A general statistical framework for vacancy and self-interstitial properties in concentrated multicomponent solids

A rigorous understanding of the thermodynamic properties of point defects, namely vacancies and self-interstitials, is crucial for the discovery and screening of structural materials in clean energy applications. In this work, we extend a previously-developed statistical framework for predicting the thermodynamics of single-site impurities to further predict the thermodynamics of self-interstitial dumbbells in an arbitrarily complex alloy. We then apply this extended framework to compute effective formation energies in fully disordered Fe-Cr and Cu-Ni alloys. Notably, we predict that some self-interstitial dumbbell types that are high-energy in pure Fe become stabilized by Cr. We additionally describe a symmetry-breaking effect, wherein high solute concentrations distort the defect free energy surface, yielding misaligned self-interstitials.

💡 Research Summary

In this work the authors develop a comprehensive statistical framework that extends their earlier single‑site impurity model to treat both vacancies and self‑interstitial atoms (SIAs) in arbitrarily complex, concentrated multicomponent alloys. The key idea is to view each lattice site σ as a collection of low‑probability micro‑states: (i) occupation by a host atom of type α, (ii) a vacancy, and (iii) a dumbbell‑type SIA formed by two atoms α and α′ aligned along a lattice vector n. The energy of each micro‑state, E(·)σ, is obtained from atomistic potentials (e.g., EAM) and is Legendre‑transformed to the grand‑canonical ensemble by subtracting the appropriate chemical potentials μ α. The probability of each micro‑state follows the Boltzmann factor p ∝ exp(−βĒ), with β = 1/kBT.

In a disordered alloy the perfect crystal symmetry is broken by the random distribution of solutes, which lifts the degeneracy of otherwise equivalent dumbbell orientations. Consequently, the number of distinct micro‑states grows combinatorially. To make the problem tractable the authors group micro‑states into symmetry‑equivalence classes defined by a lattice‑vector family ⟨n⟩ (the orbit of a chosen direction under the point group O h for BCC, or the appropriate group for FCC) and an unordered pair of alloying elements {α,α′}. Each class therefore corresponds to a unique SIA type such as ⟨110⟩ Fe‑Cr or ⟨111⟩ Cr‑Cr. The concentration of a given class, x⟨n⟩,{α,α′}, is obtained by summing the Boltzmann probabilities of all micro‑states belonging to that class; the vacancy concentration x v follows analogously.

Effective formation energies are then defined as temperature derivatives of the logarithm of these concentrations: E_form(v) = −∂ ln x v /∂β, E_form(⟨n⟩,{α,α′}) = −∂ ln x⟨n⟩,{α,α′} /∂β. These quantities can be interpreted as formation free energies of the defect subsystem, averaged over the distribution of local chemical environments. The authors derive this interpretation by partitioning the full micro‑state space M into a defect subset χ and a reference subset χ°, showing that the defect probability is proportional to exp