trunk/tengDissertation/LiquidCrystal.tex

\chapter{\label{chapt:liquidcrystal}LIQUID CRYSTAL}

\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{Computational Methodology}

A series of molecular dynamics simulations were perform to study the
phase behavior of banana shaped liquid crystals.

In each simulation, rod-like polar molecules have been represented
by polar ellipsoidal Gay-Berne (GB) particles. The four parameters
characterizing G-B potential were taken as   $\mu = 1,~ \nu = 2,
~\epsilon_{e}/\epsilon_{s}
 = 1/5$ and $\sigma_{e}/\sigma_{s} = 3$. The components of the
scaled moment of inertia $(I^{*} = I/m \sigma_{s}^{2})$ along the
major and minor axes were $I_{z}^{*} = 0.2$ and $I_{\perp}^{*} =
1.0$.  We used the reduced dipole moments $ \mu^{*} = \mu/(4 \pi
\epsilon_{fs} \sigma_{0}^{3})^{1/2}= 1.0$ for terminal dipole and
 $ \mu^{*} = \mu/(4
\pi \epsilon_{fs} \sigma_{0}^{3})^{1/2}= 0.5$ for second dipole,
where $\epsilon_{fs}$ was the permitivitty of free space. For all
simulations the position of the terminal dipole
 has been kept
 at a fixed distance $d^{*} = d/\sigma_{s} = 1.0 $ from the
centre of mass on the molecular symmetry axis. The second dipole
takes  $d^{*} = d/\sigma_{s} = 0.0 $ i.e. it is on the centre of
mass. To investigate the molecular organization behaviour due to
different dipolar orientation with respect to the symmetry axis, we
selected dipolar angle $\alpha_{d} = 0$ to model terminal outward
longitudinal dipole and $\alpha_{d} = \pi/2$ to model transverse
outward dipole where the second dipole takes  relative  anti
antiparallel orientation with respect to the first. System of
molecules having a single transverse terminal dipole has also been
studied. We ran a series of simulations to investigate the effect of
dipoles on molecular organization.

In each of the simulations 864 molecules were confined in a cubic
box with periodic boundary conditions. The run started from a
density $\rho^{*} = \rho \sigma_{0}^{3}$ = 0.01 with nonpolar
molecules loacted on the sites of FCC lattice and having parallel
orientation. This structure was not a stable structure at this
density and it was melted at a reduced temperature $T^{*} = k_{B}T/
\epsilon_{0} = 4.0$ . We used this isotropic configuration which was
both orientationally and translationally disordered, as the initial
configuration for each simulation. The dipoles were also switched on
from this point. Initial translational and angular velocities were
assigned from the gaussian distribution of velocities.

To get the ordered structure for each system of particular dipolar
angles we increased the density from $\rho^{*} = 0.01$ to $\rho_{*}
= 0.3$ with an increament size of 0.002 upto $\rho^{*} = 0.1$ and
0.01 for the rest at some higher temperature. Temperature was then
lowered in finer steps to avoid ending up with disordered glass
phase and thus to help the molecules set with more order. For each
system this process required altogether $5 \times 10^{6}$ MC cycles
for equilibration.

The torques and forces were calculated using velocity verlet
algorithm. The time step size $\delta t^{*} = \delta t/(m
\sigma_{0}^{2} / \epsilon_{0})^{1/2}$ was set at 0.0012 during the
process. The orientations of molecules were described by quaternions
instead of Eulerian angles to get the singularity-free orientational
equations of motion.

The interaction potential was truncated at a cut-off radius $r_{c} =
3.8 \sigma_{0}$. The long range dipole-dipole interaction potential
and torque were handled by the application of reaction field method
~\cite{Allen87}.

To investigate the phase structure of the model liquid crystal
family  we calculated the orientational order parameter, correlation
functions. To identify a particular phase we took configurational
snapshots at the onset of each layered phase.

The orientational order parameter for uniaxial phase was calculated
from the largest eigen value obtained by diagonalization of the
order parameter tensor

\begin{equation}
\begin{array}{lr}
Q_{\alpha \beta} = \frac{1}{2 N} \sum(3 e_{i \alpha} e_{i \beta}
- \delta_{\alpha \beta})  & \alpha, \beta = x,y,z \\
\end{array}
\end{equation}

where $e_{i \alpha}$ was the $\alpha$ th component of the unit
vector $e_{i}$ along the symmetry axis of the i th molecule.
Corresponding eigenvector gave the director which defines the
average direction of molecular alignment.

The density correlation along the director is $g(z) = < \delta
(z-z_{ij})>_{ij} / \pi R^{2} \rho $, where $z_{ij} = r_{ij} cos
\beta_{r_{ij}}$ was measured in the director frame and $R$ is the
radius of the cylindrical sampling region.


\section{Results and Conclusion}
\label{sec:results and conclusion}

Analysis of the simulation results shows that relative dipolar
orientation angle of the molecules  can give rise to rich
polymorphism of polar mesophases.

The correlation function g(z) shows layering along perpendicular
direction to the plane for a system of G-B molecules with two
transverse outward pointing dipoles in fig. \ref{fig:1}. Both the
correlation plot and the snapshot (fig. \ref{fig:4}) of their
organization indicate a bilayer phase. Snapshot for larger system of
1372 molecules also confirms bilayer structure (Fig. \ref{fig:7}).
Fig. \ref{fig:2} shows g(z) for a system of molecules having two
antiparallel longitudinal dipoles and the snapshot of their
organization shows a monolayer phase (Fig. \ref{fig:5}). Fig.
\ref{fig:3} gives g(z) for a system of G-B molecules with single
transverse outward pointing dipole and fig. \ref{fig:6} gives the
snapshot. Their organization is like a wavy antiphase (stripe
domain).  Fig. \ref{fig:8} gives the snapshot for 1372 molecules
with single transverse dipole near the end of the molecule.

\begin{figure}
\begin{center}
\epsfxsize=3in \epsfbox{fig1.ps}
\end{center}
\caption { Density projection of molecular centres (solid) and
terminal dipoles (broken) with respect to the director g(z) for a
system of G-B molecules with two transverse outward pointing
dipoles, the first dipole having $d^{*}=1.0$, $\mu^{*}=1.0$ and the
second dipole having $d^{*}=0.0$, $\mu^{*}=0.5$} \label{fig:1}
\end{figure}


\begin{figure}
\begin{center}
\epsfxsize=3in \epsfbox{fig2.ps}
\end{center}
\caption { Density projection of molecular centres (solid) and
terminal dipoles (broken) with respect to the director g(z) for a
system of G-B molecules with two antiparallel longitudinal dipoles,
the first outward pointing dipole having $d^{*}=1.0$, $\mu^{*}=1.0$
and the second dipole having $d^{*}=0.0$, $\mu^{*}=0.5$}
\label{fig:2}
\end{figure}

\begin{figure}
\begin{center}
\epsfxsize=3in \epsfbox{fig3.ps}
\end{center}
\caption {Density projection of molecular centres (solid) and
terminal
 dipoles (broken) with respect to the director g(z)
for a system of G-B molecules with single transverse outward
pointing dipole, having $d^{*}=1.0$, $\mu^{*}=1.0$} \label{fig:3}
\end{figure}

\begin{figure}
\centering \epsfxsize=2.5in \epsfbox{fig4.eps} \caption{Typical
configuration for a system of 864 G-B molecules with two transverse
dipoles, the first dipole having $d^{*}=1.0$, $\mu^{*}=1.0$ and the
second dipole having $d^{*}=0.0$, $\mu^{*}=0.5$. The white caps
indicate the location of the terminal dipole, while the orientation
of the dipoles is indicated by the blue/gold coloring.}
\label{fig:4}
\end{figure}

\begin{figure}
\begin{center}
\epsfxsize=3in \epsfbox{fig5.ps}
\end{center}
\caption {Snapshot of molecular configuration for a system of 864
G-B molecules with two antiparallel longitudinal dipoles, the first
outward pointing dipole
 having $d^{*}=1.0$, $\mu^{*}=1.0$ and the second dipole having $d^{*}=0.0$,
$\mu^{*}=0.5$ (fine lines are molecular symmetry axes and small
thick lines show terminal dipolar direction, central dipoles are not
shown).} \label{fig:5}
\end{figure}


\begin{figure}
\begin{center}
\epsfxsize=3in \epsfbox{fig6.ps}
\end{center}
\caption {Snapshot of molecular configuration for  a system of 864
G-B molecules with single transverse outward pointing dipole, having
$d^{*}=1.0$, $\mu^{*}=1.0$ (fine lines are molecular symmetry axes
and small thick lines show terminal dipolar direction).}
\label{fig:6}
\end{figure}

\begin{figure}
\begin{center}
\epsfxsize=3in \epsfbox{fig7.ps}
\end{center}
\caption {Snapshot of molecular configuration for a system of 1372
G-B molecules with two transverse outward pointing dipoles, the
first dipole having $d^{*}=1.0$, $\mu^{*}=1.0$ and the second dipole
having $d^{*}=0.0$, $\mu^{*}=0.5$(fine lines are molecular symmetry
axes and small thick lines show terminal dipolar direction,  central
dipoles are not shown).} \label{fig:7}
\end{figure}

\begin{figure}
\begin{center}
\epsfxsize=3in \epsfbox{fig8.ps}
\end{center}
\caption {Snapshot of molecular configuration for a system of 1372
G-B molecules with single transverse outward pointing dipole, having
$d^{*}=1.0$, $\mu^{*}=1.0$ (fine lines are molecular symmetry axes
and small thick lines show terminal dipolar direction).}
\label{fig:8}
\end{figure}

Starting from an isotropic configuaration of polar Gay-Berne
molecules, we could successfully simulate perfect bilayer, antiphase
and monolayer structure. To break the up-down symmetry i.e. the
nonequivalence of directions ${\bf \hat {n}}$ and ${ -\bf \hat{n}}$,
the molecules should have permanent electric or magnetic dipoles.
Longitudinal electric dipole interaction could not form polar
nematic phase as orientationally disordered phase with larger
entropy is stabler than polarly ordered phase. In fact, stronger
central dipole moment opposes polar nematic ordering more
effectively in case of rod-like molecules. However, polar ordering
like bilayer $A_{2}$, interdigitated $A_{d}$, and wavy $\tilde A$ in
smectic layers can be achieved, where adjacent layers with opposite
polarities makes bulk phase a-polar. More so, lyotropic liquid
crystals and bilayer bio-membranes can have polar layers. These
arrangements appear to get favours with the shifting of longitudinal
dipole moment to the molecular terminus, so that they can have
anti-ferroelectric dipolar arrangement giving rise to local (within
the sublayer) breaking of up-down symmetry along the director.
Transverse polarity breaks two-fold rotational symmetry, which
favours more in-plane polar order. However, the molecular origin of
these phases requires something more which are apparent from the
earlier simulation results. We have shown that to get perfect
bilayer structure in a G-B system, alongwith transverse terminal
dipole, another central dipole (or a polarizable core) is required
so that polar head and a-polar tail of Gay-Berne molecules go to
opposite directions within a bilayer. This gives some kind of
clipping interactions which forbid the molecular tail go in other
way. Moreover, we could simulate other varieties of polar smectic
phases e.g. monolayer $A_{1}$, antiphase $\tilde A$ successfully.
Apart from guiding chemical synthesization of ferroelectric,
antiferroelectric liquid crystals for technological applications,
the present study will be of scientific interest in understanding
molecular level interactions of lyotropic liquid crystals as well as
nature-designed bio-membranes.

\section{\label{liquidCrystalSection:methods}Methods}

\section{\label{liquidCrystalSection:resultDiscussion}Results and Discussion}

\section{Conclusion}
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