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.

中文翻译：

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基于二阶梯度相场理论的广义 Swift-Hohenberg 和相场-晶体方程

使用虚功率原理推导出用于二阶梯度相场理论的微力平衡。结合与自由能不平衡一致的本构关系，这种平衡产生了 Swift-Hohenberg 方程的广泛推广。当相场与守恒成分的体积分数确定时，自由能不平衡的适当增强版本产生本构关系，结合微力平衡和成分含量平衡，提供相的广泛概括 -场晶方程。还开发了系统与其环境之间的界面无结构且不能支持成分传输的情况下的热力学一致边界条件，