--- trunk/tengDissertation/LiquidCrystal.tex	2006/06/04 20:18:07	2786
+++ trunk/tengDissertation/LiquidCrystal.tex	2006/06/27 03:06:49	2896
@@ -38,13 +38,15 @@ Fig.~\cite{LCFig:SMCP}).
 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.~\cite{LCFig:SMCP}).
+Fig.~\ref{LCFig:SMCP}).
 
 \begin{figure}
 \centering
 \includegraphics[width=\linewidth]{smcp.eps}
-\caption[]
-{}
+\caption[SmCP Phase Packing] {Four possible SmCP phase packings that
+are characterized by the relative tilt direction(A and S refer an
+anticlinic tilt or a synclinic ) and the polarization orientation (A
+and F represent antiferroelectric or ferroelectric polar order).}
 \label{LCFig:SMCP}
 \end{figure}
 
@@ -76,7 +78,7 @@ of bent-core molecules\cite{Memmer2002,Orlandi2006}.
 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}.
+of bent-core molecules\cite{Memmer2002}.
 
 Experimental studies by Levelut {\it et al.}~\cite{Levelut1981}
 revealed that terminal cyano or nitro groups usually induce
@@ -98,11 +100,10 @@ molecule represented by three rigid GB particles with 
 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.
+molecule represented by three rigid GB particles with two point
+dipoles. 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}
 
@@ -181,21 +182,18 @@ ratio between \textit{end-to-end} well depth $\epsilon
 \begin{figure}
 \centering
 \includegraphics[width=\linewidth]{banana.eps}
-\caption[]{} \label{LCFig:BananaMolecule}
+\caption[Schematic representation of a typical banana shaped
+molecule]{Schematic representation of a typical banana shaped
+molecule.} \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}
+\caption[Schematic diagram showing definitions of the orientation
+vectors for a pair of Gay-Berne molecules]{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
@@ -208,6 +206,165 @@ where $\epsilon _{fs}$ is the permittivity of free spa
 \end{equation}
 where $\epsilon _{fs}$ is the permittivity of free space.
 
-\section{\label{liquidCrystalSection:methods}Methods}
+\section{Results and Discussion}
 
-\section{\label{liquidCrystalSection:resultDiscussion}Results and Discussion}
+A series of molecular dynamics simulations were perform to study the
+phase behavior of banana shaped liquid crystals. In each simulation,
+every banana shaped molecule has been represented by 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. NPTi Production runs last for 40~ns with
+time step of 20~fs.
+
+\subsection{Order Parameters}
+
+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}
+\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:p2}
+\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.
+The unit vector for the banana shaped molecule was defined by the
+principle aixs of its middle GB particle. The $P_2$ order parameters
+for the bent-core liquid crystal at different temperature is
+summarized in Table~\ref{liquidCrystal:p2} which identifies a phase
+transition temperature range.
+
+\begin{table}
+\caption{LIQUID CRYSTAL STRUCTURAL PROPERTIES AS A FUNCTION OF
+TEMPERATURE} \label{liquidCrystal:p2}
+\begin{center}
+\begin{tabular}{cccccc}
+\hline
+Temperature (K) & 420 & 440 & 460 & 480 & 600\\
+\hline
+$\langle P_2\rangle$ & 0.984 & 0.982 & 0.975 & 0.967 & 0.067\\
+\hline
+\end{tabular}
+\end{center}
+\end{table}
+
+\subsection{Structure Properties}
+
+The molecular organization obtained at temperature $T = 460K$ (below
+transition temperature) is shown in Figure~\ref{LCFigure:snapshot}.
+The diagonal view in Fig~\ref{LCFigure:snapshot}(a) shows the
+stacking of the banana shaped molecules while the side view in n
+Figure~\ref{LCFigure:snapshot}(b) demonstrates formation of a
+chevron structure. The first peak of Radial distribution function
+$g(r)$ in Fig.~\ref{LCFigure:gofrz}(a) shows the minimum distance
+for two in plane banana shaped molecules is 4.9 \AA, while the
+second split peak implies the biaxial packing. It is also important
+to show the density correlation along the director which is given by
+:
+\begin{equation}
+g(z) =\frac{1}{\pi R^{2} \rho}< \delta (z-z_{ij})>_{ij}
+\end{equation},
+where $z_{ij} = r_{ij} \dot Z$ was measured in the director frame
+and $R$ is the radius of the cylindrical sampling region. The
+oscillation in density plot along the director in
+Fig.~\ref{LCFigure:gofrz}(b) implies the existence of the layered
+structure, and the peak at 27 \AA is attribute to the defect in the
+system.
+
+\subsection{Rotational Invariants}
+
+As a useful set of correlation functions to describe
+position-orientation correlation, rotation invariants were first
+applied in a spherical symmetric system to study x-ray and light
+scatting\cite{Blum1972}. Latterly, expansion of the orientation pair
+correlation in terms of rotation invariant for molecules of
+arbitrary shape was introduce by Stone\cite{Stone1978} and adopted
+by other researchers in liquid crystal studies\cite{Berardi2003}. In
+order to study the correlation between biaxiality and molecular
+separation distance $r$, we calculate a rotational invariant
+function $S_{22}^{220} (r)$, which is given by :
+\begin{eqnarray}
+S_{22}^{220} (r) & = & \frac{1}{{4\sqrt 5 }} \left< \delta (r -
+r_{ij} )((\hat x_i  \cdot \hat x_j )^2  - (\hat x_i  \cdot \hat y_j
+)^2  - (\hat y_i  \cdot \hat x_j )^2  + (\hat y_i  \cdot \hat y_j
+)^2 ) \right. \notag \\
+ & & \left. - 2(\hat x_i  \cdot \hat y_j )(\hat y_i \cdot \hat x_j ) -
+2(\hat x_i  \cdot \hat x_j )(\hat y_i  \cdot \hat y_j )) \right>.
+\end{eqnarray}
+The long range behavior of second rank orientational correlation
+$S_{22}^{220} (r)$ in Fig~\ref{LCFigure:S22220} also confirm the
+biaxiality of the system.
+
+\begin{figure}
+\centering
+\includegraphics[width=4.5in]{snapshot.eps}
+\caption[Snapshot of the molecular organization in the layered phase
+formed at temperature T = 460K and pressure P = 1 atm]{Snapshot of
+the molecular organization in the layered phase formed at
+temperature T = 460K and pressure P = 1 atm. (a) diagonal view; (b)
+side view.} \label{LCFigure:snapshot}
+\end{figure}
+
+\begin{figure}
+\centering
+\includegraphics[width=\linewidth]{gofr_gofz.eps}
+\caption[Correlation Functions of a Bent-core Liquid Crystal System
+at Temperature T = 460K and Pressure P = 10 atm]{Correlation
+Functions of a Bent-core Liquid Crystal System at Temperature T =
+460K and Pressure P = 10 atm. (a) radial correlation function
+$g(r)$; and (b) density along the director $g(z)$.}
+\label{LCFigure:gofrz}
+\end{figure}
+
+\begin{figure}
+\centering
+\includegraphics[width=\linewidth]{s22_220.eps}
+\caption[Average orientational correlation Correlation Functions of
+a Bent-core Liquid Crystal System at Temperature T = 460K and
+Pressure P = 10 atm]{Correlation Functions of a Bent-core Liquid
+Crystal System at Temperature T = 460K and Pressure P = 10 atm. (a)
+radial correlation function $g(r)$; and (b) density along the
+director $g(z)$.} \label{LCFigure:S22220}
+\end{figure}
+
+\section{Conclusion}
+
+We have presented a simple dipolar three-site GB model for banana
+shaped molecules which are capable of forming smectic phases from
+isotropic configuration. Various order parameters and correlation
+functions were used to characterized the structural properties of
+these smectic phase. However, the forming layered structure still
+had some defects because of the mismatching between the layer
+structure spacing and the shape of simulation box. This mismatching
+can be broken by using NPTf integrator in further simulations. The
+role of terminal chain in controlling transition temperatures and
+the type of mesophase formed have been studied
+extensively\cite{Pelzl1999}. The the lack of flexibility in our
+model due to the missing terminal chains could explained the fact
+that we did not find evidence of chirality.
