Regular perturbation analysis of small amplitude oscillatory dilatation of an interface in a capillary pressure tensiometer

Abstract

The dilatational rheology of complex fluid-fluid interfaces is linked to the stability and bulk rheology of emulsions and foams. Dilatational rheology can be measured by pinning a bubble or droplet at the tip of a capillary, subjecting the interface shape to small amplitude oscillations, and recording the resulting pressure jump across the interface. The complex dilatational modulus is obtained by differentiating the interfacial stress with respect to the area change of the interface. In this paper, we perform a regular asymptotic expansion to analyze the interface response in pressure-controlled capillary pressure tensiometers to determine the dilatational modulus as a function of the measured radius of curvature. We show that small amplitude oscillatory dilation of a spherical bubble is neither stress nor strain rate controlled. The resulting dilatational modulus contains contributions from both surface tension effects as well as extra stresses. Depending on the specifics of the interface, each contribution can be a function of the dilation rate and the radius of the bubble. Thus, the radius of curvature can be used as a control parameter with which to separate surface tension and interfacial rheology effects, aiding in validation of interfacial constitutive models. We examine the limits of validity of the small amplitude assumption and provide guidelines for determining the operating limits of a capillary pressure tensiometer. Finally, we compare several existing devices, including a microtensiometer we developed previously that oscillates the pressure inside small (R ∼ 10 μm) droplets.