| 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 |
|
|
| 183 |
|
\begin{figure} |
| 184 |
|
\centering |
| 185 |
|
\includegraphics[width=\linewidth]{banana.eps} |
| 186 |
< |
\caption[]{} \label{LCFig:BananaMolecule} |
| 186 |
> |
\caption[Schematic representation of a typical banana shaped |
| 187 |
> |
molecule]{Schematic representation of a typical banana shaped |
| 188 |
> |
molecule.} \label{LCFig:BananaMolecule} |
| 189 |
|
\end{figure} |
| 190 |
|
|
| 191 |
|
%\begin{figure} |
| 216 |
|
|
| 217 |
|
A series of molecular dynamics simulations were perform to study the |
| 218 |
|
phase behavior of banana shaped liquid crystals. In each simulation, |
| 219 |
< |
every banana shaped molecule has been represented three GB particles |
| 220 |
< |
which is characterized by $\mu = 1,~ \nu = 2, |
| 219 |
> |
every banana shaped molecule has been represented by three GB |
| 220 |
> |
particles which is characterized by $\mu = 1,~ \nu = 2, |
| 221 |
|
~\epsilon_{e}/\epsilon_{s} = 1/5$ and $\sigma_{e}/\sigma_{s} = 3$. |
| 222 |
|
All of the simulations begin with same equilibrated isotropic |
| 223 |
|
configuration where 1024 molecules without dipoles were confined in |
| 275 |
|
As a useful set of correlation functions to describe |
| 276 |
|
position-orientation correlation, rotation invariants were first |
| 277 |
|
applied in a spherical symmetric system to study x-ray and light |
| 278 |
< |
scatting\cite{Blum1971}. Latterly, expansion of the orientation pair |
| 278 |
> |
scatting\cite{Blum1972}. Latterly, expansion of the orientation pair |
| 279 |
|
correlation in terms of rotation invariant for molecules of |
| 280 |
|
arbitrary shape was introduce by Stone\cite{Stone1978} and adopted |
| 281 |
< |
by other researchers in liquid crystal studies\cite{Berardi2000}. |
| 281 |
> |
by other researchers in liquid crystal studies\cite{Berardi2003}. |
| 282 |
|
|
| 283 |
< |
\begin{equation} |
| 284 |
< |
S_{22}^{220} (r) & = & \frac{1}{{4\sqrt 5 }}\left\langle {\delta (r |
| 285 |
< |
- r_{ij} )((\hat x_i \cdot \hat x_j )^2 - (\hat x_i \cdot \hat |
| 286 |
< |
y_j )^2 - (\hat y_i \cdot \hat x_j )^2 + (\hat y_i \cdot \hat |
| 287 |
< |
y_j )^2 ) - 2(\hat x_i \cdot \hat y_j )(\hat y_i \cdot \hat x_j ) |
| 288 |
< |
- 2(\hat x_i \cdot \hat x_j )(\hat y_i \cdot \hat y_j ))} |
| 289 |
< |
\right\rangle |
| 290 |
< |
\end{equation} |
| 283 |
> |
\begin{eqnarray} |
| 284 |
> |
S_{22}^{220} (r) & = & \frac{1}{{4\sqrt 5 }} \left< \delta (r - |
| 285 |
> |
r_{ij} )((\hat x_i \cdot \hat x_j )^2 - (\hat x_i \cdot \hat y_j |
| 286 |
> |
)^2 - (\hat y_i \cdot \hat x_j )^2 + (\hat y_i \cdot \hat y_j |
| 287 |
> |
)^2 ) \right. \\ |
| 288 |
> |
& & \left. - 2(\hat x_i \cdot \hat y_j )(\hat y_i \cdot \hat x_j ) - |
| 289 |
> |
2(\hat x_i \cdot \hat x_j )(\hat y_i \cdot \hat y_j )) \right> |
| 290 |
> |
\end{eqnarray} |
| 291 |
|
|
| 292 |
|
\begin{equation} |
| 293 |
|
S_{00}^{221} (r) = - \frac{{\sqrt 3 }}{{\sqrt {10} }}\left\langle |
