Stability of heterogeneous single-atom catalysts: a scaling law mapping thermodynamics to kinetics

The study addresses how to predict and enhance the stability of heterogeneous single-atom catalysts (SACs), where isolated metal atoms can sinter via diffusion, leading to deactivation. While prior work emphasized thermodynamic stability via single-atom binding energy (Ebind) to supports, the authors posit that kinetic stability governed by the diffusion activation barrier (Ea) is the key determinant of resistance to sintering. They aim to uncover a general, predictive relationship that maps thermodynamic descriptors to kinetics across diverse metal–support combinations, enabling rapid screening of SAC stability using easily computed or tabulated quantities. This is important for designing stable SACs that maximize utilization of precious metals in catalysis under realistic operating conditions.

The paper situates SAC stability within the broader context of strong metal–support interactions and common deactivation via Ostwald ripening and diffusion into support lattice sites. Previous efforts used Ebind as a descriptor for SAC stability, assuming stronger binding implies higher stability. However, a simple linear correlation between Ea and Ebind is only moderate (R2 ≈ 0.83) and particularly poor for strongly interacting supports. The Brønsted–Evans–Polanyi principle is not applicable to diffusion between similar sites with negligible initial–final energy differences. Prior ML work (e.g., O’Connor et al.) correlated Ebind with metal/support features, and studies by Mavrikakis and co-workers showed linear Ea–adsorption energy relations for adsorbates on metals, but not for metal adatom diffusion on supports. These gaps motivate discovering a more accurate scaling that includes intrinsic metal properties such as cohesive energy (Ec).

- Systems and dataset: Considered 11 transition metals (Cu, Ag, Au, Ni, Pd, Pt, Co, Rh, Ir, Fe, Ru) on nine supports: CeO2(111), CeO2(100), steps on CeO2(111), TiO2(110), MgO(100), ZnO(100), SrTiO3(100), 2H-MoS2(0001), and graphene, yielding 99 metal–support data points. Diffusion considered between adjacent identical adsorption sites on idealized surfaces without vacancies or adsorbates. - DFT calculations: Spin-polarized GGA-DFT with PBE functional and PAW potentials in VASP. DFT+U used for Ce (U=4.5 eV) and Ti (U=4.0 eV). Supercells: CeO2(111) 4×4, CeO2(100) 4×4; TiO2(110) 5×4; MgO(100) 3×3 (4 layers); ZnO(100) 2×3 (6 layers); SrTiO3(100) and LaFeO3(100) 2×2 (8 layers); graphene and MoS2 4×4. Bottom layers fixed; top layers relaxed. k-point meshes: typically 1×1×1 (oxides, graphene, MoS2), 3×3×1 (MgO), 3×2×1 (ZnO). 15 Å vacuum. The most stable single-atom adsorption configurations and relevant spin states were identified. Diffusion barriers (Ea) obtained via climbing-image NEB. Convergence: force threshold 0.05 eV/Å. - Descriptors: Primary descriptors Ebind (single-atom binding energy on support) and Ec (bulk metal cohesive energy). Explored dependence of Ebind on Ec and found weak cross-support correlation. Generated secondary/tertiary descriptors as polynomial and logarithmic combinations of Ebind and Ec, totaling 87 unique descriptors. - Machine learning: Employed ridge, LASSO, and elastic net regression for regularization and feature selection with standardized data. Training on 80% of data with 10× repeated 10-fold CV; testing on held-out 20%. Also used genetic programming (symbolic regression) with gplearn (population 5000, 100 generations, multiple seeds) to automatically search functional forms. - Stability assessment: Estimated characteristic diffusion time via transition state theory using kdiffusion = (kB T / h) exp(−Ea / kB T) and τdiffusion = 1/kdiffusion, mapping expected SAC lifetimes as functions of Ebind and Ec at different temperatures (e.g., 300 K and 1073 K). Applied to interpret experimental synthesis conditions for Pt/CeO2 in oxidative environments and effects of oxidation state (e.g., PtO2 species) and defect trapping.

- Simple linear correlation between Ea and Ebind across supports is only moderate (R2 = 0.83), insufficient for general prediction. - ML feature selection and genetic programming consistently identified (Ebind)2/Ec as the dominant descriptor for diffusion barriers of single metal atoms on supports. - Genetic programming yielded a compact model: Ea = 0.565 × (Ebind)2/Ec with testing RMSE = 0.262 eV and R2 ≈ 0.93. - Ordinary least squares using the single descriptor provided the diffusion scaling-law (DSL-SAC): Ea = 0.636 × (Ebind)2/Ec − 0.203 with testing RMSE = 0.220 eV and overall R2 = 0.946. - LASSO and elastic net models achieved testing RMSE ≈ 0.198 eV; ridge RMSE ≈ 0.264 eV, but the single-descriptor DSL-SAC offers strong accuracy and interpretability. - Quadratic dependence of Ea on Ebind for a given metal-support family is observed; introducing σ = Ebind/Ec interprets Ea ~ Ebind × σ, where σ quantifies the relative metal–support interaction strength versus bulk cohesion (σ ranges ~0.16–1.06; small σ for inert supports like graphene, large σ for reactive supports like CeO2(100)). - Validation on LaFeO3(100): Using DFT Ebind (Cu 2.40 eV, Ru 6.29 eV, Pd 2.48 eV, Pt 3.93 eV), DSL-SAC predicted Ea values of 0.69, 2.87, 0.80, 1.49 eV vs DFT barriers 0.87, 2.85, 0.85, 1.68 eV, within model RMSE. - Additional validation on ZrO2(100) further corroborated the model (details in Supplementary). - Stability mapping: Lifetimes τdiffusion as functions of Ebind and Ec at 300 K and 1073 K provide thresholds for stable operation. For example, to avoid Pt agglomeration at 1073 K for 12 h, Ebind ≳ 6.3 eV is needed; metallic Pt binding on CeO2 terraces and steps is lower (~5.5 eV), predicting aggregation, whereas oxidized PtO2 species at steps exhibit Ebind ~7.5 eV, consistent with experimental stability.

The findings demonstrate that kinetic stability of SACs, governed by adatom diffusion barriers, can be accurately predicted from thermodynamic binding energies when normalized by the metal’s cohesive energy. This maps thermodynamics to kinetics via a single, interpretable descriptor (Ebind)2/Ec, capturing both support interaction strength and intrinsic metal bonding tendencies. The scaling-law enables rapid screening across diverse supports (oxides, perovskites, 2D materials) and metals to identify stable SAC candidates at relevant temperatures. It clarifies why inert supports with low σ yield low Ea even when Ebind increases, and why reactive, defect-rich surfaces or oxidized metal species can substantially enhance stability. The approach aligns with observed experimental behaviors (e.g., Pt stabilization on ceria under oxidizing conditions) and provides a quantitative framework to anticipate sintering resistance and to guide the use of defects, dopants, or oxidation states for trapping single atoms.

Starting from DFT-calculated diffusion barriers for 99 metal–support pairs and assisted by ML, the study establishes a diffusion scaling-law for SACs: Ea correlates strongly with (Ebind)2/Ec, enabling accurate and interpretable prediction of kinetic stability from easily accessible descriptors. The model bridges thermodynamics and kinetics, validates against complex supports (e.g., LaFeO3), and offers practical maps to estimate SAC lifetimes under operating temperatures. This framework facilitates rapid, rational screening and design of stable SACs. Future work should incorporate realistic surface features (defects, dopants, hydroxylation), dynamic charge transfer and oxidation states, and environmental effects to further refine predictions and expand applicability.

- Dataset is based on idealized, defect-free surfaces (except steps), without hydroxyls, dopants, or impurities; real supports often contain such features that can alter binding and diffusion pathways. - Certain magnetic or strongly correlated oxides (e.g., LaFeO3) are computationally challenging; DFT+U and PBE approximations introduce methodological uncertainties. - The Brønsted–Evans–Polanyi principle does not apply here, and the empirical scaling form, while accurate, does not fully elucidate underlying physics; σ’s physical meaning requires further study. - Stability estimates via τdiffusion do not account for metal loading effects, atom–atom interactions at higher coverages, or environment-induced changes (e.g., oxidation state), which can extend lifetimes or modify barriers. - Model trained on a finite set of metals and supports; extrapolation beyond this chemical space should be done with caution.