Defect structures in Frank-Kasper type square-triangle tiling of multimodal cage-type mesoporous silicas

Multimodal cage-type mesoporous silicas (MCMSs) with Frank-Kasper type square-triangle tiling show a unique defect structure, so-called three-fold symmetric hexagons, or shields, which are caused by phason fluctuations in dodecagonal quasicrystals. We observed and characterized three types of configurations inside shields in both quasiperiodic and periodic 3².4.3.4 tiling of MCMSs by transmission electron microscopy (TEM). The high-resolution TEM images of the shields were well explained by polyhedral models, which are the constituents of the Frank-Kasper type tetrahedrally close-packed structures of MCMSs. Shield defects invariably formed because of mismatch in periodic and/or aperiodic square-triangle tiling, and they were also catalyzed by other defects. Multiple shields overlapped with sharing of 30° rhombus units and showed characteristic motifs in the tiling, such as defect-mediated 12-fold wheel and stripe bundle arrangements. Hence, MCMSs with square-triangle tiling would be governed by a random-tiling-like structure stabilized by entropy rather than energy, which results in defect-free tiling.