--- interfacial/interfacial.tex	2011/01/27 16:29:20	3717
+++ interfacial/interfacial.tex	2011/07/05 17:39:21	3730
@@ -22,7 +22,7 @@
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@@ -44,7 +44,22 @@ The abstract
 \begin{doublespace}
 
 \begin{abstract}
-The abstract
+
+We have developed a Non-Isotropic Velocity Scaling algorithm for
+setting up and maintaining stable thermal gradients in non-equilibrium
+molecular dynamics simulations. This approach effectively imposes
+unphysical thermal flux even between particles of different
+identities, conserves linear momentum and kinetic energy, and
+minimally perturbs the velocity profile of a system when compared with
+previous RNEMD methods. We have used this method to simulate thermal
+conductance at metal / organic solvent interfaces both with and
+without the presence of thiol-based capping agents.  We obtained
+values comparable with experimental values, and observed significant
+conductance enhancement with the presence of capping agents. Computed
+power spectra indicate the acoustic impedance mismatch between metal
+and liquid phase is greatly reduced by the capping agents and thus
+leads to higher interfacial thermal transfer efficiency.
+
 \end{abstract}
 
 \newpage
@@ -56,14 +71,643 @@ The abstract
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Introduction}
+[BACKGROUND FOR INTERFACIAL THERMAL CONDUCTANCE PROBLEM]
+Interfacial thermal conductance is extensively studied both
+experimentally and computationally, and systems with interfaces
+present are generally heterogeneous. Although interfaces are commonly
+barriers to heat transfer, it has been
+reported\cite{doi:10.1021/la904855s} that under specific circustances,
+e.g. with certain capping agents present on the surface, interfacial
+conductance can be significantly enhanced. However, heat conductance
+of molecular and nano-scale interfaces will be affected by the
+chemical details of the surface and is challenging to
+experimentalist. The lower thermal flux through interfaces is even
+more difficult to measure with EMD and forward NEMD simulation
+methods. Therefore, developing good simulation methods will be
+desirable in order to investigate thermal transport across interfaces.
 
-The intro.
+Recently, we have developed the Non-Isotropic Velocity Scaling (NIVS)
+algorithm for RNEMD simulations\cite{kuang:164101}. This algorithm
+retains the desirable features of RNEMD (conservation of linear
+momentum and total energy, compatibility with periodic boundary
+conditions) while establishing true thermal distributions in each of
+the two slabs. Furthermore, it allows more effective thermal exchange
+between particles of different identities, and thus enables extensive
+study of interfacial conductance.
+
+\section{Methodology}
+\subsection{Algorithm}
+[BACKGROUND FOR MD METHODS]
+There have been many algorithms for computing thermal conductivity
+using molecular dynamics simulations. However, interfacial conductance
+is at least an order of magnitude smaller. This would make the
+calculation even more difficult for those slowly-converging
+equilibrium methods. Imposed-flux non-equilibrium
+methods\cite{MullerPlathe:1997xw} have the flux set {\it a priori} and
+the response of temperature or momentum gradients are easier to
+measure than the flux, if unknown, and thus, is a preferable way to
+the forward NEMD methods. Although the momentum swapping approach for
+flux-imposing can be used for exchanging energy between particles of
+different identity, the kinetic energy transfer efficiency is affected
+by the mass difference between the particles, which limits its
+application on heterogeneous interfacial systems.
+
+The non-isotropic velocity scaling (NIVS)\cite{kuang:164101} approach in
+non-equilibrium MD simulations is able to impose relatively large
+kinetic energy flux without obvious perturbation to the velocity
+distribution of the simulated systems. Furthermore, this approach has
+the advantage in heterogeneous interfaces in that kinetic energy flux
+can be applied between regions of particles of arbitary identity, and
+the flux quantity is not restricted by particle mass difference.
+
+The NIVS algorithm scales the velocity vectors in two separate regions
+of a simulation system with respective diagonal scaling matricies. To
+determine these scaling factors in the matricies, a set of equations
+including linear momentum conservation and kinetic energy conservation
+constraints and target momentum/energy flux satisfaction is
+solved. With the scaling operation applied to the system in a set
+frequency, corresponding momentum/temperature gradients can be built,
+which can be used for computing transportation properties and other
+applications related to momentum/temperature gradients. The NIVS
+algorithm conserves momenta and energy and does not depend on an
+external thermostat. 
+
+\subsection{Defining Interfacial Thermal Conductivity $G$}
+For interfaces with a relatively low interfacial conductance, the bulk
+regions on either side of an interface rapidly come to a state in
+which the two phases have relatively homogeneous (but distinct)
+temperatures. The interfacial thermal conductivity $G$ can therefore
+be approximated as:
+\begin{equation}
+G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle -
+    \langle T_\mathrm{cold}\rangle \right)}
+\label{lowG}
+\end{equation}
+where ${E_{total}}$ is the imposed non-physical kinetic energy
+transfer and ${\langle T_\mathrm{hot}\rangle}$ and ${\langle
+  T_\mathrm{cold}\rangle}$ are the average observed temperature of the
+two separated phases.
+
+When the interfacial conductance is {\it not} small, two ways can be
+used to define $G$.
+
+One way is to assume the temperature is discretely different on two
+sides of the interface, $G$ can be calculated with the thermal flux
+applied $J$ and the maximum temperature difference measured along the
+thermal gradient max($\Delta T$), which occurs at the interface, as:
+\begin{equation}
+G=\frac{J}{\Delta T}
+\label{discreteG}
+\end{equation}
+
+The other approach is to assume a continuous temperature profile along
+the thermal gradient axis (e.g. $z$) and define $G$ at the point where
+the magnitude of thermal conductivity $\lambda$ change reach its
+maximum, given that $\lambda$ is well-defined throughout the space:
+\begin{equation}
+G^\prime = \Big|\frac{\partial\lambda}{\partial z}\Big|
+         = \Big|\frac{\partial}{\partial z}\left(-J_z\Big/
+           \left(\frac{\partial T}{\partial z}\right)\right)\Big|
+         = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big|
+         \Big/\left(\frac{\partial T}{\partial z}\right)^2
+\label{derivativeG}
+\end{equation}
+
+With the temperature profile obtained from simulations, one is able to
+approximate the first and second derivatives of $T$ with finite
+difference method and thus calculate $G^\prime$.
+
+In what follows, both definitions are used for calculation and comparison.
+
+[IMPOSE G DEFINITION INTO OUR SYSTEMS]
+To facilitate the use of the above definitions in calculating $G$ and
+$G^\prime$, we have a metal slab with its (111) surfaces perpendicular
+to the $z$-axis of our simulation cells. With or withour capping
+agents on the surfaces, the metal slab is solvated with organic
+solvents, as illustrated in Figure \ref{demoPic}.
+
+\begin{figure}
+\includegraphics[width=\linewidth]{demoPic}
+\caption{A sample showing how a metal slab has its (111) surface
+  covered by capping agent molecules and solvated by hexane.}
+\label{demoPic}
+\end{figure}
+
+With a simulation cell setup following the above manner, one is able
+to equilibrate the system and impose an unphysical thermal flux
+between the liquid and the metal phase with the NIVS algorithm. Under
+a stablized thermal gradient induced by periodically applying the
+unphysical flux, one is able to obtain a temperature profile and the
+physical thermal flux corresponding to it, which equals to the
+unphysical flux applied by NIVS. These data enables the evaluation of
+the interfacial thermal conductance of a surface. Figure \ref{gradT}
+is an example how those stablized thermal gradient can be used to
+obtain the 1st and 2nd derivatives of the temperature profile.
+
+\begin{figure}
+\includegraphics[width=\linewidth]{gradT}
+\caption{The 1st and 2nd derivatives of temperature profile can be
+  obtained with finite difference approximation.}
+\label{gradT}
+\end{figure}
+
+[MAY INCLUDE POWER SPECTRUM PROTOCOL]
+
+\section{Computational Details}
+\subsection{Simulation Protocol}
+In our simulations, Au is used to construct a metal slab with bare
+(111) surface perpendicular to the $z$-axis. Different slab thickness
+(layer numbers of Au) are simulated. This metal slab is first
+equilibrated under normal pressure (1 atm) and a desired
+temperature. After equilibration, butanethiol is used as the capping
+agent molecule to cover the bare Au (111) surfaces evenly. The sulfur
+atoms in the butanethiol molecules would occupy the three-fold sites
+of the surfaces, and the maximal butanethiol capacity on Au surface is
+$1/3$ of the total number of surface Au atoms[CITATION]. A series of
+different coverage surfaces is investigated in order to study the
+relation between coverage and conductance.
+
+[COVERAGE DISCRIPTION] However, since the interactions between surface
+Au and butanethiol is non-bonded, the capping agent molecules are
+allowed to migrate to an empty neighbor three-fold site during a
+simulation. Therefore, the initial configuration would not severely
+affect the sampling of a variety of configurations of the same
+coverage, and the final conductance measurement would be an average
+effect of these configurations explored in the simulations. [MAY NEED FIGURES]
+
+After the modified Au-butanethiol surface systems are equilibrated
+under canonical ensemble, Packmol\cite{packmol} is used to pack
+organic solvent molecules in the previously vacuum part of the
+simulation cells, which guarantees that short range repulsive
+interactions do not disrupt the simulations. Two solvents are
+investigated, one which has little vibrational overlap with the
+alkanethiol and plane-like shape (toluene), and one which has similar
+vibrational frequencies and chain-like shape ({\it n}-hexane). [MAY
+EXPLAIN WHY WE CHOOSE THEM]
+
+The spacing filled by solvent molecules, i.e. the gap between
+periodically repeated Au-butanethiol surfaces should be carefully
+chosen. A very long length scale for the thermal gradient axis ($z$)
+may cause excessively hot or cold temperatures in the middle of the
+solvent region and lead to undesired phenomena such as solvent boiling
+or freezing when a thermal flux is applied. Conversely, too few
+solvent molecules would change the normal behavior of the liquid
+phase. Therefore, our $N_{solvent}$ values were chosen to ensure that
+these extreme cases did not happen to our simulations. And the
+corresponding spacing is usually $35 \sim 60$\AA.
+
+The initial configurations generated by Packmol are further
+equilibrated with the $x$ and $y$ dimensions fixed, only allowing
+length scale change in $z$ dimension. This is to ensure that the
+equilibration of liquid phase does not affect the metal crystal
+structure in $x$ and $y$ dimensions. Further equilibration are run
+under NVT and then NVE ensembles.
+
+After the systems reach equilibrium, NIVS is implemented to impose a
+periodic unphysical thermal flux between the metal and the liquid
+phase. Most of our simulations are under an average temperature of
+$\sim$200K. Therefore, this flux usually comes from the metal to the
+liquid so that the liquid has a higher temperature and would not
+freeze due to excessively low temperature. This induced temperature
+gradient is stablized and the simulation cell is devided evenly into
+N slabs along the $z$-axis and the temperatures of each slab are
+recorded. When the slab width $d$ of each slab is the same, the
+derivatives of $T$ with respect to slab number $n$ can be directly
+used for $G^\prime$ calculations:
+\begin{equation}
+G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big|
+         \Big/\left(\frac{\partial T}{\partial z}\right)^2
+         = |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big|
+         \Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2
+         = |J_z|\Big|\frac{\partial^2 T}{\partial n^2}\Big|
+         \Big/\left(\frac{\partial T}{\partial n}\right)^2
+\label{derivativeG2}
+\end{equation}
+
+\subsection{Force Field Parameters}
+Our simulations include various components. Therefore, force field
+parameter descriptions are needed for interactions both between the
+same type of particles and between particles of different species.
+
+The Au-Au interactions in metal lattice slab is 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}. 
+
+Figure [REF] demonstrates how we name our pseudo-atoms of the
+molecules in our simulations.
+[FIGURE FOR MOLECULE NOMENCLATURE] 
+
+For both solvent molecules, straight chain {\it n}-hexane and aromatic
+toluene, United-Atom (UA) and All-Atom (AA) models are used
+respectively. The TraPPE-UA
+parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used
+for our UA solvent molecules. In these models, pseudo-atoms are
+located at the carbon centers for alkyl groups. By eliminating
+explicit hydrogen atoms, these models are simple and computationally
+efficient, while maintains good accuracy. However, the TraPPE-UA for
+alkanes is known to predict a lower boiling point than experimental
+values. Considering that after an unphysical thermal flux is applied
+to a system, the temperature of ``hot'' area in the liquid phase would be
+significantly higher than the average, to prevent over heating and
+boiling of the liquid phase, the average temperature in our
+simulations should be much lower than the liquid boiling point. [MORE DISCUSSION]
+For UA-toluene model, rigid body constraints are applied, so that the
+benzene ring and the methyl-CRar bond are kept rigid. This would save
+computational time.[MORE DETAILS]
+
+Besides the TraPPE-UA models, AA models for both organic solvents are
+included in our studies as well. For hexane, the OPLS-AA\cite{OPLSAA}
+force field is used. [MORE DETAILS]
+For toluene, the United Force Field developed by Rapp\'{e} {\it et
+  al.}\cite{doi:10.1021/ja00051a040} is adopted.[MORE DETAILS]
+
+The capping agent in our simulations, the butanethiol molecules can
+either use UA or AA model. The TraPPE-UA force fields includes
+parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for
+UA butanethiol model in our simulations. The OPLS-AA also provides
+parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111)
+surfaces do not have the hydrogen atom bonded to sulfur. To adapt this
+change and derive suitable parameters for butanethiol adsorbed on
+Au(111) surfaces, we adopt the S parameters from [CITATION CF VLUGT]
+and modify parameters for its neighbor C atom for charge balance in
+the molecule. Note that the model choice (UA or AA) of capping agent
+can be different from the solvent. Regardless of model choice, the
+force field parameters for interactions between capping agent and
+solvent can be derived using Lorentz-Berthelot Mixing Rule: 
+
+
+To describe the interactions between metal Au and non-metal capping
+agent and solvent particles, we refer to an adsorption study of alkyl
+thiols on gold surfaces by Vlugt {\it et
+  al.}\cite{vlugt:cpc2007154} They fitted an effective Lennard-Jones
+form of potential parameters for the interaction between Au and
+pseudo-atoms CH$_x$ and S based on a well-established and widely-used
+effective potential of Hautman and Klein[CITATION] for the Au(111)
+surface. As our simulations require the gold lattice slab to be
+non-rigid so that it could accommodate kinetic energy for thermal
+transport study purpose, the pair-wise form of potentials is
+preferred.
+
+Besides, the potentials developed from {\it ab initio} calculations by
+Leng {\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the
+interactions between Au and aromatic C/H atoms in toluene.[MORE DETAILS]
+
+However, the Lennard-Jones parameters between Au and other types of
+particles in our simulations are not yet well-established. For these
+interactions, we attempt to derive their parameters using the Mixing
+Rule. To do this, the ``Metal-non-Metal'' (MnM) interaction parameters
+for Au is first extracted from the Au-CH$_x$ parameters by applying
+the Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM''
+parameters in our simulations.
+
+\begin{table*}
+  \begin{minipage}{\linewidth}
+    \begin{center}
+      \caption{Lennard-Jones parameters for Au-non-Metal
+        interactions in our simulations.}
+      
+      \begin{tabular}{ccc}
+        \hline\hline
+        Non-metal & $\sigma$/\AA & $\epsilon$/kcal/mol \\
+        \hline
+        S    & 2.40   & 8.465   \\
+        CH3  & 3.54   & 0.2146  \\
+        CH2  & 3.54   & 0.1749  \\
+        CT3  & 3.365  & 0.1373  \\
+        CT2  & 3.365  & 0.1373  \\
+        CTT  & 3.365  & 0.1373  \\
+        HC   & 2.865  & 0.09256 \\
+        CHar & 3.4625 & 0.1680  \\
+        CRar & 3.555  & 0.1604  \\
+        CA   & 3.173  & 0.0640  \\
+        HA   & 2.746  & 0.0414  \\
+        \hline\hline
+      \end{tabular}
+      \label{MnM}
+    \end{center}
+  \end{minipage}
+\end{table*}
 
+
+\section{Results and Discussions}
+[MAY HAVE A BRIEF SUMMARY]
+\subsection{How Simulation Parameters Affects $G$}
+[MAY NOT PUT AT FIRST]
+We have varied our protocol or other parameters of the simulations in
+order to investigate how these factors would affect the measurement of
+$G$'s. It turned out that while some of these parameters would not
+affect the results substantially, some other changes to the
+simulations would have a significant impact on the measurement
+results.
+
+In some of our simulations, we allowed $L_x$ and $L_y$ to change
+during equilibrating the liquid phase. Due to the stiffness of the Au
+slab, $L_x$ 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 comparably smaller to $L_x$ and $L_y$, 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. 
+
+As stated in our computational details, the spacing filled with
+solvent molecules can be chosen within a range. This allows some
+change of solvent molecule numbers for the same Au-butanethiol
+surfaces. We did this study on our Au-butanethiol/hexane
+simulations. Nevertheless, the results obtained from systems of
+different $N_{hexane}$ did not indicate that the measurement of $G$ is
+susceptible to this parameter. For computational efficiency concern,
+smaller system size would be preferable, given that the liquid phase
+structure is not affected.
+
+Our NIVS algorithm allows change of unphysical thermal flux both in
+direction and in quantity. This feature extends our investigation of
+interfacial thermal conductance. However, the magnitude of this
+thermal flux is not arbitary if one aims to obtain a stable and
+reliable thermal gradient. A temperature profile would be
+substantially affected by noise when $|J_z|$ has a much too low
+magnitude; while an excessively large $|J_z|$ that overwhelms the
+conductance capacity of the interface would prevent a thermal gradient
+to reach a stablized steady state. NIVS has the advantage of allowing
+$J$ to vary in a wide range such that the optimal flux range for $G$
+measurement can generally be simulated by the algorithm. Within the
+optimal range, we were able to study how $G$ would change according to
+the thermal flux across the interface. For our simulations, we denote
+$J_z$ to be positive when the physical thermal flux is from the liquid
+to metal, and negative vice versa. The $G$'s measured under different
+$J_z$ is listed in Table \ref{AuThiolHexaneUA} and [REF]. These
+results do not suggest that $G$ is dependent 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
+couple $J_z$'s and do not need to test a large series of fluxes.
+
+%ADD MORE TO TABLE
+\begin{table*}
+  \begin{minipage}{\linewidth}
+    \begin{center}
+      \caption{Computed interfacial thermal conductivity ($G$ and
+        $G^\prime$) values for the Au/butanethiol/hexane interface
+        with united-atom model and different capping agent coverage
+        and solvent molecule numbers at different temperatures using a
+        range of energy fluxes.}
+      
+      \begin{tabular}{cccccc}
+        \hline\hline
+        Thiol & $\langle T\rangle$ & & $J_z$ & $G$ & $G^\prime$ \\
+        coverage (\%) & (K) & $N_{hexane}$ & (GW/m$^2$) &
+        \multicolumn{2}{c}{(MW/m$^2$/K)} \\
+        \hline
+        0.0   & 200 & 200 & 0.96 & 43.3 & 42.7 \\
+              &     &     & 1.91 & 45.7 & 42.9 \\
+              &     & 166 & 0.96 & 43.1 & 53.4 \\
+        88.9  & 200 & 166 & 1.94 & 172  & 108  \\
+        100.0 & 250 & 200 & 0.96 & 81.8 & 67.0 \\
+              &     & 166 & 0.98 & 79.0 & 62.9 \\
+              &     &     & 1.44 & 76.2 & 64.8 \\
+              & 200 & 200 & 1.92 & 129  & 87.3 \\
+              &     &     & 1.93 & 131  & 77.5 \\
+              &     & 166 & 0.97 & 115  & 69.3 \\
+              &     &     & 1.94 & 125  & 87.1 \\
+        \hline\hline
+      \end{tabular}
+      \label{AuThiolHexaneUA}
+    \end{center}
+  \end{minipage}
+\end{table*}
+
+Furthermore, we also attempted to increase system average temperatures
+to above 200K. These simulations are first equilibrated in the NPT
+ensemble under normal pressure. As stated above, the TraPPE-UA model
+for hexane tends to predict a lower boiling point. In our simulations,
+hexane had diffculty to remain in liquid phase when NPT equilibration
+temperature is higher than 250K. Additionally, the equilibrated liquid
+hexane density under 250K becomes lower than experimental value. This
+expanded liquid phase leads to lower contact between hexane and
+butanethiol as well.[MAY NEED FIGURE] And this reduced contact would
+probably be accountable for a lower interfacial thermal conductance,
+as shown in Table \ref{AuThiolHexaneUA}. 
+
+A similar study for TraPPE-UA toluene agrees with the above result as
+well. Having a higher boiling point, toluene tends to remain liquid in
+our simulations even equilibrated under 300K in NPT
+ensembles. Furthermore, the expansion of the toluene liquid phase is
+not as significant as that of the hexane. This prevents severe
+decrease of liquid-capping agent contact and the results (Table
+\ref{AuThiolToluene}) show only a slightly decreased interface
+conductance. Therefore, solvent-capping agent contact should play an
+important role in the thermal transport process across the interface
+in that higher degree of contact could yield increased conductance.
+
+[ADD SIGNS AND ERROR ESTIMATE TO TABLE]
+\begin{table*}
+  \begin{minipage}{\linewidth}
+    \begin{center}
+      \caption{Computed interfacial thermal conductivity ($G$ and
+        $G^\prime$) values for the Au/butanethiol/toluene interface at
+        different temperatures using a range of energy fluxes.}
+      
+      \begin{tabular}{cccc}
+        \hline\hline
+        $\langle T\rangle$ & $J_z$ & $G$ & $G^\prime$ \\
+        (K) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\
+        \hline
+        200 & 1.86 & 180 & 135 \\
+            & 2.15 & 204 & 113 \\
+            & 3.93 & 175 & 114 \\
+        300 & 1.91 & 143 & 125 \\
+            & 4.19 & 134 & 113 \\
+        \hline\hline
+      \end{tabular}
+      \label{AuThiolToluene}
+    \end{center}
+  \end{minipage}
+\end{table*}
+
+Besides lower interfacial thermal conductance, surfaces in relatively
+high temperatures are susceptible to reconstructions, when
+butanethiols have a full coverage on the Au(111) surface. These
+reconstructions include surface Au atoms migrated outward to the S
+atom layer, and butanethiol molecules embedded into the original
+surface Au layer. The driving force for this behavior is the strong
+Au-S interactions in our simulations. And these reconstructions lead
+to higher ratio of Au-S attraction and thus is energetically
+favorable. Furthermore, this phenomenon agrees with experimental
+results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt
+{\it et al.} had kept their Au(111) slab rigid so that their
+simulations can reach 300K without surface reconstructions. Without
+this practice, simulating 100\% thiol covered interfaces under higher
+temperatures could hardly avoid surface reconstructions. However, our
+measurement is based on assuming homogeneity on $x$ and $y$ dimensions
+so that measurement of $T$ at particular $z$ would be an effective
+average of the particles of the same type. Since surface
+reconstructions could eliminate the original $x$ and $y$ dimensional
+homogeneity, measurement of $G$ is more difficult to conduct under
+higher temperatures. Therefore, most of our measurements are
+undertaken at $<T>\sim$200K.
+
+However, when the surface is not completely covered by butanethiols,
+the simulated system is more resistent to the reconstruction
+above. Our Au-butanethiol/toluene system did not see this phenomena
+even at $<T>\sim$300K. The Au(111) surfaces have a 90\% coverage of
+butanethiols and have empty three-fold sites. These empty sites could
+help prevent surface reconstruction in that they provide other means
+of capping agent relaxation. It is observed that butanethiols can
+migrate to their neighbor empty sites during a simulation. Therefore,
+we were able to obtain $G$'s for these interfaces even at a relatively
+high temperature without being affected by surface reconstructions.
+
+\subsection{Influence of Capping Agent Coverage on $G$}
+To investigate the influence of butanethiol coverage on interfacial
+thermal conductance, a series of different coverage Au-butanethiol
+surfaces is prepared and solvated with various organic
+molecules. These systems are then equilibrated and their interfacial
+thermal conductivity are measured with our NIVS algorithm. Table
+\ref{tlnUhxnUhxnD} lists these results for direct comparison between
+different coverages of butanethiol.
+
+ With high coverage of butanethiol on the gold surface,
+the interfacial thermal conductance is enhanced
+significantly. Interestingly, a slightly lower butanethiol coverage
+leads to a moderately higher conductivity. This is probably due to
+more solvent/capping agent contact when butanethiol molecules are
+not densely packed, which enhances the interactions between the two
+phases and lowers the thermal transfer barrier of this interface.
+[COMPARE TO AU/WATER IN PAPER]
+
+
+significant conductance enhancement compared to the gold/water
+interface without capping agent and agree with available experimental
+data. This indicates that the metal-metal potential, though not
+predicting an accurate bulk metal thermal conductivity, does not
+greatly interfere with the simulation of the thermal conductance
+behavior across a non-metal interface. 
+ The results show that the two definitions used for $G$ yield
+comparable values, though $G^\prime$ tends to be smaller.
+
+
+\begin{table*}
+  \begin{minipage}{\linewidth}
+    \begin{center}
+      \caption{Computed interfacial thermal conductivity ($G$ and
+        $G^\prime$) values for the Au/butanethiol/hexane interface
+        with united-atom model and different capping agent coverage
+        and solvent molecule numbers at different temperatures using a
+        range of energy fluxes.}
+      
+      \begin{tabular}{cccccc}
+        \hline\hline
+        Thiol & $\langle T\rangle$ & & $J_z$ & $G$ & $G^\prime$ \\
+        coverage (\%) & (K) & $N_{hexane}$ & (GW/m$^2$) &
+        \multicolumn{2}{c}{(MW/m$^2$/K)} \\
+        \hline
+        0.0   & 200 & 200 & 0.96 & 43.3 & 42.7 \\
+              &     &     & 1.91 & 45.7 & 42.9 \\
+              &     & 166 & 0.96 & 43.1 & 53.4 \\
+        88.9  & 200 & 166 & 1.94 & 172  & 108  \\
+        100.0 & 250 & 200 & 0.96 & 81.8 & 67.0 \\
+              &     & 166 & 0.98 & 79.0 & 62.9 \\
+              &     &     & 1.44 & 76.2 & 64.8 \\
+              & 200 & 200 & 1.92 & 129  & 87.3 \\
+              &     &     & 1.93 & 131  & 77.5 \\
+              &     & 166 & 0.97 & 115  & 69.3 \\
+              &     &     & 1.94 & 125  & 87.1 \\
+        \hline\hline
+      \end{tabular}
+      \label{tlnUhxnUhxnD}
+    \end{center}
+  \end{minipage}
+\end{table*}
+
+\subsection{Influence of Chosen Molecule Model on $G$}
+[MAY COMBINE W MECHANISM STUDY]
+
+For the all-atom model, the liquid hexane phase was not stable under NPT
+conditions. Therefore, the simulation length scale parameters are
+adopted from previous equilibration results of the united-atom model
+at 200K. Table \ref{AuThiolHexaneAA} shows the results of these
+simulations. The conductivity values calculated with full capping
+agent coverage are substantially larger than observed in the
+united-atom model, and is even higher than predicted by
+experiments. It is possible that our parameters for metal-non-metal
+particle interactions lead to an overestimate of the interfacial
+thermal conductivity, although the active C-H vibrations in the
+all-atom model (which should not be appreciably populated at normal
+temperatures) could also account for this high conductivity. The major
+thermal transfer barrier of Au/butanethiol/hexane interface is between
+the liquid phase and the capping agent, so extra degrees of freedom
+such as the C-H vibrations could enhance heat exchange between these
+two phases and result in a much higher conductivity.
+
+\begin{table*}
+  \begin{minipage}{\linewidth}
+    \begin{center}
+      
+      \caption{Computed interfacial thermal conductivity ($G$ and
+        $G^\prime$) values for the Au/butanethiol/hexane interface
+        with all-atom model and different capping agent coverage at
+        200K using a range of energy fluxes.}
+      
+      \begin{tabular}{cccc}
+        \hline\hline
+        Thiol & $J_z$ & $G$ & $G^\prime$ \\
+        coverage (\%) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\
+        \hline
+        0.0   & 0.95 & 28.5 & 27.2 \\
+              & 1.88 & 30.3 & 28.9 \\
+        100.0 & 2.87 & 551  & 294  \\
+              & 3.81 & 494  & 193  \\
+        \hline\hline
+      \end{tabular}
+      \label{AuThiolHexaneAA}
+    \end{center}
+  \end{minipage}
+\end{table*}
+
+
+\subsection{Mechanism of Interfacial Thermal Conductance Enhancement
+  by Capping Agent}
+[MAY INTRODUCE PROTOCOL IN METHODOLOGY/COMPUTATIONAL DETAIL]
+
+
+%subsubsection{Vibrational spectrum study on conductance mechanism}
+To investigate the mechanism of this interfacial thermal conductance,
+the vibrational spectra of various gold systems were obtained and are
+shown as in the upper panel of Fig. \ref{vibration}. To obtain these
+spectra, one first runs a simulation in the NVE ensemble and collects
+snapshots of configurations; these configurations are used to compute
+the velocity auto-correlation functions, which is used to construct a
+power spectrum via a Fourier transform. The gold surfaces covered by
+butanethiol molecules exhibit an additional peak observed at a
+frequency of $\sim$170cm$^{-1}$, which is attributed to the vibration
+of the S-Au bond. This vibration enables efficient thermal transport
+from surface Au atoms to the capping agents. Simultaneously, as shown
+in the lower panel of Fig. \ref{vibration}, the large overlap of the
+vibration spectra of butanethiol and hexane in the all-atom model,
+including the C-H vibration, also suggests high thermal exchange
+efficiency. The combination of these two effects produces the drastic
+interfacial thermal conductance enhancement in the all-atom model.
+
+\begin{figure}
+\includegraphics[width=\linewidth]{vibration}
+\caption{Vibrational spectra obtained for gold in different
+  environments (upper panel) and for Au/thiol/hexane simulation in
+  all-atom model (lower panel).}
+\label{vibration}
+\end{figure}
+% MAY NEED TO CONVERT TO JPEG
+
+\section{Conclusions}
+
+
+[NECESSITY TO STUDY THERMAL CONDUCTANCE IN NANOCRYSTAL STRUCTURE]\cite{vlugt:cpc2007154}
+
 \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
+Dame. \newpage
 
 \bibliography{interfacial}
 
