Modeling of self-organization in crystal-forming MT systems: The structural mechanism of self-assembly from suprapolyhedral clusters for framework titanophosphates NaTiPO5 (SYN), NaTiPO5 (TIT), and KTiPO5 (KTP)

Ring precursor clusters are derived for the MT-system containing two different polyhedra T and M, where T stands for a TO₄ tetrahedron and M is in the general case an MO_n polyhedron with the number of vertices n ≥ 5, i.e., a pyramid, octahedron, or others. The topological representation of the precursor cluster is given in the form of two-colored graphs. The model developed for describing the formation and evolution of clusters that form periodic structures is used to search for the precursor clusters of framework phosphate structures: NaTiPO₅ (SYN), NaTiPO₅ (TIT), and KTiPO₅ (KTP). Computer methods are used to perform the full 3D reconstruction of the self-assembly of phosphate crystal structures: precursor cluster—primary chain—micro-layer—microframework (supraprecursor). New eight-polyhedral ring precursor clusters are identified. The crystal-forming polynodal 2D nets with a hierarchic superstructure are recognized: M8833+T883 for SYN (NaTiPO₅), M84634+T843+M836+T864 for TIT (NaTiPO₅), M8939+2M8339+M839+3T893+T89 for KTP (KTiPO₅), and M8484+M88+2T848 for PNAT (Na₂TiSiO₅). The coordination numbers of the clusters in all 2D nets are four.