Modeling of the self-organizing processes in crystal-forming systems: Structural mechanism of assembly of the K2Ba[Ge[6]Ge3[4]O9]2, K2Ge[6]Ge3[4]O9, and BaGe[6]Ge3[4]O9 germanates from subpolyhedral clusters

The self-assembly of the crystal structures of the K2Ba[Ge[6]Ge3[4]O9]2, K2Ge[6]Ge3[4]O9, and BaGe[6]Ge3[4]O9 germanates formed in the K-Ba-Ge-O system is modeled on a subpolyhedral level. In three topologically different three-dimensional MT frameworks (the M octahedra GeO6 and the T tetrahedra GeO4), the equivalent two-dimensional net 6464 + 664 (M: 2T) is separated, and its two-color decomposition into primary MT contours (characterizing the structure of the cluster precursor) and secondary contours (reflecting the bonds between the clusters) is performed. The structural invariant of the MT frameworks is determined in the form of an M2T4 cyclic hexapolyhedral cluster. It is demonstrated that these stable cyclic precursors of compositions K2M2T4 and BaM2T4 control all processes of the subsequent evolution of the MT crystal-forming complex and provide the formation of substructural units of a higher level with the participation of additional tetrahedra GeO4.