Hybrid-PFC: Coupling the phase-field crystal model and its amplitude-equation formulation

The phase-field crystal (PFC) model describes crystal structures on diffusive timescales through a periodic, microscopic density field. It has been proposed to model elasticity in crystal growth and encodes most of the phenomenology related to the mechanical properties of crystals like dislocation nucleation and motion, grain boundaries, and elastic or interface-energy anisotropies. To overcome limitations to small systems, previous studies introduced a coarse-grained formulation focusing on slowly varying complex amplitudes of the microscopic density field. This amplitude-PFC (APFC) model describes well elasticity and dislocations while approximating microscopic features and being limited in describing large-angle grain boundaries. We present here the foundational concepts for a hybrid multiscale PFC-APFC framework that combines the coarse-grained description of the APFC model in bulk-like crystallites while exploiting PFC resolution at dislocations, grain boundaries, and interfaces or surfaces. This is achieved by coupling the two models via an advanced discretization based on the Fourier spectral method and allowing for local solution updates. This discretization also generalizes the description of boundary conditions for PFC models. We showcase the framework capabilities through two-dimensional benchmark simulations. We also show that the proposed formulation allows for overcoming the limitations of the APFC model in describing large-angle grain boundaries.