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    • br Reaction between two truncated penta angular

    2018-11-05


    Reaction between two truncated penta-angular pyramids The procedure for visualization of reaction is the same as before. In the case of mirror-symmetry joining (Fig. 7), the atomic configuration corresponding to a perfect polyhedron (see Fig. 7d) consists of five squares, two pentagons and ten hexagons, so it buy RGFP966 could be termed a tetra5-penta2-hexa15 polyhedron. This structure was constructed in Ref. [9] on the basis of the graph theory. In the case of rotation-reflection-symmetry joining one obtains an isomer of fullerene C40 (Fig. 7h) composed of twelve pentagons and ten hexagons, so it could be termed a penta12-hexa10 polyhedron. In both cases their structure and symmetry can be described with the help of their graphs. The graphs of both polyhedrons are shown in Fig. 8; they enable us to gain some insight into the symmetry of these polyhedrons. The tetra5-penta2-hexa10 polyhedron can become more spherical if it is modified by embedding five dimers into its five hexagons lying along an equator, and so transforming into an isomer of fullerene C50.
    Reaction between two truncated hexa-angular pyramids The procedure for visualization of reaction is the same as before. In the case of mirror-symmetry joining (Fig. 9), the atomic configuration corresponding to a perfect polyhedron (Fig. 9d) consists of six squares and twenty hexagons, so it could be termed a tetra6-hexa20 polyhedron. This structure was constructed in Ref. [9] on the basis of graph theory. In the case of rotation-reflection-symmetry joining (Fig. 9) one obtains an isomer of fullerene C48 (see Fig. 9h) composed of twelve pentagons and ten hexagons, so it could be termed a penta12-hexa14 polyhedron. In both cases their structure and symmetry can be described using their graphs. The graphs of both polyhedrons are shown in Fig. 10; they enable us to gain some insight into the symmetry of these polyhedrons. The tetra5-penta2-hexa10 polyhedron can become more spherical by embedding six dimers into its six hexagons lying along an equator. This leads to transforming an isomer of fullerene C48 into an isomer of fullerene C60.
    Summary The growth of fullerenes through a series of joining reactions of cupola half fullerenes C10, C12, C16, C20, and C24 has been considered. We supposed that during the reactions new covalent bonds are formed and some old covalent bonds between the reacting atoms are splitted. The final structure of fullerenes was obtained through the use of geometric modeling. The fullerene symmetry was shown by means of graphs constructed. As to fullerenes, the geometric modeling was based on the principle “the minimum surface at the maximum volume”. In other words, a forming fullerene tends to take the form of a perfect spheroid with equal covalent bonds. The geometric modeling has shown its efficiency as a first step of a computer simulation, usually of molecular dynamics, and further theoretical analysis. The reason is that any molecular dynamics needs input data. For mini-fullerenes (up to C20) the number of possible configurations is not very large, but by passing to midi-fullerenes (C20–C60), one obtains a monstrous size of isomers. It is clear that there is no big sense in studying all of them, so it is desirable to restrict their number to the most stable configuration. In this respect, the geometric modeling allows one to imagine a possible way of growing carbon clusters from the very beginning and thereby to decrease the number of configurations worth for studying. Using geometrical modeling we obtained two families of fullerenes, each being composed of C20, C24, C32, C40, and C48 fullerenes. Both families have a layer structure. By analogy with geography, one can distinguish an equator zone, two temperate zones and two polar circles. The first family, designed in Ref. [9] on the graph basis, was termed the family of 4–6 equator fullerenes. The second family was constructed for the first time. Its progenitor C20 is a pentagonal dodecahedron, the next fullerene C24 can be realized as a twice truncated dodecahedron along one of three-fold symmetry axis. With the exception of the dodecahedron, the other fullerenes of this family can be considered similar to the previous case. Their equator zone consists of adjacent pentagons creating a zigzag; the temperate zones are formed by hexagons; each polar circle consists of an equilateral triangle, a square, a pentagon or a hexagon, these figures defining symmetry of the related fullerene. The family progenitor C20 is an exception; it has six five-fold symmetry axes, ten three-fold symmetry axes and fifteen two-fold symmetry ones. For this reason its graph is given in the form reflecting its highest symmetry.