--- trunk/tengDissertation/LiquidCrystal.tex	2006/06/21 19:46:01	2876
+++ trunk/tengDissertation/LiquidCrystal.tex	2006/07/17 20:01:05	2941
@@ -2,55 +2,52 @@ 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}
-\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\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
@@ -60,49 +57,47 @@ 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
-limitation of time scale required for phase transition and 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, 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,Orlandi2006}.
+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
-stability~\cite{Berardi1996,Satoh1996}. Molecular simulation of GB
+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
+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.
 
-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.
+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}
 
@@ -129,11 +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
@@ -181,23 +176,18 @@ 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
+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}
@@ -208,19 +198,20 @@ where $\epsilon _{fs}$ is the permittivity of free spa
 \end{equation}
 where $\epsilon _{fs}$ is the permittivity of free space.
 
-\section{Computational Methodology}
+\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 three GB particles
-which is characterized by $\mu = 1,~ \nu = 2,
+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.
+temperature and pressure. NPTi Production runs last for 40~ns with
+time step of 20~fs.
 
 \subsection{Order Parameters}
 
@@ -228,8 +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} %
@@ -237,7 +228,7 @@ parameter tensor
     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:po1}
+\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
@@ -247,54 +238,125 @@ In addition to the $P_2$ order parameter, $ R_{2,2}^2$
 \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.
+%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 are
+summarized in Table~\ref{liquidCrystal:p2} which identifies a phase
+transition temperature range.
 
-\subsection{Structure Properties}
+\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}
 
-It is more important to show the density correlation along the
-director
+\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
+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) =< \delta (z-z_{ij})>_{ij} / \pi R^{2} \rho
-\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.
+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 $\rm{\AA}$ 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{Blum1971}. 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 was introduce by Stone\cite{Stone1978} and adopted
-by other researchers in liquid crystal studies\cite{Berardi2000}.
-
+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\langle {\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 ) \\
-& & - 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\rangle
+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{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}
+\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}
 
-\section{Results and Conclusion}
-\label{sec:results and conclusion}
+\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}
 
-To investigate the molecular organization behavior due to different
-dipolar orientation and position with respect to the center of the
-molecule,
+\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.
