--- trunk/tengDissertation/LiquidCrystal.tex	2006/06/09 02:41:58	2839
+++ trunk/tengDissertation/LiquidCrystal.tex	2006/06/17 18:43:58	2867
@@ -207,7 +207,251 @@ where $\epsilon _{fs}$ is the permittivity of free spa
 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}
