Membrane dynamics and structure factor

We investigate the short time and short scale dynamics of membrane phases. The dynamics of the two point displacement correlator of the membrane is shown to be dominated by the transverse mean-square displacement of a tagged membrane point and to vary with time t as ∼t^2/3. As a result, the structure factor of oriented lamellar phases at nearly perpendicular orientations (mean membrane normal parallel to the wavevector k→) decays as a stretched exponential S(k→,t) ∼ e^-(Γ_kzt)^2/3 with Γ_kz ∼ (T/K)^1/2 (T/η)k_z³. For isotropic systems, angular averaging retains the stretched exponential decay, S(k,t) ∼ e^-(Δ_kt)^x, but with α > 2/3 and no longer universal. We also find Γ_k ∼ k^z with z < 3. Agreement with recent experimental results is elucidated.