Page 89 - Carbon Nanotubes

78 S. IHARA and S. ITOH
In the spherical forms (Le., g = 0), fs =f7 + 12. In c540 c576
C60, for example, there are no heptagons (f, = 0), so
that fs = 12. If the torus whose genius (8) is one,
fs = f7. As we mentioned in the introduction, penta-
gons and heptagons provide Gaussian positive and
negative curvatures, respectively. Therefore, penta-
gons should be located at the outermost region of the
torus and heptagons at the innermost.
Fig. 2. Optimized toroidal structures of Dunlap’s tori: (a)
2.2 Classifications of tori torus C540 and (b) torus C576; pentagons and heptagons
are shaded.
Here, the topological nature of the tori will be dis-
cussed briefly. Figure 1 shows the five possible pro-
totypes of toroidal forms that are considered to be
related to fullerenes. These structures are classified by created by the torus in our case, the properties of the
the ratios of the inner and outer diameters rj and r, , helix strongly depend on the types of the torus.
and the height of the torus, h. (Note that r, is larger
than Ti). As depicted in Fig. 1, if ri = r,, and h << ri,
and h = (r, - rj) then the toroidal forms are of type 3. TOROIDAL FORMS OF GRAPHITIC CARBON
(A). If ri < r,, and r, - h, (thus h - (r, - rj)) then the
type of the torus is of type (D). If ri - r, - h, and 3.1 Construction and properties
h = (r, - rj) then the type of the torus is (B). In these of normal tori
tori, h - ( ro - ri) and we call them normal toroidal 3.1.1 Geometric construction of tori. Possible
forms. However, if h << (r, - r;), then the type of the constructions of tori with pentagons, heptagons, and
torus is (C). Furthermore, If (ro - rj) << h, then the hexagons of carbon atoms are given independently by
type of the torus is (E). These are the elongated toroi- Dunlap[lO], Chernozatonskii[ll], and us[12-171. In
dal forms, as we can see from the definition of type ref. [18], the method-of-development map was used
(C) and (E). to define various structures of tori. For other tori, see
ref. [19].
2.3 Derivation of the helical forms By connecting the sliced parts of tubes, Dunlap
In constructing a helix, the bond lengths of the proposed toroidal structures C540 and C576r both of
hexagons substantially vary without the introduction which have six-fold rotational symmetry; both contain
of pentagons and/or heptagons. Thus, to make a gra- twelve pentagon-heptagon pairs in their equators[ 101
phitic form, it may be a good hypothesis that a heli- (See Fig. 2). Dunlap’s construction of the tori connects
cal structure will consist of pentagons, hexagons, and carbon tubules (2L,O) and (L,L) of integer L in his no-
heptagons of carbon atoms. Therefore, a helical struc- tation[lO]. The bird’s eye view of the structures of tori
ture tiled by polygons was topologically constructed Csd0 and C576 are shown in Fig. 2. This picture is use-
by cutting the torus into small pieces along the toroi- ful for understanding the difference between Dunlap’s
dal direction and replacing them, having the same to- construction and ours. Dunlap’s tori belong to the
roidal direction, but slightly displaced upwards along Type (A) according to the classification proposed in
the axis. The helix thus created contains one torus per section 2.2.
pitch without loss of generality. Because the helix is Recently, we become aware that Chernozatonskii
hypothetically proposed some structures of different
types of toroidal forrns[ll], which belong to type (B)
of our classification. He proposed toroidal forms by
creating suitable joints between tubes. See Fig. 3 of
ref. [9]. He inserts octagons or heptagons into hexa-
gons to create a negatively curved surface, as Dunlap
and we did. His tori C,,, and C440 have five-fold ro-
tational symmetry as our tori, the number of pairs of
pentagons and hexagons is ten. But the pentagons (at
the outer surface) and the heptagons (at the inner sur-
face) are located in the equator of the tori as Dunlap.
Chernozatonskii’s tori may be in the intermediate
structure between Dunlap’s and ours, but two hepta-
gons (a kind of defect) are connected to each other
(which he called an Anna saddle). Since two heptagons
are nearest neighbors, his torus would be energetically
higher and would not be thermodynamically stable, as
Fig. 1. Five possible simple prototypes of the toroidal forms the placement of the pentagons follows the isolated
of graphitic carbon. All cross-sections of the tube are square.
Here ro, ri, and h are the outer and inner radii and height pentagon rule. Other types of toroidal forms, such as
of the torus, respectively. type (C) and (E), are discussed later in section 3.2.2.