| 222 |
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barostat of 50~ps were used to equilibrate the system to desired |
| 223 |
|
temperature and pressure. |
| 224 |
|
|
| 225 |
+ |
\subsection{Order Parameters} |
| 226 |
+ |
|
| 227 |
|
To investigate the phase structure of the model liquid crystal, we |
| 228 |
|
calculated various order parameters and correlation functions. |
| 229 |
|
Particulary, the $P_2$ order parameter allows us to estimate average |
| 256 |
|
\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 |
| 263 |
< |
\end{equation}, |
| 264 |
< |
where $z_{ij} = r_{ij} cos \beta_{r_{ij}}$ was measured in the |
| 265 |
< |
director frame and $R$ is the radius of the cylindrical sampling |
| 266 |
< |
region. |
| 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 $z_{ij} = r_{ij} \dot Z$ 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 s\end{equation} |
| 292 |
+ |
|
| 293 |
|
\section{Results and Conclusion} |
| 294 |
|
\label{sec:results and conclusion} |
| 295 |
|
|
| 296 |
|
To investigate the molecular organization behavior due to different |
| 297 |
|
dipolar orientation and position with respect to the center of the |
| 298 |
|
molecule, |
| 272 |
– |
|
| 273 |
– |
|
| 274 |
– |
|
| 275 |
– |
\section{\label{liquidCrystalSection:methods}Methods} |
| 276 |
– |
|
| 277 |
– |
\section{\label{liquidCrystalSection:resultDiscussion}Results and Discussion} |
| 278 |
– |
|
| 279 |
– |
\section{Conclusion} |
