Spreading of a liquid droplet containing a bulk-soluble surfactant on a smooth solid surface is numerically studied in the sorption-controlled limit. The droplet spreads over the solid surface under the action of gravity and a dynamic contact angle larger than the advancing equilibrium contact angle. As the droplet spreads, the adsorbed surfactant is constantly redistributed along the deforming interface by advection and diffusion, leading to variations in interfacial tension along the interface. The presence of surfactant can lead to faster spreading of the droplet, and the spreading rate is affected by several factors including the induced Marangoni stresses, the contact line driving force, and the surfactant sorption rate. The dependence of the interfacial tension on monolayer concentration is modeled by the Frumkin adsorption framework which accounts for maximum surfactant packing capacity and non-ideal molecular interactions. The contact line velocity is assumed to have a power-law dependence on the deviation of the dynamic contact angle from the advancing equilibrium contact angle, similar to the functional form found experimentally for a clean droplet. The influence of surfactant on the contact line driving force is modeled by considering the equilibrium contact angle in the contact line constitutive relation to be dependent on the monolayer concentration at the contact line as dictated by a local force balance. In the lubrication limit, the evolution equations for the drop shape and the monolayer concentration are integrated using a pseudo-spectral method, and the influence of surfactant solubility on the spreading rate is examined in the sorption-controlled limit. It is found that the fastest spreading rate is achieved by droplets with O(1) values of Biot number for which the sorption rate is comparable to the rate of surface convection so that surfactant molecules transferred to the interface are effectively transported to the contact line region. The sensitivity of the numerical predictions to the contact line constitutive relation is also examined. To assess the importance of inertial effects during the initial stages of spreading (for which the lubrication approximation is not valid) a Galerkin finite element formulation is used to solve the Navier-Stokes and continuity equations for the flow field, in conjunction with the unsteady surface convective-diffusion equation for the monolayer concentration distribution. The shape of the moving interface is tracked using the method of spines. For a droplet covered with an insoluble surfactant monolayer, computational results are compared to those for spreading in the lubrication limit [1].

1.Chan, K.-Y., and Borhan, A., J. Colloid Interface Sci. 287, 233-248, 2005.