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\begin{document} |
| 30 |
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|
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\title{Simulating interfacial thermal conductance at metal-solvent |
| 32 |
< |
interfaces: the role of chemical capping agents} |
| 31 |
> |
\title{Simulating Interfacial Thermal Conductance at Metal-Solvent |
| 32 |
> |
Interfaces: the Role of Chemical Capping Agents} |
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|
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\author{Shenyu Kuang and J. Daniel |
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Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\ |
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\begin{doublespace} |
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|
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\begin{abstract} |
| 47 |
+ |
With the Non-Isotropic Velocity Scaling (NIVS) approach to Reverse |
| 48 |
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Non-Equilibrium Molecular Dynamics (RNEMD) it is possible to impose |
| 49 |
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an unphysical thermal flux between different regions of |
| 50 |
+ |
inhomogeneous systems such as solid / liquid interfaces. We have |
| 51 |
+ |
applied NIVS to compute the interfacial thermal conductance at a |
| 52 |
+ |
metal / organic solvent interface that has been chemically capped by |
| 53 |
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butanethiol molecules. Our calculations suggest that vibrational |
| 54 |
+ |
coupling between the metal and liquid phases is enhanced by the |
| 55 |
+ |
capping agents, leading to a greatly enhanced conductivity at the |
| 56 |
+ |
interface. Specifically, the chemical bond between the metal and |
| 57 |
+ |
the capping agent introduces a vibrational overlap that is not |
| 58 |
+ |
present without the capping agent, and the overlap between the |
| 59 |
+ |
vibrational spectra (metal to cap, cap to solvent) provides a |
| 60 |
+ |
mechanism for rapid thermal transport across the interface. Our |
| 61 |
+ |
calculations also suggest that this is a non-monotonic function of |
| 62 |
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the fractional coverage of the surface, as moderate coverages allow |
| 63 |
+ |
diffusive heat transport of solvent molecules that have been in |
| 64 |
+ |
close contact with the capping agent. |
| 65 |
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|
| 66 |
< |
With the Non-Isotropic Velocity Scaling algorithm (NIVS) we have |
| 67 |
< |
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 |
< |
|
| 66 |
> |
Keywords: non-equilibrium, molecular dynamics, vibrational overlap, |
| 67 |
> |
coverage dependent. |
| 68 |
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\end{abstract} |
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|
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\newpage |
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|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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|
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\section{Introduction} |
| 79 |
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Due to the importance of heat flow in nanotechnology, interfacial |
| 80 |
< |
thermal conductance has been studied extensively both experimentally |
| 81 |
< |
and computationally.\cite{cahill:793} Unlike bulk materials, nanoscale |
| 82 |
< |
materials have a significant fraction of their atoms at interfaces, |
| 83 |
< |
and the chemical details of these interfaces govern the heat transfer |
| 84 |
< |
behavior. Furthermore, the interfaces are |
| 79 |
> |
Due to the importance of heat flow (and heat removal) in |
| 80 |
> |
nanotechnology, interfacial thermal conductance has been studied |
| 81 |
> |
extensively both experimentally and computationally.\cite{cahill:793} |
| 82 |
> |
Nanoscale materials have a significant fraction of their atoms at |
| 83 |
> |
interfaces, and the chemical details of these interfaces govern the |
| 84 |
> |
thermal transport properties. Furthermore, the interfaces are often |
| 85 |
|
heterogeneous (e.g. solid - liquid), which provides a challenge to |
| 86 |
< |
traditional methods developed for homogeneous systems. |
| 86 |
> |
computational methods which have been developed for homogeneous or |
| 87 |
> |
bulk systems. |
| 88 |
|
|
| 89 |
< |
Experimentally, various interfaces have been investigated for their |
| 90 |
< |
thermal conductance. Wang {\it et al.} studied heat transport through |
| 91 |
< |
long-chain hydrocarbon monolayers on gold substrate at individual |
| 92 |
< |
molecular level,\cite{Wang10082007} Schmidt {\it et al.} studied the |
| 93 |
< |
role of CTAB on thermal transport between gold nanorods and |
| 94 |
< |
solvent,\cite{doi:10.1021/jp8051888} and Juv\'e {\it et al.} studied |
| 95 |
< |
the cooling dynamics, which is controlled by thermal interface |
| 96 |
< |
resistence of glass-embedded metal |
| 89 |
> |
Experimentally, the thermal properties of a number of interfaces have |
| 90 |
> |
been investigated. Cahill and coworkers studied nanoscale thermal |
| 91 |
> |
transport from metal nanoparticle/fluid interfaces, to epitaxial |
| 92 |
> |
TiN/single crystal oxides interfaces, and hydrophilic and hydrophobic |
| 93 |
> |
interfaces between water and solids with different self-assembled |
| 94 |
> |
monolayers.\cite{Wilson:2002uq,PhysRevB.67.054302,doi:10.1021/jp048375k,PhysRevLett.96.186101} |
| 95 |
> |
Wang {\it et al.} studied heat transport through long-chain |
| 96 |
> |
hydrocarbon monolayers on gold substrate at individual molecular |
| 97 |
> |
level,\cite{Wang10082007} Schmidt {\it et al.} studied the role of |
| 98 |
> |
cetyltrimethylammonium bromide (CTAB) on the thermal transport between |
| 99 |
> |
gold nanorods and solvent,\cite{doi:10.1021/jp8051888} and Juv\'e {\it |
| 100 |
> |
et al.} studied the cooling dynamics, which is controlled by thermal |
| 101 |
> |
interface resistance of glass-embedded metal |
| 102 |
|
nanoparticles.\cite{PhysRevB.80.195406} Although interfaces are |
| 103 |
|
normally considered barriers for heat transport, Alper {\it et al.} |
| 104 |
|
suggested that specific ligands (capping agents) could completely |
| 105 |
|
eliminate this barrier |
| 106 |
|
($G\rightarrow\infty$).\cite{doi:10.1021/la904855s} |
| 107 |
|
|
| 108 |
< |
Theoretical and computational models have also been used to study the |
| 108 |
> |
The acoustic mismatch model for interfacial conductance utilizes the |
| 109 |
> |
acoustic impedance ($Z_a = \rho_a v^s_a$) on both sides of the |
| 110 |
> |
interface.\cite{schwartz} Here, $\rho_a$ and $v^s_a$ are the density |
| 111 |
> |
and speed of sound in material $a$. The phonon transmission |
| 112 |
> |
probability at the $a-b$ interface is |
| 113 |
> |
\begin{equation} |
| 114 |
> |
t_{ab} = \frac{4 Z_a Z_b}{\left(Z_a + Z_b \right)^2}, |
| 115 |
> |
\end{equation} |
| 116 |
> |
and the interfacial conductance can then be approximated as |
| 117 |
> |
\begin{equation} |
| 118 |
> |
G_{ab} \approx \frac{1}{4} C_D v_D t_{ab} |
| 119 |
> |
\end{equation} |
| 120 |
> |
where $C_D$ is the Debye heat capacity of the hot side, and $v_D$ is |
| 121 |
> |
the Debye phonon velocity ($1/v_D^3 = 1/3v_L^3 + 2/3 v_T^3$) where |
| 122 |
> |
$v_L$ and $v_T$ are the longitudinal and transverse speeds of sound, |
| 123 |
> |
respectively. For the Au/hexane and Au/toluene interfaces, the |
| 124 |
> |
acoustic mismatch model predicts room-temperature $G \approx 87 \mbox{ |
| 125 |
> |
and } 129$ MW m$^{-2}$ K$^{-1}$, respectively. However, it is not |
| 126 |
> |
clear how one might apply the acoustic mismatch model to a |
| 127 |
> |
chemically-modified surface, particularly when the acoustic properties |
| 128 |
> |
of a monolayer film may not be well characterized. |
| 129 |
> |
|
| 130 |
> |
More precise computational models have also been used to study the |
| 131 |
|
interfacial thermal transport in order to gain an understanding of |
| 132 |
|
this phenomena at the molecular level. Recently, Hase and coworkers |
| 133 |
|
employed Non-Equilibrium Molecular Dynamics (NEMD) simulations to |
| 137 |
|
measurements for heat conductance of interfaces between the capping |
| 138 |
|
monolayer on Au and a solvent phase have yet to be studied with their |
| 139 |
|
approach. The comparatively low thermal flux through interfaces is |
| 140 |
< |
difficult to measure with Equilibrium MD or forward NEMD simulation |
| 140 |
> |
difficult to measure with Equilibrium |
| 141 |
> |
MD\cite{doi:10.1080/0026897031000068578} or forward NEMD simulation |
| 142 |
|
methods. Therefore, the Reverse NEMD (RNEMD) |
| 143 |
< |
methods\cite{MullerPlathe:1997xw,kuang:164101} would have the |
| 144 |
< |
advantage of applying this difficult to measure flux (while measuring |
| 145 |
< |
the resulting gradient), given that the simulation methods being able |
| 146 |
< |
to effectively apply an unphysical flux in non-homogeneous systems. |
| 143 |
> |
methods\cite{MullerPlathe:1997xw,kuang:164101} would be advantageous |
| 144 |
> |
in that they {\it apply} the difficult to measure quantity (flux), |
| 145 |
> |
while {\it measuring} the easily-computed quantity (the thermal |
| 146 |
> |
gradient). This is particularly true for inhomogeneous interfaces |
| 147 |
> |
where it would not be clear how to apply a gradient {\it a priori}. |
| 148 |
|
Garde and coworkers\cite{garde:nl2005,garde:PhysRevLett2009} applied |
| 149 |
|
this approach to various liquid interfaces and studied how thermal |
| 150 |
< |
conductance (or resistance) is dependent on chemistry details of |
| 151 |
< |
interfaces, e.g. hydrophobic and hydrophilic aqueous interfaces. |
| 150 |
> |
conductance (or resistance) is dependent on chemical details of a |
| 151 |
> |
number of hydrophobic and hydrophilic aqueous interfaces. And |
| 152 |
> |
recently, Luo {\it et al.} studied the thermal conductance of |
| 153 |
> |
Au-SAM-Au junctions using the same approach, comparing to a constant |
| 154 |
> |
temperature difference method.\cite{Luo20101} While this latter |
| 155 |
> |
approach establishes more ideal Maxwell-Boltzmann distributions than |
| 156 |
> |
previous RNEMD methods, it does not guarantee momentum or kinetic |
| 157 |
> |
energy conservation. |
| 158 |
|
|
| 159 |
|
Recently, we have developed a Non-Isotropic Velocity Scaling (NIVS) |
| 160 |
|
algorithm for RNEMD simulations\cite{kuang:164101}. This algorithm |
| 174 |
|
underlying mechanism for the phenomena was investigated. |
| 175 |
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|
| 176 |
|
\section{Methodology} |
| 177 |
< |
\subsection{Imposd-Flux Methods in MD Simulations} |
| 177 |
> |
\subsection{Imposed-Flux Methods in MD Simulations} |
| 178 |
|
Steady state MD simulations have an advantage in that not many |
| 179 |
|
trajectories are needed to study the relationship between thermal flux |
| 180 |
|
and thermal gradients. For systems with low interfacial conductance, |
| 198 |
|
kinetic energy fluxes without obvious perturbation to the velocity |
| 199 |
|
distributions of the simulated systems. Furthermore, this approach has |
| 200 |
|
the advantage in heterogeneous interfaces in that kinetic energy flux |
| 201 |
< |
can be applied between regions of particles of arbitary identity, and |
| 201 |
> |
can be applied between regions of particles of arbitrary identity, and |
| 202 |
|
the flux will not be restricted by difference in particle mass. |
| 203 |
|
|
| 204 |
|
The NIVS algorithm scales the velocity vectors in two separate regions |
| 205 |
< |
of a simulation system with respective diagonal scaling matricies. To |
| 206 |
< |
determine these scaling factors in the matricies, a set of equations |
| 205 |
> |
of a simulation system with respective diagonal scaling matrices. To |
| 206 |
> |
determine these scaling factors in the matrices, a set of equations |
| 207 |
|
including linear momentum conservation and kinetic energy conservation |
| 208 |
|
constraints and target energy flux satisfaction is solved. With the |
| 209 |
|
scaling operation applied to the system in a set frequency, bulk |
| 226 |
|
where ${E_{total}}$ is the total imposed non-physical kinetic energy |
| 227 |
|
transfer during the simulation and ${\langle T_\mathrm{hot}\rangle}$ |
| 228 |
|
and ${\langle T_\mathrm{cold}\rangle}$ are the average observed |
| 229 |
< |
temperature of the two separated phases. |
| 229 |
> |
temperature of the two separated phases. For an applied flux $J_z$ |
| 230 |
> |
operating over a simulation time $t$ on a periodically-replicated slab |
| 231 |
> |
of dimensions $L_x \times L_y$, $E_{total} = J_z *(t)*(2 L_x L_y)$. |
| 232 |
|
|
| 233 |
|
When the interfacial conductance is {\it not} small, there are two |
| 234 |
|
ways to define $G$. One common way is to assume the temperature is |
| 235 |
|
discrete on the two sides of the interface. $G$ can be calculated |
| 236 |
|
using the applied thermal flux $J$ and the maximum temperature |
| 237 |
|
difference measured along the thermal gradient max($\Delta T$), which |
| 238 |
< |
occurs at the Gibbs deviding surface (Figure \ref{demoPic}): |
| 238 |
> |
occurs at the Gibbs dividing surface (Figure \ref{demoPic}). This is |
| 239 |
> |
known as the Kapitza conductance, which is the inverse of the Kapitza |
| 240 |
> |
resistance. |
| 241 |
|
\begin{equation} |
| 242 |
|
G=\frac{J}{\Delta T} |
| 243 |
|
\label{discreteG} |
| 248 |
|
\caption{Interfacial conductance can be calculated by applying an |
| 249 |
|
(unphysical) kinetic energy flux between two slabs, one located |
| 250 |
|
within the metal and another on the edge of the periodic box. The |
| 251 |
< |
system responds by forming a thermal response or a gradient. In |
| 252 |
< |
bulk liquids, this gradient typically has a single slope, but in |
| 253 |
< |
interfacial systems, there are distinct thermal conductivity |
| 254 |
< |
domains. The interfacial conductance, $G$ is found by measuring the |
| 255 |
< |
temperature gap at the Gibbs dividing surface, or by using second |
| 256 |
< |
derivatives of the thermal profile.} |
| 251 |
> |
system responds by forming a thermal gradient. In bulk liquids, |
| 252 |
> |
this gradient typically has a single slope, but in interfacial |
| 253 |
> |
systems, there are distinct thermal conductivity domains. The |
| 254 |
> |
interfacial conductance, $G$ is found by measuring the temperature |
| 255 |
> |
gap at the Gibbs dividing surface, or by using second derivatives of |
| 256 |
> |
the thermal profile.} |
| 257 |
|
\label{demoPic} |
| 258 |
|
\end{figure} |
| 259 |
|
|
| 260 |
< |
The other approach is to assume a continuous temperature profile along |
| 261 |
< |
the thermal gradient axis (e.g. $z$) and define $G$ at the point where |
| 262 |
< |
the magnitude of thermal conductivity ($\lambda$) change reaches its |
| 263 |
< |
maximum, given that $\lambda$ is well-defined throughout the space: |
| 264 |
< |
\begin{equation} |
| 265 |
< |
G^\prime = \Big|\frac{\partial\lambda}{\partial z}\Big| |
| 266 |
< |
= \Big|\frac{\partial}{\partial z}\left(-J_z\Big/ |
| 267 |
< |
\left(\frac{\partial T}{\partial z}\right)\right)\Big| |
| 268 |
< |
= |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
| 269 |
< |
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
| 270 |
< |
\label{derivativeG} |
| 271 |
< |
\end{equation} |
| 260 |
> |
Another approach is to assume that the temperature is continuous and |
| 261 |
> |
differentiable throughout the space. Given that $\lambda$ is also |
| 262 |
> |
differentiable, $G$ can be defined as its gradient ($\nabla\lambda$) |
| 263 |
> |
projected along a vector normal to the interface ($\mathbf{\hat{n}}$) |
| 264 |
> |
and evaluated at the interface location ($z_0$). This quantity, |
| 265 |
> |
\begin{align} |
| 266 |
> |
G^\prime &= \left(\nabla\lambda \cdot \mathbf{\hat{n}}\right)_{z_0} \\ |
| 267 |
> |
&= \frac{\partial}{\partial z}\left(-\frac{J_z}{ |
| 268 |
> |
\left(\frac{\partial T}{\partial z}\right)}\right)_{z_0} \\ |
| 269 |
> |
&= J_z\left(\frac{\partial^2 T}{\partial z^2}\right)_{z_0} \Big/ |
| 270 |
> |
\left(\frac{\partial T}{\partial z}\right)_{z_0}^2 \label{derivativeG} |
| 271 |
> |
\end{align} |
| 272 |
> |
has the same units as the common definition for $G$, and the maximum |
| 273 |
> |
of its magnitude denotes where thermal conductivity has the largest |
| 274 |
> |
change, i.e. the interface. In the geometry used in this study, the |
| 275 |
> |
vector normal to the interface points along the $z$ axis, as do |
| 276 |
> |
$\vec{J}$ and the thermal gradient. This yields the simplified |
| 277 |
> |
expressions in Eq. \ref{derivativeG}. |
| 278 |
|
|
| 279 |
|
With temperature profiles obtained from simulation, one is able to |
| 280 |
|
approximate the first and second derivatives of $T$ with finite |
| 298 |
|
|
| 299 |
|
\begin{figure} |
| 300 |
|
\includegraphics[width=\linewidth]{gradT} |
| 301 |
< |
\caption{A sample of Au-butanethiol/hexane interfacial system and the |
| 302 |
< |
temperature profile after a kinetic energy flux is imposed to |
| 303 |
< |
it. The 1st and 2nd derivatives of the temperature profile can be |
| 304 |
< |
obtained with finite difference approximation (lower panel).} |
| 301 |
> |
\caption{A sample of Au (111) / butanethiol / hexane interfacial |
| 302 |
> |
system with the temperature profile after a kinetic energy flux has |
| 303 |
> |
been imposed. Note that the largest temperature jump in the thermal |
| 304 |
> |
profile (corresponding to the lowest interfacial conductance) is at |
| 305 |
> |
the interface between the butanethiol molecules (blue) and the |
| 306 |
> |
solvent (grey). First and second derivatives of the temperature |
| 307 |
> |
profile are obtained using a finite difference approximation (lower |
| 308 |
> |
panel).} |
| 309 |
|
\label{gradT} |
| 310 |
|
\end{figure} |
| 311 |
|
|
| 352 |
|
solvent molecules would change the normal behavior of the liquid |
| 353 |
|
phase. Therefore, our $N_{solvent}$ values were chosen to ensure that |
| 354 |
|
these extreme cases did not happen to our simulations. The spacing |
| 355 |
< |
between periodic images of the gold interfaces is $45 \sim 75$\AA. |
| 355 |
> |
between periodic images of the gold interfaces is $45 \sim 75$\AA in |
| 356 |
> |
our simulations. |
| 357 |
|
|
| 358 |
|
The initial configurations generated are further equilibrated with the |
| 359 |
|
$x$ and $y$ dimensions fixed, only allowing the $z$-length scale to |
| 371 |
|
$\sim$200K. Therefore, thermal flux usually came from the metal to the |
| 372 |
|
liquid so that the liquid has a higher temperature and would not |
| 373 |
|
freeze due to lowered temperatures. After this induced temperature |
| 374 |
< |
gradient had stablized, the temperature profile of the simulation cell |
| 375 |
< |
was recorded. To do this, the simulation cell is devided evenly into |
| 374 |
> |
gradient had stabilized, the temperature profile of the simulation cell |
| 375 |
> |
was recorded. To do this, the simulation cell is divided evenly into |
| 376 |
|
$N$ slabs along the $z$-axis. The average temperatures of each slab |
| 377 |
|
are recorded for 1$\sim$2 ns. When the slab width $d$ of each slab is |
| 378 |
|
the same, the derivatives of $T$ with respect to slab number $n$ can |
| 385 |
|
\Big/\left(\frac{\partial T}{\partial n}\right)^2 |
| 386 |
|
\label{derivativeG2} |
| 387 |
|
\end{equation} |
| 388 |
+ |
The absolute values in Eq. \ref{derivativeG2} appear because the |
| 389 |
+ |
direction of the flux $\vec{J}$ is in an opposing direction on either |
| 390 |
+ |
side of the metal slab. |
| 391 |
|
|
| 392 |
|
All of the above simulation procedures use a time step of 1 fs. Each |
| 393 |
|
equilibration stage took a minimum of 100 ps, although in some cases, |
| 406 |
|
\caption{Structures of the capping agent and solvents utilized in |
| 407 |
|
these simulations. The chemically-distinct sites (a-e) are expanded |
| 408 |
|
in terms of constituent atoms for both United Atom (UA) and All Atom |
| 409 |
< |
(AA) force fields. Most parameters are from |
| 410 |
< |
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}.} |
| 409 |
> |
(AA) force fields. Most parameters are from References |
| 410 |
> |
\protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes,TraPPE-UA.thiols} |
| 411 |
> |
(UA) and \protect\cite{OPLSAA} (AA). Cross-interactions with the Au |
| 412 |
> |
atoms are given in Table \ref{MnM}.} |
| 413 |
|
\label{demoMol} |
| 414 |
|
\end{figure} |
| 415 |
|
|
| 431 |
|
|
| 432 |
|
By eliminating explicit hydrogen atoms, the TraPPE-UA models are |
| 433 |
|
simple and computationally efficient, while maintaining good accuracy. |
| 434 |
< |
However, the TraPPE-UA model for alkanes is known to predict a slighly |
| 434 |
> |
However, the TraPPE-UA model for alkanes is known to predict a slightly |
| 435 |
|
lower boiling point than experimental values. This is one of the |
| 436 |
|
reasons we used a lower average temperature (200K) for our |
| 437 |
|
simulations. If heat is transferred to the liquid phase during the |
| 544 |
|
A series of different initial conditions with a range of surface |
| 545 |
|
coverages was prepared and solvated with various with both of the |
| 546 |
|
solvent molecules. These systems were then equilibrated and their |
| 547 |
< |
interfacial thermal conductivity was measured with our NIVS |
| 547 |
> |
interfacial thermal conductivity was measured with the NIVS |
| 548 |
|
algorithm. Figure \ref{coverage} demonstrates the trend of conductance |
| 549 |
|
with respect to surface coverage. |
| 550 |
|
|
| 551 |
|
\begin{figure} |
| 552 |
|
\includegraphics[width=\linewidth]{coverage} |
| 553 |
< |
\caption{Comparison of interfacial thermal conductivity ($G$) values |
| 554 |
< |
for the Au-butanethiol/solvent interface with various UA models and |
| 555 |
< |
different capping agent coverages at $\langle T\rangle\sim$200K.} |
| 553 |
> |
\caption{The interfacial thermal conductivity ($G$) has a |
| 554 |
> |
non-monotonic dependence on the degree of surface capping. This |
| 555 |
> |
data is for the Au(111) / butanethiol / solvent interface with |
| 556 |
> |
various UA force fields at $\langle T\rangle \sim $200K.} |
| 557 |
|
\label{coverage} |
| 558 |
|
\end{figure} |
| 559 |
|
|
| 560 |
+ |
In partially covered surfaces, the derivative definition for |
| 561 |
+ |
$G^\prime$ (Eq. \ref{derivativeG}) becomes difficult to apply, as the |
| 562 |
+ |
location of maximum change of $\lambda$ becomes washed out. The |
| 563 |
+ |
discrete definition (Eq. \ref{discreteG}) is easier to apply, as the |
| 564 |
+ |
Gibbs dividing surface is still well-defined. Therefore, $G$ (not |
| 565 |
+ |
$G^\prime$) was used in this section. |
| 566 |
|
|
| 501 |
– |
In partially covered butanethiol on the Au(111) surface, the |
| 502 |
– |
derivative definition for $G^\prime$ (Eq. \ref{derivativeG}) becomes |
| 503 |
– |
difficult to apply, as the location of maximum change of $\lambda$ |
| 504 |
– |
becomes washed out. The discrete definition (Eq. \ref{discreteG}) is |
| 505 |
– |
easier to apply, as the Gibbs dividing surface is still |
| 506 |
– |
well-defined. Therefore, $G$ (not $G^\prime$) was used in this |
| 507 |
– |
section. |
| 508 |
– |
|
| 567 |
|
From Figure \ref{coverage}, one can see the significance of the |
| 568 |
|
presence of capping agents. When even a small fraction of the Au(111) |
| 569 |
|
surface sites are covered with butanethiols, the conductivity exhibits |
| 570 |
< |
an enhancement by at least a factor of 3. This indicates the important |
| 571 |
< |
role cappping agents are playing for thermal transport at metal / |
| 572 |
< |
organic solvent surfaces. |
| 570 |
> |
an enhancement by at least a factor of 3. Capping agents are clearly |
| 571 |
> |
playing a major role in thermal transport at metal / organic solvent |
| 572 |
> |
surfaces. |
| 573 |
|
|
| 574 |
|
We note a non-monotonic behavior in the interfacial conductance as a |
| 575 |
|
function of surface coverage. The maximum conductance (largest $G$) |
| 579 |
|
between butanethiols to form. These gaps can be filled by transient |
| 580 |
|
solvent molecules. These solvent molecules couple very strongly with |
| 581 |
|
the hot capping agent molecules near the surface, and can then carry |
| 582 |
< |
(diffusively) the excess thermal energy away from the surface. |
| 582 |
> |
away (diffusively) the excess thermal energy from the surface. |
| 583 |
|
|
| 584 |
|
There appears to be a competition between the conduction of the |
| 585 |
|
thermal energy away from the surface by the capping agents (enhanced |
| 586 |
|
by greater coverage) and the coupling of the capping agents with the |
| 587 |
< |
solvent (enhanced by physical contact at lower coverages). This |
| 587 |
> |
solvent (enhanced by interdigitation at lower coverages). This |
| 588 |
|
competition would lead to the non-monotonic coverage behavior observed |
| 589 |
< |
here. |
| 589 |
> |
here. |
| 590 |
|
|
| 591 |
< |
A comparison of the results obtained from the two different organic |
| 592 |
< |
solvents can also provide useful information of the interfacial |
| 593 |
< |
thermal transport process. The deuterated hexane (UA) results do not |
| 594 |
< |
appear to be substantially different from those of normal hexane (UA), |
| 595 |
< |
given that butanethiol (UA) is non-deuterated for both solvents. The |
| 596 |
< |
UA models, even though they have eliminated C-H vibrational overlap, |
| 597 |
< |
still have significant overlap in the infrared spectra. Because |
| 598 |
< |
differences in the infrared range do not seem to produce an observable |
| 541 |
< |
difference for the results of $G$ (Figure \ref{uahxnua}). |
| 591 |
> |
Results for rigid body toluene solvent, as well as the UA hexane, are |
| 592 |
> |
within the ranges expected from prior experimental |
| 593 |
> |
work.\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406} This suggests |
| 594 |
> |
that explicit hydrogen atoms might not be required for modeling |
| 595 |
> |
thermal transport in these systems. C-H vibrational modes do not see |
| 596 |
> |
significant excited state population at low temperatures, and are not |
| 597 |
> |
likely to carry lower frequency excitations from the solid layer into |
| 598 |
> |
the bulk liquid. |
| 599 |
|
|
| 600 |
< |
\begin{figure} |
| 601 |
< |
\includegraphics[width=\linewidth]{uahxnua} |
| 602 |
< |
\caption{Vibrational spectra obtained for normal (upper) and |
| 603 |
< |
deuterated (lower) hexane in Au-butanethiol/hexane |
| 604 |
< |
systems. Butanethiol spectra are shown as reference. Both hexane and |
| 605 |
< |
butanethiol were using United-Atom models.} |
| 606 |
< |
\label{uahxnua} |
| 607 |
< |
\end{figure} |
| 608 |
< |
|
| 609 |
< |
Furthermore, results for rigid body toluene solvent, as well as other |
| 553 |
< |
UA-hexane solvents, are reasonable within the general experimental |
| 554 |
< |
ranges\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406}. This |
| 555 |
< |
suggests that explicit hydrogen might not be a required factor for |
| 556 |
< |
modeling thermal transport phenomena of systems such as |
| 557 |
< |
Au-thiol/organic solvent. |
| 558 |
< |
|
| 559 |
< |
However, results for Au-butanethiol/toluene do not show an identical |
| 560 |
< |
trend with those for Au-butanethiol/hexane in that $G$ remains at |
| 561 |
< |
approximately the same magnitue when butanethiol coverage differs from |
| 562 |
< |
25\% to 75\%. This might be rooted in the molecule shape difference |
| 563 |
< |
for planar toluene and chain-like {\it n}-hexane. Due to this |
| 564 |
< |
difference, toluene molecules have more difficulty in occupying |
| 565 |
< |
relatively small gaps among capping agents when their coverage is not |
| 566 |
< |
too low. Therefore, the solvent-capping agent contact may keep |
| 567 |
< |
increasing until the capping agent coverage reaches a relatively low |
| 568 |
< |
level. This becomes an offset for decreasing butanethiol molecules on |
| 569 |
< |
its effect to the process of interfacial thermal transport. Thus, one |
| 570 |
< |
can see a plateau of $G$ vs. butanethiol coverage in our results. |
| 600 |
> |
The toluene solvent does not exhibit the same behavior as hexane in |
| 601 |
> |
that $G$ remains at approximately the same magnitude when the capping |
| 602 |
> |
coverage increases from 25\% to 75\%. Toluene, as a rigid planar |
| 603 |
> |
molecule, cannot occupy the relatively small gaps between the capping |
| 604 |
> |
agents as easily as the chain-like {\it n}-hexane. The effect of |
| 605 |
> |
solvent coupling to the capping agent is therefore weaker in toluene |
| 606 |
> |
except at the very lowest coverage levels. This effect counters the |
| 607 |
> |
coverage-dependent conduction of heat away from the metal surface, |
| 608 |
> |
leading to a much flatter $G$ vs. coverage trend than is observed in |
| 609 |
> |
{\it n}-hexane. |
| 610 |
|
|
| 611 |
|
\subsection{Effects due to Solvent \& Solvent Models} |
| 612 |
< |
In addition to UA solvent/capping agent models, AA models are included |
| 613 |
< |
in our simulations as well. Besides simulations of the same (UA or AA) |
| 614 |
< |
model for solvent and capping agent, different models can be applied |
| 615 |
< |
to different components. Furthermore, regardless of models chosen, |
| 616 |
< |
either the solvent or the capping agent can be deuterated, similar to |
| 617 |
< |
the previous section. Table \ref{modelTest} summarizes the results of |
| 618 |
< |
these studies. |
| 612 |
> |
In addition to UA solvent and capping agent models, AA models have |
| 613 |
> |
also been included in our simulations. In most of this work, the same |
| 614 |
> |
(UA or AA) model for solvent and capping agent was used, but it is |
| 615 |
> |
also possible to utilize different models for different components. |
| 616 |
> |
We have also included isotopic substitutions (Hydrogen to Deuterium) |
| 617 |
> |
to decrease the explicit vibrational overlap between solvent and |
| 618 |
> |
capping agent. Table \ref{modelTest} summarizes the results of these |
| 619 |
> |
studies. |
| 620 |
|
|
| 621 |
|
\begin{table*} |
| 622 |
|
\begin{minipage}{\linewidth} |
| 623 |
|
\begin{center} |
| 624 |
|
|
| 625 |
< |
\caption{Computed interfacial thermal conductivity ($G$ and |
| 625 |
> |
\caption{Computed interfacial thermal conductance ($G$ and |
| 626 |
|
$G^\prime$) values for interfaces using various models for |
| 627 |
|
solvent and capping agent (or without capping agent) at |
| 628 |
< |
$\langle T\rangle\sim$200K. (D stands for deuterated solvent |
| 629 |
< |
or capping agent molecules; ``Avg.'' denotes results that are |
| 630 |
< |
averages of simulations under different $J_z$'s. Error |
| 591 |
< |
estimates indicated in parenthesis.)} |
| 628 |
> |
$\langle T\rangle\sim$200K. Here ``D'' stands for deuterated |
| 629 |
> |
solvent or capping agent molecules. Error estimates are |
| 630 |
> |
indicated in parentheses.} |
| 631 |
|
|
| 632 |
|
\begin{tabular}{llccc} |
| 633 |
|
\hline\hline |
| 634 |
< |
Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ |
| 635 |
< |
(or bare surface) & model & (GW/m$^2$) & |
| 634 |
> |
Butanethiol model & Solvent & $G$ & $G^\prime$ \\ |
| 635 |
> |
(or bare surface) & model & |
| 636 |
|
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 637 |
|
\hline |
| 638 |
< |
UA & UA hexane & Avg. & 131(9) & 87(10) \\ |
| 639 |
< |
& UA hexane(D) & 1.95 & 153(5) & 136(13) \\ |
| 640 |
< |
& AA hexane & Avg. & 131(6) & 122(10) \\ |
| 641 |
< |
& UA toluene & 1.96 & 187(16) & 151(11) \\ |
| 642 |
< |
& AA toluene & 1.89 & 200(36) & 149(53) \\ |
| 638 |
> |
UA & UA hexane & 131(9) & 87(10) \\ |
| 639 |
> |
& UA hexane(D) & 153(5) & 136(13) \\ |
| 640 |
> |
& AA hexane & 131(6) & 122(10) \\ |
| 641 |
> |
& UA toluene & 187(16) & 151(11) \\ |
| 642 |
> |
& AA toluene & 200(36) & 149(53) \\ |
| 643 |
|
\hline |
| 644 |
< |
AA & UA hexane & 1.94 & 116(9) & 129(8) \\ |
| 645 |
< |
& AA hexane & Avg. & 442(14) & 356(31) \\ |
| 646 |
< |
& AA hexane(D) & 1.93 & 222(12) & 234(54) \\ |
| 647 |
< |
& UA toluene & 1.98 & 125(25) & 97(60) \\ |
| 648 |
< |
& AA toluene & 3.79 & 487(56) & 290(42) \\ |
| 644 |
> |
AA & UA hexane & 116(9) & 129(8) \\ |
| 645 |
> |
& AA hexane & 442(14) & 356(31) \\ |
| 646 |
> |
& AA hexane(D) & 222(12) & 234(54) \\ |
| 647 |
> |
& UA toluene & 125(25) & 97(60) \\ |
| 648 |
> |
& AA toluene & 487(56) & 290(42) \\ |
| 649 |
|
\hline |
| 650 |
< |
AA(D) & UA hexane & 1.94 & 158(25) & 172(4) \\ |
| 651 |
< |
& AA hexane & 1.92 & 243(29) & 191(11) \\ |
| 652 |
< |
& AA toluene & 1.93 & 364(36) & 322(67) \\ |
| 650 |
> |
AA(D) & UA hexane & 158(25) & 172(4) \\ |
| 651 |
> |
& AA hexane & 243(29) & 191(11) \\ |
| 652 |
> |
& AA toluene & 364(36) & 322(67) \\ |
| 653 |
|
\hline |
| 654 |
< |
bare & UA hexane & Avg. & 46.5(3.2) & 49.4(4.5) \\ |
| 655 |
< |
& UA hexane(D) & 0.98 & 43.9(4.6) & 43.0(2.0) \\ |
| 656 |
< |
& AA hexane & 0.96 & 31.0(1.4) & 29.4(1.3) \\ |
| 657 |
< |
& UA toluene & 1.99 & 70.1(1.3) & 65.8(0.5) \\ |
| 654 |
> |
bare & UA hexane & 46.5(3.2) & 49.4(4.5) \\ |
| 655 |
> |
& UA hexane(D) & 43.9(4.6) & 43.0(2.0) \\ |
| 656 |
> |
& AA hexane & 31.0(1.4) & 29.4(1.3) \\ |
| 657 |
> |
& UA toluene & 70.1(1.3) & 65.8(0.5) \\ |
| 658 |
|
\hline\hline |
| 659 |
|
\end{tabular} |
| 660 |
|
\label{modelTest} |
| 662 |
|
\end{minipage} |
| 663 |
|
\end{table*} |
| 664 |
|
|
| 665 |
< |
To facilitate direct comparison, the same system with differnt models |
| 666 |
< |
for different components uses the same length scale for their |
| 667 |
< |
simulation cells. Without the presence of capping agent, using |
| 629 |
< |
different models for hexane yields similar results for both $G$ and |
| 630 |
< |
$G^\prime$, and these two definitions agree with eath other very |
| 631 |
< |
well. This indicates very weak interaction between the metal and the |
| 632 |
< |
solvent, and is a typical case for acoustic impedance mismatch between |
| 633 |
< |
these two phases. |
| 665 |
> |
To facilitate direct comparison between force fields, systems with the |
| 666 |
> |
same capping agent and solvent were prepared with the same length |
| 667 |
> |
scales for the simulation cells. |
| 668 |
|
|
| 669 |
< |
As for Au(111) surfaces completely covered by butanethiols, the choice |
| 670 |
< |
of models for capping agent and solvent could impact the measurement |
| 671 |
< |
of $G$ and $G^\prime$ quite significantly. For Au-butanethiol/hexane |
| 672 |
< |
interfaces, using AA model for both butanethiol and hexane yields |
| 639 |
< |
substantially higher conductivity values than using UA model for at |
| 640 |
< |
least one component of the solvent and capping agent, which exceeds |
| 641 |
< |
the general range of experimental measurement results. This is |
| 642 |
< |
probably due to the classically treated C-H vibrations in the AA |
| 643 |
< |
model, which should not be appreciably populated at normal |
| 644 |
< |
temperatures. In comparison, once either the hexanes or the |
| 645 |
< |
butanethiols are deuterated, one can see a significantly lower $G$ and |
| 646 |
< |
$G^\prime$. In either of these cases, the C-H(D) vibrational overlap |
| 647 |
< |
between the solvent and the capping agent is removed (Figure |
| 648 |
< |
\ref{aahxntln}). Conclusively, the improperly treated C-H vibration in |
| 649 |
< |
the AA model produced over-predicted results accordingly. Compared to |
| 650 |
< |
the AA model, the UA model yields more reasonable results with higher |
| 651 |
< |
computational efficiency. |
| 669 |
> |
On bare metal / solvent surfaces, different force field models for |
| 670 |
> |
hexane yield similar results for both $G$ and $G^\prime$, and these |
| 671 |
> |
two definitions agree with each other very well. This is primarily an |
| 672 |
> |
indicator of weak interactions between the metal and the solvent. |
| 673 |
|
|
| 674 |
< |
\begin{figure} |
| 675 |
< |
\includegraphics[width=\linewidth]{aahxntln} |
| 676 |
< |
\caption{Spectra obtained for All-Atom model Au-butanethil/solvent |
| 677 |
< |
systems. When butanethiol is deuterated (lower left), its |
| 678 |
< |
vibrational overlap with hexane would decrease significantly, |
| 679 |
< |
compared with normal butanethiol (upper left). However, this |
| 680 |
< |
dramatic change does not apply to toluene as much (right).} |
| 681 |
< |
\label{aahxntln} |
| 682 |
< |
\end{figure} |
| 674 |
> |
For the fully-covered surfaces, the choice of force field for the |
| 675 |
> |
capping agent and solvent has a large impact on the calculated values |
| 676 |
> |
of $G$ and $G^\prime$. The AA thiol to AA solvent conductivities are |
| 677 |
> |
much larger than their UA to UA counterparts, and these values exceed |
| 678 |
> |
the experimental estimates by a large measure. The AA force field |
| 679 |
> |
allows significant energy to go into C-H (or C-D) stretching modes, |
| 680 |
> |
and since these modes are high frequency, this non-quantum behavior is |
| 681 |
> |
likely responsible for the overestimate of the conductivity. Compared |
| 682 |
> |
to the AA model, the UA model yields more reasonable conductivity |
| 683 |
> |
values with much higher computational efficiency. |
| 684 |
|
|
| 685 |
< |
However, for Au-butanethiol/toluene interfaces, having the AA |
| 686 |
< |
butanethiol deuterated did not yield a significant change in the |
| 687 |
< |
measurement results. Compared to the C-H vibrational overlap between |
| 688 |
< |
hexane and butanethiol, both of which have alkyl chains, that overlap |
| 689 |
< |
between toluene and butanethiol is not so significant and thus does |
| 690 |
< |
not have as much contribution to the heat exchange |
| 691 |
< |
process. Conversely, extra degrees of freedom such as the C-H |
| 692 |
< |
vibrations could yield higher heat exchange rate between these two |
| 693 |
< |
phases and result in a much higher conductivity. |
| 685 |
> |
\subsubsection{Are electronic excitations in the metal important?} |
| 686 |
> |
Because they lack electronic excitations, the QSC and related embedded |
| 687 |
> |
atom method (EAM) models for gold are known to predict unreasonably |
| 688 |
> |
low values for bulk conductivity |
| 689 |
> |
($\lambda$).\cite{kuang:164101,ISI:000207079300006,Clancy:1992} If the |
| 690 |
> |
conductance between the phases ($G$) is governed primarily by phonon |
| 691 |
> |
excitation (and not electronic degrees of freedom), one would expect a |
| 692 |
> |
classical model to capture most of the interfacial thermal |
| 693 |
> |
conductance. Our results for $G$ and $G^\prime$ indicate that this is |
| 694 |
> |
indeed the case, and suggest that the modeling of interfacial thermal |
| 695 |
> |
transport depends primarily on the description of the interactions |
| 696 |
> |
between the various components at the interface. When the metal is |
| 697 |
> |
chemically capped, the primary barrier to thermal conductivity appears |
| 698 |
> |
to be the interface between the capping agent and the surrounding |
| 699 |
> |
solvent, so the excitations in the metal have little impact on the |
| 700 |
> |
value of $G$. |
| 701 |
|
|
| 673 |
– |
Although the QSC model for Au is known to predict an overly low value |
| 674 |
– |
for bulk metal gold conductivity\cite{kuang:164101}, our computational |
| 675 |
– |
results for $G$ and $G^\prime$ do not seem to be affected by this |
| 676 |
– |
drawback of the model for metal. Instead, our results suggest that the |
| 677 |
– |
modeling of interfacial thermal transport behavior relies mainly on |
| 678 |
– |
the accuracy of the interaction descriptions between components |
| 679 |
– |
occupying the interfaces. |
| 680 |
– |
|
| 702 |
|
\subsection{Effects due to methodology and simulation parameters} |
| 703 |
|
|
| 704 |
< |
We have varied our protocol or other parameters of the simulations in |
| 705 |
< |
order to investigate how these factors would affect the measurement of |
| 706 |
< |
$G$'s. It turned out that while some of these parameters would not |
| 707 |
< |
affect the results substantially, some other changes to the |
| 708 |
< |
simulations would have a significant impact on the measurement |
| 709 |
< |
results. |
| 704 |
> |
We have varied the parameters of the simulations in order to |
| 705 |
> |
investigate how these factors would affect the computation of $G$. Of |
| 706 |
> |
particular interest are: 1) the length scale for the applied thermal |
| 707 |
> |
gradient (modified by increasing the amount of solvent in the system), |
| 708 |
> |
2) the sign and magnitude of the applied thermal flux, 3) the average |
| 709 |
> |
temperature of the simulation (which alters the solvent density during |
| 710 |
> |
equilibration), and 4) the definition of the interfacial conductance |
| 711 |
> |
(Eqs. (\ref{discreteG}) or (\ref{derivativeG})) used in the |
| 712 |
> |
calculation. |
| 713 |
|
|
| 714 |
< |
In some of our simulations, we allowed $L_x$ and $L_y$ to change |
| 715 |
< |
during equilibrating the liquid phase. Due to the stiffness of the |
| 716 |
< |
crystalline Au structure, $L_x$ and $L_y$ would not change noticeably |
| 717 |
< |
after equilibration. Although $L_z$ could fluctuate $\sim$1\% after a |
| 718 |
< |
system is fully equilibrated in the NPT ensemble, this fluctuation, as |
| 719 |
< |
well as those of $L_x$ and $L_y$ (which is significantly smaller), |
| 720 |
< |
would not be magnified on the calculated $G$'s, as shown in Table |
| 721 |
< |
\ref{AuThiolHexaneUA}. This insensivity to $L_i$ fluctuations allows |
| 722 |
< |
reliable measurement of $G$'s without the necessity of extremely |
| 699 |
< |
cautious equilibration process. |
| 714 |
> |
Systems of different lengths were prepared by altering the number of |
| 715 |
> |
solvent molecules and extending the length of the box along the $z$ |
| 716 |
> |
axis to accomodate the extra solvent. Equilibration at the same |
| 717 |
> |
temperature and pressure conditions led to nearly identical surface |
| 718 |
> |
areas ($L_x$ and $L_y$) available to the metal and capping agent, |
| 719 |
> |
while the extra solvent served mainly to lengthen the axis that was |
| 720 |
> |
used to apply the thermal flux. For a given value of the applied |
| 721 |
> |
flux, the different $z$ length scale has only a weak effect on the |
| 722 |
> |
computed conductivities (Table \ref{AuThiolHexaneUA}). |
| 723 |
|
|
| 701 |
– |
As stated in our computational details, the spacing filled with |
| 702 |
– |
solvent molecules can be chosen within a range. This allows some |
| 703 |
– |
change of solvent molecule numbers for the same Au-butanethiol |
| 704 |
– |
surfaces. We did this study on our Au-butanethiol/hexane |
| 705 |
– |
simulations. Nevertheless, the results obtained from systems of |
| 706 |
– |
different $N_{hexane}$ did not indicate that the measurement of $G$ is |
| 707 |
– |
susceptible to this parameter. For computational efficiency concern, |
| 708 |
– |
smaller system size would be preferable, given that the liquid phase |
| 709 |
– |
structure is not affected. |
| 710 |
– |
|
| 724 |
|
\subsubsection{Effects of applied flux} |
| 725 |
< |
Our NIVS algorithm allows change of unphysical thermal flux both in |
| 726 |
< |
direction and in quantity. This feature extends our investigation of |
| 727 |
< |
interfacial thermal conductance. However, the magnitude of this |
| 728 |
< |
thermal flux is not arbitary if one aims to obtain a stable and |
| 729 |
< |
reliable thermal gradient. A temperature profile would be |
| 730 |
< |
substantially affected by noise when $|J_z|$ has a much too low |
| 731 |
< |
magnitude; while an excessively large $|J_z|$ that overwhelms the |
| 732 |
< |
conductance capacity of the interface would prevent a thermal gradient |
| 733 |
< |
to reach a stablized steady state. NIVS has the advantage of allowing |
| 734 |
< |
$J$ to vary in a wide range such that the optimal flux range for $G$ |
| 735 |
< |
measurement can generally be simulated by the algorithm. Within the |
| 736 |
< |
optimal range, we were able to study how $G$ would change according to |
| 737 |
< |
the thermal flux across the interface. For our simulations, we denote |
| 738 |
< |
$J_z$ to be positive when the physical thermal flux is from the liquid |
| 739 |
< |
to metal, and negative vice versa. The $G$'s measured under different |
| 740 |
< |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and |
| 741 |
< |
\ref{AuThiolToluene}. These results do not suggest that $G$ is |
| 742 |
< |
dependent on $J_z$ within this flux range. The linear response of flux |
| 725 |
> |
The NIVS algorithm allows changes in both the sign and magnitude of |
| 726 |
> |
the applied flux. It is possible to reverse the direction of heat |
| 727 |
> |
flow simply by changing the sign of the flux, and thermal gradients |
| 728 |
> |
which would be difficult to obtain experimentally ($5$ K/\AA) can be |
| 729 |
> |
easily simulated. However, the magnitude of the applied flux is not |
| 730 |
> |
arbitrary if one aims to obtain a stable and reliable thermal gradient. |
| 731 |
> |
A temperature gradient can be lost in the noise if $|J_z|$ is too |
| 732 |
> |
small, and excessive $|J_z|$ values can cause phase transitions if the |
| 733 |
> |
extremes of the simulation cell become widely separated in |
| 734 |
> |
temperature. Also, if $|J_z|$ is too large for the bulk conductivity |
| 735 |
> |
of the materials, the thermal gradient will never reach a stable |
| 736 |
> |
state. |
| 737 |
> |
|
| 738 |
> |
Within a reasonable range of $J_z$ values, we were able to study how |
| 739 |
> |
$G$ changes as a function of this flux. In what follows, we use |
| 740 |
> |
positive $J_z$ values to denote the case where energy is being |
| 741 |
> |
transferred by the method from the metal phase and into the liquid. |
| 742 |
> |
The resulting gradient therefore has a higher temperature in the |
| 743 |
> |
liquid phase. Negative flux values reverse this transfer, and result |
| 744 |
> |
in higher temperature metal phases. The conductance measured under |
| 745 |
> |
different applied $J_z$ values is listed in Tables 1 and 2 in the |
| 746 |
> |
supporting information. These results do not indicate that $G$ depends |
| 747 |
> |
strongly on $J_z$ within this flux range. The linear response of flux |
| 748 |
|
to thermal gradient simplifies our investigations in that we can rely |
| 749 |
< |
on $G$ measurement with only a couple $J_z$'s and do not need to test |
| 732 |
< |
a large series of fluxes. |
| 749 |
> |
on $G$ measurement with only a small number $J_z$ values. |
| 750 |
|
|
| 751 |
< |
\begin{table*} |
| 752 |
< |
\begin{minipage}{\linewidth} |
| 753 |
< |
\begin{center} |
| 754 |
< |
\caption{Computed interfacial thermal conductivity ($G$ and |
| 755 |
< |
$G^\prime$) values for the 100\% covered Au-butanethiol/hexane |
| 756 |
< |
interfaces with UA model and different hexane molecule numbers |
| 757 |
< |
at different temperatures using a range of energy |
| 758 |
< |
fluxes. Error estimates indicated in parenthesis.} |
| 759 |
< |
|
| 743 |
< |
\begin{tabular}{ccccccc} |
| 744 |
< |
\hline\hline |
| 745 |
< |
$\langle T\rangle$ & $N_{hexane}$ & Fixed & $\rho_{hexane}$ & |
| 746 |
< |
$J_z$ & $G$ & $G^\prime$ \\ |
| 747 |
< |
(K) & & $L_x$ \& $L_y$? & (g/cm$^3$) & (GW/m$^2$) & |
| 748 |
< |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 749 |
< |
\hline |
| 750 |
< |
200 & 266 & No & 0.672 & -0.96 & 102(3) & 80.0(0.8) \\ |
| 751 |
< |
& 200 & Yes & 0.694 & 1.92 & 129(11) & 87.3(0.3) \\ |
| 752 |
< |
& & Yes & 0.672 & 1.93 & 131(16) & 78(13) \\ |
| 753 |
< |
& & No & 0.688 & 0.96 & 125(16) & 90.2(15) \\ |
| 754 |
< |
& & & & 1.91 & 139(10) & 101(10) \\ |
| 755 |
< |
& & & & 2.83 & 141(6) & 89.9(9.8) \\ |
| 756 |
< |
& 166 & Yes & 0.679 & 0.97 & 115(19) & 69(18) \\ |
| 757 |
< |
& & & & 1.94 & 125(9) & 87.1(0.2) \\ |
| 758 |
< |
& & No & 0.681 & 0.97 & 141(30) & 78(22) \\ |
| 759 |
< |
& & & & 1.92 & 138(4) & 98.9(9.5) \\ |
| 760 |
< |
\hline |
| 761 |
< |
250 & 200 & No & 0.560 & 0.96 & 75(10) & 61.8(7.3) \\ |
| 762 |
< |
& & & & -0.95 & 49.4(0.3) & 45.7(2.1) \\ |
| 763 |
< |
& 166 & Yes & 0.570 & 0.98 & 79.0(3.5) & 62.9(3.0) \\ |
| 764 |
< |
& & No & 0.569 & 0.97 & 80.3(0.6) & 67(11) \\ |
| 765 |
< |
& & & & 1.44 & 76.2(5.0) & 64.8(3.8) \\ |
| 766 |
< |
& & & & -0.95 & 56.4(2.5) & 54.4(1.1) \\ |
| 767 |
< |
& & & & -1.85 & 47.8(1.1) & 53.5(1.5) \\ |
| 768 |
< |
\hline\hline |
| 769 |
< |
\end{tabular} |
| 770 |
< |
\label{AuThiolHexaneUA} |
| 771 |
< |
\end{center} |
| 772 |
< |
\end{minipage} |
| 773 |
< |
\end{table*} |
| 751 |
> |
The sign of $J_z$ is a different matter, however, as this can alter |
| 752 |
> |
the temperature on the two sides of the interface. The average |
| 753 |
> |
temperature values reported are for the entire system, and not for the |
| 754 |
> |
liquid phase, so at a given $\langle T \rangle$, the system with |
| 755 |
> |
positive $J_z$ has a warmer liquid phase. This means that if the |
| 756 |
> |
liquid carries thermal energy via diffusive transport, {\it positive} |
| 757 |
> |
$J_z$ values will result in increased molecular motion on the liquid |
| 758 |
> |
side of the interface, and this will increase the measured |
| 759 |
> |
conductivity. |
| 760 |
|
|
| 761 |
|
\subsubsection{Effects due to average temperature} |
| 762 |
|
|
| 763 |
< |
Furthermore, we also attempted to increase system average temperatures |
| 764 |
< |
to above 200K. These simulations are first equilibrated in the NPT |
| 765 |
< |
ensemble under normal pressure. As stated above, the TraPPE-UA model |
| 766 |
< |
for hexane tends to predict a lower boiling point. In our simulations, |
| 767 |
< |
hexane had diffculty to remain in liquid phase when NPT equilibration |
| 768 |
< |
temperature is higher than 250K. Additionally, the equilibrated liquid |
| 769 |
< |
hexane density under 250K becomes lower than experimental value. This |
| 770 |
< |
expanded liquid phase leads to lower contact between hexane and |
| 771 |
< |
butanethiol as well.[MAY NEED SLAB DENSITY FIGURE] |
| 772 |
< |
And this reduced contact would |
| 787 |
< |
probably be accountable for a lower interfacial thermal conductance, |
| 788 |
< |
as shown in Table \ref{AuThiolHexaneUA}. |
| 763 |
> |
We also studied the effect of average system temperature on the |
| 764 |
> |
interfacial conductance. The simulations are first equilibrated in |
| 765 |
> |
the NPT ensemble at 1 atm. The TraPPE-UA model for hexane tends to |
| 766 |
> |
predict a lower boiling point (and liquid state density) than |
| 767 |
> |
experiments. This lower-density liquid phase leads to reduced contact |
| 768 |
> |
between the hexane and butanethiol, and this accounts for our |
| 769 |
> |
observation of lower conductance at higher temperatures. In raising |
| 770 |
> |
the average temperature from 200K to 250K, the density drop of |
| 771 |
> |
$\sim$20\% in the solvent phase leads to a $\sim$40\% drop in the |
| 772 |
> |
conductance. |
| 773 |
|
|
| 774 |
< |
A similar study for TraPPE-UA toluene agrees with the above result as |
| 775 |
< |
well. Having a higher boiling point, toluene tends to remain liquid in |
| 776 |
< |
our simulations even equilibrated under 300K in NPT |
| 777 |
< |
ensembles. Furthermore, the expansion of the toluene liquid phase is |
| 778 |
< |
not as significant as that of the hexane. This prevents severe |
| 795 |
< |
decrease of liquid-capping agent contact and the results (Table |
| 796 |
< |
\ref{AuThiolToluene}) show only a slightly decreased interface |
| 797 |
< |
conductance. Therefore, solvent-capping agent contact should play an |
| 798 |
< |
important role in the thermal transport process across the interface |
| 799 |
< |
in that higher degree of contact could yield increased conductance. |
| 774 |
> |
Similar behavior is observed in the TraPPE-UA model for toluene, |
| 775 |
> |
although this model has better agreement with the experimental |
| 776 |
> |
densities of toluene. The expansion of the toluene liquid phase is |
| 777 |
> |
not as significant as that of the hexane (8.3\% over 100K), and this |
| 778 |
> |
limits the effect to $\sim$20\% drop in thermal conductivity. |
| 779 |
|
|
| 780 |
< |
\begin{table*} |
| 781 |
< |
\begin{minipage}{\linewidth} |
| 782 |
< |
\begin{center} |
| 783 |
< |
\caption{Computed interfacial thermal conductivity ($G$ and |
| 784 |
< |
$G^\prime$) values for a 90\% coverage Au-butanethiol/toluene |
| 806 |
< |
interface at different temperatures using a range of energy |
| 807 |
< |
fluxes. Error estimates indicated in parenthesis.} |
| 808 |
< |
|
| 809 |
< |
\begin{tabular}{ccccc} |
| 810 |
< |
\hline\hline |
| 811 |
< |
$\langle T\rangle$ & $\rho_{toluene}$ & $J_z$ & $G$ & $G^\prime$ \\ |
| 812 |
< |
(K) & (g/cm$^3$) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 813 |
< |
\hline |
| 814 |
< |
200 & 0.933 & 2.15 & 204(12) & 113(12) \\ |
| 815 |
< |
& & -1.86 & 180(3) & 135(21) \\ |
| 816 |
< |
& & -3.93 & 176(5) & 113(12) \\ |
| 817 |
< |
\hline |
| 818 |
< |
300 & 0.855 & -1.91 & 143(5) & 125(2) \\ |
| 819 |
< |
& & -4.19 & 135(9) & 113(12) \\ |
| 820 |
< |
\hline\hline |
| 821 |
< |
\end{tabular} |
| 822 |
< |
\label{AuThiolToluene} |
| 823 |
< |
\end{center} |
| 824 |
< |
\end{minipage} |
| 825 |
< |
\end{table*} |
| 780 |
> |
Although we have not mapped out the behavior at a large number of |
| 781 |
> |
temperatures, is clear that there will be a strong temperature |
| 782 |
> |
dependence in the interfacial conductance when the physical properties |
| 783 |
> |
of one side of the interface (notably the density) change rapidly as a |
| 784 |
> |
function of temperature. |
| 785 |
|
|
| 786 |
< |
Besides lower interfacial thermal conductance, surfaces in relatively |
| 787 |
< |
high temperatures are susceptible to reconstructions, when |
| 788 |
< |
butanethiols have a full coverage on the Au(111) surface. These |
| 789 |
< |
reconstructions include surface Au atoms migrated outward to the S |
| 790 |
< |
atom layer, and butanethiol molecules embedded into the original |
| 791 |
< |
surface Au layer. The driving force for this behavior is the strong |
| 792 |
< |
Au-S interactions in our simulations. And these reconstructions lead |
| 793 |
< |
to higher ratio of Au-S attraction and thus is energetically |
| 794 |
< |
favorable. Furthermore, this phenomenon agrees with experimental |
| 795 |
< |
results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt |
| 796 |
< |
{\it et al.} had kept their Au(111) slab rigid so that their |
| 797 |
< |
simulations can reach 300K without surface reconstructions. Without |
| 798 |
< |
this practice, simulating 100\% thiol covered interfaces under higher |
| 799 |
< |
temperatures could hardly avoid surface reconstructions. However, our |
| 800 |
< |
measurement is based on assuming homogeneity on $x$ and $y$ dimensions |
| 801 |
< |
so that measurement of $T$ at particular $z$ would be an effective |
| 802 |
< |
average of the particles of the same type. Since surface |
| 844 |
< |
reconstructions could eliminate the original $x$ and $y$ dimensional |
| 845 |
< |
homogeneity, measurement of $G$ is more difficult to conduct under |
| 846 |
< |
higher temperatures. Therefore, most of our measurements are |
| 847 |
< |
undertaken at $\langle T\rangle\sim$200K. |
| 786 |
> |
Besides the lower interfacial thermal conductance, surfaces at |
| 787 |
> |
relatively high temperatures are susceptible to reconstructions, |
| 788 |
> |
particularly when butanethiols fully cover the Au(111) surface. These |
| 789 |
> |
reconstructions include surface Au atoms which migrate outward to the |
| 790 |
> |
S atom layer, and butanethiol molecules which embed into the surface |
| 791 |
> |
Au layer. The driving force for this behavior is the strong Au-S |
| 792 |
> |
interactions which are modeled here with a deep Lennard-Jones |
| 793 |
> |
potential. This phenomenon agrees with reconstructions that have been |
| 794 |
> |
experimentally |
| 795 |
> |
observed.\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt |
| 796 |
> |
{\it et al.} kept their Au(111) slab rigid so that their simulations |
| 797 |
> |
could reach 300K without surface |
| 798 |
> |
reconstructions.\cite{vlugt:cpc2007154} Since surface reconstructions |
| 799 |
> |
blur the interface, the measurement of $G$ becomes more difficult to |
| 800 |
> |
conduct at higher temperatures. For this reason, most of our |
| 801 |
> |
measurements are undertaken at $\langle T\rangle\sim$200K where |
| 802 |
> |
reconstruction is minimized. |
| 803 |
|
|
| 804 |
|
However, when the surface is not completely covered by butanethiols, |
| 805 |
< |
the simulated system is more resistent to the reconstruction |
| 806 |
< |
above. Our Au-butanethiol/toluene system had the Au(111) surfaces 90\% |
| 807 |
< |
covered by butanethiols, but did not see this above phenomena even at |
| 808 |
< |
$\langle T\rangle\sim$300K. The empty three-fold sites not occupied by |
| 809 |
< |
capping agents could help prevent surface reconstruction in that they |
| 810 |
< |
provide other means of capping agent relaxation. It is observed that |
| 811 |
< |
butanethiols can migrate to their neighbor empty sites during a |
| 812 |
< |
simulation. Therefore, we were able to obtain $G$'s for these |
| 858 |
< |
interfaces even at a relatively high temperature without being |
| 859 |
< |
affected by surface reconstructions. |
| 805 |
> |
the simulated system appears to be more resistent to the |
| 806 |
> |
reconstruction. Our Au / butanethiol / toluene system had the Au(111) |
| 807 |
> |
surfaces 90\% covered by butanethiols, but did not see this above |
| 808 |
> |
phenomena even at $\langle T\rangle\sim$300K. That said, we did |
| 809 |
> |
observe butanethiols migrating to neighboring three-fold sites during |
| 810 |
> |
a simulation. Since the interface persisted in these simulations, we |
| 811 |
> |
were able to obtain $G$'s for these interfaces even at a relatively |
| 812 |
> |
high temperature without being affected by surface reconstructions. |
| 813 |
|
|
| 861 |
– |
|
| 814 |
|
\section{Discussion} |
| 815 |
|
|
| 816 |
< |
\subsection{Capping agent acts as a vibrational coupler between solid |
| 817 |
< |
and solvent phases} |
| 816 |
> |
The primary result of this work is that the capping agent acts as an |
| 817 |
> |
efficient thermal coupler between solid and solvent phases. One of |
| 818 |
> |
the ways the capping agent can carry out this role is to down-shift |
| 819 |
> |
between the phonon vibrations in the solid (which carry the heat from |
| 820 |
> |
the gold) and the molecular vibrations in the liquid (which carry some |
| 821 |
> |
of the heat in the solvent). |
| 822 |
> |
|
| 823 |
|
To investigate the mechanism of interfacial thermal conductance, the |
| 824 |
|
vibrational power spectrum was computed. Power spectra were taken for |
| 825 |
|
individual components in different simulations. To obtain these |
| 826 |
< |
spectra, simulations were run after equilibration, in the NVE |
| 827 |
< |
ensemble, and without a thermal gradient. Snapshots of configurations |
| 828 |
< |
were collected at a frequency that is higher than that of the fastest |
| 829 |
< |
vibrations occuring in the simulations. With these configurations, the |
| 830 |
< |
velocity auto-correlation functions can be computed: |
| 826 |
> |
spectra, simulations were run after equilibration in the |
| 827 |
> |
microcanonical (NVE) ensemble and without a thermal |
| 828 |
> |
gradient. Snapshots of configurations were collected at a frequency |
| 829 |
> |
that is higher than that of the fastest vibrations occurring in the |
| 830 |
> |
simulations. With these configurations, the velocity auto-correlation |
| 831 |
> |
functions can be computed: |
| 832 |
|
\begin{equation} |
| 833 |
|
C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle |
| 834 |
|
\label{vCorr} |
| 841 |
|
\label{fourier} |
| 842 |
|
\end{equation} |
| 843 |
|
|
| 844 |
< |
From Figure \ref{coverage}, one can see the significance of the |
| 845 |
< |
presence of capping agents. Even when a fraction of the Au(111) |
| 846 |
< |
surface sites are covered with butanethiols, the conductivity would |
| 847 |
< |
see an enhancement by at least a factor of 3. This indicates the |
| 848 |
< |
important role cappping agent is playing for thermal transport |
| 849 |
< |
phenomena on metal / organic solvent surfaces. |
| 844 |
> |
\subsection{The role of specific vibrations} |
| 845 |
> |
The vibrational spectra for gold slabs in different environments are |
| 846 |
> |
shown as in Figure \ref{specAu}. Regardless of the presence of |
| 847 |
> |
solvent, the gold surfaces which are covered by butanethiol molecules |
| 848 |
> |
exhibit an additional peak observed at a frequency of |
| 849 |
> |
$\sim$165cm$^{-1}$. We attribute this peak to the S-Au bonding |
| 850 |
> |
vibration. This vibration enables efficient thermal coupling of the |
| 851 |
> |
surface Au layer to the capping agents. Therefore, in our simulations, |
| 852 |
> |
the Au / S interfaces do not appear to be the primary barrier to |
| 853 |
> |
thermal transport when compared with the butanethiol / solvent |
| 854 |
> |
interfaces. This supports the results of Luo {\it et |
| 855 |
> |
al.}\cite{Luo20101}, who reported $G$ for Au-SAM junctions roughly |
| 856 |
> |
twice as large as what we have computed for the thiol-liquid |
| 857 |
> |
interfaces. |
| 858 |
|
|
| 893 |
– |
Interestingly, as one could observe from our results, the maximum |
| 894 |
– |
conductance enhancement (largest $G$) happens while the surfaces are |
| 895 |
– |
about 75\% covered with butanethiols. This again indicates that |
| 896 |
– |
solvent-capping agent contact has an important role of the thermal |
| 897 |
– |
transport process. Slightly lower butanethiol coverage allows small |
| 898 |
– |
gaps between butanethiols to form. And these gaps could be filled with |
| 899 |
– |
solvent molecules, which acts like ``heat conductors'' on the |
| 900 |
– |
surface. The higher degree of interaction between these solvent |
| 901 |
– |
molecules and capping agents increases the enhancement effect and thus |
| 902 |
– |
produces a higher $G$ than densely packed butanethiol arrays. However, |
| 903 |
– |
once this maximum conductance enhancement is reached, $G$ decreases |
| 904 |
– |
when butanethiol coverage continues to decrease. Each capping agent |
| 905 |
– |
molecule reaches its maximum capacity for thermal |
| 906 |
– |
conductance. Therefore, even higher solvent-capping agent contact |
| 907 |
– |
would not offset this effect. Eventually, when butanethiol coverage |
| 908 |
– |
continues to decrease, solvent-capping agent contact actually |
| 909 |
– |
decreases with the disappearing of butanethiol molecules. In this |
| 910 |
– |
case, $G$ decrease could not be offset but instead accelerated. [MAY NEED |
| 911 |
– |
SNAPSHOT SHOWING THE PHENOMENA / SLAB DENSITY ANALYSIS] |
| 912 |
– |
|
| 913 |
– |
A comparison of the results obtained from differenet organic solvents |
| 914 |
– |
can also provide useful information of the interfacial thermal |
| 915 |
– |
transport process. The deuterated hexane (UA) results do not appear to |
| 916 |
– |
be much different from those of normal hexane (UA), given that |
| 917 |
– |
butanethiol (UA) is non-deuterated for both solvents. These UA model |
| 918 |
– |
studies, even though eliminating C-H vibration samplings, still have |
| 919 |
– |
C-C vibrational frequencies different from each other. However, these |
| 920 |
– |
differences in the infrared range do not seem to produce an observable |
| 921 |
– |
difference for the results of $G$ (Figure \ref{uahxnua}). |
| 922 |
– |
|
| 859 |
|
\begin{figure} |
| 860 |
< |
\includegraphics[width=\linewidth]{uahxnua} |
| 861 |
< |
\caption{Vibrational spectra obtained for normal (upper) and |
| 862 |
< |
deuterated (lower) hexane in Au-butanethiol/hexane |
| 863 |
< |
systems. Butanethiol spectra are shown as reference. Both hexane and |
| 864 |
< |
butanethiol were using United-Atom models.} |
| 865 |
< |
\label{uahxnua} |
| 860 |
> |
\includegraphics[width=\linewidth]{vibration} |
| 861 |
> |
\caption{The vibrational power spectrum for thiol-capped gold has an |
| 862 |
> |
additional vibrational peak at $\sim $165cm$^{-1}$. Bare gold |
| 863 |
> |
surfaces (both with and without a solvent over-layer) are missing |
| 864 |
> |
this peak. A similar peak at $\sim $165cm$^{-1}$ also appears in |
| 865 |
> |
the vibrational power spectrum for the butanethiol capping agents.} |
| 866 |
> |
\label{specAu} |
| 867 |
|
\end{figure} |
| 868 |
|
|
| 869 |
< |
Furthermore, results for rigid body toluene solvent, as well as other |
| 870 |
< |
UA-hexane solvents, are reasonable within the general experimental |
| 871 |
< |
ranges\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406}. This |
| 935 |
< |
suggests that explicit hydrogen might not be a required factor for |
| 936 |
< |
modeling thermal transport phenomena of systems such as |
| 937 |
< |
Au-thiol/organic solvent. |
| 869 |
> |
Also in this figure, we show the vibrational power spectrum for the |
| 870 |
> |
bound butanethiol molecules, which also exhibits the same |
| 871 |
> |
$\sim$165cm$^{-1}$ peak. |
| 872 |
|
|
| 873 |
< |
However, results for Au-butanethiol/toluene do not show an identical |
| 874 |
< |
trend with those for Au-butanethiol/hexane in that $G$ remains at |
| 875 |
< |
approximately the same magnitue when butanethiol coverage differs from |
| 876 |
< |
25\% to 75\%. This might be rooted in the molecule shape difference |
| 877 |
< |
for planar toluene and chain-like {\it n}-hexane. Due to this |
| 878 |
< |
difference, toluene molecules have more difficulty in occupying |
| 879 |
< |
relatively small gaps among capping agents when their coverage is not |
| 880 |
< |
too low. Therefore, the solvent-capping agent contact may keep |
| 881 |
< |
increasing until the capping agent coverage reaches a relatively low |
| 882 |
< |
level. This becomes an offset for decreasing butanethiol molecules on |
| 883 |
< |
its effect to the process of interfacial thermal transport. Thus, one |
| 884 |
< |
can see a plateau of $G$ vs. butanethiol coverage in our results. |
| 885 |
< |
|
| 886 |
< |
\subsection{Influence of Chosen Molecule Model on $G$} |
| 953 |
< |
In addition to UA solvent/capping agent models, AA models are included |
| 954 |
< |
in our simulations as well. Besides simulations of the same (UA or AA) |
| 955 |
< |
model for solvent and capping agent, different models can be applied |
| 956 |
< |
to different components. Furthermore, regardless of models chosen, |
| 957 |
< |
either the solvent or the capping agent can be deuterated, similar to |
| 958 |
< |
the previous section. Table \ref{modelTest} summarizes the results of |
| 959 |
< |
these studies. |
| 873 |
> |
\subsection{Overlap of power spectra} |
| 874 |
> |
A comparison of the results obtained from the two different organic |
| 875 |
> |
solvents can also provide useful information of the interfacial |
| 876 |
> |
thermal transport process. In particular, the vibrational overlap |
| 877 |
> |
between the butanethiol and the organic solvents suggests a highly |
| 878 |
> |
efficient thermal exchange between these components. Very high |
| 879 |
> |
thermal conductivity was observed when AA models were used and C-H |
| 880 |
> |
vibrations were treated classically. The presence of extra degrees of |
| 881 |
> |
freedom in the AA force field yields higher heat exchange rates |
| 882 |
> |
between the two phases and results in a much higher conductivity than |
| 883 |
> |
in the UA force field. The all-atom classical models include high |
| 884 |
> |
frequency modes which should be unpopulated at our relatively low |
| 885 |
> |
temperatures. This artifact is likely the cause of the high thermal |
| 886 |
> |
conductance in all-atom MD simulations. |
| 887 |
|
|
| 888 |
< |
\begin{table*} |
| 889 |
< |
\begin{minipage}{\linewidth} |
| 890 |
< |
\begin{center} |
| 891 |
< |
|
| 892 |
< |
\caption{Computed interfacial thermal conductivity ($G$ and |
| 966 |
< |
$G^\prime$) values for interfaces using various models for |
| 967 |
< |
solvent and capping agent (or without capping agent) at |
| 968 |
< |
$\langle T\rangle\sim$200K. (D stands for deuterated solvent |
| 969 |
< |
or capping agent molecules; ``Avg.'' denotes results that are |
| 970 |
< |
averages of simulations under different $J_z$'s. Error |
| 971 |
< |
estimates indicated in parenthesis.)} |
| 972 |
< |
|
| 973 |
< |
\begin{tabular}{llccc} |
| 974 |
< |
\hline\hline |
| 975 |
< |
Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ |
| 976 |
< |
(or bare surface) & model & (GW/m$^2$) & |
| 977 |
< |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 978 |
< |
\hline |
| 979 |
< |
UA & UA hexane & Avg. & 131(9) & 87(10) \\ |
| 980 |
< |
& UA hexane(D) & 1.95 & 153(5) & 136(13) \\ |
| 981 |
< |
& AA hexane & Avg. & 131(6) & 122(10) \\ |
| 982 |
< |
& UA toluene & 1.96 & 187(16) & 151(11) \\ |
| 983 |
< |
& AA toluene & 1.89 & 200(36) & 149(53) \\ |
| 984 |
< |
\hline |
| 985 |
< |
AA & UA hexane & 1.94 & 116(9) & 129(8) \\ |
| 986 |
< |
& AA hexane & Avg. & 442(14) & 356(31) \\ |
| 987 |
< |
& AA hexane(D) & 1.93 & 222(12) & 234(54) \\ |
| 988 |
< |
& UA toluene & 1.98 & 125(25) & 97(60) \\ |
| 989 |
< |
& AA toluene & 3.79 & 487(56) & 290(42) \\ |
| 990 |
< |
\hline |
| 991 |
< |
AA(D) & UA hexane & 1.94 & 158(25) & 172(4) \\ |
| 992 |
< |
& AA hexane & 1.92 & 243(29) & 191(11) \\ |
| 993 |
< |
& AA toluene & 1.93 & 364(36) & 322(67) \\ |
| 994 |
< |
\hline |
| 995 |
< |
bare & UA hexane & Avg. & 46.5(3.2) & 49.4(4.5) \\ |
| 996 |
< |
& UA hexane(D) & 0.98 & 43.9(4.6) & 43.0(2.0) \\ |
| 997 |
< |
& AA hexane & 0.96 & 31.0(1.4) & 29.4(1.3) \\ |
| 998 |
< |
& UA toluene & 1.99 & 70.1(1.3) & 65.8(0.5) \\ |
| 999 |
< |
\hline\hline |
| 1000 |
< |
\end{tabular} |
| 1001 |
< |
\label{modelTest} |
| 1002 |
< |
\end{center} |
| 1003 |
< |
\end{minipage} |
| 1004 |
< |
\end{table*} |
| 888 |
> |
The similarity in the vibrational modes available to solvent and |
| 889 |
> |
capping agent can be reduced by deuterating one of the two components |
| 890 |
> |
(Fig. \ref{aahxntln}). Once either the hexanes or the butanethiols |
| 891 |
> |
are deuterated, one can observe a significantly lower $G$ and |
| 892 |
> |
$G^\prime$ values (Table \ref{modelTest}). |
| 893 |
|
|
| 1006 |
– |
To facilitate direct comparison, the same system with differnt models |
| 1007 |
– |
for different components uses the same length scale for their |
| 1008 |
– |
simulation cells. Without the presence of capping agent, using |
| 1009 |
– |
different models for hexane yields similar results for both $G$ and |
| 1010 |
– |
$G^\prime$, and these two definitions agree with eath other very |
| 1011 |
– |
well. This indicates very weak interaction between the metal and the |
| 1012 |
– |
solvent, and is a typical case for acoustic impedance mismatch between |
| 1013 |
– |
these two phases. |
| 1014 |
– |
|
| 1015 |
– |
As for Au(111) surfaces completely covered by butanethiols, the choice |
| 1016 |
– |
of models for capping agent and solvent could impact the measurement |
| 1017 |
– |
of $G$ and $G^\prime$ quite significantly. For Au-butanethiol/hexane |
| 1018 |
– |
interfaces, using AA model for both butanethiol and hexane yields |
| 1019 |
– |
substantially higher conductivity values than using UA model for at |
| 1020 |
– |
least one component of the solvent and capping agent, which exceeds |
| 1021 |
– |
the general range of experimental measurement results. This is |
| 1022 |
– |
probably due to the classically treated C-H vibrations in the AA |
| 1023 |
– |
model, which should not be appreciably populated at normal |
| 1024 |
– |
temperatures. In comparison, once either the hexanes or the |
| 1025 |
– |
butanethiols are deuterated, one can see a significantly lower $G$ and |
| 1026 |
– |
$G^\prime$. In either of these cases, the C-H(D) vibrational overlap |
| 1027 |
– |
between the solvent and the capping agent is removed (Figure |
| 1028 |
– |
\ref{aahxntln}). Conclusively, the improperly treated C-H vibration in |
| 1029 |
– |
the AA model produced over-predicted results accordingly. Compared to |
| 1030 |
– |
the AA model, the UA model yields more reasonable results with higher |
| 1031 |
– |
computational efficiency. |
| 1032 |
– |
|
| 894 |
|
\begin{figure} |
| 895 |
|
\includegraphics[width=\linewidth]{aahxntln} |
| 896 |
< |
\caption{Spectra obtained for All-Atom model Au-butanethil/solvent |
| 896 |
> |
\caption{Spectra obtained for all-atom (AA) Au / butanethiol / solvent |
| 897 |
|
systems. When butanethiol is deuterated (lower left), its |
| 898 |
< |
vibrational overlap with hexane would decrease significantly, |
| 899 |
< |
compared with normal butanethiol (upper left). However, this |
| 900 |
< |
dramatic change does not apply to toluene as much (right).} |
| 898 |
> |
vibrational overlap with hexane decreases significantly. Since |
| 899 |
> |
aromatic molecules and the butanethiol are vibrationally dissimilar, |
| 900 |
> |
the change is not as dramatic when toluene is the solvent (right).} |
| 901 |
|
\label{aahxntln} |
| 902 |
|
\end{figure} |
| 903 |
|
|
| 904 |
< |
However, for Au-butanethiol/toluene interfaces, having the AA |
| 904 |
> |
For the Au / butanethiol / toluene interfaces, having the AA |
| 905 |
|
butanethiol deuterated did not yield a significant change in the |
| 906 |
< |
measurement results. Compared to the C-H vibrational overlap between |
| 907 |
< |
hexane and butanethiol, both of which have alkyl chains, that overlap |
| 908 |
< |
between toluene and butanethiol is not so significant and thus does |
| 909 |
< |
not have as much contribution to the heat exchange |
| 1049 |
< |
process. Conversely, extra degrees of freedom such as the C-H |
| 1050 |
< |
vibrations could yield higher heat exchange rate between these two |
| 1051 |
< |
phases and result in a much higher conductivity. |
| 906 |
> |
measured conductance. Compared to the C-H vibrational overlap between |
| 907 |
> |
hexane and butanethiol, both of which have alkyl chains, the overlap |
| 908 |
> |
between toluene and butanethiol is not as significant and thus does |
| 909 |
> |
not contribute as much to the heat exchange process. |
| 910 |
|
|
| 911 |
< |
Although the QSC model for Au is known to predict an overly low value |
| 912 |
< |
for bulk metal gold conductivity\cite{kuang:164101}, our computational |
| 913 |
< |
results for $G$ and $G^\prime$ do not seem to be affected by this |
| 914 |
< |
drawback of the model for metal. Instead, our results suggest that the |
| 915 |
< |
modeling of interfacial thermal transport behavior relies mainly on |
| 916 |
< |
the accuracy of the interaction descriptions between components |
| 917 |
< |
occupying the interfaces. |
| 911 |
> |
Previous observations of Zhang {\it et al.}\cite{hase:2010} indicate |
| 912 |
> |
that the {\it intra}molecular heat transport due to alkylthiols is |
| 913 |
> |
highly efficient. Combining our observations with those of Zhang {\it |
| 914 |
> |
et al.}, it appears that butanethiol acts as a channel to expedite |
| 915 |
> |
heat flow from the gold surface and into the alkyl chain. The |
| 916 |
> |
vibrational coupling between the metal and the liquid phase can |
| 917 |
> |
therefore be enhanced with the presence of suitable capping agents. |
| 918 |
|
|
| 919 |
< |
\subsection{Role of Capping Agent in Interfacial Thermal Conductance} |
| 920 |
< |
The vibrational spectra for gold slabs in different environments are |
| 921 |
< |
shown as in Figure \ref{specAu}. Regardless of the presence of |
| 922 |
< |
solvent, the gold surfaces covered by butanethiol molecules, compared |
| 923 |
< |
to bare gold surfaces, exhibit an additional peak observed at the |
| 924 |
< |
frequency of $\sim$170cm$^{-1}$, which is attributed to the S-Au |
| 925 |
< |
bonding vibration. This vibration enables efficient thermal transport |
| 926 |
< |
from surface Au layer to the capping agents. Therefore, in our |
| 1069 |
< |
simulations, the Au/S interfaces do not appear major heat barriers |
| 1070 |
< |
compared to the butanethiol / solvent interfaces. |
| 919 |
> |
Deuterated models in the UA force field did not decouple the thermal |
| 920 |
> |
transport as well as in the AA force field. The UA models, even |
| 921 |
> |
though they have eliminated the high frequency C-H vibrational |
| 922 |
> |
overlap, still have significant overlap in the lower-frequency |
| 923 |
> |
portions of the infrared spectra (Figure \ref{uahxnua}). Deuterating |
| 924 |
> |
the UA models did not decouple the low frequency region enough to |
| 925 |
> |
produce an observable difference for the results of $G$ (Table |
| 926 |
> |
\ref{modelTest}). |
| 927 |
|
|
| 1072 |
– |
\subsubsection{Overlap of power spectrum} |
| 1073 |
– |
Simultaneously, the vibrational overlap between butanethiol and |
| 1074 |
– |
organic solvents suggests higher thermal exchange efficiency between |
| 1075 |
– |
these two components. Even exessively high heat transport was observed |
| 1076 |
– |
when All-Atom models were used and C-H vibrations were treated |
| 1077 |
– |
classically. Compared to metal and organic liquid phase, the heat |
| 1078 |
– |
transfer efficiency between butanethiol and organic solvents is closer |
| 1079 |
– |
to that within bulk liquid phase. |
| 1080 |
– |
|
| 1081 |
– |
Furthermore, our observation validated previous |
| 1082 |
– |
results\cite{hase:2010} that the intramolecular heat transport of |
| 1083 |
– |
alkylthiols is highly effecient. As a combinational effects of these |
| 1084 |
– |
phenomena, butanethiol acts as a channel to expedite thermal transport |
| 1085 |
– |
process. The acoustic impedance mismatch between the metal and the |
| 1086 |
– |
liquid phase can be effectively reduced with the presence of suitable |
| 1087 |
– |
capping agents. |
| 1088 |
– |
|
| 928 |
|
\begin{figure} |
| 929 |
< |
\includegraphics[width=\linewidth]{vibration} |
| 930 |
< |
\caption{Vibrational spectra obtained for gold in different |
| 931 |
< |
environments.} |
| 932 |
< |
\label{specAu} |
| 929 |
> |
\includegraphics[width=\linewidth]{uahxnua} |
| 930 |
> |
\caption{Vibrational power spectra for UA models for the butanethiol |
| 931 |
> |
and hexane solvent (upper panel) show the high degree of overlap |
| 932 |
> |
between these two molecules, particularly at lower frequencies. |
| 933 |
> |
Deuterating a UA model for the solvent (lower panel) does not |
| 934 |
> |
decouple the two spectra to the same degree as in the AA force |
| 935 |
> |
field (see Fig \ref{aahxntln}).} |
| 936 |
> |
\label{uahxnua} |
| 937 |
|
\end{figure} |
| 938 |
|
|
| 1096 |
– |
[MAY ADD COMPARISON OF AU SLAB WIDTHS BUT NOT MUCH TO TALK ABOUT...] |
| 1097 |
– |
|
| 939 |
|
\section{Conclusions} |
| 940 |
< |
The NIVS algorithm we developed has been applied to simulations of |
| 941 |
< |
Au-butanethiol surfaces with organic solvents. This algorithm allows |
| 942 |
< |
effective unphysical thermal flux transferred between the metal and |
| 943 |
< |
the liquid phase. With the flux applied, we were able to measure the |
| 944 |
< |
corresponding thermal gradient and to obtain interfacial thermal |
| 945 |
< |
conductivities. Under steady states, single trajectory simulation |
| 946 |
< |
would be enough for accurate measurement. This would be advantageous |
| 947 |
< |
compared to transient state simulations, which need multiple |
| 1107 |
< |
trajectories to produce reliable average results. |
| 940 |
> |
The NIVS algorithm has been applied to simulations of |
| 941 |
> |
butanethiol-capped Au(111) surfaces in the presence of organic |
| 942 |
> |
solvents. This algorithm allows the application of unphysical thermal |
| 943 |
> |
flux to transfer heat between the metal and the liquid phase. With the |
| 944 |
> |
flux applied, we were able to measure the corresponding thermal |
| 945 |
> |
gradients and to obtain interfacial thermal conductivities. Under |
| 946 |
> |
steady states, 2-3 ns trajectory simulations are sufficient for |
| 947 |
> |
computation of this quantity. |
| 948 |
|
|
| 949 |
< |
Our simulations have seen significant conductance enhancement with the |
| 950 |
< |
presence of capping agent, compared to the bare gold / liquid |
| 951 |
< |
interfaces. The acoustic impedance mismatch between the metal and the |
| 952 |
< |
liquid phase is effectively eliminated by proper capping |
| 953 |
< |
agent. Furthermore, the coverage precentage of the capping agent plays |
| 954 |
< |
an important role in the interfacial thermal transport |
| 955 |
< |
process. Moderately lower coverages allow higher contact between |
| 956 |
< |
capping agent and solvent, and thus could further enhance the heat |
| 957 |
< |
transfer process. |
| 949 |
> |
Our simulations have seen significant conductance enhancement in the |
| 950 |
> |
presence of capping agent, compared with the bare gold / liquid |
| 951 |
> |
interfaces. The vibrational coupling between the metal and the liquid |
| 952 |
> |
phase is enhanced by a chemically-bonded capping agent. Furthermore, |
| 953 |
> |
the coverage percentage of the capping agent plays an important role |
| 954 |
> |
in the interfacial thermal transport process. Moderately low coverages |
| 955 |
> |
allow higher contact between capping agent and solvent, and thus could |
| 956 |
> |
further enhance the heat transfer process, giving a non-monotonic |
| 957 |
> |
behavior of conductance with increasing coverage. |
| 958 |
|
|
| 959 |
< |
Our measurement results, particularly of the UA models, agree with |
| 960 |
< |
available experimental data. This indicates that our force field |
| 1121 |
< |
parameters have a nice description of the interactions between the |
| 1122 |
< |
particles at the interfaces. AA models tend to overestimate the |
| 959 |
> |
Our results, particularly using the UA models, agree well with |
| 960 |
> |
available experimental data. The AA models tend to overestimate the |
| 961 |
|
interfacial thermal conductance in that the classically treated C-H |
| 962 |
< |
vibration would be overly sampled. Compared to the AA models, the UA |
| 963 |
< |
models have higher computational efficiency with satisfactory |
| 964 |
< |
accuracy, and thus are preferable in interfacial thermal transport |
| 965 |
< |
modelings. Of the two definitions for $G$, the discrete form |
| 962 |
> |
vibrations become too easily populated. Compared to the AA models, the |
| 963 |
> |
UA models have higher computational efficiency with satisfactory |
| 964 |
> |
accuracy, and thus are preferable in modeling interfacial thermal |
| 965 |
> |
transport. |
| 966 |
> |
|
| 967 |
> |
Of the two definitions for $G$, the discrete form |
| 968 |
|
(Eq. \ref{discreteG}) was easier to use and gives out relatively |
| 969 |
|
consistent results, while the derivative form (Eq. \ref{derivativeG}) |
| 970 |
|
is not as versatile. Although $G^\prime$ gives out comparable results |
| 971 |
|
and follows similar trend with $G$ when measuring close to fully |
| 972 |
< |
covered or bare surfaces, the spatial resolution of $T$ profile is |
| 973 |
< |
limited for accurate computation of derivatives data. |
| 972 |
> |
covered or bare surfaces, the spatial resolution of $T$ profile |
| 973 |
> |
required for the use of a derivative form is limited by the number of |
| 974 |
> |
bins and the sampling required to obtain thermal gradient information. |
| 975 |
|
|
| 976 |
< |
Vlugt {\it et al.} has investigated the surface thiol structures for |
| 977 |
< |
nanocrystal gold and pointed out that they differs from those of the |
| 978 |
< |
Au(111) surface\cite{landman:1998,vlugt:cpc2007154}. This difference |
| 979 |
< |
might lead to change of interfacial thermal transport behavior as |
| 980 |
< |
well. To investigate this problem, an effective means to introduce |
| 981 |
< |
thermal flux and measure the corresponding thermal gradient is |
| 982 |
< |
desirable for simulating structures with spherical symmetry. |
| 976 |
> |
Vlugt {\it et al.} have investigated the surface thiol structures for |
| 977 |
> |
nanocrystalline gold and pointed out that they differ from those of |
| 978 |
> |
the Au(111) surface.\cite{landman:1998,vlugt:cpc2007154} This |
| 979 |
> |
difference could also cause differences in the interfacial thermal |
| 980 |
> |
transport behavior. To investigate this problem, one would need an |
| 981 |
> |
effective method for applying thermal gradients in non-planar |
| 982 |
> |
(i.e. spherical) geometries. |
| 983 |
|
|
| 984 |
|
\section{Acknowledgments} |
| 985 |
|
Support for this project was provided by the National Science |
| 986 |
|
Foundation under grant CHE-0848243. Computational time was provided by |
| 987 |
|
the Center for Research Computing (CRC) at the University of Notre |
| 988 |
|
Dame. |
| 989 |
+ |
|
| 990 |
+ |
\section{Supporting Information} |
| 991 |
+ |
This information is available free of charge via the Internet at |
| 992 |
+ |
http://pubs.acs.org. |
| 993 |
+ |
|
| 994 |
|
\newpage |
| 995 |
|
|
| 996 |
|
\bibliography{interfacial} |
