Capillary currents and viscous droplet spreading

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 $R_{\textit{drop}}\sim t^{1/8}$ scaling, reminiscent of viscous gravity currents (Lopez et al. 1976 J. Colloid Interface Sci. , vol. 56, pp. 460–468). The droplet is preceded by a mesoscopic fluid film that percolates through the rough substrate, its radius increasing according to $R_{\textit{film}}\sim t^{3/8}/(\log t)^{1/2}$ . To rationalise 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 J. Phys. Chem ., vol. 90, pp. 5845–5845), 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 C. R. Acad. Sci. II , vol. 299, pp. 499–503) and thereby recovers the classic spreading laws of Hoffman (1975 J. Colloid Interface Sci ., vol. 50, pp. 228–241), Voinov (1976 Fluid Dyn. , vol. 11, pp. 714–721) and Tanner (1979 J. Phys. D: Appl. Phys. vol. 12, p. 1473). For drops wide with respect to $\ell _c$ , it rationalises why millimetric, surface-tension-driven capillary currents exhibit the same spreading behaviour as relatively large-scale viscous gravity currents.