Single vacancies at Σ5, Σ9 and Σ11 grain boundaries of copper and the geometrical factors that affect their site preference

The occurrence and distribution of vacancies has implications in many technologically relevant properties of materials. Despite their importance there is no detailed account of vacancy preference at atomic sites of frequently occurring grain boundaries (GBs) of fcc Cu. Here we present a computational investigation of single vacancy site preference at three high angle low index symmetric tilt GBs of fcc Cu: Σ5(310) [100], 36.9°; Σ9(2 2‾1‾) [110], 38.9°; and Σ11(1 1‾ 3) [110], 129.5°; performed with density functional theory (DFT). Using Voronoi tessellation data we developed a symmetry quantifying parameter (σ) based on the difference in surface area between the Voronoi polyhedron and the Wigner-Seitz sphere of the corresponding atomic site. We show that this parameter accurately quantifies changes in symmetry at each atomic site relative to the symmetry at the bulk fcc lattice. Subsequently, we performed a geometrical analysis that shows that the correlation between the values of σ with the coordination number (Cn) and with the volume expansion (Vx) are different for the three GBs, and that their dependencies on the distance from the GB planes are also different. We show that for most atomic sites, a reduction in symmetry is associated with changes in Vx and this is GB dependent. The GB that has the largest distortions relative to the bulk is the Σ5, closely followed by the Σ9, while the Σ11 has a decrease in symmetry which is only ¼ of the other two. At Σ5, changes in site Cn imply changes in the site symmetry to extents that do not occur for the other two GBs.