Partial wetting by a liquid drop of a horizontal solid surface: beyond Young's equation

  • Medale, Marc (AMU, IUSTI, CNRS UMR 7343)
  • Brutin, David (AMU, IUSTI, CNRS UMR 7343)

Please login to view abstract download link

The equation that governs the equilibrium shape of a liquid drop in contact with a solid substrate remains a topical scientific question. Indeed, Young's equation determines the value of the macroscopic contact angle as a function of the surface tensions at the solid-liquid, solid-gas and gas-liquid interfaces. However, it solves for a unique equilibrium value, meanwhile most experiments report a contact angle hysteresis between advancing and receding contact lines. Finally, most of the assumptions involved in its derivation have never been clearly stated and several questions therefore remain open. What is its domain of validity? Could other physical contributions such as line tension and/or gravity have some influence on the value of the macroscopic contact angle? To contribute to answer these questions, we have derived a macroscopic model based on the physico-chemical equilibrium condition of the system, which corresponds to the minimization of the total energy of the solid-liquid-gas system. This approach leads to deal in a unified formalism with either gravity-assisted wetting (sessile drop) or opposing one (pendant drop), along with weightlessness as a particular case. The latter case leads to an analytical governing equation, which is trigonometric, thanks to the spherical cap model. On the other hand, in a gravity field assuming the solid substrate to be horizontal endows the problem with a rotational symmetry that leads to a set of six Euler-Lagrange equations to be solved numerically. The main outcomes of the proposed approach are: i) in weightlessness, line tension does influence the macroscopic contact angle in a 1/r trend (the dimensionless wetted radius), as long as r is lower than roughly one thousand. Above this value the macroscopic contact angle asymptotically tends towards the value given by the Young's equation; ii) in a gravity field the macroscopic contact angle also depends on the Bond number over a broad range of values. Thus, the range of validity of Young's equation is the one that simultaneously satisfies two conditions: i) large wetting radius (when the relative influence of line tension is negligible compared to that of surface tensions) and ii) small drop size compared to the capillary length (low Bond number). It follows that in many situations in earth gravity where these two conditions cannot be simultaneously satisfied, the surface tensions deduced from Young's equation could be incorrectly evaluated.