Diffusion-controlled phase transformation in a finite region
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Diffusion-Controlled Phase Transformation in a Finite Region
When the initial concentration is different from the equilibrium concentration value at the interface in one or both regions, diffusion starts, and phase transformtion is induced at the interface which then moves to reduce the concentration gradient. The process stops when the concentration becomes uniform at its equilibrium value in each phase. The whole system is assumed to be isolated. Hence, no mass flux crosses any outer boundary. The concentrations at the interface are assumed to be the constant equilibrium values during the entire diffusion process. Solutions are sought for the location of the interface and the concentration profile during the transition period. In the present work, only the extreme cases are solved where the diffusivity is much smaller in one phase than in the other. Hence, diffusion takes place only within one phase, although the other phase will change in dimension accordingly. The two cases considered are represented schematically in Fig. 1. In Case 1, the diffusion is negligible within phase B, and the initial concentration within phase A is higher than the equilibrium value in phase A at the interface, which is higher than the corresponding equilibrium value in phase B. The interface moves to the right so that additional phase A is formed at the expense of phase B while the overall concentration within phase A is depleted. The process stops when the concentration within phase A is at its uniform equilibrium value as dictated at the interface. This case is applicable to the growth of high-carbon austenite into ferrite after pearlite dissolution is completed during the intercritical annealing of two-phase steels 2 where the growth of austenite is controlled by carbon diffusion through the austenite phase (which is slower than in the ferrite phase). In Case 2, the diffusivity of a substitutional solute in phase A is assumed to be negligibly small, and phase A is again formed from phase B, but in this case due to the depletion of concentration within phase B. This case is applicable to the growth of austenite (Phase A ) into ferrite (Phase B) where the growth rate of austenite is controlled by the diffusion of manganese through
R. H. T I E N The importance and potential applicability of the class of diffusion-controlled phase transformations within a finite-region was discussed, and a numerical solution through a finite difference representation was given in a previous work by Tanzilli and Heckel) The system involves a moving boundary which leads to a nonlinearity, and the only available analytical solution is restricted to the case of an infinite or semi-infinite region with special initial conditions. The aim of the present work is to provide a semianalytical solution when the phase transformation occurs in a finite region. Mathematical Model Two finite regions containing different phases with constant but different initial concentrations, system including austenite and ferrite phases as an example, are brought into perfect c
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