Critical dynamics of lateral and transversal phase separations in bilayer biomembranes and surfactants.

We consider bilayer biomembranes or surfactants made of two chemically incompatible amphiphile molecules, which may laterally or transversely phase separate into macrodomains, upon variation of some suitable parameter (temperature, lateral pressure, etc.). The purpose is an extensive study of the dynamics of both lateral and transverse phase separations, when the bilayer is suddenly cooled down from a high initial temperature towards a final one very close to the spinodal point. The critical dynamics are investigated through the partial dynamic structure factors of different species. Using a two-order parameter field theory, where the two fields are the composition fluctuations of one component in the leaflets of the bilayer, combined with an extended van Hove approach that is based on two coupled Langevin equations (with noise), we exactly compute these dynamic structure factors. We first find that the dynamics is governed by two time scales. The longest one, Tau, can be related to the thermal correlation length, Xi ~ Sigma|T - T(c)|(-1/2), by Tau ~ Xi(z), with the dynamic critical exponent z = 4, where Sigma is an atomic length scale, T the absolute temperature, and T(c) its critical value. The characteristic time Tau can be interpreted as the time required for the formation of the final macrophase domains. The second time scale is rather shorter, and can be viewed as the short time during which the unlike phospholipids execute local motion. Second, we demonstrate that the dynamic structure factors obey exact scaling laws, and depend on three lengths, namely the wavelength q(-1) (q is the wave vector modulus), the correlation length Xi, and a length scale R(t) ~ t(1/z) (z = 4) representing the size of macrophase domains at time t. Of course, the two lengths Xi and R(t) coincide at the final time Tau at which the bilayer reaches its final equilibrium state. Finally, the present work must be considered as a natural extension of our previously published one dealing with the study of lateral and transverse phase separations from a static point of view.