Theory of cluster self-organization for crystal-forming systems: A model of transition from disodered to ordered systems

Abstract

The current state of ideas concerning the self-organization of crystal-forming systems where long-range order spontaneously appears in the arrangement of nanolevel structural units of any nature (micro- and macromolecules or atomic clusters) that initially existed in a dynamic state as a chaotic mixture 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) An in a crystal-forming medium is carried out. A specific set of An clusters with block-diagonal connectivity matrices is recognized. These types of clusters (S 3 0 ), having “sectional” or “hierarchic” partition, are defined as precursors of crystal structures that are capable of evolving most rapidly to give rise to a long-range order in structures. For an S 3 0 ring cluster shaped as a triangle, geometrical and topological modeling is carried out for all of the eight topologically and symmetrically possible types of S 3 1 primary chains built of S 3 0 using theory of one-dimensional symmetry groups. Thirty three structural variants of morphologically and topologically different types of S 3 2 micronets described by two-dimensional groups of symmetry are considered. Algorithms are presented for combinatorial and topological analysis to search for precursor clusters and restore a three-dimensional net of covalent and noncovalent bonds in a crystal structure by the matrix (cluster) self-assembly mechanism. The model advanced is universal. Examples of self-assembly of a series of cluster-assembled structures of AB2 alloys of the unique Friauf-Laves family (which counts in 1400 of binary and ternary compounds) are given: for MgCu2 (cF24) (with its superstructures of ZrCu 5 and MgSnCu4 types), MgZn2 (hP12), and MgNi2 (hP24) (from AB2 or A2B + B3 three-atom clusters); for ZrZn22 icosahedral structures (from a suprapolyhedral cluster built of a ZrZn16 Friauf polyhedron and two ZnZn12 icosahedra); NaCd2 (from one A cluster with 61 atoms and two B clusters with 63 atoms); and for a bimolecular compound C78H30 (which is formed of fullerene C60 and a C28H30 molecule). The scenario of formation of self-curving nets with icosahedral symmetry is considered: to form a B12 icosahedron from two isomers with n = 3, a C20 dodecahedron from two isomers with n = 5, and C60 fullerene from pentagonal clusters with n = 5.