Complex order parameter phase-field models derived from structural phase-field-crystal models

The phase-field-crystal (PFC) modeling paradigm is rapidly emerging as the model of choice when investigating materials phenomena with atomistic scale effects over diffusive time scales. Recent variants of the PFC model, so-called structural PFC (XPFC) models introduced by Greenwood et al., have further increased the capability of the method by allowing for easy access to various structural transformations in pure materials [Greenwood, Provatas, and Rottler, Phys. Rev. Lett. 105, 045702 (2010)] and binary alloys [Greenwood, Ofori-Opoku, Rottler, and Provatas, Phys. Rev. B 84, 064104 (2011)]. We present an amplitude expansion of these XPFC models, leading to a mesoscale complex order parameter, i.e., phase-field, model for two-dimensional square-triangular structures. Amplitude models retain the salient atomic scale features of the underlying PFC models, while resolving microstructures on mesoscales as traditional phase-field models. The applicability and capability of these complex-order parameter models is demonstrated with simulations of peritectic solidification and grain growth exhibiting the emergence of secondary phase structures.