Roughness (surface roughness)
Roughness (surface roughness) is the degree of unevenness of a solid surface below the size scale of its shape or waviness, but above the irregularity of crystal lattice structures. The roughness has an influence on the wettability of a solid.
How are roughness and wettability related?
Roughness affects the wettability of a solid and thus the contact angle of a liquid as well as adhesion. A rule of thumb states that roughness improves wettability at small contact angles (angle becomes smaller), and reduces it at large contact angles (angle becomes larger). The limiting contact angle of wettability, which determines the deflection in one direction or the other, is 90°. The roughness-related reduction in wetting comes into play on superhydrophobic surfaces; it is often referred to as the Lotus Effect.
Why does roughness increase wetting and adhesion in well-wettable materials?
At contact angles below 90°, wetting of the surface is energetically advantageous overall for the two phases involved. Since a rough material has a larger total surface area than a smooth one due to the tiny depressions and thus offers a larger wetting area, the wettability increases.
The use of this effect is part of everyday practical knowledge, for example when components are ground before painting or bonding to increase wetting and adhesion. For surface tension measurements using the Wilhelmy plate method, the platinum plate is roughened to optimize wetting by the sample.
How are roughness and superhydrophobicity related?
Extremely low wettability, also known as superhydrophobicity, results from the rough surface texture of a material where the smooth surface is already non-wettable. Put simply, the liquid avoids the depressions in the surface, which is enlarged by roughness, instead of filling them up. The technical term for this effect is capillary depression in contrast to capillary adhesion - in an unwettable capillary, the capillary force does not pull the liquid inward, but pushes it outward.
This wetting behavior is also known as the lotus effect since the lotus leaf is one of the least wettable non-technical surfaces. There are many industrial applications for superhydrophobic materials or coatings, for example for fouling and wetness protection or for self-cleaning textiles or building walls.
How is the influence of roughness on wetting described scientifically?
Wenzel  established a simple relationship between the roughness and the contact angle:
where θ* is the measured contact angle. This is called the apparent contact angle for a rough surface because it is the angle of the droplet contour with an apparently straight profile line. θ is the contact angle that would exist on a smooth surface of the same material and corresponds to the ideal contact angle according to Young's equation. r' is called the roughness coefficient and indicates the ratio of the total surface area to the surface geometrically projected onto a plane.
Based on the Wenzel equation, it can be seen why the critical angle for the transition from wetting enhancement to reduction is 90°. Since cos θ assumes negative values above 90°, the apparent contact angle becomes larger. However, the Wenzel equation can then be applied only to a limited extent because, at low wetting, not all the depressions of the rough surface are usually filled with liquid. Cassie and Baxter  have formulated the following relationship for a condition where only parts of the entire surface are in contact with the liquid:
Can the influence of roughness on wetting be calculated?
It seems to be obvious to calculate a "roughness-corrected" contact angle using the Wenzel formula in order not to have to investigate the roughness influence on wetting separately. In practice, however, such an approach is fraught with great uncertainties. A study carried out by KRÜSS  on the basis of surfaces with defined roughness shows that the roughness influence can deviate considerably from the behavior to be expected according to Wenzel - up to a trend reversal, in which a well-wettable material is less wetted in the rough state.
Can the roughness be measured?
Result parameters for roughness and suitable measuring methods are defined in the ISO 25178 standard. Non-destructive measurements by non-contact optical methods, for example confocal microscopy, should be emphasized. Other methods use, for example, an air stream or even vacuum and make use of the gas permeability on contact between a smooth and a rough surface.
- R. N. Wenzel, Resistance of Solid Surfaces to Wetting by Water. In Ind. Eng. Chem. 28, no. 8, 1936, pp. 988-994.
 A.B. D. Cassie, S. Baxter, Wettability of Porous Surfaces. In: Trans. Faraday Soc. 40, 1944, pp. 546-551.
 P. Swolana, Ch. Weigel, T. Geiling, L. Dittrich, D. Frese: Wetting and roughness - What to do if Wenzel does not work? KRÜSS Application Report AR295, 2020.