(147e) Statistical Mechanics of the Triple Contact Line

Paul Steen was a scholar of diverse interests, ranging from the history of science (Steen & Brutsaert, Annu. Rev. Fluid Mech. 49, 2017), to the dynamics of the gas-solid-liquid triple contact line (Bostwick & Steen, Annu. Rev. Fluid Mech. 47, 2015). His experiments and theories on mobility of the contact line inspired many scientists.

This talk considers the simpler case of the contact line at equilibrium. It describes how the static contact angle can adopt hydrophilic or hydrophobic behavior by dint of different statistics in the geometry of cavities that pit the solid surface.

To show this, calculations invoke the simplest mean-field "Ising" statistical mechanics, in which each cavity is either full or empty, while being connected to near neighbors by thin necks. The resulting theory predicts equilibrium angles for advance and recession in terms of the Young contact angle and the joint statistical distribution of two quantifiable geometrical parameters representing specific neck cross-section and specific cavity opening. It attributes contact angle hysteresis to the occurrence or absence of latent energy-releasing "first-order" phase transitions in the ensemble of adjacent cavities collectively imbibing or rejecting liquid. It calculates potential energy barriers that hysteresis erects against overcoming contact line pinning. By determining whether phase transitions occur, this ab initio analysis predicts the existence of six distinct regimes, including the non-hysteretic "dry" and "wet" Cassie-Baxter states (Cassie & Baxter, Trans. Faraday Soc. 40, 1944). It explains why two other regimes exhibit "metastability" in recession, depending on initial conditions that experiments can muster (Callies and Quere, Soft Matter 1, 2005). It also dismisses the notion of a "Wenzel state" by identifying the regimes that such state covers, and by establishing the origin of its inherent hysteresis, which the Wenzel model ignored (Wenzel, Ind. Eng. Chem. 28, 1936).

We compare predictions of our theory with measured contact angles on beds of microscopic rods (Gao & McCarthy, Langmuir 22, 2006), and for variations of the advancing angle with surface energies of the liquid (Shibuichi, et al, J. Phys. Chem. 100, 1996). We prescribe how such rods should be arranged to produce desirable properties, such as superhydrophobia or superhydrophilia. Finally, we indicate how the theory can be extended to capture electrowetting behavior.