Statistical mechanics of the triple contact line

We outline a statistical mechanics of the triple gas-solid-liquid contact line on a rough plane. The analysis regards the neighborhood of the line as a solid dotted with cavities. It adopts the simplest mean-field statistical mechanics, in which each cavity is either full or empty, while being connected to near neighbors by thin necks. The 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 first-order phase transitions among adjacent cavities, as they collectively imbibe or reject liquid. It also calculates the potential energy barriers that hysteresis erects against overcoming contact line pinning. By determining whether the phase transitions can release latent energy, this ab initio analysis distinguishes six regimes, including two metastable recession states. We compare predictions with data for superhydrophobia on microscopic rods; for hysteresis in the "Wenzel state"; and for variations of the advancing contact angle with surface energies of the liquid.