Multicomponent cluster variation method: Application to high entropy alloys

Abstract

Cluster expansion, a method commonly used in computational thermodynamics, expresses the configurational properties of alloys as a sum of products of correlation functions and cluster expansion coefficients (CECs). The CECs are defined using a basis in configurational space that is not unique and changes during the extension of the generalized Ising model from binary alloys to multicomponent alloys. In contrast to the CALPHAD formulation, this change severely restricts the development and use of CEC-based databases. To address this, a new framework is introduced where lower-order CECs remain unchanged in higher-order systems. This is achieved by choosing configurational variables (and corresponding CECs) which are not dependent on a particular choice of basis as correlation functions (CFs). Such a set is provided by cluster variables (CVs), which represent the fraction of cluster configurations of various types. A carefully selected subset of independent CVs is chosen as CFs. Completeness of the basis is demonstrated by deriving expressions for all remaining CVs in terms of these new CFs for disordered bcc, fcc, and hcp structures. A set of CECs for the bcc phase of the Nb-Ti-V-Zr system has been developed using this new formulation. Optimized CECs of all binary and ternary subsystems of the Nb-Ti-V-Zr have been used in its construction. This is a crucial step towards creating a self-consistent computational thermodynamics database. The utility of the Nb-Ti-V-Zr database was demonstrated through calculations of thermodynamic quantities, SRO parameters, and phase diagrams. Additionally, a procedure is outlined to transform existing CALPHAD databases to CEC-based databases under certain assumptions. © 2025 Elsevier Ltd