Phase-Field Model for Mixed-Mode of Growth Applied to Austenite to Ferrite Transformation
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ZENER in his pioneering work[1] proposed a diffusioncontrolled phase-transformation model, where the diffusion in the product-phase may or may not be considered with a compulsory diffusion at the parent-phase and at the interface adjoining the two phases, there is a discontinuity of the concentration. The compositions at the interface are assumed to be the result of the equilibrium. Many workers applied this ‘local-equilibrium’ approach.[2–4] However, the experimental results of rapid-solidification, the solutetrapping phenomenon raised doubts on the validity of this model. This approach can actually be considered to be oneextreme case of interface-movement which is applicable primarily when the transformation-velocity (V) is low. The other extreme is when the interface-movement is so fast that all the solute atoms due to infinite diffusivity are transferred and the boundary-layer at the interface disappears. An attempt has been made by Sietsma and Zwaag[5] with one single parameter Z to explain these two-extremes of a general transformation approach applied to solid-state transformation in Fe-C alloy, where Z is basically a ratio of diffusive-velocity to the interface-velocity, so that Z ¼ 0 for diffusional transformation and is infinite for the otherextreme. Thus, it is expected that for modeling actual phasetransformations, Z will be finite and non-zero. A more detailed state of non-equilibrium is explained by Sobolev[6,7] as a consequence of V. According to Sobolev, when V VDb VT , (where VDb is the diffusive-velocity for
AVISOR BHATTACHARYA, formerly Doctoral Student with the Department of Materials Science and Engineering, Indian Institute of Technology, Kanpur, India, is now with the Karlsruhe Institute of Technology, Institute of Applied Materials, Karlsruhe, Germany. Contact e-mail: [email protected] C.S. UPADHYAY, Professor, is with the Department of Aerospace Engineering, Indian Institute of Technology, Kanpur, India. S. SANGAL, Professor, is with the Department of Materials Science and Engineering, Indian Institute of Technology. Manuscript submitted April 4, 2014. METALLURGICAL AND MATERIALS TRANSACTIONS A
solute-transport and VT is the velocity of heat-wave) the local-equilibrium prevails and partition-coefficient(k) is equal to the equilibrium-partition coefficient (keq ). For V0) In most of the alloys, the diffusivities of solute and solvent atoms are different. In such alloys, it is possible that for even higher velocity, the more mobile solute atoms can redistribute themselves close to equilibrium, whereas the solvent atoms cannot. Therefore, the driving-force for the transformation is entirely due to nonzero K2 . In this case, K ¼ Kðcc ; VÞ ¼ K2 . Therefore, it can be noted that as the interface-velocity increases, the diffusion-potentials of the two phases no longer remain equal. Moreover, if the diffusivity of the solute in the growing product-phase is higher than that in the parent-phase, it can be assumed that solute-trapping will be absent and also if the product-phase is stoicheometric, ca ceq
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