## Introduction

The spreading of viscous droplets on solid substrates has been extensively studied over the last decades1,2,3. For droplet sizes smaller than the capillary length2, the viscocapillary regime yields self-similar asymptotic dynamics, i.e. the so-called Tanner’s law4, with the droplet radius R increasing in time t as ~t1/10. To establish this scaling, two ingredients are invoked: global volume conservation, and a local balance at the contact line between driving capillary forces and viscous dissipation in the liquid wedge. However, such a continuous description implies a finite change of the fluid velocity over a vanishing height at the contact line, and thus leads to an unphysical divergence of viscous stress and dissipation5. To solve this paradox, a microscopic molecular-like cut-off length is required, and appears through a logarithmic factor in Tanner’s law. In this spirit, theoretical and experimental investigations introduced various possible regularization mechanisms1,6,7,8,9,10,11,12,13,14, including a gravito-capillary transition15,16,17,18, surface roughness17, thermal effects19, Marangoni-driven flows20, diffusion21,22, or a slip condition at the solid substrate23. In the particular case of total wetting, the existence of a thin precursor film ahead of the contact line has been proposed as the main candidate24,25,26. However, despite tremendous efforts to measure the microscopic length, or to characterize the associated logarithmic factor, the problem is still open. Conversely, solving the free-interface dynamical evolution of a droplet-like perturbation on a thin liquid film in the lubrication approximation showed that Tanner’s law can be considered as a negligible-film-thickness limit of capillary levelling27. Such a statement was further comforted by its extension to the gravity-driven28 and elastic-bending-driven29 cases. As such, it is possible to unambiguously determine the microscopic precursor-film thickness from the spreading of any droplet in total-wetting and lubrication conditions.

In this article, we investigate droplet spreading in a near-critical phase-separated binary liquid30, with four main objectives. First and most importantly, close to a fluid-fluid critical point, an isotropic liquid belongs to the {d = 3, n = 1} universality class of the Ising model, where d and n are, respectively, the space and order-parameter dimensions, so that the results are immediately generalizable to any fluid belonging to the same universality class. Secondly, critical phenomena are often accompanied by a wetting transition at a temperature which might be either identical or distinct from the critical one3, so precursor films can also be investigated near the critical point. Thirdly, as many fluid properties vary with the proximity to a critical point according to power-law variations of the type $$\sim {\left({{\Delta }}T/{T}_{{{{{{{{\rm{c}}}}}}}}}\right)}^{\alpha }$$, with ΔT = T − Tc the temperature distance to the critical point Tc, and α some positive or negative critical exponent, the spreading dynamics may be continuously and precisely tuned by varying the temperature. Finally, our study also provides evidence for droplet spreading in a liquid environment, which was scarcely studied31.

## Results

The experimental configuration is depicted in Fig. 1. We use a water-in-oil micellar phase of microemulsion32,33, as described in detail in the Supplementary Method 1 section. Briefly, at the chosen critical composition (water, 9% wt, toluene, 79% wt, SDS, 4% wt, butanol, 17% wt), it exhibits a low critical point at Tc close to 38 C, above which the mixture separates into two phases of different micelle concentrations, Φ1 and Φ2 with Φ2 < Φ1 (see Fig. 1a). The microemulsion is enclosed in a tightly-closed fused-quartz cell of 2 mm thickness (Hellma 111-QS.10X2) which is introduced in a home-made thermally-controlled brass oven with four side-by-side windows. As working in a tight cell is mandatory with critical fluids, we use a contactless optical method to create a wetting drop at the bottom wall. Note that the microemulsion is transparent (absorption coefficient smaller than 4.6 × 10−4 cm−1) at the employed wavelength, which prevents any laser-induced heating effect. The sample is set at a temperature T > Tc and a continuous frequency-doubled Nd3+ − YAG (wavelength in vacuum λ = 532 nm, TEM00 mode) laser beam is focused on the meniscus of the phase-separated mixture using a × 10 Olympus® microscope objective (NA = 0.25). The photon momentum mismatch between the two phases, proportional to the refraction-index contrast, generates a radiation pressure, and the interface thus bends (see Fig. 1b) as a result of the balance between the latter with hydrostatic and Laplace pressures34. As the interfacial tension γ of near-critical interfaces vanishes at the critical point, with $$\gamma={\gamma }_{{{{{{{{\rm{0}}}}}}}}}{\left({{\Delta }}T/{T}_{{{{{{{{\rm{c}}}}}}}}}\right)}^{2\nu }$$, where ν = 0.63 and γ0 = 5.0 × 10−5 N/m in our case, the interfacial deformation can be made very large. When the beam propagates downwards and with sufficient power, the interface can become unstable (see Fig. 1c) due to the total reflection of light within the deformation. In this case, a jet is formed, with droplets emitted at the tip35,36. Note that the jetting power threshold can also be used to measure the interfacial tension35. The length of the jet can be tuned with the laser power to bring its tip close to the bottom of the cell, without touching it. Then, by reducing the power, the jet breaks up into many droplets due to the Rayleigh-Plateau instability. By increasing again the power, below the jetting threshold, the laser beam forces coalescence between several droplets to produce a large one which can be further pushed by radiation pressure towards a borosilicate substrate placed at the bottom of the cell (see Fig. 1d). We turn off the laser just before contact, and follow the droplet spreading using ×20 or ×50 Olympus® microscope objectives, with resolutions of 1.0 and 0.8 μm respectively, and a Phantom® VEO340L camera for the frame grabbing. Note the existence of a prewetting film on the substrate, at least up to ΔT = 15 K37.

Figure 1d displays two image sequences corresponding to the coalescence and spreading of droplets at ΔT = 8 and 1 K. The spreading time scale comparatively increases by approximately one order of magnitude for ΔT = 1 K, as a result of the vanishing interfacial tension near Tc. We also notice that the droplet volumes reduce over time, indicating the presence of evaporation, as expected for finite-size objects in an environment at thermodynamic equilibrium. We stress here that we employ the standard terminology “evaporation" all along the article, despite the outer fluid is not a “vapor" but another liquid-like system. At early stages, both profiles display large curvature gradients. Since our focus here is on the long-term asymptotic spreading behavior, we define the temporal origin t = 0 from a first experimental image where the curvature is homogeneous, except near the contact-line region, and a spherical-cap fit is valid.

Each image sequence is then treated using a custom-made automatized contour detection based on a Canny-threshold algorithm, where the droplet profiles correspond to the external maxima of the intensity gradients (see Supplementary Method 2). Spherical-cap fits allow to extract the droplet volume V(t), radius R(t), and apparent contact angle θ(t), which are then averaged using a custom-made exponentially-increasing time window to get a logarithmically-distributed data set (see Supplementary Method 2). In the inset of Fig. 2a, we plot the experimental droplet volume as a function of time, for four different values of ΔT. In all cases, the volume decreases until the droplet is fully evaporated. By using the initial volumes V0 and by extrapolating the times tf of final evaporation for all droplets, we then plot in the main panel the same data in dimensionless form, with V/V0 as a function of 1 − t/tf. We observe a data collapse onto a unique power-law behavior, with fitted exponent 1.77, which is close to the 11/7 value theoretically predicted for evaporating droplets12. In Fig. 2b, we further plot the contact radius R, normalized by its initial value R0, as a function of time, for all ΔT. A Tanner-like power law systematically emerges at intermediate times, until evaporation eventually dominates the evolution.

To model the observed spreading dynamics, we consider a large initial droplet-like interfacial perturbation profile d(r, t = 0), with r the radial coordinate, atop a flat thin-film of thickness ϵ, and describe its evolution through the profile d(r, t) at all times, in the small-slope limit within the lubrication approximation2. Therein, a horizontal Newtonian viscous flow of viscosity η is driven by the gradient of Laplace pressure. Since most of the dissipation occurs in the wedge-like region near the apparent contact line5, we further make an approximation and neglect the influence of viscous shear stresses in the surrounding phase. Note that this hypothesis would not hold if the atmospheric fluid was much more viscous than the droplet fluid31. Fortunately, this is not the case near the critical point, where both viscosities are almost equal. The evolution is then described by the axisymmetric capillary-driven thin-film equation27:

$${\partial }_{t}h+\frac{\gamma }{3\eta r}{\partial }_{r}\left[r{h}^{3}{\partial }_{r}\left(\frac{1}{r}{\partial }_{r}h+{\partial }_{r}^{2}h\right)\right]={{{{{{{\mathcal{H}}}}}}}}(1-R)f\,,$$
(1)

where h(r, t) = ϵ + d(r, t) is the free-interface height from the solid substrate, $${{{{{{{\mathcal{H}}}}}}}}(1-R)$$ is the Heaviside step function with R(t) the advancing radius of the droplet, and f(t) is an added coefficient accounting for evaporation. The latter is chosen in order to precisely mimic the experimentally-measured evaporation of the droplet (see Fig. 2a). Note that the Heaviside function ensures that the prewetting film—which is at thermodynamical equilibrium in contrast to the droplet—does not evaporate. Equation (1) is numerically integrated using a finite-element solver38. The experimental radial profiles depicted in Fig. 1d are chosen as initial profiles, after angular averaging and smoothening using fourth-order polynomials in order to avoid unphysical fluctuations related to the camera resolution and the contour-detection algorithm. As shown in Fig. 2c, d, the comparisons between the experimental and numerical evolutions reveal an excellent agreement. As the capillary velocity vcap = γ/η is independently evaluated (see ref. 32 and Supplementary Method 1 for the viscosity calibration), and typically varies between 22 and 1013 μm/s for near-critical droplets within the ΔT = 1–8 K range, the precursor-film thickness ϵ remains the only fit parameter in this comparison, and its behavior with temperature will be discussed after.

Tanner’s law1,4 can be obtained from the combination of (i) a local power balance between capillary driving and viscous damping near the contact line; and (ii) global volume conservation. The former power balance reduces to the Cox–Voinov’s law for total-wetting conditions:

$${\theta }^{3}=9\ell {{{{{{{\rm{Ca}}}}}}}}\,,$$
(2)

where $${{{{{{{\rm{Ca}}}}}}}}=\dot{R}/{v}_{{{{{{{{\rm{cap}}}}}}}}}$$ is the capillary number, and $$\ell=\ln (L/\epsilon )$$ is the logarithmic factor discussed in the introduction relating the two cut-off lengths of the problem, namely a typical macroscopic size L of the system and a microscopic length which is identified to the precursor-film thickness ϵ in total-wetting conditions.

To disentangle evaporation, through the V(t) behavior obtained in Fig. 2a, from the spreading dynamics, we introduce the following dimensionless variables:

$$\tilde{R}=R{\left[\frac{\pi }{4V(t)}\right]}^{1/3}\quad,\quad \tilde{t}={v}_{{{{{{{{\rm{cap}}}}}}}}}t{\left[\frac{\pi }{4V(t)}\right]}^{1/3}\,.$$
(3)

Tanner’s law1,4 is then written in dimensionless form as:

$${\tilde{R}}^{10}-{\tilde{{R}_{{{{{{{{\rm{0}}}}}}}}}}}^{10}=\frac{10}{9\ell }\tilde{t}\,,$$
(4)

with $$\tilde{{R}_{{{{{{{{\rm{0}}}}}}}}}}=\tilde{R}(t=0)$$. In Fig. 3a, we plot the rescaled contact radius as a function of rescaled time, for various temperatures. We systematically observe Tanner behaviors. Interestingly, from the fits to Eq. (4), we obtain increasing values of as the critical point is neared. This trend is further confirmed in Fig. 3b, where we see Cox–Voinov behaviors (see Eq. (2)) at large-enough Ca, with an identical evolution of with ΔT. The departure from Cox–Voinov’s law at low Ca is due to the evaporation-induced non-monotonic behavior of R(t) (see Fig. 2b), resulting in Ca crossing zero for finite values of θ.

Using the values obtained from the Tanner fits above, we can then represent all the data onto a single master curve, as shown in Fig. 4a. The observed collapse, over more than two orders of magnitude in time and broad ranges of material parameters, shows the surprising robustness of Tanner’s law in the vicinity of the critical point, despite the increasing roles of evaporation, gravity, and fluctuations.

Finally, Fig. 4b shows the extracted precursor-film thickness ϵ as a function of ΔT/Tc. Strikingly, over the considered temperature range, we observe a sharp decrease of ϵ from a fraction of micrometers down to a nanometer, as the critical point is approached from above. Over a decade in the considered temperature range, this behavior is consistent with the empirical power law $$\epsilon=a{\left({{\Delta }}T/{T}_{{{{{{{{\rm{c}}}}}}}}}\right)}^{2.69}$$, where a = 5.54 mm. The wetting transition in our system being located far above the largest temperature studied here37, we could have instead expected an increase of ϵ, since the precursor film is here made of the most-wetting phase3,39. Nevertheless—provided that one can extrapolate the definition of interfaces towards the critical point—the spreading parameter1 is expected to strictly vanish at that point since the two fluid phases become indistinguishable media, which might be the reason underlying the observed behavior. Beyond revealing the universality and peculiarities of viscous spreading near a critical point, our results pave the way toward further investigations closer to that point, with the aim of addressing the increasing contributions of gravity, evaporation, and eventually thermal fluctuations40,41,42.

## Methods

Supplementary Method 1 presents the near-critical properties of the binary liquid used and Supplementary Method 2 the technical details for the edge detection of interfaces and data acquisition from image analysis.