Capillary currents and viscous droplet spreading

TL;DR

This study addresses the spreading of viscous, shallow droplets on rough substrates and demonstrates that large droplets spread via capillary currents driven by edge forces, while a Darcy precursor film propagates ahead due to substrate roughness. The authors combine experiments with a capillary-current theory that enforces a quasi-equilibrium balance between hydrostatic and curvature pressures and includes bulk and edge dissipation, recovering the classical small-droplet laws ($R_1\sim V^{3/10} t^{1/10}$) and ($R_1\sim V^{3/8} t^{1/8}$) for large drops, and predicting $R_2$ scaling with a logarithmic correction ($R_2\sim t^{3/8}/\sqrt{\log t}$). The work also explains the Darcy precursor film dynamics observed ahead of droplets on rough substrates and demonstrates dynamic similarity across fluids, while extending the framework to partial wetting and smooth substrates. Overall, the capillary-current perspective unifies droplet-spreading mechanisms across scales and substrate types, with implications for coating, microfluidics, and porous-media wetting.

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.

Figures (4)

• Figure 1: Schematic of the physical system under investigation. (a) The droplet forms the rounded shape \ref{['eq:drop_profile']} of radius $R_1$, contact angle $\theta$, and maximum depth $h_1=h(0)\ll\ell_c$. In the case of total wetting, the droplet is generally preceded by a precursor film of thickness $h_2\ll h_1$ and radius $R_2\geq R_1$. (b) A small droplet ($R_1\ll\ell_c$) approximately forms a spherical cap with a maximum depth $h_1 \sim R_1\theta/2$. (c) A large droplet ($R_1\gg\ell_c$) is shaped more like a flat disk, with a depth $h_1 \sim \ell_c \tan\theta$ that is approximately constant for $r < R_1 - \ell_c$. (d) If the substrate is smooth, a microscopic precursor film arises with thickness $h_2$ on the molecular scale (e) If the substrate is rough, a Darcy precursor film arises with thickness comparable to the roughness height (in the case of our experiments, about 10 µm). The zoom box in (d) illustrates the net horizontal surface tension force $f=\sigma(1-\cos\theta)\sim\sigma\theta^2/2$ acting radially outward on the apparent contact line; such a force is present on both smooth and rough substrates. In all cases, the radius $R_1$ grows with time, while $\theta$ and $h_1$ decrease.
• Figure 2: (a) Confocal microscope scan of the surface S60, a borosilicate glass square sanded with 60 grit silicone carbide lapping compound. (b) Annotated image of a silicone oil droplet wetting the surface S100, sanded instead with 100 grit compound. Shown below are several snapshots of the spreading process, with the drop diameter $2R_1$ (blue arrows) and precursor film diameter $2R_2$ (red arrows) indicated. (c--e) Time evolution of the drop radius $R_1$, the film radius $R_2$, and their difference $\varDelta R = R_2-R_1$, for a $5$ µL silicone oil droplet ($\nu = 5$ cSt, $\sigma = 20$ mN m$^{-1}$, $\rho = 0.913$ g cm$^{-3}$) wetting both S60 and S100. The solid blue line in (c) represents the roughness-independent scaling $R_1 \sim t^{1/8}$ predicted for large droplets ($R_1\gg\ell_c$), and the dashed green line represents the scaling $R_1\sim t^{1/10}$ predicted for small droplets ($R_1\ll\ell_c$). These observations are consistent with our experiments being in the large-droplet regime. The solid red lines in (d) are fits of $R_2$ measurements, equal to $R_2(t) = At^{3/8}/[\log(Bt+C)]^{1/2}$ for surface-dependent parameters $A,B,C$. For the surface S100, the fitting coefficients are $A = 2.99$ mm s$^{-3/8}$, $B = 0.63$ s$^{-1}$, and $C = 2.62$. For the surface S60, the fitting coefficients are $A = 3.80$ mm s$^{-3/8}$, $B = 1.33$ s$^{-1}$, and $C = 3.79$. We rationalize the values of these parameters in \ref{['sec:estimates']}. We observe that neither curve adheres to the $t^{3/8}$ power law recovered from neglecting log factors in this fit. Finally, the dashed line in (d) represents the scaling $\varDelta R(t) \sim t^{1/2}$ predicted by Washburn's law of wicking, which is inconsistent with our observations. We note that previous experiments of viscous droplet spreading also report strong deviations from Washburn's law Cazabat1986Dorbolo2021. We rationalize these deviations in \ref{['sec:priordata']}.
• Figure 3: Evolution of (a) the droplet radius $R_1(t)$ and (b) the Darcy precursor film radius $R_2(t)$ for silicone oils of kinematic viscosity $\nu = 5$ cSt, $10$ cSt, and $50$ cSt on the surface S100. When non-dimensionalized with respect to the capillary length $\ell_c$ and plotted against the dimensionless time $\hat{t} = t\ell_c/u_c$, all three measurements of $R_1$ align closely with a single $R_1(\hat{t})/\ell_c \sim \hat{t}^{1/8}$ curve (blue), and all three measurements of $R_2$ align closely with a single $R_2( \hat{t}) = A \hat{t}^{3/8}/[\log(B \hat{t}+C)]^{1/2}$ curve (red), where $A = 0.10$, $B = 2.0 \times 10^{-4}$, and $C = 2.50$.
• Figure 4: (a) Downsampled version of the data recorded by Dorbolo2021 for the spreading of a Darcy precursor film ahead of a small drop of 20 cSt silicone oil on a frosted glass substrate. The radius $R_2$ of the film undergoes three distinct spreading regimes: $R_2\sim t^{3/10}$ for the intermediate time regime ($t\lesssim 10^5$ s) of present interest, $R_2\sim t^{3/20}$ after the droplet bulk sinks into the surface roughness ($10^5$ s $\lesssim t\lesssim 10^7$ s); and $R_2\sim\mathrm{const}.$ at very late times ($t\gtrsim 10^7$ s). Although Dorbolo originally inferred that the first spreading regime was driven by wicking, with deviations from Washburn's law (\ref{['sec:washburn']}) caused by irregular roughness elements, we note that the observed scaling $R_2\sim t^{3/10}$ is consistent with our predictions for overpressure-driven Darcy precursor films ahead of small drops. (b) The data recorded by deRuijter1999 for the spreading of a partially wetting, viscous drop of DBP on a smooth PET substrate, along with two fitted curves reproduced from Durian2022: the predicted behavior of the droplet if the equilibrium radius were $R_\mathrm{eq}=0.5$ cm (green); and the predicted early-time spreading behavior, independent of $R_\mathrm{eq}$ (blue). A natural extension of our work to partially wetting drops yields the evolution equation \ref{['eq:partialwetting_full']}, which reproduces the predictions of Durian2022 for both large and small drops with a purely edge-driven model. (c) Data reported by Cazabat1986 for the spreading of viscous drops of PMS on smooth glass, depicting the small-to-large-droplet transition as $R_1$ exceeds the capillary length. Also shown are the asymptotic $R_1\sim t^{1/10}$ and $R_1\sim t^{1/8}$ curves (dashed red) for both $V=1.5$ µL and $V=0.8$ µL, reproduced from Cazabat1986, and the predictions of our own model for both cases (blue). Specifically, we report the predictions of the evolution equation \ref{['eq:hoffman_u']} augmented with the three dissipation terms present in \ref{['eq:totaldissipation']}, with the fitted coefficients $\alpha=1$, $\beta=15$, $\kappa = 150$.