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The Swift-Hohenberg (SH) and Phase-Field Crystal (PFC) models are minimal yetpowerful approaches for studying phenomena such as pattern formation,collective order, and defects via smooth order parameters. They are based on afree-energy functional that includes elasticity with linear, nonlinear, andstrain-gradient contributions. The latter is peculiar to gradient elasticity(GE), a theory that accounts for elasticity effects at microscopic scales byintroducing additional characteristic lengths. While numerical simulations ofSH and PFC models display some distinctive GE features, such as theregularization of stress singularities at crystalline defects, a comprehensiveanalysis has yet to be conducted. This study addresses how GE is incorporatedinto SH and PFC models. We calculate its characteristic lengths for variouslattice symmetries in an approximated setting. We then discuss the effectiveelasticity theory encoded in SH and PFC models via numerical simulations ofdislocation stress fields and comparisons with solutions within first andsecond strain GE. We then demonstrate that effective GE characteristic lengthsdepend on the free-energy parameters, similar to the correlation length, thusunveiling how they change with the quenching depth and the phenomenologicaltemperature entering the considered models. The findings presented in thisstudy enable a thorough discussion and analysis of small-scale elasticityeffects in pattern formation using SH and PFC models and complete theelasticity analysis therein. Additionally, we provide a microscopic foundationfor GE in the context of order-disorder phase transitions.

Gradient elasticity in Swift-Hohenberg and phase-field crystal models