The pinning dynamics of a non-wetting droplet penetrating a permeable substrate

The contact line pinning mechanisms of a non-wetting droplet penetrating a permeable substrate are theoretically explained by considering the force balance of volumetric force, capillary force, and pinning and depinning forces. We propose two dimensionless numbers, Bo^∗—the ratio of the volumetric force to the capillary force, and Ct—the ratio of the depinning force to the pinning force, to establish a phase diagram that quickly determines the droplet penetration patterns. For Bo^∗ ⩽ 1, the droplet will not penetrate the substrate; for Bo^∗ > 1 and Ct ⩽ 1, the droplet will penetrate with a pinned contact line; for Bo^∗ > 1 and Ct > 1, the droplet will penetrate with contact line shrinking. Contact angle dynamics during contact line pinning and shrinking are further clarified. The time evolutions of the contact area diameter D_c, the droplet height h, the penetrated droplet volume percentage S_p, and the apparent contact angle θ are derived. We further perform a series of lattice Boltzmann simulations, and the results match well with our theoretical analysis. These theoretical and numerical results pave the way to achieve better performances of many important applications that involve droplet penetration.