We develop a constrained theory for constituent migration in bodies with microstructure described by a scalar phase field. The distinguishing features of the theory stem from a systematic treatment and characterization of the reactions needed to maintain the internal constraint given by the coincidence of the mass fraction and the phase field. We also develop boundary conditions for situations in which the interface between the body and its environment is structureless and cannot support constituent transport. In addition to yielding a new derivation of the Cahn–Hilliard equation, the theory affords an interpretation of that equation as a limiting variant of an Allen–Cahn type diffusion system arising from the unconstrained theory obtained by considering the mass fraction and the phase field as independent quantities. We corroborate that interpretation with three-dimensional numerical simulations of a recently proposed benchmark problem.

我们开发了一种约束理论，用于具有由标量相场描述的微观结构的物体中的成分迁移。该理论的显着特征源于对维持由质量分数和相场的重合给出的内部约束所需的反应的系统处理和表征。我们还为身体与其环境之间的界面无结构且不能支持成分传输的情况开发边界条件。除了产生 Cahn-Hilliard 方程的新推导外，该理论还将该方程解释为 Allen-Cahn 型扩散系统的限制变体，该系统源自无约束理论，通过考虑质量分数和相场为独立量。