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 |
tim |
2786 |
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 |
tim |
2781 |
|
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 |
tim |
2786 |
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 |
tim |
2782 |
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 |
tim |
2839 |
Fig.~\ref{LCFig:SMCP}). |
42 |
tim |
2781 |
|
43 |
tim |
2784 |
\begin{figure} |
44 |
|
|
\centering |
45 |
|
|
\includegraphics[width=\linewidth]{smcp.eps} |
46 |
|
|
\caption[] |
47 |
|
|
{} |
48 |
|
|
\label{LCFig:SMCP} |
49 |
|
|
\end{figure} |
50 |
|
|
|
51 |
tim |
2782 |
Many liquid crystal synthesis experiments suggest that the |
52 |
|
|
occurrence of polarity and chirality strongly relies on the |
53 |
tim |
2786 |
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 |
tim |
2782 |
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, |
59 |
|
|
computer simulation can provide unique insight into molecular |
60 |
|
|
ordering and phase behavior, and hence improve the development of |
61 |
|
|
new experiments and theories. In the last two decades, all-atom |
62 |
|
|
models have been adopted to investigate the structural properties of |
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 |
tim |
2786 |
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 |
tim |
2781 |
|
81 |
tim |
2784 |
Experimental studies by Levelut {\it et al.}~\cite{Levelut1981} |
82 |
|
|
revealed that terminal cyano or nitro groups usually induce |
83 |
|
|
permanent longitudinal dipole moments, which affect the phase |
84 |
|
|
behavior considerably. A series of theoretical studies also drawn |
85 |
|
|
equivalent conclusions. Monte Carlo studies of the GB potential with |
86 |
|
|
fixed longitudinal dipoles (i.e. pointed along the principal axis of |
87 |
|
|
rotation) were shown to enhance smectic phase |
88 |
|
|
stability~\cite{Berardi1996,Satoh1996}. Molecular simulation of GB |
89 |
|
|
ellipsoids with transverse dipoles at the terminus of the molecule |
90 |
|
|
also demonstrated that partial striped bilayer structures were |
91 |
|
|
developed from the smectic phase ~\cite{Berardi1996}. More |
92 |
|
|
significant effects have been shown by including multiple |
93 |
|
|
electrostatic moments. Adding longitudinal point quadrupole moments |
94 |
|
|
to rod-shaped GB mesogens, Withers \textit{et al} induced tilted |
95 |
|
|
smectic behaviour in the molecular system~\cite{Withers2003}. Thus, |
96 |
|
|
it is clear that many liquid-crystal forming molecules, specially, |
97 |
|
|
bent-core molecules, could be modeled more accurately by |
98 |
|
|
incorporating electrostatic interaction. |
99 |
|
|
|
100 |
|
|
In this chapter, we consider system consisting of banana-shaped |
101 |
|
|
molecule represented by three rigid GB particles with one or two |
102 |
|
|
point dipoles at different location. Performing a series of |
103 |
|
|
molecular dynamics simulations, we explore the structural properties |
104 |
|
|
of tilted smectic phases as well as the effect of electrostatic |
105 |
|
|
interactions. |
106 |
|
|
|
107 |
tim |
2685 |
\section{\label{liquidCrystalSection:model}Model} |
108 |
|
|
|
109 |
tim |
2784 |
A typical banana-shaped molecule consists of a rigid aromatic |
110 |
|
|
central bent unit with several rod-like wings which are held |
111 |
|
|
together by some linking units and terminal chains (see |
112 |
|
|
Fig.~\ref{LCFig:BananaMolecule}). In this work, each banana-shaped |
113 |
|
|
mesogen has been modeled as a rigid body consisting of three |
114 |
|
|
equivalent prolate ellipsoidal GB particles. The GB interaction |
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 |
tim |
2785 |
($\epsilon$) parameters. The potential between a pair of three-site |
119 |
|
|
banana-shaped molecules $a$ and $b$ is given by |
120 |
tim |
2784 |
\begin{equation} |
121 |
tim |
2785 |
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 |
tim |
2784 |
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 |
128 |
|
|
}}{{r_{ij} - \sigma (\hat u_i ,\hat u_j ,\hat r_{ij} )}}} \right)^6 |
129 |
|
|
} \right] \label{LCEquation:gb} |
130 |
|
|
\end{equation} |
131 |
|
|
where $\hat u_i,\hat u_j$ are unit vectors specifying the |
132 |
|
|
orientation of two molecules $i$ and $j$ separated by intermolecular |
133 |
|
|
vector $r_{ij}$. $\hat r_{ij}$ is the unit vector along the |
134 |
|
|
intermolecular vector. A schematic diagram of the orientation |
135 |
|
|
vectors is shown in Fig.\ref{LCFigure:GBScheme}. The functional form |
136 |
|
|
for $\sigma$ is given by |
137 |
|
|
\begin{equation} |
138 |
|
|
\sigma (\hat u_i ,\hat u_i ,\hat r_{ij} ) = \sigma _0 \left[ {1 - |
139 |
|
|
\frac{\chi }{2}\left( {\frac{{(\hat r_{ij} \cdot \hat u_i + \hat |
140 |
|
|
r_{ij} \cdot \hat u_j )^2 }}{{1 + \chi \hat u_i \cdot \hat u_j }} |
141 |
|
|
+ \frac{{(\hat r_{ij} \cdot \hat u_i - \hat r_{ij} \cdot \hat u_j |
142 |
|
|
)^2 }}{{1 - \chi \hat u_i \cdot \hat u_j }}} \right)} \right]^{ - |
143 |
|
|
\frac{1}{2}}, |
144 |
|
|
\end{equation} |
145 |
|
|
where the aspect ratio of the particles is governed by shape |
146 |
|
|
anisotropy parameter |
147 |
|
|
\begin{equation} |
148 |
|
|
\chi = \frac{{(\sigma _e /\sigma _s )^2 - 1}}{{(\sigma _e /\sigma |
149 |
|
|
_s )^2 + 1}}. |
150 |
|
|
\label{LCEquation:chi} |
151 |
|
|
\end{equation} |
152 |
|
|
Here, $\sigma_ s$ and $\sigma_{e}$ refer to the side-by-side breadth |
153 |
tim |
2785 |
and the end-to-end length of the ellipsoid, respectively. The well |
154 |
tim |
2784 |
depth parameters takes the form |
155 |
|
|
\begin{equation} |
156 |
|
|
\epsilon (\hat u_i ,\hat u_j ,\hat r_{ij} ) = \epsilon _0 \epsilon |
157 |
|
|
^v (\hat u_i ,\hat u_j )\epsilon '^\mu (\hat u_i ,\hat u_j ,\hat |
158 |
|
|
r_{ij} ) |
159 |
|
|
\end{equation} |
160 |
|
|
where $\epsilon_{0}$ is a constant term and |
161 |
|
|
\begin{equation} |
162 |
|
|
\epsilon (\hat u_i ,\hat u_j ) = \frac{1}{{\sqrt {1 - \chi ^2 (\hat |
163 |
|
|
u_i \cdot \hat u_j )^2 } }} |
164 |
|
|
\end{equation} |
165 |
|
|
and |
166 |
|
|
\begin{equation} |
167 |
|
|
\epsilon '(\hat u_i ,\hat u_j ,\hat r_{ij} ) = 1 - \frac{{\chi |
168 |
|
|
'}}{2}\left[ {\frac{{(\hat r_{ij} \cdot \hat u_i + \hat r_{ij} |
169 |
|
|
\cdot \hat u_j )^2 }}{{1 + \chi '\hat u_i \cdot \hat u_j }} + |
170 |
|
|
\frac{{(\hat r_{ij} \cdot \hat u_i - \hat r_{ij} \cdot \hat u_j |
171 |
|
|
)^2 }}{{1 - \chi '\hat u_i \cdot \hat u_j }}} \right] |
172 |
|
|
\end{equation} |
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 |
tim |
2785 |
\begin{equation} |
177 |
tim |
2784 |
\chi ' = \frac{{1 - (\epsilon _e /\epsilon _s )^{1/\mu} }}{{1 + |
178 |
|
|
(\epsilon _e /\epsilon _s )^{1/\mu} }}. |
179 |
|
|
\end{equation} |
180 |
|
|
|
181 |
|
|
\begin{figure} |
182 |
|
|
\centering |
183 |
|
|
\includegraphics[width=\linewidth]{banana.eps} |
184 |
|
|
\caption[]{} \label{LCFig:BananaMolecule} |
185 |
|
|
\end{figure} |
186 |
|
|
|
187 |
tim |
2805 |
%\begin{figure} |
188 |
|
|
%\centering |
189 |
|
|
%\includegraphics[width=\linewidth]{bananGB.eps} |
190 |
|
|
%\caption[]{} \label{LCFigure:BananaGB} |
191 |
|
|
%\end{figure} |
192 |
tim |
2784 |
|
193 |
|
|
\begin{figure} |
194 |
|
|
\centering |
195 |
|
|
\includegraphics[width=\linewidth]{gb_scheme.eps} |
196 |
|
|
\caption[]{Schematic diagram showing definitions of the orientation |
197 |
|
|
vectors for a pair of Gay-Berne molecules} |
198 |
|
|
\label{LCFigure:GBScheme} |
199 |
|
|
\end{figure} |
200 |
|
|
|
201 |
tim |
2785 |
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 |
tim |
2867 |
\section{Computational Methodology} |
212 |
|
|
|
213 |
|
|
A series of molecular dynamics simulations were perform to study the |
214 |
tim |
2870 |
phase behavior of banana shaped liquid crystals. In each simulation, |
215 |
tim |
2880 |
every banana shaped molecule has been represented by three GB |
216 |
|
|
particles which is characterized by $\mu = 1,~ \nu = 2, |
217 |
tim |
2870 |
~\epsilon_{e}/\epsilon_{s} = 1/5$ 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 |
tim |
2867 |
|
225 |
tim |
2871 |
\subsection{Order Parameters} |
226 |
|
|
|
227 |
tim |
2870 |
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 |
tim |
2867 |
\begin{equation} |
234 |
tim |
2870 |
\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 |
tim |
2867 |
\end{equation} |
242 |
tim |
2870 |
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 |
tim |
2867 |
|
259 |
tim |
2871 |
\subsection{Structure Properties} |
260 |
tim |
2867 |
|
261 |
tim |
2871 |
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 |
tim |
2870 |
\end{equation}, |
266 |
tim |
2871 |
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 |
tim |
2867 |
|
269 |
tim |
2871 |
\subsection{Rotational Invariants} |
270 |
tim |
2867 |
|
271 |
tim |
2871 |
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 |
tim |
2887 |
scatting\cite{Blum1972}. Latterly, expansion of the orientation pair |
275 |
tim |
2871 |
correlation in terms of rotation invariant for molecules of |
276 |
|
|
arbitrary shape was introduce by Stone\cite{Stone1978} and adopted |
277 |
tim |
2887 |
by other researchers in liquid crystal studies\cite{Berardi2003}. |
278 |
tim |
2871 |
|
279 |
tim |
2880 |
\begin{eqnarray} |
280 |
tim |
2882 |
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 |
tim |
2880 |
\end{eqnarray} |
287 |
tim |
2871 |
|
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 |
tim |
2876 |
\hat z_j \times \hat r_{ij} ))} \right\rangle |
292 |
|
|
\end{equation} |
293 |
tim |
2871 |
|
294 |
tim |
2867 |
\section{Results and Conclusion} |
295 |
|
|
\label{sec:results and conclusion} |
296 |
|
|
|
297 |
tim |
2870 |
To investigate the molecular organization behavior due to different |
298 |
|
|
dipolar orientation and position with respect to the center of the |
299 |
|
|
molecule, |