--- interfacial/interfacial.tex	2011/01/27 16:29:20	3717
+++ interfacial/interfacial.tex	2011/05/09 19:08:08	3725
@@ -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
@@ -57,8 +72,332 @@ 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
+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. 
+
+(wondering how much detail of algorithm should be put here...)
+
+\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
+surfaces in comparison to no capping agent present.
+
+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}. 
+
+Straight chain {\it n}-hexane and aromatic toluene are respectively
+used as solvents. For hexane, both United-Atom\cite{TraPPE-UA.alkanes}
+and All-Atom\cite{OPLSAA} force fields are used for comparison; for
+toluene, United-Atom\cite{TraPPE-UA.alkylbenzenes} force fields are
+used with rigid body constraints applied. (maybe needs more details
+about rigid body)
+
+Buatnethiol molecules are used as capping agent for some of our
+simulations. United-Atom\cite{TraPPE-UA.thiols} and All-Atom models
+are respectively used corresponding to the force field type of
+solvent.
+
+To describe the interactions between metal Au and non-metal capping
+agent and solvent, we refer to Vlugt\cite{vlugt:cpc2007154} and derive
+other interactions which are not parametrized in their work. (can add
+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
+system, the Au slab is first equilibrated without solvent under room
+pressure and a desired temperature. After the metal slab is
+equilibrated, United-Atom or All-Atom butanethiols are replicated on
+the Au surface, each occupying the (??) among three Au atoms, and is
+equilibrated under NVT ensemble. According to (CITATION), the maximal
+thiol capacity on Au surface is $1/3$ of the total number of surface
+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