--- interfacial/interfacial.tex	2011/07/05 17:39:21	3730
+++ interfacial/interfacial.tex	2011/07/05 21:30:29	3731
@@ -448,28 +448,33 @@ couple $J_z$'s and do not need to test a large series 
   \begin{minipage}{\linewidth}
     \begin{center}
       \caption{Computed interfacial thermal conductivity ($G$ and
-        $G^\prime$) values for the Au/butanethiol/hexane interface
-        with united-atom model and different capping agent coverage
-        and solvent molecule numbers at different temperatures using a
-        range of energy fluxes.}
-      
-      \begin{tabular}{cccccc}
+        $G^\prime$) 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.}
+      
+      \begin{tabular}{cccccccc}
         \hline\hline
-        Thiol & $\langle T\rangle$ & & $J_z$ & $G$ & $G^\prime$ \\
-        coverage (\%) & (K) & $N_{hexane}$ & (GW/m$^2$) &
+        $\langle T\rangle$ & & $L_x$ & $L_y$ & $L_z$ & $J_z$ &
+        $G$ & $G^\prime$ \\
+        (K) & $N_{hexane}$ & \multicolumn{3}{c}\AA & (GW/m$^2$) &
         \multicolumn{2}{c}{(MW/m$^2$/K)} \\
         \hline
-        0.0   & 200 & 200 & 0.96 & 43.3 & 42.7 \\
-              &     &     & 1.91 & 45.7 & 42.9 \\
-              &     & 166 & 0.96 & 43.1 & 53.4 \\
-        88.9  & 200 & 166 & 1.94 & 172  & 108  \\
-        100.0 & 250 & 200 & 0.96 & 81.8 & 67.0 \\
-              &     & 166 & 0.98 & 79.0 & 62.9 \\
-              &     &     & 1.44 & 76.2 & 64.8 \\
-              & 200 & 200 & 1.92 & 129  & 87.3 \\
-              &     &     & 1.93 & 131  & 77.5 \\
-              &     & 166 & 0.97 & 115  & 69.3 \\
-              &     &     & 1.94 & 125  & 87.1 \\
+        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() \\
         \hline\hline
       \end{tabular}
       \label{AuThiolHexaneUA}
@@ -500,24 +505,25 @@ 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 SIGNS AND ERROR ESTIMATE TO TABLE]
+[ADD Lxyz AND ERROR ESTIMATE TO TABLE]
 \begin{table*}
   \begin{minipage}{\linewidth}
     \begin{center}
       \caption{Computed interfacial thermal conductivity ($G$ and
-        $G^\prime$) values for the Au/butanethiol/toluene interface at
-        different temperatures using a range of energy fluxes.}
+        $G^\prime$) values for a 90\% coverage Au-butanethiol/toluene
+        interface at different temperatures using a range of energy
+        fluxes.}
       
       \begin{tabular}{cccc}
         \hline\hline
         $\langle T\rangle$ & $J_z$ & $G$ & $G^\prime$ \\
         (K) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\
         \hline
-        200 & 1.86 & 180 & 135 \\
-            & 2.15 & 204 & 113 \\
-            & 3.93 & 175 & 114 \\
-        300 & 1.91 & 143 & 125 \\
-            & 4.19 & 134 & 113 \\
+        200 & -1.86 & 180() & 135() \\
+            &  2.15 & 204() & 113() \\
+            & -3.93 & 175() & 114() \\
+        300 & -1.91 & 143() & 125() \\
+            & -4.19 & 134() & 113() \\
         \hline\hline
       \end{tabular}
       \label{AuThiolToluene}
@@ -565,54 +571,93 @@ different coverages of butanethiol.
 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.
+different coverages of butanethiol. To study the isotope effect in
+interfacial thermal conductance, deuterated UA-hexane is included as
+well.
 
- With high coverage of butanethiol on the gold surface,
-the interfacial thermal conductance is enhanced
-significantly. Interestingly, a slightly lower butanethiol coverage
-leads to a moderately higher conductivity. This is probably due to
-more solvent/capping agent contact when butanethiol molecules are
-not densely packed, which enhances the interactions between the two
-phases and lowers the thermal transfer barrier of this interface.
-[COMPARE TO AU/WATER IN PAPER]
+It 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. 
 
+From Table \ref{tlnUhxnUhxnD}, 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.
 
-significant conductance enhancement compared to the gold/water
-interface without capping agent and agree with available experimental
-data. This indicates that the metal-metal potential, though not
-predicting an accurate bulk metal thermal conductivity, does not
-greatly interfere with the simulation of the thermal conductance
-behavior across a non-metal interface. 
- The results show that the two definitions used for $G$ yield
-comparable values, though $G^\prime$ tends to be smaller.
+Interestingly, as one could observe from our results, the maximum
+conductance enhancement (largest $G$) happens while the surfaces are
+about 75\% covered with butanethiols. This again indicates that
+solvent-capping agent contact has an important role of the thermal
+transport process. Slightly lower butanethiol coverage allows small
+gaps between butanethiols to form. And these gaps could be filled with
+solvent molecules, which acts like ``heat conductors'' on the
+surface. The higher degree of interaction between these solvent
+molecules and capping agents increases the enhancement effect and thus
+produces a higher $G$ than densely packed butanethiol arrays. However,
+once this maximum conductance enhancement is reached, $G$ decreases
+when butanethiol coverage continues to decrease. Each capping agent
+molecule reaches its maximum capacity for thermal
+conductance. Therefore, even higher solvent-capping agent contact
+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.
 
+A comparison of the results obtained from differenet organic solvents
+can also provide useful information of the interfacial thermal
+transport process. The deuterated hexane (UA) results do not appear to
+be much different from those of normal hexane (UA), given that
+butanethiol (UA) is non-deuterated for both solvents. These UA model
+studies, even though eliminating C-H vibration samplings, still have
+C-C vibrational frequencies different from each other. However, these
+differences in the IR range do not seem to produce an observable
+difference for the results of $G$. [MAY NEED FIGURE]
 
+Furthermore, results for rigid body toluene solvent, as well as other
+UA-hexane solvents, are reasonable within the general experimental
+ranges[CITATIONS]. This suggests that explicit hydrogen might not be a
+required factor for modeling thermal transport phenomena of systems
+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
+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
+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
+increasing until the capping agent coverage reaches a relatively low
+level. This becomes an offset for decreasing butanethiol molecules on
+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 ($G$ and
-        $G^\prime$) values for the Au/butanethiol/hexane interface
-        with united-atom model and different capping agent coverage
-        and solvent molecule numbers at different temperatures using a
-        range of energy fluxes.}
+      \caption{Computed interfacial thermal conductivity ($G$ in
+        MW/m$^2$/K) values for the Au-butanethiol/solvent interface
+        with various UA models and different capping agent coverages
+        at $<T>\sim$200K using certain energy flux respectively.}
       
-      \begin{tabular}{cccccc}
+      \begin{tabular}{cccc}
         \hline\hline
-        Thiol & $\langle T\rangle$ & & $J_z$ & $G$ & $G^\prime$ \\
-        coverage (\%) & (K) & $N_{hexane}$ & (GW/m$^2$) &
-        \multicolumn{2}{c}{(MW/m$^2$/K)} \\
+        Thiol & & & \\
+        coverage (\%) & hexane & hexane-D & toluene \\
         \hline
-        0.0   & 200 & 200 & 0.96 & 43.3 & 42.7 \\
-              &     &     & 1.91 & 45.7 & 42.9 \\
-              &     & 166 & 0.96 & 43.1 & 53.4 \\
-        88.9  & 200 & 166 & 1.94 & 172  & 108  \\
-        100.0 & 250 & 200 & 0.96 & 81.8 & 67.0 \\
-              &     & 166 & 0.98 & 79.0 & 62.9 \\
-              &     &     & 1.44 & 76.2 & 64.8 \\
-              & 200 & 200 & 1.92 & 129  & 87.3 \\
-              &     &     & 1.93 & 131  & 77.5 \\
-              &     & 166 & 0.97 & 115  & 69.3 \\
-              &     &     & 1.94 & 125  & 87.1 \\
+        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}
@@ -666,6 +711,16 @@ two phases and result in a much higher conductivity.
 \end{table*}
 
 
+significant conductance enhancement compared to the gold/water
+interface without capping agent and agree with available experimental
+data. This indicates that the metal-metal potential, though not
+predicting an accurate bulk metal thermal conductivity, does not
+greatly interfere with the simulation of the thermal conductance
+behavior across a non-metal interface. 
+
+% The results show that the two definitions used for $G$ yield
+% comparable values, though $G^\prime$ tends to be smaller.
+
 \subsection{Mechanism of Interfacial Thermal Conductance Enhancement
   by Capping Agent}
 [MAY INTRODUCE PROTOCOL IN METHODOLOGY/COMPUTATIONAL DETAIL]
