--- interfacial/interfacial.tex	2011/07/14 19:49:12	3739
+++ interfacial/interfacial.tex	2011/07/21 00:04:26	3745
@@ -23,8 +23,8 @@
 \setlength{\belowcaptionskip}{30 pt}
 
 %\renewcommand\citemid{\ } % no comma in optional reference note
-\bibpunct{[}{]}{,}{s}{}{;}
-\bibliographystyle{aip}
+\bibpunct{[}{]}{,}{n}{}{;}
+\bibliographystyle{achemso}
 
 \begin{document}
 
@@ -193,13 +193,27 @@ gradient max($\Delta T$), which occurs at the Gibbs de
 One way is to assume the temperature is discrete on the two sides of
 the interface. $G$ can be calculated using the applied thermal flux
 $J$ and the maximum temperature difference measured along the thermal
-gradient max($\Delta T$), which occurs at the Gibbs deviding surface,
-as:
+gradient max($\Delta T$), which occurs at the Gibbs deviding surface
+(Figure \ref{demoPic}):
 \begin{equation}
 G=\frac{J}{\Delta T}
 \label{discreteG}
 \end{equation}
 
+\begin{figure}
+\includegraphics[width=\linewidth]{method}
+\caption{Interfacial conductance can be calculated by applying an
+  (unphysical) kinetic energy flux between two slabs, one located
+  within the metal and another on the edge of the periodic box.  The
+  system responds by forming a thermal response or a gradient.  In
+  bulk liquids, this gradient typically has a single slope, but in
+  interfacial systems, there are distinct thermal conductivity
+  domains.  The interfacial conductance, $G$ is found by measuring the
+  temperature gap at the Gibbs dividing surface, or by using second
+  derivatives of the thermal profile.}
+\label{demoPic}
+\end{figure}
+
 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
@@ -222,16 +236,9 @@ our simulation cells. Both with and withour capping ag
 
 To compare the above definitions ($G$ and $G^\prime$), we have modeled
 a metal slab with its (111) surfaces perpendicular to the $z$-axis of
-our simulation cells. Both with and withour capping agents on the
+our simulation cells. Both with and without capping agents on the
 surfaces, the metal slab is solvated with simple 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 the simulation cell described above, we are able to equilibrate
 the system and impose an unphysical thermal flux between the liquid
@@ -244,8 +251,10 @@ 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.}
+\caption{A sample of Au-butanethiol/hexane interfacial system and the
+  temperature profile after a kinetic energy flux is imposed to
+  it. The 1st and 2nd derivatives of the temperature profile can be
+  obtained with finite difference approximation (lower panel).}
 \label{gradT}
 \end{figure}
 
@@ -287,7 +296,7 @@ corresponding spacing is usually $35 \sim 60$\AA.
 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.
+corresponding spacing is usually $35 \sim 75$\AA.
 
 The initial configurations generated by Packmol are further
 equilibrated with the $x$ and $y$ dimensions fixed, only allowing
@@ -318,49 +327,68 @@ Our simulations include various components. Therefore,
 \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.
+Our simulations include various components. Figure \ref{demoMol}
+demonstrates the sites defined for both United-Atom and All-Atom
+models of the organic solvent and capping agent molecules in our
+simulations. Force field parameter descriptions are needed for
+interactions both between the same type of particles and between
+particles of different species.
 
+\begin{figure}
+\includegraphics[width=\linewidth]{structures}
+\caption{Structures of the capping agent and solvents utilized in
+  these simulations. The chemically-distinct sites (a-e) are expanded
+  in terms of constituent atoms for both United Atom (UA) and All Atom
+  (AA) force fields.  Most parameters are from
+  Refs. \protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} (UA) and
+  \protect\cite{OPLSAA} (AA).  Cross-interactions with the Au atoms are given
+  in Table \ref{MnM}.}
+\label{demoMol}
+\end{figure}
+
 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{demoMol} demonstrates how we name our pseudo-atoms of the
-organic solvent molecules in our simulations.
-
-\begin{figure}
-\includegraphics[width=\linewidth]{demoMol}
-\caption{Denomination of atoms or pseudo-atoms in our simulations: a)
-  UA-hexane; b) AA-hexane; c) UA-toluene; d) AA-toluene.}
-\label{demoMol}
-\end{figure}
-
 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]
+for our UA solvent molecules. In these models, sites are located at
+the carbon centers for alkyl groups. Bonding interactions, including
+bond stretches and bends and torsions, were used for intra-molecular
+sites not separated by more than 3 bonds. Otherwise, for non-bonded
+interactions, Lennard-Jones potentials are used. [MORE CITATION?]
 
+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.
+
+For UA-toluene model, the non-bonded potentials between
+inter-molecular sites have a similar Lennard-Jones formulation. For
+intra-molecular interactions, considering the stiffness of the benzene
+ring, rigid body constraints are applied for further computational
+efficiency. All bonds in the benzene ring and between the ring and the
+methyl group remain rigid during the progress of simulations.
+
 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]
+force field is used. Additional explicit hydrogen sites were
+included. Besides bonding and non-bonded site-site interactions,
+partial charges and the electrostatic interactions were added to each
+CT and HC site. For toluene, the United Force Field developed by
+Rapp\'{e} {\it et al.}\cite{doi:10.1021/ja00051a040} is
+adopted. Without the rigid body constraints, bonding interactions were
+included. For the aromatic ring, improper torsions (inversions) were
+added as an extra potential for maintaining the planar shape.
+[MORE CITATION?]
 
 The capping agent in our simulations, the butanethiol molecules can
 either use UA or AA model. The TraPPE-UA force fields includes
@@ -377,8 +405,8 @@ Lorentz-Berthelot Mixing Rule:
 interactions between capping agent and solvent can be derived using
 Lorentz-Berthelot Mixing Rule:
 \begin{eqnarray}
-\sigma_{IJ} & = & \frac{1}{2} \left(\sigma_{II} + \sigma_{JJ}\right) \\
-\epsilon_{IJ} & = & \sqrt{\epsilon_{II}\epsilon_{JJ}}
+\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\
+\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}}
 \end{eqnarray}
 
 To describe the interactions between metal Au and non-metal capping
@@ -395,40 +423,48 @@ interactions between Au and aromatic C/H atoms in tolu
 
 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]
+interactions between Au and aromatic C/H atoms in toluene. A set of
+pseudo Lennard-Jones parameters were provided for Au in their force
+fields. By using the Mixing Rule, this can be used to derive pair-wise
+potentials for non-bonded interactions between Au and non-metal sites.
 
 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''
+particles, such as All-Atom normal alkanes in our simulations are not
+yet well-established. For these interactions, we attempt to derive
+their parameters using the Mixing Rule. To do this, Au pseudo
+Lennard-Jones parameters for ``Metal-non-Metal'' (MnM) interactions
+were 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{Non-bonded interaction paramters for non-metal
-        particles and metal-non-metal interactions in our
-        simulations.}
-      
-      \begin{tabular}{cccccc}
+      \caption{Non-bonded interaction parameters (including cross
+        interactions with Au atoms) for both force fields used in this
+        work.}      
+      \begin{tabular}{lllllll}
         \hline\hline
-        Non-metal atom $I$ & $\sigma_{II}$ & $\epsilon_{II}$ & $q_I$ &
-        $\sigma_{AuI}$ & $\epsilon_{AuI}$ \\
-        (or pseudo-atom) & \AA & kcal/mol & & \AA & kcal/mol \\
+        & Site  & $\sigma_{ii}$ & $\epsilon_{ii}$ & $q_i$ &
+        $\sigma_{Au-i}$ & $\epsilon_{Au-i}$ \\
+        & & (\AA) & (kcal/mol) & ($e$) & (\AA) & (kcal/mol) \\
         \hline
-        CH3  & 3.75  & 0.1947  & -      & 3.54   & 0.2146  \\
-        CH2  & 3.95  & 0.0914  & -      & 3.54   & 0.1749  \\
-        CHar & 3.695 & 0.1003  & -      & 3.4625 & 0.1680  \\
-        CRar & 3.88  & 0.04173 & -      & 3.555  & 0.1604  \\
-        S    & 4.45  & 0.25    & -      & 2.40   & 8.465   \\
-        CT3  & 3.50  & 0.066   & -0.18  & 3.365  & 0.1373  \\
-        CT2  & 3.50  & 0.066   & -0.12  & 3.365  & 0.1373  \\
-        CTT  & 3.50  & 0.066   & -0.065 & 3.365  & 0.1373  \\
-        HC   & 2.50  & 0.030   &  0.06  & 2.865  & 0.09256 \\
-        CA   & 3.55  & 0.070   & -0.115 & 3.173  & 0.0640  \\
-        HA   & 2.42  & 0.030   &  0.115 & 2.746  & 0.0414  \\
+        United Atom (UA)
+        &CH3  & 3.75  & 0.1947  & -      & 3.54   & 0.2146  \\
+        &CH2  & 3.95  & 0.0914  & -      & 3.54   & 0.1749  \\
+        &CHar & 3.695 & 0.1003  & -      & 3.4625 & 0.1680  \\
+        &CRar & 3.88  & 0.04173 & -      & 3.555  & 0.1604  \\
+        \hline
+        All Atom (AA) 
+        &CT3  & 3.50  & 0.066   & -0.18  & 3.365  & 0.1373  \\
+        &CT2  & 3.50  & 0.066   & -0.12  & 3.365  & 0.1373  \\
+        &CTT  & 3.50  & 0.066   & -0.065 & 3.365  & 0.1373  \\
+        &HC   & 2.50  & 0.030   &  0.06  & 2.865  & 0.09256 \\
+        &CA   & 3.55  & 0.070   & -0.115 & 3.173  & 0.0640  \\
+        &HA   & 2.42  & 0.030   &  0.115 & 2.746  & 0.0414  \\
+        \hline
+        Both UA and AA 
+        & S   & 4.45  & 0.25    & -      & 2.40   & 8.465   \\
         \hline\hline
       \end{tabular}
       \label{MnM}
@@ -449,14 +485,15 @@ during equilibrating the liquid phase. Due to the stif
 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. 
+during equilibrating the liquid phase. Due to the stiffness of the
+crystalline Au structure, $L_x$ 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 of $L_x$ and $L_y$ (which is significantly smaller),
+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
@@ -483,20 +520,21 @@ $J_z$ is listed in Table \ref{AuThiolHexaneUA} and [RE
 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.
+$J_z$ is listed in Table \ref{AuThiolHexaneUA} and
+\ref{AuThiolToluene}. 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 100\% covered Au-butanethiol/hexane
         interfaces with UA model and different hexane molecule numbers
-        at different temperatures using a range of energy fluxes.}
+        at different temperatures using a range of energy
+        fluxes. Error estimates indicated in parenthesis.}
       
       \begin{tabular}{ccccccc}
         \hline\hline
@@ -505,24 +543,24 @@ couple $J_z$'s and do not need to test a large series 
         (K) & & $L_x$ \& $L_y$? & (g/cm$^3$) & (GW/m$^2$) &
         \multicolumn{2}{c}{(MW/m$^2$/K)} \\
         \hline
-        200 & 266 & No  & 0.672 & -0.96 & 102()  & 80.0() \\
-            & 200 & Yes & 0.694 &  1.92 & 129()  & 87.3() \\
-            &     & Yes & 0.672 &  1.93 & 131()  & 77.5() \\
-            &     & No  & 0.688 &  0.96 & 125()  & 90.2() \\
-            &     &     &       &  1.91 & 139()  & 101()  \\
-            &     &     &       &  2.83 & 141()  & 89.9() \\
-            & 166 & Yes & 0.679 &  0.97 & 115()  & 69.3() \\
-            &     &     &       &  1.94 & 125()  & 87.1() \\
-            &     & No  & 0.681 &  0.97 & 141()  & 77.7() \\
-            &     &     &       &  1.92 & 138()  & 98.9() \\
+        200 & 266 & No  & 0.672 & -0.96 & 102(3)    & 80.0(0.8) \\
+            & 200 & Yes & 0.694 &  1.92 & 129(11)   & 87.3(0.3) \\
+            &     & Yes & 0.672 &  1.93 & 131(16)   & 78(13)    \\
+            &     & No  & 0.688 &  0.96 & 125(16)   & 90.2(15)  \\
+            &     &     &       &  1.91 & 139(10)   & 101(10)   \\
+            &     &     &       &  2.83 & 141(6)    & 89.9(9.8) \\
+            & 166 & Yes & 0.679 &  0.97 & 115(19)   & 69(18)    \\
+            &     &     &       &  1.94 & 125(9)    & 87.1(0.2) \\
+            &     & No  & 0.681 &  0.97 & 141(30)   & 78(22)    \\
+            &     &     &       &  1.92 & 138(4)    & 98.9(9.5) \\
         \hline
-        250 & 200 & No  & 0.560 &  0.96 & 74.8() & 61.8() \\
-            &     &     &       & -0.95 & 49.4() & 45.7() \\
-            & 166 & Yes & 0.570 &  0.98 & 79.0() & 62.9() \\
-            &     & No  & 0.569 &  0.97 & 80.3() & 67.1() \\
-            &     &     &       &  1.44 & 76.2() & 64.8() \\
-            &     &     &       & -0.95 & 56.4() & 54.4() \\
-            &     &     &       & -1.85 & 47.8() & 53.5() \\
+        250 & 200 & No  & 0.560 &  0.96 & 75(10)    & 61.8(7.3) \\
+            &     &     &       & -0.95 & 49.4(0.3) & 45.7(2.1) \\
+            & 166 & Yes & 0.570 &  0.98 & 79.0(3.5) & 62.9(3.0) \\
+            &     & No  & 0.569 &  0.97 & 80.3(0.6) & 67(11)    \\
+            &     &     &       &  1.44 & 76.2(5.0) & 64.8(3.8) \\
+            &     &     &       & -0.95 & 56.4(2.5) & 54.4(1.1) \\
+            &     &     &       & -1.85 & 47.8(1.1) & 53.5(1.5) \\
         \hline\hline
       \end{tabular}
       \label{AuThiolHexaneUA}
@@ -538,7 +576,8 @@ butanethiol as well.[MAY NEED FIGURE] And this reduced
 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
+butanethiol as well.[MAY NEED SLAB DENSITY FIGURE] 
+And this reduced contact would
 probably be accountable for a lower interfacial thermal conductance,
 as shown in Table \ref{AuThiolHexaneUA}. 
 
@@ -553,26 +592,25 @@ in that higher degree of contact could yield increased
 important role in the thermal transport process across the interface
 in that higher degree of contact could yield increased conductance.
 
-[ADD ERROR ESTIMATE TO TABLE]
 \begin{table*}
   \begin{minipage}{\linewidth}
     \begin{center}
       \caption{Computed interfacial thermal conductivity ($G$ and
         $G^\prime$) values for a 90\% coverage Au-butanethiol/toluene
         interface at different temperatures using a range of energy
-        fluxes.}
+        fluxes. Error estimates indicated in parenthesis.}
       
       \begin{tabular}{ccccc}
         \hline\hline
         $\langle T\rangle$ & $\rho_{toluene}$ & $J_z$ & $G$ & $G^\prime$ \\
         (K) & (g/cm$^3$) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\
         \hline
-        200 & 0.933 & -1.86 & 180() & 135() \\
-            &       &  2.15 & 204() & 113() \\
-            &       & -3.93 & 175() & 114() \\
+        200 & 0.933 &  2.15 & 204(12) & 113(12) \\
+            &       & -1.86 & 180(3)  & 135(21) \\
+            &       & -3.93 & 176(5)  & 113(12) \\
         \hline
-        300 & 0.855 & -1.91 & 143() & 125() \\
-            &       & -4.19 & 134() & 113() \\
+        300 & 0.855 & -1.91 & 143(5)  & 125(2)  \\
+            &       & -4.19 & 135(9)  & 113(12) \\
         \hline\hline
       \end{tabular}
       \label{AuThiolToluene}
@@ -604,11 +642,11 @@ above. Our Au-butanethiol/toluene system did not see t
 
 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 $\langle T\rangle\sim$300K. The Au(111) surfaces have a 90\%
-coverage of butanethiols and have empty three-fold sites. These empty
-sites could help prevent surface reconstruction in that they provide
-other means of capping agent relaxation. It is observed that
+above. Our Au-butanethiol/toluene system had the Au(111) surfaces 90\%
+covered by butanethiols, but did not see this above phenomena even at
+$\langle T\rangle\sim$300K. The empty three-fold sites not occupied by
+capping agents 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
@@ -619,26 +657,26 @@ thermal conductivity are measured with our NIVS algori
 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. To study the isotope effect in
-interfacial thermal conductance, deuterated UA-hexane is included as
-well.
+thermal conductivity are measured with our NIVS algorithm. Figure
+\ref{coverage} demonstrates the trend of conductance change with
+respect to different coverages of butanethiol. To study the isotope
+effect in interfacial thermal conductance, deuterated UA-hexane is
+included as well.
 
 It turned out that with partial covered butanethiol on the Au(111)
-surface, the derivative definition for $G$ (Eq. \ref{derivativeG}) has
-difficulty to apply, due to the difficulty in locating the maximum of
-change of $\lambda$. Instead, the discrete definition
-(Eq. \ref{discreteG}) is easier to apply, as max($\Delta T$) can still
-be well-defined. Therefore, $G$'s (not $G^\prime$) are used for this
-section. 
+surface, the derivative definition for $G^\prime$
+(Eq. \ref{derivativeG}) was difficult to apply, due to the difficulty
+in locating the maximum of change of $\lambda$. Instead, the discrete
+definition (Eq. \ref{discreteG}) is easier to apply, as the Gibbs
+deviding surface can still be well-defined. Therefore, $G$ (not
+$G^\prime$) was used for this section.
 
-From Table \ref{tlnUhxnUhxnD}, one can see the significance of the
+From Figure \ref{coverage}, one can see the significance of the
 presence of capping agents. Even when a fraction of the Au(111)
 surface sites are covered with butanethiols, the conductivity would
 see an enhancement by at least a factor of 3. This indicates the
 important role cappping agent is playing for thermal transport
-phenomena on metal/organic solvent surfaces.
+phenomena on metal / organic solvent surfaces.
 
 Interestingly, as one could observe from our results, the maximum
 conductance enhancement (largest $G$) happens while the surfaces are
@@ -657,7 +695,8 @@ case, $G$ decrease could not be offset but instead acc
 would not offset this effect. Eventually, when butanethiol coverage
 continues to decrease, solvent-capping agent contact actually
 decreases with the disappearing of butanethiol molecules. In this
-case, $G$ decrease could not be offset but instead accelerated.
+case, $G$ decrease could not be offset but instead accelerated. [NEED
+SNAPSHOT SHOWING THE PHENOMENA]
 
 A comparison of the results obtained from differenet organic solvents
 can also provide useful information of the interfacial thermal
@@ -667,7 +706,7 @@ difference for the results of $G$. [MAY NEED FIGURE]
 studies, even though eliminating C-H vibration samplings, still have
 C-C vibrational frequencies different from each other. However, these
 differences in the infrared range do not seem to produce an observable
-difference for the results of $G$. [MAY NEED FIGURE]
+difference for the results of $G$. [MAY NEED SPECTRA FIGURE]
 
 Furthermore, results for rigid body toluene solvent, as well as other
 UA-hexane solvents, are reasonable within the general experimental
@@ -676,10 +715,10 @@ trend with those for Au-butanethiol/hexane in that $G$
 such as Au-thiol/organic solvent.
 
 However, results for Au-butanethiol/toluene do not show an identical
-trend with those for Au-butanethiol/hexane in that $G$'s remain at
+trend with those for Au-butanethiol/hexane in that $G$ remains at
 approximately the same magnitue when butanethiol coverage differs from
 25\% to 75\%. This might be rooted in the molecule shape difference
-for plane-like toluene and chain-like {\it n}-hexane. Due to this
+for planar toluene and chain-like {\it n}-hexane. Due to this
 difference, toluene molecules have more difficulty in occupying
 relatively small gaps among capping agents when their coverage is not
 too low. Therefore, the solvent-capping agent contact may keep
@@ -688,7 +727,6 @@ can see a plateau of $G$ vs. butanethiol coverage in o
 its effect to the process of interfacial thermal transport. Thus, one
 can see a plateau of $G$ vs. butanethiol coverage in our results.
 
-[NEED ERROR ESTIMATE]
 \begin{figure}
 \includegraphics[width=\linewidth]{coverage}
 \caption{Comparison of interfacial thermal conductivity ($G$) values
@@ -709,7 +747,6 @@ these studies.
 the previous section. Table \ref{modelTest} summarizes the results of
 these studies.
 
-[MORE DATA; ERROR ESTIMATE]
 \begin{table*}
   \begin{minipage}{\linewidth}
     \begin{center}
@@ -719,35 +756,35 @@ these studies.
         solvent and capping agent (or without capping agent) at
         $\langle T\rangle\sim$200K. (D stands for deuterated solvent
         or capping agent molecules; ``Avg.'' denotes results that are
-        averages of several simulations.)}
+        averages of simulations under different $J_z$'s. Error
+        estimates indicated in parenthesis.)}
       
-      \begin{tabular}{ccccc}
+      \begin{tabular}{llccc}
         \hline\hline
         Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\
         (or bare surface) & model & (GW/m$^2$) &
         \multicolumn{2}{c}{(MW/m$^2$/K)} \\
         \hline
-        UA    & UA hexane    & Avg. & 131()  & 86.5() \\
-              & UA hexane(D) & 1.95 & 153()  & 136()  \\
-              & AA hexane    & 1.94 & 135()  & 129()  \\
-              &              & 2.86 & 126()  & 115()  \\
-              & UA toluene   & 1.96 & 187()  & 151()  \\
-              & AA toluene   & 1.89 & 200()  & 149()  \\
+        UA    & UA hexane    & Avg. & 131(9)    & 87(10)    \\
+              & UA hexane(D) & 1.95 & 153(5)    & 136(13)   \\
+              & AA hexane    & Avg. & 131(6)    & 122(10)   \\
+              & UA toluene   & 1.96 & 187(16)   & 151(11)   \\
+              & AA toluene   & 1.89 & 200(36)   & 149(53)   \\
         \hline
-        AA    & UA hexane    & 1.94 & 116()  & 129()  \\
-              & AA hexane    & Avg. & 442()  & 356()  \\
-              & AA hexane(D) & 1.93 & 222()  & 234()  \\
-              & UA toluene   & 1.98 & 125()  & 96.5() \\
-              & AA toluene   & 3.79 & 487()  & 290()  \\
+        AA    & UA hexane    & 1.94 & 116(9)    & 129(8)    \\
+              & AA hexane    & Avg. & 442(14)   & 356(31)   \\
+              & AA hexane(D) & 1.93 & 222(12)   & 234(54)   \\
+              & UA toluene   & 1.98 & 125(25)   & 97(60)    \\
+              & AA toluene   & 3.79 & 487(56)   & 290(42)   \\
         \hline
-        AA(D) & UA hexane    & 1.94 & 158()  & 172()  \\
-              & AA hexane    & 1.92 & 243()  & 191()  \\
-              & AA toluene   & 1.93 & 364()  & 322()  \\
+        AA(D) & UA hexane    & 1.94 & 158(25)   & 172(4)    \\
+              & AA hexane    & 1.92 & 243(29)   & 191(11)   \\
+              & AA toluene   & 1.93 & 364(36)   & 322(67)   \\
         \hline
-        bare  & UA hexane    & Avg. & 46.5() & 49.4() \\
-              & UA hexane(D) & 0.98 & 43.9() & 43.0() \\
-              & AA hexane    & 0.96 & 31.0() & 29.4() \\
-              & UA toluene   & 1.99 & 70.1() & 65.8() \\
+        bare  & UA hexane    & Avg. & 46.5(3.2) & 49.4(4.5) \\
+              & UA hexane(D) & 0.98 & 43.9(4.6) & 43.0(2.0) \\
+              & AA hexane    & 0.96 & 31.0(1.4) & 29.4(1.3) \\
+              & UA toluene   & 1.99 & 70.1(1.3) & 65.8(0.5) \\
         \hline\hline
       \end{tabular}
       \label{modelTest}
@@ -770,13 +807,14 @@ the upper bond of experimental value range. This is pr
 interfaces, using AA model for both butanethiol and hexane yields
 substantially higher conductivity values than using UA model for at
 least one component of the solvent and capping agent, which exceeds
-the upper bond of experimental value range. This is probably due to
-the classically treated C-H vibrations in the AA model, which should
-not be appreciably populated at normal temperatures. In comparison,
-once either the hexanes or the butanethiols are deuterated, one can
-see a significantly lower $G$ and $G^\prime$. In either of these
-cases, the C-H(D) vibrational overlap between the solvent and the
-capping agent is removed. [MAY NEED FIGURE] Conclusively, the
+the general range of experimental measurement results. This is
+probably due to the classically treated C-H vibrations in the AA
+model, which should not be appreciably populated at normal
+temperatures. In comparison, once either the hexanes or the
+butanethiols are deuterated, one can see a significantly lower $G$ and
+$G^\prime$. In either of these cases, the C-H(D) vibrational overlap
+between the solvent and the capping agent is removed.
+ [MAY NEED SPECTRA FIGURE] Conclusively, the
 improperly treated C-H vibration in the AA model produced
 over-predicted results accordingly. Compared to the AA model, the UA
 model yields more reasonable results with higher computational
@@ -802,7 +840,7 @@ occupying the interfaces.
 
 \subsection{Mechanism of Interfacial Thermal Conductance Enhancement
   by Capping Agent}
-%OR\subsection{Vibrational spectrum study on conductance mechanism}
+[OR: Vibrational Spectrum Study on Conductance Mechanism]
 
 [MAY INTRODUCE PROTOCOL IN METHODOLOGY/COMPUTATIONAL DETAIL, EQN'S]
 
@@ -815,28 +853,27 @@ power spectrum via a Fourier transform.
 power spectrum via a Fourier transform.
 
 [MAY RELATE TO HASE'S]
- The gold surfaces covered by
-butanethiol molecules, compared to bare gold surfaces, exhibit an
-additional peak observed at a frequency of $\sim$170cm$^{-1}$, which
-is attributed to the vibration of the S-Au bond. This vibration
-enables efficient thermal transport from surface Au atoms to the
-capping agents. Simultaneously, as shown in the lower panel of
-Fig. \ref{vibration}, the large overlap of the vibration spectra of
-butanethiol and hexane in the all-atom model, including the C-H
-vibration, also suggests high thermal exchange efficiency. The
-combination of these two effects produces the drastic interfacial
-thermal conductance enhancement in the all-atom model.
+The gold surfaces covered by butanethiol molecules, compared to bare
+gold surfaces, exhibit an additional peak observed at the frequency of
+$\sim$170cm$^{-1}$, which is attributed to the S-Au bonding
+vibration. This vibration enables efficient thermal transport from
+surface Au layer to the capping agents.
+[MAY PUT IN OTHER SECTION] 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.
 
-[REDO. MAY NEED TO CONVERT TO JPEG]
+[NEED SEPARATE FIGURE. MAY NEED TO CONVERT TO JPEG]
 \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).}
+  environments.}
 \label{vibration}
 \end{figure}
 
-[COMPARISON OF TWO G'S; AU SLAB WIDTHS; ETC]
+[MAY ADD COMPARISON OF G AND G', AU SLAB WIDTHS, ETC]
 % The results show that the two definitions used for $G$ yield
 % comparable values, though $G^\prime$ tends to be smaller.
 
@@ -848,7 +885,7 @@ gold/liquid interfaces. The acoustic impedance mismatc
 corresponding thermal gradient and to obtain interfacial thermal
 conductivities. Our simulations have seen significant conductance
 enhancement with the presence of capping agent, compared to the bare
-gold/liquid interfaces. The acoustic impedance mismatch between the
+gold / liquid interfaces. The acoustic impedance mismatch between the
 metal and the liquid phase is effectively eliminated by proper capping
 agent. Furthermore, the coverage precentage of the capping agent plays
 an important role in the interfacial thermal transport process.
