Coarse-grained modeling of crystals by the amplitude expansion of the phase-field crystal model: an overview

TL;DR

This paper reviews the amplitude expansion of the phase-field crystal model, a mesoscale approach that bridges atomistic and continuum scales to study crystalline systems, defects, and interfaces efficiently.

Contribution

It provides a comprehensive overview of the derivation, applications, extensions, and practical aspects of the amplitude expansion of the PFC model, highlighting recent developments and future challenges.

Findings

Enables simulation of large crystalline systems with defect dynamics.

Connects atomistic details with continuum descriptions.

Discusses numerical methods and model limitations.

Abstract

Comprehensive investigations of crystalline systems often require methods bridging atomistic and continuum scales. In this context, coarse-grained mesoscale approaches are of particular interest as they allow the examination of large systems and time scales while retaining some microscopic details. The so-called Phase-Field Crystal (PFC) model conveniently describes crystals at diffusive time scales through a continuous periodic field which varies on atomic scales and is related to the atomic number density. To go beyond the restrictive atomic length scales of the PFC model, a complex amplitude formulation was first developed by Goldenfeld et al. [Phys. Rev. E 72, 020601 (2005)]. While focusing on length scales larger than the lattice parameter, this approach can describe crystalline defects, interfaces, and lattice deformations. It has been used to examine many phenomena including liquid/solid fronts, grain boundary energies, and strained films. This topical review focuses on this amplitude expansion of the PFC model and its developments. An overview of the derivation, connection to the continuum limit, representative applications, and extensions is presented. A few practical aspects, such as suitable numerical methods and examples, are illustrated as well. Finally, the capabilities and bounds of the model, current challenges, and future perspectives are addressed.