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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 |
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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 |
| 31 |
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chirality, the spontaneous polarization arises in ferroelectric (FE) |
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and antiferroelectic (AF) switching of smectic liquid crystal |
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phases, demonstrating some promising applications in second-order |
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nonlinear optical devices. The most widely investigated mesophase |
| 35 |
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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 |
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direction of the molecule in adjacent layer (see |
| 41 |
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Fig.~\ref{LCFig:SMCP}). |
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– |
|
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{smcp.eps} |
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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 |
| 62 |
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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 |
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limitation of time scales required for phase transition and the |
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length scale required for representing bulk behavior, |
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models\cite{Perram1985, Gay1981}, which are based on the observation |
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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 |
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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 |
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nematic phases, while hard rod simulation studies identified a |
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Landau point\cite{Bates2005}, at which the isotropic phase undergoes |
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a direct transition to the biaxial nematic, as well as some possible |
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liquid crystal phases\cite{Lansac2003}. Other anisotropic models |
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using Gay-Berne(GB) potential, which produce interactions that favor |
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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 |
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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 |
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possible liquid crystal phases\cite{Lansac2003}. Other anisotropic |
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models using the Gay-Berne(GB) potential, which produces |
| 74 |
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interactions that favor local alignment, give evidence of the novel |
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packing arrangements of bent-core molecules\cite{Memmer2002}. |
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|
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Experimental studies by Levelut {\it et al.}~\cite{Levelut1981} |
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revealed that terminal cyano or nitro groups usually induce |
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permanent longitudinal dipole moments, which affect the phase |
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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 |
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rotation) were shown to enhance smectic phase |
| 80 |
> |
behavior considerably. Equivalent conclusions have also been drawn |
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from a series of theoretical studies. Monte Carlo studies of the GB |
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potential with fixed longitudinal dipoles (i.e. pointed along the |
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principal axis of rotation) were shown to enhance smectic phase |
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stability~\cite{Berardi1996,Satoh1996}. Molecular simulation of GB |
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ellipsoids with transverse dipoles at the terminus of the molecule |
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also demonstrated that partial striped bilayer structures were |
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bent-core molecules, could be modeled more accurately by |
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incorporating electrostatic interaction. |
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|
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In this chapter, we consider system consisting of banana-shaped |
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In this chapter, we consider a system consisting of banana-shaped |
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molecule represented by three rigid GB particles with two point |
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dipoles. Performing a series of molecular dynamics simulations, we |
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explore the structural properties of tilted smectic phases as well |
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} \right] \label{LCEquation:gb} |
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\end{equation} |
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where $\hat u_i,\hat u_j$ are unit vectors specifying the |
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orientation of two molecules $i$ and $j$ separated by intermolecular |
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vector $r_{ij}$. $\hat r_{ij}$ is the unit vector along the |
| 129 |
< |
intermolecular vector. A schematic diagram of the orientation |
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< |
vectors is shown in Fig.\ref{LCFigure:GBScheme}. The functional form |
| 131 |
< |
for $\sigma$ is given by |
| 127 |
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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 |
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orientation vectors is shown in Fig.\ref{LCFigure:GBScheme}. The |
| 131 |
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functional form for $\sigma$ is given by |
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\begin{equation} |
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\sigma (\hat u_i ,\hat u_i ,\hat r_{ij} ) = \sigma _0 \left[ {1 - |
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\frac{\chi }{2}\left( {\frac{{(\hat r_{ij} \cdot \hat u_i + \hat |
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molecule]{Schematic representation of a typical banana shaped |
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molecule.} \label{LCFig:BananaMolecule} |
| 182 |
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\end{figure} |
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– |
|
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\begin{figure} |
| 184 |
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\centering |
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\includegraphics[width=\linewidth]{gb_scheme.eps} |
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\caption[Schematic diagram showing definitions of the orientation |
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vectors for a pair of Gay-Berne molecules]{Schematic diagram showing |
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definitions of the orientation vectors for a pair of Gay-Berne |
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molecules} \label{LCFigure:GBScheme} |
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> |
ellipsoids} \label{LCFigure:GBScheme} |
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\end{figure} |
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|
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To account for the permanent dipolar interactions, there should be |
| 192 |
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an electrostatic interaction term of the form |
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\begin{equation} |
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calculated various order parameters and correlation functions. |
| 220 |
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Particulary, the $P_2$ order parameter allows us to estimate average |
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alignment along the director axis $Z$ which can be identified from |
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< |
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} |
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|
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< |
\subsection{Structure Properties} |
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> |
\subsection{Structural Properties} |
| 270 |
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|
| 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} |
| 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 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} |
| 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 |
< |
lack of detail in. |
| 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. |
