Cohesion and coordination effects on transition metal surface energies

Abstract

Here we explore the accuracy of Stefan equation and broken-bond model semiempirical approaches to obtain surface energies on transition metals. Cohesive factors are accounted for either via the vaporization enthalpies, as proposed in Stefan equation, or via cohesive energies, as employed in the broken-bond model. Coordination effects are considered including the saturation degree, as suggested in Stefan equation, employing Coordination Numbers (CN), or as the ratio of broken bonds, according to the bond-cutting model, considering as well the square root dependency of the bond strength on CN. Further, generalized coordination numbers CN¯ are contemplated as well, exploring a total number of 12 semiempirical formulations on the three most densely packed surfaces of 3d, 4d, and 5d Transition Metals (TMs) displaying face-centered cubic (fcc), body-centered cubic (bcc), or hexagonal close-packed (hcp) crystallographic structures. Estimates are compared to available experimental surface energies obtained extrapolated to zero temperature. Results reveal that Stefan formula cohesive and coordination dependencies are only qualitative suited, but unadvised for quantitative discussion, as surface energies are highly overestimated, favoring in addition the stability of under-coordinated surfaces. Broken-bond cohesion and coordination dependencies are a suited basis for quantitative comparison, where square-root dependencies on CN to account for bond weakening are sensibly worse. An analysis using Wulff shaped averaged surface energies suggests the employment of broken-bond model using CN to gain surface energies for TMs, likely applicable to other metals.