--- trunk/mdRipple/mdripple.tex	2007/10/16 20:08:15	3263
+++ trunk/mdRipple/mdripple.tex	2007/10/18 21:07:03	3264
@@ -733,19 +733,44 @@ strength of the head group dipole moment
 strength of the head group dipole moment
 ($\mu$).\label{fig:phaseDiagram}}
 \end{figure}
+
 
+We have also computed orientational diffusion constants for the head
+groups from the relaxation of the second-order Legendre polynomial
+correlation function,
+\begin{eqnarray}
+C_{\ell}(t) & = & \langle P_{\ell}\left({\bf \mu}_{i}(t) \cdot {\bf
+\mu}_{i}(0) \right) \rangle  \\ \\
+& \approx & e^{-\ell(\ell + 1) \theta t},
+\end{eqnarray}
+of the head group dipoles.  In this last line, we have used a simple
+``Debye''-like model for the relaxation of the correlation function,
+specifically in the case when $\ell = 2$.   The computed orientational
+diffusion constants are given in table \ref{tab:relaxation}.  The
+notable feature we observe is that the orientational diffusion
+constant for the head group exhibits an order of magnitude decrease
+upon entering the rippled phase.  Our orientational correlation times
+are substantially in excess of those provided by...
+
+
 \begin{table*}
 \begin{minipage}{\linewidth}
 \begin{center}
-\caption{}
+\caption{Rotational diffusion constants for the head groups
+($\theta_h$) and molecular bodies ($\theta_b$) as a function of the
+head-to-body width ratio.  The orientational mobility of the head
+groups experiences an {\it order of magnitude decrease} upon entering
+the rippled phase, which suggests that the rippling is tied to a
+freezing out of head group orientational freedom.  Uncertainties in
+the last digit are indicated by the values in parentheses.}
 \begin{tabular}{lcc}
 \hline 
-$\sigma_h / d$ & $\theta_h (1/fs)$ & $\theta_b (1/fs)$ \\
+$\sigma_h / d$ & $\theta_h (\mu s^{-1})$ & $\theta_b (1/fs)$ \\
 \hline 
-1.20 & $2.06 \times 10^{-10} \pm 1.27 \times 10^{-12}$ & $1.75 \times 10^{-11} \pm 4.83 \times 10^{-13}$ \\
-1.28 & $1.79 \times 10^{-10} \pm 2.27 \times 10^{-12}$ & $5.52 \times 10^{-11} \pm 2.20 \times 10^{-12}$ \\
-1.35 & $2.51 \times 10^{-11} \pm 1.19 \times 10^{-12}$ & $1.95 \times 10^{-10} \pm 2.86 \times 10^{-12}$ \\
-1.41 & $2.25 \times 10^{-11} \pm 1.05 \times 10^{-12}$ & $2.42 \times 10^{-11} \pm 3.19 \times 10^{-12}$ \\
+1.20 & $0.206(1) $ & $0.0175(5) $ \\
+1.28 & $0.179(2) $ & $0.055(2)  $ \\
+1.35 & $0.025(1) $ & $0.195(3)  $ \\
+1.41 & $0.023(1) $ & $0.024(3)  $ \\
 \end{tabular}
 \label{tab:relaxation}
 \end{center}
