Tile Hamiltonians for decagonal phases

A tile Hamiltonian (TH) replaces the actual atomic interactions in a quasicrystal with effective interactions between and within tiles. We study Al–Co–Ni and Al–Co–Cu decagonal quasicrystals described as decorated hexagon-boat-star (HBS) tiles using ab initio methods. A dominant term in the TH counts the number of H, B and S tiles, favoring tilings of H and B only. In our model for Al–Co–Cu, chemical ordering of Cu and Co along tile edges defines tile edge arrowing. Unlike the edge arrowing of Penrose matching rules, however, the energetics for Al–Co–Cu do not force quasiperiodicity. Energetically favored structures resemble crystalline approximants to which the actual quasicrystalline compounds transform at low temperature.