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Comparing trunk/tengDissertation/LiquidCrystal.tex (file contents):
Revision 2839 by tim, Fri Jun 9 02:41:58 2006 UTC vs.
Revision 2867 by tim, Sat Jun 17 18:43:58 2006 UTC

+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 $I_{\perp}^{*} =
-+
+.0$.  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
-+
+.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 $r_{c} =
-+
+.8 \sigma_{0}$. 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
-+
+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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Comparing trunk/tengDissertation/LiquidCrystal.tex (file contents): Revision 2839 by tim, Fri Jun 9 02:41:58 2006 UTC vs. Revision 2867 by tim, Sat Jun 17 18:43:58 2006 UTC

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Comparing trunk/tengDissertation/LiquidCrystal.tex (file contents):
Revision 2839 by tim, Fri Jun 9 02:41:58 2006 UTC vs.
Revision 2867 by tim, Sat Jun 17 18:43:58 2006 UTC