Introduction

Wetting of rough surfaces is a complex problem which continues to attract scientists, in particular due to the emergence of new materials with controlled surface micro-, nano-, and hierarchical structure. During the past decade, the so-called ''Lotus effect,'' or surface roughness-induced superhydrophobicity and self-cleaning, became a subject of active investigation. According to early Wenzel [37] and Cassie and Baxter [9] models, there are two regimes of wetting of a rough surface: a homogeneous regime with a two-phase solid-water interface and a non-homogeneous or composite regime with a three phase solid-water-air interface (air pockets are trapped between the solid surface and water). Both models predict that surface roughness affects the water CA and can easily bring it to the extreme values close to 180° (superhydrophobicity) or close to 0° (superhydro-philicity). The studies of wetting of microstructured surfaces have concentrated on the investigation of the two regimes and the factors which affect the transition between the regimes [3, 5, 7, 16, 26-32].

Recent experimental findings and theoretical analyses made it clear that the early Wenzel [37] and Cassie and Baxter [9] models do not explain the complexity of interactions during wetting of a rough surface, which can follow several different scenarios [4, 5, 12, 14, 15, 22, 33, 36, 39]. As a result, there are several modes of wetting of a rough surface, and therefore, wetting cannot be characterized by a single number, such as the CA.

The concept of surface (or interface) energy is central for the analysis of wetting phenomena. Atoms or molecules at the surface of a solid or liquid have fewer bonds with neighboring atoms than those in the bulk. Energy is spent for breaking the bonds when a surface is created. As a result, the atoms at the surface have higher energy. This excess surface energy or surface tension, y, is measured in N/m, and it is equal to the energy needed to create a surface with the unit area. If a liquid droplet is placed on a solid surface, the liquid and solid surfaces come together under equilibrium at a characteristic angle called the static CA, h0, given by the Young equation [1, 2], a ySA _ ySL , \

ySL, ySA, and yLA are the surface energies of the solid-liquid, solid-air, and liquid-air interfaces, respectively. For a large number of combinations of materials and liquids, ySA ? yLA > ySL, which means that it is energetically profitable for a liquid to wet the solid surface rather than to have an air film separating the solid and liquid. On the other hand, for many material combinations, ySL + yLA > ySA, which means that it is energetically profitable for a solid to be in contact with air, rather than to be covered by a thin liquid film. As a result, in most situations — 1 < (ySA — ySL)/yLA) < 1, and there exists a value of the CA given by Eq. 2.1. The CA is the angle under which the liquid-air interface comes in contact with the solid surface locally, and it does not depend on the shape of the body of water.

If water CA 0° < 00 < 90°, then the surface is usually called "hydrophilic," whereas a surface with water CA 90° < 00 < 180° is usually called ''hydrophobic.''

In the ideal situation of a perfectly smooth and homogeneous surface, the static CA is a single number which corresponds to the unique equilibrium position of the solid-liquid-air contact line (the triple line). However, when the contact takes place with a rough surface, there may be multiple equilibrium positions which result in an entire spectrum of possible values of the CA. In addition, the value of the surface energy itself exhibits so-called ''adhesion hysteresis'' and can depend on whether it is measured during the approach of the two bodies or when they are taken apart. As the result, there is always the minimum value of the CA called the receding CA, 0rec, and the maximum value of the CA called the advancing CA, hadv. The difference between the advancing and receding CA is called CA hysteresis (Fig. 2.1a).

Consider now a rough solid surface with the roughness factor Rf > 1 equal to the ratio of the solid surface area to its flat projected area. When water comes in contact with such a surface, the effective values of the solid-liquid and solid-air surface energies become Rf ySL and Rf ySA (Fig. 2.1b). This leads to the Wenzel equation for the CA with a rough surface [37]

If some air is trapped between the rough solid surface and the liquid, then only the fraction 0 < fSL < 1 constitutes the solid-liquid contact interface (Fig. 2.1c). The area of the solid-liquid interface is now RffSL per unit area, and in addition, there is (1 — fSL) of the liquid-air interface under the droplet. The effective values of the solid-liquid and solid-air surface energies become RffSLySL and RffSLySA + (1 — fsL)CLA. The CA is then given by the Cassie and Baxter [9] equation cos 0 = RffsL cos 00 - 1 + fsL (2-3)

If a surface is covered by holes filled (or impregnated) with water, the contact angle is given by cos 0 = 1 + fsL (cos 00 - 1) (2-4)

This is the so-called ''impregnating'' Cassie wetting regime [32] (Fig. 2.1d).

The CA is a macroscale parameter characterizing wetting. However, hydro-phobicity/philicity is dependent upon the adhesion of water molecules to the solid. On the one hand, a high CA is a sign of low liquid-solid adhesion. On the other hand, low CA hysteresis is a sign of low liquid-solid adhesion as well. There is an argument in the literature as to whether superhydrophobicity is adequately characterized only by a high CA and whether a surface can have a high CA but at the same time strong adhesion. It is now widely believed that a surface can be superhydrophobic and at the same time strongly adhesive to water (e.g., [15]). The so-called ''petal effect'' is exhibited by a surface that has a high CA, but also a large CA hysteresis and strong adhesion to water. The phenomenon of the large

Fig. 2.1 a Schematics of a droplet on a tilted substrate showing advancing (hadv) and receding (hrec) contact angles. The difference between these angles constitutes the contact angle hysteresis. Configurations described by b the Wenzel equation for the homogeneous interface, c Cassie-Baxter equation for the composite interface with air pockets, and d the Cassie equation for the homogeneous interface

Fig. 2.1 a Schematics of a droplet on a tilted substrate showing advancing (hadv) and receding (hrec) contact angles. The difference between these angles constitutes the contact angle hysteresis. Configurations described by b the Wenzel equation for the homogeneous interface, c Cassie-Baxter equation for the composite interface with air pockets, and d the Cassie equation for the homogeneous interface

CA hysteresis and high water adhesion to rose petals (and similar surfaces), as opposed to small CA hysteresis and low adhesion to Lotus leaf, was observed by several research groups [4, 8, 10]. Bormashenko et al. [8] reported a transition between wetting regimes, e.g., the penetration of liquid into the micro/ nanostructures.

Li and Amirfazli [19] argued that since "superhydrophobicity" means a strong fear of water or lacking affinity to water, ''the claim that a superhydrophobic surface also has a high adhesive force to water is contradictory.'' Gao and McCarthy [14] pointed out that the terms ''hydrophobic/phillic'' should be defined in a more accurate way. They suggested several experiments showing that even Teflon®, which is usually considered very hydrophobic, can be, under certain conditions, considered hydrophilic, i.e., has affinity to water. They argued that the concepts of ''shear and tensile hydrophobicity'' should be used, so that the wet-tability of a surface is characterized by two numbers, advancing and receding CAs, and ''the words hydrophobic, hydrophilic, and their derivatives can and should only be considered qualitative or relative terms.'' Instead, ''shear and tensile hydrophobicity'' should be investigated, which makes wetting (''solid-liquid friction'') similar to the friction force, as it has been pointed out in the literature earlier [23]. McHale [22] noted that all solid materials, including Teflon®, are hydrophobic to some extent, if they have Young CA <180°. Therefore, it is energetically profitable for them to have contact with solid, at least to some extent. Wang and Jiang [36] suggested five superhydrophobic states (Wenzel's state, Cassie's state, so-called ''Lotus'' and ''Gecko'' states, and a transitional state between Wenzel's and Cassie's states). It may be useful also to see the transition between the Wenzel, Cassie, and dry states as a phase transition and to add the ability of a surface to bounce off a water droplet to the definition of the super-hydrophobicity [31]. In addition, there is an argument on how various definitions of the CA hysteresis are related to each other [7, 8, 10, 17, 39]. A number of wetting regimes and transitions between them have been studied since 2010 [6, 11, 13, 34]. Modern research has concentrated on the ability to switch between the wetting states by tuning the surface energy [20, 21].

The Lotus effect has been comprehensively discussed in earlier publications. The objective of this paper is to discuss various wetting modes of rough surfaces, beyond the classical Wenzel [37] and Cassie and Baxter [9] regimes in light of recent experimental data on the petal effect and strong adhesion with superhy-drophobic surfaces referred to as the ''rose petal effect.''

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