Implementation of classical nucleation and growth theories for precipitation

Three ways of implementing classical nucleation and growth theories for precipitation are presented and discussed: (i) the “mean radius approach” (particle size distribution is restricted to its mean radius and density); (ii) the “Euler-like multi-class approach” (the particle size distribution is discretized in several size classes and its time evolution is calculated evaluating the fluxes between neighboring classes); and (iii) the “Lagrange-like multi-class approach” (the particle size distribution is again discretized in several size classes, whose radius time evolution are calculated). In some simple cases, the three approaches lead to similar results, but when more complex heat treatments are involved, multi-class approaches are required. Although the Euler-like approach involves a more complex class number management, it is more adapted to the modeling of precipitate chemistry. Some examples of implementation are presented: Cu precipitation in ferrite, Al3Sc precipitation in aluminum, VC and NbVC precipitation in austenite.