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

TL;DR

This work shows that point defects in $12$-fold square–triangle quasicrystals substantially enhance configurational entropy, largely due to mixing among defect types (shield and two chiral eggs), and can thermodynamically stabilize the quasicrystal. A lattice tile model with defect penalties and line tension quantifies the entropy gain and defect statistics, while a square–shoulder particle model demonstrates that defects persist at observable concentrations and shift phase boundaries when vibrational costs are included. The results indicate that kinetic trapping is not required to explain defect prevalence; rather, equilibrium thermodynamics favors a high density of defects, which has direct consequences for the stability and dynamics of soft-matter quasicrystals. Together, these insights highlight the essential role of defect physics in accurately describing and predicting QC behavior in complex colloidal and soft-matter systems.

Abstract

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 isamplified 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.

Figures (15)

• Figure 1: Time series of defect dynamics from simulations of particles interacting via a square-shoulder potential: (a) a vacancy is introduced by removing a particle; (b) the vacancy splits into a pair of left- and right-handed egg defects; (c) the lower egg propagates and changes chirality; (d) the upper egg also propagates and flips chirality; (e) the upper egg further propagates and transforms into a shield defect. In panels (d) and (e), gray edges and particles highlight differences from the original configuration shown in (a).
• Figure 2: Schematic of the simulation geometry. (a) The five tile types allowed in the tile-based simulations: square and triangle tiles correspond to the regular tiles of the ideal quasicrystalline tiling, while the remaining three represent the only permitted defect tiles. (b) Representative snapshot from a tile-based simulation containing 289 vertices.
• Figure 3: Ensemble averaged 2D structure factor of the simulated square-triangle-defect tilings at $14068$ vertices, $\beta\gamma a=4$, and various defect costs as indicated.
• Figure 4: Boundary scaling and tile statistics in the lattice model. (a) Ensemble-averaged perimeter scaled by the square root of the tiling area as a function of defect cost. (b) Ensemble-averaged perimeter normalized by the number of vertices. (c) Area fractions of squares, triangles, shields, and eggs as a function of defect cost. Data are shown for multiple system sizes and line tensions, which are color-coded as indicated in the legend. Note that in (c) the data for all three defect types overlap.
• Figure 5: Complete list of square-triangle-rhombus decompositions of the defect tiles. In (a) we show 6 unique representations of the shield tile, while the two columns of (b) shows the 3 unique representations of the left- and right-handed egg respectively.