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1 Introduction

Metal nanoparticles (NPs) under several tens of nanometers in size have unique mechanical, optical, electronic, and catalytic properties that are different from their bulk counterparts [1–5]. Structural characterization of NPs is a key to answering many questions related to their catalytic properties, for example, what is the nature of catalytically active sites, or what are the reasons for their catalytic activity, selectivity, and stability. Various structural, geometric, and electronic characteristics were named “descriptors” of catalytic properties in transition metal catalysts, such as the number of under-coordinated sites [6, 7], or perimeter sites [7, 8], or number of (111) crystalline facets [9], or surface strain [10], or the position of the d-states relative to the Fermi level (a “d-band center” model) [11–15]. The main challenge preventing direct measurements of those characteristics is the small size of nanoparticles. Indeed, in size range of several nanometers, there are very few techniques capable of characterizing structure and electronic properties with sufficient spatial and energy resolutions. A challenge specific to catalysts is the need to monitor their structure and electronic properties in situ, during their work, also known as “in operando conditions .” That latter requirement is particularly challenging to electron microscopy and other imaging techniques that are severely limited in use by requirements of high pressure and temperature, typical for many catalytic reactions. Other structural characterization techniques such as X-ray diffraction are limited by the small size of the nanoparticles that is responsible for broadening Bragg peaks used for structural refinement. XAFS analysis methods are excellent alternatives to imaging and scattering methods due to the excellent spatial, temporal, and energy resolutions of XAFS, and due to the relative ease of its application to real-time processes in operando conditions . In recent years, it became progressively more appreciated that supported NPs are complex systems whose catalytic properties are influenced not only by the structure of the particle but also in not a small degree by its interaction with support and adsorbates. In this chapter, we review the main attributes of supported NPs that affect their catalytic activities, recent solutions to the structural characterization of NPs and describe recent advances in solving three-dimensional structure, degree of alloying, and their changes under conditions of model and real catalytic reactions.

2 Size, Shape, Strain, Support, and Composition Effects on Catalytic Properties

With the decrease of particle size, the surface to volume ratio increases, shifting the balance between the surface and the bulk energies in favor of the former. Enhanced surface energy is responsible for generating substantial surface strain [16, 17], that can be relieved by some adsorbates, e.g., hydrogen [5, 18, 19]. The enhanced surface stress causes contraction of surface metal–metal bond length [16, 20, 21]. The surface energy and surface stress thus have important influence on the elastic properties of NPs.

In nanocatalysts strain is a ubiquitous attribute of their structure. It dominates the surface and support interface regions, and, in the case of bimetallics, is also present throughout the bulk, due to the size mismatch of two types of metals [10, 16, 22–25]. The increased surface strain results in the shift of d-band center [10, 26–28] which tunes the binding energy between surface atoms and adsorbed molecules [10–12], and changes cohesive energy of surface atoms which alters the thermodynamic properties and stiffness of NPs [17, 29].

Electronic structure of NPs could also be modified through the change of particle size . Lowering the coordination number of NPs causes the tendency towards localization of the valence electrons and gap formation, hence, the transition from metallic (at large sizes) to nonmetallic (in small sizes) properties. Another consequence of decreasing particle size is thus the reduction of the width of valence band and the shift of its center of gravity towards Fermi level which leads to an increase of the adsorption energy of adsorbate and a decrease of the dissociation barriers of adsorbed molecules [13, 14, 30]. For transition metals catalysts, their properties could thus be tailored through the d-band center position relative to the Fermi energy by changing the composition (adding different metal atoms) [13, 15, 31], surface strain, support material [32–35], or adsorbate coverage [12].

NPs with various shapes expose different facets, which may have different properties with respect to catalysis in the course of the same reaction [36, 37]. In addition, different surface types contain different fractions of under-coordinated atoms (on edge and vertex), which are considered to be the active sites in many reactions [6–9]. Shapes of NPs are shown to change with size [38], however, when particles are extremely small, and they could adopt various geometries with comparable energies. The fluctuation of geometries and structures could lower the reaction barrier [39, 40]. Finally, the nanoparticles can coexist in the ordered and disordered states in the same size range, and their fractions can change in reaction condition, further hampering efforts in their characterization and thus understanding catalytic mechanisms [18].

Most heterogeneous nanocatalysts are deposited on supports, which distort the atomic structure of the interfacial layer in contact with the substrate, creating defects, strain at interface and may even change the shape of NPs [22, 32, 33]. On the other side, supports with different reducibility have different influences on the electronic structure of NPs. These structural and electronic factors control the metal/support adhesion energy, which affects chemical potentials of surface atoms and their binding ability to small adsorbates [35].

The addition of the second metal to monometallic systems was found to be an effective way to tune the properties and structures of nanocatalysts. The possible mixing patterns of bimetallic systems reported in literature vary from random alloys, core–shell, cluster-by-cluster, and other architectures, depending on the elements, their compositions and synthesis conditions [41, 42]. The introduction of the second metal could introduce strain, alter the electronic structure and, hence, the d-band center position, which greatly affects the interaction between the surface atoms and adsorbates. Many bimetallic systems were reported to change their structure and/or compositional motifs in response to the changes in the environmental conditions or in the process of catalytic reactions [43–45]. Tracking the structural changes of bimetallic systems in operando mode , that is, during real reaction conditions, while monitoring the reaction in real time, is thus quite important for revealing their working mechanisms.

3 Experimental Characterization of Nanoparticle Structure and Electronic Properties : The Uniqueness of XANES and EXAFS

X-ray diffraction (XRD), X-ray photoelectron spectroscopy (XPS) , and electron microscopy are commonly used techniques for bulk or surface structure characterization. When particle size is in nanometer range, XRD is not very helpful due to the Bragg peak broadening. Electron microscopy, on the other hand, has very high resolving power, which helps reveal the atomic and surface structure of NPs [46]. However, if used in situ, only part of the ensemble of catalytic species can be reliably detected (above the resolution limit) and reports of applications of TEM to operando studies are still scarce. X-ray absorption spectroscopy (XAS) , on the other hand, can be done in a number of sample conditions (liquid or gas, low or high temperatures and pressures) [47–52], and a number of reactor cells are available to date [53–57]. Its sensitivity to local structure and elemental specificity make it uniquely fitting for studying nanomaterials. With the development of the analytical methods for XANES and EXAFS interpretation, a lot of information related to the physical properties of nanomaterials , such as stress-induced bond length change and disorder [19, 22, 58], three-dimensional geometry of small clusters [59, 60], electronic structure modification driven by support/environment [61], and mixing pattern of bimetallic systems [45, 62, 63], could be obtained. In recent years, with the proliferation of in situ and operando methods of catalyst characterization, combination of XAFS with other techniques (IR, Raman, XRD, NMR, UV–Vis, etc.) helped illuminate cooperative phenomena at the interfaces between nanoparticle, surface species, and support under realistic working conditions [43, 50, 56, 74–88].

In the following sections, we review recent progress in structural characterization of nanocatalysts by EXAFS methods.

4 Size and Geometry of Nanocatalysts by Coordination Number Analysis

Historically, the importance of EXAFS for catalysis studies [89] was realized almost immediately after XAFS was recognized as a new method of studying local structure in 1970s [90–94]. First works used it for measuring coordination numbers of supported monometallic and heterometallic NPs to obtain their average size (only first shell/single scattering analysis was possible) [95–98]. Then, with the development of multiple scattering (in the 1990s) [99] more advanced analysis methods were developed for determination of cluster size, shape, morphology, and mixing pattern in bimetallic systems. Information about the atomic architecture (three-dimensional packing of atoms) in a representative NP can be most directly gleaned from the coordination number of first nearest neighboring metal–metal bonds. The coordination number (n AA(i)) of the ith shell with the radius R i around the absorbing atom in a monometallic cluster is defined as the average number, per absorber, of nearest neighbors within a given shell:

$$ {n}_{AA(i)}=\frac{2{N}_{\mathrm{AA}\left(\mathrm{i}\right)}}{N_A}. $$
(19.1)

Here N AA(i) is the total number of the A–A nearest neighbors within the same coordination shell, and N A is the total number of A-type atoms in the cluster. The factor of two in Eq. (19.1) is due to the fact that each atom of the A–A pair is an absorber and thus the number of these pairs should be doubled in calculating the A–A coordination numbers. Coordination numbers are obtained model-independently from data analysis of experimental EXAFS spectra. The most important information that is available via the coordination number analysis is the average particle size , and several methods are available for its determination from the EXAFS coordination numbers [100].

One such method, developed by Calvin et al. [101], assumes homogeneous spherical shape of clusters with the radius R. For atoms in the ith shell around the absorbing atom, the coordination number of the ith shell (N nano(i)) can be expressed as follows:

$$ {N}_{nano(i)}=\left[1-\frac{3}{4}\left(\frac{r_i}{R}\right)+\frac{1}{16}{\left(\frac{r_i}{R}\right)}^3\right]{N}_{bulk(i)}. $$
(19.2)

In Eq. (19.2), r i is the distance between the absorbing atom and neighboring atoms in the ith shell, and N bulk(i) is the ith shell coordination number of bulk structure. This method allows the calculation of coordination number of an arbitrary coordination shell as a function of the cluster size, which in principle, can be used to discriminate between symmetric (quasi-spherical) and asymmetric clusters if the coordination numbers of the higher-order shells are measured by EXAFS. One disadvantage of this method is that it is limited to sufficiently large clusters (with number of atoms much larger than 100) [60].

Another useful method for estimating cluster size is comparing the first nearest coordination number (N 1 ) obtained from EXAFS analysis against model structures with known geometrical characteristics. For regular polyhedra (e.g., a cuboctahedron or an icosahedron), N 1 is a function of cluster order L, which is defined as the number of spacing between adjacent atoms along the edge (see example of L = 2 in Fig. 19.1) [102]. This method developed by Montejano-Carrizales et al. [102, 103], is easy to expand to other morphologies and cluster families [59]. For example, the truncated cuboctahedral model with the (111) plane parallel to the support is most close to the morphology found in many supported metal clusters (Fig. 19.1). The relationship between N 1 and L in a truncated cuboctahedron is: [104]

Fig. 19.1
figure 1

Schematic of a truncated cuboctahedral cluster with 37 atoms (L = 2)

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$$ {N}_1=\frac{3\left(20{L}^3+21{L}^2+7L\right)}{5{L}^3+12{L}^2+10L+3}. $$
(19.3)

For truncated cuboctahedron with atom numbers of 10 (L = 1), 37 (L = 2), 92 (L = 3), and 185 (L = 4), the first nearest coordination numbers are 4.8, 7.0, 8.2, and 8.9, respectively.

Another method that is useful in the case when there is no particular symmetry known in advance, and/or nearest neighboring shells cannot be easily identified but the coordinates of atoms in the cluster are available from, e.g., first principle simulations, is the radial distribution function (RDF) method [59, 105]. This approach, proposed by Frenkel and Glasner [59], employs computer-generated cluster coordinates. The cluster-average pair radial distribution function ρ(r) is computed for a cluster of N atoms:

$$ \rho (r)=\frac{1}{N}{\displaystyle \sum_{i=1}^N{\rho}_i(r)},\ {\rho}_i(r)=\frac{d{N}_i}{d{R}_i}, $$
(19.4)

where ρ i (r) is the partial RDF for an atom i, and dN i is the number of its neighbors within the spherical shell of thickness dR i . The subsequent calculation of coordination numbers for an arbitrary coordination shell (between R 1 and R 2) is achieved by integrating the ρ(r):

$$ {n}_i={\displaystyle \underset{R_1}{\overset{R_2}{\int }}\rho (r)dr}. $$
(19.5)

Compared to the two methods described above, the RDF method enables rapid calculation of coordination numbers of clusters with arbitrary sizes and shapes. By combining electron microscopy with multiple-scattering EXAFS analysis and data modeling, several geometries with the same sequence of coordination numbers of the nearest-neighbor shells can be discriminated [106, 107]. The authors can be contacted for sharing their software program that performs RDF calculations for arbitrary cluster geometries.

The methods listed above for size estimation are strongly dependent on the knowledge of the first nearest coordination numbers that are, in turn, obtained reliably by EXAFS analysis if the bond length distribution is relatively symmetric [91, 108]. For supported nanocatalysts under working conditions, the substrate and adsorbates may induce stress, causing the bond length distribution to deviate strongly from the Gaussian shape by for example relaxing the surface atoms stronger compared to the core [16]. The asymmetric disorder in bond lengths results in an artifact of the data analysis where, if ignored, it leads to the underestimate of the coordination numbers [109]. Therefore, anharmonic corrections should be taken into account for systems with large disorders. Several methods have been proposed recently that take into account the asymmetric disorder and use structural analysis of EXAFS data to validate different theoretical models [110–112]. They are described in greater detail below.

Including multiple-scattering effects to EXAFS data analysis is another way to improve the accuracy for size determination and is also crucial for extending structural refinement of nanocatlysts beyond first nearest bond distance. From the geometrical characteristics of regular polyhedral clusters, the sequence of coordination numbers of the 1st, 2nd, 3rd, 4th, etc. nearest neighboring pairs of atoms for different types of polyhedra is unique. That uniqueness is used for comparison with EXAFS results, obtained model-independently, for the same coordination numbers (and for degeneracies of multiple-scattering paths) to determine the size, shape, structure, and, in some cases, surface orientation of NPs [59, 106, 113]. One example of such analysis is the characterization of supported Pt clusters [60, 106, 114, 115], which have been extensively studied by EXAFS to establish the relationship between cluster size, shape, and catalytic properties. The structural characteristics depend strongly on the preparation conditions and on the nature of support [6, 22, 107, 116, 117]. The demonstrations of the sensitivity of the cluster shape to the support and treatment conditions are shown in the following examples. The fully reduced γ-Al2O3 supported Pt NPs have a spherical structure through a preparative method of deposition precipitation [118] while PVP capped Pt NPs changed from spherical to flat structure after the particles were deposit on SiO2 support [119]. In another example, the 1 wt% Pt/γ-Al2O3 catalyst containing 11 Pt atoms was three-dimensional after low temperature reduction (300 °C) and changed to the raft shape with the structure similar to Pt (100) after high temperature reduction (450 °C) [120]. Interestingly, for Pt/zeolite, the three-dimensional structure was retained after high temperature treatment [121].

As an example of the multiple scattering EXAFS analysis of the cluster shape , we use the structural modeling of carbon supported Pt NPs up to the 4th/5th Pt-Pt shell [106, 114]. Figure 19.2 illustrates such analysis

Fig. 19.2
figure 2

Comparison of the average distances (up to 5NN), together with their error bars (shown as shaded rectangles), measured by EXAFS for the 10 wt% Pt/C sample and the calculated from a truncated (by (111) plane) cuboctahedron for cluster orders L up to 15. Reproduced with permission from ref. [106]. Copyright 2001 American Chemical Society

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for size and shape determination and shows that the (111)-truncated hemispherical cuboctahedron provides good approximation for the 10 wt% Pt/C sample, with a particles size of about 1.7 nm [106]. To find connections between particle shape and catalytic properties, γ-Al2O3 supported Pt NPs with various shapes but analogous average size (~1 nm) were prepared and characterized by multiple-scattering analysis (up to 4th shell) of EXAFS data in combination with microscopic tools [107]. The shape having higher percentage of undercoordinated atoms (at edge or corner sites) on the surface was found effective for lowering the onset temperature for two-propanol oxidation [6].

A more approximate method that also relies on the coordination numbers of higher shells for determining the size and shape of NPs was proposed by A. Jentys [122]. This method modeled the first five coordination numbers as a function of particle size with a hyperbolic function for particles with shapes of spheres, cubes, and distorted cubes. The coordination number of the first (n 1) or second shell (n 2) was found independent with shape and thus used to estimate the average particle size. The curves of n 1/n 3 versus particle size were found to be different for different shapes. The shape can therefore be determined by comparing the curve of experimentally determined n 1/n 3 ratio with the model n 1/n 3 graph. Such analysis was expanded by Beale and Weckhuysen to larger number of shapes [123].

5 Using EXAFS to Characterize Bimetallic Nanocatalysts

Methods of high precision synthesis of bimetallic NPs for catalysis and electrocatalysis are actively sought, due to the increased demand to minimize the use of noble metals and for rational design of catalysts with desired activity and selectivity, and increased stability [42, 124–131]. Analogously to the definition of the coordination number for a homo-metallic pair, for heterometallic bonds , the coordination number is defined as:

$$ {n}_{AB}=\frac{N_{AB}}{N_A}. $$
(19.6)

The information on the homo- and hetero-metallic coordination numbers n AA , n AB , n BA , and n BB is available from EXAFS measurements on the absorption edges of both A and B central atoms [25]. The analysis should be done for both edges concurrently, with constraints imposed on the heterometallic bonds during the fits:

$$ {n}_{AB}=\frac{x_B}{x_A}{n}_{BA},\ {R}_{AB}={R}_{BA},\ {\sigma}_{AB}^2={\sigma}_{BA}^2. $$
(19.7)

Just as in the case of monometallic catalysts, multiple-scattering analysis of bimetallic catalysts allows for measurements of coordination numbers within the first few shells [114, 132]. These parameters elucidate the intra-particle composition, such as the extent of segregation or alloying of atoms, e.g., random distribution, as opposed to the positive or negative tendency to clustering [60, 132–140]. Once the above parameters are known, the total coordination number of metal–metal (M-M) neighbors per absorbing atom can be found from: [60]

$$ {n}_{MM}={x}_A{n}_{AM}+{x}_B{n}_{BM}. $$
(19.8)

The total coordination number can be employed to determine the size and shape using the same methods applied to monometallic particles described above.

For heterogeneouse distributions , the main question in the EXAFS analysis of bimetallic NPs is to detect a certain architectrual motif, e.g., a core–shell or cluster-to-cluster [41, 42, 45]. When atoms of the type A will segregate to the surface of the nanoparticle and B—to the core, then n AM  < n BM [60]. For random alloys, the average coordination numbers n AA and n AB are in the same proportion as the bulk concentrations of these elements in the sample: [60]

$$ \frac{n_{AA}}{n_{AB}}=\frac{x_A}{x_B}. $$
(19.9)

For alloys with positive tendency to clustering of like atoms, e.g., when either the intraparticle or interparticle segregation is present, the left hand side should be larger than the right hand side:

$$ \frac{n_{AA}}{n_{AB}}>\frac{x_A}{x_B}. $$
(19.10)

For homogeneous alloys (Fig. 19.3, center) in which the atoms A and B occur with equal probability within the particle or on the surface: n AM  = n BM [60].

Fig. 19.3
figure 3

Three main types of bimetallic configuration : left: core–shell; center: alloy; right: segregated monometallic clusters. Reproduced with permission from ref. [45]. Copyright 2009 American Chemical Society

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Interpretation of the coordination numbers in heterogeneous systems in terms of a “representative cluster” can be misleading. We will discuss two commonly encountered systems that require particular care. One example is when two different NP systems differ in homogeneity of their atomic distributions, and the other—in the degree of their randomness. First, we consider a homogeneous system where all atoms have similar local environments within each NP. Two cartoons depicting two dimensional “nanoparticles” with this type of structure are shown in the Fig. 19.4a, b. Although the lattices are differently ordered, both types of atoms contain a similar number of neighbors for each element throughout the “cluster.” The similarity becomes nearly perfect when the surface to volume ratio becomes negligible, i.e., for particles larger than 4–5 nm in diameter. Such atomic arrangements are homogeneous, as there is an equal probability to find any given atom type (A or B) anywhere within the NP. An example showing the other extreme is presented in the two cartoons in the Fig. 19.4d where the atoms of each type (A or B) are segregated within different part of the NP.

Fig. 19.4
figure 4

(a, b) shows homogeneous configurations for the same 50–50 composition , characterized by a unique, non-positive values of the short range order parameter α. Different heterogeneous configurations, characterized by positive values of α, are shown in schemes (c, d). Reproduced with permission from ref. [62]. Copyright 2013, AIP Publishing LLC

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The second case we briefly discuss is when the two NP systems differ in the randomness of their atomic distributions . This characteristic will only apply to the homogeneous systems such as two NPs shown in Fig. 19.4a, b, because the two NPs shown in Fig. 19.4c, d are heterogeneous, i.e., inherently nonrandom. The cluster on the Fig. 19.4a has perfect order: both atomic types have equivalent surroundings. The cluster in Fig. 19.4b is random: for either atom (of type A or B), the probabilities of neighboring atoms being either type A or B are equal. One important consequence apparent from this simple example is the difference between the short range order and homogeneity . For example, an alloy can be homogeneous but have a “negative tendency to clustering” (i.e., short range order) and is a phenomenon frequently encountered in metallurgy [141, 142].

These two examples illustrate the importance of understanding the short range order and homogeneity of bimetallic NPs when attempting to characterize their structure. It turns out that both of these can be quantitatively expressed using J. Cowley’s short range order parameter introduced recently for bimetallic NP analysis by Frenkel, et al.: [143–145]

$$ \alpha =1-\frac{N_{AB}/{N}_{AM}}{x_B}, $$
(19.11)

where x B is the molar concentration of B-type atoms in the sample. As we show below, the Cowley parameter (α can vary in the interval between −1 and 1) can be used to investigate the degree of alloying or clustering within bimetallic NPs based on how positive/negative it is. In many cases, it can be used also as a “litmus test” demonstrating that atomic segregation, of either intra-cluster or inter-cluster type, occurred. We note that this equation has been previously employed in EXAFS studies of bulk bimetallic alloys [24] but its potential in NP studies remains unexplored .

For alloys that favor (disfavor) clustering of like atoms, α will be positive (negative). This parameter is therefore essential for studies of alloy—or core–shell, or cluster-on-cluster—NPs that can be characterized by different levels of ordering. Only after the short range order parameter is evaluated, can different models of segregation be compared. In either case, additional experimental information is needed to determine the fine detail of segregation, i.e., whether for example element A is predominantly at the surface or in the core. The analogue of the effect of compositional heterogeneity on the interpretation of the short range order within a “representative” NP is the interpretation of the size of the “representative” NP from EXAFS coordination numbers. In each case, an independent technique is needed, and in the latter case, the average particle size can be measured by electron microscopy.

We emphasize that the role of measuring and evaluating α extends beyond merely determining whether it is positive or negative. Even large negative values of α may signal segregation as there is only a finite range α min ≤ α ≤ 0 in which homogeneous systems can exist [143]. For example, α min = −1 for two dimensional AB alloys shown in Fig. 19.4a, for β-brass CuZn of bcc structure [143], but it can also be fractional, e.g., α min = −1/3 for fcc Cu0.75Au0.25 alloys [143]. Hence, if the measured value of α falls within either −1 ≤ α ≤ α min or 0 < α ≤ 1 interval, the system is heterogeneous and the segregation of atoms is evident. Finally, we note that these conclusions were obtained assuming an idealized case where all particles are equivalent and the segregation may occur only within the NP. If the bimetallic composition varies from one NP to another, even random compositional distribution may generate positive values of α, a point which is discussed in greater detail below.

6 Pitfalls and Artifacts of the Analysis

In studies of the structure of NPs, coordination numbers are the most important structural parameters that can be obtained from EXAFS analysis. Coordination numbers of the first nearest neighbors (1NN) of X-ray absorbing atoms are obtained by EXAFS analysis very reliably, and are often employed for characterizing nanoclusters in terms of their structure, size, shape, and morphology [100, 106, 114, 146]. Coordination numbers of atomic pairs in bimetallic NPs are often used to discriminate between different types of short range order in the NPs, and/or ascertain the degree of compositional homogeneity in the sample [60, 100, 147]. In this section we emphasize the pitfalls in such interpretation when nanoparticle ensembles display a broad range of sizes and compositions. We show the implications of these effects on EXAFS results and describe corrective strategies.

The values of partial coordination numbers are important for analyzing composition habits of heterometallic clusters. For example, depending on the relationship between the partial 1NN numbers and the bulk composition of the nanoalloy, the latter is characterized as either homogeneous (when average environment around each atom is approximately the same) or heterogeneous (when different regions within the sample have different compositional trends, e.g., A- rich and B-rich, or when such segregation occurs within each cluster, e.g., A-rich core and B-rich shell) [147]. For homogeneous alloys, relationships (19.9) and (19.10) can be used to describe the short range order [60]. In this section we focus on random nanoalloys (that have zero short range order) and highlight challenges in their detection by EXAFS.

We now introduce the total coordination number of metal–metal pair , or n MM which is equal to n 1 for monometallic clusters. For bulk alloys, when atoms of type A and B are distributed randomly, their partial coordination numbers are found from the overall compositions:

$$ {n}_{AA}={n}_{BA}={x}_A{n}_{MM},\ {n}_{AB}={n}_{BB}=\left(1-{x}_A\right)\;{n}_{MM}, $$
(19.12)

where the composition is defined as: x A = N A/N. Note that in random bulk alloys, n AA + n BB = n AA + n AB = n BB + n BA = n MM. In a nanocluster with random compositional distribution, more accurate relationships should be used: [100]

$$ \begin{array}{l}{n}_{AA}=\frac{N_A-1}{N-1}{n}_{MM}=\frac{N{x}_A-1}{N-1}{n}_{MM},\ {n}_{AB}=\frac{N-{N}_A}{N-1}{n}_{MM}\hfill \\ {}=\frac{N}{N-1}\left(1-{x}_A\right){n}_{MM},\kern1em {n}_{BA}=\frac{N_A}{N-1}{n}_{MM}=\frac{N}{N-1}{x}_A{n}_{MM},\hfill \\ {}\ {n}_{BB}=\frac{N_B-1}{N-1}{n}_{MM}=\left(1-\frac{N{x}_A}{N-1}\right){n}_{MM}.\hfill \end{array} $$
(19.13)

We note that in random nanoalloys , same as in the bulk random alloys, n AA + n AB = n BB + n BA = n MM but in the nanoalloys the sum of n AA and n BB is smaller than n MM:

$$ {n}_{AA}+{n}_{BB}=\frac{N-2}{N-1}{n}_{MM}. $$
(19.14)

Equations (19.13 and 19.14) are exact, and they are equivalent to Eqs. (19.1, 19.6, 19.9, and 19.12) in the limit of large total number of atoms (N) and large concentrations (x A ), as demonstrated in Fig. 19.5a. Furthermore, in the random nanoalloys , the n AA and n BB are different from n BA and n AB, respectively, while in the random bulk alloys they are the same (Eqs. 19.12, 19.13 and Fig. 19.5b).

Fig. 19.5
figure 5

(a) Exact (solid lines, Eq. 19.13) and approximate (dashed lines, Eq. 19.12) behaviors of the coordination numbers of A–A pairs for different cluster sizes. (b) Coordination numbers of different atomic pairs in random, 13-atom, cuboctahedral alloys. Solid lines correspond to exact calculations using Eq. 19.13 and dashed lines correspond to approximate calculations using Eq. 19.12. Reproduced from ref. [149] by permission of The Royal Society of Chemistry

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We have recently shown [100] that alloys with broad compositional distributions are expected to have positive values for the ensemble-average short range order despite having random intra-particle distribution of atoms. Hence, a random nanoalloy may be mistaken as a system with a core–shell motif if the NPs are not all stoichiometrically uniform. This prediction can be illustrated by the following simple example. Assume that the sample consists of two groups of bimetallic NPs. The first group consists of N particles where 30 % of all atoms are A-type and 70 %—B-type in each. The second group consists of N particles of 70 % and 30 % of A and B-type atoms, respectively. The average composition over the entire sample is then 50 % of A and 50 % of B atoms. Assume also that the distribution of atoms in each particle is random, i.e., the value of α calculated over each population is zero. Finally, assume that the geometry of all particles is the same and atoms occupy regular lattice sites. Ensemble-average calculation of the coordination numbers of AB type yields the following result: N AB = 0.3 × 0.7 N AM +0.7 × 0.3 N AM = 0.42 N AM. Hence, the ensemble-average value of α measured by EXAFS will be equal to 0.16 (Eq. 19.11), in apparent contradiction to the local randomness (α = 0) of each population. What follows is the more general demonstration of this effect [100, 148].

We assume that within each cluster, atoms of type A and B are distributed randomly, but x A is different for each nanoparticle. For simplicity, we consider a system that contains particles of the same structure and geometry. Effects of cluster size distribution [100] and asymmetric bond length disorder [109] on the apparent coordination numbers have been described separately. We distinguish between the particle-specific coordination number n AA (calculated with Eq. (19.13)) in the cluster with the concentration x A of A atoms, and the apparent (measured) coordination number \( {\tilde{n}}_{AA} \), which, in EXAFS measurement, averages the number of A nearest neighbors over all the A-type atoms in all clusters in the sample. We let the interparticle compositional distribution of x A (denoted below as simply x) be a Gaussian with standard deviation σ c and mean \( \overline{x} \):

$$ \rho (x)\propto \exp \left(-\frac{{\left(x-\overline{x}\right)}^2}{2{\sigma}_c^2}\right). $$
(19.15)

In EXAFS signal, clusters with a greater number of A atoms are weighted more than the clusters with fewer A atoms. We thus write the apparent partial coordination numbers as:

$$ {\tilde{n}}_{AA}=\frac{{\displaystyle \underset{0}{\overset{1}{\int }}\rho (x){n}_{AA}(x)xdx}}{{\displaystyle \underset{0}{\overset{1}{\int }}\rho (x)xdx}}. $$
(19.16)

Figure 19.6 shows the values of \( {\tilde{n}}_{AA} \) and \( {\tilde{n}}_{AA} \) for various values of \( \overline{x} \) and σ c calculated for a cluster containing N = 100 atoms. Cluster cartoons are added for clarity. A single cluster (the cuboctahedral shape was chosen for illustration purpose only) on the left corresponds to narrow inter-cluster compositional distribution (i.e., small σ c ). Three clusters on the right illustrate the change in composition from cluster to cluster (large σ c ). In all cases, the intra-cluster distributions are random.

Fig. 19.6
figure 6

Normalized partial coordination numbers of (a) AA and (b) AB pairs as functions of the standard deviation σc around the average cluster composition <x> for clusters of N = 100 atoms, calculated assuming a Gaussian compositional distribution. In both figures, cartoons next to the \( <x>=0.5 \) curve illustrate the difference between the narrow (one cluster on the left) and broad (three different clusters on the right) compositional distributions. Reproduced from ref. [148] by permission of The Royal Society of Chemistry

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These results indicate that the ensemble-average coordination numbers \( {\tilde{n}}_{AA} \) can be smaller for narrow compositional distributions or larger for broad distributions than the coordination numbers predicted by the equation n AA = xn MM. The reason they are smaller for narrow distributions than the nominal coordination numbers was demonstrated above (Eq. 19.13 and Fig. 19.5b). For broad distributions, the effect is due to the ensemble-averaging that favors A-rich clusters over the B-rich clusters (relative to \( {\overline{x}}_{\mathrm{A}} \)). Similar conclusions can be extended to the other partial coordination numbers. The two sets of values, \( {\tilde{n}}_{AA} \) and n AA, as well as \( {\tilde{n}}_{AB} \) and n AB, agree for \( {\sigma}_c=\sqrt{x\left(1-x\right)/N} \), for which the normal distribution coincides with binomial distribution.

In summary, partial coordination numbers in heterometallic NPs can be employed to accurately quantify the intra-particle homogeneity and short-range order for arbitrary cluster sizes and a wide range of component fractions, provided that all the clusters possess nearly identical compositions. If the intra-cluster distribution is completely random but the elemental composition varies widely from cluster to cluster, the coordination numbers measured by EXAFS will point toward either negative (\( {\tilde{n}}_{AA}<{n}_{AA} \)) or positive (\( {\tilde{n}}_{AA}>{n}_{AA} \)) short range order, which, in the latter case, can be mistaken for a core–shell motif, among other segregation scenarios, even though all clusters are completely random. The only exception when the apparent coordination numbers coincide with those in the “mean” cluster is when the compositional distribution is binomial. With the knowledge of actual compositional distribution (e.g., using energy dispersive X-ray analysis done in electron microscopy experiment) it is possible to correct apparent coordination numbers for the compositional distribution effects (Fig. 19.6).

7 Overlapping Absorption Edges

Heterometallic systems containing two or more elements with overlapping absorption edges cannot be simply analyzed by EXAFS since the EXAFS at the higher energy edge overlaps with the EXAFS extending from the lower energy edge. This is a particularly significant problem for metals that neighbor each other in the periodic table such as Re, Ir, Pt, and Au, whose L3, L2, and L1 absorption edges overlap. Unless these overlapping contributions are disentangled, extracting structural information from the data via traditional data analysis strategies is either not possible [149, 150] or difficult and/or insufficiently accurate [151].

The problem of overlapping edges in EXAFS analysis is not limited to heterometallic catalysts, of course. BaTiO3 is among the most extensively studied perovskites, yet its EXAFS studies are complicated due to the overlap of Ti K-edge and Ba L3 edge. B. Ravel et al. proposed a very original use of diffraction anomalous fine structure (DAFS) technique to deconvolute the EXAFS signals from Ti and Ba [152, 153]. Other methods have appeared recently, based on the use of the high energy resolution fluorescence detection (HERFD) that enabled separation of emission lines from different elements [85, 154–156].

Menard et al. reported a new method for deconvolution of overlapping absorption edges that is based on the use of concurrent, multiple edge analysis of EXAFS data from each edge [63]. The analysis strategy is demonstrated here for an arbitrary bimetallic composition even though, for illustration purpose only, they used notation Ir and Pt for its constituent elements. Data analysis is done by a simultaneous fit of both Ir L3 and Pt L3 edges, which involve three contributions: (1) the Ir EXAFS in the Ir L3 edge before the Pt L3 edge; (2) the Ir EXAFS in the Pt L3 edge; and (3) the Pt EXAFS in the Pt L3 edge. Because (1) and (2) describe the same coordination environments they should be constrained analytically, in the process of fitting each contribution to the experimental data. The analysis is done in r-space and is limited to nearest neighbor scattering paths, which are usually well isolated from longer scattering paths in the Fourier transforms of the EXAFS signal χ(k). In this case, the EXAFS equations that are simultaneously fit are :

$$ {\chi}_{\mathrm{Ir}\ \mathrm{edge}}\left({k}_{\mathrm{Ir}}\right)=\frac{S_{0,\mathrm{I}\mathrm{r}}^2{N}_{\mathrm{Ir}}}{k_{\mathrm{Ir}}{R}_{\mathrm{Ir}}^2}\left|{f}_{\mathrm{Ir}}^{eff}\left({k}_{\mathrm{Ir}}\right)\right| \sin \left[2{k}_{\mathrm{Ir}}{R}_{\mathrm{Ir}}-\frac{4}{3}{\sigma}_{\mathrm{Ir}}^{(3)}{k}_{\mathrm{Ir}}^3+{\delta}_{\mathrm{Ir}}\left({k}_{\mathrm{Ir}}\right)\right]{e}^{-2{\sigma}_{\mathrm{Ir}}^2{k}_{\mathrm{Ir}}^2}{e}^{-2{R}_{\mathrm{Ir}}/{\lambda}_{\mathrm{Ir}}\left({k}_{\mathrm{Ir}}\right)}, $$
(19.17)

and

$$ \begin{array}{ll}{\chi}_{\mathrm{Pt}\ \mathrm{edge}}\left({k}_{\mathrm{Pt}},{k}_{\mathrm{Ir}}\right)=& \frac{S_{0,\mathrm{P}\mathrm{t}}^2{N}_{\mathrm{Pt}}}{k_{\mathrm{Pt}}{R}_{\mathrm{Pt}}^2}\left|{f}_{\mathrm{Pt}}^{eff}\left({k}_{\mathrm{Pt}}\right)\right| \sin \left[2{k}_{\mathrm{Pt}}{R}_{\mathrm{Pt}}-\frac{4}{3}{\sigma}_{\mathrm{Pt}}^{(3)}{k}_{\mathrm{Pt}}^3+{\delta}_{\mathrm{Pt}}\left({k}_{\mathrm{Pt}}\right)\right]\hfill \\ {}& \times {e}^{-2{\sigma}_{\mathrm{Pt}}^2{k}_{\mathrm{Pt}}^2}{e}^{-2{R}_{\mathrm{Pt}}/{\lambda}_{\mathrm{Pt}}\left({k}_{\mathrm{Pt}}\right)}+\frac{A{S}_{0,\mathrm{I}\mathrm{r}}^2{N}_{\mathrm{Ir}}}{k_{\mathrm{Ir}}{R}_{\mathrm{Ir}}^2}\left|{f}_{\mathrm{Ir}}^{eff}\left({k}_{\mathrm{Ir}}\right)\right| \sin \hfill \\ {}& \times \left[2{k}_{\mathrm{Ir}}{R}_{\mathrm{Ir}}-\frac{4}{3}{\sigma}_{\mathrm{Ir}}^{(3)}{k}_{\mathrm{Ir}}^3+{\delta}_{\mathrm{Ir}}\left({k}_{\mathrm{Ir}}\right)\right]{e}^{-2{\sigma}_{\mathrm{Ir}}^2{k}_{\mathrm{Ir}}^2}{e}^{-2{R}_{\mathrm{Ir}}/{\lambda}_{\mathrm{Ir}}\left({k}_{\mathrm{Ir}}\right)}.\end{array} $$
(19.18)

The factor A = Δμ 0,Ir/Δμ 0,Pt, where Δμ 0,Ir and Δμ 0,Pt are the changes in the absorption at the edge steps, is necessary because the extraction of χ(k) includes a normalization to these edge steps. The nonlinear least squares fitting of experimental data to Eqs. (19.17 and 19.18) should be done concurrently to the overlapping L3 edges and can be achieved using available EXAFS analysis tools. In ref. [63], the interface programs Athena and Artemis were used. In practice, correction of the energy grid in k-space for the Ir EXAFS in the Pt L3 edge should be made. The correction to the threshold energy (in eV) for the Ir EXAFS in the Pt L3 edge is defined as ΔE 0,Ir − (349 + ΔE 0,Pt), where 349 eV is the difference between the empirical threshold energies. Such a large energy origin shift is necessary in this method since it accounts for a unique k = 0 reference point for the Ir EXAFS extending beyond the Pt edge when the Pt edge EXAFS is transformed to k-space. The exact value to use (here 349 eV) will depend on the E0 values that are used in the edge subtraction of the EXAFS spectra. The representative data and fits in r-space are shown in Fig. 19.7. The signature of the Ir L3 EXAFS “leaking” into the Pt L3 EXAFS is a low r feature in Fig. 19.7b.

Fig. 19.7
figure 7

Comparison of the data and fit of the Ir-Pt NPs on γ-Al2O3 under a H2 atm measured at 215 K at the Ir L3 and Pt L3 absorption edges. Fourier transform magnitude of (a) the Ir L3 data and fit, and (b) the Pt L3 data and fit with the contributions of the individual paths represented. Reproduced with permission from ref. [63]

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8 Outlook and Future Developments

With the ongoing development of X-ray absorption spectroscopy techniques, the opportunities for investigation of mechanisms of catalytic reactions employing nanoscale metal catalysts are growing and new challenges, previously ignored or overlooked, come to the surface. One such important challenge is the heterogeneity of the NP ensembles that is evident even in samples with narrow size and compositional distributions and is a common property of real catalysts with large compositional gradients. That heterogeneity, when ignored, causes artifacts in data analysis, as demonstrated above in the case of the compositional heterogeneity, and is also shown by Yevick and Frenkel [109] for the case of the structural heterogeneity due to surface relaxation in nm-scale clusters.

One possible solution is the single nanoparticle spectroscopy studies by XAS methods [157]. Current and future capabilities of X-ray spectromicroscopy based on coherence-limited imaging methods including nano-probe methods were discussed by Hitchcock and Toney [158]. These methods will benefit from the dramatic increase in brightness expected from a diffraction-limited storage ring. The applicability of nano-probe methods for spectroscopy studies of single NPs was illustrated by Y. Chu’s group. They studied the oxidation process of individual PtNi NPs by a scanning multilayer Laue lens X-ray microscope and discovered the transformation of alloyed PtNi (140 to 320 nm) to Pt/NiO core–shell and the further coalescence under thermal oxidation [159]. The scanning was performed using 10 nm step size and 30 nm focal spot size at beamline 26ID of the Advanced Photon Source at Argonne National Laboratory. This method is being developed for to the hard X-ray nanoprobe (HXN) beamline of the National Synchrotron Light Source II at BNL. Alternatively, photoemission electron microscopy was used for studying of individual Co NPs with the size of 8 nm. Significant variations in the shapes of the Co L2,3 edges of the X-ray absorption spectra between different cobalt NPs were detected and attributed to different cobalt–oxygen interactions on a particle-by-particle basis [160]. For these spectromicroscopic methods on the basis of nanoprobe, the main challenge is to keep high flux while reducing the spot size. More details on challenges and limitations of nano-probe methods for catalytic investigations were addressed recently by Frenkel and van Bokhoven [157].

Studying nanocatalysts at the single-nanoparticle level and in operando mode adds more challenges. For spectroscopic methods, the key is to increase particle sensitivity and for electron microscopic methods it’s to enable realistic working conditions. In most electron microscopy studies of catalysis, they were either investigated in an ex-situ mode, i.e., catalysts were pretreated elsewhere under controlled conditions of atmosphere, pressure and temperature, while measured under high vacuum at low temperatures, or under simulated working conditions with lower temperature and pressure compared to the real ones [46, 161, 162]. To bridge this “pressure gap”, a new mode of operation is needed, where relevant (for structural analysis) techniques can probe catalysts in the same reaction conditions. Recently, Stach and Frenkel demonstrated the advantage of using such a micro-reactor, for nanocatalysis studies at ambient temperature and pressure [104]. Basing on the idea of combining electron microscopy with spectroscopic techniques by sharing the same reactor to make sure samples/conditions under study are the same for all types of techniques [163, 164], the group discovered the complex structural dynamics of Pt/SiO2 under ethylene hydrogenation conditions that occurs at broad length scale [104].

In conclusion, with the development of XAFS instrumentation and analysis methods, the understanding of structure of nanoparticle catalysts is advanced from qualitative pictures of the mid-70s to much more quantitative ones that are capable to capture fine architectural and compositional details, and account for interparticle and intraparticle heterogeneities. More achievements are expected in the coming years, owing to the reduced beam sizes, improved energy and time resolutions, and new developments in the combinations of XAFS with complementary imaging and scattering methods in operando conditions.