| 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. |
| 10 |
> |
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 |
|
|
| 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 |
< |
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 |
| 22 |
> |
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 |
|
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 |
| 50 |
|
|
| 51 |
|
Many liquid crystal synthesis experiments suggest that the |
| 52 |
|
occurrence of polarity and chirality strongly relies on the |
| 53 |
< |
molecular structure and intermolecular interaction. From a |
| 54 |
< |
theoretical point of view, it is of fundamental interest to study |
| 55 |
< |
the structural properties of liquid crystal phases formed by |
| 53 |
> |
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 |
|
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, |
| 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 |
< |
limitation of time scale required for phase |
| 67 |
< |
transition\cite{Wilson1999} and the length scale required for |
| 68 |
< |
representing bulk behavior, the dominant models in the field of |
| 69 |
< |
liquid crystal phase behavior are generic |
| 70 |
< |
models\cite{Lebwohl1972,Perram1984, Gay1981}, which are based on the |
| 71 |
< |
observation that liquid crystal order is exhibited by a range of |
| 72 |
< |
non-molecular bodies with high shape anisotropies. Previous |
| 73 |
< |
simulation studies using hard spherocylinder dimer |
| 74 |
< |
model\cite{Camp1999} produce nematic phases, while hard rod |
| 75 |
< |
simulation studies identified a Landau point\cite{Bates2005}, at |
| 76 |
< |
which the isotropic phase undergoes a direct transition to the |
| 77 |
< |
biaxial nematic, as well as some possible liquid crystal |
| 78 |
< |
phases\cite{Lansac2003}. Other anisotropic models using |
| 79 |
< |
Gay-Berne(GB) potential, which produce interactions that favor local |
| 79 |
< |
alignment, give the evidence of the novel packing arrangements of |
| 80 |
< |
bent-core molecules\cite{Memmer2002,Orlandi2006}. |
| 66 |
> |
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 |
|
|
| 81 |
|
Experimental studies by Levelut {\it et al.}~\cite{Levelut1981} |
| 82 |
|
revealed that terminal cyano or nitro groups usually induce |
| 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 |
< |
($\epsilon$) parameters. It can can be expressed as, |
| 118 |
> |
($\epsilon$) parameters. The potential between a pair of three-site |
| 119 |
> |
banana-shaped molecules $a$ and $b$ is given by |
| 120 |
|
\begin{equation} |
| 121 |
+ |
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 |
|
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 |
| 150 |
|
\label{LCEquation:chi} |
| 151 |
|
\end{equation} |
| 152 |
|
Here, $\sigma_ s$ and $\sigma_{e}$ refer to the side-by-side breadth |
| 153 |
< |
and the end-to-end length of the ellipsoid, respectively. Twell |
| 153 |
> |
and the end-to-end length of the ellipsoid, respectively. The well |
| 154 |
|
depth parameters takes the form |
| 155 |
|
\begin{equation} |
| 156 |
|
\epsilon (\hat u_i ,\hat u_j ,\hat r_{ij} ) = \epsilon _0 \epsilon |
| 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 |
< |
\begin{eqaution} |
| 176 |
> |
\begin{equation} |
| 177 |
|
\chi ' = \frac{{1 - (\epsilon _e /\epsilon _s )^{1/\mu} }}{{1 + |
| 178 |
|
(\epsilon _e /\epsilon _s )^{1/\mu} }}. |
| 179 |
|
\end{equation} |
| 184 |
|
\caption[]{} \label{LCFig:BananaMolecule} |
| 185 |
|
\end{figure} |
| 186 |
|
|
| 187 |
< |
\begin{figure} |
| 188 |
< |
\centering |
| 189 |
< |
\includegraphics[width=\linewidth]{bananGB_grained.eps} |
| 190 |
< |
\caption[]{} \label{LCFigure:BananaGB} |
| 191 |
< |
\end{figure} |
| 187 |
> |
%\begin{figure} |
| 188 |
> |
%\centering |
| 189 |
> |
%\includegraphics[width=\linewidth]{bananGB.eps} |
| 190 |
> |
%\caption[]{} \label{LCFigure:BananaGB} |
| 191 |
> |
%\end{figure} |
| 192 |
|
|
| 193 |
|
\begin{figure} |
| 194 |
|
\centering |
| 198 |
|
\label{LCFigure:GBScheme} |
| 199 |
|
\end{figure} |
| 200 |
|
|
| 201 |
+ |
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 |
|
\section{\label{liquidCrystalSection:methods}Methods} |
| 212 |
|
|
| 213 |
|
\section{\label{liquidCrystalSection:resultDiscussion}Results and Discussion} |
