Phase-field model for solidification of a monotectic alloy with convection

In this paper we discuss two phase-field models for solidification of monotectic alloys, a situation in which a liquid phase L1 may simultaneously transform into both a new liquid phase L2 and a solid phase S via the reaction L1→L2+S. The first model uses three different phase-fields to characterize the three phases in the system and, in addition, a concentration field. This construction restricts the validity of the model to describe phase transitions within the vicinity of the monotectic temperature. In contrast, the second model distinguishes the two liquid phases by their concentration using a Cahn–Hilliard type model and employs only one phase-field to characterize the system as solid or liquid. This formulation enables the second model to represent a wider temperature range of the phase diagram including the miscibility gap where the spinodal decomposition L→L1+L2 occurs. Both our models permit the interfaces to have temperature-dependent surface energies which may induce Marangoni convection at L1–L2 interfaces in non-isothermal systems. By deriving a generalized stress tensor including stresses associated with the capillary forces on the diffuse interface, we extend the two monotectic phase-field models to account for convection in both liquid phases. Together with a generalized set of Navier–Stokes equations, we give a complete set of dynamic field equations to describe monotectic systems with fluid flow. Finally, we present numerical simulations of lamellar monotectic growth structures which exhibit wetting phenomena as well as coarsening and particle pushing.