Equilibrium Vacancy Concentrations in MPEAs

• Equilibrium vacancy concentrations in MPEAs are defined by the statistical distribution of vacancy formation energies influenced by temperature, chemical complexity, and local lattice relaxations.
• Advanced techniques such as first-principles calculations, Monte Carlo simulations, and machine learning enable accurate prediction of vacancy properties and inform rational alloy design.
• Controlling vacancy concentrations in MPEAs can optimize high-temperature stability, diffusion rates, and irradiation resistance, offering actionable insights for materials engineering.

Equilibrium vacancy concentrations in multi-principal element alloys (MPEAs)—also known as high-entropy alloys—govern key materials phenomena such as diffusion, creep, irradiation tolerance, and high-temperature microstructural stability. Accurate prediction of these concentrations requires a comprehensive treatment of the chemical complexity, configurational space, and local energetic disorder inherent to MPEAs. Recent theoretical, computational, and simulation approaches have advanced the field by bridging first-principles calculations, statistical thermodynamics, and machine learning techniques. The resulting frameworks resolve the distribution of vacancy formation energies, their dependence on temperature, chemical short-range order, and local lattice relaxations, providing insights necessary for compositional design and rational alloy optimization.

1. Thermodynamic Formulation of Equilibrium Vacancy Concentration

The equilibrium concentration of thermal (monovacancy) defects is fundamentally connected to the Gibbs energy of vacancy formation, $\Delta G_{\rm v}$. The general expression

$c_v = \exp(-\beta \Delta G_{\rm v})$

(where $c_v$ is the fractional concentration and $\beta = 1/(k_{B}T)$) is widely used across metallic systems (Mortazavifar et al., 2013, Utt et al., 2021).

In chemically complex alloys such as MPEAs, $\Delta G_{\rm v}$ is not a unique value but rather a statistical quantity, reflecting numerous local environments. For close-packed crystals, a rigorous configurational averaging must be performed, with the master formula linking $\Delta G_{\rm v}$ to partition functions: $$\Delta G_{\rm v} = \frac{p}{\rho} - \frac{1}{\beta} \sum_{i} g_i \ln \left(\frac{Z_{\rm sv, i}}{Z_{\rm s}}\right)$$ where $p$ is pressure, $\rho$ the site density, $g_i$ the coordination number of shell $i$, $Z_{\rm s}$ the single-particle partition function in the perfect lattice, and $Z_{\rm sv, i}$ the analogue with a neighboring vacancy (Mortazavifar et al., 2013). For MPEAs, the configuration integrals $Z$ must be averaged over all relevant chemical arrangements.

2. Distributions of Vacancy Formation Energies in MPEAs

In MPEAs, each lattice site is embedded in a distinct local environment, resulting in a distribution of vacancy formation energies, $E_{\rm f}$. Grand-canonical Monte Carlo (GCMC) studies confirm that for highly concentrated alloys such as (CoCrFeMn)$_{1-x}$Ni$_x$, the distribution $g(E_{\rm f})$ becomes unimodal as local configurations proliferate (Utt et al., 2021). The equilibrium vacancy concentration is then: $$c_{\rm Vac} = \int g(E_{\rm f}) \exp\left(-\frac{E_{\rm f}}{k_{\rm B} T}\right) dE_{\rm f}$$ This Maxwell–Boltzmann weighted average demonstrates that the multiplicity of local environments "smears" vacancy properties, but does not fundamentally alter the canonical thermodynamic framework.

In dilute alloys, multiple discrete peaks in $g(E_{\rm f})$ arise from distinct chemical environments. In concentrated alloys, typical of MPEAs, the vacancy concentration is only weakly dependent on composition; temperature is the primary variable affecting $c_{\rm Vac}$ (Utt et al., 2021). The robust unimodal distribution in highly disordered systems suggests that configurational entropy does not introduce additional multiplicity corrections (e.g., $1/\mathcal{N}$ factors) in vacancy statistics.

3. Atomistic Modelling, Embedded Cluster Expansions, and Machine Learning

The complexity of the local energetic environment in MPEAs requires models beyond simple mean-field or point-defect treatments. Advanced approaches include embedded cluster expansion (eCE) models coupled to Monte Carlo simulations, which decompose the total energy into chemically-resolved, symmetry-invariant tensorial descriptors and employ neural networks for predicting site energies (Lee et al., 26 Sep 2025). This architecture is trained on thousands of DFT formation energies, enabling data efficiency across 9+ element systems and giving ab initio accuracy to large-scale disorder averages.

Vacancy statistics are sampled across Monte Carlo-generated alloy configurations. The equilibrium concentration follows: $$x_{\rm Va} \approx \frac{\xi}{M}, \quad \xi = \langle \sum_\nu \exp(-\beta \Delta \Omega(\vec{\nu}))\rangle_{\text{alloy}}$$ where $\Delta \Omega(\vec{\nu})$ gives the configuration-dependent vacancy formation energy. The approach quantifies the role of short-range order and local composition on vacancy statistics at scale (Lee et al., 26 Sep 2025).

Additionally, convolutional neural networks (CNNs) trained on “pixel maps” encoding the chemical and geometric neighborhood (constructed from ab-initio or molecular statics data) allow rapid estimation of local defect and migration energies. These surrogate models enable kinetic Monte Carlo simulations of vacancy migration in MPEAs with millions of sites and extreme chemical diversity (Ponga et al., 2022).

4. Chemical and Structural Influences: Correlations and Control

Equilibrium vacancy concentrations in MPEAs are established not only by average energetic parameters but also by detailed correlations with alloy chemistry, local short-range order, and the spatial distribution of chemical bonds.

Compositional dependence is pronounced: Adding group 4 elements (Ti, Zr, Hf) systematically increases vacancy concentrations in refractory MPEAs, by up to two or three orders of magnitude, compared to group 5/6-only systems (e.g., VNbTaCrMoW) (Lee et al., 26 Sep 2025). This effect is attributed to the intrinsically lower vacancy formation energies of group 4 atoms and their propensity to enrich the atomic neighborhood of vacancies in a fully equilibrated alloy.

Ordering effects also play a significant role. Alloys with strong ordering tendencies or phase-separating binaries (e.g., Cr-W) exhibit vacancy concentrations that deviate substantially from ideal or linear interpolations. Breaking strongly bonded pairs incurs a greater energy penalty, further lowering vacancy concentrations (Lee et al., 26 Sep 2025, Jeffries et al., 11 Feb 2024). Conversely, in random solid solutions like the equiatomic Cantor alloy, vacancy thermodynamics are insensitive to chemical order (Jeffries et al., 11 Feb 2024).

5. Temperature Dependence and Kinetic Implications

The temperature dependence of equilibrium vacancy concentrations is described by Arrhenius-like relations, $x_\mathrm{Va} \propto \exp(-\Delta G_f/(k_B T))$, but the effective vacancy formation free energy, and thus activation energy, is not a scalar in MPEAs but a statistical property over local configurations (Mortazavifar et al., 2013, Lee et al., 26 Sep 2025).

The strong temperature sensitivity arises from both energetic and entropic (vibrational, electronic) terms, which can be non-monotonic or strongly environment-dependent in chemically complex alloys. Different interatomic potential parameterizations (e.g., EAM, MEAM) can yield qualitative differences in $\Delta G_{\rm v}(T)$ (Mortazavifar et al., 2013). Small changes in element ratios or short-range order can cause significant deviations in vacancy concentrations at a given temperature due to the exponential nature of the Arrhenius relation (Lee et al., 26 Sep 2025).

Kinetic consequences are profound: since diffusion in fcc and bcc metals follows a vacancy mechanism, composition-driven changes in vacancy concentration directly modulate bulk diffusivity and related phenomena such as creep, oxidation, and irradiation resistance (Thomas et al., 2020, Roy et al., 2022).

6. Modelling Approaches: Statistical Mechanics and Surrogate Models

Multiple frameworks for calculating equilibrium vacancy concentrations in MPEAs have been established:

• Cell Cluster Theory: Extends Stillinger's grand-canonical cluster expansion to chemically disordered lattices, with configuration-averaged partition functions providing the relevant free-energy terms (Mortazavifar et al., 2013).
• Maxwell–Boltzmann Statistics over Distribution Functions: Validates integration of the exponential probability over the measured/calculated distribution of vacancy formation energies $g(E_{\rm f})$ (Utt et al., 2021).
• Langmuir-like Multi-State Models: Interprets each lattice site as a $(k+1)$-state system (for $k$ element types plus the vacancy), with occupation probabilities reflecting enthalpic penalties. Global vacancy concentration is the configurational average:

$$x_v(\beta) = \langle p_{\sigma, v}(\beta) \rangle = \left\langle \frac{1}{1+\sum_\alpha \exp(\beta \mathcal{H}_\sigma^{(\alpha)})}\right\rangle$$

where $\mathcal{H}_\sigma^{(\alpha)}$ is the energy penalty for substituting atom $\alpha$ with a vacancy at site $\sigma$ (Jeffries et al., 11 Feb 2024). This model generalizes to interstitials and other point defects.

• Embedded Cluster Expansions: Site- and configuration-dependent neural network expansions for energy, efficiently sampling the phase space and providing $k$-resolved defect statistics for high-component alloys (Lee et al., 26 Sep 2025).

In practice, these frameworks are often combined: eCE or ML potentials inform kinetic (KMC/MC) or thermodynamic (GCMC) sampling; surrogate models (e.g., CNNs) rapidly provide site energies; and rigorous statistical mechanics delivers ensemble averages.

7. Implications for Materials Design and Engineering

Identification of the compositional and structural influences on equilibrium vacancy concentrations provides actionable guidance for alloy design. For high-temperature and irradiation-demanding applications (e.g., nuclear, aerospace), tailoring MPEA compositions to minimize or maximize vacancy concentrations can optimize diffusion, creep resistance, and microstructural stability.

A fundamental conclusion is that equilibrium vacancy concentrations can be modulated by controlling the fraction and distribution of low vacancy formation energy elements (notably group 4 elements), as well as the short-range chemical order in the alloy (Lee et al., 26 Sep 2025). However, strategies increasing vacancy content may entail tradeoffs, such as lower melting points or phase instability (Lee et al., 26 Sep 2025, Cuong et al., 2020). Therefore, predictive computational frameworks are essential for mapping these tradeoffs and achieving targeted performance in application-relevant environments.

In sum, equilibrium vacancy concentrations in MPEAs are a multi-scale, multi-factorial property rooted in the statistics of local energetic fluctuations, chemical order, and thermal activation. Sophisticated atomistic-through-mesoscale models, validated by simulation and experiment, are now capable of quantitatively predicting these key defect thermodynamics across the vast parametric space of chemically complex alloys.