--- interfacial/interfacial.tex	2011/07/12 22:11:11	3737
+++ interfacial/interfacial.tex	2011/07/14 19:49:12	3739
@@ -104,11 +104,11 @@ monolayer of alkylthiolate with relatively long chain 
 this phenomena at the molecular level. Recently, Hase and coworkers
 employed Non-Equilibrium Molecular Dynamics (NEMD) simulations to
 study thermal transport from hot Au(111) substrate to a self-assembled
-monolayer of alkylthiolate with relatively long chain (8-20 carbon
+monolayer of alkylthiol with relatively long chain (8-20 carbon
 atoms)\cite{hase:2010,hase:2011}. However, ensemble averaged
 measurements for heat conductance of interfaces between the capping
 monolayer on Au and a solvent phase has yet to be studied. 
-The relatively low thermal flux through interfaces is
+The comparatively low thermal flux through interfaces is
 difficult to measure with Equilibrium MD or forward NEMD simulation
 methods. Therefore, the Reverse NEMD (RNEMD) methods would have the
 advantage of having this difficult to measure flux known when studying
@@ -375,8 +375,11 @@ Lorentz-Berthelot Mixing Rule:[EQN'S]
 (UA or AA) of capping agent can be different from the
 solvent. Regardless of model choice, the force field parameters for
 interactions between capping agent and solvent can be derived using
-Lorentz-Berthelot Mixing Rule:[EQN'S]
-
+Lorentz-Berthelot Mixing Rule:
+\begin{eqnarray}
+\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
 agent and solvent particles, we refer to an adsorption study of alkyl
@@ -405,25 +408,27 @@ parameters in our simulations.
 \begin{table*}
   \begin{minipage}{\linewidth}
     \begin{center}
-      \caption{Lennard-Jones parameters for Au-non-Metal
-        interactions in our simulations.}
-      
-      \begin{tabular}{ccc}
+      \caption{Non-bonded interaction paramters for non-metal
+        particles and metal-non-metal interactions in our
+        simulations.}
+      
+      \begin{tabular}{cccccc}
         \hline\hline
-        Non-metal atom   & $\sigma$ & $\epsilon$ \\
-        (or pseudo-atom) & \AA      & kcal/mol  \\
+        Non-metal atom $I$ & $\sigma_{II}$ & $\epsilon_{II}$ & $q_I$ &
+        $\sigma_{AuI}$ & $\epsilon_{AuI}$ \\
+        (or pseudo-atom) & \AA & kcal/mol & & \AA & kcal/mol \\
         \hline
-        S    & 2.40   & 8.465   \\
-        CH3  & 3.54   & 0.2146  \\
-        CH2  & 3.54   & 0.1749  \\
-        CT3  & 3.365  & 0.1373  \\
-        CT2  & 3.365  & 0.1373  \\
-        CTT  & 3.365  & 0.1373  \\
-        HC   & 2.865  & 0.09256 \\
-        CHar & 3.4625 & 0.1680  \\
-        CRar & 3.555  & 0.1604  \\
-        CA   & 3.173  & 0.0640  \\
-        HA   & 2.746  & 0.0414  \\
+        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  \\
         \hline\hline
       \end{tabular}
       \label{MnM}
@@ -493,29 +498,31 @@ couple $J_z$'s and do not need to test a large series 
         interfaces with UA model and different hexane molecule numbers
         at different temperatures using a range of energy fluxes.}
       
-      \begin{tabular}{cccccccc}
+      \begin{tabular}{ccccccc}
         \hline\hline
-        $\langle T\rangle$ & & $L_x$ & $L_y$ & $L_z$ & $J_z$ &
-        $G$ & $G^\prime$ \\
-        (K) & $N_{hexane}$ & \multicolumn{3}{c}{(\AA)} & (GW/m$^2$) &
+        $\langle T\rangle$ & $N_{hexane}$ & Fixed & $\rho_{hexane}$ &
+        $J_z$ & $G$ & $G^\prime$ \\
+        (K) & & $L_x$ \& $L_y$? & (g/cm$^3$) & (GW/m$^2$) &
         \multicolumn{2}{c}{(MW/m$^2$/K)} \\
         \hline
-        200 & 266 & 29.86 & 25.80 & 113.1 & -0.96 &
-        102()  & 80.0() \\
-            & 200 & 29.84 & 25.81 &  93.9 &  1.92 &
-        129()  & 87.3() \\
-            &     & 29.84 & 25.81 &  95.3 &  1.93 &
-        131()  & 77.5() \\
-            & 166 & 29.84 & 25.81 &  85.7 &  0.97 &
-        115()  & 69.3() \\
-            &     &       &       &       &  1.94 &
-        125()  & 87.1() \\
-        250 & 200 & 29.84 & 25.87 & 106.8 &  0.96 &
-        81.8() & 67.0() \\
-            & 166 & 29.87 & 25.84 &  94.8 &  0.98 &
-        79.0() & 62.9() \\
-            &     & 29.84 & 25.85 &  95.0 &  1.44 &
-        76.2() & 64.8() \\
+        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() \\
+        \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() \\
         \hline\hline
       \end{tabular}
       \label{AuThiolHexaneUA}
@@ -546,7 +553,7 @@ 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 Lxyz AND ERROR ESTIMATE TO TABLE]
+[ADD ERROR ESTIMATE TO TABLE]
 \begin{table*}
   \begin{minipage}{\linewidth}
     \begin{center}
@@ -555,16 +562,17 @@ in that higher degree of contact could yield increased
         interface at different temperatures using a range of energy
         fluxes.}
       
-      \begin{tabular}{cccc}
+      \begin{tabular}{ccccc}
         \hline\hline
-        $\langle T\rangle$ & $J_z$ & $G$ & $G^\prime$ \\
-        (K) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\
+        $\langle T\rangle$ & $\rho_{toluene}$ & $J_z$ & $G$ & $G^\prime$ \\
+        (K) & (g/cm$^3$) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\
         \hline
-        200 & -1.86 & 180() & 135() \\
-            &  2.15 & 204() & 113() \\
-            & -3.93 & 175() & 114() \\
-        300 & -1.91 & 143() & 125() \\
-            & -4.19 & 134() & 113() \\
+        200 & 0.933 & -1.86 & 180() & 135() \\
+            &       &  2.15 & 204() & 113() \\
+            &       & -3.93 & 175() & 114() \\
+        \hline
+        300 & 0.855 & -1.91 & 143() & 125() \\
+            &       & -4.19 & 134() & 113() \\
         \hline\hline
       \end{tabular}
       \label{AuThiolToluene}
@@ -597,13 +605,14 @@ even at $\langle T\rangle\sim$300K. The Au(111) surfac
 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 butanethiols can
-migrate to their neighbor empty sites during a simulation. Therefore,
-we were able to obtain $G$'s for these interfaces even at a relatively
-high temperature without being affected by surface reconstructions.
+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
+butanethiols can migrate to their neighbor empty sites during a
+simulation. Therefore, we were able to obtain $G$'s for these
+interfaces even at a relatively high temperature without being
+affected by surface reconstructions.
 
 \subsection{Influence of Capping Agent Coverage on $G$}
 To investigate the influence of butanethiol coverage on interfacial
@@ -679,32 +688,15 @@ 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, MAY ALSO PUT J HERE]
-\begin{table*}
-  \begin{minipage}{\linewidth}
-    \begin{center}
-      \caption{Computed interfacial thermal conductivity ($G$) values
-        for the Au-butanethiol/solvent interface with various UA
-        models and different capping agent coverages at $\langle
-        T\rangle\sim$200K using certain energy flux respectively.}
-      
-      \begin{tabular}{cccc}
-        \hline\hline
-        Thiol & \multicolumn{3}{c}{$G$(MW/m$^2$/K)} \\
-        coverage (\%) & hexane & hexane(D) & toluene \\
-        \hline
-        0.0   & 46.5() & 43.9() & 70.1() \\
-        25.0  & 151()  & 153()  & 249()  \\
-        50.0  & 172()  & 182()  & 214()  \\
-        75.0  & 242()  & 229()  & 244()  \\
-        88.9  & 178()  & -      & -      \\
-        100.0 & 137()  & 153()  & 187()  \\
-        \hline\hline
-      \end{tabular}
-      \label{tlnUhxnUhxnD}
-    \end{center}
-  \end{minipage}
-\end{table*}
+[NEED ERROR ESTIMATE]
+\begin{figure}
+\includegraphics[width=\linewidth]{coverage}
+\caption{Comparison of interfacial thermal conductivity ($G$) values
+  for the Au-butanethiol/solvent interface with various UA models and
+  different capping agent coverages at $\langle T\rangle\sim$200K
+  using certain energy flux respectively.}
+\label{coverage}
+\end{figure}
 
 \subsection{Influence of Chosen Molecule Model on $G$}
 [MAY COMBINE W MECHANISM STUDY]
@@ -725,7 +717,9 @@ these studies.
       \caption{Computed interfacial thermal conductivity ($G$ and
         $G^\prime$) values for interfaces using various models for
         solvent and capping agent (or without capping agent) at
-        $\langle T\rangle\sim$200K.}
+        $\langle T\rangle\sim$200K. (D stands for deuterated solvent
+        or capping agent molecules; ``Avg.'' denotes results that are
+        averages of several simulations.)}
       
       \begin{tabular}{ccccc}
         \hline\hline
@@ -733,15 +727,27 @@ these studies.
         (or bare surface) & model & (GW/m$^2$) &
         \multicolumn{2}{c}{(MW/m$^2$/K)} \\
         \hline
-        UA    & AA hexane  & 1.94 & 135()  & 129()  \\
-              &            & 2.86 & 126()  & 115()  \\
-              & AA toluene & 1.89 & 200()  & 149()  \\
-        AA    & UA hexane  & 1.94 & 116()  & 129()  \\
-              & AA hexane  & 3.76 & 451()  & 378()  \\
-              &            & 4.71 & 432()  & 334()  \\
-              & AA toluene & 3.79 & 487()  & 290()  \\
-        AA(D) & UA hexane  & 1.94 & 158()  & 172()  \\
-        bare  & AA hexane  & 0.96 & 31.0() & 29.4() \\
+        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()  \\
+        \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()  \\
+        \hline
+        AA(D) & UA hexane    & 1.94 & 158()  & 172()  \\
+              & AA hexane    & 1.92 & 243()  & 191()  \\
+              & AA toluene   & 1.93 & 364()  & 322()  \\
+        \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() \\
         \hline\hline
       \end{tabular}
       \label{modelTest}
@@ -778,18 +784,21 @@ measurement results.
 
 However, for Au-butanethiol/toluene interfaces, having the AA
 butanethiol deuterated did not yield a significant change in the
-measurement results.
-. , 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.
+measurement results. Compared to the C-H vibrational overlap between
+hexane and butanethiol, both of which have alkyl chains, that overlap
+between toluene and butanethiol is not so significant and thus does
+not have as much contribution to the ``Intramolecular Vibration
+Redistribution''[CITE HASE]. Conversely, extra degrees of freedom such
+as the C-H vibrations could yield higher heat exchange rate between
+these two phases and result in a much higher conductivity.
 
-
 Although the QSC model for Au is known to predict an overly low value
-for bulk metal gold conductivity[CITE NIVSRNEMD], our computational
+for bulk metal gold conductivity\cite{kuang:164101}, our computational
 results for $G$ and $G^\prime$ do not seem to be affected by this
-drawback of the model for metal. Instead, the modeling of interfacial
-thermal transport behavior relies mainly on an accurate description of
-the interactions between components occupying the interfaces.
+drawback of the model for metal. Instead, our results suggest that the
+modeling of interfacial thermal transport behavior relies mainly on
+the accuracy of the interaction descriptions between components
+occupying the interfaces.
 
 \subsection{Mechanism of Interfacial Thermal Conductance Enhancement
   by Capping Agent}
@@ -805,6 +814,7 @@ power spectrum via a Fourier transform.
 the velocity auto-correlation functions, which is used to construct a
 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
@@ -817,7 +827,7 @@ thermal conductance enhancement in the all-atom model.
 combination of these two effects produces the drastic interfacial
 thermal conductance enhancement in the all-atom model.
 
-[MAY NEED TO CONVERT TO JPEG]
+[REDO. MAY NEED TO CONVERT TO JPEG]
 \begin{figure}
 \includegraphics[width=\linewidth]{vibration}
 \caption{Vibrational spectra obtained for gold in different
