  An arbitrary Lagrangian--Eulerian (ALE) finite element scheme for
computations of soluble surfactant droplet impingement on a horizontal surface
is presented. The numerical scheme solves the time-dependent Navier--Stokes
equations for the fluid flow, scalar convection-diffusion equation for the
surfactant transport in the bulk phase, and simultaneously, surface evolution
equations for the surfactants on the free surface and on the liquid-solid
interface. The effects of surfactants on the flow dynamics are included into
the model through the surfactant-dependent surface tension and dynamic contact
angle. In particular, the dynamic contact angle $(\theta_d)$ of the droplet is
defined in terms of the surfactant concentration at the contact line and the
equilibrium contact angle $(\theta_e^0)$ of the clean surface using the
nonlinear equation of state for surface tension. Further, the surface forces
are included in the model as the surface divergence of the surface stress
tensor that allows to incorporate the Marangoni effects without calculating the
surface gradient of the surfactant concentration on the free surface. In
addition to a mesh convergence study and validation of the numerical results
with experiments, the effects of adsorption and desorption surfactant
coefficients on the flow dynamics in wetting, partially wetting and non-wetting
droplets are studied in detail. It is observed that the effect of surfactants
are more in wetting droplets than in the non-wetting droplets. Further, the
presence of surfactants at the contact line reduces the equilibrium contact
angle further when $ \theta_e^0 $ is less than $90^\circ$, and increases it
further when $ \theta_e^0 $ is greater than $90^\circ$. The presence of
surfactants has no effect on the contact angle, when $\theta_e^0 =90^\circ$.
The numerical study clearly demonstrates that the surfactant-dependent contact
angle ...
