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

Аннотация

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 Cu_2–xAu_{2 + x}-cF4; in Mg–Cd: FR-3 for Mg₃Cd-hP8, Mg₂Cd₂-oP4, MgCd₃-hP8, and Mg_2–xCd_{2 + 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.