Coalescence of surfactant covered drops in extensional flows: Effects of the interfacial diffusivity

Abstract

Boundary integral simulations and scaling theory were employed to study the effects of insoluble surfactant surface diffusivity Ds and concentration Γ on the coalescence process of two equal-sized viscous drops. The drops underwent head-on collisions in a biaxial extensional flow, in the Stokes flow limit and low capillary numbers. The simulations were compared with the drainage time experiments of Yoon et al. [Phys. Fluids 19, 023102 (2007)10.1063/1.2409735] concerned with a polymeric system, polybutadiene (PBd) drops in a polydimethylsyloxane (PDMS) matrix, stabilized by block-copolymers acting as insoluble surfactants to explain the mechanism underneath their findings. An ad hoc equation of state, derived by mean field theory, specific for the block-copolymers in the experiments of Yoon et al., able to match the experimental surface tension data without fitting parameters, was used. We were able to reproduce the experimental drainage time data, although an additional attractive force, besides the usual van der Waals interactions, had to be introduced for high block-copolymer concentrations, probably as a result of the entropic attraction between the two facing dry brushes formed in the thin film between the two drops. According to simulations, the puzzling experimental drainage time transition for low surfactant concentrations, from high drainage time to low drainage time as Ca increases, was a consequence of the oscillating behavior of the minimum film thickness, which takes place for Marangoni numbers Ma < 5 and surface Peclet number Pes > 60. In this regard, a master curve was obtained for the scaled relative minimum film thickness attained during the oscillation as a function of Ma. This enabled to determine both the minimum value of the dimensionless attractive forces to avoid coalescence for each concentration studied and the range of Ma that favors early coalescence. The coalescence process was found very sensitive to Pes and for Pes O(100–1000) even trace amounts of surfactants can be as effective stabilizers as high surfactant concentrations. Moreover, for the polymeric system of interest, the range of Ds in which the drainage time changes from the saturation value to the clean interface value was computed as a function of the surfactant concentration. In the specific, for the PDMS/PBd system of interest the Ds range studied was O(10−12–10−5 cm2 s−1). Additionally, our scaling analysis further validates our simulations, also highlighting the effect on the drainage process of the different parameters, in particular, of the external pushing force, which is increased compared to a clean interface system, as Pes is increased or as the surfactant concentration is increased, because of the reduction in the interfacial mobility of the drop. Finally, our study suggests that matching simulations with four-roll mill drainage time experiments can be an effective method to determine block-copolymer surface diffusivity.