The ‘random’ square–triangle tiling: simulation of growth

Some alloy systems, such as Ni–Cr, V–Ni–Si and Ta–Te, have quasicrystalline phases with 12-fold symmetry. These structures may be described in terms of dodecagonal tilings by equilateral triangles and squares. The formation of quasicrystals still poses a problem, since local information is insufficient for the construction of a perfect quasiperiodic structure. The growth of real quasicrystals may be due to several mechanisms. We have simulated the growth of a quasicrystal from a melt, consisting of squares and equilateral triangles of equal edge length. We are interested in the abundancies of the vertex configurations formed, both regular and defective. Unrestricted random growth tends to result in segregation of triangles from squares. Favoring triangles to attract squares and vice versa brings about nearly perfect patterns with nearly perfect vertex abundancies, as well as realistic defect concentrations. We have also calculated the exact vertex frequencies of the ideal square–triangle tiling by relying on inflation symmetry.