An Adsorption–Desorption-Controlled Surfactant on a Deforming Droplet

Abstract

The effects of a sorption-controlled, monolayer-forming surfactant on a drop deforming in an extensional flow are studied numerically. Scaling arguments are presented for drops of 1 cm and 1 μm, indicating the applicability of these results. For all simulations, when mass transfer is slow compared to surface convection, the insoluble limit is recovered; when mass transfer is rapid, the drop behavior is the same as that for a surfactant-free drop. For a surfactant which forms a monolayer, there is an upper bound to the surface concentration, Γ∞. The surface tension reduction diverges as the surface concentration Γ approaches this limit, strongly altering the hydrodynamics.The drop deformation is studied relative to a surfactant-free drop in terms of the capillary number, Ca, the ratio of characteristic viscous stresses to surface tension. In the insoluble limit, for Γ ⪡ Γ∞, droplets deform more than in the absence of surfactants at a given Ca and break-up at lower Ca. When stable drop shapes are attained, stagnant caps form at the drop tips. Finite surfactant mass transfer rates eliminate these caps and diminish the deformation.For Γ approaching Γ∞in the insoluble limit, interfaces are strongly stressed for perturbative surface concentration gradients; Γ remains nearly uniform throughout the deformation process. Deformations are reduced at a given Ca. When stable drop shapes are attained, the surface is completely stagnated. Marangoni stresses force the surface velocity to zero to keep Γ below its upper bound. For soluble surfactants, as mass transfer rates increase, the magnitude of these stresses diminishes. Deformations change nonmonotonically with mass transfer rates and are not bounded by the limiting clean interface and insoluble limits.The drop contribution to the volume averaged stress tensor Σ is also calculated. The axial component Σzzincreases with the drop length; the radial component Σrrincreases with the drop breadth. Since the deformation is strongly influenced by the surfactant concentration and the mass transfer rates, so too is Σ.