General Theory for Capillary Waves and Surface Light Scattering

We present a general theory for capillary waves and surface quasi-elastic light scattering for an isotropic liquid interface with adsorbed surfactant. We first examine the validity of three constitutive models for isotropic interfaces in the Newtonian regime, namely those of Scriven, Goodrich, and Kramer. Scriven's constitutive model contains three interfacial constants: the equilibrium surface tension γ, the interfacial dilational viscosity ζs, and the interfacial shear viscosity ηs. Goodrich's model and Kramer's model contain an additional interfacial constant: the transverse viscosity ηN, which is the dissipative counterpart of γ. We find that while Scriven's model satisfies frame invariance, the transverse viscosity term proposed by Goodrich and Kramer violates frame invariance. We therefore conclude that ηN is unphysical and that the Scriven model represents the most general constitutive model for isotropic interfaces in the Newtonian regime. Using Scriven's model as a starting point, we calculate the stress boundary conditions for capillary waves and generalize our results to include various interfacial relaxation processes, including diffusive interchange of surfactants (both in the absence and presence of adsorption barriers) and surfactant chain reorientation and relaxation. We then derive the dispersion relation and the power spectrum for capillary waves satisfying these boundary conditions. We find that, in all cases, the transverse viscoelasticity of the interface is controlled to leading order by the unperturbed equilibrium surface tension γ0 rather than a complex surface tension γ* = γ + iωηN, which is widely used in the literature for analyzing surface light scattering results. We reanalyze surface light scattering results for a wide range of interfacial systems where unphysical results (e.g., negative dilational viscosities) have been reported in the literature and find that these unphysical results are removed when we reparametrize the transverse viscoelasticity using γ0 rather than γ*