| 100 |
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incorporating electrostatic interaction. |
| 101 |
|
|
| 102 |
|
In this chapter, we consider system consisting of banana-shaped |
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< |
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. |
| 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 |
|
|
| 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}{|c|c|c|c|c|c|} |
| 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 |
> |
|
| 281 |
> |
It is also important to show the density correlation along the |
| 282 |
> |
director which is given by : |
| 283 |
|
\begin{equation} |
| 284 |
|
g(z) =< \delta (z-z_{ij})>_{ij} / \pi R^{2} \rho |
| 285 |
|
\end{equation}, |
| 286 |
|
where $z_{ij} = r_{ij} \dot Z$ was measured in the director frame |
| 287 |
|
and $R$ is the radius of the cylindrical sampling region. |
| 288 |
|
|
| 289 |
+ |
\begin{figure} |
| 290 |
+ |
\centering |
| 291 |
+ |
\includegraphics[width=4.5in]{snapshot.eps} |
| 292 |
+ |
\caption[Snapshot of the molecular organization in the layered phase |
| 293 |
+ |
formed at temperature T = 460K and pressure P = 1 atm]{Snapshot of |
| 294 |
+ |
the molecular organization in the layered phase formed at |
| 295 |
+ |
temperature T = 460K and pressure P = 1 atm. (a) diagonal view; (b) |
| 296 |
+ |
side view.} \label{LCFigure:snapshot} |
| 297 |
+ |
\end{figure} |
| 298 |
+ |
|
| 299 |
+ |
\begin{figure} |
| 300 |
+ |
\centering |
| 301 |
+ |
\includegraphics[width=\linewidth]{gofr_gofz.eps} |
| 302 |
+ |
\caption[Correlation Functions of a Bent-core Liquid Crystal System |
| 303 |
+ |
at Temperature T = 460K and Pressure P = 10 atm]{Correlation |
| 304 |
+ |
Functions of a Bent-core Liquid Crystal System at Temperature T = |
| 305 |
+ |
460K and Pressure P = 10 atm. (a) radial correlation function |
| 306 |
+ |
$g(r)$; and (b) density along the director $g(z)$.} |
| 307 |
+ |
\label{LCFigure:gofrz} |
| 308 |
+ |
\end{figure} |
| 309 |
+ |
|
| 310 |
|
\subsection{Rotational Invariants} |
| 311 |
|
|
| 312 |
|
As a useful set of correlation functions to describe |
| 332 |
|
\hat z_j \times \hat r_{ij} ))} \right\rangle |
| 333 |
|
\end{equation} |
| 334 |
|
|
| 335 |
< |
\section{Results and Conclusion} |
| 335 |
> |
\section{Conclusion} |
| 336 |
|
To investigate the molecular organization behavior due to different |
| 337 |
|
dipolar orientation and position with respect to the center of the |
| 338 |
|
molecule, |
