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

Abstract

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.