# 2 at given finite contact angl θ an indentation in

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Fig. 2 – At given finite contact angle, θ 0 , an indentation in the substrate (S) the slopes of which are suﬃciently steep will suspend a free liquid surface. The average interfacial energy between the liquid (L) and the substrate is thus increased. of the substrate in the vicinity of the intersection is convex, the configuration is stable insofar as shifting the plane up or down will result in a restoring force due to the Laplace pressure of the liquid surface. It is diﬃcult, but possible to generalize this to indentations of arbitrary shape. One then has to deal with liquid surfaces of zero mean curvature instead of planes, and the stability requirement results in rather complicated conditions to be fulfilled by the shape of the in- dentation slopes. Nevertheless, the suspension of a free liquid surface will be possible for non-symmetrical as well as for symmetrical indentations. We will thus, in what follows, char- acterize the indentations no longer by their symmetry, but only by their lateral extension and by their steepness, or depth. It should be stressed that the general validity of the arguments for asymmetric roughness can only be conjectured here, and a stringent proof must be left to a forthcoming study. Now imagine the substrate surface to be studded with such indentations, which shall all be roughly the same size. The area of the free liquid surfaces suspended over these indentations is a certain fraction, w 0 > 0, of the projected surface of the substrate. It is easy to see that the same construction is possible with the indentations being replaced by bumps, only that w 0 must be replaced by 1 w 0 . In what follows, we will only consider indentations, but keep in mind that the concept applies to bumped surfaces as well. Let γ be the surface tension of the liquid, and σ be the tension of the ﬂat substrate/air interface. If we denote by Σ 0 the tension of the ﬂat substrate/liquid interface, we have for the effective tension [9], Σ 1 , of the interface between the indented substrate and the liquid Σ 1 (1 w 0 0 + w 0 ( γ + g 0 σ ) , (1) where g 0 1 is a geometric factor describing the total surface area of the indentation. It is defined by the total area of the indentation divided by the area of its geometrical projection. The total surface of the substrate then corresponds to (1 + ( g 0 1) w 0 ) times the projected area. With γ cos θ 0 = σ Σ 0 , it is easy to show that Σ 1 Σ 0 w 0 γ (1 + cos θ 0 ). Since w 0 , γ , and 1 + cos θ 0 are all greater than zero, this means that the indented interface has a higher effective tension than the ﬂat one, due to the freely suspended liquid surfaces. This warrants a macroscopic contact angle, θ 1 , larger than θ 0 , but does not explain the exceptionally large contact angles observed on many leaves. Let us therefore consider a hierarchical structure of the substrate: the indented surface
168 EUROPHYSICS LETTERS Fig. 3 – On a larger scale, less steepness is required for an indentation to suspend a free liquid surface due to the presence of smaller indentations.

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• Fall '19
• Surface tension, Wetting

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