  A hybrid method is developed to simulate two-phase flows with soluble
surfactants. In this method, the interface and bulk surfactant concentration
equations of diffuse-interface form, which include source terms to consider
surfactant adsorption and desorption dynamics, are solved in the entire fluid
domain by the finite difference method, while two-phase flows are solved by a
lattice Boltzmann color-gradient model, which can accurately simulate binary
fluids with unequal densities. The flow and interface surfactant concentration
fields are coupled by a modified Langmuir equation of state, which allows for
surfactant concentration beyond critical micelle concentration. The capability
and accuracy of the hybrid method are first validated by simulating three
numerical examples, including the adsorption of bulk surfactants onto the
interface of a stationary droplet, the droplet migration in a constant
surfactant gradient, and the deformation of a surfactant-laden droplet in a
simple shear flow, in which the numerical results are compared with theoretical
solutions and available literature data. Then, the hybrid method is applied to
simulate the buoyancy-driven bubble rise in a surfactant solution, in which the
influence of surfactants is identified for varying wall confinement, Eotvos
number and Biot number. It is found that surfactants exhibit a retardation
effect on the bubble rise due to the Marangoni stress that resists interface
motion, and the retardation effect weakens as the Eotvos or Biot number
increases. We further show that the weakened retardation effect at higher Biot
numbers is attributed to a decreased non-uniform effect of surfactants at the
interface.
