| 73 |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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|
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\section{Introduction} |
| 76 |
< |
Interfacial thermal conductance is extensively studied both |
| 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. |
| 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 |
< |
Heat conductance of molecular and nano-scale interfaces will be |
| 86 |
< |
affected by the chemical details of the surface. Experimentally, |
| 87 |
< |
various interfaces have been investigated for their thermal |
| 88 |
< |
conductance properties. Wang {\it et al.} studied heat transport |
| 89 |
< |
through long-chain hydrocarbon monolayers on gold substrate at |
| 90 |
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individual molecular level\cite{Wang10082007}; Schmidt {\it et al.} |
| 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 |
| 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 |
< |
commonly barriers for heat transport, Alper {\it et al.} suggested |
| 95 |
< |
that specific ligands (capping agents) could completely eliminate this |
| 96 |
< |
barrier ($G\rightarrow\infty$)\cite{doi:10.1021/la904855s}. |
| 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 |
| 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 |
| 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 has yet to be studied. |
| 108 |
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The comparatively low thermal flux through interfaces is |
| 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 |
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|
difficult to measure with Equilibrium MD or forward NEMD simulation |
| 110 |
< |
methods. Therefore, the Reverse NEMD (RNEMD) methods would have the |
| 111 |
< |
advantage of having this difficult to measure flux known when studying |
| 112 |
< |
the thermal transport across interfaces, given that the simulation |
| 113 |
< |
methods being able to effectively apply an unphysical flux in |
| 114 |
< |
non-homogeneous systems. |
| 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. |
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|
|
| 120 |
< |
Recently, we have developed the Non-Isotropic Velocity Scaling (NIVS) |
| 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 |
| 134 |
|
thermal transport across these interfaces was studied and the |
| 135 |
|
underlying mechanism for the phenomena was investigated. |
| 136 |
|
|
| 136 |
– |
[MAY ADD WHY STUDY AU-THIOL SURFACE; CITE SHAOYI JIANG] |
| 137 |
– |
|
| 137 |
|
\section{Methodology} |
| 138 |
|
\subsection{Imposd-Flux Methods in MD Simulations} |
| 139 |
< |
Steady state MD simulations has the advantage that not many |
| 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 including low conductance |
| 142 |
< |
interfaces one must have a method capable of generating or measuring |
| 143 |
< |
relatively small fluxes, compared to those required for bulk |
| 144 |
< |
conductivity. This requirement makes the calculation even more |
| 145 |
< |
difficult for those slowly-converging equilibrium |
| 146 |
< |
methods\cite{Viscardy:2007lq}. Forward methods may impose gradient, |
| 147 |
< |
but in interfacial conditions it is not clear what behavior to impose |
| 148 |
< |
at the interfacial boundaries. Imposed-flux reverse non-equilibrium |
| 149 |
< |
methods\cite{MullerPlathe:1997xw} have the flux set {\it a priori} and |
| 150 |
< |
the thermal response becomes easier to measure than the flux. Although |
| 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 to |
| 158 |
< |
non-equilibrium MD simulations is able to impose a wide range of |
| 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 |
| 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 |
< |
Given a system with thermal gradients and the corresponding thermal |
| 177 |
< |
flux, for interfaces with a relatively low interfacial conductance, |
| 178 |
< |
the bulk regions on either side of an interface rapidly come to a |
| 179 |
< |
state in which the two phases have relatively homogeneous (but |
| 180 |
< |
distinct) temperatures. The interfacial thermal conductivity $G$ can |
| 181 |
< |
therefore 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, there are two |
| 193 |
< |
ways to define $G$. |
| 194 |
< |
|
| 195 |
< |
One way is to assume the temperature is discrete on the two sides of |
| 196 |
< |
the interface. $G$ can be calculated using the applied thermal flux |
| 197 |
< |
$J$ and the maximum temperature difference measured along the thermal |
| 199 |
< |
gradient max($\Delta T$), which occurs at the Gibbs deviding surface |
| 200 |
< |
(Figure \ref{demoPic}): |
| 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 |
|
|
| 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 methods 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 have been used for calculation and |
| 236 |
< |
are compared in the results. |
| 237 |
< |
|
| 238 |
< |
To compare the above definitions ($G$ and $G^\prime$), we have modeled |
| 239 |
< |
a metal slab with its (111) surfaces perpendicular to the $z$-axis of |
| 242 |
< |
our simulation cells. Both with and without capping agents on the |
| 243 |
< |
surfaces, the metal slab is solvated with simple organic solvents, as |
| 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 |
|
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 are able to obtain a temperature profile and |
| 246 |
< |
its spatial derivatives. These quantities enable the evaluation of the |
| 247 |
< |
interfacial thermal conductance of a surface. Figure \ref{gradT} is an |
| 248 |
< |
example of how an applied thermal flux can be used to obtain the 1st |
| 253 |
< |
and 2nd derivatives of the temperature profile. |
| 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} |
| 251 |
|
\includegraphics[width=\linewidth]{gradT} |
| 259 |
|
\section{Computational Details} |
| 260 |
|
\subsection{Simulation Protocol} |
| 261 |
|
The NIVS algorithm has been implemented in our MD simulation code, |
| 262 |
< |
OpenMD\cite{Meineke:2005gd,openmd}, and was used for our |
| 263 |
< |
simulations. Different metal slab thickness (layer numbers of Au) was |
| 264 |
< |
simulated. Metal slabs were first equilibrated under atmospheric |
| 265 |
< |
pressure (1 atm) and a desired temperature (e.g. 200K). After |
| 266 |
< |
equilibration, butanethiol capping agents were placed at three-fold |
| 267 |
< |
hollow sites on the Au(111) surfaces. These sites could be either a |
| 268 |
< |
{\it fcc} or {\it hcp} site. However, Hase {\it et al.} found that |
| 269 |
< |
they are equivalent in a heat transfer process\cite{hase:2010}, so |
| 275 |
< |
they are not distinguished in our study. The maximum butanethiol |
| 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 different coverages was derived by evenly eliminating |
| 274 |
< |
butanethiols on the surfaces, and was investigated in order to study |
| 275 |
< |
the relation between coverage and interfacial conductance. |
| 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 |
|
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 would not noticeably affect the sampling of a |
| 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. [MAY NEED SNAPSHOTS] |
| 284 |
> |
configurations explored in the simulations. |
| 285 |
|
|
| 286 |
< |
After the modified Au-butanethiol surface systems were equilibrated |
| 287 |
< |
under canonical ensemble, organic solvent molecules were packed in the |
| 288 |
< |
previously empty part of the simulation cells\cite{packmol}. Two |
| 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 a planar shape (toluene), and one which has |
| 291 |
< |
similar vibrational frequencies and chain-like shape ({\it n}-hexane). |
| 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 space filled by solvent molecules, i.e. the gap between |
| 295 |
< |
periodically repeated Au-butanethiol surfaces should be carefully |
| 296 |
< |
chosen. A very long length scale for the thermal gradient axis ($z$) |
| 301 |
< |
may cause excessively hot or cold temperatures in the middle of the |
| 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 $35 \sim 75$\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 are further equilibrated with the |
| 305 |
< |
$x$ and $y$ dimensions fixed, only allowing length scale change in $z$ |
| 306 |
< |
dimension. This is to ensure that the equilibration of liquid phase |
| 307 |
< |
does not affect the metal crystal structure in $x$ and $y$ dimensions. |
| 308 |
< |
To investigate this effect, comparisons were made with simulations |
| 309 |
< |
that allow changes of $L_x$ and $L_y$ during NPT equilibration, and |
| 310 |
< |
the results are shown in later sections. After ensuring the liquid |
| 311 |
< |
phase reaches equilibrium at atmospheric pressure (1 atm), further |
| 312 |
< |
equilibration are followed under NVT and then NVE ensembles. |
| 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. After this induced |
| 320 |
< |
temperature gradient is stablized, the temperature profile of the |
| 321 |
< |
simulation cell is recorded. To do this, the simulation cell is |
| 322 |
< |
devided evenly into $N$ slabs along the $z$-axis and $N$ is maximized |
| 328 |
< |
for highest possible spatial resolution but not too many to have some |
| 329 |
< |
slabs empty most of the time. The average temperatures of each slab |
| 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: |
| 326 |
< |
\begin{equation} |
| 334 |
< |
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
| 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. And |
| 336 |
< |
each equilibration / stabilization step usually takes 100 ps, or |
| 337 |
< |
longer, if necessary. |
| 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. Figure \ref{demoMol} |
| 341 |
< |
demonstrates the sites defined for both United-Atom and All-Atom |
| 342 |
< |
models of the organic solvent and capping agent molecules in our |
| 343 |
< |
simulations. Force field parameter descriptions are needed for |
| 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 |
|
|
| 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} (UA) and |
| 362 |
< |
\protect\cite{OPLSAA} (AA). Cross-interactions with the Au atoms are given |
| 363 |
< |
in Table \ref{MnM}.} |
| 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 |
| 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 |
< |
For both solvent molecules, straight chain {\it n}-hexane and aromatic |
| 364 |
< |
toluene, United-Atom (UA) and All-Atom (AA) models are used |
| 365 |
< |
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, 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 not separated by more than 3 bonds. Otherwise, for non-bonded |
| 371 |
< |
interactions, Lennard-Jones potentials are used. [CHECK CITATION] |
| 370 |
> |
sites closer than 3 bonds. For non-bonded interactions, Lennard-Jones |
| 371 |
> |
potentials are used. |
| 372 |
|
|
| 373 |
< |
By eliminating explicit hydrogen atoms, these models are simple and |
| 374 |
< |
computationally efficient, while maintains good accuracy. However, the |
| 375 |
< |
TraPPE-UA for alkanes is known to predict a lower boiling point than |
| 376 |
< |
experimental values. Considering that after an unphysical thermal flux |
| 377 |
< |
is applied to a system, the temperature of ``hot'' area in the liquid |
| 378 |
< |
phase would be significantly higher than the average of the system, to |
| 379 |
< |
prevent over heating and boiling of the liquid phase, the average |
| 380 |
< |
temperature in our simulations should be much lower than the liquid |
| 381 |
< |
boiling point. |
| 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 model, the non-bonded potentials between |
| 384 |
< |
inter-molecular sites have a similar Lennard-Jones formulation. For |
| 385 |
< |
intra-molecular interactions, considering the stiffness of the benzene |
| 386 |
< |
ring, rigid body constraints are applied for further computational |
| 387 |
< |
efficiency. All bonds in the benzene ring and between the ring and the |
| 398 |
< |
methyl group remain rigid during the progress of simulations. |
| 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. Additional explicit hydrogen sites were |
| 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, the United Force Field developed by |
| 395 |
< |
Rapp\'{e} {\it et al.}\cite{doi:10.1021/ja00051a040} is |
| 396 |
< |
adopted. Without the rigid body constraints, bonding interactions were |
| 408 |
< |
included. For the aromatic ring, improper torsions (inversions) were |
| 409 |
< |
added as an extra potential for maintaining the planar shape. |
| 410 |
< |
[CHECK CITATION] |
| 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 Luedtke and |
| 406 |
< |
Landman\cite{landman:1998}[CHECK CITATION] |
| 407 |
< |
and modify parameters for its neighbor C |
| 408 |
< |
atom for charge balance in the molecule. Note that the model choice |
| 409 |
< |
(UA or AA) of capping agent can be different from the |
| 410 |
< |
solvent. Regardless of model choice, the force field parameters for |
| 425 |
< |
interactions between capping agent and solvent can be derived using |
| 426 |
< |
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}} |
| 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 capping |
| 417 |
< |
agent and solvent particles, we refer to an adsorption study of alkyl |
| 418 |
< |
thiols on gold surfaces by Vlugt {\it et |
| 419 |
< |
al.}\cite{vlugt:cpc2007154} They fitted an effective Lennard-Jones |
| 420 |
< |
form of potential parameters for the interaction between Au and |
| 421 |
< |
pseudo-atoms CH$_x$ and S based on a well-established and widely-used |
| 422 |
< |
effective potential of Hautman and Klein\cite{hautman:4994} for the |
| 423 |
< |
Au(111) surface. As our simulations require the gold lattice slab to |
| 424 |
< |
be non-rigid so that it could accommodate kinetic energy for thermal |
| 441 |
< |
transport study purpose, the pair-wise form of potentials is |
| 442 |
< |
preferred. |
| 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 |
< |
Besides, the potentials developed from {\it ab initio} calculations by |
| 427 |
< |
Leng {\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
| 428 |
< |
interactions between Au and aromatic C/H atoms in toluene. A set of |
| 429 |
< |
pseudo Lennard-Jones parameters were provided for Au in their force |
| 430 |
< |
fields. By using the Mixing Rule, this can be used to derive pair-wise |
| 431 |
< |
potentials for non-bonded interactions between Au and non-metal sites. |
| 432 |
< |
|
| 433 |
< |
However, the Lennard-Jones parameters between Au and other types of |
| 434 |
< |
particles, such as All-Atom normal alkanes in our simulations are not |
| 435 |
< |
yet well-established. For these interactions, we attempt to derive |
| 436 |
< |
their parameters using the Mixing Rule. To do this, Au pseudo |
| 455 |
< |
Lennard-Jones parameters for ``Metal-non-Metal'' (MnM) interactions |
| 456 |
< |
were first extracted from the Au-CH$_x$ parameters by applying the |
| 457 |
< |
Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM'' |
| 458 |
< |
parameters in our simulations. |
| 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 |
|
|
| 438 |
|
\begin{table*} |
| 439 |
|
\begin{minipage}{\linewidth} |
| 470 |
|
\end{minipage} |
| 471 |
|
\end{table*} |
| 472 |
|
|
| 473 |
< |
\subsection{Vibrational Spectrum} |
| 473 |
> |
\subsection{Vibrational Power Spectrum} |
| 474 |
> |
|
| 475 |
|
To investigate the mechanism of interfacial thermal conductance, the |
| 476 |
< |
vibrational spectrum is utilized as a complementary tool. Vibrational |
| 477 |
< |
spectra were taken for individual components in different |
| 478 |
< |
simulations. To obtain these spectra, simulations were run after |
| 479 |
< |
equilibration, in the NVE ensemble. Snapshots of configurations were |
| 480 |
< |
collected at a frequency that is higher than that of the fastest |
| 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: |
| 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 |
< |
Followed by Fourier transforms, the power spectrum can be constructed: |
| 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) = \int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt |
| 491 |
< |
\label{fourier} |
| 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} |
| 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. [NEED |
| 732 |
> |
case, $G$ decrease could not be offset but instead accelerated. [MAY NEED |
| 733 |
|
SNAPSHOT SHOWING THE PHENOMENA / SLAB DENSITY ANALYSIS] |
| 734 |
|
|
| 735 |
|
A comparison of the results obtained from differenet organic solvents |
| 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{vlugt:cpc2007154}. This difference might lead to |
| 959 |
< |
change of interfacial thermal transport behavior as well. To |
| 960 |
< |
investigate this problem, an effective means to introduce thermal flux |
| 961 |
< |
and measure the corresponding thermal gradient is desirable for |
| 962 |
< |
simulating structures with spherical symmetry. |
| 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 |
