New approach to the compound energy formalism (NACEF) part I. Thermodynamic modeling based on the sublattice model

In the CALPHAD modeling, the Compound Energy Formalism (CEF) is widely used because it is a general formalism which allows to select the most appropriate model for different kind of phases (gas phase, solid and liquid solutions, interstitial solutions, intermetallic compounds, solid oxide phases, ionic melts, phases that show chemical order-disorder transformations …). It consists in dividing the phase into sublattices and the models differ by the nature of the constituents (atoms, ions, vacancies, etc.) present in the different considered sublattices where mixing may or may not occur. For many intermetallic compounds which present a crystal structure containing many crystallographic sites on which disorder occurs, the number of parameters to be evaluated which varies exponentially with the number of sublattices becomes a problem in multicomponent systems. This yields the necessity to simplify the description and reduce the number of sublattices by combining different sites in single sublattices and this leads to that different simplifications can be used to model a phase in different systems. It is quite known that the CEF approach does not allow continuous compatibility between the different simplifications which constitutes a problem for the development of multicomponent databases.In the present paper, we present a New Approach to the Compound Energy Formalism (NACEF) which is derived mathematically from the CEF and therefore retains its full potential while providing several important improvements. As example, considering the particular case of the complex σ phase, the present work shows that NACEF makes it possible to make compatible a model based on the full description (five-sublattice model) with simplifications which can be made of it. On the other hand, it is also shown that NACEF form combined to a simple Muggianu type extrapolation allows to obtain reliable estimates of the energies of ordered configurations in multicomponent systems using those for the binary configurations obtained from DFT calculations.