Page 70 - Carbon Nanotubes

CARBON NANOTUBES:
I. GEOMETRICAL CONSIDERATIONS

R. SETTON
Centre de Recherche sur la Matiere DivisCe, CNRS, 1 B rue de la Ftrollerie,
F 45071 OrlCans Cedex 2, France
(Received 22 August 1994; accepted 15 September 1994)
Abstrdct-The geometrical conditions pertaining to closure, helicity, and interlayer distance between suc-
cessive layers with circular cross-sections in carbon tubules (nanotubes) have been examined. Both the
intralayer length of the C-C bonds and the interlayer distance between successive layers must vary with
the radius of the layers. The division into groups of the sheets in nanotubes is found to be due to the re-
ciprocal interaction of the interlayer distance variations and of the conditions required to maintain con-
stancy of the pitch angle.
Key Words-Carbon nanotubes, pitch angIe, helix angle, interlayer distance, carbon-carbon intralayer
distance.

1. INTRODUCTION folds. The sign of 8 determines the helicity of the
scroll, counterclockwise (e > 0) or clockwise (e < 0).
Carbon tubules (or nanotubes) are a new form of The consequence of the presence of scroll helicity
elemental carbon recently isolated from the soot ob- in a tubule is expected to be that any increase (de-
tained during the arc-discharge synthesis of fuller- crease) of the intralayer C-C distance G will increase
enes[ 11. High-resolution electron micrographs do not (decrease) the local length of the spiral, but not nec-
favor a scroll-like helical structure, but rather concen- essarily the mean interlayer distance, since the scroll
tric tubular shells of 2 to 50 layers, with tips closed by can easily adapt its radius of curvature to minimize,
curved, cone-shaped, or even polygonal caps. Later if necessary, any energetic strain due to a stress in the
work[2] has shown the possibility of obtaining single- local bond lengths.
shell seamless nanotubes.
Recently, the structure of some helical carbon 2.2 Screw helicity
nanotubes was examined[3], and the present work is The second type of helicity can affect both scrolls
an attempt at completing the geometrical approach to and individual cylinders. It will be present when nei-
the structural problems encountered in the case of tu-
bules with circular cross-sections. However, most of ther the a nor the b unit vector of the basic two-
the conclusions in the present work are applicable to dimensional graphite lattice of the unfolded scroll or
nanotubes with polygonal cross-sections that have also cylinder is at an angle of 0" or at a multiple of 30"
been shown to exist. with the direction of the cylinder axis (Fig. 2). The sit-
uation is complicated by the fact that whenever screw
helicity affects a seamless cylindrical layer, any change
2. THEORY in the value of G must necessarily affect the radius r
of the cylinder. It is therefore highly likely that the in-
Leaving aside the complications engendered by the terlayer distance d between two successive cylinders is
presence of end-caps and considering, therefore, only not strictly constant in any multilayered tubule with
the cylindrical part of the tubules, one must differen- a circular cross-section, since the adverse effect of the
tiate between two types of helicity. elastic strain on the sp2 orbitals and the resulting dis-
tribution of charges- which undoubtedly affect the in-
tralayer C-C distance - progressively decreases as the
2.1 Scroll helicity
The presence of scroll helicity replaces a set of con- circumference and the radius of curvature of the cyl-
centric cylinders by a single sheet rolled upon itself inders increase. As justified later, one is therefore led
to introduce a parameter 6d or 6r to characterize the
(Fig. 1). Assuming that the distance between the suc- smalI variations of r with respect to a hypothetical
cessive rolls of the scroll is constant, its cross-section nominal value likely to be encountered even if, strictly
can be conveniently represented in polar coordinates speaking, the variation 6G should also be taken into
by the Archimedean spiral:
account.
2.3 Symmetric non-helical tubules
and cylinders
where a, is the initial radius of the innermost fold In the absence of both types of helicity, the non-
and d is the (constant) distance between successive helical or symmetric tubule consists-of a set of cylin-
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