tension between the solid and fluid i . Hence, to include wetting effects in the model, it is necessary to account for the surface energies of the solid-fluid boundaries. Within the diffuse interface formulation, we use a Cahn surface free energy F$_{S}$ [ φ$_{S}$ ] = ∫ S f$_{S}$ ( φ$_{S}$ ( r ))d S, which is the integral along the solid-fluid surface, S , of the free energy per unit area, f$_{S}$ = Cφ$_{S}$ , that depends on the local value of the order parameter at the boundary, φ$_{S}$ . In this expression, the parameter C can be varied to obtain a prescribed equilibrium contact angle. Minimizing the overall free energy gives the boundary condition at the solid from which the equilibrium contact angle is related to the model parameters by the expres- sion [28] The diffuse interface model naturally allows for local slip at the contact line region. As noted by Briant and Yeomans [23] and Qian et al. [22], even though a stick boundary condition is imposed to the velocity field, the diffusive term in equation (3) allows for motion of the contact line. Such diffusive flux has been studied, e.g. , by Denniston and Robbins [29] in miscible displacements of binary fluids using molecular dynamics. In our model, the diffusion mechanism induces relaxation of the stick boundary condition in a length scale l$_{d}$ . Close to the contact line, at scales comparable to l$_{d}$ , one observes slip, while for scales much larger than this lengthscale stick is recovered. Detailed studies of this length scale have been performed by Briant and Yeomans [23] and Qian et al. [22]. C. Dimensionless numbers and units In order to relate our simulation units with experimental ones, let us briefly review the limit of the model in the lubrication regime. As explained before, in the sharp interface limit the usual continuity and Navier-Stokes equations are recovered from equations (1) and (2), while the chemical potential term in equation (2) gives the Young-Laplace condition at the interface, Δ P = − γ K , ∂φ ∂z ( x, y, z = 0) = 1 κ d f$_{S}$ d φ$_{S}$ , (5) 10