--- interfacial/interfacial.tex	2011/04/28 15:24:22	3724
+++ interfacial/interfacial.tex	2011/05/09 19:08:08	3725
@@ -44,7 +44,22 @@ The abstract version 2
 \begin{doublespace}
 
 \begin{abstract}
-The abstract version 2
+
+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
@@ -57,8 +72,29 @@ The intro.
 
 \section{Introduction}
 
-The intro.
+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.
 
+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}
 There have been many algorithms for computing thermal conductivity
@@ -97,7 +133,7 @@ external thermostat. 
 
 (wondering how much detail of algorithm should be put here...)
 
-\subsection{Simulation Parameters}
+\subsection{Force Field Parameters}
 Our simulation systems consists of metal gold lattice slab solvated by
 organic solvents. In order to study the role of capping agents in
 interfacial thermal conductance, butanethiol is chosen to cover gold
@@ -127,7 +163,10 @@ aromatic-metal interaction approximation here)
 hautman and klein's paper here and more discussion; need to put
 aromatic-metal interaction approximation here)
 
+[TABULATED FORCE FIELD PARAMETERS NEEDED]
+
 \section{Computational Details}
+\subsection{System Geometry}
 Our simulation systems consists of a lattice Au slab with the (111)
 surface perpendicular to the $z$-axis, and a solvent layer between the
 periodic Au slabs along the $z$-axis. To set up the interfacial
@@ -141,6 +180,224 @@ Au atoms. 
 
 \cite{packmol}
 
+\subsection{Simulation Parameters}
+
+When the interfacial conductance is {\it not} small, there are two
+ways to define $G$. If we assume the temperature is discretely
+different on two sides of the interface, $G$ can be calculated with
+the thermal flux applied $J$ and the temperature difference measured
+$\Delta T$ as:
+\begin{equation}
+G=\frac{J}{\Delta T}
+\label{discreteG}
+\end{equation}
+We can as well assume a continuous temperature profile along the
+thermal gradient axis $z$ and define $G$ as the change of bulk thermal
+conductivity $\lambda$ at a defined interfacial point:
+\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.
+
+\section{Results}
+\subsection{Toluene Solvent}
+
+The simulations follow a protocol similar to the previous gold/water
+interfacial systems. The results (Table \ref{AuThiolToluene}) show a
+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 solvent model is not
+particularly volatile, so the simulation cell does not expand
+significantly under higher temperature. We did not observe a
+significant conductance decrease when the temperature was increased to
+300K. 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/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*}
+
+\subsection{Hexane Solvent}
+
+Using the united-atom model, different coverages of capping agent,
+temperatures of simulations and numbers of solvent molecules were all
+investigated and Table \ref{AuThiolHexaneUA} shows the results of
+these computations. The number of hexane molecules in our simulations
+does not affect the calculations significantly. However, 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, while too
+few solvent molecules would change the normal behavior of the liquid
+phase. Our $N_{hexane}$ values were chosen to ensure that these
+extreme cases did not happen to our simulations.
+
+Table \ref{AuThiolHexaneUA} enables direct comparison between
+different coverages of capping agent, when other system parameters are
+held constant. 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]
+
+It is also noted that the overall simulation temperature is another
+factor that affects the interfacial thermal conductance. One
+possibility of this effect may be rooted in the decrease in density of
+the liquid phase. We observed that when the average temperature
+increases from 200K to 250K, the bulk hexane density becomes lower
+than experimental value, as the system is equilibrated under NPT
+ensemble. This leads to lower contact between solvent and capping
+agent, and thus lower conductivity.
+
+Conductivity values are more difficult to obtain under higher
+temperatures. This is because the Au surface tends to undergo
+reconstructions in relatively high temperatures. Surface Au atoms can
+migrate outward to reach higher Au-S contact; and capping agent
+molecules can be embedded into the surface Au layer due to the same
+driving force. This phenomenon agrees with experimental
+results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. A surface
+fully covered in capping agent is more susceptible to reconstruction,
+possibly because fully coverage prevents other means of capping agent
+relaxation, such as migration to an empty neighbor three-fold site.
+
+%MAY ADD MORE DATA 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*}
+
+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*}
+
+%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}
+% 600dpi, letter size. too large?
+
+
 \section{Acknowledgments}
 Support for this project was provided by the National Science
 Foundation under grant CHE-0848243. Computational time was provided by