Rich Ground-State Chemical Ordering in Nanoparticles: Exact Solution of a Model for Ag-Au Clusters

We show that nanoparticles can have very rich ground-state chemical order. This is illustrated by determining the chemical ordering of Ag-Au 309-atom Mackay icosahedral nanoparticles. The energy of the nanoparticles is described using a cluster expansion model, and a mixed integer programming approach is used to find the exact ground-state configurations for all stoichiometries. The chemical ordering varies widely between the different stoichiometries and displays a rich zoo of structures with nontrivial ordering.

Figure 1

A selection of ground-state configurations in a 309-atom Mackay icosahedron Ag-Au nanoparticle. The configurations exhibit a diverse range of highly ordered interior and surface structures. (a),(b) The transparent atoms represent a single layer of Ag atoms; (g),(h) they represent two layers of Au atoms. The Au atoms exhibit a strong preference for subsurface shell sites, in particular, the subsurface corners (a), followed by the subsurface edges (b). The Ag atoms show a strong preference for second shell corner sites (g),(h), as well as the central atom. The lowest energy configuration (c) (shown as a slice through the nanoparticle) has a perfectly ordered onion-shell structure. At high Au concentrations, the preference of Au atoms for corner, edge, or interior sites of the surface facets varies as a function of composition (d)–(f).

Figure 2

Number of Au atoms in each shell vs the total number of Au atoms in the nanoparticle. The structural evolution exhibits a complex interplay between the shells, with no monotonically increasing shell occupancies.

Figure 3

Energy of formation (from EMT) vs the number of Au atoms in the nanoparticle. The gray dots show configurations sampled for CE model construction. The blue line shows the convex hull of the ground-state configurations. Compositions at which the nanoparticle exhibits strong ordering (shown in Fig. 1) are marked with a circle.

Figure 4

Change in formation energy in pure nanoparticles upon the replacement of a single atom with the opposite species, for all sites in the outer three shells. This simple measure accounts for much of the variation in the ground-state structures, but is tempered at higher concentrations by a preference for heteroatomic bonding.