Effects of small concentration surfactants on the coalescence of viscous drops

Boundary integral simulations, employing Dai and Leal's code [Phys. Fluids \textbf{20, }040802 (2008)], are used to study the effects of small concentrations of insoluble surfactants C$_{s}$ on head-on collisions of two equal-sized viscous drops in a matrix of equal viscosity in a hyperbolic extensional flow, for low Reynolds numbers. The parameters were chosen to mimic the experiments of Yoon et al. [Phys. Fluids \textbf{19}, 023102 (2007)], which were performed with polymeric drops stabilized by block-copolymer insoluble surfactants in a polymer matrix, where both fluids acted as Newtonian viscous fluids. In these experiments a discontinuous transition in the coalescence process was found for low C$_{s}$ as the Capillary number Ca was increased. Thus, for Ca$>$Ca$_{c}$ a minimum surfactant concentration exists below which the system behaves like a clean interface system. Here, by varying C$_{s}$, i.e. the Marangoni number Ma, and the surface diffusivity, i.e. the interfacial Peclet number Pe$_{s}$, we explain the origin of the transition and its dependence on the parameters. The transition occurs if Pe$_{s}>$Pe$_{sc}$, Ca$>$Ca$_{c}$ and Ma$<$Ma$_{c}$.