| 211 |
|
\section{Computational Methodology} |
| 212 |
|
|
| 213 |
|
A series of molecular dynamics simulations were perform to study the |
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< |
phase behavior of banana shaped liquid crystals. |
| 215 |
< |
|
| 216 |
< |
In each simulation, rod-like polar molecules have been represented |
| 217 |
< |
by polar ellipsoidal Gay-Berne (GB) particles. The four parameters |
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< |
characterizing G-B potential were taken as $\mu = 1,~ \nu = 2, |
| 219 |
< |
~\epsilon_{e}/\epsilon_{s} |
| 220 |
< |
= 1/5$ and $\sigma_{e}/\sigma_{s} = 3$. The components of the |
| 221 |
< |
scaled moment of inertia $(I^{*} = I/m \sigma_{s}^{2})$ along the |
| 222 |
< |
major and minor axes were $I_{z}^{*} = 0.2$ and $I_{\perp}^{*} = |
| 223 |
< |
1.0$. We used the reduced dipole moments $ \mu^{*} = \mu/(4 \pi |
| 224 |
< |
\epsilon_{fs} \sigma_{0}^{3})^{1/2}= 1.0$ for terminal dipole and |
| 225 |
< |
$ \mu^{*} = \mu/(4 |
| 226 |
< |
\pi \epsilon_{fs} \sigma_{0}^{3})^{1/2}= 0.5$ for second dipole, |
| 227 |
< |
where $\epsilon_{fs}$ was the permitivitty of free space. For all |
| 228 |
< |
simulations the position of the terminal dipole |
| 229 |
< |
has been kept |
| 230 |
< |
at a fixed distance $d^{*} = d/\sigma_{s} = 1.0 $ from the |
| 231 |
< |
centre of mass on the molecular symmetry axis. The second dipole |
| 232 |
< |
takes $d^{*} = d/\sigma_{s} = 0.0 $ i.e. it is on the centre of |
| 233 |
< |
mass. To investigate the molecular organization behaviour due to |
| 234 |
< |
different dipolar orientation with respect to the symmetry axis, we |
| 235 |
< |
selected dipolar angle $\alpha_{d} = 0$ to model terminal outward |
| 236 |
< |
longitudinal dipole and $\alpha_{d} = \pi/2$ to model transverse |
| 237 |
< |
outward dipole where the second dipole takes relative anti |
| 238 |
< |
antiparallel orientation with respect to the first. System of |
| 239 |
< |
molecules having a single transverse terminal dipole has also been |
| 240 |
< |
studied. We ran a series of simulations to investigate the effect of |
| 241 |
< |
dipoles on molecular organization. |
| 242 |
< |
|
| 243 |
< |
In each of the simulations 864 molecules were confined in a cubic |
| 244 |
< |
box with periodic boundary conditions. The run started from a |
| 245 |
< |
density $\rho^{*} = \rho \sigma_{0}^{3}$ = 0.01 with nonpolar |
| 246 |
< |
molecules loacted on the sites of FCC lattice and having parallel |
| 247 |
< |
orientation. This structure was not a stable structure at this |
| 248 |
< |
density and it was melted at a reduced temperature $T^{*} = k_{B}T/ |
| 249 |
< |
\epsilon_{0} = 4.0$ . We used this isotropic configuration which was |
| 250 |
< |
both orientationally and translationally disordered, as the initial |
| 251 |
< |
configuration for each simulation. The dipoles were also switched on |
| 252 |
< |
from this point. Initial translational and angular velocities were |
| 253 |
< |
assigned from the gaussian distribution of velocities. |
| 254 |
< |
|
| 255 |
< |
To get the ordered structure for each system of particular dipolar |
| 256 |
< |
angles we increased the density from $\rho^{*} = 0.01$ to $\rho_{*} |
| 257 |
< |
= 0.3$ with an increament size of 0.002 upto $\rho^{*} = 0.1$ and |
| 258 |
< |
0.01 for the rest at some higher temperature. Temperature was then |
| 259 |
< |
lowered in finer steps to avoid ending up with disordered glass |
| 260 |
< |
phase and thus to help the molecules set with more order. For each |
| 261 |
< |
system this process required altogether $5 \times 10^{6}$ MC cycles |
| 262 |
< |
for equilibration. |
| 263 |
< |
|
| 264 |
< |
The torques and forces were calculated using velocity verlet |
| 265 |
< |
algorithm. The time step size $\delta t^{*} = \delta t/(m |
| 266 |
< |
\sigma_{0}^{2} / \epsilon_{0})^{1/2}$ was set at 0.0012 during the |
| 267 |
< |
process. The orientations of molecules were described by quaternions |
| 268 |
< |
instead of Eulerian angles to get the singularity-free orientational |
| 269 |
< |
equations of motion. |
| 270 |
< |
|
| 271 |
< |
The interaction potential was truncated at a cut-off radius $r_{c} = |
| 272 |
< |
3.8 \sigma_{0}$. The long range dipole-dipole interaction potential |
| 273 |
< |
and torque were handled by the application of reaction field method |
| 274 |
< |
~\cite{Allen87}. |
| 275 |
< |
|
| 276 |
< |
To investigate the phase structure of the model liquid crystal |
| 277 |
< |
family we calculated the orientational order parameter, correlation |
| 278 |
< |
functions. To identify a particular phase we took configurational |
| 279 |
< |
snapshots at the onset of each layered phase. |
| 280 |
< |
|
| 281 |
< |
The orientational order parameter for uniaxial phase was calculated |
| 282 |
< |
from the largest eigen value obtained by diagonalization of the |
| 283 |
< |
order parameter tensor |
| 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 |
< |
\begin{array}{lr} |
| 233 |
< |
Q_{\alpha \beta} = \frac{1}{2 N} \sum(3 e_{i \alpha} e_{i \beta} |
| 234 |
< |
- \delta_{\alpha \beta}) & \alpha, \beta = x,y,z \\ |
| 235 |
< |
\end{array} |
| 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 |
|
|
| 292 |
– |
where $e_{i \alpha}$ was the $\alpha$ th component of the unit |
| 293 |
– |
vector $e_{i}$ along the symmetry axis of the i th molecule. |
| 294 |
– |
Corresponding eigenvector gave the director which defines the |
| 295 |
– |
average direction of molecular alignment. |
| 257 |
|
|
| 258 |
< |
The density correlation along the director is $g(z) = < \delta |
| 259 |
< |
(z-z_{ij})>_{ij} / \pi R^{2} \rho $, where $z_{ij} = r_{ij} cos |
| 260 |
< |
\beta_{r_{ij}}$ was measured in the director frame and $R$ is the |
| 261 |
< |
radius of the cylindrical sampling region. |
| 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 |
< |
Analysis of the simulation results shows that relative dipolar |
| 270 |
< |
orientation angle of the molecules can give rise to rich |
| 271 |
< |
polymorphism of polar mesophases. |
| 309 |
< |
|
| 310 |
< |
The correlation function g(z) shows layering along perpendicular |
| 311 |
< |
direction to the plane for a system of G-B molecules with two |
| 312 |
< |
transverse outward pointing dipoles in fig. \ref{fig:1}. Both the |
| 313 |
< |
correlation plot and the snapshot (fig. \ref{fig:4}) of their |
| 314 |
< |
organization indicate a bilayer phase. Snapshot for larger system of |
| 315 |
< |
1372 molecules also confirms bilayer structure (Fig. \ref{fig:7}). |
| 316 |
< |
Fig. \ref{fig:2} shows g(z) for a system of molecules having two |
| 317 |
< |
antiparallel longitudinal dipoles and the snapshot of their |
| 318 |
< |
organization shows a monolayer phase (Fig. \ref{fig:5}). Fig. |
| 319 |
< |
\ref{fig:3} gives g(z) for a system of G-B molecules with single |
| 320 |
< |
transverse outward pointing dipole and fig. \ref{fig:6} gives the |
| 321 |
< |
snapshot. Their organization is like a wavy antiphase (stripe |
| 322 |
< |
domain). Fig. \ref{fig:8} gives the snapshot for 1372 molecules |
| 323 |
< |
with single transverse dipole near the end of the molecule. |
| 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 |
|
|
| 325 |
– |
\begin{figure} |
| 326 |
– |
\begin{center} |
| 327 |
– |
\epsfxsize=3in \epsfbox{fig1.ps} |
| 328 |
– |
\end{center} |
| 329 |
– |
\caption { Density projection of molecular centres (solid) and |
| 330 |
– |
terminal dipoles (broken) with respect to the director g(z) for a |
| 331 |
– |
system of G-B molecules with two transverse outward pointing |
| 332 |
– |
dipoles, the first dipole having $d^{*}=1.0$, $\mu^{*}=1.0$ and the |
| 333 |
– |
second dipole having $d^{*}=0.0$, $\mu^{*}=0.5$} \label{fig:1} |
| 334 |
– |
\end{figure} |
| 273 |
|
|
| 274 |
|
|
| 337 |
– |
\begin{figure} |
| 338 |
– |
\begin{center} |
| 339 |
– |
\epsfxsize=3in \epsfbox{fig2.ps} |
| 340 |
– |
\end{center} |
| 341 |
– |
\caption { Density projection of molecular centres (solid) and |
| 342 |
– |
terminal dipoles (broken) with respect to the director g(z) for a |
| 343 |
– |
system of G-B molecules with two antiparallel longitudinal dipoles, |
| 344 |
– |
the first outward pointing dipole having $d^{*}=1.0$, $\mu^{*}=1.0$ |
| 345 |
– |
and the second dipole having $d^{*}=0.0$, $\mu^{*}=0.5$} |
| 346 |
– |
\label{fig:2} |
| 347 |
– |
\end{figure} |
| 348 |
– |
|
| 349 |
– |
\begin{figure} |
| 350 |
– |
\begin{center} |
| 351 |
– |
\epsfxsize=3in \epsfbox{fig3.ps} |
| 352 |
– |
\end{center} |
| 353 |
– |
\caption {Density projection of molecular centres (solid) and |
| 354 |
– |
terminal |
| 355 |
– |
dipoles (broken) with respect to the director g(z) |
| 356 |
– |
for a system of G-B molecules with single transverse outward |
| 357 |
– |
pointing dipole, having $d^{*}=1.0$, $\mu^{*}=1.0$} \label{fig:3} |
| 358 |
– |
\end{figure} |
| 359 |
– |
|
| 360 |
– |
\begin{figure} |
| 361 |
– |
\centering \epsfxsize=2.5in \epsfbox{fig4.eps} \caption{Typical |
| 362 |
– |
configuration for a system of 864 G-B molecules with two transverse |
| 363 |
– |
dipoles, the first dipole having $d^{*}=1.0$, $\mu^{*}=1.0$ and the |
| 364 |
– |
second dipole having $d^{*}=0.0$, $\mu^{*}=0.5$. The white caps |
| 365 |
– |
indicate the location of the terminal dipole, while the orientation |
| 366 |
– |
of the dipoles is indicated by the blue/gold coloring.} |
| 367 |
– |
\label{fig:4} |
| 368 |
– |
\end{figure} |
| 369 |
– |
|
| 370 |
– |
\begin{figure} |
| 371 |
– |
\begin{center} |
| 372 |
– |
\epsfxsize=3in \epsfbox{fig5.ps} |
| 373 |
– |
\end{center} |
| 374 |
– |
\caption {Snapshot of molecular configuration for a system of 864 |
| 375 |
– |
G-B molecules with two antiparallel longitudinal dipoles, the first |
| 376 |
– |
outward pointing dipole |
| 377 |
– |
having $d^{*}=1.0$, $\mu^{*}=1.0$ and the second dipole having $d^{*}=0.0$, |
| 378 |
– |
$\mu^{*}=0.5$ (fine lines are molecular symmetry axes and small |
| 379 |
– |
thick lines show terminal dipolar direction, central dipoles are not |
| 380 |
– |
shown).} \label{fig:5} |
| 381 |
– |
\end{figure} |
| 382 |
– |
|
| 383 |
– |
|
| 384 |
– |
\begin{figure} |
| 385 |
– |
\begin{center} |
| 386 |
– |
\epsfxsize=3in \epsfbox{fig6.ps} |
| 387 |
– |
\end{center} |
| 388 |
– |
\caption {Snapshot of molecular configuration for a system of 864 |
| 389 |
– |
G-B molecules with single transverse outward pointing dipole, having |
| 390 |
– |
$d^{*}=1.0$, $\mu^{*}=1.0$ (fine lines are molecular symmetry axes |
| 391 |
– |
and small thick lines show terminal dipolar direction).} |
| 392 |
– |
\label{fig:6} |
| 393 |
– |
\end{figure} |
| 394 |
– |
|
| 395 |
– |
\begin{figure} |
| 396 |
– |
\begin{center} |
| 397 |
– |
\epsfxsize=3in \epsfbox{fig7.ps} |
| 398 |
– |
\end{center} |
| 399 |
– |
\caption {Snapshot of molecular configuration for a system of 1372 |
| 400 |
– |
G-B molecules with two transverse outward pointing dipoles, the |
| 401 |
– |
first dipole having $d^{*}=1.0$, $\mu^{*}=1.0$ and the second dipole |
| 402 |
– |
having $d^{*}=0.0$, $\mu^{*}=0.5$(fine lines are molecular symmetry |
| 403 |
– |
axes and small thick lines show terminal dipolar direction, central |
| 404 |
– |
dipoles are not shown).} \label{fig:7} |
| 405 |
– |
\end{figure} |
| 406 |
– |
|
| 407 |
– |
\begin{figure} |
| 408 |
– |
\begin{center} |
| 409 |
– |
\epsfxsize=3in \epsfbox{fig8.ps} |
| 410 |
– |
\end{center} |
| 411 |
– |
\caption {Snapshot of molecular configuration for a system of 1372 |
| 412 |
– |
G-B molecules with single transverse outward pointing dipole, having |
| 413 |
– |
$d^{*}=1.0$, $\mu^{*}=1.0$ (fine lines are molecular symmetry axes |
| 414 |
– |
and small thick lines show terminal dipolar direction).} |
| 415 |
– |
\label{fig:8} |
| 416 |
– |
\end{figure} |
| 417 |
– |
|
| 418 |
– |
Starting from an isotropic configuaration of polar Gay-Berne |
| 419 |
– |
molecules, we could successfully simulate perfect bilayer, antiphase |
| 420 |
– |
and monolayer structure. To break the up-down symmetry i.e. the |
| 421 |
– |
nonequivalence of directions ${\bf \hat {n}}$ and ${ -\bf \hat{n}}$, |
| 422 |
– |
the molecules should have permanent electric or magnetic dipoles. |
| 423 |
– |
Longitudinal electric dipole interaction could not form polar |
| 424 |
– |
nematic phase as orientationally disordered phase with larger |
| 425 |
– |
entropy is stabler than polarly ordered phase. In fact, stronger |
| 426 |
– |
central dipole moment opposes polar nematic ordering more |
| 427 |
– |
effectively in case of rod-like molecules. However, polar ordering |
| 428 |
– |
like bilayer $A_{2}$, interdigitated $A_{d}$, and wavy $\tilde A$ in |
| 429 |
– |
smectic layers can be achieved, where adjacent layers with opposite |
| 430 |
– |
polarities makes bulk phase a-polar. More so, lyotropic liquid |
| 431 |
– |
crystals and bilayer bio-membranes can have polar layers. These |
| 432 |
– |
arrangements appear to get favours with the shifting of longitudinal |
| 433 |
– |
dipole moment to the molecular terminus, so that they can have |
| 434 |
– |
anti-ferroelectric dipolar arrangement giving rise to local (within |
| 435 |
– |
the sublayer) breaking of up-down symmetry along the director. |
| 436 |
– |
Transverse polarity breaks two-fold rotational symmetry, which |
| 437 |
– |
favours more in-plane polar order. However, the molecular origin of |
| 438 |
– |
these phases requires something more which are apparent from the |
| 439 |
– |
earlier simulation results. We have shown that to get perfect |
| 440 |
– |
bilayer structure in a G-B system, alongwith transverse terminal |
| 441 |
– |
dipole, another central dipole (or a polarizable core) is required |
| 442 |
– |
so that polar head and a-polar tail of Gay-Berne molecules go to |
| 443 |
– |
opposite directions within a bilayer. This gives some kind of |
| 444 |
– |
clipping interactions which forbid the molecular tail go in other |
| 445 |
– |
way. Moreover, we could simulate other varieties of polar smectic |
| 446 |
– |
phases e.g. monolayer $A_{1}$, antiphase $\tilde A$ successfully. |
| 447 |
– |
Apart from guiding chemical synthesization of ferroelectric, |
| 448 |
– |
antiferroelectric liquid crystals for technological applications, |
| 449 |
– |
the present study will be of scientific interest in understanding |
| 450 |
– |
molecular level interactions of lyotropic liquid crystals as well as |
| 451 |
– |
nature-designed bio-membranes. |
| 452 |
– |
|
| 275 |
|
\section{\label{liquidCrystalSection:methods}Methods} |
| 276 |
|
|
| 277 |
|
\section{\label{liquidCrystalSection:resultDiscussion}Results and Discussion} |
