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}.

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}

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]{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.

\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.

\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}

\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}

%\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{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.
Revision:	2892
Committed:	Tue Jun 27 01:33:33 2006 UTC (19 years, 4 months ago) by tim
Content type:	application/x-tex
File size:	16542 byte(s)
Log Message:	adding r22_220
#	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	tim	2892	of bent-core molecules\cite{Memmer2002}.
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	tim	2891	molecule represented by three rigid GB particles with two point
104			dipoles. Performing a series of molecular dynamics simulations, we
105			explore the structural properties of tilted smectic phases as well
106			as the effect of electrostatic interactions.
107	tim	2784
108	tim	2685	\section{\label{liquidCrystalSection:model}Model}
109
110	tim	2784	A typical banana-shaped molecule consists of a rigid aromatic
111			central bent unit with several rod-like wings which are held
112			together by some linking units and terminal chains (see
113			Fig.~\ref{LCFig:BananaMolecule}). In this work, each banana-shaped
114			mesogen has been modeled as a rigid body consisting of three
115			equivalent prolate ellipsoidal GB particles. The GB interaction
116			potential used to mimic the apolar characteristics of liquid crystal
117			molecules takes the familiar form of Lennard-Jones function with
118			orientation and position dependent range ($\sigma$) and well depth
119	tim	2785	($\epsilon$) parameters. The potential between a pair of three-site
120			banana-shaped molecules $a$ and $b$ is given by
121	tim	2784	\begin{equation}
122	tim	2785	V_{ab}^{GB} = \sum\limits_{i \in a,j \in b} {V_{ij}^{GB} }.
123			\end{equation}
124			Every site-site interaction can can be expressed as,
125			\begin{equation}
126	tim	2784	V_{ij}^{GB} = 4\epsilon (\hat u_i ,\hat u_j ,\hat r_{ij} )\left[
127			{\left( {\frac{{\sigma _0 }}{{r_{ij} - \sigma (\hat u_i ,\hat u_j
128			,\hat r_{ij} )}}} \right)^{12} - \left( {\frac{{\sigma _0
129			}}{{r_{ij} - \sigma (\hat u_i ,\hat u_j ,\hat r_{ij} )}}} \right)^6
130			} \right] \label{LCEquation:gb}
131			\end{equation}
132			where $\hat u_i,\hat u_j$ are unit vectors specifying the
133			orientation of two molecules $i$ and $j$ separated by intermolecular
134			vector $r_{ij}$. $\hat r_{ij}$ is the unit vector along the
135			intermolecular vector. A schematic diagram of the orientation
136			vectors is shown in Fig.\ref{LCFigure:GBScheme}. The functional form
137			for $\sigma$ is given by
138			\begin{equation}
139			\sigma (\hat u_i ,\hat u_i ,\hat r_{ij} ) = \sigma _0 \left[ {1 -
140			\frac{\chi }{2}\left( {\frac{{(\hat r_{ij} \cdot \hat u_i + \hat
141			r_{ij} \cdot \hat u_j )^2 }}{{1 + \chi \hat u_i \cdot \hat u_j }}
142			+ \frac{{(\hat r_{ij} \cdot \hat u_i - \hat r_{ij} \cdot \hat u_j
143			)^2 }}{{1 - \chi \hat u_i \cdot \hat u_j }}} \right)} \right]^{ -
144			\frac{1}{2}},
145			\end{equation}
146			where the aspect ratio of the particles is governed by shape
147			anisotropy parameter
148			\begin{equation}
149			\chi = \frac{{(\sigma _e /\sigma _s )^2 - 1}}{{(\sigma _e /\sigma
150			_s )^2 + 1}}.
151			\label{LCEquation:chi}
152			\end{equation}
153			Here, $\sigma_ s$ and $\sigma_{e}$ refer to the side-by-side breadth
154	tim	2785	and the end-to-end length of the ellipsoid, respectively. The well
155	tim	2784	depth parameters takes the form
156			\begin{equation}
157			\epsilon (\hat u_i ,\hat u_j ,\hat r_{ij} ) = \epsilon _0 \epsilon
158			^v (\hat u_i ,\hat u_j )\epsilon '^\mu (\hat u_i ,\hat u_j ,\hat
159			r_{ij} )
160			\end{equation}
161			where $\epsilon_{0}$ is a constant term and
162			\begin{equation}
163			\epsilon (\hat u_i ,\hat u_j ) = \frac{1}{{\sqrt {1 - \chi ^2 (\hat
164			u_i \cdot \hat u_j )^2 } }}
165			\end{equation}
166			and
167			\begin{equation}
168			\epsilon '(\hat u_i ,\hat u_j ,\hat r_{ij} ) = 1 - \frac{{\chi
169			'}}{2}\left[ {\frac{{(\hat r_{ij} \cdot \hat u_i + \hat r_{ij}
170			\cdot \hat u_j )^2 }}{{1 + \chi '\hat u_i \cdot \hat u_j }} +
171			\frac{{(\hat r_{ij} \cdot \hat u_i - \hat r_{ij} \cdot \hat u_j
172			)^2 }}{{1 - \chi '\hat u_i \cdot \hat u_j }}} \right]
173			\end{equation}
174			where the well depth anisotropy parameter $\chi '$ depends on the
175			ratio between \textit{end-to-end} well depth $\epsilon _e$ and
176			\textit{side-by-side} well depth $\epsilon_s$,
177	tim	2785	\begin{equation}
178	tim	2784	\chi ' = \frac{{1 - (\epsilon _e /\epsilon _s )^{1/\mu} }}{{1 +
179			(\epsilon _e /\epsilon _s )^{1/\mu} }}.
180			\end{equation}
181
182			\begin{figure}
183			\centering
184			\includegraphics[width=\linewidth]{banana.eps}
185	tim	2888	\caption[Schematic representation of a typical banana shaped
186			molecule]{Schematic representation of a typical banana shaped
187			molecule.} \label{LCFig:BananaMolecule}
188	tim	2784	\end{figure}
189
190			\begin{figure}
191			\centering
192			\includegraphics[width=\linewidth]{gb_scheme.eps}
193	tim	2890	\caption[Schematic diagram showing definitions of the orientation
194			vectors for a pair of Gay-Berne molecules]{Schematic diagram showing
195			definitions of the orientation vectors for a pair of Gay-Berne
196			molecules} \label{LCFigure:GBScheme}
197	tim	2784	\end{figure}
198
199	tim	2785	To account for the permanent dipolar interactions, there should be
200			an electrostatic interaction term of the form
201			\begin{equation}
202			V_{ab}^{dp} = \sum\limits_{i \in a,j \in b} {\frac{1}{{4\pi
203			\epsilon _{fs} }}\left[ {\frac{{\mu _i \cdot \mu _j }}{{r_{ij}^3 }}
204			- \frac{{3\left( {\mu _i \cdot r_{ij} } \right)\left( {\mu _i \cdot
205			r_{ij} } \right)}}{{r_{ij}^5 }}} \right]}
206			\end{equation}
207			where $\epsilon _{fs}$ is the permittivity of free space.
208
209	tim	2891	\section{Results and Discussion}
210	tim	2867
211			A series of molecular dynamics simulations were perform to study the
212	tim	2870	phase behavior of banana shaped liquid crystals. In each simulation,
213	tim	2880	every banana shaped molecule has been represented by three GB
214			particles which is characterized by $\mu = 1,~ \nu = 2,
215	tim	2870	~\epsilon_{e}/\epsilon_{s} = 1/5$ and $\sigma_{e}/\sigma_{s} = 3$.
216			All of the simulations begin with same equilibrated isotropic
217			configuration where 1024 molecules without dipoles were confined in
218			a $160\times 160 \times 120$ box. After the dipolar interactions are
219			switched on, 2~ns NPTi cooling run with themostat of 2~ps and
220			barostat of 50~ps were used to equilibrate the system to desired
221			temperature and pressure.
222	tim	2867
223	tim	2871	\subsection{Order Parameters}
224
225	tim	2870	To investigate the phase structure of the model liquid crystal, we
226			calculated various order parameters and correlation functions.
227			Particulary, the $P_2$ order parameter allows us to estimate average
228			alignment along the director axis $Z$ which can be identified from
229			the largest eigen value obtained by diagonalizing the order
230			parameter tensor
231	tim	2867	\begin{equation}
232	tim	2870	\overleftrightarrow{\mathsf{Q}} = \frac{1}{N}\sum_i^N %
233			\begin{pmatrix} %
234			u_{ix}u_{ix}-\frac{1}{3} & u_{ix}u_{iy} & u_{ix}u_{iz} \\
235			u_{iy}u_{ix} & u_{iy}u_{iy}-\frac{1}{3} & u_{iy}u_{iz} \\
236			u_{iz}u_{ix} & u_{iz}u_{iy} & u_{iz}u_{iz}-\frac{1}{3} %
237			\end{pmatrix},
238	tim	2891	\label{lipidEq:p2}
239	tim	2867	\end{equation}
240	tim	2870	where the $u_{i\alpha}$ is the $\alpha$ element of the unit vector
241			$\mathbf{\hat{u}}_i$, and the sum over $i$ averages over the whole
242			collection of unit vectors. The $P_2$ order parameter for uniaxial
243			phase is then simply given by
244			\begin{equation}
245			\langle P_2 \rangle = \frac{3}{2}\lambda_{\text{max}}.
246			\label{lipidEq:po3}
247			\end{equation}
248	tim	2891	%In addition to the $P_2$ order parameter, $ R_{2,2}^2$ order
249			%parameter for biaxial phase is introduced to describe the ordering
250			%in the plane orthogonal to the director by
251			%\begin{equation}
252			%R_{2,2}^2 = \frac{1}{4}\left\langle {(x_i \cdot X)^2 - (x_i \cdot
253			%Y)^2 - (y_i \cdot X)^2 + (y_i \cdot Y)^2 } \right\rangle
254			%\end{equation}
255			%where $X$, $Y$ and $Z$ are axis of the director frame.
256			The unit vector for the banana shaped molecule was defined by the
257			principle aixs of its middle GB particle. The $P_2$ order parameters
258			for the bent-core liquid crystal at different temperature is
259			summarized in Table~\ref{liquidCrystal:p2} which identifies a phase
260			transition temperature range.
261	tim	2867
262	tim	2891	\begin{table}
263			\caption{LIQUID CRYSTAL STRUCTURAL PROPERTIES AS A FUNCTION OF
264			TEMPERATURE} \label{liquidCrystal:p2}
265			\begin{center}
266	tim	2892	\begin{tabular}{cccccc}
267	tim	2891	\hline
268			Temperature (K) & 420 & 440 & 460 & 480 & 600\\
269			\hline
270			$\langle P_2\rangle$ & 0.984 & 0.982 & 0.975 & 0.967 & 0.067\\
271			\hline
272			\end{tabular}
273			\end{center}
274			\end{table}
275
276	tim	2871	\subsection{Structure Properties}
277	tim	2867
278	tim	2891	The molecular organization obtained at temperature $T = 460K$ (below
279			transition temperature) is shown in Figure~\ref{LCFigure:snapshot}.
280	tim	2892	The diagonal view in Fig~\ref{LCFigure:snapshot}(a) shows the
281			stacking of the banana shaped molecules while the side view in n
282			Figure~\ref{LCFigure:snapshot}(b) demonstrates formation of a
283			chevron structure. The first peak of Radial distribution function
284			$g(r)$ in Fig.~\ref{LCFigure:gofrz}(a) shows the minimum distance
285			for two in plane banana shaped molecules is 4.9 \AA, while the
286			second split peak implies the biaxial packing. It is also important
287			to show the density correlation along the director which is given by
288			:
289	tim	2871	\begin{equation}
290	tim	2892	g(z) =\frac{1}{\pi R^{2} \rho}< \delta (z-z_{ij})>_{ij}
291	tim	2870	\end{equation},
292	tim	2871	where $z_{ij} = r_{ij} \dot Z$ was measured in the director frame
293	tim	2892	and $R$ is the radius of the cylindrical sampling region. The
294			oscillation in density plot along the director in
295			Fig.~\ref{LCFigure:gofrz}(b) implies the existence of the layered
296			structure, and the peak at 27 \AA is attribute to the defect in the
297			system.
298	tim	2867
299	tim	2891	\begin{figure}
300			\centering
301			\includegraphics[width=4.5in]{snapshot.eps}
302			\caption[Snapshot of the molecular organization in the layered phase
303			formed at temperature T = 460K and pressure P = 1 atm]{Snapshot of
304			the molecular organization in the layered phase formed at
305			temperature T = 460K and pressure P = 1 atm. (a) diagonal view; (b)
306			side view.} \label{LCFigure:snapshot}
307			\end{figure}
308
309			\begin{figure}
310			\centering
311			\includegraphics[width=\linewidth]{gofr_gofz.eps}
312			\caption[Correlation Functions of a Bent-core Liquid Crystal System
313			at Temperature T = 460K and Pressure P = 10 atm]{Correlation
314			Functions of a Bent-core Liquid Crystal System at Temperature T =
315			460K and Pressure P = 10 atm. (a) radial correlation function
316			$g(r)$; and (b) density along the director $g(z)$.}
317			\label{LCFigure:gofrz}
318			\end{figure}
319
320	tim	2871	\subsection{Rotational Invariants}
321	tim	2867
322	tim	2871	As a useful set of correlation functions to describe
323			position-orientation correlation, rotation invariants were first
324			applied in a spherical symmetric system to study x-ray and light
325	tim	2887	scatting\cite{Blum1972}. Latterly, expansion of the orientation pair
326	tim	2871	correlation in terms of rotation invariant for molecules of
327			arbitrary shape was introduce by Stone\cite{Stone1978} and adopted
328	tim	2892	by other researchers in liquid crystal studies\cite{Berardi2003}. In
329			order to study the correlation between biaxiality and molecular
330			separation distance $r$, we calculate a rotational invariant
331			function $S_{22}^{220} (r)$, which is given by :
332	tim	2880	\begin{eqnarray}
333	tim	2882	S_{22}^{220} (r) & = & \frac{1}{{4\sqrt 5 }} \left< \delta (r -
334			r_{ij} )((\hat x_i \cdot \hat x_j )^2 - (\hat x_i \cdot \hat y_j
335			)^2 - (\hat y_i \cdot \hat x_j )^2 + (\hat y_i \cdot \hat y_j
336	tim	2892	)^2 ) \right. \notag \\
337	tim	2882	& & \left. - 2(\hat x_i \cdot \hat y_j )(\hat y_i \cdot \hat x_j ) -
338	tim	2892	2(\hat x_i \cdot \hat x_j )(\hat y_i \cdot \hat y_j )) \right>.
339	tim	2880	\end{eqnarray}
340	tim	2871
341	tim	2892	%\begin{equation}
342			%S_{00}^{221} (r) = - \frac{{\sqrt 3 }}{{\sqrt {10} }}\left\langle
343			%{\delta (r - r_{ij} )((\hat z_i \cdot \hat z_j )(\hat z_i \cdot
344			%\hat z_j \times \hat r_{ij} ))} \right\rangle
345			%\end{equation}
346	tim	2871
347	tim	2891	\section{Conclusion}
348	tim	2892
349			We have presented a simple dipolar three-site GB model for banana
350			shaped molecules which are capable of forming smectic phases from
351			isotropic configuration.