Density-amplitude formulation of the phase-field crystal model for two-phase coexistence in two and three dimensions

The phase-field crystal approach describes the dynamics of the local atomic probability density on atomic length and diffusive time scales. However, the small grid spacing required to resolve the atomic length scale restricts the applicability to relatively small systems. We present a new amplitude-equation formulation that can be applied to model the evolution of a two-phase system by building upon the multiple-scale approach developed for the phase-field crystal model that explicitly incorporates a miscibility gap in the atomic density field. The new set of equations are less stiff by two orders in the spatial derivative and are found to retain acceptable accuracy even with a significantly larger grid size. Furthermore, the proposed model provides insight into the link between the phase-field-crystal model and a phase-field model in which conserved and nonconserved dynamics are coupled to describe a solid-liquid system. Models and simulation results are presented for two-dimensional hexagonal and three-dimensional body-centered cubic structures. Validation of the two-dimensional model is carried out by examining the Gibbs-Thomson effect. Simulations of growth and coarsening are performed to demonstrate the model's potential for large-scale simulations.