Foundations and Practical Implementations of the Cluster Expansion

• Published in: Journal of phase equilibria and diffusion 2017-06, Vol.38 (3), p.238-251
• Main Author: Sanchez, J. M.
• Format: Article
• Language: English
• Subjects: Alloys; Approximation; Basis functions; Ceramics; Clusters; Composites; Crystallography and Scattering Methods; Energy; Engineering Thermodynamics; Equilibrium; Glass; Heat and Mass Transfer; Ising model; Many body interactions; Mathematical models; Metallic Materials; Natural Materials; Physics; Physics and Astronomy; Thermodynamics
• ISSN: 1547-7037; 1863-7345; 1934-7243
• DOI: 10.1007/s11669-017-0521-3

Different versions of the cluster expansion are explored using the Mo-Ta system as an example. One of the objectives of this work is to establish a clear distinction between phenomenological expansions that express the energy of an alloy in the form of a generalized Ising model, i.e. with constant pair and many body interactions, and cluster expansions that use a set of complete basis functions in configurational space and define the interactions as projections of the energy onto the basis functions. For the latter case, the interactions are functions of concentration and depend, furthermore, on the full state of order of the system. Such dependence is expected since the configurational energy is shown to be a homogeneous function of degree one in the complete set of configurational variables, or correlation functions, with the interactions being the Euler derivatives of the energy with respect to the correlation functions. For the Mo-Ta system we show that, by including the concentration dependence of the interactions either explicitly or through their dependence on volume, the cluster expansion converges significantly faster than the phenomenological Ising-like models commonly used to represent the energies of disordered alloys.