Dynamic Metal–Support Interaction Dictates Cu Nanoparticle Sintering on Al₂O₃ Surfaces

Abstract

Nanoparticle sintering remains a critical challenge in heterogeneous catalysis. In this work, we present a unified deep potential (DP) model based on the Perdew–Burke–Ernzerhof approximation of density functional theory for Cu nanoparticles on three Al₂O₃ surfaces (γ-Al₂O₃(100), γ-Al₂O₃(110), and α-Al₂O₃(0001)). Using DP-accelerated simulations, we reveal that the nanoparticle size-mobility relationship strongly depends on the supporting surface. The diffusion of nanoparticles on the two γ-Al₂O₃ surfaces is almost independent of the size of the nanoparticle, while the diffusion on α-Al₂O₃(0001) decreases rapidly with increasing size. Interestingly, nanoparticles with fewer than 55 atoms diffuse several times faster on α-Al₂O₃(0001) than on γ-Al₂O₃(100) at 800 K while expected to be more sluggish based on their larger binding energy at 0 K. The diffusion on α-Al₂O₃(0001) is facilitated by dynamic metal–support interaction (MSI), where Al atoms move out of the surface plane to optimize contact with the nanoparticle and relax back to the plane as the nanoparticle moves away. In contrast, the MSI on γ-Al₂O₃(100) and on γ-Al₂O₃(110) is dominated by more stable and directional Cu–O bonds, consistent with the limited diffusion observed on these surfaces. Our extended MD simulations provide insight into the sintering processes, showing that the dispersity of the nanoparticles strongly influences the coalescence driven by nanoparticle diffusion. We observed that the coalescence of Cu₁₃ nanoparticles on α-Al₂O₃(0001) can occur in a short time (10 ns) at 800 K even with an initial internanoparticle distance increased to 3 nm, while the coalescence on the two γ-Al₂O₃ surfaces are inhibited significantly by increasing the initial internanoparticle distance. These findings demonstrate that the dynamics of the supporting surface is crucial to understanding the sintering mechanism and offer guidance for designing sinter-resistant catalysts by engineering the support morphology.

Introduction

Supported metal nanoparticles, while the cornerstone of modern heterogeneous catalysis, face critical challenges in maintaining steady catalytic performance in activity and selectivity for a long period. The overall catalytic performance of these nanoparticles is often compromised by deactivation under operando conditions, which is an inherent part of most catalytic processes, reduces the lifetime of the catalyst and severely affects industrial production. Sintering is one of the main factors in the deactivation of supported nanoparticle catalysts and is known to be governed by the metal–support interaction (MSI) that changes throughout the operational time depending on reaction conditions. − A consensus from several studies is that sintering proceeds via two pathways: Ostwald ripening (involving the migration of atomic species from smaller to larger particles), and coalescence (where entire particles move and merge). , Hu and Li in a recent work proposed the Sabatier principle for the sintering process based on a quantitative study of MSI. They found a linear relationship between the nanoparticle-support adhesion energy and the onset temperature of sintering, focusing primarily on fully relaxed, static idealized surfaces. They concluded that a stronger MSI leads to Ostwald ripening, while a weaker MSI is responsible for coalescence on homogeneous supports. However, MSI is a complex, multifaceted phenomenon that encompasses not only binding and adhesion energies at zero temperature but also the dynamic interplay between the nanoparticle and the support, which may not be fully captured by static models. It remains elusive to predict how realistic conditions determine catalyst stability when the complex interplay between nanoparticle size, support structure, and temperature is taken into account.

Supported Cu nanoparticle catalysts find extensive applications in the chemical industry, particularly in CO₂ hydrogenation and methanol synthesis. − These catalytic reactions are often structure sensitive, dictated by the Cu nanoparticle size and shape. Hence, the choice of support is crucial to achieve optimal MSI in Cu-catalyzed reactions, ensuring a stable size distribution of Cu nanoparticles that maintain activity over an extended period of operations. Al₂O₃ is one of the commonly used supports. In particular, γ-Al₂O₃ is widely preferred over α-Al₂O₃ for Cu nanoparticles due to its high specific surface area and porosity resulting in higher nanoparticle dispersion. However, thermal sintering and deactivation of supported nanoparticles is unavoidable under experimental conditions and depends on the surface termination. An experimental study showed that the sintering on reconstructed α-Al₂O₃(0001) occurs when a high coverage of Cu nanoparticles is deposited. From an atomic perspective, unravelling the morphological changes of the Cu nanoparticles and their dynamical interaction with different Al₂O₃ surfaces is of paramount importance for designing robust supported Cu nanoparticle catalysts with enhanced lifetime and activity.

Computational modeling has emerged as a powerful tool for gaining atomic insights into supported nanoparticle catalysts. Although density functional theory (DFT) has been instrumental in understanding supported nanoparticle catalysts, its high computational cost limits the studies to small, often unrealistic model systems, containing up to a few dozen metal atoms. − Moreover, supported nanoparticles are inherently dynamic under reaction conditions, so it is important to sample a set of diverse thermally accessible configurations that can influence catalytic properties. Capturing this structural fluxionality is crucial for the accurate modeling of catalytic behavior, but presents a significant computational challenge for conventional DFT approaches. Recent advances in machine learning interatomic potentials (MLIPs) have opened up exciting possibilities for modeling complex catalyst systems with ab initio accuracy at a fraction of computational cost. , By training on a diverse data set of DFT calculations, MLIPs can learn the underlying potential energy surface and enable molecular dynamics (MD) simulations of realistic supported nanoparticle models over extended time and length scales. In fact, some recent studies have made efforts to train MLIPs for a few specific combinations of metal nanoparticles and supports, , although how the dynamics of the supporting surface affects the sintering of the nanoparticles is still unclear. In addition, training robust and transferable MLIPs for supported nanoparticle systems is a nontrivial task, especially for various combinations of nanoparticle sizes and supports, requiring careful data set curation and efficient active learning strategies to sample relevant regions in the configuration space.

Here, we present a systematic workflow to train a unified deep potential (DP) model based on the Perdew–Burke–Ernzerhof (PBE) functional approximation of DFT for Cu nanoparticles supported on different surface terminations of γ- and α-Al₂O₃. Our adopted functional approximation does not include corrections for dispersion (van der Waals (vdW)) interactions, which may be important for the systems of interest here. , We will show in the Supporting Information (Section S7.3) that while vdW interactions would increase the absolute value of the calculated binding energies, they would not change the qualitative trends observed in our investigation. Our training data set is constructed by combining global optimization (GO) techniques with MD sampling that capture finite-temperature effects. The resulting DP model is capable of accurately describing supported Cu nanoparticles of different sizes at finite temperature. The model is validated against DFT-optimized global minimum configurations of supported Cu nanoparticles up to Cu₅₅ and against ab initio MD (AIMD) simulations for four selected nanoparticle sizes (Cu₁, Cu₄, Cu₁₃, and Cu₂₀) within a short time window (20 ps) on γ-Al₂O₃(100), γ-Al₂O₃(110) and α-Al₂O₃(0001). This validation allows us to explore the thermodynamic stability and sintering behavior at the atomic level. We found that the nanoparticle size-mobility relationship strongly depends on the supporting surface. The nanoparticle diffusion on the two γ-Al₂O₃ surfaces is almost independent of the nanoparticle size while the diffusion on α-Al₂O₃(0001) decreases rapidly with increasing size. Moreover, we found that Cu nanoparticles tend to diffuse faster on α-Al₂O₃(0001) than on γ-Al₂O₃(100) despite their stronger bindings in configurations that minimize potential energy. The unexpected mobility on α-Al₂O₃(0001) originates from the dynamics of the supporting surface driven by the formation of metallic bonds between the Cu and Al atoms. In the presence of a nanoparticle the closest Al atoms relax quickly out of the surface plane to optimize the contact with the nanoparticle and relax back to the surface plane as the nanoparticle diffuses away, which we call the dynamic MSI. Further MD simulations of nine supported Cu₁₃ nanoparticles reveal that coalescence can be observed on α-Al₂O₃(0001) in 10 ns at 800 K even with an initial distance between nanoparticles increased to approximately 30 Å while the coalescence on γ-Al₂O₃(100) is significantly inhibited by increasing the initial distance between nanoparticles from 15 Å to 30 Å. Besides, nanoparticles on γ-Al₂O₃(110) exhibit even more limited diffusion and coalescence. These findings suggest that surface dynamics is crucial to understanding nanoparticle diffusion and sintering, and support morphology engineering is a possible strategy for developing sinter-resistant supported metal catalysts.

Results and Discussion

Active Learning Workflow

To investigate the atomic-level structure evolution of Al₂O₃-supported Cu nanoparticles, we established an active learning workflow to train our DP models, namely, GamAlCu for γ-Al₂O₃, and AlpAlCu for α-Al₂O₃. Combining these two individual models, we created a unified DP model (UniAlCu) that is uniformly accurate for supported Cu nanoparticles on two Al₂O₃ phases (Figures S1–S2). The three models are discussed in Section S1. The active learning workflow incorporates several exploration methods implemented in the GDPy package, the details of which can be found in Section S2. Figure a shows the two exploration methods used through our active learning cycles. The GO in the flavor of genetic algorithm (GA) biases the search toward low-energy structures of supported Cu nanoparticles, providing other exploration methods with several good starting configurations. MD simulations at different temperatures sample a sheer volume of structures (γ- and α-Al₂O₃ surfaces and supported Cu nanoparticles), which are further sifted on the basis of the model deviation and geometry diversity to maintain a compact data set. Given our focus on Al₂O₃-supported Cu nanoparticles, we organized the data set into three parts shown in Figure b. The Cu-only structures, including bulk, surface and nanoparticle systems, are taken from a previous study while Cu _N /γ-Al₂O₃ and Cu _N /α-Al₂O₃ (N = number of Cu atoms in the nanoparticle) structures are actively accumulated by our active learning workflow. The γ-Al₂O₃ bulk is modeled by the structure proposed by Digne. Applications outlined in Figure c indicate that our UniAlCu model is capable of simulating both the global minimum configurations of the supported Cu nanoparticles and their dynamic behavior at finite temperature. Furthermore, nanoparticle sintering can be observed through long-time and large-scale simulations (Figure d). These applications will be demonstrated in the following sections.

1.

Schematic illustration of the active learning workflow. (a) In each active learning cycle, two exploration methods are utilized to curate a comprehensive and compact data set. The GA focuses on minimum energy configurations of supported nanoparticles. The MD samples structures at finite temperature, which are further sifted based on model deviation and geometry diversity. (b) The training data set covers Cu-only structures (bulk, surface, and nanoparticle), Al₂O₃ surfaces, and supported nanoparticles, including 147,464 structures in total. Our UniAlCu model has a root mean squared error of 0.004 eV/Atom in the energy and of 0.057 eV/Å in the force predictions. Taking supported nanoparticles on γ-Al₂O₃(100) as an example, we illustrate different applications of our UniAlCu model: (c) exploring supported nanoparticle structures at zero and finite temperatures; (d) revealing atomic details of nanoparticle sintering by MD simulations over large time and length scales. Color code: Cu: yellow, Al: cyan and O: red. (Brown is used to distinguish the second Cu nanoparticle participating in coalescence).

Minimum Energy Configurations

The configurations of metallic nanoparticles on a supporting surface depend on the balance of MSI and intraparticle binding. On Al₂O₃ surfaces, Cu nanoparticles can form bonds with Al and O atoms and can electrostatically interact with the substrate due to charge transfer. To measure the interaction strength, we calculated the binding energy (E _b) according to eq . E _b has two components, the adhesion energy of a nanoparticle on the surface and the energy cost of restructuring the gas phase nanoparticle to conform to the surface. E _b is a crucial descriptor that influences catalytic activity and stability, directly reflecting the MSI. Similar to E _b, we also evaluated the cohesive energy (E _c) as defined in eq . Since E _c uses the atomic energy of the bulk FCC Cu as a reference, it measures the stability of a nanoparticle, supported or in the gas phase, relative to the bulk. Thus, all nanoparticles have E _c greater than zero and E _c = 0 corresponds to the bulk limit.

To validate our UniAlCu model, we benchmarked E _b and E _c against DFT calculations on minimum energy configurations. The detailed values of E _b and E _c are listed in Tables S1–S3. We considered nanoparticles with a number of atoms, N, ranging from 1 to 21, and used periodic supercells to model three supporting surfaces, namely p(3 × 2) for γ-Al₂O₃(100), p(2 × 2) for γ-Al₂O₃(110), and $p(3\times 2\sqrt{3})$ for α-Al₂O₃(0001). The corresponding structures are displayed in Figure S3, which also reports the O coordination number of some Al atoms as a superscripted Latin numeral. Each Al₂O₃ system is described by a slab whose thickness is fixed by monitoring the convergence of the calculated E _b, as shown in Figure S4. Fully relaxed minimum energy configurations of supported nanoparticles are found with an active learning GA protocol that efficiently explores the potential energy landscape, − as illustrated in Figure . Figure a compares the binding energies of the nanoparticles on the three Al₂O₃ surfaces. The negative values of E _b indicate that the supported nanoparticles are always more stable than their gas-phase counterparts. On γ-Al₂O₃(100), E _b depends weakly on N, suggesting a weak size dependence of the MSI. In contrast, on γ-Al₂O₃(110), E _b takes more negative values with increasing nanoparticle size, signifying stronger size-sensitive adhesion. The stark difference in adhesion between γ-Al₂O₃(100) and γ-Al₂O₃(110) is attributed to the local coordination environments of the Al sites on these surfaces. The superior anchoring on γ-Al₂O₃(110) results from a unique surface structure that exposes Al^III sites with good affinity for Cu atoms, which, in combination with geometric matching, strengthen the adhesion of Cu nanoparticles. Similar to γ-Al₂O₃(110), E _b on α-Al₂O₃(0001) decreases with nanoparticle size with values that are intermediate between γ-Al₂O₃(100) and γ-Al₂O₃(110), where our computed E _b for a single Cu adatom agrees with a previously reported value. Undercoordinated Al sites, particularly in penta-coordinated Al^V (γ-Al₂O₃(100)) and planar Al^III (γ-Al₂O₃(110) and α-Al₂O₃(0001)) environments, play a crucial role in the initial anchoring and subsequent growth of Cu nanoparticles. Our UniAlCu model accurately captures the E _b trends across different surfaces and nanoparticle sizes, closely matching the DFT results. Despite minor quantitative deviations for some nanoparticle sizes, the model reliably reproduces the relative binding strengths, demonstrating its predictive power in ranking the MSI strength. By further analysis of the surface structures, the displacement of Al atoms on the surface is found to reflect the binding strength between the nanoparticle and the surface (Figure S5). A large displacement of surface Al atoms in the z-direction is observed on γ-Al₂O₃(110) and α-Al₂O₃(0001) while minimal displacement occurs on γ-Al₂O₃(100). In particular, this displacement increases with the nanoparticle size on α-Al₂O₃(0001) as more Al^III atoms are stabilized by the interaction with neighboring Cu atoms, leading to a more negative E _b.

2.

Global minimum configurations of Cu_1–21 on three Al₂O₃ surfaces are DFT-optimized structures obtained through a DP-accelerated GA search. N is the number of Cu atoms in the nanoparticle. The binding energy (a) and the cohesive energy (b) are compared between DP and DFT (solid and dashed lines, respectively). The structural evolution of supported Cu₁, Cu₄, Cu₁₃, and Cu₂₀ nanoparticles (c–e) reveals distinct growth patterns. The Al atoms are denoted by the superscript as their coordination numbers with O atoms. For γ-Al₂O₃(100), there are three types of Al^V on the surface and one type of Al^IV in the subsurface. For γ-Al₂O₃(110), there are two types of Al^IV and one type of Al^III on the surface. For α-Al₂O₃(0001), there is one type of Al^III on the surface and one type of Al^VI in the subsurface. Nanoparticles on γ-Al₂O₃(100) are more compact than those on γ-Al₂O₃(110) and α-Al₂O₃(0001), which is consistent with the MSI strength indicated by the binding energy.

The E _c of Cu nanoparticles on the three Al₂O₃ surfaces are shown in Figure b. Smaller nanoparticles generally exhibit larger E _c, implying a propensity to sinter into larger nanoparticles, leading to catalyst deactivation over time. On γ-Al₂O₃(100), E _c smoothly approaches the bulk limit with increasing size. Conversely, on γ-Al₂O₃(110), a sharp drop in E _c from Cu₁ to Cu₂ is followed by a plateau, suggesting similar stability for a range of intermediate sizes. Likewise, we observed a sharp drop in E _c from Cu₁ to Cu₂, followed by a smooth transition on α-Al₂O₃(0001). Notably, on all three Al₂O₃ surfaces, E_c of Cu nanoparticles is significantly lower than that of the gas phase nanoparticles, especially for small sizes. This highlights the crucial role of MSI in stabilizing dispersed nanoparticles that would be unstable in the gas phase. As size increases, the difference between supported and isolated nanoparticles decreases, consistent with the trend of E _b. Our UniAlCu model shows excellent agreement with DFT in predicting both absolute cohesive energies and size-dependent trends, further validating its accuracy.

The configurations of Cu₁, Cu₄, Cu₁₃, and Cu₂₀ nanoparticles on the three Al₂O₃ surfaces (Figure c–e) reveal distinct growth patterns. On γ-Al₂O₃(100), the nanoparticles transition from 2D planar to 3D compact geometries as they grow, driven by the increasing Cu–Cu interaction relative to the Cu-surface interaction (Figure S6). In contrast, on γ-Al₂O₃(110), the nanoparticles grow around an Al^III site up to Cu₁₀, following an epitaxial growth along the x-axis to bind to another Al^III site up to Cu₁₇, and then more Cu atoms are added along the y-axis (Figure S7). From Cu₂ to Cu₁₁ the nanoparticles include one Cu atom in the subsurface below Al^III, while from Cu₁₂ to Cu₂₁ there are two such atoms, similar to the behavior of supported Pt nanoparticles reported in a recent DFT study. On α-Al₂O₃(0001), the nanoparticles tend to stabilize around Al^III atoms by pulling them out of the surface plane and converting from planar to the trigonal pyramidal geometry of the surface AlO₃ units beneath the nanoparticle (Figure S8). In these configurations, one or more Al^III atoms can coordinate with several Cu atoms, consistent with the formation of metallic Cu–Al bonds. In the special case of Cu₁, this does not occur as the Cu–Al interaction is too weak, and the minimum energy configuration maximizes the interaction of Cu with the O atoms above a subsurface Al^VI site. The shape of the supported nanoparticles correlates with their radius of gyration (R _g) shown in Figure S9, which increases steadily with the size of the nanoparticle, indicating an increasing spread of the Cu atoms in the surface plane. For all sizes, the nanoparticles on γ-Al₂O₃(100) have a smaller R _g compared to γ-Al₂O₃(110) and α-Al₂O₃(0001), indicating a more compact shape consistent with a weaker MSI.

To further validate the UniAlCu model for Cu nanoparticles larger than those included in the training data set, we performed global optimization of Cu₂₇, Cu₃₈, Cu₄₇, and Cu₅₅ on the three Al₂O₃ surfaces. The binding and cohesive energies predicted by the model are compared to the corresponding DFT energies for sizes ranging from N = 1 to N = 55 in Figure S10, while the global minimum configurations of the nanoparticles with 27, 38, 47, and 55 Cu atoms are reported in Figure S11. Table S4 shows that while the absolute binding energy error may exceed 1 eV for the largest nanoparticles, the DFT trends are reproduced well. In particular, the errors in the cohesive energy per Cu atom are always very small, 0.02 eV/atom at most, and the model captures the effects of the supporting surfaces.

Nanoparticles at Finite Temperature

To benchmark the performance of our UniAlCu model at finite temperature, we compare the mean square displacement (MSD) of the Cu atoms (MSD_Cu defined in eq ) extracted from DPMD and AIMD simulations of Cu₁₃ on p(3 × 2) γ-Al₂O₃(100), p(2 × 2) γ-Al₂O₃(110), and $p(3\times 2\sqrt{3})$ α-Al₂O₃(0001) surfaces at 400, 800, and 1200 K (Figure a). The trajectories start from the global minimum configurations of nanoparticle plus substrate with initial atomic velocities drawn from the Maxwell–Boltzmann distribution at the simulation temperature. The trajectories run for 20 ps in the NVT ensemble with temperature controlled by a Langevin thermostat to ensure proper equilibration. For each temperature and surface orientation, we ran one AIMD trajectory and 20 independent DPMD trajectories. MSD_Cu in a time interval of 5 ps is extracted from the trajectories by window averaging (see details in Section S4.1) and reported in Figure a. In the DPMD case, the standard errors of the simulated MSD_Cu are estimated from the 20 independent trajectories and shown in the same figure as the vertical bars. Our UniAlCu model reproduces AIMD within the error bars of DPMD. At 400 K, diffusion is negligible, and displacements reflect vibrational motion. At higher temperatures (800 and 1200 K), diffusion is activated and larger MSD_Cu values, which grow approximately linearly with time, are observed on the three Al₂O₃ surfaces. This behavior is attributed to internal diffusion within the nanoparticles because no significant displacements of the center of mass (COM) are observed on the short time scale of these simulations. In agreement with the indication of E _b, the strong MSI on γ-Al₂O₃(110) hinders Cu mobility. However, at high temperature, Cu₁₃ on α-Al₂O₃(0001) shows comparable mobility to that observed on γ-Al₂O₃(100) in spite of the significantly larger E _b on the α-surface than on the γ-surface (−3.71 eV vs −1.72 eV). A similar behavior is observed in the simulations of the supported Cu₄ and Cu₂₀ nanoparticles (Figures S13 and S15). The structures in the insets of Figure a are snapshots at the elapsed time of 20 ps in the AIMD simulations at 800 K. They reveal that even in the absence of COM diffusion, the nanoparticles on γ-Al₂O₃(100) and α-Al₂O₃(0001) undergo significant structural relaxation by exploring low-energy configurations different from the ground-state configuration. Figure b,c compare the Cu–O and Cu–Al radial distribution functions (RDFs) extracted from the AIMD and DPMD simulations of Cu₁₃ on the three surfaces at 800 K. The first peak of the Cu–O and Cu–Al RDFs provides information on the bonding of the nanoparticle with the substrate. The Cu–O peak is stronger and sharper on γ-Al₂O₃(110), suggesting the formation of stronger Cu–O bonds on that surface. The Cu–Al peak is well-defined on γ-Al₂O₃(110) and on α-Al₂O₃(0001), where it is associated with metallic bonds between Cu and Al^III atoms that are pulled out of the α-Al₂O₃(0001) surface plane. Comparisons for Cu₁, Cu₄, Cu₁₃, and Cu₂₀ (Figures S16–S19) further support the good agreement between AIMD and DPMD. The UniAlCu model describes the interactions within a cutoff radius of 8 Å and may miss long-range electrostatic interactions originating from electronic charge transfer between the nanoparticles and the substrate. To estimate the importance of these effects we performed Bader charge analysis on snapshots of AIMD simulations of supported nanoparticles with up to 20 atoms. We found that the charge transfer was negligible on γ-Al₂O₃(100), small on α-Al₂O₃(0001), and more sizable on γ-Al₂O₃(110), consistent with the binding energy trends. However, all these transfers occur between atoms at close distances, such as Cu and Al atoms separated by less than 3 Å. Because this is well within the cutoff radius of the model, we infer that long-range electrostatic effects should have a minor influence on the dynamics of the nanoparticles in our systems.

3.

(a) MSD_Cu of Cu₁₃ on γ-Al₂O₃(100), γ-Al₂O₃(110) and α-Al₂O₃(0001) is evaluated by NVT simulations at 400, 800, and1200 K. The solid and dashed lines are DP and DFT results, respectively. The vertical bars indicate the standard errors from 20 independent DPMD trajectories. The insets are snapshots at the elapsed time of 20 ps in the AIMD simulations at 800 K. The light-yellow colored Cu atoms are those on top of the nanoparticle at 0 ps. It is clear that the nanoparticles on γ-Al₂O₃(100) and α-Al₂O₃(0001) undergo significant structual relaxation. (b, c) The RDFs of Cu–O and Cu–Al pairs in Cu₁₃ on three Al₂O₃ surfaces at 800 K. The solid and dashed lines are DP and DFT results, respectively. The stronger Cu–O peak for γ-Al₂O₃(110) suggests the formation of stronger Cu–O bonds. The Cu–Al peak for γ-Al₂O₃(100) and α-Al₂O₃(0001) indicates the formation of metallic bonds between Cu and Al atoms.

Additional validations of the model were performed on Cu-only systems. The melting temperature T _m of the bulk Cu and gas phase nanoparticles was estimated. The bulk melting temperature (T _m ) was obtained by simulating the coexistence of the liquid and solid phases, finding a value of 1207 K for T _m (Figure S20), close to an earlier DFT prediction and in fair agreement with the experimental value of 1358 K. We estimated the size dependence of the melting temperatures of Cu nanoparticles in vacuo from the Lindeman index, as shown in Figure S21, finding that T _m of a N-atom nanoparticle is smaller than T _m according to T _m = T _m – kN ^–1/3, where k is a constant, in good agreement with a recent study. The size dependence of the surface energy of the Cu nanoparticles exhibits a behavior (Figure S22) consistent with experimental observations for supported nanoparticles. − The surface energies of the main crystalline facets, Cu(111), Cu(100), and Cu(110) were also computed. The T = 0 K surface energies predicted by the UniAlCu model were 1.26, 1.44, and 1.52 J/m², respectively, close to the corresponding PBE values of 1.33, 1.48, 1.63 J/m².

Using the UniAlCu model, we could perform long DPMD simulations for nanoparticle sizes ranging from Cu₁ to Cu₆₇₅ to study the behavior of supported nanoparticles at finite temperature. The adopted structure models are shown in Figures S24–S26 and the details of the simulation protocol is given in Section S7.1. To validate the dynamic predictions of the model, we used configuration maps based on Smooth Overlap of Atomic Positions (SOAP) vectors , to rank the configurations visited in short AIMD trajectories and those visited in much longer DPMD trajectories that exhibit COM diffusion, finding substantial overlap of the respective maps (Figure S23). This shows that long DPMD simulations sample configurations similar to those encountered in short AIMD simulations but with displaced COM positions.

Finally, in order to extract asymptotic trends and make contact with experiment, we computed the contact area A and the adhesion energy E _adh = −E _b/A of the supported nanoparticles. The contact area A was calculated for the T = 0 K configurations of Cu _N up to N = 55 and as an average over the last 4 ns of the 20 ns-long trajectories of Cu₁₄₇, Cu₃₀₉, and Cu₆₇₅ on the three surfaces at T = 800 K. The details of the calculation are given in Section S7.2. Asymptotically A should grow with N in a way that is proportional to N ^2/3. Thus, we plot A/N ^2/3 vs N in Figure S27. This figure shows that A ∝N ^2/3 only for the three larger nanoparticles. Interestingly, A/N ^2/3 is smaller on γ-Al₂O₃(100) than on the other two surfaces, consistent with the trend of the binding energy. The figure suggests that the approximate onset of the N ^2/3 regime for the contact area should be placed between N = 55 and N = 147. The T = 0 K adhesion energy up to Cu₅₅ is reported in Figure S28. It shows a greatly reduced size dependence with increasing size on all three Al₂O₃ surfaces. However, E _adh is more meaningful for the three large nanoparticles, namely Cu₁₄₇, Cu₃₀₉, and Cu₆₇₅, for which the contact area is approximately in the asymptotic regime. E _adh of these nanoparticles was estimated from the potential energy averaged over the last 4 ns of 20 ns-long DPMD trajectories at T = 800 K. These trajectories started from the geometries generated with the Wulff construction. The estimated values of E _adh are reported in Table S8 with the corresponding contact areas. Due to the thermal fluctuations, it is difficult to extract a robust trend for E _adh vs N on the three surfaces but, generally, the average E _adh appears to increase with N. The Cu₆₇₅ nanoparticles have an approximate diameter of 2 nm. Assuming this to be close to the asymptotic regime for E _adh we can compare our E _adh on α-Al₂O₃(0001), that is 0.9 J/m², to the value of 3.2 J/m² extracted from shape measurements on this surface at T = 573 K for particles having an average diameter between 3 and 4 nm. Interestingly, our value is very close to the asymptotic PW91-DFT value for the Al-stoichiometric α-Al₂O₃(0001) surface. A possible cause of the difference between DFT estimates and experiment is the absence of vdW corrections in the DFT calculations. The vdW corrections would increase the binding energy, as shown in Figure S29 and Tables S5–S7. In the case of Cu₆₇₅ onα-Al₂O₃(0001), adding the D3 correction to the PBE-based DP model would lead to an adhesion energy of 1.7 J/m², which is closer to the experimental value of 3.2 J/m². We think, however, that the evidence favoring PBE-D3 over PBE for the systems of interest here, is inconclusive, as discussed in detail in the Supporting Information (Section S7.3). What is likely a more compelling reason for the difference between theory and experiment is that the surface supporting the nanoparticles in the experiment was not the pristine surface we used in our simulations but the reconstructed $p(\sqrt{31}\times \sqrt{31})R\pm 9^{°}$ surface. A very recent experiment showed that this reconstructed surface is rich in oxygen. This should have the effect of significantly strengthening the adhesion of a Cu nanoparticle, as suggested by calculations of the adhesion energy of a Cu overlayer on O-rich surface terminations.

We conclude that our PBE-based UniAlCu model is capable of reproducing well geometrical, local bonding, and dynamical relaxation trends observed in direct PBE-DFT calculations, suggesting that the model should also be predictive of the behavior of larger supported nanoparticles on longer time scales.

Long DPMD simulations show that the diffusion dynamics of the COM of supported nanoparticles is slow and may involve the separation of time scales between the internal dynamics of a nanoparticle and its diffusional motion. This behavior is illustrated for a Cu₁₃ nanoparticle in Figure a,b,c, which show snapshots of three successive displaced configurations along DPMD trajectories. The time evolution of the absolute value of the COM displacement is reported in Figures S34–S36 for γ-Al₂O₃(100), γ-Al₂O₃(110), and α-Al₂O₃(0001), respectively. It shows that on the two γ surfaces the COM dynamics combines fluctuations that do not contribute to diffusion with relatively large occasional jumps of about 2.5 Å on γ-Al₂O₃(100) and of about 2.0 Å on γ-Al₂O₃(110). The magnitudes of the jumps match approximately the distance between nearest-neighbor Al atoms on the two surfaces. The COM displacements result from collective atomic motions involving both the nanoparticle and the substrate rather than independent stochastic motions of the Cu atoms in contact with the substrate as assumed in simple mean field models. The time elapsed between the snapshots of Figure a,b is quite different on the two surfaces, greater than 100 ps on γ-Al₂O₃(100), and greater than 2.0 ns on γ-Al₂O₃(110), where diffusion is hampered by strong bonds between Cu and Al^III and Cu and O atoms. Diffusion on α-Al₂O₃(0001) is facilitated by dynamic MSI and is faster than on γ-Al₂O₃(100). Interestingly, the separation of time scales observed on the two γ surfaces is not evident on the α surface, as Figure S36 depicts an evolution that proceeds by a succession of small displacements on the time scale of the COM fluctuations, promoted by the motion in and out of the surface plane of the Al^III atoms adjacent to the nanoparticle. The above analysis is further corroborated by the Cu–Al and Cu–O bond correlation functions (C(t), eq ) reported for Cu₁₃ on the three surfaces (Figure d). In defining C(t), only the Al and O atoms in the top surface layer, which interact directly with the nanoparticle, were considered. In addition to O, these include Al^V₁ , Al^V₂ , Al^V₃ , and Al^IV on γ-Al₂O₃(100); Al^IV₁ , Al^IV₂ and Al^III on γ-Al₂O₃(110); Al^III and Al^VI on α-Al₂O₃(0001). Bond correlations reflect MSI in a dynamic context. The Cu–Al and Cu–O bond correlation functions decay slowly on γ-Al₂O₃(110), as expected from strong binding and slow diffusion. On γ-Al₂O₃(100), diffusion is faster and the bond correlation functions decay more rapidly. Interestingly, the Cu–O correlations decay more slowly than the Cu–Al correlations, suggesting that the diffusion is mainly controlled by Cu–O interactions. By contrast, on α-Al₂O₃(0001), the Cu–O bond correlation function decays faster than on γ-Al₂O₃(100) due to the dynamic Al^III layer that facilitates diffusion and hinders close contacts between Cu and O atoms.

4.

(a–c) Snapshots of three successive configurations with displaced COMs of Cu₁₃ on three Al₂O₃ surfaces. The light-yellow colored Cu atoms are those on the top of the nanoparticle in the first snapshot. The surface Al atoms are colored based on their heights (higher Al atoms are darker) and the subsurface Al and O atoms are painted pale. The magnitude of the elapsed time between the snapshots is quite different on three surfaces, about 100 ps on γ-Al₂O₃(100) and α-Al₂O₃(0001) while about 2.0 ns on γ-Al₂O₃(110). (d) The diffusion is controlled mainly by Cu–O interactions as the Cu–O correlations decay more slowly than the Cu–Al correlations. The rapid decay in both Cu–Al and Cu–O correlations on α-Al₂O₃(0001) elucidates the fast diffusion. (e, f) The diffusion coefficients of different nanoparticle sizes on γ- and α-Al₂O₃ surfaces. The vertical bars indicate the standard errors from 5 independent DPMD trajectories. The diffusion coefficient on α-Al₂O₃(0001) decreases significantly with nanoparticle size while it is still larger than those on γ-A₂O₃(100) and γ-Al₂O₃(110).

The COM diffusion coefficients (D _COM) defined in eq are extracted from five 10 ns-long DPMD simulations for a range of nanoparticle sizes up to Cu₁₄₇ (Figure e,f). The standard errors of the simulated D _COM are estimated from the five independent DPMD trajectories and are shown in the same figure as the vertical bars. We are unable to estimate D _COM of Cu₃₀₉ and of Cu₆₇₅ because the displacements of the COM of these large nanoparticles are negligible on the 10 ns time scale of the simulations. On the two γ-Al₂O₃ surfaces diffusion is almost independent of the nanoparticle size, with D _COM smaller on γ-Al₂O₃(110) than on γ-Al₂O₃(100), consistent with the indication of E _b. On α-Al₂O₃(0001), the diffusion is significantly faster for a single adatom and small nanoparticles, but decreases rapidly with size, while always remaining somewhat faster than on γ-Al₂O₃(100) despite the larger binding of the nanoparticles on α-Al₂O₃(0001) than on γ-Al₂O₃(100), consistent with the important role that dynamic MSI plays on the α-Al₂O₃(0001). In the case of Cu₁, we performed Climbing Image Nudged Elastic Band (CI-NEB) calculations at zero temperature to estimate the diffusion barriers between different surface sites. The diffusion barrier of Cu₁ on α-Al₂O₃(0001) is of only about 0.15 eV (Figure S37), much smaller than the barriers on γ-Al₂O₃(100) and γ-Al₂O₃(110) (Figures S38 and S39).

Nanoparticle Sintering Dynamics

After gaining insights into the static and dynamic behavior of isolated supported nanoparticles, we delve into the intricate process of nanoparticle sintering on the three Al₂O₃ surfaces considered in the previous sections. For that purpose, we strategically placed on each surface nine Cu₁₃ nanoparticles in their global minimum configuration at equal distances of approximately 15 Å (Figure S41). Then, we perform DPMD simulations controlled by a Langevin thermostat according to the following schedule. First, we equilibrate the systems at 400 K for 2 ns, then heat them at a constant rate of 0.2 K/ps for 2 ns, and finally, we run equilibrium NVT trajectories for 10 ns at 800 K. We employ the depth-first search (DFS) graph algorithm to find nanoparticles in the system and monitor Cu–Cu connectivity, nanoparticle count, minimum inter-COM distance (d _min , eq ), and minimum inter-Cu distance (d _min , eq ) throughout the simulation (Figure a). These metrics serve as indicators of possible coalescence. During the initial 2 ns at 400 K, the nanoparticle count does not change on the three surfaces, while d _min fluctuates around 15 Å, as expected from the negligible diffusion at this temperature. The initial minimum distance between Cu atoms belonging to different nanoparticles (d _min ) on the three surfaces is close to or greater than our UniAlCu model cutoff of 8 Å, indicating that interparticle interactions should be negligible and motion should be primarily controlled by MSI. As the temperature increases during the heating protocol, we observe a first coalescence event on α-Al₂O₃(0001) at a temperature of around 600 K, evidenced by a drop in d _min and d _min . Other coalescence events follow, until a single Cu₁₁₇ nanoparticle forms after approximately 6 ns at 800 K. A similar behavior is observed on γ-Al₂O₃(100) but here coalescence becomes rarer with increasing nanoparticle sizes, and three nanoparticles are left on this surface at the end of 14 ns of simulation. Coalescence is even more difficult on γ-Al₂O₃(110) due to the reduced nanoparticle mobility on this surface. Therefore, six nanoparticles are present after 6 ns, and their number does not change in the remaining 8 ns of simulation. The snapshots in Figure b suggest that sintering on α-Al₂O₃(0001) and γ-Al₂O₃(100) is driven by nanoparticle diffusion while epitaxial reconfiguration promotes coalescence between row-aligned nanoparticles on γ-Al₂O₃(110). Increasing the initial separation between the particles to 20 Å and 30 Å, respectively, for d _min , that is, 15 and 20 Å for d _min , further suppresses sintering on the three surfaces, as illustrated in Figure S42. For d _min = 30 Å, coalescence is only observed on α-Al₂O₃(0001). Compared to nanoparticles on the two γ-Al₂O₃ surfaces (Figures S43 and S44), those on α-Al₂O₃(0001) (Figure S45) exhibit more pronounced distance fluctuations, underscoring the pivotal role of diffusion in the sintering process. A previous experimental study on Pt/C catalysts demonstrated that sintering through migration and coalescence can be greatly suppressed by increasing the initial distance between nanoparticles, which is consistent with the observations from our DPMD simulations for Cu nanoparticles on Al₂O₃ surfaces.

5.

(a) Nine Cu₁₃ nanoparticles are placed on the surface with an initial internanoparticle distance of about 15 Å. During the first 2 ns at 400 K, negligible nanoparticle diffusion on all three surfaces is observed. Coalescence starts to occur as the temperature is gradually increased to 800 K. All nine Cu₁₃ nanoparticles merge into a single Cu₁₁₇ nanoparticle on α-Al₂O₃(0001), while the largest nanoparticle found after 14 ns of simulation is Cu₃₉ and Cu₂₆ on γ-Al₂O₃(100) and γ-Al₂O₃(110), respectively. (b) The snapshots exhibit that the nanoparticle diffusion is pronounced on γ-Al₂O₃(100) and α-Al₂O₃(0001) while it is significantly limited on the γ-Al₂O₃(110) surface and thus coalescence only occurs between row-aligned nanoparticles by the epitaxial reconfiguration. The overall sintering becomes more difficult with the increasing nanoparticle size since the nanoparticle mobility is reduced.

In the presence of large particles with negligible mobility, the sintering process may occur by Ostwald ripening, a mechanism in which a nanoparticle grows by incorporating adatoms diffusing on the surface. Cu₁ on γ-Al₂O₃(100) exhibits essentially zero diffusion at 800 K once it is in the global minimum structure, in which it occupies the site of a surface Al^V atom. In such a case, for large enough particles, sintering would be inhibited by reduced nanoparticle diffusion and suppression of Ostwald ripening at low coverage. However, at high coverage, when most Al^V sites are occupied, Ostwald ripening would be possible. On γ-Al₂O₃(110), Cu₁ shows a diffusion comparable to that of other nanoparticles, suggesting that Ostwald ripening should be as effective as migration and coalescence. However, in general, sintering should be slow on that surface as a result of the limited diffusion. On α-Al₂O₃(0001), Cu₁ diffuses much faster than larger nanoparticles, suggesting that sintering should not be inhibited even for nanoparticles large enough so that Ostwald ripening becomes dominant. Based on our findings, the pristine γ-Al₂O₃(110) should be the best sinter-resistant support among the three surfaces investigated, when the role of surface defects, such as steps or kinks, and surface hydroxyl groups can be neglected.

Conclusions

In this work, we have developed an active learning workflow, incorporating GA search and MD sampling, to train a robust and transferable MLIP model for supported nanoparticles, which can be used to investigate the supported nanoparticle dynamics through simulations with extended time and length scales. Our UniAlCu model for Cu nanoparticles supported on three Al₂O₃ surfaces (γ-Al₂O₃(100), γ-Al₂O₃(110), and α-Al₂O₃(0001)) achieves remarkable accuracy with root-mean-square errors of 0.004 eV/atom in energy and 0.057 eV/Å in force. With careful benchmarks on the key descriptors of supported nanoparticles (E _b, E _c, MSD_Cu, and RDF), we demonstrate that our model is capable of simulating the low-energy configurations of Cu nanoparticles on three Al₂O₃ surfaces, as well as their dynamic behavior at finite temperature, in agreement with AIMD.

With force field-like efficiency and DFT-PBE-level accuracy, our simulations reveal striking differences in Cu nanoparticle stability and mobility patterns across the three Al₂O₃ surfaces, which would be unattainable with regular DFT-based structural minimizations and AIMD. According to global minimum configurations from extensive structural searches, Cu nanoparticles bind significantly more strongly on γ-Al₂O₃(110) and α-Al₂O₃(0001) than on γ-Al₂O₃(100) surfaces due to the Cu affinity of Al^III sites. Furthermore, we found the diffusion of Cu nanoparticles on the two γ-Al₂O₃ surfaces is almost independent of the size of the nanoparticle, while the diffusion on α-Al₂O₃(0001) decreases rapidly with increasing size. The Cu nanoparticles with fewer than 55 atoms on α-Al₂O₃(0001) exhibit much faster diffusion than on γ-Al₂O₃(100) at finite temperature despite stronger bindings at 0 K. The contrast between fast diffusion and strong binding is reconciled by the dynamic MSI characterized by the bond correlation functions, where the Cu–Al and Cu–O bonds in Cu nanoparticles on α-Al₂O₃(0001) relax remarkably fast. Extended MD simulations of nine Cu₁₃ nanoparticles on different Al₂O₃ surfaces reveal distinct sintering behaviors, with complete coalescence on α-Al₂O₃(0001), moderate coalescence on γ-Al₂O₃(100), and limited nanoparticle diffusion and coalescence on γ-Al₂O₃(110) within 14 ns at 800 K.

The insights gained from this work reinforce the importance of dynamics in understanding supported nanoparticle sintering mechanisms and stress that descriptors that include dynamic effects are indispensable for the screening of sinter-resistant catalysts. In the future, our active learning protocol could be applied to investigate various combinations of metals and supports as well as more intricate systems, for example, supported nanoparticles under reaction conditions, paving the way for the rational design of sinter-resistant catalysts.

Methods

Density Functional Theory

All DFT calculations are performed with the Vienna Ab initio Simulation Package (VASP), , using PBE approximation for the exchange-correlation functional. The Projector augmented wave method (PAW)-PBE pseudopotential with a cutoff energy of 400 eV was adopted to describe the core–valence interaction of the electrons. To sample the Brilluoin zone we used Γ-centered uniform k-point meshes with a spacing of 0.04 Å^–1 in each direction of the reciprocal lattice.

Machine Learning Interatomic Potential

We trained an MLIP based on the DP architecture , using the DeePMD-kit package. , The DP scheme employs neural networks to map the local atomic environment onto the atomic energy. The local atomic environment is defined by neighboring atoms within a cutoff radius of 8 Å. The “se_e2_a” descriptor is adopted to represent the atomic local environment with an embedding net of 25 × 50 × 100. The fitting net from the descriptor to the atomic energy is 240 × 240 × 240. The details of the training and testing data sets can be found in Section S1.

Global Optimization

To sample the low-energy configurations of the supported nanoparticles, we used the GA search , implemented in ASE and interfaced with the active learning loop in GDPy, which has been successfully applied to study several catalyst surface systems in previous work. − , The search is organized into generations. The structures in the initial generation are randomly generated while the ones in the following generations are constructed by crossover and mutation. The crossover operation generates a new structure by a combination of two parent structures by cut-and-splice. The mutation operation is performed on the offspring structures by applying with equal probability either a mirror reflection or a rattle position displacement. Once a structure is generated, it is allowed to relax and its fitness is estimated from its total energy. Since the GA protocol may generate several duplicate structures, we developed a coordination-number-dependent comparison method to distinguish low-energy structures, the details of which can be found in Section S2.1. Structure sampling by GA is carried out in an active learning way. Within each cycle, the scheme searches for 10 generations with a population size of 20, thus generating 200 structures, which are added to the data set. To avoid excluding from the search good guesses for the global minimum structures, we further enlarged the search with 20 generations and a population size of 50. Then, the global minimum was found within the 50 DFT-minimized low-energy structures selected from all the DP-sampled minima.

From the global minimum configurations of the supported nanoparticles, we estimate the stability of the nanoparticles with N atoms at 0 K with two descriptors, the binding energy E _b and the cohesive energy E _c, which are defined as

$$E_{\mathrm{b}}=E_{{\mathrm{Cu}}_N/\mathrm{surf}}-E_{\mathrm{surf}}-E_{{\mathrm{Cu}}_N}$$ 1

and

$$E_{\mathrm{c}}=(E_{{\mathrm{Cu}}_N/\mathrm{surf}}-E_{\mathrm{surf}}-N\times E_{{\mathrm{Cu}}_{\mathrm{FCC}}}/4)/N$$ 2

respectively, where E _{Cu N/surf} is the potential energy of a Cu nanoparticle on an Al₂O₃ surface, E _surf is the potential energy of a pristine Al₂O₃ surface, E _{Cu N} is the potential energy of the global minimum configuration of Cu _N in the gas phase, and E _CuFCC is the total energy of an FCC Cu bulk unit cell that contains 4 atoms. The cohesive energy E _c of a Cu nanoparticle in vacuo is computed from eq with E _{Cu N/surf} set equal to E _{Cu N} and E _surf set equal to zero.

Molecular Dynamics

We utilize MD to enrich the data set with structures at finite temperature and to investigate the supported nanoparticle dynamics. All simulations are in the NVT ensemble controlled by a Langevin thermostat with a friction coefficient of 10 ps^–1 if not otherwise specified. To explore configurations at finite temperature, MD sampling includes several active learning cycles, each of which includes NVT simulations at temperatures of 300, 600, 900, and 1200 K, respectively. From the trajectories, we select the most representative structures for further learning using two filters. The first is the uncertainty of the prediction of the model, a metric that has been used in many studies. The second filter is based on diversity. For each structure, a vector is generated to represent its characteristics by adopting the SOAP descriptor, which is calculated as described in DScribe. , Subsequently, a similarity kernel is built from the SOAP vectors and a matrix decomposition algorithm is used to select the most representative structures. The MD-based active learning strategy is detailed in Section S2.2.

The mobility of Cu atoms in the supported nanoparticles is measured by their mean square displacement MSD_Cu, which is defined as

$$\mathrm{MSD}(t;r)={⟨\frac{1}{N}\sum _{i=1}^N\parallel r(t+\tau )-r(\tau ){\parallel }^2⟩}_{\tau }$$ 3

where N is the number of particles (Cu atoms), r(t) is the coordinate at time step t, τ is the time origin, and ⟨···⟩_τ denotes the average over all time origins with a lagtime of 0.1 ps. The diffusion coefficient of COM, the center of mass of a nanoparticle, is calculated from

$$D_{\mathrm{COM}}^d=\frac{1}{2d}{\lim }_{t\to \infty }\frac{\mathrm{d}}{\mathrm{d}t}{\mathrm{MSD}}_{\mathrm{C}\mathrm{O}\mathrm{M}}^d$$ 4

where d is the spatial dimension, which is equal to 2 in the present work, where diffusion occurs on the surface X–Y plane, and MSD_COM is the MSD of the COM of a nanoparticle computed using eq and the number of particles N becomes 1.

To measure the dynamic MSI we use the bond correlation functions C(t) defined by

$$C(t)={⟨\frac{{⟨h_{\mathit{ij}}(t+\tau )h_{\mathit{ij}}(\tau )⟩}_{\mathit{ij}}}{{⟨h_{\mathit{ij}}(\tau )h_{\mathit{ij}}(\tau )⟩}_{\mathit{ij}}}⟩}_{\tau }$$ 5

where h _ij = 1 if a bond exists between atoms i and j and h _ij = 0 otherwise, ⟨···⟩ _ij denotes the average over all the ij-bonds present when t = τ, the time origin, and ⟨···⟩_τ denotes the average over all time origins with a lagtime of 1 ps. In this way, the Cu–Al and Cu–O bond correlation functions were studied. Here, Cu and Al were considered bonded if their distance was less than 3.0 Å, while Cu and O were considered bonded if their distance was less than 2.2 Å. These two distances are slightly larger than the typical bond distances d _Cu–Al ≈ 2.6 Å of the bulk FCC CuAl and d _Cu–O ≈ 1.8 Å of the bulk crystalline Cu₂O. Different cutoff values were tested, finding the same trends for the correlation functions (Figure S40).

In the sintering simulations, we measure the minimum distance between the COMs of a pair of nanoparticles, which is given by

$$d_{\min }^{\mathrm{COM}}=\underset{1\le I\ne J\le N}{\min }\parallel R_I^{\mathrm{COM}}-R_J^{\mathrm{COM}}\parallel$$ 6

where R _I is the COM of the Ith nanoparticle and N is the number of nanoparticles in the system. Similarly, the minimum distance between two Cu atoms is defined by

$$d_{\min }^{\mathrm{Cu}}=\underset{1\le I\ne J\le N,i\in I,j\in J}{\min }\parallel r_i-r_j\parallel$$ 7

where r _i is the coordinate of the ith Cu atom belonging to the Ith nanoparticle.

The training and validation data set, the training input file, the VASP input files, and the global minimum configurations are available at 10.5281/zenodo.15865800.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsnano.5c04622.

• Performance analysis of the three developed models; active learning workflows; structures and energetic data of supported Cu nanoparticles on three Al₂O₃ surfaces; MSD and RDF benchmarks; validations on Cu bulk and free nanoparticles; validations on adhesion energies and dispersion corrections; sintering simulations with different initial interparticle distances (PDF)

The authors declare no competing financial interest.