733 |
|
strength of the head group dipole moment |
734 |
|
($\mu$).\label{fig:phaseDiagram}} |
735 |
|
\end{figure} |
736 |
+ |
|
737 |
+ |
We have computed translational diffusion coefficients for lipid |
738 |
+ |
molecules from the mean square displacement, |
739 |
+ |
\begin{eqnarray} |
740 |
+ |
\langle {|\left({\bf r}_{i}(t) - {\bt r}_{i}(0) \right)|}^2 \rangle \\ \\ |
741 |
+ |
& = & 6Dt |
742 |
+ |
\end{eqnarray} |
743 |
+ |
of the lipid bodies. The values of the translational diffusion |
744 |
+ |
coefficient for different head-to-tail size ratio are shown in table |
745 |
+ |
\ref{tab:relaxation}. |
746 |
+ |
|
747 |
+ |
We have also computed orientational diffusion constants for the head |
748 |
+ |
groups from the relaxation of the second-order Legendre polynomial |
749 |
+ |
correlation function, |
750 |
+ |
\begin{eqnarray} |
751 |
+ |
C_{\ell}(t) & = & \langle P_{\ell}\left({\bf \mu}_{i}(t) \cdot {\bf |
752 |
+ |
\mu}_{i}(0) \right) \rangle \\ \\ |
753 |
+ |
& \approx & e^{-\ell(\ell + 1) \theta t}, |
754 |
+ |
\end{eqnarray} |
755 |
+ |
of the head group dipoles. In this last line, we have used a simple |
756 |
+ |
``Debye''-like model for the relaxation of the correlation function, |
757 |
+ |
specifically in the case when $\ell = 2$. The computed orientational |
758 |
+ |
diffusion constants are given in table \ref{tab:relaxation}. The |
759 |
+ |
notable feature we observe is that the orientational diffusion |
760 |
+ |
constant for the head group exhibits an order of magnitude decrease |
761 |
+ |
upon entering the rippled phase. Our orientational correlation times |
762 |
+ |
are substantially in excess of those provided by... |
763 |
+ |
|
764 |
|
|
765 |
|
\begin{table*} |
766 |
|
\begin{minipage}{\linewidth} |
767 |
|
\begin{center} |
768 |
< |
\caption{} |
769 |
< |
\begin{tabular}{lcc} |
768 |
> |
\caption{Rotational diffusion constants for the head groups |
769 |
> |
($\theta_h$) and molecular bodies ($\theta_b$) as well as the |
770 |
> |
translational diffusion coefficients for the molecule as a function of |
771 |
> |
the head-to-body width ratio. The orientational mobility of the head |
772 |
> |
groups experiences an {\it order of magnitude decrease} upon entering |
773 |
> |
the rippled phase, which suggests that the rippling is tied to a |
774 |
> |
freezing out of head group orientational freedom. Uncertainties in |
775 |
> |
the last digit are indicated by the values in parentheses.} |
776 |
> |
\begin{tabular}{lccc} |
777 |
|
\hline |
778 |
< |
$\sigma_h / d$ & $\theta_h (1/fs)$ & $\theta_b (1/fs)$ \\ |
778 |
> |
$\sigma_h / d$ & $\theta_h (\mu s^{-1})$ & $\theta_b (1/fs)$ & $D ( |
779 |
> |
\times 10^{-11} m^2 s^{-1} \\ |
780 |
|
\hline |
781 |
< |
1.20 & $2.06 \times 10^{-10} \pm 1.27 \times 10^{-12}$ & $1.75 \times 10^{-11} \pm 4.83 \times 10^{-13}$ \\ |
782 |
< |
1.28 & $1.79 \times 10^{-10} \pm 2.27 \times 10^{-12}$ & $5.52 \times 10^{-11} \pm 2.20 \times 10^{-12}$ \\ |
783 |
< |
1.35 & $2.51 \times 10^{-11} \pm 1.19 \times 10^{-12}$ & $1.95 \times 10^{-10} \pm 2.86 \times 10^{-12}$ \\ |
784 |
< |
1.41 & $2.25 \times 10^{-11} \pm 1.05 \times 10^{-12}$ & $2.42 \times 10^{-11} \pm 3.19 \times 10^{-12}$ \\ |
781 |
> |
1.20 & $0.206(1) $ & $0.0175(5) $ & $0.43(1)$ \\ |
782 |
> |
1.28 & $0.179(2) $ & $0.055(2) $ & $5.91(3)$ \\ |
783 |
> |
1.35 & $0.025(1) $ & $0.195(3) $ & $3.42(1)$ \\ |
784 |
> |
1.41 & $0.023(1) $ & $0.024(3) $ & $7.16(1)$ \\ |
785 |
|
\end{tabular} |
786 |
|
\label{tab:relaxation} |
787 |
|
\end{center} |