--- trunk/tengDissertation/LiquidCrystal.tex 2006/06/04 20:18:07 2786 +++ trunk/tengDissertation/LiquidCrystal.tex 2006/06/27 03:06:49 2896 @@ -38,13 +38,15 @@ Fig.~\cite{LCFig:SMCP}). 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} @@ -76,7 +78,7 @@ of bent-core molecules\cite{Memmer2002,Orlandi2006}. 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}. +of bent-core molecules\cite{Memmer2002}. Experimental studies by Levelut {\it et al.}~\cite{Levelut1981} revealed that terminal cyano or nitro groups usually induce @@ -98,11 +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} @@ -181,21 +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 @@ -208,6 +206,165 @@ 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 +role of terminal chain in controlling transition temperatures and +the type of mesophase formed have been studied +extensively\cite{Pelzl1999}. The the lack of flexibility in our +model due to the missing terminal chains could explained the fact +that we did not find evidence of chirality.