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Although the time scale governing diffusion-limited transport of soluble species from solution onto a planar interface is well understood, the time scale governing transport onto a spherical interface is not. The time scales that have been proposed in the literature for spherical interfaces do not capture the correct asymptotic behavior for increasing bubble radius and do not capture previously reported experimental observations of the effect of concentration. This paper develops a diffusion-limited time scale that is dependent on an intrinsic length scale termed the spherical depletion depth. The time scale is determined by considering a specific example of diffusion-limited transport of surfactant species to a water-air interface and is verified using numerical simulations and experiments. This newly derived diffusion time scale will have a significant impact on our understanding of fundamental phenomena at spherical fluid-fluid and fluid-solid interfaces, especially those involving micrometer and nanometer length scales.
Alvarez et al. (Tue,) studied this question.