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\begin{document} |
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\title{Simulating interfacial thermal conductance at metal-solvent |
| 32 |
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interfaces: the role of chemical capping agents} |
| 31 |
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\title{Simulating Interfacial Thermal Conductance at Metal-Solvent |
| 32 |
> |
Interfaces: the Role of Chemical Capping Agents} |
| 33 |
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|
| 34 |
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\author{Shenyu Kuang and J. Daniel |
| 35 |
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Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\ |
| 44 |
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\begin{doublespace} |
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|
| 46 |
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\begin{abstract} |
| 47 |
< |
The abstract version 2 |
| 47 |
> |
With the Non-Isotropic Velocity Scaling (NIVS) approach to Reverse |
| 48 |
> |
Non-Equilibrium Molecular Dynamics (RNEMD) it is possible to impose |
| 49 |
> |
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 |
> |
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 |
> |
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 |
> |
|
| 66 |
> |
Keywords: non-equilibrium, molecular dynamics, vibrational overlap, |
| 67 |
> |
coverage dependent. |
| 68 |
|
\end{abstract} |
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|
| 70 |
|
\newpage |
| 76 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 77 |
|
|
| 78 |
|
\section{Introduction} |
| 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 |
+ |
computational methods which have been developed for homogeneous or |
| 87 |
+ |
bulk systems. |
| 88 |
|
|
| 89 |
< |
The intro. |
| 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 |
> |
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 |
| 134 |
> |
study thermal transport from hot Au(111) substrate to a self-assembled |
| 135 |
> |
monolayer of alkylthiol with relatively long chain (8-20 carbon |
| 136 |
> |
atoms).\cite{hase:2010,hase:2011} However, ensemble averaged |
| 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 |
| 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 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 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 |
| 161 |
> |
retains the desirable features of RNEMD (conservation of linear |
| 162 |
> |
momentum and total energy, compatibility with periodic boundary |
| 163 |
> |
conditions) while establishing true thermal distributions in each of |
| 164 |
> |
the two slabs. Furthermore, it allows effective thermal exchange |
| 165 |
> |
between particles of different identities, and thus makes the study of |
| 166 |
> |
interfacial conductance much simpler. |
| 167 |
> |
|
| 168 |
> |
The work presented here deals with the Au(111) surface covered to |
| 169 |
> |
varying degrees by butanethiol, a capping agent with short carbon |
| 170 |
> |
chain, and solvated with organic solvents of different molecular |
| 171 |
> |
properties. Different models were used for both the capping agent and |
| 172 |
> |
the solvent force field parameters. Using the NIVS algorithm, the |
| 173 |
> |
thermal transport across these interfaces was studied and the |
| 174 |
> |
underlying mechanism for the phenomena was investigated. |
| 175 |
> |
|
| 176 |
> |
\section{Methodology} |
| 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, |
| 181 |
> |
one must have a method capable of generating or measuring relatively |
| 182 |
> |
small fluxes, compared to those required for bulk conductivity. This |
| 183 |
> |
requirement makes the calculation even more difficult for |
| 184 |
> |
slowly-converging equilibrium methods.\cite{Viscardy:2007lq} Forward |
| 185 |
> |
NEMD methods impose a gradient (and measure a flux), but at interfaces |
| 186 |
> |
it is not clear what behavior should be imposed at the boundaries |
| 187 |
> |
between materials. Imposed-flux reverse non-equilibrium |
| 188 |
> |
methods\cite{MullerPlathe:1997xw} set the flux {\it a priori} and |
| 189 |
> |
the thermal response becomes an easy-to-measure quantity. Although |
| 190 |
> |
M\"{u}ller-Plathe's original momentum swapping approach can be used |
| 191 |
> |
for exchanging energy between particles of different identity, the |
| 192 |
> |
kinetic energy transfer efficiency is affected by the mass difference |
| 193 |
> |
between the particles, which limits its application on heterogeneous |
| 194 |
> |
interfacial systems. |
| 195 |
> |
|
| 196 |
> |
The non-isotropic velocity scaling (NIVS) \cite{kuang:164101} approach |
| 197 |
> |
to non-equilibrium MD simulations is able to impose a wide range of |
| 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 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 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 |
| 210 |
> |
temperature gradients can be easily established, and these can be used |
| 211 |
> |
for computing thermal conductivities. The NIVS algorithm conserves |
| 212 |
> |
momenta and energy and does not depend on an external thermostat. |
| 213 |
> |
|
| 214 |
> |
\subsection{Defining Interfacial Thermal Conductivity ($G$)} |
| 215 |
> |
|
| 216 |
> |
For an interface with relatively low interfacial conductance, and a |
| 217 |
> |
thermal flux between two distinct bulk regions, the regions on either |
| 218 |
> |
side of the interface rapidly come to a state in which the two phases |
| 219 |
> |
have relatively homogeneous (but distinct) temperatures. The |
| 220 |
> |
interfacial thermal conductivity $G$ can therefore be approximated as: |
| 221 |
> |
\begin{equation} |
| 222 |
> |
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle - |
| 223 |
> |
\langle T_\mathrm{cold}\rangle \right)} |
| 224 |
> |
\label{lowG} |
| 225 |
> |
\end{equation} |
| 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. 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 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} |
| 244 |
> |
\end{equation} |
| 245 |
> |
|
| 246 |
> |
\begin{figure} |
| 247 |
> |
\includegraphics[width=\linewidth]{method} |
| 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 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 |
> |
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 |
| 281 |
> |
difference methods and calculate $G^\prime$. In what follows, both |
| 282 |
> |
definitions have been used, and are compared in the results. |
| 283 |
> |
|
| 284 |
> |
To investigate the interfacial conductivity at metal / solvent |
| 285 |
> |
interfaces, we have modeled a metal slab with its (111) surfaces |
| 286 |
> |
perpendicular to the $z$-axis of our simulation cells. The metal slab |
| 287 |
> |
has been prepared both with and without capping agents on the exposed |
| 288 |
> |
surface, and has been solvated with simple organic solvents, as |
| 289 |
> |
illustrated in Figure \ref{gradT}. |
| 290 |
> |
|
| 291 |
> |
With the simulation cell described above, we are able to equilibrate |
| 292 |
> |
the system and impose an unphysical thermal flux between the liquid |
| 293 |
> |
and the metal phase using the NIVS algorithm. By periodically applying |
| 294 |
> |
the unphysical flux, we obtained a temperature profile and its spatial |
| 295 |
> |
derivatives. Figure \ref{gradT} shows how an applied thermal flux can |
| 296 |
> |
be used to obtain the 1st and 2nd derivatives of the temperature |
| 297 |
> |
profile. |
| 298 |
> |
|
| 299 |
> |
\begin{figure} |
| 300 |
> |
\includegraphics[width=\linewidth]{gradT} |
| 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 |
> |
|
| 312 |
> |
\section{Computational Details} |
| 313 |
> |
\subsection{Simulation Protocol} |
| 314 |
> |
The NIVS algorithm has been implemented in our MD simulation code, |
| 315 |
> |
OpenMD\cite{Meineke:2005gd,openmd}, and was used for our simulations. |
| 316 |
> |
Metal slabs of 6 or 11 layers of Au atoms were first equilibrated |
| 317 |
> |
under atmospheric pressure (1 atm) and 200K. After equilibration, |
| 318 |
> |
butanethiol capping agents were placed at three-fold hollow sites on |
| 319 |
> |
the Au(111) surfaces. These sites are either {\it fcc} or {\it |
| 320 |
> |
hcp} sites, although Hase {\it et al.} found that they are |
| 321 |
> |
equivalent in a heat transfer process,\cite{hase:2010} so we did not |
| 322 |
> |
distinguish between these sites in our study. The maximum butanethiol |
| 323 |
> |
capacity on Au surface is $1/3$ of the total number of surface Au |
| 324 |
> |
atoms, and the packing forms a $(\sqrt{3}\times\sqrt{3})R30^\circ$ |
| 325 |
> |
structure\cite{doi:10.1021/ja00008a001,doi:10.1021/cr9801317}. A |
| 326 |
> |
series of lower coverages was also prepared by eliminating |
| 327 |
> |
butanethiols from the higher coverage surface in a regular manner. The |
| 328 |
> |
lower coverages were prepared in order to study the relation between |
| 329 |
> |
coverage and interfacial conductance. |
| 330 |
> |
|
| 331 |
> |
The capping agent molecules were allowed to migrate during the |
| 332 |
> |
simulations. They distributed themselves uniformly and sampled a |
| 333 |
> |
number of three-fold sites throughout out study. Therefore, the |
| 334 |
> |
initial configuration does not noticeably affect the sampling of a |
| 335 |
> |
variety of configurations of the same coverage, and the final |
| 336 |
> |
conductance measurement would be an average effect of these |
| 337 |
> |
configurations explored in the simulations. |
| 338 |
> |
|
| 339 |
> |
After the modified Au-butanethiol surface systems were equilibrated in |
| 340 |
> |
the canonical (NVT) ensemble, organic solvent molecules were packed in |
| 341 |
> |
the previously empty part of the simulation cells.\cite{packmol} Two |
| 342 |
> |
solvents were investigated, one which has little vibrational overlap |
| 343 |
> |
with the alkanethiol and which has a planar shape (toluene), and one |
| 344 |
> |
which has similar vibrational frequencies to the capping agent and |
| 345 |
> |
chain-like shape ({\it n}-hexane). |
| 346 |
> |
|
| 347 |
> |
The simulation cells were not particularly extensive along the |
| 348 |
> |
$z$-axis, as a very long length scale for the thermal gradient may |
| 349 |
> |
cause excessively hot or cold temperatures in the middle of the |
| 350 |
> |
solvent region and lead to undesired phenomena such as solvent boiling |
| 351 |
> |
or freezing when a thermal flux is applied. Conversely, too few |
| 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 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 |
| 360 |
> |
change. This is to ensure that the equilibration of liquid phase does |
| 361 |
> |
not affect the metal's crystalline structure. Comparisons were made |
| 362 |
> |
with simulations that allowed changes of $L_x$ and $L_y$ during NPT |
| 363 |
> |
equilibration. No substantial changes in the box geometry were noticed |
| 364 |
> |
in these simulations. After ensuring the liquid phase reaches |
| 365 |
> |
equilibrium at atmospheric pressure (1 atm), further equilibration was |
| 366 |
> |
carried out under canonical (NVT) and microcanonical (NVE) ensembles. |
| 367 |
> |
|
| 368 |
> |
After the systems reach equilibrium, NIVS was used to impose an |
| 369 |
> |
unphysical thermal flux between the metal and the liquid phases. Most |
| 370 |
> |
of our simulations were done under an average temperature of |
| 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 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 |
| 379 |
> |
be directly used for $G^\prime$ calculations: \begin{equation} |
| 380 |
> |
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
| 381 |
> |
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
| 382 |
> |
= |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big| |
| 383 |
> |
\Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2 |
| 384 |
> |
= |J_z|\Big|\frac{\partial^2 T}{\partial n^2}\Big| |
| 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, |
| 394 |
> |
longer equilibration stages were utilized. |
| 395 |
> |
|
| 396 |
> |
\subsection{Force Field Parameters} |
| 397 |
> |
Our simulations include a number of chemically distinct components. |
| 398 |
> |
Figure \ref{demoMol} demonstrates the sites defined for both |
| 399 |
> |
United-Atom and All-Atom models of the organic solvent and capping |
| 400 |
> |
agents in our simulations. Force field parameters are needed for |
| 401 |
> |
interactions both between the same type of particles and between |
| 402 |
> |
particles of different species. |
| 403 |
> |
|
| 404 |
> |
\begin{figure} |
| 405 |
> |
\includegraphics[width=\linewidth]{structures} |
| 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 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 |
> |
|
| 416 |
> |
The Au-Au interactions in metal lattice slab is described by the |
| 417 |
> |
quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
| 418 |
> |
potentials include zero-point quantum corrections and are |
| 419 |
> |
reparametrized for accurate surface energies compared to the |
| 420 |
> |
Sutton-Chen potentials.\cite{Chen90} |
| 421 |
> |
|
| 422 |
> |
For the two solvent molecules, {\it n}-hexane and toluene, two |
| 423 |
> |
different atomistic models were utilized. Both solvents were modeled |
| 424 |
> |
using United-Atom (UA) and All-Atom (AA) models. The TraPPE-UA |
| 425 |
> |
parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
| 426 |
> |
for our UA solvent molecules. In these models, sites are located at |
| 427 |
> |
the carbon centers for alkyl groups. Bonding interactions, including |
| 428 |
> |
bond stretches and bends and torsions, were used for intra-molecular |
| 429 |
> |
sites closer than 3 bonds. For non-bonded interactions, Lennard-Jones |
| 430 |
> |
potentials are used. |
| 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 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 |
| 438 |
> |
NIVS simulation, the liquid in the hot slab can actually be |
| 439 |
> |
substantially warmer than the mean temperature in the simulation. The |
| 440 |
> |
lower mean temperatures therefore prevent solvent boiling. |
| 441 |
> |
|
| 442 |
> |
For UA-toluene, the non-bonded potentials between intermolecular sites |
| 443 |
> |
have a similar Lennard-Jones formulation. The toluene molecules were |
| 444 |
> |
treated as a single rigid body, so there was no need for |
| 445 |
> |
intramolecular interactions (including bonds, bends, or torsions) in |
| 446 |
> |
this solvent model. |
| 447 |
> |
|
| 448 |
> |
Besides the TraPPE-UA models, AA models for both organic solvents are |
| 449 |
> |
included in our studies as well. The OPLS-AA\cite{OPLSAA} force fields |
| 450 |
> |
were used. For hexane, additional explicit hydrogen sites were |
| 451 |
> |
included. Besides bonding and non-bonded site-site interactions, |
| 452 |
> |
partial charges and the electrostatic interactions were added to each |
| 453 |
> |
CT and HC site. For toluene, a flexible model for the toluene molecule |
| 454 |
> |
was utilized which included bond, bend, torsion, and inversion |
| 455 |
> |
potentials to enforce ring planarity. |
| 456 |
> |
|
| 457 |
> |
The butanethiol capping agent in our simulations, were also modeled |
| 458 |
> |
with both UA and AA model. The TraPPE-UA force field includes |
| 459 |
> |
parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for |
| 460 |
> |
UA butanethiol model in our simulations. The OPLS-AA also provides |
| 461 |
> |
parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) |
| 462 |
> |
surfaces do not have the hydrogen atom bonded to sulfur. To derive |
| 463 |
> |
suitable parameters for butanethiol adsorbed on Au(111) surfaces, we |
| 464 |
> |
adopt the S parameters from Luedtke and Landman\cite{landman:1998} and |
| 465 |
> |
modify the parameters for the CTS atom to maintain charge neutrality |
| 466 |
> |
in the molecule. Note that the model choice (UA or AA) for the capping |
| 467 |
> |
agent can be different from the solvent. Regardless of model choice, |
| 468 |
> |
the force field parameters for interactions between capping agent and |
| 469 |
> |
solvent can be derived using Lorentz-Berthelot Mixing Rule: |
| 470 |
> |
\begin{eqnarray} |
| 471 |
> |
\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\ |
| 472 |
> |
\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}} |
| 473 |
> |
\end{eqnarray} |
| 474 |
> |
|
| 475 |
> |
To describe the interactions between metal (Au) and non-metal atoms, |
| 476 |
> |
we refer to an adsorption study of alkyl thiols on gold surfaces by |
| 477 |
> |
Vlugt {\it et al.}\cite{vlugt:cpc2007154} They fitted an effective |
| 478 |
> |
Lennard-Jones form of potential parameters for the interaction between |
| 479 |
> |
Au and pseudo-atoms CH$_x$ and S based on a well-established and |
| 480 |
> |
widely-used effective potential of Hautman and Klein for the Au(111) |
| 481 |
> |
surface.\cite{hautman:4994} As our simulations require the gold slab |
| 482 |
> |
to be flexible to accommodate thermal excitation, the pair-wise form |
| 483 |
> |
of potentials they developed was used for our study. |
| 484 |
> |
|
| 485 |
> |
The potentials developed from {\it ab initio} calculations by Leng |
| 486 |
> |
{\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
| 487 |
> |
interactions between Au and aromatic C/H atoms in toluene. However, |
| 488 |
> |
the Lennard-Jones parameters between Au and other types of particles, |
| 489 |
> |
(e.g. AA alkanes) have not yet been established. For these |
| 490 |
> |
interactions, the Lorentz-Berthelot mixing rule can be used to derive |
| 491 |
> |
effective single-atom LJ parameters for the metal using the fit values |
| 492 |
> |
for toluene. These are then used to construct reasonable mixing |
| 493 |
> |
parameters for the interactions between the gold and other atoms. |
| 494 |
> |
Table \ref{MnM} summarizes the ``metal/non-metal'' parameters used in |
| 495 |
> |
our simulations. |
| 496 |
> |
|
| 497 |
> |
\begin{table*} |
| 498 |
> |
\begin{minipage}{\linewidth} |
| 499 |
> |
\begin{center} |
| 500 |
> |
\caption{Non-bonded interaction parameters (including cross |
| 501 |
> |
interactions with Au atoms) for both force fields used in this |
| 502 |
> |
work.} |
| 503 |
> |
\begin{tabular}{lllllll} |
| 504 |
> |
\hline\hline |
| 505 |
> |
& Site & $\sigma_{ii}$ & $\epsilon_{ii}$ & $q_i$ & |
| 506 |
> |
$\sigma_{Au-i}$ & $\epsilon_{Au-i}$ \\ |
| 507 |
> |
& & (\AA) & (kcal/mol) & ($e$) & (\AA) & (kcal/mol) \\ |
| 508 |
> |
\hline |
| 509 |
> |
United Atom (UA) |
| 510 |
> |
&CH3 & 3.75 & 0.1947 & - & 3.54 & 0.2146 \\ |
| 511 |
> |
&CH2 & 3.95 & 0.0914 & - & 3.54 & 0.1749 \\ |
| 512 |
> |
&CHar & 3.695 & 0.1003 & - & 3.4625 & 0.1680 \\ |
| 513 |
> |
&CRar & 3.88 & 0.04173 & - & 3.555 & 0.1604 \\ |
| 514 |
> |
\hline |
| 515 |
> |
All Atom (AA) |
| 516 |
> |
&CT3 & 3.50 & 0.066 & -0.18 & 3.365 & 0.1373 \\ |
| 517 |
> |
&CT2 & 3.50 & 0.066 & -0.12 & 3.365 & 0.1373 \\ |
| 518 |
> |
&CTT & 3.50 & 0.066 & -0.065 & 3.365 & 0.1373 \\ |
| 519 |
> |
&HC & 2.50 & 0.030 & 0.06 & 2.865 & 0.09256 \\ |
| 520 |
> |
&CA & 3.55 & 0.070 & -0.115 & 3.173 & 0.0640 \\ |
| 521 |
> |
&HA & 2.42 & 0.030 & 0.115 & 2.746 & 0.0414 \\ |
| 522 |
> |
\hline |
| 523 |
> |
Both UA and AA |
| 524 |
> |
& S & 4.45 & 0.25 & - & 2.40 & 8.465 \\ |
| 525 |
> |
\hline\hline |
| 526 |
> |
\end{tabular} |
| 527 |
> |
\label{MnM} |
| 528 |
> |
\end{center} |
| 529 |
> |
\end{minipage} |
| 530 |
> |
\end{table*} |
| 531 |
|
|
| 532 |
+ |
|
| 533 |
+ |
\section{Results} |
| 534 |
+ |
There are many factors contributing to the measured interfacial |
| 535 |
+ |
conductance; some of these factors are physically motivated |
| 536 |
+ |
(e.g. coverage of the surface by the capping agent coverage and |
| 537 |
+ |
solvent identity), while some are governed by parameters of the |
| 538 |
+ |
methodology (e.g. applied flux and the formulas used to obtain the |
| 539 |
+ |
conductance). In this section we discuss the major physical and |
| 540 |
+ |
calculational effects on the computed conductivity. |
| 541 |
+ |
|
| 542 |
+ |
\subsection{Effects due to capping agent coverage} |
| 543 |
+ |
|
| 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 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{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 |
+ |
|
| 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. 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 $G$) |
| 576 |
+ |
happens when the surfaces are about 75\% covered with butanethiol |
| 577 |
+ |
caps. The reason for this behavior is not entirely clear. One |
| 578 |
+ |
explanation is that incomplete butanethiol coverage allows small gaps |
| 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 |
+ |
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 interdigitation at lower coverages). This |
| 588 |
+ |
competition would lead to the non-monotonic coverage behavior observed |
| 589 |
+ |
here. |
| 590 |
+ |
|
| 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 |
+ |
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 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 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. 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 & $G$ & $G^\prime$ \\ |
| 635 |
+ |
(or bare surface) & model & |
| 636 |
+ |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 637 |
+ |
\hline |
| 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 & 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 & 158(25) & 172(4) \\ |
| 651 |
+ |
& AA hexane & 243(29) & 191(11) \\ |
| 652 |
+ |
& AA toluene & 364(36) & 322(67) \\ |
| 653 |
+ |
\hline |
| 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} |
| 661 |
+ |
\end{center} |
| 662 |
+ |
\end{minipage} |
| 663 |
+ |
\end{table*} |
| 664 |
+ |
|
| 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 |
+ |
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 |
+ |
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 |
+ |
\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 |
+ |
|
| 702 |
+ |
\subsection{Effects due to methodology and simulation parameters} |
| 703 |
+ |
|
| 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 |
+ |
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 |
+ |
|
| 724 |
+ |
\subsubsection{Effects of applied 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 small number $J_z$ values. |
| 750 |
+ |
|
| 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 |
+ |
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 |
+ |
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 |
+ |
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 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 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 |
+ |
|
| 814 |
+ |
\section{Discussion} |
| 815 |
+ |
|
| 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 |
| 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} |
| 835 |
+ |
\end{equation} |
| 836 |
+ |
The power spectrum is constructed via a Fourier transform of the |
| 837 |
+ |
symmetrized velocity autocorrelation function, |
| 838 |
+ |
\begin{equation} |
| 839 |
+ |
\hat{f}(\omega) = |
| 840 |
+ |
\int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt |
| 841 |
+ |
\label{fourier} |
| 842 |
+ |
\end{equation} |
| 843 |
+ |
|
| 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 |
+ |
|
| 859 |
+ |
\begin{figure} |
| 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 |
+ |
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 |
+ |
\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 |
+ |
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 |
+ |
|
| 894 |
+ |
\begin{figure} |
| 895 |
+ |
\includegraphics[width=\linewidth]{aahxntln} |
| 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 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 |
+ |
For the Au / butanethiol / toluene interfaces, having the AA |
| 905 |
+ |
butanethiol deuterated did not yield a significant change in the |
| 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 |
+ |
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 |
+ |
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 |
+ |
|
| 928 |
+ |
\begin{figure} |
| 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 |
+ |
|
| 939 |
+ |
\section{Conclusions} |
| 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 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 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 |
+ |
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 |
| 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.} 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. \newpage |
| 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} |
| 997 |
|
|
| 998 |
|
\end{doublespace} |
