A new approach to multi-phase formulation for the solidification of alloys

• Published in: Physica. D Vol. 241; no. 8; pp. 816 - 829
• Main Authors: Bollada, P.C., Jimack, P.K., Mullis, A.M.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 15.04.2012
• Subjects: Alloys; Crystal growth; Density; Derivatives; Dynamical systems; Entropy; Eutectic; Free energy; Lagrange multiplier; Mathematical models; Multi-phase; Phase field; Solidification; Crystal growth; Multi-phase; Lagrange multiplier; Eutectic; Phase field; Solidification
• ISSN: 0167-2789; 1872-8022
• DOI: 10.1016/j.physd.2012.01.006

This paper demonstrates that the standard approach to the modelling of multi-phase field dynamics for the solidification of alloys has three major defects and offers an alternative approach. The phase field formulation of solidification for alloys with multiple solid phases is formed by relating time derivatives of each variable of the system (e.g., phases and alloy concentration), to the variational derivative of free energy with respect to that variable, in such a way as to ensure positive local entropy production. Contributions to the free energy include the free energy density, which drives the system, and a penalty term for the phase field gradients, which ensures continuity in the variables. The phase field equations are supplemented by a constraint guaranteeing that at any point in space and time the phases sum to unity. How this constraint enters the formulation is the subject of this paper, which postulates and justifies an alternative to current methods. ► Identifies three major defects in the standard Lagrange Multiplier formulation of phase-field modelling of solidification. ► Proposes a set of attributes that a successful formulation must have. ► Identifies where the degree of freedom lies in the system to allow a formulation that satisfies the above criteria. ► Proposes alternatives that satisfy the criteria. ► Numerical results are given which support the new model.