Modeling of self-organization processes in crystal-forming systems: Symmetry and topology code for the cluster self-assembly of crystal structures of intermetallic compounds

Abstract

We performed the combinatorial and topological modeling of 1D, 2D, and 3D packs of symmetrically connected metal clusters in the form of tetrahedra А4. Three types of 1D chains with tetrahedral connectivity of 4, 6, and 8 were used to model 2D layers L-1, L-2, and L-3 and 3D frameworks FR-1, FR-2, FR-3, and FR-4. The model structures of the identified suprapolyhedral precursor clusters were used in topological analysis of crystal structures of intermetallic compounds (program package TOPOS and data bases ICSD and CRYSTMET). A match was found between the topological models of tetrahedral 3D frameworks and all types of crystal structures formed in binary systems; in Au–Cu: FR-1 for Cu3Au-cP4 (Auricupride), Cu2Au2-tP2 (Tetraauricupride), CuAu3-cP4 (Bogdanovite), and Cu2–x Au2 + x -cF4; in Mg–Cd: FR-3 for Mg3Cd-hP8, Mg2Cd2-oP4, MgCd3-hP8, and Mg2–x Cd2 + x -hP2; in Li–Hg: FR-2 for Li3Hg-cF16 and Li2Hg2-cP2 and FR-3 for LiHg3-hP8; in ternary system Li–Ag–Al: FR-2 for LiAg2Al-cF16 and Li2AgAl-cF16; and in quaternary system: framework FR-2 for LiMgPdSn-cF16. Framework FR-4 was identified in ternary intermetallic compounds A(Li2Sn2)-tI20, where A = Cu, Ag, Au. The structures of precursor nanoclusters were identified for other most abundant types of crystal structures of intermetallic compounds. For this purpose, we used the algorithms for partitioning the structural graph into nonintersecting cluster substructures and constructed the basal 3D network of the crystal structure in the form of a graph whose nodes correspond to the positions of the centers of precursor clusters. The cluster self-assembly was modeled for the following intermetallic compounds: Mg2Cu4-cF24, MgSnCu4-cF24, (ZrCu)Cu4-cF24,Mg2Zn4-hP12, (CaCu)Cu4-hP6, Cr3Si-cP8, Lu3Co(Fe3C)-cP16, Ca2Ge2(Cr2B2)-oC8, Y2Ni2(Fe2B2)-oP8, AlB2-hP3, Ca2Ge-oP12, CaHg2-hP3, Co2Ge(Ni2In)-hP6, Cs2Hg4-oI12, Ba4Po4-cF8, Mn5Ge3-hP16, and NaZn13-cF112. The symmetry and topological code of self-assembly from precursor nanoclusters was reconstituted for all crystal structure types of intermetallic compounds as: primary chain → microlayer → microframework. An abundance frequency analysis of topological and symmetry routes for the generation and evolution of precursor clusters enabled us to elucidate the crystal-formation laws in intermetallic systems on the microscopic level.