JOURNAL OF CHEMICAL PHYSICS
VOLUME 118, NUMBER 21
1 JUNE 2003
Bond-breaking bifurcation states in carbon nanotube fracture
Traian Dumitrică, Ted Belytscko, and Boris I. Yakobsona)
Center for Nanoscale Science and Technology, Department of Mechanical Engineering
and Materials Science, and Department of Chemistry, Rice University, Houston, Texas 77251
共Received 13 January 2003; accepted 3 April 2003兲
Fullerene nanotubes yield to tension in two basic ways. At high temperature 共or in the long time
limit兲 a single bond rotation creates a dislocation-dipole favored thermodynamically under large
stress. However, at low temperature 共or limited time range兲 this process remains prohibitively slow
until further increase of tension causes direct bond-breaking and brittle crack nucleation. This
instability proceeds through the formation of a distinct series of virtual defects that only exist at
larger tension and correspond to a set of shallow energy minima. The quantum mechanical
computations of the intermediate atomic structures and charge density distributions clearly indicate
a certain number of broken bonds. © 2003 American Institute of Physics.
关DOI: 10.1063/1.1577540兴
Strength of solids at given conditions 共load, temperature兲
originates from the nature of chemical bonds. Microcracks or
flaws reduce the material strength, ultimately determined by
nucleation process. Theory of nucleation of failure remains
an important topic studied for model ideal solids 共see, for
example, Refs. 1, 2兲. While macroscopic samples usually
contain imperfections, an important example of well-defined
ideal structure is naturally offered by carbon nanotubes
共CNT兲.3
The strength of nanotubes has been a subject of intense
experimental4 –9 and theoretical9–15 studies. This interest is
further stimulated by the recent advances in CNT-fiber
making,7 where mechanical properties remain in the focus.
While various causes might downgrade the moduli and
strength in a particular measurement, as better care is exercised in sample preparation, the measured strength tends to
higher values (⬃150 GPa in the very recent reports8兲. Understanding of atomistic relaxation and the theoretical
strength benchmarks is a challenge for physics of fracture
and for CNT research.
In hexagonal lattice of CNT a recognized possibility is a
dislocation yield11–14 by a Stone–Wales 共SW兲 bond
rotation.16 This rearrangement of chemical bonds has a high
activation barrier of 6 –9 eV 共Refs. 17–19兲, and therefore
requires high temperature 共or irradiation兲. Accordingly, the
yield point lowers with temperature,19 similar to the early
Monte Carlo and molecular dynamics findings for generic
Lennard-Jones crystals.1,2 While the thermally-activated
bond-rotation mechanism has been extensively studied,11–14
the failure at more realistic room temperature remains unexplored and is the focus of present work. Below we describe
such an alternative failure path that becomes possible at high
tension, while requiring no thermal activation. It proceeds by
direct bond-breaking, through a sequence of metastable
states corresponding to ‘‘lattice trapping,’’ discussed in the
phenomenology of fracture.20 Such primary states of nucle-
ation are revealed here for the first time with accurate
quantum-chemical description of a covalently-bonded
material.21 Near the failure threshold these structures are
metastable and correspond to one, two, three, etc. broken
bonds (b⫽1,2,3, . . .). Their energies are almost degenerate
and the separating barriers are low 共in contrast with the still
high SW barrier兲. Probabilistic analysis unequivocally shows
the dominance of the brittle bond-breaking path b⫽0→1
→2→3→ . . . as CNT failure mechanism at room temperature in a tensile test experiment.
Our calculations employed the AM1 model22 of
23
GAUSSIAN 98 package, which is well suited for the present
problem since it includes electron correlation effects, is selfconsistent, and practical in finding equilibrium geometries
involving a large number of atoms and especially a variety of
initial trial conditions. In addition, key features were repeated with the density functional theory 共DFT兲 with the
exchange-correlation functional of Perdew, Burke, and
Ernzerhof24 and a STO-3G basis set.
We applied incremental 3% tensile strain, aiming for
failure of both 共10,0兲 and 共5,5兲 CNT samples. The geometries (C120H20 and C130H20 in AM1, C160H20 and C180H20 in
DFT兲 were optimized under the constraint of fixed distances
between pairs of atoms located at opposite ends, so that the
equilibrium geometries are local minima on the potential energy surfaces 共PES兲. For the analysis of structural defects the
cluster approach was preferred over the periodic boundary
conditions since the latter causes unwanted long-range 共via
lattice strain兲 defect-image interactions. We choose the
H-terminated clusters to be large enough to reproduce the
bulk C–C bond properties. We also verify that AM1 and
DFT methods yield close C–C bond lengths 共1.43 Å兲, binding energy, and elastic properties. 共The differences in values
of the in-plane stiffness d 2 E/d␧ 2 are within 1%.兲
The strength of C–C bonding is quantified by Fig. 1
where it is shown that the nanotubes sustain high elongations, while the energy follows a Hooke’s parabola. The failure only occurs at ␧⬃16% for zigzag and above 24% for
a兲
Electronic mail: [email protected]
0021-9606/2003/118(21)/9485/4/$20.00
9485
© 2003 American Institute of Physics
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9486
J. Chem. Phys., Vol. 118, No. 21, 1 June 2003
Dumitrică, Belytscko, and Yakobson
FIG. 1. Energy vs strain for ideal 共10,0兲 CNT 共thick solid line兲 is intersected
by SW defect 共thin solid line, small arrow near 8%兲 or by the virtual defects
共dashed lines, large arrow near 16%兲. Vertical spacing 共3.7 eV兲 and the
minima shifts 共circles兲 for the parabolas show the energy and dilatation
added to the lattice due to these defects corresponding to the 1, 2, or 3
broken bonds, and vanishing at lower strain. Analogous fit for the armchair
tube yields another set of parabolas with higher crossover range and smaller
dilatation shift.
armchair CNT. 共As expected, since in the zigzag type the
axially oriented bonds incur a higher strain.兲
In order to see the details of fracture nucleation from our
microscopic model, with the perfect CNT system at a nearthreshold strain, we have carried systematic relaxed PES
scans under perturbation of a single central bond with increments of .02 Å or smaller. In the vicinity of threshold we
have found a series of metastable configurations and Fig.
2共a兲 presents the geometries of the first states in a 共10,0兲
CNT around 16% strain. Visually, they correspond to one,
two, and three axial broken bonds (b⫽1,2,3), respectively.
Similar configurations have been found in the 共5,5兲 type and
the emerged b⫽1 state is also shown.
The lattice-trapped states do not exist in the strain-free
material and we further studied their stability ranges. The
strain level is varied until the tiny crack either falls over into
a longer one by further bond breakage, or heals by bond
restoration, b⫺1←b→b⫹1. Fig. 2共b兲 plots the computed
formation energies as measured with respect to the perfect
CNT, ⌬E b (␧)⫽E b (␧)⫺E 0 (␧). One can see that the obtained ranges of existence increase with the number of broken bonds, from a very narrow interval in the b⫽1 case to
⌬␧⬃1% for b⫽3. The following states b⫽4 and 5, although less important in crack nucleation, are also included.
Due to their bifurcation nature, the trapped state energies
are quite close to the ideal strained lattice. For a physical
interpretation, one can formally determine the ‘‘free’’ formation energies 共without strained lattice contribution兲 of these
defects by extrapolating them into the lower strain region.
For this purpose, the computed energy-versus-strain ranges
for b⫽1,2,3 states were extended with a parabola which
employs the stiffness of ideal CNT, as shown in Fig. 1. Remarkably, we found that the near-minima energies of these
virtual defects approximately correspond to multiples of cohesive energy 共per bond兲, which justifies the broken-bonds
FIG. 2. 共a兲 Relaxed atomic structures of lattice-trapped states in a 16.1%
strained 共10,0兲 CNT display b⫽1, 2, and 3 broken bonds 共right兲 with central
interatomic separations of 1.9, 2.0, and 2.3 Å respectively, while the nearest
intact bonds retain about 1.6 Å length. In 共5,5兲 CNT a single broken bond
emerges at above 24% strain, with 1.9 Å separation while the surrounding
bonds remain shorter than 1.7 Å 共left兲. 共b兲 Formation energies ⌬E(␧)
⫽E b (␧)⫺E 0 (␧) for defect-states b⫽1⫺5 are shown as functions of strain,
to detail the crossover in Fig. 1; three gray levels indicate the ranges of
stability, with b⫽1 being the darkest, and b⫽4,5 ranges 共and some actual
data points兲 extending beyond the scope of the plot. Vertical arrows illustrate
the possibility of transition between the states in real space 共a兲 or in the
energy scale 共b兲.
association. The dilatation nature of trapped lattice states becomes also apparent in Fig. 1 where the positions of zerostrain minima are marked.25 Analysis for the armchair case
finds a similar picture where the E b (␧) curves must be
shifted toward smaller dilatation values, in order to prolong
the E 0 (␧) crossover to higher strains.
Figure 2共b兲 shows several important regularities. The
range of existence begins earlier, at lower strains, for the
larger defects 共greater b). At lower ␧ their formation energies
⌬E b (␧) are higher, but due to greater slopes the energies of
these ‘‘longer cracks’’ turn into lowest on the larger strain
side. In other words, the lattice trapped states formally
emerge in reverse order, and the shortest of them (b⫽1)
requires the highest tension. We note that by the time the
crossover of b⫽1 state with E 0 (␧) curve occurs, all following crack defects would be lower in energy. This allows for a
simple interpretation of the CNT fracture mechanism: Upon
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J. Chem. Phys., Vol. 118, No. 21, 1 June 2003
Bond breaking in carbon nanotube fracture
9487
FIG. 3. Lattice energy at different levels of applied strain ␧, presented as a function of 共a兲 overall crack size b 共or length l b ⬀b) and 共b兲 a single bond
extension. In 共a兲, a family of modified Griffith crack energy plots 关‘‘dinosaur back’’ function 共Ref. 20兲兴 shows how the lattice traps, absent at zero load (␧
⫽0), emerge as local minima under increasing tension (0⬍␧ ⬘ ⬍␧ ⬙ ⬍␧ ⵮ etc.兲. The virtual defects appear in the order reverse to their size, here b⫽3, 2, or 1
broken bonds. Consistently, the energy plots 共b兲 show how scanning a central bond length yields only the second local minimum at lower overall strain (␧
⫽15.8%) but locates two minima b⫽1, 2 at ␧⫽16.1%.
breaking of one bond, lattice can easily slide down through a
series of lattice-trapped defect states toward global failure, as
suggested by the vertical arrow in Fig. 2共b兲.
The absence of virtual defects in free lattice and the
obtained order of emergence as local minima can be explained in terms of a standard phenomenological model of a
nucleating crack. In the continuum Griffith theory20 the free
energy of the crack is a superposition of its surface energy,
proportional to the cleavage length (⬃l b ), minus elastic relaxation around the crack (⬃⫺␧ 2 l 2b ), while a dilatation effect 共in case of small sample兲 is accounted for by additional
linear term (⬃⫺␧l b ). In addition, an oscillatory component
describes the lattice discreteness yielding an energy plot
known as ‘‘dinosaur back.’’ 20 When a family of such energy
curves at increasing strain levels are plotted, as in Fig. 3共a兲,
one finds no local minima in the unloaded lattice (␧⫽0),
while at increasing strain the gradual appearance of metastable states becomes evident, starting from larger and then
the smaller b-states. To parallel this generic analysis within
our quantum-mechanical model, we have explored in Fig.
3共b兲 the PES of a strained 共10,0兲 CNT as a function of a
central bond elongation. Both energy curves 共at 15.8% and
16.1% strain兲 show presence of b⫽2 metastable states,
whereas the shoulder around 1.8 Å develops into the b⫽1
minimum only at the higher load.
An inspection of Fig. 3共b兲 shows that the barriers for
transition from perfect lattice to b⫽1 and then 2 are quite
insignificant 共10–15 meV shown兲 and thus there is no need
for thermal activation at the failure strain level. This observation brings the discussion to the comparison with the SW
yield. As mentioned above, SW transformation becomes
thermodynamically favorable at lower tension, but retains
the intrinsic activation barriers E * even at the ␧
⫽15%– 20% strain values.19 Therefore, at high temperature
(T) or long time (⌬t→⬁) limit SW mechanism dominates,
whereas in a finite duration test at lower temperature the
dislocation threshold rises 共cf. Refs. 1, 2兲 and the brittle bond
breaking sequence must prevail. At what load-experiment
duration ⌬t or rate ␧˙ is the SW relaxation more likely to
occur? What then is the primary yield event in a typical
laboratory experiment? To answer these questions we assume
an experimental situation in which the applied tension grows
from zero to the critical bond-breaking ␧ c . During this interval, a cumulative probability of SW flip can be expressed
as an integral of the Arrhenius rate
P⫽R
冕 冋
␧c
0
⬃R⌬t
exp ⫺
册
E * 共 ␧, ␹ 兲 d␧
k BT
␧˙
冋
册
E *共 ␧ c , ␹ 兲
k BT
exp ⫺
,
E *共 ␧ c , ␹ 兲
k BT
共1兲
where R⫽1015Ld nm⫺2 s⫺1 . As an example-approximation
we choose a CNT length L⫽1 ␮ m, diameter d⫽1 nm,
chirality ␹ ⫽0, and the classical potential-based activation
energy strain dependence E * (␧)⫽(6 – 28␧) eV. 19 One obtains that in near-zigzag CNTs (␧ c ⫽16.1%) SW becomes
kinetically favorable ( P⬃1) at high T or large ⌬t, e.g., in a
month-long load at room temperature. Therefore, in a room
temperature experiment with a few seconds duration, the
critical ␧ c is being reached without SW rotation and, due to
the negligible bond-breaking barriers, the brittle failure occurs.
The competition between dislocation- and bondbreaking channels can in principle be verified by comparing
MD simulations at high 共where SW rotation occurs12,14兲 and
low T, where particular caution should be observed in describing the interatomic forces 共cf. Ref. 21兲. Simulations
with nonorthogonal tight-binding indeed show dislocation
yield at T⬎2000 K, while at T⬍500 K the lattice undergoes
brittle bond breaking.26 In comparing different temperature
regimes one should keep in mind the refractory nature of
graphite 共in-plane Debye temperature ⬃2500 K, melting
point ⬃4500 K, binding energy 7.4 eV/atom兲, which essentially renders room temperature as ‘‘low.’’
Finally, the association of the found local minima with
broken-bond states becomes more evident from the picture
of electronic charge density 共Fig. 4兲, which shows a lack of
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9488
J. Chem. Phys., Vol. 118, No. 21, 1 June 2003
Dumitrică, Belytscko, and Yakobson
1
FIG. 4. A qualitative change near the bifurcation point is more evident in the
computed electronic density, here shown for a still perfect and for b⫽3
state, both at the same elongation and nearly equal total energies. A visible
thickening of the bonds along the zigzag crack edge reflects the electron
redistribution in this aromatic radical system.
bonding charge around the b⫽3 crack, as opposed to the
perfect strained lattice. The two shown configurations have
nearly equal energies, which underlines the bifurcation nature of the broken-bond states. Also visible there is an increase in electronic charge density at the crack edge, suggesting localization of nonbonding electrons, analogous with the
electronic distribution of open-ended CNTs.27
In summary, we have presented a brittle bond-breaking
CNT failure mechanism, which generally competes with the
extensively discussed11–19 Stone–Wales bond rotation dislocation path. Remarkably, the proposed failure path dominates
at low temperatures, and is more probable to take place in a
tensile-test experiment. The failure mechanism involves a
series of shallow energy minima in the highly strained CNT
lattice which can be associated with broken-bond states.
These states cease to exist in a free lattice, where they are
‘‘virtual.’’ However, at high strains these defects have well
defined structure, and their energy ranking fully overturns at
the failure threshold. The potential barriers between them are
shallow and so the lattice quickly evolves toward the energy
descend, b⫽0→1→2→3→ . . . , therefore giving rise to a
brittle crack and failure. Described lattice-trapped crack
states could be stabilized in bent nanotubes, creating an electronic structure similar to an open-end,27 of interest for field
emission study. The emerging cleave may additionally be
stabilized by termination with other chemical species 共e.g.,
H兲; their presence, conversely, must reduce the breaking
strain—a possible explanation of important sonicationchemical cutting process.28
Work supported by the Office of Naval Research, Air
Force Research Laboratory Materials Directorate, and the
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