Gravity-induced collisions of spherical drops covered with compressible surfactant

Abstract

Gravity-induced collisions of two spherical drops covered with an insoluble surfactant at low Reynolds numbers are considered. Unlike in previous collision studies, the present work accounts for nonlinear coupling between the surfactant distribution and drop hydrodynamics by solving the full unsteady convective–diffusion equation for the surfactant transport. Our method includes high-order three-dimensional multipole expansions for hydrodynamics and a Galerkin-type approach for the surfactant transport with implicit marching. The efficiency of the algorithm allows for calculating thousands of trajectories to very close contact and determining the collision efficiency (related to the critical initial horizontal offset) by trial and error. The solution is valid for arbitrary surface Péclet (Pes) and Marangoni (Ma) numbers and sets limitations on approximations used in prior work for collision-efficiency calculations. Two limiting cases are observed: at smallPesor largeMa, the variation in surfactant coverage is small, and the results for the incompressible surfactant model are recovered, while for largePesand smallMa, the collision efficiency approaches the clean-interface value. For moderate drop-size ratios (radius ratiok≤ 0.5), the results generally fall between these limits. At larger size ratios, however, the collision efficiency may even exceed the geometrical Smoluchowski limit for both drops and bubbles. Moreover, with even moderate redistribution of the surfactant, equal-sized drops can move relative to one another and collide. These novel effects do not exist for clean drops or drops covered with an incompressible surfactant, and they are due to the nonlinear coupling between surfactant dynamics and flow. This surfactant-enhanced coalescence takes place, for example, in a physical system of air bubbles in water if the surfactant surface concentration is dilute (Γ ≈ 1×10−9mol m−2, much smaller than the typical maximum-packing value of 10−5−10−6mol m−2).