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\chapter{\label{chapt:liquidcrystal}LIQUID CRYSTAL} |
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|
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\section{\label{liquidCrystalSection:introduction}Introduction} |
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|
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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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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{smcp.eps} |
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\caption[SmCP Phase Packing] {Four possible SmCP phase packings that |
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are characterized by the relative tilt direction(A and S refer an |
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anticlinic tilt or a synclinic ) and the polarization orientation (A |
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and F represent antiferroelectric or ferroelectric polar order).} |
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\label{LCFig:SMCP} |
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\end{figure} |
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|
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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\cite{Reddy2006}. |
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From a theoretical point of view, it is of fundamental interest to |
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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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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 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, have become the dominant models |
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in the field of liquid crystal phase behavior. Previous simulation |
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studies using a hard spherocylinder dimer model\cite{Camp1999} |
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produced nematic phases, while hard rod simulation studies |
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identified a direct transition to the biaxial nematic and other |
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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 |
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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. 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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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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|
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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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as the effect of electrostatic interactions. |
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|
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\section{\label{liquidCrystalSection:model}Model} |
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|
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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 ellipsoids $i$ and $j$ separated by |
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intermolecular vector $r_{ij}$. $\hat r_{ij}$ is the unit vector |
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along the inter-ellipsoid vector. A schematic diagram of the |
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orientation vectors is shown in Fig.\ref{LCFigure:GBScheme}. The |
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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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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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|
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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[Schematic representation of a typical banana shaped |
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molecule]{Schematic representation of a typical banana shaped |
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molecule.} \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]{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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ellipsoids} \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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|
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\section{Results and Discussion} |
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|
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A series of molecular dynamics simulations were perform to study the |
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phase behavior of banana shaped liquid crystals. In each simulation, |
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every banana shaped molecule has been represented by three GB |
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particles which is characterized by $\mu = 1,~ \nu = 2, |
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~\epsilon_{e}/\epsilon_{s} = 1/5$ and $\sigma_{e}/\sigma_{s} = 3$. |
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All of the simulations begin with same equilibrated isotropic |
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configuration where 1024 molecules without dipoles were confined in |
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a $160\times 160 \times 120$ box. After the dipolar interactions are |
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switched on, 2~ns NPTi cooling run with themostat of 2~ps and |
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barostat of 50~ps were used to equilibrate the system to desired |
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temperature and pressure. NPTi Production runs last for 40~ns with |
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time step of 20~fs. |
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|
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\subsection{Order Parameters} |
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|
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To investigate the phase structure of the model liquid crystal, we |
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calculated various order parameters and correlation functions. |
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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 eigenvalue obtained by diagonalizing the order parameter |
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tensor |
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\begin{equation} |
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\overleftrightarrow{\mathsf{Q}} = \frac{1}{N}\sum_i^N % |
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\begin{pmatrix} % |
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u_{ix}u_{ix}-\frac{1}{3} & u_{ix}u_{iy} & u_{ix}u_{iz} \\ |
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u_{iy}u_{ix} & u_{iy}u_{iy}-\frac{1}{3} & u_{iy}u_{iz} \\ |
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u_{iz}u_{ix} & u_{iz}u_{iy} & u_{iz}u_{iz}-\frac{1}{3} % |
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\end{pmatrix}, |
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\label{lipidEq:p2} |
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\end{equation} |
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where the $u_{i\alpha}$ is the $\alpha$ element of the unit vector |
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$\mathbf{\hat{u}}_i$, and the sum over $i$ averages over the whole |
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collection of unit vectors. The $P_2$ order parameter for uniaxial |
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phase is then simply given by |
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\begin{equation} |
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\langle P_2 \rangle = \frac{3}{2}\lambda_{\text{max}}. |
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\label{lipidEq:po3} |
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\end{equation} |
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%In addition to the $P_2$ order parameter, $ R_{2,2}^2$ order |
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%parameter for biaxial phase is introduced to describe the ordering |
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%in the plane orthogonal to the director by |
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%\begin{equation} |
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%R_{2,2}^2 = \frac{1}{4}\left\langle {(x_i \cdot X)^2 - (x_i \cdot |
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%Y)^2 - (y_i \cdot X)^2 + (y_i \cdot Y)^2 } \right\rangle |
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%\end{equation} |
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%where $X$, $Y$ and $Z$ are axis of the director frame. |
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The unit vector for the banana shaped molecule was defined by the |
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principle aixs of its middle GB particle. The $P_2$ order parameters |
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for the bent-core liquid crystal at different temperature are |
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summarized in Table~\ref{liquidCrystal:p2} which identifies a phase |
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transition temperature range. |
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|
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\begin{table} |
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\caption{LIQUID CRYSTAL STRUCTURAL PROPERTIES AS A FUNCTION OF |
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TEMPERATURE} \label{liquidCrystal:p2} |
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\begin{center} |
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\begin{tabular}{cccccc} |
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\hline |
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Temperature (K) & 420 & 440 & 460 & 480 & 600\\ |
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\hline |
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$\langle P_2\rangle$ & 0.984 & 0.982 & 0.975 & 0.967 & 0.067\\ |
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\hline |
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\end{tabular} |
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\end{center} |
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\end{table} |
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|
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\subsection{Structural Properties} |
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|
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The molecular organization obtained at temperature $T = 460K$ (below |
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transition temperature) is shown in Figure~\ref{LCFigure:snapshot}. |
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The diagonal view in Fig~\ref{LCFigure:snapshot}(a) shows the |
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stacking of the banana shaped molecules while the side view in n |
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Figure~\ref{LCFigure:snapshot}(b) demonstrates formation of a |
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chevron structure. The first peak of the radial distribution |
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function $g(r)$ in Fig.~\ref{LCFigure:gofrz}(a) shows that the |
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minimum distance for two in plane banana shaped molecules is 4.9 |
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\AA, while the second split peak implies the biaxial packing. It is |
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also important to show the density correlation along the director |
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which is given by : |
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\begin{equation} |
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g(z) =\frac{1}{\pi R^{2} \rho}< \delta (z-z_{ij})>_{ij}, |
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\end{equation} |
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where $ z_{ij} = r_{ij} \cdot \hat Z $ was measured in the |
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director frame and $R$ is the radius of the cylindrical sampling |
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region. The oscillation in density plot along the director in |
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Fig.~\ref{LCFigure:gofrz}(b) implies the existence of the layered |
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structure, and the peak at 27 \AA is attributed to a defect in the |
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system. |
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|
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\subsection{Rotational Invariants} |
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|
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As a useful set of correlation functions to describe |
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position-orientation correlation, rotation invariants were first |
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applied in a spherical symmetric system to study x-ray and light |
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scatting\cite{Blum1972}. Latterly, expansion of the orientation pair |
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correlation in terms of rotation invariant for molecules of |
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arbitrary shape has been introduced by Stone\cite{Stone1978} and |
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adopted by other researchers in liquid crystal |
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studies\cite{Berardi2003}. In order to study the correlation between |
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biaxiality and molecular separation distance $r$, we calculate a |
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rotational invariant function $S_{22}^{220} (r)$, which is given by |
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: |
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\begin{eqnarray} |
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S_{22}^{220} (r) & = & \frac{1}{{4\sqrt 5 }} \left< \delta (r - |
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r_{ij} )((\hat x_i \cdot \hat x_j )^2 - (\hat x_i \cdot \hat y_j |
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)^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 ) - |
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2(\hat x_i \cdot \hat x_j )(\hat y_i \cdot \hat y_j )) \right>. |
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\end{eqnarray} |
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The long range behavior of second rank orientational correlation |
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$S_{22}^{220} (r)$ in Fig~\ref{LCFigure:S22220} also confirm the |
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biaxiality of the system. |
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|
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\begin{figure} |
| 318 |
\centering |
| 319 |
\includegraphics[width=4.5in]{snapshot.eps} |
| 320 |
\caption[Snapshot of the molecular organization in the layered phase |
| 321 |
formed at temperature T = 460K and pressure P = 1 atm]{Snapshot of |
| 322 |
the molecular organization in the layered phase formed at |
| 323 |
temperature T = 460K and pressure P = 1 atm. (a) diagonal view; (b) |
| 324 |
side view.} \label{LCFigure:snapshot} |
| 325 |
\end{figure} |
| 326 |
|
| 327 |
\begin{figure} |
| 328 |
\centering |
| 329 |
\includegraphics[width=\linewidth]{gofr_gofz.eps} |
| 330 |
\caption[Correlation Functions of a Bent-core Liquid Crystal System |
| 331 |
at Temperature T = 460K and Pressure P = 10 atm]{Correlation |
| 332 |
Functions of a Bent-core Liquid Crystal System at Temperature T = |
| 333 |
460K and Pressure P = 10 atm. (a) radial correlation function |
| 334 |
$g(r)$; and (b) density along the director $g(z)$.} |
| 335 |
\label{LCFigure:gofrz} |
| 336 |
\end{figure} |
| 337 |
|
| 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 |
|
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\section{Conclusion} |
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|
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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. 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 |
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role of terminal chain in controlling transition temperatures and |
| 359 |
the type of mesophase formed have been studied |
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extensively\cite{Pelzl1999}. The lack of flexibility in our model |
| 361 |
due to the missing terminal chains could explain the fact that we |
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did not find evidence of chirality. |
