Cluster size dependence

To maximize the rate of a reaction, one needs the maximum exposure of metal- or catalytically active atoms to the reactants. Hence there is a great desire to stabilize small particles on catalyst supports. In the next two subsections on transition metals we will provide a detailed description of changes in the chemical reactivity of transition metals when the particle size decreases. This provides a short background to aid in understanding the effects of particle size on catalysis. In the next subsection we discuss cluster size dependence effects and in the subsections that follow we will summarize the specific effort on supported Au clusters. 

One can distinguish at least three different characteristic regions for transition metal particlesl and their catalytic activity  

We will discuss in detail cases (a) and (b). The shapes of crystallites in catagory (c) are controlled by bulk-metal energies and therefore do not require separate treatment. 

While the bulk formation energy of metallic Rh is -555 kJ/at, the most stable Rhis cluster results in a corresponding value of only -299 kJ/at. The differences may reflect the much lower average coordination number of the cluster atoms compared with the bulk. One also notes that a few selected clusters such as Rha, D h] RJ14, Dha] Rh4, D h, Rhg, Oh] Rhi3, Ih have similar formation energies, whereas the atoms in these clusters have very different coordination numbers. The planar configurations appear to be the preferred clusters at least for the smaller sized clusters. 

Quantum chemical bonding details determine the relative stability and structure of these clusters. Because of their lower stability, the small metal clusters can be expected to be generally highly active. The reactivities usually show a maximum at a particle size between three and seven metal atomsl . Three parameters appear to be important in controlling the reactivity of these clusters the coordinative unsaturation of the surface atoms, the availability of enough cluster atoms to bind with an adsorbing atom or 

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Figure 1.12 The three different types of cluster size dependence of catalytic conversion. Rates are considered normalized per exposed surface atom (schematic).

Figure 19. (a) Cluster size dependence of the rate constants for the reactions of CO2 with the large hydrated hydroxyl anions at T= 130 K O, experimental values for OH (H2O), —, calculated values for 0H (H20)n. (b) Dependence of rate constants on cluster size for the reactions of 0H (H20)n with SO2 at T = 135 K. Taken with permission from ref. 19. 

In diamond, Sahoo et al. (1983) investigated the hyperfine interaction using an unrestricted Hartree-Fock cluster method. The spin density of the muon was calculated as a function of its position in a potential well around the T site. Their value was within 10% of the experimental number. However, the energy profiles and spin densities calculated in this study were later shown to be cluster-size dependent (Estreicher et al., 1985). Estreicher et al., in their Hartree-Fock approach to the study of normal muonium in diamond (1986) and in Si (1987), found an enhancement of the spin density at the impurity over its vacuum value, in contradiction with experiment this overestimation was attributed to the neglect of correlation in the HF method. 

The cluster-size dependence has not yet been explained it may reflect an intrinsically low activity of the clusters, but it might also be a consequence of increasing removal of residual ligands such as C from the clusters with increasing severity of treatment in H2. Other possibilities include a steric effect of the support, limiting adsorption of the reactants on the metal— such an effect would be greatest for the smallest clusters. Electronic effects should not be ruled out. 

Figure 12.5 shows the cluster size dependence of X-ray emission spectra. The top, middle, and bottom curves represent the spectra measured with a laser contrast of C = 4 x 10-4 at Ar gas pressures of 60, 50, and 40 bar, respectively. Note that no X-rays were observed at an Ar gas pressure of 40 bar. According to hydrodynamic calculations (see Fig. 12.2), at 40 bar, a cluster with an average diameter of 200 nm is one order of magnitude smaller than that with an average diameter of 1.5 pm at 60 bar. Thus, in the case of the 40-bar experiment, the clusters were almost completely destroyed by the prepulse. This result demonstrates the important role of big clusters, and the validity of the nozzle design. 

Fig. 12.5. The cluster size dependence of X-ray emission spectra measured at an intensity of 3 x 1018 W/cm2, a pulse duration of 30 fs, and a laser contrast of 4 X 10" Ar = 60 bar (top curve) At = 50 bar (middle curve) Ar = 40 bar (bottom curve)...

From Fig. 27 the activity seems to increase by decreasing cluster size. In fact if we want to compare the intrinsic activity of clusters with different size the TON is not necessarily a pertinent parameter. Indeed, if the reaction rate depends on the pressure of at least one reactant, the TON would not take into account the fact that the total flux of one reactant joining the clusters is not solely given by the pressure of this reactant, but we have also to consider the flux of the molecules physisorbed on the substrate. This contribution can be up to 10 times larger than the direct flux and it is strongly cluster size dependent (see Section 4). In that case the right parameter to compare the intrinsic activity of the different clusters is the reaction probability (of NO or CO). It is equal to the consumption rate of one reactant divided by... 

Worz AS, Judai K, Abbet S, Heiz U (2003) Cluster size-dependent mechanisms of the CO-tNO reaction on small Pdn (n< = 30) clusters on oxide surfaces. J Am Chem Soc 125 7964... 

Further evidence for the validity of the frontier orbital approach derives from its success in predicting the shift (increase or decrease) in naked cluster IP upon the chemisorption of small reactant molecules. For all metal clusters examined thus far, H2 chemisorption induces an increase in cluster IP. ° This follows directly from interactions (1) and (2), since the creation of the two new metal-hydride bonding orbitals effectively removes two electrons from the cluster valence orbital manifold. Thus with resjiect to the metal cluster, H2 chemisorption can be viewed as an oxidative addition process. If a one-electron (Aufbau filling) approximation is assumed as above, the Fermi level of the cluster is shifted toward lower energy, that is, there is an increase in IP. As the cluster grows larger, the shift in IP diminishes. This is simply a manifestation of cluster-size-dependent variations in the valence orbital density of states, and is again consistent with the frontier orbital model. 

Fig. 17 Cluster size dependence of the overlap integral between the ground state wave function GS) obtained by the Lanczos method and the trial wave function [6]. In the variational wave function the...

Cluster Size Dependence of the Velocity of Sound c and the Energies Soi of the Lowest Breathing... 

The breathing mode energies Coi calculated [128] from the cluster size dependence of the sound velocity (Table IV) are considerably lower than the LDM result for small clusters. 

From the analysis of the cluster size dependence of the superfluid density (or order parameter) the following conclusions emerge ... 

Figure 10. Cluster size dependence of the binding energy of an electron in a surface state (n = 1, / = 0) on ( He)jy clusters. The localization threshold is manifested at = 5.7 x 10 and the binding energy increases with increasing the cluster radius R, according to the scaling law [ j(R) — j(oo) (X (R — reaching the flat surface binding energy Es(oo) =—OJ meV...

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