We investigate the critical nuclei morphology in phase transformation by combining two effective ingredients, with the first being the phase field modeling of the relevant energetics which has been a popular approach for phase transitions and the second being shrinking dimer dynamics and its variants for computing saddle points and transition states. In particular, the newly formulated generalized shrinking dimer dynamics is proposed by adopting the Cahn-Hilliard dynamics for the generalized gradient system. As illustrations, a couple of typical cases are considered, including a generic system modeling heterogeneous nucleation and a specific material system modeling the precipitate nucleation in FeCr alloys. While the standard shrinking dimer dynamics can be applied to study the non-conserved case of generic heterogeneous nucleation directly, the generalized shrinking dimer dynamics is efficient to compute precipitate nucleation in FeCr alloys due to the conservation of concentration. Numerical simulations are provided to demonstrate both the complex morphology associated with nucleation events and the effectiveness of generalized shrinking dimer dynamics based on phase field models.

We critically compare the practicality and accuracy of numerical approximations of phase field models and sharp interface models of solidification. Here we focus on Stefan problems, and their quasi-static variants, with applications to crystal growth. New approaches with a high mesh quality for the parametric approximations of the resulting free boundary problems and new stable discretizations of the anisotropic phase field system are taken into account in a comparison involving benchmark problems based on exact solutions of the free boundary problem.