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. 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. 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 $\text{SmCP}$. 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.~\cite{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. 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\cite{Wilson1999} and the length scale required for
representing bulk behavior, the dominant models in the field of
liquid crystal phase behavior are generic
models\cite{Lebwohl1972,Perram1984, Gay1981}, which are based on the
observation that liquid crystal order is exhibited by a range of
non-molecular bodies with high shape anisotropies. 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{\label{liquidCrystalSection:methods}Methods}

\section{\label{liquidCrystalSection:resultDiscussion}Results and Discussion}
Revision:	2785
Committed:	Fri Jun 2 21:31:49 2006 UTC (18 years, 1 month ago) by tim
Content type:	application/x-tex
File size:	10190 byte(s)
Log Message:	Adding more figures
#	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			exploited in great detail in the last two decades. Typically, these
11			mesogens consist of a rigid aromatic core and one or more attached
12			aliphatic chains. For short chain molecules, only nematic phases, in
13			which positional order is limited or absent, can be observed,
14			because the entropy of mixing different parts of the mesogens is
15			paramount to the dispersion interaction. In contrast, formation of
16			the one dimension lamellar sematic phase in rod-like molecules with
17			sufficiently long aliphatic chains has been reported, as well as the
18			segregation phenomena in disk-like molecules.
19
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			supramolecular chemistry. Unlike rods and disks, the polarity and
23			biaxiality of the banana-shaped molecules allow the molecules
24			organize into a variety of novel liquid crystalline phases which
25			show interesting material properties. Of particular interest is the
26			spontaneous formation of macroscopic chiral layers from achiral
27			banana-shaped molecules, where polar molecule orientational ordering
28			is shown within the layer plane as well as the tilted arrangement of
29			the molecules relative to the polar axis. As a consequence of
30			supramolecular chirality, the spontaneous polarization arises in
31			ferroelectric (FE) and antiferroelectic (AF) switching of smectic
32			liquid crystal phases, demonstrating some promising applications in
33	tim	2782	second-order nonlinear optical devices. The most widely investigated
34			mesophase formed by banana-shaped moleculed is the $\text{B}_2$
35			phase, which is also referred to as $\text{SmCP}$. Of the most
36			important discover in this tilt lamellar phase is the four distinct
37			packing arrangements (two conglomerates and two macroscopic
38			racemates), which depend on the tilt direction and the polar
39			direction of the molecule in adjacent layer (see
40			Fig.~\cite{LCFig:SMCP}).
41	tim	2781
42	tim	2784	\begin{figure}
43			\centering
44			\includegraphics[width=\linewidth]{smcp.eps}
45			\caption[]
46			{}
47			\label{LCFig:SMCP}
48			\end{figure}
49
50	tim	2782	Many liquid crystal synthesis experiments suggest that the
51			occurrence of polarity and chirality strongly relies on the
52			molecular structure and intermolecular interaction. From a
53			theoretical point of view, it is of fundamental interest to study
54			the structural properties of liquid crystal phases formed by
55			banana-shaped molecules and understand their connection to the
56			molecular structure, especially with respect to the spontaneous
57			achiral symmetry breaking. As a complementary tool to experiment,
58			computer simulation can provide unique insight into molecular
59			ordering and phase behavior, and hence improve the development of
60			new experiments and theories. In the last two decades, all-atom
61			models have been adopted to investigate the structural properties of
62			smectic arrangements\cite{Cook2000, Lansac2001}, as well as other
63			bulk properties, such as rotational viscosity and flexoelectric
64			coefficients\cite{Cheung2002, Cheung2004}. However, due to the
65			limitation of time scale required for phase
66			transition\cite{Wilson1999} and the length scale required for
67			representing bulk behavior, the dominant models in the field of
68			liquid crystal phase behavior are generic
69			models\cite{Lebwohl1972,Perram1984, Gay1981}, which are based on the
70			observation that liquid crystal order is exhibited by a range of
71			non-molecular bodies with high shape anisotropies. Previous
72			simulation studies using hard spherocylinder dimer
73			model\cite{Camp1999} produce nematic phases, while hard rod
74			simulation studies identified a Landau point\cite{Bates2005}, at
75	tim	2784	which the isotropic phase undergoes a direct transition to the
76	tim	2782	biaxial nematic, as well as some possible liquid crystal
77	tim	2784	phases\cite{Lansac2003}. Other anisotropic models using
78			Gay-Berne(GB) potential, which produce interactions that favor local
79			alignment, give the evidence of the novel packing arrangements of
80	tim	2782	bent-core molecules\cite{Memmer2002,Orlandi2006}.
81	tim	2781
82	tim	2784	Experimental studies by Levelut {\it et al.}~\cite{Levelut1981}
83			revealed that terminal cyano or nitro groups usually induce
84			permanent longitudinal dipole moments, which affect the phase
85			behavior considerably. A series of theoretical studies also drawn
86			equivalent conclusions. Monte Carlo studies of the GB potential with
87			fixed longitudinal dipoles (i.e. pointed along the principal axis of
88			rotation) were shown to enhance smectic phase
89			stability~\cite{Berardi1996,Satoh1996}. Molecular simulation of GB
90			ellipsoids with transverse dipoles at the terminus of the molecule
91			also demonstrated that partial striped bilayer structures were
92			developed from the smectic phase ~\cite{Berardi1996}. More
93			significant effects have been shown by including multiple
94			electrostatic moments. Adding longitudinal point quadrupole moments
95			to rod-shaped GB mesogens, Withers \textit{et al} induced tilted
96			smectic behaviour in the molecular system~\cite{Withers2003}. Thus,
97			it is clear that many liquid-crystal forming molecules, specially,
98			bent-core molecules, could be modeled more accurately by
99			incorporating electrostatic interaction.
100
101			In this chapter, we consider system consisting of banana-shaped
102			molecule represented by three rigid GB particles with one or two
103			point dipoles at different location. Performing a series of
104			molecular dynamics simulations, we explore the structural properties
105			of tilted smectic phases as well as the effect of electrostatic
106			interactions.
107
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			\caption[]{} \label{LCFig:BananaMolecule}
186			\end{figure}
187
188			\begin{figure}
189			\centering
190	tim	2785	\includegraphics[width=\linewidth]{bananGB.eps}
191	tim	2784	\caption[]{} \label{LCFigure:BananaGB}
192			\end{figure}
193
194			\begin{figure}
195			\centering
196			\includegraphics[width=\linewidth]{gb_scheme.eps}
197			\caption[]{Schematic diagram showing definitions of the orientation
198			vectors for a pair of Gay-Berne molecules}
199			\label{LCFigure:GBScheme}
200			\end{figure}
201
202	tim	2785	To account for the permanent dipolar interactions, there should be
203			an electrostatic interaction term of the form
204			\begin{equation}
205			V_{ab}^{dp} = \sum\limits_{i \in a,j \in b} {\frac{1}{{4\pi
206			\epsilon _{fs} }}\left[ {\frac{{\mu _i \cdot \mu _j }}{{r_{ij}^3 }}
207			- \frac{{3\left( {\mu _i \cdot r_{ij} } \right)\left( {\mu _i \cdot
208			r_{ij} } \right)}}{{r_{ij}^5 }}} \right]}
209			\end{equation}
210			where $\epsilon _{fs}$ is the permittivity of free space.
211
212	tim	2685	\section{\label{liquidCrystalSection:methods}Methods}
213
214			\section{\label{liquidCrystalSection:resultDiscussion}Results and Discussion}