Precipitate growth in multi-component systems with stress relaxation by diffusion and creep
The present paper is devoted to the diffusional growth of spherical precipitates having a volumetric misfit. Thus, the matrix exerts a confining pressure on the growing precipitate retarding its growth. Elastic deformation, creep and diffusion are assumed as mechanisms accommodating the misfit of the precipitate in the matrix. Based on the Thermodynamic Extremal Principle, evolution equations for the state variables (e.g. precipitate radius, chemical composition of the precipitate) are derived. The solution of these evolution equations is compared with a number of limiting cases treated in closed analytic form. Results are worked out for model parameters similar to those of a steel with 0.5% C and 2% Cr. The most important result is the evolution of the precipitate radius and its dependence on the creep strength of the matrix. If the material creeps readily, the growth rate is controlled by rapid diffusion of interstitial atoms and is, therefore, high. For a creep resistant matrix, the excess volume of the growing precipitate must be accommodated by the slow diffusion of substitutional atoms away from the precipitate. The precipitate grows correspondingly slower, in the present case by four orders of magnitude. Further results are shown for the confining pressure on the precipitate and for chemical variables.