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{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 |
+ |
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 |
+ |
\begin{equation} |
232 |
+ |
\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 |
+ |
\label{lipidEq:po1} |
239 |
+ |
\end{equation} |
240 |
+ |
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 |
+ |
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 |
+ |
|
257 |
+ |
|
258 |
+ |
The density correlation along the director is |
259 |
+ |
\begin{equation}g(z) =< \delta (z-z_{ij})>_{ij} / \pi R^{2} \rho |
260 |
+ |
\end{equation}, |
261 |
+ |
where $z_{ij} = r_{ij} cos \beta_{r_{ij}}$ was measured in the |
262 |
+ |
director frame and $R$ is the radius of the cylindrical sampling |
263 |
+ |
region. |
264 |
+ |
|
265 |
+ |
|
266 |
+ |
\section{Results and Conclusion} |
267 |
+ |
\label{sec:results and conclusion} |
268 |
+ |
|
269 |
+ |
To investigate the molecular organization behavior due to different |
270 |
+ |
dipolar orientation and position with respect to the center of the |
271 |
+ |
molecule, |
272 |
+ |
|
273 |
+ |
|
274 |
+ |
|
275 |
|
\section{\label{liquidCrystalSection:methods}Methods} |
276 |
|
|
277 |
|
\section{\label{liquidCrystalSection:resultDiscussion}Results and Discussion} |
278 |
+ |
|
279 |
+ |
\section{Conclusion} |