Transition metal nanoparticles are widely used in heterogenous catalysis for a broad variety of reactions ranging from hydrogenation to emission control, with nanoparticle sizes typically ranging from 3–20 nm. [1,2,3] While smaller particles have a higher specific surface area per gram of catalyst, they are often unstable and tend to sinter to larger particles during application. [4, 5] Interestingly, it has been reported that smaller sizes lead not only to larger catalytic surface areas and higher concentrations of various surface terminations, but could also exhibit different catalytic properties. Perhaps the most striking example is given by gold nanoparticles that have been shown to be highly active for CO oxidation while gold is otherwise not active at all. [6,7,8] These strong deviations of smaller clusters and particles from bulk-like behavior are often denoted as ‘quantum size effects’ and are due to the vanishing of the d-band into more localized, atomic-like d-states. [9, 10] The size effect has been investigated theoretically for gold [9, 11] and platinum [10] nanoparticles up to the size of 3.7 nm. The transition of the underlying electronic structure towards the bulk limit has been identified at around 2.5 nm at which point the adsorption energies of O and CO converge to the limit of extended surfaces accompanied by the formation of a d-band. Other studies of the effect of the particle size on adsorption energies included Pt [12, 13] and Pd [14, 15]. Apart from gold, copper and silver are highly interesting transition metals with applications in methanol synthesis, [16,17,18] the water–gas-shift reaction [19] and selective oxidation [20, 21]. Particle size effects have also been identified for copper experimentally (e.g. for methanol synthesis), [22] but theoretical studies exist only for smaller copper clusters (< 1 nm). [23,24,25,26,27,28,29,30,31] Similarly, experimental data for Ag demonstrated a strong size dependency for ethylene epoxidation [32, 33], but only small Ag clusters have been studied theoretically to date. [34].

In this work, we investigate the effect of particle size for all three coinage metals, Cu, Ag and Au using density functional theory (DFT) calculations. As particles that are relevant for catalysis are typically larger than 1 nm for Cu [22] Ag [35] and Au [6], we are particularly interested in particle size effects including these sizes. While particles are mostly supported in heterogeneous catalysis, we investigate free standing particles. This has the advantage of decoupling particle size effects from perturbations of a particular support in addition to being computationally more feasible. We therefore note that a direct comparison with experiments would need to account for metal-support interactions in addition to size effects. We focus on the oxygen adsorption energy, that has been shown to be a key descriptor in the hydrogenation of CO2 to methanol, [36] using cuboctahedral nanoparticles ranging from 13 to 1415 atoms (0.7–3.5 nm in diameter, see Fig. 1). The clusters are chosen such that we can systematically investigate the size effect while maintaining the shape of the cluster and the adsorption site of the oxygen. We employ the BEEF-vdW functional [37] to calculate the oxygen adsorption energy on the fcc adsorption position closest to the corner of the (111) facet of the nanoparticles such that the adsorption site is the same for all particles with the exception of M13, as shown in the insets in Fig. 1. For Cu, the effect of geometry optimization is studied explicitly. For free standing metal Cu clusters ranging from 13 to 923 atoms, a linear relation between the mean Cu-Cu distance, the chemical potential which is approximated by the average energy per atom, and the number of atoms to the power of -1/3 (see Figure S3) is observed, similarly to what has been found for other metals [13, 34, 38,39,40,41,42,43,44,45]. Figure 1 shows the adsorption energy of an oxygen atom on the Cu clusters with increasing size. The filled black circles show the results of the optimized structures. It can be seen that the adsorption energy converges to the limit of the Cu(211) surface for Cu309 and larger. As previously seen for Au55, Cu55 shows a comparatively weak adsorption energy for oxygen. Comparing these results to single point calculations, using the Cu bulk lattice constant and the copper-oxygen distance obtained from Cu(111) surface calculations, it can be seen that all adsorption energies are shifted to higher values by approximately 0.1 eV. Keeping in mind that the accuracy of the BEEF-vdW functional with respect to adsorption energies on transition metal surfaces is around 0.2 eV, [46] the constant shift implies that the effect of strain is small and similar for all clusters, which can be also seen in Figure S4 which plots the adsorption energy of oxygen on strained Cu(111) surfaces. This is in agreement with Kleis et al. [9] and Li et al. [10], who found that trends in adsorption energies are similar for relaxed and unrelaxed clusters for Au and Pt. As was shown before by Mavrikakis et al. [47] and others [48,49,50], the overall effect of strain is usually linear for up to 3% deviation [51] from the lattice constant.

Fig. 1
figure 1

Adsorption energy of oxygen (relative to ½ gas phase O2) on various fully optimized (black filled circles) and unrelaxed (red empty squares) Cu nanoparticles as a function of the number of atoms within the nanoparticles (Natoms). The upper abscissa gives the corresponding nanoparticle diameter in nm derived from spherical particles (see SI). The surface limit of the relaxed nanoparticles is given as a black dotted line and the corresponding limit for the fixed geometry as a red dotted line as calculated for adsorption on the fcc site of a Cu(211) slab with a coverage of 1/9. As insets, all Cu nanoparticles with adsorbed oxygen (black) are shown

Figure 2 compares the oxygen adsorption energy on unrelaxed Cu, Ag and Au nanoparticles, showing that for the 309-atomic cluster adsorption energies are converged to within about 0.2 eV. Interestingly, all three coinage metals follow the exact same trend, that is (1) strong adsorption on M13, (2) weakest adsorption on M55, (3) again stronger binding on M147 and (4) slow convergence from M309 and M561 towards the bulk limit. Overall, there is a systematic offset between the binding energies of a given cluster for the three metals, with binding energies generally decreasing from Cu over Ag to Au.

Fig. 2
figure 2

Adsorption energy of oxygen (relative to ½ gas phase O2) on unrelaxed Cu, Ag and Au nanoparticles versus the number of atoms. The upper abscissa shows the corresponding nanoparticle diameter. The lines are the surface limit for adsorption on the fcc-site on the fixed 211-slab of the corresponding metal

Li et al. [52] argued that quantum size effects play a significant role for platinum clusters below 2 nm. This has been attributed to a change from a broad d-band to more discrete d-states, as the extent of orbital overlap of individual atoms decreases.

So far, our study only includes geometric closed-shell cuboctahedral clusters [53], which can be generated by sequentially adding coordination shells starting from a single atom, resulting in a monotonic increase of cluster sizes by increments of about 0.4–0.5 nm. In order to investigate the size effect for smaller variations in more detail we investigated Cu nanoparticles ranging from 44 to 200 atoms as shown in Fig. 3a. These clusters have been derived from cuboctahedral Cu55, octahedral Cu146, and cuboctahedral Cu147 by adding or removing atoms to or from the fcc lattice positions to create stable intermediate clusters. In particular, the atoms with lowest coordination number were removed first or atoms were added so that the added atom has the highest possible coordination number. In case of different options for adding or removing atoms with the same coordination number, the site associated with the largest vertical distance from the top facet was chosen to minimize the direct influence on the oxygen adsorption site. For example, for cuboctahedral particles, atoms were first added to the fcc(100) sites towards the formation of octahedral particles (the coordinates of all structures are given in the SI). It can be seen that the oxygen adsorption energy varies strongly with the number of atoms and is oscillating for every 20–40 atoms. This can be explained by the so-called electronic magic numbers occurring for 58, 92 and 138 electrons, where closed-shell configurations are reached for spherical jellium clusters. [45] For the electronic configuration of coinage metals (d10s1), considering only the single s-electron per atom, particularly stable clusters are thus reached for cluster sizes Natoms = 58, 92 and 138, e.g. cluster sizes with the number of atoms equal to the number of electrons in the closed-shell jellium configurations. Larsen et al. [12] have shown that for Cu, Ag and Au, clusters are particularly stable around these numbers and demonstrated that Au clusters show an increased oxygen adsorption energy. We observe a similar behavior in the oxygen adsorption energy for Cu, which shows a maximum close to these electronic magic numbers. Quantum size effects do hence play a key role for small Cu clusters where the oxygen adsorption energies change quite drastically with the addition or removal of a few copper atoms. Note also that the weak adsorption of Cu55 as well as the rather strong adsorption of Cu147 are close to the corresponding minimum and maximum. The drastic change in oxygen adsorption also means that for example Cu147 has a strong adsorption of about 2.1 eV, but that this is reduced by 0.4 eV when going to Cu140. Atomically precise control of the particle size and size distribution would therefore be necessary if one wanted to adjust the oxygen binding energies within a narrow range. Interestingly, the behavior is very similar for Ag and Au, that follow the trends in oxygen chemisorption energies observed for copper in the 50–200 atoms range. As shown in Fig. 3b, the stability of the metal clusters follows the usual increasing trend with cluster size, further highlighting the difficulties in precise cluster synthesis and stabilization in the reported range. Synthesis techniques such as wet impregnation or colloidal synthesis are presently not able to achieve such a high atom specificity [54] and further improvement is needed for the future design and preparation of nanoparticles.

Fig. 3
figure 3

a Oxygen adsorption energy (relative to ½ O2 in the gas phase) on various Cu, Ag and Au fixed geometry nanoparticles as a function of particle size. b Chemical potential \({\mu }_{M}\) relative to the bulk chemical potential of M, \({\mu }_{CM}^{bulk}\) for M = Cu, Ag, Au as function of the number of atoms. The filled black circles correspond to the M55 cuboctahedral cluster, the M79 truncated octahedral and the M85 octahedral cluster respectively. The filled red circle corresponds to the M147 cuboctahedral cluster. The filled blue squares correspond to the octahedral M146 and truncated octahedral M140 clusters. The open circles and open squares correspond to clusters that can be derived from the parent clusters described above by adding or removing atoms in bulk fcc-positions. The insets show the atomic structure of oxygen adsorption (black atom) on octahedral Cu85 and truncated octahedral Cu79

In summary, we investigated the effect of particle size on the adsorption energy of oxygen on the coinage metals Cu, Ag and Au. We found that the effect of surface relaxation on oxygen adsorption is relatively small compared to the changes with particle size, thus allowing the use of less expensive single point calculations and thus larger particles to capture the trends in adsorption energies. Interestingly, all three metals follow the exact same trend, where adsorption is strongest for M13, weakest for M55 and approaches smoothly the bulk limit for M309/M561. In contrast, changes are quite drastic for smaller clusters for which an increase or decrease by one atom in a copper cluster changes the oxygen adsorption energy by as much as 0.1 eV. While it is interesting to note that we found the exact same trend for all three coinage metals, future work will show how this relates for other transition metals as well as how support interactions impact the oxygen adsorption energy.