Machine learning and Monte Carlo simulations of the Gibbs free energy of the Fe-C system in a magnetic field

To model the thermodynamics and kinetics of steels in high magnetic fields requires knowledge of the magnetic Gibbs free energy, G, which involves millions of energy evaluations for the potential energy landscapes as a function of the applied field. Density-functional theory (DFT) calculations provide sufficient accuracy but are computationally very demanding. To overcome this barrier, we apply the ultra-fast force field (UF³) machine learning model [1] to approximate the DFT energy landscape. A DFT database is assembled through VASP, focusing on the energies and forces as a function of magnetic field for bcc and fcc Fe(C) with different structural and magnetic configurations. The UF³ models are trained and validated on this database to quickly evaluate the energies of ensembles. The resulting UF³ models are then utilized in the subsequent Monte Carlo simulations. Thermodynamic integration is utilized to combine the simulations at different temperatures to achieve the magnetic G models for the two Fe(C) phases as a function of temperature, atomic fraction of carbon, and magnetic field. Our calculations show that the applied magnetic field of around 10 T results in a change in the transition temperature of tens of kelvins.

[1] S. R. Xie et al, arXiv:2110.00624 (2021).