Thermodynamically consistent phase-field modeling of three-phase solidification with density variation

Ice formation plays an important role in many industrial applications as well as natural phenomena. In this talk, we will present a phase-field model for the non-isothermal three-phase system that involves water, ice, and air. The water-ice phase transition is modeled by the Allen-Cahn equation and the water-air interface is tracked by the Cahn-Hilliard equation. The constitutive relations are derived based on non-negative entropy production, such that the whole set of governing equations, including the Navier-Stokes and heat equations, are consistent with the second law of thermodynamics. The three-phase mixture is treated as quasi-incompressible, i.e., the pure phases are incompressible, whereas the mixture density may evolve due to variations in composition. Our model automatically captures the curvature and pressure effects on the melting temperature, as represented by the Gibbs-Thomson and the Clausius-Clapeyron equations, respectively. Meanwhile, the volume change during phase transition is faithfully captured by the quasi-incompressibility of the mixture. In the end, we will present some numerical results on the freezing process of a water droplet deposited on a cold substrate, where a pointy tip forms due to volume expansion.