Foundations and Practical Implementations of the Cluster Expansion

Abstract

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.

Accurate representation of formation energies of crystalline alloys with many components

“…1 shows that when the atomic sizes are significantly different, the accuracy of fitting the energy of the relaxed configurations is significantly less than that of the unrelaxed configurations-this is similar to how cluster expansion behaves [8]. This can be rectified in part by introducing, for instance, the explicit dependence of the energy on the average compositiondependent equilibrium volume of the lattice [7]. Moreover, when some elements have a different ground state lattice (e.g., body-centered cubic), the model (1) may be unusable for certain compositions.…”

“…Accurate computational prediction of the mixing enthalpy and configuration entropy would be very instrumental in studying HEAs, as it is hard to experimentally explore different compositions of five or more elements due to combinatorial complexity. The state-of-the-art methodology of computationally assessing the stability of multicomponent crystalline alloys is based on cluster expansion [5,6,7,8], allowing to fit formation energies of binary systems over the entire range of compositions, ternary and quaternary systems [9,10,11,12] over, typically, some subrange of the composition range, and quinary systems at specific points of the composition range [13].…”

Chemical short-range order in derivative Cr–Ta–Ti–V–W high entropy alloys from the first-principles thermodynamic study

“…The enthalpy of mixing of an alloy can also be calculated using the Cluster Expansion method. [79][80][81][82] DH bcc mix CE ðsÞ ¼…”

Comment on “Cluster expansion and the configurational theory of alloys”

“…However in some situations there might be a relatively large non-local contribution to the property of interest, and in some such cases the use of concentrationdependent ECIs can significantly reduce the prediction error of a truncated cluster expansion [32]. Concentration-dependent ECIs may also be used to reduce the number of local clusters that must be included in a cluster expansion to reach a sufficient level of accuracy [33,34]. On the other hand, the use of concentration-dependent ECIs cannot completely resolve the well-known inability of truncated cluster expansions to predict the energies of long-period coherent superlattices, due to the anisotropy of the elastic constants [35,36].…”