Page 93 - Carbon Nanotubes

82 S. IHARA and S. ITOH

Table 1. Cohesive energies of shallow tori; the parameters
n, and n2 are the number of hexagons along the outer and
inner circle, respectively (see Fig. 6). Here N is the number
of atoms in a torus
Energy
"I n2 N (eV/atom)
3 3 288 -7.376
4 3 420 -7.369
5 3 576 -7.375
6 3 756 -7.376
7 3 960 -1.315
8 3 1188 -7.372
9 3 1440 -7.369
4 4 384 -7.378
5 4 540 -7.368
6 4 720 -7.374
7 4 924 -7.374
8 4 1152 -7.372 Fig. 9. Helically coiled form C36g: one pitch contains a
9 4 1404 -7.370 torus C,,,. (a) coil length = 12.9 A and, (b) coil length =
6 6 576 -7.382 13.23 A. The tiling pattern of heptagons in the inner ridge
7 6 780 -7.369 line is changed, though the pattern of pentagons in the outer
8 6 1008 -7.373 ridge line remains upon changing the coil length.
9 6 1260 -7.372
10 6 1536 -7.370
11 6 1836 -7.368 hexagons, and heptagons per 360 and 540 atoms in the
12 6 2160 -7.366
helical structure are the same as in the torus C3h0 and
C,,,[13,14]. By pulling the helix coil, the coil length
for helix C360 increases from 12.9 A (pitch angle a =
or twisting[29], in addition to changing the diameters 15.17 degrees, See Fig. 9 (a)) to 13.23 A (a = 19.73
(of the cross-sections) and the degree of helical ar- degrees, Fig. 9 (b)).
rangement as in straight tubes[6]; (2) a variety of ap- Because the second derivative of the cohesive en-
plications are expected because a variety of helical ergy with respect to the coil length provides the spring
structures can be formed; for instance, a helix with a constant, the spring constants of the helical structures
curved axis can form a new helix of higher order, such per pitch were estimated numerically. As shown in
as a super-coil or a super-super coil, as discussed Table 2, the spring constant for helix C360 is 25 times
below. larger than that of helix C540. We found that the he-
lix C360 is so stiff that the ring pattern changes. Al-
4.2 Properties for the helices derived though the pattern of the pentagons remains the same,
from normal tori the heptagons along the inner ridge line move their po-
The properties of optimized helical structures, sition and their pattern changes discretely with increas-
which were derived from torus C540 and C576r type ing pitch angle a (from one stable pitch angle to the
(A), (proposed by Dunlap) and torus C360, type (B), other). See Fig. 9 (a) and (b); also see Fig. 3 of ref.
(proposed by us) by molecular dynamics were com- [14]. On the contrary, helix C540 is found to be soft
pared. (see Figs. 9 (a) and 10). (Although the torus (i.e., a change in the pitch length does not change the
C576 is thermodynamically stable, helix c576 was ring pattern of the surface). Thus, helix C540 can have
found to be thermodynamically unstable[l4]. Hereaf- relatively large values of a, which corresponds to the
ter, we use helix C, to denote a helix consisting of open-coiled form and can easily transform to the
one torus (C,) in one pitch. super-coiled form without changing the ring patterns.
The diameters of the inside and outside circles, the In ref. [14], helix Close was generated from helix C360
pitch length, and the cohesive energy per atom for he- by use of Goldberg transformation, where hexagons
lices are given in Table 2. The number of pentagons, are inserted into the original helix c360. Helix Close

Table 2. Structural parameters, cohesive energies per atom, and spring constant for helices
C,,, and C,,,; here ro and ri are outer and inner diameter of a helix, respectively
diameters
Pitch length Cohesive energy Spring constant
Structure r, (nm) r, (nm) (nm) (eV/atom) (meV/nm)

Helix C360 2.26 0.78 12.9 -7.41 (-7.41 torus) 4.09
Helix C,,, 4.14 2.94 8.5 -7.39 (-7.40 torus) 0.16