A novel and predictive computational framework to calculate Gibbs energy and phase transitions under external magnetic fields

Thermodynamics is a fundamental branch of science used to describe the behavior of matter and energy in various physical systems. Its applications are governed by the different laws of thermodynamics. In material science, for example, phase diagrams form the basis of investigated alloys, including their design, development and processing. Among the available techniques for calculating phase diagrams and other thermodynamic properties in alloys, CALPHAD (Calculation of Phase Diagrams) technique is commonly used. It is more consistent, accurate and suitable for multicomponent multiphase alloys.

Whereas the thermodynamic framework required for alloy systems as well as various field variables such as temperature and pressure is well-established, those required for multicomponent material systems subjected to external fields are still lacking. Applying external magnetic fields to alloy systems can create unique microstructure to improve properties that cannot be achieved without such fields. For example, unique composite structures consisting of equiaxed and columnar grains have been reported to enhance mechanical performance.

A thorough understanding of the thermodynamics and phase diagram is vital for focused development of materials under a magnetic field via magnetic-field-assisted process. However, despite the research progress in thermodynamically modeling alloy systems such as the Fe-C and Bi-Mn systems under external magnetic field, an efficient computational framework with predictive capability is lacking. It is essential for such computational models to accurately calculate phase diagrams and thermodynamic properties for systems subjected to external magnetic fields to enhance their practical applications. The Bi-Mn system is among the commonly studied alloy systems owing to its prospective applications in various fields. However, there is no experimental validation of the effect of magnetic field on eutectoid reaction of the Bi-Mn system.

Herein, Dr. Yinping Zeng and Professor Yong Du from Central South University together with Professor Rainer Schmid-Fetzer from Clausthal University of Technology developed a novel computational framework for materials under external magnetic field. This framework was based on molecular field theory of Weiss and Heisenberg model in conjunction with CALPHAD approach. Its successful application was demonstrated with Bi-Mn system and was used to calculate Gibbs free energy, phase diagram and phase transitions under strong magnetic fields. The work is currently published in the peer-reviewed journal, Acta Materialia.

The authors derived for normalized quantities the numerical approximation of the additive Gibbs energy predicted using the proposed computational framework. The explicit functions of normalized dimensionless quantities were generated using only three parameters: atomic magnetic moment, Curie temperature and total angular momentum, without requiring any fitting to experimental phase equilibria. This not only improved the calculation accuracy but also reduced the tedious task associated with finding analytical approximations to the individual class of alloy phases, as seen in previously used methods.

The research team verified successfully the feasibility of the proposed model for the Bi-Mn system characterized by strong interactions of compounds with known magnetic fields. The four-phase equilibrium of the system proved to be a true peritectic-type formation reaction. The authors provided a deep  understanding of the microstructure and phase transition manipulation under an external magnetic field guided by thermodynamics considerations. Furthermore, the numerical approximations and the direct numerical solutions using the Weiss molecular field theory agreed well.

In summary, the study is the first worldwide to develop a robust computational framework for calculating Gibbs energy and phase transitions of the Bi-Mn system under external magnetic fields. Being a pivotal and creative scientific work contributing to materials science and engineering, the finding will provide the basis for quantitative calculations of thermodynamics and phase equilibrium in multiphase and multicomponent systems under an external magnetic field. In a statement to Advances in Engineering, the authors stated that the study paves the way for the focused development of advanced materials with unique microstructures under external fields such as the external magnetic field shown in their work.