Capillary Currents and Viscous Droplet Spreading

We present the results of a combined experimental and theoretical study of the spreading of viscous droplets over rough and smooth substrates. First, we experimentally investigate the wetting of a roughened glass surface by a viscous droplet of silicone oil, wide and shallow relative to the capillary length. The radius of the droplet grows according to a R_drop ~ t^1/8 law reminiscent of gravity currents. The droplet is preceded by a mesoscopic fluid film that percolates through the rough substrate, its radius increasing according to R_film ~ t^3/8/[log(t)]^1/2. To rationalize these observed scalings, we develop a new capillary current model for the spreading of shallow droplets with arbitrary radius on both smooth and rough surfaces. We propose that, throughout their evolution, capillary currents maintain a quasi-equilibrium balance between hydrostatic and curvature pressure, perturbed only by contact line forces. In addition to rationalizing our experimental data, our model provides new rationale for a number of scalings reported in prior work. In particular, it reveals how surface-tension-driven capillary currents can mimic the spreading behavior typically seen in viscous gravity currents at relatively large scales.