RESUMEN
The solutions of multi-phase-field models exhibit boundary layer behavior not only along the binary interfaces but also at the common contacts of three or more phases, i.e., junctions. Hence, to completely determine the asymptotic behavior of a multi-phase-field model, the inner analysis of both types of layers has to be carried out, whereas, traditionally, the junctions part is ignored. This is remedied in the current work for a phase-field model of simple grain growth in two spatial dimensions. Since the junction neighbourhoods are fundamentally different from those of the binary interfaces, pertinent matching conditions had to be derived from scratch, which is also accomplished in a detailed manner. The leading-order matching analysis of the junctions exposed the restrictions present on the interfacial arrangement at the common meeting point, while the next-to-the-leading one uncovered the law governing the instantaneous motion of the latter. In particular, it is predicted for the considered model that the Young's law is always satisfied at a triple point, whether or not it is at rest. Surprisingly, the mobilities and the curvatures of the involving interfaces as well as the driving forces on the them do not affect this result. However, they do play a significant role in determining the instantaneous velocity of the junction point. The study has opened up many new directions for future research.
RESUMEN
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.
RESUMEN
Grand-potential based multiphase-field model is extended to include surface diffusion. Diffusion is elevated in the interface through a scalar degenerate term. In contrast to the classical Cahn-Hilliard-based formulations, the present model circumvents the related difficulties in restricting diffusion solely to the interface by combining two second-order equations, an Allen-Cahn-type equation for the phase field supplemented with an obstacle-type potential and a conservative diffusion equation for the chemical potential or composition evolution. The sharp interface limiting behavior of the model is deduced by means of asymptotic analysis. A combination of surface diffusion and finite attachment kinetics is retrieved as the governing law. Infinite attachment kinetics can be achieved through a minor modification of the model, and with a slight change in the interpretation, the same model handles the cases of pure substances and alloys. Relations between model parameters and physical properties are obtained which allow one to quantitatively interpret simulation results. An extensive study of thermal grooving is conducted to validate the model based on existing theories. The results show good agreement with the theoretical sharp-interface solutions. The obviation of fourth-order derivatives and the usage of the obstacle potential make the model computationally cost-effective.
