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 response to high magnetic fields requires knowledge of the magnetic Gibbs free energy, which involves millions of energy evaluations for the potential energy landscapes as a function of the applied field in the configurational space. Although sufficiently accurate, density-functional theory (DFT) calculations would result in high computational cost, hindering its direct application.<br/>To address this challenge, we take advantage of the ultra-fast force field (UF³) method [1], a machine-learning potential that combines effective two- and three-body potentials in a cubic B-spline basis with regularized linear regression, to approximate the DFT energy landscape. We assembled a database by performing DFT calculations using the Vienna Ab initio Simulation Package (VASP). This DFT database focuses on the information of the energies and forces as a function of magnetic field for a series of bcc and fcc Fe(C) structures, throughout which both structural and magnetic configurations are varied. The UF³ potentials are trained on this database to quickly evaluate the energies of ensembles based on the structural and spin configurations, and the accuracy of the resulting UF³ models predicting energies and forces is validated.<br/>Subsequent Monte Carlo simulations take place with these machine learning models implemented. Thermodynamic integration is utilized to combine the simulations at different temperatures to achieve the magnetic Gibbs free energy models for the two Fe(C) phases as a function of temperature, atomic fraction of carbon, and magnetic field. Our calculations aim to investigate the origin of the experimentally observed shift in the transition temperature of tens of kelvins under an applied field of 10 T [2].<br/>[1] Xie, S.R., Rupp, M., and Hennig, R.G., npj Comput Mater 9, 162 (2023)<br/>[2] G. M. Ludtka, DOE technical report ORNL/TM-2005/79 (2005)