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|
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\section{\label{liquidCrystalSection:introduction}Introduction} |
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|
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Long range orientational order is one of the most fundamental |
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properties of liquid crystal mesophases. This orientational |
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anisotropy of the macroscopic phases originates in the shape |
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anisotropy of the constituent molecules. Among these anisotropy |
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mesogens, rod-like (calamitic) and disk-like molecules have been |
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exploited in great detail in the last two decades\cite{Huh2004}. |
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Typically, these mesogens consist of a rigid aromatic core and one |
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or more attached aliphatic chains. For short chain molecules, only |
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nematic phases, in which positional order is limited or absent, can |
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be observed, because the entropy of mixing different parts of the |
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mesogens is paramount to the dispersion interaction. In contrast, |
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formation of the one dimension lamellar sematic phase in rod-like |
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molecules with sufficiently long aliphatic chains has been reported, |
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as well as the segregation phenomena in disk-like molecules. |
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Rod-like (calamitic) and disk-like anisotropy liquid crystals have |
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been investigated in great detail in the last two |
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decades.\cite{Huh2004} Typically, these mesogens consist of a rigid |
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aromatic core and one or more attached aliphatic chains. For short |
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chain molecules, only nematic phases, in which positional order is |
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limited or absent, can be observed, because the entropy of mixing |
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different parts of the mesogens is larger than the dispersion |
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interaction. In contrast, formation of one dimension lamellar |
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smectic phase in rod-like molecules with sufficiently long aliphatic |
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chains has been reported, as well as the segregation phenomena in |
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disk-like molecules.\cite{McMillan1971} Recently, banana-shaped or |
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bent-core liquid crystals have became one of the most active |
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research areas in mesogenic materials and supramolecular |
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chemistry.\cite{Niori1996, Link1997, Pelzl1999} Unlike rods and |
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disks, the polarity and biaxiality of the banana-shaped molecules |
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allow the molecules organize into a variety of novel liquid |
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crystalline phases which show interesting material properties. Of |
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particular interest is the spontaneous formation of macroscopic |
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chiral layers from achiral banana-shaped molecules, where polar |
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molecule orientational ordering exhibited layered plane as well as |
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the tilted arrangement of the molecules relative to the polar axis. |
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As a consequence of supramolecular chirality, the spontaneous |
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polarization arises in ferroelectric (FE) and antiferroelectic (AF) |
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switching of smectic liquid crystal phases, demonstrating some |
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promising applications in second-order nonlinear optical devices. |
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The most widely investigated mesophase formed by banana-shaped |
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moleculed is the $\text{B}_2$ phase, which is also referred to as |
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$\text{SmCP}$.\cite{Link1997} Of the most important discoveries in |
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this tilt lamellar phase is the four distinct packing arrangements |
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(two conglomerates and two macroscopic racemates), which depend on |
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the tilt direction and the polar direction of the molecule in |
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adjacent layer (see Fig.~\ref{LCFig:SMCP}).\cite{Link1997} |
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|
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Recently, the banana-shaped or bent-core liquid crystal have became |
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one of the most active research areas in mesogenic materials and |
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supramolecular chemistry\cite{Niori1996, Link1997, Pelzl1999}. |
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Unlike rods and disks, the polarity and biaxiality of the |
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banana-shaped molecules allow the molecules organize into a variety |
25 |
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of novel liquid crystalline phases which show interesting material |
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properties. Of particular interest is the spontaneous formation of |
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macroscopic chiral layers from achiral banana-shaped molecules, |
28 |
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where polar molecule orientational ordering is shown within the |
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layer plane as well as the tilted arrangement of the molecules |
30 |
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relative to the polar axis. As a consequence of supramolecular |
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chirality, the spontaneous polarization arises in ferroelectric (FE) |
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and antiferroelectic (AF) switching of smectic liquid crystal |
33 |
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phases, demonstrating some promising applications in second-order |
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nonlinear optical devices. The most widely investigated mesophase |
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formed by banana-shaped moleculed is the $\text{B}_2$ phase, which |
36 |
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is also referred to as $\text{SmCP}$\cite{Link1997}. Of the most |
37 |
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important discover in this tilt lamellar phase is the four distinct |
38 |
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packing arrangements (two conglomerates and two macroscopic |
39 |
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racemates), which depend on the tilt direction and the polar |
40 |
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direction of the molecule in adjacent layer (see |
41 |
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Fig.~\ref{LCFig:SMCP}). |
42 |
– |
|
38 |
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\begin{figure} |
39 |
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\centering |
40 |
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\includegraphics[width=\linewidth]{smcp.eps} |
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|
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Many liquid crystal synthesis experiments suggest that the |
49 |
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occurrence of polarity and chirality strongly relies on the |
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molecular structure and intermolecular interaction\cite{Reddy2006}. |
50 |
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molecular structure and intermolecular interaction.\cite{Reddy2006} |
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From a theoretical point of view, it is of fundamental interest to |
52 |
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study the structural properties of liquid crystal phases formed by |
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banana-shaped molecules and understand their connection to the |
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ordering and phase behavior, and hence improve the development of |
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new experiments and theories. In the last two decades, all-atom |
59 |
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models have been adopted to investigate the structural properties of |
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smectic arrangements\cite{Cook2000, Lansac2001}, as well as other |
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smectic arrangements,\cite{Cook2000, Lansac2001} as well as other |
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bulk properties, such as rotational viscosity and flexoelectric |
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coefficients\cite{Cheung2002, Cheung2004}. However, due to the |
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limitation of time scale required for phase transition and the |
62 |
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coefficients.\cite{Cheung2002, Cheung2004} However, due to the |
63 |
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limitation of time scales required for phase transition and the |
64 |
|
length scale required for representing bulk behavior, |
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models\cite{Perram1985, Gay1981}, which are based on the observation |
65 |
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models,\cite{Perram1985, Gay1981} which are based on the observation |
66 |
|
that liquid crystal order is exhibited by a range of non-molecular |
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bodies with high shape anisotropies, became the dominant models in |
68 |
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the field of liquid crystal phase behavior. Previous simulation |
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studies using hard spherocylinder dimer model\cite{Camp1999} produce |
70 |
< |
nematic phases, while hard rod simulation studies identified a |
71 |
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Landau point\cite{Bates2005}, at which the isotropic phase undergoes |
72 |
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a direct transition to the biaxial nematic, as well as some possible |
73 |
< |
liquid crystal phases\cite{Lansac2003}. Other anisotropic models |
74 |
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using Gay-Berne(GB) potential, which produce interactions that favor |
75 |
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local alignment, give the evidence of the novel packing arrangements |
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< |
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 |
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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 |
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|
77 |
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Experimental studies by Levelut {\it et al.}~\cite{Levelut1981} |
78 |
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revealed that terminal cyano or nitro groups usually induce |
79 |
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permanent longitudinal dipole moments, which affect the phase |
80 |
< |
behavior considerably. A series of theoretical studies also drawn |
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equivalent conclusions. Monte Carlo studies of the GB potential with |
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fixed longitudinal dipoles (i.e. pointed along the principal axis of |
83 |
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rotation) were shown to enhance smectic phase |
84 |
< |
stability~\cite{Berardi1996,Satoh1996}. Molecular simulation of GB |
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 |
87 |
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developed from the smectic phase ~\cite{Berardi1996}. More |
87 |
> |
developed from the smectic phase.~\cite{Berardi1996} More |
88 |
|
significant effects have been shown by including multiple |
89 |
|
electrostatic moments. Adding longitudinal point quadrupole moments |
90 |
|
to rod-shaped GB mesogens, Withers \textit{et al} induced tilted |
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smectic behaviour in the molecular system~\cite{Withers2003}. Thus, |
91 |
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smectic behaviour in the molecular system.~\cite{Withers2003} Thus, |
92 |
|
it is clear that many liquid-crystal forming molecules, specially, |
93 |
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bent-core molecules, could be modeled more accurately by |
94 |
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incorporating electrostatic interaction. |
95 |
|
|
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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 |
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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 |
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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 |
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|
191 |
|
To account for the permanent dipolar interactions, there should be |
192 |
|
an electrostatic interaction term of the form |
193 |
|
\begin{equation} |
200 |
|
|
201 |
|
\section{Results and Discussion} |
202 |
|
|
203 |
< |
A series of molecular dynamics simulations were perform to study the |
203 |
> |
A series of molecular dynamics simulations were performed to study the |
204 |
|
phase behavior of banana shaped liquid crystals. In each simulation, |
205 |
|
every banana shaped molecule has been represented by three GB |
206 |
|
particles which is characterized by $\mu = 1,~ \nu = 2, |
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} % |
248 |
|
%where $X$, $Y$ and $Z$ are axis of the director frame. |
249 |
|
The unit vector for the banana shaped molecule was defined by the |
250 |
|
principle aixs of its middle GB particle. The $P_2$ order parameters |
251 |
< |
for the bent-core liquid crystal at different temperature is |
251 |
> |
for the bent-core liquid crystal at different temperature are |
252 |
|
summarized in Table~\ref{liquidCrystal:p2} which identifies a phase |
253 |
|
transition temperature range. |
254 |
|
|
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 |
< |
: |
274 |
> |
stacking of the banana shaped molecules while the side view in |
275 |
> |
Figure~\ref{LCFigure:snapshot}(b) demonstrates formation of a |
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 |
283 |
> |
g(z) =\frac{1}{\pi R^{2} \rho}< \delta (z-z_{ij})>_{ij}, |
284 |
> |
\end{equation} |
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 $\rm{\AA}$ is attributed to a defect in the |
290 |
|
system. |
291 |
|
|
292 |
|
\subsection{Rotational Invariants} |
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 |
297 |
> |
scatting.\cite{Blum1972} Latterly, expansion of the orientation pair |
298 |
|
correlation in terms of rotation invariant for molecules of |
299 |
< |
arbitrary shape was introduce by Stone\cite{Stone1978} and adopted |
300 |
< |
by other researchers in liquid crystal studies\cite{Berardi2003}. In |
301 |
< |
order to study the correlation between biaxiality and molecular |
302 |
< |
separation distance $r$, we calculate a rotational invariant |
303 |
< |
function $S_{22}^{220} (r)$, which is given by : |
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 |
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]{Correlation Functions of a Bent-core Liquid |
344 |
< |
Crystal System at Temperature T = 460K and Pressure P = 10 atm. (a) |
345 |
< |
radial correlation function $g(r)$; and (b) density along the |
353 |
< |
director $g(z)$.} \label{LCFigure:S22220} |
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 |
|
|
348 |
|
\section{Conclusion} |
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 explained the fact that we |
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. |