Capillary currents and viscous droplet spreading

TL;DR

This paper combines experiments and theory to analyze viscous droplet spreading on rough surfaces, revealing new scaling laws and a capillary current model that unify behaviors across different droplet sizes.

Contribution

It introduces a novel capillary current model for shallow droplet spreading on rough surfaces, extending classic laws to larger scales and different surface conditions.

Findings

Droplet radius scales as t^{1/8}, similar to viscous gravity currents.

Mesoscopic film radius grows as t^{3/8}/(log t)^{1/2}.

Model unifies spreading behavior across small and large droplets.

Abstract

We present the results of a combined experimental and theoretical study of the spreading of viscous droplets over rigid 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 $\ell_c$. The horizontal radius of the droplet grows according to an $R_\mathrm{drop}\sim t^{1/8}$ scaling reminiscent of viscous gravity currents (Lopez et al. 1976). The droplet is preceded by a mesoscopic fluid film that percolates through the rough substrate, its radius increasing according to $R_\mathrm{film}\sim t^{3/8}/(\log t)^{1/2}$. To rationalize these observed scalings, we develop a new 'capillary current' model for the spreading of shallow droplets with arbitrary radius on rough surfaces. Furthermore, on the basis of established similarities between droplet spreading over wetted rough and smooth substrates (Cazabat & Cohen Stuart 1986), we argue its relevance to a broader class of spreading problems. We propose that, throughout their evolution, shallow droplets maintain a quasi-equilibrium balance between hydrostatic and curvature pressure, perturbed only by unbalanced contact line forces arising along the droplet's edge. For drops with horizontal radii small with respect to $\ell_c$, our model converges to the original description of Hervet & de Gennes (1984) and thereby recovers the classic spreading laws of Hoffman (1975), Voinov (1976), and Tanner (1979). For drops wide with respect to $\ell_c$, it rationalizes why millimetric, surface-tension-driven capillary currents exhibit the same spreading behavior as relatively large-scale viscous gravity currents.