A A. Wheeler

A multi-phase-field model of eutectic and peritectic alloys: numerical simulation of growth structures

In this paper, we extend the multi-phase-field concept, recently developed to model pure systems involving grains, to multi-phase alloy systems. We derive a phase-field model in a general form which has the flexibility to model a variety of binary alloys. In particular, our new model provides a framework for describing and numerically simulating the solidification of both eutectic and peritectic systems. We report computations that exhibit a wide range of realistic phenomena, including eutectic lamellae spacing selection by the annihilation of lamellae through competitive over-growth by their neighbours as well as tip splitting of individual lamellae. Our results are consistent with the scaling predictions of the classical Jackson and Hunt theory of eutectic lamellae. With regards to peritectic growth, we report simulations that exhibit many characteristic features of the peritectic phase transition: below the peritectic temperature the peritectic phase grows preferentially along the properitectic phase by solute diffusion in the liquid until the parent phase is engulfed. The subsequent peritectic transformation continues by solid diffusion on a longer timescale.

A phase-field model of solidification with convection

We develop a phase-field model for the solidification of a pure material that includes convection in the liquid phase. The model permits the interface to have an anisotropic surface energy, and allows a quasi-incompressible thermodynamic description in which the densities in the solid and liquid phases may each be uniform. The solid phase is modeled as an extremely viscous liquid, and the formalism of irreversible thermodynamics is employed to derive the governing equations. We investigate the behavior of our model in two important simple situations corresponding to the solidification of a planar interface at constant velocity: density change flow and a shear flow. In the former case we obtain a non-equilibrium form of the Clausius–Clapeyron equation and investigate its behavior by both a direct numerical integration of the governing equations, and an asymptotic analysis corresponding to a small density difference between the two phases. In the case of a parallel shear flow we are able to obtain an exact solution which allows us to investigate its behavior in the sharp interface limit, and for large values of the viscosity ratio.

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

Abstract In this paper we discuss two phase-field models for solidification of monotectic alloys, a situation in which a liquid phase L 1 may simultaneously transform into both a new liquid phase L 2 and a solid phase S via the reaction L 1 →L 2 +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→L 1 +L 2 occurs. Both our models permit the interfaces to have temperature-dependent surface energies which may induce Marangoni convection at L 1 –L 2 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.

A phase-field model with convection: sharp-interface asymptotics

We have previously developed a phase-field model of solidification that includes convection in the melt [Physica D 135 (2000) 175]. This model represents the two phases as viscous liquids, where the putative solid phase has a viscosity much larger than the liquid phase. The object of this paper is to examine in detail a simplified version of the governing equations for this phase-field model in the sharp-interface limit to derive the interfacial conditions of the associated free-boundary problem. The importance of this analysis is that it reveals the underlying physical mechanisms built into the phase-field model in the context of a free-boundary problem and, in turn, provides a further validation of the model. In equilibrium, we recover the standard interfacial conditions including the Young–Laplace and Clausius–Clapeyron equations that relate the temperature to the pressures in the two bulk phases, the interface curvature and material parameters. In nonequilibrium, we identify boundary conditions associated with classical hydrodynamics, such as the normal mass flux condition, the no-slip condition and stress balances. We also identify the heat flux balance condition which is modified to account for the flow, interface curvature and density difference between the bulk phases. The interface temperature satisfies a nonequilibrium version of the Clausius–Clapeyron relation which includes the effects of curvature, attachment kinetics and viscous dissipation.

Phase-field modeling of multi-phase solidification

A phase-field model for a general class of multi-phase metallic alloys is now proposed which describes both multi-phase solidification phenomena as well as polycrystalline grain structures. The model serves as a computational method to simulate the motion and kinetics of multiple phase boundaries and enables the visualization of the diffusion processes and of the phase transitions in multi-phase systems. Numerical simulations are presented which illustrate the capability of the phase-field model to recover a variety of complex experimental growth structures. In particular, the phase-field model can be used to simulate microstructure evolutions in eutectic, peritectic and monotectic alloys. In addition, polycrystalline grain structures with effects such as wetting, grain growth, symmetry properties of adjacent triple junctions in thin film samples and stability criteria at multiple junctions are described by phase-field simulations.

Modelling of microstructure formation and interface dynamics

A phase-field model for a general class of binary three-phase metallic alloys is presented which describes both, multi-phase solidification phenomena as well as polycrystalline grain structures. The model serves as a computational tool to simulate the motion and kinetics of multiple phase boundaries and enables the visualization of the diffusion processes and phase transitions in multi-phase alloy systems. A selection of numerical simulation results illustrates the capability of the phase-field model to recover a variety of complex experimental growth structures. In particular, the discretized model is used to simulate the microstructure evolution in eutectic, peritectic and monotectic alloys. Moreover, the temporal development of polycrystalline grain structures with effects such as wetting, grain growth, symmetry properties of adjacent triple junctions in thin film samples and stability criteria at multiple junctions is shown in various simulations.

A Phase-Field Model of Convection With Solidification

A phase-field model for the solidification of a pure material that incorporates convection has recently been developed [Anderson, McFadden and Wheeler, Physica D, 135 (2000) pp. 175-194]. This model is a two-fluid model in which the solid phase is modeled as a sufficiently viscous fluid. The model allows for the solid and liquid phases to have different densities and hence allows for expansion or contraction flows upon solidification. In this paper we investigate numerically a simplified version of this model by considering solidification occurring between the two closelyspaced parallel plates of a Hele-Shaw cell. We assess two key aspects of the model: (1) the effect of density differences between the solid and liquid phases during dendritic growth and (2) the role played by the viscosity ratio between the solid and liquid phases.