38 |
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packing arrangements (two conglomerates and two macroscopic |
39 |
|
racemates), which depend on the tilt direction and the polar |
40 |
|
direction of the molecule in adjacent layer (see |
41 |
< |
Fig.~\cite{LCFig:SMCP}). |
41 |
> |
Fig.~\ref{LCFig:SMCP}). |
42 |
|
|
43 |
|
\begin{figure} |
44 |
|
\centering |
184 |
|
\caption[]{} \label{LCFig:BananaMolecule} |
185 |
|
\end{figure} |
186 |
|
|
187 |
< |
\begin{figure} |
188 |
< |
\centering |
189 |
< |
\includegraphics[width=\linewidth]{bananGB.eps} |
190 |
< |
\caption[]{} \label{LCFigure:BananaGB} |
191 |
< |
\end{figure} |
187 |
> |
%\begin{figure} |
188 |
> |
%\centering |
189 |
> |
%\includegraphics[width=\linewidth]{bananGB.eps} |
190 |
> |
%\caption[]{} \label{LCFigure:BananaGB} |
191 |
> |
%\end{figure} |
192 |
|
|
193 |
|
\begin{figure} |
194 |
|
\centering |
208 |
|
\end{equation} |
209 |
|
where $\epsilon _{fs}$ is the permittivity of free space. |
210 |
|
|
211 |
+ |
\section{Computational Methodology} |
212 |
+ |
|
213 |
+ |
A series of molecular dynamics simulations were perform to study the |
214 |
+ |
phase behavior of banana shaped liquid crystals. In each simulation, |
215 |
+ |
every banana shaped molecule has been represented three GB particles |
216 |
+ |
which is characterized by $\mu = 1,~ \nu = 2, |
217 |
+ |
~\epsilon_{e}/\epsilon_{s} = 1/5$ and $\sigma_{e}/\sigma_{s} = 3$. |
218 |
+ |
All of the simulations begin with same equilibrated isotropic |
219 |
+ |
configuration where 1024 molecules without dipoles were confined in |
220 |
+ |
a $160\times 160 \times 120$ box. After the dipolar interactions are |
221 |
+ |
switched on, 2~ns NPTi cooling run with themostat of 2~ps and |
222 |
+ |
barostat of 50~ps were used to equilibrate the system to desired |
223 |
+ |
temperature and pressure. |
224 |
+ |
|
225 |
+ |
To investigate the phase structure of the model liquid crystal, we |
226 |
+ |
calculated various order parameters and correlation functions. |
227 |
+ |
Particulary, the $P_2$ order parameter allows us to estimate average |
228 |
+ |
alignment along the director axis $Z$ which can be identified from |
229 |
+ |
the largest eigen value obtained by diagonalizing the order |
230 |
+ |
parameter tensor |
231 |
+ |
\begin{equation} |
232 |
+ |
\overleftrightarrow{\mathsf{Q}} = \frac{1}{N}\sum_i^N % |
233 |
+ |
\begin{pmatrix} % |
234 |
+ |
u_{ix}u_{ix}-\frac{1}{3} & u_{ix}u_{iy} & u_{ix}u_{iz} \\ |
235 |
+ |
u_{iy}u_{ix} & u_{iy}u_{iy}-\frac{1}{3} & u_{iy}u_{iz} \\ |
236 |
+ |
u_{iz}u_{ix} & u_{iz}u_{iy} & u_{iz}u_{iz}-\frac{1}{3} % |
237 |
+ |
\end{pmatrix}, |
238 |
+ |
\label{lipidEq:po1} |
239 |
+ |
\end{equation} |
240 |
+ |
where the $u_{i\alpha}$ is the $\alpha$ element of the unit vector |
241 |
+ |
$\mathbf{\hat{u}}_i$, and the sum over $i$ averages over the whole |
242 |
+ |
collection of unit vectors. The $P_2$ order parameter for uniaxial |
243 |
+ |
phase is then simply given by |
244 |
+ |
\begin{equation} |
245 |
+ |
\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. |
256 |
+ |
|
257 |
+ |
|
258 |
+ |
The density correlation along the director is |
259 |
+ |
\begin{equation}g(z) =< \delta (z-z_{ij})>_{ij} / \pi R^{2} \rho |
260 |
+ |
\end{equation}, |
261 |
+ |
where $z_{ij} = r_{ij} cos \beta_{r_{ij}}$ was measured in the |
262 |
+ |
director frame and $R$ is the radius of the cylindrical sampling |
263 |
+ |
region. |
264 |
+ |
|
265 |
+ |
|
266 |
+ |
\section{Results and Conclusion} |
267 |
+ |
\label{sec:results and conclusion} |
268 |
+ |
|
269 |
+ |
To investigate the molecular organization behavior due to different |
270 |
+ |
dipolar orientation and position with respect to the center of the |
271 |
+ |
molecule, |
272 |
+ |
|
273 |
+ |
|
274 |
+ |
|
275 |
|
\section{\label{liquidCrystalSection:methods}Methods} |
276 |
|
|
277 |
|
\section{\label{liquidCrystalSection:resultDiscussion}Results and Discussion} |
278 |
+ |
|
279 |
+ |
\section{Conclusion} |