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barostat of 50~ps were used to equilibrate the system to desired |
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temperature and pressure. |
224 |
|
|
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\subsection{Order Parameters} |
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+ |
|
227 |
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To investigate the phase structure of the model liquid crystal, we |
228 |
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calculated various order parameters and correlation functions. |
229 |
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Particulary, the $P_2$ order parameter allows us to estimate average |
256 |
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\end{equation} |
257 |
|
where $X$, $Y$ and $Z$ are axis of the director frame. |
258 |
|
|
259 |
+ |
\subsection{Structure Properties} |
260 |
|
|
261 |
< |
The density correlation along the director is |
262 |
< |
\begin{equation}g(z) =< \delta (z-z_{ij})>_{ij} / \pi R^{2} \rho |
261 |
> |
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 |
|
\end{equation}, |
266 |
< |
where $z_{ij} = r_{ij} cos \beta_{r_{ij}}$ was measured in the |
267 |
< |
director frame and $R$ is the radius of the cylindrical sampling |
268 |
< |
region. |
266 |
> |
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 |
> |
|
269 |
> |
\subsection{Rotational Invariants} |
270 |
> |
|
271 |
> |
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 |
> |
scatting\cite{Blum1971}. Latterly, expansion of the orientation pair |
275 |
> |
correlation in terms of rotation invariant for molecules of |
276 |
> |
arbitrary shape was introduce by Stone\cite{Stone1978} and adopted |
277 |
> |
by other researchers in liquid crystal studies\cite{Berardi2000}. |
278 |
|
|
279 |
+ |
\begin{equation} |
280 |
+ |
S_{22}^{220} (r) = \frac{1}{{4\sqrt 5 }}\left\langle {\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 ) - 2(\hat x_i \cdot \hat y_j )(\hat y_i \cdot \hat x_j ) - |
284 |
+ |
2(\hat x_i \cdot \hat x_j )(\hat y_i \cdot \hat y_j ))} |
285 |
+ |
\right\rangle |
286 |
+ |
\end{equation} |
287 |
|
|
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 |
+ |
\hat z_j \times \hat r_{ij} ))} \right\rangle |
292 |
+ |
\end{equation} |
293 |
+ |
|
294 |
|
\section{Results and Conclusion} |
295 |
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\label{sec:results and conclusion} |
296 |
|
|
297 |
|
To investigate the molecular organization behavior due to different |
298 |
|
dipolar orientation and position with respect to the center of the |
299 |
|
molecule, |
272 |
– |
|
273 |
– |
|
274 |
– |
|
275 |
– |
\section{\label{liquidCrystalSection:methods}Methods} |
276 |
– |
|
277 |
– |
\section{\label{liquidCrystalSection:resultDiscussion}Results and Discussion} |
278 |
– |
|
279 |
– |
\section{Conclusion} |