figure a

Yinping Zeng

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Yong Du

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Rainer Schmid-Fetzer

In order to obtain desired structure and properties of materials that are unattainable by traditional materials processes, external fields such as magnetic, electric and irradiation fields can be utilized. For example, applying an external magnetic field during solidification process can manipulate macroscopic segregation and orientation phenomena through a magneto-thermodynamics effect, which could effectively avoid defects and inclusions within material. For example, a unique composite structure in the solidified Ni-Mn-Ga alloys, which consists of columnar and equiaxed grains, was obtained through magnetic-field-assisted directional solidification (L. Hou et al., Acta Mater., 199, 2020, 383-396). It is worth mentioning that it is extremely difficult to obtain such a unique composite structure without the external magnetic field. This kind of structure enhances mechanical performance of the Ni-Mn-Ga alloys, which indicates that the combination of magnetic-field-assisted solidification and subsequent heat treatment can produce dedicated microstructure for the sake of optimizing material properties. In the case of solid-state phase transition, an excellent combination of saturation magnetization, electrical resistivity, yield strength, ultimate compressive strength and compressive plastic strain was achieved when applying a 6 T magnetic field at a heat-treatment temperature of 1200 °C for AlCoCrFeNi high entropy alloy (C. Zhao, et al., J. Alloys Compd., 820, 2020, 153407). The utilization of the magnetic field at this temperature can lead to the energy differences among ferromagnetic Bcc, paramagnetic Fcc, and \(\sigma\) phases during the phase transition, manipulating the phase fractions of these phases and thus the properties of the alloy. The research work on materials under an external magnetic field has been focusing on experimental aspects. New theories and/or methods that can describe the effect of these external fields on microstructure and properties deserve extensive investigations. Thermodynamics and phase diagram under a magnetic field are essential for focused development of materials through magnetic-field-assisted process. From a thermodynamic point of view, the external magnetic field adds an extra term to the Gibbs energy so that the phase transformation temperature or the driving force for the phase transformation is altered. The use of an external magnetic field can accelerate or delay the phase transformation kinetics and increase or decrease the nucleation site numbers for the formation of magnetic phases. The external magnetic field is an additional state variable in thermodynamics, increasing the degrees of freedom of the system which leads to a four-phase equilibrium in a binary system even at constant pressure. To the best of our knowledge, nowadays there are very limited investigations on the thermodynamic behavior of materials even under application of a constant magnetic field.

Thermodynamics with traditional field variables such as temperature and pressure is well established. Using a variety of thermodynamic databases, CALPHAD-type programs, such as Thermo-Calc, Pandat, FactSage and OpenCalphad, can calculate various phase diagrams and thermodynamic properties in multicomponent and multiphase systems. On the contrary, thermodynamic models and computational methods required for multicomponent and multiphase systems subjected to an external magnetic field are still lacking. Most recently, an attempt was made to establish a computational framework for materials under an external magnetic field (Y.P. Zeng, et al., Acta Mater., 243, 2023, 118496). This framework was based on molecular field theory of Weiss and Heisenberg model in conjunction with the CALPHAD approach. Within this computational framework, only three magnetic-related material parameters, atomic magnetic moment, total angular momentum, and Curie temperature, are needed to generate explicit functions of normalized dimensionless quantities, establishing basic equations for quantitative thermodynamic calculations of the Gibbs energy. All model equations and material parameters could be incorporated into a standard thermodynamic database file (TDB). That enables standard CALPHAD-software to calculate phase transitions and phase diagrams under an external magnetic field. The Bi-Mn and Fe-C binary phase diagrams calculated through the computational framework agree reasonably with the measured ones under external magnetic field (Y.P. Zeng, et al., Acta Mater., 225, 2022, 117595; Y.P. Zeng, et al., Acta Mater., 243, 2023, 118496). In the future, it will be necessary to apply and/or modify this computational framework along with experimental verification to multicomponent and multiphase systems.

Just like the external magnetic field, other external fields such as an external electric field can be utilized to create microstructures which are unattainable without an external field and thus enhance properties and performance of the materials. Ceramic samples can be densified to dense bulks within a short time at a furnace temperature lower than the temperature required for conventional sintering by applying an electric field. This novel electric-field-assisted sintering technique with high efficiency and low-energy consumption has revolutionized the traditional sintering technology. The role of electric field in the flash sintering behavior and microstructural evolution of eutectic Al2O3-ZrO2 composite ceramics was investigated. It was found that the applied external electric field not only promotes densification and grain growth of Al2O3-ZrO2 composite, but also improves the uniformity of microstructure of the ceramics (S. Yao, et al., Ceram. Int., 48, 2022, 36764-36772). At present, many research findings suggest that the giant piezoelectric phenomena in PMN-PT [Pb(Mg1/3Nb2/3)O3-PbTiO3] single crystals are closely related to complicated phase transitions in the Morphotropic Phase Boundary (MPB) region (Q.Y. Hu, et al., J. Alloys Compd., 953, 2023, 170118) depending on temperature, composition, electric field and so on. Therefore, phase transitions occurring between rhombohedral, tetragonal, orthorhombic and monoclinic phases, leading to T-x (Temperature—composition) and T-x-E (Temperature—composition—Electric field) phase diagrams, have been actively investigated experimentally but only a few theoretical explanations were made. In order to interpret the experimental data of electric field-induced phase transitions in piezo/ferroelectrics theoretically, it is necessary to develop a physically sound thermodynamic model of the Gibbs energy function G(T, x, E). To the best of our knowledge, there is no report on the quantitative calculation of thermodynamics and/or phase diagrams under external electric fields.

In the case of irradiation, i.e. bombardment with ions, electrons or neutrons, the evolution of morphological modification and mechanical properties of materials can also be designed. For example, TaTiNbZr refractory high entropy alloy (RHEA) film, that is composed of nano-crystalline and amorphous composite structure, was investigated using helium plasma irradiation, and the obtained results indicate that the irradiation-enhanced surface blisters were strongly dependent on helium plasma energy (G. Pu, et al., J. Nucl. Mater., 577, 2023, 154337). As for the thermodynamics and phase diagrams under irradiation, irradiation-induced mixing and irradiation-enhanced diffusion determine the extra free energy caused by irradiation which is called ballistic free energy. The irradiation can result in several orders of magnitude increase in the diffusion coefficient compared to the thermal diffusion coefficient at low temperatures, and irradiation-enhanced diffusion may also lead to phase changes and microstructural alterations, which otherwise cannot be observed (X.J. Liu, et al., J. Nucl. Mater., 451, 2014, 366-371). These authors established the phase diagrams of the U-Mo and U-Nb systems under irradiation by applying thermodynamic model using an effective free energy model (G. Martin, Phys. Rev. B., 30, 1984, 1424-1436) in which irradiation-enhanced diffusion and ballistic mixing caused by recoil implantation and cascade collision were considered. Until now, the calculations based on the concept of effective free energy can describe the experimental phenomena in the U-based systems only qualitatively since ionization processes and local effects (such as voids, bubbles and local thermal spikes) during irradiation were neglected in the effective free energy model. Consequently, a more comprehensive thermodynamic model is needed. However, we must note that irradiation “fields” show strong impact on the kinetics and diffusion in the material system; that differs significantly from external magnetic (B) or electric (E) fields which are clearly thermodynamic state (field) variables that may be incorporated into the Gibbs energy function just like temperature and pressure. Providing proper models for the explicit Gibbs energy function G(T, x, B, E) will directly enable powerful standard CALPHAD-software to include such field treatments into existing tools for computational materials design.

External fields can alter the arrangement and migration of atoms as well as the transports of substance, changing the thermodynamic and kinetic conditions of phase transformation and thus adjusting the microstructure and properties of materials. It is highly expected that thermodynamics, including effects from the external magnetic and electric fields as well as from irradiation treatment, will extend the application of computational materials science (Yong Du, Rainer Schmid-Fetzer, Jincheng Wang, Shuhong Liu, Jianchuan Wang, and Zhanpeng Jin, Computational Design of Engineering Materials: Fundamentals and Case Studies, Cambridge University Press, 2023, United Kingdom). The utilization of external fields along with related new models and simulation efforts on these fields will provide new opportunities for materials design.

Yinping Zeng

Lecturer, School of Physics and Electronics

Changsha University of Science and Technology

Changsha, Hunan 410114, China

Yong Du

Associate Editor, Journal of Phase Equilibria and Diffusion

Professor, State Key Laboratory of Powder Metallurgy

Central South University, Changsha, Hunan 410083, China

Rainer Schmid-Fetzer

Associate Editor, Journal of Phase Equilibria and Diffusion

Professor Emeritus, Institute of Metallurgy, Clausthal University of Technology

Robert-Koch-Str.42, Clausthal-Zellerfeld D-38678, Germany