Data Driven Determination in Growth of Silver from Clusters to

Jul 7, 2019 - The thresholds among atomic clusters, nanoparticles, and the bulk state have been ambiguous. A potential solution is to determine cluste...
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Letter Cite This: J. Phys. Chem. Lett. 2019, 10, 4063−4068

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Data Driven Determination in Growth of Silver from Clusters to Nanoparticles and Bulk Keisuke Takahashi*,†,‡ and Lauren Takahashi†,‡ †

Department of Chemistry, Hokkaido University, Sapporo 060-8510, Japan Center for Materials Research by Information Integration (CMI2), National Institute for Materials Science (NIMS), 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan

‡

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S Supporting Information *

ABSTRACT: The thresholds among atomic clusters, nanoparticles, and the bulk state have been ambiguous. A potential solution is to determine cluster growth toward bulk, but this is challenging to determine with experiments and computation. Data science is proposed to predict atomic cluster growth and determine the cluster−nanoparticle−bulk thresholds using Ag clusters as a prototype element. Supervised machine learning reveals that Ag cluster growth has nonlinear models where nonlinear machine learning is found to accurately predict binding energy. Unsupervised machine learning discovers three groups (cluster, semiclusters, and nanoparticles) where linear regression is used to predict the binding energy in each group. Furthermore, machine learning reveals the linear relationship between binding energy and the surface-to-volume ratio of Ag nanoparticles. This allows for a binding energy estimation of large Ag nanoparticles and a revelation of how Ag nanoparticles grow toward the bulk. Thus, data science is proposed as a powerful tool for determining cluster growth and thresholds for clusters, nanoparticles, and bulk states.

D

machine learning are proposed as an effective tool for determining and predicting such thresholds. Silver clusters are chosen as a prototype cluster for investigating the threshold of clusters, nanoparticles, and bulk as silver clusters have a wide range of applications including catalysts where the size of silver catalysts plays an important role, making the determination of such thresholds important.9−11 The structure of silver clusters has been extensively investigated as physical and chemical properties of Ag clusters are strongly coupled with their structure, making it crucial to identify the thresholds between clusters and nanoparticles.12−15 Similarly, properties of silver nanoparticles are also linked to the size and shape of nanoparticles; however, there is no clear definition of the nanoparticles in comparison to atomic clusters.7,16,17 Therefore, the thresholds of clusters, nanoparticle, and bulk in silver clusters are investigated by combining first principle calculations and machine learning. A grid based projector augmented wave (GPAW) method within density functional theory (DFT) calculations is implemented.18 The exchange correlation of Perdew−Burke− Ernzerhof (PBE) is implemented along with spin polarization calculation.19 Γ point is used for all calculations where 8 Å of vacuum is applied for all directions. The ground state structures of Ag clusters are explored using basin hop-

etermining the thresholds among atomic clusters, nanoparticles, and the bulk has been a great mystery within the field of material science as there is no clear definition of such boundaries.1,2 In general, atomic clusters consist of few to tens of hundreds of atoms where the physical and chemical properties of clusters are strongly coupled with the number of atoms and structure of the cluster.3 As the size of atomic clusters grow, one can see that the properties of atomic clusters become more constant where such a state is described as a nanoparticle.4 However, there is no clear threshold between clusters and nanoparticles. Furthermore, one can assume that the properties of nanoparticles would eventually reach the properties of the bulk state as the size of the nanoparticle increases, though it is ambiguous as to how many atoms in the nanoparticle would behave as a bulk state. Thus, it has been challenging to identify the clear definition of clusters, nanoparticles, and bulk. With the rapid development of computational algorithms such as the basin hopping algorithm and the genetic algorithm, it becomes achievable to determine the structure of clusters within first principle calculations.5,6 A series of such calculations would generate data sets of atomic clusters, though it is still challenging to calculate large clusters or nanoparticles as the number of possible structures dramatically increases as the size of the cluster increases.7 Here, if there is a trend present in the cluster data, one can consider that the prediction of how clusters grow and a clearer threshold for cluster, nanoparticle, and bulk can be uncovered and clarified in principle.8 In particular, supervised and unsupervised © 2019 American Chemical Society

Received: May 15, 2019 Accepted: July 7, 2019 Published: July 7, 2019 4063

DOI: 10.1021/acs.jpclett.9b01394 J. Phys. Chem. Lett. 2019, 10, 4063−4068

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The Journal of Physical Chemistry Letters ping.5,20,21 The binding energy per atom (Eb) of Ag clusters is calculated as eq 1: Eb =

the rest of the data is used for comparison during the prediction. Accuracy of each machine learning model evaluated by cross validation is collected in Table 1. Table 1 indicates

E[Ag n] − nE[Ag1]) n

Table 1. Comparison of Different Machine Learning Models for Predicting Binding Energy of Agn Clustersa

(1)

where n is the number of Ag atoms. Note that negative energy indicates an exothermic reaction. Two types of machine learning, supervised and unsupervised machine learning, are implemented within sciki-learn (ver. 0.17).22 Supervised machine learning algorithms are used for fitting and regression purposes within clustered data while unsupervised machine learning is used for applying probabilistic models toward determining possible physical or chemical thresholds between cluster size and binding energy. In the case of supervised machine learning, the following three regression models are applied: least-squares linear regression (LR), support vector regression (SVR), and random forest regression (RF). LR and SVR (with linear kernel) are linear models while SVR (with rbf kernel) and RF are treated as nonlinear models. Line-fitting is performed within the data when using LR SVR (with linear kernel). Additionally, the radial basis function (rbf) kernel method is applied in SVR in order to treat data as a nonlinear regression. The decision trees are constructed in RF through the ensemble method where the majority of decisions are taken as the prediction. Thus, four different types of regression models are prepared. Hyperparameters of LR, SVR (with linear kernel), SVR (with rbf kernel), and RF are also optimized. The least-squares method is used for LR. Cost(C) for SVR (with linear kernel) is set to 1 while C and Γ in SVR (with rbf kernel) are set to 1 and 0.01, respectively. The number of trees for RF is set to 1000 where the maximum depth of the random forest is expanded until all leaves become 1. LR, SVR, and RF are implemented in order to apply curve fitting for representing the relationship between the size of Ag clusters and binding energy. Each machine learning model is evaluated through cross validation where the data is randomly separated into 80% trained data and 20% test data, and the average score of 10 randomly selected trained and test sets is evaluated. The Gaussian mixture model (GMM), an unsupervised machine learning model, is implemented in order to classify the data and determine the chemical or physical thresholds that may possibly exist within the data based on the relation between cluster size and binding energy. The hyperparameter, covariance type in GMM, is set to full with each component having an independent matrix with the number of components as 3. The ground state structures of Agn (n = 2−45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 99) clusters are explored using the basin hopping algorithm. Ground state structures and binding energy per atom of Agn are collected in Supporting Information and are compared with previously reported structures, which have been found to show good agreement with previous reports of structures created using the PBE exchange correlation and the B3LYP functional and have also been found to support experimental results.12−15,23 Here, machine learning is implemented to predict the binding energy of Agn where descriptor and objective variables are set to the size of the cluster and binding energy per atom, respectively. In particular, the following four types of machine learning regression models are used: RF, SVR with rbf kernel, SVR with linear kernel, and LR. Note that RF and SVR are trained by Agn (n = 1−20, 30, 40, 50, 60, 70, 80, 90, 99) while

machine

type

score

SD

hyperparameter

RF SVR SVR LR

nonlinear nonlinear linear linear

0.88 0.81 0.26 0.24

0.17 0.12 0.47 0.75

number of tree: 1000 kernel: RBF, C: 1, γ: 0.01 kernel: linear, C: 1 least squares

a

Score is evaluated by cross validation. The size of cluster, n, is used as a descriptor. Machine: type of machine learning model. Type: linear or nonlinear model. Score: score in cross validation. SD: standard deviation. Hyperparameter: hyperparameter for each machine learning model.

that RF and SVR with rbf kernel results in high scores of 0.88 and 0.81 in cross validation, respectively, while linear based machine learning models result in low scores. Thus, the binding energy of Agn clusters against size is found to be a nonlinear model. Prediction of binding energy is performed using trained RF and SVR(rbf kernel). Predicted binding energy by RF and SVR is shown in Figure 1 along with the calculated binding energy

Figure 1. Predicted binding energy of Agn (n = 1−99) clusters using trained RF, SVR(rbf kernel), SVR(linear kernel), and LR. The size of cluster, n, is used as a descriptor. DFT test and train indicate the calculated binding energy using density functional theory calculation for train and test data, respectively.

by DFT. Note that Agn (n = 21−29, 31−39,41−45, 55, 65, 75, 85) in DFT is test data which is not trained by RF and SVR in Figure 1. Figure 1 shows that RF accurately predicts binding energy where the average error is 0.5% (0.7% for test data) when compared to those by DFT. On the other hand, prediction by SVR roughly follows the trends of DFT results with an average error of 2.4% while the average error for test data is 4.9% when compared to the DFT results. Needless to say, linear machine learning, SVR(linear) and LR, failed to predict binding energy. As a result, one can consider that nonlinear models, RF and SVR, can be used to estimate the binding energy of Agn clusters using the size of cluster. The geometrical information on the Ag clusters is then considered for predicting the binding energy. In particular, the average distance per atom of each cluster is evaluated. Further details regarding the distance information are collected in Supporting Information. Here, cluster size and average 4064

DOI: 10.1021/acs.jpclett.9b01394 J. Phys. Chem. Lett. 2019, 10, 4063−4068

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The Journal of Physical Chemistry Letters distance per atom in the cluster are set as descriptor variables while the binding energy per atom is set as the object variable. The accuracy of each machine learning model evaluated with the two descriptor variables is collected in Table 2. Table 2 Table 2. Comparison of Different Machine Learning Models for Predicting Binding Energy of Agn Clustersa machine

type

score

SD

hyperparameter

RF SVR SVR LR

nonlinear nonlinear linear linear

0.92 0.81 0.50 0.85

0.11 0.12 0.40 0.10

number of tree: 1000 kernel: RBF, C:1, γ: 0.01 lernel: linear, C:1 least squares

a

Score is evaluated by cross validation. The size of cluster, n, and corresponding average distance per atom in cluster are used as descriptors. Machine: type of machine learning model. Type: linear or nonlinear model. Score: score in cross validation. SD: standard deviation. Hyperparameter: hyperparameter for each machine learning model.

illustrates that the cross validation scores for predicting binding energy are greatly improved in the cases of linear machine learning SVR(Linear) and LR. Similar to the prediction shown in Agn (n = 2−45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 99), prediction of binding energy using the size of cluster and average distance per atom in cluster is attempted. Here, Agn (n = 21−29, 31−39,41−45, 55, 65, 75, 85) is used as the test data while Agn (n = 2−45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 99) is used as the training data. Figure 2a shows that RF and LR predict the binding energy of the Ag cluster. Extrapolation of predicting the binding energy is performed using LR where Agn (n = 147, 309, 561) species are estimated as shown in Figure 2b. From the figure, it becomes apparent that LR is unable to accurately reflect the binding energy−cluster size relationship once the cluster size surpasses 100 atoms. This can be considered to be due to binding energy becoming less constant as cluster size increases. One can consider that there may be different models within the data, thereby leading to the implementation of unsupervised machine learning. Unsupervised machine learning, Gaussian mixture model (GMM), is implemented in order to find the hidden patterns in the binding energies of Agn (n = 2−45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 99). Within the GMM, the number of mixture components is set to 3 where each covariance is independent. The results predicted using GMM are shown in Figure 3. Figure 2 shows that Ag clusters are classified by the size of cluster into the following three groups: Ag2−Ag5, Ag6-Ag20, and Ag21−Ag99. The difference between the first and second groups can be considered to originate in the change of structure and binding energy. In particular, the structures of Ag2−Ag5 have a two-dimensional structure while the transition from a two-dimensional structure to a three-dimensional structure is observed between Ag6 and Ag7. Moreover, a large change of binding energy is observed in Ag2−Ag5 while the change of binding energy is relatively large. This transition point between Ag5 and Ag6 reflects findings in previous studies which have reported a transition from preferred molecular O2 adsorption to dissociative O2 adsorption between Ag5 and Ag6 as well as reported trends where the adsorption energy increases as the cluster size increases.14 These results therefore demonstrate that GMM was able to distinguish groups through changes in reactivity which are supported by previous studies. From these findings, one can consider that the first group

Figure 2. Predicted binding energy of Agn (n = 1−99) clusters using trained RF, SVR(rbf kernel), SVR(linear kernel), and LR where prediction for (a) size 2−99 and (b) after size is above 99, particularly, n = 147, 309, and 561 are shown. The size of cluster, n, and corresponding average distance per atom in cluster are used as descriptors. DFT-test and DFT-train indicate the calculated binding energy using density functional theory calculation for test and train data, respectively.

Figure 3. Three classified groups in binding energy of Agn by GMM. Atomic model of Ag cluster at each threshold is also displayed.

(Ag2−Ag5) has uniqueness in structure, energy, and oxidation and can therefore be described as clusters while the second group (Ag6−Ag20) can be defined as semiclusters. Thus, GMM proposed two regions which can be described as cluster and semicluster regions. The threshold between atomic clusters and nanoparticles has been ambiguous in the field of atomic clusters. Here, GMM proposes that there is a threshold of semiclusters and nanoparticles between Ag20 and Ag21 as shown in Figure 3. At 4065

DOI: 10.1021/acs.jpclett.9b01394 J. Phys. Chem. Lett. 2019, 10, 4063−4068

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The Journal of Physical Chemistry Letters

predicted. GMM determines that the Ag clusters behave as nanoparticles when the size of the Ag cluster is above 21 as shown in Figure 3; therefore, Agn (n = 21−45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 99, 147, 309, 561) are investigated. The LR model reveals that there is a linear relationship between binding energy and the surface-to-volume ratio which is calculated by the number of surface atoms divided by atoms not exposed at the surface. Therefore, descriptor and objective variables in the LR model are set to the surface-to-volume ratio and binding energy, respectively, where the average score in cross validation is 0.84%. In order to evaluate the generalization ability (predicting extrapolation) of the trained LR models, Ag561 is removed from the trained data where the LR models predict that the binding energy is −2.05 eV while the calculated binding energy is −2.13 eV, which is a 3.7% error. Thus, the constructed LR models demonstrate a good generalization ability. Prediction of binding energy against the surface-to-volume ratio is performed using the LR model in order to understand how Ag nanoparticles grow toward the bulk state. 1000 evenly spaced numbers between 0 and 10 are first generated as input variables representing the surface-to-volume ratio. Then, the generated 1000 surface-to-volume ratio variables are given to the trained LR models which returned the corresponding binding energy as shown in Figure 5a. The details of size, surface-to-volume ratio, and predicted binding energies are

this threshold, there is a change in structure where clusters grow toward a pyramid structure in Ag6−Ag20 while clusters grow in a round and circular shape after Ag21. This can be described in that a threshold between a semicluster and nanoparticles can be supported by the structure change. Hence, GMM reveals the thresholds between clusters and the nanoparticle region. A least-squares linear regression (LR) model is implemented in each Ag2−Ag5, Ag6−Ag20, and Ag21−Ag99 group classified by GGM shown in Figure 2. Three different LR models, namely, LR1, LR2, and LR3, are constructed where the descriptor variable for each model is set to the size of cluster [LR1, Agn (n = 2−5); LR2, Agn (n = 6−20); LR3, Agn (n = 20−45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 99)] and the objective variable is set to corresponding binding energy for all cases. The average scores of the LR2 and LR3 models in cross validation are 0.92% and 0.84%, respectively. Note that cross validation is not applicable in the LR1 model as there are only 4 data points. Prediction of binding energy using each LR model is then performed where the predicted binding energy by LR and calculated binding energy by DFT are shown in Figure 4. The

Figure 4. Prediction of the binding energy of Ag clusters using the following 3 different LR models based on classified group by GMM: LR1, Agn (n = 2−5); LR2, Agn (n = 6−20); LR3, Agn (n = 20−45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 99). DFT is the calculated binding energy using density functional theory calculation.

details of the predicted binding energy are collected in Supporting Information. Figure 4 shows that each LR model accurately estimates the binding energy where the average errors of binding energy by LR1, LR2, and LR3 are 0.4%, 0.01%, 0.06%, respectively. Hence, it is revealed that three groups (clusters, semiclusters, nanoparticles) classified by GMM enable the implementation of the LR model which has an advantage in estimating extrapolation compared to the nonlinear machine learning model. Growth of nanoparticles toward the bulk is explored using machine learning. Here, binding energies of large Agn (n = 147, 309, 561) clusters are calculated using density functional theory in order to improve the diversity of data where the structure of Ag309 has good agreement with the previously reported structure calculated by effective medium theory potential.7 The structure and binding energies of Agn (n = 147, 309, 561) are collected in Supporting Information. In general, the properties of nanoparticles are defined to be constant compared to atomic clusters.4 If one can understand the properties that are consistent for nanoparticles, then the growth of nanoparticles toward the bulk can, in principle, be

Figure 5. (a) Prediction of the binding energy against surface/volume of Agn using linear regression model. DFT is the calculated binding energy of Agn (n = 21−45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 99,147, 309, 561) using density functional theory calculation. (b) Surface/ volume against corresponding number of atoms in Agn (n = 21−45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 99, 147, 309, 561, 1007, 2209, 6087, 12731, 37349). The numbers within the parentheses represent the corresponding surface-to-volume ratio. 4066

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listed in Supporting Information. Figure 5a shows that the predicted binding energy has good agreement with the calculated binding energy by DFT. Interestingly, the binding energy reaches −2.10 eV with a surface-to-volume ratio of 0.01 while the binding energy of bulk Ag is −2.31 eV. The reason for this is that technically the binding energy of Ag nanoparticles does not reach the bulk binding energy as nanoparticles always have a surface where the binding energy of bulk Ag is calculated in periodic boundary condition in all directions. This can be explained by Figure 5b which demonstrates the number of atoms in Ag clusters against the surface-to-volume ratio. In order to observe how the surfaceto-volume ratio changes upon the increase of size, atomic models of Agn (n = 1007, 2209, 6087, 12731, 37349) are constructed which have a surface-to-volume ratio of 0.48, 0.29, 0.19, 0.14, and 0.10, respectively. In other words, one can observe that the change in the surface-to-volume ratio against the size of clusters drastically becomes small; thus, reaching the bulk energy can be considered as an infinity matter. Hence, the threshold between nanoparticles and bulk can be estimated using binding energy and the surface-to-volume ratio. More importantly, the linear relationship between binding energy and the surface-to-volume ratio estimates the binding energy of Ag nanoparticles using LR models. As a result, this would allow for the estimation of binding energy of large Ag nanoparticles; thus, growth of a nanoparticle toward the bulk state can be well-understood. Data science is implemented in order to reveal how Ag clusters grow toward nanoparticle and bulk states. Supervised machine learning demonstrates that the growth of Ag clusters in terms of binding energy is nonlinear matter. In particular, RF and SVR(rbf kernel) can accurately estimate the binding energy of Ag clusters. Unsupervised machine learning, GMM, classifies Agn clusters into three groups, clusters, semiclusters and nanoparticles, where linear regression can be used with each of the classified groups to predict binding energy. In addition, LR discovers that there is a linear relationship between binding energy and the surface-to-volume ratio in Ag nanoparticles. With the use of LR models, the binding energy of large Ag nanoparticles is achieved, resulting in an understanding of how Ag nanoparticles grow toward bulk. Fitting and regression based on supervised machine learning can hint toward new insight into how clusters would grow or details of interpolation within cluster data while unsupervised machine learning gives insight into how binding energy is distributed in a probabilistic model, hinting toward classification of the hidden groups in clusters. Thus, data science is proposed to be an effective approach to estimate and predict how clusters grow and to discover the threshold for clusters, nanoparticles, and the bulk state.

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Letter

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Keisuke Takahashi: 0000-0002-9328-1694 Lauren Takahashi: 0000-0001-9922-8889 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

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REFERENCES

This work is funded by Japan Science and Technology Agency (JST) CREST Grants JPMJCR17P2, JSPS KAKENHI Grantin-Aid for Young Scientists (B), and JP17K14803, and Materials Research by Information Integration (MI 2I) Initiative project of the Support Program for Starting Up Innovation Hub from JST.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.9b01394. Calculated and predicted binding energies, average distances per atom in cluster, ground state structure of Ag clusters, structure of large Ag nanoparticles estimated by EMT potential, and true and predicted regression models used in Table 1 (PDF) 4067

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The Journal of Physical Chemistry Letters (17) Benedetti, F.; Luches, P.; Spadaro, M. C.; Gasperi, G.; D’Addato, S.; Valeri, S.; Boscherini, F. Structure and Morphology of Silver Nanoparticles on the (111) Surface of Cerium Oxide. J. Phys. Chem. C 2015, 119, 6024−6032. (18) Mortensen, J. J.; Hansen, L. B.; Jacobsen, K. W. Real-space Grid Implementation of the Projector Augmented Wave Method. Phys. Rev. B: Condens. Matter Mater. Phys. 2005, 71, 035109. (19) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865. (20) Wales, D. J.; Doye, J. P. Global Optimization by Basin-hopping and the Lowest Energy Structures of Lennard-Jones Clusters Containing up to 110 Atoms. J. Phys. Chem. A 1997, 101, 5111−5116. (21) Takahashi, K.; Isobe, S.; Ohnuki, S. Chemisorption of Hydrogen on Fe Clusters through Hybrid Bonding Mechanisms. Appl. Phys. Lett. 2013, 102, 113108. (22) Pedregosa, F.; Varoquaux, G.; Gramfort, A.; Michel, V.; Thirion, B.; Grisel, O.; Blondel, M.; Prettenhofer, P.; Weiss, R.; Dubourg, V. Scikit-learn: Machine learning in Python. J. Mach. Learn. Res. 2011, 12, 2825−2830. (23) He, Y.; Zeng, T. First-principles Study and Model of Dielectric Functions of Silver Nanoparticles. J. Phys. Chem. C 2010, 114, 18023−18030.

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