A criterion to assess the impact of confined volumes on surfactant transport to liquid-fluid interfaces

When dissolved surfactant adsorbs at an interface, the bulk concentration decreases. If the initial concentration is low or the interfacial area large, the concentration decrease can be significant, and the solution depleted. Although depletion is not a new phenomenon, properly accounting for it requires a global mass conservation constraint in addition to a mass transport model. The emergence of new applications involving adsorption in finite volumes and with large surface areas, including micro- and nanoscale droplet formation, has introduced new scenarios in which depletion can be significant but complex to analyze. The purpose of this paper is to develop simple criteria to allow practitioners in these applications to rapidly and easily assess the potential impact of depletion. We use a global mass balance to show that two dimensionless parameters fully describe the role of depletion in both equilibrium surface properties and timescales to reach equilibrium. The dimensionless parameters represent the potential mass lost to the interface, denoted ζ, and the surface activity of the surfactant, denoted f. Characteristic transport timescales are shown to be a function of the finite geometry. A scaling analysis is developed for the case of surfactant dissolved inside a spherical drop, and compared with that of a finite spherical shell. The analyses developed here lead to simple criteria that are useful even when the surfactant properties are not well characterized or a full transport analysis is difficult. The criteria can be generalized to adsorption at solid surfaces.