Extending the mean-field microkinetics for an accurate and efficient modeling of complex heterogeneous catalyst surfaces

Abstract

Mean-field microkinetic models derived from quantum-mechanical energetics are widely used to predict catalytic rates. However, they often treat different facets as independent and neglect surface crowding and diffusion of intermediates between site types, effects that matter on nanoparticles and are expensive to capture with stochastic simulations. Here we show that an extended mean-field framework can treat a catalyst nanoparticle as a single interacting system by coupling multiple site types through surface diffusion while accounting for surface crowding. Benchmarks for the platinum-catalyzed water-gas shift reaction on single-crystal surfaces, nanoparticles and platinum-ruthenium alloys reproduce reaction rates, preferred active sites, pathways and activation energies from kinetic Monte Carlo at a cost comparable to conventional mean-field models. The results also challenge the common assumption that sufficiently large particles can be represented as independent collections of single-crystal facets, even near 100 micrometers. The framework enables efficient microkinetic simulations and high-throughput catalyst screening under working conditions.

The study presents a fast model that predicts catalyst nanoparticle performance while accounting for surface crowding and diffusion between facets. It matches detailed simulations at far lower cost, enabling rapid catalyst screening

Introduction

The discovery of new catalytic materials lies at the heart of heterogeneous catalysis, which accounts for 80% of industrial catalytic processes¹. Advances in materials synthesis, surface characterization, and theoretical simulation have offered many fundamental insights into the reaction mechanisms on the well-defined catalyst surfaces, promoting catalyst discovery from a blind ‘trial-and-error’ practice to a process of rational design². In particular, theoretical simulations, which provide an efficient way to correlate the electronic/geometrical structures of a material with its physical/chemical properties, have become an indispensable tool in the research of heterogeneous catalysts in recent years.

In theoretical treatments, supported metal catalysts are typically regarded as crystalline particles that combine independent single crystal facets, with the total activity of the catalyst represented as a weighted sum of ensembles of constituent facets and the activity determined by the most active facet³. Consequently, the investigation of complex supported metal catalysts can be reduced to a simple process of searching for the most active facet of metal crystals. The calculation of activity often comprises two stages. Initially, the energetic parameters for each elementary reaction are calculated. Subsequently, the microkinetic modeling (MKM), which is constructed on top of these energetic parameters, is solved. The former stage is typically conducted using a density functional theory (DFT) method within the generalized gradient approximation (GGA), while the latter often employs reaction rate equations (RRE) under the mean-field (MF) approximation⁴. Due to its efficiency, simplicity and interpretability, the combination of GGA-MF-MKM has become the most popular theoretical tool for revealing trends in catalytic activity of metals^3,5.

Despite its notable achievements, the precision of GGA-MF-MKM is sometimes insufficient for an accurate prediction of thermodynamic and kinetic properties. In particular, the absence of lateral interactions between adsorbates and the absence of kinetic coupling between different active sites represent two significant sources of error, which can result in qualitative errors in some important cases^6–10. While high-fidelity MKM methods, such as kinetic Monte Carlo (KMC), which incorporate explicit lattice information can, in principle, address these limitations^11–17, they are often infeasible from a computational perspective due to the high complexity of an explicit lattice simulation. Therefore, there is a clear need, such as in the cases of rational design and fast screening of catalysts under ‘operando’ conditions, for efficient yet accurate models that can provide qualitatively correct predictions without significantly increasing the computational complexity.

In order to enhance the performance of conventional MF-MKM, a considerable amount of effort has been devoted to the inclusion of lateral interactions among adsorbates within the MF framework^18–21. The MF approaches assume that the adsorbates are well-mixed, resulting in an effective field that is described by a set of averaged quantities, i.e., the coverages of the adsorbed species. This field interacts with each of the adsorbates individually, which makes their adsorption energies coverage-dependent. Consequently, the RRE is also coverage-dependent, and thus should be solved self-consistently²². In the interests of efficiency, however, the coverage effect on the energetic parameters is often estimated by assuming a coverage for the most abundant surface intermediate (MASI)^23,24.

Compared to the absence of lateral interactions, the absence of inter-site coupling is a considerably more complex issue in MF-MKM. In conventional MF-MKM treatments, a catalyst nanoparticle is often approximated as an ensemble of independent single-crystal facets, and the particle-level kinetics is obtained by combining the facet-resolved contributions^3,25. It is well-recognized that supported metal nanoparticles expose a variety of active-site motifs in distinct chemical environments (e.g., terraces, steps/kinks, edges/corners, and alloy- or defect-induced sites)^26,27. Moreover, the low diffusion barriers permit rapid movement of adsorbates across metallic particles, thereby enabling them to select the preferred active sites for different chemical transformations⁸. Consequently, inter-site kinetic coupling can introduce new reaction pathways that intersect the energetic profiles of disparate active sites. Such coupling is an important aspect of the synergy effect that should be accounted for in simulations of nanocatalysts⁸. However, the interconnection of reaction networks on different active sites significantly increases the complexity of RRE, particularly when the lateral interactions are present. Therefore, its application on realistic systems is highly challenging, although some initial attempts have been encouraged for simple model systems^28–30. However, to the best of our knowledge, there is still a lack of a general, scalable workflow for realistic nanoparticles/alloys that (a) systematically constructs morphology-dependent coupling information (e.g., site counts, inter-site pair degeneracies, and neighbor statistics) and (b) reduces the resulting multisite MF-MKM system to a form that remains computationally tractable for nanoparticle-scale and high-throughput applications.

In this work, we present an extended MF-MKM approach (hereafter referred to as the XMF approach), designed to describe complex catalyst nanoparticles accurately, yet efficiently. XMF treats a nanoparticle as an integrated, interacting entity, explicitly accounting for both lateral interactions among adsorbates and diffusion-mediated kinetic coupling among different sites. To achieve high efficiency without sacrificing the key physics, XMF adopts two approximations that are routinely invoked in the Langmuir-Hinshelwood-Hougen-Watson (LHHW) kinetic formalism⁵, whose rate expressions have been widely successful in rationalizing and predicting catalytic kinetics^31–34. First, motivated by the LHHW rationale that catalyst kinetics are largely governed by the MASI coverage, we assume that the coverage dependence of energetic parameters is dominated by the MASI. This avoids exhaustive DFT calculations of all pairwise interactions, a simplification routinely adopted in both coverage-dependent MF and KMC treatments. Second, we assume that the MASI is in quasi-equilibrium with its corresponding gas-phase reactant, consistent with the LHHW rationale that elementary steps preceding the rate-determining step can often be treated as near-equilibrated^35,36. These assumptions pave the way for the application of the coverage-dependent MF approach to complex heterogeneous surfaces. Three levels of methods have been developed within the XMF approach, according to their different coupling treatments. The accuracy and efficiency of all XMF variants have been examined in the Pt-catalyzed water-gas shift (WGS) reaction, demonstrating their applicability to realistic systems such as nanocatalysts and alloys.

Results and discussion

The XMF approach

In XMF, a nanoparticle is represented by an explicit site-connectivity graph (neighbor statistics between site types), which couples the site-type coverages and energetic parameters in a connectivity-aware manner. Figure 1a summarizes the four steps involved in the application of levels of methods in the XMF approach.

Fig. 1. XMF workflow and nanoparticle site-connectivity representation.

a Schematic diagram of the XMF approach that involves 4 steps. b Delineation of active sites on a Pt₄₀₄ nanoparticle and the connectivity between these sites. Here, t0 and t1 correspond to the terrace sites on the surfaces of Pt(100) and Pt(111), respectively; te corresponds to the edge sites on the Pt(211) surface; and tc corresponds to the corner sites. The t0, t1, te and tc sites are colored in cyan, blue, pink and purple, respectively. The average numbers for the first nearest neighboring (1NN) sites and the second nearest neighboring (2NN) sites of each type of active sites are shown in the respective pie charts.

(1) The surface sites of a catalyst nanoparticle are classified into site types according to their local chemical environments. The MASI coverages on all site types are then determined simultaneously from a single connectivity-aware mean-field description of the entire particle. Here, we take the adsorption of CO, an important MASI in C₁ chemistry³⁷, on a Pt₄₀₄ particle that contains 404 surface atoms as an example to illustrate the procedure. As shown in Fig. 1b, the active sites on the Pt₄₀₄ particle are divided into four types: t0, t1, te, and tc. While t0 and t1 correspond to the terrace sites on the surfaces of Pt(100) and Pt(111), respectively, te and tc refer to the edge sites on the Pt(211) surface, and the corner sites, respectively. The CO adsorption energy on each type of the active sites and the CO-CO interaction energy are then calculated using the corresponding periodic model (for a more detailed description of the modeling and calculations, see Supplementary Note 1 of the Supplementary Information (SI)).

For CO adsorbed on site A of α-type, the adsorption can be considered as a two-state system, where the CO molecule can be either adsorbed (*) or in the gas phase (g). At equilibrium, the occupancy probability of this two-state system is given by the Fermi-Dirac (F-D) distribution²⁸, which therefore represents the equilibrium distribution between the adsorbed and gas-phase states,

$${\theta }_{CO-A^{*}}^{F-D}=\frac{\exp \left(-\frac{\mathrm{\Delta }{G}_{CO-A^{*}}}{k_{B}T}\right)}{1+\exp \left(-\frac{\mathrm{\Delta }{G}_{CO-A^{*}}}{k_{B}T}\right)}$$ 1

Here $k_B$ is the Boltzmann constant, T is the temperature, and ${\theta }_{CO-A^{*}}^{F-D}$ is the CO coverage calculated from the Fermi-Dirac distribution, and $\mathrm{\Delta }{G}_{CO-A^{*}}={\mu }_{CO-{\alpha }^{*}}-{\mu }_{CO\left(g\right)}-{\mu }_{{\alpha }^{*}}$ is the free energy difference for the CO adsorption on site A. For simplicity, we assume that the energetic influence on the CO adsorption on site A is linearly dependent on the CO coverage at its neighboring site B. Thus, the coverage-dependent CO adsorption energy at site A can be evaluated by accumulating the effects of all its neighboring sites,

$$\mathrm{\Delta }{\mathrm{G}}_{CO-A^{*}}=\mathrm{\Delta }{G}_{CO-{\alpha }^{*}}^0+{\sum }_B{\omega }_{CO-A^{*}}^{CO-B^{*}}{\theta }_{CO-B^{*}}^{F-D},$$ 2

where ${\omega }_{CO-A^{*}}^{CO-B^{*}}$ is the repulsive interaction and $\mathrm{\Delta }{G}_{CO-{\alpha }^{*}}^0$ is the CO adsorption free energy on α-type sites at zero coverage. As illustrated in Fig. 1b, different sites of an identical type can be located in different chemical environments. For example, on the (111)-facet colored in blue, the outer t1 sites have both te and t1 neighbors, whereas the inner t1 sites have only t1 neighbors. Due to different interaction effects from neighboring sites, the CO coverage at the outer and inner t1 sites differs, despite the identical adsorption energies. In the MF treatment, it was proposed that the influences of neighbors on the same site type can be averaged²⁸. Consequently, we also assume the CO coverage belonging to the same site type to be the same. Equation 2 is then simplified as,

$$\mathrm{\Delta }{\mathrm{G}}_{CO-{\alpha }^{*}}=\mathrm{\Delta }{G}_{CO-{\alpha }^{*}}^0+{\sum }_{\beta }\overline{N_{\alpha \beta }}{\omega }_{CO-{\alpha }^{*}}^{CO-{\beta }^{*}}{\theta }_{CO-{\beta }^{*}}^{F-D},$$ 3

where the subscript β covers all types of sites and $\overline{N_{\alpha \beta }}$ is the average number of β-type neighboring sites defined as,

$$\overline{N_{\alpha \beta }}=\frac{{\sum }_C{\delta }_{\beta C}}{N_{\alpha }}$$ 4

Here $N_{\alpha }$ is the total number of α-type sites, $\delta$ is the Kronecker $\delta$ function, and the subscript C covers all neighboring sites of all α-type sites. The corresponding average number of neighbors for each site type is shown in pie charts in Fig. 1b. For example, t1-type sites have on average 2.40 te neighbors and 3.60 t1 neighbors in the first nearest neighboring (1NN) shell, while they have 0.30 tc neighbors, 2.70 te neighbors, and 1.50 t1 neighbors in the second nearest neighboring (2NN) shell. By substituting Eq. 3 into Eq. 1, we obtain four coupled equations for the CO coverage on these four site types, and then the CO coverage on each site type can be solved numerically. Consequently, the quasi-equilibrium CO coverage on each site type, as determined in this manner by the connectivity, corresponds to the particle-level self-consistent solution from a single coupled (nonlinear) solve across all site types, without requiring an outer self-consistent iteration over the MASI coverage (as in coverage-dependent MF-SCF treatments²²), rather than the value predicted by conventional MF-MKM for each isolated facet (see Supplementary Note 2 in the SI for details).

(2) Coverage-dependent energetic parameters for the whole reaction network on different types of sites are calculated according to the CO coverage determined in the first step. Then the forward and backward reaction rates are evaluated for each elementary reaction. Similar to the case of CO-CO interaction, the effect of CO coverage on the surface intermediate I adsorbed on sites of α-type can be estimated by,

$$\mathrm{\Delta }{\mathrm{G}}_{I-{\alpha }^{*}}=\mathrm{\Delta }{G}_{I-{\alpha }^{*}}^0+{\sum }_{\beta }\overline{N_{\alpha \beta }}{\omega }_{I-{\alpha }^{*}}^{CO-{\beta }^{*}}{\theta }_{CO-{\beta }^{*}}^{F-D},$$ 5

where $\mathrm{\Delta }{G}_{I-{\alpha }^{*}}$ corresponds to the coverage-dependent adsorption free energy of intermediate I on sites of α-type, while $\mathrm{\Delta }{G}_{I-{\alpha }^{*}}^0$ refers to the corresponding value at the limit of zero CO coverage. Here in Eq. 5, ${\omega }_{I-{\alpha }^{*}}^{CO-{\beta }^{*}}$ is the repulsion between intermediate I and CO adsorbed on sites of α- and β-types, respectively.

Once all the energetic parameters are known, the reaction rate constants can be calculated via transition state theory (TST)^38–40. For a single-site reaction, i, (e.g., the desorption of CO) occurring on the site of α-type, the forward reaction rate ${\vec{r}}_i\left(t\right)$ is calculated by,

$${\vec{r}}_i\left(t\right)=k_i{\theta }_{CO-{\alpha }^{*}}\left(t\right).$$ 6

The reaction rate constant $k_i$ in Eq. 6 is given by the Arrhenius equation,

$$k_i=A_i{e}^{\frac{-\mathrm{\Delta }{\mathrm{G}}_{CO-{\alpha }^{*}}}{RT}},$$ 7

where $A_i$ is the pre-exponential factor. For a double-site reaction, j, whose reactants are R₁ and R₂ adsorbed on a pair of neighboring sites A and B belonging to types α and β, the forward reaction rate ${\vec{r}}_j\left(t\right)$ is given by,

$${\vec{r}}_j\left(t\right)=k_j\overline{N_{\alpha \beta }}{\theta }_{R_1-{\alpha }^{*}}\left(t\right){\theta }_{R_2-{\beta }^{*}}\left(t\right).$$ 8

The reaction rate constant $k_j$ in Eq. 8 is evaluated by TST,

$$k_j=A_j{e}^{\frac{-\mathrm{\Delta }{G}_j^a}{RT}},$$ 9

where $\mathrm{\Delta }{G}_j^a$ is the energy barrier of the forward reaction, given by,

$$\mathrm{\Delta }{G}_j^a=\mathrm{\Delta }{\mathrm{G}}_{TS-{\alpha \beta }^{*}}-\mathrm{\Delta }{\mathrm{G}}_{R_1-{\alpha }^{*}}-\mathrm{\Delta }{\mathrm{G}}_{R_2-{\beta }^{*}},$$ 10

where $\mathrm{\Delta }{G}_{TS-{\alpha }^{*}{\beta }^{*}}$ is the free adsorption energy of the transition state on the pair of A and B sites.

(3) We now construct the RRE network, introducing further simplifications through additional approximations. As discussed above, on a heterogeneous surface, both the electronic structure and geometric symmetry of a site affect kinetic properties such as the coverage of an intermediate and the reaction rate constants. The conventional MF approximation assumes that all sites with identical energetic parameters at the zero-coverage limit are kinetically equivalent. This reduces the RRE network from one equation per individual site to one equation per each site type. Assuming the surface contains m distinct site types and n possible surface adsorbates, the total number of independent adsorbate coverages to be solved is $m\times n$, leading to $m\times n$ coupled ordinary differential equations (ODEs). The coverage of empty sites is determined by the normalization condition, which introduces m algebraic constraints. Specifically, for each intermediate I adsorbed on the α-type sites, the ODE describing its time evolution is given by:

$$\frac{d{\theta }_{I-{\alpha }^{*}}\left(t\right)}{dt}={\sum }_j{v}_j^{I-{\alpha }^{*}}\left({\vec{r}}_j\left(t\right)-{\stackrel{⃖}{r}}_j\left(t\right)\right),$$ 11

where ${\theta }_{I-{\alpha }^{*}}\left(t\right)$ denotes the corresponding coverage, $v_j^{I-{\alpha }^{*}}$ is the stoichiometric number, ${\vec{r}}_j\left(t\right)$ and ${\stackrel{⃖}{r}}_j\left(t\right)$ are the forward and backward reaction rates, respectively, for the j-th elementary reaction. These ODEs are coupled through dual-site reactions and inter-site diffusion of adsorbates. The coverage of empty sites of the α-type is determined by,

$${\theta }_{*-{\alpha }^{*}}\left(t\right)=1-{\sum }_I{\theta }_{I-{\alpha }^{*}}\left(t\right),$$ 12

where the sum runs over all possible adsorbates I. This formulation, comprising $m\times n$ ODEs (Eq. 11) and m algebraic constraints (Eq. 12), is hereafter referred to as the Level 1 method, denoted XMF-1. Clearly, the computational complexity of the XMF-1 method grows rapidly with increasing numbers of site types or elementary steps, posing challenges for numerical integration.

It is well established that diffusion of surface adsorbates is often much faster than chemical reactions. We therefore adopt the “fast-diffusion limit”, further assuming that all adsorbates equilibrate across site types before any net chemical reaction takes place^15,41. This assumption leads to $\left(m-1\right)\times n$ equilibrium conditions of the form:

$$I_{{\alpha }^{*}}+{*}_{{\beta }^{*}}\rightleftharpoons {*}_{{\alpha }^{*}}+I_{{\beta }^{*}},$$ 13

where $I_{{\alpha }^{*}}$ and ${*}_{{\alpha }^{*}}$ denote, respectively, an α-type site occupied by adsorbate I or vacant. Equation 13 results in $\left(m-1\right)\times n$ additional algebraic relations,

$$\frac{{\theta }_{*-{\alpha }^{*}}\left(t\right){\theta }_{I-{\beta }^{*}}\left(t\right)}{{\theta }_{I-{\alpha }^{*}}\left(t\right){\theta }_{*-{\beta }^{*}}\left(t\right)}=K_{\alpha \beta },$$ 14

where $K_{\alpha \beta }$ is the equilibrium constant determined by the enthalpic contribution associated with the diffusion process described by Eq. 13. As shown in Eq. 14, for a given adsorbate I, its coverage on any site type can be expressed in terms of its coverage on a particular type of sites, e.g., the α-type site, and the corresponding empty-site coverage. Therefore, only n independent adsorbate coverages remain, and the number of ODEs is reduced to n. Here, we chose the global coverage of each adsorbate as the independent variable:

$${\theta }_I^{global}\left(t\right)=\frac{{\sum }_{\alpha }{N}_{\alpha }{\theta }_{I-{\alpha }^{*}}\left(t\right)}{N_s},$$ 15

where $N_s$ is the total number of surface sites and the subscript $\alpha$ covers all possible site types. The RRE of the global coverage can then be expressed as a weighted sum of the site-specific equations:

$$\frac{d{\theta }_I^{global}}{dt}={\sum }_{\alpha }\frac{N_{\alpha }}{N_s}\frac{d{\theta }_{I-{\alpha }^{*}}\left(t\right)}{dt}={\sum }_{\alpha }\frac{N_{\alpha }}{N_s}{\sum }_j{v}_j^{I-{\alpha }^{*}}\left({\vec{r}}_j\left(t\right)-{\stackrel{⃖}{r}}_j\left(t\right)\right).$$ 16

This Level 2 method, denoted XMF-2, thus contains n ODEs (Eq. 16) and $m\times \left(n+1\right)-n$ algebraic equations (Eqs. 14 and 12), offering a significant reduction in complexity compared to XMF-1.

In the Level 3 method, i.e., XMF-3, we introduce a further approximation: the ratio of the empty-site coverages across different site types is governed solely by the MASI:

$$\frac{{\theta }_{*-{\alpha }^{*}}\left(t\right)}{{\theta }_{*-{\beta }^{*}}\left(t\right)}\approx \frac{1-{\theta }_{CO-{\alpha }^{*}}^{F-D}}{1-{\theta }_{CO-{\beta }^{*}}^{F-D}}.$$ 17

Here we use the CO coverages calculated by the Fermi-Dirac statistics (${\theta }_{CO-{\alpha }^{*}}^{F-D}$ and ${\theta }_{CO-{\beta }^{*}}^{F-D}$), rather than the full time-dependent variables (${\theta }_{CO-{\alpha }^{*}}\left(t\right)$ and ${\theta }_{CO-{\beta }^{*}}\left(t\right)$). Substituting Eq. 17 into Eq. 14 yields, for any adsorbate I:

$$\frac{{\theta }_{I-{\beta }^{*}}\left(t\right)}{{\theta }_{I-{\alpha }^{*}}\left(t\right)}=K_{\alpha \beta }\frac{1-{\theta }_{CO-{\beta }^{*}}^{F-D}}{1-{\theta }_{CO-{\alpha }^{*}}^{F-D}},$$ 18

which is a time-independent proportionality constant. This relation allows elimination of all but one coverage variable per adsorbate from the kinetic network. As a result, the RRE system in XMF-3 reduces to just n ODEs (for the global coverage, Eq. 16) and a single algebraic normalization equation (Eq. 12), matching the computational cost of conventional MF-MKM on a homogeneous periodic surface. We emphasize here that, on one hand, XMF-2 and XMF-3 adopt the “fast-diffusion limit” assumption to eliminate explicit diffusion terms and reduce model complexity; on the other hand, they enforce diffusion-equilibrium constraints that couple coverages across different site types, thereby effectively retaining inter-site coupling.

(4) The RREs are ultimately solved numerically and the kinetic properties are calculated accordingly. When only the steady state of the reaction system is of interest, where the coverage of all adsorbates becomes time-invariant,

$$\frac{d{\theta }_I^{global}\left(t\right)}{dt}=0,$$ 19

the ODEs are reduced to a system of algebraic equations. In the XMF approach, coverage-dependent energetic parameters are evaluated based on the MASI coverage predicted from its quasi-equilibrium relation with the gas phase obtained from Step (1), and are then held fixed during the subsequent kinetic solve. The steady-state RREs therefore form a fixed-parameter nonlinear algebraic system, which can be solved efficiently using standard scientific computing libraries (e.g., SciPy⁴²). To enforce the site-normalization constraints, however, the coverages of all adsorbates, including the MASI, are retained as unknown variables in the algebraic system and are determined by the nonlinear root finder.

After the nonlinear solver converges for this fixed-parameter problem, we perform a posteriori consistency check of the quasi-equilibrium MASI closure by comparing the MASI coverage obtained from the RRE solution with the quasi-equilibrium MASI coverage used to parameterize the energetic terms. In most cases, the two values agree closely, indicating that the quasi-equilibrium MASI assumption is self-consistent under the given conditions. If a substantial mismatch is observed, this signals that the quasi-equilibrium MASI assumption may no longer be justified, and the frozen-parameter XMF approximation can lose accuracy. In such cases, one may refine the calculation by introducing an outer self-consistent parameter-update loop (MF‑SCF style²², see Supplementary Note 2 in the SI), i.e., iterating between updating the MASI-dependent energetic parameters and re-solving the RREs until the input and output MASI coverages agree within a chosen tolerance, or alternatively resort to a higher-level method such as KMC.

Analogous to simulations on homogeneous periodic surfaces, all three levels of the XMF methods enable kinetic analyses under varying reaction conditions to compute key observables such as apparent activation energies and reaction orders. Furthermore, advanced diagnostic tools—including degree of rate control (DRC) and degree of thermodynamic rate control (DTRC) analyses^43–45, can also be performed. For detailed demonstrations of the hierarchical XMF methods in practice, we refer the reader to Supplementary Note 2 in the SI.

We conclude that the key contribution of XMF is to introduce a morphology-derived site-connectivity closure (neighbor statistics) for the MASI-driven interaction field and to provide a hierarchy of diffusion-equilibrium reductions (XMF-1/2/3) that preserves inter-site coupling while reducing the number of dynamical variables from O(m × n) to O(n), thereby enabling nanoparticle-scale and high-throughput applications, as shown below.

Applications to single crystal surfaces

As a natural extension of the conventional MF methods, the XMF approach can be readily applied to periodic single crystal surfaces. In these simple cases, there is only one type of sites, so all XMF methods reduce to XMF-3, where we only need to calculate the CO (i.e., the MASI) coverage via the Fermi-Dirac distribution before solving the RREs. The performance of XMF-3 was evaluated by simulating the WGS reaction on three prototypical Pt surfaces: Pt(111), Pt(100), and Pt(211).

Figure 2 summarizes the performance of the XMF-3 method in calculating the reaction rate and its temperature variation, i.e., the apparent activation energy. The results obtained from the KMC simulations are taken as the reference, while the results of the conventional MF method (which neglects the adsorbate-adsorbate interactions) are included for comparison.

Fig. 2. Benchmarking extended and conventional mean-field microkinetic models against kinetic Monte Carlo on Pt single-crystal surfaces.

Arrhenius plots for the WGS reaction on (a) Pt(111), b Pt(100), and c Pt(211) surfaces. CO coverages calculated using different methods for (d) Pt(111), e Pt(100), and f Pt(211) surfaces. DRC/DTRC analysis results for the elementary steps of CO adsorption, H₂O dissociative adsorption, COOH* formation and COOH* decomposition on the Pt(211) surface at (g) 525 K and h 675 K. Source data are provided as a Source Data file.

Figure 2a–c show the turnover frequencies (TOFs), which correspond to the reaction rates per site, for the Pt(111), Pt(100), and Pt(211) surfaces, respectively. Although the XMF-3 method tends to overestimate the TOFs, in all cases it provides comparable results to the reference values from the KMC simulations. In contrast, the conventional MF method performs poorly in predicting the TOFs, particularly at low temperatures. In the calculations of the apparent activation energy, the XMF-3 method yields values of 120, 136, and 81 kJ/mol for the Pt(111), Pt(100), and Pt(211) surfaces, respectively. These are in close agreement with the KMC simulation results, being 101, 118, and 83 kJ/mol for the Pt(111), Pt(100), and Pt(211) surfaces, respectively. The conventional MF method, however, fails completely in predicting the apparent activation energies, giving the corresponding values of 356, 414 and 371 kJ/mol, due to its unbalanced description of the high and low temperature regions. In addition, both the XMF-3 and KMC methods lead to the conclusion that the Pt(211) surface with the lowest apparent activation energy is the most active site, whereas the conventional MF method predicts that the Pt(111) surface is the most active site.

Figure 2d–f summarize the surface coverages of CO, the MASI in the WGS reaction, on Pt(111), Pt(100), and Pt(211) surfaces, respectively, to help understand the performance of the different methods. The KMC simulations show that the CO coverages remain below 0.66 in all cases in the temperature range from 525 K to 675 K. In the XMF-3 simulations, the CO coverages calculated by the Fermi-Dirac distribution (denoted as F-D in Fig. 2d–f) closely match those obtained in kinetic simulations, verifying the validity of the quasi-equilibrium assumption for CO adsorption/desorption. The CO coverages in the XMF-3 simulations differ slightly from the KMC results for the Pt(100) surface, while the values on Pt(111) and Pt(211) surfaces match well the KMC results. In contrast, the conventional MF method has a strong tendency to overestimate the CO coverages, leading to values close to 1.00 in most cases due to the absence of adsorbate-adsorbate interactions. This unphysical surface poisoning would have indicated that Pt surfaces are completely inactive under WGS reaction conditions, in contrast to the KMC reference.

The adsorbate-adsorbate interactions also modulate the reaction mechanism. We analyzed these influences on four elementary steps, including CO adsorption (Equation 20), H₂O dissociative adsorption (Eq. 21), COOH* formation (Eq. 22), and COOH* decomposition (Eq. 23), to locate the key step for the activity enhancement on the most active site of the Pt(211) surface. The DRC analysis was performed to analyze the influence of the transition state energies for the latter three steps, while the DTRC analysis was applied to analyze the influence of the reaction enthalpy for the CO adsorption step. The results are plotted in Fig. 2g, h. According to the KMC results, the RDS is H₂O decomposition at 525 K, which shifts to COOH* decomposition as the temperature increases to 675 K. The increase in CO adsorption energy apparently inhibits the reaction at 525 K when the surface becomes poisoned, but this effect diminishes at 675 K when the surface becomes dynamic and remains active. The XMF-3 method captures this behavior qualitatively, although it slightly underestimates the effect of the CO adsorption energy due to the lack of explicit spatial correlations^6,7. In contrast, the conventional MF method, lacking any lateral interactions between adsorbates, significantly overestimates the CO poisoning effect and incorrectly suggests that the adsorption energy of CO controls the reaction rate at both 525 K and 675 K, leading to qualitative errors in identifying the key enhancement step for activity prediction. Similar discrepancies are found for describing WGS on the Pt(111) surface (see Supplementary Note 2 in the SI for details).

In the calculations on single crystal surfaces, XMF-3 faithfully reproduces the KMC predictions in active sites, CO coverages, and RDS evolution, whereas the conventional MF method introduces qualitative errors that could misdirect catalyst optimization efforts. This underscores the critical importance of incorporating adsorbate-adsorbate interactions in microkinetic modeling. Moreover, XMF-3 proves to be a computationally efficient alternative to self-consistent coverage-dependent MF methods^46,47, as demonstrated by the identical results produced for both methods when applied to these single crystal surfaces. A detailed discussion can be found in Supplementary Note 2 of the SI.

Application to a simple dual-site system

On metal surfaces, diffusion barriers are typically low, enabling adsorbates to migrate rapidly across nanoparticles and access the most favorable active sites for specific elementary steps. This kinetic coupling between distinct site types creates new reaction pathways that effectively bridge the energetic landscapes of different surface facets, a key manifestation of the synergistic effects commonly observed in nanocatalysis. Accurate modeling of such systems therefore requires explicit treatment of inter-site communication⁸. However, conventional MF treatment often views a metal particle as a mere superposition of independent single crystal surfaces³. This “decoupled” view precludes any redistribution of reaction fluxes across sites and thus fails to capture kinetic coupling. In contrast, the XMF framework explicitly allows reaction pathways to shift dynamically among site types, making it well-suited to model such cooperative effects. To demonstrate this capability, we constructed a minimal dual-site model consisting of the Pt(111) and Pt(211) surfaces and investigated the mechanism of the WGS reaction on this dual-site composite surface. Results are shown in Fig. 3.

Fig. 3. A simplified Pt(111)/Pt(211) dual-site model illustrating diffusion-mediated inter-site coupling.

a Arrhenius plot on the dual-site system consisting of Pt(111) and Pt(211) surfaces. b DRC analysis of the RDS for KMC and the XMF methods at 675 K. CO coverages on (c) the Pt(111) site and d the Pt(211) site of the dual-site system. e Site preference for different elementary steps. f The major inter-facet reaction pathway on the dual-site system. Source data are provided as a Source Data file.

Figure 3a presents Arrhenius plots for the dual-site system obtained using various methods. The KMC simulations demonstrate that kinetic coupling has a significant impact on the kinetic properties of the dual-site system, increasing the TOF and changing the apparent activation energy compared to the Pt(111) or Pt(211) surface alone. The XMF approach at all three levels yields identical predictions for both TOF and the apparent activation energy. While the XMF methods systematically overestimate the reaction rates by approximately a factor of two relative to KMC, this bias has little impact on the apparent activation energy, which is predicted as 111, 111 and 110 kJ/mol for XMF-1, XMF-2 and XMF-3, respectively, closely matching the KMC reference value of 103 kJ/mol. Critically, all three XMF methods correctly identify the dual-site system as more active than either constituent facet alone, consistent with KMC.

In stark contrast, the conventional “MF-SUM” approach, which simply sums the rates from independent Pt(111) and Pt(211) surfaces, assumes no inter-site communication. Consequently, the more active Pt(211) sites dominate the overall kinetics. At low temperatures, where coupling is weak, MF-SUM yields TOFs comparable to XMF. However, at higher temperatures—where kinetic coupling becomes significant—MF-SUM severely underestimates the TOF. The apparent agreement between MF-SUM and KMC at high temperatures is fortuitous and morphology-dependent, arising from error cancellation in this specific model; it is not expected to generalize across different nanoparticle shapes or compositions. Moreover, by neglecting coupling, MF-SUM incorrectly reduces the apparent activation energy to 81 kJ/mol, the value for isolated Pt(211), failing to reproduce the elevated barrier observed in both KMC and XMF.

Kinetic coupling also alters the RDS. Given that Pt(211) exhibits a much higher intrinsic TOF than Pt(111) (see Fig. 2a, c), a decoupled model would predict the RDS of the dual-site system to mirror that of Pt(211). However, according to the DRC analysis shown in Fig. 3b, the RDS of the coupled dual-site system is actually H₂O dissociative adsorption at 675 K, which differs from the RDS of the Pt(211) surface, i.e., COOH* decomposition. Although XMF slightly overestimates the degree of rate control for this step compared to KMC, it correctly captures the qualitative shift in the RDS, underscoring its ability to model emergent mechanistic behavior.

Figure 3c, d compare CO coverages on the dual-site system. KMC shows that CO coverages on Pt(111) and Pt(211) sites within the coupled system closely match those on the corresponding isolated single crystals. Similarly, all XMF variants, and even the simple Fermi-Dirac estimate, predict nearly identical coverages, corroborating the KMC observation. This consistency implies that the activity enhancement stems not from modified intrinsic site properties, but from the emergence of a new reaction channel enabled by kinetic coupling.

Figure 3e presents the site preference for the H₂O decomposition and COOH* decomposition steps. The data show that the maximum net reaction rate of H₂O decomposition occurs on two neighboring Pt(211) sites (labelled as te*te*), while COOH* decomposition occurs mainly on two neighboring Pt(111) sites (labelled as t1*t1*). Based on these observations, it can be concluded that the most favorable reaction pathway for the dual-site system involves both Pt(211) and Pt(111) sites. As illustrated in Fig. 3f, H₂O first dissociates on the Pt(211) sites to form OH*, which then reacts with the adsorbed CO* on the boundary of Pt(111)/Pt(211) to form the COOH* species. This intermediate subsequently diffuses to the Pt(111) sites, where it decomposes to CO₂ and H*. Finally, H* recombines and desorbs as H₂ from any available site.

The dual-site model, though simplified, serves as a paradigmatic example of how kinetic coupling between distinct active sites can reshape the reaction network. The analysis of this model system demonstrates that kinetic coupling can alter fundamental kinetics regarding the reaction pathway, which, in turn, changes properties such as the RDS, the TOF, and the apparent activation energy. These properties are central to rational catalyst design, and their accurate prediction demands theoretical frameworks that account for inter-site communication. The XMF methods successfully capture these cooperative effects and reproduce the trends of KMC far more faithfully than conventional MF approaches, where kinetic coupling is disregarded.

Applications to nanoparticles and alloys

The accuracy and efficiency of the XMF approach were further evaluated by applying it to complex models of nanoparticles and alloys, which more closely resemble realistic catalysts. Figure 4a compares the apparent activation energies for Pt nanoparticles of varying size and shape (see Supplementary Fig. 8) as predicted by both XMF and KMC simulations. While particle size and shape influence the computed TOFs, they have little effect on apparent activation energies. For instance, the TOF of the Pt₅₂₃₄ nanoparticle is 3.47 times higher than that of Pt₁₆₃₈, yet their apparent activation energies are nearly identical, being 108 and 112 kJ/mol, respectively. This behavior arises because changes in size and shape primarily alter the relative abundance of edge sites, which modulate the net rate of the RDS and thus the TOF. However, since this effect is relatively consistent across different temperatures and, therefore, it does not alter the temperature dependence of the TOF, i.e., the apparent activation energy. Consistent with observations for the dual-site model, the predicted apparent activation energies for Pt nanoparticles are ~110 kJ/mol, in excellent agreement with the experimental value of 107 ± 6 kJ/mol⁴⁸. This is notably higher than the value of 81 kJ/mol observed for the isolated Pt(211) surface. Both KMC and XMF simulations reproduce this trend, whereas the conventional MF-SUM treatment, which neglects kinetic coupling, fails to predict the correct apparent activation energy.

Fig. 4. XMF application to nanoparticles and alloys.

a The apparent activation energies for Pt nanoparticles of varying sizes and shapes. The experimental measurements are represented by the gray shaded area. b DRC analysis of the WGS reaction on the Pt₄₀₄ nanoparticle at 675 K. c Ratios of the Pt(211) (te) and Pt(111) (t1) sites for Wulff-structured particles. Size-dependence of the TOF for Wulff-structured Pt nanoparticles at (d) 525 K and e 675 K. The dashed lines indicate the TOF for the Pt(211) and the Pt(111) crystalline surfaces. f The activities of the Pt-Ru alloys. Source data are provided as a Source Data file.

Figure 4b presents a DRC analysis for the Pt₄₀₄ nanoparticle at 675 K. The RDS is identified as H₂O dissociation occurring at Pt(211)-type edge sites (te) and their neighboring sites. Importantly, adjacent Pt(111) (t1) and Pt(100) (t0) terrace sites also contribute significantly to the net rate of this step. This occurs because the activation barrier for H₂O dissociation is governed almost entirely by the adsorption site of H₂O /OH* (i.e., the te site), while the identity of the neighboring site, whether empty or occupied by H*, has negligible influence on the barrier. Consequently, each te site effectively “activates” all its neighbors for H₂O decomposition.

As the diameter of nanoparticles increases, the number of terrace sites increases much more rapidly than the number of edge sites. Therefore, it is commonly assumed that terrace sites dominate the catalytic activity in sufficiently large particles. This assumption is particularly relevant in the study of nanocrystals with well-controlled shapes that expose a specific facet. To evaluate the validity of this assumption, we analyzed the size dependence of the TOF for Pt particles with a Wulff structure. For particles with a Wulff structure, the ratio of a specific site type is determined by its surface energy. However, for small particles, the site ratio must be adjusted to form a closed surface for a particle with a given diameter, leading to non-monotonic fluctuations in the ratio of sites. In this study, Wulff-structured particles ranging from 4 nm to 100,000 nm (100 μm) in diameter were constructed using Pt(211) edge sites (te), Pt(111) (t1), and Pt(100) (t0) terrace sites. Figure 4c presents the ratios of te and t1 sites to the total number of sites on the Wulff-structured particles. The ratio of t0 is small due to its higher surface energy and is therefore not included in Fig. 4c. It can be seen that the ratios of te and t1 sites fluctuate for small particles, while they converge to the values determined by the respective surface energies as the diameter exceeds 500 nm. The converged ratios of te and t1 sites are 0.25 and 0.66, respectively.

Simulation using KMC at nanoparticles up to 100 μm is impractical, highlighting the advantage of XMF for large-scale modeling. Results of the XMF approach at 525 K and 675 K are presented in Fig. 4d and e, respectively. The results of the conventional MF-SUM method are also included for comparison. At 525 K (Fig. 4d), the smallest Pt₇₀₀ particle exhibits the highest TOF, which decreases with increasing size up to ~10 nm before rising again toward 500 nm. Notably, XMF and MF-SUM yield nearly identical results, indicating that kinetic coupling is negligible at lower temperatures. In contrast, at 675 K (Fig. 4e), XMF predicts TOFs approximately three times higher than MF-SUM across all particle sizes. This enhancement stems from kinetic coupling between te and t1 sites, which boosts activity beyond what would be expected from the isolated Pt(211) surface, even though te sites constitute only a small fraction of the total surface. The TOF trend closely follows the te site ratio, confirming that these edge sites govern the RDS at both temperatures. Thus, while the conventional view correctly identifies active sites as key determinants of activity, it fails to predict the additional enhancement from inter-site cooperation, a phenomenon captured only by methods like XMF that account for kinetic coupling. For further details, see Supplementary Note 3 of the SI.

Finally, we evaluated the XMF approach for alloy catalysts. Six Pt-Ru alloys were selected as examples, which have been considered as candidate catalysts for Pt-catalyzed WGS reaction⁶. Figure 4f shows the calculated TOFs for different alloys, while the results for the Pt(111) surface are also presented for comparison. The results suggest that the XMF approach produces the TOFs that are highly similar to the KMC results, although it overestimates the TOFs for Ru₂@Pt and Ru₃@Pt alloys by approximately one order of magnitude. This discrepancy arises because CO binds more strongly to Ru than to Pt, leading to pronounced CO poisoning on Ru-rich ensembles⁶. XMF captures the essence of the spatial correlations between different active sites when only a single Ru is exposed on the surface, yielding reliable activity predictions. However, for clusters containing multiple adjacent Ru sites, XMF underestimates cooperative CO adsorption effects, thereby underestimating poisoning and overpredicting activity for Ru₂@Pt and Ru₃@Pt alloys. Despite this limitation, XMF successfully identifies the Ru_1-3@Pt alloy as the most active candidate, demonstrating its utility for rapid catalyst screening. A more detailed discussion on Ru-Pt alloy behavior is provided in ref. ⁶ and Supplementary Note 3 in the SI.

Scope and limitations

XMF is designed for a broad class of heterogeneous catalytic systems for which a mean-field description is appropriate and LHHW-type kinetic reasoning has proven successful^31–34. In particular, XMF builds upon the LHHW paradigm by incorporating three key assumptions: (i) a mean-field treatment of surface processes, (ii) a MASI-dominated description of coverage effects in which the dominant coverage dependence of energetic parameters is controlled by the MASI, and (iii) a quasi-equilibrium relation between the MASI and its corresponding gas-phase reactants. Within this scope, XMF provides a more faithful and predictive realization of the LHHW spirit for realistic catalyst nanoparticles. Unlike conventional approaches, it simultaneously incorporates lateral adsorbate-adsorbate interactions and diffusion-mediated inter-facet coupling in a single, connectivity-aware multisite formulation.

A central advantage of XMF lies in its convenient treatment of nanoparticle or alloy surfaces as interacting, integrated entities rather than ensembles of isolated facets. By explicitly leveraging topological connectivity, i.e., which site types are adjacent to one another, it couples coverages and reaction kinetics across site types. This enables XMF to capture inter-site (cross-facet) correlations at the site-type level: the coverage on a given site type depends not only on its intrinsic properties but also on its connectivity to other site types on the whole nanoparticle, differing from the values predicted for isolated single-crystal surfaces. Critically, XMF achieves this without resorting to explicit lattice sampling or stochastic simulation. As a result, it allows for the rapid evaluation of large ensembles of catalysts that share identical composition but differ in morphology. For instance, in our implementation, we screened 1000 distinct Pt-Ru alloy configurations in under 1500 s of wall-clock time on a single CPU core, successfully capturing connectivity-induced coupling and correlation effects across site types (see Supplementary Fig. 10 and Supplementary Table 17 in the SI). Such capabilities are inaccessible to conventional “MF-SUM” approaches, which inherently neglect inter-facet communication.

Nevertheless, like all mean-field-based methods, XMF assumes statistical equivalence among sites of the same type. Consequently, it cannot resolve local fluctuations, ordering phenomena, or spatially inhomogeneous adsorbate distributions within a single facet (e.g., CO islanding⁴⁹, phase separation⁵⁰, or pattern formation⁵¹). When such intra-facet spatial correlations dominate the kinetics, more detailed methods (e.g, KMC^9,52,53) or higher-order MF approximations (e.g., pair/cluster approximations⁵⁴) may be necessary. This limitation is not unique to XMF; it is shared by all conventional MF-MKM approaches, including those that incorporate coverage-dependent energetics.

Furthermore, strongly directional or saturating interactions, such as hydrogen bonding or dipole-dipole coupling, introduce additional complexity, as their magnitude and sign depend on local geometry, orientation, and coordination⁵⁵. In practice, such effects are difficult to parameterize accurately, even within lattice-resolved KMC, and may require atomistic simulations (e.g., ab initio molecular dynamics⁵⁶) or specialized coarse-grained models for quantitative treatment⁵⁷.

Transient dynamics can also be studied within XMF, since the framework is formulated as a coupled rate-equation (ODE) network that can be integrated over time to yield coverages and rates as functions of time. The MASI quasi-equilibrium treatment is used to parameterize coverage-dependent energetic/kinetic terms. This approach is most reliable when the MASI equilibrates rapidly compared to other surface intermediates. However, if the MASI itself evolves on a similar timescale to those intermediates and its instantaneous coverage strongly influences the energetics, the resulting transient predictions may suffer from non-negligible errors, necessitating a more advanced treatment.

Finally, XMF is not intended to model strongly far-from-equilibrium dynamical phenomena, such as temporal oscillations, bistability, or chaotic behavior, which often emerge from the coupling of surface reactions with mass transport (reaction-diffusion effects) and/or dynamic catalyst restructuring on mesoscopic length and time scales^58,59. These regimes remain among the most challenging frontiers in theoretical catalysis, where highly accurate methods (e.g., atomistic KMC) struggle to reach the required spatiotemporal scales, while efficient continuum models often lack the resolution to capture essential local correlations^12,14. Although XMF does not fully address these extreme scenarios, it represents a significant step toward efficient, physics-informed modeling of heterogeneous catalysts as integrated, interacting entities, rather than as disconnected facets, and provides a practical foundation upon which future multiscale or hybrid approaches can be built.

Concluding remarks

In this work, an extended MF approach, dubbed XMF, has been developed for efficient yet accurate microkinetic simulations of complex heterogeneous catalyst surfaces and nanoparticles. Rather than treating a nanoparticle as a mere collection of independent facets whose kinetics are linearly combined, XMF models it as an integrated, interacting entity. Our approach incorporates the effects of adsorbate-adsorbate interactions at the MF level and allows diffusion-mediated inter-site coupling to be considered at three levels of approximation, leading to a hierarchy of the XMF methods. The performance of the XMF methods in predicting key kinetic properties, such as reaction rates, preferred active sites, RDS, optimal reaction pathways, as well as apparent activation energies, has been evaluated using the Pt-catalyzed WGS reaction as a benchmark. The consistency and accuracy of the approximations underlying XMF have been clearly demonstrated across a range of model systems, including single crystal surfaces, Pt nanoparticles, and Pt-Ru alloys. In all cases, XMF reproduces the reference results from KMC simulations with high fidelity, while remaining computationally feasible even in regimes where KMC is out of reach. Although XMF, like all MF approaches, neglects spatial correlations among identical active sites, which can reduce quantitative accuracy in certain scenarios, it consistently yields qualitatively correct predictions, as evidenced by its successful application to Pt nanoparticles. In stark contrast, conventional MF methods that ignore either adsorbate–adsorbate interactions or kinetic coupling between site types often produce qualitative errors in key kinetic properties. These findings underscore the superior capability of XMF in modeling complex catalytic surfaces, significantly outperforming traditional MF treatments. Critically, XMF achieves this enhanced accuracy with its computational complexity comparable to that of conventional MF methods. Hence, XMF is well positioned to serve as a powerful tool for elucidating catalyst behavior and enabling high-throughput catalyst screening.

Methods

The Pt-catalyzed WGS reaction

In this work, the WGS reaction, which plays an important role in hydrogen regeneration and purification^60,61, is selected as an example to illustrate our approach and to test its accuracy and efficiency. The mechanism of the WGS reaction has been extensively studied by both experimental and computational methods^62–64. It is generally accepted that the WGS reaction on the Pt catalysts follows the carboxylation mechanism^6,8,63, which consists of the following five elementary reactions:

$$\mathrm{CO}\left(\mathrm{g}\right)+*\rightleftharpoons {\mathrm{CO}}^{*}$$ 20

$${\mathrm{H}}_2\mathrm{O}\left(\mathrm{g}\right)+*+*\rightleftharpoons {\mathrm{OH}}^{*}+{\mathrm{H}}^{*}$$ 21

$${\mathrm{CO}}^{*}+{\mathrm{OH}}^{*}\rightleftharpoons {\mathrm{COOH}}^{*}+*$$ 22

$${\mathrm{COOH}}^{*}+*\rightleftharpoons {\mathrm{CO}}_2\left(\mathrm{g}\right)+*+{\mathrm{H}}^{*}$$ 23

$${\mathrm{H}}^{*}+{\mathrm{H}}^{*}\rightleftharpoons {\mathrm{H}}_2\left(\mathrm{g}\right)+*+*$$ 24

In the above equations, empty sites are indicated by asterisks, surface adsorbates by asterisk superscripts, and gas-phase species by the letter ‘g’. Here we have ignored the adsorbed state of CO₂ and H₂, which is consistent with the settings in our previous studies^6,8. Four additional diffusion processes are explicitly included in the KMC simulations:

$${\mathrm{CO}}^{*}+*\rightleftharpoons *+{\mathrm{CO}}^{*}$$ 25

$${\mathrm{OH}}^{*}+*\rightleftharpoons *+{\mathrm{OH}}^{*}$$ 26

$${\mathrm{COOH}}^{*}+*\rightleftharpoons *+{\mathrm{COOH}}^{*}$$ 27

$${\mathrm{H}}^{*}+*\rightleftharpoons {\mathrm{H}}^{*}+*$$ 28

In kinetic simulations of nanoparticles, chemical reactions 20-24 and diffusions 25-28 are extended to all site types, resulting in a complex reaction network. For instance, this leads to a total of 33 chemical reactions and 12 diffusion processes for the Pt₄₀₄ nanoparticle depicted in Fig. 1.

DFT computational details

In this work, two types of model catalysts, Pt nanoparticles and Pt-Ru alloys, were used to investigate the performance of the XMF approach on complex heterogeneous surfaces. On Pt nanoparticles, active sites on the surfaces are divided into four types, t0, t1, te, and tc as illustrated in Fig. 1b. For more detailed description of the Pt particles and the distribution of different types of these sites, we refer to ref. ⁸ and Section S3 in the SI. For Pt-Ru alloy catalysts, we studied Ru clusters embedded on the Pt(111) surface. For the Ru_x@Pt catalysts, x surface Pt atoms were replaced by Ru atoms, while for the Ru_1-y@Pt catalysts, one surface Pt atom and y subsurface Pt atoms were replaced by Ru atoms. Further details of Pt-Ru alloy catalysts can be found in ref. ⁶ and in Section Supplementary Note 3 in the SI.

Density functional theory (DFT) calculations were carried out by using the Vienna ab initio simulation package (VASP)^65,66. The electron-core interaction was described by the projector-augmented wave (PAW) scheme, while the electronic states were expanded on a plane wave basis with a cutoff of 400 eV for kinetic energy^67,68. The electron exchange-correlation for all gaseous and surface species, including adsorbates and transition states, was treated by the Bayesian error estimation functional with van der Waals corrections (BEEF-vdW)⁶⁹.

The non-periodic calculations for gaseous species were performed in cubic supercells with an edge length of 10 Å in order to eliminate the unphysical lateral interaction between neighboring supercells. All atoms in the non-periodic systems were relaxed during the geometrical optimizations. The Monkhorst-Pack grid for the k-point was set to (1× 1 × 1). For DFT calculations on extended surfaces, the Pt(100), Pt(111), Pt(211) surfaces and the corner model were described using a (3 × 3) slab model with four layers (Supplementary Fig. 1). Two bottommost layers were kept fixed, while two topmost layers and the adsorbates were relaxed during geometrical optimizations. The calculated lattice constant of Pt bulk was found to be 3.924 Å, which is in good agreement with the experimental value of 3.92 Å. In this work, a (4 × 4 × 1) Monkhorst-Pack grid was applied to integrate the first Brillouin zone, and a 15 Å vacuum layer perpendicular to the surface slabs along the z-direction was adopted. The transition states (TS) were located by the Climbing-Image Nudged Elastic Band (CI-NEB) method and further confirmed in harmonic analysis by the existence of a single imaginary frequency along the reaction coordinate^70–72. The adsorbates and transition states were optimized until all forces were below 0.03 eV/Å. Harmonic analysis was used to calculate the zero-point energy (ZPE) and entropy corrections for all species, except that the entropy correction for the adsorption of H₂O was evaluated by using the Campbell-Sellers equation⁷³. The repulsive interactions between adsorbed CO and all surface species, including adsorbates and transition states, were calculated to determine the effects of CO coverage on the thermodynamic parameters. Further details are summarized in Supplementary Note 1 in the SI.

In this work, the applicability of the extended mean-field (XMF) approach was tested not only on Pt surfaces but also on Pt nanoparticles, as well as alloy systems. The alloys tested were the Pt-Ru alloy systems. In the DFT calculations, the same settings were used for the Pt-Ru alloys as for the Pt(111) surface.

Kinetic simulations

After obtaining the necessary energetic parameters, the KMC simulations were performed using the standard KMC module in the XPK package⁷⁴. In the KMC simulations, 100 × 100 periodic lattice grids were employed to model the Pt(111) and the Pt(100) surfaces, while the Pt(211) surface was modeled by a lattice containing 750 surface sites. The Pt nanoparticles were modeled by a coarse-grained lattice containing all surface sites of the nanoparticles. The Pt-Ru alloy was modeled by a 30 × 30 lattice grid, with the Ru@Pt, Ru₂@Pt and Ru₃@Pt models containing one, two and three surface Ru atomic lattice sites, respectively, while Ru_1-1@Pt, Ru_1-2@Pt and Ru_1-3@Pt models containing one lattice point representing the surface Ru atomic site, and one, two and three subsurface Ru atomic lattice sites, respectively (Fig. 4f). We emphasize again, for simplicity, the KMC simulations were carried out under a coarse-grained site assumption, i.e. each surface intermediate could only occupy its most stable adsorption site (top, bridge, or hollow sites) correlated to a surface atom; and the atom is occupied once one of the correlated sites is occupied, and the others cannot be occupied due to the repulsive interactions between intermediates.

The apparent activation energy was obtained by an Arrhenius plot in the temperature range of 525 K to 675 K. In the kinetic simulations of Pt crystalline surfaces and Pt nanoparticle catalysts, the partial pressures for the gaseous CO and H₂O were set as 0.020 bar and 0.033 bar, respectively, while the partial pressures of CO₂ and H₂ were zero. In the investigation of kinetics of the Pt-Ru alloys, the partial pressures of CO and H₂O were set to be 0.032 bar and 0.075 bar, respectively, while the partial pressures of CO₂ and H₂ were set to be zero. These parameters are consistent with the experimental setting reported by Duke et al.⁷⁵.

It is well known that the rate of adsorption and desorption processes is much higher than that of other kinetically relevant reactions, which may cause the problem of time scale separation¹⁵. In this work, we proportionally scaled down the rate coefficient of adsorption and desorption processes without changing the thermodynamic properties, to make the reaction rate for adsorption and desorption processes 100 to 1000 times larger than the other elementary reactions.

In some cases, conventional MF-MKM simulations and coverage-dependent MF-MKM were performed for comparison. In all kinetic simulations, the thermodynamic parameters involved are identical. A Python script was used to perform the XMF simulations. An example of the XMF simulation script can be found in the file “Supplementary information-Python code”. For further details on the kinetic simulations, we refer to Supplementary Note 2 in the SI.

Author contributions

Conceptualization: X.X., T.S., Methodology: T.S., Y.W., Software: T.S., Y.W., Investigation: Y.W., Y.Y., T.S., Formal analysis: T.S., Y.W., Y.Y., Data curation: T.S., Y.W., Visualization: Y.W., T.S., Writing: T.S., Y.W., X.X., Supervision: X.X., Funding acquisition: X.X., T.S., Y.W. and T.S. contributed equally to this work.

Peer review

Peer review information

Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work. A peer review file is available.

Data availability

The structural data generated in this study have been deposited in the Zenodo database with accession code⁷⁶ https://doi.org/10.5281/zenodo.18728312. Source data are provided with this paper in the accompanying Source Data file. Source data are provided with this paper.

Code availability

Python code implementation of the XMF methods for the Pt₄₀₄ nanoparticle has been deposited to the Code Ocean⁷⁷ (https://codeocean.com/capsule/5988616/tree/v1).

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at https://doi.org/10.1038/s41467-026-70896-0.