Generalized Swift–Hohenberg and phase-field-crystal equations based on a second-gradient phase-field theory

The principle of virtual power is used derive a microforce balance for a second-gradient phase-field theory. In conjunction with constitutive relations consistent with a free-energy imbalance, this balance yields a broad generalization of the Swift–Hohenberg equation. When the phase field is identified with the volume fraction of a conserved constituent, a suitably augmented version of the free-energy imbalance yields constitutive relations which, in conjunction with the microforce balance and the constituent content balance, delivers a broad generalization of the phase-field-crystal equation. Thermodynamically consistent boundary conditions for situations in which the interface between the system and its environment is structureless and cannot support constituent transport are also developed, as are energy decay relations that ensue naturally from the thermodynamic structure of the theory.
source:https://link.springer.com/article/10.1007%2Fs11012-020-01228-9