping agent, compared to the bare |
957 |
< |
gold/liquid interfaces. The acoustic impedance mismatch between the |
958 |
< |
metal and the liquid phase is effectively eliminated by proper capping |
950 |
> |
The NIVS algorithm has been applied to simulations of |
951 |
> |
butanethiol-capped Au(111) surfaces in the presence of organic |
952 |
> |
solvents. This algorithm allows the application of unphysical thermal |
953 |
> |
flux to transfer heat between the metal and the liquid phase. With the |
954 |
> |
flux applied, we were able to measure the corresponding thermal |
955 |
> |
gradients and to obtain interfacial thermal conductivities. Under |
956 |
> |
steady states, 2-3 ns trajectory simulations are sufficient for |
957 |
> |
computation of this quantity. |
958 |
> |
|
959 |
> |
Our simulations have seen significant conductance enhancement in the |
960 |
> |
presence of capping agent, compared with the bare gold / liquid |
961 |
> |
interfaces. The acoustic impedance mismatch between the metal and the |
962 |
> |
liquid phase is effectively eliminated by a chemically-bonded capping |
963 |
|
agent. Furthermore, the coverage precentage of the capping agent plays |
964 |
< |
an important role in the interfacial thermal transport process. |
964 |
> |
an important role in the interfacial thermal transport |
965 |
> |
process. Moderately low coverages allow higher contact between capping |
966 |
> |
agent and solvent, and thus could further enhance the heat transfer |
967 |
> |
process, giving a non-monotonic behavior of conductance with |
968 |
> |
increasing coverage. |
969 |
|
|
970 |
< |
Our measurement results, particularly of the UA models, agree with |
971 |
< |
available experimental data. This indicates that our force field |
857 |
< |
parameters have a nice description of the interactions between the |
858 |
< |
particles at the interfaces. AA models tend to overestimate the |
970 |
> |
Our results, particularly using the UA models, agree well with |
971 |
> |
available experimental data. The AA models tend to overestimate the |
972 |
|
interfacial thermal conductance in that the classically treated C-H |
973 |
< |
vibration would be overly sampled. Compared to the AA models, the UA |
974 |
< |
models have higher computational efficiency with satisfactory |
975 |
< |
accuracy, and thus are preferable in interfacial thermal transport |
976 |
< |
modelings. |
973 |
> |
vibrations become too easily populated. Compared to the AA models, the |
974 |
> |
UA models have higher computational efficiency with satisfactory |
975 |
> |
accuracy, and thus are preferable in modeling interfacial thermal |
976 |
> |
transport. |
977 |
|
|
978 |
< |
Vlugt {\it et al.} has investigated the surface thiol structures for |
979 |
< |
nanocrystal gold and pointed out that they differs from those of the |
980 |
< |
Au(111) surface\cite{vlugt:cpc2007154}. This difference might lead to |
981 |
< |
change of interfacial thermal transport behavior as well. To |
982 |
< |
investigate this problem, an effective means to introduce thermal flux |
983 |
< |
and measure the corresponding thermal gradient is desirable for |
984 |
< |
simulating structures with spherical symmetry. |
978 |
> |
Of the two definitions for $G$, the discrete form |
979 |
> |
(Eq. \ref{discreteG}) was easier to use and gives out relatively |
980 |
> |
consistent results, while the derivative form (Eq. \ref{derivativeG}) |
981 |
> |
is not as versatile. Although $G^\prime$ gives out comparable results |
982 |
> |
and follows similar trend with $G$ when measuring close to fully |
983 |
> |
covered or bare surfaces, the spatial resolution of $T$ profile |
984 |
> |
required for the use of a derivative form is limited by the number of |
985 |
> |
bins and the sampling required to obtain thermal gradient information. |
986 |
|
|
987 |
+ |
Vlugt {\it et al.} have investigated the surface thiol structures for |
988 |
+ |
nanocrystalline gold and pointed out that they differ from those of |
989 |
+ |
the Au(111) surface.\cite{landman:1998,vlugt:cpc2007154} This |
990 |
+ |
difference could also cause differences in the interfacial thermal |
991 |
+ |
transport behavior. To investigate this problem, one would need an |
992 |
+ |
effective method for applying thermal gradients in non-planar |
993 |
+ |
(i.e. spherical) geometries. |
994 |
|
|
995 |
|
\section{Acknowledgments} |
996 |
|
Support for this project was provided by the National Science |
997 |
|
Foundation under grant CHE-0848243. Computational time was provided by |
998 |
|
the Center for Research Computing (CRC) at the University of Notre |
999 |
< |
Dame. \newpage |
999 |
> |
Dame. |
1000 |
> |
\newpage |
1001 |
|
|
1002 |
|
\bibliography{interfacial} |
1003 |
|
|