Lambda transitions in materials science: Recent advances in CALPHAD and first-principles modelling

This paper provides a comprehensive overview of state-of-the-art computational techniques to thermodynamically model magnetic and chemical order-disorder transitions. Recent advances as well as limitations of various approaches to these so-called lambda transitions are examined in detail, focussing on calphad models and first-principles methods based on density functional theory (DFT). On the one hand empirical implementations -based on the Inden-Hillert-Jarl formalism -are investigated, including a detailed interpretation of the relevant parameters, physical limiting cases and potential extensions. In addition, Bragg-Williams-based approaches as well as cluster-variation methods of chemical order-disorder transitions are discussed. On the other hand, it is shown how magnetic contributions can be introduced based on various microscopic model Hamiltonians (Hubbard model, Heisenberg model and beyond) in combination with DFT-computed parameters. As a result of the investigation we were able to indicate similarities between the treatment of chemical and magnetic degrees of freedom as well as the treatment within the calphad and DFT approaches. Potential synergy effects resulting from this overlap have been derived and alternative approaches have been suggested, in order to improve future thermodynamic modelling of lambda transitions.