Modification of Cluster Variation Method Entropy Functional for Binary fcc Phases using Tetrahedron Approximation
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ORIGINAL ARTICLE
Modification of Cluster Variation Method Entropy Functional for Binary fcc Phases using Tetrahedron Approximation Rajendra Prasad Gorrey1
•
Vikas Jindal1 • Bandikatla Nageswara Sarma1 • Shrikant Lele1
Received: 21 January 2020 / Accepted: 24 October 2020 Ó The Indian Institute of Metals - IIM 2020
Abstract The binary prototype phase diagrams for fcc structures calculated with the tetrahedron, tetrahedron–octahedron and quadruple tetrahedron approximations of the cluster variation method and Monte-Carlo simulations (MCS) for first neighbor interactions show significant differences in the computed triple points of the A1, L10 and L12 phases.The finite number of terms retained in the cluster variation method entropy functional in different approximations leads to differences in the accuracy of the approximation. In general, larger cluster approximations and MCS results are accepted to be more accurate, but are associated with large computational burden. The multiplicities of the basic clusters in the entropy functional for tetrahedron–octahedron approximation were earlier modified to reproduce many of the results of larger clusters with lower computational burden. In the present communication, a similar approach is adopted for the case of tetrahedron approximation for binary fcc phases. This approximation has a smaller number of variables, but yields the results which are in good agreement with the best known results. Keywords Cluster variation method Entropy functional A1 phase L10 phase L12 phase
& Rajendra Prasad Gorrey [email protected] 1
Department of Metallurgical Engineering, Indian Institute of Technology (BHU), Varanasi 221005, India
1 Introduction Computational thermodynamics provides an integrated framework for representing thermodynamics and phase equilibria of alloy systems and is widely used based on the CALPHAD method [1]. Another promising approach for this is cluster expansion–cluster variation methods (CE– CVM) [2, 3], which can account for short-range order (SRO) in the description of Gibbs energies of phases. In general, accuracy of the thermodynamic description improves with larger basic clusters in CE–CVM, which is associated with rapidly increasing algebraic complexity and computational burden. In a binary fcc ordering prototype phase diagram, approximations using tetrahedron (T) and tetrahedron–octahedron (TO) basic clusters yield different triple points involving A1, L10 and L12 phases for the same first neighbor pair interactions with the latter being relatively closer to the results of Monte Carlo simulations (MCS) [4, 5]. Finel used different combinations of approximations for the disordered phase and ordered phases, referred to as mixed CVM, namely quadruple tetrahedron (QT)–TO [6] and 13–14 point clusters–TO (CVM - (13–14 pt)–TO) [7] for disordered and ordered phases, respectively, for exclusive first neighbor pair interactions and showed that their results are in good agreement with the best known MCS results. Tepesch et al. [8] showed that the
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