--- trunk/tengDissertation/LiquidCrystal.tex	2006/06/04 20:18:07	2786
+++ trunk/tengDissertation/LiquidCrystal.tex	2006/06/22 22:19:02	2880
@@ -38,7 +38,7 @@ Fig.~\cite{LCFig:SMCP}).
 packing arrangements (two conglomerates and two macroscopic
 racemates), which depend on the tilt direction and the polar
 direction of the molecule in adjacent layer (see
-Fig.~\cite{LCFig:SMCP}).
+Fig.~\ref{LCFig:SMCP}).
 
 \begin{figure}
 \centering
@@ -184,11 +184,11 @@ ratio between \textit{end-to-end} well depth $\epsilon
 \caption[]{} \label{LCFig:BananaMolecule}
 \end{figure}
 
-\begin{figure}
-\centering
-\includegraphics[width=\linewidth]{bananGB.eps}
-\caption[]{} \label{LCFigure:BananaGB}
-\end{figure}
+%\begin{figure}
+%\centering
+%\includegraphics[width=\linewidth]{bananGB.eps}
+%\caption[]{} \label{LCFigure:BananaGB}
+%\end{figure}
 
 \begin{figure}
 \centering
@@ -208,6 +208,94 @@ where $\epsilon _{fs}$ is the permittivity of free spa
 \end{equation}
 where $\epsilon _{fs}$ is the permittivity of free space.
 
-\section{\label{liquidCrystalSection:methods}Methods}
+\section{Computational Methodology}
 
-\section{\label{liquidCrystalSection:resultDiscussion}Results and Discussion}
+A series of molecular dynamics simulations were perform to study the
+phase behavior of banana shaped liquid crystals. In each simulation,
+every banana shaped molecule has been represented by three GB
+particles which is characterized by $\mu = 1,~ \nu = 2,
+~\epsilon_{e}/\epsilon_{s} = 1/5$ and $\sigma_{e}/\sigma_{s} = 3$.
+All of the simulations begin with same equilibrated isotropic
+configuration where 1024 molecules without dipoles were confined in
+a $160\times 160 \times 120$ box. After the dipolar interactions are
+switched on, 2~ns NPTi cooling run with themostat of 2~ps and
+barostat of 50~ps were used to equilibrate the system to desired
+temperature and pressure.
+
+\subsection{Order Parameters}
+
+To investigate the phase structure of the model liquid crystal, we
+calculated various order parameters and correlation functions.
+Particulary, the $P_2$ order parameter allows us to estimate average
+alignment along the director axis $Z$ which can be identified from
+the largest eigen value obtained by diagonalizing the order
+parameter tensor
+\begin{equation}
+\overleftrightarrow{\mathsf{Q}} = \frac{1}{N}\sum_i^N %
+    \begin{pmatrix} %
+    u_{ix}u_{ix}-\frac{1}{3} & u_{ix}u_{iy} & u_{ix}u_{iz} \\
+    u_{iy}u_{ix} & u_{iy}u_{iy}-\frac{1}{3} & u_{iy}u_{iz} \\
+    u_{iz}u_{ix} & u_{iz}u_{iy} & u_{iz}u_{iz}-\frac{1}{3} %
+    \end{pmatrix},
+\label{lipidEq:po1}
+\end{equation}
+where the $u_{i\alpha}$ is the $\alpha$ element of the unit vector
+$\mathbf{\hat{u}}_i$, and the sum over $i$ averages over the whole
+collection of unit vectors. The $P_2$ order parameter for uniaxial
+phase is then simply given by
+\begin{equation}
+\langle P_2 \rangle = \frac{3}{2}\lambda_{\text{max}}.
+\label{lipidEq:po3}
+\end{equation}
+In addition to the $P_2$ order parameter, $ R_{2,2}^2$ order
+parameter for biaxial phase is introduced to describe the ordering
+in the plane orthogonal to the director by
+\begin{equation}
+R_{2,2}^2  = \frac{1}{4}\left\langle {(x_i  \cdot X)^2  - (x_i \cdot
+Y)^2  - (y_i  \cdot X)^2  + (y_i  \cdot Y)^2 } \right\rangle
+\end{equation}
+where $X$, $Y$ and $Z$ are axis of the director frame.
+
+\subsection{Structure Properties}
+
+It is more important to show the density correlation along the
+director
+\begin{equation}
+g(z) =< \delta (z-z_{ij})>_{ij} / \pi R^{2} \rho
+\end{equation},
+where $z_{ij} = r_{ij} \dot Z$ was measured in the director frame
+and $R$ is the radius of the cylindrical sampling region.
+
+\subsection{Rotational Invariants}
+
+As a useful set of correlation functions to describe
+position-orientation correlation, rotation invariants were first
+applied in a spherical symmetric system to study x-ray and light
+scatting\cite{Blum1971}. Latterly, expansion of the orientation pair
+correlation in terms of rotation invariant for molecules of
+arbitrary shape was introduce by Stone\cite{Stone1978} and adopted
+by other researchers in liquid crystal studies\cite{Berardi2000}.
+
+\begin{eqnarray}
+S_{22}^{220} (r) & = & \frac{1}{{4\sqrt 5 }}\left\langle {\delta (r
+- r_{ij} )((\hat x_i  \cdot \hat x_j )^2  - (\hat x_i  \cdot \hat
+y_j )^2  - (\hat y_i  \cdot \hat x_j )^2  + (\hat y_i  \cdot \hat
+y_j
+)^2 ) \right.\\
+& & \left.- 2(\hat x_i  \cdot \hat y_j )(\hat y_i \cdot \hat x_j ) -
+2(\hat x_i  \cdot \hat x_j )(\hat y_i  \cdot \hat y_j ))}
+\right\rangle
+\end{eqnarray}
+
+\begin{equation}
+S_{00}^{221} (r) =  - \frac{{\sqrt 3 }}{{\sqrt {10} }}\left\langle
+{\delta (r - r_{ij} )((\hat z_i  \cdot \hat z_j )(\hat z_i  \cdot
+\hat z_j  \times \hat r_{ij} ))} \right\rangle
+\end{equation}
+
+\section{Results and Conclusion}
+\label{sec:results and conclusion}
+
+To investigate the molecular organization behavior due to different
+dipolar orientation and position with respect to the center of the
+molecule,