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We present a mesoscale field theory unifying the modeling of growth, elasticity, and dislocations in quasicrystals. The theory is based on the amplitudes entering their density-wave representation. We introduce a free energy functional for complex amplitudes and assume nonconserved dissipative dynamics to describe their evolution. Elasticity, including phononic and phasonic deformations, along with defect nucleation and motion, emerges self-consistently by prescribing only the symmetry of quasicrystals. Predictions on the formation of semicoherent interfaces and dislocation kinematics are given. Published by the American Physical Society2024 

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 (2005 Phys. Rev. E 72 020601). 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.