| 116 |
|
potential used to mimic the apolar characteristics of liquid crystal |
| 117 |
|
molecules takes the familiar form of Lennard-Jones function with |
| 118 |
|
orientation and position dependent range ($\sigma$) and well depth |
| 119 |
< |
($\epsilon$) parameters. It can can be expressed as, |
| 119 |
> |
($\epsilon$) parameters. The potential between a pair of three-site |
| 120 |
> |
banana-shaped molecules $a$ and $b$ is given by |
| 121 |
|
\begin{equation} |
| 122 |
+ |
V_{ab}^{GB} = \sum\limits_{i \in a,j \in b} {V_{ij}^{GB} }. |
| 123 |
+ |
\end{equation} |
| 124 |
+ |
Every site-site interaction can can be expressed as, |
| 125 |
+ |
\begin{equation} |
| 126 |
|
V_{ij}^{GB} = 4\epsilon (\hat u_i ,\hat u_j ,\hat r_{ij} )\left[ |
| 127 |
|
{\left( {\frac{{\sigma _0 }}{{r_{ij} - \sigma (\hat u_i ,\hat u_j |
| 128 |
|
,\hat r_{ij} )}}} \right)^{12} - \left( {\frac{{\sigma _0 |
| 151 |
|
\label{LCEquation:chi} |
| 152 |
|
\end{equation} |
| 153 |
|
Here, $\sigma_ s$ and $\sigma_{e}$ refer to the side-by-side breadth |
| 154 |
< |
and the end-to-end length of the ellipsoid, respectively. Twell |
| 154 |
> |
and the end-to-end length of the ellipsoid, respectively. The well |
| 155 |
|
depth parameters takes the form |
| 156 |
|
\begin{equation} |
| 157 |
|
\epsilon (\hat u_i ,\hat u_j ,\hat r_{ij} ) = \epsilon _0 \epsilon |
| 174 |
|
where the well depth anisotropy parameter $\chi '$ depends on the |
| 175 |
|
ratio between \textit{end-to-end} well depth $\epsilon _e$ and |
| 176 |
|
\textit{side-by-side} well depth $\epsilon_s$, |
| 177 |
< |
\begin{eqaution} |
| 177 |
> |
\begin{equation} |
| 178 |
|
\chi ' = \frac{{1 - (\epsilon _e /\epsilon _s )^{1/\mu} }}{{1 + |
| 179 |
|
(\epsilon _e /\epsilon _s )^{1/\mu} }}. |
| 180 |
|
\end{equation} |
| 187 |
|
|
| 188 |
|
\begin{figure} |
| 189 |
|
\centering |
| 190 |
< |
\includegraphics[width=\linewidth]{bananGB_grained.eps} |
| 190 |
> |
\includegraphics[width=\linewidth]{bananGB.eps} |
| 191 |
|
\caption[]{} \label{LCFigure:BananaGB} |
| 192 |
|
\end{figure} |
| 193 |
|
|
| 199 |
|
\label{LCFigure:GBScheme} |
| 200 |
|
\end{figure} |
| 201 |
|
|
| 202 |
+ |
To account for the permanent dipolar interactions, there should be |
| 203 |
+ |
an electrostatic interaction term of the form |
| 204 |
+ |
\begin{equation} |
| 205 |
+ |
V_{ab}^{dp} = \sum\limits_{i \in a,j \in b} {\frac{1}{{4\pi |
| 206 |
+ |
\epsilon _{fs} }}\left[ {\frac{{\mu _i \cdot \mu _j }}{{r_{ij}^3 }} |
| 207 |
+ |
- \frac{{3\left( {\mu _i \cdot r_{ij} } \right)\left( {\mu _i \cdot |
| 208 |
+ |
r_{ij} } \right)}}{{r_{ij}^5 }}} \right]} |
| 209 |
+ |
\end{equation} |
| 210 |
+ |
where $\epsilon _{fs}$ is the permittivity of free space. |
| 211 |
+ |
|
| 212 |
|
\section{\label{liquidCrystalSection:methods}Methods} |
| 213 |
|
|
| 214 |
|
\section{\label{liquidCrystalSection:resultDiscussion}Results and Discussion} |
