--- trunk/tengDissertation/LiquidCrystal.tex	2006/06/27 02:42:30	2895
+++ trunk/tengDissertation/LiquidCrystal.tex	2006/07/17 20:01:05	2941
@@ -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}
@@ -52,7 +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
@@ -62,44 +57,43 @@ 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}.
+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
+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}
@@ -208,7 +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,
@@ -227,8 +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} %
@@ -256,7 +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.
 
@@ -274,27 +266,27 @@ $\langle P_2\rangle$ & 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
+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 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
+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 attribute to the defect in the
+structure, and the peak at 27 $\rm{\AA}$ is attributed to a defect in the
 system.
 
 \subsection{Rotational Invariants}
@@ -302,13 +294,14 @@ 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 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 :
+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
@@ -347,10 +340,9 @@ Pressure P = 10 atm]{Correlation Functions of a Bent-c
 \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}
+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}
@@ -363,4 +355,8 @@ lack of detail in.
 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
-lack of detail in.
+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.