Coherent equilibrium in alloys containing spherical precipitates
1995, Acta Metallurgica et Materialia
https://doi.org/10.1016/0956-7151(94)00398-2Abstract
Abstrac~Coherent equilibrium is investigated theoretically in alloys containing monodispersed spherical precipitates. The alloy system is subjected to a Dirichlet tessellation by constructing a Voronoi polyhedron around each particle, and for the purpose of calculation each polyhedron is approximated as a sphere. The basic unit of the microstructure is thus a cell consisting of concentric spheres, with the outer sphere occupied by the majority (matrix) phase. The stresses in each cell are derived analytically using as one of the boundary conditions the requirement that the displacements at the outer boundary vanish. Isotropic elasticity is assumed throughout. The equilibrium volume fraction and the coherent solubilities of the two phases are calculated as a function of the overall concentration of solute in the alloy. The relative importance of the lattice mismatch between the two phases, their elastic constants, and the curvatures of their free energies of mixing, is investigated theoretically. It is predicted that the equilibrium concentrations of both phases always increase with increasing volume fraction. The consequences of the theory on the Ni-Al and Cu-Co alloy phase diagrams are explored. Much larger effects are predicted for Ni-AI alloys than for CuzCo alloys, despite the fact that the lattice mismatch is over three times as large in the latter system. The reason is that the difference between the solute concentrations of the two phases is much bigger in CuzCo alloys. It is concluded that the influence of coherency effects will always be much greater for solid solutions in equilibrium with intermediate phases than for terminal solid solutions. Data on the solubility of Ni3AI in Ni-A1 alloys, which support the theory, are presented and discussed.
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