--- trunk/tengDissertation/LiquidCrystal.tex	2006/06/19 17:55:26	2869
+++ trunk/tengDissertation/LiquidCrystal.tex	2006/06/20 13:46:10	2870
@@ -211,245 +211,67 @@ phase behavior of banana shaped liquid crystals.
 \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
+phase behavior of banana shaped liquid crystals. In each simulation,
+every banana shaped molecule has been represented three GB particles
+which is characterized by $\mu = 1,~ \nu = 2,
+~\epsilon_{e}/\epsilon_{s} = 1/5$ and $\sigma_{e}/\sigma_{s} = 3$.
+All of the simulations begin with same equilibrated isotropic
+configuration where 1024 molecules without dipoles were confined in
+a $160\times 160 \times 120$ box. After the dipolar interactions are
+switched on, 2~ns NPTi cooling run with themostat of 2~ps and
+barostat of 50~ps were used to equilibrate the system to desired
+temperature and pressure.
 
+To investigate the phase structure of the model liquid crystal, we
+calculated various order parameters and correlation functions.
+Particulary, the $P_2$ order parameter allows us to estimate average
+alignment along the director axis $Z$ which can be identified from
+the largest eigen value obtained by diagonalizing 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}
+\overleftrightarrow{\mathsf{Q}} = \frac{1}{N}\sum_i^N %
+    \begin{pmatrix} %
+    u_{ix}u_{ix}-\frac{1}{3} & u_{ix}u_{iy} & u_{ix}u_{iz} \\
+    u_{iy}u_{ix} & u_{iy}u_{iy}-\frac{1}{3} & u_{iy}u_{iz} \\
+    u_{iz}u_{ix} & u_{iz}u_{iy} & u_{iz}u_{iz}-\frac{1}{3} %
+    \end{pmatrix},
+\label{lipidEq:po1}
 \end{equation}
+where the $u_{i\alpha}$ is the $\alpha$ element of the unit vector
+$\mathbf{\hat{u}}_i$, and the sum over $i$ averages over the whole
+collection of unit vectors. The $P_2$ order parameter for uniaxial
+phase is then simply given by
+\begin{equation}
+\langle P_2 \rangle = \frac{3}{2}\lambda_{\text{max}}.
+\label{lipidEq:po3}
+\end{equation}
+In addition to the $P_2$ order parameter, $ R_{2,2}^2$ order
+parameter for biaxial phase is introduced to describe the ordering
+in the plane orthogonal to the director by
+\begin{equation}
+R_{2,2}^2  = \frac{1}{4}\left\langle {(x_i  \cdot X)^2  - (x_i \cdot
+Y)^2  - (y_i  \cdot X)^2  + (y_i  \cdot Y)^2 } \right\rangle
+\end{equation}
+where $X$, $Y$ and $Z$ are axis of the director frame.
 
-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.
+The density correlation along the director is
+\begin{equation}g(z) =< \delta (z-z_{ij})>_{ij} / \pi R^{2} \rho
+\end{equation},
+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.
+To investigate the molecular organization behavior due to different
+dipolar orientation and position with respect to the center 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}