| 38 |
|
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 |
| 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} |
