| 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} |
| 212 |
|
\end{equation} |
| 213 |
|
where $\epsilon _{fs}$ is the permittivity of free space. |
| 214 |
|
|
| 215 |
< |
\section{\label{liquidCrystalSection:methods}Methods} |
| 215 |
> |
\section{Computational Methodology} |
| 216 |
|
|
| 217 |
< |
\section{\label{liquidCrystalSection:resultDiscussion}Results and Discussion} |
| 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 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 |
| 224 |
> |
a $160\times 160 \times 120$ box. After the dipolar interactions are |
| 225 |
> |
switched on, 2~ns NPTi cooling run with themostat of 2~ps and |
| 226 |
> |
barostat of 50~ps were used to equilibrate the system to desired |
| 227 |
> |
temperature and pressure. |
| 228 |
> |
|
| 229 |
> |
\subsection{Order Parameters} |
| 230 |
> |
|
| 231 |
> |
To investigate the phase structure of the model liquid crystal, we |
| 232 |
> |
calculated various order parameters and correlation functions. |
| 233 |
> |
Particulary, the $P_2$ order parameter allows us to estimate average |
| 234 |
> |
alignment along the director axis $Z$ which can be identified from |
| 235 |
> |
the largest eigen value obtained by diagonalizing the order |
| 236 |
> |
parameter tensor |
| 237 |
> |
\begin{equation} |
| 238 |
> |
\overleftrightarrow{\mathsf{Q}} = \frac{1}{N}\sum_i^N % |
| 239 |
> |
\begin{pmatrix} % |
| 240 |
> |
u_{ix}u_{ix}-\frac{1}{3} & u_{ix}u_{iy} & u_{ix}u_{iz} \\ |
| 241 |
> |
u_{iy}u_{ix} & u_{iy}u_{iy}-\frac{1}{3} & u_{iy}u_{iz} \\ |
| 242 |
> |
u_{iz}u_{ix} & u_{iz}u_{iy} & u_{iz}u_{iz}-\frac{1}{3} % |
| 243 |
> |
\end{pmatrix}, |
| 244 |
> |
\label{lipidEq:po1} |
| 245 |
> |
\end{equation} |
| 246 |
> |
where the $u_{i\alpha}$ is the $\alpha$ element of the unit vector |
| 247 |
> |
$\mathbf{\hat{u}}_i$, and the sum over $i$ averages over the whole |
| 248 |
> |
collection of unit vectors. The $P_2$ order parameter for uniaxial |
| 249 |
> |
phase is then simply given by |
| 250 |
> |
\begin{equation} |
| 251 |
> |
\langle P_2 \rangle = \frac{3}{2}\lambda_{\text{max}}. |
| 252 |
> |
\label{lipidEq:po3} |
| 253 |
> |
\end{equation} |
| 254 |
> |
In addition to the $P_2$ order parameter, $ R_{2,2}^2$ order |
| 255 |
> |
parameter for biaxial phase is introduced to describe the ordering |
| 256 |
> |
in the plane orthogonal to the director by |
| 257 |
> |
\begin{equation} |
| 258 |
> |
R_{2,2}^2 = \frac{1}{4}\left\langle {(x_i \cdot X)^2 - (x_i \cdot |
| 259 |
> |
Y)^2 - (y_i \cdot X)^2 + (y_i \cdot Y)^2 } \right\rangle |
| 260 |
> |
\end{equation} |
| 261 |
> |
where $X$, $Y$ and $Z$ are axis of the director frame. |
| 262 |
> |
|
| 263 |
> |
\subsection{Structure Properties} |
| 264 |
> |
|
| 265 |
> |
It is more important to show the density correlation along the |
| 266 |
> |
director |
| 267 |
> |
\begin{equation} |
| 268 |
> |
g(z) =< \delta (z-z_{ij})>_{ij} / \pi R^{2} \rho |
| 269 |
> |
\end{equation}, |
| 270 |
> |
where $z_{ij} = r_{ij} \dot Z$ was measured in the director frame |
| 271 |
> |
and $R$ is the radius of the cylindrical sampling region. |
| 272 |
> |
|
| 273 |
> |
\subsection{Rotational Invariants} |
| 274 |
> |
|
| 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{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{Berardi2003}. |
| 282 |
> |
|
| 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 |
| 294 |
> |
{\delta (r - r_{ij} )((\hat z_i \cdot \hat z_j )(\hat z_i \cdot |
| 295 |
> |
\hat z_j \times \hat r_{ij} ))} \right\rangle |
| 296 |
> |
\end{equation} |
| 297 |
> |
|
| 298 |
> |
\section{Results and Conclusion} |
| 299 |
> |
\label{sec:results and conclusion} |
| 300 |
> |
|
| 301 |
> |
To investigate the molecular organization behavior due to different |
| 302 |
> |
dipolar orientation and position with respect to the center of the |
| 303 |
> |
molecule, |
