The Use of Cluster Expansions To Predict the Structures and

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The Use of Cluster Expansions to Predict the Structures and Properties of Surfaces and Nanostructured Materials Liang Cao, Chenyang Li, and Tim Mueller J. Chem. Inf. Model., Just Accepted Manuscript • DOI: 10.1021/acs.jcim.8b00413 • Publication Date (Web): 18 Sep 2018 Downloaded from http://pubs.acs.org on September 18, 2018

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Journal of Chemical Information and Modeling

The Use of Cluster Expansions to Predict the Structures and Properties of Surfaces and Nanostructured Materials Liang Cao,1 Chenyang Li,1 and Tim Mueller1, *

1

Department of Materials Science and Engineering, Johns Hopkins University, Baltimore, Maryland 21218, United States

*Email: [email protected]

ABSTRACT The construction of cluster expansions parameterized by first-principles calculations is a powerful tool for calculating properties of materials. In this perspective, we discuss the application of cluster expansions to surfaces and nanomaterials. We review the fundamentals of the cluster expansion formalism and how machine learning is used to improve the predictive accuracy of cluster expansions. We highlight several representative applications of cluster expansions to surfaces and nanomaterials, demonstrating how cluster expansions help researchers build structure-property relationships and enable rational design to accelerate the discovery of new materials. Potential applications and future challenges of cluster expansions are also discussed.

Keywords: materials informatics, machine learning, cluster expansion, Bayesian, nanomaterials, nanoparticles, surfaces, alloys 1 ACS Paragon Plus Environment

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INTRODUCTION Cluster expansions1 are generalized Ising models2 that account for many-body interactions. In the fields of materials science, physical chemistry and condensed matter physics, cluster expansions are widely used to study materials that exhibit substitutional disorder, in which some crystalline sites can be occupied by more than one type of atom. In 1973, Baal3 did pioneering work of calculating the order-disorder phase diagrams in an fcc binary alloy using the cluster expansion formalism, and in the following years cluster expansions have been successfully used to study a variety of bulk (i.e. periodic in three dimensions) crystalline materials.3-25 In recent years, advances in the development of nanostructured materials have driven the application of cluster expansions to low-dimensional systems.25-60 Due to the limitations of experimental methods and the computational cost of studying low-symmetry materials, structureproperty relationships for surfaces and nanostructured materials remain relatively poorly understood. However, the continuing increase in available computational power and advances in machine learning that significantly reduce the cost of generating cluster expansions have made it feasible in recent years to use cluster expansions to study low-dimensional systems at experimentally relevant length scales. In this perspective, we present the evolution and improvement of the use of cluster expansions to predict the structures and properties of various material systems, with a focus on surfaces and nanostructured materials. In the following section we present the basics of the cluster expansion approach and the role of machine learning in training cluster expansions. In subsequent sections we review recent advances for the use of cluster expansions on surfaces25-42 and nanoparticles4360

. We then conclude by discussing some of the current opportunities and challenges in this field. 2 ACS Paragon Plus Environment

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OVERVIEW OF CLUSTER EXPANSIONS Basic theory of cluster expansions The origins of cluster expansions can be traced back to the early 1950s, when Kikuchi developed the cluster variation method to study order-disorder phenomena in Ising lattices.61 In 1984, Sanchez et al.1 developed a general formalism for cluster expansions as a complete basis set expansion. Here, we review the basic idea of the cluster expansion based primarily on the work of Sanchez et al.1 In cluster expansions for systems with substitutional disorder, the “spin” variables used to describe the single atomic magnetic moments in an Ising model2 are replaced by discrete “site” variables that indicate which atom (or vacancy) is present at a certain site. In Figure 1, we show a straightforward view of how to assign site variables in a simple two-dimensional Pt-Ni alloy on a square lattice. Each site in space lattice is assigned a site variable, s j , where s j = 1 indicates it is occupied by a Ni atom, and s j = − 1 indicates it is occupied by a Pt atom (Figure 1a, c).

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Figure 1. An illustration of site variables and examples of clusters in a simple two-dimensional square lattice. (a) The values assigned to site variables could represent (b) atomic spins as in the Ising model or (c) the element occupying each site. (d) Examples of clusters in a cluster expansion. The one atom marked as “A” is a single-site cluster, the two atoms marked as “B” are in a nearest-neighbor two-site cluster, the two atoms marked as “C” are in a second-nearestneighbor two-site cluster, and the three atoms marked as “D” are in a three-site cluster.

For each site variable, an orthonormal single-site basis is defined,1 such that: Nj

∑ Θ (s )Θ b

j

b'

s j =1

(s j ) = δ bb '

Nj

(1)

where s j is the site variable for the j th site, N j represents the number of values this variable may take, Θ b ( s j ) is the b

th

basis function for that site and δ bb ' is the Kronecker delta. The

tensor product of all single-site bases produces a basis of “cluster functions”, Φ b (s) . Each cluster function can be defined by a single vector b,


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Φb (s) = ∏ Θbj (s j ),

(2)

j

where s is the set of all site variables and bj is the index of the basis function used at site j. The single-site basis is chosen so that one of the basis functions is the constant “1”, and as a result each cluster function Φ b (s) only depends on s j for the subset of sites for which Θb j ( s j ) is not 1. This subset of sites is known as a cluster, and each cluster function represents an interaction between the sites in the cluster. The number of sites in a cluster could range from 1 to all the sites in the material. Examples of clusters are provided in Figure 1d. In two-component systems there are only two single-site basis functions, and it is convenient to define the second single-site basis function as Θ2 ( s j ) = s j . As a result the cluster functions in Equation (2) are simply products of the site variables for the sites in the corresponding clusters. A property (such as energy) of the material can be expressed as a linear expansion of cluster functions,

F (s) = V0 + ∑VbΦb (s)

(3)

b

where the unknown coefficients Vb are known as effective cluster interactions (ECI), V0 is a constant term representing the “empty” cluster, s is the vector of site variables, and Φ b is the

b th cluster function. The sum in Equation (3) is over all possible cluster functions. When all possible cluster functions are included in the cluster expansions, Equation (3) is exact. However, the ECIs for clusters that contain a large number of sites or the sites that are far apart are usually negligible. Therefore, the cluster expansions are usually truncated to a sum over finite numbers of cluster functions with little loss of accuracy. The symmetry of the system may be used to



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further reduce the total number of terms in the expansion by making use of the fact that symmetrically equivalent cluster functions must have identical ECI. Although they are lattice models, cluster expansions trained on structures in which the atomic positions have been relaxed will then predict the energies of relaxed materials; i.e. there is no implicit assumption that the atoms stay on their ideal lattice sites. Cluster expansions are typically capable of calculating the energies of millions of atomic arrangements per minute on a single processor with a level of accuracy that is within 10 meV / atom of density functional theory (DFT)

42, 51-56, 62, 63

, which is the level of accuracy required to distinguish between the

energies of competing ground states for many alloy systems.64 Because of this combination of speed and accuracy, cluster expansions are often used to search the space of possible atomic arrangements for optimal (e.g. ground state) structures, used in Metropolis Monte Carlo65 simulations to calculate thermodynamic averages, or used in kinetic Monte Carlo (KMC)66, 67 simulations to simulate the kinetic evolution of a system.

Machine learning approach to cluster expansions The cost of building a cluster expansion model comes primarily from the cost of generating the training data (usually by DFT) to which the ECIs are fitted. This is particularly problematic for surfaces and nanoparticles, as the loss of translational symmetry typically increases both the amount of training data that must be generated to reach a given level of accuracy and the cost of generating each data point. Thus, significant effort has been put into developing approaches to generating cluster expansions in a way that minimizes prediction error for a given training set size. Machine learning approaches such as cross validation,6 active learning,6,


68, 69

feature

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selection,47, 49, 50, 70-73 and regularization23, 70, 73 are all well established in this field and commonly used to address this problem. A common approach to creating cluster expansions is to use “cluster selection”, in which a set of small, compact clusters is selected for inclusion in the expansion in a way that minimizes the expected prediction error of the expansion. Typically a form of cross-validation (CV), in which the predictive accuracy of the expansion is evaluated on subsets of the training data that are left out of the fitting procedure, is used to estimate the prediction error.6 It is common for leave-oneout cross validation to be used for computational efficiency,6 but other forms of cross-validation are often used to reduce the risk of overfitting the data.74-77

Methods such as genetic

algorithms47, 49, 50 may be used to search the space of possible subsets of clusters to include to identify ones with low cross-validation error. The ECI for the included clusters are then determined using a least-squares fit. In 2009, Mueller and Ceder73 applied machine learning concepts to develop a Bayesian approach to fitting cluster expansions that explicitly incorporates physical insights into the nature of ECIs via a prior probability distribution with adjustable parameters. The prior probability distribution serves as an educated guess of the likelihood of ECI values before calculating the property values. For example, the prior probability distribution could be used to express the insight that ECI are typically more likely to be on the order of meV / atom than keV / atom.73 The exact shape of the prior probability distribution may be determined using cross-validation. Many common forms of fitting cluster expansions can be expressed in the Bayesian framework.73 For example, cluster selection (Figure 2a), the prior probability distributions are delta functions for cluster functions that are excluded and uniformly constant functions for cluster functions that are included. In the “compressive sensing”71 and “LASSO”72 approaches, 7 ACS Paragon Plus Environment


a Laplace prior distribution over the ECI values is used (Figure 2b). This is equivalent to ℓ1 regularization of the ECI, in which a term proportional to the sum of the absolute values of the ECI is used to penalize a candidate set of ECI values.71 The use of the ℓ1 norm produces sparse cluster expansions with relatively few non-zero ECI.73, 78 This can speed up the evaluation of cluster expansions, although typically the computational cost of generating training data is greater than the cost of using the resulting cluster expansions. Alternatively, Gaussian prior probability distributions, resulting in ℓ2 regularization, can be used. Although this approach typically results in more non-zero ECIs than ℓ1 regularization, it has the advantages of being simpler to implement due to the use of a prior probability distribution with no derivative discontinuity, and the results are invariant with respect to a rotation of the single-site basis.78 The use of prior probability distributions is most effective when the prior probability distributions are physically motivated.73 For example, the magnitudes for ECI typically decline as the number of sites in the cluster and/or the maximum distance between sites increases, as atoms interact more strongly when they are close to each other. This observation and other physically-motivated observations can be directly incorporated into the prior probability distributions (Figure 2c). Because there are relatively few parameters to be optimized when using leave-one-out cross-validation to determine the shape of a physically motivated prior probability distribution, the risk of overfitting the data when using this approach is significantly lower than when using cluster selection.73

Using such physically-motivated priors can

significantly reduce the prediction error for a given training set size.52, 73 In surfaces and nanoparticles, there are far more distinct and significant ECIs than in the corresponding bulk material due to the loss of periodicity. Clusters that are symmetrically equivalent in the bulk may become merely congruent in a surface or nanoparticle (Figure 3a, b). 8 ACS Paragon Plus Environment

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Because such congruent clusters are not symmetrically equivalent, they won’t necessarily have the same ECI. However it can be expected that their ECI will be similar to each other, and a Gaussian prior probability distribution can be used to couple the ECI for congruent clusters (Figure 3c), accelerating the convergence of the cluster expansion.73

Figure 2. Bayesian interpretation of (a) least-squares fit, (b) LASSO / compressive sensing, and (c) physically motivated Gaussian prior probability distributions under the assumptions that ECI typically become smaller as the distance between sites in the cluster becomes larger. V is the ECI value and P(V) is the prior probability density for values of V. The dark vertical arrows indicate Dirac delta functions.



Figure 3. The illustrations of symmetric and congruent clusters near (a) a surface on a square lattice and (b) a nanoparticle with a fixed shape. Sites with the same color in each figure are symmetrically equivalent. The cluster of sites marked with “a” is symmetrically equivalent to the cluster marked with “b”, as are “d” and “e”. Clusters “a” and “c” are congruent, as are “d” and “f”. (c) Prior probability distribution of the difference between the ECIs for congruent clusters. For instance, V1 and V2 could be the ECIs for cluster “a” and “c”, or “d” and “f”.

CLUSTER EXPANSIONS OF EXTENDED SURFACES The use of cluster expansions on surfaces starts from the predictions of surface energies of alloys.25-27, 29 In 1999, Sadigh et al.26 investigated the phase stability of monolayer Pd-Au alloys on Ru(0001) using a cluster expansion parametrized by DFT calculations. A schematic for this approach is provided in Figure 4a. A similar approach has been used to study monolayer Pt– Ti(111) alloy subsurface,35 and polymorphism of 2D close-packed triangular boron.36 Subsequently, cluster expansions were used to study slabs in which more than one layer could have substitutional disorder (Figure 4b), including the phase diagram of CoAl(100) binary alloy surfaces,29 surface segregation and ordering of Cu3Pt(111) surfaces,31 and the ordering configurations of eight ternary MXene alloys (2D transition metal carbides and nitrides) with


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multi-layers slabs.40 Cluster expansions have also been used to predict the atomic structures of SiGe nanowires as a function of composition and diameter.33, 39, 41 Cluster expansions have also been used to predict the arrangements of surface adsorbates either by considering only the arrangement of adsorbates on a monolayer of surface sites (Figure 4c) or by accounting for substitutional disorder in the underlying material as well (Figure 4d, e). As an early example of the former approach, in 2003 Sluiter et al. used a cluster expansion to investigate the ground-state atomic order of adsorbed hydrogen on a monolayer of graphene as a function of hydrogen coverage.28 The application of cluster expansions to predict the atomic order and/or binding energies of surface adsorbates has continued on the Ir(100)-(5×1) surface with adsorbed hydrogen32 and Pt(111) surfaces with adsorbed oxygen34 and hydrogen.37 More recently, kinetic Monte Carlo (KMC) simulations based on a cluster expansion of surface adsorbates have been used to determine the rate-determining step for HCl oxidation on RuO2(110).38



Figure 4. Schematic illustrations of slab models of surfaces. Green and blue spheres represent different species that could occupy each site. Black spheres represent the sites not included in cluster expansion models. (a) Only the outmost layer and (b) all layers of the slab are modeled in the cluster expansion in vacuum. (c) Only adsorbates are included in the cluster expansion. (d) Only the outmost layer and (e) all layers of the slabs are included in the cluster expansion with adsorbates on the surface. In (c-e), red and white spheres represent adsorbates and vacancies. 12 ACS Paragon Plus Environment

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Surface segregation driven by adsorbates Surface segregation, in which one element preferentially migrates to surface sites, can be driven by surface adsorbates in different chemical environments (e.g. oxidizing or reducing conditions).79 In 2005, Han et al.30 did pioneering work to develop a coupled binary cluster expansion of a Pt/Ru surface with adsorbed oxygen to model surface segregation in Pt–Ru(111) surfaces in an oxidizing environment. They built a coupled binary cluster expansion of the Pt– Ru(111) surface in which the outermost metal layer could be occupied by Pt or Ru atoms, and on top of this surface was one monolayer of sites that could be occupied by oxygen atoms or vacancies. A schematic of this approach is shown in Figure 4d. The cross-validation error for this cluster expansion was 15 meV / atom with respect to DFT. They screened through oxygen chemical potential (or equivalently oxygen partial pressure), temperature, and bulk Ru composition to build a phase diagram (Figure 5a) and adsorption isotherm (Figure 5b) of Pt– Ru(111) surfaces with adsorbed oxygen. Because oxygen binds more strongly with Ru than Pt (Figure 5a), they predicted that in oxidizing environments, Ru segregates to the surface, leaving Pt atoms to form small islands within the Ru-rich surfaces (Figure 5c, d).



Figure 5. (a) Phase diagram and ground-state structures of the O/Pt-Ru(111) surfaces. The points marked by squares are combinations of oxygen and Ru surface compositions at which the total energy was calculated. The filled squares represent the ground states that make up the convex hull and are stable against decomposition into other structures. Pictures of four of these structures are inserted. (b) Calculated oxygen coverage on a pure Pt(111) surface at T=726 K as a function of oxygen chemical potential. The chemical potential (bottom scale) has an arbitrary reference. An approximate oxygen partial pressure (top scale) is given for guidance. The inset shows the stable surface structure at θO=0.25. (c, d) Monte Carlo simulations and surface structure evolutions as a function of oxygen chemical potential µO, temperature T, and bulk Ru composition CRu .The conditions are (c) T=600 K, and (d) T=1050 K. Ru is assumed to be in equilibrium with bulk Pt-Ru alloys with (d) having a much higher bulk Ru concentration. The gold, black, and light cyan spheres represent Pt, Ru, and O atoms, respectively. Reproduced with permission from ref 30. Copyright 2005 American Physical Society.


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Application and rational design Cluster expansions on extended surfaces are of particular use in the field of alloy catalysis, as the binding energies of adsorbates are often good descriptors of catalytic activities,80, 81 and these binding energies depend strongly on the arrangement of atoms near the alloy surface.82 In a single cluster expansion, it is possible to rapidly evaluate the energies of different arrangements (and coverages) of surface adsorbates and near-surface atoms. Through Monte Carlo simulations, it is thus possible to identify equilibrium surface structures in different chemical environments and assess adsorbate binding energies (and hence catalytic activities) of these structures. The cluster expansion approach can also be used for the rational design of alloy catalysts, by searching through the space of possible surface structures to identify those that are expected to be highly active and, ideally, stable. We have previously applied this approach to the design of Pt3Ni surfaces for the oxygen reduction reaction (ORR).42

Pt and Pt-based alloys have been extensively studied by

experiments53, 83-85 and theoretical calculations53, 84, 86 due to their high catalytic activity for the ORR.53, 83 It has been shown that the oxygen binding energy, ∆EO, is a good descriptor of the catalytic activity of an fcc (111) surface, through the construction of a “volcano” plot (Figure 6b).87 To investigate how atomic order will affect ∆EO we built a coupled binary cluster expansion using the physically motivated Bayesian approach on symmetrical 9-layer slabs with a monolayer of adsorbate sites that could be occupied by an oxygen atom or a vacancy.42 We explicitly modeled substitutional Pt/Ni disorder on all layers of the slab to determine how subsurface atomic order affects surface order and adsorbate binding energies (Figure 4e). The resulting cluster expansion had 329 distinct ECIs. To reduce the computational cost of generating an expansion with so many distinct ECIs we used the physically motivated Bayesian approach, 15 ACS Paragon Plus Environment


including the coupling of congruent clusters. This machine-learning-based approach significantly enhanced the accuracy of the cluster expansion, leading to a leave-one-out cross-validation error of 2.8 meV / atom relative to DFT. The cluster expansion revealed that Pt3Ni(111) surfaces in thermodynamic equilibrium have substantial subsurface disorder (Figure 6a), leading to the predicted ORR current on different surface sites that varies by approximately three orders of magnitude (Figure 6b). Both the predicted layer-by-layer composition profile and the predicted ORR enhancement of the Pt3Ni(111) surface compared to the pure Pt(111) surface42 are in very good agreement with experiments.83 The ECIs reveal that the Pt/Ni atoms in the second layer that are 2nd-nearest to an adsorbed oxygen atom have a much more significant contribution to ∆EO than those in the second layer that are nearest to the adsorbed oxygen (Figure 6c). Using this cluster expansion in Metropolis Monte Carlo simulations, we predicted ORR activity on Pt3Ni(111) would be maximized by a metastable Pt3Ni(111) surface with a three-layer Pt skin, which is close to the peak of the volcano plot (representing the hypothetical ideal surface) and would likely be durable against Ni dissolution.88


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Figure 6. Rational design of Pt– –Ni(111) surfaces with optimized ORR activity. (a) Top view of the second layer (the first layer is pure Pt) of a Pt3Ni surface taken from a Monte Carlo simulation at 400 K, showing the gas-phase oxygen binding energy (∆EO) at every fcc site with 1/144 ML coverage referenced to the peak of volcano plot in (b). Large gray and green spheres represent the Pt and Ni atoms, respectively. The small spheres represent the oxygen atoms. (b) Volcano plot showing the range of ∆EO in (a) and corresponding predicted ORR activity. (c) Effective cluster interactions for pair clusters containing one oxygen fcc binding site (yellow sphere) and one Pt/Ni site within the outmost three layers. Blue indicates a positive ECI (stronger oxygen binding when Pt is present), and red indicates a negative ECI (stronger oxygen binding when Ni is present).

CLUSTER EXPANSIONS OF NANOPARTICLES In recent years researchers have witnessed a dramatic increase in the design and synthesis of nanoscale materials for applications including electrocatalysis,48,

50, 51, 53-55

photocatalysis,89

batteries,90 data storage,91 and hydrogen storage.44, 78, 92 The atomic structures of nanoparticles can be modified through a variety of experimental treatments including doping,53 annealing,88 and electrochemical erosion,93 but atomic-scale structure-property relationships in nanoparticles are not fully understood in part because of the limitations of experimental and computational methods. Due to their small sizes and structural variety it is difficult to experimentally



characterize the atomic structures of nanoparticles, and it can be infeasible to directly use firstprinciples (e.g. DFT) to calculate the properties of large (greater than about 2~3 nm in diameter) nanoparticles, as the computational cost of DFT calculations grows rapidly with the diameter of a nanoparticle. If DFT scales as O(N3) with the number of electrons in the system,94 then the cost of using DFT to calculate the properties of a particle with diameter d scales as O(d9). It quickly becomes prohibitively expensive to calculate the properties of a single particle, much less explore the space of possible nanoparticle structures. In this section, we review how these problems can be addressed using cluster expansions.

Nanoparticles with fixed size and shape The application of cluster expansions to nanoparticles with fixed size and shape is straightforward, as illustrated in Figure 7a. In this approach, all training data and predicted nanoparticle structures must have the same size and shape. The size of the nanoparticle that can be modeled is limited by the computational cost of generating the training data; for data generated using DFT, all applications of this approach to date have been on particles with fewer than 300 atoms. When constructing a cluster expansion for a nanoparticle of fixed shape and size, it is necessary to account for the fact that nanoparticles lack translational symmetry but may have other symmetries (e.g. five-fold rotational) that do not exist in slabs or bulk materials.


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Figure 7. Schematics (top row) and example structures (bottom row) of cluster expansion models of (a) nanoparticle with fixed shape; (b, c) nanoparticle with changeable shape; (d, e) surface adsorbates on (d) a nanoparticle with fixed shape and (e) a nanoparticle with changeable shape. The colors in a circle on the top row represent which species can occupy that site. Blue, green and white represent species A, species B and a vacancy, and red represents an adsorbate. Models are usually constructed on 3-dimensional lattices instead of the 2-dimensional lattices shown here.

The initial practice of cluster expansion on nanoparticles of fixed size and shape was done by Yuge. In 2010 Yuge built a DFT-parametrized cluster expansion, in which the relevant clusters were selected by genetic algorithm, on cuboctahedral 55-atom Pt28Rh27 nanoparticles to investigate surface segregation.43 Subsequently, cluster expansions on 55-atom cuboctahedral nanoparticles were used to model the concentration effects of Pt-Rh nanoparticles,57 electronic structures (d-band) of surface Pt atoms on Pt28Rh27 nanoparticles,47 and ground state structures of Pt-Pd nanoparticles.45 Mueller demonstrated that due to quantum finite-size effects, the accuracy of a cluster expansion on 201-atom Au-Pd cuboctahedral nanoparticles could be significantly improved by using concentration-dependent ECIs.52 These concentration-dependent ECIs were



incorporated into the Bayesian cluster expansion by replacing the constant term V0 (“empty” cluster) with V0,n (n represents the number of gold atoms in the nanoparticle) in Equation (3), and adding one more “physical insight” into the prior probability distribution: the concentrationdependent ECIs should vary smoothly with composition (i.e. V0,n and V0, n +1 should be similar). It was demonstrated that the concentration-dependent ECIs reduced the required training set size by about a factor of 5 compared with cluster selection method, and a factor of 2 compared with the physically motivated Bayesian approach with concentration-independent ECIs (Figure 2 and Figure 3).

Nanoparticles with varied size and shape Nanoparticles used in many applications53, 55, 56, 84, 85, 95, 96 have diameters larger than the ~2-3 nm that is currently accessible using DFT, and they typically have a variety of shapes and sizes. Cluster expansions on nanoparticles with fixed size and shape, reviewed in previous subsection, cannot be used to investigate the structure-property relationships of such nanoparticles. Wei and Chou addressed this problem by using cluster expansions to predict the surface energies of elemental metals. These surface energies were then used to predict equilibrium crystal shapes for particles of any size (assuming negligible contributions to the surface energy from edges and vertices) using a Wulff construction.58 Chepulskii et al. modified a bulk Fe-Pt cluster expansion by adjusting the ECI for single-site clusters in the first and second layers from the surface to study the equilibrium order parameter and surface segregation of cuboctahedral Fe-Pt nanoparticles with different particle sizes (2.5 nm to 7.0 nm).59, 60 More recently, Teeriniemi et al.49 have developed an approach in which the ECI are coordination-dependent; i.e. the 20 ACS Paragon Plus Environment

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coordination numbers of the sites are used to determine whether two congruent clusters have the same ECI. This approach makes it possible to use a cluster expansion trained on small particles to predict the properties of larger particles, provided the same types of clusters (with corresponding coordination numbers) exist in both particles. They have used this method to investigate the geometry- and size-dependent properties of Fe-Ni nanoparticles of different shapes (fcc, bcc, and icosahedral). An alternative to the above approaches is to build a cluster expansion that treats the vacuum outside of nanoparticles as an extension of the lattice, in which each site is occupied by a vacancy.44, 53-55 A schematic of this approach is shown in Figure 7b, c. As the size and shape of the nanoparticle changes, so do the subsets of vacant and occupied sites on the lattice. This approach can significantly increase the number of ECI that need to be fit due to the addition of a new species (vacancies); for example, to model a binary A-B alloy using this method it would be necessary to create a ternary A-B-Vacancy cluster expansion (Figure 7c). Such cluster expansions can be trained on a library of small nanoparticles and used to predict the properties of larger ones; e.g. DFT calculations on ~2 nm nanoparticles can be used to predict the properties of 4~10 nm nanoparticles. To improve the accuracy of the cluster expansion in the limit of large particles, bulk materials (periodic in three dimensions) can be included in the training set. If small particles with large quantum finite size effects are included in the training set, the cluster expansion can have poor predictive ability for larger particles that more closely resemble a bulk system. This problem is particularly severe if very small particles (with less than roughly 100 atoms) are included in the training set,53-56, 97 but the exact sizes at which quantum finite size effects are significant depend on the material being studied.97



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In 2010, Mueller and Ceder44 introduced the above strategy to investigate the size effect on hydrogen release from 2~10 nm sodium alanate (NaAlH4) nanoparticles. In this work, the structure of the nanoparticle was represented using a binary cluster expansion, where site -

variables were assigned a value of +1 if the particle (e.g. Na+ or AlH4 ) was present at the corresponding site and −1 if vacuum was present at the site (Figure 7b). The cluster expansion was used to determine the equilibrium crystal shapes and size-dependent surface energies for nanoparticles of NaAlH4 and its decomposition products. These surface energies were combined with bulk thermodynamic data to generate a phase diagram as a function of temperature and NaAlH4 particle size (Figure 8). The cluster expansion predicts that below a certain size sodium alanate nanoparticles will decompose in a single step to NaH, Al, and H2 (from region A directly to region C, Figure 8), in good agreement with experimental observations.44, 98

Figure 8. Phase diagram for sodium alanate (NaAlH4) nanoparticles as a function of the size and temperature. In region A NaAlH4 particles are stable, in region B they decompose to Na3AlH6 (shown), Al, and H2, and in region C they decompose to NaH (shown), Al, and H2. Orange represents Na, blue represents Al or AlHx, and pink represents H.


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The above approach may also be used to model nanoparticles that have internal substitutional disorder while allowing the size and shape of the particle to vary (Figure 7c). This is particularly useful in the field of alloy nanocatalysts, as such nanoparticles often have substitutional disorder among the elements of the alloy and are too large to be directly modeled using DFT calculations. For example, Pt-Ni alloys nanoparticles have been widely studied for the oxygen reduction reaction (ORR) in fuel cells, but they suffer from Ni dissolution in cell operating conditions. Mo-doped octahedral Pt3Ni nanoparticles (with the size of about 4.2 nm in edge length) have shown highly improved activity and stability.53 To understand the Mo-doping effects, we built a quaternary Pt-Ni-Mo-Vacancy cluster expansion (i.e. the sites in Figure 7c could be occupied by Pt, Ni, Mo, or Vacancy) that was trained on small Pt-Ni-Mo nanoparticles of varying shape and size. Since there are four possible values for the site variables for each site (Pt/Ni/Mo/Vacancy), there are many more ECIs (e.g. a total of 374 symmetrically distinct cluster functions) for the quaternary cluster expansion than the binary cluster expansion (e.g. a total of 24 symmetrically distinct cluster functions with the same truncated distances). Thus, it is difficult to fit ECIs using the cluster selection method. However, the Bayesian approach makes it feasible to fit them all with a reasonable training set size (about 200 structures) and a leave-one-out cross-validation error of 3.9 meV / atom with respect to DFT. This cluster expansion revealed that the Mo dopant prefers positions near the nanoparticle edges (Figure 9), where it can stabilize the sites that are most vulnerable to dissolution.



Figure 9. The results of the Pt-Ni-Mo-Vacancy cluster expansion. The average site occupancies of (a, b) the second layer of (a) a Ni1175Pt3398 particle and (b) a Mo73Ni1143Pt3357 particle at 170 °C in vacuum (reproduced from ref 53. Copyright 2015 American Association for the Advancement of Science) and (c, d) the first layer of (c) a Ni1175Pt3398 particle and (d) a Mo73Ni1143Pt3357 particle in oxidizing conditions (reproduced from ref 54. Copyright 2016 American Chemical Society) as determined by Monte Carlo simulations. Small spheres represent the atoms in the outer layer. (e, f, reproduced with permission from ref 55. Copyright 2018 American Chemical Society) Snapshots of (e) a Pt4495Ni1680 particle and (f) a Pt4323Ni1766Mo86 particle after evolution under KMC simulations at 27 °C. The insets at the bottom right show middle cross-sections of the particles. Gray, blue, and red spheres represent Pt, Ni, and Mo atoms, respectively.


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Nanoparticles with dopants and adsorbates As nanoparticles typically do not exist in a vacuum, it is important to investigate how adsorbates interact with nanoparticles. Adsorbates can drive surface segregation, change surface energies (and hence equilibrium particle shapes), and affect the kinetic evolution of particle structure. In some applications, such as catalysis, it is critically important to know the adsorption energies of different species at different sites on the particle surface. For these reasons, there has recently been increased attention on developing nanoparticle cluster expansions that account for the interactions between the particles and potential adsorbates. In 2012, Wang et al. demonstrated how a cluster expansion could be used to model configurational thermodynamics of alloy nanoparticles with adsorbates by building a cluster expansion of 55-atom cuboctahedral Pd-Rh nanoparticles with adsorbed oxygen.46 This approach, illustrated in Figure 7d, makes it possible to investigate the interplay between atomic order within the nanoparticle and oxygen adsorbed on the particle surface. More recently, to better understand the hydrogen evolution reaction (HER) on Pt nanoparticles, Tan et al.51 constructed a cluster expansion of a 55-atom cuboctahedral Pt nanoparticle with surface hydrogen adsorbates (*H) (Figure 7d with Pt atoms excluded from cluster expansion). Through this cluster expansion they were able to identify the atomic configurations of the adsorbed hydrogen as well as calculate the hydrogen adsorption energy, which has been shown to be an important descriptor of activity for the HER.99,

100

The predicted hydrogen adsorption isotherms and the cyclic

voltammogram are in agreement with experiment observations (Figure 10a, b), similar to the previous work on the Pt(111) surface.37 Using a micro-kinetic model, nearest-neighbor (100)facet bridge site pairs and edge sites combining with (100) facet sites were found to contribute most to HER activity. With this data, they were able to estimate the effect of particle size on 25 ACS Paragon Plus Environment


HER activity, showing an adverse size effect for HER activity for sizes below 2 nm (Figure 10d) due to the increasing ratio of facet sites to edge sites.

Figure 10. (a) Calculated hydrogen adsorption isotherm on Pt55 at 298 K. Contributions to the total coverage (black curve) from each *H species are shown. (b) The derived cyclic voltammetry diagram. ܳ଴ is the total charge accumulated per Pt mass (or surface area) when all adsorption sites are occupied. ܷ ᇱ is the voltage cycling rate. (c) The lowest-energy configuration at 0 V. Gray and cyan spheres represent Pt and H, respectively. (d) Activity trend versus cuboctahedral particle size. Exchange current is normalized against the particle mass and surface area for ݅ and ݆଴ , respectively. Reproduced with permission from ref 51. Copyright 2015 American Chemical Society.


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The above examples assumed that the nanoparticles have fixed size and shape. Removing this assumption makes the problem of including adsorbates in a nanoparticle cluster expansion significantly more challenging. We have addressed this problem for Mo-doped Pt3Ni nanoparticles by introducing coordination-number-dependent and metal-specific (*OH for surface Pt/Ni, and 3*O for surface Mo) correction terms into the cluster expansion.54 These terms, determined by using a small set of DFT calculations for different types of adsorption sites, account for the possible presence of *O and *OH. Our calculations indicate that Mo (Mo4+/Mo6+) atoms prefer surface sites with low coordination numbers, such as edge, and vertex sites, under oxidizing conditions (Figure 9d). Mo doping was found to stabilize Pt3Ni nanoparticles by increasing surface vacancy formation energies of Pt/Ni and reducing the concentration of Ni on vulnerable surface sites with low coordination. Because real-world nanoparticles are unlikely to be in thermodynamic equilibrium,55 we have used this approach in kinetic Monte Carlo simulations to model the structural evolution of nanoparticles. These calculations reveal that surface Mo oxides may also pin the motion of surface Pt atoms, protecting sub-surface atoms from exposure to the surface and subsequent dissolution (Figure 9e, f).55 The cluster expansions discussed above, with implicit *OH/*O on the surface, have limitations: they do not directly account for the interactions between surface adsorbates and subsurface atoms, and the uncertainty in the magnitude of the correction terms is about one order of magnitude higher than the leave-one-out cross-validation error of a clean nanoparticle. In a study of adsorption energies on Au nanoparticles, Eom et al. have demonstrated an alternative approach in which each nanoparticle is divided into three types of sites: adsorption sites (which could be occupied by an adsorbate or vacancy), surface sites, and sub-surface sites.50 In their work the latter two types of sites are always occupied by gold. Clusters that have the same types 27 ACS Paragon Plus Environment


of sites in the same geometric arrangement are assigned the same ECI. This approximation makes it possible to calculate adsorption energies on particles of a variety of shapes and sizes. Using this approach, they studied the catalytic activity of CO2 reduction and the hydrogen evolution reaction on Au nanoparticles with realistic size (about 9 nm).50 To study the structures of Pt-Cu alloy nanoparticles under oxidizing conditions, we developed an approach that fully accounts for the variety of local environments in nanoparticles.56 In this approach each site on the fcc lattice can be occupied by either Pt, Cu, O, or a vacancy, with the oxygen atoms only allowed on surface sites in the training set and Monte Carlo simulations (Figure 7e). This approach allows us to calculate the energies of nanoparticles with different sizes, shapes, and compositions under different temperatures and oxygen chemical potentials. To accommodate the large number of ECI that needed to be fit in this expansion, again the physically motivated Bayesian approach was used, resulting in a leave-one-out cross-validation error of 4.5 meV / atom. An implicit assumption of this approach is that oxygen only adsorbs on the fcc sites of the nanoparticle surface. Despite this approximation, it was found that the explicit inclusion of oxygen in the cluster expansion significantly improved the predicted surface composition when compared to experimental measurements (Figure 11). We also found that the cluster-expansion-predicted shapes and lattice parameters were in good agreement with experiments.56 These results demonstrate that explicitly including adsorbates in cluster expansions can greatly bridge the gap between cluster expansion prediction and experimental characterization and significantly improve the accuracy of the predicted atomic structures of alloy nanoparticles at experimentally relevant sizes. This approach could also be used to predict adsorbate binding energies on nanocatalysts, but it may be difficult to extend this approach to


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model more complex adsorbates (e.g. *COOH) because of the need to account for adsorbate orientations at different types of sites.

Figure 11. (a) Experimental and cluster-expansion-predicted surface Pt compositions of Pt-Cu nanoparticles in vacuum and an oxidizing environment as a function of overall bulk concentration. (b) Equilibrium structures of Pt3Cu in vacuum (left) and in oxidizing environments (right). Gray, blue and red spheres represent Pt, Cu and O, respectively. Reproduced with permission from ref 56. Copyright 2018 American Chemical Society.

SUMMARY AND OUTLOOK Due to the expense of directly using DFT to model nanoparticles, the development of advanced cluster expansion models that are consistent with experiments is of critical importance for determining structure-property relationships and accelerating the discovery and design of surfaces and nanostructured materials. Researchers have made great progress in modeling these 29 ACS Paragon Plus Environment


challenging systems by developing increasingly sophisticated applications of the cluster expansion methodology. Extra cluster functions can be included to account for the loss of translational symmetry,42,

52

the presence of adsorbates,42,

56

or the interactions between a

material and the surrounding vacuum.53-56 A limiting factor for the application of cluster expansions to surfaces and nanoparticles is the computational cost of generating the training data, which is a result of the relatively large number of distinct ECIs required to model these systems and the cost of running DFT calculations on low-symmetry systems. Machine learning provides a valuable tool for mitigating this increased computational cost; a physically motivated Bayesian approach to fitting cluster expansions reduces the cost of generating a cluster expansion for a simple 2-nm binary alloy nanoparticle by about a factor of five.52 By incorporating physical insights into the process of fitting the ECI, machine learning makes it feasible to model highly complex systems such as ternary alloy nanoparticles with experimentally relevant shapes and sizes. Although cluster expansions of surfaces and nanoparticles have been developed to account for increasingly more realistic conditions, there are still several challenges remaining. On extended surfaces, the lattice parameter of the surface is determined by the underlying bulk material. However, the bulk material is not included in a typical slab cluster expansion, so the lattice parameter for the slab must be set to a fixed value representing a single bulk phase. This works well when the composition and phase of the underlying bulk material is well known (e.g. in Pt3Ni 42) but makes it challenging to generate a cluster expansion for a system in which the bulk material can have a range of compositions and lattice parameters. This is particularly important in catalysis, as it is known that the strain effect (lattice parameters) can dramatically affect catalytic properties.85, 101


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There is also a need for better modeling of the interaction between the material and its environment. Adsorbates may be present at a variety of surface sites or even penetrate into subsurface layers,102 which is particularly challenging to model for nanoparticle cluster expansions in which the particle shape and size is allowed to change. In addition, larger molecules may have a variety of orientations that must be accounted for when they are included in cluster expansions. To date, most surface cluster expansions have assumed that the surface exists in vacuum or gas phase, where the interaction with the environment is well-approximated by the inclusion of adsorbates on the surface. However, many materials exist in liquid solutions, which can significantly change the properties of the surface and adsorbed molecules. These systems would best be modeled by accounting for solvation effects when generating the training data, but this would further increase the cost of generating a surface cluster expansion. There are significant opportunities for the application of cluster expansions to kinetic modeling of nanoparticles. To avoid melting and agglomeration, nanoparticles are typically synthesized and processed at relatively low temperatures.

As a result, the particles may not reach

thermodynamic equilibrium,55 making it critically important to understand the kinetic evolution of nanoparticles.

Cluster expansions are well suited to modeling growth, dissolution, and

diffusion of nanoparticles using kinetic Monte Carlo, provided the positions of the atoms can be mapped to a well-defined lattice. For situations in which this is not possible, such as liquids or amorphous solids, a force field model would be more appropriate, although these are typically more expensive and/or less accurate than cluster expansions for systems on lattices.103 Cluster expansions have been on the forefront of the application of machine learning concepts to atomistic modeling, with numerous machine learning concepts well established in the field and routinely used to solve materials problems.6, 23, 47, 49, 68-73 It will be necessary to build upon 31 ACS Paragon Plus Environment


this success to address the above challenges and model increasingly complex problems with high accuracy. With continued improvements in the cost and accuracy of cluster expansions, they will become an increasingly valuable tool for understanding and designing surfaces and nanoscale materials.

AUTHOR INFORMATION Corresponding Author *Email: [email protected]

Notes The authors declare no competing financial interests.

ACKNOWLEDGMENTS We acknowledge the support from National Science Foundation (NSF) through award No. CHE1437396. Atomic-scale structural images were generated using VESTA.104

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for Table of Contents use only

The Use of Cluster Expansions to Predict the Structures and Properties of Surfaces and Nanostructured Materials Liang Cao,1 Chenyang Li,1 and Tim Mueller1, *

1

Department of Materials Science and Engineering, Johns Hopkins University, Baltimore, Maryland 21218, United States

*Email: [email protected]


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The Use of Cluster Expansions To Predict the Structures and

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