Phase-field modeling of interface dynamics

Abstract The phase-field approach provides both a theoretical framework and a powerful computational tool to model the dynamics of a rich variety of physical interfaces that impact materials behavior, from phase boundaries to cracks. It is also being increasingly used to model biological interfaces that control shape changes of cells, tissues, and organs in health and disease. By representing interfaces as spatially diffuse entities, the approach naturally handles evolving geometries and topological changes without explicit front tracking, enabling diverse physical processes to be integrated into a unified framework. This article provides an introduction to the basic physical concepts and mathematical techniques used to construct phase-field models following three complementary approaches: a bottom-up approach, restricted in applicability, but conceptually important, which derives the phase-field equations from a more microscopic model; a partially bottom-up commonly used approach, which derives the equations from a physically motivated coarse-grained free-energy functional; and a more top-down mathematical approach that formulates phase-field equations only requiring that they that map onto a known set of sharp-interface equations—such as the Stefan problem of solidification or the continuum theory of linear elastic fracture mechanics—in a computationally tractable limit where the diffuse interface width is small compared to the characteristic scale of the interface pattern of interest. Selected applications are discussed that illustrate, in a historical context, how these complementary approaches have been used to model interface dynamics in varied contexts, from antiphase boundaries to solidification fronts to brittle cracks. Graphical abstract From ordering kinetics to solidification to fracture: phase-field modeling captures the rich landscape of interface dynamics across materials processes.