Spreading and splashing of drops impacting rough substrates

Here, we present experimental results of water and ethanol drops of radii R, density ρ and interfacial tension coefficient σ, impacting with a velocity V over different types of sandpapers containing particles of characteristic diameter ε embedded in their surfaces. It is shown that the transition from spreading to splashing at normal atmospheric conditions can be classified depending on the value of the parameter ε /Hₜ We_ε =We(ε /R), with We=ρ V² R/σ the Weber number and Hₜ indicating the initial thickness of the thin film – the lamella – which is ejected along the substrate once the drop touches the solid. When We_ε 1 and the liquid wets the substrate, the critical value of the Weber number above which the drop splashes, Wec, can be predicted using the results in Gordillo & Riboux (J. Fluid Mech., vol. 871, 2019, R3) once the angle the advancing rim forms with the substrate, α, is expressed as a decreasing function of the static advancing contact angle. The calculated values of Wec for the case of water drops impacting over rough substrates are smaller than the corresponding ones for smooth substrates, in agreement with experimental observations. Moreover, if the liquid does not wet the substrate, it is also shown that the splash velocity can be predicted using the theory for superhydrophobic substrates in Quintero, Riboux & Gordillo (J. Fluid Mech., vol. 870, 2019, 175–188). For those cases in which We_ε 1 and the liquid wets the substrate, we demonstrate that the critical Weber number for splashing decreases with ε as Wec∝ (Rcos θ ₀ /ε )3/5, with θ ₀ the value of the Young contact angle.