--- stokes/stokes.tex	2011/12/08 01:34:21	3775
+++ stokes/stokes.tex	2011/12/11 03:22:47	3780
@@ -43,25 +43,17 @@ Notre Dame, Indiana 46556}
 \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.
+  We present a new method for introducing stable nonequilibrium
+  velocity and temperature gradients in molecular dynamics simulations
+  of heterogeneous systems. This method conserves the linear momentum
+  and total energy of the system and improves previous Reverse
+  Non-Equilibrium Molecular Dynamics (RNEMD) methods and maintains
+  thermal velocity distributions. It also avoid thermal anisotropy
+  occured in NIVS simulations by using isotropic velocity scaling on
+  the molecules in specific regions of a system. To test the method,
+  we have computed the thermal conductivity and shear viscosity of
+  model liquid systems as well as the interfacial frictions of a
+  series of  metal/liquid interfaces.
 
 \end{abstract}
 
@@ -218,8 +210,6 @@ thermal anisotropy when applying a momentum flux.
 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
@@ -228,7 +218,6 @@ section will illustrate this effect.
 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,
@@ -389,13 +378,14 @@ solid-liquid interface.
 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.
+solid-liquid interface. [MAY INCLUDE A SNAPSHOT IN FIGURE]
 
 \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.}
+  profile of a solid-liquid interface simulation. An example of
+  Au/hexane interfaces is shown. Calculation for the left side is 
+  illustrated. The right side is similar to the left side.}
 \label{slipLength}
 \end{figure}
 
@@ -407,462 +397,355 @@ data.
 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.
+\section{Results and Discussions}
+\subsection{Lennard-Jones fluid}
+Our orthorhombic simulation cell of Lennard-Jones fluid has identical
+parameters to our previous work\cite{kuang:164101} to facilitate
+comparison. Thermal conductivitis and shear viscosities were computed
+with the algorithm applied to the simulations. The results of thermal
+conductivity are compared with our previous NIVS algorithm. However,
+since the NIVS algorithm could produce temperature anisotropy for
+shear viscocity computations, these results are instead compared to
+the momentum swapping approaches. Table \ref{LJ} lists these
+calculations with various fluxes in reduced units.
 
-\subsection{Effects due to capping agent coverage}
+\begin{table*}
+  \begin{minipage}{\linewidth}
+    \begin{center}
 
-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. 
+      \caption{Thermal conductivity ($\lambda^*$) and shear viscosity
+        ($\eta^*$) (in reduced units) of a Lennard-Jones fluid at
+        ${\langle T^* \rangle = 0.72}$ and ${\rho^* = 0.85}$ computed
+        at various momentum fluxes.  The new method yields similar
+        results to previous RNEMD methods. All results are reported in
+        reduced unit. Uncertainties are indicated in parentheses.}
+      
+      \begin{tabular}{cccccc}
+        \hline\hline
+        \multicolumn{2}{c}{Momentum Exchange} &
+        \multicolumn{2}{c}{$\lambda^*$} &
+        \multicolumn{2}{c}{$\eta^*$} \\
+        \hline
+        Swap Interval $t^*$ & Equivalent $J_z^*$ or $j_z^*(p_x)$ &
+        NIVS & This work & Swapping & This work \\
+        \hline
+        0.116 & 0.16  & 7.30(0.10) & 6.25(0.23) & 3.57(0.06) & 3.53(0.16)\\
+        0.232 & 0.09  & 6.95(0.09) & 8.02(0.56) & 3.64(0.05) & 3.43(0.17)\\
+        0.463 & 0.047 & 7.19(0.07) & 6.48(0.15) & 3.52(0.16) & 3.51(0.08)\\
+        0.926 & 0.024 & 7.19(0.28) & 7.02(0.13) & 3.72(0.05) & 3.79(0.11)\\
+        1.158 & 0.019 & 7.98(0.33) & 7.37(0.23) & 3.42(0.06) & 3.53(0.06)\\
+        \hline\hline
+      \end{tabular}
+      \label{LJ}
+    \end{center}
+  \end{minipage}
+\end{table*}
+
+\subsubsection{Thermal conductivity}
+Our thermal conductivity calculations with this method yields
+comparable results to the previous NIVS algorithm. This indicates that
+the thermal gradients rendered using this method are also close to
+previous RNEMD methods. Simulations with moderately higher thermal
+fluxes tend to yield more reliable thermal gradients and thus avoid
+large errors, while overly high thermal fluxes could introduce side
+effects such as non-linear temperature gradient response or
+inadvertent phase transitions.
 
-\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}
+Since the scaling operation is isotropic in this method, one does not
+need extra care to ensure temperature isotropy between the $x$, $y$
+and $z$ axes, while thermal anisotropy might happen if the criteria
+function for choosing scaling coefficients does not perform as
+expected. Furthermore, this method avoids inadvertent concomitant
+momentum flux when only thermal flux is imposed, which could not be
+achieved with swapping or NIVS approaches. The thermal energy exchange
+in swapping ($\vec{p}_i$ in slab ``c'' with $\vec{p}_j$ in slab ``h'')
+or NIVS (total slab momentum conponemts $P^\alpha$ scaled to $\alpha
+P^\alpha$) would not obtain this result unless thermal flux vanishes
+(i.e. $\vec{p}_i=\vec{p}_j$ or $\alpha=1$ which are trivial to apply a
+thermal flux). In this sense, this method contributes to having
+minimal perturbation to a simulation while imposing thermal flux.
 
-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.
+\subsubsection{Shear viscosity}
+Table \ref{LJ} also compares our shear viscosity results with momentum
+swapping approach. Our calculations show that our method predicted
+similar values for shear viscosities to the momentum swapping
+approach, as well as the velocity gradient profiles. Moderately larger
+momentum fluxes are helpful to reduce the errors of measured velocity
+gradients and thus the final result. However, it is pointed out that
+the momentum swapping approach tends to produce nonthermal velocity
+distributions.\cite{Maginn:2010}
 
-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.
+To examine that temperature isotropy holds in simulations using our
+method, we measured the three one-dimensional temperatures in each of
+the slabs (Figure \ref{tempXyz}). Note that the $x$-dimensional
+temperatures were calculated after subtracting the effects from bulk
+velocities of the slabs. The one-dimensional temperature profiles
+showed no observable difference between the three dimensions. This
+ensures that isotropic scaling automatically preserves temperature
+isotropy and that our method is useful in shear viscosity
+computations.
 
-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.
+\begin{figure}
+\includegraphics[width=\linewidth]{tempXyz}
+\caption{Unlike the previous NIVS algorithm, the new method does not
+  produce a thermal anisotropy. No temperature difference between
+  different dimensions were observed beyond the magnitude of the error
+  bars. Note that the two ``hotter'' regions are caused by the shear
+  stress, as reported by Tenney and Maginn\cite{Maginn:2010}, but not
+  an effect that only observed in our methods.}
+\label{tempXyz}
+\end{figure}
 
-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.
+Furthermore, the velocity distribution profiles are tested by imposing
+a large shear stress into the simulations. Figure \ref{vDist}
+demonstrates how our method is able to maintain thermal velocity
+distributions against the momentum swapping approach even under large
+imposed fluxes. Previous swapping methods tend to deplete particles of
+positive velocities in the negative velocity slab (``c'') and vice
+versa in slab ``h'', where the distributions leave a notch. This
+problematic profiles become significant when the imposed-flux becomes
+larger and diffusions from neighboring slabs could not offset the
+depletion. Simutaneously, abnormal peaks appear corresponding to
+excessive velocity swapped from the other slab. This nonthermal
+distributions limit applications of the swapping approach in shear
+stress simulations. Our method avoids the above problematic
+distributions by altering the means of applying momentum
+fluxes. Comparatively, velocity distributions recorded from
+simulations with our method is so close to the ideal thermal
+prediction that no observable difference is shown in Figure
+\ref{vDist}. Conclusively, our method avoids problems happened in
+previous RNEMD methods and provides a useful means for shear viscosity
+computations.
 
-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.
+\begin{figure}
+\includegraphics[width=\linewidth]{velDist}
+\caption{Velocity distributions that develop under the swapping and
+  our methods at high flux. These distributions were obtained from
+  Lennard-Jones simulations with $j_z^*(p_x)\sim 0.4$ (equivalent to a
+  swapping interval of 20 time steps). This is a relatively large flux
+  to demonstrate the nonthermal distributions that develop under the
+  swapping method. Distributions produced by our method are very close
+  to the ideal thermal situations.}
+\label{vDist}
+\end{figure}
 
-\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.
+\subsection{Bulk SPC/E water}
+Since our method was in good performance of thermal conductivity and
+shear viscosity computations for simple Lennard-Jones fluid, we extend
+our applications of these simulations to complex fluid like SPC/E
+water model. A simulation cell with 1000 molecules was set up in the
+same manner as in \cite{kuang:164101}. For thermal conductivity
+simulations, measurements were taken to compare with previous RNEMD
+methods; for shear viscosity computations, simulations were run under
+a series of temperatures (with corresponding pressure relaxation using
+the isobaric-isothermal ensemble[CITE NIVS REF 32]), and results were
+compared to available data from Equilibrium MD methods[CITATIONS].
 
+\subsubsection{Thermal conductivity}
+Table \ref{spceThermal} summarizes our thermal conductivity
+computations under different temperatures and thermal gradients, in
+comparison to the previous NIVS results\cite{kuang:164101} and
+experimental measurements\cite{WagnerKruse}. Note that no appreciable
+drift of total system energy or temperature was observed when our
+method is applied, which indicates that our algorithm conserves total
+energy even for systems involving electrostatic interactions.
+
+Measurements using our method established similar temperature
+gradients to the previous NIVS method. Our simulation results are in
+good agreement with those from previous simulations. And both methods
+yield values in reasonable agreement with experimental
+values. Simulations using moderately higher thermal gradient or those
+with longer gradient axis ($z$) for measurement seem to have better
+accuracy, from our results.
+
 \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.}
+      \caption{Thermal conductivity of SPC/E water under various
+        imposed thermal gradients. Uncertainties are indicated in
+        parentheses.}
       
-      \begin{tabular}{llccc}
+      \begin{tabular}{ccccc}
         \hline\hline
-        Butanethiol model & Solvent & $G$ & $G^\prime$ \\
-        (or bare surface) & model & \multicolumn{2}{c}{(MW/m$^2$/K)} \\
+        $\langle T\rangle$ & $\langle dT/dz\rangle$ & \multicolumn{3}{c}
+        {$\lambda (\mathrm{W m}^{-1} \mathrm{K}^{-1})$} \\
+        (K) & (K/\AA) & This work & Previous NIVS\cite{kuang:164101} &
+        Experiment\cite{WagnerKruse} \\
         \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) \\
+        300 & 0.8 & 0.815(0.027)     & 0.770(0.008) & 0.61 \\
+        318 & 0.8 & 0.801(0.024)     & 0.750(0.032) & 0.64 \\
+            & 1.6 & 0.766(0.007)     & 0.778(0.019) &      \\
+            & 0.8 & 0.786(0.009)\footnote{Simulation with $L_z$
+                     twice as long.} &              &      \\
         \hline\hline
       \end{tabular}
-      \label{modelTest}
+      \label{spceThermal}
     \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.
+\subsubsection{Shear viscosity}
+The improvement our method achieves for shear viscosity computations
+enables us to apply it on SPC/E water models. The series of
+temperatures under which our shear viscosity calculations were carried
+out covers the liquid range under normal pressure. Our simulations
+predict a similar trend of $\eta$ vs. $T$ to EMD results we refer to
+(Table \ref{spceShear}). Considering subtlties such as temperature or
+pressure/density errors in these two series of measurements, our
+results show no significant difference from those with EMD
+methods. Since each value reported using our method takes only one
+single trajectory of simulation, instead of average from many
+trajectories when using EMD, our method provides an effective means
+for shear viscosity computations.
 
-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.
+\begin{table*}
+  \begin{minipage}{\linewidth}
+    \begin{center}
+      
+      \caption{Computed shear viscosity of SPC/E water under different
+        temperatures. Results are compared to those obtained with EMD
+        method[CITATION]. Uncertainties are indicated in parentheses.}
+      
+      \begin{tabular}{cccc}
+        \hline\hline
+        $T$ & $\partial v_x/\partial z$ & \multicolumn{2}{c}
+        {$\eta (\mathrm{mPa}\cdot\mathrm{s})$} \\
+        (K) & (10$^{10}$s$^{-1}$) & This work & Previous simulations[CITATION]\\
+        \hline
+        273 &  & 1.218(0.004) &  \\
+            &  & 1.140(0.012) &  \\
+        303 &  & 0.646(0.008) &  \\
+        318 &  & 0.536(0.007) &  \\
+            &  & 0.510(0.007) &  \\
+            &  &  &  \\
+        333 &  & 0.428(0.002) &  \\
+        363 &  & 0.279(0.014) &  \\
+            &  & 0.306(0.001) &  \\
+        \hline\hline
+      \end{tabular}
+      \label{spceShear}
+    \end{center}
+  \end{minipage}
+\end{table*}
 
-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.
+[MAY COMBINE JZPX AND JZKE TO RUN ONE JOB BUT GET ETA=F(T)]
+[PUT RESULTS AND FIGURE HERE IF IT WORKS]
+\subsection{Interfacial frictions and slip lengths}
+An attractive aspect of our method is the ability to apply momentum
+and/or thermal flux in nonhomogeneous systems, where molecules of
+different identities (or phases) are segregated in different
+regions. We have previously studied the interfacial thermal transport
+of a series of metal gold-liquid
+surfaces\cite{kuang:164101,kuang:AuThl}, and attemptions have been
+made to investigate the relationship between this phenomenon and the
+interfacial frictions.
+
+Table \ref{etaKappaDelta} includes these computations and previous
+calculations of corresponding interfacial thermal conductance. For
+bare Au(111) surfaces, slip boundary conditions were observed for both
+organic and aqueous liquid phases, corresponding to previously
+computed low interfacial thermal conductance. Instead, the butanethiol
+covered Au(111) surface appeared to be sticky to the organic liquid
+molecules in our simulations. We have reported conductance enhancement
+effect for this surface capping agent,\cite{kuang:AuThl} and these
+observations have a qualitative agreement with the thermal conductance
+results. This agreement also supports discussions on the relationship
+between surface wetting and slip effect and thermal conductance of the
+interface.[CITE BARRAT, GARDE]
 
-\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 $G$.
+\begin{table*}
+  \begin{minipage}{\linewidth}
+    \begin{center}
+      
+      \caption{Computed interfacial friction coefficient values for
+        interfaces with various components for liquid and solid
+        phase. Error estimates are indicated in parentheses.}
+      
+      \begin{tabular}{llcccccc}
+        \hline\hline
+        Solid & Liquid & $T$ & $j_z(p_x)$ & $\eta_{liquid}$ & $\kappa$
+        & $\delta$ & $G$\footnote{References \cite{kuang:AuThl} and
+          \cite{kuang:164101}.} \\
+        surface & molecules & K & MPa & mPa$\cdot$s & Pa$\cdot$s/m & nm
+        & MW/m$^2$/K \\
+        \hline
+        Au(111) & hexane & 200 & 1.08 & 0.20() & 5.3$\times$10$^4$() &
+        3.7 & 46.5 \\
+                &        &     & 2.15 & 0.14() & 5.3$\times$10$^4$() &
+        2.7 &      \\
+        Au-SC$_4$H$_9$ & hexane & 200 & 2.16 & 0.29() & $\infty$ & 0 &
+        131 \\
+                       &        &     & 5.39 & 0.32() & $\infty$ & 0 &
+            \\
+        \hline
+        Au(111) & toluene & 200 & 1.08 & 0.72() & 1.?$\times$10$^5$() &
+        4.6 & 70.1 \\
+                &         &     & 2.16 & 0.54() & 1.?$\times$10$^5$() &
+        4.9 &      \\
+        Au-SC$_4$H$_9$ & toluene & 200 & 5.39 & 0.98() & $\infty$ & 0
+        & 187 \\
+                       &         &     & 10.8 & 0.99() & $\infty$ & 0
+        &     \\
+        \hline
+        Au(111) & water & 300 & 1.08 & 0.40() & 1.9$\times$10$^4$() &
+        20.7 & 1.65 \\
+                &       &     & 2.16 & 0.79() & 1.9$\times$10$^4$() &
+        41.9 &      \\
+        \hline
+        ice(basal) & water & 225 & 19.4 & 15.8() & $\infty$ & 0 & \\
+        \hline\hline
+      \end{tabular}
+      \label{etaKappaDelta}
+    \end{center}
+  \end{minipage}
+\end{table*}
 
-\subsection{Effects due to methodology and simulation parameters}
+An interesting effect alongside the surface friction change is
+observed on the shear viscosity of liquids in the regions close to the
+solid surface. Note that $\eta$ measured near a ``slip'' surface tends
+to be smaller than that near a ``stick'' surface. This suggests that
+an interface could affect the dynamic properties on its neighbor
+regions. It is known that diffusions of solid particles in liquid
+phase is affected by their surface conditions (stick or slip
+boundary).[CITE SCHMIDT AND SKINNER] Our observations could provide
+support to this phenomenon.
 
-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.
+In addition to these previously studied interfaces, we attempt to
+construct ice-water interfaces and the basal plane of ice lattice was
+first studied. In contrast to the Au(111)/water interface, where the
+friction coefficient is relatively small and large slip effect
+presents, the ice/liquid water interface demonstrates strong
+interactions and appears to be sticky. The supercooled liquid phase is
+an order of magnitude viscous than measurements in previous
+section. It would be of interst to investigate the effect of different
+ice lattice planes (such as prism surface) on interfacial friction and
+corresponding liquid viscosity.
 
-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 demonstrate the validity of our method in RNEMD
+computations of thermal conductivity and shear viscosity in atomic and
+molecular liquids. Our method maintains thermal velocity distributions
+and avoids thermal anisotropy in previous NIVS shear stress
+simulations, as well as retains attractive features of previous RNEMD
+methods. There is no {\it a priori} restrictions to the method to be
+applied in various ensembles, so prospective applications to
+extended-system methods are possible.
 
-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.
+Furthermore, using this method, investigations can be carried out to
+characterize interfacial interactions. Our method is capable of
+effectively imposing both thermal and momentum flux accross an
+interface and thus facilitates studies that relates dynamic property
+measurements to the chemical details of an interface.
 
-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.
+Another attractive feature of our method is the ability of
+simultaneously imposing thermal and momentum flux in a
+system. potential researches that might be benefit include complex
+systems that involve thermal and momentum gradients. For example, the
+Soret effects under a velocity gradient would be of interest to
+purification and separation researches.
 
-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
@@ -875,4 +758,3 @@ Dame. 
 
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