Page 95 - Carbon Nanotubes

84 S. IHARA and S. ITOH

Fig. 14. TEM picture of a single layered helix-shaped structure: a 1.3-nrn diameter helix coils around the
3.6 nrn tube (after C.H. Kiang ef al.).

dius of about 18 nm, pitch about 30 nm, has ten gra- than those of theoretically predicted ones. However,
phitic layered tubes (diameter of the innermost tube it should be noted that in ref. [14], we have pointed
is about 2.5 nm.). out the existence of the larger helices. We provide an
C.-H. Kiang et a1.[33] reported that the single- example of a large helical form: the helix Close can be
layered coiled tubes were obtained by co-vaporizing generated from helix C3m using the Goldberg algo-
cobalt with carbon in an arc fullerene generator. A rithm, as larger tori were derived from genetic one
single-layered helical structure with radii of curvature such as torus Clzo[ 141.
as small as 20 nm was seen. These helically coiled Zhang et al. [34] also provide a molecular model by
forms tend to bundle together. In the soot obtained connecting tubes at angle 30" bends by introducing the
with sulfur-containing anodes, they also found the required pairs of pentagons and hexagons in the hex-
1.3-nm diameter tube coil around the 3.6 nm tube (see agonal network. Their method is quite similar to Dun-
Fig. 14). This kind of structure was theoretically pro- lap's way of creating torus c540. They combined the
posed in ref. [14]. (12 + 9n,0) tube and (7 + 5n,7 + 5n) tube in the Dun-
By close analysis of diffraction pattern of catalyst- lap's notation, and also combined (9n,0) tube to
grown coiled tube, X. B. Zhang et a1.[34] reported the (5n,5n) tube to show the feasibility of the multi-
larger angular bends and their number to be about 12 layered helical forms. Combination of these tubes,
per helix turn (the magnitude of angular bends is however, leads to toroidal forms whose connection is
about 30 degrees), and this is the essential in determin- quite similar to that of torus c540. But the tori of
ing the geometry. Their smallest observed helixhas a Zhang et al. (see Fig. 15) can easily be turned into he-
radius of about 8 nm. Thus, their sizes are much larger lices by regularly rotating the azimuth of the succes-
sive lines connecting pentagon and heptagons with
small energy, as helix C540 as shown in ref. [14]. (To-
rus C540 can transform to helix C540 without rebond-
ing the carbon atoms on the network with small
energy, as we showed in section 4.2.) See also ref. [35].
Good semi-quantitative agreements are found in
diffraction patterns and proposed models obtained by
molecular-dynamics[ 141, because the results of the ex-
periments[3 1-34] are consistent with the atomic mod-
els proposed by us[ 141. However, in the present state
of high-resolution electron microscopy, taking into ac-
count, moreover, the number of sheets and the com-
plicated geometry of the helix, it seems unlikely to
directly visualize the pentagon-hexagon pairs.

5. CONCLUSION
We showed the possible existence of various forms
of helically coiled and toroidal structures based on en-
ergetic and thermodynamic stability considerations.
Though the formation process of these structures is
not the subject of this work, the variety of patterns in
the outer and inner surface of the structures indicates
that there exist many different forms of stable cage
carbon structures[l0-19]. The molecules in a one-
I
dimensional chain, or a two-dimensional plane, or a
Fig. 15. Tori of Zhang et al.; a 30-degree connections of three-dimensional supermolecule are possible extended
tubes: a (12,O) segment and a (7,7) segment (After Zhang structures of tori with rich applications.
et al.). Many different coiled forms of stable cage carbon