The influence of inter-phase mass transfer on the drainage of partially-mobile liquid films between drops undergoing a constant interaction force

Abstract

The equations describing the drainage of a partially mobile liquid film separating two drops under a constant interaction force (Yiantsios and Davis, 1990; Chesters, 1991) are extended to include inter-phase solute transfer and the resulting Marangoni forces. In the limit of gentle interactions and small variations in solute concentration, a suitable transformation of variables reduces the number of parameters entering the equations to four: a transformed partition coefficient P, Peclet numbers in each phase and a Marangoni number Ma=Δ σ/ σa ′2, in which Δσ denotes the variation in interfacial tension corresponding to the difference in solute concentration between the phases and a′ the radius of the draining film, normalized with the equivalent radius of the drops ( a′≪1). Numerical solutions are presented for both positive and negative values of Ma (corresponding to solute transfer both to and from the drops) for fixed, physically pertinent values of the others parameters, including a large Peclet number for which the diffusion boundary layer within the drop is thin, thereby somewhat simplifying the equations to be solved. In accordance with experimental indications, the acceleration of drainage by Marangoni effects in the case of D→ C transfer is found to be immense, final drainage rates rising by two orders of magnitude for concentration differences of only a few percent. These effects are associated with an intensification of the dimple. For C→ D transfer, dimple formation is suppressed and initial drainage rates greatly reduced.