Page 75 - Carbon Nanotubes

64 R. SETTON
successive sheets have nearly constant helicity suggests 2.8 Influence of the innermost cylinder
the following mechanism: once the ith (topmost) sheet If the template effect of Endo and Kroto[9] does
is formed, C, species in the gas phase (n = 3,1,2. . . , indeed exist and the growth of the tubule occurs out-
in order of decreasing abundance[6]) condense onto wards, a knowledge of the Characteristics of the inner-
the outer surface of the tube, probably at a distance most tube is then of paramount importance in any
slightly larger than 0.34 nm[7], in registry with the un- attempt at modelizing the structure of a symmetric
derlying substrate and in sp2 hybridization, since this nanotube[ 111. By reversing eqn (17), possible values
configuration is the one that best minimizes the energy for P1 and Q, can be obtained from
of the system[8]. The ith sheet thus constitutes the
template for the construction of the (i + 1)th sheet[9], 3P; + (212 = (16a2/3) [(rl/G)2 * Wr], (26)
and the helicity of the as-yet-unclosed cylinder during
its formation is the same as that of the underlying sub- and the inequalities (8) and (9), provided rl has been
strate. This process continues until the C atoms on ei- determined (say, by HRTEM - high resolution trans-
ther side of the cut form the C-C bonds that close mission electron microscopy), and G is arbitrarily
the cylinder. If the geometric characteristics of these fixed, say, at G = 0.142 nm. Although the task may
bonds are too different from 120" and G = 0.142 nm, become formidable if r1 and/or 6r/r are large, the
the pitch angle will change to the immediately higher number of possibilities is strictly limited, especially
or lower value, thus ensuring closure. Once the cylin- since selected area electron diffraction (SAED) can
der is closed, and even if closure only involved a single provide narrower limits by giving a first approxima-
row of hexagons, the following rows of the cylindri- tion of the pitch angle[3].
cal sheet will necessarily have the same pitch angle, at
least as long as the sheet is growing over an existing
substrate. 3. CONCLUSION
It is clear that a large number of parameters influ-
ence the formation of the sheets in a tubule and that
2.7 Smallest inner tube diameter
Liu and Cowley[3] have reported the observation their relative importance is still unknown, as is also the
of innermost cylinders with diameters as small as 0.7 cause of the occurrence of the defects responsible for
or 1.3 nm. The first value is only slightly larger than the eventual polygonization of the sheets. Although
0.68 nm, the well known diameter of the Cs0 mol- the model presented here highlights the necessity of in-
ecule[lO], while the second is in fair agreement with cluding, as one of the parameters, the uncertainty 6r
1.36 nm, the calculated diameter Cm. As shown in or 6d on the separation of successive cylindrical sheets,
it is impossibIe to predict with absolute certainty the
Table 5, there are 5 (PI, Q1) doublets which, with G = final characteristics of any of these sheets, symmetric
0.142 nm, give rl = 0.35 nm +3%, and 14 (P1,Q1)
doublets (not all are given in Table 4) for the larger or not, on the basis of the characteristics of thepre-
of the two values within k5%. It is interesting to find vious one. Nevertheless, a number of features of their
that there is, in both cases, the possibility of forming structure, such as the presence or absence of helicity,
a non-helical cylindrical sheet with or = Oo, namely and the presence of groups of sheets with nearly the
same angle of pitch, can be explained and quantified.
(P,,Q,) = (10,0), rl = 0.339 nm, and (Pl,QI) =
(20,0), rl = 0.678 nm, with the first row of C atoms
corresponding to one of the equatorial cuts of C60 or REFERENCES
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