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

TL;DR

This work introduces a dichotomic-based, $N$-phase phase-field (DBPF) model that eliminates simplex constraints by representing an $N$-phase partition with $N-1$ independent binary phase fields and an energy interpolating binary Ginzburg-Landau energies. The dynamics are derived via the Onsager principle, yielding a linear, decoupled, energy-dissipative $N$-phase Cahn-Hilliard system with degenerate mobility, and a rigorous framework for ensuring mechanic, energetic, algebraic, and dynamic consistency. Sharp-interface analysis confirms asymptotic compatibility with sharp-interface theory, including the Neumann triangle condition at triple junctions. A Mobility Operator Splitting (MOS) based, second-order, linear, energy-stable scheme is developed and demonstrated through extensive numerical simulations, reproducing liquid lenses and complex emulsions in agreement with experiments. The approach provides a scalable, robust framework for simulating multiphase flows with many components and opens avenues for applications in materials science and fluid dynamics.

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.

Figures (15)

• Figure 1: (a) Configuration of a phase-field vector $(c_1,c_2,c_3)$ representing ternary phase. The bulks of the three phases in CBPF model are characterized by $\bar{\mathcal{B}}_1:(1,0,0)$, $\bar{\mathcal{B}}_2:(0,1,0)$ and $\bar{\mathcal{B}}_3:(0,0,1)$. (b) Configuration of a phase-field vector $(\psi,\varphi)$ representing ternary phase. The bulks of the three phases in DBPF model are characterized by ${\mathcal{B}}_1:(-1,\cdot)$, ${\mathcal{B}}_2:(1,-1)$ and ${\mathcal{B}}_3:(1,1)$.
• Figure 2: Sketch of dichotomic representation (a) vs. characteristic representation (b) for ternary phases. ${\mathcal{B}}^i$ and $\bar{\mathcal{B}}^i$$(i=1,2,3)$ represent the bulk phases (the red portion) while ${\mathcal{I}}^{ik}$ and $\bar{\mathcal{I}}^{ik}$$(1\leqslant i<k\leqslant 3)$ represent the interfaces (the blue portion) in the respective representations. In this case, there is a nonlinear correspondence between the two sets of phase variables $(\phi_1,\phi_2)$and $(c_1,c_2,c_3)$.
• Figure 3: The sketch of the outer region, inner region and the triple junction region.
• Figure 4: Sketch of the interfaces and apparent contact angles in the outer region (left panel: macroscopic view) and triple junction region (right panel: microscopic view). In both plots, $\theta_{23}$, $\theta_{12}$, and $\theta_{13}$ represent the apparent contact angles at the triple junction.
• Figure 5: Auxiliary triangle for matching in the triple junction region.

Theorems & Definitions (8)

• Remark 2.1
• Lemma 2.1
• Corollary 2.1
• Example 2.1
• Remark 3.1
• Theorem 4.1
• Corollary 4.1
• Theorem 4.2