Phase-field modeling of liquid droplet migration in a temperature gradient

Abstract

The migration of liquid droplets in a solid phase caused by temperature gradient zone melting (TGZM) is simulated by employing a quantitative phase-field (PF) model proposed by Echebarria et al. The PF simulation results are compared with the predictions of an analytical model that describes the droplet migration for both static and dynamic conditions, allowing the direct solution of the time dependent migration velocity of a liquid droplet that is initially located at an arbitrary position in the mushy zone. For dynamic conditions as e.g. during directional solidification, criteria for the critical pulling velocity and critical droplet position are suggested and validated by the PF simulations. When the pulling velocity is lower than the critical pulling velocity, the droplet will migrate through the moving liquidus into the bulk liquid. The droplet velocity gradually increases as it is approaching liquidus. On the other hand, when a pulling velocity higher than the critical pulling velocity is imposed, the droplet will travel through the moving solidus into the fully solid region while the droplet velocity decreases with time. The droplets initially located above the critical position migrate toward liquidus, while the others sink into the bulk solid. The effect of the temperature gradient on the droplet migration kinetics is investigated by both PF simulations and analytical predictions. The results confirm that the upward droplet migration velocity increases, while the time needed for a liquid droplet to move through the entire mushy zone decreases with increasing temperature gradient. The PF simulation results compare well with the analytical predictions.