--- trunk/tengDissertation/LiquidCrystal.tex	2006/06/01 20:14:11	2784
+++ trunk/tengDissertation/LiquidCrystal.tex	2006/06/02 21:31:49	2785
@@ -116,8 +116,13 @@ orientation and position dependent range ($\sigma$) an
 potential used to mimic the apolar characteristics of liquid crystal
 molecules takes the familiar form of Lennard-Jones function with
 orientation and position dependent range ($\sigma$) and well depth
-($\epsilon$) parameters. It can can be expressed as,
+($\epsilon$) parameters. The potential between a pair of three-site
+banana-shaped molecules $a$ and $b$ is given by
 \begin{equation}
+V_{ab}^{GB}  = \sum\limits_{i \in a,j \in b} {V_{ij}^{GB} }.
+\end{equation}
+Every site-site interaction can can be expressed as,
+\begin{equation}
 V_{ij}^{GB}  = 4\epsilon (\hat u_i ,\hat u_j ,\hat r_{ij} )\left[
 {\left( {\frac{{\sigma _0 }}{{r_{ij}  - \sigma (\hat u_i ,\hat u_j
 ,\hat r_{ij} )}}} \right)^{12}  - \left( {\frac{{\sigma _0
@@ -146,7 +151,7 @@ and the end-to-end length of the ellipsoid, respective
 \label{LCEquation:chi}
 \end{equation}
 Here, $\sigma_ s$ and $\sigma_{e}$ refer to the side-by-side breadth
-and the end-to-end length of the ellipsoid, respectively. Twell
+and the end-to-end length of the ellipsoid, respectively. The well
 depth parameters takes the form
 \begin{equation}
 \epsilon (\hat u_i ,\hat u_j ,\hat r_{ij} ) = \epsilon _0 \epsilon
@@ -169,7 +174,7 @@ ratio between \textit{end-to-end} well depth $\epsilon
 where the well depth anisotropy parameter $\chi '$ depends on the
 ratio between \textit{end-to-end} well depth $\epsilon _e$ and
 \textit{side-by-side} well depth $\epsilon_s$,
-\begin{eqaution}
+\begin{equation}
 \chi ' = \frac{{1 - (\epsilon _e /\epsilon _s )^{1/\mu} }}{{1 +
 (\epsilon _e /\epsilon _s )^{1/\mu} }}.
 \end{equation}
@@ -182,7 +187,7 @@ ratio between \textit{end-to-end} well depth $\epsilon
 
 \begin{figure}
 \centering
-\includegraphics[width=\linewidth]{bananGB_grained.eps}
+\includegraphics[width=\linewidth]{bananGB.eps}
 \caption[]{} \label{LCFigure:BananaGB}
 \end{figure}
 
@@ -194,6 +199,16 @@ vectors for a pair of Gay-Berne molecules}
 \label{LCFigure:GBScheme}
 \end{figure}
 
+To account for the permanent dipolar interactions, there should be
+an electrostatic interaction term of the form
+\begin{equation}
+V_{ab}^{dp}  = \sum\limits_{i \in a,j \in b} {\frac{1}{{4\pi
+\epsilon _{fs} }}\left[ {\frac{{\mu _i  \cdot \mu _j }}{{r_{ij}^3 }}
+- \frac{{3\left( {\mu _i  \cdot r_{ij} } \right)\left( {\mu _i \cdot
+r_{ij} } \right)}}{{r_{ij}^5 }}} \right]}
+\end{equation}
+where $\epsilon _{fs}$ is the permittivity of free space.
+
 \section{\label{liquidCrystalSection:methods}Methods}
 
 \section{\label{liquidCrystalSection:resultDiscussion}Results and Discussion}
