Vacancy defects in square–triangle tilings and their implications for quasicrystals formed by square-shoulder particles

Almost all observed square-triangle quasicrystals in soft-matter systems contain a large number of point-like defects, yet the role these defects play in stabilizing the quasicrystal phase remains poorly understood. In this work, we investigate the thermodynamic role of such defects in the widely observed 12-fold symmetric square-triangle quasicrystal. We develop a new Monte Carlo simulation to compute the configurational entropy of square-triangle tilings augmented to contain two types of irregular hexagons as defect tiles. We find that the introduction of defects leads to a notable entropy gain, with each defect contributing considerably more than a conventional vacancy in a periodic crystal. Intriguingly, the entropy gain is not simply due to individual defect types but is amplified by their combinatorial mixing. We then apply our findings to a microscopic model of core-corona particles interacting via a square-shoulder potential. By combining the configurational entropy with vibrational free-energy calculations, we predict the equilibrium defect concentration and confirm that the quasicrystalline phase contains a higher concentration of point defects than a typical periodic crystal. These results provide a new understanding of the prominence of observed defects in soft-matter quasicrystals.