A hybrid lattice Boltzmann and finite difference method for droplet dynamics with insoluble surfactants

Droplet dynamics in microfluidic applications is significantly influenced by surfactants. It remains a research challenge to model and simulate droplet behavior including deformation, breakup and coalescence, especially in the confined microfluidic environment. Here we propose a hybrid method to simulate interfacial flows with insoluble surfactants. The immiscible two-phase flow is solved by an improved lattice Boltzmann color-gradient model that incorporates a Marangoni stress resulting from non-uniform interfacial tension, while the convection-diffusion equation which describes the evolution of surfactant concentration in the entire fluid domain is solved by a finite difference method. The lattice Boltzmann and finite difference simulations are coupled through an equation of state, which describes how surfactant concentration influences interfacial tension. Our method is first validated for the surfactant-laden droplet deformation in a three- dimensional (3D) extensional flow and a 2D shear flow, and then applied to investigate the effect of surfactants on droplet dynamics in a 3D shear flow. Numerical results show that, at low capillary numbers, surfactants increase droplet deformation, due to reduced interfacial tension by the average surfactant concentration, and non-uniform effects from non-uniform capillary pressure and Marangoni stresses. The role of surfactants on critical capillary number (Cacr) of droplet breakup is investigated for various confinements (defined as the ratio of droplet diameter to wall separation) and Reynolds numbers. For clean droplets, Cacr first decreases and then increases with confinement, and the minimum value of Cacr is reached at the confinement of 0.5; for surfactant-laden droplets, Cacr exhibits the same variation in trend for the confinements lower than 0.7, but for higher confinements, Cacr is almost a constant. The presence of surfactants decreases Cacr for each confinement, and the decrease is also attributed to the reduction in average interfacial tension and non-uniform effects, which are found to prevent droplet breakup at low confinements but promote breakup at high confinements. In either clean or surfactant-laden cases, Cacr first keeps almost unchanged and then decreases with increasing Reynolds number, and a higher confinement or Reynolds number favors ternary breakup. Finally, we study the collision of two equal-sized droplets in a shear flow in both surfactant-free and surfactant-contaminated systems with the same effective capillary numbers. It is identified that the non-uniform effects in near-contact interfacial region immobilize the interfaces when two droplets are approaching each other and thus inhibit their coalescence