Page 74 - Carbon Nanotubes

Carbon nanotubes: I. Geometrical considerations 63
fore, highly unlikely) or d > 0.348 nm (Le., at least 4% Table 3. Computed characteristics of a 28-sheet tubule show-
above the value of graphite and also unlikely). If both ing the division into groups of sheets
conditions are met, the pitch angle is 24.18", or 6.59",
or 13.00". If only condition (25) above is operative, i (pi, Qi) ri (nm) cyo
the successive values of a will be nearly but not strictly 1 (14~0) 0.475 0
equai, and will be different from these three values. 2 (24,O) 0.814 0
Two examples illustrate these points: 3 (34,2) 1.153 1.94
4 (44,4) 1.494 3 .oo
(a) the unbroken series of 21 cylinders with a = 6.59", 5 (54,6) 1.834 3.67
r, = 0.6825 nm, G = 0.142 nm, d = 0.341 nm, 6d/d = 6 (64,8) 2.175 4.13
&1.5%, with Qi:Pi 2:10, PI = 20, Pz, = 220, and 7 (74,101 2.516 4.46
=
q:p = 2:10 in all cases; 8 (84,121 2.857 4.72
9 (94,12) 3.195 4.22
(b) the five successive cylindrical sheets with r, = 10 (104~4) 3.566 4.44
10.22 nm, G =0.142 nm, d =0.34 nm, 6d/d = f IS%, 11 (1 14,16) 3.877 4.63
and the characteristics shown in Table 2. A "single- 12 (123,31) 4.214 8.28
helix angle of about 3"" was observed in the selected 13 (133,33) 4.555 8.15
area electron diffraction pattern of a four-sheet sym- 14 (143,35) 4.896 8.04
metric helical tube[3], but the small variations of a 1s (1S3,37) 5.237 7.95
7.87
5.578
shown in Table 2 would obviously not have been vis- 16 (163,39)
ible in the diffractogram shown in the reference. 17 (171,61) 5.919 11.64
18 (1 8 1,63) 6.259 11.36
19 (191,65) 6.599 11.12
2.6 Sheet groups 20 (202,56) 6.935 9.09
It has also been stated[3] that "the helix angle (of 21 (212,SS) 7.276 8.98
a multiple-sheet helical tube) changes about every three 22 (222,60) 7.617 8.87
to five sheets" so that the gross structure of the tube is 23 (232,62) 7.958 8.77
constituted by successive groups of cylindrical sheets, 24 (239,9 1) 8.296 12.40
with the helix angles increasing by a constant value 25 (249,93) 8.635 12.17
from each group to the next. In light of the arguments 26 (259,95) 8.975 11.96
9.315
(269,97)
11.76
27
developed in the preceding paragraphs, and except for 28 (279,99) 9.655 11.56
the specific cases a = O", 6.59", 13.00", 24.18", the
pitch angle of the sheets within a group is more likely
to be nearly constant rather than strictly constant. This
is illustrated by the values in Table 3 showing the char- ues between groups 2 and 3, and 5 and 6, the suc-
acteristics of the 28 sheets in a tube whose inner and cessive values of (a) increase by a nearly constant
outer radii are, respectively, equal to 0.475 and 9.66 amount, namely, about 3 to 4". The question then
nm, calculated on the basis of rl = 0.476 nm, d = arises as to the reason causing the modification of Q!
0.34 nm, G = 0.142 nm, 6d/d = +0.8%. Since a is not between two successive groups, such as the one occur-
strictly constant within each group, it is more appro- ring between i = 11 and i = 12. If (P12, Q12) had been
priate to consider a mean value (a) of the pitch an- (124,18) instead of (123,31), rI2 would have been
gle for each group, as shown in Table 4. Only in the 4.218 nm with a = 4.78" (Le., within 6d/d = +1.0%
first group of sheets (i = 1 to 2) is a truly constant instead of kO.8% imposed for the values in Table 3).
(a = 0"). For all the other groups, the increments Hence, during the actual synthesis of the tube, the end
( p9 q) of the (Pi, doublets are constant and equal of each group must be determined by the tolerance in
Q,)
to (10,2). In fact, it is because these increments are the G values accepted by the spz orbitals compatible
identical that the group pitch angles are nearly con- with the need to close the cylindrical sheet.
stant when the values of Pi and Qi are sufficiently Although the physical growth mechanism of the cy-
Iarge, which is not yet the case in the second group. lindrical sheets is not yet fully known, the fact that
The above series also illustrates an experimentally
established fact[3]: Except for the change of mean vd-
Table 4. Group characteristics of the 28-sheet nanotube
described in Table 3
Table 2. Computed characteristics of a five-sheet symmet-
ric tubule with a nearly constant pitch angle Grow Range of i P, 4 (a">
Qi)
I ri (nm) (Pi, (Yo 1 1 to 2 10,o 0
2 3 to 8 10,2 3.65
1 10.22 (301,29) 3.18 3 9 to 11 10,2 4.43
2 10.56 (31 1,31) 3.29 4 12 to 16 10,2 8.06
3 10.90 (321,31) 3.19 5 17 to 19 10,2 11.37
4 11.24 (33 1,33) 3.29 6 20 to 23 10,2 8.93
5 11.58 (341,33) 3.20 I 24 to 28 10,2 11.97