--- 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.
