trunk/gb-bilayer/bilayer.tex

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\title{Bilayer phase and antiphase formation in polar liquid crystals}

\author{J. Saha\footnote{Current Address: Department of Physics, Visva-Bharati
University, Santiniketan - 731235, West Bengal, India}  and
J. Daniel Gezelter\footnote{e-mail: gezelter@nd.edu}}

\affiliation{Department of Chemistry and Biochemistry, University of
Notre Dame, Notre Dame, Indiana 46556}

\date{\today}

\begin{abstract}
We present results of a series of Molecular Dynamics simulations on
the molecular organization of systems of ellipsoidal Gay-Berne
molecules containing two fixed dipole moments. The effects of relative
dipolar orientations and positions on the generation of bilayer
lamellar phase, antiphases and monolayer phases has been studied.  We
report on the structural features of the phases formed by molecules of
both liquid crystalline and biological interest.
\end{abstract}

\pacs{61.30}

\keywords{Gay-Berne molecule, two dipoles, molecular dynamics, bilayer
phase, antiphase} 

\maketitle

\newpage
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\section{Introduction}
\label{sec:intro}

Long range orientational order is the most fundamental property of
liquid crystal mesophases, while positional order is limited or
absent. This orientational anisotropy of the macroscopic phases
originates in the anisotropy of the constituent molecules. For the
existence of orientational ordering, these molecules typically have
highly non-spherical structure with some degree of rigidity. Liquid
crystalline compounds typically possess cylindrically symmetric rod or
disc-like rigid core structures and usually have flexible substituents
associated with these central regions. In nematic phases, rod-like
molecules are orientationally ordered with isotropic distributions of
molecular centers of mass. In smectic phases, the molecules arrange
themselves into layers with their long (symmetry) axis normal
($S_{A}$) or tilted ($S_{C}$) with respect to the layer planes. The
layers themselves are two dimensional liquids.

\subsection{Previous Experimental Work}

Experimental and theoretical studies of smectic liquid crystals
include a range of polymorphic variations on the basic smectic
organization. The behaviour of the $S_{A}$ phase can be explained with
theoretical models mainly based on geometric factors and van der Waals
interactions.  However, these simple models are insufficient to
describe liquid crystal phases which exhibit more complex polymorphic
nature. X-ray diffraction studies have shown that the ratio between
lamellar spacing ($s$) and molecular length ($l$) can take
significantly different values.\cite{Leadbetter77,Gray84} Typical
$S_{A}$ phases have $s/l$ ratios on the order of $0.8$ , whereas for
some compounds e.g. 4-alkyl-4'-cyanobiphenyls the $s/l$ ratio is on
the order of $1.4$. Experiments show that depending on dipole
delocalization within the molecules, $s$ can take values ranging from
the length of a single molecule to twice the molecular
length~\cite{Leadbetter77,Hardouin80}. Extensive experimental studies
reveal that compounds of the $S_{A}$ type which show a variety of
phases like monolayers ($S_{A1}$), uniform bilayers ($S_{A2}$),
partial bilayers ($S_{\tilde A}$) and interdigitated bilayers
($S_{A_{d}}$), usually have a terminal cyano or nitro group. These
classes of liquid crystal and in particular, lyotropic liquid crystals
(those exhibiting liquid crystal phase transition as a function of
water concentration) which form surfactant and lipid membranes, often
have polar head groups or zwitterionic charge separated groups that
result in strong dipolar interactions.\cite{Collings97} Apart from the
uniaxial $S_{A}$ phases mentioned above, compounds having permanent
dipole moments can also give rise to reentrant smectic, biaxial
($S_{C}$) and ferroelectric phases. Because of their versatile
polymorphic nature, these liquid crystalline materials have important
technological applications in addition to their immense relevance to
biological systems.\cite{Collings97}

Experimental studies by Levelut {\it et
al.}~\cite{Levelut81a,Levelut81b} revealed that terminal cyano or
nitro groups usually induce permanent longitudinal dipole moments on
the molecules.  Strong lateral dipole $(C=O)$ and terminal transverse
dipoles can also effect the phase behaviour considerably. Many
liquid-crystal forming molecules of biological interest
(e.g. phospholipids) could be modelled more accurately with {\em
transverse} or angled dipole moments at the terminus of the molecule.
In particular, the dipole moment of phosphatidylcholine (PC) head
groups are typically oriented perpendicular to the molecular axis,
while the dipole of phosphatidylethanolamine (PE) head groups are
tilted relative to the molecular axis.  Moreover, there is strong
indication from the molecular structure of liquid crystalline
molecules that form these phases, that in adddition to the terminal
dipole, it is advantageous to have the presence of aromatic $\pi$
bonds and/or other dipoles near the core which can be easily polarized
by the strong electron withdrawing properties of the terminal group,
resulting in the various $S_{A}$
phases.~\cite{Gray84,Jeu83,Levelut81a,Levelut81b}

\subsection{Previous theoretical work}

Theoretical models of polar smectics using generic molecular field
descriptions were studied by Photinos, Saupe and
others.~\cite{Photinos76,Vanakaras98} Prost coupled two different
order parameters (the density wave and dipolar ordering of molecules
along the director) and using a mean field approch showed that
competition between them was responsible for polar liquid crystal
behaviour.~\cite{Prost84,Prost80} Comparing some polar liquid crystal
compounds, de~Jeu inferred that variation of dipole correlation with
molecular structure could affect the phase behaviour of polar liquid
crystals.~\cite{Jeu83} A number of other analytical approaches to
these phases have been presented in the
literature,~\cite{Meyer76,Dowell85,Indekeu86,Baus89,Netz92} but most
have been too simplified to confirm any more than the qualitative
behavior of these phases.  Therefore, it seems that a molecular-scale
simulation approach will be required for a more complete understanding
of bilayer and antiphases.

Levesque {\em et al.} presented a hard rod model which exhibited
monolayers and further indicated that the very symmetric nature of the
potential was responsible for their failure to generate true
bilayer.~\cite{Levesque93} Reproduction of small domains of bilayers
was reported for transverse dipoles but these were not always present
in their simulations.
 
The Gay-Berne potential has seen widespread use in the liquid crystal
community to describe this anisotropic phase
behavior.~\cite{Gay81,Berne72,Kushick76,Luckhurst90,Perram96} It is an
appropriate model for simulation of these systems because fairly rigid
liquid crystal-forming molecules maintain their rod-like or disc-like
shapes, which produce interactions that favor local alignment.  In its
original form, the Gay-Berne potential was a computationally efficient
{\em single} site model for the interactions of rigid ellipsoidal
molecules.~\cite{Gay81} It can be thought of as a modification of the
Gaussian overlap model originally described by Berne and
Pechukas.~\cite{Berne72} The potential is constructed in the familiar
form of the Lennard-Jones function using orientation-dependent
$\sigma$ and $\epsilon$ parameters. The functional form for the
potential is given in section 2.  Luckhurst has given a particularly
good explanation of the choice of the Gay-Berne parameters $\mu$ and
$\nu$ for modeling non-polar liquid crystal molecules.

Although there have been a large number of studies of the phase
behaviour of the {\em non-polar} Gay-Berne potential using Monte Carlo
and Molecular Dynamics techniques,~\cite{Zannonibook2000} there has
been comparatively little work done on an important class of liquid
crystal forming molecules which have {\em dipolar} interactions.
There have been some preliminary Monte Carlo studies of the Gay-Berne
potential with fixed longitudinal dipoles (i.e. pointed along the
principal axis of rotation).~\cite{Berardi96,Satoh96} There have also
been some recent molecular dynamics simulations on polar Gay-Berne
models by Pasterny {\em et al.}~\cite{Pasterny2000} Zannoni's group
has studied the phase behavior of Gay-Berne ellipsoids with
longitudinal and transverse dipoles both at the midpoint and terminus
of the molecule.~\cite{Berardi99} Their exhaustive simulation with a
model potential comprising both the attractive-repulsive G-B
interaction and a single dipolar interaction exhibited partial striped
bilayer structures.  

MISSING: AYTON AND VOTH WORK ON GB with terminal LJ spheres.

Simulations of small domains of local bilayers for nCB {WHAT IS NCB}
have been reported recently.~\cite{Fukunaga2004}

Since none of these simulations, which considered molecules with
single dipoles and studied the effects of either the terminal or the
central dipoles seperately, were able to reproduce perfect bilayer
arrangement the molecular origin of these liquid crystal phases is not
very well understood.  There is, of course, a vast literature of
all-atom, or coarse-grained simulation models for lipid bilayers, but
typical system sizes with these models allow only relatively small
patches of single or double bilayers to be studied.  In this work, we
present a model that is simple enough to allow us to probe the
equilibrated phase behavior at a number of different conditions, while
still maintaining enough molecular-scale realism to be useful as a
predictive tool.

To mimic the terminal dipolar interaction coupled with polar cores we
considered systems comprising Gay-Berne particles with an embedded
terminal dipole and another weaker central dipole. Performing a series
of molecular dynamics simulations, we studied the structural
properties of these phases in systems of prolate ellipsoidal particles
each having relatively two dipole moments oriented perpendicularly
with respect to their respective molecular symmetry axes.  In this
paper we report the generation of bilayer, monolayer and wavy
antiphase structures.  To our knowledge, the present simulation work
is the first of its kind which could generate the bilayer liquid
crystalline phase successfully along with other important
experimentally-observed phases.

\section{Model}
\label{sec:model}

In this work, rod-like polar molecules are modelled as prolate
ellipsoidal Gay-Berne (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}
\begin{array}{ll}
V^{GB}_{ij}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}) = 
4 \epsilon({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}})
  & \left[ \left(
\frac{\sigma_{o}}{r - \sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf
\hat{r}})+\sigma_{o}} \right)^{12} \right. \\ \\
& - \left. \left(
\frac{\sigma_{o}}{r - \sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf
\hat{r}})+\sigma_{o}} 
\right)^{6}
 \right], \\
\end{array}
\label{eq:gb}
\end {equation}
where ${\bf \hat{u}_{i},\hat{u}_{j}}$ are unit vectors specifying the
orientation of two molecules $i$ and $j$ separated by intermolecular
vector ${\bf r}$. ${\bf \hat{r}}$ is the unit vector along the
intermolecular vector. 

The functional form for $\sigma$ is given by 
\begin {equation}
\begin{array}{ll}
\sigma ({\bf \hat{u}_{i}, \hat{u}_{j},\hat{r}}) = \sigma_{0}
 &  \left[ 1- \frac {\chi}{2} \left( \frac{({\bf \hat{r}}.{\bf \hat{u}_{i}}+{\bf \hat{r}}.
{\bf \hat {u}_{j}})^2}{1+\chi
 ({\bf \hat{u}_{i}}.{\bf \hat{u}_{j}})} \right. \right. \\ \\
& \left.\left. + \frac {({\bf \hat{r}}.{\bf \hat{u}_{i
}}-{\bf \hat{r}}.
{\bf \hat{u}_{j}})^2}{1-\chi ({\bf \hat{u}_{i}}.{\bf \hat{u}_{j}})} \right)
 \right ]^{-1/2}
\end{array}
\end {equation}

The aspect ratio of the particles is governed by shape anisotropy
parameter
\begin {equation}
\begin{array}{rcl}
\chi & = & \frac
{(\sigma_{e}/\sigma_{s})^2-1}{(\sigma_{e}/\sigma_{s})^2+1} \\ \\
\end{array}
\label{eq:chi}
\end {equation}
Here, the subscript $s$ indicates the {\it side-by-side} configuration
where $\sigma$ has its minimum value, $\sigma_{s}$, and where the
potential well is $\epsilon_{s}$ deep.  The subscript $e$ refers to
the {\it end-to-end} configuration where $\sigma$ has its maximum
value, $\sigma_{e}$, and where the well depth, $\epsilon_{e}$ is
somewhat smaller than in the side-by-side configuration.  For prolate
ellipsoids, we have
\begin{equation}
\begin{array}{rcl}
\sigma_{s} & < & \sigma_{e} \\
\epsilon_{s} & > & \epsilon_{e}
\end{array}
\end{equation}
where, $\sigma_{e}$ is the measure of the length and $\sigma_{s}$ is
the breadth of a molecule.  The shape anisotropy parameter $\chi$ has
a functional dependence on the length to breadth ratio (i.e. the
aspect ratio of the particles.).

The functional form of well depth is
\begin {equation}
\epsilon({\bf \hat{u}_{i},\hat{u}_{j},\hat{r}}) = \epsilon_{0}
 \epsilon^{\nu}({\bf \hat{u}_{i}.\hat{u}_{j}})
 \epsilon^{\prime\mu}({\bf \hat{u}_{i},\hat{u}_{j},\hat{r}})
\end {equation}
where $\epsilon_{0}$ is a constant term and
\begin {equation}
\epsilon ({\bf \hat{u}_{i},\hat{u}_{j}}) =
 [1-\chi^{2}({\bf \hat{u}_{i}.\hat{u}_{j}})^{2}]^{-1/2}
\end {equation}
and
\begin {equation}
\epsilon^{\prime} ({\bf \hat{u}_{i}, \hat{u}_{j},\hat{r}})
 =  1- \frac {\chi^{'}}{2} \left
 ( \frac {({\bf \hat{r}}.{\bf \hat{u}_{i}}+{\bf\hat{r}}.
{\bf \hat{u}_{j}})^2}{1+\chi^{\prime}
 ({\bf \hat{u}_{i}}.{\bf \hat{u}_{j}})} +\frac {({\bf\hat{ r}}.{\bf\hat{
 u}_{i}}-{\bf \hat{r}}.
{\bf \hat{u}_{j}})^2}{1-\chi^{\prime} ({\bf \hat{u}_{i}}.{\bf
\hat{u}_{j}})} \right) 
 \end {equation}
where the well depth anisotropy parameter $\chi\prime$ can be expressed as
\begin {equation}
\chi^{\prime} = \frac {1-(\epsilon_{e}/\epsilon_{s})^{1/\mu}}{1+(\epsilon_{e}/
\epsilon_{s})^{1/\mu}}.
\end {equation}


As the molecules have two dipoles, for each pair of them (4 pairs for
two interacting molecules), there should be an electrostatic
interaction term of the form

\begin {equation}
U_{dd} =   \frac { \mu^{*}_{id} \mu^{*}_{jd}}{r_{d}^{3}} 
 \left [({\bf \hat{u}_{id}.\hat{u}_{jd}}) - 3 ({\bf \hat{u}_{id}. \hat{r}_{d}})(
{\bf \hat{u}_{jd}. \hat{r}_{d}}) \right ] 
\end {equation}

where reduced dipole moment

\begin{equation}
\mu_{d}^{*} = \frac {\mu_{d}^{2}}
{(4 \pi {\it \epsilon} \epsilon_{s} \sigma_{s}^{3})^{1/2}}
\end{equation}

$\it {\epsilon}$ is the permitivity of the free space and $r_{d}$ is
the unit vector along the vector joining the two dipoles. $\bf \hat{u}_{id}$
 and
$\bf \hat{u}_{jd}$ are the direction of the unit vectors along the direction of
the dipoles situated on molecules i and j respectively.
 

So the total interaction potential for a pair of polar molecules is
the sum of attractive-repulsive term and dipole-dople interaction term,
which can be expressed as 

\begin {equation}
U_{ij} = U_{GB} + (U_{dd})_{1st~ dipole} + (U_{dd})_{2nd~ dipole}
\end {equation}

To simulate systems of dipolar Gay-Berne
particles with different relative dipolar orientations and positions, we
used this model potential.

\begin{figure}
\begin{center}
\epsfxsize=3in
\epsfbox{system_sketch.eps}
\end{center}
\caption{The molecular models studied in this work.  All are prolate
Gay-Berne ellipsoids which have point dipoles embedded centrally or
terminally within the molecular bodies.  System A is a molecular-scale
model for XXXX, System B is a model of YYYY, and System C could be
considered a model for phosphatidylcholine (PC) lipids.  Details on
the dipolar locations and strengths are given in the text.}
\label{fig:gbdp}
\end{figure}


\section{Computational Methodology}
We performed a series of extensive Molecular Dynamics (MD) simulations to 
study 
the phase behaviour of a family of polar 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.

\begin{acknowledgments}
Support for this project was provided by the National Science
Foundation under grant CHE-0134881. Computation time was provided by
the Notre Dame High Performance Computing Cluster and the Notre Dame
Bunch-of-Boxes (B.o.B) computer cluster (NSF grant DMR-0079647).
\end{acknowledgments}


\bibliography{bilayer}

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\end {document}
 

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