Superhydrophobic frictions

• superhydrophobic
• friction
• drops
• dissipation
• velocity

Despite its low viscosity, water running down tilted solids is surprisingly lazy. Its worm’s pace arises from contact line, whose presence induces pining and magnifies viscous dissipation, which contributes to slow down and even stop the liquid (1, 2). In contrast, water on superhydrophobic (SH) materials move at unrivaled speeds, owing to the conjunction of minimized pining and maximized contact angle. While drops on tilted plastic or glass immediately reach a velocity of typically 1 cm/s (3, 4), water on nonwetting materials speeds up by decimeter-size or meter-size distances (5⇓⇓–8) before reaching a terminal speed U as high as a few meters per second. In such Galileo-like experiments, the drop is subjected to an acceleration g sinα, denoting g the acceleration of gravity and α the tilting angle, possibly diminished by the (weak) pining on the solid (9⇓⇓⇓⇓⇓–15). The stationary regime of descent is observed when the weight is balanced by the friction acting on the moving drop, a resistance that remains to be characterized on SH materials. Our aim in this paper is to deduce the nature of this friction from direct measurements, contrasting with previous studies performed in transient regimes (12, 13) or inside rotating SH cylinders (16). In all of the latter studies, friction was assumed to be simply viscous, which we question in this paper.

The substrates in our experiments are long brass bars rendered water-repellent by a spray of colloidal suspension of hydrophobic silica nanobeads in isopropanol (Glaco Mirror Coat Zero; Soft99). The resulting texture imaged by atomic force microscopy is shown in Fig. 1 A and B both in the plane of the material and perpendicular to it. The surface exhibits cavities and bumps at the scale of 100 nm, and the rms roughness deduced from atomic force microscopy (AFM) pictures is 35 ± 5 nm, which is comparable to the mean size of the silica nanobeads and to the average top-to-bottom distance in Fig. 1B. This simple and reproducible treatment allows us to coat long solids (around 2.5 m), a necessary condition for reaching the terminal velocity U of water drops (SI Appendix, Fig. S1). Starting with an initial acceleration g sinα, a typical distance of U²/2g sinα is needed to reach U, that is for U = 1 m/s and sinα = 0.1, about 1 m—a length significantly smaller than that of our inclines.

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Fig. 1.

Drops running down SH materials. (A) AFM picture of the material used in our experiments: x and y are the coordinates in the plane, and h is the depth. (B) AFM cross-sectional view along the red dotted line in A, around y = 200 nm. (C) Water drop descending a tilted SH plate in the regime where it reached its terminal velocity U = 66 cm/s. We superimpose colored images separated by 20 ms for a volume Ω = 100 µL, an equatorial radius R ∼ 3.7 mm and a tilt α = 2°. (See also Movie S1.) (D) Terminal velocity U of water–glycerol mixtures (Ω = 100 µL) as a function of their viscosity η. The tilt angle here is α = 2.3°, and the dashed line represents Eq. 3.

Advancing and receding angles of water are θ_a = 171 ± 2° and θ_r = 165 ± 2°, with the high angles and low hysteresis Δθ = θ_a − θ_r typical of SH materials (17). Due to the hydrophobic nature of the coating, water remains upon the roughness and contacts a mixture of solid and air, as evidenced by the silvery aspect of its base. Using the Cassie formula, we can deduce from contact angles the proportion ϕ of solid/water contact and find ϕ ∼ 3.5 ± 2.0%, a value much smaller than unity. Aging of our materials is quite slow, and the quality of nonwetting is regularly controlled. If degraded or damaged, the surface is simply regenerated by a new treatment.

An example experiment is shown in Fig. 1C, where we superimpose colored images of a water drop (volume Ω = 100 µL, surface tension γ = 72 mN/m, viscosity η = 1 mPa∙s, and density ρ = 1,000 kg/m³) running down a SH plate tilted by 2°. This figure is extracted from a high-speed movie shot at 2,000 frames per second. We first notice that the drop has reached its terminal velocity (here U = 66 cm/s). In addition, despite its high mobility, the drop keeps a quasi-static shape: it is slightly flattened by gravity as expected from its size, above the capillary length a = (γ/ρg)^1/2 (a³ ∼ 20 μL), and its front–rear symmetry evidences the small value of the hysteresis Δθ. Denoting R as the equatorial drop radius, the adhesion force opposing the motion scales as γR (cos θ_r − cos θ_a), which reduces to γR sin θ Δθ at small Δθ (3, 5). For Ω = 100 μL, adhesion of water on our materials (θ ∼ 168°, Δθ ∼ 6°) is small compared with the gravity force ρgΩsinα for α > 0.2°. Thus, the drop terminal speed directly results from a balance between the projected weight ρgΩsinα and the (unknown) friction force F(U).

The aim of this paper is to characterize the function F(U) and consequently to understand what fixes the terminal velocity of drops. Contrasting with the case of partial wetting, where friction is dominated by viscous effects around the contact line (2), the friction on SH materials was up to now assumed to arise from viscous effects inside the liquid (12⇓⇓⇓–16). We first check the influence of viscosity η by using water/glycerol mixtures, which provides variations from η = 1 mPa∙s to η = 1,490 mPa∙s. Fig. 1D presents the terminal speed U of drops as a function of η, for a tilt α equal to 2.3°. At large η, U strongly decreases as the liquid gets more viscous, which highlights the dominant role of viscosity in the resistance to motion. In contrast, viscous effects for η < 10 mPa∙s (including the important case of water) tend to become marginal, as seen from the tendency of the data to plateau. In what follows, we discuss these two successive regimes of friction.

Focusing first on the viscous case, we plot in Fig. 2A the terminal speed U of drops as a function of the slope sinα, for η = 110 mPa∙s and two volumes (Ω = 100 μL and 200 μL).

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Fig. 2.

Mobility of viscous drops on SH materials tilted by an angle α. (A) Terminal velocity U as a function of the sine of α for a water–glycerol mixture with viscosity η ∼ 110 mPa∙s and Ω ∼ 100 µL (blue circles) or Ω ∼ 200 µL (red squares). (See also Movie S2.) (B) Terminal velocity U of viscous drops as a function of 1/η, the inverse of the viscosity, for α = 2.3° and Ω ∼ 100 µL. The linear fit (dashed line) has a slope of 3.35 mN/m.

Fig. 2A shows that speed is linear in slope sinα, as expected in a viscous regime (18⇓–20). Indeed, denoting δ as the characteristic scale of velocity gradients, the viscous force varies as ηUΣ/δ, denoting Σ as the surface area of the drop base. For puddles, the volume Ω and surface area Σ are simply proportional to each other (Ω ∼ Σa ∼ πR²a), and velocity gradients develop across the thickness of the puddles (δ ∼ a). Hence, the balance of the viscous force with the weight ρgaΣsinα yields:U∼γηsin α.[1]The velocity is expected to be linear in sinα and independent of the volume Ω, as observed in Fig. 2A and also in SI Appendix, Fig. S2. Eq. 1 also predicts that U decreases hyperbolically with the viscosity η, which we check in Fig. 2B for α = 2.3°. U is observed to increase linearly with 1/η, and the slope deduced from the fit (dashed line in the figure), 3.35 mN/m, nicely compares to γsinα ∼ 2.9 mN/m. This allows us to evaluate a prefactor of 0.85 in the scaling formula of the force, which finally writes F ∼ 0.85 ηUΩ/a².

As seen in Fig. 1D, a purely viscous friction does not describe the behavior at small η. These large deviations are confirmed by plotting the terminal velocity of water drops (η = 1 mPa∙s) as a function of the slope sinα (Fig. 3A). Instead of the linear behavior reported in Fig. 2A, we now observe a concave curve that highlights the existence of a supplementary friction.

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Fig. 3.

Water drop mobility on SH inclines. In all plots, the drop volume is Ω = 100 μL, and dashed lines show Eq. 3. (A) Terminal velocity U as a function of the sine of the sliding angle α for water drops. (B) Terminal velocity U as a function of sinα for water drops in an atmosphere of air (light blue circles) or neon (red circles).

Water runs down at velocities of typically 1 m/s, comparable to that of raindrops, which suggests an aerodynamical drag. The Reynolds number in air Re = 2ρ_aRU/η_a (defined with the diameter 2R of the drop and with the air viscosity η_a = 18 µPa∙s and density ρ_a = 1 kg/m³) is typically between 100 and 1,000. The ratio between viscous friction in water (that scales as ηUΣ/a) and inertial friction in air (that scales as ρ_aU²Σ) is ρ_aUa/η, a quantity independent of the drop volume. For water, this number is of order unity, confirming the aerodynamical origin of the additional friction. More generally, the drag force in air can be written:F≈ρaCd(Re)SU2/2,[2]where S is a surface area and the drag coefficient C_d is a function of Re. In our Reynolds range, the drag results from a skin friction, that is, the friction at stake in the air boundary layer around the drop. This statement has two consequences: (i) The relevant surface area S in Eq. 2 is the top surface of the drop, whose area scales as Σ. (ii) The drag coefficient is C_d(Re) = 2/Re^1/2, as deduced by balancing Eq. 2 with the viscous friction in the air boundary layer whose thickness is given by δ ∼ (η_aR/ρ_aU)^1/2 ∼ R/Re^1/2. Adding viscous and aerodynamical frictions, we get an equation for the terminal velocity of the drop:ρga sin α≈xηU/a+yρaU2/Re1/2,[3]where the first prefactor x can be extracted from Fig. 2 (x ∼ 0.85), and where the second prefactor y has to be determined. As seen in Fig. 1D, a unique value of y (y ∼ 34, dashed line) allows us to adjust data at all viscosities, from pure water to pure glycerol. The high value of y can be explained by the fact that nonwetting drops rotate as they move (SI Appendix, Fig. S3), which is known for rolling spheres to increase C_d by a factor of ∼10 compared with sliding spheres (21). Moreover, assuming a disk shape for the drop, we may underestimate the top surface S where the boundary layer develops. The adjustment of our data by Eq. 3 seems robust: using the same prefactors, we can deduce the relationship between velocity and slope, and there again the model nicely adjusts the data with water (Fig. 3A). The term varying as U^3/2 in Eq. 3 dominates over the viscous linear term at large velocity, so that we expect U to asymptotically vary as sin^2/3α, a simple way to explain the concavity of the curve in Fig. 3A, while the viscous term dictates a linear behavior at small tilt.

As an implication of Eq. 3, we expect the drop velocity to depend on the nature of the surrounding air. It is challenging to test this dependency, but we tried it by using neon, whose viscosity η_a = 29 µPa∙s and density ρ_a = 0.9 kg/m³ make its kinematic viscosity η_a/ρ_a larger by 60% than that of air. We performed the experiment described in Fig. 1C in a closed transparent box filled either with air or with neon. Fig. 3B shows our results for water drops with Ω = 100 µL, the terminal velocity U being plotted as a function of the slope sinα. Owing to the box size, the experimental range of accessible slopes to reach terminal velocity is limited, which reveals the small offset invisible in Fig. 3A and due to the residual adhesion of water (drops move if the substrate is tilted by more than 0.3°). However, we clearly distinguish data obtained in air from that in neon: drops systematically move slower in the latter case, showing the existence of a larger friction, in agreement with Eq. 3, where an increase of the kinematic viscosity of the surrounding gas should induce such an effect. The colored dashes in the figure show the predictions of Eq. 3 with the parameters of the respective gas and the same prefactors as previously (x = 0.85, y = 34). The fits are found to be satisfactory, showing in particular the same weak, yet significant, differences between the two systems.

Water moving along SH materials develop original dynamical behaviors, compared with the usual cases where the drop velocity is rather fixed by viscous effects close to the contact line. For water-repellent materials, such effects are negligible, and viscosity is found to be only a weak correction to aerodynamical effects. The ratio between viscous and aerodynamical forces in Eq. 3, yρ_aUa/xηRe^1/2, is typically 5 in our experiments, implying that aerodynamical drag is the main force opposing the drop motion—an original situation in wetting dynamics. Assuming it is dominant, we can deduce the plateau velocity in Fig. 1D (corresponding to “large” slopes or small viscosities) by simply balancing the weight with the aerodynamical force in Eq. 3:U∼( γRρg⁡sin2⁡αηaρay2)1/3.[4]This speed is independent of the liquid viscosity and expected to be ∼2 m/s for a tilt angle of 10°, in good agreement with observations. Such a high velocity shows that the mobility of water is increased by a factor of at least 100 on repellent materials compared with usual ones—reflecting the conjunction of highly reduced adhesion and highly reduced friction, both arising from the air trapped in the texture. About this thin layer, we can wonder whether its presence could also generate a significant (and specific) friction. Such a friction is promoted by the thinness of the air film and by the existence of a slip at the water/air interface, at the drop base. As seen in Fig. 1 B and C, the roughness has comparable wavelength and height h, which sets a slip length of order h (22, 23). This means that the slip velocity U_s scales as Uh/a, and thus that the stress in the air film is of order η_aU_s/h ∼ η_aU/a. If this stress were balanced by the drop weight, the resulting velocity would depend on the air viscosity, in qualitative agreement with Fig. 3B, but be proportional to the slope, in strong disagreement with Fig. 3A. More fundamentally, this additional friction is found to be smaller by a factor η/η_a ∼ 50 than the viscous force in water and typically 100 times smaller than the aerodynamical force in Eq. 3, showing that the air film, so crucial in SH states, has a negligible impact on the friction.

Drops on SH materials are highly mobile, and what limits their mobility depends on their viscosity. Viscous liquids are slowed by the dissipation in the bulk, a consequence of the rolling motion accompanying the drop translation. In contrast, the internal dissipation in water, of low viscosity, is weaker than the aerodynamical resistance. This minimizes the role of the substrate, whose influence can be however revealed by a closer view. (i) A solid substrate generates a small (yet measurable) contact angle hysteresis, which can stop the liquid at small tilt and slightly lower the speed at larger tilt. (ii) Less trivially, the drag coefficient needed to fit the results is significantly larger than that for a raindrop in pure translation, as also observed for rotating objects along planes (21). Indeed, even if viscosity is only a small correction to aerodynamical effects, it induces rotation in water (SI Appendix, Fig. S3 and Movie S3), which in turn increases the drag coefficient. (iii) In the same vein, even a marginal viscous friction due to the no-slip boundary condition at the substrate can impact the drop shape. We considered here situations where this shape remains quasi-static. However, at higher substrate tilt, that is, at higher drop velocity U, the capillary number Ca = ηU/γ can become large enough to imply changes in the drop shape. Water subjected to a viscous force will elongate, which is indeed what we find when substrates are inclined by ∼20° or more. The critical capillary number at which such deformations are found is on the order of 10⁻², a relevant value for a dynamical wetting transition (24)—even if this problem remains to be discussed in nonwetting situations. The use of repellent materials in this limit remains highly valuable, since we do not observe any continuous deposition, owing to the high speed of dewetting on SH materials. Of course, drop deformation might in turn impact the friction law, which remains to be described. More generally, the universality of our model (i.e., the fact that the detail of the solid texture does not seem to matter in Eq. 3) should be explored in the future, when technology will allow us to microfabricate the long, controlled substrates needed for such studies. The case of smaller water drops would also deserve a dedicated study. First, we expect their motion to be highly sensitive to the hysteretic adhesion, that now writes (γR²/a) sin θ Δθ and thus can become comparable to the weight ρR³gsinα at small radius R. However, even in an ideal situation without hysteresis, we anticipate a modification of both viscous and aerodynamical frictions, due to the spherical shape. The rolling motion of a sphere minimizes its internal dissipation (18), while the air skin drag friction now applies on a typical surface area R². Hence, in the case of a dominant air friction, the terminal velocity U should scale as R (ρ²g²sin² α/ρ_aη_a)^1/3, an expression quite different from that for larger drops (Eq. 4)—and, remarkably, more sensitive to gravity, despite the size reduction.