Surface energy and its anisotropy of hexagonal close-packed metals

The surface energy and its anisotropy of 13 hexagonal close-packed (HCP) metals have been investigated via a broken-bond based geometric model. The model can assess arbitrary orientations which are difficult to construct in atomistic simulations. Using only three material dependent parameters, our results are in good agreement with the majority of reported experimental values. An exception occurs in the cases of divalent sp metals, namely Mg, Zn and Cd, for which the calculated values are lower by a factor of 2. For all 13 metals, the stereographic projections of surface energy demonstrate strong six-fold symmetries with a global minimum on (0001) pole, whereas the actual projection patterns are unique for every element. The overall anisotropy is found to be 15% to 21%. The equilibrium crystal shape of HCP metals is found to be a truncated hexagonal bi-prism, with the (0001) facets always shown, but the bi-prismatic facets vary from one metal to another. The detailed anisotropy of surface energy is found to be largely determined by an anharmonicity factor η. The results of metals possessing comparatively low η, namely Be, Sc, Ti, Y, Zr and Hf, are in better agreement with experimental findings. We believe the surface energy anisotropy of these elements is more representative for HCP metals.