Size-dependent diffusion of supported metal nanoclusters: mean-field-type treatments and beyond for faceted clusters

Nanostructured systems are intrinsically metastable and subject to coarsening. For supported 3D metal nanoclusters (NCs), coarsening can involve NC diffusion across the support and subsequent coalescence (as an alternative to Ostwald ripening). When used as catalysts, this leads to deactivation. The dependence of diffusivity, D_N, on NC size, N (in atoms), controls coarsening kinetics. Traditional mean-field (MF) theory for D_Nversus N assumes that NC diffusion is mediated by independent random hopping of surface adatoms with low coordination, and predicts that D_N ∼ hN^-4/3n_eq. Here, h = ν exp[-E_d/(k_BT)] denotes the hop rate, and n_eq = exp[-E_form/(k_BT)] the density of those adatoms. The adatom formation energy, E_form, approaches a finite large-N limit, as does the effective barrier, E_eff = E_d + E_form, for NC diffusion. This MF theory is critically assessed for a realistic stochastic atomistic model for diffusion of faceted fcc metal NCs with a {100} facet epitaxially attached to a (100) support surface. First, the MF formulation is refined to account for distinct densities and hop rates for surface adatoms on different facets and along the base contact line, and to incorporate the exact values of E_form and n_eqversus N for our model. MF theory then captures the occurrence of local minima in D_Nversus N at closed-shell sizes, as shown by KMC simulation. However, the MF treatment also displays fundamental shortcomings due to the feature that diffusion of faceted NCs is actually dominated by a cooperative multi-step process involving disassembling and reforming of outer layers on side facets. This mechanism leads to an E_eff which is well above MF values, and which increases with N, features captured by a beyond-MF treatment.