--- trunk/tengDissertation/LiquidCrystal.tex	2006/06/21 19:46:01	2876
+++ trunk/tengDissertation/LiquidCrystal.tex	2006/06/27 01:33:33	2892
@@ -43,8 +43,10 @@ Fig.~\ref{LCFig:SMCP}).
 \begin{figure}
 \centering
 \includegraphics[width=\linewidth]{smcp.eps}
-\caption[]
-{}
+\caption[SmCP Phase Packing] {Four possible SmCP phase packings that
+are characterized by the relative tilt direction(A and S refer an
+anticlinic tilt or a synclinic ) and the polarization orientation (A
+and F represent antiferroelectric or ferroelectric polar order).}
 \label{LCFig:SMCP}
 \end{figure}
 
@@ -76,7 +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
@@ -98,11 +100,10 @@ molecule represented by three rigid GB particles with 
 incorporating electrostatic interaction.
 
 In this chapter, we consider system consisting of banana-shaped
-molecule represented by three rigid GB particles with one or two
-point dipoles at different location. Performing a series of
-molecular dynamics simulations, we explore the structural properties
-of tilted smectic phases as well as the effect of electrostatic
-interactions.
+molecule represented by three rigid GB particles with two point
+dipoles. Performing a series of molecular dynamics simulations, we
+explore the structural properties of tilted smectic phases as well
+as the effect of electrostatic interactions.
 
 \section{\label{liquidCrystalSection:model}Model}
 
@@ -181,21 +182,18 @@ ratio between \textit{end-to-end} well depth $\epsilon
 \begin{figure}
 \centering
 \includegraphics[width=\linewidth]{banana.eps}
-\caption[]{} \label{LCFig:BananaMolecule}
+\caption[Schematic representation of a typical banana shaped
+molecule]{Schematic representation of a typical banana shaped
+molecule.} \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]{gb_scheme.eps}
-\caption[]{Schematic diagram showing definitions of the orientation
-vectors for a pair of Gay-Berne molecules}
-\label{LCFigure:GBScheme}
+\caption[Schematic diagram showing definitions of the orientation
+vectors for a pair of Gay-Berne molecules]{Schematic diagram showing
+definitions of the orientation vectors for a pair of Gay-Berne
+molecules} \label{LCFigure:GBScheme}
 \end{figure}
 
 To account for the permanent dipolar interactions, there should be
@@ -208,12 +206,12 @@ where $\epsilon _{fs}$ is the permittivity of free spa
 \end{equation}
 where $\epsilon _{fs}$ is the permittivity of free space.
 
-\section{Computational Methodology}
+\section{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 three GB particles
-which is characterized by $\mu = 1,~ \nu = 2,
+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
@@ -237,7 +235,7 @@ parameter tensor
     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}
+\label{lipidEq:p2}
 \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
@@ -247,54 +245,107 @@ In addition to the $P_2$ order parameter, $ R_{2,2}^2$
 \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.
+%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.
+The unit vector for the banana shaped molecule was defined by the
+principle aixs of its middle GB particle. The $P_2$ order parameters
+for the bent-core liquid crystal at different temperature is
+summarized in Table~\ref{liquidCrystal:p2} which identifies a phase
+transition temperature range.
 
+\begin{table}
+\caption{LIQUID CRYSTAL STRUCTURAL PROPERTIES AS A FUNCTION OF
+TEMPERATURE} \label{liquidCrystal:p2}
+\begin{center}
+\begin{tabular}{cccccc}
+\hline
+Temperature (K) & 420 & 440 & 460 & 480 & 600\\
+\hline
+$\langle P_2\rangle$ & 0.984 & 0.982 & 0.975 & 0.967 & 0.067\\
+\hline
+\end{tabular}
+\end{center}
+\end{table}
+
 \subsection{Structure Properties}
 
-It is more important to show the density correlation along the
-director
+The molecular organization obtained at temperature $T = 460K$ (below
+transition temperature) is shown in Figure~\ref{LCFigure:snapshot}.
+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.
 
+\begin{figure}
+\centering
+\includegraphics[width=4.5in]{snapshot.eps}
+\caption[Snapshot of the molecular organization in the layered phase
+formed at temperature T = 460K and pressure P = 1 atm]{Snapshot of
+the molecular organization in the layered phase formed at
+temperature T = 460K and pressure P = 1 atm. (a) diagonal view; (b)
+side view.} \label{LCFigure:snapshot}
+\end{figure}
+
+\begin{figure}
+\centering
+\includegraphics[width=\linewidth]{gofr_gofz.eps}
+\caption[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:gofrz}
+\end{figure}
+
 \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
+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{Berardi2000}.
-
+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\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 ) \\
-& & - 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
+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}
 
-\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}
+%\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}
+\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.
