Consideration of the geometric and thermodynamic properties of the Wulff construction

By considering the geometrical characteristics of the Wulff construction and taking this as the approximate structure of nanoparticles, the average coordination number (CN), Z _c(r), as a function of both size and shape, is given, with r being the particle radius. It is found that the Z _c(r)/Z _b0 value of the Wulff construction not only approaches that of bulk structures, such as body-centered cubic and simple cubic, at larger r but also approaches that of cluster structures, such as an icosahedron and a cuboctahedron, at smaller r, where Z _b0 is the bulk inner CN. Based on the Z _c(r) function established in this work, the cohesive energy and melting points of nanoparticles for structural stability consideration are modeled. The model predictions are in good agreement with experimental observations, confirming that taking the Wulff construction as the nanoparticle structure is valid, although it leads to small error.