Surfactants in the two gradient theory of phase transitions

Marco Cicalese; Tim Heilmann; Marco Cicalese; Tim Heilmann

doi:10.3934/mine.2025022

Mathematics in Engineering

2025, Volume 7, Issue 4: 522-552. doi: 10.3934/mine.2025022

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Research article

Surfactants in the two gradient theory of phase transitions

Marco Cicalese ^{1
,
,},
Tim Heilmann ^{1
,
,}

Zentrum Mathematik – M7, Technische Universität München, Boltzmannstrasse 3, 85748 Garching, Germany

Received: 08 November 2024 Revised: 18 July 2025 Accepted: 28 August 2025 Published: 08 September 2025

We investigate the influence of surfactants on stabilizing the formation of interfaces in solid-solid phase transitions. The analysis focuses on singularly perturbed van der Waals-Cahn-Hillard-type energies for gradient vector fields, supplemented with a term that accounts for the interaction between the surfactant and the solid. Assuming the potential term to have only two rank-$ 1 $ connected wells, we prove that the effective energy for the formation of an interface decreases when the surfactant segregate to the interface.
- phase transitions,
- surfactants,
- $ \Gamma $-convergence
Citation: Marco Cicalese, Tim Heilmann. Surfactants in the two gradient theory of phase transitions[J]. Mathematics in Engineering, 2025, 7(4): 522-552. doi: 10.3934/mine.2025022

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Abstract

We investigate the influence of surfactants on stabilizing the formation of interfaces in solid-solid phase transitions. The analysis focuses on singularly perturbed van der Waals-Cahn-Hillard-type energies for gradient vector fields, supplemented with a term that accounts for the interaction between the surfactant and the solid. Assuming the potential term to have only two rank-$ 1 $ connected wells, we prove that the effective energy for the formation of an interface decreases when the surfactant segregate to the interface.

References

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