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incorporating electrostatic interaction. |
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|
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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 |
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point dipoles at different location. Performing a series of |
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molecular dynamics simulations, we explore the structural properties |
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of tilted smectic phases as well as the effect of electrostatic |
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interactions. |
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> |
molecule represented by three rigid GB particles with two point |
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dipoles. Performing a series of molecular dynamics simulations, we |
105 |
> |
explore the structural properties of tilted smectic phases as well |
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> |
as the effect of electrostatic interactions. |
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|
|
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\section{\label{liquidCrystalSection:model}Model} |
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|
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\end{equation} |
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where $\epsilon _{fs}$ is the permittivity of free space. |
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|
|
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\section{Computational Methodology} |
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\section{Results and Discussion} |
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|
|
211 |
|
A series of molecular dynamics simulations were perform to study the |
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|
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} \\ |
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|
u_{iz}u_{ix} & u_{iz}u_{iy} & u_{iz}u_{iz}-\frac{1}{3} % |
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|
\end{pmatrix}, |
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< |
\label{lipidEq:po1} |
238 |
> |
\label{lipidEq:p2} |
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|
\end{equation} |
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where the $u_{i\alpha}$ is the $\alpha$ element of the unit vector |
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|
$\mathbf{\hat{u}}_i$, and the sum over $i$ averages over the whole |
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|
\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, |