Aperiodic approximants bridging quasicrystals and modulated structures

Aperiodic crystals constitute a class of materials that includes quasicrystals and modulated structures. Although these two categories share a common foundation in the concept of superspace, the relationship between them has remained enigmatic.

In this study, we find aperiodic tilings composed of large and small hexagons, and parallelograms. These tilings have metallic-mean self-similar structures. One of the important properties is that the tilings serve as aperiodic approximants for the honeycomb lattice: as the metallic mean increases, the size of honeycomb domains bounded by the parallelograms also increases, and the whole tiling converges to the honeycomb lattice. This suggests that the tilings have not only quasicrystalline but also modulated structures. In fact, the tilings have the self-similarity, which is a characteristic of quasicrystals. Additionally, the satellite peaks in the structure factors appear, which are characteristics of modulated structures. The superspace analysis of the tilings is applied to Engel's colloidal simulation of modulated structures in Ref. [1]: its molecular structure and structure factors are described by the tiling theory. Furthermore, we discover the unique tiles in a terpolymer/homopolymer blend system. These results clarify that quasicrystals and modulated structures are bridged in terms of superspace.



[1] M. Engel, Phys. Rev. Lett. 106, 095504 (2011).