  We derive a continuum sharp-interface model for moving contact lines with
soluble surfactants in a thermodynamically consistent framework. The model
consists of the isothermal two-phase incompressible Navier-Stokes equations for
the fluid dynamic and the bulk\slash surface convection-diffusion equations for
the surfactant transportation. The interface condition, the slip boundary
condition, the dynamic contact angle condition, and the adsorption\slash
desorption condition are derived based on the principle of the total free
energy dissipation. In particular, we recover classical adsorption isotherms
from different forms of the surface free energy. The model is then numerically
solved in two spatial dimensions. We present an Eulerian weak formulation for
the Navier-Stokes equations together with an arbitrary Lagrangian-Eulerian weak
formulation for the surfactant transport equations. Finite element
approximations are proposed to discretize the two weak formulations on the
moving mesh. The resulting numerical method is shown to conserve the total mass
of the surfactants exactly. By using the proposed model and its numerical
method, we investigate the droplet spreading and migration in the presence of
surfactants and study their dependencies on various dimensionless adsorption
parameters.
