trunk/tengDissertation/LiquidCrystal.tex

\chapter{\label{chapt:liquidcrystal}LIQUID CRYSTAL}

\section{\label{liquidCrystalSection:introduction}Introduction}

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\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.

Recently, the banana-shaped or bent-core liquid crystal have became
one of the most active research areas in mesogenic materials and
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}).

\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}

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,Orlandi2006}.

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 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.

\section{\label{liquidCrystalSection:model}Model}

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}

\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]{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}
\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{Computational Methodology}

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.

\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: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.

\subsection{Structure Properties}

It is more important to show the density correlation along the
director
\begin{equation}
g(z) =< \delta (z-z_{ij})>_{ij} / \pi R^{2} \rho
\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.

\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}.

\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. \\
 & & \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}

\begin{equation}
S_{00}^{221} (r) =  - \frac{{\sqrt 3 }}{{\sqrt {10} }}\left\langle
{\delta (r - r_{ij} )((\hat z_i  \cdot \hat z_j )(\hat z_i  \cdot
\hat z_j  \times \hat r_{ij} ))} \right\rangle
\end{equation}

\section{Results and Conclusion}
\label{sec:results and conclusion}

To investigate the molecular organization behavior due to different
dipolar orientation and position with respect to the center of the
molecule,
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#	User	Rev	Content
1	tim	2685	\chapter{\label{chapt:liquidcrystal}LIQUID CRYSTAL}
2
3			\section{\label{liquidCrystalSection:introduction}Introduction}
4
5	tim	2781	Long range orientational order is one of the most fundamental
6			properties of liquid crystal mesophases. This orientational
7			anisotropy of the macroscopic phases originates in the shape
8			anisotropy of the constituent molecules. Among these anisotropy
9			mesogens, rod-like (calamitic) and disk-like molecules have been
10	tim	2786	exploited in great detail in the last two decades\cite{Huh2004}.
11			Typically, these mesogens consist of a rigid aromatic core and one
12			or more attached aliphatic chains. For short chain molecules, only
13			nematic phases, in which positional order is limited or absent, can
14			be observed, because the entropy of mixing different parts of the
15			mesogens is paramount to the dispersion interaction. In contrast,
16			formation of the one dimension lamellar sematic phase in rod-like
17			molecules with sufficiently long aliphatic chains has been reported,
18			as well as the segregation phenomena in disk-like molecules.
19	tim	2781
20			Recently, the banana-shaped or bent-core liquid crystal have became
21			one of the most active research areas in mesogenic materials and
22	tim	2786	supramolecular chemistry\cite{Niori1996, Link1997, Pelzl1999}.
23			Unlike rods and disks, the polarity and biaxiality of the
24			banana-shaped molecules allow the molecules organize into a variety
25			of novel liquid crystalline phases which show interesting material
26			properties. Of particular interest is the spontaneous formation of
27			macroscopic chiral layers from achiral banana-shaped molecules,
28			where polar molecule orientational ordering is shown within the
29			layer plane as well as the tilted arrangement of the molecules
30			relative to the polar axis. As a consequence of supramolecular
31			chirality, the spontaneous polarization arises in ferroelectric (FE)
32			and antiferroelectic (AF) switching of smectic liquid crystal
33			phases, demonstrating some promising applications in second-order
34			nonlinear optical devices. The most widely investigated mesophase
35			formed by banana-shaped moleculed is the $\text{B}_2$ phase, which
36			is also referred to as $\text{SmCP}$\cite{Link1997}. Of the most
37	tim	2782	important discover in this tilt lamellar phase is the four distinct
38			packing arrangements (two conglomerates and two macroscopic
39			racemates), which depend on the tilt direction and the polar
40			direction of the molecule in adjacent layer (see
41	tim	2839	Fig.~\ref{LCFig:SMCP}).
42	tim	2781
43	tim	2784	\begin{figure}
44			\centering
45			\includegraphics[width=\linewidth]{smcp.eps}
46	tim	2888	\caption[SmCP Phase Packing] {Four possible SmCP phase packings that
47			are characterized by the relative tilt direction(A and S refer an
48			anticlinic tilt or a synclinic ) and the polarization orientation (A
49			and F represent antiferroelectric or ferroelectric polar order).}
50	tim	2784	\label{LCFig:SMCP}
51			\end{figure}
52
53	tim	2782	Many liquid crystal synthesis experiments suggest that the
54			occurrence of polarity and chirality strongly relies on the
55	tim	2786	molecular structure and intermolecular interaction\cite{Reddy2006}.
56			From a theoretical point of view, it is of fundamental interest to
57			study the structural properties of liquid crystal phases formed by
58	tim	2782	banana-shaped molecules and understand their connection to the
59			molecular structure, especially with respect to the spontaneous
60			achiral symmetry breaking. As a complementary tool to experiment,
61			computer simulation can provide unique insight into molecular
62			ordering and phase behavior, and hence improve the development of
63			new experiments and theories. In the last two decades, all-atom
64			models have been adopted to investigate the structural properties of
65			smectic arrangements\cite{Cook2000, Lansac2001}, as well as other
66			bulk properties, such as rotational viscosity and flexoelectric
67			coefficients\cite{Cheung2002, Cheung2004}. However, due to the
68	tim	2786	limitation of time scale required for phase transition and the
69			length scale required for representing bulk behavior,
70			models\cite{Perram1985, Gay1981}, which are based on the observation
71			that liquid crystal order is exhibited by a range of non-molecular
72			bodies with high shape anisotropies, became the dominant models in
73			the field of liquid crystal phase behavior. Previous simulation
74			studies using hard spherocylinder dimer model\cite{Camp1999} produce
75			nematic phases, while hard rod simulation studies identified a
76			Landau point\cite{Bates2005}, at which the isotropic phase undergoes
77			a direct transition to the biaxial nematic, as well as some possible
78			liquid crystal phases\cite{Lansac2003}. Other anisotropic models
79			using Gay-Berne(GB) potential, which produce interactions that favor
80			local alignment, give the evidence of the novel packing arrangements
81			of bent-core molecules\cite{Memmer2002,Orlandi2006}.
82	tim	2781
83	tim	2784	Experimental studies by Levelut {\it et al.}~\cite{Levelut1981}
84			revealed that terminal cyano or nitro groups usually induce
85			permanent longitudinal dipole moments, which affect the phase
86			behavior considerably. A series of theoretical studies also drawn
87			equivalent conclusions. Monte Carlo studies of the GB potential with
88			fixed longitudinal dipoles (i.e. pointed along the principal axis of
89			rotation) were shown to enhance smectic phase
90			stability~\cite{Berardi1996,Satoh1996}. Molecular simulation of GB
91			ellipsoids with transverse dipoles at the terminus of the molecule
92			also demonstrated that partial striped bilayer structures were
93			developed from the smectic phase ~\cite{Berardi1996}. More
94			significant effects have been shown by including multiple
95			electrostatic moments. Adding longitudinal point quadrupole moments
96			to rod-shaped GB mesogens, Withers \textit{et al} induced tilted
97			smectic behaviour in the molecular system~\cite{Withers2003}. Thus,
98			it is clear that many liquid-crystal forming molecules, specially,
99			bent-core molecules, could be modeled more accurately by
100			incorporating electrostatic interaction.
101
102			In this chapter, we consider system consisting of banana-shaped
103			molecule represented by three rigid GB particles with one or two
104			point dipoles at different location. Performing a series of
105			molecular dynamics simulations, we explore the structural properties
106			of tilted smectic phases as well as the effect of electrostatic
107			interactions.
108
109	tim	2685	\section{\label{liquidCrystalSection:model}Model}
110
111	tim	2784	A typical banana-shaped molecule consists of a rigid aromatic
112			central bent unit with several rod-like wings which are held
113			together by some linking units and terminal chains (see
114			Fig.~\ref{LCFig:BananaMolecule}). In this work, each banana-shaped
115			mesogen has been modeled as a rigid body consisting of three
116			equivalent prolate ellipsoidal GB particles. The GB interaction
117			potential used to mimic the apolar characteristics of liquid crystal
118			molecules takes the familiar form of Lennard-Jones function with
119			orientation and position dependent range ($\sigma$) and well depth
120	tim	2785	($\epsilon$) parameters. The potential between a pair of three-site
121			banana-shaped molecules $a$ and $b$ is given by
122	tim	2784	\begin{equation}
123	tim	2785	V_{ab}^{GB} = \sum\limits_{i \in a,j \in b} {V_{ij}^{GB} }.
124			\end{equation}
125			Every site-site interaction can can be expressed as,
126			\begin{equation}
127	tim	2784	V_{ij}^{GB} = 4\epsilon (\hat u_i ,\hat u_j ,\hat r_{ij} )\left[
128			{\left( {\frac{{\sigma _0 }}{{r_{ij} - \sigma (\hat u_i ,\hat u_j
129			,\hat r_{ij} )}}} \right)^{12} - \left( {\frac{{\sigma _0
130			}}{{r_{ij} - \sigma (\hat u_i ,\hat u_j ,\hat r_{ij} )}}} \right)^6
131			} \right] \label{LCEquation:gb}
132			\end{equation}
133			where $\hat u_i,\hat u_j$ are unit vectors specifying the
134			orientation of two molecules $i$ and $j$ separated by intermolecular
135			vector $r_{ij}$. $\hat r_{ij}$ is the unit vector along the
136			intermolecular vector. A schematic diagram of the orientation
137			vectors is shown in Fig.\ref{LCFigure:GBScheme}. The functional form
138			for $\sigma$ is given by
139			\begin{equation}
140			\sigma (\hat u_i ,\hat u_i ,\hat r_{ij} ) = \sigma _0 \left[ {1 -
141			\frac{\chi }{2}\left( {\frac{{(\hat r_{ij} \cdot \hat u_i + \hat
142			r_{ij} \cdot \hat u_j )^2 }}{{1 + \chi \hat u_i \cdot \hat u_j }}
143			+ \frac{{(\hat r_{ij} \cdot \hat u_i - \hat r_{ij} \cdot \hat u_j
144			)^2 }}{{1 - \chi \hat u_i \cdot \hat u_j }}} \right)} \right]^{ -
145			\frac{1}{2}},
146			\end{equation}
147			where the aspect ratio of the particles is governed by shape
148			anisotropy parameter
149			\begin{equation}
150			\chi = \frac{{(\sigma _e /\sigma _s )^2 - 1}}{{(\sigma _e /\sigma
151			_s )^2 + 1}}.
152			\label{LCEquation:chi}
153			\end{equation}
154			Here, $\sigma_ s$ and $\sigma_{e}$ refer to the side-by-side breadth
155	tim	2785	and the end-to-end length of the ellipsoid, respectively. The well
156	tim	2784	depth parameters takes the form
157			\begin{equation}
158			\epsilon (\hat u_i ,\hat u_j ,\hat r_{ij} ) = \epsilon _0 \epsilon
159			^v (\hat u_i ,\hat u_j )\epsilon '^\mu (\hat u_i ,\hat u_j ,\hat
160			r_{ij} )
161			\end{equation}
162			where $\epsilon_{0}$ is a constant term and
163			\begin{equation}
164			\epsilon (\hat u_i ,\hat u_j ) = \frac{1}{{\sqrt {1 - \chi ^2 (\hat
165			u_i \cdot \hat u_j )^2 } }}
166			\end{equation}
167			and
168			\begin{equation}
169			\epsilon '(\hat u_i ,\hat u_j ,\hat r_{ij} ) = 1 - \frac{{\chi
170			'}}{2}\left[ {\frac{{(\hat r_{ij} \cdot \hat u_i + \hat r_{ij}
171			\cdot \hat u_j )^2 }}{{1 + \chi '\hat u_i \cdot \hat u_j }} +
172			\frac{{(\hat r_{ij} \cdot \hat u_i - \hat r_{ij} \cdot \hat u_j
173			)^2 }}{{1 - \chi '\hat u_i \cdot \hat u_j }}} \right]
174			\end{equation}
175			where the well depth anisotropy parameter $\chi '$ depends on the
176			ratio between \textit{end-to-end} well depth $\epsilon _e$ and
177			\textit{side-by-side} well depth $\epsilon_s$,
178	tim	2785	\begin{equation}
179	tim	2784	\chi ' = \frac{{1 - (\epsilon _e /\epsilon _s )^{1/\mu} }}{{1 +
180			(\epsilon _e /\epsilon _s )^{1/\mu} }}.
181			\end{equation}
182
183			\begin{figure}
184			\centering
185			\includegraphics[width=\linewidth]{banana.eps}
186	tim	2888	\caption[Schematic representation of a typical banana shaped
187			molecule]{Schematic representation of a typical banana shaped
188			molecule.} \label{LCFig:BananaMolecule}
189	tim	2784	\end{figure}
190
191	tim	2805	%\begin{figure}
192			%\centering
193			%\includegraphics[width=\linewidth]{bananGB.eps}
194			%\caption[]{} \label{LCFigure:BananaGB}
195			%\end{figure}
196	tim	2784
197			\begin{figure}
198			\centering
199			\includegraphics[width=\linewidth]{gb_scheme.eps}
200			\caption[]{Schematic diagram showing definitions of the orientation
201			vectors for a pair of Gay-Berne molecules}
202			\label{LCFigure:GBScheme}
203			\end{figure}
204
205	tim	2785	To account for the permanent dipolar interactions, there should be
206			an electrostatic interaction term of the form
207			\begin{equation}
208			V_{ab}^{dp} = \sum\limits_{i \in a,j \in b} {\frac{1}{{4\pi
209			\epsilon _{fs} }}\left[ {\frac{{\mu _i \cdot \mu _j }}{{r_{ij}^3 }}
210			- \frac{{3\left( {\mu _i \cdot r_{ij} } \right)\left( {\mu _i \cdot
211			r_{ij} } \right)}}{{r_{ij}^5 }}} \right]}
212			\end{equation}
213			where $\epsilon _{fs}$ is the permittivity of free space.
214
215	tim	2867	\section{Computational Methodology}
216
217			A series of molecular dynamics simulations were perform to study the
218	tim	2870	phase behavior of banana shaped liquid crystals. In each simulation,
219	tim	2880	every banana shaped molecule has been represented by three GB
220			particles which is characterized by $\mu = 1,~ \nu = 2,
221	tim	2870	~\epsilon_{e}/\epsilon_{s} = 1/5$ and $\sigma_{e}/\sigma_{s} = 3$.
222			All of the simulations begin with same equilibrated isotropic
223			configuration where 1024 molecules without dipoles were confined in
224			a $160\times 160 \times 120$ box. After the dipolar interactions are
225			switched on, 2~ns NPTi cooling run with themostat of 2~ps and
226			barostat of 50~ps were used to equilibrate the system to desired
227			temperature and pressure.
228	tim	2867
229	tim	2871	\subsection{Order Parameters}
230
231	tim	2870	To investigate the phase structure of the model liquid crystal, we
232			calculated various order parameters and correlation functions.
233			Particulary, the $P_2$ order parameter allows us to estimate average
234			alignment along the director axis $Z$ which can be identified from
235			the largest eigen value obtained by diagonalizing the order
236			parameter tensor
237	tim	2867	\begin{equation}
238	tim	2870	\overleftrightarrow{\mathsf{Q}} = \frac{1}{N}\sum_i^N %
239			\begin{pmatrix} %
240			u_{ix}u_{ix}-\frac{1}{3} & u_{ix}u_{iy} & u_{ix}u_{iz} \\
241			u_{iy}u_{ix} & u_{iy}u_{iy}-\frac{1}{3} & u_{iy}u_{iz} \\
242			u_{iz}u_{ix} & u_{iz}u_{iy} & u_{iz}u_{iz}-\frac{1}{3} %
243			\end{pmatrix},
244			\label{lipidEq:po1}
245	tim	2867	\end{equation}
246	tim	2870	where the $u_{i\alpha}$ is the $\alpha$ element of the unit vector
247			$\mathbf{\hat{u}}_i$, and the sum over $i$ averages over the whole
248			collection of unit vectors. The $P_2$ order parameter for uniaxial
249			phase is then simply given by
250			\begin{equation}
251			\langle P_2 \rangle = \frac{3}{2}\lambda_{\text{max}}.
252			\label{lipidEq:po3}
253			\end{equation}
254			In addition to the $P_2$ order parameter, $ R_{2,2}^2$ order
255			parameter for biaxial phase is introduced to describe the ordering
256			in the plane orthogonal to the director by
257			\begin{equation}
258			R_{2,2}^2 = \frac{1}{4}\left\langle {(x_i \cdot X)^2 - (x_i \cdot
259			Y)^2 - (y_i \cdot X)^2 + (y_i \cdot Y)^2 } \right\rangle
260			\end{equation}
261			where $X$, $Y$ and $Z$ are axis of the director frame.
262	tim	2867
263	tim	2871	\subsection{Structure Properties}
264	tim	2867
265	tim	2871	It is more important to show the density correlation along the
266			director
267			\begin{equation}
268			g(z) =< \delta (z-z_{ij})>_{ij} / \pi R^{2} \rho
269	tim	2870	\end{equation},
270	tim	2871	where $z_{ij} = r_{ij} \dot Z$ was measured in the director frame
271			and $R$ is the radius of the cylindrical sampling region.
272	tim	2867
273	tim	2871	\subsection{Rotational Invariants}
274	tim	2867
275	tim	2871	As a useful set of correlation functions to describe
276			position-orientation correlation, rotation invariants were first
277			applied in a spherical symmetric system to study x-ray and light
278	tim	2887	scatting\cite{Blum1972}. Latterly, expansion of the orientation pair
279	tim	2871	correlation in terms of rotation invariant for molecules of
280			arbitrary shape was introduce by Stone\cite{Stone1978} and adopted
281	tim	2887	by other researchers in liquid crystal studies\cite{Berardi2003}.
282	tim	2871
283	tim	2880	\begin{eqnarray}
284	tim	2882	S_{22}^{220} (r) & = & \frac{1}{{4\sqrt 5 }} \left< \delta (r -
285			r_{ij} )((\hat x_i \cdot \hat x_j )^2 - (\hat x_i \cdot \hat y_j
286			)^2 - (\hat y_i \cdot \hat x_j )^2 + (\hat y_i \cdot \hat y_j
287			)^2 ) \right. \\
288			& & \left. - 2(\hat x_i \cdot \hat y_j )(\hat y_i \cdot \hat x_j ) -
289			2(\hat x_i \cdot \hat x_j )(\hat y_i \cdot \hat y_j )) \right>
290	tim	2880	\end{eqnarray}
291	tim	2871
292			\begin{equation}
293			S_{00}^{221} (r) = - \frac{{\sqrt 3 }}{{\sqrt {10} }}\left\langle
294			{\delta (r - r_{ij} )((\hat z_i \cdot \hat z_j )(\hat z_i \cdot
295	tim	2876	\hat z_j \times \hat r_{ij} ))} \right\rangle
296			\end{equation}
297	tim	2871
298	tim	2867	\section{Results and Conclusion}
299			\label{sec:results and conclusion}
300
301	tim	2870	To investigate the molecular organization behavior due to different
302			dipolar orientation and position with respect to the center of the
303			molecule,