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Modelling of pattern formation in solidification with the phase-field method

By: Mathis Plapp
From: Ecole Polytechnique, Palaiseau, France
At: Complexo Interdisciplinar, Anfiteatro
[2009-11-04] 12:00

The microstructures formed during the solidification of metallic alloys have been studied by metallurgists for decades because of their influence on the mechanical properties of the finished material. To predict which type of structure and charactieristic length scales are selected under given growth conditions is a classic problem of pattern formation outside of equilibrium. Complex morphologies such as dendrites or cellular structures arise from a subtle interplay between instabilities linked to the transport of heat and solute and the dynamics of the solid-liquid interfaces. The phase-field method has emerged as a powerful and elegant tool to simulate numerically such moving boundary problems. It describes the moving interfaces as diffuse but narrow kinks of a smooth time-dependent scalar field that satisfies an equation of motion obtained from a phenomenological free energy functional. With recent improvements both in model design and numerical algorithms, fully quantitative three-dimensional simulations can be carried out on the scale of a few microstructural units for experimentally relevant growth conditions. This has made it possible to directly compare simulations and experiments, and to critically assess existing models and theories of pattern formation in solidification. After an introduction to the phase-field method and the matched asymptotic analysis that is crucial to obtain quantitative results, its capabilities will be illustrated by results on dendritic growth as well as on the directional solidification of dilute and eutectic binary alloys. Furthermore, the generalization of the method to other moving boundary problems will be discussed.