| 43 |
|
\begin{figure} |
| 44 |
|
\centering |
| 45 |
|
\includegraphics[width=\linewidth]{smcp.eps} |
| 46 |
< |
\caption[] |
| 47 |
< |
{} |
| 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 |
|
|
| 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}. |
| 81 |
> |
of bent-core molecules\cite{Memmer2002}. |
| 82 |
|
|
| 83 |
|
Experimental studies by Levelut {\it et al.}~\cite{Levelut1981} |
| 84 |
|
revealed that terminal cyano or nitro groups usually induce |
| 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 |
| 105 |
< |
interactions. |
| 103 |
> |
molecule represented by three rigid GB particles with two point |
| 104 |
> |
dipoles. Performing a series of molecular dynamics simulations, we |
| 105 |
> |
explore the structural properties of tilted smectic phases as well |
| 106 |
> |
as the effect of electrostatic interactions. |
| 107 |
|
|
| 108 |
|
\section{\label{liquidCrystalSection:model}Model} |
| 109 |
|
|
| 182 |
|
\begin{figure} |
| 183 |
|
\centering |
| 184 |
|
\includegraphics[width=\linewidth]{banana.eps} |
| 185 |
< |
\caption[]{} \label{LCFig:BananaMolecule} |
| 185 |
> |
\caption[Schematic representation of a typical banana shaped |
| 186 |
> |
molecule]{Schematic representation of a typical banana shaped |
| 187 |
> |
molecule.} \label{LCFig:BananaMolecule} |
| 188 |
|
\end{figure} |
| 189 |
|
|
| 187 |
– |
%\begin{figure} |
| 188 |
– |
%\centering |
| 189 |
– |
%\includegraphics[width=\linewidth]{bananGB.eps} |
| 190 |
– |
%\caption[]{} \label{LCFigure:BananaGB} |
| 191 |
– |
%\end{figure} |
| 192 |
– |
|
| 190 |
|
\begin{figure} |
| 191 |
|
\centering |
| 192 |
|
\includegraphics[width=\linewidth]{gb_scheme.eps} |
| 193 |
< |
\caption[]{Schematic diagram showing definitions of the orientation |
| 194 |
< |
vectors for a pair of Gay-Berne molecules} |
| 195 |
< |
\label{LCFigure:GBScheme} |
| 193 |
> |
\caption[Schematic diagram showing definitions of the orientation |
| 194 |
> |
vectors for a pair of Gay-Berne molecules]{Schematic diagram showing |
| 195 |
> |
definitions of the orientation vectors for a pair of Gay-Berne |
| 196 |
> |
molecules} \label{LCFigure:GBScheme} |
| 197 |
|
\end{figure} |
| 198 |
|
|
| 199 |
|
To account for the permanent dipolar interactions, there should be |
| 206 |
|
\end{equation} |
| 207 |
|
where $\epsilon _{fs}$ is the permittivity of free space. |
| 208 |
|
|
| 209 |
< |
\section{Computational Methodology} |
| 209 |
> |
\section{Results and Discussion} |
| 210 |
|
|
| 211 |
|
A series of molecular dynamics simulations were perform to study the |
| 212 |
|
phase behavior of banana shaped liquid crystals. In each simulation, |
| 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} |
| 238 |
> |
\label{lipidEq:p2} |
| 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 |
| 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. |
| 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 |
> |
The unit vector for the banana shaped molecule was defined by the |
| 257 |
> |
principle aixs of its middle GB particle. The $P_2$ order parameters |
| 258 |
> |
for the bent-core liquid crystal at different temperature is |
| 259 |
> |
summarized in Table~\ref{liquidCrystal:p2} which identifies a phase |
| 260 |
> |
transition temperature range. |
| 261 |
|
|
| 262 |
+ |
\begin{table} |
| 263 |
+ |
\caption{LIQUID CRYSTAL STRUCTURAL PROPERTIES AS A FUNCTION OF |
| 264 |
+ |
TEMPERATURE} \label{liquidCrystal:p2} |
| 265 |
+ |
\begin{center} |
| 266 |
+ |
\begin{tabular}{cccccc} |
| 267 |
+ |
\hline |
| 268 |
+ |
Temperature (K) & 420 & 440 & 460 & 480 & 600\\ |
| 269 |
+ |
\hline |
| 270 |
+ |
$\langle P_2\rangle$ & 0.984 & 0.982 & 0.975 & 0.967 & 0.067\\ |
| 271 |
+ |
\hline |
| 272 |
+ |
\end{tabular} |
| 273 |
+ |
\end{center} |
| 274 |
+ |
\end{table} |
| 275 |
+ |
|
| 276 |
|
\subsection{Structure Properties} |
| 277 |
|
|
| 278 |
< |
It is more important to show the density correlation along the |
| 279 |
< |
director |
| 278 |
> |
The molecular organization obtained at temperature $T = 460K$ (below |
| 279 |
> |
transition temperature) is shown in Figure~\ref{LCFigure:snapshot}. |
| 280 |
> |
The diagonal view in Fig~\ref{LCFigure:snapshot}(a) shows the |
| 281 |
> |
stacking of the banana shaped molecules while the side view in n |
| 282 |
> |
Figure~\ref{LCFigure:snapshot}(b) demonstrates formation of a |
| 283 |
> |
chevron structure. The first peak of Radial distribution function |
| 284 |
> |
$g(r)$ in Fig.~\ref{LCFigure:gofrz}(a) shows the minimum distance |
| 285 |
> |
for two in plane banana shaped molecules is 4.9 \AA, while the |
| 286 |
> |
second split peak implies the biaxial packing. It is also important |
| 287 |
> |
to show the density correlation along the director which is given by |
| 288 |
> |
: |
| 289 |
|
\begin{equation} |
| 290 |
< |
g(z) =< \delta (z-z_{ij})>_{ij} / \pi R^{2} \rho |
| 290 |
> |
g(z) =\frac{1}{\pi R^{2} \rho}< \delta (z-z_{ij})>_{ij} |
| 291 |
|
\end{equation}, |
| 292 |
|
where $z_{ij} = r_{ij} \dot Z$ was measured in the director frame |
| 293 |
< |
and $R$ is the radius of the cylindrical sampling region. |
| 293 |
> |
and $R$ is the radius of the cylindrical sampling region. The |
| 294 |
> |
oscillation in density plot along the director in |
| 295 |
> |
Fig.~\ref{LCFigure:gofrz}(b) implies the existence of the layered |
| 296 |
> |
structure, and the peak at 27 \AA is attribute to the defect in the |
| 297 |
> |
system. |
| 298 |
|
|
| 299 |
+ |
\begin{figure} |
| 300 |
+ |
\centering |
| 301 |
+ |
\includegraphics[width=4.5in]{snapshot.eps} |
| 302 |
+ |
\caption[Snapshot of the molecular organization in the layered phase |
| 303 |
+ |
formed at temperature T = 460K and pressure P = 1 atm]{Snapshot of |
| 304 |
+ |
the molecular organization in the layered phase formed at |
| 305 |
+ |
temperature T = 460K and pressure P = 1 atm. (a) diagonal view; (b) |
| 306 |
+ |
side view.} \label{LCFigure:snapshot} |
| 307 |
+ |
\end{figure} |
| 308 |
+ |
|
| 309 |
+ |
\begin{figure} |
| 310 |
+ |
\centering |
| 311 |
+ |
\includegraphics[width=\linewidth]{gofr_gofz.eps} |
| 312 |
+ |
\caption[Correlation Functions of a Bent-core Liquid Crystal System |
| 313 |
+ |
at Temperature T = 460K and Pressure P = 10 atm]{Correlation |
| 314 |
+ |
Functions of a Bent-core Liquid Crystal System at Temperature T = |
| 315 |
+ |
460K and Pressure P = 10 atm. (a) radial correlation function |
| 316 |
+ |
$g(r)$; and (b) density along the director $g(z)$.} |
| 317 |
+ |
\label{LCFigure:gofrz} |
| 318 |
+ |
\end{figure} |
| 319 |
+ |
|
| 320 |
|
\subsection{Rotational Invariants} |
| 321 |
|
|
| 322 |
|
As a useful set of correlation functions to describe |
| 323 |
|
position-orientation correlation, rotation invariants were first |
| 324 |
|
applied in a spherical symmetric system to study x-ray and light |
| 325 |
< |
scatting\cite{Blum1971}. Latterly, expansion of the orientation pair |
| 325 |
> |
scatting\cite{Blum1972}. Latterly, expansion of the orientation pair |
| 326 |
|
correlation in terms of rotation invariant for molecules of |
| 327 |
|
arbitrary shape was introduce by Stone\cite{Stone1978} and adopted |
| 328 |
< |
by other researchers in liquid crystal studies\cite{Berardi2000}. |
| 329 |
< |
|
| 328 |
> |
by other researchers in liquid crystal studies\cite{Berardi2003}. In |
| 329 |
> |
order to study the correlation between biaxiality and molecular |
| 330 |
> |
separation distance $r$, we calculate a rotational invariant |
| 331 |
> |
function $S_{22}^{220} (r)$, which is given by : |
| 332 |
|
\begin{eqnarray} |
| 333 |
< |
S_{22}^{220} (r) & = & \frac{1}{{4\sqrt 5 }}\left\langle {\delta (r |
| 334 |
< |
- r_{ij} )((\hat x_i \cdot \hat x_j )^2 - (\hat x_i \cdot \hat |
| 335 |
< |
y_j )^2 - (\hat y_i \cdot \hat x_j )^2 + (\hat y_i \cdot \hat |
| 336 |
< |
y_j |
| 337 |
< |
)^2 ) \right.\\ |
| 338 |
< |
& & \left.- 2(\hat x_i \cdot \hat y_j )(\hat y_i \cdot \hat x_j ) - |
| 286 |
< |
2(\hat x_i \cdot \hat x_j )(\hat y_i \cdot \hat y_j ))} |
| 287 |
< |
\right\rangle |
| 333 |
> |
S_{22}^{220} (r) & = & \frac{1}{{4\sqrt 5 }} \left< \delta (r - |
| 334 |
> |
r_{ij} )((\hat x_i \cdot \hat x_j )^2 - (\hat x_i \cdot \hat y_j |
| 335 |
> |
)^2 - (\hat y_i \cdot \hat x_j )^2 + (\hat y_i \cdot \hat y_j |
| 336 |
> |
)^2 ) \right. \notag \\ |
| 337 |
> |
& & \left. - 2(\hat x_i \cdot \hat y_j )(\hat y_i \cdot \hat x_j ) - |
| 338 |
> |
2(\hat x_i \cdot \hat x_j )(\hat y_i \cdot \hat y_j )) \right>. |
| 339 |
|
\end{eqnarray} |
| 340 |
|
|
| 341 |
< |
\begin{equation} |
| 342 |
< |
S_{00}^{221} (r) = - \frac{{\sqrt 3 }}{{\sqrt {10} }}\left\langle |
| 343 |
< |
{\delta (r - r_{ij} )((\hat z_i \cdot \hat z_j )(\hat z_i \cdot |
| 344 |
< |
\hat z_j \times \hat r_{ij} ))} \right\rangle |
| 345 |
< |
\end{equation} |
| 341 |
> |
%\begin{equation} |
| 342 |
> |
%S_{00}^{221} (r) = - \frac{{\sqrt 3 }}{{\sqrt {10} }}\left\langle |
| 343 |
> |
%{\delta (r - r_{ij} )((\hat z_i \cdot \hat z_j )(\hat z_i \cdot |
| 344 |
> |
%\hat z_j \times \hat r_{ij} ))} \right\rangle |
| 345 |
> |
%\end{equation} |
| 346 |
|
|
| 347 |
< |
\section{Results and Conclusion} |
| 297 |
< |
\label{sec:results and conclusion} |
| 347 |
> |
\section{Conclusion} |
| 348 |
|
|
| 349 |
< |
To investigate the molecular organization behavior due to different |
| 350 |
< |
dipolar orientation and position with respect to the center of the |
| 351 |
< |
molecule, |
| 349 |
> |
We have presented a simple dipolar three-site GB model for banana |
| 350 |
> |
shaped molecules which are capable of forming smectic phases from |
| 351 |
> |
isotropic configuration. |
