title: Curvature Effects in the Analysis of Pendant Bubble Data: Comparison of Numerical Solutions, Asymptotic Arguments, and Data.
abstract: The pendant bubble method is commonly used to measure the evolution of the surface tension of surfactant solutions. Initially, the bubble interface is free of adsorbed surfactant. As time progresses, surfactant diffuses to the interface, adsorbs, and reduces the surface tension. The surface tension is assumed to be in equilibrium with the instantaneous surface concentration. Therefore, surface tension data are analyzed in terms of interfacial thermodynamics and mass transfer models in order to infer the mechanisms which determine the surfactant transport. Diffusion from the bulk solution to the bubble can be approximated as diffusion to a spherical interface. Approximating this process as diffusion to a plane introduces significant errors into the data analysis. Mass transfer to a sphere differs from that to a plane; the equilibration of the spherical interface is more rapid simply because of geometry. The failure to account for this effect in the interpretation of pendant bubble data can lead to serious errors in the transport coefficients for the surfactants. In the diffusion-controlled limit, surfactant diffuses to the sublayer immediately adjacent to the interface and adsorbs in local equilibrium according to the adsorption isotherm. There is a closed-form solution for Fick's law describing adsorption to a sphere in an infinite solution which reduces to the Ward and Tordai solution when the bubble radius is large. This equation, along with the adsorption isotherm relating the surface concentration and the sublayer concentration, must be solved numerically in order to solve for the time evolution of the surface concentration. At early times, the adsorption isotherm can be expanded about the clean interface state. At long times, small departures from the equilibrium state can be assumed. In these limits, asymptotic expansions can be obtained. The short- and long-time expansions are found in this study for adsorption to a sphere and compared to those obtained previously for adsorption to a planar interface. In particular, the long-time asymptote for adsorption to a sphere is proportional to t(-3/2); this asymptote differs significantly from that for adsorption to a plane, which goes as t(-1/2). The full solution for adsorption to a sphere is compared to the Ward and Tordai solution for adsorption to a planar interface. From a comparison of the full solutions, it is established that curvature cannot be neglected unless the ratio of the adsorption depth to the bubble radius is negligible. This ratio can be calculated a priori from equilibrium isotherm parameters. Using constants which describe the surfactant C(12)E(8), for which curvature plays a strong role in the surfactant adsorption dynamics, the short- and long-time solutions for adsorption to the interface are compared to the full solutions and to dynamic surface tension data to infer the range of validity of the approximations. Copyright 2001 Academic Press.
authors.affiliation: Department of Chemical Engineering, The Johns Hopkins University, Baltimore, Maryland, 21218-2689
authors.forename: James K.
authors.initials: JK
authors.lastname: Ferri
authors.forename: Shi Yow
authors.initials: SY
authors.lastname: Lin
authors.forename: Kathleen J.
authors.initials: KJ
authors.lastname: Stebe