NANO LETTERS
A Kinetic Model for Time-Dependent Fracture of Carbon Nanotubes
2004 Vol. 4, No. 6 1139-1142
Tan Xiao† School of Electrical and Electronic Engineering, Nanyang Technological UniVersity, Singapore 639798
Yu Ren and Kin Liao* School of Mechanical and Production Engineering, Nanyang Technological UniVersity, Singapore 639798 Received February 17, 2004
ABSTRACT Based on the classical kinetic concept of solid fracture and a strain concentration concept, a model is proposed for predicting time-dependent fracture of carbon nanotubes. The time-dependent fracture behavior of a zigzag type single-walled carbon nanotube with a range of preexisting cracks under tension is studied by molecular mechanics simulations and a numerical scheme using crack front strain energy concentration. Results of the study quantitatively agree with a recent study on fatigue of aligned single-walled carbon nanotube bundles. It is found that the coefficient of strain energy concentration increases as a crack grows and the time-to-failure of the carbon nanotube is dominated by the lifetimes of a few bonds after initial bond dissociation when load is large while more bonds contribute to the overall lifetime when applied load is small, resulting in the logarithm of time-to-failure of carbon nanotubes being approximately linearly related to applied stress.
Harnessing the exceptional mechanical properties of carbon nanotubes (CNTs) for high performance composites is one of the most anticipated applications for this new class of nanosized reinforcement materials. As a novel ultrastrong reinforcement for nanocomposites, CNTs have already drawn numerous studies on their elastic and plastic behaviors in recent years,1 and there have been increasing interests in studying the phenomenon of CNT fracture. Yakobson et al.2 studied the tensile fracture and C-chain unraveling in CNTs by molecular dynamics (MD) simulations, suggesting that CNTs have an extremely large breaking strain (>30%). Contrastively, Belytschko et al.3 argued, also based on results of MD simulations, that CNT fracture depends primarily on the inflection point on the curve of interatomic potential of the C-C bond and predicted that their fracture strain ranged between 10 and 19%. So far, predictions of CNT failure, in which the atomic structure is always assumed defect-free at the beginning, are limited to yielding and fracture behaviors at relatively high strain levels (>5%) and hypothesizing the onset and propagation of defects such as the 5/7/7/5 type4,5 or the diamond-like bonding transitions,6 and C-C bond breakage.2 All these hypotheses apply to transient behaviors of CNTs under critical conditions. However, in actual applications CNTs will most likely be loaded at much lower * Corresponding author. E-mail:
[email protected]. † Present address: Department of Physics, Shantou University, Shantou 515063, P.R. China. 10.1021/nl049731d CCC: $27.50 Published on Web 05/06/2004
© 2004 American Chemical Society
strains. A recent study on the fatigue behavior of singlewalled carbon nanotube (SWNT) bundles indicates that SWNT fractures do occur after sustaining sufficient number of load cycles, at low strain (stress) levels.7 Apparently, the 5/7/7/5 type and the diamond-like defects are high-strain preferred5,6 and will not arise at low strains. A new model is proposed herewith to predict the time-dependent fracture of CNTs under tensile load within the range of strains anticipated in structural applications. Zhurkov8 had examined time-to-failure of a broad range of solid materials under static fatigue and propounded an empirical equation relating lifetime, τ, to applied stress, σ, and temperature, T, of the solid, on the basis of the kinetic concept of fracture,9 τ ) τ0 exp
(
)
U0 - γσ kT
(1)
where k is the Boltzmann constant, τ0 the reciprocal of the natural oscillation frequency of atoms in the solid, U0 the atomic binding energy, and γ is related to the disorientation of the molecular structure of a solid. It is indicated by eq 1 “the existence of a direct connection between the kinetics of fracture of solids under the action of mechanical stress and the rupture of interatomic bonds”.8 A similar equation was also proposed to represent the lifetime of receptorligand bonds in a model for specific cell-cell adhesion10
Figure 1. Schematic diagram of cracking modes of a zigzag-type carbon nanotube. The number of broken bonds is denoted by parameter i. The crack front C-C bond with strain energy ui is represented in bold.
and for predicting CNT yielding behavior.11 These examples have manifested the potential validity of the kinetic concept for describing bond fracture of CNTs. Based on the aforementioned kinetic fracture concept, we suppose that a CNT will rupture along the direction where the C-C bond is having comparatively larger strain energy, as a result of the atomic thermal motion and the preference of bond dissociation under strain, although the stress it is experiencing may be smaller than the critical value for rupture. The process of CNT fracture is described in a statistical scheme where the probability of bond breakage is only determined by two parameters, the energy gap of the bonds before and after the dissociation and temperature. To illustrate the idea, we consider the time-dependent fracture of a zigzag-type SWNT subject to axial tension, as shown in Figure 1, where the processes of onset and propagation of an atomic-sized crack (shaded region) in the nanotube are illustrated. The two C-C bonds at the crack front (represented in bold) oriented parallel to the axis are sustaining greatest strain. We identify each of the discrete crack geometries with i, the number of broken C-C bonds within the crack, and denote u0 and ui as the strain energy associated with the C-C bond of an intact nanotube, and an ith mode crack front C-C bond oriented parallel to the tube axis, respectively. In light of Zhurkov’s model, we postulate that the lifetime, ti, of a strained ith mode crack front C-C bond in a CNT is predicted by ti ) τ0 exp
(
)
U0 - ui kT
(2)
where U0 is the bond dissociation energy and ui represents the strain energy of the bond near absolute zero before dissociation, such that U0 - ui is the energy barrier. Belytschko et al.3 suggested that the fracture behavior of CNTs is almost independent of the atomic binding energy like the U0 in eq 1, but depends on the inflection point on the C-C bond potential energy versus strain curve, implying that fracture of the nanotube starts once the applied tensile force reaches its peak value, so the U0 in eq 2 is the corresponding bond dissociation energy at the inflection point. Briefly, the interpretation of eq 2 is that at certain temperature, T, two bound atoms are in random thermal motion described by the Maxwell-Boltzmann distribution. It is probable, at some instance, when the atoms acquire 1140
Figure 2. Total energy change per atom, Ei, of a (18, 0) zigzag carbon nanotube versus axial tensile strain, . Circles are simulation results and dashed line and solid lines are the regression curves by eq 3a for intact nanotubes and for nanotubes with a crack, respectively. Only the results of the first five cracking modes are shown.
sufficient kinetic energy that the energy barrier is overcome, resulting in bond dissociation. It is manifested in eq 2 that C-C bond breakage of CNTs depends on thermal motion of the atom (in terms of temperature T) and on tensile force (in term of strain energy ui), which lowers the energy barrier. Apparently the bonds at the crack front (Figure 1) have the highest ui and thus the lowest energy barrier to be overcome by thermal fluctuation for dissociation to occur. At present, direct experimental determination of ui is not possible due to CNTs’ extremely small size, although tensile experiments have been performed on individual multiwalled CNT (MWNT) and SWNT bundles.12,13 Alternatively, the strain energy can be estimated through molecular mechanics (MM) calculations. Different from the previous studies4-6 in which detailed processes of bond dissociation were analyzed using MD methods, the MM method is sufficient in this study for the calculation of energy state before and after bond breakage. In Figure 2, simulation results of the total strain energy per atom of a zigzag (18, 0) SWNT with discrete cracks for the first five cracking modes (i ) 0-4) subjected to tensile load is shown.14 It is seen that the slopes of the curves gradually decrease with an increase of the number of broken C-C bonds, suggesting softening of the nanotube during the fracture process. The energies of a single crack front C-C bond versus the strain of the nanotube for the first five cracking modes are shown in Figure 3. On the abscissa, the strain of the nanotube is used instead of the strain of the C-C bond for the purpose of direct comparison with results shown in Figure 2. It is seen that the strain energy of the crack front C-C bond (solid lines) is greater than that of the intact nanotube (dashed line). Moreover, it increases from one mode to the next as more C-C bonds are being broken within the crack, and progressively approaches a limit, indicating a loss of sensitivity as the crack grows. Owing to the discrete atomic structures of CNTs, we propose “strain energy concentration” for CNT fracture analysis. The strain energies, Ei, for the nanotube and ui for the crack front C-C bond in Figures 2 Nano Lett., Vol. 4, No. 6, 2004
Figure 3. Energy change, ui, of a crack front C-C bond in a (18, 0) carbon nanotube versus axial tensile strain, . Circles are simulation results and dashed and solid lines are the regression curves by eq 3b for intact nanotubes and nanotubes with a crack, respectively. Only the results of the first five cracking modes are shown.
and 3 can be approximately fitted into quadratic functions at small strains,15 such that Ei ) ai2, ui ) bi2
(3a,b)
The tensile force per unit length (of the nanotube circumference), Ti, applied at the ends of the CNT is the first-order derivative of the strain energy Ti )
∂Ei ) 2ai ∂
(4)
For SWNTs, different cross-sectional areas have been adopted in the literature for calculations due to a lack of translational invariance in the radial direction.16-18 Taking into account CNT bundles as solid ensembles in a composite matrix, we approximate the cross-sectional area, A, of a CNT by π(r + h)2, in which r is the radius of the nanotube and 2h the gap between two adjacent nanotubes. Using eq 4, the stress at the end of the nanotube is
Figure 4. Time-to-failure of (18, 0) carbon nanotubes versus applied stress, σ. Dashed lines are lifetime curves of crack front C-C bond; solid lines are time-to-failure curves of intact nanotubes (i ) 0) and of nanotubes with preexisting flaws (i > 0). Only the results of the first six cracking modes are shown. Solid circles are results of fatigue experiment from ref 7.
local strain energy distribution as a result of a flaw in the CNT. With γi the lifetime, ti, of a bond and the time-tofailure of a nanotube can be determined. Of course, the path of crack propagation and morphology of fracture are dependent on the chirality of a CNT because of the isotropic nature of graphene structure from which it forms. Multiple crack branches developed from a common origin are found on a graphene sheet when it is loaded along the direction perpendicular to C-C bonds, based on molecular simulations,19 thus the complex fracture mode of an armchair nanotube renders the analysis of its time-dependent behavior more complicated and will be dealt with in a subsequent paper. For simplicity we consider a zigzag-type nanotube (18, 0). MD simulations reveal that a crack expands along the circumference of a zigzag SWNT perpendicular to the loading direction, thus the time-to-failure of an intact nanotube, t, is the summation of the lifetime, ti, of each individual C-C bond around its circumference 17
σ)
2πrTi 4air ) A (r + h)2
(5)
Substituting eq 5 into eq 3b, the strain energy of the bond is ui ) γiσ2
(6)
where γi is the coefficient of strain energy concentration and it reads γi )
bi(r + h)4 16ai2r2
(7)
Here, γi is a function of the cracking mode, radial dimension, and chirality of the nanotube. It characterizes the nonuniform Nano Lett., Vol. 4, No. 6, 2004
t)
17
ti ) ∑τ0 exp ∑ i)0 i)0
(
)
U0 - γiσ2 kT
(8)
In the present case there should be 18 C-C bonds to be broken for complete separation. According to ref 3, the strain at the inflection point on the curve of the potential energy of C-C bond, which is believed to be the critical point for bond dissociation, is 19%, and the corresponding bond dissociation energy, U0, is 1.84 × 10-19 J. The reciprocal of the fluctuation frequency of atoms, τ0, is usually taken as 10-13 sec, the Boltzmann constant k is 1.38 × 10-23 J/K, and room temperature is assumed to be 300 K. Using r of 1.4 nm, 2h of 0.34 nm (the representative distance between two adjacent graphite layers20), and simulation results of ai and bi in eq 7, γi are obtained for a range of crack sizes. The increase of γi as the crack grows leads to a rapid decline of C-C bond lifetime, ti, in eq 8. The results of time-to1141
failure are shown in Figure 4: the dashed lines are lifetime curves of crack front C-C bonds with i broken bonds within a crack (i ) 0-5) and solid lines are time-to-failure curves of a nanotube with a crack of the ith mode (i ) 0 to 5). Thus time-to-failure for a zigzag (18, 0) SWNT with a preexisted defect of ith mode is the summation from ti to t17, according to eq 8, i.e., each solid line in Figure 4 is the superposition of the dashed lines underneath. It is seen that the ith solid curve (of the nanotube) is very close to the dashed curve of ti when the stress is relatively large, implying that the time-to-fracture of a nanotube is dominated by the lifetimes of a few C-C bonds after initial bond dissociation. When the stress is low, more bonds contribute their lifetime to the total time-to-failure of the nanotube, resulting in a flattening of the solid curve, so that the stress versus logarithm of time-to-failure curve is approximately linear. At the moment, experimental data on time-dependent fracture (static fatigue) of individual CNTs are not available for direct comparison with our proposed life prediction model. The closest data to date is that by Ren et al.7 on fatigue of aligned SWNT bundles embedded in epoxy matrix under cyclic tensile load, and the stress-life data of SWNTs from that study, indicated by solid circles, is included in Figure 4 for comparison. Here we do not intend to address issues such as frequency effect, SWNT size effect, and multiple SWNT fractures, which will be dealt with in subsequent studies, although it is realized that all these factors do exercise influence, to various extents, on the lifetime of a material undergoing long-term loading. Nonetheless, the data from Ren et al.7 do fall within the range of predications of the current kinetic model. It is seen from Figure 4 that experimental data are scattered in a zone where the SWNTs are with preexisting cracks with from 1 to 5 broken C-C bonds. Intact SWNT (i ) 0) has much longer lifetime than that of the fatigue-tested specimens at a specific load level. As the length of the specimen is 40 mm or so, the embedded SWNT bundle very likely contains preexisting flaws.7 Experimental observations21,22 reveal that the raw CNTs may contain numerous defects on their walls. A level of 5% surface defect site density was measured23 in purified SWNTs even after the samples were pretreated to 1273 K. Although defects can be annealed by graphitization at higher temperatures,24 few defects still remain, especially in long CNTs produced for material reinforcement purpose. These defects, including atomic vacancies and dislocations, will result in discontinuity and nonuniformity of strain energy transfer and thus influence the fatigue strength of the specimen. Additinoally, the matrix material itself may have some influence on the fatigue performance of SWNTs. For instance, Zhou and Shi25 have examined the mechanical properties of SWNTs and MWNTs with or without hydrogen storage by MD simulations. Their results showed that the strength of CNTs is reduced by the presence of hydrogen molecules because H-C bonds tend to replace C-C bonds in the hydrogen environment. Despite these aforementioned
1142
complications, the present simple model seems to give reasonable predictions for the available data. In this proposed model, the only parameters that need to be determined are ai and bi from molecular simulations. For CNTs with limited number of defects or crack geometry, a comprehensive lifeprediction map could be developed in the future. In summary, we have proposed a life prediction scheme for SWNTs under time-dependent loading, based on the concept of bond strain energy concentration and the classical kinetic concept of bond fracture. Although only a specific case (zigzag-type SWNT) is analyzed, the approach can be applied to CNTs with different chiralities and with multiple walls, or to other nanostructures, provided that the path of crack propagation is deterministic. Only several discrete cracking modes are expected to dominate the lifetime of a nanotube because the crack develops in an accelerative manner and the subsequent modes have less effect on the overall accumulated times for crack propagation. Also, the numerical results suggest an approximate linear relationship between the logarithm of time-to-failure and the applied stress for SWNTs. References (1) Yakobson, B. I.; Avouris, Ph. Top. Appl. Phys. 2001, 80, 287. (2) Yakobson, B. I.; Campbell, M. P.; Brabec, C. J.; Bernholc, J. Comput. Mater. Sci. 1997, 8, 341. (3) Belytschko, T.; Xiao, S. P.; Schatz, G. C.; Ruoff, R. S. Phys. ReV. B 2002, 65, 235430. (4) Zhang, P.; Lammert, P. E.; Crespi, V. H. Phys. ReV. Lett. 1998, 81, 5346. (5) Nardelli, M. B.; Yakobson, B. I.; Bernholc, J. Phys. ReV. B 1998, 57, R4277. (6) Srivastava, D.; Menon, M.; Cho, K. Phys. ReV. Lett. 1999, 83, 2973. (7) Ren, Y.; Li, F.; Cheng, H. M.; Liao, K. Carbon 2003, 77, 2177. (8) Zhurkov, S. N. Int. J. Fracture Mechanics 1965, 1, 311. (9) Tobolsky, A.; Eyring, H. J. Chem. Phys. 1943, 11, 125. (10) Bell, G. I. Science 1978, 200, 618. (11) Samsonidze, G. G.; Yakobson, B. I. Phys. ReV. Lett. 2002, 88, 065501. (12) Yu, M. F.; Lourie, O.; Dyer, M.; Moloni, K.; Kelly, T.; Ruoff, R. S. Science 2002, 287, 637. (13) Yu, M. F.; Files, B. S.; Arepalli, S.; Ruoff, R. S. Phys. ReV. Lett. 2000, 84, 5552. (14) Hyperchem, the molecular modeling software developed by Hypercube, Inc. (15) Xiao, T.; Liao, K. Phys. ReV. B 2002, 66, 153407. (16) Yakobson, B. I.; Brabec, C. J.; Bernholc, J. Phys. ReV. Lett. 1996, 76, 2511. (17) Lu, J. P. Phys. ReV. Lett. 1997, 79, 1297. (18) Corwell, C. F.; Wille, L. T. Solid State Commun. 1997, 101, 555. (19) Omeltchenko, A.; Yu, J.; Kalia, R. K.; Vashishta, P. Phys. ReV. Lett. 1977, 78, 2148. (20) Physics of Graphite; Kelly, B. T.; Applied Science: London, 1981; p 3. (21) Andrew, R.; Jacques, D.; Qian, D.; Dickey, E. C. Carbon 2001, 39, 1681. (22) Chiang, I. W.; Brinson, B. E.; Huang, A. Y.; Willis, P. A.; Bronikowski, M. J.; Margrave, J. L.; Smalley, R. E.; Hauge, R. H. J. Phys. Chem. B 2001, 105, 8297. (23) Mawhinney, D. B.; Naumenko, V.; Kuznetsova, A.; Yates, J. T., Jr.; Liu, J.; Smalley, R. E. Chem. Phys. Lett. 2000, 324, 213. (24) Bom, D.; Andrews, R.; Jacques, D.; Anthony, J.; Chen, B.; Meier, M. S.; Selegue, J. P. Nano Lett. 2002, 2, 615. (25) Zhou, L. G.; Shi, S. Q. Philos. Mag. A 2002, 82, 3201.
NL049731D
Nano Lett., Vol. 4, No. 6, 2004
