Page 100 - Carbon Nanotubes

90 A. FONSECA al.
et

Fig. 5. Model structure of the (9,O)-(5,5) curved nanotubule ended by two half C,, caps (a) and of the
(12,O)-(7,7) curved nanotubule (b). A knee angle of 36” is observed in both models.

knee if the five and seven membered rings are nanotubule series,-the diameter of the graphite layer
removed. By folding the knee to its three-dimensional n is equal to 6.9n0A, within 2%, so that the interlayer
shape, the real angle expands to 36”. At the present distance is 3.46 A. For this interlayer distance, the
time, it is not known whether this discrepancy is an smallest possible knee is (5,O)-( 3,3), with a diameter
artefact of the ball and stick model based on sp2 of 3.99 A (Table l), because the two smaller ones
bonds or whether strain relaxation around the knee could not give layers at the graphitic distance. Note
demands the 6” angle increase. Electron diffraction that all the n=O tubules are probably unstable due
and imaging data[8-101 have not so far allowed to their excessive strain energy[ 161.
assessment of the true value of the knee angle in The nanotube connections whose diameter differ-
polygonized nanotubes. ences are different from the 3.8% value characteristic
The diameters (D,) of the perfectly graphitized to the (9n,0)-( 5n,5n) series, will tend to that value
series are Dnl= 15naln. and Dn,,=9nufi/x, respec- with increasing the graphite layer order n.
tively, for the “perpendicular” and “parallel” straight The inner (outer) diameter of the observed curved
segments. D,,, is 3.8% larger than DnL. This diameter or coiled nanotubules produced jy the catalytic
difference is larger than the 1% characterising the method[8] varies from 20 to 100 A (150 to 200 A),
which corresponds to the graphite layer order
(12,O)-(7,7) ideal connection of Dunlap[ 12,131, but 31n115 (Table 1).
it is independent of n. A few percent diameter differ-
ence can easily be accommodated by bond relaxation 2.3 Description of a perfectly graphitizable chiral
over some distance away from the knee. tubule knee series
Table 1 gives the characteristics of classes of Among all the chiral nanotubules connectable by
bent tubules built on (9n + x, 0)-( 5n + y, 5n + y) knee a knee, the series (8n,n)-(6n,4n), with n an integer, is
connections. In these classes, y=(x/2) f 1, and the perfectly graphitizable. For that series, the diameter
integers x and y are selected to give small relative of the graphite layer of order n is equal to 6.7% 4,
diameter differences, so that within 1%, so that the interlayer distance is 3.38 A.
Moreover, the two chiral tubules are connected by a
3
barn. = (Dn// -Dnl)lDn//, Dnl= ( 5n + Y) 4 ~ pentagon-heptagon knee, with the equatorial plane
and Dn,,=(9n+x)u$/n. passing through the pentagon and heptagon as for
the (9n,0)-( 5n,5n) series. On the plane graphene con-
The first layers of the connections leading to the struction, the two chiral tubules are connected at an
minimum diameter difference are also described in angle of 30”. As for the (9n,O)-(Sn,Sn) series, 36” is
Table 1. For the general case, contrary to the observed from our ball and stick molecular model
(9n,O)-(Sn,Sn) knees, the largest side of the knee is constructed with rigid sp2 triangular bonds (Fig. 6).
not always the parallel side (9n+x, 0). As seen in
Table 1, the connections (7,O)-(4,4) and (14,O)-(8,s) 2.4 Constructing a torus with (9n,0)-(5n,5n)
are, from the diameter difference point of view, as knees
ideal as the (12,O)-(7,7) described by Dunlap[ 12,131. Building up a torus using the (9,O)-( $5) knee
For the perfectly graphitizable (9n,0)-( 5n,5n) [Fig. 7(a)] is compatible with the 36” knee angle.