On the phase field based model for the crystalline transition and nucleation within the Lagrange multiplier framework

Understanding the complexity of the nucleation and transition between the crystalline and quasicrystalline is significant because the structural incommensurability is anisotropic and of significance in revealing material properties. This paper reports the two- and three-dimensional nucleation and transition investigation from quasicrystals to crystals using a phase field method. The investigation starts with the Lifshitz–Petrich (LP) model, which is derived from the Landau theory for the exploration of critical nuclei. By the variational derivation, we construct two phase field models with tenth- and eighth-order, which provide the possibility of exploring the transition under stable phase states. In order to dissipate the original Lifshitz–Petrich energy, we apply a Lagrange multiplier method to modify the two models and solve them by the Fourier spectral method. Whereas the nonlinearity leads to expensive computational burden and extra stiffness, the designed algorithm can effectively avoid the numerical oscillations caused by rigidity and keep an O(Nlog⁡N) computational complexity, where N is the mesh grid size. To further demonstrate the robustness and advantages of the proposed method for handling phase-field modeling of crystalline structures, we compare its performance with other methods for constructing unconditionally stable methods. Our method can be directly implemented on a GPU for acceleration and achieves multiple times faster performance compared to CPU-only alternatives. Various numerical tests have been used to validate that our method works well for revealing the transition between different stable states during the nucleation process.