Droplet Shrinkage in Phase-Field Approaches: A Comparison of the Allen-Cahn and the Cahn-Hilliard Models.

Diffuse interface methods for multiphase flow simulations often exhibit nonphysical droplet or bubble shrinkage, particularly when based on the Cahn-Hilliard equation. This well-known artifact introduces a critical radius below which droplets vanish, thereby limiting the fidelity of simulations involving small-scale structures. In this study we examine a conservative Allen-Cahn model and demonstrate, through both analytical and numerical investigations, that it inherently avoids this shrinkage behavior. We show that the conservative Allen-Cahn model enforces an exact local balance of interfacial terms, enabling the stable preservation of droplets with radii larger than the interface thickness, regardless of the initial droplet size or domain geometry. Our analysis also provides a different theoretical treatment of the shrinkage phenomenon in the Cahn-Hilliard equation, distinguishing itself from earlier works by providing a rigorous explanation of why conservative Allen-Cahn models avoid this problem. The analysis reveals that conservative Allen-Cahn formulations achieve exact cancellation of curvature-driven terms through a local geometric force balance, while Cahn-Hilliard models exhibit systematic shrinkage due to unbalanced curvature effects that create driving forces for mass transport. Theoretical and numerical findings show that there is a shrinkage relation between interface radius r_{I} and time t in the form of r_{I}^{3}(t)∼-t, which leads to a power law t_{f}∝r_{0}^{3} for the disappearance time t_{f} versus initial radius r_{0}. Numerical experiments across a wide range of droplet sizes corroborate the theoretical predictions. These results position the conservative Allen-Cahn model as a robust and accurate phase-field approach, particularly well suited for applications demanding the resolution of fine features or long-time integration, such as multiphase transport in porous media.