General Theory for Capillary Waves and Surface Light Scattering

Abstract

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*.

Contact mechanics for randomly rough surfaces

“…The surface roughness power spectrum due to capillary waves is of the form (see Appendix A) [16,17,18] …”

“…[9] the "diffusion equation" (17) was solved analytically with the boundary conditions P (0, ζ) = 0 and P (σ Y , ζ) = 0. Qualitative we may state that with increasing magnification the contact area diffuses over the σ = 0 boundary into no-contact, and over the σ = σ Y boundary into plastic contact.…”

Dynamics at the air-water interface revealed by evanescent wave light scattering

“…In fact, in the phenomenological approach applied to SQELS [8] a large number of parameters must be fitted. Although the surface viscoelastic moduli can be extracted from the analysis of the spectrum P (ω) or of the correlation function, Cicuta [9] has shown that reliable fits can be obtained only for a limited range of viscoelastic moduli [10]: for large value of dilatational modulus (e.g. at high polymer interface concentration) no fitting procedure is stable.…”

“…Advances in the understanding of capillary wave phenomena by SQELS investigations were reviewed in Langevin's monograph [1], by Earnshaw [8] and Cicuta [9]. The surface tension γ, the dilatational modulus , and the dilatational viscosity can be extracted by using a dispersion relation, which describes both the dilatational (inplane) and transverse waves [10]. The transverse contribution strongly dominates the intensity in SQELS.…”

Surface fluctuations of liquids confined on flat and patterned solid substrates

“…Rheological properties can also be inferred from the Brownian motion of tracers added to a liquid, which is the principle of micro-rheology techniques [2]. Other experiments use the measurement of the fluctuations of free surfaces, which are corrugated owing to thermal energy [3][4][5][6][7][8]. The way a surface fluctuates not only reveals surface tension effects, but also the bulk rheological properties of the medium underneath.…”