--- trunk/tengDissertation/LiquidCrystal.tex 2006/06/02 21:31:49 2785 +++ trunk/tengDissertation/LiquidCrystal.tex 2006/06/27 02:42:30 2895 @@ -7,51 +7,54 @@ exploited in great detail in the last two decades. Typ anisotropy of the macroscopic phases originates in the shape anisotropy of the constituent molecules. Among these anisotropy mesogens, rod-like (calamitic) and disk-like molecules have been -exploited in great detail in the last two decades. Typically, these -mesogens consist of a rigid aromatic core and one or more attached -aliphatic chains. For short chain molecules, only nematic phases, in -which positional order is limited or absent, can be observed, -because the entropy of mixing different parts of the mesogens is -paramount to the dispersion interaction. In contrast, formation of -the one dimension lamellar sematic phase in rod-like molecules with -sufficiently long aliphatic chains has been reported, as well as the -segregation phenomena in disk-like molecules. +exploited in great detail in the last two decades\cite{Huh2004}. +Typically, these mesogens consist of a rigid aromatic core and one +or more attached aliphatic chains. For short chain molecules, only +nematic phases, in which positional order is limited or absent, can +be observed, because the entropy of mixing different parts of the +mesogens is paramount to the dispersion interaction. In contrast, +formation of the one dimension lamellar sematic phase in rod-like +molecules with sufficiently long aliphatic chains has been reported, +as well as the segregation phenomena in disk-like molecules. Recently, the banana-shaped or bent-core liquid crystal have became one of the most active research areas in mesogenic materials and -supramolecular chemistry. Unlike rods and disks, the polarity and -biaxiality of the banana-shaped molecules allow the molecules -organize into a variety of novel liquid crystalline phases which -show interesting material properties. Of particular interest is the -spontaneous formation of macroscopic chiral layers from achiral -banana-shaped molecules, where polar molecule orientational ordering -is shown within the layer plane as well as the tilted arrangement of -the molecules relative to the polar axis. As a consequence of -supramolecular chirality, the spontaneous polarization arises in -ferroelectric (FE) and antiferroelectic (AF) switching of smectic -liquid crystal phases, demonstrating some promising applications in -second-order nonlinear optical devices. The most widely investigated -mesophase formed by banana-shaped moleculed is the $\text{B}_2$ -phase, which is also referred to as $\text{SmCP}$. Of the most +supramolecular chemistry\cite{Niori1996, Link1997, Pelzl1999}. +Unlike rods and disks, the polarity and biaxiality of the +banana-shaped molecules allow the molecules organize into a variety +of novel liquid crystalline phases which show interesting material +properties. Of particular interest is the spontaneous formation of +macroscopic chiral layers from achiral banana-shaped molecules, +where polar molecule orientational ordering is shown within the +layer plane as well as the tilted arrangement of the molecules +relative to the polar axis. As a consequence of supramolecular +chirality, the spontaneous polarization arises in ferroelectric (FE) +and antiferroelectic (AF) switching of smectic liquid crystal +phases, demonstrating some promising applications in second-order +nonlinear optical devices. The most widely investigated mesophase +formed by banana-shaped moleculed is the $\text{B}_2$ phase, which +is also referred to as $\text{SmCP}$\cite{Link1997}. Of the most important discover in this tilt lamellar phase is the four distinct packing arrangements (two conglomerates and two macroscopic racemates), which depend on the tilt direction and the polar direction of the molecule in adjacent layer (see -Fig.~\cite{LCFig:SMCP}). +Fig.~\ref{LCFig:SMCP}). \begin{figure} \centering \includegraphics[width=\linewidth]{smcp.eps} -\caption[] -{} +\caption[SmCP Phase Packing] {Four possible SmCP phase packings that +are characterized by the relative tilt direction(A and S refer an +anticlinic tilt or a synclinic ) and the polarization orientation (A +and F represent antiferroelectric or ferroelectric polar order).} \label{LCFig:SMCP} \end{figure} Many liquid crystal synthesis experiments suggest that the occurrence of polarity and chirality strongly relies on the -molecular structure and intermolecular interaction. From a -theoretical point of view, it is of fundamental interest to study -the structural properties of liquid crystal phases formed by +molecular structure and intermolecular interaction\cite{Reddy2006}. +From a theoretical point of view, it is of fundamental interest to +study the structural properties of liquid crystal phases formed by banana-shaped molecules and understand their connection to the molecular structure, especially with respect to the spontaneous achiral symmetry breaking. As a complementary tool to experiment, @@ -62,22 +65,20 @@ limitation of time scale required for phase smectic arrangements\cite{Cook2000, Lansac2001}, as well as other bulk properties, such as rotational viscosity and flexoelectric coefficients\cite{Cheung2002, Cheung2004}. However, due to the -limitation of time scale required for phase -transition\cite{Wilson1999} and the length scale required for -representing bulk behavior, the dominant models in the field of -liquid crystal phase behavior are generic -models\cite{Lebwohl1972,Perram1984, Gay1981}, which are based on the -observation that liquid crystal order is exhibited by a range of -non-molecular bodies with high shape anisotropies. Previous -simulation studies using hard spherocylinder dimer -model\cite{Camp1999} produce nematic phases, while hard rod -simulation studies identified a Landau point\cite{Bates2005}, at -which the isotropic phase undergoes a direct transition to the -biaxial nematic, as well as some possible liquid crystal -phases\cite{Lansac2003}. Other anisotropic models using -Gay-Berne(GB) potential, which produce interactions that favor local -alignment, give the evidence of the novel packing arrangements of -bent-core molecules\cite{Memmer2002,Orlandi2006}. +limitation of time scale required for phase transition and the +length scale required for representing bulk behavior, +models\cite{Perram1985, Gay1981}, which are based on the observation +that liquid crystal order is exhibited by a range of non-molecular +bodies with high shape anisotropies, became the dominant models in +the field of liquid crystal phase behavior. Previous simulation +studies using hard spherocylinder dimer model\cite{Camp1999} produce +nematic phases, while hard rod simulation studies identified a +Landau point\cite{Bates2005}, at which the isotropic phase undergoes +a direct transition to the biaxial nematic, as well as some possible +liquid crystal phases\cite{Lansac2003}. Other anisotropic models +using Gay-Berne(GB) potential, which produce interactions that favor +local alignment, give the evidence of the novel packing arrangements +of bent-core molecules\cite{Memmer2002}. Experimental studies by Levelut {\it et al.}~\cite{Levelut1981} revealed that terminal cyano or nitro groups usually induce @@ -99,11 +100,10 @@ molecule represented by three rigid GB particles with incorporating electrostatic interaction. In this chapter, we consider system consisting of banana-shaped -molecule represented by three rigid GB particles with one or two -point dipoles at different location. Performing a series of -molecular dynamics simulations, we explore the structural properties -of tilted smectic phases as well as the effect of electrostatic -interactions. +molecule represented by three rigid GB particles with two point +dipoles. Performing a series of molecular dynamics simulations, we +explore the structural properties of tilted smectic phases as well +as the effect of electrostatic interactions. \section{\label{liquidCrystalSection:model}Model} @@ -182,21 +182,18 @@ ratio between \textit{end-to-end} well depth $\epsilon \begin{figure} \centering \includegraphics[width=\linewidth]{banana.eps} -\caption[]{} \label{LCFig:BananaMolecule} +\caption[Schematic representation of a typical banana shaped +molecule]{Schematic representation of a typical banana shaped +molecule.} \label{LCFig:BananaMolecule} \end{figure} \begin{figure} \centering -\includegraphics[width=\linewidth]{bananGB.eps} -\caption[]{} \label{LCFigure:BananaGB} -\end{figure} - -\begin{figure} -\centering \includegraphics[width=\linewidth]{gb_scheme.eps} -\caption[]{Schematic diagram showing definitions of the orientation -vectors for a pair of Gay-Berne molecules} -\label{LCFigure:GBScheme} +\caption[Schematic diagram showing definitions of the orientation +vectors for a pair of Gay-Berne molecules]{Schematic diagram showing +definitions of the orientation vectors for a pair of Gay-Berne +molecules} \label{LCFigure:GBScheme} \end{figure} To account for the permanent dipolar interactions, there should be @@ -209,6 +206,161 @@ where $\epsilon _{fs}$ is the permittivity of free spa \end{equation} where $\epsilon _{fs}$ is the permittivity of free space. -\section{\label{liquidCrystalSection:methods}Methods} +\section{Results and Discussion} -\section{\label{liquidCrystalSection:resultDiscussion}Results and Discussion} +A series of molecular dynamics simulations were perform to study the +phase behavior of banana shaped liquid crystals. In each simulation, +every banana shaped molecule has been represented by three GB +particles which is characterized by $\mu = 1,~ \nu = 2, +~\epsilon_{e}/\epsilon_{s} = 1/5$ and $\sigma_{e}/\sigma_{s} = 3$. +All of the simulations begin with same equilibrated isotropic +configuration where 1024 molecules without dipoles were confined in +a $160\times 160 \times 120$ box. After the dipolar interactions are +switched on, 2~ns NPTi cooling run with themostat of 2~ps and +barostat of 50~ps were used to equilibrate the system to desired +temperature and pressure. NPTi Production runs last for 40~ns with +time step of 20~fs. + +\subsection{Order Parameters} + +To investigate the phase structure of the model liquid crystal, we +calculated various order parameters and correlation functions. +Particulary, the $P_2$ order parameter allows us to estimate average +alignment along the director axis $Z$ which can be identified from +the largest eigen value obtained by diagonalizing the order +parameter tensor +\begin{equation} +\overleftrightarrow{\mathsf{Q}} = \frac{1}{N}\sum_i^N % + \begin{pmatrix} % + u_{ix}u_{ix}-\frac{1}{3} & u_{ix}u_{iy} & u_{ix}u_{iz} \\ + u_{iy}u_{ix} & u_{iy}u_{iy}-\frac{1}{3} & u_{iy}u_{iz} \\ + u_{iz}u_{ix} & u_{iz}u_{iy} & u_{iz}u_{iz}-\frac{1}{3} % + \end{pmatrix}, +\label{lipidEq:p2} +\end{equation} +where the $u_{i\alpha}$ is the $\alpha$ element of the unit vector +$\mathbf{\hat{u}}_i$, and the sum over $i$ averages over the whole +collection of unit vectors. The $P_2$ order parameter for uniaxial +phase is then simply given by +\begin{equation} +\langle P_2 \rangle = \frac{3}{2}\lambda_{\text{max}}. +\label{lipidEq:po3} +\end{equation} +%In addition to the $P_2$ order parameter, $ R_{2,2}^2$ order +%parameter for biaxial phase is introduced to describe the ordering +%in the plane orthogonal to the director by +%\begin{equation} +%R_{2,2}^2 = \frac{1}{4}\left\langle {(x_i \cdot X)^2 - (x_i \cdot +%Y)^2 - (y_i \cdot X)^2 + (y_i \cdot Y)^2 } \right\rangle +%\end{equation} +%where $X$, $Y$ and $Z$ are axis of the director frame. +The unit vector for the banana shaped molecule was defined by the +principle aixs of its middle GB particle. The $P_2$ order parameters +for the bent-core liquid crystal at different temperature is +summarized in Table~\ref{liquidCrystal:p2} which identifies a phase +transition temperature range. + +\begin{table} +\caption{LIQUID CRYSTAL STRUCTURAL PROPERTIES AS A FUNCTION OF +TEMPERATURE} \label{liquidCrystal:p2} +\begin{center} +\begin{tabular}{cccccc} +\hline +Temperature (K) & 420 & 440 & 460 & 480 & 600\\ +\hline +$\langle P_2\rangle$ & 0.984 & 0.982 & 0.975 & 0.967 & 0.067\\ +\hline +\end{tabular} +\end{center} +\end{table} + +\subsection{Structure Properties} + +The molecular organization obtained at temperature $T = 460K$ (below +transition temperature) is shown in Figure~\ref{LCFigure:snapshot}. +The diagonal view in Fig~\ref{LCFigure:snapshot}(a) shows the +stacking of the banana shaped molecules while the side view in n +Figure~\ref{LCFigure:snapshot}(b) demonstrates formation of a +chevron structure. The first peak of Radial distribution function +$g(r)$ in Fig.~\ref{LCFigure:gofrz}(a) shows the minimum distance +for two in plane banana shaped molecules is 4.9 \AA, while the +second split peak implies the biaxial packing. It is also important +to show the density correlation along the director which is given by +: +\begin{equation} +g(z) =\frac{1}{\pi R^{2} \rho}< \delta (z-z_{ij})>_{ij} +\end{equation}, +where $z_{ij} = r_{ij} \dot Z$ was measured in the director frame +and $R$ is the radius of the cylindrical sampling region. The +oscillation in density plot along the director in +Fig.~\ref{LCFigure:gofrz}(b) implies the existence of the layered +structure, and the peak at 27 \AA is attribute to the defect in the +system. + +\subsection{Rotational Invariants} + +As a useful set of correlation functions to describe +position-orientation correlation, rotation invariants were first +applied in a spherical symmetric system to study x-ray and light +scatting\cite{Blum1972}. Latterly, expansion of the orientation pair +correlation in terms of rotation invariant for molecules of +arbitrary shape was introduce by Stone\cite{Stone1978} and adopted +by other researchers in liquid crystal studies\cite{Berardi2003}. In +order to study the correlation between biaxiality and molecular +separation distance $r$, we calculate a rotational invariant +function $S_{22}^{220} (r)$, which is given by : +\begin{eqnarray} +S_{22}^{220} (r) & = & \frac{1}{{4\sqrt 5 }} \left< \delta (r - +r_{ij} )((\hat x_i \cdot \hat x_j )^2 - (\hat x_i \cdot \hat y_j +)^2 - (\hat y_i \cdot \hat x_j )^2 + (\hat y_i \cdot \hat y_j +)^2 ) \right. \notag \\ + & & \left. - 2(\hat x_i \cdot \hat y_j )(\hat y_i \cdot \hat x_j ) - +2(\hat x_i \cdot \hat x_j )(\hat y_i \cdot \hat y_j )) \right>. +\end{eqnarray} +The long range behavior of second rank orientational correlation +$S_{22}^{220} (r)$ in Fig~\ref{LCFigure:S22220} also confirm the +biaxiality of the system. + +\begin{figure} +\centering +\includegraphics[width=4.5in]{snapshot.eps} +\caption[Snapshot of the molecular organization in the layered phase +formed at temperature T = 460K and pressure P = 1 atm]{Snapshot of +the molecular organization in the layered phase formed at +temperature T = 460K and pressure P = 1 atm. (a) diagonal view; (b) +side view.} \label{LCFigure:snapshot} +\end{figure} + +\begin{figure} +\centering +\includegraphics[width=\linewidth]{gofr_gofz.eps} +\caption[Correlation Functions of a Bent-core Liquid Crystal System +at Temperature T = 460K and Pressure P = 10 atm]{Correlation +Functions of a Bent-core Liquid Crystal System at Temperature T = +460K and Pressure P = 10 atm. (a) radial correlation function +$g(r)$; and (b) density along the director $g(z)$.} +\label{LCFigure:gofrz} +\end{figure} + +\begin{figure} +\centering +\includegraphics[width=\linewidth]{s22_220.eps} +\caption[Average orientational correlation Correlation Functions of +a Bent-core Liquid Crystal System at Temperature T = 460K and +Pressure P = 10 atm]{Correlation Functions of a Bent-core Liquid +Crystal System at Temperature T = 460K and Pressure P = 10 atm. (a) +radial correlation function $g(r)$; and (b) density along the +director $g(z)$.} \label{LCFigure:S22220} +\end{figure} + +\section{Conclusion} + +We have presented a simple dipolar three-site GB model for banana +shaped molecules which are capable of forming smectic phases from +isotropic configuration. Various order parameters and correlation +functions were used to characterized the structural properties of +these smectic phase. However, the forming layered structure still +had some defects because of the mismatching between the layer +structure spacing and the shape of simulation box. This mismatching +can be broken by using NPTf integrator in further simulations. The +lack of detail in.