--- trunk/tengDissertation/LiquidCrystal.tex 2006/06/29 23:00:35 2909 +++ trunk/tengDissertation/LiquidCrystal.tex 2006/07/17 20:01:05 2941 @@ -4,7 +4,7 @@ decades\cite{Huh2004}. Typically, these mesogens consi 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 +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 @@ -12,10 +12,10 @@ disk-like molecules\cite{McMillan1971}. Recently, bana 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 +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 +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 @@ -29,11 +29,11 @@ $\text{SmCP}$\cite{Link1997}. Of the most important di 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 +$\text{SmCP}$.\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}. +adjacent layer (see Fig.~\ref{LCFig:SMCP}).\cite{Link1997} \begin{figure} \centering @@ -47,7 +47,7 @@ molecular structure and intermolecular interaction\cit Many liquid crystal synthesis experiments suggest that the occurrence of polarity and chirality strongly relies on the -molecular structure and intermolecular interaction\cite{Reddy2006}. +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 @@ -57,22 +57,22 @@ smectic arrangements\cite{Cook2000, Lansac2001}, as we ordering and phase behavior, and hence improve the development of new experiments and theories. In the last two decades, all-atom models have been adopted to investigate the structural properties of -smectic arrangements\cite{Cook2000, Lansac2001}, as well as other +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 +coefficients.\cite{Cheung2002, Cheung2004} However, due to 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 +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, 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 +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}. +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 @@ -81,14 +81,14 @@ stability~\cite{Berardi1996,Satoh1996}. Molecular simu 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 +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 -developed from the smectic phase ~\cite{Berardi1996}. More +developed from the smectic phase.~\cite{Berardi1996} More significant effects have been shown by including multiple electrostatic moments. Adding longitudinal point quadrupole moments to rod-shaped GB mesogens, Withers \textit{et al} induced tilted -smectic behaviour in the molecular system~\cite{Withers2003}. Thus, +smectic behaviour in the molecular system.~\cite{Withers2003} Thus, it is clear that many liquid-crystal forming molecules, specially, bent-core molecules, could be modeled more accurately by incorporating electrostatic interaction. @@ -200,7 +200,7 @@ A series of molecular dynamics simulations were perfor \section{Results and Discussion} -A series of molecular dynamics simulations were perform to study the +A series of molecular dynamics simulations were performed 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, @@ -248,7 +248,7 @@ for the bent-core liquid crystal at different temperat %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 +for the bent-core liquid crystal at different temperature are summarized in Table~\ref{liquidCrystal:p2} which identifies a phase transition temperature range. @@ -271,7 +271,7 @@ stacking of the banana shaped molecules while the side 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 +stacking of the banana shaped molecules while the side view in Figure~\ref{LCFigure:snapshot}(b) demonstrates formation of a chevron structure. The first peak of the radial distribution function $g(r)$ in Fig.~\ref{LCFigure:gofrz}(a) shows that the @@ -280,13 +280,13 @@ g(z) =\frac{1}{\pi R^{2} \rho}< \delta (z-z_{ij})>_{ij 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}, +g(z) =\frac{1}{\pi R^{2} \rho}< \delta (z-z_{ij})>_{ij}, +\end{equation} 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 attributed to a defect in the +structure, and the peak at 27 $\rm{\AA}$ is attributed to a defect in the system. \subsection{Rotational Invariants} @@ -294,11 +294,11 @@ scatting\cite{Blum1972}. Latterly, expansion of the or 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 +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 +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 : @@ -357,6 +357,6 @@ extensively\cite{Pelzl1999}. The lack of flexibility i 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 +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.