Theoretical and Numerical Study of Growth in Multi-Component Alloys
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DIFFUSION-DRIVEN growth of precipitates is a classical problem in phase transformations in materials. In binary alloys, it has been studied by Zener,[1] Frank,[2] and Ham.[3] The classical result from Zener[1] for the case of binary alloys is that the growth of a precipitate into a supersaturated matrix in the diffusion-controlled regime can be described through a scaling law, where the square of the displacement of the interface is directly proportional to time, and the proportionality constant gs is characteristic of the chosen supersaturation as well as the diffusivity. Our aim in this paper is to extend the classical Zener solution in binary alloys to a generic multi-component alloy for any given arbitrary diffusivity matrix. Here, in this paper, we derive closed-form analytical expressions for the prediction of the composition profiles of the elements in the matrix in the scaling regime as well as the parabolic growth exponent. These results are validated and benchmarked using numerical simulation methods which are described in the following paragraphs. At the theoretical level, the difference between binary and multi-component systems is in the way diffusion is treated. Fickian diffusion in a binary phase requires the knowledge of interdiffusivity; however, in a multicomponent phase (with, say K components), it is
ARKA LAHIRI, T.A. ABINANDANAN, and ABHIK CHOUDHURY are with the Department of Materials Engineering, Indian Institute of Science, Bangalore 560012, India. Contact e-mails: [email protected], [email protected] Manuscript submitted January 31, 2017.
METALLURGICAL AND MATERIALS TRANSACTIONS A
characterized by a (symmetric, [ðK 1Þ ðK 1Þ] diffusivity matrix with [KðK 1Þ=2] elements, of which (K 1) are diagonal elements). This manifests itself in a key difference between binary and multi-component systems: in binary alloys, the matrix and precipitate compositions are fixed by the temperature; in multi-component alloys, however, the chosen tie-line (i.e., matrix and precipitate compositions at the interface selected during growth) may be (a) different from the thermodynamic tie-line containing the alloy composition, and (b) the selection is dependent on the relative solute diffusivities. This can be appreciated by comparing the Stefan problems in binary and multi-component systems. Particularly, the Stefan condition in a binary alloy, which relates the velocity of the interface to the diffusion gradients writes as, @ca ; v caeq cbeq ¼ D j @r interface
½1
where caeq and cbeq are the equilibrium compositions of matrix and precipitate phases, respectively. v, D, and r denote the velocity, the solute diffusivity in the matrix (we assume the gradients in the precipitate to be small, for brevity), and the spatial coordinate, respectively. Here, fixing the far-field composition and the temperature, the velocity v of the interface is the only unknown in the problem, which is determined by solving the diffusion equation in conjunction with the Stefan condition and the far-field bounda
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