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Structural Characteristics of Gas Hydrates Within the Framework of Generalized Crystallography A

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Structural Characteristics of Gas Hydrates Within the Framework of Generalized Crystallography A Crystallography Reports, Vol. 48, No. 3, 2003, pp. 347–350. Translated from Kristallografiya, Vol. 48, No. 3, 2003, pp. 391–394. Original Russian Text Copyright © 2003 by Talis. THEORY OF CRYSTAL STRUCTURES Structural Characteristics of Gas Hydrates within the Framework of Generalized Crystallography A. L. Talis Russian Institute of Synthesis of Mineral Raw Materials, Aleksandrov, Vladimir oblast, 601650 Russia e-mail: [email protected] Received September 16, 2002 Abstract—It is shown that the symmetry of the {5, 3, 3} polytope (four-dimensional dodecahedron) embedded into E4 enables one to derive all the polyhedra–cavities that make up gas hydrates from the {5, 3} dodecahedron. Consideration is given to the allomorphic embedding of the {5, 3} subgraph into the incidence graph of the Desargues configuration 103, which determines the mechanism of incorporation of guest molecules into the polyhedra–cavities of gas hydrates and their escape from these polyhedra at the symmetry level. The relation- ships obtained may be considered as a basis for an a priori derivation of the determined (periodic and aperiodic) structures of gas hydrates. © 2003 MAIK “Nauka/Interperiodica”. The impossibility of dividing the three-dimensional pentagonal “cap” of the icosahedron into a hexagonal Euclidean space E3 into regular tetrahedra determines 2π one (if a wedge with an angle ------ is inserted), with the the structural characteristics of tetrahedrally coordi- 5 nated aqueous frameworks of gas hydrates that cannot disclination interactions being determined by the prod- be adequately reflected within the framework of classi- uct of the corresponding elements of group Y'. The cal crystallography. For example, the polyhedra–cavi- 2π introduction of two, three, and four Ð------ disclinations ties D, T, P, and H (dodecahedron, tetra-, penta-, and 5 hexadecahedra), the constituent fragments of the struc- into an icosahedron along its fivefold axes results in the tures of gas hydrates, are determined as dual to the Z12, formation of the Z14, Z15, and Z16 polyhedra [2]. Z14, Z15, and Z16 polyhedra (icosahedron and Frank– Kasper polyhedra) [1, 2]; the D' and E polyhedra–cavi- The introduction of a disclination quadruplet (con- 2π 2π ties (Fig. 1) have not been determined as yet [3]. The sisting of two Ð------ and two ------ disclinations) into the symmetry basis of the structural mechanism of the 5 5 incorporation of guest molecules into polyhedra–cavi- vertices of a “rhombus” built by two neighboring trian- ties and their escape from these cavities is also gular faces of the icosahedron results in the flipping of unknown. Below, we show that the specific structural the diagonal in this rhombus (Fig. 1). To the quadruplet features of gas hydrates can be adequately described q or the disclination 2π there corresponds the element only within the framework of the generalized crystal- y(n, 2π) = Ð1 of group Y' and the rotation by 2π along lography [4–7]. the Möbius band [6]. Thus, introducing three 2π discli- nations into the icosahedron (which preserve the sub- The division into regular tetrahedra that are impos- group C3 of its group Yh) and demanding the maximum sible in E3 can be made in a four-dimensional icosahe- point symmetry, we arrive at the twelve-vertex polyhe- dron–polytope {3, 3, 5} whose 120 vertices belong to 3q 3 6 3 dron Z12 /D3h = [4 , 5 , 6 ] with the symmetry group the three-dimensional sphere S3 embedded into E4. The D3h and twenty triangular faces. In this case, four edges {3, 3, 5} polytope and the dual {5, 3, 3} polytope (four- meet at three vertices Z123q/D , five edges at six verti- dimensional dodecahedron) are mapped onto them- 3h ces, and six edges at three vertices; a polyhedron dual selves by the group Y' × Y' of order 14 400. If group Y to this polyhedron is the polyhedron–cavity D' (Fig. 1). of icosahedron rotation consists of 60 g n ϕ elements, ( , ) Along each of the three straight lines connecting pairs where n is the unit vector directed along the icosahe- ϕ of the vertices of the upper and lower hexagonal caps in dron axis around which the rotation by an angle takes π ϕ ϕ 2d 12 2 −2 place, then group Y' consists of 120 y(n, ) and y(n, + Z14 = Z12 /D6d = [5 , 6 ], one may introduce two ------ . 2π) = Ðy(n, ϕ) elements uniquely corresponding to ±ϕ 5 π disclinations rearrange the diagonals of three tet- disclinations introduced into the icosahedron along the 2 ragons consisting of halves of the triangular faces. As a 2π vector n. For example, the element y n, Ð------ corre- result, a 20-vertex polyhedron Z146d, 3q/D = 5 6h (Z122d)6d, 3q/D = [512, 68] is formed, to which the 2π 6h sponds to the disclination Ð------ , which transforms the polyhedron–cavity E is dual (Fig. 1). 5 1063-7745/03/4803-0347 $24.00 © 2003 MAIK “Nauka/Interperiodica” 348 TALIS 1 q 4 3 2 q q q 2* 3* 4* q 1* 3q 3 6 3 6d, 3q 12 8 Z12 /D3h = [4 , 5 , 6 ] Z14 /D6h = [5 , 6 ] D' E 3q 6d, 3q Fig. 1. Graphs of the Z12 /D3h and Z14 /D6h polyhedra and the dual polyhedra–cavities D' and E. The vertices where six edges meet are indicated by filled circles; the vertices where four edges meet, by open squares; the 2π disclination is denoted by letter q in the center of a tetragon, where this disclination can reject and introduce edges shown by dashed and double lines, respectively. 6d, 3q Some of the “back lines” in Z14 /D6h are not shown, the filled vertices in its “equatorial belt” arise due to the introduction of 2π − ------ disclinations along the straight lines 22*, 33*, and 44* in Z14. 5 3q 3 6 In fact, the polyhedron symbols Z12 /D3h = [4 , 5 , into a tetrahedrally coordinated 20-vertex cluster with 3 2d 6d, 3q 12 8 equal edges that leaves no space for a guest molecule. 6 ] and (Z12 ) /D6h = [5 , 6 ] set the disclinations and the corresponding elements of group Y', which A dodecahedron is a {5, 3} map on a sphere (at each determine the derivation of these polyhedra from the vertex, three pentagons meet); its Petrie polygon is a icosahedron. The symmetry of the dual polyhedron is decacycle—the sequence of edges in which each two the same as the symmetry of the initial one, and, there- (and not three) edges belong to one face [7]. Six such fore, all the polyhedra–cavities D', T, P, H, and E can decacycles contain all 30 edges and 20 vertices of the be derived from the dodecahedron D (with the symme- dodecahedron and are isomorphous to the {10, 3}5 map 2π try group Y ) using the Ð------ and 2π disclinations, whose on the nonoriented surface (at each vertex, three deca- h 5 cycles meet, the vertices separated by five edges of Pet- interaction is determined by the products of the corre- rie’s polygon are considered to be equivalent) [8]. The sponding elements of Y'. {10, 3}5 map also contains pentacycles, which are its minimum cycles (Fig. 2a). It is possible to consider as an energetically favor- able transformation such a transformation of the cluster The {10, 3} map is allomorphic to the {10, 3}' that would keep the maximum subgraph containing all 5 5 map (Fig. 2b) and differs from the latter only in its min- its vertices, with all the other conditions being the imum cycles—hexacycles [8]. The transition from same. Since the polyhedra–cavities are derived from π the dodecahedron, the symmetry basis of the structural {10, 3}5 to {10, 3}5' and back is provided by 2 discli- mechanism of incorporation of guest molecules into a nations. Thus, if, for example, in the decacycle polyhedron–cavity (and escape from it) can be deter- 13'65'21'36'74' (the primed figures numerate black ver- mined from the transformation of the dodecahedron tices) divided into four pentacycles the edge 8'10 is CRYSTALLOGRAPHY REPORTS Vol. 48 No. 3 2003 STRUCTURAL CHARACTERISTICS OF GAS HYDRATES 349 (a) 1 1 2 4 3 4 3 12345678910 1 10 6 10 2 8 7 5 9 7 6 q 3 4 5 6 5 6 9 7 8 10 8 5 6 7 3 4 2 3 2 8 9 1 1 10 (b) {5, 3} Table of incidence {5, 3} 10 9 1 3 4 3 9 8 12345678910 2 1 6 10 8 1 2 4 10 6 3 7 5 4 2 8 5 2 7 5 5 6 6 7 4 7 2 5 7 9 1 8 8 9 4 10 3 6 10 1 3 9 6 6 {103} {10} + {10/3} Table of incidence {103} Fig. 2. Dodecahedron and the allomorphic twenty-vertex diamond cluster, their incidence graphs and corresponding incidence tables: (a) in the bichromatic {5, 3} graph of the dodecahedron, six arrows connect the vertices of the same color; the flipping of the edges by a 2π disclination q results in the appearance of a hexacycle in the decacycle; the incidence table of the {5, 3} graph coincides with the incidence table of the Desargues configuration 103 if six arrows are rejected; (b) a twenty-vertex diamond cluster whose bichromatic graph is embedded into the bichromatic graphs {10} + {10/3} set by the incidence table of the Desargues con- figuration 103. The nondiamond bonds are shown by dashed and dash-dotted lines. The filled circles in the incidence tables corre- spond to the cluster edges; the empty circles correspond to the bonds that are not edges.
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