We report on an interesting formulation of a phase-field model which incorporates a description of individual phases and particles with preserved volume evolving in a system of multiple phases such that the interfacial energy decreases. In our model, an antiforcing free energy density is defined to fulfill constraints on selected volume fractions by counterbalancing phase changes. Phases are defined as regions with energy bearing boundaries that may differ in their physical states, i.e., the regions may be distinguished in structure crystal transformations, in composition alloys, mixtures of fluids, or in the orientation of the crystal lattice grains. The method allows one to simulate the formation of equilibrium crystal shapes and of the migration of inert particles and phases in microstructures. We show two- and three-dimensional simulations of bubble ensembles and foam textures and demonstrate the excellent agreement of crystal morphology configurations with analytical results.
Phase-eld model for multiphase systems with preserved volume fractions
Phys. Rev. E, Band 78, Seiten 011604-1-011604-7 (2008)