Abstract
Cluster expansions have proven a very useful tool to model thermodynamics and kinetics of substitutional alloys in metallic, ionic, and even covalently bonded systems. Cluster expansions are usually obtained with the structure inversion method in which the energies, or other relevant property, of a set of structures are used to obtain expansion coefficients. The expansion coefficients are multipliers of correlation functions which pertain to clusters of sites on the parent lattice. There are significant practical issues associated with obtaining a cluster expansion, such as selecting which structures and especially which correlation functions are required for an adequate description of the energy. While these issues are significant for binary alloys, they become much more daunting when dealing with multicomponent alloys. Moreover, oftentimes interest is not limited to the energetics of the thermodynamic equilibrium state, but the evolution of quenched alloys with time is just as important. The treatment of diffusion within the context of cluster expansions is then another challenge. The article describes a formal method for utilizing cluster expansions for transition states as occur during vacancy mediated diffusion in substitutional alloys. The methods are illustrated with some applications to the prediction of initial coherent precipitates in Al-Cu and Al-Mg-Si alloys.
Highlights
For a multicomponent alloy the completeness criterion can be shown to require that if a certain correlation function is included in a Cluster expansions (CEs), (a) all correlations associated with that cluster must be included, and (b) all correlations associated with the subclusters must be included
It has been shown that the definition of the correlation functions is a crucial aspect of cluster expansions
For multicomponent alloys, the current definition of the correlation functions allows inheritance, i.e. effective cluster interactions from constituent systems can be directly re-used. This assures that a good description of the alloy energetics in constituent systems is carried forth in more complex alloys
Summary
Cluster expansions (CEs) have become a ubiquitous tool in computational alloy theory[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21] and even in seemingly completely unrelated fields such and electronic band gap engineering and protein sequencing.[22,23,24,25] It is used to express thermodynamic and other properties as function of the configurational order of alloys. This article is an invited paper selected from presentations at the Hume-Rothery Award Symposium on ââMulticomponent Alloy Metallurgy, the Bridge from Materials Science to Materials Engineering,ââ during TMS 2015, held March 15-19, 2015, in Orlando, FL, and has been expanded from the original presentation. This symposium was held in honor of the 2015 Hume-Rothery award recipient, William Boettinger, for outstanding contributions to thermodynamics and kinetics of metallurgical systems and their application to the understanding of alloy microstructures and the relationship to processing conditions
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