  Quasicrystals and their periodic approximants are complex phases, which have
by now been observed in many metallic alloys, soft matter systems, and particle
simulations. In recent experiments of thin-film perovskites on solid
substrates, the type of complex phase was found to change depending on
thermodynamic conditions and the type of substrate used. Here, we investigate
the effect of a substrate on the relative stability of a two-dimensional model
quasicrystal and its approximants. Our numerical methods are molecular dynamics
simulations and free energy calculations that take into account phason flips
explicitly. For weak substrates, we observe an incommensurate-commensurate
transition, in which a continuous series of QC approximants locks into a
discrete number of approximants. For stronger substrates, an enhancement of the
stability of the dodecagonal quasicrystal and a variants of square lattices
were found. All phenomena can be explained by the interplay of the model system
with the substrate. Our results demonstrate that designing novel complex
periodic and quasiperiodic structures by choice of suitable substrates is a
promising strategy.

Soft Condensed Matter;Computational Physics