Capillary waves at a gas-liquid phase interface

Abstract

A first-order perturbation analysis of capillary waves at a gas-liquid phase interface is carried out. We recognize in this analysis that a surfactant may be present that will be transferred to and from the phase interface as a function of time. The Newtonian surface fluid model is used to describe the surface source of momentum and the jump momentum balance; Levich's model for surface convection has been adopted for the surface source in the jump mass balance for surfactant. We do not assume equilibrium to be instantaneously established between the surface and the substrate in the sense that the surface concentration of surfactant is not required to be a function merely of the concentration of surfactant in the bulk phase at the interface. The results indicate that the surface shear viscosity and surface dilational viscosity are equally important. To date only Mann's data for this geometry present the wavelength and damping factor as functions of frequency. Unfortunately our analysis and all other analyses presently available do not appear to be applicable to his data, because he introduced a mechanical probe in the interface. If we assume that Mann's correction for the effect of his mechanical probe is sufficient, then there appears to be a question concerning his value for surface tension. The three models studied give a much better description of his data, if one uses a value for surface tension smaller than that he reported. If we assume both that Mann has fully accounted for the presence of his mechanical probe in reporting his experimental data and that the values of surface tension he gave are in error, then we can say that the analysis presented here for the damping of capillary waves is superior to any other analysis previously available. With these assumptions, we would be encouraged to say that the use of the Newtonian surface fluid model for surface stress and the Levich model for surface convection are apparently justified in this situation. It would also seem that equilibrium is not instantaneously established between the surface and the substrate, in the sense that the surface concentration of surfactant cannot be assumed to be a function merely of the concentration of surfactant in the bulk phase at the interface. However, these conclusions must still be considered tentative in view of the scatter in Man's experimental data.