Perspectives on Computational Catalysis for Metal Nanoparticles

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Perspective

Perspectives on Computational Catalysis for Metal Nanoparticles Mikkel Jørgensen, and Henrik Grönbeck ACS Catal., Just Accepted Manuscript • DOI: 10.1021/acscatal.9b02228 • Publication Date (Web): 19 Aug 2019 Downloaded from pubs.acs.org on August 19, 2019

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Perspectives on Computational Catalysis for Metal Nanoparticles Mikkel Jørgensen∗ and Henrik Grönbeck∗ Department of Physics and Competence Centre for Catalysis, Chalmers University of Technology, 412 96 Göteborg, Sweden E-mail: [email protected]; [email protected] Abstract Understanding reaction kinetics over metal-nanoparticles is central in technical catalyst design. In this perspective, we compare computational methods to analyze and model reaction kinetics on metal nanoparticles. We discuss energy-barrier and Sabatier analysis, mean-field microkinetic modeling, and kinetic Monte Carlo simulations. By explicit simulations we show that reactions on metal-nanoparticles are characterized by longrange kinetic couplings, which requires methods that consider coupled site-assemblies. In the light of these observations, extended model surfaces may not capture the complexity of nanoparticle-kinetics, and arguments about catalytic performance relying on single energy barriers may be insufficient. Keywords: Nanoparticle catalysis, Microkinetic modeling, Kinetic Monte Carlo, Density Functional Theory, Acetylene hydrogenation, Single atom alloy.

Introduction Catalysis is a key technology in modern society, enabling production of chemicals and efficient aftertreatment systems. Catalysis is, moreover, crucial in the development of new 1

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sustainable energy systems. Despite extensive use of catalysts, there are still reactions that are inefficient with respect to activity and selectivity, and in many cases it would be advantageous to operate the reactions at less demanding temperature and pressure conditions. Heterogeneous catalysts are commonly realized as nanoparticles supported on high surface area oxides. The structural complexity of these systems is one reason for the general challenge to unravel the mechanistic details in the reaction kinetics. However, to design successful catalysts, it is important to understand the kinetic behavior of supported nanoparticles. Such an understanding is paved by an iterative interplay between experimental and theoretical insights. Computational methods can provide crucial atomic-scale information for rational catalyst development. The accuracy of electronic structure calculations has now reached a level, where reaction kinetics of complete catalytic cycles accurately can be simulated directly from first principles. 1 This kind of work is presently mainly performed on extended surfaces, and it has become crucial to explore methods to simulate reaction kinetics over nanoparticles. In particular, as many reactions have been measured to be structure sensitive 2,3 and depend significantly on particle shape and size 4 . Because of the structural complexity of nanoparticles, it would be advantageous to know when a full kinetic model is required and when a simple comparison of reaction energy barriers is sufficient. Atomistic model systems in heterogeneous catalysis has over the decades moved from extended flat terraces 5,6 to stepped surfaces and nanoparticles 7 [Figure 1 (a)]. The use of single crystal surfaces allows atomistic precision in the experimental characterization, which, for example, enables detailed understanding of the role of step sites for certain reactions. 8 The complexity in the reaction kinetics is increased when going from flat to stepped surfaces. On a flat surface, all atoms are equivalent and the reactants can adsorb, diffuse, and react in a limited number of ways. Thus, the reaction is described by a fairly simple potential energy surface, although adsorbate-adsorbate interactions makes the situation less straightforward under reactions conditions. The complexity is enhanced for stepped surfaces, where different types of atoms give the possibility for the reaction to occur on different types

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of sites. The number of different sites is further increased for nanoparticles. Taking the example of CO oxidation with the reactants CO and O, the possible adsorption energies on a truncated octahedron is sketched in Figure 1 (b). The adsorption energy of both CO and O scales linearly with the coordination number (or the d-band center), which gives a linear correlation between the two adsorption energies. The strongest adsorption occurs for corner sites, whereas the weakest are obtained for the (111) sites. The facet-sites next to edge-sites yield different adsorption energies than sites in the inner facet. Thus, there are three distinct points on the line belonging to the (111) facet. A (111) site on a nanoparticle can be geometrically close to a corner site. Thus, a high adsorption energy of one adsorbate is spatially connected to a low adsorption energy of another adsorbate. This results in multiple possible combinations for the adsorption energies, represented by the shaded area in the figure. An additional spread in adsorption energies is given also in this case by adsorbate-adsorbate interactions. The spread makes it important to consider not only the sites that are present in the system, but also how they are connected. We refer to this as the site-assembly.

Figure 1: (a) Model systems with different levels of complexity. (b) Adsorption energies of CO and O shown for the Pt nanoparticle in part a. Special sites are highlighted, and the shaded area indicates schematically the spread from the specific site-arrangement and the adsorbate-adsorbate interactions. The close to continuous range of adsorption energies on a nanoparticle can lead to a large number of possible reaction paths, even for relatively symmetric nanoparticles. To illustrate 3


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this, we consider a simple Langmuir-Hinshelwood reaction with adsorbate diffusion:

A(g) + ∗

↔

A∗

(R1)

B(g) + ∗

↔

B∗

(R2)

A∗ + B ∗

→

AB(g) + 2∗

(R3)

A∗ + ∗

↔

∗ + A∗

(diffusion)

(R4)

B∗ + ∗

↔

∗ + B∗

(diffusion).

(R5)

For a nanoparticle, the number of sites that yield different energies can be quite large as compared to extended surfaces. For a nanoparticle with eight distinct types of sites, this simple Langmuir-Hinshelwood scheme may lead to several hundreds of different reaction pathways. Thus, the reaction energy landscape may only be feasible to compute directly for some of the reaction paths. While the number of possible reaction paths is large, reactions at steady-state often follow one dominant mechanism. This reaction mechanism can be predicted by the principle of least resistance, 9 which states that a reaction at steady-state will follow the lowest energy path. This can cause different sites to become dominant at different reaction conditions, 10 and lead to particle-size and shape effects. 10,11 The issue is, of course, that the main reaction path is not known a priori. As the number of possible reaction paths is large for a nanoparticle, typically detailed kinetic models are not considered. Instead, reductionistic strategies are applied, where a few reaction energies are used as descriptors for catalytic performance. 1,12–15 Thus, nanoparticles have rarely been considered from a systems theory approach, considering the diffusionmediated dynamic interplay between different sites. Accounting for the complexity meditated by diffusion may be the next step to enhance the understanding of heterogeneous catalysis. This perspective discusses the merits of different approaches to understand reaction kinetics over nanoparticles. Our main conclusion is that for nanoparticles where adsorbate diffusion is facile, it is preferable to apply a method where kinetic couplings between all sites

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are considered explicitly. Here, we define kinetic couplings as the process where multiple different sites communicate by adsorbate diffusion, yielding non-linearly additive rates. The perspective starts by discussing how reaction kinetics is governed by the chemical master equation, and thereafter we briefly introduce three methods for investigating kinetics. 1) Energy and Sabatier analysis which implicitly assume mean-field kinetics, 2) Kinetic models in the Mean-Field Approximation (MFA), and 3) kinetic Monte Carlo simulations. We present our recently developed Scaling Relations Monte Carlo (SRMC) 10 method, which is suitable for studies of metal nanoparticles. Results are presented from studies of CO oxidation over Pt, and acetylene hydrogenation over Pd/Cu single-atom alloys. Finally, we discuss the implications of the findings, and how to infer simple design strategies from systemic insights. Our results show that it is important to perform explicit simulations on nanoparticles. SRMC provides such a possibility, and with further development, the method may become a useful tool to study reaction kinetics of nanoparticles.

Modeling reaction kinetics Reaction kinetics can be viewed as transitions between different chemical states 16 proceeding in time. This is described by the chemical master equation, 17 which models the timeevolution for the probabilities of observing certain states. The chemical master equation for a state α is: dPα X = Wαβ Pβ − Wβα Pα , dt β

(1)

where P is the probability, and W is the transition rate between states. The sum runs over all states β that are connected to α by elementary reaction steps. By solving the chemical master equation, the kinetics is revealed. However, there are an astronomical number of different states, even for simple systems and reactions, which renders (1) unsolvable by direct techniques. Despite the fact that CO oxidation over RuO2 have been solved directly, 18 the kinetics must generally be solved by simplified approaches. 5


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Energy and Sabatier analysis Analyzing reaction energies can reveal important information about the possible reaction paths. Such isolated-site energy analysis have previously been used extensively in catalyst screening studies, 1,12 owing to the low computational cost. The energy analysis can be translated to kinetic predicitons by a so-called Sabatier analysis, 13 which provide an upper bound of the reaction rate. 19 The Sabatier analysis is of the mean-field type, because it works with average adsorbate-coverages. The Sabatier activity is found by assuming optimal coverages for each reaction step, and taking the slowest reaction step as the rate-determining step. The rate-constants are generally calculated in harmonic transition state theory, 20 neglecting molecular-entropy, 19 which is likely a fair approximation for this level of detail. While an upper bound on the Turnover Frequency (TOF) is valuable for initial screening studies, a lower bound is also interesting, due to various adsorbate-poisoning mechanisms. We note that alloys may be difficult to target within Sabatier analysis, although a recently established relation between surface structure and adsorption energies could reduce this issue. 21,22 The advantages of performing an energy or Sabatier analysis are that the results are relatively simple to analyze, and initial catalyst screening studies are highly feasible with this approach. Sabatier analysis can be successful when one step is clearly rate-determining on a specific site, or when only a few sites are active, such as for Au in CO oxidation. 19,23,24 A pitfall with the method can, however, be that a reaction path must be assumed which is non-intuitive for nanoparticles. Thus, a simple free energy diagram may reveal important information about the reaction mechanisms and kinetic bottlenecks, but it could be illusive when there are multiple types of sites available.

Explicit Mean-Field Kinetic Models Mean-field (MF) microkinetic models are approximate solutions to the chemical master equation (1). MF kinetics simplifies the complex problem of the huge number of states in (1) by 6


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working with surface coverages. The problem is in this way reduced to a system of coupled differential equations in the coverages; with one equation for each intermediate species: (

dθx X = Gxi ri θ~ dt i

) ,

(2)

Nspecies

where the sum runs over each reaction step in the model, Gxi is the number of species x ~ While created in reaction i, and ri is the rate of reaction i that depends on the coverages θ. the coverages provide a huge reduction in complexity, MF kinetics simultaneously assumes a random adsorbate-distribution. This prohibits strong adsorbate-adsorbate interactions and too heterogeneous surfaces. Other central assumptions are that the system is assumed to be large and that fluctuations from the mean coverages are small. Several MF microkinetic models have been developed for catalysis during the past decades, 14,15,19,25–29 describing for example ammonia synthesis, hydrogenation reactions and oxidation reactions. Although adsorbate-adsorbate interactions strictly are prohibited in MF kinetics, some form of repulsive interaction is often included, which is necessary to obtain reasonable coverages and rates. Similarly, adsorbate pairing can be important to describe reactions on oxide surfaces, which has been achieved following a quasi-chemical approach, 30 resulting in good experimental agreement. 31 Bimetallic alloy surfaces have also been investigated with the MF approach, where bifunctional mechanisms were found to be important. 32,33 Bifunctional mechanisms move the modeling one step towards nanoparticle catalysis with several reaction paths, however, a fully coupled nanoparticle can, in principle, be multifunctional. While these additions to MF kinetics are strictly in conflict with the MF assumptions, such approaches have been demonstrated to yield valuable insights. To improve systematically over MF models, while keeping the computational cost low, an alternative method is cluster mean-field (CMF) for kinetic lattice-gas models. 34,35 In the CMF approach, a small cluster on the lattice is considered explicitly, whereas the surrounding sites are treated as a mean-field-cloud of adsorbates interacting with the cluster.

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This approach allows for a systematic improvement over the MF kinetics by increasing the cluster-size; when one site is considered, MF kinetics is recovered. 35 An advantage of performing mean-field investigations is that information of kinetics is obtained using the simple measures of adsorbate-coverages. Additional advantages include the low computational cost and ease of analysis, for example using the degree of rate control. 36–38 A disadvantage is that the assumptions of mean-field kinetics likely are not well-suited to simulate nanoparticles with multiple types of sites. Moreover, it is not possible to simulate specific site-assemblies in the mean-field approach, because all sites are modeled as nearestneighbors. Thus, effects of particle shape and size are difficult to assess when analyzing mean-field results.

Kinetic Monte Carlo Kinetic Monte Carlo (kMC) is a stochastic method to solve the chemical master equation. KMC achieves this by transitioning the system between several configurations, and updating time accordingly. A simple algorithm is the first reaction method, 17 where initially a set of events are identified. For CO oxidation, these events could include CO adsorption on site 1 and CO∗ + O∗ → CO∗2 on site 2 and 3. For each possible event i, a time of occurrence is generated: tocc = tgen − i

ln u , ki

(3)

where tgen is the time where the event is generated, ki is the rate constant of the event, and u is a random uniform number on ]0, 1]. Multiple software packages are available to perform kMC of catalytic reactions. 39–44 Equation (3) implies that fast events are executed with the highest probability and have the shortest time-step. This is a considerable challenge in kMC, because diffusion is often several orders of magnitudes faster than chemical reactions. Thus, temporal acceleration schemes are necessary to compute activities and selectivities. Acceleration can be achieved

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by taking advantage of the fact that fast events often are quasi-equilibrated. In simple cases, it is sufficient to slow down fast events, 17 while for more realistic reaction networks, schemes have been invented to periodically slow down quasi-equilibrated reactions based on their relative rates. 45–48 KMC has been applied to study reaction kinetics over extended surfaces for multiple catalytic reactions. 49–54 An extension is to investigate finite-sized systems with multiple available reaction energies. Reactions over nanoparticles have in the past been simulated using schematic reaction mechanisms (2A+B2 → 2AB), including the possibility for adsorbate diffusion. 55–57 Although these schematic models do not consider the detailed reaction energy landscape, they demonstrate the effect of multiple sites: the kinetics is not simply a sum of the kinetics for the different facets. In addition to schematic kinetics, simulations have been performed that distinguish edges and different facets, 58 and spill-over effects from metaloxide supports have been emulated using simplified energy landscapes. 42 All these models capture some characteristic features of nanoparticles, but have not attempted to describe the complex reaction energy landscape of small nanoparticles. 59–63 The benefit of performing kMC simulations is that the evolution of reactions can be followed in space and time, using specific particle shapes and sizes. This, however, also makes data analysis less straightforward. Another disadvantage of kMC simulations is that the computational complexity and cost is significantly higher than in MF approaches. It has been argued that the MFA should coincide with kMC simulations for extended surfaces, in the limit of fast diffusion and absence of adsorbate-adsorbate interactions. 54,64,65 However, to reach absolute agreement, special artificial mixing steps appears to be required. 54 While the correspondence of MFA and kMC has intuitive appeal for extended surfaces, the MFA assumptions break down on nanoparticles, because nanoparticles contain multiple energetically different sites and are of finite size. Thus, the MFA provides indications about the kinetics over nanoparticles, whereas kMC is suitable for more direct insights.

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Scaling Relations Monte Carlo for nanoparticle catalysis Performing Scaling Relations Monte Carlo The reaction energy landscape on adsorbate covered nanoparticles is complex, prohibiting direct computation of all adsorption energies and reaction barriers. One efficient solution to the problem is the use of well-established scaling relations that have emerged from early work on adsorbate bonding to metal surfaces. 66,67 One example is the d-band model where the adsorption energy is related the center of the metal d-band. 68 As the d-band center depends on the coordination, adsorbate energies can also be correlated with coordination numbers or generalized coordination numbers. 59,60 An alternative descriptor is the metalsite stability, which has been designed to capture effects of alloying. 61 Having the adsorption energies, Brønsted-Evans-Polanyi (BEP) relations can be used to obtain the energy barriers. These relations are utilized in the Scaling Relations Monte Carlo (SRMC) method. 10 The relevant energies are obtained with Density Functional Theory 69,70 (DFT), and rate constants are modeled within Harmonic Transition State Theory 20 (TST). The SRMC approach for nanoparticles, involves the following steps: • The adsorption energies are obtained using a descriptor. The relation between the adsorption energy and the descriptor is calculated using relevant model systems, for example flat and stepped surfaces. The descriptor should hold simultaneously for both nanoparticles and extended surfaces. • Adsorbate-adsorbate interactions are implemented in a neighborhood around each site. • BEP relation-slopes are used to obtain the reaction energy barriers as a function of the adsorption energies. This means that the barriers vary between facets, edges, and corners. Moreover, adsorbate-adsorbate interactions affect the barriers. Using BEP relations circumvents calculating each reaction energy barrier explicitly for all sites including adsorbate-adsorbate interactions. 10


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• Diffusion barriers are calculated, and should obey thermodynamic consistency, such that the diffusion from a high- to a low-energy site is exothermic. • The site in the kMC simulation may be coarse-grained to entail both ontop, bridge, and hollow positions. Note that the energies still relate to the preferred geometrical adsorption site. One possible descriptor for the entire energy landscape is the conventional coordination number (CN), which reflects the number of neighbors for an ontop site. Another more detailed choice is the generalized coordination number 59,60 (CN), which is a normalized sum over CN for the first nearest-neighbors to a site. Thus, CN is likely more appropriate than CN for bridge and hollow sites. In addition to these two geometric descriptors, recently the metal-site stability (∆EM ) was introduced. 61 ∆EM reflects how stable a site is in the system, and is calculated by removing the considered site and evaluating the energy change. It is important to choose descriptors that sufficiently capture the characteristics of the reaction energy landscape. For example, there is a gradual change of adsorption energies between a fcc(111) facet and edge, which could be important to make adsorbates diffuse towards low energy sites on the particle. Figure 2 compares CN, CN, and ∆EM for the reaction energy landscape of a Pd/Cu single-atom alloy nanoparticle. The CN predicts distinct reaction energies on (111), (100), edges, and corners. However, it does not differentiate between a (111) site next to the edge and a (111) site in the center of the facet. In this basic form, CN does not recognize that the Pd site yields a different reactivity than Cu, when placed in the center of a (100) microfacet. The CN captures the effects of outer and inner facet sites, however, it does not directly describe the alloying. Instead ∆EM captures both the alloying and the finite size effects. This is a consequence of the fact that ∆EM is not a purely geometric descriptor, involving more energy calculations than CN and CN.

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Figure 2: Descriptors of reaction energy landscapes compared for a 201 atom Pd/Cu singleatom alloy nanoparticle (a). Coordination numbers (b), Generalized coordination numbers (c), and metal-site stability (d).

Insights from fully coupled kinetics over nanoparticles We have previously performed SRMC simulations for CO oxidation over Pt 10,11,71,72 and acetylene hydrogenation over Pd/Cu single-atom alloys 73 (SAA). These studies provide insights into the fundamental differences between extended surfaces and nanoparticles. The most obvious difference is that nanoparticles contain multiple different sites, which can lead to long-range kinetic couplings. In the following, we discuss effects of particle size, 10 shape, 11 strain, 71,72 and alloying, 73 and how selectivity and TOF are affected by kinetic couplings. CO oxidation was modeled with a simple Langmuir-Hinshelwood mechanism:

CO(g) + ∗

↔

CO∗

(R6)

O2 (g) + 2∗

↔

2O∗

(R7)

CO∗ + O∗

→

CO2 (g) + 2∗

(R8)

A∗ + ∗

↔

∗ + A∗

(R9)

(adsorbate diffusion).

Figure 3 shows the TOF as a function of temperature, comparing Pt(111) to a truncated octahedral nanoparticle of 3.5 nm in diameter. Below 500 K, the reaction rate is zero, and above 500 K the reaction increases rapidly for both systems. This is because CO desorption occurs above 500 K, which leaves free sites for O2 to dissociate and the reaction 12


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Figure 3: Kinetic Monte Carlo simulations of CO oxidation over Pt. TOF as a function of temperature for a 3.5 nm octahedral nanoparticle (blue) and an extended Pt(111) surface (red). Pressures: 20 mbar CO, and 10 mbar O2 . to proceed. In both cases, CO desorption starts from the (111) facets. For the nanoparticle, the TOF increases faster than for the extended surface. This can partly be understood from the principle of least resistance, 9 because nanoparticles have multiple types of sites present. Figure 3 clearly shows that nanoparticles can enable a higher TOF than extended surfaces; despite the fact that the reaction energy barrier for CO∗ + O∗ → CO2 (g) is lowest on Pt(111). This simple example shows that the reaction mechanism on nanoparticles is strongly influenced by kinetic couplings between different sites or regions on the nanoparticle. On flat surfaces, similar effects can potentially be obtained in electrocatalysis by applying an oscillating electric field. 74 To analyze kinetic couplings on nanoparticles, we followed a biology-inspired approach and selectively disabled all adsorption, desorption, diffusion, and reactions on specific regions of the particle. This analysis provides central information about the role of the disabled region. We performed separate simulations, excluding the (100) and (111) facets and investigated the response of the edges to disabling the facets. Edges are most relevant to

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Figure 4: Kinetic Monte Carlo simulations of CO oxidation over Pt nanoparticles. (a) TOF of edges shown for: a full particle (black), a particle with disabled (100) facets, and a particle with disabled (111) facets. (b) TOF for each CN at 500 K, shown for multiple particle-shapes. Pressures: 20 mbar CO, and 10 mbar O2 . analyze because the TOF is highest on these sites after the light-off. The high TOF of edges may be surprising as the reaction energy barrier is higher on edges as compared to facets. However, CO and O are thermodynamically more stable on the edges, where the reaction proceeds. Figure 4 (a) shows the TOF of the edges when the (100) sites and the (111) are disabled. Note the different scale for the TOF in the figure as compared to Figure 3 (the light-off is at 500 K in both figures). The TOF of the edges respond unfavorably to the disabled facets, where the (111) facets give a large decrease in TOF. This is explained by the reaction mechanism on nanoparticles. O2 dissociates preferentially on the (111) facets, because dissociation requires a free pair of neighbor sites, and the (111) facet has most free sites. After O2 dissociation, O diffuses to the edges, where CO is strongly bound and CO2 formation occurs. In this manner, different regions on the nanoparticle catalyze different elementary steps, and adsorbate diffusion enables an increased TOF as compared to extended surfaces. The fact that different particle regions can enable different elementary steps makes particle shape a non-intuitive parameter in predicting catalytic activity. The catalytic performance of a site depends on the specific site-assembly. This is illustrated in Figure 4 (b), which shows the TOF at 500 K for multiple particle shapes as a function CN. In the present simulations, CN determines all reaction energies, including barriers and adsorption energies. 14


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Thus, each CN corresponds to a distinct site-type. There is no obvious correlation between CN and the TOF. This is interesting as the reaction energy barrier for CO2 formation is a linear function of CN. The operating temperature determines the CN of the most active sites. Increasing the temperature will make sites with low CN more active on average. Similarly, varying pressures lead to a change in the dominant site, which makes TOF depend on particle diameter. 10 The examples show that for nanoparticles, the reaction energy barriers cannot necessarily be used as descriptors for catalytic activity without considering the entire particle geometry. These results highlight that a systemic approach is necessary to fully understand catalysis on metal nanoparticles.

Figure 5: Kinetic Monte Carlo simulations of CO oxidation over Pt. (a) TOF of extended Pt(111) as a function of temperature for different strains (b) TOF of a truncated octahedron Pt nanoparticle of 3.5 nm in diameter, for different full-particle strains. Pressures: 20 mbar CO, and 10 mbar O2 . Supported nanoparticles generally contain sites that are strained. The strain can be internal owing to grain-boundaries or originate from the support. The question of how strain affects catalytic activity has previously been explored for extended surfaces. 75,76 These investigations reveal that strain decreases reaction rates rather linearly on extended surfaces. A 5% compression of Pt destabilizes CO∗ and O∗ by about 0.5 eV, and lowers the barrier of CO2 formation by about 0.2 eV. However, the lower reaction energy barrier does not

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necessarily imply a lower light-off temperature or higher reaction rate. For nanoparticles, kinetic couplings render the situation more difficult to analyze. 71,72 Figure 5 compares the response of TOF to strain for extended surfaces and a nanoparticle. Pt(111) responds with a lower TOF to a 5% compression, which is reasonable as the strain destabilizes the adsorbates. As both the CO and O adsorption energies scale similarly with strain, the strain does not significantly change the relative coverage. Similarly, a 5% expansion also lowers the TOF slightly, due to the increased barrier for CO∗ + O∗ → CO2 (g). As both compressive and expansive strain lowers the activity, Pt(111) is close to optimal for CO oxidation 19 at the investigated reaction conditions. On nanoparticles, the compressive strain lowers the TOF somewhat, but not enough to extinct the reaction. This is because the strain is insufficient to make CO and O desorb from the lower-coordinated sites. Expansive strain increases the nanoparticle-TOF. Thus, strain influences Pt(111) and nanoparticles differently. Acetylene hydrogenation to ethylene is an important reaction, for example, in polymer fabrication, and can be considered a model for more complex hydrogenation reactions. Typically, Pd-based catalysts are applied for the reaction, however, pure Pd can easily lead to overhydrogenation of ethylene to ethane, yielding a selectivity problem. Alloying is a viable solution to increase the selectivity, where recently Single-Atom Alloys (SAAs) have been shown to yield excellent selectivity. 77–80 SAAs have a low Pd-concentration of isolated atoms embedded in a larger host-metal surface, because a larger Pd concentration may lower selectivity. Extended surfaces have mostly been studied for this reaction, and these results have been used to rationalize the selectivity of nanoparticles. 77 To investigate how selectivity differs between extended surfaces and nanoparticles, we performed simulations of acetylene hydrogenation over SAA Pd/Cu(111) surfaces and nanoparticles. The reaction was modeled

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by a simple Horiuti-Polanyi mechanism as

H2 (g) + 2∗

↔

2H∗

(R10)

C2 H2 (g) + ∗

↔

C2 H∗2

(R11)

C2 H∗2 + H∗

↔

CHCH∗2 + ∗

(R12)

CHCH∗2 + H∗

↔

C2 H∗4 + ∗

(R13)

C2 H∗4

↔

C2 H4 (g) + ∗

(R14)

C2 H∗4 + H∗

↔

C2 H∗5 + ∗

(R15)

C2 H∗5 + H∗

→

C2 H6 (g) + 2∗

(R16)

A∗ + ∗

↔

∗ + A∗

(R17)

(adsorbate diffusion).

Figure 6 (a) compares the selectivity for a Pd/Cu(111) surface and nanoparticle with a Pd/Cu surface-ratio of about 1%. Pd/Cu(111) yields a selectivity close to 100%, and the nanoparticle has a significantly lower selectivity. This is owing to edges and corners that stabilize ethylene, which promotes further hydrogenation of ethylene to ethane. Thus, to keep the selectivity high, the low-coordinated Cu sites should be minimized. The Pd atom in the Cu host particle can be placed in different positions, and the effect of the Pd site-placement is shown in Figure 6 (b). The selectivity is lower than 40% for each site-placement. There is no apparent correlation between the relative particle stability and selectivity. Moreover, the selectivity varies differently with temperature, depending on the Pd-placement. The non-linear behavior of the nanoparticles in acetylene hydrogenation is attributed to a complex reaction mechanism. For the particle in Figure 6 (a), C2 H4 is preferentially hydrogenated to C2 H5 on the corners and edges, whereas C2 H5 hydrogenation to C2 H6 is most facile over the Pd site and Cu(111). The observations vary between the different Pd site-placements. 73

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Figure 6: (a) Selectivity in acetylene hydrogenation as a function of temperature for the extended Pd/Cu(111) single-atom alloy surface, and a Pd/Cu single-atom alloy nanoparticle. (b) Effect of Pd site-placement on selectivity for nanoparticles. Relative energies for the nanoparticles are indicated. Pressures: 1 mbar C2 H2 , and 10 mbar H2 and C2 H4 . Adapted with permission from J. Am. Chem. Soc. 2019, 141, 8541. Copyright 2019 American Chemical Society.

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Simple design strategies from complex data The examples of CO oxidation and acetylene hydrogenation illustrate how nanoparticles should be understood as a site-assembly (a combined set of sites with a special arrangement), rather than isolated sites. This is a consequence of significant long-range kinetic couplings between sites, which are mediated by fast adsorbate-diffusion. To improve catalyst design, it is important to understand which reaction steps are hindering the reaction. The degree of rate control 36,37 (χi ), of an elementary step (i), quantifies the sensitivity of a kinetic model to the particular elementary step. χi is the derivative of the TOF with respect to the forward rate constant of the step, keeping the equilibrium constant fixed: χi =

∂lnR ∂lnki

.

(4)

Ki

A step with χi = 1 is the rate-determining step, whereas a small χi shows that the rate does not depend on the step. For χi < 0, the step is hindering the reaction. In order to use a degree of rate control analysis for nanoparticles that include kinetic couplings, it could be useful to expand the concept of χ to be specific for each site. That is to calculate χi for each elementary step and each type of site. For example, in CO oxidation, likely O2 dissociation will have a positive χ on the (111) facets, whereas it will be smaller on the edges that are CO covered. Similarly, the CO2 formation step would likely have a larger χ on the edges than on the (111) facets, because the reaction occurs on edges. To improve the activity of nanoparticle catalysts, the principle of least resistance 9 provides an important concept. As each elementary step can proceed where most efficient, it can be beneficial to have multiple available types of sites. For CO oxidation on Pt, it is advantageous to combine some large (111) facets for O2 dissociation with edges that can keep CO adsorbed at larger temperatures. For a technical CO-oxidation catalyst that should to operate over a range of reaction conditions, our results show that it could be beneficial to have a wide distribution of particle shapes and sizes. To improve the rate of acetylene

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hydrogenation over Pd/Cu SAA nanoparticles, the number of Pd sites should be increased to facilitate H2 dissociation. Reaching a high selectivity is a considerable challenge for nanoparticles. This is again owing to the principle of least resistance, because inherently selective sites may couple to strongly non-selective sites. Thus, we infer that selectivity should be achieved by minimizing the abundance of sites that potentially can reduce the selectivity. In this respect, selectivity is difficult to achieve on nanoparticles, where there are multiple types of sites present. Selectivity on nanoparticles could be improved either by designing particle shape, size, alloying, or alternatively by poisoning non-selective sites. 81 In acetylene hydrogenation over Pd/Cu SAA nanoparticles, the edges and corners lower the selectivity, because they bind ethylene strongly. Thus, a possible design strategy for selectivity of this reaction is to fabricate relatively large icosahedral particles.

Conclusions Based on the results presented herein, we argue that to understand and predict catalytic reactions over nanoparticles, it is preferable to adopt a systemic viewpoint, which considers the specific site-assembly and allows for kinetic couplings. For nanoparticles the active site could, in fact, be viewed as a collection of sites. This implies that extended surfaces in general are questionable as model systems for nanoparticles. A characteristic feature of nanoparticles is the wide possible range of adsorption energies (Figure 1). Screening studies may benefit from explicitly considering nanoparticles, as the energy differences within a nanoparticle may be larger than the differences between different elements. The accuracy needed in the reaction energy landscape highly depends on the targeted question. However, owing to the wide range of energies on nanoparticles, contributions to the energy below 0.1 eV should generally not be crucial. We have discussed isolated-site and Sabatier analyses, which provide a cheap and useful

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framework for initial screening studies, where an overview of different materials is needed. These approaches require chemical intuition, and should be used where the main reaction path is well-established. Explicit mean-field models are not strictly applicable to nanoparticles, yet such models may provide information about the most active sites. However, one major limitation of the mean-field approach is that a specific particle size and geometry cannot be simulated in an explicit manner. The scaling relations Monte Carlo approach is a promising compromise between computational cost and accuracy. This scheme captures the most important features of the reaction energy landscape, and it simulates the kinetics for a specific particle geometry. This allows for understanding nanoparticles as a system, and goes beyond the reduction of explaining catalysis as reactions occurring on isolated sites. We believe that it is important to simulate nanoparticles explicitly, and with some future development, the SRMC approach can become a valuable tool. To improve the performance of the SRMC method, it may be necessary to lift the coarse graining of the adsorption site and explicitly treat ontop, bridge, and hollow positions. It is probably also necessary to develop increasingly accurate scaling relations and improve the description of adsorbate-adsorbate interactions together with molecular entropies. More challenging, but highly valuable would be development of computational methods to capture phase changes of catalytic particles such as oxide formation. Our simulations show that the simulated catalytic performance is strongly affected by kinetic couplings. Nanoparticles appear to be multifunctional as the different elementary steps proceed over different sites. Therefore, one interesting feature of kinetics over nanoparticles is that it inherently breaks the linear scaling relations. Here, we have shown that this applies even for monometallic nanoparticles. Alloy nanoparticles break the scaling relations to a larger extent, as the energy landscape is more corrugated. It is our hope that the presented systemic perspectives on nanoparticle catalysis could aid in future rational design of catalysts.

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Acknowledgement Financial support is acknowledged from the Chalmers Excellence Initiative Nanoscience and Nanotechnology and the Swedish Research Council (2016-05234). The calculations were performed at PDC (Stockholm) and C3SE (Göteborg) via a SNIC grant. The Competence Centre for Catalysis (KCK) is hosted by Chalmers University of Technology and is financially supported by the Swedish Energy Agency and the member companies AB Volvo, ECAPS AB, Johnson Matthey AB, Preem AB, Scania CV AB, Umicore Denmark ApS and Volvo Car Corporation AB.

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Perspectives on Computational Catalysis for Metal Nanoparticles

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