Abstract
In this paper first it is shown for several geometries that classical similarity solutions for particle growth exist if and only if the Stefan problem is well-posed in the sense of being mass conserving. The extension of the similarity solutions to multicomponent alloys, which makes the problem nonlinear, is illustrated by the application to a hypothetic alloy with realistic input values. The similarity solutions are based on the assumption of local equilibrium at the interface. In the second part, the assumption of local equilibrium is relaxed using a first-order interface reaction. The influence of the interface reaction on the movement of the interface and on the interface concentrations is evaluated using Finite Difference calculations. A Newton scheme is used to solve the nonlinear problem.
Highlights
In the thermal processing of both ferrous and non-ferrous alloys, homogenization of the as-cast microstructure by annealing at such a high temperature that unwanted precipitates are fully dissolved is required to obtain a microstructure suited to undergo heavy plastic deformation
In this study we describe particle growth as a Stefan problem, i.e. a diffusion equation with a moving sharp interface interface between the particle and its surrounding diffusive phase
Since metallic alloys often contain secondary particles in the form of plates, needles and spherical particles, analytic solutions for several geometries have been constructed for the growth of particles
Summary
In the thermal processing of both ferrous and non-ferrous alloys, homogenization of the as-cast microstructure by annealing at such a high temperature that unwanted precipitates are fully dissolved is required to obtain a microstructure suited to undergo heavy plastic deformation. Next to viewing particle dissolution and growth as a Stefan problem with a sharp interface, diffuse-interface models, such as the phase-field method, the Cahn-Hilliard equation, are presented with the appropriate references for the metallurgical literature. We apply a first-order interface reaction for dissolution and growth of particles in multi-component alloys, and present a numerical solution of this nonlinear problem, which is the second innovation of this paper. The similarity solutions are used as an initial guess for the numerical solution of the nonlinear problem
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