--- trunk/tengDissertation/LiquidCrystal.tex	2006/06/27 01:33:33	2892
+++ trunk/tengDissertation/LiquidCrystal.tex	2006/07/06 17:44:16	2924
@@ -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} %
@@ -255,7 +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.
 
@@ -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
+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 \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.
