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    • In this contribution we treat the growth

    2018-10-24

    In this contribution, we treat the growth of fullerenes as a series of joining reactions of cupola half-fullerenes C10, C12, C16, C20, and C24[9] through the use of the geometrical modeling.
    Reaction between two base-truncated triangular pyramids The atomic configurations corresponding to reaction between two base-truncated triangular pyramids C10 are presented in Fig. 1. At first two molecules C10 are moving towards each other (Fig. 1a). Then the atoms, marked in black, interact with each other producing a compound (Fig. 1b). New covalent bonds (heavy-black solid lines) have formed in this process, whereas the old covalent bonds between the reacting atoms (light-grey dashed lines) have splitted. As a result, a distorted polyhedron has formed (Fig. 1c), then it relaxes into a perfect polyhedron (Fig. 1d). The surface of its atomic configuration consists of three squares, three hexagons and six pentagons so it has been termed a (tetra-hexa)3-penta6 polyhedron [10]. This structure together with its consistent electronic one was obtained in Ref. [10] on a basis of a new mathematic concept of fullerenes. According to this concept, a fullerene has any shape composed of atoms, each endothelin receptor antagonist having three nearest neighbors, which can be inscribed into a spherical, ellipsoidal, or similar surface. We have examined the case when the lower cupola is a mirror copy of the upper one. However, there is another case when the lower cupola is a rotary reflection of the upper one (Fig. 1e–h). Here the reacting atoms and the broken covalent bonds are the same (Fig. 1a, b, e, f), but due to changing the symmetry at first a distorted dodecahedron is formed (Fig. 1g). Then it relaxes into a perfect dodecahedron (Fig. 1h). To make clear the symmetry of the obtained fullerenes it is necessary to turn to their graphs (Fig. 2). It can be assumed that the most stable fullerenes will have the form close to a spherical one. It is apparent that the dodecahedron is more stable than the (tetra-hexa)3-penta6 polyhedron. However, the latter can become more spherical if it is modified by embedding three dimers into its three hexagons [2]. In doing so it transforms into a C26 fullerene.
    Reaction between two truncated triangular pyramids Similar to the previous reasoning, let us consider the atomic configurations corresponding to the reaction between two truncated triangular pyramids C12. As before, we have two joinings, mirror-symmetry and rotation-reflection-symmetry ones (Fig. 3). The first case (see Fig. 3d) results in the atomic configuration corresponding to a perfect polyhedron that consists of three equilateral triangles, three squares, and nine hexagons, so competition could be named a tri2-tetra3-hexa9 polyhedron. This structure was constructed in Ref. [9] on the basis of the graph theory. In the second case (see Fig. 3h) an isomer of fullerene C24 considered in Ref. [1] is obtained; it is a truncated dodecahedron. The symmetry of both polyhedrons is shown in Fig. 4. It is apparent that the truncated dodecahedron is more stable than the tri2-tetra3-hexa9 polyhedron. However, the latter can become more spherical if it is modified by embedding three dimers into its three hexagons. As a result, C30 fullerene is obtained.
    Reaction between two truncated tetra-angular pyramids The procedure for visualization of reaction is the same as before. In the case of mirror-symmetry joining (Fig. 5), the atomic configuration corresponding to a perfect polyhedron (see Fig. 5d) consists of six squares and twelve hexagons, so it could be termed a tetra6-hexa12 polyhedron. This structure was constructed in Ref. [9] on the basis of the graph theory. In the case of rotation-reflection-symmetry joining an isomer of fullerene C32 is obtained (Fig. 5h); it is composed of two squares, eight pentagons and eight hexagons, so it could be termed a tetra2-(penta-hexa)8 polyhedron. In both cases their structure and symmetry can be described by application of their graphs. The graphs of both polyhedrons are shown in Fig. 6; they enable us to gain some insight into the symmetry of these polyhedrons. The tetra6-hexa12 polyhedron can become more spherical if it is modified by embedding four dimers into its four hexagons lying along an equator or a meridian. This leads to the formation of C40 fullerene.