We study numerically the dynamics of deformable drops in the presence of surfactant species both on the drop-matrix interfaces and in the bulk fluids using a novel 3D adaptive finite-element method. The method is based on unstructured adaptive triangulated and tetrahedral meshes that discretize the interfaces and the bulk respectively, and on an efficient parallelization of the numerical solvers. We use this method to investigate the effects of surfactants on drop-drop interactions in shear flow. The simulations account for surfactant effects through a nonlinear Langmuir equation of state and through adsorption/desorption laws describing the transport between bulk and interface. Van der Waals forces responsible for coalescence are included. For clean drops (no surfactant), our simulations confirm (for the first time to our knowledge) a well known theoretical result  for the dependence of the critical capillary number-below which coalescence occurs-on the drop radius with an exponent -4/9. Our results reveal a non-monotonic dependence of the critical capillary number Cac on the surface coverage of surfactant. Marangoni stresses prevent drop approach thus decreasing Cac with respect to the clean-drop case. However, at large coverages close to the maximum packing of surfactant molecules, surfactant redistribution is prohibited (the surfactant is nearly incompressible) and thus the effect of Marangoni stresses is weakened, leading to an increase of Cac. In some cases, Cac at high coverages is even larger than in the clean-drop case: surfactant near-incompressibility hinders drop deformation and thus coalescence can occur at higher capillary number. Finally, our results also reveal a non-monotonic dependence of Cac on surfactant solubility in the bulk. At moderate surfactant concentration, diffusion in the bulk decreases surfactant redistribution on the interface and thus weakens Marangoni stresses resulting in higher Cac than in the insoluble case. However, when the surfactant bulk concentration is large, high adsorption fluxes maintain a higher surface concentration in equilibrium than for the insoluble case, thus resulting in larger drop deformation and in lower Cac.
Institute for Mathematics and Its Applications>IMA Preprints Series
Zhou, H.; Cristini, V.; Macosko, C.W..
Numerical simulation of deformable drops with soluble surfactant: Pair interactions and coalescence in shear flow.
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