Letter pubs.acs.org/JPCL
Predicting Catalytic Activity of Nanoparticles by a DFT-Aided Machine-Learning Algorithm Ryosuke Jinnouchi* and Ryoji Asahi Toyota Central R&D Laboratories, Inc., 41-1 Yokomichi, Nagakute, Aichi 480-1192, Japan S Supporting Information *
ABSTRACT: Catalytic activities are often dominated by a few specific surface sites, and designing active sites is the key to realize highperformance heterogeneous catalysts. The great triumphs of modern surface science lead to reproduce catalytic reaction rates by modeling the arrangement of surface atoms with well-defined single-crystal surfaces. However, this method has limitations in the case for highly inhomogeneous atomic configurations such as on alloy nanoparticles with atomic-scale defects, where the arrangement cannot be decomposed into single crystals. Here, we propose a universal machine-learning scheme using a local similarity kernel, which allows interrogation of catalytic activities based on local atomic configurations. We then apply it to direct NO decomposition on RhAu alloy nanoparticles. The proposed method can efficiently predict energetics of catalytic reactions on nanoparticles using DFT data on single crystals, and its combination with kinetic analysis can provide detailed information on structures of active sites and size- and composition-dependent catalytic activities.
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acquire statistically meaningful kinetic information for NPs with various sizes, shapes, and compositions.17 To overcome the difficulties, descriptor-based approaches have been extensively studied in the theoretical community.18−22 In these approaches, easily calculable descriptors instead of the rigorously calculated activation free energies are used to predict the catalytic activities. A key in these approaches is to find good descriptors. For accurate predictions, descriptors need to possess strong correlations with the activation free energies or binding energies of reaction intermediates with the catalytic surfaces, which are linearly related to the activation free energies through Brønsted−Evans−Polanyi (BEP) relations.8,12 The predicted activation free energies of the formations and removals of reaction intermediates can then be used to evaluate catalytic activities by kinetic analyses like Sabatier analysis, where the net catalytic reactions are judged to be dominated by a formation or removal step with the highest activation free energy.2,8,9,13,16 Two types of descriptors were suggested in past studies: electronic and geometrical descriptors. The d-band center7,18 is, for example, a well-known electronic descriptor, and the generalized coordination number (GCN)13,19 is a recently suggested geometrical descriptor. The former was constructed on the basis of a simple tight-binding model describing metal− adsorbate interactions, and the latter was constructed by extending an empirical rule that states that an atom with fewer
esigning heterogeneous catalysts that facilitate specific chemical reactions to overcome energy and environmental issues has been one of the greatest challenges in science and industry. In most cases, catalytic nanoparticles (NPs) are widely used because of their high surface-area-to-volume ratio saving costly catalytic metals such as rhodium and platinum. Their surfaces are composed of a variety of surface sites, and their catalytic activities are often dominated by a few specific active sites.1−3 Identifications of active sites are challenging tasks that require combined experimental and theoretical analyses on practically applicable NPs and model catalysts.2−15 Although modern experimental techniques eventually realize direct observations of detailed surface structures of NPs and amounts of reaction intermediates,11,14,15 the obtainable information is still incomplete for understanding whole pictures of reaction mechanisms on inhomogeneous surface sites. Single-crystal (SC) surfaces with welldefined surface sites have been, therefore, widely used to clarify contributions from each surface site,2−9,12 and kinetic analyses using density functional theory (DFT) are essential for clarification.2,3,16 Such combined analyses have provided valuable information on active sites for a wide variety of catalytic reactions.2,3,5−7,10,12 There are, however, still limitations in the combined approaches. Surfaces of NPs are composed of a variety of geometries, part of which cannot be exactly described by any SC surfaces, and the structural gaps often cause difficulties in extrapolating information on SC surfaces to NPs, especially for the case of alloy NPs. Although DFT calculations on NPs can provide detailed kinetic information, their computational cost is still too expensive to © 2017 American Chemical Society
Received: August 1, 2017 Accepted: August 24, 2017 Published: August 24, 2017 4279
DOI: 10.1021/acs.jpclett.7b02010 J. Phys. Chem. Lett. 2017, 8, 4279−4283
Letter
The Journal of Physical Chemistry Letters
Figure 1. (A) Schematic of the algorithm. (B) Mean absolute error σ in the predicted binding energies of O on the Rh(1−x)Aux SC surfaces as a function of the number of used training data Ndata that equals the number of basis sets Nbasis in this calculation. (C−E) Predicted binding energies (Eb,NML, Eb,OML, and Eb,NOML) of (c) N, (d) O, and (e) NO on Rh(1−x)Aux SCs and NPs as a function of those obtained by DFT (Eb,NDFT, Eb,ODFT, and Eb,NODFT). R2 indicates the correlation coefficient. (F) Predicted formation energies (ΔEML) of Rh(1−x)Aux SCs and NPs from the pure Rh and Au bulks as a function of those obtained by DFT (ΔEDFT). The numbers of training and test data set are shown in Table S13 in the Supporting Information.
the local structural similarity between the Ith and Jth unrelaxed intermediates on the surfaces is evaluated as KIJ. For the evaluation, a similarity kernel called the smooth overlap atomic position (SOAP)23 is used. The SOAP KIJ consists of overlap integrals between three-dimensional atomic distributions within a cutoff radius Rcut from the Ith surface site and those from the Jth surface site, and it is defined to approach unity (or zero) when the two atomic distributions overlap more strongly (or less strongly). The binding energy Eb,I of the Ith relaxed reaction intermediate is assumed to be described by a linear function of KIJ as Eb,I = ∑J wJKIJ, and the regression coefficients wJ are determined to reproduce the stored DFT training data Eb,IDFT by using a Bayesian linear regression method. In the extrapolating step, an unrelaxed NP model is constructed from the bulk geometry, and the local similarity between the Kth intermediate on the NP and Jth intermediate on the SC surface is evaluated as KKJ by the SOAP. By using KKJ and the regression coefficients wJ, the binding energy Eb,K of the Kth relaxed intermediate on the NP is predicted as Eb,K = ∑J wJKKJ. When the local structure of the Kth unrelaxed intermediate with the surrounding local environments on the NP is exactly the same as that of the Ith unrelaxed intermediate on the SC surface, Eb,K equals Eb,IDFT because KKJ equals KIJ. This extreme situation indicates that the accuracy of the prediction can be systematically improved by increasing the number of DFT data to cover all of the possible local structures that appear on the NPs if the binding energies are assumed to be dominated by the local structure. It should be also noted that this algorithm is applicable to any properties dominated by the local structures. Interaction among metal atoms in NPs is such a property, and the method was used to predict stable atomistic structures of NPs in this study. Details of the method are also described in section S1 in the Supporting Information. The present machine-learning scheme was applied to the direct decomposition of NO on Rh(1−x)Aux alloy NPs to
coordination numbers binds more strongly with other atoms and molecules. More recently, mixtures of two types of descriptors were also suggested on the basis of simple physical models20 or machine-learning schemes. 21,22 There are, however, severe limitations in accuracies and applicability of the methods using the physics-based descriptors. In any approache, the correlations between the descriptors and binding energies include empirical parameters, and they need to be determined by “learning” DFT data prepared before the predictions. The determined functions possess a certain level of universality beyond the small number (typically 10−100) of stored DFT data as long as physical rules that authenticate the descriptors are effective. Unfortunately, these rules have many limitations, and their accuracies and universality fall into a trade-off situation. It should also be noted that most of these methods require quite expensive DFT calculations on NPs to obtain reasonable descriptors and/or the training data. Here, we report an alternative method using a machinelearning scheme that can resolve the problems described above. The method relies on the fact that catalytic active sites are determined by their local atomic configurations; in other words, two sites having a similar atomic configuration should give similar catalytic activity. The schematic of the proposed algorithm is illustrated in Figure 1A, and its outlines are explained below. Details of the basic equations and parametrizations are described in section S1 in the Supporting Information. The algorithm is composed of two steps: learning DFT data on SC surfaces and extrapolating the lessons to NPs. In the learning step, DFT data on SC surfaces are stored in a database. The data consist of geometrical information on unrelaxed surface sites of SC surfaces determined from the bulk geometry and binding energies of reaction intermediates on the relaxed surface sites. Here, the word “relaxed” means geometries and energies obtained by optimizing the intermediate and surface structures by the DFT calculations. Next, 4280
DOI: 10.1021/acs.jpclett.7b02010 J. Phys. Chem. Lett. 2017, 8, 4279−4283
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The Journal of Physical Chemistry Letters demonstrate its usefulness. Efficient NO removal technology under lean-burn conditions has been intensely studied to satisfy increasingly strict emission requirements for gasoline and diesel engines, and catalytic converters based on the direct decomposition of NO without any reducing agent, namely, 2NO → N2 + O2, would be ideal for small passenger vehicles.12,24 Several pure metal catalysts are known to activate the NO dissociation, which is the first step of the NO decomposition, but those catalyst surfaces are known to be poisoned by oxygen atoms generated by the dissociation because of their too strong binding energies with the oxygens.25 Past studies indicate that alloying the catalysts with Au can enhance the oxygen desorption,26−28 and the results encourage development of the bifunctional Rh(1−x)Aux alloy catalyst. Figure 1B−F summarizes results of the predictions on binding energies of three reaction intermediates N, O, and NO of the direct decomposition of NO and formation energies of Rh(1−x)Aux alloys from the pure Rh and Au bulks. For predictions of the binding energies, DFT data on Rh(1−x)Aux (x = 0.0−1.0) SC surfaces with randomly distributed Au atoms were stored as training data and used for calculating KKJ and for determining wJ. For the predictions of formation energies, DFT data on Rh(1−x)Aux bulks were also used. Parts of the stored data were used as test data to check the applicability of the determined regression coefficients to SCs other than the training data. To check the accuracy of the predictions on NPs, DFT calculations were executed on Rh(1−x)Aux alloy NPs with a diameter of 2 nm, and the results were compared with the predicted results. Details of the data preparation and used structural models are described in section S2 in the Supporting Information. In Figure 1B, mean absolute errors σ in the binding energies of O for the test data on SC surfaces are shown as functions of the number of the training data Ndata. As expected, the error σ can be decreased by increasing Ndata for any Rcut. The reachable accuracy, however, depends on Rcut; the longer the Rcut, the smaller the reachable σ (see also Figure S16 in the Supporting Information). When Rcut is short, the dimension of the variables composing the SOAP KIJ is small because the number of atoms inside of the radius Rcut is few. In such a situation, the number of linearly independent SOAPs decreases, and the feature space spanned by the SOAP loses the flexibility. After careful examinations, Rcut and other hyperparameters were determined as shown in Tables S1 and S2 in the Supporting Information. Figure 1C−F summarizes the binding energies and formation energies predicted by using the parameters. The test DFT data on both SC surfaces and NPs are reproduced with mean absolute errors of about 100 meV for the binding energies and 20 meV for the formation energies. The generated regression models were applied to predict the catalytic activity of the direct decomposition of NO. In the predictions, the NPs were modeled by truncated octahedral particles with diameters of 2.0−5.0 nm, as shown in Figure S17. The stable atomic distributions inside of the NPs were determined by Monte Carlo simulations at 500 K in vacuum using the formation energies given by the regression model. The catalytic activities on the determined NPs were calculated by Sabatier analysis using the predicted binding energies and BEP relations similarly to the past studies.2,8,12,16 Details of the Monte Carlo simulations and Sabatier analysis are described in section S3 in the Supporting Information. The predicted catalytic activities (TOFs) are plotted as functions of the Au atomic ratio x in Figure 2. At any diameter, the catalytic activity has a volcano-type correlation versus x, while both the maximal
Figure 2. Predicted turnover frequencies (TOFs) per surface site at 500 K and a structure of the active site on Rh(1−x)Aux.
catalytic activity (TOFmax) and its x value (xmax) increase with the decrease in the diameter d (see also Figure S20 in the Supporting Information). The origin of the size dependence of xmax is clarified by analyzing atomic distributions in the NPs. As shown in Figure S20A in the Supporting Information, the Au atomic ratio xmaxsurf on the particle surface giving TOFmax is always higher than the net Au atomic ratio xmax and less strongly depends on the diameter d. The results mean the presence of surface segregations of Au atoms. The surface segregations also appear in atomic distributions shown in Figure 3, where Au atoms are segregated at corners and edges of NPs at low x and gradually cover the whole surfaces with the increase in x. The surface segregation generates the size dependence of xmax shown in Figures 2 or S20A. If all of the introduced Au atoms are segregated on the surfaces, the number of surface Au atoms should linearly depend on xd3, while the number of total surface atoms is proportional to d2. The Au atomic ratio on the surface, therefore, should be proportional to xd, indicating that on the smaller NPs, the higher xmax is necessary for achieving the optimal surface composition xmaxsurf. The origin of the size dependence of the maximal catalytic activity TOFmax shown in Figures 2 and S20B is clarified by examining the distributions of local catalytic activity. Figure 3, for example, clearly indicates that the alloyed corners of NPs shown in Figure 2 are the active sites. Because the total number of corners on the truncated octahedral NPs does not depend on the diameter d, as shown in Figure S17, the surface density of the active corner sites linearly depends on d−2, and as shown in Figure S20B, the calculated TOFmax, indeed, obeys this trend. The mechanism of the activity enhancement at the alloyed corner sites can be clarified by examining the local energetics. As indicated by Figure 4A, free energies of all of the intermediate and transition states at the active site are lifted up (or pulled down) from those on the Rh(211) (or Au(211)) SC surface, and the free energy diagram is significantly flattened. The results indicate that the intermediates moderately bind with the active site, and the rates of the formations and removals of the intermediates are balanced on the basis of the Sabatier principle. It should be noted that, similarly to other catalytic reactions, the Sabatier principle that appears in our analyses relies significantly on so-called scaling relations,2,16 which ensure that all intermediates are stabilized when one of them is stabilized. The scaling relations are, for example, illustrated by maps of binding energies of N, O, and NO on the Rh(1−x)Aux NP shown in Figure 4C−E, where all intermediates bind more strongly with Rh-rich sites. More quantitatively, the scaling relations are shown in Figure S21 in the Supporting 4281
DOI: 10.1021/acs.jpclett.7b02010 J. Phys. Chem. Lett. 2017, 8, 4279−4283
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Figure 3. Distributions of atoms and the local activation free energy ΔGs* on the Rh(1−x)Aux NP with d = 3 nm.
the intermediates and lateral interactions among the intermediates, as discussed in section S3, the same scheme can also be applied to the predictions of properties necessary for going beyond these approximations, such as the diffusion barriers and binding energies including the lateral interactions. We, therefore, believe that the proposed data-driven method is a useful and efficient approach applicable to a wide variety of catalytic properties.
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ASSOCIATED CONTENT
* Supporting Information S
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.7b02010. Computational methods and additional tables and figures showing SOAP parameters, surface models, geometries, parameters, and structures, basis set, training data, and test data information, formation energies, error data, nuclear motion contributions, coefficients in Brønsted− Evans−Polanyi relationships, atomic ratios, turnover frequencies, and binding energies (PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Ryosuke Jinnouchi: 0000-0002-0822-1161
Figure 4. (A) Free energy diagrams of the direct decomposition of NO at 500 K on the active site of the Rh(1−x)Aux NP with x = 0.19 and d = 3 nm (Rh578Au133 in the chemical formula) and on Rh(211) and Au(211) SC surfaces. Circles and triangles indicate intermediate and transition states. (B−E) Atomic distributions and binding energies of N, O, and NO with the surface sites on the Rh(1−x)Aux NP with x = 0.19 and d = 5 nm.
Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors appreciate helpful advice on computational methods given by Dr. Kazutoshi Miwa and Dr. Nobuko Ohba at Toyota Central R&D Laboratories., Inc. The authors also appreciate fruitful discussion on the machine-learning algorithms with Dr. Gábor Csányi at the University of Cambridge and Dr. Michele Ceriotti at Ecole Polytechnique Fédérale de Lausanne.
Information. By the scaling relations, the free energies of all intermediates move in a same direction, as illustrated in Figure 4A, and this constraint generates the volcano-type trend in Figure 2. In summary, the binding energies of atoms and molecules with the NPs and formation energies of NPs were shown to be predicted by the machine-learning scheme within practically useful accuracies. The method was applied to the direct decomposition of NO on the Rh(1−x)Aux NPs, and the application showed that the kinetic analysis using the predicted binding energies can provide useful information on surface segregations, size and composition dependences of catalytic activities, and detailed structures of active sites. Although the simple Sabatier analysis adopted in this study still includes many approximations, such as neglects of surface diffusions of
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REFERENCES
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