--- trunk/tengDissertation/LiquidCrystal.tex	2006/05/31 04:29:44	2781
+++ trunk/tengDissertation/LiquidCrystal.tex	2006/06/27 03:07:14	2897
@@ -1,47 +1,370 @@
 \chapter{\label{chapt:liquidcrystal}LIQUID CRYSTAL}
 
 \section{\label{liquidCrystalSection:introduction}Introduction}
-% liquid crystal
 
 Long range orientational order is one of the most fundamental
 properties of liquid crystal mesophases. This orientational
 anisotropy of the macroscopic phases originates in the shape
 anisotropy of the constituent molecules. Among these anisotropy
 mesogens, rod-like (calamitic) and disk-like molecules have been
-exploited in great detail in the last two decades. Typically, these
-mesogens consist of a rigid aromatic core and one or more attached
-aliphatic chains. For short chain molecules, only nematic phases, in
-which positional order is limited or absent, can be observed,
-because the entropy of mixing different parts of the mesogens is
-paramount to the dispersion interaction. In contrast, formation of
-the one dimension lamellar sematic phase in rod-like molecules with
-sufficiently long aliphatic chains has been reported, as well as the
-segregation phenomena in disk-like molecules.
+exploited in great detail in the last two decades\cite{Huh2004}.
+Typically, these mesogens consist of a rigid aromatic core and one
+or more attached aliphatic chains. For short chain molecules, only
+nematic phases, in which positional order is limited or absent, can
+be observed, because the entropy of mixing different parts of the
+mesogens is paramount to the dispersion interaction. In contrast,
+formation of the one dimension lamellar sematic phase in rod-like
+molecules with sufficiently long aliphatic chains has been reported,
+as well as the segregation phenomena in disk-like molecules.
 
-% banana shaped
 Recently, the banana-shaped or bent-core liquid crystal have became
 one of the most active research areas in mesogenic materials and
-supramolecular chemistry. Unlike rods and disks, the polarity and
-biaxiality of the banana-shaped molecules allow the molecules
-organize into a variety of novel liquid crystalline phases which
-show interesting material properties. Of particular interest is the
-spontaneous formation of macroscopic chiral layers from achiral
-banana-shaped molecules, where polar molecule orientational ordering
-is shown within the layer plane as well as the tilted arrangement of
-the molecules relative to the polar axis. As a consequence of
-supramolecular chirality, the spontaneous polarization arises in
-ferroelectric (FE) and antiferroelectic (AF) switching of smectic
-liquid crystal phases, demonstrating some promising applications in
-second-order nonlinear optical devices.
+supramolecular chemistry\cite{Niori1996, Link1997, Pelzl1999}.
+Unlike rods and disks, the polarity and biaxiality of the
+banana-shaped molecules allow the molecules organize into a variety
+of novel liquid crystalline phases which show interesting material
+properties. Of particular interest is the spontaneous formation of
+macroscopic chiral layers from achiral banana-shaped molecules,
+where polar molecule orientational ordering is shown within the
+layer plane as well as the tilted arrangement of the molecules
+relative to the polar axis. As a consequence of supramolecular
+chirality, the spontaneous polarization arises in ferroelectric (FE)
+and antiferroelectic (AF) switching of smectic liquid crystal
+phases, demonstrating some promising applications in second-order
+nonlinear optical devices. The most widely investigated mesophase
+formed by banana-shaped moleculed is the $\text{B}_2$ phase, which
+is also referred to as $\text{SmCP}$\cite{Link1997}. Of the most
+important discover in this tilt lamellar phase is the four distinct
+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.~\ref{LCFig:SMCP}).
 
-The most widely investigated mesophase formed by banana-shaped
-moleculed is the $\text{B}_2$ phase, which is also known as
-$\text{SmCP}$.
+\begin{figure}
+\centering
+\includegraphics[width=\linewidth]{smcp.eps}
+\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}
 
-%Previous Theoretical Studies
+Many liquid crystal synthesis experiments suggest that the
+occurrence of polarity and chirality strongly relies on the
+molecular structure and intermolecular interaction\cite{Reddy2006}.
+From a theoretical point of view, it is of fundamental interest to
+study the structural properties of liquid crystal phases formed by
+banana-shaped molecules and understand their connection to the
+molecular structure, especially with respect to the spontaneous
+achiral symmetry breaking. As a complementary tool to experiment,
+computer simulation can provide unique insight into molecular
+ordering and phase behavior, and hence improve the development of
+new experiments and theories. In the last two decades, all-atom
+models have been adopted to investigate the structural properties of
+smectic arrangements\cite{Cook2000, Lansac2001}, as well as other
+bulk properties, such as rotational viscosity and flexoelectric
+coefficients\cite{Cheung2002, Cheung2004}. However, due to the
+limitation of time scale required for phase transition and the
+length scale required for representing bulk behavior,
+models\cite{Perram1985, Gay1981}, which are based on the observation
+that liquid crystal order is exhibited by a range of non-molecular
+bodies with high shape anisotropies, became the dominant models in
+the field of liquid crystal phase behavior. Previous simulation
+studies using hard spherocylinder dimer model\cite{Camp1999} produce
+nematic phases, while hard rod simulation studies identified a
+Landau point\cite{Bates2005}, at which the isotropic phase undergoes
+a direct transition to the biaxial nematic, as well as some possible
+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}.
 
+Experimental studies by Levelut {\it et al.}~\cite{Levelut1981}
+revealed that terminal cyano or nitro groups usually induce
+permanent longitudinal dipole moments, which affect the phase
+behavior considerably. A series of theoretical studies also drawn
+equivalent conclusions. Monte Carlo studies of the GB potential with
+fixed longitudinal dipoles (i.e. pointed along the principal axis of
+rotation) were shown to enhance smectic phase
+stability~\cite{Berardi1996,Satoh1996}. Molecular simulation of GB
+ellipsoids with transverse dipoles at the terminus of the molecule
+also demonstrated that partial striped bilayer structures were
+developed from the smectic phase ~\cite{Berardi1996}. More
+significant effects have been shown by including multiple
+electrostatic moments. Adding longitudinal point quadrupole moments
+to rod-shaped GB mesogens, Withers \textit{et al} induced tilted
+smectic behaviour in the molecular system~\cite{Withers2003}. Thus,
+it is clear that many liquid-crystal forming molecules, specially,
+bent-core molecules, could be modeled more accurately by
+incorporating electrostatic interaction.
+
+In this chapter, we consider system consisting of banana-shaped
+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}
 
-\section{\label{liquidCrystalSection:methods}Methods}
+A typical banana-shaped molecule consists of a rigid aromatic
+central bent unit with several rod-like wings which are held
+together by some linking units and terminal chains (see
+Fig.~\ref{LCFig:BananaMolecule}). In this work, each banana-shaped
+mesogen has been modeled as a rigid body consisting of three
+equivalent prolate ellipsoidal 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. The potential between a pair of three-site
+banana-shaped molecules $a$ and $b$ is given by
+\begin{equation}
+V_{ab}^{GB}  = \sum\limits_{i \in a,j \in b} {V_{ij}^{GB} }.
+\end{equation}
+Every site-site interaction can can be expressed as,
+\begin{equation}
+V_{ij}^{GB}  = 4\epsilon (\hat u_i ,\hat u_j ,\hat r_{ij} )\left[
+{\left( {\frac{{\sigma _0 }}{{r_{ij}  - \sigma (\hat u_i ,\hat u_j
+,\hat r_{ij} )}}} \right)^{12}  - \left( {\frac{{\sigma _0
+}}{{r_{ij}  - \sigma (\hat u_i ,\hat u_j ,\hat r_{ij} )}}} \right)^6
+} \right] \label{LCEquation:gb}
+\end{equation}
+where $\hat u_i,\hat u_j$ are unit vectors specifying the
+orientation of two molecules $i$ and $j$ separated by intermolecular
+vector $r_{ij}$. $\hat r_{ij}$ is the unit vector along the
+intermolecular vector. A schematic diagram of the orientation
+vectors is shown in Fig.\ref{LCFigure:GBScheme}. The functional form
+for $\sigma$ is given by
+\begin{equation}
+\sigma (\hat u_i ,\hat u_i ,\hat r_{ij} ) = \sigma _0 \left[ {1 -
+\frac{\chi }{2}\left( {\frac{{(\hat r_{ij}  \cdot \hat u_i  + \hat
+r_{ij}  \cdot \hat u_j )^2 }}{{1 + \chi \hat u_i  \cdot \hat u_j }}
++ \frac{{(\hat r_{ij}  \cdot \hat u_i  - \hat r_{ij}  \cdot \hat u_j
+)^2 }}{{1 - \chi \hat u_i  \cdot \hat u_j }}} \right)} \right]^{ -
+\frac{1}{2}},
+\end{equation}
+where the aspect ratio of the particles is governed by shape
+anisotropy parameter
+\begin{equation}
+\chi  = \frac{{(\sigma _e /\sigma _s )^2  - 1}}{{(\sigma _e /\sigma
+_s )^2  + 1}}.
+\label{LCEquation:chi}
+\end{equation}
+Here, $\sigma_ s$ and $\sigma_{e}$ refer to the side-by-side breadth
+and the end-to-end length of the ellipsoid, respectively. The well
+depth parameters takes the form
+\begin{equation}
+\epsilon (\hat u_i ,\hat u_j ,\hat r_{ij} ) = \epsilon _0 \epsilon
+^v (\hat u_i ,\hat u_j )\epsilon '^\mu (\hat u_i ,\hat u_j ,\hat
+r_{ij} )
+\end{equation}
+where $\epsilon_{0}$ is a constant term and
+\begin{equation}
+\epsilon (\hat u_i ,\hat u_j ) = \frac{1}{{\sqrt {1 - \chi ^2 (\hat
+u_i  \cdot \hat u_j )^2 } }}
+\end{equation}
+and
+\begin{equation}
+\epsilon '(\hat u_i ,\hat u_j ,\hat r_{ij} ) = 1 - \frac{{\chi
+'}}{2}\left[ {\frac{{(\hat r_{ij}  \cdot \hat u_i  + \hat r_{ij}
+\cdot \hat u_j )^2 }}{{1 + \chi '\hat u_i  \cdot \hat u_j }} +
+\frac{{(\hat r_{ij}  \cdot \hat u_i  - \hat r_{ij}  \cdot \hat u_j
+)^2 }}{{1 - \chi '\hat u_i  \cdot \hat u_j }}} \right]
+\end{equation}
+where the well depth anisotropy parameter $\chi '$ depends on the
+ratio between \textit{end-to-end} well depth $\epsilon _e$ and
+\textit{side-by-side} well depth $\epsilon_s$,
+\begin{equation}
+\chi ' = \frac{{1 - (\epsilon _e /\epsilon _s )^{1/\mu} }}{{1 +
+(\epsilon _e /\epsilon _s )^{1/\mu} }}.
+\end{equation}
 
-\section{\label{liquidCrystalSection:resultDiscussion}Results and Discussion}
+\begin{figure}
+\centering
+\includegraphics[width=\linewidth]{banana.eps}
+\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]{gb_scheme.eps}
+\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
+an electrostatic interaction term of the form
+\begin{equation}
+V_{ab}^{dp}  = \sum\limits_{i \in a,j \in b} {\frac{1}{{4\pi
+\epsilon _{fs} }}\left[ {\frac{{\mu _i  \cdot \mu _j }}{{r_{ij}^3 }}
+- \frac{{3\left( {\mu _i  \cdot r_{ij} } \right)\left( {\mu _i \cdot
+r_{ij} } \right)}}{{r_{ij}^5 }}} \right]}
+\end{equation}
+where $\epsilon _{fs}$ is the permittivity of free space.
+
+\section{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 lack of flexibility in our model
+due to the missing terminal chains could explained the fact that we
+did not find evidence of chirality.