Phase-field simulation of freezing water droplet

When a water droplet freezes on a cold plate, a pointy tip forms as the result of volume expansion. In this dissertation, we will introduce a quasi-compressible phase-field model that deals with this non-isothermal three-phase system involving water, ice, and air. The water-ice phase transition and the water-air fluid interface are handled by the Allen-Cahn and the Cahn-Hilliard equations, respectively. The governing equations, including the two phase-field equations, the Navier-Stokes equations, and the energy equation, are designed such that the non-negative entropy production is guaranteed. These equations are then solved by finite-element methods using the open-source deal.ii library. Our model reproduces the Gibbs-Thomson and Clausius-Clapeyron equations, which establish the dependence of the melting temperature on interface curvature and pressure, respectively. Furthermore, the built-in quasi-compressibility accurately accounts for the volume change due to the water-ice density contrast during the phase transition. With proper parameters, our simulation captures the pointy tip of the frozen droplet with good agreement with the experiment.