A floor and a ceiling for the advancing contact angle

Abstract

Reported maxima of the advancing contact angle scatter from $87^\circ$ to $147^\circ$ in a single systematic study, and liquids on surfaces with static angles as low as $5^\circ$ reach dynamic maxima near $90^\circ$. We show that both observations follow from the hinged motion of the free surface near a moving contact line. The local Stokes solution for a wedge rotating about its contact line has two singular angles of opposite character: at $90^\circ$ the hinged motion generates no wall shear and the rotation is free, and at $θ_h = 128.73^\circ$, where $\tan 2α= 2α$, a resonance with the $r^2$ eigensolution arrests the rotation through an $r^2 \ln r$ term. Between the two angles lies a band that a transient advancing angle cannot leave while the interface near the contact line remains a quasi-steady wedge. A compilation of 68 liquid-solid systems from nine sources and five configurations confirms the band and its two exits: systems not limited by the apparatus, with $θ_s \le 40^\circ$, reach $87^\circ$ to $119^\circ$ regardless of chemistry, and every exceedance of $θ_h$ occurs either where chemistry places the static angle above the band or where the flow leaves the quasi-steady regime. The record of a waterline on a vertical wall exhibits the band within a single unsteady experiment, together with a distinct quieting of the local contact angle at the crossing of $90^\circ$.