Nucleation and growth of second-phase precipitates under quenching and annealing

The Langer–Schwartz equations for precipitation are formulated to calculate nucleation, growth and coarsening of second-phase precipitates under non-isothermal situations. A field-theoretic steady-state nucleation rate model is used in the analysis. The field-theoretic nucleation rate is compared with the classical nucleation rate relation. The results of our model calculations are compared with experiments on the precipitation of cobalt particles in Cu–Co alloys under isothermal conditions. This integrated model is then used to simulate a body-centered cubic to hexagonal closed-packed phase quenching and subsequent annealing steps in the hcp phase of a dilute zirconium alloy, where the mean precipitate size, the number density and the degree of supersaturation are calculated as a function of time. The influence of cooling rate on the aforementioned parameters is evaluated. A lower cooling rate results in larger precipitates with a smaller number density in concordance with observations.