Vibrations of microemulsion droplets and vesicles with compressible surface layer

The surface vibration spectra of Liquid droplets with flexible interfaces, like microemulsion droplets or vesicles, are studied. As distinct from the previous theories, we proceed with exact solutions of hydrodynamic equation s for incompressible bulk fluids inside and outside the droplet. The dynami cal equations for the interface are those obtained by Lebedev and Muratov [ JETP 68, 1011 (1989)] but with the improved continuity equation for the sur face layer. Within the Helfrich's concept of the interfacial elasticity and taking into account the compressibility of the surface layer, the exact eq uation is obtained for the frequencies of the droplet vibrations. The equat ion describes uniformly a broad region of frequencies from the lowest, almo st purely relaxation modes, up to the modes determined mainly by the change of the area per molecule of the layer. The dispersion laws for some of the modes are obtained analytically in the Limits of large and small penetrati on depths of the corresponding waves. Our analysis corrects the previous re sults concerning the relaxation modes, the capillary wave frequency and the frequency of the mode connected with the fluctuations of molecules in the surface layer. An additional mode of this kind is obtained for almost incom pressible layers. In the region corresponding to large penetration depths, a couple of modes exist with frequencies depending both on the surface elas ticity and compressibility. In the limit df infinite compressibility of the layer, the lower of the two modes disappears. The conditions necessary for the existence of all the modes were specified. Some representative numeric al solutions of the obtained equation are presented as depending on various values of the model parameters including those for realistic microemulsion systems. [S1063-651X(98)06812-3].