--- trunk/mattDisertation/collections/initalSimulations.tex	2003/09/08 20:58:12	752
+++ trunk/mattDisertation/collections/initalSimulations.tex	2003/09/09 16:35:31	753
@@ -10,31 +10,33 @@ side spacing of $\approx\mbox{8\AA}$ (Fig.~\ref{fig:25
 
 The 25 lipid system (scheme I) was initially started from an ordered
 configuration. The lipids were aligned in a 5x5 square with a side by
-side spacing of $\approx\mbox{8\AA}$ (Fig.~\ref{fig:25lipidInit}) The
+side spacing of $\sim \mbox{8\AA}$ (Fig.~\ref{fig:25lipidInit}) The
 box was then filled with water in an FCC lattice. The temperature was
-then equilibrated to 300~K by resampling the atomic velocitiues from a
+then equilibrated to 300~K by resampling the atomic velocities from a
 Maxwell-Boltzmann distribution. Once the temperature had been set, the
-system was allowed to progress for $\approx 20$~ns.
+system was allowed to progress for $\sim 20$~ns.
 
 \begin{figure}
-\parbox{\linewidth}{Here is the initial 25 lipid configuration}
-\caption[The initial configuration of scheme I]{The starting configuration of scheme I.}
-\label{fig:25lipidinit}
+\centering
+\includegraphics[width=\linewidth]{5x5-initial_pre.eps}
+\caption[The initial configuration of scheme I]{The starting configuration of scheme I after thermalization.}
+\label{fig:25lipidInit}
 \end{figure}
 
 The 50 lipid system (scheme II) was initially started from a randomly
-generarted configuration. A system box was initially filled with water
+generated configuration. A system box was initially filled with water
 in an FCC lattice. The 50 lipid molecules were then sequentially
-placed in the box with a random orientaion and position. If the lipid
+placed in the box with a random orientation and position. If the lipid
 did not overlap with a previously placed one, its position was
 accepted, and the next lipid was placed. Once all 50 lipids were
-positioned, any waters overlaping with the lipids were removed
+positioned, any waters overlapping with the lipids were removed
 (Fig.~\ref{fig:r50Init}. The system was then equilibrated to 300~K in
 the same manner as scheme I.
 
 \begin{figure}
-\parbox{\linewidth}{Here is the initial random 50 lipid configuration}
-\caption[The initial configuration of scheme II]{The starting configuration of scheme II.}
+\centering
+\includegraphics[width=\linewidth]{r50-initial_pre.eps}
+\caption[The initial configuration of scheme II]{The starting configuration of scheme II after thermalization}
 \label{fig:r50Init}
 \end{figure}
 
@@ -42,13 +44,112 @@ Scheme I was found to split into two leaves within the
 \label{subSec:initSimsResults}
 
 Scheme I was found to split into two leaves within the first
-5~ns. However, as the simulation progressed, it became appaarent that
+5~ns. However, as the simulation progressed, it became apparent that
 the system was frustrated in its confined box. After 15~ns, the two
-leaveswere still unable to form a bilayer. Instead, thew were only
+leaves were still unable to form a bilayer. Instead, thew were only
 able to form two skewed layers (Fig.~\ref{fig:25lipidFinal}).
 
-\begin{figure} 
-\parbox{\linewidth}{Here is the final 25 lipid configuration} 
-\caption[The final configuration of scheme I]{The final configuration of scheme I.}
+\begin{figure}
+\centering
+\includegraphics[width=\linewidth]{5x5-montage_pre.eps}
+\caption[The time evolution of scheme I]{The time evolution of scheme I.}
 \label{fig:25lipidFinal}
 \end{figure}
+
+Scheme II behaved similarly. Within the first 2~ns the system
+aggregated into lipid and water regions. After $\sim 5$~ns the lipid
+regions had become micelles. Over the course of the next 10~ns the
+micelles underwent little change. At 15~ns simulation time, we switched
+the ensemble to isobaric-isothermal, NPT. The new integrator had just
+been written, and allowed form isometric scaling of the simulation
+box. The new integrator allowed the system to relax more, and over the
+next 10~ns several of the micelles started to merge. However, no
+bilayer was formed. Fig.~\ref{fig:r50Final} shows the progression of
+the simulation.
+
+\begin{figure}
+\centering
+\includegraphics[width=\linewidth]{r50-montage_pre.eps}
+\caption[The time evolution of scheme II]{The time evolution of scheme II.}
+\label{fig:25lipidFinal}
+\end{figure}
+
+\begin{figure}
+\centering
+	\subfigure[The self correlation of the phospholipid head groups. $g_{\text{Head,Head}}(r)$ is on the top, the bottom chart is the $\langle \cos \omega \rangle_{\text{Head,Head}}(r)$.]{%
+		\label{fig:5x5HHCorr}%
+		\includegraphics[angle=-90,width=0.49\linewidth]{all5x5-HEAD-HEAD.epsi}%
+		}
+	\subfigure[The $g_{\text{CH}_2\text{,CH}_2}(r)$ for the tail chains]{%
+		\label{fig:5x5CCg}%
+		\includegraphics[angle=-90,width=0.49\linewidth]{all5x5-CH2-CH2.epsi}}
+	\subfigure%
+	[The pair correlations between the head groups and the water]{%
+		\label{fig:5x5HXCorr}%
+		\includegraphics[angle=-90,width=0.49\linewidth]{all5x5-HEAD-X.epsi}}
+\caption[Scheme I pair correlations]{The pair correlation functions for scheme I.}
+\label{fig:5x5PairCorrs}
+\end{figure}
+
+Structural information about the simulations were calculated via the
+equations in Sec.~\ref{sec:analysis}. Fig.~\ref{fig:5x5HHCorr} shows
+$g_{\text{Head,Head}}(r)$ and $\langle \cos \omega
+\rangle_{\text{Head,Head}}(r)$ for scheme II. The
+first peak in the $g(r)$ at 4.03~$\mbox{\AA}$ is the nearest neighbor
+separation of the heads of two lipids. This corresponds to a maximum
+in the $\langle \cos \omega \rangle(r)$ which means that the two
+neighbors on the same leaf have their dipoles aligned. The broad peak
+at 6.5~$\mbox{\AA}$ is the inter-surface spacing. Here, there is a
+corresponding anti-alignment in the angular correlation. This means
+that although the dipoles are aligned on the same monolayer, the
+dipoles will orient themselves to be anti-aligned on a opposite facing
+monolayer. With this information, the two peaks at 7.0~$\mbox{\AA}$
+and 7.4~$\mbox{\AA}$ are head groups on the same monolayer, and they
+are the second nearest neighbors to the head group. The peak is likely
+a split peak because of the small statistics of this system. Finally,
+the peak at 8.0~$\mbox{\AA}$ is likely the second nearest neighbor on
+the opposite monolayer because of the anti-alignment evident in the
+angular correlation.
+
+Fig.~\ref{fig:5x5CCg} shows the radial distribution function for the
+$\text{CH}_2$ unified atoms. The spacing of the atoms along the tail
+chains accounts for the regularly spaced sharp peaks, but the broad
+underlying peak with its maximum at 4.6~$\mbox{\AA}$ is the
+distribution of chain-chain distances between two lipids. The final
+figure, Fig.~\ref{fig:5x5HXCorr}, includes the correlation functions
+between the Head group and the SSD atoms. The peak in $g(r)$ at
+3.6~$\mbox{\AA}$ is the distance of closest approach between the two,
+and $\langle \cos \omega \rangle(r)$ shows that the SSD atoms will
+align their dipoles with the head groups at close distance. However,
+as one increases the distance, the SSD atoms are no longer aligned.
+
+\begin{figure}
+\centering
+	\subfigure[The self correlation of the phospholipid head groups.]{%
+		\label{fig:r50HHCorr}%
+		\includegraphics[angle=-90,width=0.49\linewidth]{r50-HEAD-HEAD.epsi}%
+		}
+	\subfigure%
+	[The pair correlations between the head groups and the water]{%
+		\label{fig:r50HXCorr}%
+		\includegraphics[angle=-90,width=0.49\linewidth]{r50-HEAD-X.epsi}%
+		}
+	\subfigure[The $g_{\text{CH}_2\text{,CH}_2}(r)$ for the tail chains]{%
+		\label{fig:r50CCg}%
+		\includegraphics[angle=-90,width=0.49\linewidth]{r50-CH2-CH2.epsi}}
+
+\caption[Scheme II pair correlations]{The pair correlation functions for scheme II}
+\label{fig:r50PairCorrs}
+\end{figure}
+
+Fig.~ \ref{fig:r50HHCorr}, \ref{fig:r50HXCorr}, and \ref{fig:r50CCg}
+are the same correlation functions for scheme II as for scheme I. What
+is most interesting to note, is the high degree of similarity between
+the correlation functions between systems. Even though scheme I formed
+a skewed bilayer and scheme II is still in the micelle stage, both
+have an inter-surface spacing of 6.5~$\mbox{\AA}$ with the same
+characteristic anti-alignment between surfaces. Not as surprising, is
+the consistency of the closest packing statistics between
+systems. Namely, a head-head closest approach distance of
+4~$\mbox{\AA}$, and similar findings for the chain-chain and
+head-water distributions as in scheme I.