Phase-Field Modeling of Eutectic Growth in a Ti-Fe System with Multiple Nuclei and Misorientations
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THE phase-field method has emerged as a powerful tool to predict microstructure evolution in many material systems like coherent solid-state phase transitions.[1,2] It is a genuine representation of the original free-boundary problem in sharp-interface limit as the interface thickness tends to zero. In the phase-field concept, the interfacial conditions are avoided by introducing a set of smooth variables, the so-called phase-field variables, which characterize time and spacial evolutions of bulk phases in the underlying system. An elastic phase-field model, which include from the elastic energy and elastic misfit in lamellar eutectic growth, has already been developed in Reference 3. Here it was assumed that the free energy of the system consists of a chemical free energy based on previous models for multiphase and two-phase growth[4–7] and an elastic free energy that includes the misfit stress between eutectic lamellae. Since the misfit stresses and strains arising during the eutectic solidification would relax in the liquid, the study of the influence of elastic interactions on eutectic growth is relevant only in solid–solid interactions where the alloy is completely solidified. On the other hand, in real material systems, grains typically have different orientations. Therefore, the orientation free energy has to be included into the elastic model of Reference 3 to assess the influence of the misfit stresses on eutectic growth in the case of solid-solid interactions. This model can be applied to simulate more realistic growth in Ti-Fe systems as observed in References 8 through 10. Several phase-field models for crystal growth with different orientations have already been developed.[11,12] The first model is based on the work of Kobayashi et al.,[13] ZOHREH EBRAHIMI, is with the Department of Mechanical Engineering, Payame Noor University, PO BOX 19395-3697, Tehran, Iran. Contact e-mail: [email protected] JOAO LUIZ LOPEZ REZENDE, Researcher, is with the Department of Ferrous Metallurgy, RWTH Aachen University, Aachen, 52056, Germany. H. EMMERICH, Professor, is with the Bayreuth University, Bayreuth, Germany. Manuscript submitted March 2, 2012. Article published online December 11, 2012 METALLURGICAL AND MATERIALS TRANSACTIONS A
where a nonconserved orientation field was introduced to specify the orientation of crystals, whereas the solid-liquid phases were distinguished with a single phase-field variable. This method has been applied more recently by Warren et al. to model polycrystalline materials.[14] Similar free energies are used by Gra´na´sy et al. to establish the nucleation and growth in binary alloys.[15] In this article, the previous elastic phase-field model[3] is first adopted to the growth of multiple eutectic nuclei with arbitrary orientations and then applied to eutectic growth in Ti-Fe system. To satisfy different orientations, we formulate the orientation field and orientation energy for binary crystal nucleation as proposed in Reference 15 and for polycrystalline eutectic nucleation in Reference 16. This ext
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