Abstract
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.
Highlights
The Swift–Hohenberg and phase-field-crystal equations describe a multitude of processes involving spatiotemporal pattern formation
Our goal in this paper is to develop a broadly applicable thermodynamically consistent framework that subsumes the Swift–Hohenberg and phase-fieldcrystal equations, accompanied by suitable boundary conditions and associated relations that guarantee that appropriate measures of total energy be nonincreasing along solution paths
The microforce balance (23) that arises from the principle of virtual power (10) in conjunction with the thermodynamically compatible constitutive relations (51) and (52) yields a family of evolution equations that encompasses and broadly generalizes the classical Swift–Hohenberg equation
Summary
The Swift–Hohenberg and phase-field-crystal equations describe a multitude of processes involving spatiotemporal pattern formation. 5, we pose the free-energy imbalance for situations where the phase field is not a conserved order parameter and use that imbalance to determine thermodynamically admissible constitutive relations for the microstress, hypermicrostress, and internal microforce density in terms of the phase field, its first and second spatial gradients, and its temporal rate. 7 and 8, we consider the consequences of identifying the phase field with the volume fraction of a single, independent, mobile constituent, augment our theory to account for the transport effects, use the free-energy imbalance to determine thermodynamically admissible constitutive relations for the microstress, hypermicrostress, internal microforce density, and constituent flux, and arrive our generalization of the phase-field-crystal equation.
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