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. |
214 |
> |
phase behavior of banana shaped liquid crystals. In each simulation, |
215 |
> |
every banana shaped molecule has been represented by three GB |
216 |
> |
particles 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 |
< |
In each simulation, rod-like polar molecules have been represented |
217 |
< |
by polar ellipsoidal Gay-Berne (GB) particles. The four parameters |
218 |
< |
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. |
225 |
> |
\subsection{Order Parameters} |
226 |
|
|
227 |
< |
In each of the simulations 864 molecules were confined in a cubic |
228 |
< |
box with periodic boundary conditions. The run started from a |
229 |
< |
density $\rho^{*} = \rho \sigma_{0}^{3}$ = 0.01 with nonpolar |
230 |
< |
molecules loacted on the sites of FCC lattice and having parallel |
231 |
< |
orientation. This structure was not a stable structure at this |
232 |
< |
density and it was melted at a reduced temperature $T^{*} = k_{B}T/ |
233 |
< |
\epsilon_{0} = 4.0$ . We used this isotropic configuration which was |
234 |
< |
both orientationally and translationally disordered, as the initial |
235 |
< |
configuration for each simulation. The dipoles were also switched on |
236 |
< |
from this point. Initial translational and angular velocities were |
237 |
< |
assigned from the gaussian distribution of velocities. |
238 |
< |
|
239 |
< |
To get the ordered structure for each system of particular dipolar |
240 |
< |
angles we increased the density from $\rho^{*} = 0.01$ to $\rho_{*} |
241 |
< |
= 0.3$ with an increament size of 0.002 upto $\rho^{*} = 0.1$ and |
242 |
< |
0.01 for the rest at some higher temperature. Temperature was then |
243 |
< |
lowered in finer steps to avoid ending up with disordered glass |
244 |
< |
phase and thus to help the molecules set with more order. For each |
245 |
< |
system this process required altogether $5 \times 10^{6}$ MC cycles |
246 |
< |
for equilibration. |
247 |
< |
|
248 |
< |
The torques and forces were calculated using velocity verlet |
249 |
< |
algorithm. The time step size $\delta t^{*} = \delta t/(m |
250 |
< |
\sigma_{0}^{2} / \epsilon_{0})^{1/2}$ was set at 0.0012 during the |
251 |
< |
process. The orientations of molecules were described by quaternions |
252 |
< |
instead of Eulerian angles to get the singularity-free orientational |
253 |
< |
equations of motion. |
254 |
< |
|
255 |
< |
The interaction potential was truncated at a cut-off radius $r_{c} = |
256 |
< |
3.8 \sigma_{0}$. The long range dipole-dipole interaction potential |
257 |
< |
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. |
227 |
> |
To investigate the phase structure of the model liquid crystal, we |
228 |
> |
calculated various order parameters and correlation functions. |
229 |
> |
Particulary, the $P_2$ order parameter allows us to estimate average |
230 |
> |
alignment along the director axis $Z$ which can be identified from |
231 |
> |
the largest eigen value obtained by diagonalizing the order |
232 |
> |
parameter tensor |
233 |
> |
\begin{equation} |
234 |
> |
\overleftrightarrow{\mathsf{Q}} = \frac{1}{N}\sum_i^N % |
235 |
> |
\begin{pmatrix} % |
236 |
> |
u_{ix}u_{ix}-\frac{1}{3} & u_{ix}u_{iy} & u_{ix}u_{iz} \\ |
237 |
> |
u_{iy}u_{ix} & u_{iy}u_{iy}-\frac{1}{3} & u_{iy}u_{iz} \\ |
238 |
> |
u_{iz}u_{ix} & u_{iz}u_{iy} & u_{iz}u_{iz}-\frac{1}{3} % |
239 |
> |
\end{pmatrix}, |
240 |
> |
\label{lipidEq:po1} |
241 |
> |
\end{equation} |
242 |
> |
where the $u_{i\alpha}$ is the $\alpha$ element of the unit vector |
243 |
> |
$\mathbf{\hat{u}}_i$, and the sum over $i$ averages over the whole |
244 |
> |
collection of unit vectors. The $P_2$ order parameter for uniaxial |
245 |
> |
phase is then simply given by |
246 |
> |
\begin{equation} |
247 |
> |
\langle P_2 \rangle = \frac{3}{2}\lambda_{\text{max}}. |
248 |
> |
\label{lipidEq:po3} |
249 |
> |
\end{equation} |
250 |
> |
In addition to the $P_2$ order parameter, $ R_{2,2}^2$ order |
251 |
> |
parameter for biaxial phase is introduced to describe the ordering |
252 |
> |
in the plane orthogonal to the director by |
253 |
> |
\begin{equation} |
254 |
> |
R_{2,2}^2 = \frac{1}{4}\left\langle {(x_i \cdot X)^2 - (x_i \cdot |
255 |
> |
Y)^2 - (y_i \cdot X)^2 + (y_i \cdot Y)^2 } \right\rangle |
256 |
> |
\end{equation} |
257 |
> |
where $X$, $Y$ and $Z$ are axis of the director frame. |
258 |
|
|
259 |
< |
The orientational order parameter for uniaxial phase was calculated |
282 |
< |
from the largest eigen value obtained by diagonalization of the |
283 |
< |
order parameter tensor |
259 |
> |
\subsection{Structure Properties} |
260 |
|
|
261 |
+ |
It is more important to show the density correlation along the |
262 |
+ |
director |
263 |
|
\begin{equation} |
264 |
< |
\begin{array}{lr} |
265 |
< |
Q_{\alpha \beta} = \frac{1}{2 N} \sum(3 e_{i \alpha} e_{i \beta} |
266 |
< |
- \delta_{\alpha \beta}) & \alpha, \beta = x,y,z \\ |
267 |
< |
\end{array} |
290 |
< |
\end{equation} |
264 |
> |
g(z) =< \delta (z-z_{ij})>_{ij} / \pi R^{2} \rho |
265 |
> |
\end{equation}, |
266 |
> |
where $z_{ij} = r_{ij} \dot Z$ was measured in the director frame |
267 |
> |
and $R$ is the radius of the cylindrical sampling region. |
268 |
|
|
269 |
< |
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. |
269 |
> |
\subsection{Rotational Invariants} |
270 |
|
|
271 |
< |
The density correlation along the director is $g(z) = < \delta |
272 |
< |
(z-z_{ij})>_{ij} / \pi R^{2} \rho $, where $z_{ij} = r_{ij} cos |
273 |
< |
\beta_{r_{ij}}$ was measured in the director frame and $R$ is the |
274 |
< |
radius of the cylindrical sampling region. |
271 |
> |
As a useful set of correlation functions to describe |
272 |
> |
position-orientation correlation, rotation invariants were first |
273 |
> |
applied in a spherical symmetric system to study x-ray and light |
274 |
> |
scatting\cite{Blum1972}. Latterly, expansion of the orientation pair |
275 |
> |
correlation in terms of rotation invariant for molecules of |
276 |
> |
arbitrary shape was introduce by Stone\cite{Stone1978} and adopted |
277 |
> |
by other researchers in liquid crystal studies\cite{Berardi2003}. |
278 |
|
|
279 |
+ |
\begin{eqnarray} |
280 |
+ |
S_{22}^{220} (r) & = & \frac{1}{{4\sqrt 5 }} \left< \delta (r - |
281 |
+ |
r_{ij} )((\hat x_i \cdot \hat x_j )^2 - (\hat x_i \cdot \hat y_j |
282 |
+ |
)^2 - (\hat y_i \cdot \hat x_j )^2 + (\hat y_i \cdot \hat y_j |
283 |
+ |
)^2 ) \right. \\ |
284 |
+ |
& & \left. - 2(\hat x_i \cdot \hat y_j )(\hat y_i \cdot \hat x_j ) - |
285 |
+ |
2(\hat x_i \cdot \hat x_j )(\hat y_i \cdot \hat y_j )) \right> |
286 |
+ |
\end{eqnarray} |
287 |
|
|
288 |
+ |
\begin{equation} |
289 |
+ |
S_{00}^{221} (r) = - \frac{{\sqrt 3 }}{{\sqrt {10} }}\left\langle |
290 |
+ |
{\delta (r - r_{ij} )((\hat z_i \cdot \hat z_j )(\hat z_i \cdot |
291 |
+ |
\hat z_j \times \hat r_{ij} ))} \right\rangle |
292 |
+ |
\end{equation} |
293 |
+ |
|
294 |
|
\section{Results and Conclusion} |
295 |
|
\label{sec:results and conclusion} |
305 |
– |
|
306 |
– |
Analysis of the simulation results shows that relative dipolar |
307 |
– |
orientation angle of the molecules can give rise to rich |
308 |
– |
polymorphism of polar mesophases. |
296 |
|
|
297 |
< |
The correlation function g(z) shows layering along perpendicular |
298 |
< |
direction to the plane for a system of G-B molecules with two |
299 |
< |
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. |
324 |
< |
|
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} |
335 |
< |
|
336 |
< |
|
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 |
< |
|
453 |
< |
\section{\label{liquidCrystalSection:methods}Methods} |
454 |
< |
|
455 |
< |
\section{\label{liquidCrystalSection:resultDiscussion}Results and Discussion} |
456 |
< |
|
457 |
< |
\section{Conclusion} |
297 |
> |
To investigate the molecular organization behavior due to different |
298 |
> |
dipolar orientation and position with respect to the center of the |
299 |
> |
molecule, |