Undulations and Dynamic Structure Factor of Membranes

We investigate the dynamic structure factor $S(q,t)$ of membrane phases at large wave numbers $q$ which are sensitive to single membrane dynamics. Considering an ensemble of membrane plaquettes at random orientations, the effect of membrane thermal undulations is calculated. Their statistics is modeled by bending energy and standard hydrodynamic dispersion law. We predict a stretched exponential relaxation $S(q,t)\sim e^{-({\gamma }_{q}t{)}^{2/3}}$ with a relaxation rate ${\gamma }_q\propto {\kappa }^{-1\mathrm{}/2}{q}^3$ where $\kappa$ is the bending modulus. Our results are in good agreement with recent dynamic light scattering spectra from dilute sponge phases obtained by Freyssingeas, Roux, and Nallet [(unpublished)].