Fractal Model for Wettability of Rough Surfaces

This paper presents a fractal model to describe wettability on multiscale randomly rough surfaces. Hydrophobic or superhydrophobic surfaces, produced by processes such as electrodeposition and etching, lead to the creation of random roughness at multiple length scales on the surface. This paper considers the description of such surfaces with a fractal asperity model based on the Weierstrass-Mandelbrot (W-M) function, where the fractal parameters are uniquely determined from a power spectrum of the surface. By use of this description, a model is presented to evaluate the apparent contact angle in the different wetting regimes. The model is predictive in that it does not use any empirical or correlatory fitting of parameters to experimental data. Experimental validation of the model predictions is presented on various hydrophobic and superhydrophobic surfaces generated on several materials under different processing conditions. The contact angle is found be strongly dependent on the range of asperity length scale and weakly dependent on the fractal dimension for a surface with stable Cassie state. Based on the fractal description, desired surface roughness characteristics for generating superhydrophobicity on a particular substrate are also derived.