# Shall bear larger indentations on a larger scale as

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shall bear larger indentations on a larger scale as well, as is depicted in fig. 3. Its effective interfacial tension with the liquid, Σ 2 1 , w 1 , g 1 , g 0 , σ ), can be expressed in straightforward analogy to eq. (1), yielding Σ 2 (1 w 1 1 + w 1 ( γ + g 1 σ (1 + ( g 0 1) w 0 )) . (2) If we now consider substrates with indentations the walls of which are covered with indenta- tions the walls of which are covered with indentations, and so on, it turns out that we can write down a recursion relation for the interfacial tension in a very compact form if we consider the cosine of the macroscopic contact angle: cos θ n +1 = (1 w n ) cos θ n w n , (3) where n numbers the generation of the indentation hierarchy. Larger n corresponds to larger length scale. Note that the geometry factors, g n , of the indentations do all cancel out. According to eq. (3), cos θ n +1 cos θ n = w n (1 + cos θ n ) < 0, so that the sequence represented by eq. (3) is monotonic. Since its elements are furthermore restricted to the stripe defined by 1 < cos θ n < 1, it must be convergent, and its limit represents the macroscopic contact angle at infinite length scale. It is obtained as the fixed point of the return map expressed by eq. (3) and reads cos θ limit = 1, representing a contact angle of π . Consequently, the surface can indeed be macroscopically non-wet, provided there is an appropriate roughness topography, and the microscopic contact angle is finite. If θ 0 = 0, there is no way to design indentations which suspend a free liquid surface. It is important to stress the fact that θ 0 must only be finite, but need not exceed π/ 2, as in former models. If we define α n := π θ n , we can investigate the scaling behaviour of θ n at large scales. Since cos θ n α 2 n / 2 1, we have α n +1 α n 1 w n , such that α converges exponentially to zero. Given the large span of available length scales, between the molecular scale of, say, protein tertiary structures and the capillary length of the probing liquid (2.7 mm for water), there is plenty of room for α to come as close to zero as experimentally observed for plant leaves: For the strikingly repellent surfaces of Nelumbo nucifera L. Druce or Brassica oleracea L. , α exp 20 [1]. If the steepness of the indentations as well as w n and g n are similar at all length scales, n , a fractal surface results. We can estimate the Hausdorff dimension of such a surface if we consider the areal fraction φ = (1 + ( g 1) w ) 1 which is “lost” when smoothing out one generation of indentations. If we denote by b the change in length scale from one generation
S. Herminghaus : Roughness-induced non-wetting 169 to the next, we have D H = 2 ln b ln φ ln b = 2 + ln(1 + ( g 1) w ) ln b . (4) Since the contact angle at the ﬂat material, θ 0 , is usually not too far from π/ 2, such a surface exhibits steep structures also at large scales. However, this is not observed on the leaves of most of the above-mentioned plants (aside from Ginkgo biloba L. ), and is consequently not a good model for those leaves. Furthermore, it will become clear below that a fractal surface like the one described above is not necessary for a surface to be liquid repellent.

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• Fall '19
• Surface tension, Wetting

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