Stability of Diverse Dodecagonal Quasicrystals in T‑Shaped Liquid Crystalline Molecules

Quasicrystals are intriguing ordered structures characterized by quasiperiodic translational and noncrystallographic rotational symmetry. The tiling of different geometric units such as triangles and squares in two-dimensional space can result in a great variety of quasicrystals that could be realized by the self-assembly of liquid crystalline molecules. In this study, we introduce three self-similar dodecagonal tilings, including a novel diamond–square–triangle pattern, composed of triangular and quadrangular tiles, and examine their thermodynamic stability by using the self-consistent field theory applied to T-shaped liquid crystalline molecules. Specifically, we detail the inflation rules for the construction of these dodecagonal tilings, analyze their self-similarity, and show that these tilings can be viewed as projections of higher-dimensional periodic lattice points with projection windows. Using these dodecagonal tilings as initial configurations of the SCFT results in solutions corresponding to quasicrystals that could form from T-shaped liquid crystalline molecules. The relative stability of these aperiodic phases is analyzed to obtain design rules that could stabilize quasicrystals. Meanwhile, we provide a criterion for distinguishing three dodecagonal quasicrystals and their approximants by analyzing their diffraction peaks. These findings shed new light on the discovery of new quasicrystals in soft materials.