A multiphase-field model for bubble and dendrite co-growth considering gas-liquid density difference

Gas porosity formation during solidification is influenced by bubble-dendrite interactions and the significant gas-liquid density difference, which induces an expansion flow during bubble growth. However, existing phase-field models generally neglect the expansion flow or are only restricted to bubble or dendrite growth. To address this limitation, a multiphase-field model for bubble and dendrite co-growth is developed, incorporating gas-liquid density difference and expansion flow into the diffusion equation. The solid-liquid (S-L) and gas-liquid (G-L) transformations, the partitioning and diffusion of alloy solute and dissolved gas species, as well as bubble-dendrite interactions, are considered in this model. The driving force for bubble growth is derived from the chemical potential difference, which provides a bridge connecting supersaturation and pressure driving force based on Sievert's law. Numerical results regarding growth velocity, concentration distribution, and contact angle are validated against analytical solutions. Simulations elucidating the influence of density differences reveal that expansion flow retards bubble growth kinetics compared to the pure diffusion-controlled regime. Furthermore, the co-evolution of dendrites and a bubble during directional solidification is simulated. The results show that the bubble grows by absorbing surrounding local gas supersaturation and subsequently stops growing due to entrapment by the dendritic front, which is consistent with experimental observations. This work offers a robust framework for the quantitative prediction of microstructural evolution and gas porosity formation.