| 2 |
|
|
| 3 |
|
\section{\label{liquidCrystalSection:introduction}Introduction} |
| 4 |
|
|
| 5 |
+ |
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 |
+ |
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\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.~\ref{LCFig:SMCP}). |
| 42 |
+ |
|
| 43 |
+ |
\begin{figure} |
| 44 |
+ |
\centering |
| 45 |
+ |
\includegraphics[width=\linewidth]{smcp.eps} |
| 46 |
+ |
\caption[SmCP Phase Packing] {Four possible SmCP phase packings that |
| 47 |
+ |
are characterized by the relative tilt direction(A and S refer an |
| 48 |
+ |
anticlinic tilt or a synclinic ) and the polarization orientation (A |
| 49 |
+ |
and F represent antiferroelectric or ferroelectric polar order).} |
| 50 |
+ |
\label{LCFig:SMCP} |
| 51 |
+ |
\end{figure} |
| 52 |
+ |
|
| 53 |
+ |
Many liquid crystal synthesis experiments suggest that the |
| 54 |
+ |
occurrence of polarity and chirality strongly relies on the |
| 55 |
+ |
molecular structure and intermolecular interaction\cite{Reddy2006}. |
| 56 |
+ |
From a theoretical point of view, it is of fundamental interest to |
| 57 |
+ |
study the structural properties of liquid crystal phases formed by |
| 58 |
+ |
banana-shaped molecules and understand their connection to the |
| 59 |
+ |
molecular structure, especially with respect to the spontaneous |
| 60 |
+ |
achiral symmetry breaking. As a complementary tool to experiment, |
| 61 |
+ |
computer simulation can provide unique insight into molecular |
| 62 |
+ |
ordering and phase behavior, and hence improve the development of |
| 63 |
+ |
new experiments and theories. In the last two decades, all-atom |
| 64 |
+ |
models have been adopted to investigate the structural properties of |
| 65 |
+ |
smectic arrangements\cite{Cook2000, Lansac2001}, as well as other |
| 66 |
+ |
bulk properties, such as rotational viscosity and flexoelectric |
| 67 |
+ |
coefficients\cite{Cheung2002, Cheung2004}. However, due to the |
| 68 |
+ |
limitation of time scale required for phase transition and the |
| 69 |
+ |
length scale required for representing bulk behavior, |
| 70 |
+ |
models\cite{Perram1985, Gay1981}, which are based on the observation |
| 71 |
+ |
that liquid crystal order is exhibited by a range of non-molecular |
| 72 |
+ |
bodies with high shape anisotropies, became the dominant models in |
| 73 |
+ |
the field of liquid crystal phase behavior. Previous simulation |
| 74 |
+ |
studies using hard spherocylinder dimer model\cite{Camp1999} produce |
| 75 |
+ |
nematic phases, while hard rod simulation studies identified a |
| 76 |
+ |
Landau point\cite{Bates2005}, at which the isotropic phase undergoes |
| 77 |
+ |
a direct transition to the biaxial nematic, as well as some possible |
| 78 |
+ |
liquid crystal phases\cite{Lansac2003}. Other anisotropic models |
| 79 |
+ |
using Gay-Berne(GB) potential, which produce interactions that favor |
| 80 |
+ |
local alignment, give the evidence of the novel packing arrangements |
| 81 |
+ |
of bent-core molecules\cite{Memmer2002,Orlandi2006}. |
| 82 |
+ |
|
| 83 |
+ |
Experimental studies by Levelut {\it et al.}~\cite{Levelut1981} |
| 84 |
+ |
revealed that terminal cyano or nitro groups usually induce |
| 85 |
+ |
permanent longitudinal dipole moments, which affect the phase |
| 86 |
+ |
behavior considerably. A series of theoretical studies also drawn |
| 87 |
+ |
equivalent conclusions. Monte Carlo studies of the GB potential with |
| 88 |
+ |
fixed longitudinal dipoles (i.e. pointed along the principal axis of |
| 89 |
+ |
rotation) were shown to enhance smectic phase |
| 90 |
+ |
stability~\cite{Berardi1996,Satoh1996}. Molecular simulation of GB |
| 91 |
+ |
ellipsoids with transverse dipoles at the terminus of the molecule |
| 92 |
+ |
also demonstrated that partial striped bilayer structures were |
| 93 |
+ |
developed from the smectic phase ~\cite{Berardi1996}. More |
| 94 |
+ |
significant effects have been shown by including multiple |
| 95 |
+ |
electrostatic moments. Adding longitudinal point quadrupole moments |
| 96 |
+ |
to rod-shaped GB mesogens, Withers \textit{et al} induced tilted |
| 97 |
+ |
smectic behaviour in the molecular system~\cite{Withers2003}. Thus, |
| 98 |
+ |
it is clear that many liquid-crystal forming molecules, specially, |
| 99 |
+ |
bent-core molecules, could be modeled more accurately by |
| 100 |
+ |
incorporating electrostatic interaction. |
| 101 |
+ |
|
| 102 |
+ |
In this chapter, we consider system consisting of banana-shaped |
| 103 |
+ |
molecule represented by three rigid GB particles with one or two |
| 104 |
+ |
point dipoles at different location. Performing a series of |
| 105 |
+ |
molecular dynamics simulations, we explore the structural properties |
| 106 |
+ |
of tilted smectic phases as well as the effect of electrostatic |
| 107 |
+ |
interactions. |
| 108 |
+ |
|
| 109 |
|
\section{\label{liquidCrystalSection:model}Model} |
| 110 |
|
|
| 111 |
< |
\section{\label{liquidCrystalSection:methods}Methods} |
| 111 |
> |
A typical banana-shaped molecule consists of a rigid aromatic |
| 112 |
> |
central bent unit with several rod-like wings which are held |
| 113 |
> |
together by some linking units and terminal chains (see |
| 114 |
> |
Fig.~\ref{LCFig:BananaMolecule}). In this work, each banana-shaped |
| 115 |
> |
mesogen has been modeled as a rigid body consisting of three |
| 116 |
> |
equivalent prolate ellipsoidal GB particles. The GB interaction |
| 117 |
> |
potential used to mimic the apolar characteristics of liquid crystal |
| 118 |
> |
molecules takes the familiar form of Lennard-Jones function with |
| 119 |
> |
orientation and position dependent range ($\sigma$) and well depth |
| 120 |
> |
($\epsilon$) parameters. The potential between a pair of three-site |
| 121 |
> |
banana-shaped molecules $a$ and $b$ is given by |
| 122 |
> |
\begin{equation} |
| 123 |
> |
V_{ab}^{GB} = \sum\limits_{i \in a,j \in b} {V_{ij}^{GB} }. |
| 124 |
> |
\end{equation} |
| 125 |
> |
Every site-site interaction can can be expressed as, |
| 126 |
> |
\begin{equation} |
| 127 |
> |
V_{ij}^{GB} = 4\epsilon (\hat u_i ,\hat u_j ,\hat r_{ij} )\left[ |
| 128 |
> |
{\left( {\frac{{\sigma _0 }}{{r_{ij} - \sigma (\hat u_i ,\hat u_j |
| 129 |
> |
,\hat r_{ij} )}}} \right)^{12} - \left( {\frac{{\sigma _0 |
| 130 |
> |
}}{{r_{ij} - \sigma (\hat u_i ,\hat u_j ,\hat r_{ij} )}}} \right)^6 |
| 131 |
> |
} \right] \label{LCEquation:gb} |
| 132 |
> |
\end{equation} |
| 133 |
> |
where $\hat u_i,\hat u_j$ are unit vectors specifying the |
| 134 |
> |
orientation of two molecules $i$ and $j$ separated by intermolecular |
| 135 |
> |
vector $r_{ij}$. $\hat r_{ij}$ is the unit vector along the |
| 136 |
> |
intermolecular vector. A schematic diagram of the orientation |
| 137 |
> |
vectors is shown in Fig.\ref{LCFigure:GBScheme}. The functional form |
| 138 |
> |
for $\sigma$ is given by |
| 139 |
> |
\begin{equation} |
| 140 |
> |
\sigma (\hat u_i ,\hat u_i ,\hat r_{ij} ) = \sigma _0 \left[ {1 - |
| 141 |
> |
\frac{\chi }{2}\left( {\frac{{(\hat r_{ij} \cdot \hat u_i + \hat |
| 142 |
> |
r_{ij} \cdot \hat u_j )^2 }}{{1 + \chi \hat u_i \cdot \hat u_j }} |
| 143 |
> |
+ \frac{{(\hat r_{ij} \cdot \hat u_i - \hat r_{ij} \cdot \hat u_j |
| 144 |
> |
)^2 }}{{1 - \chi \hat u_i \cdot \hat u_j }}} \right)} \right]^{ - |
| 145 |
> |
\frac{1}{2}}, |
| 146 |
> |
\end{equation} |
| 147 |
> |
where the aspect ratio of the particles is governed by shape |
| 148 |
> |
anisotropy parameter |
| 149 |
> |
\begin{equation} |
| 150 |
> |
\chi = \frac{{(\sigma _e /\sigma _s )^2 - 1}}{{(\sigma _e /\sigma |
| 151 |
> |
_s )^2 + 1}}. |
| 152 |
> |
\label{LCEquation:chi} |
| 153 |
> |
\end{equation} |
| 154 |
> |
Here, $\sigma_ s$ and $\sigma_{e}$ refer to the side-by-side breadth |
| 155 |
> |
and the end-to-end length of the ellipsoid, respectively. The well |
| 156 |
> |
depth parameters takes the form |
| 157 |
> |
\begin{equation} |
| 158 |
> |
\epsilon (\hat u_i ,\hat u_j ,\hat r_{ij} ) = \epsilon _0 \epsilon |
| 159 |
> |
^v (\hat u_i ,\hat u_j )\epsilon '^\mu (\hat u_i ,\hat u_j ,\hat |
| 160 |
> |
r_{ij} ) |
| 161 |
> |
\end{equation} |
| 162 |
> |
where $\epsilon_{0}$ is a constant term and |
| 163 |
> |
\begin{equation} |
| 164 |
> |
\epsilon (\hat u_i ,\hat u_j ) = \frac{1}{{\sqrt {1 - \chi ^2 (\hat |
| 165 |
> |
u_i \cdot \hat u_j )^2 } }} |
| 166 |
> |
\end{equation} |
| 167 |
> |
and |
| 168 |
> |
\begin{equation} |
| 169 |
> |
\epsilon '(\hat u_i ,\hat u_j ,\hat r_{ij} ) = 1 - \frac{{\chi |
| 170 |
> |
'}}{2}\left[ {\frac{{(\hat r_{ij} \cdot \hat u_i + \hat r_{ij} |
| 171 |
> |
\cdot \hat u_j )^2 }}{{1 + \chi '\hat u_i \cdot \hat u_j }} + |
| 172 |
> |
\frac{{(\hat r_{ij} \cdot \hat u_i - \hat r_{ij} \cdot \hat u_j |
| 173 |
> |
)^2 }}{{1 - \chi '\hat u_i \cdot \hat u_j }}} \right] |
| 174 |
> |
\end{equation} |
| 175 |
> |
where the well depth anisotropy parameter $\chi '$ depends on the |
| 176 |
> |
ratio between \textit{end-to-end} well depth $\epsilon _e$ and |
| 177 |
> |
\textit{side-by-side} well depth $\epsilon_s$, |
| 178 |
> |
\begin{equation} |
| 179 |
> |
\chi ' = \frac{{1 - (\epsilon _e /\epsilon _s )^{1/\mu} }}{{1 + |
| 180 |
> |
(\epsilon _e /\epsilon _s )^{1/\mu} }}. |
| 181 |
> |
\end{equation} |
| 182 |
|
|
| 183 |
< |
\section{\label{liquidCrystalSection:resultDiscussion}Results and Discussion} |
| 183 |
> |
\begin{figure} |
| 184 |
> |
\centering |
| 185 |
> |
\includegraphics[width=\linewidth]{banana.eps} |
| 186 |
> |
\caption[Schematic representation of a typical banana shaped |
| 187 |
> |
molecule]{Schematic representation of a typical banana shaped |
| 188 |
> |
molecule.} \label{LCFig:BananaMolecule} |
| 189 |
> |
\end{figure} |
| 190 |
> |
|
| 191 |
> |
%\begin{figure} |
| 192 |
> |
%\centering |
| 193 |
> |
%\includegraphics[width=\linewidth]{bananGB.eps} |
| 194 |
> |
%\caption[]{} \label{LCFigure:BananaGB} |
| 195 |
> |
%\end{figure} |
| 196 |
> |
|
| 197 |
> |
\begin{figure} |
| 198 |
> |
\centering |
| 199 |
> |
\includegraphics[width=\linewidth]{gb_scheme.eps} |
| 200 |
> |
\caption[]{Schematic diagram showing definitions of the orientation |
| 201 |
> |
vectors for a pair of Gay-Berne molecules} |
| 202 |
> |
\label{LCFigure:GBScheme} |
| 203 |
> |
\end{figure} |
| 204 |
> |
|
| 205 |
> |
To account for the permanent dipolar interactions, there should be |
| 206 |
> |
an electrostatic interaction term of the form |
| 207 |
> |
\begin{equation} |
| 208 |
> |
V_{ab}^{dp} = \sum\limits_{i \in a,j \in b} {\frac{1}{{4\pi |
| 209 |
> |
\epsilon _{fs} }}\left[ {\frac{{\mu _i \cdot \mu _j }}{{r_{ij}^3 }} |
| 210 |
> |
- \frac{{3\left( {\mu _i \cdot r_{ij} } \right)\left( {\mu _i \cdot |
| 211 |
> |
r_{ij} } \right)}}{{r_{ij}^5 }}} \right]} |
| 212 |
> |
\end{equation} |
| 213 |
> |
where $\epsilon _{fs}$ is the permittivity of free space. |
| 214 |
> |
|
| 215 |
> |
\section{Computational Methodology} |
| 216 |
> |
|
| 217 |
> |
A series of molecular dynamics simulations were perform to study the |
| 218 |
> |
phase behavior of banana shaped liquid crystals. In each simulation, |
| 219 |
> |
every banana shaped molecule has been represented by three GB |
| 220 |
> |
particles which is characterized by $\mu = 1,~ \nu = 2, |
| 221 |
> |
~\epsilon_{e}/\epsilon_{s} = 1/5$ and $\sigma_{e}/\sigma_{s} = 3$. |
| 222 |
> |
All of the simulations begin with same equilibrated isotropic |
| 223 |
> |
configuration where 1024 molecules without dipoles were confined in |
| 224 |
> |
a $160\times 160 \times 120$ box. After the dipolar interactions are |
| 225 |
> |
switched on, 2~ns NPTi cooling run with themostat of 2~ps and |
| 226 |
> |
barostat of 50~ps were used to equilibrate the system to desired |
| 227 |
> |
temperature and pressure. |
| 228 |
> |
|
| 229 |
> |
\subsection{Order Parameters} |
| 230 |
> |
|
| 231 |
> |
To investigate the phase structure of the model liquid crystal, we |
| 232 |
> |
calculated various order parameters and correlation functions. |
| 233 |
> |
Particulary, the $P_2$ order parameter allows us to estimate average |
| 234 |
> |
alignment along the director axis $Z$ which can be identified from |
| 235 |
> |
the largest eigen value obtained by diagonalizing the order |
| 236 |
> |
parameter tensor |
| 237 |
> |
\begin{equation} |
| 238 |
> |
\overleftrightarrow{\mathsf{Q}} = \frac{1}{N}\sum_i^N % |
| 239 |
> |
\begin{pmatrix} % |
| 240 |
> |
u_{ix}u_{ix}-\frac{1}{3} & u_{ix}u_{iy} & u_{ix}u_{iz} \\ |
| 241 |
> |
u_{iy}u_{ix} & u_{iy}u_{iy}-\frac{1}{3} & u_{iy}u_{iz} \\ |
| 242 |
> |
u_{iz}u_{ix} & u_{iz}u_{iy} & u_{iz}u_{iz}-\frac{1}{3} % |
| 243 |
> |
\end{pmatrix}, |
| 244 |
> |
\label{lipidEq:po1} |
| 245 |
> |
\end{equation} |
| 246 |
> |
where the $u_{i\alpha}$ is the $\alpha$ element of the unit vector |
| 247 |
> |
$\mathbf{\hat{u}}_i$, and the sum over $i$ averages over the whole |
| 248 |
> |
collection of unit vectors. The $P_2$ order parameter for uniaxial |
| 249 |
> |
phase is then simply given by |
| 250 |
> |
\begin{equation} |
| 251 |
> |
\langle P_2 \rangle = \frac{3}{2}\lambda_{\text{max}}. |
| 252 |
> |
\label{lipidEq:po3} |
| 253 |
> |
\end{equation} |
| 254 |
> |
In addition to the $P_2$ order parameter, $ R_{2,2}^2$ order |
| 255 |
> |
parameter for biaxial phase is introduced to describe the ordering |
| 256 |
> |
in the plane orthogonal to the director by |
| 257 |
> |
\begin{equation} |
| 258 |
> |
R_{2,2}^2 = \frac{1}{4}\left\langle {(x_i \cdot X)^2 - (x_i \cdot |
| 259 |
> |
Y)^2 - (y_i \cdot X)^2 + (y_i \cdot Y)^2 } \right\rangle |
| 260 |
> |
\end{equation} |
| 261 |
> |
where $X$, $Y$ and $Z$ are axis of the director frame. |
| 262 |
> |
|
| 263 |
> |
\subsection{Structure Properties} |
| 264 |
> |
|
| 265 |
> |
It is more important to show the density correlation along the |
| 266 |
> |
director |
| 267 |
> |
\begin{equation} |
| 268 |
> |
g(z) =< \delta (z-z_{ij})>_{ij} / \pi R^{2} \rho |
| 269 |
> |
\end{equation}, |
| 270 |
> |
where $z_{ij} = r_{ij} \dot Z$ was measured in the director frame |
| 271 |
> |
and $R$ is the radius of the cylindrical sampling region. |
| 272 |
> |
|
| 273 |
> |
\subsection{Rotational Invariants} |
| 274 |
> |
|
| 275 |
> |
As a useful set of correlation functions to describe |
| 276 |
> |
position-orientation correlation, rotation invariants were first |
| 277 |
> |
applied in a spherical symmetric system to study x-ray and light |
| 278 |
> |
scatting\cite{Blum1972}. Latterly, expansion of the orientation pair |
| 279 |
> |
correlation in terms of rotation invariant for molecules of |
| 280 |
> |
arbitrary shape was introduce by Stone\cite{Stone1978} and adopted |
| 281 |
> |
by other researchers in liquid crystal studies\cite{Berardi2003}. |
| 282 |
> |
|
| 283 |
> |
\begin{eqnarray} |
| 284 |
> |
S_{22}^{220} (r) & = & \frac{1}{{4\sqrt 5 }} \left< \delta (r - |
| 285 |
> |
r_{ij} )((\hat x_i \cdot \hat x_j )^2 - (\hat x_i \cdot \hat y_j |
| 286 |
> |
)^2 - (\hat y_i \cdot \hat x_j )^2 + (\hat y_i \cdot \hat y_j |
| 287 |
> |
)^2 ) \right. \\ |
| 288 |
> |
& & \left. - 2(\hat x_i \cdot \hat y_j )(\hat y_i \cdot \hat x_j ) - |
| 289 |
> |
2(\hat x_i \cdot \hat x_j )(\hat y_i \cdot \hat y_j )) \right> |
| 290 |
> |
\end{eqnarray} |
| 291 |
> |
|
| 292 |
> |
\begin{equation} |
| 293 |
> |
S_{00}^{221} (r) = - \frac{{\sqrt 3 }}{{\sqrt {10} }}\left\langle |
| 294 |
> |
{\delta (r - r_{ij} )((\hat z_i \cdot \hat z_j )(\hat z_i \cdot |
| 295 |
> |
\hat z_j \times \hat r_{ij} ))} \right\rangle |
| 296 |
> |
\end{equation} |
| 297 |
> |
|
| 298 |
> |
\section{Results and Conclusion} |
| 299 |
> |
\label{sec:results and conclusion} |
| 300 |
> |
|
| 301 |
> |
To investigate the molecular organization behavior due to different |
| 302 |
> |
dipolar orientation and position with respect to the center of the |
| 303 |
> |
molecule, |
