--- trunk/tengDissertation/LiquidCrystal.tex	2006/04/03 18:07:54	2685
+++ trunk/tengDissertation/LiquidCrystal.tex	2006/06/09 02:41:58	2839
@@ -2,8 +2,212 @@
 
 \section{\label{liquidCrystalSection:introduction}Introduction}
 
+Long range orientational order is one of the most fundamental
+properties of liquid crystal mesophases. This orientational
+anisotropy of the macroscopic phases originates in the shape
+anisotropy of the constituent molecules. Among these anisotropy
+mesogens, rod-like (calamitic) and disk-like molecules have been
+exploited in great detail in the last two decades\cite{Huh2004}.
+Typically, these mesogens consist of a rigid aromatic core and one
+or more attached aliphatic chains. For short chain molecules, only
+nematic phases, in which positional order is limited or absent, can
+be observed, because the entropy of mixing different parts of the
+mesogens is paramount to the dispersion interaction. In contrast,
+formation of the one dimension lamellar sematic phase in rod-like
+molecules with sufficiently long aliphatic chains has been reported,
+as well as the segregation phenomena in disk-like molecules.
+
+Recently, the banana-shaped or bent-core liquid crystal have became
+one of the most active research areas in mesogenic materials and
+supramolecular chemistry\cite{Niori1996, Link1997, Pelzl1999}.
+Unlike rods and disks, the polarity and biaxiality of the
+banana-shaped molecules allow the molecules organize into a variety
+of novel liquid crystalline phases which show interesting material
+properties. Of particular interest is the spontaneous formation of
+macroscopic chiral layers from achiral banana-shaped molecules,
+where polar molecule orientational ordering is shown within the
+layer plane as well as the tilted arrangement of the molecules
+relative to the polar axis. As a consequence of supramolecular
+chirality, the spontaneous polarization arises in ferroelectric (FE)
+and antiferroelectic (AF) switching of smectic liquid crystal
+phases, demonstrating some promising applications in second-order
+nonlinear optical devices. The most widely investigated mesophase
+formed by banana-shaped moleculed is the $\text{B}_2$ phase, which
+is also referred to as $\text{SmCP}$\cite{Link1997}. Of the most
+important discover in this tilt lamellar phase is the four distinct
+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.~\ref{LCFig:SMCP}).
+
+\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\cite{Reddy2006}.
+From a theoretical point of view, it is of fundamental interest to
+study the structural properties of liquid crystal phases formed by
+banana-shaped molecules and understand their connection to the
+molecular structure, especially with respect to the spontaneous
+achiral symmetry breaking. As a complementary tool to experiment,
+computer simulation can provide unique insight into molecular
+ordering and phase behavior, and hence improve the development of
+new experiments and theories. In the last two decades, all-atom
+models have been adopted to investigate the structural properties of
+smectic arrangements\cite{Cook2000, Lansac2001}, as well as other
+bulk properties, such as rotational viscosity and flexoelectric
+coefficients\cite{Cheung2002, Cheung2004}. However, due to the
+limitation of time scale required for phase transition and the
+length scale required for representing bulk behavior,
+models\cite{Perram1985, Gay1981}, which are based on the observation
+that liquid crystal order is exhibited by a range of non-molecular
+bodies with high shape anisotropies, became the dominant models in
+the field of liquid crystal phase behavior. Previous 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 direct transition to the biaxial nematic, as well as some possible
+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}.
+
+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. 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
+}}{{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. The well
+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{equation}
+\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.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}
+
+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}
