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 |
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 |
< |
\section{\label{liquidCrystalSection:methods}Methods} |
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:resultDiscussion}Results and Discussion} |
211 |
> |
\section{Computational Methodology} |
212 |
> |
|
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 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 |
> |
|
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{equation} |
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 ) - 2(\hat x_i \cdot \hat y_j )(\hat y_i \cdot \hat x_j ) |
284 |
> |
- 2(\hat x_i \cdot \hat x_j )(\hat y_i \cdot \hat y_j ))} |
285 |
> |
\right\rangle |
286 |
> |
\end{equation} |
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, |