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 - |
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 ) - 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} |
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 |