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Gradient-Stable Algorithm for the Dynamics of the Phase Field in Modeling of Directional Solidification in Si–As

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Abstract

The main provisions of the phase-field method for concentrated solutions and the problems of numerical modeling of a thermodynamically consistent model of the dynamics of the order parameter for phase transformations of the first kind in the framework of the phase-field approach are briefly considered. The model is based on extended irreversible thermodynamics, which uses Gibbs potentials of real systems to describe the state of the system. Numerical algorithms for modeling phase-field equations for both locally equilibrium and locally nonequilibrium models of relaxation dynamics are derived from the gradient-stability condition, which guarantees a decrease in the free energy of the system during relaxation. The estimates of the eigenvalues of the Hessian matrix in the lattice space allow us to construct a gradient-stable algorithm for modeling the time evolution of the phase-field equations in real thermodynamic systems, despite the existing differences in Gibbs energies. The paper analyzes gradient-stable numerical schemes for stability to small deviations for phase field equations in the Fourier approximation. The numerical calculations performed for the SiAs solution demonstrate the stability of the obtained algorithm both with an increase in the time step by several orders of magnitude, and with an increase in the driving forces due to severe supercooling of the system.

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ACKNOWLEDGMENTS

The authors are grateful to A.V. Obukhov for a number of valuable remarks on numerical modeling.

Funding

This study was supported in part by the Russian Foundation for Basic Research and by the administration of the Usmurt Republic (research project no. 18-42-180002).

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Authors and Affiliations

  1. Udmurt State University, Izhevsk, Russia

    V. G. Lebedev, A. A. Obukhov & V. P. Bovin

  2. Udmurt Federal Research Center, Ural Brunch, Russian Academy of Sciences, 426067, Izhevsk, Russia

    V. G. Lebedev & V. I. Lad’yanov

Authors
  1. V. G. Lebedev
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  2. A. A. Obukhov
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  3. V. P. Bovin
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  4. V. I. Lad’yanov
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Corresponding author

Correspondence to V. G. Lebedev.

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Translated by N. Wadhwa

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Lebedev, V.G., Obukhov, A.A., Bovin, V.P. et al. Gradient-Stable Algorithm for the Dynamics of the Phase Field in Modeling of Directional Solidification in Si–As. Tech. Phys. 67, 700–706 (2022). https://doi.org/10.1134/S1063784222100048

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