--- trunk/tengDissertation/LiquidCrystal.tex 2006/06/27 01:33:33 2892 +++ trunk/tengDissertation/LiquidCrystal.tex 2006/06/29 23:00:35 2909 @@ -2,44 +2,39 @@ Long range orientational order is one of the most fund \section{\label{liquidCrystalSection:introduction}Introduction} -Long range orientational order is one of the most fundamental -properties of liquid crystal mesophases. This orientational -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\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. +Rod-like (calamitic) and disk-like anisotropy liquid crystals have +been investigated 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 larger than the dispersion +interaction. In contrast, formation of one dimension lamellar +smectic phase in rod-like molecules with sufficiently long aliphatic +chains has been reported, as well as the segregation phenomena in +disk-like molecules\cite{McMillan1971}. Recently, banana-shaped or +bent-core liquid crystals have became one of the most active +research areas in mesogenic materials and 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 exhibited layered 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 discoveries 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.~\ref{LCFig:SMCP})\cite{Link1997}. -Recently, the banana-shaped or bent-core liquid crystal have became -one of the most active research areas in mesogenic materials and -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.~\ref{LCFig:SMCP}). - \begin{figure} \centering \includegraphics[width=\linewidth]{smcp.eps} @@ -65,28 +60,27 @@ limitation of time scale required for phase transition 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 and the +limitation of time scales 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}. +bodies with high shape anisotropies, have become the dominant models +in the field of liquid crystal phase behavior. Previous simulation +studies using a hard spherocylinder dimer model\cite{Camp1999} +produced nematic phases, while hard rod simulation studies +identified a direct transition to the biaxial nematic and other +possible liquid crystal phases\cite{Lansac2003}. Other anisotropic +models using the Gay-Berne(GB) potential, which produces +interactions that favor local alignment, give 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 permanent longitudinal dipole moments, which affect the phase -behavior considerably. A series of theoretical studies also drawn -equivalent conclusions. Monte Carlo studies of the GB potential with -fixed longitudinal dipoles (i.e. pointed along the principal axis of -rotation) were shown to enhance smectic phase +behavior considerably. Equivalent conclusions have also been drawn +from a series of theoretical studies. Monte Carlo studies of the GB +potential with fixed longitudinal dipoles (i.e. pointed along the +principal axis of rotation) were shown to enhance smectic phase stability~\cite{Berardi1996,Satoh1996}. Molecular simulation of GB ellipsoids with transverse dipoles at the terminus of the molecule also demonstrated that partial striped bilayer structures were @@ -99,7 +93,7 @@ In this chapter, we consider system consisting of bana bent-core molecules, could be modeled more accurately by incorporating electrostatic interaction. -In this chapter, we consider system consisting of banana-shaped +In this chapter, we consider a system consisting of banana-shaped 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 @@ -130,11 +124,11 @@ orientation of two molecules $i$ and $j$ separated by } \right] \label{LCEquation:gb} \end{equation} where $\hat u_i,\hat u_j$ are unit vectors specifying the -orientation of two molecules $i$ and $j$ separated by intermolecular -vector $r_{ij}$. $\hat r_{ij}$ is the unit vector along the -intermolecular vector. A schematic diagram of the orientation -vectors is shown in Fig.\ref{LCFigure:GBScheme}. The functional form -for $\sigma$ is given by +orientation of two ellipsoids $i$ and $j$ separated by +intermolecular vector $r_{ij}$. $\hat r_{ij}$ is the unit vector +along the inter-ellipsoid vector. A schematic diagram of the +orientation vectors is shown in Fig.\ref{LCFigure:GBScheme}. The +functional form for $\sigma$ is given by \begin{equation} \sigma (\hat u_i ,\hat u_i ,\hat r_{ij} ) = \sigma _0 \left[ {1 - \frac{\chi }{2}\left( {\frac{{(\hat r_{ij} \cdot \hat u_i + \hat @@ -186,16 +180,14 @@ molecule.} \label{LCFig:BananaMolecule} molecule]{Schematic representation of a typical banana shaped molecule.} \label{LCFig:BananaMolecule} \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]{Schematic diagram showing definitions of the orientation vectors for a pair of Gay-Berne -molecules} \label{LCFigure:GBScheme} +ellipsoids} \label{LCFigure:GBScheme} \end{figure} - To account for the permanent dipolar interactions, there should be an electrostatic interaction term of the form \begin{equation} @@ -218,7 +210,8 @@ temperature and pressure. 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. +temperature and pressure. NPTi Production runs last for 40~ns with +time step of 20~fs. \subsection{Order Parameters} @@ -226,8 +219,8 @@ the largest eigen value obtained by diagonalizing the 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 +the largest eigenvalue obtained by diagonalizing the order parameter +tensor \begin{equation} \overleftrightarrow{\mathsf{Q}} = \frac{1}{N}\sum_i^N % \begin{pmatrix} % @@ -273,29 +266,54 @@ $\langle P_2\rangle$ & 0.984 & 0.982 & 0.975 & 0.967 & \end{center} \end{table} -\subsection{Structure Properties} +\subsection{Structural 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 -: +chevron structure. The first peak of the radial distribution +function $g(r)$ in Fig.~\ref{LCFigure:gofrz}(a) shows that 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 +where $ z_{ij} = r_{ij} \cdot \hat 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 +structure, and the peak at 27 \AA is attributed to a 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 has been introduced 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} @@ -317,35 +335,28 @@ $g(r)$; and (b) density along the director $g(z)$.} \label{LCFigure:gofrz} \end{figure} -\subsection{Rotational Invariants} +\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]{Average orientational correlation Correlation +Functions of a Bent-core Liquid Crystal System at Temperature T = +460K and Pressure P = 10 atm.} \label{LCFigure:S22220} +\end{figure} -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} - -%\begin{equation} -%S_{00}^{221} (r) = - \frac{{\sqrt 3 }}{{\sqrt {10} }}\left\langle -%{\delta (r - r_{ij} )((\hat z_i \cdot \hat z_j )(\hat z_i \cdot -%\hat z_j \times \hat r_{ij} ))} \right\rangle -%\end{equation} - \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. +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 lack of flexibility in our model +due to the missing terminal chains could explain the fact that we +did not find evidence of chirality.