Three-position model of segregations and critical grain sizes in nanocrystalline alloys

A three-position thermodynamic model of grain-boundary segregations in binary and ternary alloys is proposed that accounts for the finite grain size and the energetic inequivalence of sites at the boundary. The classical McLean and Fowler isotherms are generalized to the case of three types of sites (i.e., with negative, near-zero, and positive segregation energy). The model parameters are matched to the first moments of the skew-normal spectra of segregation energies E s obtained from atomistic and machine-learning calculations. For concentrated systems and ternary alloys, the equilibrium concentrations are obtained by minimization of the free energy of a system consisting of the bulk and grain boundaries, and the critical grain sizes are determined. The critical size L cr (1) is the grain size at which the grain interior is practically purified, the bulk concentration falling to nearly zero, while the boundary concentration undergoes a sharp break; L cr (2) is the grain size at which the integral degree of demixing is maximal. Both depend on temperature and on the average composition. In the dilute weak-segregation approximation, analytical expressions are also given for two characteristic sizes, L c (1) (half-depletion radius) and L c (2) , which locate a smooth crossover in this case. It is shown that, in the presence of sites with a tendency to antisegregation, the average boundary concentration in the low-temperature limit tends to a value lower than unity; with decreasing grain size, the impurity is exhausted first in the bulk and then in the sites with segregation energies of smaller absolute value. In ternary systems with competition for the same sites, “locking” effects of the components appear and the concentration–grain-size dependences can become non-monotonic. Size-effect analysis is performed for Ni–Cu and Ni–(Pd, Pt) alloys. A generalized phase diagram of the nanocrystalline Ni–Cu alloy is constructed that demonstrates the suppression of decomposition and the stabilization of the solid solution upon grain-size reduction. The model reveals qualitative features of segregation size effects that are inaccessible within the single-site approximation and can serve as a basis for the analysis of thermodynamic stability of nanocrystalline alloys.