| 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 |
|
|
| 78 |
|
liquid crystal phases\cite{Lansac2003}. Other anisotropic models |
| 79 |
|
using Gay-Berne(GB) potential, which produce interactions that favor |
| 80 |
|
local alignment, give the evidence of the novel packing arrangements |
| 81 |
< |
of bent-core molecules\cite{Memmer2002,Orlandi2006}. |
| 81 |
> |
of bent-core molecules\cite{Memmer2002}. |
| 82 |
|
|
| 83 |
|
Experimental studies by Levelut {\it et al.}~\cite{Levelut1981} |
| 84 |
|
revealed that terminal cyano or nitro groups usually induce |
| 100 |
|
incorporating electrostatic interaction. |
| 101 |
|
|
| 102 |
|
In this chapter, we consider system consisting of banana-shaped |
| 103 |
< |
molecule represented by three rigid GB particles with one or two |
| 104 |
< |
point dipoles at different location. Performing a series of |
| 105 |
< |
molecular dynamics simulations, we explore the structural properties |
| 106 |
< |
of tilted smectic phases as well as the effect of electrostatic |
| 105 |
< |
interactions. |
| 103 |
> |
molecule represented by three rigid GB particles with two point |
| 104 |
> |
dipoles. Performing a series of molecular dynamics simulations, we |
| 105 |
> |
explore the structural properties of tilted smectic phases as well |
| 106 |
> |
as the effect of electrostatic interactions. |
| 107 |
|
|
| 108 |
|
\section{\label{liquidCrystalSection:model}Model} |
| 109 |
|
|
| 182 |
|
\begin{figure} |
| 183 |
|
\centering |
| 184 |
|
\includegraphics[width=\linewidth]{banana.eps} |
| 185 |
< |
\caption[]{} \label{LCFig:BananaMolecule} |
| 185 |
> |
\caption[Schematic representation of a typical banana shaped |
| 186 |
> |
molecule]{Schematic representation of a typical banana shaped |
| 187 |
> |
molecule.} \label{LCFig:BananaMolecule} |
| 188 |
|
\end{figure} |
| 186 |
– |
|
| 187 |
– |
%\begin{figure} |
| 188 |
– |
%\centering |
| 189 |
– |
%\includegraphics[width=\linewidth]{bananGB.eps} |
| 190 |
– |
%\caption[]{} \label{LCFigure:BananaGB} |
| 191 |
– |
%\end{figure} |
| 192 |
– |
|
| 189 |
|
\begin{figure} |
| 190 |
|
\centering |
| 191 |
|
\includegraphics[width=\linewidth]{gb_scheme.eps} |
| 192 |
< |
\caption[]{Schematic diagram showing definitions of the orientation |
| 193 |
< |
vectors for a pair of Gay-Berne molecules} |
| 194 |
< |
\label{LCFigure:GBScheme} |
| 192 |
> |
\caption[Schematic diagram showing definitions of the orientation |
| 193 |
> |
vectors for a pair of Gay-Berne molecules]{Schematic diagram showing |
| 194 |
> |
definitions of the orientation vectors for a pair of Gay-Berne |
| 195 |
> |
molecules} \label{LCFigure:GBScheme} |
| 196 |
|
\end{figure} |
| 200 |
– |
|
| 197 |
|
To account for the permanent dipolar interactions, there should be |
| 198 |
|
an electrostatic interaction term of the form |
| 199 |
|
\begin{equation} |
| 204 |
|
\end{equation} |
| 205 |
|
where $\epsilon _{fs}$ is the permittivity of free space. |
| 206 |
|
|
| 207 |
< |
\section{\label{liquidCrystalSection:methods}Methods} |
| 207 |
> |
\section{Results and Discussion} |
| 208 |
|
|
| 209 |
< |
\section{\label{liquidCrystalSection:resultDiscussion}Results and Discussion} |
| 209 |
> |
A series of molecular dynamics simulations were perform to study the |
| 210 |
> |
phase behavior of banana shaped liquid crystals. In each simulation, |
| 211 |
> |
every banana shaped molecule has been represented by three GB |
| 212 |
> |
particles which is characterized by $\mu = 1,~ \nu = 2, |
| 213 |
> |
~\epsilon_{e}/\epsilon_{s} = 1/5$ and $\sigma_{e}/\sigma_{s} = 3$. |
| 214 |
> |
All of the simulations begin with same equilibrated isotropic |
| 215 |
> |
configuration where 1024 molecules without dipoles were confined in |
| 216 |
> |
a $160\times 160 \times 120$ box. After the dipolar interactions are |
| 217 |
> |
switched on, 2~ns NPTi cooling run with themostat of 2~ps and |
| 218 |
> |
barostat of 50~ps were used to equilibrate the system to desired |
| 219 |
> |
temperature and pressure. NPTi Production runs last for 40~ns with |
| 220 |
> |
time step of 20~fs. |
| 221 |
> |
|
| 222 |
> |
\subsection{Order Parameters} |
| 223 |
> |
|
| 224 |
> |
To investigate the phase structure of the model liquid crystal, we |
| 225 |
> |
calculated various order parameters and correlation functions. |
| 226 |
> |
Particulary, the $P_2$ order parameter allows us to estimate average |
| 227 |
> |
alignment along the director axis $Z$ which can be identified from |
| 228 |
> |
the largest eigen value obtained by diagonalizing the order |
| 229 |
> |
parameter tensor |
| 230 |
> |
\begin{equation} |
| 231 |
> |
\overleftrightarrow{\mathsf{Q}} = \frac{1}{N}\sum_i^N % |
| 232 |
> |
\begin{pmatrix} % |
| 233 |
> |
u_{ix}u_{ix}-\frac{1}{3} & u_{ix}u_{iy} & u_{ix}u_{iz} \\ |
| 234 |
> |
u_{iy}u_{ix} & u_{iy}u_{iy}-\frac{1}{3} & u_{iy}u_{iz} \\ |
| 235 |
> |
u_{iz}u_{ix} & u_{iz}u_{iy} & u_{iz}u_{iz}-\frac{1}{3} % |
| 236 |
> |
\end{pmatrix}, |
| 237 |
> |
\label{lipidEq:p2} |
| 238 |
> |
\end{equation} |
| 239 |
> |
where the $u_{i\alpha}$ is the $\alpha$ element of the unit vector |
| 240 |
> |
$\mathbf{\hat{u}}_i$, and the sum over $i$ averages over the whole |
| 241 |
> |
collection of unit vectors. The $P_2$ order parameter for uniaxial |
| 242 |
> |
phase is then simply given by |
| 243 |
> |
\begin{equation} |
| 244 |
> |
\langle P_2 \rangle = \frac{3}{2}\lambda_{\text{max}}. |
| 245 |
> |
\label{lipidEq:po3} |
| 246 |
> |
\end{equation} |
| 247 |
> |
%In addition to the $P_2$ order parameter, $ R_{2,2}^2$ order |
| 248 |
> |
%parameter for biaxial phase is introduced to describe the ordering |
| 249 |
> |
%in the plane orthogonal to the director by |
| 250 |
> |
%\begin{equation} |
| 251 |
> |
%R_{2,2}^2 = \frac{1}{4}\left\langle {(x_i \cdot X)^2 - (x_i \cdot |
| 252 |
> |
%Y)^2 - (y_i \cdot X)^2 + (y_i \cdot Y)^2 } \right\rangle |
| 253 |
> |
%\end{equation} |
| 254 |
> |
%where $X$, $Y$ and $Z$ are axis of the director frame. |
| 255 |
> |
The unit vector for the banana shaped molecule was defined by the |
| 256 |
> |
principle aixs of its middle GB particle. The $P_2$ order parameters |
| 257 |
> |
for the bent-core liquid crystal at different temperature is |
| 258 |
> |
summarized in Table~\ref{liquidCrystal:p2} which identifies a phase |
| 259 |
> |
transition temperature range. |
| 260 |
> |
|
| 261 |
> |
\begin{table} |
| 262 |
> |
\caption{LIQUID CRYSTAL STRUCTURAL PROPERTIES AS A FUNCTION OF |
| 263 |
> |
TEMPERATURE} \label{liquidCrystal:p2} |
| 264 |
> |
\begin{center} |
| 265 |
> |
\begin{tabular}{cccccc} |
| 266 |
> |
\hline |
| 267 |
> |
Temperature (K) & 420 & 440 & 460 & 480 & 600\\ |
| 268 |
> |
\hline |
| 269 |
> |
$\langle P_2\rangle$ & 0.984 & 0.982 & 0.975 & 0.967 & 0.067\\ |
| 270 |
> |
\hline |
| 271 |
> |
\end{tabular} |
| 272 |
> |
\end{center} |
| 273 |
> |
\end{table} |
| 274 |
> |
|
| 275 |
> |
\subsection{Structure Properties} |
| 276 |
> |
|
| 277 |
> |
The molecular organization obtained at temperature $T = 460K$ (below |
| 278 |
> |
transition temperature) is shown in Figure~\ref{LCFigure:snapshot}. |
| 279 |
> |
The diagonal view in Fig~\ref{LCFigure:snapshot}(a) shows the |
| 280 |
> |
stacking of the banana shaped molecules while the side view in n |
| 281 |
> |
Figure~\ref{LCFigure:snapshot}(b) demonstrates formation of a |
| 282 |
> |
chevron structure. The first peak of Radial distribution function |
| 283 |
> |
$g(r)$ in Fig.~\ref{LCFigure:gofrz}(a) shows the minimum distance |
| 284 |
> |
for two in plane banana shaped molecules is 4.9 \AA, while the |
| 285 |
> |
second split peak implies the biaxial packing. It is also important |
| 286 |
> |
to show the density correlation along the director which is given by |
| 287 |
> |
: |
| 288 |
> |
\begin{equation} |
| 289 |
> |
g(z) =\frac{1}{\pi R^{2} \rho}< \delta (z-z_{ij})>_{ij} |
| 290 |
> |
\end{equation}, |
| 291 |
> |
where $z_{ij} = r_{ij} \dot Z$ was measured in the director frame |
| 292 |
> |
and $R$ is the radius of the cylindrical sampling region. The |
| 293 |
> |
oscillation in density plot along the director in |
| 294 |
> |
Fig.~\ref{LCFigure:gofrz}(b) implies the existence of the layered |
| 295 |
> |
structure, and the peak at 27 \AA is attribute to the defect in the |
| 296 |
> |
system. |
| 297 |
> |
|
| 298 |
> |
\subsection{Rotational Invariants} |
| 299 |
> |
|
| 300 |
> |
As a useful set of correlation functions to describe |
| 301 |
> |
position-orientation correlation, rotation invariants were first |
| 302 |
> |
applied in a spherical symmetric system to study x-ray and light |
| 303 |
> |
scatting\cite{Blum1972}. Latterly, expansion of the orientation pair |
| 304 |
> |
correlation in terms of rotation invariant for molecules of |
| 305 |
> |
arbitrary shape was introduce by Stone\cite{Stone1978} and adopted |
| 306 |
> |
by other researchers in liquid crystal studies\cite{Berardi2003}. In |
| 307 |
> |
order to study the correlation between biaxiality and molecular |
| 308 |
> |
separation distance $r$, we calculate a rotational invariant |
| 309 |
> |
function $S_{22}^{220} (r)$, which is given by : |
| 310 |
> |
\begin{eqnarray} |
| 311 |
> |
S_{22}^{220} (r) & = & \frac{1}{{4\sqrt 5 }} \left< \delta (r - |
| 312 |
> |
r_{ij} )((\hat x_i \cdot \hat x_j )^2 - (\hat x_i \cdot \hat y_j |
| 313 |
> |
)^2 - (\hat y_i \cdot \hat x_j )^2 + (\hat y_i \cdot \hat y_j |
| 314 |
> |
)^2 ) \right. \notag \\ |
| 315 |
> |
& & \left. - 2(\hat x_i \cdot \hat y_j )(\hat y_i \cdot \hat x_j ) - |
| 316 |
> |
2(\hat x_i \cdot \hat x_j )(\hat y_i \cdot \hat y_j )) \right>. |
| 317 |
> |
\end{eqnarray} |
| 318 |
> |
The long range behavior of second rank orientational correlation |
| 319 |
> |
$S_{22}^{220} (r)$ in Fig~\ref{LCFigure:S22220} also confirm the |
| 320 |
> |
biaxiality of the system. |
| 321 |
> |
|
| 322 |
> |
\begin{figure} |
| 323 |
> |
\centering |
| 324 |
> |
\includegraphics[width=4.5in]{snapshot.eps} |
| 325 |
> |
\caption[Snapshot of the molecular organization in the layered phase |
| 326 |
> |
formed at temperature T = 460K and pressure P = 1 atm]{Snapshot of |
| 327 |
> |
the molecular organization in the layered phase formed at |
| 328 |
> |
temperature T = 460K and pressure P = 1 atm. (a) diagonal view; (b) |
| 329 |
> |
side view.} \label{LCFigure:snapshot} |
| 330 |
> |
\end{figure} |
| 331 |
> |
|
| 332 |
> |
\begin{figure} |
| 333 |
> |
\centering |
| 334 |
> |
\includegraphics[width=\linewidth]{gofr_gofz.eps} |
| 335 |
> |
\caption[Correlation Functions of a Bent-core Liquid Crystal System |
| 336 |
> |
at Temperature T = 460K and Pressure P = 10 atm]{Correlation |
| 337 |
> |
Functions of a Bent-core Liquid Crystal System at Temperature T = |
| 338 |
> |
460K and Pressure P = 10 atm. (a) radial correlation function |
| 339 |
> |
$g(r)$; and (b) density along the director $g(z)$.} |
| 340 |
> |
\label{LCFigure:gofrz} |
| 341 |
> |
\end{figure} |
| 342 |
> |
|
| 343 |
> |
\begin{figure} |
| 344 |
> |
\centering |
| 345 |
> |
\includegraphics[width=\linewidth]{s22_220.eps} |
| 346 |
> |
\caption[Average orientational correlation Correlation Functions of |
| 347 |
> |
a Bent-core Liquid Crystal System at Temperature T = 460K and |
| 348 |
> |
Pressure P = 10 atm]{Correlation Functions of a Bent-core Liquid |
| 349 |
> |
Crystal System at Temperature T = 460K and Pressure P = 10 atm. (a) |
| 350 |
> |
radial correlation function $g(r)$; and (b) density along the |
| 351 |
> |
director $g(z)$.} \label{LCFigure:S22220} |
| 352 |
> |
\end{figure} |
| 353 |
> |
|
| 354 |
> |
\section{Conclusion} |
| 355 |
> |
|
| 356 |
> |
We have presented a simple dipolar three-site GB model for banana |
| 357 |
> |
shaped molecules which are capable of forming smectic phases from |
| 358 |
> |
isotropic configuration. Various order parameters and correlation |
| 359 |
> |
functions were used to characterized the structural properties of |
| 360 |
> |
these smectic phase. However, the forming layered structure still |
| 361 |
> |
had some defects because of the mismatching between the layer |
| 362 |
> |
structure spacing and the shape of simulation box. This mismatching |
| 363 |
> |
can be broken by using NPTf integrator in further simulations. The |
| 364 |
> |
role of terminal chain in controlling transition temperatures and |
| 365 |
> |
the type of mesophase formed have been studied |
| 366 |
> |
extensively\cite{Pelzl1999}. The lack of flexibility in our model |
| 367 |
> |
due to the missing terminal chains could explained the fact that we |
| 368 |
> |
did not find evidence of chirality. |
