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\chapter{\label{chapt:liquidcrystal}LIQUID CRYSTAL} |
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\section{\label{liquidCrystalSection:introduction}Introduction} |
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% liquid crystal |
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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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sufficiently long aliphatic chains has been reported, as well as the |
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segregation phenomena in disk-like molecules. |
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% banana shaped |
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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. Unlike rods and disks, the polarity and |
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supramolecular chirality, the spontaneous polarization arises in |
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ferroelectric (FE) and antiferroelectic (AF) switching of smectic |
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liquid crystal phases, demonstrating some promising applications in |
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second-order nonlinear optical devices. |
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second-order nonlinear optical devices. The most widely investigated |
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mesophase formed by banana-shaped moleculed is the $\text{B}_2$ |
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phase, which is also referred to as $\text{SmCP}$. Of the most |
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important discover in this tilt lamellar phase is the four distinct |
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packing arrangements (two conglomerates and two macroscopic |
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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 |
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Fig.~\cite{LCFig:SMCP}). |
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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 known as |
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$\text{SmCP}$. |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{smcp.eps} |
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\caption[] |
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{} |
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\label{LCFig:SMCP} |
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\end{figure} |
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%Previous Theoretical Studies |
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Many liquid crystal synthesis experiments suggest that the |
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occurrence of polarity and chirality strongly relies on the |
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molecular structure and intermolecular interaction. From a |
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theoretical point of view, it is of fundamental interest to study |
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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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molecular structure, especially with respect to the spontaneous |
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achiral symmetry breaking. As a complementary tool to experiment, |
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computer simulation can provide unique insight into molecular |
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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 |
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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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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 |
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transition\cite{Wilson1999} and the length scale required for |
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representing bulk behavior, the dominant models in the field of |
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liquid crystal phase behavior are generic |
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models\cite{Lebwohl1972,Perram1984, Gay1981}, which are based on the |
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observation that liquid crystal order is exhibited by a range of |
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non-molecular bodies with high shape anisotropies. Previous |
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simulation studies using hard spherocylinder dimer |
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model\cite{Camp1999} produce nematic phases, while hard rod |
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simulation studies identified a Landau point\cite{Bates2005}, at |
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which the isotropic phase undergoes a direct transition to the |
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biaxial nematic, as well as some possible liquid crystal |
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phases\cite{Lansac2003}. Other anisotropic models using |
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Gay-Berne(GB) potential, which produce interactions that favor local |
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alignment, give the evidence of the novel packing arrangements of |
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bent-core molecules\cite{Memmer2002,Orlandi2006}. |
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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 |
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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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developed from the smectic phase ~\cite{Berardi1996}. More |
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significant effects have been shown by including multiple |
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electrostatic moments. Adding longitudinal point quadrupole moments |
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to rod-shaped GB mesogens, Withers \textit{et al} induced tilted |
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smectic behaviour in the molecular system~\cite{Withers2003}. Thus, |
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it is clear that many liquid-crystal forming molecules, specially, |
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bent-core molecules, could be modeled more accurately by |
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incorporating electrostatic interaction. |
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In this chapter, we consider system consisting of banana-shaped |
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molecule represented by three rigid GB particles with one or two |
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point dipoles at different location. Performing a series of |
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molecular dynamics simulations, we explore the structural properties |
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of tilted smectic phases as well as the effect of electrostatic |
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interactions. |
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\section{\label{liquidCrystalSection:model}Model} |
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A typical banana-shaped molecule consists of a rigid aromatic |
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central bent unit with several rod-like wings which are held |
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together by some linking units and terminal chains (see |
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Fig.~\ref{LCFig:BananaMolecule}). In this work, each banana-shaped |
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mesogen has been modeled as a rigid body consisting of three |
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equivalent prolate ellipsoidal GB particles. The GB interaction |
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potential used to mimic the apolar characteristics of liquid crystal |
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molecules takes the familiar form of Lennard-Jones function with |
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orientation and position dependent range ($\sigma$) and well depth |
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($\epsilon$) parameters. The potential between a pair of three-site |
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banana-shaped molecules $a$ and $b$ is given by |
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\begin{equation} |
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V_{ab}^{GB} = \sum\limits_{i \in a,j \in b} {V_{ij}^{GB} }. |
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\end{equation} |
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Every site-site interaction can can be expressed as, |
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\begin{equation} |
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V_{ij}^{GB} = 4\epsilon (\hat u_i ,\hat u_j ,\hat r_{ij} )\left[ |
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{\left( {\frac{{\sigma _0 }}{{r_{ij} - \sigma (\hat u_i ,\hat u_j |
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,\hat r_{ij} )}}} \right)^{12} - \left( {\frac{{\sigma _0 |
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}}{{r_{ij} - \sigma (\hat u_i ,\hat u_j ,\hat r_{ij} )}}} \right)^6 |
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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 |
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intermolecular vector. A schematic diagram of the orientation |
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vectors is shown in Fig.\ref{LCFigure:GBScheme}. The functional form |
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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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r_{ij} \cdot \hat u_j )^2 }}{{1 + \chi \hat u_i \cdot \hat u_j }} |
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+ \frac{{(\hat r_{ij} \cdot \hat u_i - \hat r_{ij} \cdot \hat u_j |
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)^2 }}{{1 - \chi \hat u_i \cdot \hat u_j }}} \right)} \right]^{ - |
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\frac{1}{2}}, |
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\end{equation} |
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where the aspect ratio of the particles is governed by shape |
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anisotropy parameter |
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\begin{equation} |
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\chi = \frac{{(\sigma _e /\sigma _s )^2 - 1}}{{(\sigma _e /\sigma |
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_s )^2 + 1}}. |
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\label{LCEquation:chi} |
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\end{equation} |
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Here, $\sigma_ s$ and $\sigma_{e}$ refer to the side-by-side breadth |
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and the end-to-end length of the ellipsoid, respectively. The well |
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depth parameters takes the form |
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\begin{equation} |
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\epsilon (\hat u_i ,\hat u_j ,\hat r_{ij} ) = \epsilon _0 \epsilon |
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^v (\hat u_i ,\hat u_j )\epsilon '^\mu (\hat u_i ,\hat u_j ,\hat |
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r_{ij} ) |
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\end{equation} |
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where $\epsilon_{0}$ is a constant term and |
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\begin{equation} |
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\epsilon (\hat u_i ,\hat u_j ) = \frac{1}{{\sqrt {1 - \chi ^2 (\hat |
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u_i \cdot \hat u_j )^2 } }} |
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\end{equation} |
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and |
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\begin{equation} |
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\epsilon '(\hat u_i ,\hat u_j ,\hat r_{ij} ) = 1 - \frac{{\chi |
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'}}{2}\left[ {\frac{{(\hat r_{ij} \cdot \hat u_i + \hat r_{ij} |
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\cdot \hat u_j )^2 }}{{1 + \chi '\hat u_i \cdot \hat u_j }} + |
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\frac{{(\hat r_{ij} \cdot \hat u_i - \hat r_{ij} \cdot \hat u_j |
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)^2 }}{{1 - \chi '\hat u_i \cdot \hat u_j }}} \right] |
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\end{equation} |
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where the well depth anisotropy parameter $\chi '$ depends on the |
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ratio between \textit{end-to-end} well depth $\epsilon _e$ and |
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\textit{side-by-side} well depth $\epsilon_s$, |
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\begin{equation} |
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\chi ' = \frac{{1 - (\epsilon _e /\epsilon _s )^{1/\mu} }}{{1 + |
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(\epsilon _e /\epsilon _s )^{1/\mu} }}. |
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\end{equation} |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{banana.eps} |
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\caption[]{} \label{LCFig:BananaMolecule} |
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\end{figure} |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{bananGB.eps} |
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\caption[]{} \label{LCFigure:BananaGB} |
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\end{figure} |
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\begin{figure} |
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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} |
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\label{LCFigure:GBScheme} |
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\end{figure} |
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To account for the permanent dipolar interactions, there should be |
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an electrostatic interaction term of the form |
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\begin{equation} |
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V_{ab}^{dp} = \sum\limits_{i \in a,j \in b} {\frac{1}{{4\pi |
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\epsilon _{fs} }}\left[ {\frac{{\mu _i \cdot \mu _j }}{{r_{ij}^3 }} |
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- \frac{{3\left( {\mu _i \cdot r_{ij} } \right)\left( {\mu _i \cdot |
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r_{ij} } \right)}}{{r_{ij}^5 }}} \right]} |
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\end{equation} |
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where $\epsilon _{fs}$ is the permittivity of free space. |
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\section{\label{liquidCrystalSection:methods}Methods} |
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\section{\label{liquidCrystalSection:resultDiscussion}Results and Discussion} |
