Page 379 - Fiber Fracture

ATOMIC TRANSFORMATIONS 36 1

However, when very large strains are involved, the continuum description is no
longer adequate and one needs a fully atomistic picture. In order to identify the first
stages of the mechanical yield of carbon nanotubes, ab initio and classical molecular
dynamics simulations were performed at high temperatures, so that defect formation
could be observed during the simulation times, which are of the order of several picosec-
onds for quantum molecular dynamics and several nanoseconds for classical dynamics.
These simulations uncovered the dominant strain release mechanisms as well as their
energetics (Buongiorno Nardelli et al., 1998a,b). Beyond a critical value of the tension,
the system releases its excess strain via a spontaneous formation of topological defects.
The first defect to form corresponds to a 90” rotation of a C-C bond about its center,
the so-called Stone-Wales transformation (Stone and Wales, 1986), which produces two
pentagons and two heptagons coupled in pairs (5-7-7-9, see Fig. 3. Static calculations
under fixed dilation show a crossover in the stability of this defect configuration with
respect to the ideal hexagonal network. The crossover is observed at about 5% tensile
strain in (55) and (10,lO) armchair tubes. This implies that the (5-7-7-5) defect is
effective in releasing the excess strain energy in a tube under tensile strain. Moreover,
through its dynamical evolution, this defect acts as a nucleation center for the formation
of dislocations in the ideal graphite network and can lead to plastic behavior.
After the dominant defect process was identified, the energetics of the defect
formation was determined through static calculations, in order to obtain reliable
estimates of defect processes at low temperatures. Fig. 4 plots the formation energies
of a (5-7-7-5) defect in an armchair (53) tube, as obtained in ab initio calculations.
Calculations for other armchair tubes show that the crossover value is only weakly
dependent on their diameters and always falls in the range of 5-6%.
Subsequent experiments (Walters et al., 1999; Yu et al., 2000) indeed find that
nanotubes fail at strains of up to a little over 5%. Since the most recent measurements
of the Young modulus of nanotubes give about an exceptionally large value of 1.25 TPa
(Krishnan et a]., 1998), nanotubes are among the strongest materials known. Indeed, a
direct measurement of breaking strengths of nanotube ‘ropes’ gave values ranging up to
52 GPa (Yu et al., 2000). Furthermore, the computed activation energies for the bond
rotation transformation are very high (cf. Fig. 5), indicating that perfect, defect-free
nanotubes could be kinetically stable at even greater strain values. However, due to
the high temperature at which the growth of nanotubes occurs, defects will form for
thermodynamic reasons and then remain frozen in. For a material at thermal equilibrium,
the number of defects of a particular type is given by Nsitese(-GF’kT), where Nsites is the
number of potential sites and GF is the Gibbs free energy of formation of the defect.
The entropic contributions are usually a small part of GF; onc can thus use the energy of
formation to obtain a lower bound. For single-walled tubes, which are grown at - 1500
K, the above formula suggests that even at zero strain there might be point defects every
few tenths of a mm. The presence of the frozen-in defects certainly limits the ultimate
strength of nanotubes and thus some of their proposed uses.
The appearance of a (5-7-7-5) defect can be interpreted as the nucleation of a
degenerate dislocation loop in the planar hexagonal network of the graphite sheet. The
configuration of this primary dipole is a (5-7) core attached to an inverted (7-5) core.
The (5-7) defect behaves thus as a single edge dislocation in the graphitic plane. Once