Near-contact motion of surfactant-covered spherical drops

Abstract

A lubrication analysis is presented for the near-contact axisymmetric motion of spherical drops covered with an insoluble non-diffusing surfactant. Detailed results are presented for the surfactant distribution, the interfacial velocity, and the gap width between the drop surfaces. The effect of surfactant is characterized by a dimensionless force parameter: the external force normalized by Marangoni stresses. Critical values of the force parameter have been established for drop coalescence and separation. Surfactant-covered drops are stable to rapid coalescence for external forces less than 4πkTac₀, where c₀ is the surfactant concentration at the edge of the near-contact region and a is the reduced drop radius.

For subcritical forces, the behaviour of surfactant-covered drops is described by two time scales: a fast time scale characteristic of near-contact motion between drops with clean interfaces and a slow time scale associated with rigid particles. The surfactant distribution evolves on the short time scale until Marangoni stresses approximately balance the external force. Supercritical values of the external force cannot be balanced; coalescence and separation occur on the fast time scale. The coalescence time normalized by the result for drops with clean interfaces is independent of the viscosity ratio and initial gap width.

Under subcritical force conditions, a universal long-time behaviour is attained on the slow time scale. At long times, the surfactant distribution scales with the near-contact region and the surface velocity is directed inward which impedes the drop approach and accelerates their separation compared to rigid particles. For drops pressed together with a sufficiently large subcritical force, a shrinking surfactant-free clean spot forms.

Surfactant-covered drops exhibit an elastic response to unsteady external forces because of energy stored in the surfactant distribution.