Vibrations of microemulsion droplets and vesicles with compressible surface layer

Abstract

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 equations for incompressible bulk fluids inside and outside the droplet. The dynamical equations for the interface are those obtained by Lebedev and Muratov @JETP 68, 1011 ~1989!# but with the improved continuity equation for the surface layer. Within the Helfrich’s concept of the interfacial elasticity and taking into account the compressibility of the surface layer, the exact equation is obtained for the frequencies of the droplet vibrations. The equation describes uniformly a broad region of frequencies from the lowest, almost 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 penetration depths of the corresponding waves. Our analysis corrects the previous results 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 incompressible layers. In the region corresponding to large penetration depths, a couple of modes exist with frequencies depending both on the surface elasticity and compressibility. In the limit of 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 numerical 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#