A novel phase-field model for $N$-phase problems: modeling, asymptotic analysis and numerical simulations

Abstract: The classical phase-field modeling approaches for multiphase problems represent each phase using a regularized characteristic function, which necessarily introduces a simplex constraint for the phase-field variables. Additionally, the consistency requirement for phase-field modeling brings difficulties to the construction of nonlinear potentials in the energy functionals, posing significant challenges for classical phase-field modeling and its numerical methods for problems involving many phases. In this work, by adopting a dichotomic approach to represent multiphase, we propose a novel phase-field modeling framework without simplex constraint,in which the free energy is interpolated from the classical two-phase Ginzburg-Landau free energies. We systematically establish the interpolation rules and explicitly construct the interpolation functions, rendering the consistency properties of the model. The proposed model enjoys an energy dissipation property and is shown to be asymptotically consistent with its sharp interface limit, with the Neumann triangle condition recovered at the triple junction.Based on a mobility operator splitting technique, we develop a linear, decoupled, and energy stable scheme for efficiently solving the system of phase-field equations. The numerical stability and accuracy, as well as the consistency properties of the model, are validated through a large number of numerical examples. In particular, the model demonstrates its success in several benchmark simulations for multiphase problems, such as the formation of liquid lenses between two stratified fluids, the generation of double emulsions and Janus emulsions, showing good agreement with experimental observations.