Scaling laws for slippage on superhydrophobic fractal surfaces

TL;DR

This paper investigates how the complex geometry of fractal superhydrophobic surfaces influences fluid slippage, revealing that fractal dimension significantly affects drag reduction and that contact angle alone is insufficient to predict slip behavior.

Contribution

It introduces a scaling law for effective slip length on fractal surfaces and validates it through numerical simulations, highlighting the impact of fractal geometry on hydrodynamic friction.

Findings

Slippage depends strongly on fractal dimension.

Fractal surfaces exhibit less slip than regular patterned surfaces.

Effective contact angle does not reliably predict slip on fractal surfaces.

Abstract

We study the slippage on hierarchical fractal superhydrophobic surfaces, and find an unexpected rich behavior for hydrodynamic friction on these surfaces. We develop a scaling law approach for the effective slip length, which is validated by numerical resolution of the hydrodynamic equations. Our results demonstrate that slippage does strongly depend on the fractal dimension, and is found to be always smaller on fractal surfaces as compared to surfaces with regular patterns. This shows that in contrast to naive expectations, the value of effective contact angle is not sufficient to infer the amount of slippage on a fractal surface: depending on the underlying geometry of the roughness, strongly superhydrophobic surfaces may in some cases be fully inefficient in terms of drag reduction. Finally, our scaling analysis can be directly extended to the study of heat transfer at fractal surfaces, in order to estimate the Kapitsa surface resistance on patterned surfaces, as well as to the question of trapping of diffusing particles by patchy hierarchical surfaces, in the context of chemoreception.