## New Research In

### Physical Sciences

### Social Sciences

#### Featured Portals

#### Articles by Topic

### Biological Sciences

#### Featured Portals

#### Articles by Topic

- Agricultural Sciences
- Anthropology
- Applied Biological Sciences
- Biochemistry
- Biophysics and Computational Biology
- Cell Biology
- Developmental Biology
- Ecology
- Environmental Sciences
- Evolution
- Genetics
- Immunology and Inflammation
- Medical Sciences
- Microbiology
- Neuroscience
- Pharmacology
- Physiology
- Plant Biology
- Population Biology
- Psychological and Cognitive Sciences
- Sustainability Science
- Systems Biology

# Superhydrophobic frictions

Edited by Laurent Limat, CNRS, and accepted by Editorial Board Member John D. Weeks March 5, 2019 (received for review November 26, 2018)

## Significance

The matchless mobility of water on superhydrophobic materials is often considered as the hallmark of water repellency. The friction of drops is indeed found to be unusual: instead of observing classical friction such as due to contact line and liquid viscosity, we show that the main source of resistance to the water motion arises from the air around it. This explains why the drop velocity becomes quasi-independent of its viscosity at low viscosity and more generally why the mobility is so high. We also discuss the case of more viscous liquids whose rolling motion induces a bulk viscous dissipation that caps the mobility.

## Abstract

Contrasting with its sluggish behavior on standard solids, water is extremely mobile on superhydrophobic materials, as shown, for instance, by the continuous acceleration of drops on tilted water-repellent leaves. For much longer substrates, however, drops reach a terminal velocity that results from a balance between weight and friction, allowing us to question the nature of this friction. We report that the relationship between force and terminal velocity is nonlinear. This is interpreted by showing that classical sources of friction are minimized, so that the aerodynamical resistance to motion becomes dominant, which eventually explains the matchless mobility of water. Our results are finally extended to viscous liquids, also known to be unusually quick on these materials.

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. 1*B*. 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 *U*^{2}/2*g* 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.

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. 1*C*, 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/m^{3}) 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/ρ*g*)^{1/2} (*a*^{3} ∼ 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 γ*R* (cos θ_{r} − cos θ_{a}), which reduces to γ*R* 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 ρ*g*Ωsinα for α > 0.2°. Thus, the drop terminal speed directly results from a balance between the projected weight ρ*g*Ωsinα 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. 1*D* 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. 2*A* 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).

Fig. 2*A* 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 η*U*Σ/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 ∼ Σ*a* ∼ π*R*^{2}*a*), and velocity gradients develop across the thickness of the puddles (δ ∼ *a*). Hence, the balance of the viscous force with the weight ρ*ga*Σsinα yields:*A* and also in *SI Appendix*, Fig. S2. Eq. **1** also predicts that *U* decreases hyperbolically with the viscosity η, which we check in Fig. 2*B* 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 η*U*Ω/*a*^{2}.

As seen in Fig. 1*D*, 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. 3*A*). Instead of the linear behavior reported in Fig. 2*A*, we now observe a concave curve that highlights the existence of a supplementary friction.

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_{a}*RU*/η_{a} (defined with the diameter 2*R* of the drop and with the air viscosity η_{a} = 18 µPa∙s and density ρ_{a} = 1 kg/m^{3}) is typically between 100 and 1,000. The ratio between viscous friction in water (that scales as η*U*Σ/*a*) and inertial friction in air (that scales as ρ_{a}*U*^{2}Σ) is ρ_{a}*Ua*/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:*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_{a}*R/*ρ_{a}*U*)^{1/2} ∼ *R/Re*^{1/2}. Adding viscous and aerodynamical frictions, we get an equation for the terminal velocity of the drop:*x* can be extracted from Fig. 2 (*x* ∼ 0.85), and where the second prefactor *y* has to be determined. As seen in Fig. 1*D*, 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. 3*A*). 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^{2/3}α, a simple way to explain the concavity of the curve in Fig. 3*A*, 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/m^{3} make its kinematic viscosity η_{a}/ρ_{a} larger by 60% than that of air. We performed the experiment described in Fig. 1*C* in a closed transparent box filled either with air or with neon. Fig. 3*B* 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. 3*A* 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**, *y*ρ_{a}*Ua*/*x*η*Re*^{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. 1*D* (corresponding to “large” slopes or small viscosities) by simply balancing the weight with the aerodynamical force in Eq. **3**:*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_{a}*U*_{s}/*h* ∼ η_{a}*U*/*a*. If this stress were balanced by the drop weight, the resulting velocity would depend on the air viscosity, in qualitative agreement with Fig. 3*B*, but be proportional to the slope, in strong disagreement with Fig. 3*A*. 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* = η*U*/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 (γ*R*^{2}/*a*) sin θ Δθ and thus can become comparable to the weight ρ*R*^{3}*g*sinα 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 *R*^{2}. Hence, in the case of a dominant air friction, the terminal velocity *U* should scale as *R* (ρ^{2}*g*^{2}sin^{2} α/ρ_{a}η_{a})^{1/3}, an expression quite different from that for larger drops (Eq. **4**)—and, remarkably, more sensitive to gravity, despite the size reduction.

## Acknowledgments

We thank Philippe Bourrianne, Antoine Fosset, and Evan Spruijt for their help in the experiments and Emmanuel du Pontavice for valuable discussions.

## Footnotes

↵

^{1}T.M. and P.S.R. contributed equally to this work.- ↵
^{2}To whom correspondence should be addressed. Email: david.quere{at}espci.fr.

Author contributions: T.M., P.S.R., and D.Q. designed research; T.M. and P.S.R. performed the experiments; T.M., P.S.R., C.C., and D.Q. analyzed data; and T.M., P.S.R., C.C., and D.Q. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. L.L. is a guest editor invited by the Editorial Board.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1819979116/-/DCSupplemental.

Published under the PNAS license.

## References

- ↵
- Dussan VEB,
- Chow RTP

- ↵
- ↵
- ↵
- ↵
- ↵
- Miwa M,
- Nakajima A,
- Fujishima A,
- Hashimoto K,
- Watanabe T

- ↵
- ↵
- ↵
- Pilat DW, et al.

- ↵
- ↵
- Hao P,
- Lv C,
- Yao Z,
- He F

- ↵
- Olin P,
- Lindström SB,
- Pettersson T,
- Wågberg L

- ↵
- Timonen JVI,
- Latikka M,
- Ikkala O,
- Ras RHA

- ↵
- Nguyen TV,
- Takahashi H,
- Matsumoto K,
- Shimoyama I

- ↵
- Qiao S, et al.

- ↵
- Burkarter E,
- Berlim LS,
- Schreiner WH,
- Saul CK

- ↵
- ↵
- ↵
- ↵
- Hodges SR,
- Jensen OE,
- Rallison JM

- ↵
- Jan CD,
- Chen JC

- ↵
- ↵
- ↵

## Citation Manager Formats

## Sign up for Article Alerts

## Article Classifications

- Physical Sciences
- Physics