| 2 |
|
|
| 3 |
|
\section{\label{liquidCrystalSection:introduction}Introduction} |
| 4 |
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|
| 5 |
< |
Long range orientational order is one of the most fundamental |
| 6 |
< |
properties of liquid crystal mesophases. This orientational |
| 7 |
< |
anisotropy of the macroscopic phases originates in the shape |
| 8 |
< |
anisotropy of the constituent molecules. Among these anisotropy |
| 9 |
< |
mesogens, rod-like (calamitic) and disk-like molecules have been |
| 10 |
< |
exploited in great detail in the last two decades\cite{Huh2004}. |
| 11 |
< |
Typically, these mesogens consist of a rigid aromatic core and one |
| 12 |
< |
or more attached aliphatic chains. For short chain molecules, only |
| 13 |
< |
nematic phases, in which positional order is limited or absent, can |
| 14 |
< |
be observed, because the entropy of mixing different parts of the |
| 15 |
< |
mesogens is paramount to the dispersion interaction. In contrast, |
| 16 |
< |
formation of the one dimension lamellar sematic phase in rod-like |
| 17 |
< |
molecules with sufficiently long aliphatic chains has been reported, |
| 18 |
< |
as well as the segregation phenomena in disk-like molecules. |
| 5 |
> |
Rod-like (calamitic) and disk-like anisotropy liquid crystals have |
| 6 |
> |
been investigated in great detail in the last two |
| 7 |
> |
decades\cite{Huh2004}. Typically, these mesogens consist of a rigid |
| 8 |
> |
aromatic core and one or more attached aliphatic chains. For short |
| 9 |
> |
chain molecules, only nematic phases, in which positional order is |
| 10 |
> |
limited or absent, can be observed, because the entropy of mixing |
| 11 |
> |
different parts of the mesogens is larger than the dispersion |
| 12 |
> |
interaction. In contrast, formation of one dimension lamellar |
| 13 |
> |
smectic phase in rod-like molecules with sufficiently long aliphatic |
| 14 |
> |
chains has been reported, as well as the segregation phenomena in |
| 15 |
> |
disk-like molecules\cite{McMillan1971}. Recently, banana-shaped or |
| 16 |
> |
bent-core liquid crystals have became one of the most active |
| 17 |
> |
research areas in mesogenic materials and supramolecular |
| 18 |
> |
chemistry\cite{Niori1996, Link1997, Pelzl1999}. Unlike rods and |
| 19 |
> |
disks, the polarity and biaxiality of the banana-shaped molecules |
| 20 |
> |
allow the molecules organize into a variety of novel liquid |
| 21 |
> |
crystalline phases which show interesting material properties. Of |
| 22 |
> |
particular interest is the spontaneous formation of macroscopic |
| 23 |
> |
chiral layers from achiral banana-shaped molecules, where polar |
| 24 |
> |
molecule orientational ordering exhibited layered plane as well as |
| 25 |
> |
the tilted arrangement of the molecules relative to the polar axis. |
| 26 |
> |
As a consequence of supramolecular chirality, the spontaneous |
| 27 |
> |
polarization arises in ferroelectric (FE) and antiferroelectic (AF) |
| 28 |
> |
switching of smectic liquid crystal phases, demonstrating some |
| 29 |
> |
promising applications in second-order nonlinear optical devices. |
| 30 |
> |
The most widely investigated mesophase formed by banana-shaped |
| 31 |
> |
moleculed is the $\text{B}_2$ phase, which is also referred to as |
| 32 |
> |
$\text{SmCP}$\cite{Link1997}. Of the most important discoveries in |
| 33 |
> |
this tilt lamellar phase is the four distinct packing arrangements |
| 34 |
> |
(two conglomerates and two macroscopic racemates), which depend on |
| 35 |
> |
the tilt direction and the polar direction of the molecule in |
| 36 |
> |
adjacent layer (see Fig.~\ref{LCFig:SMCP})\cite{Link1997}. |
| 37 |
|
|
| 20 |
– |
Recently, the banana-shaped or bent-core liquid crystal have became |
| 21 |
– |
one of the most active research areas in mesogenic materials and |
| 22 |
– |
supramolecular chemistry\cite{Niori1996, Link1997, Pelzl1999}. |
| 23 |
– |
Unlike rods and disks, the polarity and biaxiality of the |
| 24 |
– |
banana-shaped molecules allow the molecules organize into a variety |
| 25 |
– |
of novel liquid crystalline phases which show interesting material |
| 26 |
– |
properties. Of particular interest is the spontaneous formation of |
| 27 |
– |
macroscopic chiral layers from achiral banana-shaped molecules, |
| 28 |
– |
where polar molecule orientational ordering is shown within the |
| 29 |
– |
layer plane as well as the tilted arrangement of the molecules |
| 30 |
– |
relative to the polar axis. As a consequence of supramolecular |
| 31 |
– |
chirality, the spontaneous polarization arises in ferroelectric (FE) |
| 32 |
– |
and antiferroelectic (AF) switching of smectic liquid crystal |
| 33 |
– |
phases, demonstrating some promising applications in second-order |
| 34 |
– |
nonlinear optical devices. The most widely investigated mesophase |
| 35 |
– |
formed by banana-shaped moleculed is the $\text{B}_2$ phase, which |
| 36 |
– |
is also referred to as $\text{SmCP}$\cite{Link1997}. Of the most |
| 37 |
– |
important discover in this tilt lamellar phase is the four distinct |
| 38 |
– |
packing arrangements (two conglomerates and two macroscopic |
| 39 |
– |
racemates), which depend on the tilt direction and the polar |
| 40 |
– |
direction of the molecule in adjacent layer (see |
| 41 |
– |
Fig.~\ref{LCFig:SMCP}). |
| 42 |
– |
|
| 38 |
|
\begin{figure} |
| 39 |
|
\centering |
| 40 |
|
\includegraphics[width=\linewidth]{smcp.eps} |
| 60 |
|
smectic arrangements\cite{Cook2000, Lansac2001}, as well as other |
| 61 |
|
bulk properties, such as rotational viscosity and flexoelectric |
| 62 |
|
coefficients\cite{Cheung2002, Cheung2004}. However, due to the |
| 63 |
< |
limitation of time scale required for phase transition and the |
| 63 |
> |
limitation of time scales required for phase transition and the |
| 64 |
|
length scale required for representing bulk behavior, |
| 65 |
|
models\cite{Perram1985, Gay1981}, which are based on the observation |
| 66 |
|
that liquid crystal order is exhibited by a range of non-molecular |
| 67 |
< |
bodies with high shape anisotropies, became the dominant models in |
| 68 |
< |
the field of liquid crystal phase behavior. Previous simulation |
| 69 |
< |
studies using hard spherocylinder dimer model\cite{Camp1999} produce |
| 70 |
< |
nematic phases, while hard rod simulation studies identified a |
| 71 |
< |
Landau point\cite{Bates2005}, at which the isotropic phase undergoes |
| 72 |
< |
a direct transition to the biaxial nematic, as well as some possible |
| 73 |
< |
liquid crystal phases\cite{Lansac2003}. Other anisotropic models |
| 74 |
< |
using Gay-Berne(GB) potential, which produce interactions that favor |
| 75 |
< |
local alignment, give the evidence of the novel packing arrangements |
| 81 |
< |
of bent-core molecules\cite{Memmer2002}. |
| 67 |
> |
bodies with high shape anisotropies, have become the dominant models |
| 68 |
> |
in the field of liquid crystal phase behavior. Previous simulation |
| 69 |
> |
studies using a hard spherocylinder dimer model\cite{Camp1999} |
| 70 |
> |
produced nematic phases, while hard rod simulation studies |
| 71 |
> |
identified a direct transition to the biaxial nematic and other |
| 72 |
> |
possible liquid crystal phases\cite{Lansac2003}. Other anisotropic |
| 73 |
> |
models using the Gay-Berne(GB) potential, which produces |
| 74 |
> |
interactions that favor local alignment, give evidence of the novel |
| 75 |
> |
packing arrangements of bent-core molecules\cite{Memmer2002}. |
| 76 |
|
|
| 77 |
|
Experimental studies by Levelut {\it et al.}~\cite{Levelut1981} |
| 78 |
|
revealed that terminal cyano or nitro groups usually induce |
| 79 |
|
permanent longitudinal dipole moments, which affect the phase |
| 80 |
< |
behavior considerably. A series of theoretical studies also drawn |
| 81 |
< |
equivalent conclusions. Monte Carlo studies of the GB potential with |
| 82 |
< |
fixed longitudinal dipoles (i.e. pointed along the principal axis of |
| 83 |
< |
rotation) were shown to enhance smectic phase |
| 80 |
> |
behavior considerably. Equivalent conclusions have also been drawn |
| 81 |
> |
from a series of theoretical studies. Monte Carlo studies of the GB |
| 82 |
> |
potential with fixed longitudinal dipoles (i.e. pointed along the |
| 83 |
> |
principal axis of rotation) were shown to enhance smectic phase |
| 84 |
|
stability~\cite{Berardi1996,Satoh1996}. Molecular simulation of GB |
| 85 |
|
ellipsoids with transverse dipoles at the terminus of the molecule |
| 86 |
|
also demonstrated that partial striped bilayer structures were |
| 93 |
|
bent-core molecules, could be modeled more accurately by |
| 94 |
|
incorporating electrostatic interaction. |
| 95 |
|
|
| 96 |
< |
In this chapter, we consider system consisting of banana-shaped |
| 96 |
> |
In this chapter, we consider a system consisting of banana-shaped |
| 97 |
|
molecule represented by three rigid GB particles with two point |
| 98 |
|
dipoles. Performing a series of molecular dynamics simulations, we |
| 99 |
|
explore the structural properties of tilted smectic phases as well |
| 124 |
|
} \right] \label{LCEquation:gb} |
| 125 |
|
\end{equation} |
| 126 |
|
where $\hat u_i,\hat u_j$ are unit vectors specifying the |
| 127 |
< |
orientation of two molecules $i$ and $j$ separated by intermolecular |
| 128 |
< |
vector $r_{ij}$. $\hat r_{ij}$ is the unit vector along the |
| 129 |
< |
intermolecular vector. A schematic diagram of the orientation |
| 130 |
< |
vectors is shown in Fig.\ref{LCFigure:GBScheme}. The functional form |
| 131 |
< |
for $\sigma$ is given by |
| 127 |
> |
orientation of two ellipsoids $i$ and $j$ separated by |
| 128 |
> |
intermolecular vector $r_{ij}$. $\hat r_{ij}$ is the unit vector |
| 129 |
> |
along the inter-ellipsoid vector. A schematic diagram of the |
| 130 |
> |
orientation vectors is shown in Fig.\ref{LCFigure:GBScheme}. The |
| 131 |
> |
functional form for $\sigma$ is given by |
| 132 |
|
\begin{equation} |
| 133 |
|
\sigma (\hat u_i ,\hat u_i ,\hat r_{ij} ) = \sigma _0 \left[ {1 - |
| 134 |
|
\frac{\chi }{2}\left( {\frac{{(\hat r_{ij} \cdot \hat u_i + \hat |
| 180 |
|
molecule]{Schematic representation of a typical banana shaped |
| 181 |
|
molecule.} \label{LCFig:BananaMolecule} |
| 182 |
|
\end{figure} |
| 189 |
– |
|
| 183 |
|
\begin{figure} |
| 184 |
|
\centering |
| 185 |
|
\includegraphics[width=\linewidth]{gb_scheme.eps} |
| 186 |
|
\caption[Schematic diagram showing definitions of the orientation |
| 187 |
|
vectors for a pair of Gay-Berne molecules]{Schematic diagram showing |
| 188 |
|
definitions of the orientation vectors for a pair of Gay-Berne |
| 189 |
< |
molecules} \label{LCFigure:GBScheme} |
| 189 |
> |
ellipsoids} \label{LCFigure:GBScheme} |
| 190 |
|
\end{figure} |
| 198 |
– |
|
| 191 |
|
To account for the permanent dipolar interactions, there should be |
| 192 |
|
an electrostatic interaction term of the form |
| 193 |
|
\begin{equation} |
| 210 |
|
a $160\times 160 \times 120$ box. After the dipolar interactions are |
| 211 |
|
switched on, 2~ns NPTi cooling run with themostat of 2~ps and |
| 212 |
|
barostat of 50~ps were used to equilibrate the system to desired |
| 213 |
< |
temperature and pressure. |
| 213 |
> |
temperature and pressure. NPTi Production runs last for 40~ns with |
| 214 |
> |
time step of 20~fs. |
| 215 |
|
|
| 216 |
|
\subsection{Order Parameters} |
| 217 |
|
|
| 219 |
|
calculated various order parameters and correlation functions. |
| 220 |
|
Particulary, the $P_2$ order parameter allows us to estimate average |
| 221 |
|
alignment along the director axis $Z$ which can be identified from |
| 222 |
< |
the largest eigen value obtained by diagonalizing the order |
| 223 |
< |
parameter tensor |
| 222 |
> |
the largest eigenvalue obtained by diagonalizing the order parameter |
| 223 |
> |
tensor |
| 224 |
|
\begin{equation} |
| 225 |
|
\overleftrightarrow{\mathsf{Q}} = \frac{1}{N}\sum_i^N % |
| 226 |
|
\begin{pmatrix} % |
| 266 |
|
\end{center} |
| 267 |
|
\end{table} |
| 268 |
|
|
| 269 |
< |
\subsection{Structure Properties} |
| 269 |
> |
\subsection{Structural Properties} |
| 270 |
|
|
| 271 |
|
The molecular organization obtained at temperature $T = 460K$ (below |
| 272 |
|
transition temperature) is shown in Figure~\ref{LCFigure:snapshot}. |
| 273 |
|
The diagonal view in Fig~\ref{LCFigure:snapshot}(a) shows the |
| 274 |
|
stacking of the banana shaped molecules while the side view in n |
| 275 |
|
Figure~\ref{LCFigure:snapshot}(b) demonstrates formation of a |
| 276 |
< |
chevron structure. The first peak of Radial distribution function |
| 277 |
< |
$g(r)$ in Fig.~\ref{LCFigure:gofrz}(a) shows the minimum distance |
| 278 |
< |
for two in plane banana shaped molecules is 4.9 \AA, while the |
| 279 |
< |
second split peak implies the biaxial packing. It is also important |
| 280 |
< |
to show the density correlation along the director which is given by |
| 281 |
< |
: |
| 276 |
> |
chevron structure. The first peak of the radial distribution |
| 277 |
> |
function $g(r)$ in Fig.~\ref{LCFigure:gofrz}(a) shows that the |
| 278 |
> |
minimum distance for two in plane banana shaped molecules is 4.9 |
| 279 |
> |
\AA, while the second split peak implies the biaxial packing. It is |
| 280 |
> |
also important to show the density correlation along the director |
| 281 |
> |
which is given by : |
| 282 |
|
\begin{equation} |
| 283 |
|
g(z) =\frac{1}{\pi R^{2} \rho}< \delta (z-z_{ij})>_{ij} |
| 284 |
|
\end{equation}, |
| 285 |
< |
where $z_{ij} = r_{ij} \dot Z$ was measured in the director frame |
| 286 |
< |
and $R$ is the radius of the cylindrical sampling region. The |
| 287 |
< |
oscillation in density plot along the director in |
| 285 |
> |
where $ z_{ij} = r_{ij} \cdot \hat Z $ was measured in the |
| 286 |
> |
director frame and $R$ is the radius of the cylindrical sampling |
| 287 |
> |
region. The oscillation in density plot along the director in |
| 288 |
|
Fig.~\ref{LCFigure:gofrz}(b) implies the existence of the layered |
| 289 |
< |
structure, and the peak at 27 \AA is attribute to the defect in the |
| 289 |
> |
structure, and the peak at 27 \AA is attributed to a defect in the |
| 290 |
|
system. |
| 291 |
|
|
| 292 |
+ |
\subsection{Rotational Invariants} |
| 293 |
+ |
|
| 294 |
+ |
As a useful set of correlation functions to describe |
| 295 |
+ |
position-orientation correlation, rotation invariants were first |
| 296 |
+ |
applied in a spherical symmetric system to study x-ray and light |
| 297 |
+ |
scatting\cite{Blum1972}. Latterly, expansion of the orientation pair |
| 298 |
+ |
correlation in terms of rotation invariant for molecules of |
| 299 |
+ |
arbitrary shape has been introduced by Stone\cite{Stone1978} and |
| 300 |
+ |
adopted by other researchers in liquid crystal |
| 301 |
+ |
studies\cite{Berardi2003}. In order to study the correlation between |
| 302 |
+ |
biaxiality and molecular separation distance $r$, we calculate a |
| 303 |
+ |
rotational invariant function $S_{22}^{220} (r)$, which is given by |
| 304 |
+ |
: |
| 305 |
+ |
\begin{eqnarray} |
| 306 |
+ |
S_{22}^{220} (r) & = & \frac{1}{{4\sqrt 5 }} \left< \delta (r - |
| 307 |
+ |
r_{ij} )((\hat x_i \cdot \hat x_j )^2 - (\hat x_i \cdot \hat y_j |
| 308 |
+ |
)^2 - (\hat y_i \cdot \hat x_j )^2 + (\hat y_i \cdot \hat y_j |
| 309 |
+ |
)^2 ) \right. \notag \\ |
| 310 |
+ |
& & \left. - 2(\hat x_i \cdot \hat y_j )(\hat y_i \cdot \hat x_j ) - |
| 311 |
+ |
2(\hat x_i \cdot \hat x_j )(\hat y_i \cdot \hat y_j )) \right>. |
| 312 |
+ |
\end{eqnarray} |
| 313 |
+ |
The long range behavior of second rank orientational correlation |
| 314 |
+ |
$S_{22}^{220} (r)$ in Fig~\ref{LCFigure:S22220} also confirm the |
| 315 |
+ |
biaxiality of the system. |
| 316 |
+ |
|
| 317 |
|
\begin{figure} |
| 318 |
|
\centering |
| 319 |
|
\includegraphics[width=4.5in]{snapshot.eps} |
| 335 |
|
\label{LCFigure:gofrz} |
| 336 |
|
\end{figure} |
| 337 |
|
|
| 338 |
< |
\subsection{Rotational Invariants} |
| 338 |
> |
\begin{figure} |
| 339 |
> |
\centering |
| 340 |
> |
\includegraphics[width=\linewidth]{s22_220.eps} |
| 341 |
> |
\caption[Average orientational correlation Correlation Functions of |
| 342 |
> |
a Bent-core Liquid Crystal System at Temperature T = 460K and |
| 343 |
> |
Pressure P = 10 atm]{Average orientational correlation Correlation |
| 344 |
> |
Functions of a Bent-core Liquid Crystal System at Temperature T = |
| 345 |
> |
460K and Pressure P = 10 atm.} \label{LCFigure:S22220} |
| 346 |
> |
\end{figure} |
| 347 |
|
|
| 322 |
– |
As a useful set of correlation functions to describe |
| 323 |
– |
position-orientation correlation, rotation invariants were first |
| 324 |
– |
applied in a spherical symmetric system to study x-ray and light |
| 325 |
– |
scatting\cite{Blum1972}. Latterly, expansion of the orientation pair |
| 326 |
– |
correlation in terms of rotation invariant for molecules of |
| 327 |
– |
arbitrary shape was introduce by Stone\cite{Stone1978} and adopted |
| 328 |
– |
by other researchers in liquid crystal studies\cite{Berardi2003}. In |
| 329 |
– |
order to study the correlation between biaxiality and molecular |
| 330 |
– |
separation distance $r$, we calculate a rotational invariant |
| 331 |
– |
function $S_{22}^{220} (r)$, which is given by : |
| 332 |
– |
\begin{eqnarray} |
| 333 |
– |
S_{22}^{220} (r) & = & \frac{1}{{4\sqrt 5 }} \left< \delta (r - |
| 334 |
– |
r_{ij} )((\hat x_i \cdot \hat x_j )^2 - (\hat x_i \cdot \hat y_j |
| 335 |
– |
)^2 - (\hat y_i \cdot \hat x_j )^2 + (\hat y_i \cdot \hat y_j |
| 336 |
– |
)^2 ) \right. \notag \\ |
| 337 |
– |
& & \left. - 2(\hat x_i \cdot \hat y_j )(\hat y_i \cdot \hat x_j ) - |
| 338 |
– |
2(\hat x_i \cdot \hat x_j )(\hat y_i \cdot \hat y_j )) \right>. |
| 339 |
– |
\end{eqnarray} |
| 340 |
– |
|
| 341 |
– |
%\begin{equation} |
| 342 |
– |
%S_{00}^{221} (r) = - \frac{{\sqrt 3 }}{{\sqrt {10} }}\left\langle |
| 343 |
– |
%{\delta (r - r_{ij} )((\hat z_i \cdot \hat z_j )(\hat z_i \cdot |
| 344 |
– |
%\hat z_j \times \hat r_{ij} ))} \right\rangle |
| 345 |
– |
%\end{equation} |
| 346 |
– |
|
| 348 |
|
\section{Conclusion} |
| 349 |
|
|
| 350 |
|
We have presented a simple dipolar three-site GB model for banana |
| 351 |
|
shaped molecules which are capable of forming smectic phases from |
| 352 |
< |
isotropic configuration. |
| 352 |
> |
isotropic configuration. Various order parameters and correlation |
| 353 |
> |
functions were used to characterized the structural properties of |
| 354 |
> |
these smectic phase. However, the forming layered structure still |
| 355 |
> |
had some defects because of the mismatching between the layer |
| 356 |
> |
structure spacing and the shape of simulation box. This mismatching |
| 357 |
> |
can be broken by using NPTf integrator in further simulations. The |
| 358 |
> |
role of terminal chain in controlling transition temperatures and |
| 359 |
> |
the type of mesophase formed have been studied |
| 360 |
> |
extensively\cite{Pelzl1999}. The lack of flexibility in our model |
| 361 |
> |
due to the missing terminal chains could explain the fact that we |
| 362 |
> |
did not find evidence of chirality. |
