Depinning of three-dimensional drops from wettability defects

Substrate defects crucially influence the onset of sliding drop motion under lateral driving. A finite force is necessary to overcome the pinning influence even of microscale heterogeneities. The depinning dynamics of three-dimensional drops is studied for hydrophilic and hydrophobic wettability defects using a long-wave evolution equation for the film thickness profile. The model is studied employing effective algorithms for the parameter continuation of pinned steady drops and for the time simulation of the dynamics of sliding drops that perform a stick-slip motion. The discussion focuses on common features and significant differences of the depinning process for three-dimensional and two-dimensional drops. Copyright c © EPLA, 2009 Introduction. – Drops sliding along a solid substrate under the influence of a lateral force are a very common physical phenomenon. The force might be gravity for drops on an inclined or vertical wall, centrifugal forces for drops on a rotating disk or external shear for drops in an ambient flow. Note that lateral gradients in wettability, temperature or electrical fields can as well drive sliding motion. For smooth homogeneous substrates, an arbitrarily small driving force results in drops that move with constant velocity and shape [1–3]. Larger driving forces may lead to shape instabilities, e.g., trailing cusps may evolve that periodically emit small satellite drops [1,4]. Real substrates, however, are normally not smooth. They are rough or have local chemical or topographical defects. Even microscopic defects can have a strong influence on the drop dynamics. The heterogeneities may cause stick-slip motion [5,6] or roughening [7,8] of moving contact lines, and are thought to be responsible for contact angle hysteresis [9–12]. A local variation of the driving force (e.g., electrostatic field or temperature gradient) may play the same role as a substrate defect. The present paper focuses on the depinning of threedimensional (3d) drops from hydrophobic and hydrophilic line defects that pin them at their front and back, respectively: A hydrophobic defect is less wettable for the drop that therefore has to be forced to pass over it. On the contrary, a hydrophilic defect is more wettable and (a)E-mail: Philippe.Beltrame@physik.uni-augsburg.de the drop has to be forced to leave it as sketched in fig. 1. A recent theoretical study of the depinning dynamics of two-dimensional (2d) drops corresponding to 3d ridges with imposed transverse translational symmetry finds stick-slip motion beyond depinning [13,14]. The present work is based on a thin film evolution equation in long-wave approximation [15,16] that incorporates wettability in the form of a disjoining pressure [9]. It models the effective molecular interactions between the substrate and the free surface of the liquid, e.g., long-range apolar van der Waals interactions and short-range polar electrostatic or entropic interactions [17]. With the proper choice of terms, such a disjoining pressure describes drops of partially wetting liquid with a small equilibrium contact angle that coexist with an ultra-thin precursor film. An advantage of such a model is the absence of a contact line singularity. Incorporating wettability in such a way allows one to study the influence of chemical substrate heterogeneities or defects by a spatial modulation of the involved material parameters. Note, however, that they have to vary on length scales much larger than the film thickness to be consistent with the long-wave approximation [18]. The presented analysis of the behaviour of 3d drops is based on a study of i) steady drops and their stability and ii) the stick-slip motion of droplets beyond depinning. Both are based on recently developed effective algorithms for the continuation of pinned steady drops described by a partial differential equation and the time simulation of the dynamics of sliding drops [19].