Thermodynamics of Binary bcc and fcc Phases for Exclusive Second-Neighbour Pair Interactions Using Cluster Variation Method: Analytical Solutions

Abstract

Background: In the framework of cluster expansion—cluster variation methods (CE - CVM), the equilibrium values of the microscopic state variables called correlation functions (CFs) can be obtained by minimization of the Helmholtz energy. This involves solving the nonlinear equilibrium equations using numerical techniques. As an exception, Guggenheim obtained an analytical solution for the pair CF in the pair approximation of quasi-chemical theory for a binary alloy. Results: In this communication, analytical solutions for the CFs for the case of exclusive second neighbour pair interactions in binary A2 & B32 phases and A1 & L11 phases respectively under tetrahedron and tetrahedron–octahedron approximations of CVM are obtained as functions of the corresponding energy coefficient, temperature, and composition. Results obtained by results in the previous section and a few new results: Further, analytical expressions for the phase boundaries in phase separating and ordering systems have been obtained. Apart from these, thermodynamic quantities such as heat capacities have also been evaluated. In addition, the effect of composition and temperature dependence of the energy coefficient on the phase boundary is also discussed. Application: For the selected composition, the solution of CF reduces to a simple rational function. Taking this as a guide, the coefficients of the polynomials used to approximate the CFs are expressed as rational functions, which was discussed elsewhere. © 2021, The Indian Institute of Metals - IIM.