Page 88 - Carbon Nanotubes

HELICALLY COILED AND TOROIDAL CAGE FORMS
OF GRAPHITIC CARBON

SIGEO IHARA and SATOSHI ITOH
Central Research Laboratory, Hitachi Ltd., Kokubunji, Tokyo 185, Japan
(Received 22 August 1994; accepted in revised form 10 February 1995)
Abstract-Toroidal forms for graphitic carbon are classified into five possible prototypes by the ratios
of their inner and outer diameters, and the height of the torus. Present status of research of helical and
toroidal forms, which contain pentagons, hexagons, and heptagons of carbon atoms, are reviewed. By
molecular-dynamics simulations, we studied the length and width dependence of the stability of the elon-
gated toroidal structures derived from torus C240 and discuss their relation to nanotubes. The atomic ar-
rangements of the structures of the helically coiled forms of the carbon cage for the single layer, which
are found to be thermodynamically stable, are compared to those of the experimental helically coiled forms
of single- and multi-layered graphitic forms that have recently been experimentally observed.
Key Words-Carbon, molecular dynamics, torus, helix, graphitic forms.

1. INTRODUCTION The toroidal and helical forms that we consider
here are created as such examples; these forms have
Due, in part, to the geometrical uniqueness of their quite interesting geometrical properties that may lead
cage structure and, in part, to their potentially tech- to interesting electrical and magnetic properties, as
nological use in various fields, fullerenes have been the well as nonlinear optical properties. Although the
focus of very intense research[l]. Recently, higher
numbers of fullerenes with spherical forms have been method of the simulations through which we evaluate
the reality of the structure we have imagined is omit-
available[2]. It is generally recognized that in the ful- ted, the construction of toroidal forms and their prop-
lerene, C60, which consists of pentagons and hexa- erties, especially their thermodynamic stability, are
gons formed by carbon atoms, pentagons play an discussed in detail. Recent experimental results on to-
essential role in creating the convex plane. This fact
was used in the architecture of the geodesic dome in- roidal and helically coiled forms are compared with
vented by Robert Buckminster Fuller[3], and in tra- theoretical predictions.
ditional bamboo art[4] (‘toke-zaiku’,# for example).
By wrapping a cylinder with a sheet of graphite, we 2. TOPOLOGY OF TOROIDAL
can obtain a carbon nanotube, as experimentally ob- AND HELICAL FORMS
served by Iijima[S]. Tight binding calculations indicate
that if the wrapping is charged (i.e., the chirality of the 2.1 Tiling rule for cage structure
surface changes), the electrical conductivity changes: of graphitic carbon
the material can behave as a semiconductor or metal Because of the sp2 bonding nature of carbon at-
depending on tube diameter and chirality[6]. oms, the atoms on a graphite sheet should be con-
In the study of the growth of the tubes, Iijima nected by the three bonds. Therefore, we consider how
found that heptagons, seven-fold rings of carbon at- to tile the hexagons created by carbon atoms on the
oms, appear in the negatively curved surface. Theo- toroidal surfaces. Of the various bonding lengths that
retically, it is possible to construct a crystal with only can be taken by carbon atoms, we can tile the toroi-
a negatively curved surface, which is called a minimal dal surface using only hexagons. Such examples are
surface[7]. However, such surfaces of carbon atoms provided by Heilbonner[8] and Miyazakif91. However,
are yet to be synthesized. The positively curved sur- the side lengths of the hexagons vary substantially. If
face is created by insertion of pentagons into a hex- we restrict the side length to be almost constant as in
agonal sheet, and a negatively curved surface is created graphite, we must introduce, at least, pentagons and
by heptagons. Combining these surfaces, one could, heptagons.
in principle, put forward a new form of carbon, hav- Assuming that the surface consists of pentagons,
ing new features of considerable technological inter- hexagons, and heptagons, we apply Euler’s theorem.
est by solving the problem of tiling the surface with Because the number of hexagons is eliminated by a
pentagons, heptagons, and hexagons. kind of cancellation, the relation thus obtained con-
tains only the number of pentagons and heptagons:
#At the Ooishi shrine of Ako in Japan, a geodesic dome fs - f, = 12(1-g), where fs stands for the number of
made of bamboo with three golden balls, which was the sym- pentagons, f, the number of heptagons, and g is the
bol called “Umajirushi” used by a general named Mori Mis-
aemon’nojyo Yoshinari at the battle of Okehazama in 1560, genius (the number of topological holes) of the
has been kept in custody. (See ref. 141). surface.
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