| 44 |
|
\begin{doublespace} |
| 45 |
|
|
| 46 |
|
\begin{abstract} |
| 47 |
< |
|
| 48 |
< |
With the Non-Isotropic Velocity Scaling algorithm (NIVS) we have |
| 49 |
< |
developed, an unphysical thermal flux can be effectively set up even |
| 50 |
< |
for non-homogeneous systems like interfaces in non-equilibrium |
| 51 |
< |
molecular dynamics simulations. In this work, this algorithm is |
| 52 |
< |
applied for simulating thermal conductance at metal / organic solvent |
| 53 |
< |
interfaces with various coverages of butanethiol capping |
| 54 |
< |
agents. Different solvents and force field models were tested. Our |
| 55 |
< |
results suggest that the United-Atom models are able to provide an |
| 56 |
< |
estimate of the interfacial thermal conductivity comparable to |
| 57 |
< |
experiments in our simulations with satisfactory computational |
| 58 |
< |
efficiency. From our results, the acoustic impedance mismatch between |
| 59 |
< |
metal and liquid phase is effectively reduced by the capping |
| 60 |
< |
agents, and thus leads to interfacial thermal conductance |
| 61 |
< |
enhancement. Furthermore, this effect is closely related to the |
| 62 |
< |
capping agent coverage on the metal surfaces and the type of solvent |
| 63 |
< |
molecules, and is affected by the models used in the simulations. |
| 64 |
< |
|
| 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 the acoustic |
| 54 |
> |
impedance mismatch between the metal and liquid phases is |
| 55 |
> |
effectively reduced by the capping agents, leading to a greatly |
| 56 |
> |
enhanced conductivity at the interface. Specifically, the chemical |
| 57 |
> |
bond between the metal and the capping agent introduces a |
| 58 |
> |
vibrational overlap that is not present without the capping agent, |
| 59 |
> |
and the overlap between the vibrational spectra (metal to cap, cap |
| 60 |
> |
to solvent) provides a mechanism for rapid thermal transport across |
| 61 |
> |
the interface. Our calculations also suggest that this is a |
| 62 |
> |
non-monotonic function of the fractional coverage of the surface, as |
| 63 |
> |
moderate coverages allow convective heat transport of solvent |
| 64 |
> |
molecules that have been in close contact with the capping agent. |
| 65 |
|
\end{abstract} |
| 66 |
|
|
| 67 |
|
\newpage |
| 73 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 74 |
|
|
| 75 |
|
\section{Introduction} |
| 76 |
< |
Due to the importance of heat flow in nanotechnology, interfacial |
| 77 |
< |
thermal conductance has been studied extensively both experimentally |
| 78 |
< |
and computationally.\cite{cahill:793} Unlike bulk materials, nanoscale |
| 79 |
< |
materials have a significant fraction of their atoms at interfaces, |
| 80 |
< |
and the chemical details of these interfaces govern the heat transfer |
| 81 |
< |
behavior. Furthermore, the interfaces are |
| 76 |
> |
Due to the importance of heat flow (and heat removal) in |
| 77 |
> |
nanotechnology, interfacial thermal conductance has been studied |
| 78 |
> |
extensively both experimentally and computationally.\cite{cahill:793} |
| 79 |
> |
Nanoscale materials have a significant fraction of their atoms at |
| 80 |
> |
interfaces, and the chemical details of these interfaces govern the |
| 81 |
> |
thermal transport properties. Furthermore, the interfaces are often |
| 82 |
|
heterogeneous (e.g. solid - liquid), which provides a challenge to |
| 83 |
< |
traditional methods developed for homogeneous systems. |
| 83 |
> |
computational methods which have been developed for homogeneous or |
| 84 |
> |
bulk systems. |
| 85 |
|
|
| 86 |
< |
Experimentally, various interfaces have been investigated for their |
| 87 |
< |
thermal conductance. Cahill and coworkers studied nanoscale thermal |
| 86 |
> |
Experimentally, the thermal properties of a number of interfaces have |
| 87 |
> |
been investigated. Cahill and coworkers studied nanoscale thermal |
| 88 |
|
transport from metal nanoparticle/fluid interfaces, to epitaxial |
| 89 |
< |
TiN/single crystal oxides interfaces, to hydrophilic and hydrophobic |
| 89 |
> |
TiN/single crystal oxides interfaces, and hydrophilic and hydrophobic |
| 90 |
|
interfaces between water and solids with different self-assembled |
| 91 |
|
monolayers.\cite{Wilson:2002uq,PhysRevB.67.054302,doi:10.1021/jp048375k,PhysRevLett.96.186101} |
| 92 |
< |
Wang {\it et al.} studied heat transport through |
| 93 |
< |
long-chain hydrocarbon monolayers on gold substrate at individual |
| 94 |
< |
molecular level,\cite{Wang10082007} Schmidt {\it et al.} studied the |
| 95 |
< |
role of CTAB on thermal transport between gold nanorods and |
| 96 |
< |
solvent,\cite{doi:10.1021/jp8051888} and Juv\'e {\it et al.} studied |
| 97 |
< |
the cooling dynamics, which is controlled by thermal interface |
| 98 |
< |
resistence of glass-embedded metal |
| 92 |
> |
Wang {\it et al.} studied heat transport through long-chain |
| 93 |
> |
hydrocarbon monolayers on gold substrate at individual molecular |
| 94 |
> |
level,\cite{Wang10082007} Schmidt {\it et al.} studied the role of |
| 95 |
> |
cetyltrimethylammonium bromide (CTAB) on the thermal transport between |
| 96 |
> |
gold nanorods and solvent,\cite{doi:10.1021/jp8051888} and Juv\'e {\it |
| 97 |
> |
et al.} studied the cooling dynamics, which is controlled by thermal |
| 98 |
> |
interface resistance of glass-embedded metal |
| 99 |
|
nanoparticles.\cite{PhysRevB.80.195406} Although interfaces are |
| 100 |
|
normally considered barriers for heat transport, Alper {\it et al.} |
| 101 |
|
suggested that specific ligands (capping agents) could completely |
| 115 |
|
difficult to measure with Equilibrium |
| 116 |
|
MD\cite{doi:10.1080/0026897031000068578} or forward NEMD simulation |
| 117 |
|
methods. Therefore, the Reverse NEMD (RNEMD) |
| 118 |
< |
methods\cite{MullerPlathe:1997xw,kuang:164101} would have the |
| 119 |
< |
advantage of applying this difficult to measure flux (while measuring |
| 120 |
< |
the resulting gradient), given that the simulation methods being able |
| 121 |
< |
to effectively apply an unphysical flux in non-homogeneous systems. |
| 118 |
> |
methods\cite{MullerPlathe:1997xw,kuang:164101} would be advantageous |
| 119 |
> |
in that they {\it apply} the difficult to measure quantity (flux), |
| 120 |
> |
while {\it measuring} the easily-computed quantity (the thermal |
| 121 |
> |
gradient). This is particularly true for inhomogeneous interfaces |
| 122 |
> |
where it would not be clear how to apply a gradient {\it a priori}. |
| 123 |
|
Garde and coworkers\cite{garde:nl2005,garde:PhysRevLett2009} applied |
| 124 |
|
this approach to various liquid interfaces and studied how thermal |
| 125 |
< |
conductance (or resistance) is dependent on chemistry details of |
| 126 |
< |
interfaces, e.g. hydrophobic and hydrophilic aqueous interfaces. |
| 125 |
> |
conductance (or resistance) is dependent on chemical details of a |
| 126 |
> |
number of hydrophobic and hydrophilic aqueous interfaces. {\bf And |
| 127 |
> |
Luo {\it et al.} studied the thermal conductance of Au-SAM-Au |
| 128 |
> |
junctions using the same approach, with comparison to a constant |
| 129 |
> |
temperature difference method\cite{Luo20101}. While this latter |
| 130 |
> |
approach establishes more thermal distributions compared to the |
| 131 |
> |
former RNEMD methods, it does not guarantee momentum or kinetic |
| 132 |
> |
energy conservations.} |
| 133 |
|
|
| 134 |
|
Recently, we have developed a Non-Isotropic Velocity Scaling (NIVS) |
| 135 |
|
algorithm for RNEMD simulations\cite{kuang:164101}. This algorithm |
| 143 |
|
The work presented here deals with the Au(111) surface covered to |
| 144 |
|
varying degrees by butanethiol, a capping agent with short carbon |
| 145 |
|
chain, and solvated with organic solvents of different molecular |
| 146 |
< |
properties. Different models were used for both the capping agent and |
| 147 |
< |
the solvent force field parameters. Using the NIVS algorithm, the |
| 148 |
< |
thermal transport across these interfaces was studied and the |
| 149 |
< |
underlying mechanism for the phenomena was investigated. |
| 146 |
> |
properties. {\bf To our knowledge, few previous MD inverstigations |
| 147 |
> |
have been found to address to these systems yet.} Different models |
| 148 |
> |
were used for both the capping agent and the solvent force field |
| 149 |
> |
parameters. Using the NIVS algorithm, the thermal transport across |
| 150 |
> |
these interfaces was studied and the underlying mechanism for the |
| 151 |
> |
phenomena was investigated. |
| 152 |
|
|
| 153 |
|
\section{Methodology} |
| 154 |
< |
\subsection{Imposd-Flux Methods in MD Simulations} |
| 154 |
> |
\subsection{Imposed-Flux Methods in MD Simulations} |
| 155 |
|
Steady state MD simulations have an advantage in that not many |
| 156 |
|
trajectories are needed to study the relationship between thermal flux |
| 157 |
|
and thermal gradients. For systems with low interfacial conductance, |
| 175 |
|
kinetic energy fluxes without obvious perturbation to the velocity |
| 176 |
|
distributions of the simulated systems. Furthermore, this approach has |
| 177 |
|
the advantage in heterogeneous interfaces in that kinetic energy flux |
| 178 |
< |
can be applied between regions of particles of arbitary identity, and |
| 178 |
> |
can be applied between regions of particles of arbitrary identity, and |
| 179 |
|
the flux will not be restricted by difference in particle mass. |
| 180 |
|
|
| 181 |
|
The NIVS algorithm scales the velocity vectors in two separate regions |
| 182 |
< |
of a simulation system with respective diagonal scaling matricies. To |
| 183 |
< |
determine these scaling factors in the matricies, a set of equations |
| 182 |
> |
of a simulation system with respective diagonal scaling matrices. To |
| 183 |
> |
determine these scaling factors in the matrices, a set of equations |
| 184 |
|
including linear momentum conservation and kinetic energy conservation |
| 185 |
|
constraints and target energy flux satisfaction is solved. With the |
| 186 |
|
scaling operation applied to the system in a set frequency, bulk |
| 203 |
|
where ${E_{total}}$ is the total imposed non-physical kinetic energy |
| 204 |
|
transfer during the simulation and ${\langle T_\mathrm{hot}\rangle}$ |
| 205 |
|
and ${\langle T_\mathrm{cold}\rangle}$ are the average observed |
| 206 |
< |
temperature of the two separated phases. |
| 206 |
> |
temperature of the two separated phases. For an applied flux $J_z$ |
| 207 |
> |
operating over a simulation time $t$ on a periodically-replicated slab |
| 208 |
> |
of dimensions $L_x \times L_y$, $E_{total} = J_z *(t)*(2 L_x L_y)$. |
| 209 |
|
|
| 210 |
|
When the interfacial conductance is {\it not} small, there are two |
| 211 |
|
ways to define $G$. One common way is to assume the temperature is |
| 212 |
|
discrete on the two sides of the interface. $G$ can be calculated |
| 213 |
|
using the applied thermal flux $J$ and the maximum temperature |
| 214 |
|
difference measured along the thermal gradient max($\Delta T$), which |
| 215 |
< |
occurs at the Gibbs deviding surface (Figure \ref{demoPic}). This is |
| 215 |
> |
occurs at the Gibbs dividing surface (Figure \ref{demoPic}). This is |
| 216 |
|
known as the Kapitza conductance, which is the inverse of the Kapitza |
| 217 |
|
resistance. |
| 218 |
|
\begin{equation} |
| 225 |
|
\caption{Interfacial conductance can be calculated by applying an |
| 226 |
|
(unphysical) kinetic energy flux between two slabs, one located |
| 227 |
|
within the metal and another on the edge of the periodic box. The |
| 228 |
< |
system responds by forming a thermal response or a gradient. In |
| 229 |
< |
bulk liquids, this gradient typically has a single slope, but in |
| 230 |
< |
interfacial systems, there are distinct thermal conductivity |
| 231 |
< |
domains. The interfacial conductance, $G$ is found by measuring the |
| 232 |
< |
temperature gap at the Gibbs dividing surface, or by using second |
| 233 |
< |
derivatives of the thermal profile.} |
| 228 |
> |
system responds by forming a thermal gradient. In bulk liquids, |
| 229 |
> |
this gradient typically has a single slope, but in interfacial |
| 230 |
> |
systems, there are distinct thermal conductivity domains. The |
| 231 |
> |
interfacial conductance, $G$ is found by measuring the temperature |
| 232 |
> |
gap at the Gibbs dividing surface, or by using second derivatives of |
| 233 |
> |
the thermal profile.} |
| 234 |
|
\label{demoPic} |
| 235 |
|
\end{figure} |
| 236 |
|
|
| 269 |
|
|
| 270 |
|
\begin{figure} |
| 271 |
|
\includegraphics[width=\linewidth]{gradT} |
| 272 |
< |
\caption{A sample of Au-butanethiol/hexane interfacial system and the |
| 273 |
< |
temperature profile after a kinetic energy flux is imposed to |
| 274 |
< |
it. The 1st and 2nd derivatives of the temperature profile can be |
| 275 |
< |
obtained with finite difference approximation (lower panel).} |
| 272 |
> |
\caption{A sample of Au (111) / butanethiol / hexane interfacial |
| 273 |
> |
system with the temperature profile after a kinetic energy flux has |
| 274 |
> |
been imposed. Note that the largest temperature jump in the thermal |
| 275 |
> |
profile (corresponding to the lowest interfacial conductance) is at |
| 276 |
> |
the interface between the butanethiol molecules (blue) and the |
| 277 |
> |
solvent (grey). First and second derivatives of the temperature |
| 278 |
> |
profile are obtained using a finite difference approximation (lower |
| 279 |
> |
panel).} |
| 280 |
|
\label{gradT} |
| 281 |
|
\end{figure} |
| 282 |
|
|
| 323 |
|
solvent molecules would change the normal behavior of the liquid |
| 324 |
|
phase. Therefore, our $N_{solvent}$ values were chosen to ensure that |
| 325 |
|
these extreme cases did not happen to our simulations. The spacing |
| 326 |
< |
between periodic images of the gold interfaces is $45 \sim 75$\AA. |
| 326 |
> |
between periodic images of the gold interfaces is $45 \sim 75$\AA in |
| 327 |
> |
our simulations. |
| 328 |
|
|
| 329 |
|
The initial configurations generated are further equilibrated with the |
| 330 |
|
$x$ and $y$ dimensions fixed, only allowing the $z$-length scale to |
| 342 |
|
$\sim$200K. Therefore, thermal flux usually came from the metal to the |
| 343 |
|
liquid so that the liquid has a higher temperature and would not |
| 344 |
|
freeze due to lowered temperatures. After this induced temperature |
| 345 |
< |
gradient had stablized, the temperature profile of the simulation cell |
| 346 |
< |
was recorded. To do this, the simulation cell is devided evenly into |
| 345 |
> |
gradient had stabilized, the temperature profile of the simulation cell |
| 346 |
> |
was recorded. To do this, the simulation cell is divided evenly into |
| 347 |
|
$N$ slabs along the $z$-axis. The average temperatures of each slab |
| 348 |
|
are recorded for 1$\sim$2 ns. When the slab width $d$ of each slab is |
| 349 |
|
the same, the derivatives of $T$ with respect to slab number $n$ can |
| 374 |
|
\caption{Structures of the capping agent and solvents utilized in |
| 375 |
|
these simulations. The chemically-distinct sites (a-e) are expanded |
| 376 |
|
in terms of constituent atoms for both United Atom (UA) and All Atom |
| 377 |
< |
(AA) force fields. Most parameters are from |
| 378 |
< |
Refs. \protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes,TraPPE-UA.thiols} |
| 377 |
> |
(AA) force fields. Most parameters are from References |
| 378 |
> |
\protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes,TraPPE-UA.thiols} |
| 379 |
|
(UA) and \protect\cite{OPLSAA} (AA). Cross-interactions with the Au |
| 380 |
|
atoms are given in Table \ref{MnM}.} |
| 381 |
|
\label{demoMol} |
| 399 |
|
|
| 400 |
|
By eliminating explicit hydrogen atoms, the TraPPE-UA models are |
| 401 |
|
simple and computationally efficient, while maintaining good accuracy. |
| 402 |
< |
However, the TraPPE-UA model for alkanes is known to predict a slighly |
| 402 |
> |
However, the TraPPE-UA model for alkanes is known to predict a slightly |
| 403 |
|
lower boiling point than experimental values. This is one of the |
| 404 |
|
reasons we used a lower average temperature (200K) for our |
| 405 |
|
simulations. If heat is transferred to the liquid phase during the |
| 518 |
|
|
| 519 |
|
\begin{figure} |
| 520 |
|
\includegraphics[width=\linewidth]{coverage} |
| 521 |
< |
\caption{Comparison of interfacial thermal conductivity ($G$) values |
| 522 |
< |
for the Au-butanethiol/solvent interface with various UA models and |
| 523 |
< |
different capping agent coverages at $\langle T\rangle\sim$200K.} |
| 521 |
> |
\caption{The interfacial thermal conductivity ($G$) has a |
| 522 |
> |
non-monotonic dependence on the degree of surface capping. This |
| 523 |
> |
data is for the Au(111) / butanethiol / solvent interface with |
| 524 |
> |
various UA force fields at $\langle T\rangle \sim $200K.} |
| 525 |
|
\label{coverage} |
| 526 |
|
\end{figure} |
| 527 |
|
|
| 528 |
< |
In partially covered surfaces, the derivative definition for $G^\prime$ (Eq. \ref{derivativeG}) becomes difficult to apply, as the location of maximum change of $\lambda$ becomes washed out. The discrete definition (Eq. \ref{discreteG}) is easier to apply, as the Gibbs dividing surface is still well-defined. Therefore, $G$ (not $G^\prime$) was used in this section. |
| 528 |
> |
In partially covered surfaces, the derivative definition for |
| 529 |
> |
$G^\prime$ (Eq. \ref{derivativeG}) becomes difficult to apply, as the |
| 530 |
> |
location of maximum change of $\lambda$ becomes washed out. The |
| 531 |
> |
discrete definition (Eq. \ref{discreteG}) is easier to apply, as the |
| 532 |
> |
Gibbs dividing surface is still well-defined. Therefore, $G$ (not |
| 533 |
> |
$G^\prime$) was used in this section. |
| 534 |
|
|
| 535 |
< |
From Figure \ref{coverage}, one can see the significance of the presence of capping agents. When even a small fraction of the Au(111) surface sites are covered with butanethiols, the conductivity exhibits an enhancement by at least a factor of 3. Cappping agents are clearly playing a major role in thermal transport at metal / organic solvent surfaces. |
| 535 |
> |
From Figure \ref{coverage}, one can see the significance of the |
| 536 |
> |
presence of capping agents. When even a small fraction of the Au(111) |
| 537 |
> |
surface sites are covered with butanethiols, the conductivity exhibits |
| 538 |
> |
an enhancement by at least a factor of 3. Capping agents are clearly |
| 539 |
> |
playing a major role in thermal transport at metal / organic solvent |
| 540 |
> |
surfaces. |
| 541 |
|
|
| 542 |
< |
We note a non-monotonic behavior in the interfacial conductance as a function of surface coverage. The maximum conductance (largest $G$) happens when the surfaces are about 75\% covered with butanethiol caps. The reason for this behavior is not entirely clear. One explanation is that incomplete butanethiol coverage allows small gaps between butanethiols to form. These gaps can be filled by transient solvent molecules. These solvent molecules couple very strongly with the hot capping agent molecules near the surface, and can then carry away (diffusively) the excess thermal energy from the surface. |
| 542 |
> |
We note a non-monotonic behavior in the interfacial conductance as a |
| 543 |
> |
function of surface coverage. The maximum conductance (largest $G$) |
| 544 |
> |
happens when the surfaces are about 75\% covered with butanethiol |
| 545 |
> |
caps. The reason for this behavior is not entirely clear. One |
| 546 |
> |
explanation is that incomplete butanethiol coverage allows small gaps |
| 547 |
> |
between butanethiols to form. These gaps can be filled by transient |
| 548 |
> |
solvent molecules. These solvent molecules couple very strongly with |
| 549 |
> |
the hot capping agent molecules near the surface, and can then carry |
| 550 |
> |
away (diffusively) the excess thermal energy from the surface. |
| 551 |
|
|
| 552 |
< |
There appears to be a competition between the conduction of the thermal energy away from the surface by the capping agents (enhanced by greater coverage) and the coupling of the capping agents with the solvent (enhanced by interdigitation at lower coverages). This competition would lead to the non-monotonic coverage behavior observed here. |
| 552 |
> |
There appears to be a competition between the conduction of the |
| 553 |
> |
thermal energy away from the surface by the capping agents (enhanced |
| 554 |
> |
by greater coverage) and the coupling of the capping agents with the |
| 555 |
> |
solvent (enhanced by interdigitation at lower coverages). This |
| 556 |
> |
competition would lead to the non-monotonic coverage behavior observed |
| 557 |
> |
here. |
| 558 |
|
|
| 559 |
< |
Results for rigid body toluene solvent, as well as the UA hexane, are within the ranges expected from prior experimental work.\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406} This suggests that explicit hydrogen atoms might not be required for modeling thermal transport in these systems. C-H vibrational modes do not see significant excited state population at low temperatures, and are not likely to carry lower frequency excitations from the solid layer into the bulk liquid. |
| 559 |
> |
Results for rigid body toluene solvent, as well as the UA hexane, are |
| 560 |
> |
within the ranges expected from prior experimental |
| 561 |
> |
work.\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406} This suggests |
| 562 |
> |
that explicit hydrogen atoms might not be required for modeling |
| 563 |
> |
thermal transport in these systems. C-H vibrational modes do not see |
| 564 |
> |
significant excited state population at low temperatures, and are not |
| 565 |
> |
likely to carry lower frequency excitations from the solid layer into |
| 566 |
> |
the bulk liquid. |
| 567 |
|
|
| 568 |
< |
The toluene solvent does not exhibit the same behavior as hexane in that $G$ remains at approximately the same magnitude when the capping coverage increases from 25\% to 75\%. Toluene, as a rigid planar molecule, cannot occupy the relatively small gaps between the capping agents as easily as the chain-like {\it n}-hexane. The effect of solvent coupling to the capping agent is therefore weaker in toluene except at the very lowest coverage levels. This effect counters the coverage-dependent conduction of heat away from the metal surface, leading to a much flatter $G$ vs. coverage trend than is observed in {\it n}-hexane. |
| 568 |
> |
The toluene solvent does not exhibit the same behavior as hexane in |
| 569 |
> |
that $G$ remains at approximately the same magnitude when the capping |
| 570 |
> |
coverage increases from 25\% to 75\%. Toluene, as a rigid planar |
| 571 |
> |
molecule, cannot occupy the relatively small gaps between the capping |
| 572 |
> |
agents as easily as the chain-like {\it n}-hexane. The effect of |
| 573 |
> |
solvent coupling to the capping agent is therefore weaker in toluene |
| 574 |
> |
except at the very lowest coverage levels. This effect counters the |
| 575 |
> |
coverage-dependent conduction of heat away from the metal surface, |
| 576 |
> |
leading to a much flatter $G$ vs. coverage trend than is observed in |
| 577 |
> |
{\it n}-hexane. |
| 578 |
|
|
| 579 |
|
\subsection{Effects due to Solvent \& Solvent Models} |
| 580 |
< |
In addition to UA solvent and capping agent models, AA models have also been included in our simulations. In most of this work, the same (UA or AA) model for solvent and capping agent was used, but it is also possible to utilize different models for different components. We have also included isotopic substitutions (Hydrogen to Deuterium) to decrease the explicit vibrational overlap between solvent and capping agent. Table \ref{modelTest} summarizes the results of these studies. |
| 580 |
> |
In addition to UA solvent and capping agent models, AA models have |
| 581 |
> |
also been included in our simulations. In most of this work, the same |
| 582 |
> |
(UA or AA) model for solvent and capping agent was used, but it is |
| 583 |
> |
also possible to utilize different models for different components. |
| 584 |
> |
We have also included isotopic substitutions (Hydrogen to Deuterium) |
| 585 |
> |
to decrease the explicit vibrational overlap between solvent and |
| 586 |
> |
capping agent. Table \ref{modelTest} summarizes the results of these |
| 587 |
> |
studies. |
| 588 |
|
|
| 589 |
|
\begin{table*} |
| 590 |
|
\begin{minipage}{\linewidth} |
| 593 |
|
\caption{Computed interfacial thermal conductance ($G$ and |
| 594 |
|
$G^\prime$) values for interfaces using various models for |
| 595 |
|
solvent and capping agent (or without capping agent) at |
| 596 |
< |
$\langle T\rangle\sim$200K. (D stands for deuterated solvent |
| 597 |
< |
or capping agent molecules; ``Avg.'' denotes results that are |
| 598 |
< |
averages of simulations under different applied thermal flux values $(J_z)$. Error |
| 599 |
< |
estimates are indicated in parentheses.)} |
| 596 |
> |
$\langle T\rangle\sim$200K. Here ``D'' stands for deuterated |
| 597 |
> |
solvent or capping agent molecules; ``Avg.'' denotes results |
| 598 |
> |
that are averages of simulations under different applied |
| 599 |
> |
thermal flux $(J_z)$ values. Error estimates are indicated in |
| 600 |
> |
parentheses.} |
| 601 |
|
|
| 602 |
|
\begin{tabular}{llccc} |
| 603 |
|
\hline\hline |
| 632 |
|
\end{minipage} |
| 633 |
|
\end{table*} |
| 634 |
|
|
| 635 |
< |
To facilitate direct comparison between force fields, systems with the same capping agent and solvent were prepared with the same length scales for the simulation cells. |
| 635 |
> |
To facilitate direct comparison between force fields, systems with the |
| 636 |
> |
same capping agent and solvent were prepared with the same length |
| 637 |
> |
scales for the simulation cells. |
| 638 |
|
|
| 639 |
< |
On bare metal / solvent surfaces, different force field models for hexane yield similar results for both $G$ and $G^\prime$, and these two definitions agree with each other very well. This is primarily an indicator of weak interactions between the metal and the solvent, and is a typical case for acoustic impedance mismatch between these two phases. |
| 639 |
> |
On bare metal / solvent surfaces, different force field models for |
| 640 |
> |
hexane yield similar results for both $G$ and $G^\prime$, and these |
| 641 |
> |
two definitions agree with each other very well. This is primarily an |
| 642 |
> |
indicator of weak interactions between the metal and the solvent, and |
| 643 |
> |
is a typical case for acoustic impedance mismatch between these two |
| 644 |
> |
phases. |
| 645 |
|
|
| 646 |
< |
For the fully-covered surfaces, the choice of force field for the capping agent and solvent has a large impact on the calulated values of $G$ and $G^\prime$. The AA thiol to AA solvent conductivities are much larger than their UA to UA counterparts, and these values exceed the experimental estimates by a large measure. The AA force field allows significant energy to go into C-H (or C-D) stretching modes, and since these modes are high frequency, this non-quantum behavior is likely responsible for the overestimate of the conductivity. |
| 646 |
> |
For the fully-covered surfaces, the choice of force field for the |
| 647 |
> |
capping agent and solvent has a large impact on the calculated values |
| 648 |
> |
of $G$ and $G^\prime$. The AA thiol to AA solvent conductivities are |
| 649 |
> |
much larger than their UA to UA counterparts, and these values exceed |
| 650 |
> |
the experimental estimates by a large measure. The AA force field |
| 651 |
> |
allows significant energy to go into C-H (or C-D) stretching modes, |
| 652 |
> |
and since these modes are high frequency, this non-quantum behavior is |
| 653 |
> |
likely responsible for the overestimate of the conductivity. Compared |
| 654 |
> |
to the AA model, the UA model yields more reasonable conductivity |
| 655 |
> |
values with much higher computational efficiency. |
| 656 |
|
|
| 576 |
– |
The similarity in the vibrational modes available to solvent and capping agent can be reduced by deuterating one of the two components. Once either the hexanes or the butanethiols are deuterated, one can see a significantly lower $G$ and $G^\prime$ (Figure \ref{aahxntln}). Compared to the AA model, the UA model yields more reasonable conductivity values with much higher computational efficiency. |
| 577 |
– |
|
| 578 |
– |
\begin{figure} |
| 579 |
– |
\includegraphics[width=\linewidth]{aahxntln} |
| 580 |
– |
\caption{Spectra obtained for all-atom (AA) Au / butanethiol / solvent |
| 581 |
– |
systems. When butanethiol is deuterated (lower left), its |
| 582 |
– |
vibrational overlap with hexane decreases significantly. Since aromatic molecules and the butanethiol are vibrationally dissimilar, the change is not as dramatic when toluene is the solvent (right).} |
| 583 |
– |
\label{aahxntln} |
| 584 |
– |
\end{figure} |
| 585 |
– |
|
| 586 |
– |
For the Au / butanethiol / toluene interfaces, having the AA butanethiol deuterated did not yield a significant change in the measured conductance. Compared to the C-H vibrational overlap between hexane and butanethiol, both of which have alkyl chains, the overlap between toluene and butanethiol is not as significant and thus does not contribute as much to the heat exchange process. The presence of extra degrees of freedom in the AA force field for toluene yields higher heat exchange rates between the two phases and results in a much higher conductivity than in the UA force field. |
| 587 |
– |
|
| 657 |
|
\subsubsection{Are electronic excitations in the metal important?} |
| 658 |
< |
Because they lack electronic excitations, the QSC and related embedded atom method (EAM) models for gold are known to predict unreasonably low values for bulk conductivity ($\lambda$).\cite{kuang:164101,ISI:000207079300006,Clancy:1992} If the conductance between the phases ($G$) is governed primarily by phonon excitation (and not electronic degrees of freedom), one would expect a classical model to capture most of the interfacial thermal conductance. Our results for $G$ and $G^\prime$ indicate that this is indeed the case, and suggest that the modeling of interfacial thermal transport depends primarily on the description of the interactions between the various components at the interface. When the metal is chemically capped, the primary barrier to thermal conductivity appears to be the interface between the capping agent and the surrounding solvent, so the excitations in the metal have little impact on the value of $G$. |
| 658 |
> |
Because they lack electronic excitations, the QSC and related embedded |
| 659 |
> |
atom method (EAM) models for gold are known to predict unreasonably |
| 660 |
> |
low values for bulk conductivity |
| 661 |
> |
($\lambda$).\cite{kuang:164101,ISI:000207079300006,Clancy:1992} If the |
| 662 |
> |
conductance between the phases ($G$) is governed primarily by phonon |
| 663 |
> |
excitation (and not electronic degrees of freedom), one would expect a |
| 664 |
> |
classical model to capture most of the interfacial thermal |
| 665 |
> |
conductance. Our results for $G$ and $G^\prime$ indicate that this is |
| 666 |
> |
indeed the case, and suggest that the modeling of interfacial thermal |
| 667 |
> |
transport depends primarily on the description of the interactions |
| 668 |
> |
between the various components at the interface. When the metal is |
| 669 |
> |
chemically capped, the primary barrier to thermal conductivity appears |
| 670 |
> |
to be the interface between the capping agent and the surrounding |
| 671 |
> |
solvent, so the excitations in the metal have little impact on the |
| 672 |
> |
value of $G$. |
| 673 |
|
|
| 674 |
|
\subsection{Effects due to methodology and simulation parameters} |
| 675 |
|
|
| 676 |
< |
START HERE |
| 676 |
> |
We have varied the parameters of the simulations in order to |
| 677 |
> |
investigate how these factors would affect the computation of $G$. Of |
| 678 |
> |
particular interest are: 1) the length scale for the applied thermal |
| 679 |
> |
gradient (modified by increasing the amount of solvent in the system), |
| 680 |
> |
2) the sign and magnitude of the applied thermal flux, 3) the average |
| 681 |
> |
temperature of the simulation (which alters the solvent density during |
| 682 |
> |
equilibration), and 4) the definition of the interfacial conductance |
| 683 |
> |
(Eqs. (\ref{discreteG}) or (\ref{derivativeG})) used in the |
| 684 |
> |
calculation. |
| 685 |
|
|
| 686 |
< |
We have varied our protocol or other parameters of the simulations in order to investigate how these factors would affect the computation of $G$. |
| 686 |
> |
Systems of different lengths were prepared by altering the number of |
| 687 |
> |
solvent molecules and extending the length of the box along the $z$ |
| 688 |
> |
axis to accomodate the extra solvent. Equilibration at the same |
| 689 |
> |
temperature and pressure conditions led to nearly identical surface |
| 690 |
> |
areas ($L_x$ and $L_y$) available to the metal and capping agent, |
| 691 |
> |
while the extra solvent served mainly to lengthen the axis that was |
| 692 |
> |
used to apply the thermal flux. For a given value of the applied |
| 693 |
> |
flux, the different $z$ length scale has only a weak effect on the |
| 694 |
> |
computed conductivities (Table \ref{AuThiolHexaneUA}). |
| 695 |
|
|
| 597 |
– |
We allowed $L_x$ and $L_y$ to change during equilibrating the liquid phase. Due to the stiffness of the crystalline Au structure, $L_x$ and $L_y$ would not change noticeably after equilibration. Although $L_z$ could fluctuate $\sim$1\% after a system is fully equilibrated in the NPT ensemble, this fluctuation, as well as those of $L_x$ and $L_y$ (which is significantly smaller), would not be magnified on the calculated $G$'s, as shown in Table \ref{AuThiolHexaneUA}. This insensivity to $L_i$ fluctuations allows reliable measurement of $G$'s without the necessity of extremely cautious equilibration process. |
| 598 |
– |
|
| 599 |
– |
As stated in our computational details, the spacing filled with |
| 600 |
– |
solvent molecules can be chosen within a range. This allows some |
| 601 |
– |
change of solvent molecule numbers for the same Au-butanethiol |
| 602 |
– |
surfaces. We did this study on our Au-butanethiol/hexane |
| 603 |
– |
simulations. Nevertheless, the results obtained from systems of |
| 604 |
– |
different $N_{hexane}$ did not indicate that the measurement of $G$ is |
| 605 |
– |
susceptible to this parameter. For computational efficiency concern, |
| 606 |
– |
smaller system size would be preferable, given that the liquid phase |
| 607 |
– |
structure is not affected. |
| 608 |
– |
|
| 696 |
|
\subsubsection{Effects of applied flux} |
| 697 |
< |
Our NIVS algorithm allows change of unphysical thermal flux both in |
| 698 |
< |
direction and in quantity. This feature extends our investigation of |
| 699 |
< |
interfacial thermal conductance. However, the magnitude of this |
| 700 |
< |
thermal flux is not arbitary if one aims to obtain a stable and |
| 701 |
< |
reliable thermal gradient. A temperature profile would be |
| 702 |
< |
substantially affected by noise when $|J_z|$ has a much too low |
| 703 |
< |
magnitude; while an excessively large $|J_z|$ that overwhelms the |
| 704 |
< |
conductance capacity of the interface would prevent a thermal gradient |
| 705 |
< |
to reach a stablized steady state. NIVS has the advantage of allowing |
| 706 |
< |
$J$ to vary in a wide range such that the optimal flux range for $G$ |
| 707 |
< |
measurement can generally be simulated by the algorithm. Within the |
| 708 |
< |
optimal range, we were able to study how $G$ would change according to |
| 622 |
< |
the thermal flux across the interface. For our simulations, we denote |
| 623 |
< |
$J_z$ to be positive when the physical thermal flux is from the liquid |
| 624 |
< |
to metal, and negative vice versa. The $G$'s measured under different |
| 625 |
< |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and |
| 626 |
< |
\ref{AuThiolToluene}. These results do not suggest that $G$ is |
| 627 |
< |
dependent on $J_z$ within this flux range. The linear response of flux |
| 628 |
< |
to thermal gradient simplifies our investigations in that we can rely |
| 629 |
< |
on $G$ measurement with only a couple $J_z$'s and do not need to test |
| 630 |
< |
a large series of fluxes. |
| 697 |
> |
The NIVS algorithm allows changes in both the sign and magnitude of |
| 698 |
> |
the applied flux. It is possible to reverse the direction of heat |
| 699 |
> |
flow simply by changing the sign of the flux, and thermal gradients |
| 700 |
> |
which would be difficult to obtain experimentally ($5$ K/\AA) can be |
| 701 |
> |
easily simulated. However, the magnitude of the applied flux is not |
| 702 |
> |
arbitrary if one aims to obtain a stable and reliable thermal gradient. |
| 703 |
> |
A temperature gradient can be lost in the noise if $|J_z|$ is too |
| 704 |
> |
small, and excessive $|J_z|$ values can cause phase transitions if the |
| 705 |
> |
extremes of the simulation cell become widely separated in |
| 706 |
> |
temperature. Also, if $|J_z|$ is too large for the bulk conductivity |
| 707 |
> |
of the materials, the thermal gradient will never reach a stable |
| 708 |
> |
state. |
| 709 |
|
|
| 710 |
+ |
Within a reasonable range of $J_z$ values, we were able to study how |
| 711 |
+ |
$G$ changes as a function of this flux. In what follows, we use |
| 712 |
+ |
positive $J_z$ values to denote the case where energy is being |
| 713 |
+ |
transferred by the method from the metal phase and into the liquid. |
| 714 |
+ |
The resulting gradient therefore has a higher temperature in the |
| 715 |
+ |
liquid phase. Negative flux values reverse this transfer, and result |
| 716 |
+ |
in higher temperature metal phases. The conductance measured under |
| 717 |
+ |
different applied $J_z$ values is listed in Tables |
| 718 |
+ |
\ref{AuThiolHexaneUA} and \ref{AuThiolToluene}. These results do not |
| 719 |
+ |
indicate that $G$ depends strongly on $J_z$ within this flux |
| 720 |
+ |
range. The linear response of flux to thermal gradient simplifies our |
| 721 |
+ |
investigations in that we can rely on $G$ measurement with only a |
| 722 |
+ |
small number $J_z$ values. |
| 723 |
+ |
|
| 724 |
|
\begin{table*} |
| 725 |
|
\begin{minipage}{\linewidth} |
| 726 |
|
\begin{center} |
| 727 |
< |
\caption{Computed interfacial thermal conductivity ($G$ and |
| 728 |
< |
$G^\prime$) values for the 100\% covered Au-butanethiol/hexane |
| 729 |
< |
interfaces with UA model and different hexane molecule numbers |
| 730 |
< |
at different temperatures using a range of energy |
| 731 |
< |
fluxes. Error estimates indicated in parenthesis.} |
| 727 |
> |
\caption{In the hexane-solvated interfaces, the system size has |
| 728 |
> |
little effect on the calculated values for interfacial |
| 729 |
> |
conductance ($G$ and $G^\prime$), but the direction of heat |
| 730 |
> |
flow (i.e. the sign of $J_z$) can alter the average |
| 731 |
> |
temperature of the liquid phase and this can alter the |
| 732 |
> |
computed conductivity.} |
| 733 |
|
|
| 734 |
|
\begin{tabular}{ccccccc} |
| 735 |
|
\hline\hline |
| 736 |
< |
$\langle T\rangle$ & $N_{hexane}$ & Fixed & $\rho_{hexane}$ & |
| 736 |
> |
$\langle T\rangle$ & $N_{hexane}$ & $\rho_{hexane}$ & |
| 737 |
|
$J_z$ & $G$ & $G^\prime$ \\ |
| 738 |
< |
(K) & & $L_x$ \& $L_y$? & (g/cm$^3$) & (GW/m$^2$) & |
| 738 |
> |
(K) & & (g/cm$^3$) & (GW/m$^2$) & |
| 739 |
|
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 740 |
|
\hline |
| 741 |
< |
200 & 266 & No & 0.672 & -0.96 & 102(3) & 80.0(0.8) \\ |
| 742 |
< |
& 200 & Yes & 0.694 & 1.92 & 129(11) & 87.3(0.3) \\ |
| 743 |
< |
& & Yes & 0.672 & 1.93 & 131(16) & 78(13) \\ |
| 744 |
< |
& & No & 0.688 & 0.96 & 125(16) & 90.2(15) \\ |
| 745 |
< |
& & & & 1.91 & 139(10) & 101(10) \\ |
| 746 |
< |
& & & & 2.83 & 141(6) & 89.9(9.8) \\ |
| 654 |
< |
& 166 & Yes & 0.679 & 0.97 & 115(19) & 69(18) \\ |
| 655 |
< |
& & & & 1.94 & 125(9) & 87.1(0.2) \\ |
| 656 |
< |
& & No & 0.681 & 0.97 & 141(30) & 78(22) \\ |
| 657 |
< |
& & & & 1.92 & 138(4) & 98.9(9.5) \\ |
| 741 |
> |
200 & 266 & 0.672 & -0.96 & 102(3) & 80.0(0.8) \\ |
| 742 |
> |
& 200 & 0.688 & 0.96 & 125(16) & 90.2(15) \\ |
| 743 |
> |
& & & 1.91 & 139(10) & 101(10) \\ |
| 744 |
> |
& & & 2.83 & 141(6) & 89.9(9.8) \\ |
| 745 |
> |
& 166 & 0.681 & 0.97 & 141(30) & 78(22) \\ |
| 746 |
> |
& & & 1.92 & 138(4) & 98.9(9.5) \\ |
| 747 |
|
\hline |
| 748 |
< |
250 & 200 & No & 0.560 & 0.96 & 75(10) & 61.8(7.3) \\ |
| 749 |
< |
& & & & -0.95 & 49.4(0.3) & 45.7(2.1) \\ |
| 750 |
< |
& 166 & Yes & 0.570 & 0.98 & 79.0(3.5) & 62.9(3.0) \\ |
| 751 |
< |
& & No & 0.569 & 0.97 & 80.3(0.6) & 67(11) \\ |
| 752 |
< |
& & & & 1.44 & 76.2(5.0) & 64.8(3.8) \\ |
| 753 |
< |
& & & & -0.95 & 56.4(2.5) & 54.4(1.1) \\ |
| 665 |
< |
& & & & -1.85 & 47.8(1.1) & 53.5(1.5) \\ |
| 748 |
> |
250 & 200 & 0.560 & 0.96 & 75(10) & 61.8(7.3) \\ |
| 749 |
> |
& & & -0.95 & 49.4(0.3) & 45.7(2.1) \\ |
| 750 |
> |
& 166 & 0.569 & 0.97 & 80.3(0.6) & 67(11) \\ |
| 751 |
> |
& & & 1.44 & 76.2(5.0) & 64.8(3.8) \\ |
| 752 |
> |
& & & -0.95 & 56.4(2.5) & 54.4(1.1) \\ |
| 753 |
> |
& & & -1.85 & 47.8(1.1) & 53.5(1.5) \\ |
| 754 |
|
\hline\hline |
| 755 |
|
\end{tabular} |
| 756 |
|
\label{AuThiolHexaneUA} |
| 758 |
|
\end{minipage} |
| 759 |
|
\end{table*} |
| 760 |
|
|
| 761 |
+ |
The sign of $J_z$ is a different matter, however, as this can alter |
| 762 |
+ |
the temperature on the two sides of the interface. The average |
| 763 |
+ |
temperature values reported are for the entire system, and not for the |
| 764 |
+ |
liquid phase, so at a given $\langle T \rangle$, the system with |
| 765 |
+ |
positive $J_z$ has a warmer liquid phase. This means that if the |
| 766 |
+ |
liquid carries thermal energy via convective transport, {\it positive} |
| 767 |
+ |
$J_z$ values will result in increased molecular motion on the liquid |
| 768 |
+ |
side of the interface, and this will increase the measured |
| 769 |
+ |
conductivity. |
| 770 |
+ |
|
| 771 |
|
\subsubsection{Effects due to average temperature} |
| 772 |
|
|
| 773 |
< |
Furthermore, we also attempted to increase system average temperatures |
| 774 |
< |
to above 200K. These simulations are first equilibrated in the NPT |
| 775 |
< |
ensemble under normal pressure. As stated above, the TraPPE-UA model |
| 776 |
< |
for hexane tends to predict a lower boiling point. In our simulations, |
| 777 |
< |
hexane had diffculty to remain in liquid phase when NPT equilibration |
| 778 |
< |
temperature is higher than 250K. Additionally, the equilibrated liquid |
| 779 |
< |
hexane density under 250K becomes lower than experimental value. This |
| 780 |
< |
expanded liquid phase leads to lower contact between hexane and |
| 781 |
< |
butanethiol as well.[MAY NEED SLAB DENSITY FIGURE] |
| 782 |
< |
And this reduced contact would |
| 685 |
< |
probably be accountable for a lower interfacial thermal conductance, |
| 686 |
< |
as shown in Table \ref{AuThiolHexaneUA}. |
| 773 |
> |
We also studied the effect of average system temperature on the |
| 774 |
> |
interfacial conductance. The simulations are first equilibrated in |
| 775 |
> |
the NPT ensemble at 1 atm. The TraPPE-UA model for hexane tends to |
| 776 |
> |
predict a lower boiling point (and liquid state density) than |
| 777 |
> |
experiments. This lower-density liquid phase leads to reduced contact |
| 778 |
> |
between the hexane and butanethiol, and this accounts for our |
| 779 |
> |
observation of lower conductance at higher temperatures as shown in |
| 780 |
> |
Table \ref{AuThiolHexaneUA}. In raising the average temperature from |
| 781 |
> |
200K to 250K, the density drop of $\sim$20\% in the solvent phase |
| 782 |
> |
leads to a $\sim$40\% drop in the conductance. |
| 783 |
|
|
| 784 |
< |
A similar study for TraPPE-UA toluene agrees with the above result as |
| 785 |
< |
well. Having a higher boiling point, toluene tends to remain liquid in |
| 786 |
< |
our simulations even equilibrated under 300K in NPT |
| 787 |
< |
ensembles. Furthermore, the expansion of the toluene liquid phase is |
| 788 |
< |
not as significant as that of the hexane. This prevents severe |
| 789 |
< |
decrease of liquid-capping agent contact and the results (Table |
| 694 |
< |
\ref{AuThiolToluene}) show only a slightly decreased interface |
| 695 |
< |
conductance. Therefore, solvent-capping agent contact should play an |
| 696 |
< |
important role in the thermal transport process across the interface |
| 697 |
< |
in that higher degree of contact could yield increased conductance. |
| 784 |
> |
Similar behavior is observed in the TraPPE-UA model for toluene, |
| 785 |
> |
although this model has better agreement with the experimental |
| 786 |
> |
densities of toluene. The expansion of the toluene liquid phase is |
| 787 |
> |
not as significant as that of the hexane (8.3\% over 100K), and this |
| 788 |
> |
limits the effect to $\sim$20\% drop in thermal conductivity (Table |
| 789 |
> |
\ref{AuThiolToluene}). |
| 790 |
|
|
| 791 |
+ |
Although we have not mapped out the behavior at a large number of |
| 792 |
+ |
temperatures, is clear that there will be a strong temperature |
| 793 |
+ |
dependence in the interfacial conductance when the physical properties |
| 794 |
+ |
of one side of the interface (notably the density) change rapidly as a |
| 795 |
+ |
function of temperature. |
| 796 |
+ |
|
| 797 |
|
\begin{table*} |
| 798 |
|
\begin{minipage}{\linewidth} |
| 799 |
|
\begin{center} |
| 800 |
< |
\caption{Computed interfacial thermal conductivity ($G$ and |
| 801 |
< |
$G^\prime$) values for a 90\% coverage Au-butanethiol/toluene |
| 802 |
< |
interface at different temperatures using a range of energy |
| 803 |
< |
fluxes. Error estimates indicated in parenthesis.} |
| 800 |
> |
\caption{When toluene is the solvent, the interfacial thermal |
| 801 |
> |
conductivity is less sensitive to temperature, but again, the |
| 802 |
> |
direction of the heat flow can alter the solvent temperature |
| 803 |
> |
and can change the computed conductance values.} |
| 804 |
|
|
| 805 |
|
\begin{tabular}{ccccc} |
| 806 |
|
\hline\hline |
| 820 |
|
\end{minipage} |
| 821 |
|
\end{table*} |
| 822 |
|
|
| 823 |
< |
Besides lower interfacial thermal conductance, surfaces in relatively |
| 824 |
< |
high temperatures are susceptible to reconstructions, when |
| 825 |
< |
butanethiols have a full coverage on the Au(111) surface. These |
| 826 |
< |
reconstructions include surface Au atoms migrated outward to the S |
| 827 |
< |
atom layer, and butanethiol molecules embedded into the original |
| 828 |
< |
surface Au layer. The driving force for this behavior is the strong |
| 829 |
< |
Au-S interactions in our simulations. And these reconstructions lead |
| 830 |
< |
to higher ratio of Au-S attraction and thus is energetically |
| 831 |
< |
favorable. Furthermore, this phenomenon agrees with experimental |
| 832 |
< |
results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt |
| 833 |
< |
{\it et al.} had kept their Au(111) slab rigid so that their |
| 834 |
< |
simulations can reach 300K without surface reconstructions. Without |
| 835 |
< |
this practice, simulating 100\% thiol covered interfaces under higher |
| 836 |
< |
temperatures could hardly avoid surface reconstructions. However, our |
| 837 |
< |
measurement is based on assuming homogeneity on $x$ and $y$ dimensions |
| 838 |
< |
so that measurement of $T$ at particular $z$ would be an effective |
| 839 |
< |
average of the particles of the same type. Since surface |
| 742 |
< |
reconstructions could eliminate the original $x$ and $y$ dimensional |
| 743 |
< |
homogeneity, measurement of $G$ is more difficult to conduct under |
| 744 |
< |
higher temperatures. Therefore, most of our measurements are |
| 745 |
< |
undertaken at $\langle T\rangle\sim$200K. |
| 823 |
> |
Besides the lower interfacial thermal conductance, surfaces at |
| 824 |
> |
relatively high temperatures are susceptible to reconstructions, |
| 825 |
> |
particularly when butanethiols fully cover the Au(111) surface. These |
| 826 |
> |
reconstructions include surface Au atoms which migrate outward to the |
| 827 |
> |
S atom layer, and butanethiol molecules which embed into the surface |
| 828 |
> |
Au layer. The driving force for this behavior is the strong Au-S |
| 829 |
> |
interactions which are modeled here with a deep Lennard-Jones |
| 830 |
> |
potential. This phenomenon agrees with reconstructions that have been |
| 831 |
> |
experimentally |
| 832 |
> |
observed.\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt |
| 833 |
> |
{\it et al.} kept their Au(111) slab rigid so that their simulations |
| 834 |
> |
could reach 300K without surface |
| 835 |
> |
reconstructions.\cite{vlugt:cpc2007154} Since surface reconstructions |
| 836 |
> |
blur the interface, the measurement of $G$ becomes more difficult to |
| 837 |
> |
conduct at higher temperatures. For this reason, most of our |
| 838 |
> |
measurements are undertaken at $\langle T\rangle\sim$200K where |
| 839 |
> |
reconstruction is minimized. |
| 840 |
|
|
| 841 |
|
However, when the surface is not completely covered by butanethiols, |
| 842 |
< |
the simulated system is more resistent to the reconstruction |
| 843 |
< |
above. Our Au-butanethiol/toluene system had the Au(111) surfaces 90\% |
| 844 |
< |
covered by butanethiols, but did not see this above phenomena even at |
| 845 |
< |
$\langle T\rangle\sim$300K. The empty three-fold sites not occupied by |
| 846 |
< |
capping agents could help prevent surface reconstruction in that they |
| 847 |
< |
provide other means of capping agent relaxation. It is observed that |
| 848 |
< |
butanethiols can migrate to their neighbor empty sites during a |
| 849 |
< |
simulation. Therefore, we were able to obtain $G$'s for these |
| 756 |
< |
interfaces even at a relatively high temperature without being |
| 757 |
< |
affected by surface reconstructions. |
| 842 |
> |
the simulated system appears to be more resistent to the |
| 843 |
> |
reconstruction. Our Au / butanethiol / toluene system had the Au(111) |
| 844 |
> |
surfaces 90\% covered by butanethiols, but did not see this above |
| 845 |
> |
phenomena even at $\langle T\rangle\sim$300K. That said, we did |
| 846 |
> |
observe butanethiols migrating to neighboring three-fold sites during |
| 847 |
> |
a simulation. Since the interface persisted in these simulations, we |
| 848 |
> |
were able to obtain $G$'s for these interfaces even at a relatively |
| 849 |
> |
high temperature without being affected by surface reconstructions. |
| 850 |
|
|
| 759 |
– |
|
| 851 |
|
\section{Discussion} |
| 852 |
|
|
| 853 |
< |
\subsection{Capping agent acts as a vibrational coupler between solid |
| 854 |
< |
and solvent phases} |
| 853 |
> |
The primary result of this work is that the capping agent acts as an |
| 854 |
> |
efficient thermal coupler between solid and solvent phases. One of |
| 855 |
> |
the ways the capping agent can carry out this role is to down-shift |
| 856 |
> |
between the phonon vibrations in the solid (which carry the heat from |
| 857 |
> |
the gold) and the molecular vibrations in the liquid (which carry some |
| 858 |
> |
of the heat in the solvent). |
| 859 |
> |
|
| 860 |
|
To investigate the mechanism of interfacial thermal conductance, the |
| 861 |
|
vibrational power spectrum was computed. Power spectra were taken for |
| 862 |
|
individual components in different simulations. To obtain these |
| 863 |
< |
spectra, simulations were run after equilibration, in the NVE |
| 864 |
< |
ensemble, and without a thermal gradient. Snapshots of configurations |
| 865 |
< |
were collected at a frequency that is higher than that of the fastest |
| 866 |
< |
vibrations occuring in the simulations. With these configurations, the |
| 867 |
< |
velocity auto-correlation functions can be computed: |
| 863 |
> |
spectra, simulations were run after equilibration in the |
| 864 |
> |
microcanonical (NVE) ensemble and without a thermal |
| 865 |
> |
gradient. Snapshots of configurations were collected at a frequency |
| 866 |
> |
that is higher than that of the fastest vibrations occurring in the |
| 867 |
> |
simulations. With these configurations, the velocity auto-correlation |
| 868 |
> |
functions can be computed: |
| 869 |
|
\begin{equation} |
| 870 |
|
C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle |
| 871 |
|
\label{vCorr} |
| 878 |
|
\label{fourier} |
| 879 |
|
\end{equation} |
| 880 |
|
|
| 881 |
< |
|
| 785 |
< |
\subsubsection{The role of specific vibrations} |
| 881 |
> |
\subsection{The role of specific vibrations} |
| 882 |
|
The vibrational spectra for gold slabs in different environments are |
| 883 |
|
shown as in Figure \ref{specAu}. Regardless of the presence of |
| 884 |
< |
solvent, the gold surfaces covered by butanethiol molecules, compared |
| 885 |
< |
to bare gold surfaces, exhibit an additional peak observed at the |
| 886 |
< |
frequency of $\sim$170cm$^{-1}$, which is attributed to the S-Au |
| 887 |
< |
bonding vibration. This vibration enables efficient thermal transport |
| 888 |
< |
from surface Au layer to the capping agents. Therefore, in our |
| 889 |
< |
simulations, the Au/S interfaces do not appear major heat barriers |
| 890 |
< |
compared to the butanethiol / solvent interfaces. |
| 891 |
< |
|
| 892 |
< |
\subsubsection{Overlap of power spectrum} |
| 893 |
< |
Simultaneously, the vibrational overlap between butanethiol and |
| 894 |
< |
organic solvents suggests higher thermal exchange efficiency between |
| 799 |
< |
these two components. Even exessively high heat transport was observed |
| 800 |
< |
when All-Atom models were used and C-H vibrations were treated |
| 801 |
< |
classically. Compared to metal and organic liquid phase, the heat |
| 802 |
< |
transfer efficiency between butanethiol and organic solvents is closer |
| 803 |
< |
to that within bulk liquid phase. |
| 804 |
< |
|
| 805 |
< |
Furthermore, our observation validated previous |
| 806 |
< |
results\cite{hase:2010} that the intramolecular heat transport of |
| 807 |
< |
alkylthiols is highly effecient. As a combinational effects of these |
| 808 |
< |
phenomena, butanethiol acts as a channel to expedite thermal transport |
| 809 |
< |
process. The acoustic impedance mismatch between the metal and the |
| 810 |
< |
liquid phase can be effectively reduced with the presence of suitable |
| 811 |
< |
capping agents. |
| 884 |
> |
solvent, the gold surfaces which are covered by butanethiol molecules |
| 885 |
> |
exhibit an additional peak observed at a frequency of |
| 886 |
> |
$\sim$165cm$^{-1}$. We attribute this peak to the S-Au bonding |
| 887 |
> |
vibration. This vibration enables efficient thermal coupling of the |
| 888 |
> |
surface Au layer to the capping agents. Therefore, in our simulations, |
| 889 |
> |
the Au / S interfaces do not appear to be the primary barrier to |
| 890 |
> |
thermal transport when compared with the butanethiol / solvent |
| 891 |
> |
interfaces. {\bf This confirms the results from Luo {\it et |
| 892 |
> |
al.}\cite{Luo20101}, which reported $G$ for Au-SAM junctions |
| 893 |
> |
generally twice larger than what we have computed for the |
| 894 |
> |
thiol-liquid interfaces.} |
| 895 |
|
|
| 896 |
|
\begin{figure} |
| 897 |
|
\includegraphics[width=\linewidth]{vibration} |
| 898 |
< |
\caption{Vibrational spectra obtained for gold in different |
| 899 |
< |
environments.} |
| 898 |
> |
\caption{The vibrational power spectrum for thiol-capped gold has an |
| 899 |
> |
additional vibrational peak at $\sim $165cm$^{-1}$. Bare gold |
| 900 |
> |
surfaces (both with and without a solvent over-layer) are missing |
| 901 |
> |
this peak. A similar peak at $\sim $165cm$^{-1}$ also appears in |
| 902 |
> |
the vibrational power spectrum for the butanethiol capping agents.} |
| 903 |
|
\label{specAu} |
| 904 |
|
\end{figure} |
| 905 |
|
|
| 906 |
< |
\subsubsection{Isotopic substitution and vibrational overlap} |
| 906 |
> |
Also in this figure, we show the vibrational power spectrum for the |
| 907 |
> |
bound butanethiol molecules, which also exhibits the same |
| 908 |
> |
$\sim$165cm$^{-1}$ peak. |
| 909 |
> |
|
| 910 |
> |
\subsection{Overlap of power spectra} |
| 911 |
|
A comparison of the results obtained from the two different organic |
| 912 |
|
solvents can also provide useful information of the interfacial |
| 913 |
< |
thermal transport process. The deuterated hexane (UA) results do not |
| 914 |
< |
appear to be substantially different from those of normal hexane (UA), |
| 915 |
< |
given that butanethiol (UA) is non-deuterated for both solvents. The |
| 916 |
< |
UA models, even though they have eliminated C-H vibrational overlap, |
| 917 |
< |
still have significant overlap in the infrared spectra. Because |
| 918 |
< |
differences in the infrared range do not seem to produce an observable |
| 919 |
< |
difference for the results of $G$ (Figure \ref{uahxnua}). |
| 913 |
> |
thermal transport process. In particular, the vibrational overlap |
| 914 |
> |
between the butanethiol and the organic solvents suggests a highly |
| 915 |
> |
efficient thermal exchange between these components. Very high |
| 916 |
> |
thermal conductivity was observed when AA models were used and C-H |
| 917 |
> |
vibrations were treated classically. The presence of extra degrees of |
| 918 |
> |
freedom in the AA force field yields higher heat exchange rates |
| 919 |
> |
between the two phases and results in a much higher conductivity than |
| 920 |
> |
in the UA force field. {\bf Due to the classical models used, this |
| 921 |
> |
even includes those high frequency modes which should be unpopulated |
| 922 |
> |
at our relatively low temperatures. This artifact causes high |
| 923 |
> |
frequency vibrations accountable for thermal transport in classical |
| 924 |
> |
MD simulations.} |
| 925 |
|
|
| 926 |
+ |
The similarity in the vibrational modes available to solvent and |
| 927 |
+ |
capping agent can be reduced by deuterating one of the two components |
| 928 |
+ |
(Fig. \ref{aahxntln}). Once either the hexanes or the butanethiols |
| 929 |
+ |
are deuterated, one can observe a significantly lower $G$ and |
| 930 |
+ |
$G^\prime$ values (Table \ref{modelTest}). |
| 931 |
+ |
|
| 932 |
|
\begin{figure} |
| 933 |
+ |
\includegraphics[width=\linewidth]{aahxntln} |
| 934 |
+ |
\caption{Spectra obtained for all-atom (AA) Au / butanethiol / solvent |
| 935 |
+ |
systems. When butanethiol is deuterated (lower left), its |
| 936 |
+ |
vibrational overlap with hexane decreases significantly. Since |
| 937 |
+ |
aromatic molecules and the butanethiol are vibrationally dissimilar, |
| 938 |
+ |
the change is not as dramatic when toluene is the solvent (right).} |
| 939 |
+ |
\label{aahxntln} |
| 940 |
+ |
\end{figure} |
| 941 |
+ |
|
| 942 |
+ |
For the Au / butanethiol / toluene interfaces, having the AA |
| 943 |
+ |
butanethiol deuterated did not yield a significant change in the |
| 944 |
+ |
measured conductance. Compared to the C-H vibrational overlap between |
| 945 |
+ |
hexane and butanethiol, both of which have alkyl chains, the overlap |
| 946 |
+ |
between toluene and butanethiol is not as significant and thus does |
| 947 |
+ |
not contribute as much to the heat exchange process. |
| 948 |
+ |
|
| 949 |
+ |
Previous observations of Zhang {\it et al.}\cite{hase:2010} indicate |
| 950 |
+ |
that the {\it intra}molecular heat transport due to alkylthiols is |
| 951 |
+ |
highly efficient. Combining our observations with those of Zhang {\it |
| 952 |
+ |
et al.}, it appears that butanethiol acts as a channel to expedite |
| 953 |
+ |
heat flow from the gold surface and into the alkyl chain. The |
| 954 |
+ |
acoustic impedance mismatch between the metal and the liquid phase can |
| 955 |
+ |
therefore be effectively reduced with the presence of suitable capping |
| 956 |
+ |
agents. |
| 957 |
+ |
|
| 958 |
+ |
Deuterated models in the UA force field did not decouple the thermal |
| 959 |
+ |
transport as well as in the AA force field. The UA models, even |
| 960 |
+ |
though they have eliminated the high frequency C-H vibrational |
| 961 |
+ |
overlap, still have significant overlap in the lower-frequency |
| 962 |
+ |
portions of the infrared spectra (Figure \ref{uahxnua}). Deuterating |
| 963 |
+ |
the UA models did not decouple the low frequency region enough to |
| 964 |
+ |
produce an observable difference for the results of $G$ (Table |
| 965 |
+ |
\ref{modelTest}). |
| 966 |
+ |
|
| 967 |
+ |
\begin{figure} |
| 968 |
|
\includegraphics[width=\linewidth]{uahxnua} |
| 969 |
< |
\caption{Vibrational spectra obtained for normal (upper) and |
| 970 |
< |
deuterated (lower) hexane in Au-butanethiol/hexane |
| 971 |
< |
systems. Butanethiol spectra are shown as reference. Both hexane and |
| 972 |
< |
butanethiol were using United-Atom models.} |
| 969 |
> |
\caption{Vibrational power spectra for UA models for the butanethiol |
| 970 |
> |
and hexane solvent (upper panel) show the high degree of overlap |
| 971 |
> |
between these two molecules, particularly at lower frequencies. |
| 972 |
> |
Deuterating a UA model for the solvent (lower panel) does not |
| 973 |
> |
decouple the two spectra to the same degree as in the AA force |
| 974 |
> |
field (see Fig \ref{aahxntln}).} |
| 975 |
|
\label{uahxnua} |
| 976 |
|
\end{figure} |
| 977 |
|
|
| 978 |
|
\section{Conclusions} |
| 979 |
< |
The NIVS algorithm we developed has been applied to simulations of |
| 980 |
< |
Au-butanethiol surfaces with organic solvents. This algorithm allows |
| 981 |
< |
effective unphysical thermal flux transferred between the metal and |
| 982 |
< |
the liquid phase. With the flux applied, we were able to measure the |
| 983 |
< |
corresponding thermal gradient and to obtain interfacial thermal |
| 984 |
< |
conductivities. Under steady states, single trajectory simulation |
| 985 |
< |
would be enough for accurate measurement. This would be advantageous |
| 986 |
< |
compared to transient state simulations, which need multiple |
| 849 |
< |
trajectories to produce reliable average results. |
| 979 |
> |
The NIVS algorithm has been applied to simulations of |
| 980 |
> |
butanethiol-capped Au(111) surfaces in the presence of organic |
| 981 |
> |
solvents. This algorithm allows the application of unphysical thermal |
| 982 |
> |
flux to transfer heat between the metal and the liquid phase. With the |
| 983 |
> |
flux applied, we were able to measure the corresponding thermal |
| 984 |
> |
gradients and to obtain interfacial thermal conductivities. Under |
| 985 |
> |
steady states, 2-3 ns trajectory simulations are sufficient for |
| 986 |
> |
computation of this quantity. |
| 987 |
|
|
| 988 |
< |
Our simulations have seen significant conductance enhancement with the |
| 989 |
< |
presence of capping agent, compared to the bare gold / liquid |
| 988 |
> |
Our simulations have seen significant conductance enhancement in the |
| 989 |
> |
presence of capping agent, compared with the bare gold / liquid |
| 990 |
|
interfaces. The acoustic impedance mismatch between the metal and the |
| 991 |
< |
liquid phase is effectively eliminated by proper capping |
| 992 |
< |
agent. Furthermore, the coverage precentage of the capping agent plays |
| 991 |
> |
liquid phase is effectively eliminated by a chemically-bonded capping |
| 992 |
> |
agent. Furthermore, the coverage percentage of the capping agent plays |
| 993 |
|
an important role in the interfacial thermal transport |
| 994 |
< |
process. Moderately lower coverages allow higher contact between |
| 995 |
< |
capping agent and solvent, and thus could further enhance the heat |
| 996 |
< |
transfer process. |
| 994 |
> |
process. Moderately low coverages allow higher contact between capping |
| 995 |
> |
agent and solvent, and thus could further enhance the heat transfer |
| 996 |
> |
process, giving a non-monotonic behavior of conductance with |
| 997 |
> |
increasing coverage. |
| 998 |
|
|
| 999 |
< |
Our measurement results, particularly of the UA models, agree with |
| 1000 |
< |
available experimental data. This indicates that our force field |
| 863 |
< |
parameters have a nice description of the interactions between the |
| 864 |
< |
particles at the interfaces. AA models tend to overestimate the |
| 999 |
> |
Our results, particularly using the UA models, agree well with |
| 1000 |
> |
available experimental data. The AA models tend to overestimate the |
| 1001 |
|
interfacial thermal conductance in that the classically treated C-H |
| 1002 |
< |
vibration would be overly sampled. Compared to the AA models, the UA |
| 1003 |
< |
models have higher computational efficiency with satisfactory |
| 1004 |
< |
accuracy, and thus are preferable in interfacial thermal transport |
| 1005 |
< |
modelings. Of the two definitions for $G$, the discrete form |
| 1002 |
> |
vibrations become too easily populated. Compared to the AA models, the |
| 1003 |
> |
UA models have higher computational efficiency with satisfactory |
| 1004 |
> |
accuracy, and thus are preferable in modeling interfacial thermal |
| 1005 |
> |
transport. |
| 1006 |
> |
|
| 1007 |
> |
Of the two definitions for $G$, the discrete form |
| 1008 |
|
(Eq. \ref{discreteG}) was easier to use and gives out relatively |
| 1009 |
|
consistent results, while the derivative form (Eq. \ref{derivativeG}) |
| 1010 |
|
is not as versatile. Although $G^\prime$ gives out comparable results |
| 1011 |
|
and follows similar trend with $G$ when measuring close to fully |
| 1012 |
< |
covered or bare surfaces, the spatial resolution of $T$ profile is |
| 1013 |
< |
limited for accurate computation of derivatives data. |
| 1012 |
> |
covered or bare surfaces, the spatial resolution of $T$ profile |
| 1013 |
> |
required for the use of a derivative form is limited by the number of |
| 1014 |
> |
bins and the sampling required to obtain thermal gradient information. |
| 1015 |
|
|
| 1016 |
< |
Vlugt {\it et al.} has investigated the surface thiol structures for |
| 1017 |
< |
nanocrystal gold and pointed out that they differs from those of the |
| 1018 |
< |
Au(111) surface\cite{landman:1998,vlugt:cpc2007154}. This difference |
| 1019 |
< |
might lead to change of interfacial thermal transport behavior as |
| 1020 |
< |
well. To investigate this problem, an effective means to introduce |
| 1021 |
< |
thermal flux and measure the corresponding thermal gradient is |
| 1022 |
< |
desirable for simulating structures with spherical symmetry. |
| 1016 |
> |
Vlugt {\it et al.} have investigated the surface thiol structures for |
| 1017 |
> |
nanocrystalline gold and pointed out that they differ from those of |
| 1018 |
> |
the Au(111) surface.\cite{landman:1998,vlugt:cpc2007154} This |
| 1019 |
> |
difference could also cause differences in the interfacial thermal |
| 1020 |
> |
transport behavior. To investigate this problem, one would need an |
| 1021 |
> |
effective method for applying thermal gradients in non-planar |
| 1022 |
> |
(i.e. spherical) geometries. |
| 1023 |
|
|
| 1024 |
|
\section{Acknowledgments} |
| 1025 |
|
Support for this project was provided by the National Science |
| 1026 |
|
Foundation under grant CHE-0848243. Computational time was provided by |
| 1027 |
|
the Center for Research Computing (CRC) at the University of Notre |
| 1028 |
|
Dame. |
| 1029 |
+ |
|
| 1030 |
+ |
\section{Supporting Information} |
| 1031 |
+ |
This information is available free of charge via the Internet at |
| 1032 |
+ |
http://pubs.acs.org. |
| 1033 |
+ |
|
| 1034 |
|
\newpage |
| 1035 |
|
|
| 1036 |
|
\bibliography{interfacial} |
