Combinatorial Topological Analysis Research Articles

Cluster Self-Organization of Intermetallic Systems: Clusters-Precursors K15, K6, K5, and K4 for the Self-Assembly of Crystal Structures Pu31Rh20-tI204, Pu20Os12-tI32, (Pu4Co)2(Pu4)-tI28, (Ti4Ni)2(Bi4)-tI28, and Bi4-tI8

Using the ToposPro software package, a combinatorial-topological analysis and modeling of the self-assembly of the following crystal structures with space group I4/mcm are realized: Pu31Rh20-tI204: a = 11.076 Å, c = 36.933 Å, V = 4530.86 Å3, Pu20Os12-tI32: a = 10.882 Å, c = 5.665 Å, V = 670.8 Å3. (Pu4Co)2 (Pu4)-tI28: a = 10.475 Å, c = 5.340 Å, V = 585.9Å3. (Ti4Ni)2(Bi4)-tI28: a = 10.554 Å, c = 4.814 Å, V = 536.2Å3, Bi4-tI8: a = 8.518 Å, c = 4.164 Å, V = 302.15 Å3. For the crystal structure of Pu31Rh20-tI204, 113 variants of the cluster representation of the 3D atomic network with the following number of structural units are established: 4 (14 variants), 5 (61 variants), and 6 (38 variants). A variant of the self-assembly of the crystal structure with the participation of three types of framework-forming polyhedra is considered: K15 = Pu@14(Rh2Pu5)2 with symmetry –42m, double pyramids K10 = (Rh@Pu4)2 with symmetry 4, and octahedra K6 = 0@8(Rh2Pu6) with symmetry mmm and spacers Rh. For the crystal structure of Pu20Os12-tI32, framework-forming pyramid-shaped polyhedra K5 = 0@OsPu4 with symmetry 4, as well as spacers Pu and Os, are defined. For the crystal structure (Ti4Ni)2(Bi4), frame-forming pyramids K5 = 0@Ti4Ni and tetrahedra K4 = 0@Bi4) are defined. For the crystal structure (Pu4Co)2(Pu4)-tI28, frame-forming pyramids K5 = 0@ Pu4Co and tetrahedra K4 = 0@Pu4 are defined. For the crystal structure of Bi4-tI8, frame-forming tetrahedra K4 = 0@Bi4 are defined. The symmetric and topological code of self-assembly processes of 3D structures is reconstructed from clusters-precursors in the following form: primary chain → layer → framework.

Cluster Self-Organization of Intermetallic Systems: Clusters-Precursors K4, K6, and K7 for the Self-Assembly of Crystal Structures Y20Cu20Mg64-oC104, Y20Cu20Mg52-oC92, and Y3(NiAl3)Ge2-hP9

Using computer methods (ToposPro software package), a combinatorial topological analysis and modeling of the self-assembly of the following crystal structures are carried out: Y20Cu20Mg64-oC104 (a = 4.136 Å, b = 19.239 Å, c = 29.086 Å, V = 2314.45 Å3, Cmcm), Y20Cu20Mg52-oC92 (a = 4.097 Å, b = 19.279 Å, c = 25.790 Å, V = 2037.30 Å3, Cmcm), and Y3(NiAl3)Ge2-hP9 (a = b = 6.948 Å, c = 4.156 Å, V = 173.78 5 Å3, P-62m). For the Y20Cu20Mg64-oC104 crystal structure, 52 variants of the cluster representation of the 3D atomic network with the 3, 4, and 5 structural units are established. Four crystallographically independent structural units in the form of a tetrahedron are determined: tetrahedron K4 = 0@CuMg3, tetrahedron K4 = 0@YMg3, tetrahedron K4 = 0@YCuMg2, and a supratetrahedron K6 = 0@YCu2Mg3. A variant of self-assembly with the participation of hexamers from six linked structural units is considered (K4B+ K4C)(K4A+ K6)(K4B+ K4C). For the Y20Cu20Mg64-oC92 crystal structure, 27 variants of cluster representation of the 3D atomic mesh with 3, 4, and 5 structural units are established. Three crystallographically independent structural units in the form of a tetrahedron are determined: tetrahedron K4 = 0@YCuMg2, cluster K6 = 0@6(Y2Mg4) in the form of double tetrahedrons YMg3, and a nine-atom supratetrahedron K9 = Mg@Y2Cu2Mg4 consisting of two YMg2Cu and two YMg3 tetrahedrons. A variant of the self-assembly involving trimers of three structural units K4+ K6+ K9 is considered. For the Y3(NiAl3)Ge2-hP9 crystal structure, eight variants of decomposition of the 3D atomic mesh into cluster structures with the participation of two structural units are established. A variant of the self-assembly with the participation of packing generatrices of seven-atom clusters-precursors K7 = 0@Y3(NiAl3) with the participation of Ge atoms-spacers is considered. The symmetry and topological code of the self-assembly processes of 3D-structures is reconstructed from clusters-precursor in the following form: primary chain → layer → framework.

Cluster Self-Organization of Intermetallic Systems: K3, K4, K5, K6, and K13 Clusters-Precursors for the Self-Assembly of U8Ni10Al36-mC54, U20Ni26-mC46, and U8Co8-cI16 Crystal Structures

Using computer methods (the ToposPro software package), a combinatorial topological analysis and modeling of the self-assembly of U8Ni10Al36-mC54 (a = 15.5470 Å, b = 4.0610 Å, c = 16.4580 Å, β = 120.00°, V = 899.89 Å3, C m), U20Ni26-mC46 (a = 7.660 Å, b = 13.080 Å, c = 7.649 Å, β = 108.88°, V = 725.26 Å3, C2/m), and U8Co8-cI16 (a = 6.343 Å, V = 255.20 Å3, I 213) are carried out. For the U8Ni10Al36-mC54 crystal structure, 960 variants of the cluster representation of the 3D atomic grid with the number of structural units 5, 6, and 7 are established. Six crystallographically independent structural units in the form of a pyramid K5 = 0@Al(U2Al2), pyramid K6A = 0@U(NiAl4), and pyramid K6B = 0@U(NiAl4), as well as rings K3A = 0@NiAl2, K3B = 0@NiAl2, and K3C = 0@Al3, are determined. For the U20Ni26-mC46 crystal structure, the structural units K5 = Ni(Ni2U2) and icosahedra K13= Ni@Ni6U6 are defined. For the crystal structure U2Co2-cI16, the structural units—tetrahedra K4 = U2Co2—are defined. The symmetry and topological code of the processes of self-assembly of 3D structures from clusters-precursors are reconstructed in the following form: primary chain → layer → framework.

Cluster Self-Organization of Intermetallic Systems: Cluster Precursors K3, K4, K5, K7, and K8 for the Self-Assembly of Lu66Te24-mC90, Te4Lu28-oC32, Lu3(TeLu3)Lu2-hP9, and Lu4Te4-cF8 Crystal Structures

With the help of computer methods (ToposPro software package), a combinatorial topological analysis and modeling of the self-assembly of Lu4Te4-oF8 (Fm-3m, V = 211.0 Å3), Te4Lu28-oC32 (Cmcm, V = 908.3 Å3), Lu3(TeLu3)Lu2-hP9 (P-62m, V = 908.3 Å3), and Lu66Te24-mC90 (C12/m1, V = 2467.2 Å3) crystal structures are carried out. For the crystal structure of Lu4Te4-oF8, cluster precursors K8 = 0@Te4Lu4 with symmetry –43m; for Te4Lu28-oC32, tetrahedral cluster precursors K4 = 0@Lu4 and K4 = 0@TeLu3 with symmetry 2 and m; and for Lu3(TeLu3)Lu2, cluster precursors K7 = 0@Lu3(TeLu3) with symmetry 3m and spacers Lu are established. For the crystal structure of Lu66Te24-mC90, pyramid-shaped cluster precursors K5 = 0@Lu5 with symmetry 2, tetrahedra K4 = 0@Lu4 with symmetry 2, tetrahedra K4 = 0@TeLu3, and tetrahedra K4 = 0@Te2Lu2 are established, and rings K3 = 0@TeLu2 are involved in the formation of supraclusters-trimers. The symmetry and topological code of the processes of self-assembly of 3D structures from cluster precursors is reconstructed in the following form: primary chain → layer → framework.

Cluster Self-Organization of Intermetallic Systems: Cluster Precursors K4, K5, and K9 for the Self-Assembly of Zr72P36-oS108, Zr18Ni22-tI40, and Zr4Ni4-oS8 Crystal Structures

Using computer methods (ToposPro software package), a combinatorial topological analysis and modeling of the self-assembly of Zr72P36-oS108 (a = 29.509 Å, b = 19.063 Å, c = 3.607 Å, V = 2029.49 Å3, Cmmm), Zr18Ni22-tI40 (a = b = 9.880 Å, c = 6.610 Å, V = 645.23 Å3, I4/m, and Zr4Ni4-oS8 (a = 3.271 Å, b = 9.931 Å, c = 4.107 Å, V = 133.43 Å3, Cmcm) crystal structures are carried out. For the crystal structure of Zr72P36-oS108, 40 variants of the cluster representation of the 3D atomic net with the number of structural units 5, 6, and 7 are established. Structural units in the form of a pyramid K5 = 0@PZr4, tetrahedron K4 = 0@Zr4, and supratetrahedron K9 = Zr(Zr4P4) of four connected tetrahedra. For the crystal structure of Zr18Ni22-tI40 also defined supratetrahedra K9 = Ni(Zr4Ni4) are defined. For the crystal structure of Zr4Ni4-oS8, the tetrahedral cluster precursor K4 = Zr2Ni2 is defined. The symmetry and topological code of the processes of self-assembly of 3D structures from cluster precursors is reconstructed in the following form: primary chain → layer → framework.

Cluster Self-Organization of Intermetallic Systems: K66 and K130 Clusters for the Self-Assembly of the K78In160-hP238 Crystal Structure and K17 Clusters for the Self-Assembly of the K8In11-hR114 Crystal Structure

The combinatorial topological analysis and modeling of self-assembly of the K78In160-hP238 (space group P-3m1, a = b = 17.211, c = 28.888 A, V = 7410 A3) and K8In11-hR114 (space group R-3c, a = b = 10.021 A, c = 50.891 A, V = 4426 A3) crystal structures are performed by computer methods (the ToposPro software package). Icosahedra In12 (with the -3m symmetry) are the templates on which three-layer clusters K130 = 0@12(In12)@30(In12K18)@86(K20In66) with the diameter of 17 A are formed. The two-layer cluster K66 formed on Friauf polyhedra K(K4In12) (with 3m symmetry) has a chemical composition of shells K@16(K4In12)@49(K16In33) and a diameter of 14 A. The K130 and K66 nanoclusters are framework-forming and involved in the formation of 2D layers A and B, respectively, thus forming a three-layer B–A–B package. The In12 clusters (with the -3m symmetry), In8 hexagonal bipyramids (with the 3m symmetry), and K spacer atoms are located in the voids of the layer of the K66 clusters. The three-layer package thickness corresponds to the value of the translation vector modulus c = 28.888 A. The framework structure is formed by linking the three-layer B–A–B packages in the [001] direction. The cluster precursor K17 = 0@In11K6 is installed for K8In11-hR114 in the form of a triangular In5 bipyramid on six faces of which In atoms are located with six bound potassium atoms. The microlayer is formed when the primary chains are bound in the (001) plane with a shift. The localization of the K spacer atoms takes place in the layer. The microframework of the structure is formed when the microlayers are bound with a shift.

Cluster-Precursors and Self-Assembly Li36Ca4Sn24-oS64 and LiMgEu2Sn3-oS28 Crystalline Structures

Using computer methods (the ToposPro software package), the combinatorial-topological analysis and modeling of self-assembly of the crystal structure of Li36Ca4Sn24-oS64 (space group (sg) Cmcm, a = 4.640 A, b = 27.112 A, c = 11.491 A, V = 1445.5 A3) and LiMgEu2Sn3-oS28 (sg Cmcm, a = 4.782 A, b = 20.717 A, c = 7.743 A, V = 767.1 A3). For the Li36Ca4Sn24 intermetallic compound a new type of 11-atom cluster K11 is installed, formed from double pentagonal rings: K11 = 0@11 (Li5)Ca(Sn5). The maximum cluster symmetry K11 and primary strand of translationally related K11 clusters corresponds to noncrystallographic symmetry 5m. Primary chains of linked K11 clusters preserving only symmetry m are located in the direction [100] and the distance between the centers of the clusters determines the length of vector a = 4.640 A. The layer is formed when the primary chains located antiparallel are bound. The tetrahedral clusters are located between the primary chains K4 = 0@4 (Li3Sn), forming chains in the direction [100]. The symmetry and topological code of the self-assembly processes of the 3D structure of the LiMgEu intermetallic compound LiMgEu2Sn3-oS28 is reconstructed from clusters K4 = 0@4 (LiMgEuSn) and K3 = 0@3 (Sn2Eu). The layer is formed when the parallel chains of K4 + K3 are bound.

Cluster Self-Organization of Intermetallic Systems: Role of K5 = 0@5, K9 = 1@8 and K11 = 0@11 Clusters in the Self-Assembly of Crystal Structures

The combinatorial-topological analysis and simulation of self-assembly of the Na96Hg36-hR132 (space group R-3c, a = b = 9.228 A, c = 52.6380 A, V = 3881.91 A3) and Na12Hg8-tP20 (space group P42/mnm, a = b = 8.520, c = 7.800 A, V = 566.2 A3) crystal structures are conducted by computer-based methods (ToposPro program package). Polyhedral cluster precursors K11 = 0@11(Na8Hg3) and K9 = Hg@Na8 are first determined for the intermetallic compound Na96Hg36-hR132, while polyhedral cluster precursors K5 = 0@Na3Hg2 are determined for the intermetallic compound Na12Hg8-tP20. The symmetry and topology code of the self-assembly processes of the 3D structure Na96Hg36-hR132 from the nanocluster precursors K11 and K9, and of the 3D structure Na12Hg8-tP20 from the K5 clusters are reconstructed as the primary chain $$S_{{\text{3}}}^{1}$$ → layer $$S_{{\text{3}}}^{2}$$ → frame $$S_{{\text{3}}}^{3}.$$ The structural analysis of all known intermetallic compounds is conducted, and numerous examples of the assembly of their structures from the K5, K9, and K11 clusters are found.

Cluster Self-Organization of Intermetallic Systems: Three-Layer Icosahedral Nanoclusters K132 = 0@12(In6Tl6)@30(In6Na6K18)@90(In72Na12K6) and K116 = 0@12(In6Tl6)@26(In12K14)@78(In36Tl20K12) for the Self-Assembly of the K52Na12Tl36In122-hP224 Crystalline Structure

The TOPOS software package is used for the combinatorial topological analysis and simulation of the self-assembly of the K52Na12Tl36In122-hP224 (spatial group P-3m1, a = b = 16.909, c = 28.483 A, V = 7 052 A3) crystalline structure. A total of 1649 variants of the cluster representation of a 3D atomic lattice with the number of structural units ranging from 4 to 10 are found. Two frame-forming three-layer icosahedral nanoclusters are detected: K132 and K116 with symmetry g = –3m and a diameter of 17 A. The chemical composition of the shells of the three-layer 132-atom nanocluster K132 is 0@12(In6Tl6)@30(In6Na6K18)@90(In72Na12K6) and that of the 116-atom nanocluster K116 is (0@12(In6Tl6)@26(In12K14)@78(In36Tl20K12). The symmetrical and topological code of the self-assembly of the 3D Na12K52Tl36In122 structure from the precursor K132 and K116 nanoclusters is reconstructed in the following form: primary chain → layer → frame. The framework’s voids are occupied by spacer atoms K and Tl.

Modeling Self-Organization Processes in Crystal-Forming Systems: New Two-Layer Cluster–Precursor K44 = 0@8(Na2In6)@36(In6Cd6K6)2 for the Self-Assembly of the K23Na8Cd12In48–hP91 Crystal Structure

Using computer methods (the ToposPro software package), the combinatorial-topological analysis and modeling of the self-assembly of the K23Na8Cd12In48–hP91(a = b = 17.114 A, c = 10.442 A, group P6/mmm) crystal structure are carried out. The chemically different precursor clusters 0@8(Na2In6) and 0@K2In6 in the form of hexagonal bipyramids are established. The centers of Na2In6 clusters occupy positions 1a with a symmetry of 6/mmm. The centers of the K2In6–A and K2In6–B clusters occupy positions 2c with symmetry –6m2, and 3g positions with mmm symmetry. The Na2In6 clusters are the templates on the surface of which atomic shells of 36 atoms are formed. The composition of the two-layer cluster is K44 = 0@8(Na2In6)@36(In6Cd6K6)2. Layer formation occurs upon the K44 clusters binding to the K2Cd6–A clusters. The symmetry and topological code of the self-assembly of the 3D structures from the K44 suprapolyhedral precursors with the participation of the K2Cd6 polyhedral clusters, as well as Na and K spacer atoms, are reconstructed.

Cluster Self-Organization of Intermetallic Systems. New Cluster Presursor (InNa5)(AuAu5) and Primary Chain with the 5m Symmetry for the Self-Assembly of the Na32Au44In24–oP100 Crystal Structure

The combinatorial-topological analysis and simulation of the self-assembly of the Na32Au44In24–oP100 (space group P bcm, a = 5.483 A, b = 24.519 A, c = 14.573 A, V = 1895 A3) crystal structure are conducted by the computer-based methods (TOPOS program package). A new type of the 12-atom K12 cluster formed from doubled pentagonal pyramids—AuAu5 and InNa5—is established. The maximal symmetry of the K12 cluster and the primary chain of translationally bound K12 clusters correspond to the noncrystallographic 5m symmetry. The symmetry and topological codes of the Na32Au44In24–oP100 3D structure’s self-assembly processes from the K12 nanocluster precursors are reconstructed in the following form: primary chain → microlayer → micro-framework. The primary chains of bound K12 clusters with the m symmetry are located in the direction [100], and the distance between cluster centers determines the vector’s value: a = 5.483 A. There are four nonparallel primary chains in the primary chain’s local environment. The chains from InAu atoms and NaAu2In2 clusters are located in the 2D layer between the primary chains. The distance between the equivalent chains in the direction [001] determined the vector’s value: c = 14.573 A. In the 3D framework in the direction [010], the distance between the equivalent 2D layers determined the vector’s value: b = 24.519 A.

Cluster Self-Organization of Intermetallic Systems: 0@12(Ga12)@24(Na12Ga12)@72(Rb4Na8Ga60) 108-Atom Three-Layer Icosahedral Cluster and 0@12(Ga12)@32(Na20Ga12) 44-Atom Two-Layer Icosahedral Cluster for Rb24Na200Ga696-oF920 Crystal Structure Self-Assembly

The combinatorial-topological analysis and simulation of the self-assembly of the Rb24Na200Ga696-oF920 (space group Fmmm, V = 17 837 A3) crystal structure are conducted by the computer-based methods (TOPOS program package). The number of options for the cluster representation of the 3D atomic framework with the number of structural units ranging from 4 to 12 came to 9565 variants. Two framework-forming icosahedral clusters—ico-K108 and ico-K44—are determined. The ico-K108 three-layer 108-atom nanocluster has the 0@12(Ga12)@24(Na12Ga12)@72(Rb4Na8Ga60) chemical composition of shells, a diameter of 17 A, and symmetry g = mmm. The ico-K44 two-layer 44-atom nanocluster has the 0@12(Ga12)@32(Na20Ga12) chemical composition of shells, a diameter of 11 A, and symmetry g = 2/m. The symmetry and topological codes of the Rb24Na200Ga696–oF920 3D structure’s self-assembly processes from the iсo-K108 and ico-K44 nanocluster precursors are reconstructed in the form primary chain → microlayer → microframework. The Rb spacer atoms and related groups of Ga atoms in the form of chains are located in the large voids of the 3D framework.

Symmetrical and Topological Self-Assembly Code of the Crystalline Structure of a New Aluminosilicate Zeolite ISC-1 from Templated t-plg Suprapolyhedral Precursors

In 2008, V.Ya. Shevchenko and S.V. Krivovichev built the zeolites series related to paulingite based on the inorganic gene concept and predicted a new zeolite named ISC-1 (Institute of Silicate Chemistry-1) [1]. The structure and composition of ISC-1 are described in detail in [2]. The found chemical formula of the new zeolite ISC-1 is Na14K24Al38Si202O48 · nH2O. Further research on the principles of assembly of zeolites and prediction of another previously unknown zeolite, ISC-2 (Institute of Silicate Chemistry-2), and the conditions of its formation are presented in [3–5]. The combinatorial-topological analysis of the crystal structure of the new aluminosilicate zeolite ISC-1 with cubic cell parameters а = 25.039 A, V = 15 699 A3, and spatial group Im $$\bar {3}$$ m is performed by computer-based methods (ToposPro software package) [6]. The topological type of the framework composed of bonded Т–(Si,Al)O4 tetrahedra is characterized by a combination of polyhedral tilings: t-grc (48 T-atoms), t-pau (32 T-atoms), t-plg (30 T-atoms),t-opr (16 T- atoms), and t-oto (16 T- atoms). A framework-forming precursor for zeolites of 30 T-tetrahedra, which corresponds to the t-plg tile and contains an organic template Me2-DABCO (N,N′-dimethyl-1,4-diazabicyclo[2.2.2]octane), is established by the complete decomposition of the 3D atomic lattice into cluster structures. t-plg nanoclusters with the symmetry g = $$\bar {3}$$ m are characterized by 4-, 6-, and 8-rings and the n-hedral symbol [46. 62. 86]. Na-spacers statistically occupy neighboring positions in the 8-ring and between the 4‑rings of the neighboring t-plg clusters. The basic 3D lattice type indicative of t-plg clusters center-of-gravity positions correspond to a simple cubic lattice with CN = 6. The self-assembly code of the 3D structure from complementary bonded nanoclusters-precursors is simulated in its entirety: primary chain → microlayer → framework. The doubled distance between t-plg clusters centers corresponds to the cubic cell translation vector a = 25.039 A.

Modeling Self-Organization Processes in Crystal Forming Systems. Symmetry and Topology Codes of Cluster Self-Assembly of Crystal Structure of Na44Tl7 (Na6Tl)

The geometrical and topological analysis of the crystal structure of intermetallide Na44Tl7 (Na6Tl, a = 24.154 A, V = 14091.8 A3, space group F-43m) is carried out. The algorithms of the combinatorial-topological analysis, which ensure the recovery of the symmetrical and topological code (program) of the cluster self-assembly of the crystal structure of an intermetallide, are developed. The topological type of the basic 3D network for two types of cluster precursors corresponds to a simple cubic 3D network P c with CN = 6 and basic 2D network of type 44. There are eight cluster precursors in the unit cell: four K86 and four K50. The cluster precursor K86 made from 86 atoms is formed from eight icosahedra i-TlNa12 linked by the apices. The center of the cluster precursor K86 is at the position 4a (0, 0, 0) with the point symmetry g = –43m. The 50-atomic cluster precursor K50 consists of six i-TlNa12 icosahedra. The center of the cluster precursor K50 is in the partial position 4b (0, 0, 0) with the point symmetry g = –43m. The symmetrical and topological codes of the self-assembly of 3D structures from nanocluster precursors K86 and K50 are reconstructed in the following form: primary chain → microlayer → microframework.

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

This is a review over the state of art in self-organization in crystal-forming systems, where a longrange order appears spontaneously in the arrangement of nanoscale building units of any nature (atomic clusters and molecules), which initially existed in a dynamic state as a chaotic mixture. The focus is on bimolecular compounds where building units are M-octahedra M(O, OH, H2O) and T-tetrahedra T(O, OH)4 bonded by OH–O hydrogen bonds and/or bridging oxygen atoms O. The combinatorial–topological modeling of 1D and 2D packings of suprapolyhedral clusters (MT)n has been carried out. These packings are symmetrically and topologically conceivable types of molecular MT clusters S03, from which chains S13 and 2D microlayers S23 are formed. The theoretically derived model MT-clusters (MT)n are compared with known types of precursor clusters in the molecular and framework MT-structures of polyvalent metal sulfates and their analogues. Combinatorial–topological analysis algorithms are provided to restore, from structure data, the convergent matrix crystal structure self-assembly code in the form of the sequence of significant elementary events. Crystal structure modeling involves combinatorial–topological analysis methods, whose underlying idea is the recogniition of packing-forming precursor nanoclusters and the construction of the relevant basal 2D and 3D networks in the form of graphs where nodes correspond to the positions of their centroids. The model is universal and has first been used to model the cluster self-assembly of molecular MT structures for V(2+)(H2O)4(SO4) (V-Rozenite), V(4+)O(H2O)5(SO4)(H2O) (SYN1), V(4+)O(H2O)5(SO4) (Orthominasragrite, Minasragrite, Anorthominasragrite), V(4+)O(H2O)3(SO4) (Bobjonesite); framework MT structures V(4+)O(SO4)(H2(SO4) (SYN2), V(4+)O(SO4) (Pauflerite and SYN3), V(4+)2O3(SO4)2 (SYN4), V(4+)2O3(SeO4)2 (SYN5), Fe(3+)2(SO4)3 (LIPHOS), and Fe(3+)2(SO4)3 (NASICON); and for templated framework MT structures (NH4)2[Ga2(PO4)2F2] (KTP), (NH4)2[Ga2(PO4)2(HF)F2] (p-KTP), TlMo2(PO4)3 (SYN06), and K2Mg2(SO4)3 (Langbeinite). Frequency analysis of topological and symmetry pathways in the formation and evolution of clusters (from primary chain S13 through microlayer S23 to microframework S33) elucidates new crystal-formation trends in diverse chemical systems at the microscopic level.

Modeling of self-organization processes in crystal-forming systems: Symmetry and topology code for cluster self-assembly of crystal structures in MT-structures of polyvalent sulfates and phosphates

We consider the supramolecular chemistry of polyvalent metal sulfates and phosphates built of M(n+)(O, OH, H2O)6 polyhedral clusters with octahedral O-coordination (M-polyhedra) and SO4 and PO4 polyhedral clusters with tetrahedral O-coordination (T-polyhedra). These compounds form molecular (insular), chain, layered, or framework 3D MT structures by means of directed (bridging) hydrogen bonds O–H…O and/or bridging covalent O-bonds. Combinatorial–topological analysis algorithms are provided to restore, from structure data, the convergent matrix crystal structure self-assembly code. Cluster self-assembly modeling has been performed for the following sulfates: MgSO4 (Pnma and Cmcm), MgSO4 ∙ 1H2O (Kieserite), and CuSO4 ∙ 3H2O (Bonattite) having 3D MT strucutres; MgSO4 ∙ 2H2O (Sanderite), MgSO4 · 2.5H2O, MnSO4 · 3H2O, and MnSO4(OH) · 2H2O having 2D MT structures; MgSO4 · 4H2O (Cranswickite) and MgSO4 · 5H2O (Pentahydrite) having 1D MT structures; and MgSO4 · 4H2O (Starkeyite) having a 0D MT structure, for the following phosphates: (VO)(HPO4) (P-1), MgSO4 · H2O (Kieserite), and CuSO4 · 3H2O (Bonattite) having 3D MT structures; (VO)(HPO4)(H2O)0.5 (P121/c1 and Pmmn), (VO)(HPO4)(H2O)2 (P4/nmm and P121/c1), V(3+)(PO4)(H2O)2 (P121/n1), V(3+)(PO4)(H2O)4 (C12/c1), and Mg(HPO4)(H2O)3 (Newberyite, Pbca) having 2D MT structures; phosphite (VO(H2O)2)(HPO3)(H2O)3 (P41) and phosphate (VO)(HPO4)(H2O)4 (P-1) having 1D MT structures. For all compounds, the symmetry and combinatorial codes of cluster self-assembly code in the form of the following sequence: precursor nanocluster S30 → primary chain S31→ microlayer S32→ microframework S33. At the stage of primary chain S31 formation, in all of MT crystal structures considered, precursor clusters are four-polyhedral MTMT chains, transor cis, and their derivatives with additional O-bonds between polyhedra and ring cluster M2T2 (the result of coiling a cis-chain into a ring). Frequency analysis of topological and symmetry pathways in the formation and evolution of MT-clusters (from primary chain S31 through microlayer S32 to microframework S33) elucidates new crystal-formation trends in diverse chemical systems at the microscopic level.

Modeling of self-organization processes in crystal-forming systems: Templated precursor nanoclusters T48 and the self-assembly of crystal structures of 15-crown-5, Na-FAU, 18-crown-6, Na-EMT, and Ca,Ba-TSC zeolites

The combinatorial and topological modeling of packings of symmetry-related nanoclusters T48 (diameter: ∼16 A, symmetry $$\bar 43m$$ ) is performed. The packings are 1D claims S 3 1 and 2D microlayers S 3 2 , which give rise to 3D structures S 3 3 . For three of the five framework structures, correspondence was established with zeolite FAU (Faujasite, Fd-3m, Z = 4 T48), EMT (P63/mmc, Z = 2 T48), and TSC (Tschortnerite $$Fm\bar 3m$$ , Z = 8 T48). Combinatorial topological analysis methods (the ToposPro program package) were used to model the steps of zeolite crystal structure self-assembly. The underlying idea of these methods consists in that the basal 3D network of the zeolite structure is designed as a graph, where the nodes correspond to the centroid positions of clusters T48. For FAU, the basal 3D network corresponds to the copper cubic strucutre (CN = 12); for EMT, it corresponds to the hexagonal (H) strucutre of magnesium (CN = 12); and for TSC, to the cubic structure of polonium (CN = 6). The symmetry and topology code of zeolite strucutre formation has been restored in the form of the sequence of significant elementary events that characterize the shortest (rapidest) program of convergent cluster self-assembly. The functional role played by 15-crown-5 and 18-crown-6 template molecules has been established to consist in the stabilization of the primary chains and microlayers of FAU and EMT frameworks and the relevant network of diamond and graphite topological types. The model for 3D structure self-assembly from clusters T48 is capable of explaining the morphogenesis of single crystals to form regular octahedra in FAU, hexagonal prisms in EMT, and regular cubes in TSC.

Symmetry and topology code (program) of crystal structure cluster self-assembly for molecular and framework compounds

This is a review over the state of art in self-organization in crystal-forming systems where a long-range order appears spontaneously in the arrangement of nanoscale structural units of any nature (atomic clusters, molecules, metal clusters), which initially existed in a dynamic state as a chaotic mixture. Combinatorial topological analysis algorithms are provided to restore, from structure data, the convergent matrix crystal structure self-assembly code in the form of the sequence of significant elementary events. The model is universal and has first been used to model the cluster self-assembly of crystal structures for all elemental compounds that are formed at the solution-solid interface and from a gas phase, namely: Cu-cF4 (FCC), Fe-cI2 (BCC), Mg-hP2 (HCP), Hg-hR1, B12-hR12, Ga-oC8,C6-hP4 (graphite, GRA),C6-cF8 (Diamond, DIA), Sn-tI4, N2-cP8, P4-mC16, As6-hR2, O3-oP24 (Ozone), S3-hP9, Se3-hP6, Po-cP1, F2-mC8, and Cl2-oC8. Cluster modeling and crystal-chemical analysis are carried out for a large crystal-chemical family NaCl-cF8. The dimorphs of TeO2-oP24 (tellurite, TEL) and TeO2-tP12 (paratellurite, PAR) and the icosahedral structures of WAl12-cI26 and sillenite Bi12SiO20-cI66 are considered. Frequency analysis of topological and symmetry pathways in the formation and evolution of clusters (from primary chain S31 through microlayer S32 to microframework S33) elucidates new crystal-formation trends in diverse chemical systems at the microscopic level.

Cluster self-organization theory for crystal-forming systems: The geometrical and topological model of formation, selection, and evolution of precursor nanoclusters of molecular and framework compounds

This review concerns the contemporary state of the problem of self-organization in crystal-forming systems, where a long-range order appears spontaneously in the arrangement of nanoscale structural units of any nature (atomic clusters and molecules), which initially existed as a chaotic mixture. Examples are provided where combinatorial topological analysis algorithms are used to restore, from structure data, the convergent matrix self-assembly code of crystal structures in the form of the sequence of significant elementary events e i . For a cyclic six-node cluster S 3 0 , the geometrical and topological modeling of various self-organization levels of hierarchic structures was carried out for six types of primary chains S 3 1 , fifteen types of networks S 3 2 , and thirty types of frameworks S 3 3 . The model is universal and has been used to model the self-assembly of the following crystal structures: monomolecular compound S6 and bimolecular compound S6 + S10, ozone O3, benzene C6H6, cubane C8H8, Zn4O4 (NaCl structure type), carbon oxides C6-GRA (graphite), C6-DIA (diamond), and C6-LON (lonsdaleite), boron nitrides B3N3-GRB, B3N3-DIA, and B3N3-LON, Ni3As3-NIC, B6(OH)16, zeolite K3(Al2Si4O12(OH)-LIT, Na2ZrSi2O7-PKL (parakeldyshite), La3Ga5GeO14 (LGG), and La3GaGe5O16 (LAN). It is for the first time that structural invariants are recognized in topologically different crystal structures of chemical systems. Bifurcations in the evolution pathways of precursor clusters (structural branching points) have been determined for the formation of three-dimensional periodic structures. Frequency analysis carried out for the topological and symmetry pathways in the formation and evolution of clusters (primary chain S 3 1 -microlayer S 3 2 -microframework S 3 3 ) elucidated new crystal-formation trends in diverse chemical systems at the microscopic level.

Geometrical and topological model of formation, selection, and evolution of precursor nanoclusters in the convergent matrix self-assembly mechanism of crystal formation

Combinatorial-topological analysis algorithms have been developed to reconstruct the code of convergent matrix self-assembly in the form of a sequence of significant elementary events ei proceeding from structure data in a crystal-forming system that contains diverse structural units (atoms, molecules, and polyheral clusters). Mechanisms of selection of suprastructural units with maximal values of their connectivity indices to form chains from precursor clusters are considered. Two special cluster formation mechanisms are recognized. One is a linear mechanism where the number of particles is constantly increasing. The other is a discrete mechanism with an obligatory transiton of the evolving system to the matrix (complementary) selfassembly mode. Structure self-assembly follows the principle of the maximal filling of the crystal space and, accordingly, obeys the requirement for the maximal degree of complementary connectivity of precursor nanoclusters at all evolution steps. The model developed in this way has been used to interpret the formation and structural-topology specifics of carbon allotropes (fullerenes C60 and C70) and polymorphs of metals, H2O (ice), silica, hydrido- and chlorosiloxanes, MEP clathrate family, some aluminosilicates (feldspar and zeolites), and C204H756Cl36Mn84O462 (V = 20992 A).