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Droplet shrinkage in phase-field approaches: A comparison of the Allen-Cahn and the Cahn-Hilliard models

Reza Haghani1, Carl Fredrik Berg1,*, and Eirik Grude Flekkøy2

  • *Contact author: carl.f.berg@ntnu.no
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Phys. Rev. E 113, 055107 – Published 18 May, 2026

DOI: https://doi.org/10.1103/y6sf-3lf2

Abstract

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 rI and time t in the form of rI3(t)∼−t, which leads to a power law tf∝r03 for the disappearance time tf versus initial radius r0. 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.

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Phys. Rev. E 113, 055107– Published 18 May, 2026
  • Received 27 June 2025
  • Accepted 10 March 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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