| 45 |
|
|
| 46 |
|
\begin{abstract} |
| 47 |
|
|
| 48 |
< |
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 |
|
|
| 65 |
|
\end{abstract} |
| 66 |
|
|
| 73 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 74 |
|
|
| 75 |
|
\section{Introduction} |
| 74 |
– |
[BACKGROUND FOR INTERFACIAL THERMAL CONDUCTANCE PROBLEM] |
| 76 |
|
Interfacial thermal conductance is extensively studied both |
| 77 |
< |
experimentally and computationally, and systems with interfaces |
| 78 |
< |
present are generally heterogeneous. Although interfaces are commonly |
| 79 |
< |
barriers to heat transfer, it has been |
| 80 |
< |
reported\cite{doi:10.1021/la904855s} that under specific circustances, |
| 81 |
< |
e.g. with certain capping agents present on the surface, interfacial |
| 82 |
< |
conductance can be significantly enhanced. However, heat conductance |
| 83 |
< |
of molecular and nano-scale interfaces will be affected by the |
| 84 |
< |
chemical details of the surface and is challenging to |
| 85 |
< |
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. |
| 77 |
> |
experimentally and computationally\cite{cahill:793}, due to its |
| 78 |
> |
importance in nanoscale science and technology. Reliability of |
| 79 |
> |
nanoscale devices depends on their thermal transport |
| 80 |
> |
properties. Unlike bulk homogeneous materials, nanoscale materials |
| 81 |
> |
features significant presence of interfaces, and these interfaces |
| 82 |
> |
could dominate the heat transfer behavior of these |
| 83 |
> |
materials. Furthermore, these materials are generally heterogeneous, |
| 84 |
> |
which challenges traditional research methods for homogeneous |
| 85 |
> |
systems. |
| 86 |
|
|
| 87 |
+ |
Heat conductance of molecular and nano-scale interfaces will be |
| 88 |
+ |
affected by the chemical details of the surface. Experimentally, |
| 89 |
+ |
various interfaces have been investigated for their thermal |
| 90 |
+ |
conductance properties. Wang {\it et al.} studied heat transport |
| 91 |
+ |
through long-chain hydrocarbon monolayers on gold substrate at |
| 92 |
+ |
individual molecular level\cite{Wang10082007}; Schmidt {\it et al.} |
| 93 |
+ |
studied the role of CTAB on thermal transport between gold nanorods |
| 94 |
+ |
and solvent\cite{doi:10.1021/jp8051888}; Juv\'e {\it et al.} studied |
| 95 |
+ |
the cooling dynamics, which is controlled by thermal interface |
| 96 |
+ |
resistence of glass-embedded metal |
| 97 |
+ |
nanoparticles\cite{PhysRevB.80.195406}. Although interfaces are |
| 98 |
+ |
commonly barriers for heat transport, Alper {\it et al.} suggested |
| 99 |
+ |
that specific ligands (capping agents) could completely eliminate this |
| 100 |
+ |
barrier ($G\rightarrow\infty$)\cite{doi:10.1021/la904855s}. |
| 101 |
+ |
|
| 102 |
+ |
Theoretical and computational models have also been used to study the |
| 103 |
+ |
interfacial thermal transport in order to gain an understanding of |
| 104 |
+ |
this phenomena at the molecular level. Recently, Hase and coworkers |
| 105 |
+ |
employed Non-Equilibrium Molecular Dynamics (NEMD) simulations to |
| 106 |
+ |
study thermal transport from hot Au(111) substrate to a self-assembled |
| 107 |
+ |
monolayer of alkylthiolate with relatively long chain (8-20 carbon |
| 108 |
+ |
atoms)\cite{hase:2010,hase:2011}. However, ensemble averaged |
| 109 |
+ |
measurements for heat conductance of interfaces between the capping |
| 110 |
+ |
monolayer on Au and a solvent phase has yet to be studied. |
| 111 |
+ |
The relatively low thermal flux through interfaces is |
| 112 |
+ |
difficult to measure with Equilibrium MD or forward NEMD simulation |
| 113 |
+ |
methods. Therefore, the Reverse NEMD (RNEMD) methods would have the |
| 114 |
+ |
advantage of having this difficult to measure flux known when studying |
| 115 |
+ |
the thermal transport across interfaces, given that the simulation |
| 116 |
+ |
methods being able to effectively apply an unphysical flux in |
| 117 |
+ |
non-homogeneous systems. |
| 118 |
+ |
|
| 119 |
|
Recently, we have developed the Non-Isotropic Velocity Scaling (NIVS) |
| 120 |
|
algorithm for RNEMD simulations\cite{kuang:164101}. This algorithm |
| 121 |
|
retains the desirable features of RNEMD (conservation of linear |
| 122 |
|
momentum and total energy, compatibility with periodic boundary |
| 123 |
|
conditions) while establishing true thermal distributions in each of |
| 124 |
< |
the two slabs. Furthermore, it allows more effective thermal exchange |
| 125 |
< |
between particles of different identities, and thus enables extensive |
| 126 |
< |
study of interfacial conductance. |
| 124 |
> |
the two slabs. Furthermore, it allows effective thermal exchange |
| 125 |
> |
between particles of different identities, and thus makes the study of |
| 126 |
> |
interfacial conductance much simpler. |
| 127 |
|
|
| 128 |
+ |
The work presented here deals with the Au(111) surface covered to |
| 129 |
+ |
varying degrees by butanethiol, a capping agent with short carbon |
| 130 |
+ |
chain, and solvated with organic solvents of different molecular |
| 131 |
+ |
properties. Different models were used for both the capping agent and |
| 132 |
+ |
the solvent force field parameters. Using the NIVS algorithm, the |
| 133 |
+ |
thermal transport across these interfaces was studied and the |
| 134 |
+ |
underlying mechanism for this phenomena was investigated. |
| 135 |
+ |
|
| 136 |
+ |
[MAY ADD WHY STUDY AU-THIOL SURFACE; CITE SHAOYI JIANG] |
| 137 |
+ |
|
| 138 |
|
\section{Methodology} |
| 139 |
< |
\subsection{Algorithm} |
| 140 |
< |
[BACKGROUND FOR MD METHODS] |
| 141 |
< |
There have been many algorithms for computing thermal conductivity |
| 142 |
< |
using molecular dynamics simulations. However, interfacial conductance |
| 143 |
< |
is at least an order of magnitude smaller. This would make the |
| 144 |
< |
calculation even more difficult for those slowly-converging |
| 145 |
< |
equilibrium methods. Imposed-flux non-equilibrium |
| 139 |
> |
\subsection{Imposd-Flux Methods in MD Simulations} |
| 140 |
> |
For systems with low interfacial conductivity one must have a method |
| 141 |
> |
capable of generating relatively small fluxes, compared to those |
| 142 |
> |
required for bulk conductivity. This requirement makes the calculation |
| 143 |
> |
even more difficult for those slowly-converging equilibrium |
| 144 |
> |
methods\cite{Viscardy:2007lq}. |
| 145 |
> |
Forward methods impose gradient, but in interfacail conditions it is |
| 146 |
> |
not clear what behavior to impose at the boundary... |
| 147 |
> |
Imposed-flux reverse non-equilibrium |
| 148 |
|
methods\cite{MullerPlathe:1997xw} have the flux set {\it a priori} and |
| 149 |
< |
the response of temperature or momentum gradients are easier to |
| 150 |
< |
measure than the flux, if unknown, and thus, is a preferable way to |
| 151 |
< |
the forward NEMD methods. Although the momentum swapping approach for |
| 152 |
< |
flux-imposing can be used for exchanging energy between particles of |
| 153 |
< |
different identity, the kinetic energy transfer efficiency is affected |
| 154 |
< |
by the mass difference between the particles, which limits its |
| 113 |
< |
application on heterogeneous interfacial systems. |
| 149 |
> |
the thermal response becomes easier to |
| 150 |
> |
measure than the flux. Although M\"{u}ller-Plathe's original momentum |
| 151 |
> |
swapping approach can be used for exchanging energy between particles |
| 152 |
> |
of different identity, the kinetic energy transfer efficiency is |
| 153 |
> |
affected by the mass difference between the particles, which limits |
| 154 |
> |
its application on heterogeneous interfacial systems. |
| 155 |
|
|
| 156 |
< |
The non-isotropic velocity scaling (NIVS)\cite{kuang:164101} approach in |
| 157 |
< |
non-equilibrium MD simulations is able to impose relatively large |
| 158 |
< |
kinetic energy flux without obvious perturbation to the velocity |
| 159 |
< |
distribution of the simulated systems. Furthermore, this approach has |
| 156 |
> |
The non-isotropic velocity scaling (NIVS)\cite{kuang:164101} approach to |
| 157 |
> |
non-equilibrium MD simulations is able to impose a wide range of |
| 158 |
> |
kinetic energy fluxes without obvious perturbation to the velocity |
| 159 |
> |
distributions of the simulated systems. Furthermore, this approach has |
| 160 |
|
the advantage in heterogeneous interfaces in that kinetic energy flux |
| 161 |
|
can be applied between regions of particles of arbitary identity, and |
| 162 |
< |
the flux quantity is not restricted by particle mass difference. |
| 162 |
> |
the flux will not be restricted by difference in particle mass. |
| 163 |
|
|
| 164 |
|
The NIVS algorithm scales the velocity vectors in two separate regions |
| 165 |
|
of a simulation system with respective diagonal scaling matricies. To |
| 166 |
|
determine these scaling factors in the matricies, a set of equations |
| 167 |
|
including linear momentum conservation and kinetic energy conservation |
| 168 |
< |
constraints and target momentum/energy flux satisfaction is |
| 169 |
< |
solved. With the scaling operation applied to the system in a set |
| 170 |
< |
frequency, corresponding momentum/temperature gradients can be built, |
| 171 |
< |
which can be used for computing transportation properties and other |
| 172 |
< |
applications related to momentum/temperature gradients. The NIVS |
| 132 |
< |
algorithm conserves momenta and energy and does not depend on an |
| 133 |
< |
external thermostat. |
| 168 |
> |
constraints and target energy flux satisfaction is solved. With the |
| 169 |
> |
scaling operation applied to the system in a set frequency, bulk |
| 170 |
> |
temperature gradients can be easily established, and these can be used |
| 171 |
> |
for computing thermal conductivities. The NIVS algorithm conserves |
| 172 |
> |
momenta and energy and does not depend on an external thermostat. |
| 173 |
|
|
| 174 |
|
\subsection{Defining Interfacial Thermal Conductivity $G$} |
| 175 |
|
For interfaces with a relatively low interfacial conductance, the bulk |
| 187 |
|
T_\mathrm{cold}\rangle}$ are the average observed temperature of the |
| 188 |
|
two separated phases. |
| 189 |
|
|
| 190 |
< |
When the interfacial conductance is {\it not} small, two ways can be |
| 191 |
< |
used to define $G$. |
| 190 |
> |
When the interfacial conductance is {\it not} small, there are two |
| 191 |
> |
ways to define $G$. |
| 192 |
|
|
| 193 |
< |
One way is to assume the temperature is discretely different on two |
| 194 |
< |
sides of the interface, $G$ can be calculated with the thermal flux |
| 195 |
< |
applied $J$ and the maximum temperature difference measured along the |
| 196 |
< |
thermal gradient max($\Delta T$), which occurs at the interface, as: |
| 193 |
> |
One way is to assume the temperature is discrete on the two sides of |
| 194 |
> |
the interface. $G$ can be calculated using the applied thermal flux |
| 195 |
> |
$J$ and the maximum temperature difference measured along the thermal |
| 196 |
> |
gradient max($\Delta T$), which occurs at the Gibbs deviding surface, |
| 197 |
> |
as: |
| 198 |
|
\begin{equation} |
| 199 |
|
G=\frac{J}{\Delta T} |
| 200 |
|
\label{discreteG} |
| 215 |
|
|
| 216 |
|
With the temperature profile obtained from simulations, one is able to |
| 217 |
|
approximate the first and second derivatives of $T$ with finite |
| 218 |
< |
difference method and thus calculate $G^\prime$. |
| 218 |
> |
difference methods and thus calculate $G^\prime$. |
| 219 |
|
|
| 220 |
< |
In what follows, both definitions are used for calculation and comparison. |
| 220 |
> |
In what follows, both definitions have been used for calculation and |
| 221 |
> |
are compared in the results. |
| 222 |
|
|
| 223 |
< |
[IMPOSE G DEFINITION INTO OUR SYSTEMS] |
| 224 |
< |
To facilitate the use of the above definitions in calculating $G$ and |
| 225 |
< |
$G^\prime$, we have a metal slab with its (111) surfaces perpendicular |
| 226 |
< |
to the $z$-axis of our simulation cells. With or withour capping |
| 227 |
< |
agents on the surfaces, the metal slab is solvated with organic |
| 187 |
< |
solvents, as illustrated in Figure \ref{demoPic}. |
| 223 |
> |
To compare the above definitions ($G$ and $G^\prime$), we have modeled |
| 224 |
> |
a metal slab with its (111) surfaces perpendicular to the $z$-axis of |
| 225 |
> |
our simulation cells. Both with and withour capping agents on the |
| 226 |
> |
surfaces, the metal slab is solvated with simple organic solvents, as |
| 227 |
> |
illustrated in Figure \ref{demoPic}. |
| 228 |
|
|
| 229 |
|
\begin{figure} |
| 230 |
|
\includegraphics[width=\linewidth]{demoPic} |
| 233 |
|
\label{demoPic} |
| 234 |
|
\end{figure} |
| 235 |
|
|
| 236 |
< |
With a simulation cell setup following the above manner, one is able |
| 237 |
< |
to equilibrate the system and impose an unphysical thermal flux |
| 238 |
< |
between the liquid and the metal phase with the NIVS algorithm. Under |
| 239 |
< |
a stablized thermal gradient induced by periodically applying the |
| 240 |
< |
unphysical flux, one is able to obtain a temperature profile and the |
| 241 |
< |
physical thermal flux corresponding to it, which equals to the |
| 242 |
< |
unphysical flux applied by NIVS. These data enables the evaluation of |
| 243 |
< |
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. |
| 236 |
> |
With the simulation cell described above, we are able to equilibrate |
| 237 |
> |
the system and impose an unphysical thermal flux between the liquid |
| 238 |
> |
and the metal phase using the NIVS algorithm. By periodically applying |
| 239 |
> |
the unphysical flux, we are able to obtain a temperature profile and |
| 240 |
> |
its spatial derivatives. These quantities enable the evaluation of the |
| 241 |
> |
interfacial thermal conductance of a surface. Figure \ref{gradT} is an |
| 242 |
> |
example how those applied thermal fluxes can be used to obtain the 1st |
| 243 |
> |
and 2nd derivatives of the temperature profile. |
| 244 |
|
|
| 245 |
|
\begin{figure} |
| 246 |
|
\includegraphics[width=\linewidth]{gradT} |
| 249 |
|
\label{gradT} |
| 250 |
|
\end{figure} |
| 251 |
|
|
| 214 |
– |
[MAY INCLUDE POWER SPECTRUM PROTOCOL] |
| 215 |
– |
|
| 252 |
|
\section{Computational Details} |
| 253 |
|
\subsection{Simulation Protocol} |
| 254 |
< |
In our simulations, Au is used to construct a metal slab with bare |
| 255 |
< |
(111) surface perpendicular to the $z$-axis. Different slab thickness |
| 256 |
< |
(layer numbers of Au) are simulated. This metal slab is first |
| 257 |
< |
equilibrated under normal pressure (1 atm) and a desired |
| 258 |
< |
temperature. After equilibration, butanethiol is used as the capping |
| 259 |
< |
agent molecule to cover the bare Au (111) surfaces evenly. The sulfur |
| 260 |
< |
atoms in the butanethiol molecules would occupy the three-fold sites |
| 261 |
< |
of the surfaces, and the maximal butanethiol capacity on Au surface is |
| 262 |
< |
$1/3$ of the total number of surface Au atoms[CITATION]. A series of |
| 263 |
< |
different coverage surfaces is investigated in order to study the |
| 264 |
< |
relation between coverage and conductance. |
| 254 |
> |
The NIVS algorithm has been implemented in our MD simulation code, |
| 255 |
> |
OpenMD\cite{Meineke:2005gd,openmd}, and was used for our |
| 256 |
> |
simulations. Different slab thickness (layer numbers of Au) were |
| 257 |
> |
simulated. Metal slabs were first equilibrated under atmospheric |
| 258 |
> |
pressure (1 atm) and a desired temperature (e.g. 200K). After |
| 259 |
> |
equilibration, butanethiol capping agents were placed at three-fold |
| 260 |
> |
sites on the Au(111) surfaces. The maximum butanethiol capacity on Au |
| 261 |
> |
surface is $1/3$ of the total number of surface Au |
| 262 |
> |
atoms\cite{vlugt:cpc2007154}. A series of different coverages was |
| 263 |
> |
investigated in order to study the relation between coverage and |
| 264 |
> |
interfacial conductance. |
| 265 |
|
|
| 266 |
< |
[COVERAGE DISCRIPTION] However, since the interactions between surface |
| 267 |
< |
Au and butanethiol is non-bonded, the capping agent molecules are |
| 268 |
< |
allowed to migrate to an empty neighbor three-fold site during a |
| 269 |
< |
simulation. Therefore, the initial configuration would not severely |
| 270 |
< |
affect the sampling of a variety of configurations of the same |
| 271 |
< |
coverage, and the final conductance measurement would be an average |
| 272 |
< |
effect of these configurations explored in the simulations. [MAY NEED FIGURES] |
| 266 |
> |
The capping agent molecules were allowed to migrate during the |
| 267 |
> |
simulations. They distributed themselves uniformly and sampled a |
| 268 |
> |
number of three-fold sites throughout out study. Therefore, the |
| 269 |
> |
initial configuration would not noticeably affect the sampling of a |
| 270 |
> |
variety of configurations of the same coverage, and the final |
| 271 |
> |
conductance measurement would be an average effect of these |
| 272 |
> |
configurations explored in the simulations. [MAY NEED FIGURES] |
| 273 |
|
|
| 274 |
< |
After the modified Au-butanethiol surface systems are equilibrated |
| 275 |
< |
under canonical ensemble, Packmol\cite{packmol} is used to pack |
| 276 |
< |
organic solvent molecules in the previously vacuum part of the |
| 277 |
< |
simulation cells, which guarantees that short range repulsive |
| 278 |
< |
interactions do not disrupt the simulations. Two solvents are |
| 279 |
< |
investigated, one which has little vibrational overlap with the |
| 244 |
< |
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] |
| 274 |
> |
After the modified Au-butanethiol surface systems were equilibrated |
| 275 |
> |
under canonical ensemble, organic solvent molecules were packed in the |
| 276 |
> |
previously empty part of the simulation cells\cite{packmol}. Two |
| 277 |
> |
solvents were investigated, one which has little vibrational overlap |
| 278 |
> |
with the alkanethiol and a planar shape (toluene), and one which has |
| 279 |
> |
similar vibrational frequencies and chain-like shape ({\it n}-hexane). |
| 280 |
|
|
| 281 |
< |
The spacing filled by solvent molecules, i.e. the gap between |
| 281 |
> |
The space filled by solvent molecules, i.e. the gap between |
| 282 |
|
periodically repeated Au-butanethiol surfaces should be carefully |
| 283 |
|
chosen. A very long length scale for the thermal gradient axis ($z$) |
| 284 |
|
may cause excessively hot or cold temperatures in the middle of the |
| 323 |
|
same type of particles and between particles of different species. |
| 324 |
|
|
| 325 |
|
The Au-Au interactions in metal lattice slab is described by the |
| 326 |
< |
quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
| 326 |
> |
quantum Sutton-Chen (QSC) formulation\cite{PhysRevB.59.3527}. The QSC |
| 327 |
|
potentials include zero-point quantum corrections and are |
| 328 |
|
reparametrized for accurate surface energies compared to the |
| 329 |
|
Sutton-Chen potentials\cite{Chen90}. |
| 330 |
|
|
| 331 |
< |
Figure [REF] demonstrates how we name our pseudo-atoms of the |
| 332 |
< |
molecules in our simulations. |
| 300 |
< |
[FIGURE FOR MOLECULE NOMENCLATURE] |
| 331 |
> |
Figure \ref{demoMol} demonstrates how we name our pseudo-atoms of the |
| 332 |
> |
organic solvent molecules in our simulations. |
| 333 |
|
|
| 334 |
+ |
\begin{figure} |
| 335 |
+ |
\includegraphics[width=\linewidth]{demoMol} |
| 336 |
+ |
\caption{Denomination of atoms or pseudo-atoms in our simulations: a) |
| 337 |
+ |
UA-hexane; b) AA-hexane; c) UA-toluene; d) AA-toluene.} |
| 338 |
+ |
\label{demoMol} |
| 339 |
+ |
\end{figure} |
| 340 |
+ |
|
| 341 |
|
For both solvent molecules, straight chain {\it n}-hexane and aromatic |
| 342 |
|
toluene, United-Atom (UA) and All-Atom (AA) models are used |
| 343 |
|
respectively. The TraPPE-UA |
| 369 |
|
parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) |
| 370 |
|
surfaces do not have the hydrogen atom bonded to sulfur. To adapt this |
| 371 |
|
change and derive suitable parameters for butanethiol adsorbed on |
| 372 |
< |
Au(111) surfaces, we adopt the S parameters from [CITATION CF VLUGT] |
| 373 |
< |
and modify parameters for its neighbor C atom for charge balance in |
| 374 |
< |
the molecule. Note that the model choice (UA or AA) of capping agent |
| 375 |
< |
can be different from the solvent. Regardless of model choice, the |
| 376 |
< |
force field parameters for interactions between capping agent and |
| 377 |
< |
solvent can be derived using Lorentz-Berthelot Mixing Rule: |
| 372 |
> |
Au(111) surfaces, we adopt the S parameters from Luedtke and |
| 373 |
> |
Landman\cite{landman:1998} and modify parameters for its neighbor C |
| 374 |
> |
atom for charge balance in the molecule. Note that the model choice |
| 375 |
> |
(UA or AA) of capping agent can be different from the |
| 376 |
> |
solvent. Regardless of model choice, the force field parameters for |
| 377 |
> |
interactions between capping agent and solvent can be derived using |
| 378 |
> |
Lorentz-Berthelot Mixing Rule:[EQN'S] |
| 379 |
|
|
| 380 |
|
|
| 381 |
|
To describe the interactions between metal Au and non-metal capping |
| 384 |
|
al.}\cite{vlugt:cpc2007154} They fitted an effective Lennard-Jones |
| 385 |
|
form of potential parameters for the interaction between Au and |
| 386 |
|
pseudo-atoms CH$_x$ and S based on a well-established and widely-used |
| 387 |
< |
effective potential of Hautman and Klein[CITATION] for the Au(111) |
| 388 |
< |
surface. As our simulations require the gold lattice slab to be |
| 389 |
< |
non-rigid so that it could accommodate kinetic energy for thermal |
| 387 |
> |
effective potential of Hautman and Klein\cite{hautman:4994} for the |
| 388 |
> |
Au(111) surface. As our simulations require the gold lattice slab to |
| 389 |
> |
be non-rigid so that it could accommodate kinetic energy for thermal |
| 390 |
|
transport study purpose, the pair-wise form of potentials is |
| 391 |
|
preferred. |
| 392 |
|
|
| 410 |
|
|
| 411 |
|
\begin{tabular}{ccc} |
| 412 |
|
\hline\hline |
| 413 |
< |
Non-metal & $\sigma$/\AA & $\epsilon$/kcal/mol \\ |
| 413 |
> |
Non-metal atom & $\sigma$ & $\epsilon$ \\ |
| 414 |
> |
(or pseudo-atom) & \AA & kcal/mol \\ |
| 415 |
|
\hline |
| 416 |
|
S & 2.40 & 8.465 \\ |
| 417 |
|
CH3 & 3.54 & 0.2146 \\ |
| 497 |
|
\hline\hline |
| 498 |
|
$\langle T\rangle$ & & $L_x$ & $L_y$ & $L_z$ & $J_z$ & |
| 499 |
|
$G$ & $G^\prime$ \\ |
| 500 |
< |
(K) & $N_{hexane}$ & \multicolumn{3}{c}\AA & (GW/m$^2$) & |
| 500 |
> |
(K) & $N_{hexane}$ & \multicolumn{3}{c}{(\AA)} & (GW/m$^2$) & |
| 501 |
|
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 502 |
|
\hline |
| 503 |
|
200 & 266 & 29.86 & 25.80 & 113.1 & -0.96 & |
| 592 |
|
reconstructions could eliminate the original $x$ and $y$ dimensional |
| 593 |
|
homogeneity, measurement of $G$ is more difficult to conduct under |
| 594 |
|
higher temperatures. Therefore, most of our measurements are |
| 595 |
< |
undertaken at $<T>\sim$200K. |
| 595 |
> |
undertaken at $\langle T\rangle\sim$200K. |
| 596 |
|
|
| 597 |
|
However, when the surface is not completely covered by butanethiols, |
| 598 |
|
the simulated system is more resistent to the reconstruction |
| 599 |
|
above. Our Au-butanethiol/toluene system did not see this phenomena |
| 600 |
< |
even at $<T>\sim$300K. The Au(111) surfaces have a 90\% coverage of |
| 600 |
> |
even at $\langle T\rangle\sim$300K. The Au(111) surfaces have a 90\% coverage of |
| 601 |
|
butanethiols and have empty three-fold sites. These empty sites could |
| 602 |
|
help prevent surface reconstruction in that they provide other means |
| 603 |
|
of capping agent relaxation. It is observed that butanethiols can |
| 657 |
|
butanethiol (UA) is non-deuterated for both solvents. These UA model |
| 658 |
|
studies, even though eliminating C-H vibration samplings, still have |
| 659 |
|
C-C vibrational frequencies different from each other. However, these |
| 660 |
< |
differences in the IR range do not seem to produce an observable |
| 660 |
> |
differences in the infrared range do not seem to produce an observable |
| 661 |
|
difference for the results of $G$. [MAY NEED FIGURE] |
| 662 |
|
|
| 663 |
|
Furthermore, results for rigid body toluene solvent, as well as other |
| 683 |
|
\begin{table*} |
| 684 |
|
\begin{minipage}{\linewidth} |
| 685 |
|
\begin{center} |
| 686 |
< |
\caption{Computed interfacial thermal conductivity ($G$ in |
| 687 |
< |
MW/m$^2$/K) values for the Au-butanethiol/solvent interface |
| 688 |
< |
with various UA models and different capping agent coverages |
| 689 |
< |
at $<T>\sim$200K using certain energy flux respectively.} |
| 686 |
> |
\caption{Computed interfacial thermal conductivity ($G$) values |
| 687 |
> |
for the Au-butanethiol/solvent interface with various UA |
| 688 |
> |
models and different capping agent coverages at $\langle |
| 689 |
> |
T\rangle\sim$200K using certain energy flux respectively.} |
| 690 |
|
|
| 691 |
|
\begin{tabular}{cccc} |
| 692 |
|
\hline\hline |
| 693 |
< |
Thiol & & & \\ |
| 694 |
< |
coverage (\%) & hexane & hexane-D & toluene \\ |
| 693 |
> |
Thiol & \multicolumn{3}{c}{$G$(MW/m$^2$/K)} \\ |
| 694 |
> |
coverage (\%) & hexane & hexane(D) & toluene \\ |
| 695 |
|
\hline |
| 696 |
< |
0.0 & 46.5 & 43.9 & 70.1 \\ |
| 697 |
< |
25.0 & 151 & 153 & 249 \\ |
| 698 |
< |
50.0 & 172 & 182 & 214 \\ |
| 699 |
< |
75.0 & 242 & 229 & 244 \\ |
| 700 |
< |
88.9 & 178 & - & - \\ |
| 701 |
< |
100.0 & 137 & 153 & 187 \\ |
| 696 |
> |
0.0 & 46.5() & 43.9() & 70.1() \\ |
| 697 |
> |
25.0 & 151() & 153() & 249() \\ |
| 698 |
> |
50.0 & 172() & 182() & 214() \\ |
| 699 |
> |
75.0 & 242() & 229() & 244() \\ |
| 700 |
> |
88.9 & 178() & - & - \\ |
| 701 |
> |
100.0 & 137() & 153() & 187() \\ |
| 702 |
|
\hline\hline |
| 703 |
|
\end{tabular} |
| 704 |
|
\label{tlnUhxnUhxnD} |
| 709 |
|
\subsection{Influence of Chosen Molecule Model on $G$} |
| 710 |
|
[MAY COMBINE W MECHANISM STUDY] |
| 711 |
|
|
| 712 |
< |
For the all-atom model, the liquid hexane phase was not stable under NPT |
| 713 |
< |
conditions. Therefore, the simulation length scale parameters are |
| 714 |
< |
adopted from previous equilibration results of the united-atom model |
| 715 |
< |
at 200K. Table \ref{AuThiolHexaneAA} shows the results of these |
| 716 |
< |
simulations. The conductivity values calculated with full capping |
| 717 |
< |
agent coverage are substantially larger than observed in the |
| 718 |
< |
united-atom model, and is even higher than predicted by |
| 678 |
< |
experiments. It is possible that our parameters for metal-non-metal |
| 679 |
< |
particle interactions lead to an overestimate of the interfacial |
| 680 |
< |
thermal conductivity, although the active C-H vibrations in the |
| 681 |
< |
all-atom model (which should not be appreciably populated at normal |
| 682 |
< |
temperatures) could also account for this high conductivity. The major |
| 683 |
< |
thermal transfer barrier of Au/butanethiol/hexane interface is between |
| 684 |
< |
the liquid phase and the capping agent, so extra degrees of freedom |
| 685 |
< |
such as the C-H vibrations could enhance heat exchange between these |
| 686 |
< |
two phases and result in a much higher conductivity. |
| 712 |
> |
In addition to UA solvent/capping agent models, AA models are included |
| 713 |
> |
in our simulations as well. Besides simulations of the same (UA or AA) |
| 714 |
> |
model for solvent and capping agent, different models can be applied |
| 715 |
> |
to different components. Furthermore, regardless of models chosen, |
| 716 |
> |
either the solvent or the capping agent can be deuterated, similar to |
| 717 |
> |
the previous section. Table \ref{modelTest} summarizes the results of |
| 718 |
> |
these studies. |
| 719 |
|
|
| 720 |
+ |
[MORE DATA; ERROR ESTIMATE] |
| 721 |
|
\begin{table*} |
| 722 |
|
\begin{minipage}{\linewidth} |
| 723 |
|
\begin{center} |
| 724 |
|
|
| 725 |
|
\caption{Computed interfacial thermal conductivity ($G$ and |
| 726 |
< |
$G^\prime$) values for the Au/butanethiol/hexane interface |
| 727 |
< |
with all-atom model and different capping agent coverage at |
| 728 |
< |
200K using a range of energy fluxes.} |
| 726 |
> |
$G^\prime$) values for interfaces using various models for |
| 727 |
> |
solvent and capping agent (or without capping agent) at |
| 728 |
> |
$\langle T\rangle\sim$200K.} |
| 729 |
|
|
| 730 |
< |
\begin{tabular}{cccc} |
| 730 |
> |
\begin{tabular}{ccccc} |
| 731 |
|
\hline\hline |
| 732 |
< |
Thiol & $J_z$ & $G$ & $G^\prime$ \\ |
| 733 |
< |
coverage (\%) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 732 |
> |
Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ |
| 733 |
> |
(or bare surface) & model & (GW/m$^2$) & |
| 734 |
> |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 735 |
|
\hline |
| 736 |
< |
0.0 & 0.95 & 28.5 & 27.2 \\ |
| 737 |
< |
& 1.88 & 30.3 & 28.9 \\ |
| 738 |
< |
100.0 & 2.87 & 551 & 294 \\ |
| 739 |
< |
& 3.81 & 494 & 193 \\ |
| 736 |
> |
UA & AA hexane & 1.94 & 135() & 129() \\ |
| 737 |
> |
& & 2.86 & 126() & 115() \\ |
| 738 |
> |
& AA toluene & 1.89 & 200() & 149() \\ |
| 739 |
> |
AA & UA hexane & 1.94 & 116() & 129() \\ |
| 740 |
> |
& AA hexane & 3.76 & 451() & 378() \\ |
| 741 |
> |
& & 4.71 & 432() & 334() \\ |
| 742 |
> |
& AA toluene & 3.79 & 487() & 290() \\ |
| 743 |
> |
AA(D) & UA hexane & 1.94 & 158() & 172() \\ |
| 744 |
> |
bare & AA hexane & 0.96 & 31.0() & 29.4() \\ |
| 745 |
|
\hline\hline |
| 746 |
|
\end{tabular} |
| 747 |
< |
\label{AuThiolHexaneAA} |
| 747 |
> |
\label{modelTest} |
| 748 |
|
\end{center} |
| 749 |
|
\end{minipage} |
| 750 |
|
\end{table*} |
| 751 |
|
|
| 752 |
+ |
To facilitate direct comparison, the same system with differnt models |
| 753 |
+ |
for different components uses the same length scale for their |
| 754 |
+ |
simulation cells. Without the presence of capping agent, using |
| 755 |
+ |
different models for hexane yields similar results for both $G$ and |
| 756 |
+ |
$G^\prime$, and these two definitions agree with eath other very |
| 757 |
+ |
well. This indicates very weak interaction between the metal and the |
| 758 |
+ |
solvent, and is a typical case for acoustic impedance mismatch between |
| 759 |
+ |
these two phases. |
| 760 |
|
|
| 761 |
< |
significant conductance enhancement compared to the gold/water |
| 762 |
< |
interface without capping agent and agree with available experimental |
| 763 |
< |
data. This indicates that the metal-metal potential, though not |
| 764 |
< |
predicting an accurate bulk metal thermal conductivity, does not |
| 765 |
< |
greatly interfere with the simulation of the thermal conductance |
| 766 |
< |
behavior across a non-metal interface. |
| 761 |
> |
As for Au(111) surfaces completely covered by butanethiols, the choice |
| 762 |
> |
of models for capping agent and solvent could impact the measurement |
| 763 |
> |
of $G$ and $G^\prime$ quite significantly. For Au-butanethiol/hexane |
| 764 |
> |
interfaces, using AA model for both butanethiol and hexane yields |
| 765 |
> |
substantially higher conductivity values than using UA model for at |
| 766 |
> |
least one component of the solvent and capping agent, which exceeds |
| 767 |
> |
the upper bond of experimental value range. This is probably due to |
| 768 |
> |
the classically treated C-H vibrations in the AA model, which should |
| 769 |
> |
not be appreciably populated at normal temperatures. In comparison, |
| 770 |
> |
once either the hexanes or the butanethiols are deuterated, one can |
| 771 |
> |
see a significantly lower $G$ and $G^\prime$. In either of these |
| 772 |
> |
cases, the C-H(D) vibrational overlap between the solvent and the |
| 773 |
> |
capping agent is removed. [MAY NEED FIGURE] Conclusively, the |
| 774 |
> |
improperly treated C-H vibration in the AA model produced |
| 775 |
> |
over-predicted results accordingly. Compared to the AA model, the UA |
| 776 |
> |
model yields more reasonable results with higher computational |
| 777 |
> |
efficiency. |
| 778 |
|
|
| 779 |
< |
% The results show that the two definitions used for $G$ yield |
| 780 |
< |
% comparable values, though $G^\prime$ tends to be smaller. |
| 779 |
> |
However, for Au-butanethiol/toluene interfaces, having the AA |
| 780 |
> |
butanethiol deuterated did not yield a significant change in the |
| 781 |
> |
measurement results. |
| 782 |
> |
. , so extra degrees of freedom |
| 783 |
> |
such as the C-H vibrations could enhance heat exchange between these |
| 784 |
> |
two phases and result in a much higher conductivity. |
| 785 |
|
|
| 786 |
+ |
|
| 787 |
+ |
Although the QSC model for Au is known to predict an overly low value |
| 788 |
+ |
for bulk metal gold conductivity[CITE NIVSRNEMD], our computational |
| 789 |
+ |
results for $G$ and $G^\prime$ do not seem to be affected by this |
| 790 |
+ |
drawback of the model for metal. Instead, the modeling of interfacial |
| 791 |
+ |
thermal transport behavior relies mainly on an accurate description of |
| 792 |
+ |
the interactions between components occupying the interfaces. |
| 793 |
+ |
|
| 794 |
|
\subsection{Mechanism of Interfacial Thermal Conductance Enhancement |
| 795 |
|
by Capping Agent} |
| 796 |
< |
[MAY INTRODUCE PROTOCOL IN METHODOLOGY/COMPUTATIONAL DETAIL] |
| 796 |
> |
%OR\subsection{Vibrational spectrum study on conductance mechanism} |
| 797 |
|
|
| 798 |
+ |
[MAY INTRODUCE PROTOCOL IN METHODOLOGY/COMPUTATIONAL DETAIL, EQN'S] |
| 799 |
|
|
| 729 |
– |
%subsubsection{Vibrational spectrum study on conductance mechanism} |
| 800 |
|
To investigate the mechanism of this interfacial thermal conductance, |
| 801 |
|
the vibrational spectra of various gold systems were obtained and are |
| 802 |
|
shown as in the upper panel of Fig. \ref{vibration}. To obtain these |
| 803 |
|
spectra, one first runs a simulation in the NVE ensemble and collects |
| 804 |
|
snapshots of configurations; these configurations are used to compute |
| 805 |
|
the velocity auto-correlation functions, which is used to construct a |
| 806 |
< |
power spectrum via a Fourier transform. The gold surfaces covered by |
| 737 |
< |
butanethiol molecules exhibit an additional peak observed at a |
| 738 |
< |
frequency of $\sim$170cm$^{-1}$, which is attributed to the vibration |
| 739 |
< |
of the S-Au bond. This vibration enables efficient thermal transport |
| 740 |
< |
from surface Au atoms to the capping agents. Simultaneously, as shown |
| 741 |
< |
in the lower panel of Fig. \ref{vibration}, the large overlap of the |
| 742 |
< |
vibration spectra of butanethiol and hexane in the all-atom model, |
| 743 |
< |
including the C-H vibration, also suggests high thermal exchange |
| 744 |
< |
efficiency. The combination of these two effects produces the drastic |
| 745 |
< |
interfacial thermal conductance enhancement in the all-atom model. |
| 806 |
> |
power spectrum via a Fourier transform. |
| 807 |
|
|
| 808 |
+ |
The gold surfaces covered by |
| 809 |
+ |
butanethiol molecules, compared to bare gold surfaces, exhibit an |
| 810 |
+ |
additional peak observed at a frequency of $\sim$170cm$^{-1}$, which |
| 811 |
+ |
is attributed to the vibration of the S-Au bond. This vibration |
| 812 |
+ |
enables efficient thermal transport from surface Au atoms to the |
| 813 |
+ |
capping agents. Simultaneously, as shown in the lower panel of |
| 814 |
+ |
Fig. \ref{vibration}, the large overlap of the vibration spectra of |
| 815 |
+ |
butanethiol and hexane in the all-atom model, including the C-H |
| 816 |
+ |
vibration, also suggests high thermal exchange efficiency. The |
| 817 |
+ |
combination of these two effects produces the drastic interfacial |
| 818 |
+ |
thermal conductance enhancement in the all-atom model. |
| 819 |
+ |
|
| 820 |
+ |
[MAY NEED TO CONVERT TO JPEG] |
| 821 |
|
\begin{figure} |
| 822 |
|
\includegraphics[width=\linewidth]{vibration} |
| 823 |
|
\caption{Vibrational spectra obtained for gold in different |
| 825 |
|
all-atom model (lower panel).} |
| 826 |
|
\label{vibration} |
| 827 |
|
\end{figure} |
| 754 |
– |
% MAY NEED TO CONVERT TO JPEG |
| 828 |
|
|
| 829 |
+ |
[COMPARISON OF TWO G'S; AU SLAB WIDTHS; ETC] |
| 830 |
+ |
% The results show that the two definitions used for $G$ yield |
| 831 |
+ |
% comparable values, though $G^\prime$ tends to be smaller. |
| 832 |
+ |
|
| 833 |
|
\section{Conclusions} |
| 834 |
+ |
The NIVS algorithm we developed has been applied to simulations of |
| 835 |
+ |
Au-butanethiol surfaces with organic solvents. This algorithm allows |
| 836 |
+ |
effective unphysical thermal flux transferred between the metal and |
| 837 |
+ |
the liquid phase. With the flux applied, we were able to measure the |
| 838 |
+ |
corresponding thermal gradient and to obtain interfacial thermal |
| 839 |
+ |
conductivities. Our simulations have seen significant conductance |
| 840 |
+ |
enhancement with the presence of capping agent, compared to the bare |
| 841 |
+ |
gold/liquid interfaces. The acoustic impedance mismatch between the |
| 842 |
+ |
metal and the liquid phase is effectively eliminated by proper capping |
| 843 |
+ |
agent. Furthermore, the coverage precentage of the capping agent plays |
| 844 |
+ |
an important role in the interfacial thermal transport process. |
| 845 |
|
|
| 846 |
+ |
Our measurement results, particularly of the UA models, agree with |
| 847 |
+ |
available experimental data. This indicates that our force field |
| 848 |
+ |
parameters have a nice description of the interactions between the |
| 849 |
+ |
particles at the interfaces. AA models tend to overestimate the |
| 850 |
+ |
interfacial thermal conductance in that the classically treated C-H |
| 851 |
+ |
vibration would be overly sampled. Compared to the AA models, the UA |
| 852 |
+ |
models have higher computational efficiency with satisfactory |
| 853 |
+ |
accuracy, and thus are preferable in interfacial thermal transport |
| 854 |
+ |
modelings. |
| 855 |
|
|
| 856 |
< |
[NECESSITY TO STUDY THERMAL CONDUCTANCE IN NANOCRYSTAL STRUCTURE]\cite{vlugt:cpc2007154} |
| 856 |
> |
Vlugt {\it et al.} has investigated the surface thiol structures for |
| 857 |
> |
nanocrystal gold and pointed out that they differs from those of the |
| 858 |
> |
Au(111) surface\cite{vlugt:cpc2007154}. This difference might lead to |
| 859 |
> |
change of interfacial thermal transport behavior as well. To |
| 860 |
> |
investigate this problem, an effective means to introduce thermal flux |
| 861 |
> |
and measure the corresponding thermal gradient is desirable for |
| 862 |
> |
simulating structures with spherical symmetry. |
| 863 |
|
|
| 864 |
+ |
|
| 865 |
|
\section{Acknowledgments} |
| 866 |
|
Support for this project was provided by the National Science |
| 867 |
|
Foundation under grant CHE-0848243. Computational time was provided by |
