2.1. Quasi-equilibrium geometry

The basic geometry of our system is depicted in figure 1. We suppose for simplicity that the apparent contact angle $\theta$ satisfies $\theta \lesssim \pi /4$ (so that $\sin \theta \approx \tan \theta \approx \theta$ ) and that the droplet is axisymmetric. Here and below, we fix non-dimensionalised units such that

(2.4) \begin{equation} \frac {\sigma }{\rho g} = \frac {\sigma }{\mu } = 1,\qquad p_{\textit{atm.}} = 0. \end{equation}

Equivalently, we non-dimensionalise by the length, velocity and time scales

(2.5) \begin{align} \ell _c = \sqrt {\sigma /\rho g},\qquad u_c = \sigma /\mu ,\qquad \tau _c = \ell _c/u_c, \end{align}

and we tare the pressure such that $p_{\textit{atm.}}=0$ .

To a good approximation, after a momentary equilibration period when the droplet is first placed on a substrate (Biance, Clanet & Quéré Reference Biance, Clanet and Quéré2004; Eddi, Winkels & Snoeijer Reference Eddi, Winkels and Snoeijer2013), the instantaneous droplet shape is determined by a balance of hydrostatic and curvature pressures. Such a balance is well understood to underlie the leading-order droplet geometry, and much of the following argument has appeared, for instance, in the work of Tanner (Reference Tanner1979) and Voinov (Reference Voinov1999).

We write $r$ for the radial coordinate and $h(r)$ for the height of the droplet’s surface. To leading order in $h/\ell _c\lesssim 1$ , the pressure immediately inside the droplet satisfies $p(r) = -\sigma \boldsymbol{\nabla} ^2 h$ . Balancing hydrostatic and curvature pressures of the surface requires that

(2.6) \begin{align} \sigma \partial _z\boldsymbol{\nabla} ^2 h = \sigma \frac {\partial _r\boldsymbol{\nabla} ^2 h}{\partial _r h} = \rho g. \end{align}

By non-dimensionalising and enforcing axisymmetry, one may find the solution

(2.7) \begin{equation} h(r) = \frac {I_0(R_1) - I_0(r)}{I_1(R_1)}\,\tan \theta , \end{equation}

which has an associated total droplet volume

(2.8) \begin{equation} V = \int _0^{R_1} 2\pi r h(r)\,\text{d}r = \left (\pi R_1^2\frac {I_0(R_1)}{I_1(R_1)} - 2\pi R_1\right )\tan \theta . \end{equation}

Here, $R_1=R_1/\ell _c$ is the dimensionless horizontal droplet radius, $\theta$ is the apparent contact angle at the droplet–film interface and $I_0$ and $I_1$ are modified Bessel functions. As one might expect, this height profile always describes a rounded shape with radial extent $R_1$ . Its qualitative behaviour is best understood by investigating the small- and large-droplet limits. The Bessel functions $I_0$ and $I_1$ have the following asymptotic behaviour for small and large arguments:

(2.9) \begin{equation} \begin{aligned} x\ll 1\quad &\implies \quad I_0(x)\sim 1 + x^2/4,\quad I_1(x)\sim x/2,\hspace {2em}\\ x\gg 1\quad &\implies \quad I_0(x)\sim I_1(x)\sim e^x/\sqrt {2\pi x}. \end{aligned} \end{equation}

As evident from these asymptotic expressions, the qualitative shape of the droplet greatly depends on the relative magnitudes of $R_1$ and $\ell _c$ . When $R_1\ll \ell _c$ and $\theta \lesssim \pi /4$ , the profile (2.7) asymptotes to that of a spherical cap with radius of curvature $R_{\textit{cap}}\sim R_1/\theta$ and maximum height $h_1\sim R_1\theta /2$ , as illustrated in figure 1(b). When $R_1\gg \ell _c$ , the profile asymptotes to that of a flat puddle of height $h_1\sim \ell _c\tan \theta$ , except in the boundary region $|R_1-r|\lesssim \ell _c$ , where it decays to zero (figure 1 c).

The profile (2.7) interpolates smoothly between these two limits. In particular, if the radius of an initially small droplet grows to exceed $\ell _c$ , the droplet transitions through a sequence of increasingly flattened spherical shapes until reaching the pancake-like form depicted in figure 1(c) (Cazabat & Cohen Stuart Reference Cazabat and Cohen Stuart1986; Ehrhard Reference Ehrhard1993). The precise cross-over radius is of the order of $\ell _c$ , but has been reported to depend weakly on the droplet volume (Cazabat & Cohen Stuart Reference Cazabat and Cohen Stuart1986).

2.2. Existing models for total wetting by small droplets

The modern understanding of small droplet spreading ( $R_1\ll \ell _c$ ) over a flat substrate began with the experimental work of Hoffman (Reference Hoffman1975), who observed the scaling $\dot {R}\sim \theta ^3$ . An initial attempt to rationalise Hoffman’s findings was provided by Voinov (Reference Voinov1976) and Tanner (Reference Tanner1979), based on a lubrication approximation of the droplet dynamics. Their work rationalised Tanner’s laws (sometimes known as the Hoffman–Voinov–Tanner laws) for the spreading of a small droplet on a rough substrate: $R\sim V^{3/10}t^{1/10}$ and $\theta \sim V^{1/10}t^{-3/10}$ . A more complete description of the droplet motion – and the first to correctly account for the precursor film – was subsequently provided by Hervet & de Gennes (Reference Hervet and de Gennes1984). They argue that small droplets maintain the shape of a spherical cap (see figure 1(b)) as they are slowly stretched outward by contact line forces. In turn, the rate of spreading can be deduced from the droplet’s internal energy balance.

The argument of Hervet & de Gennes proceeds as follows. Suppose the droplet boundary advances at a speed $U$ . On one hand, there is a force $f=\sigma (1-\cos \theta )\approx \sigma \theta ^2/2\gt 0$ per unit arclength along the apparent contact line – arising from unbalanced surface tension forces when the droplet meets the surface at an angle $\theta \lesssim \pi /4$ (see figure 1(d)) – which does work $w\sim \sigma \theta ^2 U/2$ per unit arclength. Hervet & de Gennes argue that this work is dissipated primarily along the outer edge of the droplet. To model this process, they define a radial coordinate $\chi = R_1-r$ measuring the distance (inward) from the edge. In order to ensure that the droplet surface (at $z = \theta \chi$ ) travels with velocity $U$ , a linear shear profile $u(\chi ,z)\sim Uz/\theta \chi$ is adopted. The resulting energy dissipation rate is not integrable, so the authors fix upper and lower cutoff lengths in $\chi$ , say, $L$ and $a$ , respectively. Physically, one expects $L\lesssim \ell _c$ and for $a$ to be on the molecular scale; indeed, one generally needs a microscopic cutoff length in order to regularise dissipation near a moving contact line (Huh & Scriven Reference Huh and Scriven1971). The energy dissipation per unit arclength is thus

(2.10) \begin{align} \int _{a}^L\int _0^{\theta \chi } \mu \left (\frac {\partial u}{\partial z}\right )^2\,\text{d}z\,\text{d}\chi = \int _{\ell _0}^L\frac {\mu U^2}{\theta \chi }\,\text{d}\chi = \frac {\mu }{\theta }\log (L/a)U^2. \end{align}

Writing $\ell _D=\log (L/a)$ , they thus deduce the evolution equation $\dot {R}_1\sim U \sim \theta ^3/2\ell _D$ , consistent with the predictions of Hoffman (Reference Hoffman1975), Voinov (Reference Voinov1976) and Tanner (Reference Tanner1979).

Hervet & de Gennes went a step further and estimated the cutoff lengths $L$ and $a$ based on the micro-structure of the droplet-film transition on a smooth substrate. They thus find a logarithmic correction to Hoffman’s scaling, of the form

(2.11) \begin{equation} \theta ^3 \sim \dot {R}_1\log \left ([\dot {R}_1]^{2/3}\right ). \end{equation}

The work of Eggers & Stone (Reference Eggers and Stone2004) has since offered a slight correction to the scaling coefficients predicted by Hervet & de Gennes, and separate work by Hocking & Rivers (Reference Hocking and Rivers1982), Hocking (Reference Hocking1983), Cox (Reference Cox1986a , Reference Coxb ), Snoeijer (Reference Snoeijer2006), Chan et al. (Reference Chan, Srivastava, Marchand, Andreotti, Biferale, Toschi and Snoeijer2013) and Luo & Gao (Reference Luo and Gao2025) has clarified the local structure of a moving contact line. The physical model furnished by the spreading laws of Hervet & de Gennes and subsequent work on the contact line problem remains well supported by a variety of experimental investigations of liquid spreading on smooth substrates (Ngan & Dussan V. Reference Ngan and Dussan V.1982; Chen Reference Chen1988; Chen & Wada Reference Chen and Wada1989; Dussan V., Ramé & Garoff Reference Dussan V., Ramé and Garoff1991; Marsh, Garoff & Dussan V. Reference Marsh, Garoff and Dussan V.1993; Ramé & Garoff Reference Ramé and Garoff1996).

The work of Cazabat & Cohen Stuart (Reference Cazabat and Cohen Stuart1986) showed that the same spreading laws apply to the spreading of small droplets on rough substrates, where the Darcy precursor film plays a role similar to the precursor film on a smooth substrate. Their findings are consistent with the more general theoretical treatment of Starov et al. (Reference Starov, Kostvintsev, Sobolev, Velarde and Zhdanov2002a , Reference Starov, Kosvintsev, Sobolev, Velarde and Zhdanovb , Reference Starov, Zhdanov, Kosvintsev, Sobolev and Velarde2003) of small droplet spreading on rough substrates, and subsequent experimental investigations of the same (Grzelakowski et al. Reference Grzelakowski, Ben Jazia, Lebeau, Vonna, Dupuis and Haidara2009; Gambaryan-Roisman Reference Gambaryan-Roisman2014; Dorbolo Reference Dorbolo2021). Notably, spreading on deep, porous substrates ( $h_2\gtrsim h,\ell _c$ ) does not yield the same universal behaviour (Davis & Hocking Reference Davis and Hocking1998; Starov et al. Reference Starov, Zhdanov and Velarde2002c , Reference Starov, Zhdanov, Kosvintsev, Sobolev and Velarde2003; Frank & Perré Reference Frank and Perré2012; Chebbi Reference Chebbi2021). In the same vein, we note that the logarithmic correction (2.11) to Hoffman’s scaling depends on the microstructure of the droplet-film transition on a smooth surface. To our knowledge, evidence for such a logarithmic correction has yet to be observed for spreading on rough substrates.

2.3. Existing models for total wetting by large droplets

For large-droplet spreading ( $R\gg \ell _c$ ), the most widely accepted model is that of Lopez et al. (Reference Lopez, Miller and Ruckenstein1976), who apply a lubrication approximation to model the interior of the droplet, away from the apparent contact line. The height profile $h$ thus satisfies

(2.12) \begin{equation} \partial _t h(r) = -\frac {1}{3\mu }\boldsymbol{\nabla }\boldsymbol{\cdot }\big (h^3\boldsymbol{\nabla }(\sigma \boldsymbol{\nabla} ^2 h - \rho g h)\big ). \end{equation}

Lopez et al. proceed by supposing that surface tension is negligible in the relatively flat droplet interior (see figure 1(c)). The resulting physics is then exactly that of a viscous gravity current, as discussed in § 1, giving rise to the scaling $R_1\sim V^{3/8}t^{1/8}$ .

Subsequent work has adapted this gravity-driven model in various ways. Ehrhard & Davis (Reference Ehrhard and Davis1991) suggested that gravitational overpressure is resisted by edge-localised energy dissipation, rather than the bulk dissipation of Lopez et al. (Reference Lopez, Miller and Ruckenstein1976), and so recovered a scaling $R_1\sim t^{1/7}$ . Subsequent authors have argued that bulk dissipation should be dominant, and thus that the widely observed $R_1\sim t^{1/8}$ scaling should emerge (Hocking Reference Hocking1992, Reference Hocking1994; Bonn et al. Reference Bonn, Eggers, Indekeu, Meunier and Rolley2009). In a different direction, Voinov (Reference Voinov1995, Reference Voinov1999) suggested perturbatively correcting this gravity-driven model to better account for the effect of surface tension near the edge of the droplet. Voinov posits two distinct regimes in the droplet: a gravity-driven interior, like that of Lopez et al., and a quasi-static meniscus region. The two regimes are connected by asymptotically matching height profiles at their interface, and Voinov uses the contact angle predicted by the fully quasi-equilibrium droplet profile (2.7) as a boundary condition for the outer region.

It has been observed by Cazabat & Cohen Stuart (Reference Cazabat and Cohen Stuart1986) that, if a droplet’s radius $R_1$ grows to exceed the capillary length during the course of spreading, it transitions rapidly from the small-droplet scaling $R_1\sim t^{1/10}$ to the large-droplet scaling $R_1\sim t^{1/8}$ . Ehrhard (Reference Ehrhard1993) observed similar behaviour, but argued that their data support the $R_1\sim t^{1/7}$ large-droplet scaling derived by Ehrhard & Davis (Reference Ehrhard and Davis1991). In either case, according to the current understanding of these two regimes, such a transition would entail a rapid change from a quasi-equilibrium, edge-driven flow to a dynamic, overpressure-driven flow. We revisit this problem in § 5, where we see that the model presented here rationalises the results of Cazabat & Cohen Stuart (Reference Cazabat and Cohen Stuart1986) with an edge-driven spreading mechanism valid for both small and large drops.

2.4. Existing models for partial wetting on smooth substrates

While the total wetting regime remains our primary focus, it is worth outlining the close connection between the spreading of totally wetting droplets (with spreading parameter $S\gt 0$ ) and the relaxation to equilibrium of partially wetting droplets (with $S\lt 0$ ). Droplets in both regimes exhibit a leading-order geometry of the form described in § 2.1, as well as unbalanced contact line forces that drive the droplet spreading. These similarities suggest that the present model can be adapted to describe the relaxation to equilibrium of partially wetting droplets, a case we return to in § 5.

Partially wetting droplets do not generally give rise to precursor films (Bonn et al. Reference Bonn, Eggers, Indekeu, Meunier and Rolley2009), so are marked by a well-defined contact line where the liquid, substrate, and ambient vapour all meet. Such droplets tend toward a static equilibrium as they spread, in which the drop meets the substrate at a contact angle $\theta _{\textit{eq}}\gt 0$ prescribed by Young’s law (2.3). Various models have been proposed to rationalise the spreading of partially wetting drops as they relax into equilibrium (de Gennes, Hua & Levinson Reference de Gennes, Hua and Levinson1990; Brochard-Wyart & Gennes Reference Brochard-Wyart and de Gennes1992; Petrov & Petrov Reference Petrov and Petrov1992; de Ruijter et al. Reference de Ruijter, De Coninck, Blake, Clarke and Rankin1997, Reference de Ruijter, De Coninck and Oshanin1999; Eggers Reference Eggers2005). The most recent and comprehensive work on the subject is that of Durian et al. (Reference Durian, Dillavou, Markin, Portales, Maldonado, Irvine, Arratia and Durian2022), which extends the model of de Ruijter et al. (Reference de Ruijter, De Coninck and Oshanin1999) to account for droplets of arbitrary horizontal radius.

Consistent with prior literature, Durian et al. posit that spreading is driven by either capillary or gravitational forces, depending on the horizontal radius of the drop. As in our own model, however, they posit that spreading is resisted by a combination of bulk and edge-localised energy dissipation, and thereby recover smooth transitions between previously observed scaling laws. For small droplets, they find an equation of the form

(2.13) \begin{equation} \left (\alpha R_1^9/V^3 + \kappa R_1^6/V^2\right )\dot {R}_1 = 1 - (R_1/R_{1,\textit{eq}})^6 \end{equation}

for system-dependent parameters $\alpha ,\kappa \gt 0$ . For large droplets, they similarly find

(2.14) \begin{equation} \left (\alpha R_1^7/V^3 + \kappa R_1^4/V^2\right )\dot {R}_1 = 1 - (R_1/R_{1,\textit{eq}})^4. \end{equation}

A key point in the derivation of Durian et al. is that the form of the edge-localised energy dissipation in the droplet is not the same as that prescribed by Hervet & de Gennes (Reference Hervet and de Gennes1984) for totally wetting droplets. Following de Ruijter et al. (Reference de Ruijter, De Coninck and Oshanin1999), the edge-localised energy dissipation for a partially wetting droplet is posited to take the form

(2.15) \begin{equation} D_{\textit{edge}}\sim \kappa R_1 (\dot {R}_1)^2, \end{equation}

independent of the contact angle $\theta$ . By comparison, recall that Hervet & de Gennes suggest the form $D_{\textit{edge}}\sim \beta R_1 (\dot {R}_1)^2/\theta$ for totally wetting droplets. We note that the form (2.15) is consistent with resistance generated by corrugation of the contact line.

We revisit the question of partial wetting in § 5, where we show that, by adapting our model to account for substrate-dependent contact line forces, one can interpolate between the small-droplet prediction (2.13) and the large-droplet prediction (2.14) using a single edge-driven mechanism.

2.5. Washburn’s law for precursor films on rough substrates

Precursor films on rough substrates – which we refer to as Darcy precursor films – can be understood as viscous flows through a shallow, porous medium (Gambaryan-Roisman Reference Gambaryan-Roisman2014). Such flows are governed by Darcy’s law (Darcy Reference Darcy1856; Whitaker Reference Whitaker1986):

(2.16) \begin{equation} {\overline {u}} = -\frac {k}{\mu \phi }\left (\boldsymbol{\nabla }p - f\right ). \end{equation}

Here, $\overline {u}$ is the local mean velocity of the fluid, $k$ is the permeability of the medium, $\phi$ is the porosity (i.e. void fraction) of the medium and $f$ is the volumetric force within the precursor film.

Volume conservation requires that ${\overline {u}} \propto 1/r$ , and matching ${\overline {u}}|_{R_2}=\dot {R}_2$ yields ${\overline {u}} = R_2\dot {R}_2/r$ . The volumetric force $f$ is prescribed by the surface tension acting on the inner and outer boundaries of the precursor film. To calculate it, we model the film as a ring of independent, radial strands of fluid, each subtending an angle $\text{d}\theta$ . The total tension applied to such a strand is $S R_2\,\text{d}\theta$ , where $S$ is the roughness-dependent spreading parameter of our fluid over the given substrate. The tension-per-unit-radius is thus $S R_2(R_2-R_1)^{-1}\,\text{d}\theta$ . If we suppose that the tension is uniformly distributed across the cross section of fluid at radius $r$ , then the volumetric force is

(2.17) \begin{equation} f = \frac {SR_2(R_2-R_1)^{-1}\,\text{d}\theta }{h_2r\,\text{d}\theta } = \frac {\sigma sR_2/r}{R_2-R_1}, \end{equation}

where we define $s = S/h_2\sigma$ for convenience. Finally, if one neglects the pressure gradient in (2.16), one finds the following variant of Washburn’s law (Washburn Reference Washburn1921):

(2.18) \begin{equation} \dot {R}_2 = \frac {ks}{\phi }\,\frac {1}{R_2-R_1},\qquad \Delta R = R_2-R_1\sim \sqrt {ks t/\phi }, \end{equation}

after making the approximation $\dot {R}_2\gg \dot {R}_1$ and adopting our non-dimensionalisation. In practice, while the particular scaling $\Delta R\sim t^{1/2}$ can be achieved with certain combinations of liquids and substrates, previous authors have observed a range of exponents between $0.25$ and $0.5$ (Cazabat & Cohen Stuart Reference Cazabat and Cohen Stuart1986; Dorbolo Reference Dorbolo2021). We will build upon this model through consideration of gravitational overpressure from the droplet bulk, which will play a critical role in rationalising our experimental results. In § 5, we revisit the problem in order to demonstrate how the $R_2\sim t^{3/10}$ spreading reported by Dorbolo (Reference Dorbolo2021) can be rationalised with the same model.

3.1. Methodology

Rough surfaces were made by hand-lapping $75\times 75$ mm $^2$ square sections of borosilicate glass on a slurry of course-grain silicone-carbide lapping compound and water. Two types of lapping compound with different grain sizes were used to make surfaces of different roughness: the first surface (S60) was sanded with 60 grit (or 250 ${\unicode{x03BC}}\textrm{m}$ ) compound, and the second surface (S100) was sanded with 100 grit (120 ${\unicode{x03BC}}\textrm{m}$ ). The resulting surfaces were visibly ‘frosted’ (or opaque). The surfaces are characterised by their permeability $k$ and porosity (i.e. void fraction) $\phi$ . It is shown that S60 and S100 have $k/\phi = 3.28\times 10^{-5}\,\textrm{cm}^2$ and $k/\phi = 2.03\times 10^{-5}$ cm $^2$ , respectively, implying the pore size decreases as the grit of the lapping compound increases (or the average size of the lapping aggregate decreases). A laser-scanning confocal microscope (VK-X250, Keyence) was used to visualise the topography of the roughened glass substrates. A sample two-dimensional projection of a microscope scan for S60 is shown in figure 2(a). Surface asperities exhibit a hierarchy of length scales from tens of nanometres to hundreds of micrometres. The areal footprint of the large-scale asperities observed in figure 2(a) for S60 are comparable to the derived pore size $k/\phi = 3.28\times 10^{-5}$ cm $^2$ . The root-mean-squared (r.m.s.) roughness height for the S60 surface measured using the confocal microscope was 7.6 ${\unicode{x03BC}}\textrm{m}$ and the average lateral length scale (an auto-correlation length) was 56.7 ${\unicode{x03BC}}\textrm{m}$ . Due to limitations in equipment availability, we were unable to make the same measurement for S100. Since the lapping process was the same, however, we can scale by the ratio of (square root) permeabilities, $0.79\approx 45$ ${\unicode{x03BC}}\textrm{m}$ $/$ 57 ${\unicode{x03BC}}\textrm{m}$ , to estimate that the r.m.s. roughness height is 6.0 ${\unicode{x03BC}}\textrm{m}$ and the lateral length scale is 44.9 ${\unicode{x03BC}}\textrm{m}$ for S100.

3.2. Experimental results

In our primary experiments, a $V = 5$ ${\unicode{x03BC}}\textrm{l}$ drop of silicone oil (63148-62-9, Millipore Sigma), with $\sigma = 20$ mN m $^{-1}$ , $\rho = 0.913$ g cm $^{-3}$ , $\mu = 4.6$ mPa s (or $\nu =\mu /\rho = 5.0$ cSt) and initial radius $R_1(t=0)\approx 3.0$ mm, was gently placed on the roughened glass substrate. Here, the capillary length $\ell _c$ is 1.5 mm and the capillary velocity is $u_c=\sigma /\mu = 4.3$ m s $^{-1}$ . We note that the silicone oils we use are non-volatile at normal (ambient) temperature and pressure, with ‘no detectable vapour pressure’ (European Centre for Ecotoxicology and Toxicology of Chemicals (ECETOC) 2011), and they totally wet borosilicate glass (Dorbolo Reference Dorbolo2021). A secondary series of experiments with liquids of higher viscosity is reported in § 3.3, and supports the conclusions drawn from our primary experiments.

The shape of the drop as it wet the substrate was monitored using a 4 Mpixel digital camera (Lumix GH5, Panasonic) at an image acquisition rate of 1 Hz, with a diffuse spot light for backlit illumination. The camera was positioned at an acute angle above the horizontal plane to promote detection of the droplet bulk from reflection. A sample image of the droplet on S60 is shown in figure 2(b) with the droplet radius $R_1$ and precursor film radius $R_2$ labelled. An edge detection algorithm developed in MATLAB (MathWorks) was used to extract the droplet bulk radius $R_1$ and precursor film radius $R_2$ from each image. Namely, the image background is first subtracted to enhance contrast with the spreading liquid. Images are then binarised using the method of Otsu (Reference Otsu1979), which adequately discriminates the bright precursor film from the relatively dark background, yielding an estimate of $R_2$ . A similar procedure is used to extract the droplet radius $R_1$ , after inverting the image to isolate the darker droplet bulk. Measurements of the droplet diameter $2R_1$ and precursor film diameter $2R_2$ are annotated in blue and red, respectively, on the sample snapshots in figure 2(b).

At least five experimental trials, each consisting of 600 images taken over 600 s, were performed for both roughened glass surfaces. The average values of $R_1(t)$ and $R_2(t)$ across the repeat trials are shown in figures 2(c) and 2(d). Error bars represent the standard deviation in the measurements of $R_1(t)$ and $R_2(t)$ across the repeated trials. These repeatability errors are no larger than $\pm 5$ % for all times $t$ . Sample images of the spreading droplet on S60 at different times $t$ are shown above figures 2(c) and 2(d). This imaging setup is ideal for capturing droplet spreading on our opaque, roughened surfaces, where there is clear contrast between the droplet, film and substrate, as illustrated in figure 2. However, it is less effective for visualising droplet spreading on smooth, transparent glass substrates, where contrast is significantly reduced. Measurements of $R_2$ for spreading on a smooth substrate require techniques with much larger spatial resolution, given that $h_2 \sim \mathcal{O}(10$ Å) in this case. We do not pursue such measurements here, as there are several previous investigations of droplet spreading on smooth substrates using methods such as ellipsometry or X-ray reflectivity that can accurately resolve both radii (Bascom, Cottingdon & Singleterry Reference Bascom, Cottingdon, Singleterry and Fowkes1964; Daillant et al. Reference Daillant, Benattar, Bosio and Leger1988; Cazabat et al. Reference Cazabat, Gerdes, Valignat and Villette1997; Popescu et al. Reference Popescu, Oshanin, Dietrich and Cazabat2012). In § 5, we will revisit existing datasets of Cazabat & Cohen Stuart (Reference Cazabat and Cohen Stuart1986) and de Ruijter et al. (Reference de Ruijter, De Coninck and Oshanin1999) for droplet spreading over smooth substrates.

Figure 2(c) demonstrates the evolution of $R_1$ with time, and figure 2(d) shows the evolution of $R_2$ . Measurements for the surfaces of different roughness are represented with different symbols (black triangles for S100, black circles for S60). For all observed times $t$ , the droplet radius $R_1$ is large ( $R_1 \gg \ell _c = 1.5$ mm). Consistent with previous studies (Lopez et al. Reference Lopez, Miller and Ruckenstein1976; Cazabat & Cohen Stuart Reference Cazabat and Cohen Stuart1986; Voinov Reference Voinov1999), fits of $R_1(t)$ in this large-droplet regime demonstrate a power-law trend with an exponent of $1/8$ , independent of the surface roughness. This fit is shown with a solid blue curve in figure 2(c). A dashed green curve with a power-law exponent of $1/10$ , the predicted scaling for small droplets (Voinov Reference Voinov1976; Tanner Reference Tanner1979; Hervet & de Gennes Reference Hervet and de Gennes1984), is shown for comparison.

For all values of $t$ and $k$ , $R_2$ exhibits excellent agreement with a function of the form $R_2(t) = At^{3/8}/[\log (Bt+C)]^{1/2}$ . These fits are shown with red solid curves in figure 2(d) for both surfaces. For the surface S100, the parameters $A = 2.99\;\text{mm}\;\text{s}^{-3/8}$ , $B = 0.63 \;\text{s}^{-1}$ and $C = 2.62$ produce good overlap with the measurements; the relative r.m.s. deviation between the measurements and the fit is 0.68 %. For the surface S60, the fitting coefficients $A = 3.80 \;\text{mm}\;\text{s}^{-3/8}$ , $B = 1.33 \;\text{s}^{-1}$ and $C = 3.79$ yield a r.m.s. deviation of 0.39 %. We note that the numerical value of $C$ is negligible beyond $t\gtrsim 10$ s; we rationalise the observed scalings of $R_1$ and $R_2$ in § 4 and the observed values of $A$ and $B$ in Appendix A. For the sake of comparison, we show in figure 2(e) that neither set of experiments satisfies the $\Delta R = R_2-R_1\sim t^{1/2}$ scaling predicted by the classic version of Washburn’s law (see § 2.5).

A few features of figures 2(c) and 2(d) merit further comment. First, we note that the droplet and film radii can only be clearly distinguished beyond $t\sim 5$ – $10$ s, indicating that the Darcy precursor film is not fully developed until that time. Similarly, the droplet depth becomes comparable to the roughness scale when $t\gtrsim 400$ s for S100 and when $t\gtrsim 100$ s for S60. We believe these limitations account for the slight deviations of both $R_1(t)$ and $R_2(t)$ from predictions for $t\lesssim 5$ s and those of $R_1(t)$ for $t\gtrsim 400$ s. Next, the evolution of $R_2$ depends on the substrate roughness $k$ ; as the substrate roughness decreases from S60 to S100, $R_2$ also decreases for all $t$ . Physically, this trend reflects the fact that resistance to flow decreases with greater substrate permeability. Finally, figure 2(d) highlights the need for the $(\log t)^{1/2}$ term in the denominator of the scaling for $R_2(t)$ ; the spreading is not well described by a $t^{3/8}$ scaling.

3.3. Dynamic similarity

We here present the results of a secondary series of experiments, in which we compare the spreading rates of silicone oils of different viscosities on the same rough substrate S100. We perform experiments using $\nu = 10$ and $\nu =50$ cSt silicone oil, to complement our data for $\nu = 5$ cSt silicone oil. The material properties of all three fluids are listed in table 1. The same experimental setup is used, but with longer acquisition times. For the $\nu = 10$ cSt silicone oil, 600 images are collected at an acquisition rate of 0.5 frames per second, equivalent to 20 min of data collection. For the $\nu = 50$ cSt silicone oil, 600 images are collected at an acquisition rate of 0.1 frames per second, equivalent to 100 minutes of data collection.

Table 1.

Material properties of the different test fluids in § 3.3.

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 $\nu = 5$ , $10$ and $50$ cSt on the surface S100. When non-dimensionalised 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 $C = 2.50$ .

Figures 3(a) and 3(b) illustrate the evolution of the droplet radius $R_1(t)$ and the Darcy precursor film radius $R_2(t)$ , respectively, for silicone oils of viscosity $\nu = 5$ , $10$ and $50$ cSt on the roughened surface S100. When the radii $R_1(t)$ and $R_2(t)$ are normalised with respect to the capillary length $\ell _c$ and plotted with respect to a dimensionless time $\hat {t} =t \ell _c/u_c$ , the measurements overlap closely, indicating that the spreading dynamics is dynamically similar. As before, the droplet radius exhibits a $R_1(\hat {t})/\ell _c\sim \hat {t}^{1/8}$ scaling, and the Darcy precursor film fits follows the trend ${R}_2( \hat {t})/\ell _c = A \hat {t}^{3/8}/[\log (B \hat {t}+C)]^{1/2}$ . Fits of the data yield dimensionless fitting coefficients of $A = 0.10$ , $B = 2.0 \times 10^{-4}$ and $C = 2.50$ . Because all three experiments are dynamically similar to the $\nu = 5$ cSt case, the discussion in Appendix A can be used to rationalise these coefficients.

4.1. Dynamics of the droplet edge

We first derive an expression for the speed at which the droplet’s edge advances, using a simple scaling argument. The present section simplifies and generalises the argument of Hervet & de Gennes (Reference Hervet and de Gennes1984) presented in § 2.2.

We suppose here that viscous dissipation is not necessarily confined to the vicinity of the droplet edge, as posited by Hervet & de Gennes (Reference Hervet and de Gennes1984), but that some component of the dissipation may occur throughout the droplet bulk. The total dissipation may thus be expressed as

(4.1) \begin{align} D = D_{\textit{edge}} + D_{\textit{bulk}}. \end{align}

Lifting the assumption of edge-localised dissipation is natural in the Stokes-flow limit, where one expects instantaneous equilibration of the flow field throughout the droplet bulk; indeed, splitting the total energy dissipation into ‘edge’ and ‘bulk’ contributions was previously suggested by de Ruijter et al. (Reference de Ruijter, De Coninck and Oshanin1999) and Durian et al. (Reference Durian, Dillavou, Markin, Portales, Maldonado, Irvine, Arratia and Durian2022) for the case of partial wetting, and bulk dissipation was assumed in the gravity-driven model of Lopez et al. (Reference Lopez, Miller and Ruckenstein1976). For a simple, concrete flow field that exhibits the properties we seek, one could imagine that, away from the edges, a large droplet would evolve as a slow Poiseuille-type current of the form

(4.2) \begin{align} u(r,z) \sim U\frac {r}{R_1}\frac {z}{h_1}\left (2-\frac {z}{h_1}\right ). \end{align}

We emphasise that we do not prescribe such a flow profile. The model we present relies only on scaling arguments, and thus rationalises the observed spreading rates of viscous droplets without detailed knowledge of the droplets’ flow profiles.

Suppose $\varPhi$ is the average local rate of energy dissipation throughout the droplet bulk, and $U\sim \dot {R}_1$ is the rate of spreading. Then the bulk dissipation rate is given by

(4.3) \begin{equation} D_{\textit{bulk}} \sim \alpha V\varPhi \sim \alpha \mu h_1 R_1^2 \frac {U^2}{h_1^2} \sim \frac {\alpha \mu R_1^2I_1(R_1)U^2}{(I_0(R_1)-1)\tan \theta }, \end{equation}

for some constant $\alpha \gt 0$ . The above formula is deduced by substituting $V\sim h_1R_1^2$ and $h_1 = h|_{r=0}$ , noting that $V\sim h_1R_1^2$ holds for both small and large droplets. We remark that a similar result would follow if we instead used the full expression (2.8) to estimate the volume in (4.3); in particular, the asymptotic limits of our model would remain unchanged.

An expression for $D_{\textit{edge}}$ can be derived using the analytical argument of Hervet & de Gennes (Reference Hervet and de Gennes1984), presented in § 2.2. For the sake of completeness, however, we illustrate how the form of $D_{\textit{edge}}$ can be deduced from a similar scaling argument as that of $D_{\textit{bulk}}$ . If dissipation occurs within a distance $L$ of the edge, then it occurs over the solid of rotation carved out by a triangle of height $H=L\tan \theta$ and area $A = {1}/{2}L^2\tan \theta$ ; this solid has a volume $V_{\textit{edge}}\sim R_1HL$ , and we expect the total dissipation within it to take the form

(4.4) \begin{equation} D_{\textit{edge}}\sim \beta V_{\textit{edge}}\varPhi _{\textit{edge}} \sim \beta \mu R_1HL\frac {U^2}{H^2} = \beta \mu R_1\frac {U^2}{\tan \theta }, \end{equation}

for some constant $\beta \gt 0$ . Setting $\beta = \log (L/a)$ recovers the result of Hervet and de Gennes (see § 2.2). We remark that, a priori, there might also be a contribution of the form (2.15) proposed by de Ruijter et al. (Reference de Ruijter, De Coninck and Oshanin1999) and Durian et al. (Reference Durian, Dillavou, Markin, Portales, Maldonado, Irvine, Arratia and Durian2022), dependent only on the spreading speed $U$ and the arclength $2\pi R_1$ of the apparent contact line and independent of the apparent contact angle $\theta$ . Such a contribution might correspond to micro-physics at the droplet–film interface, at length scales $\ell \ll L$ ; for the analogous case of partially wetting drops on rough substrates, for instance, one expects resistance to arise from microscopic corrugation of the contact line (de Gennes et al. Reference de Gennes, Brochard-Wyart and Quere2003). In all, the edge-localised dissipation would take the more general form

(4.5) \begin{align} D_{\textit{edge}}\sim \beta \mu R_1\frac {U^2}{\tan \theta } + \kappa \mu R_1 U^2, \end{align}

for some constant $\kappa \gt 0$ . In any case, the total energy dissipation $D$ takes the form

(4.6) \begin{equation} D = D_{\textit{bulk}} + D_{\textit{edge}} \sim \left (\alpha \frac {R_1^2I_1(R_1)}{(I_0(R_1)-1)\tan \theta } + \beta \frac {R_1}{\tan \theta } + \kappa R_1\right )\mu U^2. \end{equation}

One can simplify this expression in our limits of interest. First, we assume that $\beta R_1/\tan \theta \gg \kappa R_1$ , because $\theta \ll 1$ . This assumption can otherwise be justified by considering that, in the small-droplet limit, the $\beta$ term must be dominant in order to recover the widely corroborated scaling predictions of Hoffman (Reference Hoffman1975), Voinov (Reference Voinov1976) and Tanner (Reference Tanner1979). Secondly, the remaining contribution of $D_{\textit{edge}}$ does not change the asymptotic form of $D$ in either the large- or small-droplet limit. In the limit $R_1\gg \ell _c$ , the $\alpha$ term is quadratic in $R_1$ and thus dominates the edge dissipation. In the limit $R_1\ll \ell _c$ , both the $\alpha$ and the $\beta$ terms are linear in $R_1$ , and one can simply augment $\alpha \mapsto \alpha +\beta$ to account for both. Since we are ultimately interested in these asymptotic limits, we proceed as though edge dissipation is accounted for in the value of $\alpha \gt 0$ for small droplets.

The work done in moving the contact line is $W \sim \sigma (1-\cos \theta ) R_1U$ . The droplet’s internal energy balance is given by $D\sim W$ , which upon rearrangement yields

(4.7) \begin{equation} \frac {R_1I_1(R_1)}{I_0(R_1)-1}\,\dot {R}_1 = 2\alpha ^{-1}(\tan \theta - \sin \theta )\sim \alpha ^{-1}\theta ^3, \end{equation}

for $\theta \lesssim \pi /4$ and the same constant $\alpha \gt 0$ as above. We employ the asymptotic formulas (2.9) to deduce the small- and large-drop limits of this result. For small droplets ( $R_1\ll \ell _c$ ), we have $R_1I_1(R_1)/[I_0(R_1)-1]\to 2=\textrm{const}.$ , so (4.7) recovers the $\dot {R}_1\sim \theta ^3$ scaling of Hoffman (Reference Hoffman1975). For large droplets, $R_1I_1(R_1)/[I_0(R_1)-1]\to R_1$ , so (4.7) yields the modified scaling $\dot {R}_1\sim \theta ^3/R_1$ .

Using the expression (2.8) for the droplet volume to eliminate $\theta$ , we find

(4.8) \begin{equation} \dot {R}_1 =\alpha ^{-1}\,\frac {I_0(R_1)-1}{R_1I_1(R_1)}\,\Bigg (\frac {V/\pi }{R_1^2\frac{I_0(R_1)}{I_1(R_1)} - 2R_1}\Bigg )^{3}. \end{equation}

We note that the liquid volume that drains into the precursor film (over either smooth or rough substrates) is negligible over the time scales of interest. Indeed, because the roughness (or microscopic) scale $h_2$ is small relative to the droplet depth $h_1$ , the liquid volume contained within the film is negligible with respect to that within the droplet,

(4.9) \begin{equation} V_{\textit{film}} = h_2R_2^2\ll h_1R_1^2\sim V, \end{equation}

noting that the scaling $V\sim h_1R_1^2$ holds for both small and large droplets. Provided (4.9) holds, the change in droplet volume $V$ is negligible, so (4.8) yields an expression for the evolution of the droplet radius. We note that the system could alternatively be closed by an equation of the form $\dot {V} = -\dot {V}_{\textit{film}}=-2\pi \phi h_1(R_1\dot {R}_1 - R_2\dot {R}_2),$ where $\phi$ is the porosity of the rough substrate (and $\phi =1$ for smooth substrates). We do not pursue this direction further at present, and instead take $V$ to be constant.

The physical picture we propose can be summarised as follows, for both large and small drops. At all times, hydrostatic and curvature pressure gradients are in balance throughout the droplet, except in the immediate vicinity of the apparent contact line. At the apparent contact line, interfacial forces act to stretch the droplet uniformly outward. The work done by these interfacial forces is dissipated inside the droplet, and the associated energy balance determines the rate of spreading.

The basic physical picture is common to both small and large droplets; different scalings arise in the two limits only because the relationship between $V$ , $\theta$ and $R_1$ changes according to (2.8). In the small-droplet limit $R_1\ll \ell _c$ , the asymptotic formulas (2.9) show that the right-hand side of (4.8) scales as $V^3/R_1^{9}$ , so the evolution equation reduces to the $R_1(t) \sim V^{3/10}t^{1/10}$ scaling of Hoffman (Reference Hoffman1975), Voinov (Reference Voinov1976) and Tanner (Reference Tanner1979). For large droplets, the right-hand side of (4.8) scales as $V^3/R_1^{7}$ , which yields a scaling $R_1(t)\sim V^{3/8}t^{1/8}$ familiar from viscous gravity currents. We summarise the small- and large-droplet limits of our model in table 2, in both one- and two-dimensional geometries. Since the derivation presented here depends on scaling arguments, one should expect (4.8) to hold in both limits, but not necessarily to yield exact predictions for the small-to-large-droplet transition. We return to the latter question in § 5.

Table 2.

Scalings predicted by our model in various limits of interest, for both one-dimensional and axisymmetric two-dimensional systems. Here, $R_1$ , $h_1$ and $\theta$ are the horizontal radius, depth and apparent contact angle of the droplet bulk, $R_2$ is the radius of the Darcy precursor film on a rough substrate and $\Delta R=R_2-R_1$ . The scalings reported for $\dot {R}_1$ , $R_1$ , $h_1$ and $\theta$ are predicted to hold for droplets totally wetting either smooth or rough substrates, but we clarify that we do not predict the spreading rate of a microscopic precursor film on a smooth substrate. The $\dot {R}_1\sim \theta ^3$ scaling of Hoffman (Reference Hoffman1975) and Hervet & de Gennes (Reference Hervet and de Gennes1984) arises in the small-droplet limit in both one- and two-dimensional systems. The $R_1\sim V^{3/10}t^{1/10}$ and $\theta \sim V^{1/10}t^{-3/10}$ laws of Voinov (Reference Voinov1976) and Tanner (Reference Tanner1979) arise in the small-droplet limit in two dimensions. The $R_1\sim V^{3/7}t^{1/7}$ scaling of Tanner (Reference Tanner1979) and McHale et al. (Reference McHale, Newton, Rowan and Banerjee1995) arises in the small-droplet limit in one dimension. The $\Delta R\sim t^{1/2}$ law of Washburn (Reference Washburn1921) arises in the limit of a tension-driven film dynamics. The $R_1\sim V^{3/8}t^{1/8}$ scaling observed by Lopez et al. (Reference Lopez, Miller and Ruckenstein1976) arises for large droplets in two dimensions.

We note that our predicted $R_1\sim V^{3/8}t^{1/8}$ scaling for large droplets carries the same prefactor as that of gravity currents. Indeed, reintroducing units to our scaling yields

(4.10) \begin{align} R_1 \sim \ell _c\big(V/\ell _c^3\big)^{3/8}(t u_c/\ell _c)^{1/8} = \ell _c^{-1/4}u_c^{1/8}V^{3/8}t^{1/8} = (\rho g/\mu )^{1/8}V^{3/8}t^{1/8}. \end{align}

Notably, despite the dynamics of our model being driven by interfacial forces, $\sigma$ necessarily drops out from the final scaling. We can rationalise this phenomenon as follows. Recall that the horizontal contact line force per unit arclength is $f\sim \sigma \theta ^2$ . If one fixes the droplet radius $R_1$ and depth $h_1$ , then the equilibrium droplet profile (2.7) shows that the contact angle decreases with increasing surface tension, as $\theta \sim h_1/\ell _c\sim h_1(\rho g/\sigma )^{1/2}$ . Thus, the total interfacial force scales as

(4.11) \begin{align} f\sim \sigma \theta ^2\sim \sigma (h_1/\ell _c)^2\sim \rho g h_1^2, \end{align}

independent of $\sigma$ . As a consequence, when $R_1\gg \ell _c$ , one cannot distinguish our edge-driven ‘capillary current’ model from traditional gravity currents on the basis of $R_1(t)$ alone. Specifically, capillary currents (for which $\textit{Bo}\lesssim 1$ ) that are wide with respect to the capillary length obey precisely the same surface-tension-independent spreading law (1.1) satisfied by viscous gravity currents (for which $\textit{Bo}\gg 1$ ). We distinguish between the two models in the following subsection, by testing their respective predictions for the evolution of a Darcy precursor film when a droplet spreads on a rough substrate.

4.2. Dynamics of a Darcy precursor film on a rough substrate

We model the Darcy precursor film as a viscous, two-dimensional flow through a homogeneous, porous medium, subject to Darcy’s law (2.16). Here, we modify the statement of Washburn’s law (2.18) to account for the pressure gradient across the precursor film, associated with overpressure in the droplet bulk. Similar calculations were carried out by Clarke et al. (Reference Clarke, Blake, Carruthers and Woodward2002) and Starov et al. (Reference Starov, Kostvintsev, Sobolev, Velarde and Zhdanov2002a , Reference Starov, Zhdanov, Kosvintsev, Sobolev and Velarde2003) for small droplets on deep and shallow porous substrates, respectively, but we here give a more detailed account of the time-evolving overpressure within the droplet.

Following the same argument as § 2.5, we know that the local mean velocity $\overline {u}$ and volumetric force $f$ within the precursor film must satisfy ${\overline {u}},f\propto 1/r$ . These are two of the three terms appearing in Darcy’s law (2.16), so we deduce that the third term must have the same dependence, i.e. $\boldsymbol{\nabla }p\propto 1/r$ . Matching the pressure conditions at the leading and trailing edge of the film, $p|_{r=R_2} = 0$ and $p|_{r=R_1}=-\sigma \boldsymbol{\nabla} ^2h(R_1)$ , we find

(4.12) \begin{align} p = -\sigma \big(\boldsymbol{\nabla} ^2 h\big)(R_1)\,\frac {\log (r/R_2)}{\log (R_1/R_2)} =\frac {\sigma V/\pi }{R_1^2 -2R_1\frac{I_1(R_1)}{I_0(R_1)}}\,\frac {\log (r/R_2)}{\log (R_1/R_2)}. \end{align}

Inserting this expression into (2.16) and using the expression (2.17) for $f$ , we deduce a modified evolution equation for the radius of the precursor film

(4.13) \begin{align} R_2\dot {R}_2 = \frac {k}{\phi }\left (\frac {V/\pi }{(R_1^2-2R_1\frac{I_1(R_1)}{I_0(R_1)})\log (R_2/R_1)} + \frac {sR_2}{R_2-R_1}\right ). \end{align}

We note that our theory is based on the assumption of a precursor film, in which case $\Delta R=R_2-R_1\gt 0$ .

Depending on surface chemistry, the system obeys one of two different dynamics. First, in the limit where surface tension dominates (roughly, if $\sigma s\gg \rho g\ell _c$ ), we recover the classic variant of Washburn’s law, derived in § 2.5. In the limit where hydrostatic pressure dominates ( $\sigma s\ll \rho g\ell _c$ ), we approximate $\log (R_2/R_1)\sim \log R_2$ in order to write

(4.14) \begin{equation} \log (R_2) R_2\dot {R}_2\sim \frac {k}{\phi }\frac {V}{R^2_1 - 2R_1\frac{I_1(R_1)}{I_0(R_1)}}. \end{equation}

In the large-droplet limit $R_1\gg \ell _c$ , for instance, we find $\log (R_2) R_2^2 \sim (k/\phi )\alpha ^{1/4}V^{1/4} t^{3/4}$ . In general, if $y^n\log y = x$ , then $y = \exp (n^{-1}W(nx))$ , where $W$ is the Lambert $W$ -function (Corless et al. Reference Corless, Gonnet, Hare, Jeffrey and Knuth1996). Since $W(x)\sim \log x - \log \log x$ for large arguments, we find

(4.15) \begin{equation} R_2(t) \sim \frac {(k/\phi )^{1/2}\alpha ^{1/8}V^{1/8}t^{3/8}}{\sqrt {\log [(k/\phi )\alpha ^{1/4}V^{1/4}t^{3/4}]}}\sim \frac {t^{3/8}}{\sqrt {\log {t}}}. \end{equation}

Our experimental results in § 3 are consistent with this scaling, and would appear to disqualify simpler power-law scalings. The small-droplet limit follows similarly, giving $R_2\sim t^{3/10}/(\log {t})^{1/2}$ . All scalings are reported in table 2.

We note that the logarithmic correction to our predicted $R_2$ scaling is necessary to rationalise our experimental results in § 3. This correction arises from the circular geometry of the system, as it does in the scalings found for coalescence of circular droplets in the work of Eggers, Lister & Stone (Reference Eggers, Lister and Stone1999). Indeed, the one-dimensional equivalent of our system – corresponding to the ‘fluid stripe’ experiments of McHale et al. (Reference McHale, Newton, Rowan and Banerjee1995) – gives a modified power law with no logarithmic factors.

Finally, we remark that the $R_2\sim t^{3/8}/(\log t)^{1/2}$ scaling regime is not consistent with the gravity-driven hypothesis introduced by Lopez et al. (Reference Lopez, Miller and Ruckenstein1976) and currently understood to underlie the $R_1\sim V^{3/8}t^{1/8}$ scaling of large droplets (Bonn et al. Reference Bonn, Eggers, Indekeu, Meunier and Rolley2009; Popescu et al. Reference Popescu, Oshanin, Dietrich and Cazabat2012). Gravity-driven spreading implies a horizontal pressure gradient across the droplet radius, and thus a relatively low pressure $p|_{r=R_1}\ll \rho gh_1$ at the edge. By contrast, the $R_2\sim t^{3/8}/(\log t)^{1/2}$ scaling regime arises owing to hydrostatic overpressure $p|_{r=R_1}\approx \rho gh_1$ at the droplet edge, consistent with our quasi-equilibrium model. In demonstrating the existence of an $R_2\sim t^{3/8}/(\log t)^{1/2}$ film spreading regime, our experiments in § 3 would thus appear to distinguish the large-droplet limit of our edge-driven model from gravity currents.