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  • In this contribution we consider the geometric modeling

    2018-10-24

    In this contribution we consider the geometric modeling of middle-size fullerenes growth beginning with C32; namely C34, C36, C38, C40, C42, C44, C46, C48, C50, C52, C54, C56, C58, and C60. We have studied their growth at first obtaining their graphs, what is simpler, and then designing their structure on the basis of the graphs obtained. The aim of the study is to find perfect fullerenes.
    The initial perfect fullerene C32 consists of six squares and twelve hexagons (Fig. 1) so it butein was named a tetra6-hexa12 polyhedron. It has D4h symmetry. The fullerene was predicted together with its graph in Ref. [14], but the structure was not given. . Starting with this fullerene, it is possible to obtain its direct descendants with the help of the mechanism of dimer embedding into a hexagon. We have emphasized in our previous papers that drawing the axonometric projections of fullerenes is a rather tedious procedure, but it allows avoiding many mistakes in subsequent reasoning. Constructing the graphs of fullerenes is easier than drawing the axonometric projections. Taking as a basis the structure and graph of fullerene C32, we have obtained the fullerenes from C32 to C44 (Figs. 1 and 2). To gain a better understanding of the mechanism of dimer embedding, its main features are given in the form of schematic representation (Fig. 3). Let us analyze these figures. From the configurations shown it follows that the first embedding, which transforms fullerene C32 into fullerene C34, influences deeply only one of the hexagons and two of its square neighbors. This hexagon transforms into two adjacent pentagons and its square neighbors become pentagons; the fullerene C34 loses four-fold symmetry. It becomes an imperfect fullerene with C1 symmetry. At that, a cell which contains four pentagons appears in the fullerene. The second imbedding transforms fullerene C34 into fullerene C36. Similar to the previous case, one of two remaining hexagons transforms into two adjacent pentagons, its square neighbor into a pentagon, and its pentagon neighbor into a hexagon. As a result, the more symmetric fullerene C36 with two-fold symmetry is obtained. The fullerene is semi-perfect with two cells of three adjacent pentagons and belongs to C2 symmetry. The third embedding leads to the transition from fullerene C36 to fullerene C38. It eliminates one more hexagon and two its neighbors, a pentagon and a square, but in return creates an adjacent hexagon of another local orientation and a new pentagon. Again the fullerene becomes less symmetric. As fullerene C34, fullerene C38 belongs to C1 symmetry. At last the fourth embedding restores D4h symmetry. The perfect fullerene C40 obtained could be named a tetra2-penta8-hexa12 polyhedron where every two adjacent pentagons have the form of a bow tie. Now the fullerene C40 is up against the problem that it can grow only at an angle to its main axis of symmetry, for example, by the way shown in Figs. 2 and 3. It is connected with the fact that embedding can be realized only normal to a direction along which a hexagon has two neighboring mutually antithetic pentagons. During further growth fullerenes C42, C44, C46, and C48 (Figs. 2 and 3) are obtained. Similar to the first stage, the fullerenes with an odd number of embedded dimers are imperfect (C42 and C48, C1 symmetry), fullerene C44 having two embedded dimers is semi-perfect (C2 symmetry) and fullerene C48 having four embedded dimers is perfect (four-fold D4 symmetry). It should be emphasized that in this case the number of embedded dimers is equal to the degree of symmetry. The structure of all the fullerenes obtained at the first and the second stages has one common feature. For visualization of this feature it is convenient to use the system of coordinates where the axis z, or the main axis of symmetry, passes through the centers of two squares. It is easy to verify that each square surrounded with four hexagons forms a cluster. The clusters are separated by other atoms creating an equator. It is worth to note that all the equator atoms are former dimer atoms.