Supported metal catalysts are crucial in various industries, and strong metal-support interactions are vital for their stability. Single-atom catalysts (SACs) offer the advantage of maximizing the use of often precious metals, but their stability is a key challenge. Sintering and diffusion into the support are major deactivation pathways. While previous studies focused on the binding energy (Ebind) as a descriptor of SAC stability, assuming stronger binding implies higher stability, this study argues that the diffusion activation barrier (Ea) is a more relevant factor. Calculating Ea is computationally expensive, particularly for complex supports like certain oxides. Therefore, this research aims to find a correlation between Ea and easily computable parameters like Ebind and Ec to facilitate high-throughput screening of SACs. The study uses DFT calculations on 11 transition metals on various supports (CeO2, TiO2, MgO, ZnO, SrTiO3, MoS2, graphene, and stepped CeO2(111)) to create a dataset of 99 data points. Machine learning techniques are employed to identify correlations between Ea, Ebind, and Ec, aiming to establish a universal scaling relation for predicting SAC stability.

The literature highlights the importance of strong metal-support interactions in heterogeneous catalysis and the emerging field of single-atom catalysts (SACs). Several studies have attempted to correlate SAC stability with the binding energy (Ebind) of the metal atom to the support, suggesting that stronger binding leads to greater stability. However, these studies primarily focus on thermodynamic aspects. Existing research on metal atom diffusion on supports is limited in its systematic exploration of the thermodynamic and kinetic aspects of SAC stability. The need to identify correlations between diffusion activation barriers and binding strength for predicting SAC stability across a wide range of metal-support combinations motivated this research. The authors review previous work on describing Ebind using parameters such as Ec.

The study utilizes density functional theory (DFT) calculations at the generalized gradient approximation (GGA) level with the Perdew–Burke–Ernzerhof (PBE) functional to compute Ebind and Ea for the diffusion of metal adatoms between adjacent similar sites on idealized support surfaces. The most stable adsorption configurations and spin states are considered for each metal-support pair. Three machine-learning (ML) algorithms—ridge, LASSO, and elastic net regression—are applied to identify correlations between Ea, Ebind, and Ec (cohesive energy of the bulk metal). The hypothesis space is populated by including polynomial and logarithmic terms of the primary descriptors as secondary descriptors and pair interactions as tertiary descriptors. The models are trained using 80% of the dataset, with the remaining 20% used for testing. Tenfold cross-validation with ten repeats is performed for hyperparameter tuning to prevent overfitting. Root mean square error (RMSE) and R² values are used to evaluate model performance. Genetic programming (GP) is also used to explore the hypothesis space without manual descriptor combination. The best model from GP is further refined using ordinary least squares fitting. The characteristic time of diffusion (τdiffusion) is estimated using the Arrhenius equation to assess SAC stability at different temperatures. The stability of isolated Pd and Pt atoms on LaFeO3(100) is used to validate the developed model. DFT calculations are performed using the Vienna ab initio simulation package (VASP), employing the projector-augmented wave method and the PBE exchange-correlation functional. For ceria and titania systems, the DFT + U approach is used. The climbing image nudged-elastic band algorithm is employed to determine the transition states for metal atom diffusion.

The study reveals that a simple linear correlation between Ea and Ebind is insufficient to accurately predict SAC stability across various metal-support combinations. Machine learning algorithms, including ridge, LASSO, and elastic net regression, along with genetic programming, consistently identify (Ebind)²/Ec as the most significant descriptor for the diffusion barrier of single-metal atoms. The resulting diffusion scaling-law (DSL-SAC) model, Ea = 0.636 × (Ebind)²/Ec − 0.203, exhibits a high R² value (0.946) and low RMSE (0.220 eV) on the testing set. The DSL-SAC model accurately predicts the diffusion barriers of single Cu, Ru, Pd, and Pt atoms on LaFeO3(100). The ratio σ = Ebind/Ec acts as a correction factor to Ebind, reflecting the degree of metal-support interaction. Low σ values indicate a larger correction and lower Ea relative to what would be expected from a simple proportionality with Ebind. The analysis of SAC lifetime (τdiffusion) based on the Arrhenius equation reveals the strong temperature dependence of SAC stability. The minimum required Ebind to achieve desired SAC lifetimes at different temperatures is predicted. For example, the high stability of experimentally observed Pt/CeO2 SACs prepared at 1073 K for 12 hours is explained by the high binding energy of PtO2 on CeO2 steps (7.5 eV), rather than elemental Pt.

The findings demonstrate that the kinetic stability of SACs, represented by Ea, is more accurately described by considering both the thermodynamic stability (Ebind) and the intrinsic reactivity of the metal (Ec), rather than simply relying on Ebind alone. The DSL-SAC model provides a simple, interpretable, and generalizable approach for high-throughput screening of SAC stability. The model's success validates the importance of considering the kinetic aspect (diffusion barrier) alongside the thermodynamic aspect (binding energy) in understanding and predicting SAC stability. The model's limitations (idealized surfaces, absence of defects, and the simplified consideration of SAC lifetime) are acknowledged, suggesting avenues for future research. The study's approach offers a powerful strategy for rationally designing stable SACs.

This study presents a diffusion scaling-law (DSL-SAC) model, Ea = 0.636 × (Ebind)²/Ec − 0.203, which accurately predicts the diffusion activation barrier of single-metal atoms on various supports. This model improves upon previous thermodynamic approaches by incorporating both binding energy and cohesive energy, providing a simple and efficient tool for screening the stability of SACs. Future work could focus on refining the model by considering the effects of surface defects, dopants, and hydroxyl groups. The DSL-SAC model represents a significant advancement in the rational design of stable and efficient SACs.

The model focuses on idealized surfaces without considering defects, hydroxyl groups, or dopants, which can significantly influence metal atom diffusion and sintering. The simplified treatment of SAC lifetime, considering only the characteristic diffusion time, doesn't account for metal loading effects, especially for weakly bonded systems. The model's accuracy may be limited for complex supports or under specific reaction conditions not included in the dataset. The model primarily focuses on single transition metal atoms, and its applicability to other types of SACs requires further investigation.