A phase-field-informed interfacial area-tensor model for immiscible biphasic flows and casting applications

The evolution of interfacial morphology in immiscible biphasic materials results from the combined effects of convection, deformation, capillary relaxation, coalescence, and topological rearrangements. Direct phase-field simulations can resolve these mechanisms at the microscopic scale, but their computational cost prevents their systematic use in process-scale simulations. This work proposes a micro–macro framework in which the diffuse interfacial geometry provided by a phase-field model is projected onto a second-order interfacial area tensor. The trace of this tensor measures the interfacial area density, while its deviatoric part characterizes the anisotropy of the interface-normal distribution. Periodic phase-field simulations are performed under simple shear and incompressible planar extension for different values of the dimensionless interface thickness ε and the phase-field Péclet number Pe. The resulting tensor histories are used to identify a reduced objective evolution law involving only three macroscopic parameters: the anisotropy relaxation time τ ₐ , the interfacial-area relaxation time τ q , and the equilibrium interfacial area density Qeq . A single parameter set is identified simultaneously from the shear and extension histories by minimizing the discrepancies associated with both the tensor deviator and its trace. Response surfaces are subsequently constructed to relate the identified macroscopic parameters to (ε ,Pe) . The approach is finally assessed in a simplified two-dimensional casting-filling application. The reduced evolution law is integrated along Lagrangian trajectories reconstructed by backward tracing, providing a spatial distribution of the interfacial area and orientation throughout the filled cavity. Direct phase-field calculations are also performed along representative trajectories located in divergent and convergent flow regions. The microscopic realizations exhibit fluctuations related to random initialization and discrete topological events, whereas the deterministic macroscopic model provides an effective morphological state around which these realizations are distributed. Although the macroscopic model cannot reproduce every microscopic overshoot or realization-specific anisotropy, it captures the dominant orientation, interfacial area level, and dependence on the accumulated flow history. The proposed framework therefore provides a computationally efficient connection between resolved phase-field microstructures and process-scale predictions of interfacial morphology.