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\begin{document} |
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|
| 45 |
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|
| 46 |
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\begin{abstract} |
| 47 |
|
|
| 48 |
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We have developed a Non-Isotropic Velocity Scaling algorithm for |
| 49 |
< |
setting up and maintaining stable thermal gradients in non-equilibrium |
| 50 |
< |
molecular dynamics simulations. This approach effectively imposes |
| 51 |
< |
unphysical thermal flux even between particles of different |
| 52 |
< |
identities, conserves linear momentum and kinetic energy, and |
| 53 |
< |
minimally perturbs the velocity profile of a system when compared with |
| 54 |
< |
previous RNEMD methods. We have used this method to simulate thermal |
| 55 |
< |
conductance at metal / organic solvent interfaces both with and |
| 56 |
< |
without the presence of thiol-based capping agents. We obtained |
| 57 |
< |
values comparable with experimental values, and observed significant |
| 58 |
< |
conductance enhancement with the presence of capping agents. Computed |
| 59 |
< |
power spectra indicate the acoustic impedance mismatch between metal |
| 60 |
< |
and liquid phase is greatly reduced by the capping agents and thus |
| 61 |
< |
leads to higher interfacial thermal transfer efficiency. |
| 48 |
> |
With the Non-Isotropic Velocity Scaling algorithm (NIVS) we have |
| 49 |
> |
developed, an unphysical thermal flux can be effectively set up even |
| 50 |
> |
for non-homogeneous systems like interfaces in non-equilibrium |
| 51 |
> |
molecular dynamics simulations. In this work, this algorithm is |
| 52 |
> |
applied for simulating thermal conductance at metal / organic solvent |
| 53 |
> |
interfaces with various coverages of butanethiol capping |
| 54 |
> |
agents. Different solvents and force field models were tested. Our |
| 55 |
> |
results suggest that the United-Atom models are able to provide an |
| 56 |
> |
estimate of the interfacial thermal conductivity comparable to |
| 57 |
> |
experiments in our simulations with satisfactory computational |
| 58 |
> |
efficiency. From our results, the acoustic impedance mismatch between |
| 59 |
> |
metal and liquid phase is effectively reduced by the capping |
| 60 |
> |
agents, and thus leads to interfacial thermal conductance |
| 61 |
> |
enhancement. Furthermore, this effect is closely related to the |
| 62 |
> |
capping agent coverage on the metal surfaces and the type of solvent |
| 63 |
> |
molecules, and is affected by the models used in the simulations. |
| 64 |
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|
| 65 |
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\end{abstract} |
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|
| 73 |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 74 |
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|
| 75 |
|
\section{Introduction} |
| 76 |
< |
[BACKGROUND FOR INTERFACIAL THERMAL CONDUCTANCE PROBLEM] |
| 77 |
< |
Interfacial thermal conductance is extensively studied both |
| 78 |
< |
experimentally and computationally, and systems with interfaces |
| 79 |
< |
present are generally heterogeneous. Although interfaces are commonly |
| 80 |
< |
barriers to heat transfer, it has been |
| 81 |
< |
reported\cite{doi:10.1021/la904855s} that under specific circustances, |
| 82 |
< |
e.g. with certain capping agents present on the surface, interfacial |
| 83 |
< |
conductance can be significantly enhanced. However, heat conductance |
| 82 |
< |
of molecular and nano-scale interfaces will be affected by the |
| 83 |
< |
chemical details of the surface and is challenging to |
| 84 |
< |
experimentalist. The lower thermal flux through interfaces is even |
| 85 |
< |
more difficult to measure with EMD and forward NEMD simulation |
| 86 |
< |
methods. Therefore, developing good simulation methods will be |
| 87 |
< |
desirable in order to investigate thermal transport across interfaces. |
| 76 |
> |
Due to the importance of heat flow in nanotechnology, interfacial |
| 77 |
> |
thermal conductance has been studied extensively both experimentally |
| 78 |
> |
and computationally.\cite{cahill:793} Unlike bulk materials, nanoscale |
| 79 |
> |
materials have a significant fraction of their atoms at interfaces, |
| 80 |
> |
and the chemical details of these interfaces govern the heat transfer |
| 81 |
> |
behavior. Furthermore, the interfaces are |
| 82 |
> |
heterogeneous (e.g. solid - liquid), which provides a challenge to |
| 83 |
> |
traditional methods developed for homogeneous systems. |
| 84 |
|
|
| 85 |
< |
Recently, we have developed the Non-Isotropic Velocity Scaling (NIVS) |
| 85 |
> |
Experimentally, various interfaces have been investigated for their |
| 86 |
> |
thermal conductance. Wang {\it et al.} studied heat transport through |
| 87 |
> |
long-chain hydrocarbon monolayers on gold substrate at individual |
| 88 |
> |
molecular level,\cite{Wang10082007} Schmidt {\it et al.} studied the |
| 89 |
> |
role of CTAB on thermal transport between gold nanorods and |
| 90 |
> |
solvent,\cite{doi:10.1021/jp8051888} and Juv\'e {\it et al.} studied |
| 91 |
> |
the cooling dynamics, which is controlled by thermal interface |
| 92 |
> |
resistence of glass-embedded metal |
| 93 |
> |
nanoparticles.\cite{PhysRevB.80.195406} Although interfaces are |
| 94 |
> |
normally considered barriers for heat transport, Alper {\it et al.} |
| 95 |
> |
suggested that specific ligands (capping agents) could completely |
| 96 |
> |
eliminate this barrier |
| 97 |
> |
($G\rightarrow\infty$).\cite{doi:10.1021/la904855s} |
| 98 |
> |
|
| 99 |
> |
Theoretical and computational models have also been used to study the |
| 100 |
> |
interfacial thermal transport in order to gain an understanding of |
| 101 |
> |
this phenomena at the molecular level. Recently, Hase and coworkers |
| 102 |
> |
employed Non-Equilibrium Molecular Dynamics (NEMD) simulations to |
| 103 |
> |
study thermal transport from hot Au(111) substrate to a self-assembled |
| 104 |
> |
monolayer of alkylthiol with relatively long chain (8-20 carbon |
| 105 |
> |
atoms).\cite{hase:2010,hase:2011} However, ensemble averaged |
| 106 |
> |
measurements for heat conductance of interfaces between the capping |
| 107 |
> |
monolayer on Au and a solvent phase have yet to be studied with their |
| 108 |
> |
approach. The comparatively low thermal flux through interfaces is |
| 109 |
> |
difficult to measure with Equilibrium MD or forward NEMD simulation |
| 110 |
> |
methods. Therefore, the Reverse NEMD (RNEMD) |
| 111 |
> |
methods\cite{MullerPlathe:1997xw,kuang:164101} would have the |
| 112 |
> |
advantage of applying this difficult to measure flux (while measuring |
| 113 |
> |
the resulting gradient), given that the simulation methods being able |
| 114 |
> |
to effectively apply an unphysical flux in non-homogeneous systems. |
| 115 |
> |
Garde and coworkers\cite{garde:nl2005,garde:PhysRevLett2009} applied |
| 116 |
> |
this approach to various liquid interfaces and studied how thermal |
| 117 |
> |
conductance (or resistance) is dependent on chemistry details of |
| 118 |
> |
interfaces, e.g. hydrophobic and hydrophilic aqueous interfaces. |
| 119 |
> |
|
| 120 |
> |
Recently, we have developed a Non-Isotropic Velocity Scaling (NIVS) |
| 121 |
|
algorithm for RNEMD simulations\cite{kuang:164101}. This algorithm |
| 122 |
|
retains the desirable features of RNEMD (conservation of linear |
| 123 |
|
momentum and total energy, compatibility with periodic boundary |
| 124 |
|
conditions) while establishing true thermal distributions in each of |
| 125 |
< |
the two slabs. Furthermore, it allows more effective thermal exchange |
| 126 |
< |
between particles of different identities, and thus enables extensive |
| 127 |
< |
study of interfacial conductance. |
| 125 |
> |
the two slabs. Furthermore, it allows effective thermal exchange |
| 126 |
> |
between particles of different identities, and thus makes the study of |
| 127 |
> |
interfacial conductance much simpler. |
| 128 |
|
|
| 129 |
+ |
The work presented here deals with the Au(111) surface covered to |
| 130 |
+ |
varying degrees by butanethiol, a capping agent with short carbon |
| 131 |
+ |
chain, and solvated with organic solvents of different molecular |
| 132 |
+ |
properties. Different models were used for both the capping agent and |
| 133 |
+ |
the solvent force field parameters. Using the NIVS algorithm, the |
| 134 |
+ |
thermal transport across these interfaces was studied and the |
| 135 |
+ |
underlying mechanism for the phenomena was investigated. |
| 136 |
+ |
|
| 137 |
|
\section{Methodology} |
| 138 |
< |
\subsection{Algorithm} |
| 139 |
< |
[BACKGROUND FOR MD METHODS] |
| 140 |
< |
There have been many algorithms for computing thermal conductivity |
| 141 |
< |
using molecular dynamics simulations. However, interfacial conductance |
| 142 |
< |
is at least an order of magnitude smaller. This would make the |
| 143 |
< |
calculation even more difficult for those slowly-converging |
| 144 |
< |
equilibrium methods. Imposed-flux non-equilibrium |
| 145 |
< |
methods\cite{MullerPlathe:1997xw} have the flux set {\it a priori} and |
| 146 |
< |
the response of temperature or momentum gradients are easier to |
| 147 |
< |
measure than the flux, if unknown, and thus, is a preferable way to |
| 148 |
< |
the forward NEMD methods. Although the momentum swapping approach for |
| 149 |
< |
flux-imposing can be used for exchanging energy between particles of |
| 150 |
< |
different identity, the kinetic energy transfer efficiency is affected |
| 151 |
< |
by the mass difference between the particles, which limits its |
| 152 |
< |
application on heterogeneous interfacial systems. |
| 138 |
> |
\subsection{Imposd-Flux Methods in MD Simulations} |
| 139 |
> |
Steady state MD simulations have an advantage in that not many |
| 140 |
> |
trajectories are needed to study the relationship between thermal flux |
| 141 |
> |
and thermal gradients. For systems with low interfacial conductance, |
| 142 |
> |
one must have a method capable of generating or measuring relatively |
| 143 |
> |
small fluxes, compared to those required for bulk conductivity. This |
| 144 |
> |
requirement makes the calculation even more difficult for |
| 145 |
> |
slowly-converging equilibrium methods.\cite{Viscardy:2007lq} Forward |
| 146 |
> |
NEMD methods impose a gradient (and measure a flux), but at interfaces |
| 147 |
> |
it is not clear what behavior should be imposed at the boundaries |
| 148 |
> |
between materials. Imposed-flux reverse non-equilibrium |
| 149 |
> |
methods\cite{MullerPlathe:1997xw} set the flux {\it a priori} and |
| 150 |
> |
the thermal response becomes an easy-to-measure quantity. Although |
| 151 |
> |
M\"{u}ller-Plathe's original momentum swapping approach can be used |
| 152 |
> |
for exchanging energy between particles of different identity, the |
| 153 |
> |
kinetic energy transfer efficiency is affected by the mass difference |
| 154 |
> |
between the particles, which limits its application on heterogeneous |
| 155 |
> |
interfacial systems. |
| 156 |
|
|
| 157 |
< |
The non-isotropic velocity scaling (NIVS)\cite{kuang:164101} approach in |
| 158 |
< |
non-equilibrium MD simulations is able to impose relatively large |
| 159 |
< |
kinetic energy flux without obvious perturbation to the velocity |
| 160 |
< |
distribution of the simulated systems. Furthermore, this approach has |
| 157 |
> |
The non-isotropic velocity scaling (NIVS) \cite{kuang:164101} approach |
| 158 |
> |
to non-equilibrium MD simulations is able to impose a wide range of |
| 159 |
> |
kinetic energy fluxes without obvious perturbation to the velocity |
| 160 |
> |
distributions of the simulated systems. Furthermore, this approach has |
| 161 |
|
the advantage in heterogeneous interfaces in that kinetic energy flux |
| 162 |
|
can be applied between regions of particles of arbitary identity, and |
| 163 |
< |
the flux quantity is not restricted by particle mass difference. |
| 163 |
> |
the flux will not be restricted by difference in particle mass. |
| 164 |
|
|
| 165 |
|
The NIVS algorithm scales the velocity vectors in two separate regions |
| 166 |
|
of a simulation system with respective diagonal scaling matricies. To |
| 167 |
|
determine these scaling factors in the matricies, a set of equations |
| 168 |
|
including linear momentum conservation and kinetic energy conservation |
| 169 |
< |
constraints and target momentum/energy flux satisfaction is |
| 170 |
< |
solved. With the scaling operation applied to the system in a set |
| 171 |
< |
frequency, corresponding momentum/temperature gradients can be built, |
| 172 |
< |
which can be used for computing transportation properties and other |
| 173 |
< |
applications related to momentum/temperature gradients. The NIVS |
| 132 |
< |
algorithm conserves momenta and energy and does not depend on an |
| 133 |
< |
external thermostat. |
| 169 |
> |
constraints and target energy flux satisfaction is solved. With the |
| 170 |
> |
scaling operation applied to the system in a set frequency, bulk |
| 171 |
> |
temperature gradients can be easily established, and these can be used |
| 172 |
> |
for computing thermal conductivities. The NIVS algorithm conserves |
| 173 |
> |
momenta and energy and does not depend on an external thermostat. |
| 174 |
|
|
| 175 |
< |
\subsection{Defining Interfacial Thermal Conductivity $G$} |
| 176 |
< |
For interfaces with a relatively low interfacial conductance, the bulk |
| 177 |
< |
regions on either side of an interface rapidly come to a state in |
| 178 |
< |
which the two phases have relatively homogeneous (but distinct) |
| 179 |
< |
temperatures. The interfacial thermal conductivity $G$ can therefore |
| 180 |
< |
be approximated as: |
| 175 |
> |
\subsection{Defining Interfacial Thermal Conductivity ($G$)} |
| 176 |
> |
|
| 177 |
> |
For an interface with relatively low interfacial conductance, and a |
| 178 |
> |
thermal flux between two distinct bulk regions, the regions on either |
| 179 |
> |
side of the interface rapidly come to a state in which the two phases |
| 180 |
> |
have relatively homogeneous (but distinct) temperatures. The |
| 181 |
> |
interfacial thermal conductivity $G$ can therefore be approximated as: |
| 182 |
|
\begin{equation} |
| 183 |
< |
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle - |
| 183 |
> |
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle - |
| 184 |
|
\langle T_\mathrm{cold}\rangle \right)} |
| 185 |
|
\label{lowG} |
| 186 |
|
\end{equation} |
| 187 |
< |
where ${E_{total}}$ is the imposed non-physical kinetic energy |
| 188 |
< |
transfer and ${\langle T_\mathrm{hot}\rangle}$ and ${\langle |
| 189 |
< |
T_\mathrm{cold}\rangle}$ are the average observed temperature of the |
| 190 |
< |
two separated phases. |
| 187 |
> |
where ${E_{total}}$ is the total imposed non-physical kinetic energy |
| 188 |
> |
transfer during the simulation and ${\langle T_\mathrm{hot}\rangle}$ |
| 189 |
> |
and ${\langle T_\mathrm{cold}\rangle}$ are the average observed |
| 190 |
> |
temperature of the two separated phases. |
| 191 |
|
|
| 192 |
< |
When the interfacial conductance is {\it not} small, two ways can be |
| 193 |
< |
used to define $G$. |
| 194 |
< |
|
| 195 |
< |
One way is to assume the temperature is discretely different on two |
| 196 |
< |
sides of the interface, $G$ can be calculated with the thermal flux |
| 197 |
< |
applied $J$ and the maximum temperature difference measured along the |
| 157 |
< |
thermal gradient max($\Delta T$), which occurs at the interface, as: |
| 192 |
> |
When the interfacial conductance is {\it not} small, there are two |
| 193 |
> |
ways to define $G$. One common way is to assume the temperature is |
| 194 |
> |
discrete on the two sides of the interface. $G$ can be calculated |
| 195 |
> |
using the applied thermal flux $J$ and the maximum temperature |
| 196 |
> |
difference measured along the thermal gradient max($\Delta T$), which |
| 197 |
> |
occurs at the Gibbs deviding surface (Figure \ref{demoPic}): |
| 198 |
|
\begin{equation} |
| 199 |
< |
G=\frac{J}{\Delta T} |
| 199 |
> |
G=\frac{J}{\Delta T} |
| 200 |
|
\label{discreteG} |
| 201 |
|
\end{equation} |
| 202 |
|
|
| 203 |
+ |
\begin{figure} |
| 204 |
+ |
\includegraphics[width=\linewidth]{method} |
| 205 |
+ |
\caption{Interfacial conductance can be calculated by applying an |
| 206 |
+ |
(unphysical) kinetic energy flux between two slabs, one located |
| 207 |
+ |
within the metal and another on the edge of the periodic box. The |
| 208 |
+ |
system responds by forming a thermal response or a gradient. In |
| 209 |
+ |
bulk liquids, this gradient typically has a single slope, but in |
| 210 |
+ |
interfacial systems, there are distinct thermal conductivity |
| 211 |
+ |
domains. The interfacial conductance, $G$ is found by measuring the |
| 212 |
+ |
temperature gap at the Gibbs dividing surface, or by using second |
| 213 |
+ |
derivatives of the thermal profile.} |
| 214 |
+ |
\label{demoPic} |
| 215 |
+ |
\end{figure} |
| 216 |
+ |
|
| 217 |
|
The other approach is to assume a continuous temperature profile along |
| 218 |
|
the thermal gradient axis (e.g. $z$) and define $G$ at the point where |
| 219 |
< |
the magnitude of thermal conductivity $\lambda$ change reach its |
| 219 |
> |
the magnitude of thermal conductivity ($\lambda$) change reaches its |
| 220 |
|
maximum, given that $\lambda$ is well-defined throughout the space: |
| 221 |
|
\begin{equation} |
| 222 |
|
G^\prime = \Big|\frac{\partial\lambda}{\partial z}\Big| |
| 227 |
|
\label{derivativeG} |
| 228 |
|
\end{equation} |
| 229 |
|
|
| 230 |
< |
With the temperature profile obtained from simulations, one is able to |
| 230 |
> |
With temperature profiles obtained from simulation, one is able to |
| 231 |
|
approximate the first and second derivatives of $T$ with finite |
| 232 |
< |
difference method and thus calculate $G^\prime$. |
| 232 |
> |
difference methods and calculate $G^\prime$. In what follows, both |
| 233 |
> |
definitions have been used, and are compared in the results. |
| 234 |
|
|
| 235 |
< |
In what follows, both definitions are used for calculation and comparison. |
| 235 |
> |
To investigate the interfacial conductivity at metal / solvent |
| 236 |
> |
interfaces, we have modeled a metal slab with its (111) surfaces |
| 237 |
> |
perpendicular to the $z$-axis of our simulation cells. The metal slab |
| 238 |
> |
has been prepared both with and without capping agents on the exposed |
| 239 |
> |
surface, and has been solvated with simple organic solvents, as |
| 240 |
> |
illustrated in Figure \ref{gradT}. |
| 241 |
|
|
| 242 |
< |
[IMPOSE G DEFINITION INTO OUR SYSTEMS] |
| 243 |
< |
To facilitate the use of the above definitions in calculating $G$ and |
| 244 |
< |
$G^\prime$, we have a metal slab with its (111) surfaces perpendicular |
| 245 |
< |
to the $z$-axis of our simulation cells. With or withour capping |
| 246 |
< |
agents on the surfaces, the metal slab is solvated with organic |
| 247 |
< |
solvents, as illustrated in Figure \ref{demoPic}. |
| 242 |
> |
With the simulation cell described above, we are able to equilibrate |
| 243 |
> |
the system and impose an unphysical thermal flux between the liquid |
| 244 |
> |
and the metal phase using the NIVS algorithm. By periodically applying |
| 245 |
> |
the unphysical flux, we obtained a temperature profile and its spatial |
| 246 |
> |
derivatives. Figure \ref{gradT} shows how an applied thermal flux can |
| 247 |
> |
be used to obtain the 1st and 2nd derivatives of the temperature |
| 248 |
> |
profile. |
| 249 |
|
|
| 250 |
|
\begin{figure} |
| 190 |
– |
\includegraphics[width=\linewidth]{demoPic} |
| 191 |
– |
\caption{A sample showing how a metal slab has its (111) surface |
| 192 |
– |
covered by capping agent molecules and solvated by hexane.} |
| 193 |
– |
\label{demoPic} |
| 194 |
– |
\end{figure} |
| 195 |
– |
|
| 196 |
– |
With a simulation cell setup following the above manner, one is able |
| 197 |
– |
to equilibrate the system and impose an unphysical thermal flux |
| 198 |
– |
between the liquid and the metal phase with the NIVS algorithm. Under |
| 199 |
– |
a stablized thermal gradient induced by periodically applying the |
| 200 |
– |
unphysical flux, one is able to obtain a temperature profile and the |
| 201 |
– |
physical thermal flux corresponding to it, which equals to the |
| 202 |
– |
unphysical flux applied by NIVS. These data enables the evaluation of |
| 203 |
– |
the interfacial thermal conductance of a surface. Figure \ref{gradT} |
| 204 |
– |
is an example how those stablized thermal gradient can be used to |
| 205 |
– |
obtain the 1st and 2nd derivatives of the temperature profile. |
| 206 |
– |
|
| 207 |
– |
\begin{figure} |
| 251 |
|
\includegraphics[width=\linewidth]{gradT} |
| 252 |
< |
\caption{The 1st and 2nd derivatives of temperature profile can be |
| 253 |
< |
obtained with finite difference approximation.} |
| 252 |
> |
\caption{A sample of Au-butanethiol/hexane interfacial system and the |
| 253 |
> |
temperature profile after a kinetic energy flux is imposed to |
| 254 |
> |
it. The 1st and 2nd derivatives of the temperature profile can be |
| 255 |
> |
obtained with finite difference approximation (lower panel).} |
| 256 |
|
\label{gradT} |
| 257 |
|
\end{figure} |
| 258 |
|
|
| 214 |
– |
[MAY INCLUDE POWER SPECTRUM PROTOCOL] |
| 215 |
– |
|
| 259 |
|
\section{Computational Details} |
| 260 |
|
\subsection{Simulation Protocol} |
| 261 |
< |
In our simulations, Au is used to construct a metal slab with bare |
| 262 |
< |
(111) surface perpendicular to the $z$-axis. Different slab thickness |
| 263 |
< |
(layer numbers of Au) are simulated. This metal slab is first |
| 264 |
< |
equilibrated under normal pressure (1 atm) and a desired |
| 265 |
< |
temperature. After equilibration, butanethiol is used as the capping |
| 266 |
< |
agent molecule to cover the bare Au (111) surfaces evenly. The sulfur |
| 267 |
< |
atoms in the butanethiol molecules would occupy the three-fold sites |
| 268 |
< |
of the surfaces, and the maximal butanethiol capacity on Au surface is |
| 269 |
< |
$1/3$ of the total number of surface Au atoms[CITATION]. A series of |
| 270 |
< |
different coverage surfaces is investigated in order to study the |
| 271 |
< |
relation between coverage and conductance. |
| 272 |
< |
|
| 273 |
< |
[COVERAGE DISCRIPTION] However, since the interactions between surface |
| 274 |
< |
Au and butanethiol is non-bonded, the capping agent molecules are |
| 275 |
< |
allowed to migrate to an empty neighbor three-fold site during a |
| 276 |
< |
simulation. Therefore, the initial configuration would not severely |
| 234 |
< |
affect the sampling of a variety of configurations of the same |
| 235 |
< |
coverage, and the final conductance measurement would be an average |
| 236 |
< |
effect of these configurations explored in the simulations. [MAY NEED FIGURES] |
| 261 |
> |
The NIVS algorithm has been implemented in our MD simulation code, |
| 262 |
> |
OpenMD\cite{Meineke:2005gd,openmd}, and was used for our simulations. |
| 263 |
> |
Metal slabs of 6 or 11 layers of Au atoms were first equilibrated |
| 264 |
> |
under atmospheric pressure (1 atm) and 200K. After equilibration, |
| 265 |
> |
butanethiol capping agents were placed at three-fold hollow sites on |
| 266 |
> |
the Au(111) surfaces. These sites are either {\it fcc} or {\it |
| 267 |
> |
hcp} sites, although Hase {\it et al.} found that they are |
| 268 |
> |
equivalent in a heat transfer process,\cite{hase:2010} so we did not |
| 269 |
> |
distinguish between these sites in our study. The maximum butanethiol |
| 270 |
> |
capacity on Au surface is $1/3$ of the total number of surface Au |
| 271 |
> |
atoms, and the packing forms a $(\sqrt{3}\times\sqrt{3})R30^\circ$ |
| 272 |
> |
structure\cite{doi:10.1021/ja00008a001,doi:10.1021/cr9801317}. A |
| 273 |
> |
series of lower coverages was also prepared by eliminating |
| 274 |
> |
butanethiols from the higher coverage surface in a regular manner. The |
| 275 |
> |
lower coverages were prepared in order to study the relation between |
| 276 |
> |
coverage and interfacial conductance. |
| 277 |
|
|
| 278 |
< |
After the modified Au-butanethiol surface systems are equilibrated |
| 279 |
< |
under canonical ensemble, Packmol\cite{packmol} is used to pack |
| 280 |
< |
organic solvent molecules in the previously vacuum part of the |
| 281 |
< |
simulation cells, which guarantees that short range repulsive |
| 282 |
< |
interactions do not disrupt the simulations. Two solvents are |
| 283 |
< |
investigated, one which has little vibrational overlap with the |
| 284 |
< |
alkanethiol and plane-like shape (toluene), and one which has similar |
| 245 |
< |
vibrational frequencies and chain-like shape ({\it n}-hexane). [MAY |
| 246 |
< |
EXPLAIN WHY WE CHOOSE THEM] |
| 278 |
> |
The capping agent molecules were allowed to migrate during the |
| 279 |
> |
simulations. They distributed themselves uniformly and sampled a |
| 280 |
> |
number of three-fold sites throughout out study. Therefore, the |
| 281 |
> |
initial configuration does not noticeably affect the sampling of a |
| 282 |
> |
variety of configurations of the same coverage, and the final |
| 283 |
> |
conductance measurement would be an average effect of these |
| 284 |
> |
configurations explored in the simulations. |
| 285 |
|
|
| 286 |
< |
The spacing filled by solvent molecules, i.e. the gap between |
| 287 |
< |
periodically repeated Au-butanethiol surfaces should be carefully |
| 288 |
< |
chosen. A very long length scale for the thermal gradient axis ($z$) |
| 289 |
< |
may cause excessively hot or cold temperatures in the middle of the |
| 286 |
> |
After the modified Au-butanethiol surface systems were equilibrated in |
| 287 |
> |
the canonical (NVT) ensemble, organic solvent molecules were packed in |
| 288 |
> |
the previously empty part of the simulation cells.\cite{packmol} Two |
| 289 |
> |
solvents were investigated, one which has little vibrational overlap |
| 290 |
> |
with the alkanethiol and which has a planar shape (toluene), and one |
| 291 |
> |
which has similar vibrational frequencies to the capping agent and |
| 292 |
> |
chain-like shape ({\it n}-hexane). |
| 293 |
> |
|
| 294 |
> |
The simulation cells were not particularly extensive along the |
| 295 |
> |
$z$-axis, as a very long length scale for the thermal gradient may |
| 296 |
> |
cause excessively hot or cold temperatures in the middle of the |
| 297 |
|
solvent region and lead to undesired phenomena such as solvent boiling |
| 298 |
|
or freezing when a thermal flux is applied. Conversely, too few |
| 299 |
|
solvent molecules would change the normal behavior of the liquid |
| 300 |
|
phase. Therefore, our $N_{solvent}$ values were chosen to ensure that |
| 301 |
< |
these extreme cases did not happen to our simulations. And the |
| 302 |
< |
corresponding spacing is usually $35 \sim 60$\AA. |
| 301 |
> |
these extreme cases did not happen to our simulations. The spacing |
| 302 |
> |
between periodic images of the gold interfaces is $45 \sim 75$\AA. |
| 303 |
|
|
| 304 |
< |
The initial configurations generated by Packmol are further |
| 305 |
< |
equilibrated with the $x$ and $y$ dimensions fixed, only allowing |
| 306 |
< |
length scale change in $z$ dimension. This is to ensure that the |
| 307 |
< |
equilibration of liquid phase does not affect the metal crystal |
| 308 |
< |
structure in $x$ and $y$ dimensions. Further equilibration are run |
| 309 |
< |
under NVT and then NVE ensembles. |
| 304 |
> |
The initial configurations generated are further equilibrated with the |
| 305 |
> |
$x$ and $y$ dimensions fixed, only allowing the $z$-length scale to |
| 306 |
> |
change. This is to ensure that the equilibration of liquid phase does |
| 307 |
> |
not affect the metal's crystalline structure. Comparisons were made |
| 308 |
> |
with simulations that allowed changes of $L_x$ and $L_y$ during NPT |
| 309 |
> |
equilibration. No substantial changes in the box geometry were noticed |
| 310 |
> |
in these simulations. After ensuring the liquid phase reaches |
| 311 |
> |
equilibrium at atmospheric pressure (1 atm), further equilibration was |
| 312 |
> |
carried out under canonical (NVT) and microcanonical (NVE) ensembles. |
| 313 |
|
|
| 314 |
< |
After the systems reach equilibrium, NIVS is implemented to impose a |
| 315 |
< |
periodic unphysical thermal flux between the metal and the liquid |
| 316 |
< |
phase. Most of our simulations are under an average temperature of |
| 317 |
< |
$\sim$200K. Therefore, this flux usually comes from the metal to the |
| 314 |
> |
After the systems reach equilibrium, NIVS was used to impose an |
| 315 |
> |
unphysical thermal flux between the metal and the liquid phases. Most |
| 316 |
> |
of our simulations were done under an average temperature of |
| 317 |
> |
$\sim$200K. Therefore, thermal flux usually came from the metal to the |
| 318 |
|
liquid so that the liquid has a higher temperature and would not |
| 319 |
< |
freeze due to excessively low temperature. This induced temperature |
| 320 |
< |
gradient is stablized and the simulation cell is devided evenly into |
| 321 |
< |
N slabs along the $z$-axis and the temperatures of each slab are |
| 322 |
< |
recorded. When the slab width $d$ of each slab is the same, the |
| 323 |
< |
derivatives of $T$ with respect to slab number $n$ can be directly |
| 324 |
< |
used for $G^\prime$ calculations: |
| 325 |
< |
\begin{equation} |
| 326 |
< |
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
| 319 |
> |
freeze due to lowered temperatures. After this induced temperature |
| 320 |
> |
gradient had stablized, the temperature profile of the simulation cell |
| 321 |
> |
was recorded. To do this, the simulation cell is devided evenly into |
| 322 |
> |
$N$ slabs along the $z$-axis. The average temperatures of each slab |
| 323 |
> |
are recorded for 1$\sim$2 ns. When the slab width $d$ of each slab is |
| 324 |
> |
the same, the derivatives of $T$ with respect to slab number $n$ can |
| 325 |
> |
be directly used for $G^\prime$ calculations: \begin{equation} |
| 326 |
> |
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
| 327 |
|
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
| 328 |
|
= |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big| |
| 329 |
|
\Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2 |
| 332 |
|
\label{derivativeG2} |
| 333 |
|
\end{equation} |
| 334 |
|
|
| 335 |
+ |
All of the above simulation procedures use a time step of 1 fs. Each |
| 336 |
+ |
equilibration stage took a minimum of 100 ps, although in some cases, |
| 337 |
+ |
longer equilibration stages were utilized. |
| 338 |
+ |
|
| 339 |
|
\subsection{Force Field Parameters} |
| 340 |
< |
Our simulations include various components. Therefore, force field |
| 341 |
< |
parameter descriptions are needed for interactions both between the |
| 342 |
< |
same type of particles and between particles of different species. |
| 340 |
> |
Our simulations include a number of chemically distinct components. |
| 341 |
> |
Figure \ref{demoMol} demonstrates the sites defined for both |
| 342 |
> |
United-Atom and All-Atom models of the organic solvent and capping |
| 343 |
> |
agents in our simulations. Force field parameters are needed for |
| 344 |
> |
interactions both between the same type of particles and between |
| 345 |
> |
particles of different species. |
| 346 |
|
|
| 347 |
+ |
\begin{figure} |
| 348 |
+ |
\includegraphics[width=\linewidth]{structures} |
| 349 |
+ |
\caption{Structures of the capping agent and solvents utilized in |
| 350 |
+ |
these simulations. The chemically-distinct sites (a-e) are expanded |
| 351 |
+ |
in terms of constituent atoms for both United Atom (UA) and All Atom |
| 352 |
+ |
(AA) force fields. Most parameters are from |
| 353 |
+ |
Refs. \protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes,TraPPE-UA.thiols} (UA) and \protect\cite{OPLSAA} (AA). Cross-interactions with the Au atoms are given in Table \ref{MnM}.} |
| 354 |
+ |
\label{demoMol} |
| 355 |
+ |
\end{figure} |
| 356 |
+ |
|
| 357 |
|
The Au-Au interactions in metal lattice slab is described by the |
| 358 |
|
quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
| 359 |
|
potentials include zero-point quantum corrections and are |
| 360 |
|
reparametrized for accurate surface energies compared to the |
| 361 |
< |
Sutton-Chen potentials\cite{Chen90}. |
| 361 |
> |
Sutton-Chen potentials.\cite{Chen90} |
| 362 |
|
|
| 363 |
< |
Figure [REF] demonstrates how we name our pseudo-atoms of the |
| 364 |
< |
molecules in our simulations. |
| 365 |
< |
[FIGURE FOR MOLECULE NOMENCLATURE] |
| 301 |
< |
|
| 302 |
< |
For both solvent molecules, straight chain {\it n}-hexane and aromatic |
| 303 |
< |
toluene, United-Atom (UA) and All-Atom (AA) models are used |
| 304 |
< |
respectively. The TraPPE-UA |
| 363 |
> |
For the two solvent molecules, {\it n}-hexane and toluene, two |
| 364 |
> |
different atomistic models were utilized. Both solvents were modeled |
| 365 |
> |
using United-Atom (UA) and All-Atom (AA) models. The TraPPE-UA |
| 366 |
|
parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
| 367 |
< |
for our UA solvent molecules. In these models, pseudo-atoms are |
| 368 |
< |
located at the carbon centers for alkyl groups. By eliminating |
| 369 |
< |
explicit hydrogen atoms, these models are simple and computationally |
| 370 |
< |
efficient, while maintains good accuracy. However, the TraPPE-UA for |
| 371 |
< |
alkanes is known to predict a lower boiling point than experimental |
| 311 |
< |
values. Considering that after an unphysical thermal flux is applied |
| 312 |
< |
to a system, the temperature of ``hot'' area in the liquid phase would be |
| 313 |
< |
significantly higher than the average, to prevent over heating and |
| 314 |
< |
boiling of the liquid phase, the average temperature in our |
| 315 |
< |
simulations should be much lower than the liquid boiling point. [MORE DISCUSSION] |
| 316 |
< |
For UA-toluene model, rigid body constraints are applied, so that the |
| 317 |
< |
benzene ring and the methyl-CRar bond are kept rigid. This would save |
| 318 |
< |
computational time.[MORE DETAILS] |
| 367 |
> |
for our UA solvent molecules. In these models, sites are located at |
| 368 |
> |
the carbon centers for alkyl groups. Bonding interactions, including |
| 369 |
> |
bond stretches and bends and torsions, were used for intra-molecular |
| 370 |
> |
sites closer than 3 bonds. For non-bonded interactions, Lennard-Jones |
| 371 |
> |
potentials are used. |
| 372 |
|
|
| 373 |
+ |
By eliminating explicit hydrogen atoms, the TraPPE-UA models are |
| 374 |
+ |
simple and computationally efficient, while maintaining good accuracy. |
| 375 |
+ |
However, the TraPPE-UA model for alkanes is known to predict a slighly |
| 376 |
+ |
lower boiling point than experimental values. This is one of the |
| 377 |
+ |
reasons we used a lower average temperature (200K) for our |
| 378 |
+ |
simulations. If heat is transferred to the liquid phase during the |
| 379 |
+ |
NIVS simulation, the liquid in the hot slab can actually be |
| 380 |
+ |
substantially warmer than the mean temperature in the simulation. The |
| 381 |
+ |
lower mean temperatures therefore prevent solvent boiling. |
| 382 |
+ |
|
| 383 |
+ |
For UA-toluene, the non-bonded potentials between intermolecular sites |
| 384 |
+ |
have a similar Lennard-Jones formulation. The toluene molecules were |
| 385 |
+ |
treated as a single rigid body, so there was no need for |
| 386 |
+ |
intramolecular interactions (including bonds, bends, or torsions) in |
| 387 |
+ |
this solvent model. |
| 388 |
+ |
|
| 389 |
|
Besides the TraPPE-UA models, AA models for both organic solvents are |
| 390 |
< |
included in our studies as well. For hexane, the OPLS-AA\cite{OPLSAA} |
| 391 |
< |
force field is used. [MORE DETAILS] |
| 392 |
< |
For toluene, the United Force Field developed by Rapp\'{e} {\it et |
| 393 |
< |
al.}\cite{doi:10.1021/ja00051a040} is adopted.[MORE DETAILS] |
| 390 |
> |
included in our studies as well. The OPLS-AA\cite{OPLSAA} force fields |
| 391 |
> |
were used. For hexane, additional explicit hydrogen sites were |
| 392 |
> |
included. Besides bonding and non-bonded site-site interactions, |
| 393 |
> |
partial charges and the electrostatic interactions were added to each |
| 394 |
> |
CT and HC site. For toluene, a flexible model for the toluene molecule |
| 395 |
> |
was utilized which included bond, bend, torsion, and inversion |
| 396 |
> |
potentials to enforce ring planarity. |
| 397 |
|
|
| 398 |
< |
The capping agent in our simulations, the butanethiol molecules can |
| 399 |
< |
either use UA or AA model. The TraPPE-UA force fields includes |
| 398 |
> |
The butanethiol capping agent in our simulations, were also modeled |
| 399 |
> |
with both UA and AA model. The TraPPE-UA force field includes |
| 400 |
|
parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for |
| 401 |
|
UA butanethiol model in our simulations. The OPLS-AA also provides |
| 402 |
|
parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) |
| 403 |
< |
surfaces do not have the hydrogen atom bonded to sulfur. To adapt this |
| 404 |
< |
change and derive suitable parameters for butanethiol adsorbed on |
| 405 |
< |
Au(111) surfaces, we adopt the S parameters from [CITATION CF VLUGT] |
| 406 |
< |
and modify parameters for its neighbor C atom for charge balance in |
| 407 |
< |
the molecule. Note that the model choice (UA or AA) of capping agent |
| 408 |
< |
can be different from the solvent. Regardless of model choice, the |
| 409 |
< |
force field parameters for interactions between capping agent and |
| 410 |
< |
solvent can be derived using Lorentz-Berthelot Mixing Rule: |
| 403 |
> |
surfaces do not have the hydrogen atom bonded to sulfur. To derive |
| 404 |
> |
suitable parameters for butanethiol adsorbed on Au(111) surfaces, we |
| 405 |
> |
adopt the S parameters from Luedtke and Landman\cite{landman:1998} and |
| 406 |
> |
modify the parameters for the CTS atom to maintain charge neutrality |
| 407 |
> |
in the molecule. Note that the model choice (UA or AA) for the capping |
| 408 |
> |
agent can be different from the solvent. Regardless of model choice, |
| 409 |
> |
the force field parameters for interactions between capping agent and |
| 410 |
> |
solvent can be derived using Lorentz-Berthelot Mixing Rule: |
| 411 |
> |
\begin{eqnarray} |
| 412 |
> |
\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\ |
| 413 |
> |
\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}} |
| 414 |
> |
\end{eqnarray} |
| 415 |
|
|
| 416 |
+ |
To describe the interactions between metal (Au) and non-metal atoms, |
| 417 |
+ |
we refer to an adsorption study of alkyl thiols on gold surfaces by |
| 418 |
+ |
Vlugt {\it et al.}\cite{vlugt:cpc2007154} They fitted an effective |
| 419 |
+ |
Lennard-Jones form of potential parameters for the interaction between |
| 420 |
+ |
Au and pseudo-atoms CH$_x$ and S based on a well-established and |
| 421 |
+ |
widely-used effective potential of Hautman and Klein for the Au(111) |
| 422 |
+ |
surface.\cite{hautman:4994} As our simulations require the gold slab |
| 423 |
+ |
to be flexible to accommodate thermal excitation, the pair-wise form |
| 424 |
+ |
of potentials they developed was used for our study. |
| 425 |
|
|
| 426 |
< |
To describe the interactions between metal Au and non-metal capping |
| 427 |
< |
agent and solvent particles, we refer to an adsorption study of alkyl |
| 428 |
< |
thiols on gold surfaces by Vlugt {\it et |
| 429 |
< |
al.}\cite{vlugt:cpc2007154} They fitted an effective Lennard-Jones |
| 430 |
< |
form of potential parameters for the interaction between Au and |
| 431 |
< |
pseudo-atoms CH$_x$ and S based on a well-established and widely-used |
| 432 |
< |
effective potential of Hautman and Klein[CITATION] for the Au(111) |
| 433 |
< |
surface. As our simulations require the gold lattice slab to be |
| 434 |
< |
non-rigid so that it could accommodate kinetic energy for thermal |
| 435 |
< |
transport study purpose, the pair-wise form of potentials is |
| 436 |
< |
preferred. |
| 426 |
> |
The potentials developed from {\it ab initio} calculations by Leng |
| 427 |
> |
{\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
| 428 |
> |
interactions between Au and aromatic C/H atoms in toluene. However, |
| 429 |
> |
the Lennard-Jones parameters between Au and other types of particles, |
| 430 |
> |
(e.g. AA alkanes) have not yet been established. For these |
| 431 |
> |
interactions, the Lorentz-Berthelot mixing rule can be used to derive |
| 432 |
> |
effective single-atom LJ parameters for the metal using the fit values |
| 433 |
> |
for toluene. These are then used to construct reasonable mixing |
| 434 |
> |
parameters for the interactions between the gold and other atoms. |
| 435 |
> |
Table \ref{MnM} summarizes the ``metal/non-metal'' parameters used in |
| 436 |
> |
our simulations. |
| 437 |
|
|
| 353 |
– |
Besides, the potentials developed from {\it ab initio} calculations by |
| 354 |
– |
Leng {\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
| 355 |
– |
interactions between Au and aromatic C/H atoms in toluene.[MORE DETAILS] |
| 356 |
– |
|
| 357 |
– |
However, the Lennard-Jones parameters between Au and other types of |
| 358 |
– |
particles in our simulations are not yet well-established. For these |
| 359 |
– |
interactions, we attempt to derive their parameters using the Mixing |
| 360 |
– |
Rule. To do this, the ``Metal-non-Metal'' (MnM) interaction parameters |
| 361 |
– |
for Au is first extracted from the Au-CH$_x$ parameters by applying |
| 362 |
– |
the Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM'' |
| 363 |
– |
parameters in our simulations. |
| 364 |
– |
|
| 438 |
|
\begin{table*} |
| 439 |
|
\begin{minipage}{\linewidth} |
| 440 |
|
\begin{center} |
| 441 |
< |
\caption{Lennard-Jones parameters for Au-non-Metal |
| 442 |
< |
interactions in our simulations.} |
| 443 |
< |
|
| 444 |
< |
\begin{tabular}{ccc} |
| 441 |
> |
\caption{Non-bonded interaction parameters (including cross |
| 442 |
> |
interactions with Au atoms) for both force fields used in this |
| 443 |
> |
work.} |
| 444 |
> |
\begin{tabular}{lllllll} |
| 445 |
|
\hline\hline |
| 446 |
< |
Non-metal & $\sigma$/\AA & $\epsilon$/kcal/mol \\ |
| 446 |
> |
& Site & $\sigma_{ii}$ & $\epsilon_{ii}$ & $q_i$ & |
| 447 |
> |
$\sigma_{Au-i}$ & $\epsilon_{Au-i}$ \\ |
| 448 |
> |
& & (\AA) & (kcal/mol) & ($e$) & (\AA) & (kcal/mol) \\ |
| 449 |
|
\hline |
| 450 |
< |
S & 2.40 & 8.465 \\ |
| 451 |
< |
CH3 & 3.54 & 0.2146 \\ |
| 452 |
< |
CH2 & 3.54 & 0.1749 \\ |
| 453 |
< |
CT3 & 3.365 & 0.1373 \\ |
| 454 |
< |
CT2 & 3.365 & 0.1373 \\ |
| 455 |
< |
CTT & 3.365 & 0.1373 \\ |
| 456 |
< |
HC & 2.865 & 0.09256 \\ |
| 457 |
< |
CHar & 3.4625 & 0.1680 \\ |
| 458 |
< |
CRar & 3.555 & 0.1604 \\ |
| 459 |
< |
CA & 3.173 & 0.0640 \\ |
| 460 |
< |
HA & 2.746 & 0.0414 \\ |
| 450 |
> |
United Atom (UA) |
| 451 |
> |
&CH3 & 3.75 & 0.1947 & - & 3.54 & 0.2146 \\ |
| 452 |
> |
&CH2 & 3.95 & 0.0914 & - & 3.54 & 0.1749 \\ |
| 453 |
> |
&CHar & 3.695 & 0.1003 & - & 3.4625 & 0.1680 \\ |
| 454 |
> |
&CRar & 3.88 & 0.04173 & - & 3.555 & 0.1604 \\ |
| 455 |
> |
\hline |
| 456 |
> |
All Atom (AA) |
| 457 |
> |
&CT3 & 3.50 & 0.066 & -0.18 & 3.365 & 0.1373 \\ |
| 458 |
> |
&CT2 & 3.50 & 0.066 & -0.12 & 3.365 & 0.1373 \\ |
| 459 |
> |
&CTT & 3.50 & 0.066 & -0.065 & 3.365 & 0.1373 \\ |
| 460 |
> |
&HC & 2.50 & 0.030 & 0.06 & 2.865 & 0.09256 \\ |
| 461 |
> |
&CA & 3.55 & 0.070 & -0.115 & 3.173 & 0.0640 \\ |
| 462 |
> |
&HA & 2.42 & 0.030 & 0.115 & 2.746 & 0.0414 \\ |
| 463 |
> |
\hline |
| 464 |
> |
Both UA and AA |
| 465 |
> |
& S & 4.45 & 0.25 & - & 2.40 & 8.465 \\ |
| 466 |
|
\hline\hline |
| 467 |
|
\end{tabular} |
| 468 |
|
\label{MnM} |
| 470 |
|
\end{minipage} |
| 471 |
|
\end{table*} |
| 472 |
|
|
| 473 |
+ |
\subsection{Vibrational Power Spectrum} |
| 474 |
|
|
| 475 |
+ |
To investigate the mechanism of interfacial thermal conductance, the |
| 476 |
+ |
vibrational power spectrum was computed. Power spectra were taken for |
| 477 |
+ |
individual components in different simulations. To obtain these |
| 478 |
+ |
spectra, simulations were run after equilibration, in the NVE |
| 479 |
+ |
ensemble, and without a thermal gradient. Snapshots of configurations |
| 480 |
+ |
were collected at a frequency that is higher than that of the fastest |
| 481 |
+ |
vibrations occuring in the simulations. With these configurations, the |
| 482 |
+ |
velocity auto-correlation functions can be computed: |
| 483 |
+ |
\begin{equation} |
| 484 |
+ |
C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle |
| 485 |
+ |
\label{vCorr} |
| 486 |
+ |
\end{equation} |
| 487 |
+ |
The power spectrum is constructed via a Fourier transform of the |
| 488 |
+ |
symmetrized velocity autocorrelation function, |
| 489 |
+ |
\begin{equation} |
| 490 |
+ |
\hat{f}(\omega) = |
| 491 |
+ |
\int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt |
| 492 |
+ |
\label{fourier} |
| 493 |
+ |
\end{equation} |
| 494 |
+ |
|
| 495 |
|
\section{Results and Discussions} |
| 496 |
< |
[MAY HAVE A BRIEF SUMMARY] |
| 496 |
> |
In what follows, how the parameters and protocol of simulations would |
| 497 |
> |
affect the measurement of $G$'s is first discussed. With a reliable |
| 498 |
> |
protocol and set of parameters, the influence of capping agent |
| 499 |
> |
coverage on thermal conductance is investigated. Besides, different |
| 500 |
> |
force field models for both solvents and selected deuterated models |
| 501 |
> |
were tested and compared. Finally, a summary of the role of capping |
| 502 |
> |
agent in the interfacial thermal transport process is given. |
| 503 |
> |
|
| 504 |
|
\subsection{How Simulation Parameters Affects $G$} |
| 397 |
– |
[MAY NOT PUT AT FIRST] |
| 505 |
|
We have varied our protocol or other parameters of the simulations in |
| 506 |
|
order to investigate how these factors would affect the measurement of |
| 507 |
|
$G$'s. It turned out that while some of these parameters would not |
| 510 |
|
results. |
| 511 |
|
|
| 512 |
|
In some of our simulations, we allowed $L_x$ and $L_y$ to change |
| 513 |
< |
during equilibrating the liquid phase. Due to the stiffness of the Au |
| 514 |
< |
slab, $L_x$ and $L_y$ would not change noticeably after |
| 515 |
< |
equilibration. Although $L_z$ could fluctuate $\sim$1\% after a system |
| 516 |
< |
is fully equilibrated in the NPT ensemble, this fluctuation, as well |
| 517 |
< |
as those comparably smaller to $L_x$ and $L_y$, would not be magnified |
| 518 |
< |
on the calculated $G$'s, as shown in Table \ref{AuThiolHexaneUA}. This |
| 519 |
< |
insensivity to $L_i$ fluctuations allows reliable measurement of $G$'s |
| 520 |
< |
without the necessity of extremely cautious equilibration process. |
| 513 |
> |
during equilibrating the liquid phase. Due to the stiffness of the |
| 514 |
> |
crystalline Au structure, $L_x$ and $L_y$ would not change noticeably |
| 515 |
> |
after equilibration. Although $L_z$ could fluctuate $\sim$1\% after a |
| 516 |
> |
system is fully equilibrated in the NPT ensemble, this fluctuation, as |
| 517 |
> |
well as those of $L_x$ and $L_y$ (which is significantly smaller), |
| 518 |
> |
would not be magnified on the calculated $G$'s, as shown in Table |
| 519 |
> |
\ref{AuThiolHexaneUA}. This insensivity to $L_i$ fluctuations allows |
| 520 |
> |
reliable measurement of $G$'s without the necessity of extremely |
| 521 |
> |
cautious equilibration process. |
| 522 |
|
|
| 523 |
|
As stated in our computational details, the spacing filled with |
| 524 |
|
solvent molecules can be chosen within a range. This allows some |
| 545 |
|
the thermal flux across the interface. For our simulations, we denote |
| 546 |
|
$J_z$ to be positive when the physical thermal flux is from the liquid |
| 547 |
|
to metal, and negative vice versa. The $G$'s measured under different |
| 548 |
< |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and [REF]. These |
| 549 |
< |
results do not suggest that $G$ is dependent on $J_z$ within this flux |
| 550 |
< |
range. The linear response of flux to thermal gradient simplifies our |
| 551 |
< |
investigations in that we can rely on $G$ measurement with only a |
| 552 |
< |
couple $J_z$'s and do not need to test a large series of fluxes. |
| 548 |
> |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and |
| 549 |
> |
\ref{AuThiolToluene}. These results do not suggest that $G$ is |
| 550 |
> |
dependent on $J_z$ within this flux range. The linear response of flux |
| 551 |
> |
to thermal gradient simplifies our investigations in that we can rely |
| 552 |
> |
on $G$ measurement with only a couple $J_z$'s and do not need to test |
| 553 |
> |
a large series of fluxes. |
| 554 |
|
|
| 446 |
– |
%ADD MORE TO TABLE |
| 555 |
|
\begin{table*} |
| 556 |
|
\begin{minipage}{\linewidth} |
| 557 |
|
\begin{center} |
| 558 |
|
\caption{Computed interfacial thermal conductivity ($G$ and |
| 559 |
< |
$G^\prime$) values for the Au/butanethiol/hexane interface |
| 560 |
< |
with united-atom model and different capping agent coverage |
| 561 |
< |
and solvent molecule numbers at different temperatures using a |
| 562 |
< |
range of energy fluxes.} |
| 559 |
> |
$G^\prime$) values for the 100\% covered Au-butanethiol/hexane |
| 560 |
> |
interfaces with UA model and different hexane molecule numbers |
| 561 |
> |
at different temperatures using a range of energy |
| 562 |
> |
fluxes. Error estimates indicated in parenthesis.} |
| 563 |
|
|
| 564 |
< |
\begin{tabular}{cccccc} |
| 564 |
> |
\begin{tabular}{ccccccc} |
| 565 |
|
\hline\hline |
| 566 |
< |
Thiol & $\langle T\rangle$ & & $J_z$ & $G$ & $G^\prime$ \\ |
| 567 |
< |
coverage (\%) & (K) & $N_{hexane}$ & (GW/m$^2$) & |
| 566 |
> |
$\langle T\rangle$ & $N_{hexane}$ & Fixed & $\rho_{hexane}$ & |
| 567 |
> |
$J_z$ & $G$ & $G^\prime$ \\ |
| 568 |
> |
(K) & & $L_x$ \& $L_y$? & (g/cm$^3$) & (GW/m$^2$) & |
| 569 |
|
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 570 |
|
\hline |
| 571 |
< |
0.0 & 200 & 200 & 0.96 & 43.3 & 42.7 \\ |
| 572 |
< |
& & & 1.91 & 45.7 & 42.9 \\ |
| 573 |
< |
& & 166 & 0.96 & 43.1 & 53.4 \\ |
| 574 |
< |
88.9 & 200 & 166 & 1.94 & 172 & 108 \\ |
| 575 |
< |
100.0 & 250 & 200 & 0.96 & 81.8 & 67.0 \\ |
| 576 |
< |
& & 166 & 0.98 & 79.0 & 62.9 \\ |
| 577 |
< |
& & & 1.44 & 76.2 & 64.8 \\ |
| 578 |
< |
& 200 & 200 & 1.92 & 129 & 87.3 \\ |
| 579 |
< |
& & & 1.93 & 131 & 77.5 \\ |
| 580 |
< |
& & 166 & 0.97 & 115 & 69.3 \\ |
| 581 |
< |
& & & 1.94 & 125 & 87.1 \\ |
| 571 |
> |
200 & 266 & No & 0.672 & -0.96 & 102(3) & 80.0(0.8) \\ |
| 572 |
> |
& 200 & Yes & 0.694 & 1.92 & 129(11) & 87.3(0.3) \\ |
| 573 |
> |
& & Yes & 0.672 & 1.93 & 131(16) & 78(13) \\ |
| 574 |
> |
& & No & 0.688 & 0.96 & 125(16) & 90.2(15) \\ |
| 575 |
> |
& & & & 1.91 & 139(10) & 101(10) \\ |
| 576 |
> |
& & & & 2.83 & 141(6) & 89.9(9.8) \\ |
| 577 |
> |
& 166 & Yes & 0.679 & 0.97 & 115(19) & 69(18) \\ |
| 578 |
> |
& & & & 1.94 & 125(9) & 87.1(0.2) \\ |
| 579 |
> |
& & No & 0.681 & 0.97 & 141(30) & 78(22) \\ |
| 580 |
> |
& & & & 1.92 & 138(4) & 98.9(9.5) \\ |
| 581 |
> |
\hline |
| 582 |
> |
250 & 200 & No & 0.560 & 0.96 & 75(10) & 61.8(7.3) \\ |
| 583 |
> |
& & & & -0.95 & 49.4(0.3) & 45.7(2.1) \\ |
| 584 |
> |
& 166 & Yes & 0.570 & 0.98 & 79.0(3.5) & 62.9(3.0) \\ |
| 585 |
> |
& & No & 0.569 & 0.97 & 80.3(0.6) & 67(11) \\ |
| 586 |
> |
& & & & 1.44 & 76.2(5.0) & 64.8(3.8) \\ |
| 587 |
> |
& & & & -0.95 & 56.4(2.5) & 54.4(1.1) \\ |
| 588 |
> |
& & & & -1.85 & 47.8(1.1) & 53.5(1.5) \\ |
| 589 |
|
\hline\hline |
| 590 |
|
\end{tabular} |
| 591 |
|
\label{AuThiolHexaneUA} |
| 601 |
|
temperature is higher than 250K. Additionally, the equilibrated liquid |
| 602 |
|
hexane density under 250K becomes lower than experimental value. This |
| 603 |
|
expanded liquid phase leads to lower contact between hexane and |
| 604 |
< |
butanethiol as well.[MAY NEED FIGURE] And this reduced contact would |
| 604 |
> |
butanethiol as well.[MAY NEED SLAB DENSITY FIGURE] |
| 605 |
> |
And this reduced contact would |
| 606 |
|
probably be accountable for a lower interfacial thermal conductance, |
| 607 |
|
as shown in Table \ref{AuThiolHexaneUA}. |
| 608 |
|
|
| 617 |
|
important role in the thermal transport process across the interface |
| 618 |
|
in that higher degree of contact could yield increased conductance. |
| 619 |
|
|
| 503 |
– |
[ADD SIGNS AND ERROR ESTIMATE TO TABLE] |
| 620 |
|
\begin{table*} |
| 621 |
|
\begin{minipage}{\linewidth} |
| 622 |
|
\begin{center} |
| 623 |
|
\caption{Computed interfacial thermal conductivity ($G$ and |
| 624 |
< |
$G^\prime$) values for the Au/butanethiol/toluene interface at |
| 625 |
< |
different temperatures using a range of energy fluxes.} |
| 624 |
> |
$G^\prime$) values for a 90\% coverage Au-butanethiol/toluene |
| 625 |
> |
interface at different temperatures using a range of energy |
| 626 |
> |
fluxes. Error estimates indicated in parenthesis.} |
| 627 |
|
|
| 628 |
< |
\begin{tabular}{cccc} |
| 628 |
> |
\begin{tabular}{ccccc} |
| 629 |
|
\hline\hline |
| 630 |
< |
$\langle T\rangle$ & $J_z$ & $G$ & $G^\prime$ \\ |
| 631 |
< |
(K) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 630 |
> |
$\langle T\rangle$ & $\rho_{toluene}$ & $J_z$ & $G$ & $G^\prime$ \\ |
| 631 |
> |
(K) & (g/cm$^3$) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 632 |
|
\hline |
| 633 |
< |
200 & 1.86 & 180 & 135 \\ |
| 634 |
< |
& 2.15 & 204 & 113 \\ |
| 635 |
< |
& 3.93 & 175 & 114 \\ |
| 636 |
< |
300 & 1.91 & 143 & 125 \\ |
| 637 |
< |
& 4.19 & 134 & 113 \\ |
| 633 |
> |
200 & 0.933 & 2.15 & 204(12) & 113(12) \\ |
| 634 |
> |
& & -1.86 & 180(3) & 135(21) \\ |
| 635 |
> |
& & -3.93 & 176(5) & 113(12) \\ |
| 636 |
> |
\hline |
| 637 |
> |
300 & 0.855 & -1.91 & 143(5) & 125(2) \\ |
| 638 |
> |
& & -4.19 & 135(9) & 113(12) \\ |
| 639 |
|
\hline\hline |
| 640 |
|
\end{tabular} |
| 641 |
|
\label{AuThiolToluene} |
| 663 |
|
reconstructions could eliminate the original $x$ and $y$ dimensional |
| 664 |
|
homogeneity, measurement of $G$ is more difficult to conduct under |
| 665 |
|
higher temperatures. Therefore, most of our measurements are |
| 666 |
< |
undertaken at $<T>\sim$200K. |
| 666 |
> |
undertaken at $\langle T\rangle\sim$200K. |
| 667 |
|
|
| 668 |
|
However, when the surface is not completely covered by butanethiols, |
| 669 |
|
the simulated system is more resistent to the reconstruction |
| 670 |
< |
above. Our Au-butanethiol/toluene system did not see this phenomena |
| 671 |
< |
even at $<T>\sim$300K. The Au(111) surfaces have a 90\% coverage of |
| 672 |
< |
butanethiols and have empty three-fold sites. These empty sites could |
| 673 |
< |
help prevent surface reconstruction in that they provide other means |
| 674 |
< |
of capping agent relaxation. It is observed that butanethiols can |
| 675 |
< |
migrate to their neighbor empty sites during a simulation. Therefore, |
| 676 |
< |
we were able to obtain $G$'s for these interfaces even at a relatively |
| 677 |
< |
high temperature without being affected by surface reconstructions. |
| 670 |
> |
above. Our Au-butanethiol/toluene system had the Au(111) surfaces 90\% |
| 671 |
> |
covered by butanethiols, but did not see this above phenomena even at |
| 672 |
> |
$\langle T\rangle\sim$300K. The empty three-fold sites not occupied by |
| 673 |
> |
capping agents could help prevent surface reconstruction in that they |
| 674 |
> |
provide other means of capping agent relaxation. It is observed that |
| 675 |
> |
butanethiols can migrate to their neighbor empty sites during a |
| 676 |
> |
simulation. Therefore, we were able to obtain $G$'s for these |
| 677 |
> |
interfaces even at a relatively high temperature without being |
| 678 |
> |
affected by surface reconstructions. |
| 679 |
|
|
| 680 |
|
\subsection{Influence of Capping Agent Coverage on $G$} |
| 681 |
|
To investigate the influence of butanethiol coverage on interfacial |
| 682 |
|
thermal conductance, a series of different coverage Au-butanethiol |
| 683 |
|
surfaces is prepared and solvated with various organic |
| 684 |
|
molecules. These systems are then equilibrated and their interfacial |
| 685 |
< |
thermal conductivity are measured with our NIVS algorithm. Table |
| 686 |
< |
\ref{tlnUhxnUhxnD} lists these results for direct comparison between |
| 687 |
< |
different coverages of butanethiol. |
| 685 |
> |
thermal conductivity are measured with our NIVS algorithm. Figure |
| 686 |
> |
\ref{coverage} demonstrates the trend of conductance change with |
| 687 |
> |
respect to different coverages of butanethiol. To study the isotope |
| 688 |
> |
effect in interfacial thermal conductance, deuterated UA-hexane is |
| 689 |
> |
included as well. |
| 690 |
|
|
| 691 |
< |
With high coverage of butanethiol on the gold surface, |
| 692 |
< |
the interfacial thermal conductance is enhanced |
| 693 |
< |
significantly. Interestingly, a slightly lower butanethiol coverage |
| 694 |
< |
leads to a moderately higher conductivity. This is probably due to |
| 695 |
< |
more solvent/capping agent contact when butanethiol molecules are |
| 696 |
< |
not densely packed, which enhances the interactions between the two |
| 697 |
< |
phases and lowers the thermal transfer barrier of this interface. |
| 698 |
< |
[COMPARE TO AU/WATER IN PAPER] |
| 691 |
> |
\begin{figure} |
| 692 |
> |
\includegraphics[width=\linewidth]{coverage} |
| 693 |
> |
\caption{Comparison of interfacial thermal conductivity ($G$) values |
| 694 |
> |
for the Au-butanethiol/solvent interface with various UA models and |
| 695 |
> |
different capping agent coverages at $\langle T\rangle\sim$200K |
| 696 |
> |
using certain energy flux respectively.} |
| 697 |
> |
\label{coverage} |
| 698 |
> |
\end{figure} |
| 699 |
|
|
| 700 |
+ |
It turned out that with partial covered butanethiol on the Au(111) |
| 701 |
+ |
surface, the derivative definition for $G^\prime$ |
| 702 |
+ |
(Eq. \ref{derivativeG}) was difficult to apply, due to the difficulty |
| 703 |
+ |
in locating the maximum of change of $\lambda$. Instead, the discrete |
| 704 |
+ |
definition (Eq. \ref{discreteG}) is easier to apply, as the Gibbs |
| 705 |
+ |
deviding surface can still be well-defined. Therefore, $G$ (not |
| 706 |
+ |
$G^\prime$) was used for this section. |
| 707 |
|
|
| 708 |
< |
significant conductance enhancement compared to the gold/water |
| 709 |
< |
interface without capping agent and agree with available experimental |
| 710 |
< |
data. This indicates that the metal-metal potential, though not |
| 711 |
< |
predicting an accurate bulk metal thermal conductivity, does not |
| 712 |
< |
greatly interfere with the simulation of the thermal conductance |
| 713 |
< |
behavior across a non-metal interface. |
| 586 |
< |
The results show that the two definitions used for $G$ yield |
| 587 |
< |
comparable values, though $G^\prime$ tends to be smaller. |
| 708 |
> |
From Figure \ref{coverage}, one can see the significance of the |
| 709 |
> |
presence of capping agents. Even when a fraction of the Au(111) |
| 710 |
> |
surface sites are covered with butanethiols, the conductivity would |
| 711 |
> |
see an enhancement by at least a factor of 3. This indicates the |
| 712 |
> |
important role cappping agent is playing for thermal transport |
| 713 |
> |
phenomena on metal / organic solvent surfaces. |
| 714 |
|
|
| 715 |
+ |
Interestingly, as one could observe from our results, the maximum |
| 716 |
+ |
conductance enhancement (largest $G$) happens while the surfaces are |
| 717 |
+ |
about 75\% covered with butanethiols. This again indicates that |
| 718 |
+ |
solvent-capping agent contact has an important role of the thermal |
| 719 |
+ |
transport process. Slightly lower butanethiol coverage allows small |
| 720 |
+ |
gaps between butanethiols to form. And these gaps could be filled with |
| 721 |
+ |
solvent molecules, which acts like ``heat conductors'' on the |
| 722 |
+ |
surface. The higher degree of interaction between these solvent |
| 723 |
+ |
molecules and capping agents increases the enhancement effect and thus |
| 724 |
+ |
produces a higher $G$ than densely packed butanethiol arrays. However, |
| 725 |
+ |
once this maximum conductance enhancement is reached, $G$ decreases |
| 726 |
+ |
when butanethiol coverage continues to decrease. Each capping agent |
| 727 |
+ |
molecule reaches its maximum capacity for thermal |
| 728 |
+ |
conductance. Therefore, even higher solvent-capping agent contact |
| 729 |
+ |
would not offset this effect. Eventually, when butanethiol coverage |
| 730 |
+ |
continues to decrease, solvent-capping agent contact actually |
| 731 |
+ |
decreases with the disappearing of butanethiol molecules. In this |
| 732 |
+ |
case, $G$ decrease could not be offset but instead accelerated. [MAY NEED |
| 733 |
+ |
SNAPSHOT SHOWING THE PHENOMENA / SLAB DENSITY ANALYSIS] |
| 734 |
|
|
| 735 |
< |
\begin{table*} |
| 736 |
< |
\begin{minipage}{\linewidth} |
| 737 |
< |
\begin{center} |
| 738 |
< |
\caption{Computed interfacial thermal conductivity ($G$ and |
| 739 |
< |
$G^\prime$) values for the Au/butanethiol/hexane interface |
| 740 |
< |
with united-atom model and different capping agent coverage |
| 741 |
< |
and solvent molecule numbers at different temperatures using a |
| 742 |
< |
range of energy fluxes.} |
| 743 |
< |
|
| 599 |
< |
\begin{tabular}{cccccc} |
| 600 |
< |
\hline\hline |
| 601 |
< |
Thiol & $\langle T\rangle$ & & $J_z$ & $G$ & $G^\prime$ \\ |
| 602 |
< |
coverage (\%) & (K) & $N_{hexane}$ & (GW/m$^2$) & |
| 603 |
< |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 604 |
< |
\hline |
| 605 |
< |
0.0 & 200 & 200 & 0.96 & 43.3 & 42.7 \\ |
| 606 |
< |
& & & 1.91 & 45.7 & 42.9 \\ |
| 607 |
< |
& & 166 & 0.96 & 43.1 & 53.4 \\ |
| 608 |
< |
88.9 & 200 & 166 & 1.94 & 172 & 108 \\ |
| 609 |
< |
100.0 & 250 & 200 & 0.96 & 81.8 & 67.0 \\ |
| 610 |
< |
& & 166 & 0.98 & 79.0 & 62.9 \\ |
| 611 |
< |
& & & 1.44 & 76.2 & 64.8 \\ |
| 612 |
< |
& 200 & 200 & 1.92 & 129 & 87.3 \\ |
| 613 |
< |
& & & 1.93 & 131 & 77.5 \\ |
| 614 |
< |
& & 166 & 0.97 & 115 & 69.3 \\ |
| 615 |
< |
& & & 1.94 & 125 & 87.1 \\ |
| 616 |
< |
\hline\hline |
| 617 |
< |
\end{tabular} |
| 618 |
< |
\label{tlnUhxnUhxnD} |
| 619 |
< |
\end{center} |
| 620 |
< |
\end{minipage} |
| 621 |
< |
\end{table*} |
| 735 |
> |
A comparison of the results obtained from differenet organic solvents |
| 736 |
> |
can also provide useful information of the interfacial thermal |
| 737 |
> |
transport process. The deuterated hexane (UA) results do not appear to |
| 738 |
> |
be much different from those of normal hexane (UA), given that |
| 739 |
> |
butanethiol (UA) is non-deuterated for both solvents. These UA model |
| 740 |
> |
studies, even though eliminating C-H vibration samplings, still have |
| 741 |
> |
C-C vibrational frequencies different from each other. However, these |
| 742 |
> |
differences in the infrared range do not seem to produce an observable |
| 743 |
> |
difference for the results of $G$ (Figure \ref{uahxnua}). |
| 744 |
|
|
| 745 |
< |
\subsection{Influence of Chosen Molecule Model on $G$} |
| 746 |
< |
[MAY COMBINE W MECHANISM STUDY] |
| 745 |
> |
\begin{figure} |
| 746 |
> |
\includegraphics[width=\linewidth]{uahxnua} |
| 747 |
> |
\caption{Vibrational spectra obtained for normal (upper) and |
| 748 |
> |
deuterated (lower) hexane in Au-butanethiol/hexane |
| 749 |
> |
systems. Butanethiol spectra are shown as reference. Both hexane and |
| 750 |
> |
butanethiol were using United-Atom models.} |
| 751 |
> |
\label{uahxnua} |
| 752 |
> |
\end{figure} |
| 753 |
|
|
| 754 |
< |
For the all-atom model, the liquid hexane phase was not stable under NPT |
| 755 |
< |
conditions. Therefore, the simulation length scale parameters are |
| 756 |
< |
adopted from previous equilibration results of the united-atom model |
| 757 |
< |
at 200K. Table \ref{AuThiolHexaneAA} shows the results of these |
| 758 |
< |
simulations. The conductivity values calculated with full capping |
| 759 |
< |
agent coverage are substantially larger than observed in the |
| 632 |
< |
united-atom model, and is even higher than predicted by |
| 633 |
< |
experiments. It is possible that our parameters for metal-non-metal |
| 634 |
< |
particle interactions lead to an overestimate of the interfacial |
| 635 |
< |
thermal conductivity, although the active C-H vibrations in the |
| 636 |
< |
all-atom model (which should not be appreciably populated at normal |
| 637 |
< |
temperatures) could also account for this high conductivity. The major |
| 638 |
< |
thermal transfer barrier of Au/butanethiol/hexane interface is between |
| 639 |
< |
the liquid phase and the capping agent, so extra degrees of freedom |
| 640 |
< |
such as the C-H vibrations could enhance heat exchange between these |
| 641 |
< |
two phases and result in a much higher conductivity. |
| 754 |
> |
Furthermore, results for rigid body toluene solvent, as well as other |
| 755 |
> |
UA-hexane solvents, are reasonable within the general experimental |
| 756 |
> |
ranges\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406}. This |
| 757 |
> |
suggests that explicit hydrogen might not be a required factor for |
| 758 |
> |
modeling thermal transport phenomena of systems such as |
| 759 |
> |
Au-thiol/organic solvent. |
| 760 |
|
|
| 761 |
+ |
However, results for Au-butanethiol/toluene do not show an identical |
| 762 |
+ |
trend with those for Au-butanethiol/hexane in that $G$ remains at |
| 763 |
+ |
approximately the same magnitue when butanethiol coverage differs from |
| 764 |
+ |
25\% to 75\%. This might be rooted in the molecule shape difference |
| 765 |
+ |
for planar toluene and chain-like {\it n}-hexane. Due to this |
| 766 |
+ |
difference, toluene molecules have more difficulty in occupying |
| 767 |
+ |
relatively small gaps among capping agents when their coverage is not |
| 768 |
+ |
too low. Therefore, the solvent-capping agent contact may keep |
| 769 |
+ |
increasing until the capping agent coverage reaches a relatively low |
| 770 |
+ |
level. This becomes an offset for decreasing butanethiol molecules on |
| 771 |
+ |
its effect to the process of interfacial thermal transport. Thus, one |
| 772 |
+ |
can see a plateau of $G$ vs. butanethiol coverage in our results. |
| 773 |
+ |
|
| 774 |
+ |
\subsection{Influence of Chosen Molecule Model on $G$} |
| 775 |
+ |
In addition to UA solvent/capping agent models, AA models are included |
| 776 |
+ |
in our simulations as well. Besides simulations of the same (UA or AA) |
| 777 |
+ |
model for solvent and capping agent, different models can be applied |
| 778 |
+ |
to different components. Furthermore, regardless of models chosen, |
| 779 |
+ |
either the solvent or the capping agent can be deuterated, similar to |
| 780 |
+ |
the previous section. Table \ref{modelTest} summarizes the results of |
| 781 |
+ |
these studies. |
| 782 |
+ |
|
| 783 |
|
\begin{table*} |
| 784 |
|
\begin{minipage}{\linewidth} |
| 785 |
|
\begin{center} |
| 786 |
|
|
| 787 |
|
\caption{Computed interfacial thermal conductivity ($G$ and |
| 788 |
< |
$G^\prime$) values for the Au/butanethiol/hexane interface |
| 789 |
< |
with all-atom model and different capping agent coverage at |
| 790 |
< |
200K using a range of energy fluxes.} |
| 788 |
> |
$G^\prime$) values for interfaces using various models for |
| 789 |
> |
solvent and capping agent (or without capping agent) at |
| 790 |
> |
$\langle T\rangle\sim$200K. (D stands for deuterated solvent |
| 791 |
> |
or capping agent molecules; ``Avg.'' denotes results that are |
| 792 |
> |
averages of simulations under different $J_z$'s. Error |
| 793 |
> |
estimates indicated in parenthesis.)} |
| 794 |
|
|
| 795 |
< |
\begin{tabular}{cccc} |
| 795 |
> |
\begin{tabular}{llccc} |
| 796 |
|
\hline\hline |
| 797 |
< |
Thiol & $J_z$ & $G$ & $G^\prime$ \\ |
| 798 |
< |
coverage (\%) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 797 |
> |
Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ |
| 798 |
> |
(or bare surface) & model & (GW/m$^2$) & |
| 799 |
> |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 800 |
|
\hline |
| 801 |
< |
0.0 & 0.95 & 28.5 & 27.2 \\ |
| 802 |
< |
& 1.88 & 30.3 & 28.9 \\ |
| 803 |
< |
100.0 & 2.87 & 551 & 294 \\ |
| 804 |
< |
& 3.81 & 494 & 193 \\ |
| 801 |
> |
UA & UA hexane & Avg. & 131(9) & 87(10) \\ |
| 802 |
> |
& UA hexane(D) & 1.95 & 153(5) & 136(13) \\ |
| 803 |
> |
& AA hexane & Avg. & 131(6) & 122(10) \\ |
| 804 |
> |
& UA toluene & 1.96 & 187(16) & 151(11) \\ |
| 805 |
> |
& AA toluene & 1.89 & 200(36) & 149(53) \\ |
| 806 |
> |
\hline |
| 807 |
> |
AA & UA hexane & 1.94 & 116(9) & 129(8) \\ |
| 808 |
> |
& AA hexane & Avg. & 442(14) & 356(31) \\ |
| 809 |
> |
& AA hexane(D) & 1.93 & 222(12) & 234(54) \\ |
| 810 |
> |
& UA toluene & 1.98 & 125(25) & 97(60) \\ |
| 811 |
> |
& AA toluene & 3.79 & 487(56) & 290(42) \\ |
| 812 |
> |
\hline |
| 813 |
> |
AA(D) & UA hexane & 1.94 & 158(25) & 172(4) \\ |
| 814 |
> |
& AA hexane & 1.92 & 243(29) & 191(11) \\ |
| 815 |
> |
& AA toluene & 1.93 & 364(36) & 322(67) \\ |
| 816 |
> |
\hline |
| 817 |
> |
bare & UA hexane & Avg. & 46.5(3.2) & 49.4(4.5) \\ |
| 818 |
> |
& UA hexane(D) & 0.98 & 43.9(4.6) & 43.0(2.0) \\ |
| 819 |
> |
& AA hexane & 0.96 & 31.0(1.4) & 29.4(1.3) \\ |
| 820 |
> |
& UA toluene & 1.99 & 70.1(1.3) & 65.8(0.5) \\ |
| 821 |
|
\hline\hline |
| 822 |
|
\end{tabular} |
| 823 |
< |
\label{AuThiolHexaneAA} |
| 823 |
> |
\label{modelTest} |
| 824 |
|
\end{center} |
| 825 |
|
\end{minipage} |
| 826 |
|
\end{table*} |
| 827 |
|
|
| 828 |
+ |
To facilitate direct comparison, the same system with differnt models |
| 829 |
+ |
for different components uses the same length scale for their |
| 830 |
+ |
simulation cells. Without the presence of capping agent, using |
| 831 |
+ |
different models for hexane yields similar results for both $G$ and |
| 832 |
+ |
$G^\prime$, and these two definitions agree with eath other very |
| 833 |
+ |
well. This indicates very weak interaction between the metal and the |
| 834 |
+ |
solvent, and is a typical case for acoustic impedance mismatch between |
| 835 |
+ |
these two phases. |
| 836 |
|
|
| 837 |
< |
\subsection{Mechanism of Interfacial Thermal Conductance Enhancement |
| 838 |
< |
by Capping Agent} |
| 839 |
< |
[MAY INTRODUCE PROTOCOL IN METHODOLOGY/COMPUTATIONAL DETAIL] |
| 837 |
> |
As for Au(111) surfaces completely covered by butanethiols, the choice |
| 838 |
> |
of models for capping agent and solvent could impact the measurement |
| 839 |
> |
of $G$ and $G^\prime$ quite significantly. For Au-butanethiol/hexane |
| 840 |
> |
interfaces, using AA model for both butanethiol and hexane yields |
| 841 |
> |
substantially higher conductivity values than using UA model for at |
| 842 |
> |
least one component of the solvent and capping agent, which exceeds |
| 843 |
> |
the general range of experimental measurement results. This is |
| 844 |
> |
probably due to the classically treated C-H vibrations in the AA |
| 845 |
> |
model, which should not be appreciably populated at normal |
| 846 |
> |
temperatures. In comparison, once either the hexanes or the |
| 847 |
> |
butanethiols are deuterated, one can see a significantly lower $G$ and |
| 848 |
> |
$G^\prime$. In either of these cases, the C-H(D) vibrational overlap |
| 849 |
> |
between the solvent and the capping agent is removed (Figure |
| 850 |
> |
\ref{aahxntln}). Conclusively, the improperly treated C-H vibration in |
| 851 |
> |
the AA model produced over-predicted results accordingly. Compared to |
| 852 |
> |
the AA model, the UA model yields more reasonable results with higher |
| 853 |
> |
computational efficiency. |
| 854 |
|
|
| 855 |
+ |
\begin{figure} |
| 856 |
+ |
\includegraphics[width=\linewidth]{aahxntln} |
| 857 |
+ |
\caption{Spectra obtained for All-Atom model Au-butanethil/solvent |
| 858 |
+ |
systems. When butanethiol is deuterated (lower left), its |
| 859 |
+ |
vibrational overlap with hexane would decrease significantly, |
| 860 |
+ |
compared with normal butanethiol (upper left). However, this |
| 861 |
+ |
dramatic change does not apply to toluene as much (right).} |
| 862 |
+ |
\label{aahxntln} |
| 863 |
+ |
\end{figure} |
| 864 |
|
|
| 865 |
< |
%subsubsection{Vibrational spectrum study on conductance mechanism} |
| 866 |
< |
To investigate the mechanism of this interfacial thermal conductance, |
| 867 |
< |
the vibrational spectra of various gold systems were obtained and are |
| 868 |
< |
shown as in the upper panel of Fig. \ref{vibration}. To obtain these |
| 869 |
< |
spectra, one first runs a simulation in the NVE ensemble and collects |
| 870 |
< |
snapshots of configurations; these configurations are used to compute |
| 871 |
< |
the velocity auto-correlation functions, which is used to construct a |
| 872 |
< |
power spectrum via a Fourier transform. The gold surfaces covered by |
| 873 |
< |
butanethiol molecules exhibit an additional peak observed at a |
| 683 |
< |
frequency of $\sim$170cm$^{-1}$, which is attributed to the vibration |
| 684 |
< |
of the S-Au bond. This vibration enables efficient thermal transport |
| 685 |
< |
from surface Au atoms to the capping agents. Simultaneously, as shown |
| 686 |
< |
in the lower panel of Fig. \ref{vibration}, the large overlap of the |
| 687 |
< |
vibration spectra of butanethiol and hexane in the all-atom model, |
| 688 |
< |
including the C-H vibration, also suggests high thermal exchange |
| 689 |
< |
efficiency. The combination of these two effects produces the drastic |
| 690 |
< |
interfacial thermal conductance enhancement in the all-atom model. |
| 865 |
> |
However, for Au-butanethiol/toluene interfaces, having the AA |
| 866 |
> |
butanethiol deuterated did not yield a significant change in the |
| 867 |
> |
measurement results. Compared to the C-H vibrational overlap between |
| 868 |
> |
hexane and butanethiol, both of which have alkyl chains, that overlap |
| 869 |
> |
between toluene and butanethiol is not so significant and thus does |
| 870 |
> |
not have as much contribution to the heat exchange |
| 871 |
> |
process. Conversely, extra degrees of freedom such as the C-H |
| 872 |
> |
vibrations could yield higher heat exchange rate between these two |
| 873 |
> |
phases and result in a much higher conductivity. |
| 874 |
|
|
| 875 |
+ |
Although the QSC model for Au is known to predict an overly low value |
| 876 |
+ |
for bulk metal gold conductivity\cite{kuang:164101}, our computational |
| 877 |
+ |
results for $G$ and $G^\prime$ do not seem to be affected by this |
| 878 |
+ |
drawback of the model for metal. Instead, our results suggest that the |
| 879 |
+ |
modeling of interfacial thermal transport behavior relies mainly on |
| 880 |
+ |
the accuracy of the interaction descriptions between components |
| 881 |
+ |
occupying the interfaces. |
| 882 |
+ |
|
| 883 |
+ |
\subsection{Role of Capping Agent in Interfacial Thermal Conductance} |
| 884 |
+ |
The vibrational spectra for gold slabs in different environments are |
| 885 |
+ |
shown as in Figure \ref{specAu}. Regardless of the presence of |
| 886 |
+ |
solvent, the gold surfaces covered by butanethiol molecules, compared |
| 887 |
+ |
to bare gold surfaces, exhibit an additional peak observed at the |
| 888 |
+ |
frequency of $\sim$170cm$^{-1}$, which is attributed to the S-Au |
| 889 |
+ |
bonding vibration. This vibration enables efficient thermal transport |
| 890 |
+ |
from surface Au layer to the capping agents. Therefore, in our |
| 891 |
+ |
simulations, the Au/S interfaces do not appear major heat barriers |
| 892 |
+ |
compared to the butanethiol / solvent interfaces. |
| 893 |
+ |
|
| 894 |
+ |
Simultaneously, the vibrational overlap between butanethiol and |
| 895 |
+ |
organic solvents suggests higher thermal exchange efficiency between |
| 896 |
+ |
these two components. Even exessively high heat transport was observed |
| 897 |
+ |
when All-Atom models were used and C-H vibrations were treated |
| 898 |
+ |
classically. Compared to metal and organic liquid phase, the heat |
| 899 |
+ |
transfer efficiency between butanethiol and organic solvents is closer |
| 900 |
+ |
to that within bulk liquid phase. |
| 901 |
+ |
|
| 902 |
+ |
Furthermore, our observation validated previous |
| 903 |
+ |
results\cite{hase:2010} that the intramolecular heat transport of |
| 904 |
+ |
alkylthiols is highly effecient. As a combinational effects of these |
| 905 |
+ |
phenomena, butanethiol acts as a channel to expedite thermal transport |
| 906 |
+ |
process. The acoustic impedance mismatch between the metal and the |
| 907 |
+ |
liquid phase can be effectively reduced with the presence of suitable |
| 908 |
+ |
capping agents. |
| 909 |
+ |
|
| 910 |
|
\begin{figure} |
| 911 |
|
\includegraphics[width=\linewidth]{vibration} |
| 912 |
|
\caption{Vibrational spectra obtained for gold in different |
| 913 |
< |
environments (upper panel) and for Au/thiol/hexane simulation in |
| 914 |
< |
all-atom model (lower panel).} |
| 697 |
< |
\label{vibration} |
| 913 |
> |
environments.} |
| 914 |
> |
\label{specAu} |
| 915 |
|
\end{figure} |
| 699 |
– |
% MAY NEED TO CONVERT TO JPEG |
| 916 |
|
|
| 917 |
+ |
[MAY ADD COMPARISON OF AU SLAB WIDTHS BUT NOT MUCH TO TALK ABOUT...] |
| 918 |
+ |
|
| 919 |
|
\section{Conclusions} |
| 920 |
+ |
The NIVS algorithm we developed has been applied to simulations of |
| 921 |
+ |
Au-butanethiol surfaces with organic solvents. This algorithm allows |
| 922 |
+ |
effective unphysical thermal flux transferred between the metal and |
| 923 |
+ |
the liquid phase. With the flux applied, we were able to measure the |
| 924 |
+ |
corresponding thermal gradient and to obtain interfacial thermal |
| 925 |
+ |
conductivities. Under steady states, single trajectory simulation |
| 926 |
+ |
would be enough for accurate measurement. This would be advantageous |
| 927 |
+ |
compared to transient state simulations, which need multiple |
| 928 |
+ |
trajectories to produce reliable average results. |
| 929 |
|
|
| 930 |
+ |
Our simulations have seen significant conductance enhancement with the |
| 931 |
+ |
presence of capping agent, compared to the bare gold / liquid |
| 932 |
+ |
interfaces. The acoustic impedance mismatch between the metal and the |
| 933 |
+ |
liquid phase is effectively eliminated by proper capping |
| 934 |
+ |
agent. Furthermore, the coverage precentage of the capping agent plays |
| 935 |
+ |
an important role in the interfacial thermal transport |
| 936 |
+ |
process. Moderately lower coverages allow higher contact between |
| 937 |
+ |
capping agent and solvent, and thus could further enhance the heat |
| 938 |
+ |
transfer process. |
| 939 |
|
|
| 940 |
< |
[NECESSITY TO STUDY THERMAL CONDUCTANCE IN NANOCRYSTAL STRUCTURE]\cite{vlugt:cpc2007154} |
| 940 |
> |
Our measurement results, particularly of the UA models, agree with |
| 941 |
> |
available experimental data. This indicates that our force field |
| 942 |
> |
parameters have a nice description of the interactions between the |
| 943 |
> |
particles at the interfaces. AA models tend to overestimate the |
| 944 |
> |
interfacial thermal conductance in that the classically treated C-H |
| 945 |
> |
vibration would be overly sampled. Compared to the AA models, the UA |
| 946 |
> |
models have higher computational efficiency with satisfactory |
| 947 |
> |
accuracy, and thus are preferable in interfacial thermal transport |
| 948 |
> |
modelings. Of the two definitions for $G$, the discrete form |
| 949 |
> |
(Eq. \ref{discreteG}) was easier to use and gives out relatively |
| 950 |
> |
consistent results, while the derivative form (Eq. \ref{derivativeG}) |
| 951 |
> |
is not as versatile. Although $G^\prime$ gives out comparable results |
| 952 |
> |
and follows similar trend with $G$ when measuring close to fully |
| 953 |
> |
covered or bare surfaces, the spatial resolution of $T$ profile is |
| 954 |
> |
limited for accurate computation of derivatives data. |
| 955 |
|
|
| 956 |
+ |
Vlugt {\it et al.} has investigated the surface thiol structures for |
| 957 |
+ |
nanocrystal gold and pointed out that they differs from those of the |
| 958 |
+ |
Au(111) surface\cite{landman:1998,vlugt:cpc2007154}. This difference |
| 959 |
+ |
might lead to change of interfacial thermal transport behavior as |
| 960 |
+ |
well. To investigate this problem, an effective means to introduce |
| 961 |
+ |
thermal flux and measure the corresponding thermal gradient is |
| 962 |
+ |
desirable for simulating structures with spherical symmetry. |
| 963 |
+ |
|
| 964 |
|
\section{Acknowledgments} |
| 965 |
|
Support for this project was provided by the National Science |
| 966 |
|
Foundation under grant CHE-0848243. Computational time was provided by |
