Mater. Res. Soc. Symp. Proc. Vol. 901E © 2006 Materials Research Society
0901-Ra16-34-Rb16-34.1
Nanocones - a different form of carbon with unique properties
H. Heiberg-Andersen1 , G. Helgesen1 , K. Knudsen1 , J.P. Pinheiro1 , E. Svåsand1,2 and
A.T. Skjeltorp1,2
1
2
Institute for Energy Technology, POB 40, N-2027 Kjeller, Norway
Physics Department, University of Oslo, POB 1048 Blindern, N-0316 Oslo, Norway
ABSTRACT
It is possible to make perfect conical carbon nanostructures fundamentally different
from the other nanocarbon materials, notably buckyballs and nanotubes. Carbon cones
are realized in five distinctly different forms. They consist of curved graphite sheets formed
as open cones with one to five carbon pentagons at the tip with successively smaller cone
angles, respectively. The nucleation and physics of nanocones has been relatively little
explored until now. We present here the key facts and latest results on this ”5’th form of
carbon”.
INTRODUCTION
There are two periodic forms of carbon: diamond and graphite. In a flat graphene sheet,
three of the four most energetic electrons from each carbon atom are bound in so-called
sp2 hybrid orbitals. The optimum angle between the associated bonds, usually called σbond, is 120◦ . Therefore, the carbon atoms of an undisturbed graphene sheet are arranged
in a hexagonal network. However, each atom contributes also a p-electron which there
is no place left for in the low-lying sp2 states. Consequently, the electronic properties of
graphitic structures are almost exclusively determined by the so-called π-orbitals occupied
by these electrons. This virtual separation of the roles of the different electrons has been
a highly useful concept in organic chemistry. In the simplest molecular orbital theory
of plane sp2 hybridized carbon networks, the Hückel model [1], the π-orbitals are ruled
solely by the interactions between adjacent carbon atoms. The predictive power of this
simple model sparked off extensive research in spectral graph theory, as the energies and
LCAO (Linear Combination of Atomic Orbitals) coefficients of the Hückel orbitals are
determined by the graph defined by the σ-bonds.
With the advent of the fullerenes came the question about the meaning of σ- and πorbitals for curved graphene sheets; see [2] for an overview of this discussion. It turns out
that the ordering of the valence states is largely unaffected by the curvature itself; there
is good agreement between the Hückel or Tight-Binding [3,4] calculations and ab initio
results in this respect [5–7]. The π-orbitals and the σ-bonds are thus useful concepts also in
the study of curved graphene sheets. However, to produce a stable curvature, the topology
of the graphene sheet must be altered. As a result, the electronic properties of curved
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and flat graphite are significantly different. An open-ended nanotube can be imagined
as a sheet rolled into a cylinder, but fullerenes and nanocones with only hexagonal faces
are simply impossible. This can be deduced from Euler’s Theorem, which states that a
convex polyhedron with f faces, n vertices and e edges satisfy the equation f + n − e = 2.
By consensus the term nanocone imply a distinction from a helically wound graphene
sheet, just like a nanotube is understood to be different from a scrolled graphene sheet.
To construct a nanocone, it is most convenient to cut 1 to 5 sectors of 60◦ disclination
angle from a flat graphene sheet, and join the resulting dangling bonds. Demanding all
bond lengths equal, the new sheet is then a cone with apex angle ϕ given by
µ
sin
¶
2π − n · π/3
ϕ
,
=
2π
2
(1)
where n = 1, 2, ..., 5 is the number of sectors removed from the flat sheet. Other facial
combinations than hexagons and pentagons are considered relatively unlikely, since they
will effect larger and more abundant deviations from the optimum σ-bond angle of 120◦ .
From this point of view, the five apex angles allowed by (1), namely 112.9◦ , 83.6◦ , 60.0◦ ,
38.9◦ and 19.2◦ , correspond to 1, 2, 3, 4 and 5 pentagons at the tip, respectively.
The first observation of carbon nanocones were reported by Ge and Sattler in 1994 [8],
a short time after the first theoretical works [9,10] on this new form of carbon. In the
hot vapor phase, open-ended cones were synthesized together with cone-shaped fullerenes
and nanotubes. All the observed cones had apex angles close to 19.2◦ , which is consistent
with 5 pentagons at the tip of an otherwise hexagonal graphene sheet, or a fullerene cone
with respectively 5 and 7 pentagons at the tip and base ends. The number of synthesized
cones was small, but still remarkable, since the cone is an utterly seldom guest among the
reaction products. Even more sensational was the subsequent finding that the geometry
of the protein core shell of the HIV-virus was consistent with the topology of a fullerene
cone [11,12]. Three years later, a more efficient method to produce nanocones were found
by accident. Under pyrolysis of heavy oil in a cycle known as Kværner’s Carbon-Black
& Hydrogen Process (KCBP) [13], large quantities of open-ended cones with all the five
possible apex angles surprisingly appeared [14]. With samples from this process as a
testing ground for theoretical predictions, our group is working to unveil the chemiphysical
properties and nucleation mechanisms of carbon nanocones.
RESULTS AND DISCUSSION
In addition to the milestones mentioned above, several interesting results on carbon
nanocones were obtained by various research groups through the 1990’s. See [15] for a
recent review of relevant works. Due to the still mysterious nucleation process, only a few
groups have been in possession of cone samples until now, so the experimental literature
on cones is very limited. Most of the theoretical works are concerned with the electronic
effects of the pentagonal faces, when these are regarded as defects in a periodic graphene
sheet. However, the cones produced by the KCBP are finite molecular structures, so their
true atomic arrangements is a more immediate issue. Without these, we cannot perform
realistic calculations of the electronic properties.
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Figure 1. SEM image of a sample from the Kværner’s Carbon-Black & Hydrogen Process.
Determination of stable cone topologies
Figure 1 shows a Scanning Electron Microscopy (SEM) image from a KCBP sample.
In agreement with the conclusions drawn in [14], our collected scans shows a discrete
distribution of apex angles, which is clear evidence of the underlying nanocone topologies
described above. Quite recently, natural occurrences of helically wound graphitic cones
were found [16] in the Central Metasedimentary Belt of the Canadian Greenville province.
The measured apex angles of these cones were scattered over values far off those allowed
by (1). We are therefore convinced that a typical cone produced by the KCBP is not
helical, but consists of stacked graphene sheets with identical tip topologies.
For an apex angle corresponding to multiple pentagons at the tip, the next question is
then: Which of the many possible configurations of the pentagons gives the most stable
cone? The answer is three-faceted. According to the old valence-bond theory, which
has experienced a moderate renaissance in the study of fullerenes [17,18], it is important
to keep the π-bonds outside the already stressed pentagons. Historically, this classical
approach lost its momentum with the first observation of Jahn-Teller distortions in the
cyclopentadienyl radical. According to the superior molecular orbital theory, we should
primarily be concerned with the topologies of closed-shell systems, as these are stable
against Jahn-Teller distortions and chemically inert. For fullerenes and cones, there is
fortunately no conflict between the demands of closed shells and absence of double bonds
in the pentagons. A fullerene or cone is said to have a Fries Kekulé structure if the
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topology allows a bond alternation pattern where each face is one of the three types
shown below.
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One of the main achievements in the field of fullerenes, is the graph-theoretical proof [19]
of the prior hypothesis that all fullerenes with a Fries Kekulé structure have closed Hückel
shells. Outside this class, where the well-known C60 molecule is the smallest member, there
are very few closed-shell fullerenes. For the cones, the situation is slightly different. We
have recently proved [20] that all open-ended conjugated cones where the configuration of
the pentagons is consistent with a Fries Kekulé structure, and each CC-bond is part of a
hexagon, have closed Hückel shells. Due to this extension, the fraction of stable topologies
is larger for open-ended cones than for fullerenes. For a given number of pentagons, which
of these stable topologies is most favorable? This remaining ambiguity is easily resolved
by the fact that increasing distance between the pentagons leads to larger and more
localized bond stress at the resulting cusps. The tightest non-adjacent configuration of
Figure 2. The predicted tip topologies of the most stable cones.
0901-Ra16-34-Rb16-34.5
the pentagons, as shown for the different cone tips in Figure 2, is consistent with a Fries
Kekulé structure and should therefore give the most stable cone. Experimental efforts are
initiated to test this prediction against the KCBP samples.
Our ab initio calculations of the exact geometry and electronic structure of cones with
the tip topologies of Figure 2 show something striking: The valence orbitals of the cones
with 4 or 5 pentagons are displaced off the tip [21]. Since comparative Hückel calculations
give the same result, this must be a shear topological effect. However, the existing results
on eigenvectors of topological matrices are insufficient for its explanation.
Three unique series of radicals, cations and anions
Earlier theoretical works on nanocones have not been particularly concerned with symmetry. Consequently, an unique feature of the cones has hitherto been overlooked: Removal of the outer ring of carbon atoms from a conjugated cone with 1, 3, or 5 pentagons
at the tip gives a radical with the same symmetry. This trick cannot be carried out for
the more well-known nanotubes, which are either open-ended or capped with hemispheres
containing 6 pentagons. However, this observation is hardly of interest unless we can determine if these radicals and their associated anions or cations are likely to exist or not.
Figure 3. DFT simulations of the total electron density (left) and Coulomb potential
(right) of a stable conic anion consisting of 125 carbon atoms. The curvature originates
from a single pentagon, and the structure is viewed from the concave side.
The graph-theoretical techniques used in the stability proofs for the conjugated cones do
not apply to radicals. Fortunately then, the molecular graphs of radicals and conjugated
0901-Ra16-34-Rb16-34.6
Figure 4. DFT simulation of the two degenerate HOMO orbitals of the conic anion of
Figure 3.
cones with 1, 3, or 5 pentagons configured as in Figure 2 have certain invariants that
provide a link between their spectra. Through this link we have determined the critical
eigenvalue bounds for three infinite series of radicals [22]: The series with 1 or 5 pentagons
at the tip has one vacancy in the last bonding Hückel orbital, while the last electron is
forced into the first anti-bonding orbital for the series with 3 pentagons at the tip. None
of the three series contains radicals with un-bonding orbitals. As far as the Hückel theory
applies, these results imply three associated series of stable conic anions and cations,
obtained respectively by addition and removal of a single electron from each radical.
Figure 3 shows accurate Density Functional Theory simulations of the electron density
and the Coulomb potential on the concave side of a conic anion consisting of 125 carbon
atoms with one pentagon at the tip. Figure 4 shows the degenerate Highest Occupied
Molecular Orbital (HOMO). The calculations were carried out with the renown Amsterdam Density Functional (ADF) package [23–25], using triple zeta doubly polarized basis
sets. The exchange functional of Becke [26] was combined with the correlation functional
of Lee, Yang and Parr [27,28]. This choice is usually abbreviated as BLYP. So far, our
electronic structure calculations on radicals from this series invariably give HOMO orbitals
of E1 type. We are currently not able to decide whether or not this is a true invariant
property of the molecular graphs.
CONCLUSIONS
Our SEM images confirm the reported [14] discrete distribution of apex angles for the
cones produced in Kværner’s Carbon-Black & Hydrogen Process. With this signature
of nanocone topology firmly established, we have determined the most favorable configu-
0901-Ra16-34-Rb16-34.7
rations of multiple pentagons at the tips by graph-theoretical techniques. By exploiting
invariant features of the molecular graphs of conjugated cones and radicals with 1, 3, or
5 pentagons we have unveiled three unique classes of conic radicals and associated stable
cations and anions. If a cone tip has 4 or 5 pentagons, in the most favorable configuration, the valence orbitals are displaced off the cone tip. This effect is of purely topological
origin, but yet not understood.
ACKNOWLEDGEMENTS
We appreciate funding of this work by the Research Council of Norway under the
programmes Energy for the Future, Project No. 149351/431, and NANOMAT, Project
No. 163570/S10. In addition, we have received support from the Research Council of
Norway (Programme for Supercomputing) through a grant of computing time.
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