Advances of and by phase-field modelling in condensed-matter physics

Phase-field modelling is still a young discipline in condensed-matter physics, which established itself for the class of systems that can be characterised by domains of different phases separated by a distinct interface. Driven out of equilibrium, their dynamics result in the evolution of those interfaces which might develop into well defined-structures with characteristic length scales at the nano-, micro- or meso-scale. Since the material properties of such systems are to a large extent determined by those small-scale structures, acquiring a precise understanding of the mechanisms that drive the interfacial dynamics is a great challenge for scientists in this field. Phase-field modelling is an approach that allows this challenge to be tackled in a simulation-based manner. This review provides a critical overview of the conceptual background of the phase-field method, the most relevant fields of condensed-matter physics that have been approached using phase-field modelling, as well as the respective model formulations and the insight gained so far via their simulation and analysis. Moreover, we discuss directions of further development and the quality of the scientific contributions to be expected from those.