For instance, there are non-isomorphic fullerenes C Note that only one form of C 60 , buckminsterfullerene, has no pair of adjacent pentagons the smallest such fullerene. To further illustrate the growth, there are ,, non-isomorphic fullerenes C , 15,, of which have no adjacent pentagons. Optimized structures of many fullerene isomers are published and listed on the web. Heterofullerenes have heteroatoms substituting carbons in cage or tube-shaped structures. They were discovered in  and greatly expand the overall fullerene class of compounds.
Notable examples include boron, nitrogen azafullerene , oxygen, and phosphorus derivatives. Trimetasphere carbon nanomaterials were discovered by researchers at Virginia Tech and licensed exclusively to Luna Innovations. This class of novel molecules comprises 80 carbon atoms C 80 forming a sphere which encloses a complex of three metal atoms and one nitrogen atom. These fullerenes encapsulate metals which puts them in the subset referred to as metallofullerenes.
Trimetaspheres have the potential for use in diagnostics as safe imaging agents , therapeutics  and in organic solar cells. Carbon nanotubes are cylindrical fullerenes. These tubes of carbon are usually only a few nanometres wide, but they can range from less than a micrometer to several millimeters in length.
They often have closed ends, but can be open-ended as well. There are also cases in which the tube reduces in diameter before closing off. Their unique molecular structure results in extraordinary macroscopic properties, including high tensile strength , high electrical conductivity , high ductility , high heat conductivity , and relative chemical inactivity as it is cylindrical and "planar" — that is, it has no "exposed" atoms that can be easily displaced. One proposed use of carbon nanotubes is in paper batteries , developed in by researchers at Rensselaer Polytechnic Institute.
Buckyballs and carbon nanotubes have been used as building blocks for a great variety of derivatives and larger structures, such as . After the discovery of C60, many fullerenes have been synthesized or studied theoretically by molecular modeling methods in which some or all the carbon atoms are replaced by other elements. Inorganic nanotubes , in particular, have attracted much attention.
A type of buckyball which uses boron atoms, instead of the usual carbon, was predicted and described in The B 80 structure, with each atom forming 5 or 6 bonds, is predicted to be more stable than the C 60 buckyball. However, this work has been subject to much criticism by quantum chemists   as it was concluded that the predicted I h symmetric structure was vibrationally unstable and the resulting cage undergoes a spontaneous symmetry break, yielding a puckered cage with rare T h symmetry symmetry of a volleyball.
There is an additional atom in the center of each six-member ring, bonded to each atom surrounding it.
By employing a systematic global search algorithm, later it was found that the previously proposed B80 fullerene is not global minimum for 80 atom boron clusters and hence can not be found in nature. Inorganic carbon-free fullerene-type structures have been built with the di sulfides of molybdenum MoS 2 , long used as a graphite-like lubricant, tungsten WS 2 , titanium TiS 2 and niobium NbS 2. Below is a table of main closed carbon fullerenes synthesized and characterized so far, with their CAS number when known.
In the table, "Num. Both are specified for the most experimentally abundant form s. When C 76 or C 82 crystals are grown from toluene solution they have a monoclinic symmetry. The crystal structure contains toluene molecules packed between the spheres of the fullerene. However, evaporation of the solvent from C 76 transforms it into a face-centered cubic form. Schlegel diagrams are often used to clarify the 3D structure of closed-shell fullerenes, as 2D projections are often not ideal in this sense. In mathematical terms, the combinatorial topology that is, the carbon atoms and the bonds between them, ignoring their positions and distances of a closed-shell fullerene with a simple sphere-like mean surface orientable , genus zero can be represented as a convex polyhedron ; more precisely, its one-dimensional skeleton, consisting of its vertices and edges.
The Schlegel diagram is a projection of that skeleton onto one of the faces of the polyhedron, through a point just outside that face; so that all other vertices project inside that face. The Schlegel diagram of a closed fullerene is a graph that is planar and 3-regular or "cubic"; meaning that all vertices have degree 3. A closed fullerene with sphere-like shell must have at least some cycles that are pentagons or heptagons. Similar constraints exist if the fullerene has heptagonal seven-atom cycles.
Open fullerenes, like carbon nanotubes and graphene, can consist entirely of hexagonal rings.
In theory, a long nanotube with ends joined to form a closed torus -like sheet could also consist entirely of hexagons. Since each carbon atom is connected to only three neighbors, instead of the usual four, it is customary to describe those bonds as being a mixture of single and double covalent bonds. So-called endohedral fullerenes have ions or small molecules incorporated inside the cage atoms. In the early s, the chemical and physical properties of fullerenes were a hot topic in the field of research and development. Popular Science discussed possible uses of fullerenes graphene in armor.
In the field of nanotechnology , heat resistance and superconductivity are some of the more heavily studied properties. There are many calculations that have been done using ab-initio quantum methods applied to fullerenes. Results of such calculations can be compared with experimental results. Fullerene is an unusual reactant in many organic reactions such as the Bingel reaction discovered in Researchers have been able to increase the reactivity of fullerenes by attaching active groups to their surfaces. Buckminsterfullerene does not exhibit " superaromaticity ": that is, the electrons in the hexagonal rings do not delocalize over the whole molecule.
A spherical fullerene of n carbon atoms has n pi-bonding electrons, free to delocalize. These should try to delocalize over the whole molecule. This has been shown to be the case using quantum chemical modelling, which showed the existence of strong diamagnetic sphere currents in the cation. As a result, C 60 in water tends to pick up two more electrons and become an anion. The n C 60 described below may be the result of C 60 trying to form a loose metallic bond.
Under high pressure and temperature, buckyballs collapse to form various one-, two-, or three-dimensional carbon frameworks. Such treatment converts fullerite into a nanocrystalline form of diamond which has been reported to exhibit remarkable mechanical properties. Fullerenes are stable, but not totally unreactive. The sp 2 -hybridized carbon atoms, which are at their energy minimum in planar graphite , must be bent to form the closed sphere or tube, which produces angle strain.
The characteristic reaction of fullerenes is electrophilic addition at 6,6-double bonds, which reduces angle strain by changing sp 2 -hybridized carbons into sp 3 -hybridized ones. This decrease in bond angles allows for the bonds to bend less when closing the sphere or tube, and thus, the molecule becomes more stable. Other atoms can be trapped inside fullerenes to form inclusion compounds known as endohedral fullerenes. An unusual example is the egg-shaped fullerene Tb 3 N C 84 , which violates the isolated pentagon rule.
Fullerenes are soluble in many organic solvents , such as toluene , chlorobenzene , and 1,2,3-trichloropropane. Still, fullerenes are the only known allotrope of carbon that can be dissolved in common solvents at room temperature. Solutions of pure buckminsterfullerene have a deep purple color. Solutions of C 70 are a reddish brown. The higher fullerenes C 76 to C 84 have a variety of colors. Millimeter-sized crystals of C 60 and C 70 , both pure and solvated, can be grown from benzene solution.
The Mathematics and Topology of Fullerenes / Edition 1
In , researchers from the University of Vienna demonstrated that wave-particle duality applied to molecules such as fullerene. Fullerenes are normally electrical insulators, but when crystallized with alkali metals, the resultant compound can be conducting or even superconducting. Some fullerenes e. C 76 , C 78 , C 80 , and C 84 are inherently chiral because they are D 2 -symmetric, and have been successfully resolved. Research efforts are ongoing to develop specific sensors for their enantiomers.
Two theories have been proposed to describe the molecular mechanisms that make fullerenes. In researchers discovered that asymmetrical fullerenes formed from larger structures settle into stable fullerenes. The synthesized substance was a particular metallofullerene consisting of 84 carbon atoms with two additional carbon atoms and two yttrium atoms inside the cage. The process produced approximately micrograms.
However, they found that the asymmetrical molecule could theoretically collapse to form nearly every known fullerene and metallofullerene. Minor perturbations involving the breaking of a few molecular bonds cause the cage to become highly symmetrical and stable. This insight supports the theory that fullerenes can be formed from graphene when the appropriate molecular bonds are severed.
According to the IUPAC , to name a fullerene, one must cite the number of member atoms for the rings which comprise the fullerene, its symmetry point group in the Schoenflies notation , and the total number of atoms. For example, buckminsterfullerene C 60 is systematically named C 60 - I h [5,6]fullerene. The name of the point group should be retained in any derivative of said fullerene, even if that symmetry is lost by the derivation. To indicate the position of substituted or attached elements, the fullerene atoms are usually numbered in spiral pathway, usually starting with the ring on one of the main axes.
If the structure of the fullerene does not allow such numbering, another starting atom was chosen to still achieve a spiral path sequence.
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The symmetry D 5h 6 means that this is the isomer where the C 5 axis goes through a pentagon surrounded by hexagons rather than pentagons. C 60 - I h [5,6]fullerene Carbon numbering. C 70 - D 5h 6 [5,6]fullerene Carbon numbering. C 70 - D 5h 6 [5,6]fullerene Non-equivalent bonds shown by different colours.
C 71 -PCBM, [1,2]-isomer. If the mesh has other element s substituted for one or more carbons, the compound is named a heterofullerene. If a double bond is replaced by a methylene bridge —CH 2 — , the resulting structure is a homofullerene. If an atom is fully deleted and missing valences saturated with hydrogen atoms, it is a norfullerene.
When bonds are removed both sigma and pi , the compound becomes secofullerene ; if some new bonds are added in an unconventional order, it is a cyclofullerene. Fullerene production generally starts by producing fullerene-rich soot. Although great care is being taken to ensure the correctness of all entries, we cannot accept any liability that may arise from the presence, absence or incorrectness of any particular information on this website.
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The Mathematics and Topology of Fullerenes | SpringerLink
International Symposium on Clusters and Nanomaterials. Event listing ID:. Event website:. Theory and Computation for 2D Materials.
Molecules and molecular compounds are often modeled by molecular graph which is a representation of the structural formula of a chemical compound in terms of graph theory. Recently, the Max-min rodeg index Mm sde which is vertex degree-based topological index has attracted attention and gained popularity. This index give the best predictor for enthalpy of vaporization and standard enthalpy of vaporization in the set of octane isomers and also for log water activity coefficient in the set of polychlorobiphenyles.
A fullerene graph is a cubic planar graph whose faces are pentagons and hexagons. In this study, the Max-min rodeg index of fullerenes and link of fullerenes is computed. Moreover, it is presented exact expressions for the Max-min rodeg index of bridge graphs.
Alizadeh, Y. Wiener dimension: Fundamental properties and 5, 0 -nanotubical fullerenes.
Passar bra ihop
Computing Schultz polynomial, Schultz index of C60 fullerene by gap program. Digest Journal of Nanomaterials and Biostructures, 4 1 , Azari, M. Zagreb indices of bridge and chain graphs. De, N. F-index of bridge and chain graphs. Malaysian Journal of Fundamental and Applied Sciences, 12 3 , Ghorbani, M. On Wiener index of special case of link of fullerenes.
Optoelectronics and Advanced Materials-Rapid Communications, 4 4 , On the forgotten topological index. Iranian Journal of Mathematical Chemistry, 8 3 , Gutman, I. A property of the simple topological index.