*4.1. Phase-Field Method*

Phase-field method is based on the description of diffuse-interface model, and the concept of diffuse interface is derived from van der Waals [76] and Cahn and Hilliard [77]. In contrast to conventional sharp-interface models (field variables are discontinuous at interface), the interface in the diffuse-interface model is represented by a series of consecutive values [18]. Assigning different values to different phases (for example, γ' strengthening phase and γ matrix phase in nickel-based superalloys), the interface region is implicitly given [78]. Such a method makes it possible to analyze the evolution of complex microstructures, including precipitation, dissolution, coarsening, connection, and topological inversion of γ' strengthening phases [79,80]. Additionally, the physical characteristics, i.e., the γ/γ' lattice misfit, elastic constants and dislocation densities, and the mechanical behavior, i.e., heterogeneous elasticity and plastic deformation, of superalloys can be well incorporated in the mathematical framework of the phase-field method. The phase-field method has become an excellent technique to study internal stresses and plastic deformation and illustrate the mechanisms for the microstructure evolution in nickel-based superalloys.

The application of the phase-field method in the simulation of microstructural evolution was initially based on an elastic framework, where the mechanical stress, σ*A*, the γ/γ' lattice misfit, δ, and the difference of mechanical properties between γ and γ' phases, Δ*G*, were incorporated in the model to reveal the rafting behavior [14,15,26]. The elasto-plastic frameworks, which take into account the plastic deformation in γ matrices during creep, were later developed [8,12,19].

Superalloys are generally multiphase and/or multicomponent materials, which require large number of field variables or physical quantities in the phase-field method in addition to mechanical deformation (elastic and elasto-plastic deformation). They provide a practical application of the phase-field theory in the understanding of the microstructure evolution of multiphase, polycrystal, and multicomponent materials [81]. The combination of the phase-field model and the crystal plasticity likely opens a new avenue for the study of the mechanical deformation and microstructure evolution of superalloys under concurrent action of thermal and mechanical loading.
