5 Enabling Materials By Dimensionality: From 0D to 3D Carbon-Based. . .
143
mesoscopic and continuum models to integrate different levels of simulation
for climbing the “dimensional ladder”. This approach is not surprisingly called
multiscale analysis.
5.2 0D Carbon Materials: The Fullerenes
Pristine C 60 or buckminsterfullerene is a 0D all-carbon structure (molecule) made
of 20 hexagonal and 12 pentagonal faces, with a carbon atom at the vertices of each
polygon interconnected by bonds along each edge [25]. This amounts to 60 carbon
atom vertices, connected via 90 covalent bonds among single (60) and double (30)
bonds. C 60 resembles a soccer ball (see bottom right panel of Fig. 5.4) and is
shaped as a truncated icosahedron. Carbon atoms in C 60 fullerene are equivalent,
and their s and p outer shells are sp 2 hybridized (or almost like that if one refers
this hybridization to planar-only geometries). The p orbitals, which are orthogonal
to the hypothetical sphere inscribed in the fullerene, form π -type bonds.
While theoretical predictions [26] of the existence and stability of buckminsterfullerene were already put forward, the first isolated and characterized structure
belonging to the fullerines’ family was synthesized in 1985 by R.F. Curl, H.W. Kroto
and R.E. Smalley [27]. Fullerene is the by-product of two synthesis methods, that
is, the arc and the combustion methods, and sublimates below 800 ◦ C. They own
their name to Buckminster Fuller, an American architect universally known for his
innovative design of the geodesic domes shaped as buckyballs.
Nevertheless, some evidence can be found that its geometrical shape was already
postulated by Archimedes among his thirteen solids. To the best of author’s
knowledge, the oldest picture of the fullerene’s structure dates back to the Italian
renaissance painter and mathematician Piero della Francesca, who drew the shape
of C 60 in his book Libellus de quinque corporibus regularibus (see Fig. 5.5a). A
further rendition of the truncated icosahedron, ascribed to Leonardo da Vinci, can
be found in the book De Divina Proportione, written by Luca Pacioli around the end
of the fifteenth century (see Fig. 5.5b).
The existence and stability of C 60 are confirmed by mass spectrometry experiments, which show discrete peaks corresponding to molecules with the exact
mass of 30 to 90 carbon atoms (see Fig. 5.6a [28]), making buckminsterfullerene
one the most abundant molecules among the fullerenes’ family. Furthermore, we
report in Fig. 5.6c the relative cohesive energies of carbon architectures at different
dimensionality, including linear chains, rings, 2D flakes and fullerenes, which are
sketched in Fig. 5.6b. We notice that fullerenes are more stable than rings and planar
flakes for n > 20, due to the absence of edges with respect to other 2D and 1D
structures. Nevertheless, due to the constant positive curvature, which induces stress
into the structure, fullerenes are not totally unreactive at odds with the planar sp 2
carbon net of graphene. They have been actually functionalized by several chemical
elements, such as hydrogen [31], fluorine, bromine and chlorine [32], both in the
143
mesoscopic and continuum models to integrate different levels of simulation
for climbing the “dimensional ladder”. This approach is not surprisingly called
multiscale analysis.
5.2 0D Carbon Materials: The Fullerenes
Pristine C 60 or buckminsterfullerene is a 0D all-carbon structure (molecule) made
of 20 hexagonal and 12 pentagonal faces, with a carbon atom at the vertices of each
polygon interconnected by bonds along each edge [25]. This amounts to 60 carbon
atom vertices, connected via 90 covalent bonds among single (60) and double (30)
bonds. C 60 resembles a soccer ball (see bottom right panel of Fig. 5.4) and is
shaped as a truncated icosahedron. Carbon atoms in C 60 fullerene are equivalent,
and their s and p outer shells are sp 2 hybridized (or almost like that if one refers
this hybridization to planar-only geometries). The p orbitals, which are orthogonal
to the hypothetical sphere inscribed in the fullerene, form π -type bonds.
While theoretical predictions [26] of the existence and stability of buckminsterfullerene were already put forward, the first isolated and characterized structure
belonging to the fullerines’ family was synthesized in 1985 by R.F. Curl, H.W. Kroto
and R.E. Smalley [27]. Fullerene is the by-product of two synthesis methods, that
is, the arc and the combustion methods, and sublimates below 800 ◦ C. They own
their name to Buckminster Fuller, an American architect universally known for his
innovative design of the geodesic domes shaped as buckyballs.
Nevertheless, some evidence can be found that its geometrical shape was already
postulated by Archimedes among his thirteen solids. To the best of author’s
knowledge, the oldest picture of the fullerene’s structure dates back to the Italian
renaissance painter and mathematician Piero della Francesca, who drew the shape
of C 60 in his book Libellus de quinque corporibus regularibus (see Fig. 5.5a). A
further rendition of the truncated icosahedron, ascribed to Leonardo da Vinci, can
be found in the book De Divina Proportione, written by Luca Pacioli around the end
of the fifteenth century (see Fig. 5.5b).
The existence and stability of C 60 are confirmed by mass spectrometry experiments, which show discrete peaks corresponding to molecules with the exact
mass of 30 to 90 carbon atoms (see Fig. 5.6a [28]), making buckminsterfullerene
one the most abundant molecules among the fullerenes’ family. Furthermore, we
report in Fig. 5.6c the relative cohesive energies of carbon architectures at different
dimensionality, including linear chains, rings, 2D flakes and fullerenes, which are
sketched in Fig. 5.6b. We notice that fullerenes are more stable than rings and planar
flakes for n > 20, due to the absence of edges with respect to other 2D and 1D
structures. Nevertheless, due to the constant positive curvature, which induces stress
into the structure, fullerenes are not totally unreactive at odds with the planar sp 2
carbon net of graphene. They have been actually functionalized by several chemical
elements, such as hydrogen [31], fluorine, bromine and chlorine [32], both in the