  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 inherently includes elasticity effects. This study
addresses how gradient elasticity (GE), a theory that accounts for elasticity
effects at microscopic scales by introducing additional characteristic lengths,
is incorporated into SH and PFC models. After presenting the fundamentals of
these theories and models, we first calculate the characteristic lengths for
various lattice symmetries in an approximated setting. We then discuss
numerical simulations of stress fields at dislocations and comparisons with
analytic solutions within first and second strain-gradient elasticity.
Effective GE characteristic lengths for the elastic fields induced by
dislocations are found to depend on the free-energy parameters in the same
manner as the phase correlation length, thus unveiling how they change with the
quenching depth. The findings presented in this study enable a thorough
discussion and analysis of small-scale elasticity effects in pattern formation
and crystalline systems using SH and PFC models and, importantly, complete the
elasticity analysis therein. Additionally, we provide a microscopic foundation
for GE in the context of order-disorder phase transitions.
