The Great Reduction of a Carbon Nanotube’s Mechanical Performance by a Few Topological Defects Liyan Zhu,†,‡,§ Jinlan Wang,*,§,⊥ and Feng Ding*,† †
Institute of Textiles and Clothing, Hong Kong Polytechnic University, Hong Kong, China School of Physics and Electronic & Electrical Engineering and Jiangsu Key Construction Laboratory of Modern Measurement Technology and Intelligent Systems, Huaiyin Normal University, Huai’an, Jiangsu 223300, China § Department of Physics, Southeast University, Nanjing, 211189, China ⊥ Synergetic Innovation Center for Quantum Effects and Applications (SICQEA), Hunan Normal University, Changsha 410081, China ‡
S Supporting Information *
ABSTRACT: It is widely believed that carbon nanotubes (CNTs) can be employed to produce superstrong materials with tensile strengths of up to 50 GPa. Numerous efforts have, however, led to CNT fibers with maximum strengths of only a few GPa. Here we report that, due to different mechanical responses to the tensile loading of disclination topological defects in the CNT walls, a few of these topological defects are able to greatly decrease the strength of the CNTs, by up to an order of magnitude. This study reveals that even nearly perfect CNTs cannot be used to build exceptionally strong materials, and therefore synthesizing flawless CNTs is essential for utilizing the ideal strength of CNTs. KEYWORDS: carbon nanotubes, tensile strength, disclination topological defects, density functional tight binding
C
most promising approaches is to spin large numbers of CNTs into macroscopic fibers, namely, CNT fibers. Since a CNT fiber contains nearly 100% CNTs, its ideal mechanical performance should be very close to that of an individual CNT, as has been proven by the thousands of years of spinning technology used in the production of cotton yarns.14 It was even predicted that CNT fibers should be strong enough to meet the requirement of building a space elevator, where very light, strong cables with tensile strength up to 50−60 GPa are required.15 In addition, CNTs are very stable even at a temperature of 2000 °C, and so a CNT fiber might also maintain its mechanical performance at a very high temperature. Considering that the highest recorded strength of fiber materials is only ∼6.0 GPa,16 such an intriguing potential has stimulated numerous studies and investments into developing CNT fibers.17−21 It is surprising that improvements in CNT fiber strength have been very limited even after more than 10 years of study. Although the highest recorded strength of a very short CNT fiber is ∼9 GPa,17 most values reported are around 1−1.5 GPa or even
arbon nanotubes (CNTs) are a form of two-dimensional graphene rolled up in one-dimensional structures and exhibit extremely high values of Young’s modulus and tensile strength. The effective Young’s modulus of a CNT, almost independent of its chirality or diameter, was both predicted and measured to be ∼1.0 TPa.1−3 In contrast, its values of ideal tensile strength and critical strain, which are ∼100 GPa and ∼20−25%, respectively, are expected to rely on its chiral angle.4 These predictions of impressive mechanical performance have been well supported by many experimental measurements on individual CNTs.5−9 For example, Yu et al. reported an average value of 1002 GPa for the Young’s modulus of SWCNTs through direct tensile loading tests.6 Recently, two groups reported the elongation to failure of ultralong CNTs up to 14%10 and 17%,11 respectively, which is very close to the theoretical limit, ∼20−23%.12 Combining the lightness of carbon atoms, such enormous values of tensile strength and critical strain imply that CNTs might be used to build the strongest material in nature. Their superior mechanical performance makes CNTs one of the most promising materials for many applications.13 While a single CNT is not suitable for most applications because of its small size, technologies to assemble large numbers of CNTs into macroscopic materials are urgently required. One of the © 2016 American Chemical Society
Received: May 16, 2016 Accepted: June 1, 2016 Published: June 2, 2016 6410
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ACS Nano lower.22 This large gap between the theoretical strength and experimental values have been attributed mainly to the existence of numerous CNT ends, poor CNT alignment, and catalyst particles, voids, and irregular entanglements inside the fibers.23 It was predicted that, if these factors could be eliminated, the strength of CNT fibers might be expected to be as high as ∼70 GPa.23 It is important to note that all these considerations and estimations on the mechanical performance of CNT fibers presume that all the CNTs in the fiber have good mechanical properties similar to those of the perfect CNTs. Experimentally, for CNT fibers pulled directly from CNT carpets/forests, when the CNTs are examined, they are found to have numerous defects.24−27 One type of defect that has been extensively studied is a point defect, such as an atomic vacancy in the CNT wall.28 However, systematic investigations demonstrate that point defects can reduce the strength of CNTs to only 60% of that of perfect CNTs.6,29 More importantly, CNTs with reconstructed vacancies have almost the ideal strength of perfect CNTs.29 In this study, we focus on the effect of topological defects in a CNT wall upon the mechanical performance of the tube via a self-consistent charge density functional tight-binding (SCCDFTB) method as implemented in the DFTB+ package.30,31 It is found that, due to different mechanical responses to the tensile loading of disclinations, a few disclination topological defects in the cylindrical hexagonal network may lead to a sharp turn or kink in the tube wall, which will affect the strain distribution around the area of the turn or kink immensely. Such a defect can
greatly reduce the mechanical performance of the CNT by up to 1 order of magnitude.
RESULTS AND DISCUSSION In contrast to point defects, a disclination topological defect, i.e., pentagon and heptagon, in a CNT or graphene wall not only changes the local structure around it but will also affect the overall shape of the CNT or graphene. A pentagon or a heptagon in an sp2 carbon network can turn a flat graphene layer into a cone-like shape or a warped structure, respectively (see Figure S1 in the Supporting Information). Under tensile loading, such a change in the overall shape of the CNT or graphene must lead to an uneven distribution of tensile strain in the CNT/graphene wall. To demonstrate this, we plot the strain distribution in a strained graphene nanoribbon with either a pentagon or a heptagon in the wall (Figure 1a and c). Apparently, the local area around the pentagon is not affected by the tensile stress, and most C−C bonds are free of strain. In contrast, the heptagon serves as a stress concentration center, with all the C−C bonds around it highly stretched. A CNT that is formed by rolling a graphene nanoribbon containing a pentagon and a heptagon (5|7) usually has a sharp turn or kink in the wall. The strain distribution in such a CNT under external loading is presented in Figure 1f. Similarly, the stress concentrates greatly in the area close to the heptagon, whereas the C−C bonds around the pentagon are nearly unaffected. In sharp contrast, the strain distributions in the areas far from the turn, i.e., the two straight segments shown in Figure 1f,
Figure 1. Strain distributions in graphene/CNT with disclination topological defects. Strained graphene nanoribbons with a pentagon (a) and a heptagon (c) under a tensile stress applied along the length direction. The corresponding strain distributions along the blue, black, and red dashed lines are plotted in panels (b) and (d), respectively. Schematics of a SWCNT with one pair of 5|7 defects interposed without (e) and with (f) tensile loading. 6411
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Figure 2. Breaking of a perfect CNT versus a CNT with topological defects. (a) Energy−strain curve as a function of tensile strain in a perfect CNT (11, 0) under tensile loading. The two insets are a ∼9.8% strained CNT (top left) and a fractured CNT (bottom right), respectively. (b) The 5% strained (6, 6)-(11, 0)-35.81° CNT in a supercell of the study. The strains of the C−C bonds along red and black lines shown in (b) are plotted in panel (c). (d) Strain energy and tensile stress versus elongation of a (6, 6)-(11, 0)-35.81° CNT. (e−h) Snapshots of the strained CNT (6, 6)-(11, 0)-35.81° during the fracture process. The pentagon and the heptagon are highlighted as blue and red balls. Snapshots of the fracture process of CNTs with two and three pairs of 5|7 can be seen in Figures S3 and S4 in the Supporting Information.
Hence the area around the pentagon makes no contribution to resisting the external loading. As the applied tensile loading increases, the bonds of the tube near the topological defects break in a sequential manner. A crack is initiated first by breaking two side bonds of the heptagon and then gradually propagates from the heptagon to the pentagon step by step (see Figure 2e → f → g → h). Figure 2d presents the strain energy and stress versus tensile strain for a (6, 6)-(11, 0)-35.81° CNT. In the beginning, the strain energy increases with tensile strain in a parabolic way. A pair of C−C bonds of the heptagon (Figure 2e) then breaks at ∼7% tensile strain. After the rupture of the two C−C bonds, the C−C bonds in the vicinity of the crack are able to relax and the accumulated strain energy is released as seen in Figure 2d. Then, more strain energy gradually builds up and leads to the rupture of another two C−C bonds on both sides of the crack. Such a procedure is repeated several times until the whole CNT breaks into two fragments. The stress required to break a pair of C−C bonds decreases almost linearly during the whole fracture process. In other words, the stress required to break the first two C−C bonds determines the ultimate strength of a CNT with topological defects present in the wall. The failure stress as a function of the turning or kink angle and diameter is presented in Figure 3a and the inset therein. It can be clearly seen that there is a remarkable decrease in failure stress as the turning angle increases. The failure stress of a perfect CNT is ∼100 GPa, which is close to the frequently reported theoretical value.4 If the CNT wall has one adjacent pentagon−heptagon pair along its axial direction, it can maintain its straight shape; that is, the turning angle is 0°. The failure stress of such CNTs is roughly ∼90.0 GPa, which is lower by only 10% from that of the
are very uniform. Because the external load is evenly distributed, the strains of the bonds in straight segments of a CNT are much lower than those around the heptagon. This implies that the sharp turn is a weak point in the CNT. With one pentagon and one heptagon incorporated into the CNT, the turning angle (θ shown in Figure 1e) varies between 0 and ∼30 degrees, depending upon the relative spacing between the pentagon and heptagon. More pentagons and heptagons in the wall can lead to larger turning angles. For example, a relaxed CNT with a turning angle between ∼30 and ∼60 degrees contains at least two pairs of 5|7, and three pairs of 5|7 can lead to a large turning angle from ∼60 to ∼90 deg (see Figure S2 in the Supporting Information). For simplicity, we use the notation (n1, m1)-(n2, m2)-θ to denote a CNT with a few disclination topological defects, where (n1, m1) and (n2, m2) are the chirality indices of the straight CNT segments and θ is the turning angle due to the presence of pentagons and heptagons in the CNT wall. Due to the high symmetry of the perfect CNT, when under tensile loading, all of the bonds along equivalent orientations in the wall are uniformly strained. As shown in Figure 2a, the bonds that are parallel to the tube axis in a perfect CNT (8, 0) are evenly stretched. Therefore, to fracture a perfect CNT, one has to break all of the bonds around a circle simultaneously, which corresponds to a massive tensile stress of ∼100 GPa and a very high critical strain (>20%). Figure 2b depicts the geometry of a (6, 6)-(11, 0)-35.81° tube under a 5% tensile strain. During the elongation process, the strain is nearly uniformly distributed in the two straight segments of the tube. A strain gradient, however, is seen along the circle going from the heptagon to the pentagon. Stress is highly concentrated in the area around the heptagon, but the C−C bonds close to the pentagon are nearly intact. 6412
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Figure 3. Greatly reduced mechanical properties of CNTs with a few topological defects. The failure stress, σ (a), of CNT as a function of the turning angle of the CNT wall and the diameter of the CNT (inset). Equivalent breaking strain, ε (b), and toughness, T (c), as a function of turning angle and diameter (inset). Dashed dark green curves in panels (b) and (c) correspond to the fitting curve for equivalent breaking strain and toughness: ε = 36.55 × θ−0.57 and T = 43.72 × θ−1, respectively. (d) Strain distribution in the area around the turn in the (6, 6)-(11, 0)-35.81° CNT (red dashed line) and (6, 6)-(11, 0)-92.53° CNT (blue dashed line) under a tensile stress of 20 GPa.
perfect CNTs. When the turning angle further increases up to ∼30°, ∼60°, or ∼90°, the tensile stress drops quickly to ∼45, ∼35, and ∼25 GPa, respectively. It is important to note that the failure stress is not sensitive to the change of the diameter of the CNT, as shown in the inset of Figure 3a. For CNTs containing one pair of 5|7 defects in the wall and with a turning angle of ∼35°, the failure stress decreases slowly as the diameter increases (see the solid red circles in the inset of Figure 3a). Figure 3b presents the critical strain of the CNTs with a few pairs of disclination topological defects. In agreement with many previous studies, a perfect CNT breaks at a critical strain of >20%. With one pair of 5|7 in the wall, the critical strain will be drastically reduced to only ∼12%. This shows the significant impact of topological defects on the mechanical performance of CNTs. Recently, millimeter-sized CNTs have been reported to elongate up to 15% or higher.11 So, we can conclude that these CNTs observed experimentally must be topological defect free. Similar to the failure stress, the critical failure strain of CNTs undergoes a drastic reduction when the turning angle is increased (see Figure 3b). For example, the failure strain of a CNT with a 30° turn in the wall is only ∼5%. Toughness, defined as the energy required to break a CNT, which can be roughly estimated as the product of the failure stress and critical strain, must undergo a more drastic reduction when the turning angle becomes larger. Figure 3c shows the toughness of the CNT as a function of the turning angle. Even for the CNTs with zero tuning angles, the toughness is greatly reduced by ∼60% if one pair of 5|7 defects is interposed. For CNTs with large turning angles, their toughness is only a small fraction of that of the perfect ones. As toughness is critical for many applications in strong materials, we have to pay particular attention to the effects of topological defects in real applications. To have an insightful understanding of why the failure stress strongly depends on the turning or kink angle but is insensitive to
the variation of the diameter, we take (6, 6)-(11, 0)-35.81° and (6, 6)-(11, 0)-92.53° CNTs, whose diameters are the same, as examples to have a close look at the process of the failure. Under a tensile stress of 20 GPa (Figure 3d), the C−C bonds on both sides of the heptagons of the (6, 6)-(11, 0)-35.81° and (6, 6)-(11, 0)-92.53° CNTs are elongated by 5% and 24%, respectively, which is due to the stress concentration in a CNT with larger turning angle (Figure 3d). Therefore, the failure stress will undergo a remarkable decrease with increasing turning angle. To understand why the failure stress depends weakly on the CNT diameter, we plot the average strain distribution of three CNTs, (3, 3)-(5, 0)-34.78°, (5, 5)-(9, 0)-35.96°, and (7, 7)-(13, 0)-35.81°, with a similar turning angle of ∼35 degrees in Figure 4. The diameters of the three CNTs are 4.11, 6.93, and
Figure 4. Distribution of normalized strain in CNT (3, 3)-(5, 0)34.78°, (5, 5)-(9, 0)-35.96°, and (7, 7)-(13, 0)-35.81° under a tensile stress of 20 GPa. The carbon atom is colored by the average strain of its associated three bonds.
9.70 Å, respectively. The strain distributions are very similar in all three samples. As the load under at the first C−C bond around the heptagon of the tube controls the failure stress, the similar strain distribution around the heptagon indicates a similar failure stress. For CNTs with smaller diameters, the pentagon is closer to the heptagon, which can effectively compensate the concentration of stress around the heptagon. Therefore, the higher separation between the pentagon and heptagon in larger CNTs 6413
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Figure 5. Breaking process of a CNT fiber by atomic simulation. (a) CNT fiber model consisting of two CNTs, (8, 0)-(9, 0)-0° (green) and CNT (5, 5)-(9, 0)-35.96° (blue). (b) Strain energy and tensile stress versus strain of the CNT fiber. (b) Length of the longest C−C bonds in the heptagon of the (5,5)-(9,0) CNT versus strain. Snapshots of the CNT fiber before (d) and after (e) the fracture of the (5,5)-(9,0) CNT.
will follow easily because all the tensile load will be acting on one CNT. So we can draw a conclusion that the tensile strength of a defective CNT in the CNT fiber behaves very similarly to that of a freestanding one and the weakest CNT in the fiber plays a crucial role in the failure process of CNT. To conclude, our study demonstrates that just a few pairs of topological defects in the wall of a CNT can result in a dramatic decrease in the tensile strength of the CNTs. In a CNT wall, the area around a pentagon is nearly free of stress, while the area around a heptagon is subjected to a high stress concentration. Therefore, a CNT with well-separated pentagons and heptagons presents a sharp turn in the wall, and the tensile loading is concentrated mostly in the vicinity of the heptagon. Due to the strain concentration effect, the critical strain of a CNT with topological defects will be greatly reduced by more than 50%, down to ∼12% or less, and the larger the turning angles, the higher the failure stress/critical strain reduction. For CNTs with topological defects and turning angles of ∼0°, ∼30°, ∼60°, and ∼90°, their values of failure stress/critical strain/toughness are only 90%/50%/40%, 50%/20%/6%, 35%/12%/3%, and 22%/10%/1.7% of those of a perfect CNT, respectively. Such a large reduction of the mechanical performance of CNTs caused by the presence of a few topological defects indicates that highly perfect CNTs are required for extreme applications that utilize the exceptional strength of CNTs.
makes the failure stress a little less. Actually, this kind of compensative interaction between pentagon and heptagon has already been observed in graphene, which enhances the strength of graphene with grain boundaries with increasing tilt angles.32 In the above discussions, our focus is placed on mechanical performances of a single CNT only. To connect our results with realistic materials, e.g., CNT fibers, we further investigated the effect of topological defects on the tensile strength of CNT fibers/bundles. Two CNTs with different turning angles, (8, 0)-(9, 0)-0° and (5, 5)-(9, 0)-35.96°, are selected and were spun into a smallest CNT fiber (see Figure 5a). It can be clearly seen that the strain energy of the fiber increases with the strain in a parabolic manner (see Figure 5b), which is similar to a single CNT. At the strain of 8.27%, a sudden drop in strain energy indicates the fracture of the fiber. Figure 5d and e show snapshots of the structure of the fiber before and after fracture, respectively. The fiber fracture occurs at the CNT with a larger turning angle, namely, CNT (5, 5)-(9, 0)-35.96°. To gain further insights into why the CNT with larger turning angle breaks first, we plot the largest bond length of C−C bonds in the heptagon in both CNTs since the heptagon is the weakest point during CNT elongation. As shown in Figure 5c, the largest bond length of C−C bonds in the heptagon in CNT (8, 0)-(9, 0)-0 increases with the strain smoothly and slowly. However, the largest bond length of C−C bonds in the heptagon in CNT (5, 5)-(9, 0)-35.96° suddenly increased from 1.58 Å to 1.89 Å at the strain of 5.5%. This suggests that the concentrated stress around the vicinity of the heptagon in CNT (5, 5)-(9, 0)-35.96° leads to a bond fracture. As a result, the fracture of the CNT certainly begins from CNT (5, 5)-(9, 0)-35.96°, and the complete fracture of the CNT occurs at a strain of 8.27%. Moreover, the fracture of the (5, 5)-(9,0)-35.96° CNT in the fiber is a brittle process, which is different from that of a single CNT (see Figure 2e−h). This is due to the lowered structural symmetry of a twisted CNT in the fiber. The failure strain of the fiber is close to that of the single CNT (5, 5)-(9, 0)-35.96°, i.e., 8.63%. However, the failure stress of the fiber, 62 GPa, is slightly larger than that of a single CNT (5, 5)-(9, 0)-35.96°, 52 GPa. After the breakage of the first CNT, the failure of the another one
METHODS All calculations were performed using the SCC-DFTB method as implemented in the DFTB+ package.30,31 Sufficient vacuum space (at least 15 Å) is kept in the x and y directions to make the interaction between two adjacent images negligible. Geometry optimization was carried out using a conjugate gradient algorithm until force on each ion was smaller than 1.0 × 10−4 eV/Å. The convergence criterion of energy for the self-consistent-field calculation was chosen to be 1.0 × 10−5 eV. Convergence tests on k-point sampling reveal that the 1 × 1 × 1 Monkhorst−Pack grid is adequate. The structure was gradually stretched in a minimization step of 0.1 Å along the z direction. The failure stress (σ) is defined as
σ= 6414
Fmax πDh
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ACS Nano where Fmax is the maximum force to make the CNT rupture; D is the diameter of the relatively smaller constituent CNT, although the diameters of two constituent tubes are chosen to be as close as possible; and h is the wall thickness of the CNT, chosen to be 3.4 Å. To validate the above method, we calculate the failure stress of perfect CNTs. The calculated results are 101.82 and 105.88 GPa for perfect CNTs (8, 0) and (7, 7), respectively, which are in good agreement with the most frequently reported values.1,4
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.6b03231. Failure stress of CNTs with and without topological defects; snapshots of a strained graphene nanoribbon with an isolate pentagon; geometries of CNTs with a few pairs of 5|7 interposed; snapshots of the fracture process of CNTs with two and three pairs of 5|7 (PDF)
AUTHOR INFORMATION Corresponding Authors
*E-mail (J. Wang):
[email protected]. *E-mail (F. Ding):
[email protected]. Notes
The authors declare no competing financial interest.
ACKNOWLEDGMENTS This work was supported by the NSFC (21525311, 21173040, 21373045, 11504122) and by Jiangsu (BK20130016) and SRFDP (2013009, 2110029) in China. REFERENCES (1) Yakobson, B. I.; Brabec, C. J.; Bernholc, J. Nanomechanics of Carbon Tubes: Instabilities Beyond Linear Response. Phys. Rev. Lett. 1996, 76, 2511. (2) Lu, J. P. Elastic Properties of Carbon Nanotubes and Nanoropes. Phys. Rev. Lett. 1997, 79, 1297. (3) Hernandez, E.; Goze, C.; Bernier, P.; Rubio, A. Elastic Properties of C and BxCyNz Composite Nanotubes. Phys. Rev. Lett. 1998, 80, 4502. (4) Belytschko, T.; Xiao, S. P.; Schatz, G. C.; Ruoff, R. S. Atomistic Simulations of Nanotube Fracture. Phys. Rev. B: Condens. Matter Mater. Phys. 2002, 65, 235430. (5) Lourie, O.; Wagner, H. D. Evaluation of Young’s Modulus of Carbon Nanotubes by Micro-Raman Spectroscopy. J. Mater. Res. 1998, 13, 2418−2422. (6) Yu, M.-F.; Files, B. S.; Arepalli, S.; Ruoff, R. S. Tensile Loading of Ropes of Single Wall Carbon Nanotubes and Their Mechanical Properties. Phys. Rev. Lett. 2000, 84, 5552. (7) Yu, M.-F.; Lourie, O.; Dyer, M. J.; Moloni, K.; Kelly, T. F.; Ruoff, R. S. Strength and Breaking Mechanism of Multiwalled Carbon Nanotubes under Tensile Load. Science 2000, 287, 637−640. (8) Wong, E. W.; Sheehan, P. E.; Lieber, C. M. Nanobeam Mechanics: Elasticity, Strength, and Toughness of Nanorods and Nanotubes. Science 1997, 277, 1971−1975. (9) Treacy, M. M. J.; Ebbesen, T. W.; Gibson, J. M. Exceptionally High Young’s Modulus Observed for Individual Carbon Nanotubes. Nature 1996, 381, 678−680. (10) Chang, C.-C.; Hsu, I. K.; Aykol, M.; Hung, W.-H.; Chen, C.-C.; Cronin, S. B. A New Lower Limit for the Ultimate Breaking Strain of Carbon Nanotubes. ACS Nano 2010, 4, 5095−5100. (11) Zhang, R.; Wen, Q.; Qian, W.; Su, D. S.; Zhang, Q.; Wei, F. Superstrong Ultralong Carbon Nanotubes for Mechanical Energy Storage. Adv. Mater. 2011, 23, 3387−3391. (12) Dumitrica, T.; Hua, M.; Yakobson, B. I. Symmetry-, Time-, and Temperature-Dependent Strength of Carbon Nanotubes. Proc. Natl. Acad. Sci. U. S. A. 2006, 103, 6105−6109. 6415
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