A new magnetic-coupled binary phase-field crystal model and its efficient fully discrete decoupled energy-stable scheme

Despite extensive research on ferromagnetic materials and binary alloys, the interplay between magnetic fields and binary microstructure formation has not yet been systematically modeled within the phase-field crystal (PFC) framework. In this work, we propose a novel magnetic-coupled binary PFC model which, to the best of our knowledge, is the first to integrate ferromagnetic ordering and binary atomic density evolution within a unified formulation. The resulting system consists of three strongly coupled and highly nonlinear partial differential equations: two mass-conserved Allen–Cahn type equations for the binary density fields, and a third Allen-Cahn type equation governing the magnetization field dynamics, derived via a Ginzburg–Landau approach. To solve this challenging system efficiently, we further develop a fully discrete scheme that combines the Invariant Energy Quadratization (IEQ) and Zero-Energy-Contribution (ZEC) approaches for time discretization with a Fourier spectral method for spatial discretization. The proposed scheme is linear, fully decoupled, unconditionally energy stable, and second-order accurate in time, enabling robust and efficient simulation of magneto-structural dynamics. Extensive two- and three-dimensional numerical experiments demonstrate the method’s capability to capture magnetically driven phenomena such as crystal growth and phase transitions. This work not only establishes a new theoretical framework for modeling magnetic field-coupled binary systems but also provides an effective computational tool for simulating complex ferromagnetic material behavior.