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 presentation, 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. In the end, we will discuss the effects of the water-ice-air tri-junction conditions and other parameters on the final shape of the frozen droplet.