Adaptive mesh computation of polycrystalline pattern formation using a renormalization-group reduction of the phase-field crystal model

Abstract

We implement an adaptive mesh algorithm for calculating the space and time dependence of the atomic density field in microscopic material processes. Our numerical approach uses the systematic renormalization-group formulation of a phase-field crystal model of a pure material to provide the underlying equations for the complex amplitude of the atomic density field--a quantity that is spatially uniform except near topological defects, grain boundaries, and other lattice imperfections. Our algorithm employs a hybrid formulation of the amplitude equations, combining Cartesian and polar decompositions of the complex amplitude. We show that this approach leads to an acceleration by three orders of magnitude in model calculations of polycrystalline grain growth in two dimensions.

Defect stability in phase-field crystal models: Stacking faults and partial dislocations

“…Existing modeling methods typically operate either exclusively on atomic scales or on meso-and macroscopic scales. Phase-field crystal models on the other hand provide a framework that combines atomic length scales and mesoscopic/diffusive time scales [1][2][3] , with the potential to reach mesoscopic lengths through systematic multi-scale expansion methods [4][5][6][7][8] . The PFC approach naturally incorporates elasticity, plasticity, and effects of local crystal orientation into a relatively simple atomic-level continuum theory 1,2 .…”

Phase-field-crystal models and mechanical equilibrium

“…The combination of the two can describe dislocations and grain boundaries, since the phase can be discontinuous when the magnitude goes to zero. This approach allows for larger length and time scales, and as shown by Athreya et al [27] can be numerically implemented using efficient multigrid methods. It is also very useful for studies in which the crystal orientation is almost the same everywhere (except near dislocations) as in the case of heteroepitaxial systems [28][29][30][31].…”

Amplitude expansion of the binary phase-field-crystal model

“…Deformations of the crystal lattice can be represented by spatial variations in the phase of the amplitude. Using this amplitude representation, Athreya et al [28] were able to apply adaptive mesh refinement to simulate grain growth on micron scales while simultaneously resolving atomic scale structures at interfaces. This remarkable achievement suggests that the development of amplitude expansions is very promising for computational materials research.…”