--- trunk/tengDissertation/LiquidCrystal.tex	2006/06/29 23:00:35	2909
+++ trunk/tengDissertation/LiquidCrystal.tex	2006/07/17 15:28:44	2938
@@ -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}