| 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 |
| 40 |
|
direction of the molecule in adjacent layer (see |
| 41 |
< |
Fig.~\cite{LCFig:SMCP}). |
| 41 |
> |
Fig.~\ref{LCFig:SMCP}). |
| 42 |
|
|
| 43 |
|
\begin{figure} |
| 44 |
|
\centering |
| 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 |
| 184 |
|
\caption[]{} \label{LCFig:BananaMolecule} |
| 185 |
|
\end{figure} |
| 186 |
|
|
| 187 |
< |
\begin{figure} |
| 188 |
< |
\centering |
| 189 |
< |
\includegraphics[width=\linewidth]{bananGB.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 |
| 208 |
|
\end{equation} |
| 209 |
|
where $\epsilon _{fs}$ is the permittivity of free space. |
| 210 |
|
|
| 211 |
< |
\section{\label{liquidCrystalSection:methods}Methods} |
| 211 |
> |
\section{Computational Methodology} |
| 212 |
|
|
| 213 |
< |
\section{\label{liquidCrystalSection:resultDiscussion}Results and Discussion} |
| 213 |
> |
A series of molecular dynamics simulations were perform to study the |
| 214 |
> |
phase behavior of banana shaped liquid crystals. In each simulation, |
| 215 |
> |
every banana shaped molecule has been represented by three GB |
| 216 |
> |
particles 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 |
> |
|
| 225 |
> |
\subsection{Order Parameters} |
| 226 |
> |
|
| 227 |
> |
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 |
> |
\begin{equation} |
| 234 |
> |
\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 |
> |
\end{equation} |
| 242 |
> |
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 |
> |
|
| 259 |
> |
\subsection{Structure Properties} |
| 260 |
> |
|
| 261 |
> |
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 |
> |
\end{equation}, |
| 266 |
> |
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 |
> |
|
| 269 |
> |
\subsection{Rotational Invariants} |
| 270 |
> |
|
| 271 |
> |
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 |
> |
\begin{eqnarray} |
| 280 |
> |
S_{22}^{220} (r) & = & \frac{1}{{4\sqrt 5 }} \left< \delta (r - |
| 281 |
> |
r_{ij} )((\hat x_i \cdot \hat x_j )^2 - (\hat x_i \cdot \hat y_j |
| 282 |
> |
)^2 - (\hat y_i \cdot \hat x_j )^2 + (\hat y_i \cdot \hat y_j |
| 283 |
> |
)^2 ) \right. \\ |
| 284 |
> |
& & \left. - 2(\hat x_i \cdot \hat y_j )(\hat y_i \cdot \hat x_j ) - |
| 285 |
> |
2(\hat x_i \cdot \hat x_j )(\hat y_i \cdot \hat y_j )) \right> |
| 286 |
> |
\end{eqnarray} |
| 287 |
> |
|
| 288 |
> |
\begin{equation} |
| 289 |
> |
S_{00}^{221} (r) = - \frac{{\sqrt 3 }}{{\sqrt {10} }}\left\langle |
| 290 |
> |
{\delta (r - r_{ij} )((\hat z_i \cdot \hat z_j )(\hat z_i \cdot |
| 291 |
> |
\hat z_j \times \hat r_{ij} ))} \right\rangle |
| 292 |
> |
\end{equation} |
| 293 |
> |
|
| 294 |
> |
\section{Results and Conclusion} |
| 295 |
> |
\label{sec:results and conclusion} |
| 296 |
> |
|
| 297 |
> |
To investigate the molecular organization behavior due to different |
| 298 |
> |
dipolar orientation and position with respect to the center of the |
| 299 |
> |
molecule, |
