Page 384 - Fiber Fracture

366 J. Bemholc et al.

ab initio vs. Classical MD
lo '1

- graphene, classical MD
H (5,5)tube, ab initio
-5 ' - graphene, ab initio J
I
0 5 10
Strain (%)
Fig. 8. Formation energy of the off-axis (5-7-7-5) defect in (n,O) tubes of various diameters. In (n,O)
tubes, D = 0.078~ nm. Inset: formation energy of the off-axis (5-7-7-5) defect in (10,m) tubes of different
chiralities. Note that in (n,m) tubes the chiral angle x = arctan[&/(2n +m)] is zero in zigzag tubes and
30" in armchair tubes. All data refer to 10% strain.

nanotube, the same C-C bond will be parallel to the applied tension, which is already
the minimum energy configuration for the strained bond. The formation of the Stone-
Wales defect is then limited to rotation of the bonds oriented 120" with respect to the
tube axis. Our analysis shows that the formation energy of these defects is strongly
dependent on curvature and thereby on the diameter of the tube. In fact, a number of
different behaviors have been observed when constructing the brittle vs. ductile map of
stress response of carbon nanotubes. The results of static energetics calculations and
molecular dynamics simulations for (n,O) tubes of various diameters D at 10% strain
are summarized in Fig. 8. Remarkably, the formation energy of the off-axis (5-7-7-5)
defect (obtained via the rotation of the C-C bond oriented 120" wrt. tube axis) shows
a crossover with respect to the diameter, and it is negative for (n,O) tubes with n < 14
(D < 1.1 nm). Similarly, the formation energy of this defect in chiral tubes of the
(10,m) family (chosen as a particular example) is always negative, although it changes
with the chiral angle x. This result implies that the bond-rotation transformation is
still efficient in releasing the strain energy of the tube. This effect is clearly due to the
variation in curvature, which in the small-diameter tubes makes the process energetically
advantageous. Therefore, above a critical value of the curvature a plastic behavior is
always possible and the tubes can be ductile.
From these calculations one can identify the full range of elastic responses in strained
carbon nanotubes. In particular, under high-strain and low-temperature conditions, all