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[]
{}
\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[]{} \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 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{Blum1971}. 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{Berardi2000}.

\begin{eqnarray}
S_{22}^{220} (r) & = & \frac{1}{{4\sqrt 5 }}\left\langle {\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 ) \\
& & - 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\rangle
\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,
Revision:	2876
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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			\caption[]
47			{}
48			\label{LCFig:SMCP}
49			\end{figure}
50
51	tim	2782	Many liquid crystal synthesis experiments suggest that the
52			occurrence of polarity and chirality strongly relies on the
53	tim	2786	molecular structure and intermolecular interaction\cite{Reddy2006}.
54			From a theoretical point of view, it is of fundamental interest to
55			study the structural properties of liquid crystal phases formed by
56	tim	2782	banana-shaped molecules and understand their connection to the
57			molecular structure, especially with respect to the spontaneous
58			achiral symmetry breaking. As a complementary tool to experiment,
59			computer simulation can provide unique insight into molecular
60			ordering and phase behavior, and hence improve the development of
61			new experiments and theories. In the last two decades, all-atom
62			models have been adopted to investigate the structural properties of
63			smectic arrangements\cite{Cook2000, Lansac2001}, as well as other
64			bulk properties, such as rotational viscosity and flexoelectric
65			coefficients\cite{Cheung2002, Cheung2004}. However, due to the
66	tim	2786	limitation of time scale required for phase transition and the
67			length scale required for representing bulk behavior,
68			models\cite{Perram1985, Gay1981}, which are based on the observation
69			that liquid crystal order is exhibited by a range of non-molecular
70			bodies with high shape anisotropies, became the dominant models in
71			the field of liquid crystal phase behavior. Previous simulation
72			studies using hard spherocylinder dimer model\cite{Camp1999} produce
73			nematic phases, while hard rod simulation studies identified a
74			Landau point\cite{Bates2005}, at which the isotropic phase undergoes
75			a direct transition to the biaxial nematic, as well as some possible
76			liquid crystal phases\cite{Lansac2003}. Other anisotropic models
77			using Gay-Berne(GB) potential, which produce interactions that favor
78			local alignment, give the evidence of the novel packing arrangements
79			of bent-core molecules\cite{Memmer2002,Orlandi2006}.
80	tim	2781
81	tim	2784	Experimental studies by Levelut {\it et al.}~\cite{Levelut1981}
82			revealed that terminal cyano or nitro groups usually induce
83			permanent longitudinal dipole moments, which affect the phase
84			behavior considerably. A series of theoretical studies also drawn
85			equivalent conclusions. Monte Carlo studies of the GB potential with
86			fixed longitudinal dipoles (i.e. pointed along the principal axis of
87			rotation) were shown to enhance smectic phase
88			stability~\cite{Berardi1996,Satoh1996}. Molecular simulation of GB
89			ellipsoids with transverse dipoles at the terminus of the molecule
90			also demonstrated that partial striped bilayer structures were
91			developed from the smectic phase ~\cite{Berardi1996}. More
92			significant effects have been shown by including multiple
93			electrostatic moments. Adding longitudinal point quadrupole moments
94			to rod-shaped GB mesogens, Withers \textit{et al} induced tilted
95			smectic behaviour in the molecular system~\cite{Withers2003}. Thus,
96			it is clear that many liquid-crystal forming molecules, specially,
97			bent-core molecules, could be modeled more accurately by
98			incorporating electrostatic interaction.
99
100			In this chapter, we consider system consisting of banana-shaped
101			molecule represented by three rigid GB particles with one or two
102			point dipoles at different location. Performing a series of
103			molecular dynamics simulations, we explore the structural properties
104			of tilted smectic phases as well as the effect of electrostatic
105			interactions.
106
107	tim	2685	\section{\label{liquidCrystalSection:model}Model}
108
109	tim	2784	A typical banana-shaped molecule consists of a rigid aromatic
110			central bent unit with several rod-like wings which are held
111			together by some linking units and terminal chains (see
112			Fig.~\ref{LCFig:BananaMolecule}). In this work, each banana-shaped
113			mesogen has been modeled as a rigid body consisting of three
114			equivalent prolate ellipsoidal GB particles. The GB interaction
115			potential used to mimic the apolar characteristics of liquid crystal
116			molecules takes the familiar form of Lennard-Jones function with
117			orientation and position dependent range ($\sigma$) and well depth
118	tim	2785	($\epsilon$) parameters. The potential between a pair of three-site
119			banana-shaped molecules $a$ and $b$ is given by
120	tim	2784	\begin{equation}
121	tim	2785	V_{ab}^{GB} = \sum\limits_{i \in a,j \in b} {V_{ij}^{GB} }.
122			\end{equation}
123			Every site-site interaction can can be expressed as,
124			\begin{equation}
125	tim	2784	V_{ij}^{GB} = 4\epsilon (\hat u_i ,\hat u_j ,\hat r_{ij} )\left[
126			{\left( {\frac{{\sigma _0 }}{{r_{ij} - \sigma (\hat u_i ,\hat u_j
127			,\hat r_{ij} )}}} \right)^{12} - \left( {\frac{{\sigma _0
128			}}{{r_{ij} - \sigma (\hat u_i ,\hat u_j ,\hat r_{ij} )}}} \right)^6
129			} \right] \label{LCEquation:gb}
130			\end{equation}
131			where $\hat u_i,\hat u_j$ are unit vectors specifying the
132			orientation of two molecules $i$ and $j$ separated by intermolecular
133			vector $r_{ij}$. $\hat r_{ij}$ is the unit vector along the
134			intermolecular vector. A schematic diagram of the orientation
135			vectors is shown in Fig.\ref{LCFigure:GBScheme}. The functional form
136			for $\sigma$ is given by
137			\begin{equation}
138			\sigma (\hat u_i ,\hat u_i ,\hat r_{ij} ) = \sigma _0 \left[ {1 -
139			\frac{\chi }{2}\left( {\frac{{(\hat r_{ij} \cdot \hat u_i + \hat
140			r_{ij} \cdot \hat u_j )^2 }}{{1 + \chi \hat u_i \cdot \hat u_j }}
141			+ \frac{{(\hat r_{ij} \cdot \hat u_i - \hat r_{ij} \cdot \hat u_j
142			)^2 }}{{1 - \chi \hat u_i \cdot \hat u_j }}} \right)} \right]^{ -
143			\frac{1}{2}},
144			\end{equation}
145			where the aspect ratio of the particles is governed by shape
146			anisotropy parameter
147			\begin{equation}
148			\chi = \frac{{(\sigma _e /\sigma _s )^2 - 1}}{{(\sigma _e /\sigma
149			_s )^2 + 1}}.
150			\label{LCEquation:chi}
151			\end{equation}
152			Here, $\sigma_ s$ and $\sigma_{e}$ refer to the side-by-side breadth
153	tim	2785	and the end-to-end length of the ellipsoid, respectively. The well
154	tim	2784	depth parameters takes the form
155			\begin{equation}
156			\epsilon (\hat u_i ,\hat u_j ,\hat r_{ij} ) = \epsilon _0 \epsilon
157			^v (\hat u_i ,\hat u_j )\epsilon '^\mu (\hat u_i ,\hat u_j ,\hat
158			r_{ij} )
159			\end{equation}
160			where $\epsilon_{0}$ is a constant term and
161			\begin{equation}
162			\epsilon (\hat u_i ,\hat u_j ) = \frac{1}{{\sqrt {1 - \chi ^2 (\hat
163			u_i \cdot \hat u_j )^2 } }}
164			\end{equation}
165			and
166			\begin{equation}
167			\epsilon '(\hat u_i ,\hat u_j ,\hat r_{ij} ) = 1 - \frac{{\chi
168			'}}{2}\left[ {\frac{{(\hat r_{ij} \cdot \hat u_i + \hat r_{ij}
169			\cdot \hat u_j )^2 }}{{1 + \chi '\hat u_i \cdot \hat u_j }} +
170			\frac{{(\hat r_{ij} \cdot \hat u_i - \hat r_{ij} \cdot \hat u_j
171			)^2 }}{{1 - \chi '\hat u_i \cdot \hat u_j }}} \right]
172			\end{equation}
173			where the well depth anisotropy parameter $\chi '$ depends on the
174			ratio between \textit{end-to-end} well depth $\epsilon _e$ and
175			\textit{side-by-side} well depth $\epsilon_s$,
176	tim	2785	\begin{equation}
177	tim	2784	\chi ' = \frac{{1 - (\epsilon _e /\epsilon _s )^{1/\mu} }}{{1 +
178			(\epsilon _e /\epsilon _s )^{1/\mu} }}.
179			\end{equation}
180
181			\begin{figure}
182			\centering
183			\includegraphics[width=\linewidth]{banana.eps}
184			\caption[]{} \label{LCFig:BananaMolecule}
185			\end{figure}
186
187	tim	2805	%\begin{figure}
188			%\centering
189			%\includegraphics[width=\linewidth]{bananGB.eps}
190			%\caption[]{} \label{LCFigure:BananaGB}
191			%\end{figure}
192	tim	2784
193			\begin{figure}
194			\centering
195			\includegraphics[width=\linewidth]{gb_scheme.eps}
196			\caption[]{Schematic diagram showing definitions of the orientation
197			vectors for a pair of Gay-Berne molecules}
198			\label{LCFigure:GBScheme}
199			\end{figure}
200
201	tim	2785	To account for the permanent dipolar interactions, there should be
202			an electrostatic interaction term of the form
203			\begin{equation}
204			V_{ab}^{dp} = \sum\limits_{i \in a,j \in b} {\frac{1}{{4\pi
205			\epsilon _{fs} }}\left[ {\frac{{\mu _i \cdot \mu _j }}{{r_{ij}^3 }}
206			- \frac{{3\left( {\mu _i \cdot r_{ij} } \right)\left( {\mu _i \cdot
207			r_{ij} } \right)}}{{r_{ij}^5 }}} \right]}
208			\end{equation}
209			where $\epsilon _{fs}$ is the permittivity of free space.
210
211	tim	2867	\section{Computational Methodology}
212
213			A series of molecular dynamics simulations were perform to study the
214	tim	2870	phase behavior of banana shaped liquid crystals. In each simulation,
215			every banana shaped molecule has been represented three GB particles
216			which is characterized by $\mu = 1,~ \nu = 2,
217			~\epsilon_{e}/\epsilon_{s} = 1/5$ and $\sigma_{e}/\sigma_{s} = 3$.
218			All of the simulations begin with same equilibrated isotropic
219			configuration where 1024 molecules without dipoles were confined in
220			a $160\times 160 \times 120$ box. After the dipolar interactions are
221			switched on, 2~ns NPTi cooling run with themostat of 2~ps and
222			barostat of 50~ps were used to equilibrate the system to desired
223			temperature and pressure.
224	tim	2867
225	tim	2871	\subsection{Order Parameters}
226
227	tim	2870	To investigate the phase structure of the model liquid crystal, we
228			calculated various order parameters and correlation functions.
229			Particulary, the $P_2$ order parameter allows us to estimate average
230			alignment along the director axis $Z$ which can be identified from
231			the largest eigen value obtained by diagonalizing the order
232			parameter tensor
233	tim	2867	\begin{equation}
234	tim	2870	\overleftrightarrow{\mathsf{Q}} = \frac{1}{N}\sum_i^N %
235			\begin{pmatrix} %
236			u_{ix}u_{ix}-\frac{1}{3} & u_{ix}u_{iy} & u_{ix}u_{iz} \\
237			u_{iy}u_{ix} & u_{iy}u_{iy}-\frac{1}{3} & u_{iy}u_{iz} \\
238			u_{iz}u_{ix} & u_{iz}u_{iy} & u_{iz}u_{iz}-\frac{1}{3} %
239			\end{pmatrix},
240			\label{lipidEq:po1}
241	tim	2867	\end{equation}
242	tim	2870	where the $u_{i\alpha}$ is the $\alpha$ element of the unit vector
243			$\mathbf{\hat{u}}_i$, and the sum over $i$ averages over the whole
244			collection of unit vectors. The $P_2$ order parameter for uniaxial
245			phase is then simply given by
246			\begin{equation}
247			\langle P_2 \rangle = \frac{3}{2}\lambda_{\text{max}}.
248			\label{lipidEq:po3}
249			\end{equation}
250			In addition to the $P_2$ order parameter, $ R_{2,2}^2$ order
251			parameter for biaxial phase is introduced to describe the ordering
252			in the plane orthogonal to the director by
253			\begin{equation}
254			R_{2,2}^2 = \frac{1}{4}\left\langle {(x_i \cdot X)^2 - (x_i \cdot
255			Y)^2 - (y_i \cdot X)^2 + (y_i \cdot Y)^2 } \right\rangle
256			\end{equation}
257			where $X$, $Y$ and $Z$ are axis of the director frame.
258	tim	2867
259	tim	2871	\subsection{Structure Properties}
260	tim	2867
261	tim	2871	It is more important to show the density correlation along the
262			director
263			\begin{equation}
264			g(z) =< \delta (z-z_{ij})>_{ij} / \pi R^{2} \rho
265	tim	2870	\end{equation},
266	tim	2871	where $z_{ij} = r_{ij} \dot Z$ was measured in the director frame
267			and $R$ is the radius of the cylindrical sampling region.
268	tim	2867
269	tim	2871	\subsection{Rotational Invariants}
270	tim	2867
271	tim	2871	As a useful set of correlation functions to describe
272			position-orientation correlation, rotation invariants were first
273			applied in a spherical symmetric system to study x-ray and light
274			scatting\cite{Blum1971}. Latterly, expansion of the orientation pair
275			correlation in terms of rotation invariant for molecules of
276			arbitrary shape was introduce by Stone\cite{Stone1978} and adopted
277			by other researchers in liquid crystal studies\cite{Berardi2000}.
278
279	tim	2876	\begin{eqnarray}
280			S_{22}^{220} (r) & = & \frac{1}{{4\sqrt 5 }}\left\langle {\delta (r
281			- r_{ij} )((\hat x_i \cdot \hat x_j )^2 - (\hat x_i \cdot \hat
282			y_j )^2 - (\hat y_i \cdot \hat x_j )^2 + (\hat y_i \cdot \hat
283			y_j )^2 ) \\
284			& & - 2(\hat x_i \cdot \hat y_j )(\hat y_i \cdot \hat x_j ) -
285	tim	2871	2(\hat x_i \cdot \hat x_j )(\hat y_i \cdot \hat y_j ))}
286			\right\rangle
287	tim	2876	\end{eqnarray}
288	tim	2871
289			\begin{equation}
290			S_{00}^{221} (r) = - \frac{{\sqrt 3 }}{{\sqrt {10} }}\left\langle
291			{\delta (r - r_{ij} )((\hat z_i \cdot \hat z_j )(\hat z_i \cdot
292	tim	2876	\hat z_j \times \hat r_{ij} ))} \right\rangle
293			\end{equation}
294	tim	2871
295	tim	2867	\section{Results and Conclusion}
296			\label{sec:results and conclusion}
297
298	tim	2870	To investigate the molecular organization behavior due to different
299			dipolar orientation and position with respect to the center of the
300			molecule,