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Revision 2839 by tim, Fri Jun 9 02:41:58 2006 UTC

+\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}.
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+Typically, these mesogens consist of a rigid aromatic core and one
-+
+or more attached aliphatic chains. For short chain molecules, only
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+nematic phases, in which positional order is limited or absent, can
-+
+be observed, because the entropy of mixing different parts of the
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+mesogens is paramount to the dispersion interaction. In contrast,
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+formation of the one dimension lamellar sematic phase in rod-like
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+molecules with sufficiently long aliphatic chains has been reported,
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+as well as the segregation phenomena in disk-like molecules.
-+
-+
+Recently, the banana-shaped or bent-core liquid crystal have became
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+one of the most active research areas in mesogenic materials and
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+supramolecular chemistry\cite{Niori1996, Link1997, Pelzl1999}.
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+Unlike rods and disks, the polarity and biaxiality of the
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+banana-shaped molecules allow the molecules organize into a variety
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+of novel liquid crystalline phases which show interesting material
-+
+properties. Of particular interest is the spontaneous formation of
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+macroscopic chiral layers from achiral banana-shaped molecules,
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+where polar molecule orientational ordering is shown within the
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+layer plane as well as the tilted arrangement of the molecules
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+relative to the polar axis. As a consequence of supramolecular
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+chirality, the spontaneous polarization arises in ferroelectric (FE)
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+and antiferroelectic (AF) switching of smectic liquid crystal
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+phases, demonstrating some promising applications in second-order
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+nonlinear optical devices. The most widely investigated mesophase
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+formed by banana-shaped moleculed is the $\text{B}_2$ phase, which
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+is also referred to as $\text{SmCP}$\cite{Link1997}. Of the most
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+important discover in this tilt lamellar phase is the four distinct
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+packing arrangements (two conglomerates and two macroscopic
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+racemates), which depend on the tilt direction and the polar
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+direction of the molecule in adjacent layer (see
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+Fig.~\ref{LCFig:SMCP}).
-+
-+
+\begin{figure}
-+
+\centering
-+
+\includegraphics[width=\linewidth]{smcp.eps}
-+
+\caption[]
-+
+{}
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+\label{LCFig:SMCP}
-+
+\end{figure}
-+
-+
+Many liquid crystal synthesis experiments suggest that the
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+occurrence of polarity and chirality strongly relies on the
-+
+molecular structure and intermolecular interaction\cite{Reddy2006}.
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+From a theoretical point of view, it is of fundamental interest to
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+study the structural properties of liquid crystal phases formed by
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+banana-shaped molecules and understand their connection to the
-+
+molecular structure, especially with respect to the spontaneous
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+achiral symmetry breaking. As a complementary tool to experiment,
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+computer simulation can provide unique insight into molecular
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+ordering and phase behavior, and hence improve the development of
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+new experiments and theories. In the last two decades, all-atom
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+models have been adopted to investigate the structural properties of
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+smectic arrangements\cite{Cook2000, Lansac2001}, as well as other
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+bulk properties, such as rotational viscosity and flexoelectric
-+
+coefficients\cite{Cheung2002, Cheung2004}. However, due to the
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+limitation of time scale required for phase transition and the
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+length scale required for representing bulk behavior,
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+models\cite{Perram1985, Gay1981}, which are based on the observation
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+that liquid crystal order is exhibited by a range of non-molecular
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+bodies with high shape anisotropies, became the dominant models in
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+the field of liquid crystal phase behavior. Previous simulation
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+studies using hard spherocylinder dimer model\cite{Camp1999} produce
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+nematic phases, while hard rod simulation studies identified a
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+Landau point\cite{Bates2005}, at which the isotropic phase undergoes
-+
+a direct transition to the biaxial nematic, as well as some possible
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+liquid crystal phases\cite{Lansac2003}. Other anisotropic models
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+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}
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+revealed that terminal cyano or nitro groups usually induce
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+permanent longitudinal dipole moments, which affect the phase
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+behavior considerably. A series of theoretical studies also drawn
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+equivalent conclusions. Monte Carlo studies of the GB potential with
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+fixed longitudinal dipoles (i.e. pointed along the principal axis of
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+rotation) were shown to enhance smectic phase
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+stability~\cite{Berardi1996,Satoh1996}. Molecular simulation of GB
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+ellipsoids with transverse dipoles at the terminus of the molecule
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+also demonstrated that partial striped bilayer structures were
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+developed from the smectic phase ~\cite{Berardi1996}. More
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+significant effects have been shown by including multiple
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+electrostatic moments. Adding longitudinal point quadrupole moments
-+
+to rod-shaped GB mesogens, Withers \textit{et al} induced tilted
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+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
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+molecule represented by three rigid GB particles with one or two
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+point dipoles at different location. Performing a series of
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+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
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+orientation and position dependent range ($\sigma$) and well depth
-+
+($\epsilon$) parameters. The potential between a pair of three-site
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+banana-shaped molecules $a$ and $b$ is given by
-+
+\begin{equation}
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+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
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+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
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+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
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+ratio between \textit{end-to-end} well depth $\epsilon _e$ and
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+\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}

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Comparing trunk/tengDissertation/LiquidCrystal.tex (file contents): Revision 2685 by tim, Mon Apr 3 18:07:54 2006 UTC vs. Revision 2839 by tim, Fri Jun 9 02:41:58 2006 UTC

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Comparing trunk/tengDissertation/LiquidCrystal.tex (file contents):
Revision 2685 by tim, Mon Apr 3 18:07:54 2006 UTC vs.
Revision 2839 by tim, Fri Jun 9 02:41:58 2006 UTC