Asymptotic analysis of multi-phase-field models: A thorough consideration of binary interfaces

Although multi-phase-field models are applied extensively to simulate various pattern formations, their asymptotic analysis is not typically performed at a level of rigor common to their scalar counterparts. Most of the time, arguments given, such as for the justification of the selection of the bulk phases or the phasal composition of the interfaces between them, are only heuristic in nature. In particular, the reduction of the multi-phase-field models to two-phase ones, so as to ascertain the dynamical laws captured by them, can only be termed as hand waving, at best. It is also common to land the starting point of the analysis directly at a point where the binary interfaces have already formed and continue therefrom with the prediction of their instantaneous evolution. However, exactly how a given initial filling transitions to a state characterized by the presence of bulk phases separated by internal layers, and with what distribution, is rarely addressed. Moreover, a detailed and systematic study, focused on the numerical realization of the asymptotics predicted laws, has never been reported before for multi-phase-field models. In the current article, endorsing against these undesirabilities of the common presentations, a full-fledged asymptotic analysis of a multi-grain-growth phase-field model is put forth and numerically verified. However, the consideration is only limited to the analysis of binary interfaces; that of junctions (triple points, quadruple points, etc.) is deferred to a later work.