--- trunk/tengDissertation/LiquidCrystal.tex	2006/06/01 18:06:33	2783
+++ trunk/tengDissertation/LiquidCrystal.tex	2006/06/01 20:14:11	2784
@@ -1,7 +1,6 @@
 \chapter{\label{chapt:liquidcrystal}LIQUID CRYSTAL}
 
 \section{\label{liquidCrystalSection:introduction}Introduction}
-% liquid crystal
 
 Long range orientational order is one of the most fundamental
 properties of liquid crystal mesophases. This orientational
@@ -18,7 +17,6 @@ segregation phenomena in disk-like molecules.
 sufficiently long aliphatic chains has been reported, as well as the
 segregation phenomena in disk-like molecules.
 
-% banana shaped
 Recently, the banana-shaped or bent-core liquid crystal have became
 one of the most active research areas in mesogenic materials and
 supramolecular chemistry. Unlike rods and disks, the polarity and
@@ -41,7 +39,14 @@ Fig.~\cite{LCFig:SMCP}).
 direction of the molecule in adjacent layer (see
 Fig.~\cite{LCFig:SMCP}).
 
-%general banana-shaped molecule modeling
+\begin{figure}
+\centering
+\includegraphics[width=\linewidth]{smcp.eps}
+\caption[]
+{}
+\label{LCFig:SMCP}
+\end{figure}
+
 Many liquid crystal synthesis experiments suggest that the
 occurrence of polarity and chirality strongly relies on the
 molecular structure and intermolecular interaction. From a
@@ -67,14 +72,128 @@ which the isotropic phase undergoes a transition direc
 simulation studies using hard spherocylinder dimer
 model\cite{Camp1999} produce nematic phases, while hard rod
 simulation studies identified a Landau point\cite{Bates2005}, at
-which the isotropic phase undergoes a transition directly to the
+which the isotropic phase undergoes a direct transition to the
 biaxial nematic, as well as some possible liquid crystal
-phases\cite{Lansac2003}. Other anisotropic models using Gay-Berne
-potential give the evidence of the novel packing arrangement of
+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}.
 
+Experimental studies by Levelut {\it et al.}~\cite{Levelut1981}
+revealed that terminal cyano or nitro groups usually induce
+permanent longitudinal dipole moments, which affect the phase
+behavior considerably. A series of theoretical studies also drawn
+equivalent conclusions. Monte Carlo studies of the GB potential with
+fixed longitudinal dipoles (i.e. pointed along the principal axis of
+rotation) were shown to enhance smectic phase
+stability~\cite{Berardi1996,Satoh1996}. Molecular simulation of GB
+ellipsoids with transverse dipoles at the terminus of the molecule
+also demonstrated that partial striped bilayer structures were
+developed from the smectic phase ~\cite{Berardi1996}. More
+significant effects have been shown by including multiple
+electrostatic moments. Adding longitudinal point quadrupole moments
+to rod-shaped GB mesogens, Withers \textit{et al} induced tilted
+smectic behaviour in the molecular system~\cite{Withers2003}. Thus,
+it is clear that many liquid-crystal forming molecules, specially,
+bent-core molecules, could be modeled more accurately by
+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.
+
 \section{\label{liquidCrystalSection:model}Model}
 
+A typical banana-shaped molecule consists of a rigid aromatic
+central bent unit with several rod-like wings which are held
+together by some linking units and terminal chains (see
+Fig.~\ref{LCFig:BananaMolecule}). In this work, each banana-shaped
+mesogen has been modeled as a rigid body consisting of three
+equivalent prolate ellipsoidal GB particles. The GB interaction
+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,
+\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
+}}{{r_{ij}  - \sigma (\hat u_i ,\hat u_j ,\hat r_{ij} )}}} \right)^6
+} \right] \label{LCEquation:gb}
+\end{equation}
+where $\hat u_i,\hat u_j$ are unit vectors specifying the
+orientation of two molecules $i$ and $j$ separated by intermolecular
+vector $r_{ij}$. $\hat r_{ij}$ is the unit vector along the
+intermolecular vector. A schematic diagram of the orientation
+vectors is shown in Fig.\ref{LCFigure:GBScheme}. The functional form
+for $\sigma$ is given by
+\begin{equation}
+\sigma (\hat u_i ,\hat u_i ,\hat r_{ij} ) = \sigma _0 \left[ {1 -
+\frac{\chi }{2}\left( {\frac{{(\hat r_{ij}  \cdot \hat u_i  + \hat
+r_{ij}  \cdot \hat u_j )^2 }}{{1 + \chi \hat u_i  \cdot \hat u_j }}
++ \frac{{(\hat r_{ij}  \cdot \hat u_i  - \hat r_{ij}  \cdot \hat u_j
+)^2 }}{{1 - \chi \hat u_i  \cdot \hat u_j }}} \right)} \right]^{ -
+\frac{1}{2}},
+\end{equation}
+where the aspect ratio of the particles is governed by shape
+anisotropy parameter
+\begin{equation}
+\chi  = \frac{{(\sigma _e /\sigma _s )^2  - 1}}{{(\sigma _e /\sigma
+_s )^2  + 1}}.
+\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
+depth parameters takes the form
+\begin{equation}
+\epsilon (\hat u_i ,\hat u_j ,\hat r_{ij} ) = \epsilon _0 \epsilon
+^v (\hat u_i ,\hat u_j )\epsilon '^\mu (\hat u_i ,\hat u_j ,\hat
+r_{ij} )
+\end{equation}
+where $\epsilon_{0}$ is a constant term and
+\begin{equation}
+\epsilon (\hat u_i ,\hat u_j ) = \frac{1}{{\sqrt {1 - \chi ^2 (\hat
+u_i  \cdot \hat u_j )^2 } }}
+\end{equation}
+and
+\begin{equation}
+\epsilon '(\hat u_i ,\hat u_j ,\hat r_{ij} ) = 1 - \frac{{\chi
+'}}{2}\left[ {\frac{{(\hat r_{ij}  \cdot \hat u_i  + \hat r_{ij}
+\cdot \hat u_j )^2 }}{{1 + \chi '\hat u_i  \cdot \hat u_j }} +
+\frac{{(\hat r_{ij}  \cdot \hat u_i  - \hat r_{ij}  \cdot \hat u_j
+)^2 }}{{1 - \chi '\hat u_i  \cdot \hat u_j }}} \right]
+\end{equation}
+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}
+\chi ' = \frac{{1 - (\epsilon _e /\epsilon _s )^{1/\mu} }}{{1 +
+(\epsilon _e /\epsilon _s )^{1/\mu} }}.
+\end{equation}
+
+\begin{figure}
+\centering
+\includegraphics[width=\linewidth]{banana.eps}
+\caption[]{} \label{LCFig:BananaMolecule}
+\end{figure}
+
+\begin{figure}
+\centering
+\includegraphics[width=\linewidth]{bananGB_grained.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}
+\end{figure}
+
 \section{\label{liquidCrystalSection:methods}Methods}
 
 \section{\label{liquidCrystalSection:resultDiscussion}Results and Discussion}
