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Quantitative nondiagonal phase field modeling of pearlite growth involving multi-diffusion paths

Kai Wang

Guillaume Boussinot

Efim A. Brener

Robert Spatschek*

  • Mechanics of Functional Materials Division, Institute of Materials Science, Technical University of Darmstadt, 64287 Darmstadt, Germany and Institute of Energy Materials and Devices IMD-1, Forschungszentrum Jülich GmbH, 52428 Jülich, Germany

  • Access e.V., Intzestrasse 5, 52072 Aachen, Germany

  • Institute of Energy Materials and Devices IMD-1, Forschungszentrum Jülich GmbH, 52428 Jülich, Germany and JARA-ENERGY, 52428 Jülich, Germany

  • *Contact author: r.spatschek@fz-juelich.de
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Phys. Rev. Materials 10, 083402 – Published 24 August, 2026

DOI: https://doi.org/10.1103/27bn-shc7

Abstract

Despite extensive research on pearlite formation, the rate-controlling mechanisms of pearlite growth remain under debate. Phase field modeling has become a key tool for investigating such transformations; however, in solid-state systems where diffusion occurs both within individual phases and along their interfaces, the commonly assumed local-equilibrium boundary conditions are rarely satisfied, leading to artificial interface effects in conventional diagonal formulations. Here, we present a thermodynamically consistent nondiagonal three-phase field model formulated within the grand-potential framework to investigate pearlite growth. By introducing an Onsager-symmetric cross-coupling term between conserved and nonconserved order parameters, the model inherently recovers the correct equilibrium boundary conditions regardless of the active diffusion pathways, including bulk diffusion and interfacial diffusion. We systematically analyze four diffusion scenarios: (i) diffusion in austenite; (ii) diffusion in austenite, ferrite, and cementite; (iii) diffusion in austenite and ferrite; and (iv) diffusion in austenite combined with interfacial diffusion along austenite/ferrite and austenite/cementite boundaries. The model's quantitative performance is benchmarked against available analytical solutions for each case. Overall, the proposed framework provides a robust and thermodynamically sound approach for elucidating how different diffusion pathways control pearlite growth kinetics.

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Phys. Rev. Materials 10, 083402– Published 24 August, 2026
  • Received 16 January 2026
  • Accepted 30 July 2026
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