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Two recipes for repelling hot water

Although a hydrophobic microtexture at a solid surface most often reflects rain owing to the presence of entrapped air within the texture, it is much more challenging to repel hot water. As it contacts a colder material, hot water generates condensation within the cavities at the solid surface, which eventually builds bridges between the substrate and the water, and thus destroys repellency. Here we show that both “small” (~100 nm) and “large” (~10 µm) model features do reflect hot drops at any drop temperature and in the whole range of explored impact velocities. Hence, we can define two structural recipes for repelling hot water: drops on nanometric features hardly stick owing to the miniaturization of water bridges, whereas kinetics of condensation in large features is too slow to connect the liquid to the solid at impact.


Supplementary Figure 1. Repellency for another drop radius.
Coefficient of restitution e as a function of the temperature difference ∆T for substrates A, B and C and water drops with radius R ≈ 1.1 mm and velocity V ≈ 40 cm s -1 . Coloured lines show the model (equation (5)) for the three regimes: t > tr (surface C), where no condensation adhesion occurs; t < tr (surface A), where condensation is immediate; t ≈ tr (surface B), where the transition from bouncing to sticking is observed for ∆T » 17°C. Error bars represent uncertainty of the measurement. Supplementary Figure 4. Influence of hygrometry. Water drop coefficient of restitution e as a function of the water/substrate temperature difference ∆T for two hygrometries, on sample A for R ≈ 1.4 mm: RH = 30% and V ≈ 36 cm s -1 (green symbols); RH = 60% and V ≈ 40 cm s -1 (red symbols). Error bars represent uncertainty of the measurement.

Supplementary Discussion
We performed our experiments on samples A, B and C with a smaller water drop (R = 1.10 ± 0.05 mm), at an impact velocity V = 40 ± 5 cm s -1 (Supplementary Figure 1). The characteristics of the impact are close to that reported in the Fig. 2d,e of the accompanying paper: the restitution coefficient e of the shock is quite unsensitive to water temperature for sample C, it slightly decreases with sample A, and it evidences a sharp transition to sticking with sample B for ∆T ≈ 17°C. There again, data are convincingly fitted by equation (5)  with solid lines. The adjustable parameter for the condensation time t (in equation (1)) is the same as for the larger radius, i.e. a = 8, which confirms the robustness of our model.
We also tested higher impact velocities, namely V = 59 ± 5 cm s -1 and V = 86 ± 5 cm s -1 -to be compared with V = 40 ± 5 cm s -1 in the accompanying paper. At smaller V, the contact time increases, which complicates the analysis; at larger V, water can splash, which also changes the physics. These experiments were carried out with two radii, R = 1.10 ± 0.05 mm and R = 1.40 ± 0.05 mm. As shown in Supplementary Figures 2 and 3, results are quite similar to that reported in the accompanying paper, and the model (equation (5), solid lines) convincingly fits the data at all explored values of V and R.
In order to quantify the influence of ambient humidity on bouncing, we performed an experiment with sample A (the sample having the smallest texture, that is, likely to be filled by small amounts of condensed water) at a hygrometry RH of both 30% and 60%, and for an atmospheric temperature To = 20 ± 1°C. We plot in Supplementary Figure 4 the coefficient of restitution e as a function of the excess temperature DT for R = 1.4 mm. The two series of data are found to be nearly superimposed, and both well described by the model (equation (5), drawn with solid lines), showing that repellency can resist significant variations of hygrometry. The slight shift between the two series of data arises from small differences in the impact velocity, that is, V = 40 ± 4 cm s -1 for RH = 60% and V = 36 ± 4 cm s -1 for RH = 30%. A faster impact generates a less elastic shock (whatever the drop temperature), which explains the slight difference between both curves.
As shown in the Fig. 4 of the accompanying paper, our model allows us to predict a phase diagram for homothetic arrays of hydrophobic pillars, confirming the existence of two bounds in pillar heights h1 and h2 below and above which hot water always bounces. The two recipes for repelling hot water respectively correspond to h < h1 and h > h2. For "short" pillars, the probability n(∆T)rp 2 of finding a nucleus in a cell being small, h1 is obtained by writing e = 0 in equation (3) for ∆Tm = 75°C (boiling point of water since the substrate temperature is around 25°C). This yields: (6) h1 = [e0MV 2 /4πgRm 2 rn(∆Tm)] 1/2 For typical values of the parameters, we expect that h1 is on the order of 100 nm. Similarly, h2 is obtained by writing e = 0 in equation (5) for ∆Tm = 75°C. For tall pillars, we have n(∆Tm) > 1/rp 2 , which yields: (7) h2 = [trD∆csat(∆Tm)/ar] 1/2 [1 -(e0MV 2 /4πgRm 2 ) 1/2 ] 1/4 For typical values of the parameters, h2 is ~4 µm. We can further simplify the model by noticing that the correction to 1 in the bracket in equation (7) is ~0.13 -so that the first term only in the equation can be used for roughly estimating the value of h2.
The two limits given by equations (6) and (7), drawn with a dotted line in Fig. 4, are analytical.
We find in particular that h1 depends on the coefficient of restitution e0 as e0 1/2 , and that e0 only appears as a small correction in the expression of h2. Both variations h1(e0) and h2(e0) are weak, which explains that taking a unique, average value e0 = 0.2 in the model allows us to draw a phase diagram (Fig. 4) valid for samples having slightly different e0.

Supplementary Note 1
We reported that the contact radius Rc can take the form Rc ~ 2(RVt) 1/2 at small time. This expression is valid for t < 2R/V, that is, around 7 ms in our situation -a time longer than the 5 ms needed for the drop to reach its maximum spreading radius (Rc = Rm). Besides, no rim is formed at maximal deformation, a consequence of the low Weber number (We ≈ 3). This means that the thickness z of the pancake remains constant spatially so that the retraction speed can be correctly approximated by (g/rz) 1/2 .