Dynamics of wetting with nonideal surfaces. The single defect problem

Under static conditions, the macroscopic contact angle θ between a (partially wetting) liquid, a solid, and air, lies between two limiting values θr (receding) and θa (advancing). If we go beyond these limits (e.g., θ=θa+ε, ε>0) the contact line moves with a certain macroscopic velocity U(ε). In the present paper, we discuss U(ε) (at small ε) for a very special situation where the contact line interacts only with one defect at a time. (This could be achieved inside a very thin capillary, of radius smaller than the average distance between defects.) Using earlier results on the elasticity and dynamics of the contact line in ideal conditions, we can describe the motions around ‘‘smooth’’ defects (where the local wettability does not change abruptly from point to point). For the single defect problem in a capillary, two nonequivalent experiments can be performed: (a) the force F is imposed (e.g., by the weight of the liquid column in the capillary). Here we define ε=(F−Fm)/Fm, where Fm is the maximum pinning force which one defect can provide. We are led to a time averaged velocity Ū∼ε1/2. (b) The velocity U is imposed (e.g., by moving a horizontal column with a piston). Here the threshold force is not at F=Fm, but at a lower value F=FU —obtained when the contact line, after moving through the defect, leaves it abruptly. Defining ε̄=(F̄−FU)/FU, where F̄ is the time average of the force, we find here U∼ε̄3/2. These conclusions are strictly restricted to the single defect problem (and to smooth defects). In practical situations, the contact line couples simultaneously to many defects: the resulting averages probably suppress the distinction between fixed force and fixed velocity.