group/stokes/stokes.tex

\documentclass[11pt]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{setspace}
\usepackage{endfloat}
\usepackage{caption}
%\usepackage{tabularx}
\usepackage{graphicx}
\usepackage{multirow}
%\usepackage{booktabs}
%\usepackage{bibentry}
%\usepackage{mathrsfs}
%\usepackage[ref]{overcite}
\usepackage[square, comma, sort&compress]{natbib}
\usepackage{url}
\pagestyle{plain} \pagenumbering{arabic} \oddsidemargin 0.0cm
\evensidemargin 0.0cm \topmargin -21pt \headsep 10pt \textheight
9.0in \textwidth 6.5in \brokenpenalty=10000

% double space list of tables and figures
\AtBeginDelayedFloats{\renewcommand{\baselinestretch}{1.66}}
\setlength{\abovecaptionskip}{20 pt}
\setlength{\belowcaptionskip}{30 pt}

%\renewcommand\citemid{\ } % no comma in optional reference note
\bibpunct{[}{]}{,}{n}{}{;}
\bibliographystyle{achemso}

\begin{document}

\title{ENTER TITLE HERE}

\author{Shenyu Kuang and J. Daniel
Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\
Department of Chemistry and Biochemistry,\\
University of Notre Dame\\
Notre Dame, Indiana 46556}

\date{\today}

\maketitle 

\begin{doublespace}

\begin{abstract}
  REPLACE ABSTRACT HERE
  With the Non-Isotropic Velocity Scaling (NIVS) approach to Reverse
  Non-Equilibrium Molecular Dynamics (RNEMD) it is possible to impose
  an unphysical thermal flux between different regions of
  inhomogeneous systems such as solid / liquid interfaces.  We have
  applied NIVS to compute the interfacial thermal conductance at a
  metal / organic solvent interface that has been chemically capped by
  butanethiol molecules.  Our calculations suggest that coupling
  between the metal and liquid phases is enhanced by the capping
  agents, leading to a greatly enhanced conductivity at the interface.
  Specifically, the chemical bond between the metal and the capping
  agent introduces a vibrational overlap that is not present without
  the capping agent, and the overlap between the vibrational spectra
  (metal to cap, cap to solvent) provides a mechanism for rapid
  thermal transport across the interface. Our calculations also
  suggest that this is a non-monotonic function of the fractional
  coverage of the surface, as moderate coverages allow diffusive heat
  transport of solvent molecules that have been in close contact with
  the capping agent.

\end{abstract}

\newpage

%\narrowtext

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                          BODY OF TEXT
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}
[REFINE LATER, ADD MORE REF.S]
Imposed-flux methods in Molecular Dynamics (MD)
simulations\cite{MullerPlathe:1997xw} can establish steady state
systems with a set applied flux vs a corresponding gradient that can
be measured. These methods does not need many trajectories to provide
information of transport properties of a given system. Thus, they are
utilized in computing thermal and mechanical transfer of homogeneous
or bulk systems as well as heterogeneous systems such as liquid-solid
interfaces.\cite{kuang:AuThl}

The Reverse Non-Equilibrium MD (RNEMD) methods adopt constraints that
satisfy linear momentum and total energy conservation of a system when
imposing fluxes in a simulation. Thus they are compatible with various
ensembles, including the micro-canonical (NVE) ensemble, without the
need of an external thermostat. The original approaches by
M\"{u}ller-Plathe {\it et
  al.}\cite{MullerPlathe:1997xw,ISI:000080382700030} utilize simple
momentum swapping for generating energy/momentum fluxes, which is also
compatible with particles of different identities. Although simple to
implement in a simulation, this approach can create nonthermal
velocity distributions, as discovered by Tenney and
Maginn\cite{Maginn:2010}. Furthermore, this approach to kinetic energy
transfer between particles of different identities is less efficient
when the mass difference between the particles becomes significant,
which also limits its application on heterogeneous interfacial
systems.

Recently, we developed a different approach, using Non-Isotropic
Velocity Scaling (NIVS) \cite{kuang:164101} algorithm to impose
fluxes. Compared to the momentum swapping move, it scales the velocity
vectors in two separate regions of a simulated system with respective
diagonal scaling matrices. These matrices are determined by solving a
set of equations including linear momentum and kinetic energy
conservation constraints and target flux satisfaction. This method is
able to effectively impose a wide range of kinetic energy fluxes
without obvious perturbation to the velocity distributions of the
simulated systems, regardless of the presence of heterogeneous
interfaces. We have successfully applied this approach in studying the
interfacial thermal conductance at metal-solvent
interfaces.\cite{kuang:AuThl}

However, the NIVS approach limits its application in imposing momentum
fluxes. Temperature anisotropy can happen under high momentum fluxes,
due to the nature of the algorithm. Thus, combining thermal and
momentum flux is also difficult to implement with this
approach. However, such combination may provide a means to simulate
thermal/momentum gradient coupled processes such as freeze
desalination. Therefore, developing novel approaches to extend the
application of imposed-flux method is desired.

In this paper, we improve the NIVS method and propose a novel approach
to impose fluxes. This approach separate the means of applying
momentum and thermal flux with operations in one time step and thus is
able to simutaneously impose thermal and momentum flux. Furthermore,
the approach retains desirable features of previous RNEMD approaches
and is simpler to implement compared to the NIVS method. In what
follows, we first present the method to implement the method in a
simulation. Then we compare the method on bulk fluids to previous
methods. Also, interfacial frictions are computed for a series of
interfaces.

\section{Methodology}
Similar to the NIVS methodology,\cite{kuang:164101} we consider a
periodic system divided into a series of slabs along a certain axis
(e.g. $z$). The unphysical thermal and/or momentum flux is designated
from the center slab to one of the end slabs, and thus the center slab
would have a lower temperature than the end slab (unless the thermal
flux is negative). Therefore, the center slab is denoted as ``$c$''
while the end slab as ``$h$''.

To impose these fluxes, we periodically apply separate operations to
velocities of particles {$i$} within the center slab and of particles
{$j$} within the end slab:
\begin{eqnarray}
\vec{v}_i & \leftarrow & c\cdot\left(\vec{v}_i - \langle\vec{v}_c
  \rangle\right) + \left(\langle\vec{v}_c\rangle + \vec{a}_c\right) \\
\vec{v}_j & \leftarrow & h\cdot\left(\vec{v}_j - \langle\vec{v}_h
  \rangle\right) + \left(\langle\vec{v}_h\rangle + \vec{a}_h\right)
\end{eqnarray}
where $\langle\vec{v}_c\rangle$ and $\langle\vec{v}_h\rangle$ denotes
the instantaneous bulk velocity of slabs $c$ and $h$ respectively
before an operation occurs. When a momentum flux $\vec{j}_z(\vec{p})$
presents, these bulk velocities would have a corresponding change
($\vec{a}_c$ and $\vec{a}_h$ respectively) according to Newton's
second law:
\begin{eqnarray}
M_c \vec{a}_c & = & -\vec{j}_z(\vec{p}) \Delta t \\
M_h \vec{a}_h & = &  \vec{j}_z(\vec{p}) \Delta t
\end{eqnarray}
where
\begin{eqnarray}
M_c & = & \sum_{i = 1}^{N_c} m_i \\
M_h & = & \sum_{j = 1}^{N_h} m_j
\end{eqnarray}
and $\Delta t$ is the interval between two operations.

The above operations conserve the linear momentum of a periodic
system. To satisfy total energy conservation as well as to impose a
thermal flux $J_z$, one would have
[SUPPORT INFO MIGHT BE NECESSARY TO PUT EXTRA MATH IN]
\begin{eqnarray}
K_c - J_z\Delta t & = & c^2 (K_c - \frac{1}{2}M_c \langle\vec{v}_c
\rangle^2) + \frac{1}{2}M_c (\langle \vec{v}_c \rangle + \vec{a}_c)^2 \\
K_h + J_z\Delta t & = & h^2 (K_h - \frac{1}{2}M_h \langle\vec{v}_h
\rangle^2) + \frac{1}{2}M_h (\langle \vec{v}_h \rangle + \vec{a}_h)^2
\end{eqnarray}
where $K_c$ and $K_h$ denotes translational kinetic energy of slabs
$c$ and $h$ respectively before an operation occurs. These
translational kinetic energy conservation equations are sufficient to
ensure total energy conservation, as the operations applied do not
change the potential energy of a system, given that the potential
energy does not depend on particle velocity.

The above sets of equations are sufficient to determine the velocity
scaling coefficients ($c$ and $h$) as well as $\vec{a}_c$ and
$\vec{a}_h$. Note that two roots of $c$ and $h$ exist
respectively. However, to avoid dramatic perturbations to a system,
the positive roots (which are closer to 1) are chosen. Figure
\ref{method} illustrates the implementation of this algorithm in an
individual step.

\begin{figure}
\includegraphics[width=\linewidth]{method}
\caption{Illustration of the implementation of the algorithm in a
  single step. Starting from an ideal velocity distribution, the
  transformation is used to apply both thermal and momentum flux from
  the ``c'' slab to the ``h'' slab. As the figure shows, the thermal
  distributions preserve after this operation.}
\label{method}
\end{figure}

By implementing these operations at a certain frequency, a steady
thermal and/or momentum flux can be applied and the corresponding
temperature and/or momentum gradients can be established.

This approach is more computationaly efficient compared to the
previous NIVS method, in that only quadratic equations are involved,
while the NIVS method needs to solve a quartic equations. Furthermore,
the method implements isotropic scaling of velocities in respective
slabs, unlike the NIVS, where an extra criteria function is necessary
to choose a set of coefficients that performs the most isotropic
scaling. More importantly, separating the momentum flux imposing from
velocity scaling avoids the underlying cause that NIVS produced
thermal anisotropy when applying a momentum flux.
%NEW METHOD DOESN'T CAUSE UNDESIRED CONCOMITENT MOMENTUM FLUX WHEN
%IMPOSING A THERMAL FLUX

The advantages of the approach over the original momentum swapping
approach lies in its nature to preserve a Gaussian
distribution. Because the momentum swapping tends to render a
nonthermal distribution, when the imposed flux is relatively large,
diffusion of the neighboring slabs could no longer remedy this effect,
and nonthermal distributions would be observed. Results in later
section will illustrate this effect.
%NEW METHOD (AND NIVS) HAVE LESS PERTURBATION THAN MOMENTUM SWAPPING

\section{Computational Details}
The algorithm has been implemented in our MD simulation code,
OpenMD\cite{Meineke:2005gd,openmd}. We compare the method with
previous RNEMD methods or equilibrium MD methods in homogeneous fluids
(Lennard-Jones and SPC/E water). And taking advantage of the method,
we simulate the interfacial friction of different heterogeneous
interfaces (gold-organic solvent and gold-SPC/E water and ice-liquid
water).

\subsection{Simulation Protocols}
The systems to be investigated are set up in a orthorhombic simulation
cell with periodic boundary conditions in all three dimensions. The
$z$ axis of these cells were longer and was set as the gradient axis
of temperature and/or momentum. Thus the cells were divided into $N$
slabs along this axis, with various $N$ depending on individual
system. The $x$ and $y$ axis were usually of the same length in
homogeneous systems or close to each other where interfaces
presents. In all cases, before introducing a nonequilibrium method to
establish steady thermal and/or momentum gradients for further
measurements and calculations, canonical ensemble with a Nos\'e-Hoover
thermostat\cite{hoover85} and microcanonical ensemble equilibrations
were used to prepare systems ready for data
collections. Isobaric-isothermal equilibrations are performed before
this for SPC/E water systems to reach normal pressure (1 bar), while
similar equilibrations are used for interfacial systems to relax the
surface tensions.

While homogeneous fluid systems can be set up with random
configurations, our interfacial systems needs extra steps to ensure
the interfaces be established properly for computations. The
preparation and equilibration of butanethiol covered gold (111)
surface and further solvation and equilibration process is described
as in reference \cite{kuang:AuThl}. 

As for the ice/liquid water interfaces, the basal surface of ice
lattice was first constructed. Hirsch {\it et
  al.}\cite{doi:10.1021/jp048434u} explored the energetics of ice
lattices with different proton orders. We refer to their results and
choose the configuration of the lowest energy after geometry
optimization as the unit cells of our ice lattices. Although
experimental solid/liquid coexistant temperature near normal pressure
is 273K, Bryk and Haymet's simulations of ice/liquid water interfaces
with different models suggest that for SPC/E, the most stable
interface is observed at 225$\pm$5K. Therefore, all our ice/liquid
water simulations were carried out under 225K. To have extra
protection of the ice lattice during initial equilibration (when the
randomly generated liquid phase configuration could release large
amount of energy in relaxation), a constraint method (REF?) was
adopted until the high energy configuration was relaxed.
[MAY ADD A FIGURE HERE FOR BASAL PLANE, MAY INCLUDE PRISM IF POSSIBLE]

\subsection{Force Field Parameters}
For comparison of our new method with previous work, we retain our
force field parameters consistent with the results we will compare
with. The Lennard-Jones fluid used here for argon , and reduced unit
results are reported for direct comparison purpose.

As for our water simulations, SPC/E model is used throughout this work
for consistency. Previous work for transport properties of SPC/E water
model is available\cite{Bedrov:2000,10.1063/1.3330544,Medina2011} so
that unnecessary repetition of previous methods can be avoided.

The Au-Au interaction parameters in all simulations are described by
the quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The
QSC potentials include zero-point quantum corrections and are
reparametrized for accurate surface energies compared to the
Sutton-Chen potentials.\cite{Chen90} For gold/water interfaces, the
Spohr potential was adopted\cite{ISI:000167766600035} to depict
Au-H$_2$O interactions.

The small organic molecules included in our simulations are the Au
surface capping agent butanethiol and liquid hexane and toluene. The
United-Atom
models\cite{TraPPE-UA.thiols,TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes}
for these components were used in this work for better computational
efficiency, while maintaining good accuracy. We refer readers to our
previous work\cite{kuang:AuThl} for further details of these models,
as well as the interactions between Au and the above organic molecule
components.

\subsection{Thermal conductivities}
When $\vec{j}_z(\vec{p})$ is set to zero and a target $J_z$ is set to
impose kinetic energy transfer, the method can be used for thermal
conductivity computations. Similar to previous RNEMD methods, we
assume linear response of the temperature gradient with respect to the
thermal flux in general case. And the thermal conductivity ($\lambda$)
can be obtained with the imposed kinetic energy flux and the measured
thermal gradient:
\begin{equation}
J_z = -\lambda \frac{\partial T}{\partial z}
\end{equation}
Like other imposed-flux methods, the energy flux was calculated using
the total non-physical energy transferred (${E_{total}}$) from slab
``c'' to slab ``h'', which is recorded throughout a simulation, and
the time for data collection $t$:
\begin{equation}
J_z = \frac{E_{total}}{2 t L_x L_y}
\end{equation}
where $L_x$ and $L_y$ denotes the dimensions of the plane in a
simulation cell perpendicular to the thermal gradient, and a factor of
two in the denominator is present for the heat transport occurs in
both $+z$ and $-z$ directions. The temperature gradient
${\langle\partial T/\partial z\rangle}$ can be obtained by a linear
regression of the temperature profile, which is recorded during a
simulation for each slab in a cell. For Lennard-Jones simulations,
thermal conductivities are reported in reduced units
(${\lambda^*=\lambda \sigma^2 m^{1/2} k_B^{-1}\varepsilon^{-1/2}}$).

\subsection{Shear viscosities}
Alternatively, the method can carry out shear viscosity calculations
by switching off $J_z$. One can specify the vector
$\vec{j}_z(\vec{p})$ by choosing the three components
respectively. For shear viscosity simulations, $j_z(p_z)$ is usually
set to zero. Although for isotropic systems, the direction of
$\vec{j}_z(\vec{p})$ on the $xy$ plane should not matter, the ability
of arbitarily specifying the vector direction in our method provides
convenience in anisotropic simulations.

Similar to thermal conductivity computations, linear response of the
momentum gradient with respect to the shear stress is assumed, and the
shear viscosity ($\eta$) can be obtained with the imposed momentum
flux (e.g. in $x$ direction) and the measured gradient:
\begin{equation}
j_z(p_x) = -\eta \frac{\partial v_x}{\partial z}
\end{equation}
where the flux is similarly defined:
\begin{equation}
j_z(p_x) = \frac{P_x}{2 t L_x L_y}
\end{equation}
with $P_x$ being the total non-physical momentum transferred within
the data collection time. Also, the velocity gradient
${\langle\partial v_x/\partial z\rangle}$ can be obtained using linear
regression of the $x$ component of the mean velocity, $\langle
v_x\rangle$, in each of the bins. For Lennard-Jones simulations, shear
viscosities are reported in reduced units
(${\eta^*=\eta\sigma^2(\varepsilon m)^{-1/2}}$).

\subsection{Interfacial friction and Slip length}
While the shear stress results in a velocity gradient within bulk
fluid phase, its effect at a solid-liquid interface could vary due to
the interaction strength between the two phases. The interfacial
friction coefficient $\kappa$ is defined to relate the shear stress
(e.g. along $x$-axis) and the relative fluid velocity tangent to the
interface:
\begin{equation}
j_z(p_x)|_{interface} = \kappa\Delta v_x|_{interface}
\end{equation}
Under ``stick'' boundary condition, $\Delta v_x|_{interface}
\rightarrow 0$, which leads to $\kappa\rightarrow\infty$. However, for
``slip'' boundary condition at the solid-liquid interface, $\kappa$
becomes finite. To characterize the interfacial boundary conditions,
slip length ($\delta$) is defined using $\kappa$ and the shear
viscocity of liquid phase ($\eta$):
\begin{equation}
\delta = \frac{\eta}{\kappa}
\end{equation}
so that $\delta\rightarrow 0$ in the ``no-slip'' boundary condition,
and depicts how ``slippery'' an interface is. Figure \ref{slipLength}
illustrates how this quantity is defined and computed for a
solid-liquid interface.

\begin{figure}
\includegraphics[width=\linewidth]{defDelta}
\caption{The slip length $\delta$ can be obtained from a velocity
  profile of a solid-liquid interface. An example of Au/hexane
  interfaces is shown.}
\label{slipLength}
\end{figure}

In our method, a shear stress can be applied similar to shear
viscosity computations by applying an unphysical momentum flux
(e.g. $j_z(p_x)$). A corresponding velocity profile can be obtained as
shown in Figure \ref{slipLength}, in which the velocity gradients
within liquid phase and velocity difference at the liquid-solid
interface can be measured respectively. Further calculations and
characterizations of the interface can be carried out using these
data.
[MENTION IN RESULTS THAT ETA OBTAINED HERE DOES NOT NECESSARILY EQUAL
TO BULK VALUES]

\section{Results}
[L-J COMPARED TO RNEMD NIVS; WATER COMPARED TO RNEMD NIVS AND EMD;
SLIP BOUNDARY VS STICK BOUNDARY; ICE-WATER INTERFACES]

There are many factors contributing to the measured interfacial
conductance; some of these factors are physically motivated
(e.g. coverage of the surface by the capping agent coverage and
solvent identity), while some are governed by parameters of the
methodology (e.g. applied flux and the formulas used to obtain the
conductance). In this section we discuss the major physical and
calculational effects on the computed conductivity.

\subsection{Effects due to capping agent coverage}

A series of different initial conditions with a range of surface
coverages was prepared and solvated with various with both of the
solvent molecules. These systems were then equilibrated and their
interfacial thermal conductivity was measured with the NIVS
algorithm. Figure \ref{coverage} demonstrates the trend of conductance
with respect to surface coverage. 

\begin{figure}
\includegraphics[width=\linewidth]{coverage}
\caption{The interfacial thermal conductivity ($G$) has a
  non-monotonic dependence on the degree of surface capping.  This
  data is for the Au(111) / butanethiol / solvent interface with
  various UA force fields at $\langle T\rangle \sim $200K.}
\label{coverage}
\end{figure}

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.

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.  Capping agents are clearly
playing a major role in thermal transport at metal / organic solvent
surfaces.

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.

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.

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.

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.

\subsection{Effects due to Solvent \& Solvent Models}
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.

\begin{table*}
  \begin{minipage}{\linewidth}
    \begin{center}
      
      \caption{Computed interfacial thermal conductance ($G$ and
        $G^\prime$) values for interfaces using various models for
        solvent and capping agent (or without capping agent) at
        $\langle T\rangle\sim$200K.  Here ``D'' stands for deuterated
        solvent or capping agent molecules. Error estimates are
        indicated in parentheses.}
      
      \begin{tabular}{llccc}
        \hline\hline
        Butanethiol model & Solvent & $G$ & $G^\prime$ \\
        (or bare surface) & model & \multicolumn{2}{c}{(MW/m$^2$/K)} \\
        \hline
        UA    & UA hexane    & 131(9)    & 87(10)    \\
              & UA hexane(D) & 153(5)    & 136(13)   \\
              & AA hexane    & 131(6)    & 122(10)   \\
              & UA toluene   & 187(16)   & 151(11)   \\
              & AA toluene   & 200(36)   & 149(53)   \\
        \hline
        AA    & UA hexane    & 116(9)    & 129(8)    \\
              & AA hexane    & 442(14)   & 356(31)   \\
              & AA hexane(D) & 222(12)   & 234(54)   \\
              & UA toluene   & 125(25)   & 97(60)    \\
              & AA toluene   & 487(56)   & 290(42)   \\
        \hline
        AA(D) & UA hexane    & 158(25)   & 172(4)    \\
              & AA hexane    & 243(29)   & 191(11)   \\
              & AA toluene   & 364(36)   & 322(67)   \\
        \hline
        bare  & UA hexane    & 46.5(3.2) & 49.4(4.5) \\
              & UA hexane(D) & 43.9(4.6) & 43.0(2.0) \\
              & AA hexane    & 31.0(1.4) & 29.4(1.3) \\
              & UA toluene   & 70.1(1.3) & 65.8(0.5) \\
        \hline\hline
      \end{tabular}
      \label{modelTest}
    \end{center}
  \end{minipage}
\end{table*}

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.

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.

For the fully-covered surfaces, the choice of force field for the
capping agent and solvent has a large impact on the calculated 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.  Compared
to the AA model, the UA model yields more reasonable conductivity
values with much higher computational efficiency.

\subsubsection{Are electronic excitations in the metal important?}
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 $G$.

\subsection{Effects due to methodology and simulation parameters}

We have varied the parameters of the simulations in order to
investigate how these factors would affect the computation of $G$.  Of
particular interest are: 1) the length scale for the applied thermal
gradient (modified by increasing the amount of solvent in the system),
2) the sign and magnitude of the applied thermal flux, 3) the average
temperature of the simulation (which alters the solvent density during
equilibration), and 4) the definition of the interfacial conductance
(Eqs. (\ref{discreteG}) or (\ref{derivativeG})) used in the
calculation.

Systems of different lengths were prepared by altering the number of
solvent molecules and extending the length of the box along the $z$
axis to accomodate the extra solvent.  Equilibration at the same
temperature and pressure conditions led to nearly identical surface
areas ($L_x$ and $L_y$) available to the metal and capping agent,
while the extra solvent served mainly to lengthen the axis that was
used to apply the thermal flux.  For a given value of the applied
flux, the different $z$ length scale has only a weak effect on the
computed conductivities.

\subsubsection{Effects of applied flux}
The NIVS algorithm allows changes in both the sign and magnitude of
the applied flux.  It is possible to reverse the direction of heat
flow simply by changing the sign of the flux, and thermal gradients
which would be difficult to obtain experimentally ($5$ K/\AA) can be
easily simulated.  However, the magnitude of the applied flux is not
arbitrary if one aims to obtain a stable and reliable thermal gradient.
A temperature gradient can be lost in the noise if $|J_z|$ is too
small, and excessive $|J_z|$ values can cause phase transitions if the
extremes of the simulation cell become widely separated in
temperature.  Also, if $|J_z|$ is too large for the bulk conductivity
of the materials, the thermal gradient will never reach a stable
state.  

Within a reasonable range of $J_z$ values, we were able to study how
$G$ changes as a function of this flux.  In what follows, we use
positive $J_z$ values to denote the case where energy is being
transferred by the method from the metal phase and into the liquid.
The resulting gradient therefore has a higher temperature in the
liquid phase.  Negative flux values reverse this transfer, and result
in higher temperature metal phases.  The conductance measured under
different applied $J_z$ values is listed in Tables 2 and 3 in the
supporting information. These results do not indicate that $G$ depends
strongly on $J_z$ within this flux range. The linear response of flux
to thermal gradient simplifies our investigations in that we can rely
on $G$ measurement with only a small number $J_z$ values.

The sign of $J_z$ is a different matter, however, as this can alter
the temperature on the two sides of the interface. The average
temperature values reported are for the entire system, and not for the
liquid phase, so at a given $\langle T \rangle$, the system with
positive $J_z$ has a warmer liquid phase.  This means that if the
liquid carries thermal energy via diffusive transport, {\it positive}
$J_z$ values will result in increased molecular motion on the liquid
side of the interface, and this will increase the measured
conductivity.

\subsubsection{Effects due to average temperature}

We also studied the effect of average system temperature on the
interfacial conductance.  The simulations are first equilibrated in
the NPT ensemble at 1 atm.  The TraPPE-UA model for hexane tends to
predict a lower boiling point (and liquid state density) than
experiments.  This lower-density liquid phase leads to reduced contact
between the hexane and butanethiol, and this accounts for our
observation of lower conductance at higher temperatures.  In raising
the average temperature from 200K to 250K, the density drop of
$\sim$20\% in the solvent phase leads to a $\sim$40\% drop in the
conductance.

Similar behavior is observed in the TraPPE-UA model for toluene,
although this model has better agreement with the experimental
densities of toluene.  The expansion of the toluene liquid phase is
not as significant as that of the hexane (8.3\% over 100K), and this
limits the effect to $\sim$20\% drop in thermal conductivity.

Although we have not mapped out the behavior at a large number of
temperatures, is clear that there will be a strong temperature
dependence in the interfacial conductance when the physical properties
of one side of the interface (notably the density) change rapidly as a
function of temperature.

Besides the lower interfacial thermal conductance, surfaces at
relatively high temperatures are susceptible to reconstructions,
particularly when butanethiols fully cover the Au(111) surface. These
reconstructions include surface Au atoms which migrate outward to the
S atom layer, and butanethiol molecules which embed into the surface
Au layer. The driving force for this behavior is the strong Au-S
interactions which are modeled here with a deep Lennard-Jones
potential. This phenomenon agrees with reconstructions that have been
experimentally
observed.\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}.  Vlugt
{\it et al.} kept their Au(111) slab rigid so that their simulations
could reach 300K without surface
reconstructions.\cite{vlugt:cpc2007154} Since surface reconstructions
blur the interface, the measurement of $G$ becomes more difficult to
conduct at higher temperatures.  For this reason, most of our
measurements are undertaken at $\langle T\rangle\sim$200K where
reconstruction is minimized.

However, when the surface is not completely covered by butanethiols,
the simulated system appears to be more resistent to the
reconstruction. Our Au / butanethiol / toluene system had the Au(111)
surfaces 90\% covered by butanethiols, but did not see this above
phenomena even at $\langle T\rangle\sim$300K.  That said, we did
observe butanethiols migrating to neighboring three-fold sites during
a simulation.  Since the interface persisted in these simulations, we
were able to obtain $G$'s for these interfaces even at a relatively
high temperature without being affected by surface reconstructions.

\section{Discussion}
[COMBINE W. RESULTS]
The primary result of this work is that the capping agent acts as an
efficient thermal coupler between solid and solvent phases.  One of
the ways the capping agent can carry out this role is to down-shift
between the phonon vibrations in the solid (which carry the heat from
the gold) and the molecular vibrations in the liquid (which carry some
of the heat in the solvent).

To investigate the mechanism of interfacial thermal conductance, the
vibrational power spectrum was computed. Power spectra were taken for
individual components in different simulations. To obtain these
spectra, simulations were run after equilibration in the
microcanonical (NVE) ensemble and without a thermal
gradient. Snapshots of configurations were collected at a frequency
that is higher than that of the fastest vibrations occurring in the
simulations. With these configurations, the velocity auto-correlation
functions can be computed:
\begin{equation}
C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle
\label{vCorr} 
\end{equation}
The power spectrum is constructed via a Fourier transform of the
symmetrized velocity autocorrelation function,
\begin{equation}
  \hat{f}(\omega) =
  \int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt
\label{fourier} 
\end{equation}

\subsection{The role of specific vibrations}
The vibrational spectra for gold slabs in different environments are
shown as in Figure \ref{specAu}. Regardless of the presence of
solvent, the gold surfaces which are covered by butanethiol molecules
exhibit an additional peak observed at a frequency of
$\sim$165cm$^{-1}$.  We attribute this peak to the S-Au bonding
vibration. This vibration enables efficient thermal coupling of the
surface Au layer to the capping agents. Therefore, in our simulations,
the Au / S interfaces do not appear to be the primary barrier to
thermal transport when compared with the butanethiol / solvent
interfaces.  This supports the results of Luo {\it et
  al.}\cite{Luo20101}, who reported $G$ for Au-SAM junctions roughly
twice as large as what we have computed for the thiol-liquid
interfaces.

\begin{figure}
\includegraphics[width=\linewidth]{vibration}
\caption{The vibrational power spectrum for thiol-capped gold has an
  additional vibrational peak at $\sim $165cm$^{-1}$.  Bare gold
  surfaces (both with and without a solvent over-layer) are missing
  this peak.   A similar peak at  $\sim $165cm$^{-1}$ also appears in
  the vibrational power spectrum for the butanethiol capping agents.}
\label{specAu}
\end{figure}

Also in this figure, we show the vibrational power spectrum for the
bound butanethiol molecules, which also exhibits the same
$\sim$165cm$^{-1}$ peak.

\subsection{Overlap of power spectra}
A comparison of the results obtained from the two different organic
solvents can also provide useful information of the interfacial
thermal transport process.  In particular, the vibrational overlap
between the butanethiol and the organic solvents suggests a highly
efficient thermal exchange between these components.  Very high
thermal conductivity was observed when AA models were used and C-H
vibrations were treated classically. The presence of extra degrees of
freedom in the AA force field yields higher heat exchange rates
between the two phases and results in a much higher conductivity than
in the UA force field. The all-atom classical models include high
frequency modes which should be unpopulated at our relatively low
temperatures.  This artifact is likely the cause of the high thermal
conductance in all-atom MD simulations.

The similarity in the vibrational modes available to solvent and
capping agent can be reduced by deuterating one of the two components
(Fig. \ref{aahxntln}).  Once either the hexanes or the butanethiols
are deuterated, one can observe a significantly lower $G$ and
$G^\prime$ values (Table \ref{modelTest}).

\begin{figure}
\includegraphics[width=\linewidth]{aahxntln}
\caption{Spectra obtained for all-atom (AA) Au / butanethiol / solvent
  systems. When butanethiol is deuterated (lower left), its
  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).}
\label{aahxntln}
\end{figure}

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. 

Previous observations of Zhang {\it et al.}\cite{hase:2010} indicate
that the {\it intra}molecular heat transport due to alkylthiols is
highly efficient.  Combining our observations with those of Zhang {\it
  et al.}, it appears that butanethiol acts as a channel to expedite
heat flow from the gold surface and into the alkyl chain.  The
vibrational coupling between the metal and the liquid phase can
therefore be enhanced with the presence of suitable capping agents.

Deuterated models in the UA force field did not decouple the thermal
transport as well as in the AA force field.  The UA models, even
though they have eliminated the high frequency C-H vibrational
overlap, still have significant overlap in the lower-frequency
portions of the infrared spectra (Figure \ref{uahxnua}).  Deuterating
the UA models did not decouple the low frequency region enough to
produce an observable difference for the results of $G$ (Table
\ref{modelTest}). 

\begin{figure}
\includegraphics[width=\linewidth]{uahxnua}
\caption{Vibrational power spectra for UA models for the butanethiol
  and hexane solvent (upper panel) show the high degree of overlap
  between these two molecules, particularly at lower frequencies.
  Deuterating a UA model for the solvent (lower panel) does not
  decouple the two spectra to the same degree as in the AA force
  field (see Fig \ref{aahxntln}).}
\label{uahxnua}
\end{figure}

\section{Conclusions}
The NIVS algorithm has been applied to simulations of
butanethiol-capped Au(111) surfaces in the presence of organic
solvents. This algorithm allows the application of unphysical thermal
flux to transfer heat between the metal and the liquid phase. With the
flux applied, we were able to measure the corresponding thermal
gradients and to obtain interfacial thermal conductivities. Under
steady states, 2-3 ns trajectory simulations are sufficient for
computation of this quantity.

Our simulations have seen significant conductance enhancement in the
presence of capping agent, compared with the bare gold / liquid
interfaces. The vibrational coupling between the metal and the liquid
phase is enhanced by a chemically-bonded capping agent. Furthermore,
the coverage percentage of the capping agent plays an important role
in the interfacial thermal transport process. Moderately low coverages
allow higher contact between capping agent and solvent, and thus could
further enhance the heat transfer process, giving a non-monotonic
behavior of conductance with increasing coverage.

Our results, particularly using the UA models, agree well with
available experimental data.  The AA models tend to overestimate the
interfacial thermal conductance in that the classically treated C-H
vibrations become too easily populated. Compared to the AA models, the
UA models have higher computational efficiency with satisfactory
accuracy, and thus are preferable in modeling interfacial thermal
transport.

Of the two definitions for $G$, the discrete form
(Eq. \ref{discreteG}) was easier to use and gives out relatively
consistent results, while the derivative form (Eq. \ref{derivativeG})
is not as versatile. Although $G^\prime$ gives out comparable results
and follows similar trend with $G$ when measuring close to fully
covered or bare surfaces, the spatial resolution of $T$ profile
required for the use of a derivative form is limited by the number of
bins and the sampling required to obtain thermal gradient information.

Vlugt {\it et al.} have investigated the surface thiol structures for
nanocrystalline gold and pointed out that they differ from those of
the Au(111) surface.\cite{landman:1998,vlugt:cpc2007154} This
difference could also cause differences in the interfacial thermal
transport behavior. To investigate this problem, one would need an
effective method for applying thermal gradients in non-planar
(i.e. spherical) geometries. 

\section{Acknowledgments}
Support for this project was provided by the National Science
Foundation under grant CHE-0848243. Computational time was provided by
the Center for Research Computing (CRC) at the University of Notre
Dame. 

\newpage

\bibliography{stokes}

\end{doublespace}
\end{document}

Revision:	3775
Committed:	Thu Dec 8 01:34:21 2011 UTC (12 years, 9 months ago) by skuang
Content type:	application/x-tex
File size:	43488 byte(s)
Log Message:	a rought draft of simulation details and a figure added.