| 733 |
|
strength of the head group dipole moment |
| 734 |
|
($\mu$).\label{fig:phaseDiagram}} |
| 735 |
|
\end{figure} |
| 736 |
+ |
|
| 737 |
|
|
| 738 |
+ |
We have also computed orientational diffusion constants for the head |
| 739 |
+ |
groups from the relaxation of the second-order Legendre polynomial |
| 740 |
+ |
correlation function, |
| 741 |
+ |
\begin{eqnarray} |
| 742 |
+ |
C_{\ell}(t) & = & \langle P_{\ell}\left({\bf \mu}_{i}(t) \cdot {\bf |
| 743 |
+ |
\mu}_{i}(0) \right) \rangle \\ \\ |
| 744 |
+ |
& \approx & e^{-\ell(\ell + 1) \theta t}, |
| 745 |
+ |
\end{eqnarray} |
| 746 |
+ |
of the head group dipoles. In this last line, we have used a simple |
| 747 |
+ |
``Debye''-like model for the relaxation of the correlation function, |
| 748 |
+ |
specifically in the case when $\ell = 2$. The computed orientational |
| 749 |
+ |
diffusion constants are given in table \ref{tab:relaxation}. The |
| 750 |
+ |
notable feature we observe is that the orientational diffusion |
| 751 |
+ |
constant for the head group exhibits an order of magnitude decrease |
| 752 |
+ |
upon entering the rippled phase. Our orientational correlation times |
| 753 |
+ |
are substantially in excess of those provided by... |
| 754 |
+ |
|
| 755 |
+ |
|
| 756 |
|
\begin{table*} |
| 757 |
|
\begin{minipage}{\linewidth} |
| 758 |
|
\begin{center} |
| 759 |
< |
\caption{} |
| 759 |
> |
\caption{Rotational diffusion constants for the head groups |
| 760 |
> |
($\theta_h$) and molecular bodies ($\theta_b$) as a function of the |
| 761 |
> |
head-to-body width ratio. The orientational mobility of the head |
| 762 |
> |
groups experiences an {\it order of magnitude decrease} upon entering |
| 763 |
> |
the rippled phase, which suggests that the rippling is tied to a |
| 764 |
> |
freezing out of head group orientational freedom. Uncertainties in |
| 765 |
> |
the last digit are indicated by the values in parentheses.} |
| 766 |
|
\begin{tabular}{lcc} |
| 767 |
|
\hline |
| 768 |
< |
$\sigma_h / d$ & $\theta_h (1/fs)$ & $\theta_b (1/fs)$ \\ |
| 768 |
> |
$\sigma_h / d$ & $\theta_h (\mu s^{-1})$ & $\theta_b (1/fs)$ \\ |
| 769 |
|
\hline |
| 770 |
< |
1.20 & $2.06 \times 10^{-10} \pm 1.27 \times 10^{-12}$ & $1.75 \times 10^{-11} \pm 4.83 \times 10^{-13}$ \\ |
| 771 |
< |
1.28 & $1.79 \times 10^{-10} \pm 2.27 \times 10^{-12}$ & $5.52 \times 10^{-11} \pm 2.20 \times 10^{-12}$ \\ |
| 772 |
< |
1.35 & $2.51 \times 10^{-11} \pm 1.19 \times 10^{-12}$ & $1.95 \times 10^{-10} \pm 2.86 \times 10^{-12}$ \\ |
| 773 |
< |
1.41 & $2.25 \times 10^{-11} \pm 1.05 \times 10^{-12}$ & $2.42 \times 10^{-11} \pm 3.19 \times 10^{-12}$ \\ |
| 770 |
> |
1.20 & $0.206(1) $ & $0.0175(5) $ \\ |
| 771 |
> |
1.28 & $0.179(2) $ & $0.055(2) $ \\ |
| 772 |
> |
1.35 & $0.025(1) $ & $0.195(3) $ \\ |
| 773 |
> |
1.41 & $0.023(1) $ & $0.024(3) $ \\ |
| 774 |
|
\end{tabular} |
| 775 |
|
\label{tab:relaxation} |
| 776 |
|
\end{center} |
