Theory of cluster self-organization of crystal-forming systems: geometrical-topological modeling of nanocluster precursors with a hierarchical structure

The current state of the art related with the self-organization of crystal-forming systems, where long-range order spontaneously appears in the arrangement of structural units of any nature (micro- and macromolecules or atomic clusters), is considered. Three partially overlapping stages of self-organization of a system accepted in physical models of "order-disorder" kinetic transitions are matched to those used in supramolecular chemistry. An algorithmically constructed model of transition from disordered to hierarchically ordered systems is considered. The geometrical and topological modeling of density fluctuations of n-atomic species (clusters) A(n) in a crystal-forming medium is carried out. Clusters A(n) of a higher level of the system self-organization (the time moment t(i)) were determined as assemblies of specially selected clusters at the lower level (the time moment t((i-1))). Such clusters are built of equivalent clusters and, therefore, have a hierarchical structure; i.e., the simplest clusters are integrated into the clusters of the next higher level. Also, the mechanism of self-assembly of symmetrically and topologically different chains and microlayers (in the form of planar nets) from cyclic clusters A(n) is considered in the model system. The 45 obtained nets correspond to 11 uninodal Shubnikov nets and new binodal nets. Algorithms are presented for combinatorial and topological analysis to search for precursor clusters and restore a three-dimensional net of covalent and non-covalent bonds in a crystal structure by the matrix (cluster) self-assembly mechanism. The developed model is universal. The mechanism of self-assembly has been modeled for O-3 (ozone), C6H6 (benzene), C2H2 (acetylene), NaAlSi3O8 (albite), K2.25Na0.31Ca2.25Ba1.44 (Al11.5Si30.5O84H1.4) (H2O)(25) (paulingite, PAU), B(OH)(3), H3B3O6, H2SeO3, the Friauf-Laves structure family(which counts in 1,400 of binary and ternary compounds): MgCu2 (cF24), MgZn2 (hP12), and MgNi2 (hP24), the icosahedral structures: B-12, C20H20, C-60, ZrZn22 (cF184), and NaCd2 (cF1192) Samson Phase.