Gradient elasticity in Swift-Hohenberg and phase-field crystal models
The Swift-Hohenberg (SH) and Phase-Field Crystal (PFC) models are minimal yet powerful approaches for studying phenomena such as pattern formation, collective order, and defects via smooth order parameters. They are based on a free-energy functional that includes elasticity with linear, nonlinear, and strain-gradient contributions. The latter is peculiar to gradient elasticity (GE), a theory that accounts for elasticity effects at microscopic scales by introducing additional characteristic lengths. While numerical simulations of SH and PFC models display some distinctive GE features, such as the regularization of stress singularities at crystalline defects, a comprehensive analysis has yet to be conducted. This study addresses how GE is incorporated into SH and PFC models. We calculate its characteristic lengths for various lattice symmetries in an approximated setting. We then discuss the effective elasticity theory encoded in SH and PFC models via numerical simulations of dislocation stress fields and comparisons with solutions within first and second strain GE. We then demonstrate that effective GE characteristic lengths depend on the free-energy parameters, similar to the correlation length, thus unveiling how they change with the quenching depth and the phenomenological temperature entering the considered models. The findings presented in this study enable a thorough discussion and analysis of small-scale elasticity effects in pattern formation using SH and PFC models and complete the elasticity analysis therein. Additionally, we provide a microscopic foundation for GE in the context of order-disorder phase transitions.