Multiscale Problems in Solidification Processes

Our objective is to describe solidification phenomena in alloy systems. In the classical approach, balance equations in the phases are coupled to conditions on the phase boundaries which are modelled as moving hypersurfaces. The Gibbs-Thomson condition ensures that the evolution is consistent with thermodynamics. We present a derivation of that condition by defining the motion via a localized gradient flow of the entropy. Another general framework for modelling solidification of alloys with multiple phases and components is based on the phase field approach. The phase boundary motion is then given by a system of Allen-Cahn type equations for order parameters. In the sharp interface limit, i.e., if the smallest length scale ± related to the thickness of the diffuse phase boundaries converges to zero, a model with moving boundaries is recovered. In the case of two phases it can even be shown that the approximation of the sharp interface model by the phase field model is of second order in ±. Nowadays it is not possible to simulate the microstructure evolution in a whole workpiece. We present a two-scale model derived by homogenization methods including a mathematical justification by an estimate of the model error

A Two-Scale Method for Liquid-Solid Phase Transitions with Dendritic Microstructure

A two-scale model for liquid-solid phase transitions with equiaxed dendritic microstructure for binary material with slow solute diffusion is presented. The model consists of a macroscopic energy transport equation, coupled with local cell problems describing the evolution of the microstructure and the microsegregation. It is derived by an asymptotic expansion of a sharp interface model with Gibbs-Thomson effect. A discretization of the model leading to a two-scale method for such problems is presented, and a numerical example is given