--- trunk/tengDissertation/LiquidCrystal.tex	2006/04/03 18:07:54	2685
+++ trunk/tengDissertation/LiquidCrystal.tex	2006/06/29 23:00:35	2909
@@ -2,8 +2,361 @@
 
 \section{\label{liquidCrystalSection:introduction}Introduction}
 
+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}.
+
+\begin{figure}
+\centering
+\includegraphics[width=\linewidth]{smcp.eps}
+\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\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,
+computer simulation can provide unique insight into molecular
+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
+bulk properties, such as rotational viscosity and flexoelectric
+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
+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
+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. 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
+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,
+it is clear that many liquid-crystal forming molecules, specially,
+bent-core molecules, could be modeled more accurately by
+incorporating electrostatic interaction.
+
+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
+as the effect of electrostatic interactions.
+
 \section{\label{liquidCrystalSection:model}Model}
 
-\section{\label{liquidCrystalSection:methods}Methods}
+A typical banana-shaped molecule consists of a rigid aromatic
+central bent unit with several rod-like wings which are held
+together by some linking units and terminal chains (see
+Fig.~\ref{LCFig:BananaMolecule}). In this work, each banana-shaped
+mesogen has been modeled as a rigid body consisting of three
+equivalent prolate ellipsoidal GB particles. The GB interaction
+potential used to mimic the apolar characteristics of liquid crystal
+molecules takes the familiar form of Lennard-Jones function with
+orientation and position dependent range ($\sigma$) and well depth
+($\epsilon$) parameters. The potential between a pair of three-site
+banana-shaped molecules $a$ and $b$ is given by
+\begin{equation}
+V_{ab}^{GB}  = \sum\limits_{i \in a,j \in b} {V_{ij}^{GB} }.
+\end{equation}
+Every site-site interaction can can be expressed as,
+\begin{equation}
+V_{ij}^{GB}  = 4\epsilon (\hat u_i ,\hat u_j ,\hat r_{ij} )\left[
+{\left( {\frac{{\sigma _0 }}{{r_{ij}  - \sigma (\hat u_i ,\hat u_j
+,\hat r_{ij} )}}} \right)^{12}  - \left( {\frac{{\sigma _0
+}}{{r_{ij}  - \sigma (\hat u_i ,\hat u_j ,\hat r_{ij} )}}} \right)^6
+} \right] \label{LCEquation:gb}
+\end{equation}
+where $\hat u_i,\hat u_j$ are unit vectors specifying the
+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
+r_{ij}  \cdot \hat u_j )^2 }}{{1 + \chi \hat u_i  \cdot \hat u_j }}
++ \frac{{(\hat r_{ij}  \cdot \hat u_i  - \hat r_{ij}  \cdot \hat u_j
+)^2 }}{{1 - \chi \hat u_i  \cdot \hat u_j }}} \right)} \right]^{ -
+\frac{1}{2}},
+\end{equation}
+where the aspect ratio of the particles is governed by shape
+anisotropy parameter
+\begin{equation}
+\chi  = \frac{{(\sigma _e /\sigma _s )^2  - 1}}{{(\sigma _e /\sigma
+_s )^2  + 1}}.
+\label{LCEquation:chi}
+\end{equation}
+Here, $\sigma_ s$ and $\sigma_{e}$ refer to the side-by-side breadth
+and the end-to-end length of the ellipsoid, respectively. The well
+depth parameters takes the form
+\begin{equation}
+\epsilon (\hat u_i ,\hat u_j ,\hat r_{ij} ) = \epsilon _0 \epsilon
+^v (\hat u_i ,\hat u_j )\epsilon '^\mu (\hat u_i ,\hat u_j ,\hat
+r_{ij} )
+\end{equation}
+where $\epsilon_{0}$ is a constant term and
+\begin{equation}
+\epsilon (\hat u_i ,\hat u_j ) = \frac{1}{{\sqrt {1 - \chi ^2 (\hat
+u_i  \cdot \hat u_j )^2 } }}
+\end{equation}
+and
+\begin{equation}
+\epsilon '(\hat u_i ,\hat u_j ,\hat r_{ij} ) = 1 - \frac{{\chi
+'}}{2}\left[ {\frac{{(\hat r_{ij}  \cdot \hat u_i  + \hat r_{ij}
+\cdot \hat u_j )^2 }}{{1 + \chi '\hat u_i  \cdot \hat u_j }} +
+\frac{{(\hat r_{ij}  \cdot \hat u_i  - \hat r_{ij}  \cdot \hat u_j
+)^2 }}{{1 - \chi '\hat u_i  \cdot \hat u_j }}} \right]
+\end{equation}
+where the well depth anisotropy parameter $\chi '$ depends on the
+ratio between \textit{end-to-end} well depth $\epsilon _e$ and
+\textit{side-by-side} well depth $\epsilon_s$,
+\begin{equation}
+\chi ' = \frac{{1 - (\epsilon _e /\epsilon _s )^{1/\mu} }}{{1 +
+(\epsilon _e /\epsilon _s )^{1/\mu} }}.
+\end{equation}
 
-\section{\label{liquidCrystalSection:resultDiscussion}Results and Discussion}
+\begin{figure}
+\centering
+\includegraphics[width=\linewidth]{banana.eps}
+\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]{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
+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}
+V_{ab}^{dp}  = \sum\limits_{i \in a,j \in b} {\frac{1}{{4\pi
+\epsilon _{fs} }}\left[ {\frac{{\mu _i  \cdot \mu _j }}{{r_{ij}^3 }}
+- \frac{{3\left( {\mu _i  \cdot r_{ij} } \right)\left( {\mu _i \cdot
+r_{ij} } \right)}}{{r_{ij}^5 }}} \right]}
+\end{equation}
+where $\epsilon _{fs}$ is the permittivity of free space.
+
+\section{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 eigenvalue 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{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 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}  \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
+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}
+\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]{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}
+
+\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 lack of flexibility in our model
+due to the missing terminal chains could explain the fact that we
+did not find evidence of chirality.