CC BY
12
DOI: 10.5862/JPM.248.7 UDC: 539.21
A.I. Melker, T.V. Vorobyeva
Peter the Great St. Petersburg Polytechnic University
FUSION REACTIONS OF CUPOLA HALF FULLERENES
Reactions of cupola half fullerenes C10, C12, C16, C20 and C24 with each other are considered on the basis of Arrhenius's postulate. It means that at first there forms an intermediate compound and only afterwards a usual chemical reaction is going on. We supposed that during the reactions new covalent bonds are formed and some old covalent bonds between the reacting atoms are destroyed. The final structure of a fullerene is obtained through the use of geometric modeling. As applied to fullerenes, geometric modeling supposes that a forming fullerene tends to take the appearance of a perfect spheroid with equal covalent bonds. The graphs describing the process are constructed.
CUPOLA HALF FULLERENE, FUSION REACTION, GRAPH, MODELING.
1. Introduction
Up to now the fullerene-formation mechanism is a controversial point. Research suggests that fullerene assemblage originates of individual atoms and C2 dimers, and, probably, of very small clusters. In Ref. [1, 2], we have exhaustively investigated a dimer mechanism of fullerene growing. According to it, a carbon dimer embeds either into a hexagon or a pentagon of an initial fullerene. This leads to stretching and breaking the covalent bonds which are parallel to arising tensile forces. In both cases there arises a new atomic configuration and there is a mass increase of two carbon atoms. However, the above-stated mechanisms of fullerene growth are not unique. Fullerenes can be imagined to grow by reacting with each other, similar to a bubble growth in the soap solution.
This possibility was demonstrated by the example of such reactions as
С + С
С24 + С4
С
С30 + С - c36>
and
С30 + С30 — С60'
through the use of a new molecular dynamics that takes into consideration both atomic and electronic degrees of freedom simultaneously, especially the excited electronic states created by electronic transitions [4 — 7]. Fullerenes and nanotubes are formed at high temperatures and
the new molecular dynamics, termed 'charged-bond' molecular dynamics, accounts for this factor properly. At first this molecular dynamics was developed as a rather sophisticated design, but later it obtained a strict theoretical basis [8].
Any molecular dynamics needs input data. For mini-fullerenes (up to C20) the number of possible configurations is not very large, but as one passes 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. In this respect, it makes sense to use geometric modeling as a first step of a computer simulation and further theoretical analysis [9]. We suppose that the geometric modeling will allow us to envision a possible way of growing carbon clusters from the very beginning and thereby to decrease the number of configurations being worthy of further study.
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 [5] through the use of the geometrical modeling.
2. Reaction between two base-truncated triangular pyramids
The atomic configurations corresponding to reaction
С10 + С10 -
(С10 CJ — С2
—>
Fig. 1. Joining of two half fullerenes C10 with the mirror symmetry (a — d) and the rotation-reflection symmetry (e — h): (a, e) Separate carbon cupolas C10; (b, f) Intermediate compound; (c, g) Distorted polyhedron formed;
(d) (Tetra-hexa)3-penta6 polyhedron C20; (h) Dodecahedron C20 after relaxation; Black and light-grey balls are reacting and neutral atoms, respectively; thin black solid and dashed lines are covalent bonds; light-grey dashed lines are old covalent bonds to be destroyed; heavy-black solid lines are new covalent bonds
between two base-truncated triangular pyramids C10 are presented in Fig. 1. At first two molecules C10 are moving towards each other (Fig. 1, a). Then the atoms, marked in black, interact with each other producing a compound (Fig. 1, b). 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. 1, c), then it relaxes into a perfect polyhedron (Fig. 1, d). 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
[1]. This structure together with its consistent electronic one was obtained in Ref. [1] on a basis of a new mathematic concept of fullerenes. According to this concept, a fullerene has any shape composed of atoms, each atom 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. 1, e — h). Here the reacting atoms and the broken covalent bonds are the same (Fig. 1, a, b, e, f), but due to changing the sym-
Fig. 2. Graphs of two isomers of fullerene C20: (tetra-hexa)3-penta6 polyhedron (a) and dodecahedron (b); heavy-black lines are new covalent bonds
J Ч:
Fig. 3. Joining of two half fullerenes C12. This caption is almost identical to that of Fig. 1, with the following differences: (a, e) Separate carbon cupolas C12; (d) Tri2-tetra3-hexa9 polyhedron C24; (h) Truncated dodecahedron C after relaxation
metry at first a distorted dodecahedron is formed (Fig. 1, g). Then it relaxes into a perfect dodecahedron (Fig. 1, h).
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 [7]. 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
C,2 + C,2 -
(C,2 C„) - C
'24
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. 3, d) results in the atomic configuration corresponding to a perfect polyhedron that consists of three equilateral triangles, three squares, and nine hexagons, so it could be named a tri2-tetra3-hexa9 polyhedron. This structure was constructed in Ref. [5] on the basis of the graph theory. In the second case (see Fig. 3, h) an isomer of fullerene C24 considered in Ref. [6] is obtained; it is a truncated dodecahedron.
The symmetry of both polyhedrons is shown
Fig. 4. Graphs of two isomers of fullerene C24: Tri2-tetra3-hexa9 polyhedron (a) and fullerene obtained by truncating two opposite vertices of a dodecahedron (b) [6]
Fig. 5. Mirror-symmetry joining of two half fullerenes C16. This caption is almost identical to those of Figs. 1, 3 with the following differences: (a, e) Separate carbon cupolas C16; (d) Tetra6-hexa12 polyhedron C32 and (h) Tetra2-(penta-hexa)
polyhedron C32 after relaxation
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
C16 + C16 ^ (C16C16) - C32
is the same as before. In the case of mirror-symmetryj oining ( Fig .5),the atomic configuration corresponding to a perfect polyhedron (see
Fig. 5, d) consists of six squares and twelve hexagons, so it could be termed a tetra6-hexa12 polyhedron. This structure was constructed in Ref. [5] on the basis of the graph theory. In the case of rotation-reflection-symmetry joining an isomer of fullerene C32 is obtained (Fig. 5, h); 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
Fig. 6. Graphs of two isomers of fullerene C32: Tetra6-hexa12 polyhedron (a) and Tetra2-(penta-hexa)8 polyhedron (b)
embedding four dimers into its four hexagons lying along an equator or a meridian. This leads to the formation
of C40 fullerene.
Reaction between two truncated penta-angular pyramids
C20 + C20 ^
(C20 C20^
The procedure for visualization of reaction
0) ^ C40
is the same as before. In the case of mirror-symmetry joining (Fig. 7), the atomic configuration corresponding to a perfect polyhedron (see Fig. 7, d) consists of five squares, two pentagons and ten hexagons, so it could be termed a tetra5-penta2-hexa15 polyhedron. This structure was constructed
in Ref. [5] on the basis of the graph theory. In the case of rotation-reflection-symmetry joining (Fig. 11) one obtains an isomer of fullerene C40 (Fig. 7, h) 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.
Fig. 7. Joining of two half fullerenes C20. This caption is almost identical to those of Figs. 1, 3, 5 with the following difference: (a, e) Separate carbon cupolas C20; (d) Tetra5-penta2-hexa15 polyhedron C40 and (h) Penta12-hexa10
polyhedron C40 after relaxation
Fig. 8. The graphs of two C40 fullerene isomers: tetra5-penta2-hexa15 polyhedron (a) and penta -hexa polyhedron (b)
Reaction between two truncated hexa-angular pyramids
The procedure for visualization of reaction
C24 + C24 ^
(C24 C24) ^ c
'48
is the same as before. In the case of mirror-symmetry joining (Fig. 9), the atomic configuration corresponding to a perfect polyhedron (Fig. 9, d) consists of six squares and twenty hexagons, so it could be termed a tetra6-hexa20 polyhedron. This structure was constructed in Ref. [5] on the basis of the graph theory. In the case of rotation-reflection-symmetry joining (Fig. 9) one obtains an isomer of fullerene C48 (see Fig. 9, h) 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 [21]. 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
Fig. 9. Joining of two half fullerenes C24. This caption is almost identical to those of Figs. 1, 3, 5, 7 with the following differences: (a, e) Separate carbon cupolas C24; (d) Tetra6-hexa20 polyhedron C48 and (h) Penta12-hexa14 polyhedron C48
after relaxation
Fig. 10. Graphs of two isomers of fullerenes C48: tetra6-hexa20 polyhedron (a) and penta^-hexa^ polyhedron (b)
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 half 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. [5] 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.
REFERENCES
[1] A.I. Melker, V. Lonch, Atomic and electronic structure of mini-fullerenes: from four to twenty, Materials Physics and Mechanics. 13(1) (2012) 22-26.
[2] A.I. Melker, Possible ways of forming mini-fullerenes and their graphs, Materials Physics and Mechanics. 20 (1) (2014) 1-11.
[3] F. Harary, Graph Theory, Addison-Wesley Publishing, Reading, 1969.
[4] A.I. Melker, S.A. Starovoitov, T.V. Vorobyeva, Classification of mini-fullerenes on graph basis, Materials Physics and Mechanics. 20(1) (2014) 12-17.
[5] A.I. Melker, M.A. Krupina, Designing mini-fullerenes and their relatives on graph basis, Materials Physics and Mechanics. 20 (1) (2014) 18-24.
[6] M.A. Krupina, A.I. Melker, S.A. Starovoitov, T.V. Vorobyeva, Structure and graphs of midi-fullerenes, Proceedings of NDTCS' 2015.16 (2015) 23-26.
[7] A.I. Melker, Growth of midi-fullerenes from twenty to sixty, Proceedings of NDTCS' 2015. 16 (2015) 34-37.
[8] C. Piskoti, J. Yarger, A. zettl, C36, a new
carbon solid, Nature. No. 393 (1998) 771-774.
[9] A.I. Melker, S.N. Romanov, D.A. Kornilov, Computer simulation of formation of carbon fullerenes, Materials Physics and Mechanics. 2(1) (2000) 42-50.
[10] D.A. Kornilov, A.I. Melker, S.N. Romanov, New molecular dynamics predicts fullerene formation, Proceedings of SPIE. 4348 (2001) 146-153.
[11] D.A. Kornilov, A.I. Melker, S.N. Romanov, Conformation transitions in fullerenes at non-zero temperatures, Proceedings of SPIE. 5127 (2003) 81-85.
[12] A.I. Melker, Fullerenes and nanotubes: molecular dynamics study, Proceedings of SPIE. 5400 (2004) 54-64.
[13] A.I. Melker, M.A. Vorobyeva, Electronic theory of molecule vibrations, Proceedings of SPIE. 6253 (2006) 6253-05.
[14] A.I. Melker, Dynamics of Condensed Matter, Vol. 2, Collisions and Branchings, St. Petersburg, St. Petersburg Academy of Sciences on Strength Problems, 2010, 342 p.
THE AUTHORS
MELKER Alexander I.
Peter the Great St. Petersburg Polytechnic University
29 Politekhnicheskaya St., St. Petersburg, 195261, Russian Federation
newton@imop.spbstu.ru
VoroBYEVA Tatiana V.
Peter the Great St. Petersburg Polytechnic University 29 Politekhnicheskaya St., St. Petersburg, 195261, Russian Federation newton@imop.spbstu. ru
Мелькер А.И., Воробьева Т.В. РЕАКЦИИ СИНТЕЗА КУПОЛООБРАЗНЫХ ПОЛУ-ФУЛЛЕРЕНОВ.
На основе постулата Аррениуса рассмотрены реакции синтеза куполообразных полуфуллере-нов С10, С12, С16, С20 и С24 друг с другом. Согласно постулату, вначале образуется промежуточное соединение, а затем проходит обычная химическая реакция. Предполагается, что во время реакции возникают новые ковалентные связи, а старые ковалентные связи между атомами, вступающими в реакцию, разрушаются. Конечная структура фуллерена получена с помощью геометрического моделирования. В случае фуллеренов геометрическое моделирование предполагает, что образующийся фуллерен стремится принять форму совершенного сфероида с равными ковалентными связями. Построены графы, отражающие этот процесс.
КУПОЛООБРАЗНЫЙ ПОЛУФУЛЛЕРЕН, РЕАКЦИЯ СИНТЕЗА, ГРАФ, МОДЕЛИРОВАНИЕ.
СПИСОК ЛИТЕРАТУРЫ
[1] Melker A.I., Lonch V. Atomic and electronic structure of mini-fullerenes: from four to twenty // Materials Physics and Mechanics. 2012. Vol. 13. No. 1. Pp. 22-26.
[2] Melker A.I. Possible ways of forming mini-fullerenes and their graphs // Materials Physics and Mechanics. 2014. Vol. 20. No. 1. Pp. 1-11.
[3] Harary Frank. Graph Theory. Addison-Wesley Publishing, Reading, 1969. 300 p.
[4] Melker A.I., Starovoitov S.A., Vorobyeva T.V. Classification of mini-fullerenes on graph basis // Materials Physics and Mechanics. 2014. Vol. 20. No. 1. Pp. 12-17.
[5] Melker A.I., Krupina M.A. Designing mini-fullerenes and their relatives on graph basis // Materials Physics and Mechanics. 2014. Vol. 20. No. 1. Pp. 18-24.
[6] Krupina M.A., Melker A.I., Starovoitov S.A., Vorobyeva T.V. Structure and graphs of midi-fullerenes // Proceedings of NDTCS' 2015. 2015. Vol. 16. Pp. 23-26.
[7] Melker A.I. Growth of midi-fullerenes from twenty to sixty // Proceedings of NDTCS' 2015. 2015. Vol. 16. Pp. 34-37.
[8] Piskoti C., Yarger J., zettl A. C36, a
new carbon solid // Nature. 1998. No. 393. Pp. 771-774.
[9] Melker A.I., Romanov S.N., Kornilov D.A. Computer simulation of formation of carbon fullerenes // Materials Physics and Mechanics. 2000. Vol. 2. No. 1. Pp. 42-50.
[10] Kornilov D.A., Melker A.I., Romanov S.N. New molecular dynamics predicts fullerene formation // Proceedings of SPIE. 2001. Vol. 4348. Pp. 146-153.
[11] Kornilov D.A., Melker A.I., Romanov S.N. Conformation transitions in fullerenes at nonzero temperatures // Proceedings of SPIE. 2003. Vol. 5127. Pp. 81-85.
[12] Melker A.I. Fullerenes and nanotubes: molecular dynamics study // Proceedings of SPIE. 2004. Vol. 5400. Pp. 54-64.
[13] Melker A.I., Vorobyeva M.A. Electronic theory of molecule vibrations// Proceedings of SPIE. 2006. Vol. 6253. P. 6253-05.
[14] мелькер А.и. Динамика конденсированных сред. Ч. II. Столкновения и ветвления. СПб.: Санкт-Петербургская академия наук по проблемам прочности, 2014. 342 с.
СВЕДЕНИЯ ОБ АВТОРАХ
мЕлькЕр Александр Иосифович — доктор физико-математических наук, профессор кафедры «Механика и процессы управления» Санкт-Петербургского политехнического университета Петра Великого.
195251, Российская Федерация, г. Санкт-Петербург, Политехническая ул., 29. пеМопШшор.зрЪзШ. ги
ВОРОБЬЕВА Татьяна Владимировна — кандидат физико-математических наук, доцент кафедры экспериментальной физики Санкт-Петербургского политехнического университета Петра Великого. 195251, Российская Федерация, г. Санкт-Петербург, Политехническая ул., 29 newton@imop.spbstu.ru
© Санкт-Петербургский политехнический университет Петра Великого, 2016
