--- trunk/tengDissertation/LiquidCrystal.tex	2006/06/26 21:59:54	2891
+++ trunk/tengDissertation/LiquidCrystal.tex	2006/06/27 02:42:30	2895
@@ -78,7 +78,7 @@ of bent-core molecules\cite{Memmer2002,Orlandi2006}.
 liquid crystal phases\cite{Lansac2003}. Other anisotropic models
 using Gay-Berne(GB) potential, which produce interactions that favor
 local alignment, give the evidence of the novel packing arrangements
-of bent-core molecules\cite{Memmer2002,Orlandi2006}.
+of bent-core molecules\cite{Memmer2002}.
 
 Experimental studies by Levelut {\it et al.}~\cite{Levelut1981}
 revealed that terminal cyano or nitro groups usually induce
@@ -218,7 +218,8 @@ temperature and pressure.
 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.
+temperature and pressure. NPTi Production runs last for 40~ns with
+time step of 20~fs.
 
 \subsection{Order Parameters}
 
@@ -263,7 +264,7 @@ TEMPERATURE} \label{liquidCrystal:p2}
 \caption{LIQUID CRYSTAL STRUCTURAL PROPERTIES AS A FUNCTION OF
 TEMPERATURE} \label{liquidCrystal:p2}
 \begin{center}
-\begin{tabular}{|c|c|c|c|c|c|}
+\begin{tabular}{cccccc}
 \hline
 Temperature (K) & 420 & 440 & 460 & 480 & 600\\
 \hline
@@ -277,14 +278,48 @@ transition temperature) is shown in Figure~\ref{LCFigu
 
 The molecular organization obtained at temperature $T = 460K$ (below
 transition temperature) is shown in Figure~\ref{LCFigure:snapshot}.
-
-It is also important to show the density correlation along the
-director which is given by :
+The diagonal view in Fig~\ref{LCFigure:snapshot}(a) shows the
+stacking of the banana shaped molecules while the side view in n
+Figure~\ref{LCFigure:snapshot}(b) demonstrates formation of a
+chevron structure. The first peak of Radial distribution function
+$g(r)$ in Fig.~\ref{LCFigure:gofrz}(a) shows the minimum distance
+for two in plane banana shaped molecules is 4.9 \AA, while the
+second split peak implies the biaxial packing. It is also important
+to show the density correlation along the director which is given by
+:
 \begin{equation}
-g(z) =< \delta (z-z_{ij})>_{ij} / \pi R^{2} \rho
+g(z) =\frac{1}{\pi R^{2} \rho}< \delta (z-z_{ij})>_{ij}
 \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.
+and $R$ is the radius of the cylindrical sampling region. The
+oscillation in density plot along the director in
+Fig.~\ref{LCFigure:gofrz}(b) implies the existence of the layered
+structure, and the peak at 27 \AA is attribute to the defect in the
+system.
+
+\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{Blum1972}. 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{Berardi2003}. In
+order to study the correlation between biaxiality and molecular
+separation distance $r$, we calculate a rotational invariant
+function $S_{22}^{220} (r)$, which is given by :
+\begin{eqnarray}
+S_{22}^{220} (r) & = & \frac{1}{{4\sqrt 5 }} \left< \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. \notag \\
+ & & \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>.
+\end{eqnarray}
+The long range behavior of second rank orientational correlation
+$S_{22}^{220} (r)$ in Fig~\ref{LCFigure:S22220} also confirm the
+biaxiality of the system.
 
 \begin{figure}
 \centering
@@ -307,32 +342,25 @@ $g(r)$; and (b) density along the director $g(z)$.}
 \label{LCFigure:gofrz}
 \end{figure}
 
-\subsection{Rotational Invariants}
+\begin{figure}
+\centering
+\includegraphics[width=\linewidth]{s22_220.eps}
+\caption[Average orientational correlation Correlation Functions of
+a Bent-core Liquid Crystal System at Temperature T = 460K and
+Pressure P = 10 atm]{Correlation Functions of a Bent-core Liquid
+Crystal System at Temperature T = 460K and Pressure P = 10 atm. (a)
+radial correlation function $g(r)$; and (b) density along the
+director $g(z)$.} \label{LCFigure:S22220}
+\end{figure}
 
-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{Blum1972}. 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{Berardi2003}.
-
-\begin{eqnarray}
-S_{22}^{220} (r) & = & \frac{1}{{4\sqrt 5 }} \left< \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>
-\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{Conclusion}
-To investigate the molecular organization behavior due to different
-dipolar orientation and position with respect to the center of the
-molecule,
+
+We have presented a simple dipolar three-site GB model for banana
+shaped molecules which are capable of forming smectic phases from
+isotropic configuration. Various order parameters and correlation
+functions were used to characterized the structural properties of
+these smectic phase. However, the forming layered structure still
+had some defects because of the mismatching between the layer
+structure spacing and the shape of simulation box. This mismatching
+can be broken by using NPTf integrator in further simulations. The
+lack of detail in.
