Multiscale-Failure Criteria of Carbon Nanotube Systems Under Biaxial Tension–Torsion

IOP PUBLISHING NANOTECHNOLOGY Nanotechnology 18 (2007) 485715 (7pp) doi:10.1088/0957-4484/18/48/485715 Multiscale-failure criteria of carbon nanotube systems under biaxial tension–torsion

Byeong-Woo Jeong1,2, Jang-Keun Lim3 and Susan B Sinnott1,4

1 Department of Materials Science and Engineering, University of Florida, Gainesville, FL 32611-6400, USA 2 School of Mechanical Engineering, Yeungjin College, Taegu 702-721, Korea 3 Department of Mechanical Design and Production Engineering, Hanyang University, Seoul 133-791, Korea

E-mail: [email protected]ﬂ.edu

Received 18 July 2007, in final form 8 October 2007 Published 1 November 2007 Online at stacks.iop.org/Nano/18/485715 Abstract The failure criteria for carbon nanotube system fracture under biaxial tensileÐtorsional loads are developed based on a multiscale approach that adopts continuum mechanics models to describe atomistic predictions of failure from molecular dynamics simulations. The failure strength or envelope of carbon nanotube systems under this type of loading is significantly different from what occurs under uniaxial tensile loading and, importantly, is different from the predictions of failure criteria for macroscopic objects. The failure criteria developed here can be used to design carbon nanotube-based devices and materials, such as nanoelectromechanical systems and nanocomposites, which undergo biaxial tensileÐtorsional loading. (Some figures in this article are in colour only in the electronic version)

1. Introduction biaxial tensile and torsional loading, especially concerning the details of failure responses or failure criteria. The unique electrical and mechanical properties of carbon When stress is applied to a part in a uniaxial manner, nanotubes (CNTs) have attracted considerable interest and stress and strength can be compared directly to estimate sparked discussion of their potential use in applications such whether or not the part will fail. This comparison is relatively as nanoelectromechanical systems (NEMS) [1] and nanotube simple because there is only one value of stress and strength. composite materials [2]. In these and similar applications, However, the problem becomes more complex when the stress biaxial tensile and torsional loads on the CNTs are widely state is multiaxial. In such cases there are a multitude expected to occur. For instance, CNTs may be used as drive of stresses but only one signiﬁcant value for strength, and shafts [3], torsion bar springs [4], and torsional actuators [5] this requires that failure be characterized using multiaxial that can experience torsion as well as tension. Thus, strength (or failure) criteria [15]. Under multiaxial loading understanding the mechanical responses of CNTs undergoing conditions, the details of failure at the micromechanical and this type of loading is important in optimizing their use in new nanomechanical levels are so incomplete that the failure materials and devices. While numerous studies have examined process cannot be followed analytically [15]. Thus, failure some aspect of the mechanical responses of CNTs, such criteria for macroscopic objects have evolved from attempts as their strength [6Ð9], buckling instability [10, 11], elastic to develop analytical or empirical macromechanical models to modulus [12, 13], and twist induced by tension [14] under describe experimental observations of failure under multiaxial uniaxial loading conditions, there is much that is still unknown loading [15]. Such failure criteria use the concept of ‘a failure about other aspects of the mechanical responses of CNTs in surface’ or ‘a failure envelope’ generated by plotting principal 4 Author to whom any correspondence should be addressed. stress components in principal material axes [15].

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7 PHYSICAL REVIEW B 76, 094114 ͑2007͒

Atomistic and multiscale analyses of brittle fracture in crystal lattices

Sulin Zhang,1,2,* Ting Zhu,3 and Ted Belytschko4 1Department of Mechanical Engineering, University of Arkansas, 204 Mechanical Building, Fayetteville, Arkansas 72701, USA 2Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA 3Woodruff School of Mechanical Engineering, Georgia Institute and Technology, Atlanta, Georgia 30332, USA 4Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60201, USA ͑Received 28 April 2007; revised manuscript received 17 July 2007; published 27 September 2007͒ Applicability of the Griffith criterion ͓A. A. Griffith, Philos. Trans. R. Soc. London, Ser. 221, 163 ͑1920͒; S. Zhang, S. L. Meilke, R. Khare, D. Troya, R. S. Ruoff, G. C. Schatz, and T. Belytschko, Phys. Rev. B 71, 115403 ͑2005͔͒ for predicting the onset of crack extension in crystal lattices is systematically evaluated using atomistic and multiscale simulations with a focus on the effects of crack size and lattice discreteness. An atomistic scheme is developed to determine the true Griffith load defined by the thermodynamic energy balance of crack extension for both finite-sized and semi-infinite crack models. For a model monolayer lattice, we identify a characteristic crack length ͑about ten lattice spacings͒ below which the Griffith fracture stress markedly overestimates the true Griffith load. Through a stability analysis of crack-tip bond separation, the athermal ͑nonthermally activated͒ loads of instantaneous fracture are determined, thereby yielding the estimated lattice trapping range. Our simulations show that the strength of lattice trapping depends on the interaction range of the interatomic force fields. Using the reaction pathway exploration method, we determine the minimum energy paths of bond breaking and healing at a crack tip, giving a more precise estimate of the lattice trapping range. The activation energy barriers governing the rate of kinetic crack extension are extracted from the minimum energy paths. Implications concerning the distinction between the athermal and Griffith fracture loads are discussed. Based on these results, a general criterion is established to predict the onset of crack growth in crystal lattices. In addition to taking into account the lattice trapping effect, this criterion is applicable to a large spectrum of crack sizes.

DOI: 10.1103/PhysRevB.76.094114 PACS number͑s͒: 61.72.Ϫy, 61.50.Ah, 62.20.Ϫx, 68.65.Ϫk

I. INTRODUCTION length 2a embedded in an infinitely large, linear elastic medium, subject to remotely applied uniform tension, the Predicting the failure strength of nanostructured materials energy-balance criterion yields a critical stress1 often involves quantum mechanical calculations or atomistic models with empirical force fields. While these numerical ␴Ј = ͱ2Y␥ /␲a ͑2͒ methods have been useful to elucidate the failure mecha- G s nisms at the atomic level, their prohibitive computational for plane-stress condition, where Y denotes Young’s modu- cost becomes a major concern for specimens of realistic size. lus. To distinguish the critical stress given by Eq. ͑2͒ from In contrast, the fracture criteria1,2 established within the the true Griffith load given by the energy-balance criterion of framework of continuum fracture mechanics have been Eq. ͑1͒, the critical stress determined by Eq. ͑2͒ is hereafter widely used to predict the critical conditions for the onset of referred to as the Griffith fracture stress. crack extension in continua. If such continuum-based frac- The Griffith fracture stress has been widely used to pre- ture criteria were applicable to nanostructured materials, the dict the onset of crack extension in continua, yet it suffers aforementioned computational burden could often be deficiencies when applied to specimens with nanosized avoided. Thus, it is both fundamentally and practically criti- cracks. For extremely short cracks ͑a→0͒, the Griffith frac- cal to evaluate the applicability of these fracture criteria to ture stress may exceed the theoretical strength ␴ of the crystal lattices with the consideration of flaw size and lattice th perfect lattice, which, of course, is nonphysical. In address- discreteness. ing this issue, Gao et al.4 suggested that a characteristic The fundamental fracture criterion for brittle continua is crack length can be identified as a* ϰY␥ /␴2 , below which the energy-balance criterion by Griffith,1,3 which holds when s th the Griffith fracture stress overestimates the true fracture there is no generation or motion of dislocations or other dis- stress and the material becomes flaw insensitive. Pugno and sipation mechanisms, such as void nucleation. The Griffith Ruoff5 developed a quantized fracture mechanics ͑QFM͒ criterion states that a crack meets the critical growth condi- theory, where the classical stress intensity factor is redefined tion when the net change in the total energy of the system by considering an infinitesimal crack extension at the crack ⌬E vanishes upon crack extension by an infinitesimal dis- tip. With a predetermined geometric parameter, the QFM tance ⌬a: theory predicts satisfactory results on the fracture strengths ⌬ ͑ ␥ ͒⌬ ͑ ͒ of nanostructured materials. Recently, Mattoni et al.6 pro- E = G −2 s a =0, 1 posed a modified Griffith condition, where Young’s modulus ␥ where G is the elastic energy release rate and s is the sur- and surface energy density are taken to be strain dependent. face energy density which measures the fracture resistance of Despite yielding an improved agreement between the pre- the material. For a model system with a central crack of dicted Griffith fracture stress and the failure load, these

1098-0121/2007/76͑9͒/094114͑10͒ 094114-1 ©2007 The American Physical Society ZHANG, ZHU, AND BELYTSCHKO PHYSICAL REVIEW B 76, 094114 ͑2007͒ computational schemes developed in this work are general ACKNOWLEDGMENTS and can be applied to study the energetics and kinetics of ductile fracture involving dislocation nucleation or motion at S.Z. acknowledges the support from the National Science a crack tip. Brittle fracture by bond breaking generally pre- Foundation ͑NSF͒ Grant No. CMMI-0600661. T.Z. acknowl- vails at low temperatures. There exists a critical temperature edges the support from NSF Grant No. CMMI-0653769 and at which ductile fracture by dislocation motions prevails AFOSR/MURI Grant No. F49620-02-1-0382. T.B. acknowl- and brittle-to-ductile transition occurs. A study of such phe- edges the support from NSF Grant No. CMMI-0304472 and nomena in systems of ﬂat monolayer lattices and curved from the NASA University Research, Engineering and Tech- monolayers of single-walled carbon nanotubes is currently nology Institute on Bio Inspired Materials ͑BIMat͒ under under way. Award No. NCC-1-02037.

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094114-10 Chemical Physics Letters 446 (2007) 128–132 www.elsevier.com/locate/cplett

The eﬀects of extensive pitting on the mechanical properties of carbon nanotubes

Steven L. Mielke a,*, Sulin Zhang b,1, Roopam Khare b, Rodney S. Ruoﬀ b, Ted Belytschko b, George C. Schatz a

a Department of Chemistry, Northwestern University, Evanston, IL 60208-3113, USA b Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208-3111, USA

Received 12 June 2007; in ﬁnal form 8 August 2007 Available online 15 August 2007

Abstract

As previously demonstrated, a single hole is sufficient to markedly reduce the fracture strength of a carbon nanotube (CNT). Herein we present calculations exploring the effects of multiple holes on the modulus, fracture strength, and fracture strain of CNTs. The modulus decreases sharply and approximately linearly as a function of the pitting density. A few holes cause a decrease in the failure strain but extensive pitting leads to higher failure strains. These results suggest that the unusually low modulus measurements and high failure strains reported in the experiments of Yu et al. [Science 287 (2000) 637] were a consequence of purification induced oxidative pitting. Ó 2007 Elsevier B.V. All rights reserved.

1. Introduction studies of arc-discharge-grown MWCNTs, and only the outermost shell was load bearing for those samples. The Quantum mechanical (QM) calculations [1–6] pre- experiment of Yu et al. [7] reported 19 fracture strength dict that defect-free carbon nanotubes (CNTs) have measurements ranging from 11 to 63 GPa (with a mean Young’s modulus values of 1 TPa, fracture strengths of value of 28 GPa), four modulus measurements of 950, 100 GPa, and failure strains of 20–30% depending on 470, 335, and 274 GPa, and four failure strain measure- their chirality, with generally good agreement existing ments, one below 3% and three between 11% and 13%. between the predictions of tight binding [1,2,5], semiempir- The later experiment of Ding et al. [11] reported data for ical [3,4,6], and density functional theory [5,6] methods. A 14 MWCNTs with fracture strengths ranging from 10 to number of experimental studies [7–12] have been conducted 66 GPa (with a mean value of 24 GPa), modulus measure- for CNT fracture, but agreement both between the experiments ranging from 620 to 1200 GPa (with a mean value of mental results and between theory and experiment is lim- 955 GPa), and failure strains ranging from 1.0% to 6.3% ited. Some of the experiments [8–10] involved multiwalled (with a mean value of 2.6%). CNTs (MWCNTs) for which more than one shell is load The discrepancy between theory and experiment for the bearing, and this complicates experimental–theoretical fracture strengths has received the most attention, and comparisons. Two experiments [7,11] reported fracture early theories [13] focused on stress-induced Stone–Wales defects [14] as the likely cause of the low strength measure- * Corresponding author. Fax: +847 491 7713. ments. It was eventually realized that even though Stone– E-mail addresses: [email protected] (S.L. Mielke), Wales defects become energetically favorable at higher r-ruoff@northwestern.edu (R.S. Ruoff), [email protected] strains, the transformation barriers remain sufficiently high (T. Belytschko), [email protected] (G.C. Schatz). 1 Present address: Department of Mechanical Engineering, 204 Mechan- [4,15] that such processes could not explain fracture mea- ical Engineering Building, University of Arkansas, Fayetteville, AR 72701, surements at room temperature. It was subsequently USA. observed [6,16,17] that the MWCNTs used in the

0009-2614/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2007.08.033 132 S.L. Mielke et al. / Chemical Physics Letters 446 (2007) 128–132 the CNT fracture measurements of Yu et al. [7]. In fact, [10] A.H. Barber, R. Andrews, L.S. Schadler, H.D. Wagner, Appl. Phys. plausible pitting densities of 0.1–0.2 lead to the range of Lett. 87 (2005) 203106. failure stresses of 10–30 GPa, which corresponds to the [11] W. Ding, L. Calabri, K.M. Kohlhaas, X. Chen, D.A. Dikin, R.S. Ruoff, Exp. Mech. 47 (2007) 25. failure stresses most commonly observed in the experi- [12] J.Y. Huang et al., Nature 439 (2006) 281. ments. Alternative theories involving stress-induced [13] B.I. Yakobson, Appl. Phys. Lett. 72 (1998) 918. Stone–Wales defects [13] cannot explain the modulus mea- [14] A.J. Stone, D.J. Wales, Chem. Phys. Lett. 128 (1986) 501. surements and are not consistent with the much lower fail- [15] Q. Zhao, M.B. Nardelli, J. Bernholc, Phys. Rev. B 65 (2002) 144105. ure strains reported in the Ding et al. [11] measurements of [16] S. Zhang, S.L. Mielke, R. Khare, D. Troya, R.S. Ruoff, G.C. Schatz, T. Belytschko, Phys. Rev. B 71 (2005) 115403. unpurified CNTs. [17] S.L. Mielke, T. Belytschko, G.C. Schatz, Annu. Rev. Phys. Chem. 58 (2007) 185. Acknowledgement [18] D.T. Colbert et al., Science 266 (1994) 1218. [19] T. Belytschko, S.P. Xiao, G.C. Schatz, R.S. Ruoff, Phys. Rev. B 65 (2002) 235430. We gratefully acknowledge the grant support from the [20] D.R. Olander, W. Siekhaus, R. Jones, J.A. Schwarz, J. Chem. Phys. NASA University Research, Engineering, and Technology 57 (1972) 408. Institute on Bio Inspired Materials (BIMat) under Award [21] R.T. Yang, C. Wong, Science 214 (1981) 437. No. NCC-1-02037. [22] R.T. Yang, C. Wong, J. Chem. Phys. 75 (1981) 4471. [23] S.M. Lee, Y.H. Lee, Y.G. Hwang, J.R. Hahn, H. Kang, Phys. Rev. Lett. 82 (1999) 217. References [24] A.G. Rinzler et al., Appl. Phys. A 67 (1998) 29. [25] R.C. Haddon, J. Sippel, A.G. Rinzler, F. Papadimitrakopoulos, MRS [1] E. Hernańdez, C. Goze, P. Bernier, A. Rubio, Phys. Rev. Lett. 80 Bull. 29 (2004) 252. (1998) 4502. [26] R. Khare, S.L. Mielke, J.T. Paci, S. Zhang, R. Ballarini, G.C. Schatz, [2] T. Ozaki, Y. Iwasa, T. Mitani, Phys. Rev. Lett. 84 (2000) 1712. T. Belytschko, Phys. Rev. B 75 (2007) 075412. [3] D. Troya, S.L. Mielke, G.C. Schatz, Chem. Phys. Lett. 382 (2003) [27] D.W. Brenner, O.A. Shenderova, J.A. Harrison, S.J. Stuart, B. Ni, 133. S.B. Sinnott, J. Phys.: Condens. Mat. 14 (2002) 783. [4] T. Dumitrica, T. Belytschko, B.I. Yakobson, J. Chem. Phys. 118 [28] O.A. Shenderova, D.W. Brenner, A. Omeltchenko, X. Su, L.H. Yang, (2003) 9485. Phys. Rev. B 61 (2000) 3877. [5] S. Ogata, Y. Shibutani, Phys. Rev. B 68 (2003) 165409. [29] J.P. Watt, G.F. Davies, R.J. O’Connor, Rev. Geophys. 14 (1976) 541. [6] S.L. Mielke et al., Chem. Phys. Lett. 390 (2004) 413. [30] N.M. Pugno, Appl. Phys. Lett. 90 (2007) 043106. [7] M.-F. Yu, O. Lourie, M.J. Dyer, K. Moloni, T.F. Kelly, R.S. Ruoff, [31] M.F. Thorpe, P.N. Sen, J. Acoust. Soc. Am. 77 (1985) 1674. Science 287 (2000) 637. [32] R. Hill, J. Mech. Phys. Solids 13 (1965) 213. [8] B.G. Demczyk, Y.M. Wang, J. Cumings, M. Hetman, W. Han, A. [33] J.D. Eshelby, Proc. Roy. Soc. Lond. A 241 (1957) 376. Zettl, R.O. Ritchie, Mater. Sci. Eng. A 334 (2002) 173. [34] W. Xia, M.F. Thorpe, Phys. Rev. A 38 (1988) 2650. [9] A.H. Barber, I. Kaplan-Ashiri, S.R. Cohen, R. Tenne, H.D. Wagner, [35] A.R. Day, K.A. Snyder, E.J. Garboczi, M.F. Thorpe, J. Mech. Phys. Compos. Sci. Technol. 65 (2005) 2380. Solids 40 (1992) 1031. Acta Materialia 55 (2007) 5269–5279 www.elsevier.com/locate/actamat

The role of defects in the design of space elevator cable: From nanotube to megatube

Nicola M. Pugno *

Department of Structural Engineering and Geotechnics, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Received 5 April 2007; received in revised form 29 May 2007; accepted 29 May 2007 Available online 25 July 2007

Abstract

Researchers are claiming that the feasibility of space elevator cable is now realistic, thanks to carbon nanotube technology, proposing its realization within a decade. However, the current view of basing the design of the megacable on the theoretical strength of a single carbon nanotube is naı¨ve, as has recently been emphasized. In this paper the role of thermodynamically unavoidable atomistic defects with different size and shape is quantified on brittle fracture, fatigue and elasticity, for nanotubes and nanotube bundles. Nonasymptotic regimes, elastic plasticity, rough cracks, finite domains and size effects are also discussed. The results are compared with atomistic simulations and nanotensile tests of carbon nanotubes. Key simple formulas for the design of a flaw-tolerant space elevator megacable are reported, suggesting that it would need a taper ratio (for uniform stress) of about two orders of magnitude larger than currently proposed. 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Fracture; Scaling; Nanocomposite; Stress rupture; Toughness

1. Introduction chronous orbit, is 63 GPa/(1300 kg m3), corresponding to 63 GPa if low carbon density is assumed for the cable. A space elevator basically consists of a cable attached to It is only recently, after the discovery of carbon nanotubes the Earth’s surface for carrying payloads into space [1].If [4], that such large failure stresses have been measured the cable is long enough, i.e. around 150 Mm (a value that experimentally, during tensile tests on ropes composed of can be reduced by a counterweight), the centrifugal forces single-walled [5] or multiwalled [6–8] carbon nanotubes, exceed the gravity of the cable that will work under tension both of which were expected to have an ideal strength of [2]. The elevator would stay fixed geosynchronously; once 100 GPa. Note that for steel (density 7900 kg m3, max- sent far enough, climbers would be accelerated by the imum strength 5 GPa) the maximum stress expected in the Earth’s rotational energy. A space elevator would revolu- cable would be 383 GPa, whereas for Kevlar (density tionize the methodology for carrying payloads into space 1440 kg m3, strength 3.6 GPa) it would be 70 GPa, both at low cost, but its design is very challenging. The most crit- much higher than their respective strengths [3]. ical component in the space elevator design is undoubtedly However, an optimized cable design must consider a the cable [3], which requires a material with very high uniform tensile stress profile rather than a constant cross- strength and low density. sectional area [2]. Accordingly, the cable could be built If we consider a cable with constant cross-section and a from any material simply by using a sufficiently large taper vanishing tension at the planet surface, the maximum ratio, i.e., the ratio of the maximum (at the geosynchro- stress–density ratio for the Earth, reached at the geosyn- nous orbit) to the minimum (at the Earth’s surface) cross-sectional area. For example, for steel and Kevlar 33 8 * Tel.: +39 011 564 4902; fax: +39 011 564 4899. huge and unrealistic taper ratios of 10 and 2.6 · 10 , E-mail address: [email protected] respectively, would be required, whereas for carbon

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[26] Wang QZ. Simple formulae for the stress-concentration factor for Seifert G, et al. On the mechanical behavior of WS2 nanotubes under two- and three-dimensional holes in ﬁnite domains. J Strain Anal axial tension and compression. Proceedings of the national academy 2002;73:259–64. of science USA 2006;103:523–8. PHYSICAL REVIEW B 76, 064120 ͑2007͒

Ab initio calculation of ideal strength and phonon instability of graphene under tension

Fang Liu School of Applied Mathematics, Central University of Finance and Economics, Beijing 100081, China

Pingbing Ming LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China

Ju Li* Department of Materials Science and Engineering, Ohio State University, Columbus, Ohio 43210, USA ͑Received 14 May 2007; revised manuscript received 16 June 2007; published 28 August 2007͒ Graphene-based sp2-carbon nanostructures such as carbon nanotubes and nanofibers can fail near their ideal strengths due to their exceedingly small dimensions. We have calculated the phonon spectra of graphene as a function of uniaxial tension by density functional perturbation theory to assess the first occurrence of phonon instability on the strain path, which controls the strength of a defect-free crystal at 0 K. Uniaxial tensile strain is applied in the x ͑nearest-neighbor͒ and y ͑second nearest-neighbor͒ directions, related to tensile deformation of zigzag and armchair nanotubes, respectively. The Young’s modulus E=1050 GPa and Poisson’s ratio ␯ =0.186 from our small-strain results are in good agreement with previous calculations. We find that in both x and y uniaxial tensions, phonon instabilities occur near the center of the Brillouin zone, at ͑␧xx=0.194, ␴xx =110 GPa, ␧yy=−0.016͒ and ͑␧yy=0.266, ␴yy=121 GPa, ␧xx=−0.027͒, respectively. Both soft phonons are longitudinal elastic waves in the pulling direction, suggesting that brittle cleavage fracture may be an inherent behavior of graphene and carbon nanotubes at low temperatures. We also predict that a phonon band gap will appear in highly stretched graphene, which could be a useful spectroscopic signature for highly stressed carbon nanotubes.

DOI: 10.1103/PhysRevB.76.064120 PACS number͑s͒: 63.20.Dj, 62.20.Ϫx, 81.05.Uw, 81.07.De

I. MOTIVATION tubes ͑MWCNTs͒ inside an atomic force microscope, Falvo 1,2 et al. estimated that 16% tensile strain can be achieved in The ideal strength is the highest achievable strength of a 13 defect-free crystal at 0 K. Even though a conventional mate- local regions of some MWCNTs without breaking them. Yu rial deforms or fractures at macroscopic stresses far below its et al. measured the tensile response of single-walled carbon ͑ ͒ 14 ideal strength, the ideal strength is nonetheless a crucial the- nanotube SWCNT ropes and inferred a mean breaking strength of 30 GPa, 3% of their mean Young’s modulus of oretical parameter, because it fundamentally characterizes 15 the nature of chemical bonding in that crystal.3,4 The Peierls- 1002 GPa. Ding et al. measured the fracture strengths and Nabarro model of dislocation,5,6 for instance, relies on the moduli of arc-grown MWCNTs. The outer-shell fracture Frenkel model of ideal strength, because defects such as strength was estimated to range from 10 to 66 GPa, and the cracks and dislocations work like levers, amplifying the far- Young’s modulus from 620 to 1200 GPa. Demczyk et al. field stress to near ideal strength levels inside the defect core conducted room-temperature pulling and bending tests on in order to move.7 It is thus not surprising that the study of MWCNT of diameter of 12.5 nm and measured an astonishing 150 GPa failure strength, which is 17% of its Young’s ideal strength can tell us a lot about why some materials 16 ͑such as diamond͒ are intrinsically brittle, while others ͑such modulus, E=900 GPa. The authors noted there is no nar- ͒ 4 rowing of the nanotubes immediately before failure ͑the de- as copper are intrinsically ductile. ͒ The ideal strength becomes even more important with the formation was elastic and reversible , and the nanotubes fail progress of nanotechnology. Recent experiments on “as essentially defect-free materials.” The 150 GPa strength 8 9 10 value, however, seems to be inconsistent with their reported nanocrystals, nanoporous materials, nanopillars, and 17 nanoindentation11 have revealed a host of ultrastrength phe- 5% strain prior to failure. Barber et al. measured the tensile nomena, defined by internal stress levels broadly and persis- strength of MWCNTs produced by chemical vapor deposi- tently rising up to a significant fraction of the ideal strength. tion and fitted the data to the Weibull-Poisson distribution. A Ultrastrength materials typically have geometric features characteristic strength value of 109 GPa was obtained. around or less than L ϳ102 nm. To put this in perspective, Márquez-Lucero et al. reported similar high strengths in car- C 18 computers one can buy off the shelf now have chips with bon nanotubes and nanofibers. These recent experiments 65 nm strained silicon features.12 At such material length indicate that ultratensile strength can indeed be achieved in scales, the population dynamics of defects is fundamentally an important component of nanotechnology. different from that in the macroscale material, leading to It is reasonable to speculate that graphene-based carbon Ͼ size-dependent mechanical behavior at L LC, which, how- nanotubes, nanofibers, etc., hold record or near-record ideal Ͻ ever, starts to level off at L LC due to the intrinsic upper tensile strength among all materials. It is well known that the bound, the ideal strength. electronic structure and mechanical properties of carbon Carbon nanotube is an ultimate example of small-size, nanotubes are similar to those of flat graphene, aside from ultrastrength material. By bending multiwalled carbon nano- quantum confinement effect.19 For this reason, in this paper

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064120-7 THE JOURNAL OF CHEMICAL PHYSICS 127, 074708 ͑2007͒

Tight-binding molecular dynamics study of the role of defects on carbon nanotube moduli and failure ͒ Richard W. Haskins,a Robert S. Maier, Robert M. Ebeling, Charles P. Marsh, Dustin L. Majure, Anthony J. Bednar, Charles R. Welch, and Bruce C. Barker U.S. Army Engineer Research and Development Center, Vicksburg, Mississippi 39180-6133 USA David T. Wu Chemical Engineering Department, Colorado School of Mines, Golden, Colorado 80401-1887 and Chemistry Department, Colorado School of Mines, Golden, Colorado 80401-1887 USA ͑Received 10 August 2006; accepted 13 June 2007; published online 20 August 2007͒

We performed tight-binding molecular dynamics on single-walled carbon nanotubes with and without a variety of defects to study their effect on the nanotube modulus and failure through bond rupture. For a pristine ͑5,5͒ nanotube, Young’s modulus was calculated to be ϳ1.1 TPa, and brittle rupture occurred at a strain of 17% under quasistatic loading. The predicted modulus is consistent with values from experimentally derived thermal vibration and pull test measurements. The defects studied consist of moving or removing one or two carbon atoms, and correspond to a 1.4% defect density. The occurrence of a Stone-Wales defect does not signiﬁcantly affect Young’s modulus, but failure occurs at 15% strain. The occurrence of a pair of separated vacancy defects lowers Young’s modulus by ϳ160 GPa and the critical or rupture strain to 13%. These defects apparently act independently, since one of these defects alone was independently determined to lower Young’s modulus by ϳ90 GPa, also with a critical strain of 13%. When the pair of vacancy defects adjacent, however, Young’s modulus is lowered by only ϳ100 GPa, but with a lower critical strain of 11%. In all cases, there is noticeable strain softening, for instance, leading to an ϳ250 GPa drop in the apparent secant modulus at 10% strain. When a chiral ͑10,5͒ nanotube with a vacancy defect was subjected to tensile strain, failure occurred through a continuous spiral-tearing mechanism that maintained a high level of stress ͑2.5 GPa͒ even as the nanotube unraveled. Since the statistical likelihood of defects occurring near each other increases with nanotube length, these studies may have important implications for interpreting the experimental distribution of moduli and critical strains. © 2007 American Institute of Physics. ͓DOI: 10.1063/1.2756832͔

I. INTRODUCTION mote intertube and matrix bonding will alter the mechanical properties of individual tubes in a manner roughly analogous Pristine carbon nanotubes ͑CNT’s͒ offer tremendous po- to defects.1,3 tential improvements in tensile strength over that of conven- The prediction of CNT failure modes is an important tional materials for structures. With a tensile strength many issue for both experimental and simulation work. Experi- times that of even very high strength ͑1.4 GPa͒ steels, but at mental measurements of mechanical properties of individual only 1/6–1/3 the density, CNTs could be used for a broad CNT’s have a large variance because of the inherent difﬁ- variety of applications in which strength-to-weight ratios are culty in manipulating, characterizing, and measuring the be- important, such as for lightweight, quickly erectable infra- havior of objects at the nanometer scale ͑see reviews by Qian structure; airframes; bridges; and armor. But before these et al.4 and Yu5͒. In contrast, the simulation of individual practical uses can be realized, macromaterials will need to be CNT mechanical properties is less demanding than experi- fabricated from CNT’s. These macromaterials will involve mental determination as relatively few atoms are involved. macroscale quantities of CNT’s, often adjoined with other The simulation results can be used to verify and further un- materials. In these nanocomposite materials, individual derstand experimental results; see, for example, results from 1 CNT’s could represent a strain relief mechanism or a load- Kaplan-Ashiri et al.6,7 on Young’s modulus. carrying member. The prediction and characterization of Different groups have used numerical simulations to pre- CNT failure modes are therefore of fundamental importance dict Young’s modulus of CNT’s with generally good agree- in the design of structures and composite materials using ment between independent studies and available experimen- CNT’s. A major factor that can affect CNT failure is the tal data. Sears and Batra8 found that the values of Young’s presence of defects that result from production or fabrication, modulus reported by ten different research groups between the environment they have been exposed to, or prior stress. 1992 and 2002 varied between 0.94 and 1.24 TPa. Note that Furthermore, the functionalization of CNT’s ͑Ref. 2͒ to pro- the value of Young’s modulus depends on the value taken for the tube wall thickness, which is not a well-deﬁned quantity ͒ 9 a Electronic mail: [email protected] ͑see, for example, the discussion by Srivastava et al. ͒. Sears

Downloaded 18 Nov 2007 to 140.109.112.41. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp 074708-10 Haskins et al. J. Chem. Phys. 127, 074708 ͑2007͒

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Downloaded 18 Nov 2007 to 140.109.112.41. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp IOP PUBLISHING NANOTECHNOLOGY Nanotechnology 18 (2007) 335702 (7pp) doi:10.1088/0957-4484/18/33/335702 Atomistic simulations of the mechanical properties of ‘super’ carbon nanotubes

VRColuci1, N M Pugno2,SODantas3,DSGalvao˜ 1 and A Jorio4

1 Instituto de F´ısica ‘Gleb Wataghin’, Universidade Estadual de Campinas, Unicamp 13083-970, Campinas, S˜ao Paulo, Brazil 2 Department of Structural Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy 3 Departamento de F´ısica, ICE, Universidade Federal de Juiz de Fora, 36036-330 Juiz de Fora MG, Brazil 4 Departamento de F´ısica, Universidade Federal de Minas Gerais, 30123-970 Belo Horizonte MG, Brazil

E-mail: coluci@iﬁ.unicamp.br

Received 3 May 2007, in final form 6 June 2007 Published 25 July 2007 Online at stacks.iop.org/Nano/18/335702 Abstract The mechanical properties of the so-called ‘super’ carbon nanotubes (STs) are investigated using classical molecular dynamics simulations. The STs are built from single-walled carbon nanotubes (SWCNTs) connected by Y-like junctions forming an ordered carbon nanotube network that is then rolled into a seamless cylinder. We observed that the ST behaviour under tensile tests is similar to the one presented by fishing nets. This interesting behaviour provides a way to vary the accessible channels to the inner parts of STs by applying an external mechanical load. The Young’s modulus is dependent on the ST chirality and it inversely varies with the ST radius. Smaller reduction of breaking strain values due to temperature increase is predicted for zigzag STs compared to SWCNTs. The results show that, for STs with radius ∼5 nm, the junctions between the constituent SWCNTs play an important role in the fracture process. The Young’s modulus and tensile strength were estimated for hierarchical higher-order STs using scaling laws related to the ST fractal dimension. The obtained mechanical properties suggest that STs may be used in the development of new porous, flexible, and high-strength materials. (Some figures in this article are in colour only in the electronic version)

1. Introduction can be generated from an ordered carbon network based on the honeycomb symmetry, generically named super-graphene, Many attempts have been made in order to develop procedures which is heuristically constructed by replacing the carbon– to controllably assemble a large number of single-walled carbon bonds of the graphene architecture by single-walled carbon nanotubes (SWCNTs) in terms of position and carbon nanotubes (SWCNTs) and the carbon atoms by Y- orientation [1–6]. The achievement of such procedures like junctions. The associated STs can be then generated would allow the fabrication of ordered SWCNT networks by rolling up super-graphene sheets. Similarly to a (n, m) representing a breakthrough in the ‘bottom-up’ manufacturing SWCNT [8], [N, M] STs with different chiralities can be approach. These ordered networks would open possibilities to constructed. The STs are represented as [N, M]@(n, m) and design new materials with desirable electronic and mechanical are characterized by the (n, m) SWCNT used to form them, properties. the necessary junctions to join consecutive SWCNTs, and Recently, the structure of the so-called ‘super’ carbon the distance between these junctions. The ST construction nanotubes (STs) was proposed [7] (ﬁgure 1). This structure is not limited to carbon nanotubes and to the honeycomb

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7 Downloaded By: [Academia Sinica] At: 03:54 19 November 2007 Crepnigato.Eal [email protected] Email: author. *Corresponding C–C a rotating by formed 19, dipole 16, dislocation 12, 5/7–7/5 11, a 90 [8, as by defects viewed bond (SW) be and Stone–Wales can dominates of which deformation motion 23], plastic and temperatures, structures nucleation elevated open-ring by breaking at large proceeds whereas bond of 21], via formation 15, mechanisms fracture 10, stress-mediated active brittle [9, brittle with involves 16], cleavage temperatures, often 12, [7]: low 11, and 8, routes prevails At [7, distinct flow temperature. plastic two by or high 22] take mediated high- At 21, may 15, [6–24]. 13, ultra-stiff, CNTs theoretically [9–11, and of fracture as [1–5] deformation of experimentally the (CNTs) study both the loads, CNTs to nanotubes efforts of considerable mechanics carbon directed the have nanocomposites of in fibres applications flexibility promising The Introduction 1. 567–574 2007, August 8, No. 87, Vol. Letters Magazine Philosophical akdkn omto taot2000 carbon single-walled about which at in with reduction formation 5] radial [4, 15-fold kink and until CNTs elongation marked attention 280% superplastic less underwent much deformation of (SWCNT) received the nanotube discovery have have contrast, In modelling, temperatures recent CNTs. continuum high simulations the of based at fracture atomistic [25] CNTs brittle rule of empirical the Cauchy–Born mechanisms study 15], to to 24], 13, performed 22, 12, been 21, 10, 17, 11, 9, [10 [6, calculations first-principles tmcgoer n nreiso abnnntb necking nanotube carbon of energetics and geometry Atomic z essaymti ekn.Terslspoieaqatttv ai o the for basis quantitative a symmetric CNTs. deformed provide of diameter. superplastically modes results of deformation simulations The tube dynamics competing necking. dislocation the strain. in by asymmetric tensile of manifested high versus reveals separation reduction as at and glide defects, nucleation favoured subsequent stepwise SW dislocation energetically affect of are critically of dislocations a motions lead Pre-existing energetics nucleation that dislocation strain-dependent through dipoles such involves dislocation that the 5/7 necking which of of Quantification glide (CNTs), quantized spiral and nanotubes plastic to defects incipient carbon the (SW) probe Stone–Wales to in performed were deformation simulations mechanics Molecular odufSho fMcaia niern,GogaIsiueo Technology, of Institute Georgia Engineering, Mechanical of School Woodruff ( eevd4Fbur 07 cetdi eie om2 ac 2007 March 23 form revised in accepted 2007; February 4 Received xesv xeiet 13 n ueia iuain,rnigfrom ranging simulations, numerical and [1–3] experiments Extensive . y eateto ehnclEgneig nvriyArkansas, University Engineering, Mechanical of Department SN05-89pitIS 3233 online 1362-3036 print/ISSN 0950-0839 ISSN rass771 aetvle USA Fayetteville, 72701, Arkansas .ZHANG* S. eri 03,Alna USA Atlanta, 30332, Georgia O:10.1080/09500830701370799 DOI: http://www.tandf.co.uk/journals hlspia aaieLetters Magazine Philosophical , y n .ZHU T. and .Weesi-iueprmnshave experiments in-situ Whereas C. ß 07Tyo Francis & Taylor 2007 z ) Downloaded By: [Academia Sinica] At: 03:54 19 November 2007 1]GG asnde ..Smoiz n ..Ykbo,Py.Rv Lett. Rev. Phys. Yakobson, B.I. and Samsonidze G.G. Samsonidze, G.G. [14] 1]S gt n .Siuai hs e.B Rev. B Phys. Rev. Shibutani, Phys. Y. Bernholc, and J. Ogata and S. Yakobson [13] B.I. Nardelli, M.B. [12] 1]MB adli ..Ykbo n .Brhl,Py.Rv Lett. Rev. Phys. Bernholc, J. and Yakobson B.I. Nardelli, M.B. [11] 1]SL ile .Toa .Zhang, S. Troya, D. Mielke, S.L. [10] prlgiet uti h lsi lwo Nsa ihtmeaue.Activation temperatures. have high CNTs bent at Mori in CNTs dislocation by dislocations 5/7 may calculated against of of been CNTs compete flow steps recently slip in will plastic consecutive deformation process between the barriers plastic a sustain energy the the Such to the control that [4]. glide evaluate that note climb spiral to motion Also dislocation necessary flow. and by also plastic nucleation proceed complete of is dislocation a rate it of For kinetic CNTs, barriers dislocations. in the of energy glide activities activation for and dislocation formation basis of the description of quantitative forces of driving a formulation dynamic a the provide CNT. deformed guide on also superplastically this can dislocations of and simulations governing results between dynamics model, These mechanics dislocation interactions CNTs. constitutive the elastic and curved studied continuum the formation of further on surface neck have cylindrical based asymmetric We process versus CNTs. competing symmetric stepwise in i.e. modes a competing propagation necking, two through identified necking have atomic-scale quantized we Specifically, to of diameter. lead tube can the in processes reduction these CNTs; in defects References support the No. acknowledges award Zhu under T. F49620-02-1-0382. grant Manager). Grant Foundation AFOSR/MURI Program Science from Cooper, National V. acknowledges the (Clark Zhang from 0600661 S. discussions. support helpful grant for the Belytschko Ted Professor thank We Acknowledgements atomic the using CNTs study. in present sequential climb) the of and from barriers slip obtained energy (both CNTs, configurations activation motion in the and flow compute plastic nucleation large to dislocation the underway of currently understanding full is a work obtain to order In considered. 8 .DmtiaadBI aosn pl hs Lett. Phys. Appl. Yakobson, B.I. and Dumitrica T. [8] 9 .Kae ..Mek,JT Paci, J.T. Mielke, S.L. U.S.A. Khare, Sci. R. Acad. Natn. [9] Proc. Yakobson, B.I. and Hua M. Dumitrica, T. [7] 6 .Dmtia .Bltck n ..Ykbo,J hm Phys. chem. J. Yakobson, B.I. and Belytschko T. Wang, Dumitrica, Z.Q. T. Chen, [6] S. Huang, J.Y. [5] 4 ..Hag .Ce,ZF Ren, Z.F. Moloni, Chen, K. S. Dyer, Huang, M.J. J.Y. Lourie, [4] O. Casavant, Yu, M.J. M.F. Ericson, [3] L.M. Cumings, Walters, J. D.A. Wang, [2] Y.M. Demczyk, B.G. [1] ecnld ycmetn httepeetsuyfcsso h thermo- the on focuses study present the that commenting by conclude We (2002). tmcgoer n nreiso abnnntb necking nanotube carbon of energetics and geometry Atomic tal. et tal et tal. et tal. et tal. et hs e.Lett. Rev. Phys. , 2] hr nyoedsoaindpl was dipole dislocation one only where [29], . hs e.B Rev. Phys. , hm hs Lett. Phys. Chem. , Nature , 68 tal. et 649(2003). 165409 tal. et tal. et ae.Si n.A Eng. Sci. Mater. , 439 Science , pl hs Lett. Phys. Appl. , 84 8 (2006). 281 75 75(2004). 2775 97 742(2007). 075412 751(2006). 075501 390 287 57 1 (2004). 413 3 (2000). 637 118 47 (1998). R4277 81 66(1998). 4656 334 45(2003). 9485 74 103 7 (2002). 173 83(1999). 3803 15(2006). 6105 88 065501 573 Carbon 45 (2007) 1769–1776 www.elsevier.com/locate/carbon

Molecular mechanics modeling of carbon nanotube fracture

W.H. Duan a, Q. Wang a,*, K.M. Liew b, X.Q. He b

a Department of Mechanical and Manufacturing Engineering, University of Manitoba, Winnipeg, Manitoba, Canada R3T 5V6 b Department of Building and Construction, City University of Hong Kong, Hong Kong, China

Received 13 March 2007; accepted 4 May 2007 Available online 24 May 2007

Abstract

The fracture of carbon nanotubes (CNTs) is studied in this paper. Molecular mechanics models that incorporate the modified Morse potential and reactive empirical bond-order potential are developed to envisage the fracture behavior of perfect CNTs. The tensile strength, fracture strain, and fracture angle under tension are discussed, and special attention is paid to the effects of tube chirality. Expli- cit expressions for the fracture solutions for achiral carbon nanotubes are presented, but only numerical results are available for chiral carbon nanotubes. The predicted results of the present model are in good agreement with existing data and those of molecular mechanics simulations via the Materials Studio software package, which indicates the effectiveness of the developed models. 2007 Elsevier Ltd. All rights reserved.

1. Introduction as Stone-Wales transformation [14,15], ‘‘lattice-trapped’’ states [16], single vacancy [17–19], and its derivative point Carbon nanotubes (CNTs), which were discovered by defects [20]. These defects can signiﬁcantly reduce the Iijima [1], are carbon macromolecules in a periodic hexag- strength of CNTs. However, based on the classical molec- onal arrangement with a cylindrical shell shape. As the ular dynamics simulations, tight-binding, and ab initio cal- strength of CNTs is of great interest, the atomic mecha- culations, Yakobson et al. [21] concluded that the strength nisms of CNT failure have been extensively investigated of CNTs is highly dependent on their strain level, strain both experimentally and theoretically [2–7]. rate, temperature, and chirality. The fracture strain In experiments, atomic force microscopy (AFM) or appears to range from 30% to 40% at room temperature. transmission electron microscopy (TEM) is usually At high temperatures, CNTs can sustain extensive elonga- employed to measure the breaking strain and observe the tion with a very high strain rate [16], up to 280%, which fracture patterns of CNTs [8–12]. The reported experimen- was reported by Huang et al. [12], due to the possibility tal values of the breaking strain are quite diverse, and of plastic yield. With a decrease in temperature, the tensile range from 13% [8], 17% [9], 30% [10], and 50% [11] to strength with brittle fracture will increase and has no per- 280% [12] due to the variability of the samples and mea- ceptible dependence on the strain rate. Despite these surement conditions. There are two major perspectives on insights, some challenging questions still remain, such as such a scattered range of experimental data. Belytschko the dependence of the fracture modes on the chirality of et al. [13] suggested that the fracture strain, which ranged CNTs. from 2% to 13% [8], can be attributed to defects in CNTs, In their analysis of the tensile strength and strain of because these values are lower than those for pristine CNTs, Yakobson et al. [21] pointed out that the continuum CNTs that are generated by molecular mechanics sim- elasticity theory cannot predict any special features, ulations. There are diﬀerent types of defects in CNTs, such because under tension, the CNT remains structurally stable, and preserves a straight cylinder geometry. How- * Corresponding author. Fax: +1 204 275 7507. ever, extensive molecular dynamics simulations are inappli- E-mail address: [email protected] (Q. Wang). cable to a system on a larger scale because of the expensive

[33] Sears A, Batra RC. Macroscopic properties of carbon nanotubes tension using molecular dynamics simulation. Acta Mater 2004;52(9): from molecular-mechanics simulations. Phys Rev B 2004;69(23): 2521–7. 235406-1–10. [36] Mielke SL, Troya D, Zhang S, Li J-L, Xiao S, Car R, et al. The role [34] Ogata S, Shibutani Y. Ideal tensile strength and band gap of vacancy defects and holes in the fracture of carbon nanotubes. of single-walled carbon nanotubes. Phys Rev B 2003;68(16):165409- Chem Phys Lett 2004;390(4–6):413–20. 1–4. [37] Chang TC, Guo WL, Guo XM. Buckling of multiwalled carbon [35] Liew KM, He XQ, Wong CH. On the study of elastic and nanotubes under axial compression and bending via a molecular plastic properties of multi-walled carbon nanotubes under axial mechanics model. Phys Rev B 2005;72(6):064101-1–11. Composite Structures 79 (2007) 581–589 www.elsevier.com/locate/compstruct

The eﬀect of Stone–Wales defect on the tensile behavior and fracture of single-walled carbon nanotubes

K.I. Tserpes a,*, P. Papanikos b

a Laboratory of Structural Mechanics, Department of Rural and Surveying Engineering, National Technical University of Athens, Zografou Campus, 9 Iroon Polytechniou St., 15780 Athens, Greece b Department of Product and Systems Design Engineering, University of the Aegean, Ermoupolis, Syros 84100, Greece

Available online 18 April 2006

Abstract

The effectiveness of carbon nanotubes as reinforcements in the next generation of composites is designated by their mechanical behavior as standalone units. One of the most commonly present topological defects, whose effect on the mechanical behavior of carbon nanotubes needs to be clarified, is the Stone–Wales (SW) defect. In this paper, the effect of SW defect on the tensile behavior and fracture of armchair, zigzag and chiral single-walled carbon nanotubes (SWCNTs) was studied using an atomistic-based progressive fracture model. The model uses the finite element method for analyzing the structure of SWCNTs and the modified Morse interatomic potential for describing the nonlinear force-field of the C–C bonds. In all cases examined, the SW defect serves as nucleation site for fracture. Its effect on the tensile behavior of the SWCNTs depends solely on nanotube chirality. In armchair SWCNTs, contrary to zigzag ones, a significant reduction in failure stress and failure strain was predicted; ranging from 18% to 25% and from 30% to 41%, respectively. In chiral SWCNTs, the effect of the defect is between those of the armchair and zigzag SWCNTs, depending on chiral angle. The stiffness of the nanotubes was not affected. The nanotube size was found to play a minimal role in the tensile behavior of SW-defected SWCNTs; only in cases of very small nanotube diameters, where the fraction of defect area to the nanotube area is high, was a larger decrease in the failure stress predicted. Ó 2006 Elsevier Ltd. All rights reserved.

Keywords: Carbon nanotubes; Finite element analysis; Interatomic potential; Progressive fracture analysis; Stone–Wales defect

1. Introduction presence of vacancy defects significantly reduces the failure stress and failure strain of CNTs providing an explanation Due to their extraordinary mechanical properties, car- for the extant theoretical–experimental discrepancies. bon nanotubes (CNTs) are being considered as the ideal The SW defect is the most important defective structure reinforcing material for the next generation of composites. in CNTs. Investigations have shown that besides the effect In this role, CNTs’ performance is designated by their that may have on the mechanical behavior of CNTs, SW mechanical behavior as stand alone units. So far, in the defect also affects their electronic, magnetic and hybridiza- majority of studies, CNTs have been treated as defect-free tion characteristics. Because of its multiple effect, the SW materials. However, experimental observations [1] have defect demonstrates several utilities. For example, the tran- revealed that topological defects, such as the Stone–Wales sition in the Y junction, contemplated in CNT based (SW) defect, and vacancy defects, are commonly present. molecular electronics, is achieved through the incorpora- The presence of defects in CNTs is consolidated by the tion of many SW defects either by design or otherwise recent findings of Mielke et al. [2] who predicted that the [3]. Similarly, a transition of nanotubes from one diameter to another can be achieved by locating a few SW defects * Corresponding author. Tel.: +30 210 9886743; fax: +30 210 8254177. strategically in the transition region. In addition, when E-mail address: [email protected] (K.I. Tserpes). nanotubes are used as fibers in nanocomposites, interfacial

30% to 41.4%, respectively. On the other hand, in zigzag SWCNTs, the SW defect formed by the 90° rotation of a longitudinal bond does no affect their tensile behavior at all, while the defect formed by the 90° rotation of a diago- nal bond reduces the failure stress and failure strain of the nanotubes by about 3% and 15%, respectively. The effect of a SW defect in chiral SWCNTs is between those of the armchair and zigzag SWCNTs according to the chiral angle. For the SW-defected (16,8) tube considered in this work, a 15.18% reduction was predicted in the failure stress and a 32.4% in failure strain. The stiffness of the SWCNTs was not affected by the presence of SW defects. The nanotube size was found to play a minimal role on the tensile behavior of SW-defected SWCNTs; only in cases of small nanotube diameters, where the fraction of defect area to the nanotube area is high, was a larger decrease in the failure stress predicted. Additionally to the fulfillment of the basic scope described in the previous paragraph, this paper intends also to give support to the belief that continuum mechanics methods combined to the appropriate physics can be an efficient computational tool for modeling the structure of CNTs and predicting their mechanical behavior. The PFM used to accomplish the current study treats CNTs as space-frame structures in order to enable use of finite elements for modeling their response to mechanical load- Fig. 8. Predicted fracture evolution in the zigzag (20,0) tube containing a ing. The FE method has lent the PFM the ability to model SW defect: (a) type I defect and (b) type II defect. nanotube systems with a very large number of atoms subjected to complex mechanical loading conditions in small CPU times. For example, the analysis of the (18,18) tube (1200 atoms, 120 load steps) took about 120 s on a Pen- tiumÒ 4 CPU 3.20 GHz, 1.0 GB RAM personal computer. This ability is the main advantage of atomistic-based continuum mechanics approaches over the classical atomistic modeling approaches, such as MD simulations. The modified Morse interatomic potential used to describe the nonlinear force-field of the C–C bonds, may not be appropriate for describing fracture evolution, since it does not take into account many-body interactions as well as reconfiguration of bonds, but gives correct predictions for the fracture initiation and the behavior of the nanotube prior to fracture. Besides, the specific potential has been extensively used in the literature for predicting Fig. 9. Predicted fracture evolution in the chiral (16,8) tube containing a the mechanical properties and behavior of CNTs with suc- SW defect. cess. Nevertheless, it is in the authors’ interests to incorporate many-body potentials in the PFM and compare their performance in various cases against the pairwise modified 6. Conclusions Morse potential.

In the present paper, a study was accomplished on the References effect of SW defect on the tensile behavior and fracture of SWCNTs. It was found that the SW defect serves as [1] Ebbesen TW, Takada T. Topological and sp3 defect structures in nucleation site for fracture. Its effect on the tensile behavior nanotubes. Carbon 1995;33(7):937–78. of SWCNTs concerns the failure stress and failure strain [2] Mielke SL, Troya D, Zhang S, Li J-L, Xiao S, Car R, et al. The role of vacancy defects and holes in the fracture of carbon nanotubes. and depends solely on nanotube chirality. In armchair Chem Phys Lett 2004;390:413–20. SWCNTs, there is a significant reduction in failure stress [3] Yao Z, Postma HWC, Balents L, Dekker C. Carbon nanotube and failure strain ranging from 18% to 25% and from intramolecular junctions. Nature (London) 1999;402:273–6. PHYSICAL REVIEW B 76, 024104 ͑2007͒

Hydrogen sorption in defective hexagonal BN sheets and BN nanotubes

S. A. Shevlin*,† and Z. X. Guo*,‡ Department of Materials, Queen Mary, University of London, Mile End Road, London E1 4NS, United Kingdom ͑Received 2 March 2007; revised manuscript received 2 May 2007; published 11 July 2007͒ We perform ab initio simulations on the interaction of molecular hydrogen with the native and substitutional defects of single hexagonal boron-nitride sheets and small-diameter ͑8,0͒ nanotubes. We ﬁnd that the adsorption of molecular hydrogen on both types of structure is endothermic with respect to dissociation, with the small-diameter nanotube possessing the smaller barrier. Although chemisorption along the tube axis is energetically preferred, the barrier for dissociation is lower for chemisorption across the tube axis, implying that chemisorbed hydrogen can be kinetically trapped in a higher energy state. Dopants that maximize the localization of the highest occupied molecular orbital and lowest unoccupied molecular orbital states maximize hydrogen binding energies. Carbon dopants do not enhance H2 binding in contrast to the literature, whereas silicon dopants for nitrogen provide H2 binding energies of 0.8 eV, at the upper end of the range required to meet DOE targets for hydrogen storage. The formation energy of most defects is reduced with increasing curvature except for the carbon substitutionals. Vacancies reduce the barriers for H2 dissociation for the planar sheets but not for strongly curved nanotubes. The surface stress induced by nanotube curvature boosts the hydrogen storage capabilities of vacancies compared to the sheet, with the nitrogen vacancy chemisorbing 4H and allowing a H2 molecule to enter the interior of the tube. The hydrogen binding properties of boron-nitride systems are strongly dependent on the defects and dopants present. Pretreating of these systems so as to partially remove nitrogen should enhance H2 adsorption properties.

DOI: 10.1103/PhysRevB.76.024104 PACS number͑s͒: 81.07.De, 71.15.Mb, 68.43.Bc

I. INTRODUCTION nanotube ͑BNT͒, analogous to carbon nanotubes. Unfortu- Energy is a major issue for the world today. Energy con- nately, in these systems, the binding of molecular hydrogen to the boron-nitride substrate is very weak, about sumption is intimately linked with CO2 emission, a major 15 human contributor to undesirable climate change. Several −0.09 eV/H2. For practical vehicular hydrogen storage ap- energy solutions are available, including large increases in plications, the molecular binding energy should be of order energy efficiency, energy generation by renewable sources, −0.2 to −0.7 eV/H2. and increasing usage of nuclear power. However, fossil fuels There are several papers that discuss the simulation of with a high-energy density will still be a dominant feature in hydrogen storage mechanisms in BN nanosystems. Mårlid et al.16 performed density functional theory local density ap- the energy economy in the near future. If CO2 emission is to ͑ ͒ be rapidly reduced, there should be a switchover to a clean proximation DFT-LDA simulations on the adsorption of atomic hydrogen and fluorine on a h-BN cluster. They found energy carrier, of which hydrogen is particularly powerful.1,2 that adsorption on boron atoms causes a local sp2 to sp3 One of the main problems limiting the use of hydrogen for 1–4 transformation but that adsorption on nitrogen does not oc- energy applications is the difficulty of storing it safely. A cur. Wu et al.17 considered the chemisorption of isolated hy- promising series of hydrogen storage materials are systems drogen atoms on an ͑8,0͒ BNT with DFT-LDA, finding that that possess a fullerene structural motif, such as carbon or hydrogen prefers to chemisorb along the tube axis in an arm- boron-nitride nanotubes. Hydrogen adsorption on these sys- chair chain structure with alternating adsorption on B and N 5,6 tems can be modified via external pressure or transition atoms. The largest adsorption energy was found to corre- 7–9 ϳ metal doping. spond to a 50% H2 coverage, storing 4wt%H2, a result Boron-nitride nanotubes are particularly attractive be- also confirmed by the work of Han et al.18 with DFT-PW-91. cause, as opposed to carbon nanotubes, their electronic prop- Wu et al.19 modeled the effects of tube deformation on hy- erties are independent of helicity, diameter, and number of drogen atom adsorption, finding that increasing deformation walls, and as well they have a strong tendency to form zig- increases the adsorption energy of H atoms on N. Jhi and 10,11 20 zag nanotubes. The hydrogen storage capacity of multi- Kwon found from DFT-PBE simulation that H2 physisorbs wall BN nanotubes has been measured to be 1.8–2.6 wt %,12 weakly onto h-BN and BNTs and does not dissociate, with a at a pressure of 10 MPa at room temperature, with 70% of molecular binding energy larger than for the equivalent ad- the hydrogen chemisorbed to the nanotube and the rest phy- sorption on graphite or carbon nanotubes. The addition of sisorbed. Compared to carbon systems, boron nitride can carbon dopants or Stone-Wales defects increased the H2 store hydrogen at elevated temperatures.12,13 The addition of binding energy. Modifications of the sp2 binding induced by metal nanoparticles such as platinum causes the creation of these types of defects were determined to be responsible for voids and other point defects in the tubes that boost hydro- the increase in binding strength. Wu et al.21 performed DFT- gen storage capacity to approximately 4.2 wt %,14 close to PW-91 simulations on the effects of several native defects on the US DOE targets for vehicular hydrogen storage for 2007. hydrogen dissociation, using the nudged elastic band ͑NEB͒ ͑ ͒ Boron nitride can exist in a hexagonal graphitelike struc- method to determine the minimum energy path MEP of H2 ture ͑h-BN͒ that can be rolled up to form a boron-nitride dissociation. They found that the barrier for hydrogen disso-

1098-0121/2007/76͑2͒/024104͑11͒ 024104-1 ©2007 The American Physical Society HYDROGEN SORPTION IN DEFECTIVE HEXAGONAL BN… PHYSICAL REVIEW B 76, 024104 ͑2007͒ reducing the energetic penalty for atoms surrounding the va- quences for the usage of these materials for hydrogen storage cancy to adopt a locally sp3 structure allowing the twofold and other applications such as nanoscale electronics. atoms bordering the vacancy to adsorb more hydrogen and catalyzing spillover hydrogen storage. ACKNOWLEDGMENTS In conclusion, we have shown via detailed simulation that the hydrogen binding properties of nanostructured boron- This work was supported by the EPSRC under the UK nitride systems are strongly dependent on the defects and Sustainable Hydrogen Energy Consortium ͑GR/S26965/01 dopants present, with both covalent and noncovalent binding and EP/E040071/1͒ and a Platform Grant ͑GR/S52636/01 behaviors observed. This variation has important conse- and EP/E046193/1͒.

*Corresponding authors. ͑2004͒. †[email protected] 20 S.-H. Jhi and Y.-K. Kwon, Phys. Rev. B 69, 245407 ͑2005͒. ‡[email protected] 21 X. Wu, J. Yang, J. G. Hou, and Q. Zhu, J. Chem. Phys. 124, 1 The Energy White Paper: Our energy future—creating a low car- 054706 ͑2006͒. bon economy, Department of Trade and Industry, UK, 2003. 22 G. Kresse and J. Furthmüller, http://cms.mpi.univie.ac.at/VASP 2 V. M. Vishyakov, Vacuum 80, 1053 ͑2006͒. 23 J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. 3 Y. Song, Z. X. Guo, and R. Yang, Phys. Rev. B 69, 094205 Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev. B 46, 6671 ͑2004͒. ͑1992͒. 4 L. Schlapbach and A. Züttel, Nature ͑London͒ 414, 353 ͑2001͒. 24 W. Paskowicz, J. B. Pelka, M. Knapp, T. Szyszko, and S. Pod- 5 O. Gülseren, T. Yildirim, and S. Ciraci, Phys. Rev. Lett. 87, siadlo, Appl. Phys. A: Mater. Sci. Process. 75, 431 ͑2002͒. 116802 ͑2001͒. 25 D. C. Langreth, M. Dion, H. Rydberg, E. Schröder, P. Hyldgaard, 6 X. Wu, J. Yang, J. G. Hou, and Q. Zhu, Phys. Rev. B 69, 153411 and B. I. Lundqvist, Int. J. Quantum Chem. 101, 599 ͑2004͒. ͑2004͒. 26 J. Henkelmann and H. Jónsson, J. Chem. Phys. 113, 9978 ͑2000͒. 7 Y. Zhao, Y-H. Kim, A. C. Dillon, M. J. Heben, and S. B. Zhang, 27 G. Henkelman, B. P. Uberuaga, and H. Jónsson, J. Chem. Phys. Phys. Rev. Lett. 94, 155504 ͑2005͒. 113, 9901 ͑2000͒. 8 T. Yildirim and S. Ciraci, Phys. Rev. Lett. 94, 175501 ͑2005͒. 28 T. M. Schmidt, R. J. Baierle, P. Piquini, and A. Fazzio, Phys. Rev. 9 S. A. Shevlin and Z. X. Guo, Appl. Phys. Lett. 89, 153104 B 67, 113407 ͑2003͒. ͑2006͒. 29 S.-H. Jhi, Phys. Rev. B 74, 155424 ͑2006͒. 10 H. J. Xiang, J. L. Yang, J. G. Hou, and Q. S. Zhu, Phys. Rev. B 30 R. Q. Zhang, S. T. Lee, C.-K. Law, W.-K. Li, and B. K. Teo., 68, 035427 ͑2003͒. Chem. Phys. Lett. 364, 251 ͑2002͒. 11 M. Terauchi, M. Tanaka, K. Suzuki, A. Ogino, and K. Kimura, 31 S. Guerini, T. Kar, and P. Piquini, Eur. Phys. J. B 38, 515 ͑2004͒. Chem. Phys. Lett. 324, 359 ͑2000͒. 32 A. V. Krasheninkov, P. O. Lehtinen, A. S. Foster, and R. M. 12 R. Z. Ma, Y. Bando, H. W. Zhu, T. Sato, C. L. Xu, and D. H. Wu, Nieminen, Chem. Phys. Lett. 418, 132 ͑2006͒. J. Am. Chem. Soc. 124, 7862 ͑2002͒. 33 A. V. Krasheninnikov, K. Nordlund, and J. Keinonen, Phys. Rev. 13 R. Z. Ma, Y. Bando, T. Sato, D. Golberg, H. W. Zhu, C. L. Xu, B 65, 165423 ͑2002͒. and D. H. Wu, Appl. Phys. Lett. 81, 5225 ͑2002͒. 34 S. Mielke, D. Troya, S. Zhang, J. L. Li, S. Xiao, R. Car, R. S. 14 C. Tang, Y. Bando, X. Ding, S. Qi, and D. Golberg., J. Am. Ruoff, G. C. Schatz, and T. Belytschko, Chem. Phys. Lett. 390, Chem. Soc. 124, 14550 ͑2002͒. 413 ͑2004͒. 15 S.-H. Jhi and Y.-K. Kwon, Phys. Rev. B 69, 245407 ͑2004͒. 35 A. V. Krasheninnikov, F. Banhart, J. X. Li, A. S. Foster, and R. 16 B. Mårlid, K. Larsson, and J.-O. Carlsson, J. Phys. Chem. 103, M. Nieminen, Phys. Rev. B 72, 125428 ͑2005͒. 7637 ͑1999͒. 36 J. Li, G. Zhou, H. Liu, and W. Duan, Chem. Phys. Lett. 426, 148 17 X. Wu, J. Yang, J. G. Hou, and Q. Zhu, J. Chem. Phys. 121, 8481 ͑2006͒. ͑2004͒. 37 Y.-H. Kim, Y. Zhao, A. Williamson, M. J. Heben, and S. B. 18 S. S. Han, S. H. Lee, J. K. Kang, and H. M. Lee, Phys. Rev. B 72, Zhang, Phys. Rev. Lett. 96, 016102 ͑2006͒. 113402 ͑2005͒. 38 S. M. Lee, K. Hyeok, Y. Lee, G. Seifert, and T. Frauenheim, J. 19 X. Wu, J. Yang, J. Hou, and Q. Zhu, Phys. Rev. B 69, 153411 Am. Chem. Soc. 123, 5059 ͑2001͒.

024104-11 Computational Materials Science 40 (2007) 147–158 www.elsevier.com/locate/commatsci

Atomistic-continuum and ab initio estimation of the elastic moduli of single-walled carbon nanotubes

Karthick Chandraseker, Subrata Mukherjee *

Department of Theoretical and Applied Mechanics, Kimball Hall, Cornell University, Ithaca, NY 14853, United States

Received 4 October 2006; accepted 29 November 2006

Abstract

This paper deals with the calculation of elastic moduli and stress–strain curves for single-walled carbon nanotubes (SWNTs) using a computationally efficient, atomistically enriched continuum analysis. This approach is adopted to estimate shear and Young’s moduli and obtain stress–strain curves for carbon nanotubes (CNTs) subject to coupled extension and twist deformations. This accounts for the effects of natural extension-twist coupling [K. Chandraseker, S. Mukherjee, ASME J. Appl. Mech. 73 (2) (2006) 315–326; K. Chand- raseker, S. Mukherjee, Y.X. Mukherjee, Int. J. Solids Struct. 43 (2006) 7128–7144] on SWNT constitutive properties. The constitutive properties are evaluated by assuming a cylindrical reference configuration [Chandraseker and Mukherjee, 2006; Chandraseker et al., 2006] rather than a planar graphene sheet [P. Zhang, Y. Huang, P.H. Geubelle, P.A. Klein, K.C. Hwang, Int. J. Solids Struct. 39 (2002) 3893–3906; M. Arroyo, T. Belytschko, Phys. Rev. B 69 (2004) 115415] thereby allowing for the anisotropy and change in strain energy that results from the finite deformation required to roll up a graphene sheet into a nanotube [M. Arroyo, T. Belytschko, Phys. Rev. B 69 (2004) 115415]. The Tersoff–Brenner multi-body empirical interatomic potential for carbon [J. Tersoff, Phys. Rev. B 37 (1988) 6991–7000; D.W. Brenner, Phys. Rev. B 42 (1990) 9458–9471] is used to model the C–C bond energies in this work. This enables exact analytic evaluation of the derivatives of the strain energy density rather than a numerical approach. Consistent values obtained corresponding to these material properties indicate that they do not depend strongly on the chirality of the nanotube [R. Saito, G. Dressel- haus, M.S. Dresselhaus, Physical Properties of Carbon Nanotubes, Imperial College Press, London, 1998 [7]]. The relative magnitudes of the Young’s and shear moduli obtained from this approach fall within the well known range in classical elasticity theory in most cases, and the computed values for the moduli agree well with existing experimental results and atomistic studies that employ the same interatomic potential. Further, in the present work, the moduli are also evaluated using a more accurate, albeit computationally expensive, ab initio density-functional-theoretic (DFT) approach (see for e.g., [D. Sańcez-Portal, E. Artacho, J.M. Soler, A. Rubio, P. Ordejoń, Phys. Rev. B 59 (1999) 12678; K.N. Kudin, G.E. Scuseria, B.I. Yakobson, Phys. Rev. B 64 (2001) 235406]). A comparison between these values and the ones from the atomistic-continuum analysis brings to notice some of the advantages and limitations of both these approaches. 2006 Elsevier B.V. All rights reserved.

Keywords: Quasicontinuum; Atomistic-continuum; Cauchy–Born rule; Membrane; Nonlinear elasticity; Ab initio calculations

1. Introduction tions of brittle fracture [10] and computational predictions of the onset of plastic deformations [11,12]. Characteriza- Carbon nanotubes are known to possess a remarkable tion of the constitutive properties of carbon nanotubes ability to sustain large elastic deformations without deve- through elastic moduli such as Young’s modulus, Poisson’s loping lattice defects, in spite of some experimental observa- ratio, or the ﬂexural rigidity has been the subject of several studies [8,9,13–15]. Several computational approaches have been adopted to extract elastic constitutive properties from * Corresponding author. Tel.: +1 607 255 7143; fax: +1 607 255 2011. atomistic models, such as tight-binding models [19], ab E-mail address: [email protected] (S. Mukherjee). initio calculations [8,9,20], or use of analytic potentials

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Failure strength of brittle materials containing nanovoids

Mariella Ippolito,1,2 Alessandro Mattoni,2 Nicola Pugno,3 and Luciano Colombo1,2,* 1Department of Physics, University of Cagliari, Cittadella Universitaria, I-09042 Monserrato, Cagliari, Italy 2Sardinian Laboratory for Computational Materials Science (SLACS, CNR-INFM), c/o Department of Physics, University of Cagliari, Cittadella Universitaria, I-09042 Monserrato (Ca), Italy 3Department of Structural Engineering and Geotechnics, Politecnico di Torino, I-10138 Torino, Italy ͑Received 4 August 2006; revised manuscript received 30 March 2007; published 15 June 2007͒ By means of atomistic simulations, we investigate the failure strength in plane strain conditions of a brittle solid containing nanosized stress concentrators, i.e., a straight crack, a cylindrical hole, or a spherical hole. We ﬁnd that the failure strength of the defected solid strongly depends on the defect size, in contrast with the predictions of standard elasticity theory. A high strength reduction due to voids as large as few atoms is observed. Such results have been included in two analytical failure criteria, namely, the average stress criterion and the point stress criterion. Both models introduce a length scale typical of the system, tailored at describing the process zone near the nanovoids. We provide a numerical estimate for this length scale, which is found to be speciﬁc for any defect, and we reconcile atomistic results to continuum into a coherent picture.

DOI: 10.1103/PhysRevB.75.224110 PACS number͑s͒: 62.25.ϩg, 62.20.Mk, 81.40.Np

I. INTRODUCTION theory predicts that the failure from a void ͑as it is the case of cylindrical or spherical holes͒ takes place when the maxi- ␴ 5 Defects such as cracks and voids affect the mechanical mum local stress equals the ideal material strength th. Both behavior of brittle solids since they modify the overall alternative continuum approaches ͑for cracks and voids͒ are strength of the material. Sometimes such defects are un- based on linear elasticity and they unlikely work at the avoidable because they form during materials synthesis and nanoscale. Their possible weaknesses could, in principle, be processing such as, e.g., ceramic sintering. On the other side, due to the failure of at least one of the three underlying voids may be introduced into the material by design in order ͑constitutive͒ hypotheses they rely on: either continuum me- to obtain specific properties. This is the case of porous ma- chanics, elasticity, or linearity. terials where pores at a suitable concentration are used to In order to improve classical continuum models, modern control the thermal or acoustic isolation, the impact energy theories of fracture are generally formulated so as to incor- absorption, and many other properties.1 In any case, such porate into their formalism a suitable material length scale ␭: inhomogeneities are of great relevance on the mechanical this key quantity is aimed at describing a process zone close response of the system, since they enhance the local stress to the crack tip where at least one of the above constitutive and they possibly may initiate failure. In addition, as the hypotheses fails. The characteristic length scale is typically technological demand for extremely high strength materials given by increases ͑as well as the development of nanoscale devices or machines͒, defects as small as a few nanometers cannot be 2K2 ␭ ϳ c . ͑1͒ neglected. As an example, it has been recently found that ␲␴2 th even one- or two-atom vacancy defect can reduce the failure strength of carbon nanotubes by an amount of 26%.2,3 We The interpretation of the length ␭ is not unique and it could will show that sizable strength reduction due to voids as be related to the existence of either a plastic zone ͑i.e., the large as few atoms are observed in bulk ␤-SiC as well. mechanical response is beyond pure elasticity͒, a cohesive The strength of materials containing cracks and voids is zone ͑linearity is lost͒, or a discrete unit for crack advance- traditionally described according to stress intensification or ment ͑continuum hypothesis is no longer applicable͒. stress concentration arguments, respectively.4,5 The need of In the framework of brittle fracture formalism, the char- different approaches is motivated, according to linear elastic acteristic length ␭ has been incorporated in four different fracture mechanics ͑LEFM͒, by the mathematical divergence models. The point stress criterion9,10 ͑PSC͒ assumes that the ␴ of the stress field near the crack tip. Following LEFM, load- failure occurs if the stress becomes equal to th at a suitable ing produces a 1/ͱx singularity at the crack tip ͑where x is distance l from the notch, corresponding to lϳ␭/4. An al- the distance from the crack tip along the plane of the crack͒ ternative approach is the average stress criterion9,10 ͑ASC͒ and a critical stress equal to zero is expected. As a conse- according to which the failure occurs if the mean value of the quence, a straightforward prediction of failure stress as stress along a line ͑or a surface, or a volume͒ starting at the ␴ uniquely based on local stress criteria cannot be applied. The notch root is equal to th; the length l of such a line is in this critical stress of the cracked body is therefore calculated by case as large as ␭. Furthermore, the equivalent linear elastic analyzing the stress singularity at the crack tip: the failure fracture mechanics ͑equivalent LEFM͒4 assumes the exis- takes place when the stress intensity factor K is equal to the tence of a crack at the root of the notch ͑i.e., the effective 6,7 ͒ material fracture toughness Kc. This criterion relies on the crack length is longer than its original size : the failure is energy balance of the Griffith theory.8 In contrast, elasticity predicted to occur when this effective crack reaches the criti-

1098-0121/2007/75͑22͒/224110͑7͒ 224110-1 ©2007 The American Physical Society FAILURE STRENGTH OF BRITTLE MATERIALS… PHYSICAL REVIEW B 75, 224110 ͑2007͒ teristic length actually depends on the defect type. Its value ACKNOWLEDGMENTS is estimated to be 0.60 nm for cracks, 0.66 nm for cylindrical holes, and 0.79 nm for spherical voids. The larger process This work has been funded by MIUR under projects zones associated with holes are interpreted as an evidence of “PROMOMAT” and PON “CyberSar.” We also acknowledge a higher ﬂaw-tolerance regime for holes and/or voids than computational support by CASPUR ͑Rome, Italy͒ computing for cracks. center.

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224110-7 RAPID COMMUNICATIONS

PHYSICAL REVIEW B 75, 201405͑R͒͑2007͒

Modeling of fracture of carbon nanotubes with vacancy defect

Q. Wang, W. H. Duan, and N. L. Richards Department of Mechanical and Manufacturing Engineering, University of Manitoba, Winnipeg, Manitoba, Canada R3T 5V6

K. M. Liew Department of Building and Construction, City University of Hong Kong, Hong Kong, China ͑Received 27 February 2007; published 31 May 2007; corrected 6 June 2007͒ The fracture of achiral carbon nanotubes ͑CNTs͒ with one atomic vacancy is studied. Nonuniform variation of the fracture strain is observed from molecular mechanics simulations, and an elastic shell model is developed to describe the research findings. Hardening and softening domains near the edges of the tubes are specifically positioned. The effectiveness of the continuum mechanics model is further verified by molecular mechanics simulations via the MATERIALS STUDIO software package. In addition, the dependence of hardening or softening domains on the length and diameter of the CNTs is investigated. The rupture progress of defected carbon nanotubes with the vacancy at different locations is observed from molecular mechanics simulations, and adequate physical interpretations of the block-tearing fracture mode are provided.

DOI: 10.1103/PhysRevB.75.201405 PACS number͑s͒: 62.20.Mk, 31.15.Qg, 62.25.ϩg, 81.07.De

Carbon nanotubes ͑CNTs͒ can be viewed as one ͑or more͒ some extent in all molecular mechanics simulations, findings graphite sheet͑s͒ rolled into a seamless tube. The way the on the nonuniform variation of the fracture strain with re- graphite sheet is wrapped is represented by a pair of indices spect to the location of the vacancy have seldom appeared; ͑n,m͒ that is called the chirality. When the indices are set as we have observed this from molecular mechanics ͑MM͒ n=m and m=0, the nanotubes are called “armchair” and simulations. Such an interesting and significant phenomenon “zigzag,” respectively, or simply achiral CNTs. Existing needs to be investigated through an effective model. In ad- studies have shown that CNTs exhibit superior mechanical dition, a study of the rupture progress of CNTs is also indis- properties over other known materials and hold substantial pensable to enable a thorough understanding of the fracture promise as fibers in composites and other devices,1–3 since of vacancy-defected CNTs. they were discovered in 1991.4 However, the superior me- In this Rapid Communication, an elastic shell model is chanical properties hold only for perfect CNTs. When CNTs developed to investigate the variation of the fracture strain of have defects in the atomic network, degradation of the out- vacancy-defected CNTs with respect to the location of the standing mechanical properties has been identified.5–7 defect. The prediction of the fracture strain from the con- Defects in CNTs can be initiated at the stage of CNT tinuum mechanics model is verified by MM simulations. The growth and purification, or during device or composite observation of a distinct rupture progress of the defected production8 by chemical treatment or by irradiation.9 There CNTs is also reported and interpreted via continuum me- are different types of defects in CNTs, such as Stone-Wales chanics. defects, bond-breaking defects, and single vacancies and In our MM simulations, the force field used to model the their derivative point defects.10 Among these types of de- interatomic interactions in MATERIALS STUDIO is a fects, the effect of atomic vacancies has been studied condensed-phase optimized molecular potential for atomistic extensively.5–7,11,12 A single atomic vacancy in a CNT can be simulation studies ͑COMPASS͒ force field,15 which is the simulated by removing a single atom and the three associated first ab initio force field that was parametrized and validated bonds. The configuration of the single vacancy is referred to using condensed-phase properties, and has been proved to be as a nonreconstructed defect. This configuration is meta- applicable in describing the mechanical properties of CNTs. stable but can survive for macroscopic times at low In MM calculations, the configurations are optimized at zero temperatures.13 On the other hand, the removal of carbon temperature to avoid thermal effects. Our MM simulations ␧ atoms from the hexagonal network of the CNT creates a show that the fracture strains pristine for a pristine armchair number of carbon atoms with unsaturated valence orbitals. ͑10,10͒ and zigzag ͑15,0͒ CNT are 41% and 34%, respec- The excess energy arising from the unsaturated valence or- tively. The fracture strains are in good agreement with the bitals promotes reconstructions local to the vacancy.14 The predictions by Yakobson et al.16 and Mielke et al.,11 which effects of both the nonreconstructed and reconstructed are 42% for a ͑11,3͒ CNT and 30% for a ͑5,5͒ CNT. Figure atomic vacancy defects on mechanical properties of CNTs, 1 shows the strain energy per atom versus the strain using the especially the fracture of defected CNTs, have been investi- potential function for ͑15,0͒ and ͑10,10͒ CNTs with single gated. Sammalkorpi et al.5 studied how the Young’s modulus vacancies at the centers of the tubes and the length to diam- and tensile strength of CNTs with vacancy defects depend on eter ratio L/d=10 that are subjected to an axial tensile load- the defect characteristics by employing molecular mechanics ing and fixed at the two boundaries. In the simulations of simulations and a simple elastic rod theory. Their results defected CNTs, the single vacancy is located at the center of showed that the tensile strength and fracture strain of a the tubes to avoid edge effects. The bond breaking can be single-walled CNT decrease by nearly a factor of 2 if a non- determined from the drop of the strain energy due to the reconstructed vacancy is present. Although the available release of energy at the corresponding fracture. Hence, the fracture strain for defected CNTs was found to decrease to occurrence of the first bond breaking can be observed at a

WANG et al. PHYSICAL REVIEW B 75, 201405͑R͒͑2007͒

0.28 (5%)

f 0.26 ε L=24.87nm 0.24 L=10.11nm (a) 0.22 0 0.05 0.1 0.15 0.2 (20%) η

0.24 L=24.11nm (40%) 0.23 L=15.59nm f ε 0.22 0.21 (b) 0.2 0 0.05 0.1 0.15 0.2 (75%) η FIG. 3. Fracture strains versus the location of the defect for ͑a͒ armchair ͑20,20͒ and ͑b͒ zigzag ͑30,0͒ CNTs with single vacan- (100%) cies. Discrepancy in location of the hardening and softening domains is obviously seen from the peak locations in the two figures. (a) (b) ͑ ͒ The hardening domains of the 20,20 CNT with L=24.87 and FIG. 4. ͑Color online͒ Rupture progress of a ͑10,10͒ CNT with 10.11 nm cover about 3% and 7% of the length, respectively. The a defect located ͑a͒ around the center of the CNT and ͑b͒ near the ͑ ͒ hardening domains of the 30,0 CNT with L=24.11 and 15.59 nm end of the CNT. Fracture occurs at the center of the tube in case ͑a͒. cover about 2% and 4% of the length, respectively. The dependence For the tube in case ͑b͒, an edge block-tearing phenomenon is ob- of hardening or softening domains on the length and diameter of the served due to the force redistribution and defect-location effects. CNT is discussed in text. engineering perspective. From our MM simulations, the frac- to the edge as shown in Fig. 4͑b͒. The special fracture mode ture at the final step exhibits a sudden failure, showing a is of significance as knowledge of the simultaneous fracture brittle fracture process.6 However, when the defect is at the at the defect location and the edge of CNTs will enable more edge of the tube, the rupture progress reveals a completely understanding of the mechanical behavior of the materials, different route. With increase of the intensity of the cross- especially the fracture of the edge portion of a CNT when sectional breaking during the rupture process, the occurrence interacting with another substrate. of bond breaking at the edge is observed. The bond breaking In summary, an elastic shell model is developed to study initiated at the edge propagates toward the defect location, the fracture of achiral CNTs with one atomic vacancy. With leading to the fracture mode of a block-tearing configuration the developed model, hardening and softening effects are seen in Fig. 4͑b͒. Such a fracture mode, different from the found and the corresponding effective domains are observed. one shown in Fig. 4͑a͒ when the defect is at the center of the MM simulations are conducted to verify the effectiveness of tube, is due to force redistribution and defect-location ef- the developed model. In addition, the dependence of the fects. From elastic beam theory, the force on the tube will hardening or softening domains on the length and diameter redistribute at the partially broken cross section, leading to of CNTs is investigated. A block-tearing fracture mode is an induced moment in the direction of the defect. The in- observed via MM simulations when the defect is close to the duced moment will thus stimulate a tensile stress on the edge edge of the tubes. A physical interpretation of the phenom- at the position opposite to the defect. In addition to the in- enon is provided. duced tensile stress on the edge, the bending rigidity of the beam element under tension becomes smaller when the de- Q.W. appreciates support from the Canada Research Chair fect location is toward the edge. As such, the induced tensile ͑CRC͒ program from the Canadian government and the Uni- stress undergoes a larger deformation on the edge when the versity of Manitoba. The work described in this paper was defect is closer to the edge. Thus, the edge block-tearing supported by the City University of Hong Kong Strategic phenomenon is observed only when the defect is very close Research Grant ͑Project No. 7002080͒.

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201405-4 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2007; 70:913–933 Published online 2 November 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1895

A bridging domain and strain computation method for coupled atomistic–continuum modelling of solids

Sulin Zhang1,2, Roopam Khare1, Qiang Lu1 and Ted Belytschko1,∗,†

1Department of Mechanical Engineering, 2145 N. Sheridan Rd., Northwestern University, Evanston, IL 60208-3111, U.S.A. 2Department of Mechanical Engineering, 204 Mechanical Engineering Building, University of Arkansas, Fayetteville, AR 72701, U.S.A.

SUMMARY We present a multiscale method that couples atomistic models with continuum mechanics. The method is based on an overlapping domain-decomposition scheme. Constraints are imposed by a Lagrange multiplier method to enforce displacement compatibility in the overlapping subdomain in which atomistic and continuum representations overlap. An efficient version of the method is developed for cases where the continuum can be modelled as a linear elastic material. An iterative scheme is utilized to optimize the coupled configuration. Conditions for the regularity of the constrained matrices are determined. A method for computing strain in atomistic models and handshake domains is formulated based on a moving least-square approximation which includes both extensional and angle-bending terms. It is shown that this method exactly computes the linear strain field. Applications to the fracture of defected single-layer atomic sheets and nanotubes are given. Copyright q 2006 John Wiley & Sons, Ltd.

Received 10 August 2006; Accepted 30 August 2006

KEY WORDS: bridging domain; graphene sheets; moving least square; molecular mechanics

1. INTRODUCTION

Modelling of material failure often involves phenomena simultaneously occurring at multiple length scales. On the one hand, such behaviour cannot be described by continuum mechanics without

∗Correspondence to: Ted Belytschko, Department of Mechanical Engineering, 2145 N. Sheridan Rd., Northwestern University, Evanston, IL 60208-3111, U.S.A. †E-mail: [email protected]

Contract/grant sponsor: NASA University Research, Engineering and Technology Institute on Bio Inspired Materials (BIMat); contract/grant number: NCC-1-02037

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Copyright q 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 70:913–933 DOI: 10.1002/nme Materials Science and Engineering A 454–455 (2007) 170–177

A molecular-mechanics based ﬁnite element model for strength prediction of single wall carbon nanotubes M. Meo ∗, M. Rossi Material Research Center, Department of Mechanical Engineering, University of Bath, Bath, Ba2 7AY, UK Received 1 August 2005; accepted 1 November 2006

Abstract The aim of this work was to develop a finite element model based on molecular mechanics to predict the ultimate strength and strain of single wallet carbon nanotubes (SWCNT). The interactions between atoms was modelled by combining the use of non-linear elastic and torsional elastic spring. In particular, with this approach, it was tried to combine the molecular mechanics approach with finite element method without providing any not-physical data on the interactions between the carbon atoms, i.e. the CC-bond inertia moment or Young’s modulus definition. Mechanical properties as Young’s modulus, ultimate strength and strain for several CNTs were calculated. Further, a stress–strain curve for large deformation (up to 70%) is reported for a nanotube Zig-Zag (9,0). The results showed that good agreement with the experimental and numerical results of several authors was obtained. A comparison of the mechanical properties of nanotubes with same diameter and different chirality was carried out. Finally, the influence of the presence of defects on the strength and strain of a SWNT was also evaluated. In particular, the stress–strain curve a nanotube with one-vacancy defect was evaluated and compared with the curve of a pristine one, showing a reduction of the ultimate strength and strain for the defected nanotube. The FE model proposed demonstrate to be a reliable tool to simulate mechanical behaviour of carbon nanotubes both in the linear elastic field and the non-linear elastic field. © 2007 Published by Elsevier B.V.

Keywords: Carbon nanotubes; Molecular mechanics; Young’s modulus; Mechanical properties; Finite element analysis

1. Introduction ratio around 0.14–0.28 was reported depending on the approach and the energy potential used. Further, experimental data of 15 Since their discovery carbon nanotubes [1] have attracted SWNT bundles under tensile load showed that, the Young’s considerable attention in scientific communities. This is partly modulus ranged from 0.32 up to 1.47 TPa with an average of due to their remarkable mechanical, electrical and thermal 1.02 TPa. The tensile strength ranged from 13 to 53 GPa [4].In properties. In particular, material composites such as carbon nan- the case of MWNTs a Young’s modulus of 0.9 TPa was estimated otube, nanoparticle-reinforced polymers and metals have shown by conducting pulling and bending tests [5]. potentially wide application. Computational simulation for predicting mechanical prop- Specifically to mechanical properties, single wall nanotubes erties of CNTs has been recognised to be a powerful tool to (SWNTs) have the highest Young’s modulus about 1 TPa, if overcome the difficulties arising from the measurements of normalized to their diameter, and this is one of the main reason nanoscale dimensions. Several approaches can be used to eval- why carbon nanotubes (CNTs) have attracted much interest for uate the mechanical properties of SWNT and MWNT [6]. Xiao low weight structural composites [2]. et al. [8] found a tensile strength for Armchair (126.2 GPa) A detailed summary of CNTs mechanical properties can be and Zig-Zag (94.5 GPa) with a maximum strain of 23.1% and found in [3]. A Young’s modulus for SWNTs and multi wall nan- 15.6–17.5%, respectively. Natsuki and Endo [12], predicted the otubes (MWNTs) was reported to be 1.25 TPa, while a Poisson maximum stress to be around 70 GPa at 11% of strain for the Zig- Zag nanotube and 88 GPa at 15% for Armchair. Sun [13] found a tensile strength from 77 GPa (Zig-Zag) up to 101 GPa (Arm- ∗ Corresponding author. Tel.: +44 1234 750111x5220; fax: +44 1234 752149. chair). Further, an independence of the tensile strength from E-mail address: [email protected] (M. Meo). nanotube diameter was found.

[16] M. Sammalkorpi, A. Krasheninnikov, A. Kuronen, K. Nordlund, K. Kaski, [22] A.K. Rappe’, C.J. Casewit, Molecular Mecahnics Across Chemistry, Uni- Phys. Rev. B 70 (2004) 245416. versity Science Books ed., University Science Books, Suasalito, CA, [17] S.L. Mielke, D. Troya, S. Zhang, Li Je-Luen, S. Xiao, R. Car, R.S. Ruoff, 1997. G.C. Schatz, T. Beliytschko, Chem. Phys. Lett. 390 (2004) 413–420. [23] K.L. Lau, M. Chipara, H.Y. Ling, D. Hui, Composites B: Eng. 35 (2004) [18] L. Nasdala, G. Ernst, Comput. Mater. Sci. 33 (2005) 443–458. 95–101. [19] M.S. Dresselhaus, G. Dresselhaus, R. Saito, Carbon 33 (7) (1995) 883–891. [24] G.M. Odegard, T.S. Gates, L.M. Nicholson, K.E. Wise, Compos. Sci. Tech- [20] J. Koloczek, K. Young-Kyun, A. Burian, J. Alloys Compd. 328 (2001) nol. 62 (2002) 1869–1880. 222–225. [25] ANSYS Inc. Theory Manual, SAS IP Inc., 2003. [21] K. Machida, Principles of Molecular Mechanics, Wiley ed., Wiley and [26] K.M. Liew, X.Q. He, C.H. Wong, Acta Mater. 52 (2004) 2521–2527. Kodansha, 1999. Using theory and computation to model INAUGURAL ARTICLE nanoscale properties

George C. Schatz*

Department of Chemistry, Northwestern University, Evanston, IL 60208-3113

This contribution is part of the special series of Inaugural Articles by members of the National Academy of Sciences elected on May 3, 2005.

Contributed by George C. Schatz, March 9, 2007 (sent for review February 20, 2007)

This article provides an overview of the use of theory and com- sometimes be synthesized with molecular perfection, and when this putation to describe the structural, thermodynamic, mechanical, is the case, the resulting properties of these structures are often and optical properties of nanoscale materials. Nanoscience pro- remarkable, including carbon nanotubes that are 10 times stronger vides important opportunities for theory and computation to lead than steel, nanoscale particles that can enhance the spectroscopic in the discovery process because the experimental tools often properties of nearby molecules by many orders of magnitude, provide an incomplete picture of the structure and/or function of nanoscale wires in which electrons move ballistically, etc. nanomaterials, and theory can often fill in missing features crucial Experiments on nanoscale objects are often fraught with uncer- to understanding what is being measured. However, there are tainty due to the difficulty of fabricating and manipulating these important challenges to using theory as well, as the systems of objects at length scales below Ϸ10 nm. In many experiments, the interest are usually too large, and the time scales too long, for a device that holds or measures the nanoscale structure can also purely atomistic level theory to be useful. At the same time, produce irreversible degradation. Thus, electron microscopy can continuum theories that are appropriate for describing larger-scale produce defects in the structure being measured, optical measure- (micrometer) phenomena are often not accurate for describing the ments on the smallest metal nanoparticles are subject to serious nanoscale. Despite these challenges, there has been important heating effects that can melt or destroy the particle, and atomic progress in a number of areas, and there are exciting opportunities force microscopy studies of soft materials often lead to structural that we can look forward to as the capabilities of computational reorganization. In addition, the highest resolution measurements of facilities continue to expand. Some specific applications that are structure, such as electron microscopy measurements, require that discussed in this paper include: self-assembly of supramolecular the nanostructure be removed from its natural environment, such structures, the thermal properties of nanoscale molecular systems as from solution, and placed on a grid under circumstances where (DNA melting and nanoscale water meniscus formation), the me- aggregation and restructuring can occur. In situ measurements, for chanical properties of carbon nanotubes and diamond crystals, and example, of the structure of a self-assembled monolayer on a the optical properties of silver and gold nanoparticles. colloidal particle in solution or defects in diamond crystals under high stress, are either not possible or are of limited resolution. Thus, CHEMISTRY ͉ ͉ ͉ ͉ molecular dynamics nanomaterials nanoparticle plasmon there is a huge role for theory in ‘‘filling in the gaps’’ in our self-assembly understanding of phenomena at the nanoscale, and to predict new properties and phenomena. anoscience deals with the behavior of matter on length scales This paper is designed to provide an overview of the use of Nwhere a large number of atoms play a role, but where the theoretical methods to describe nanoscience problems. A number system is still small enough that the material does not behave like of examples are provided where theory has been used for nanoscale bulk matter. For example, a 5-nm gold particle, which contains on problems, and I hope to use these examples to illustrate some of the the order of 105 atoms, absorbs light strongly at 520 nm, whereas possibilities for getting theory to work, and also some of the existing bulk gold is reflective at this wavelength and small clusters of gold challenges. This is a field where no one type of theory can be used atoms have absorption at shorter wavelengths. The special prop- in all cases, and where the marriage of theories associated with erties associated with nanoscale systems like this have provided different length scales is still somewhat rocky. Thus, some of the both challenges and opportunities for the use of theory and problems will be addressed by pushing traditional atomistic theo- computation to play a role in the discovery process. The challenges ries, such as electronic structure theory, to systems that are much arise from the fact that most theories that describe the properties larger than they have been traditionally calibrated for, and for which of matter by using first-principles approaches in which all atoms serious approximations need to be introduced to calculate useful (and even all electrons in these atoms) are explicitly described are numbers. In other areas, one tries to push continuum theories down close to or (more often) beyond their capability to describe such to smaller length scales than they were originally developed for, and systems, even using the largest computer available. At the same again one needs to introduce approximations to make these pro- time, continuum theories, which play such a useful role for mi- duce useful results. Another approach involves multiscale theories, crometer-scale systems, are sometimes incapable of describing in which the atomistic and continuum scales are matched together. nanoscale properties due to incomplete incorporation of the un- Alternatively, one can use coarse-grained models, in which one derlying physics in the size-dependent materials parameters. How- attempts to describe the effective properties of small groups of ever, the opportunities for theory to play a role are significant, as atoms that are contained within a larger nanoscale structure. nanoscience often provides the smallest systems that are amenable to study using methods that involve macroscopic manipulation of nanoscale structures. Thus, it is possible to measure the structure Author contributions: G.C.S. wrote the paper. and thermal properties of a single supramolecular assembly, ob- The author declares no conflict of interest. serve the thermal properties of a single nanodroplet, or study the Abbreviations: PA, peptide amphiphile; AFM, atomic force microscopy; QM, quantum mechanical properties of a single carbon nanotube or light scatter- mechanics; MM, molecular mechanics; CM, continuum mechanics; UNCD, ultrananocrys- ing from a single metal nanoparticle. Nanoscale structures often talline diamond; SERS, surface-enhanced Raman spectroscopy. have properties that are familiar in much larger systems, such as *E-mail: [email protected]. defects and thermal instability; however, nanoscale structures can © 2007 by The National Academy of Sciences of the USA www.pnas.org͞cgi͞doi͞10.1073͞pnas.0702187104 PNAS ͉ April 24, 2007 ͉ vol. 104 ͉ no. 17 ͉ 6885–6892 I am grateful to my present and former students and postdocs who have Excellence, Nanoscale Science and Engineering Center, Materials Research contributed to the projects described in this paper, including Christine Science and Engineering Center, and National Aeronautics and Space Aikens, Karen Drukker, Encai Hao, Joonkyun Jang, Lasse Jensen, Hai Administration Biologically Inspired Materials/University Research, Engi- Long, Steven Mielke, Jeff Paci, Stefan Tsonchev, Guosheng Wu, Linlin neering, and Technology Institute centers at Northwestern University, the Zhao, and Shengli Zou. I would also like to acknowledge the contributions Network for Computational Nanotechnology center at Purdue University of several of my colleagues to these projects, including Chad Mirkin, (West Lafayette, IN). This research was also supported by National Science Richard Van Duyne, Sam Stupp, Mark Ratner, Jochen Autschbach, Ted Foundation NIRT, CRC, and CHE grants, and the Air Force Office of Belytschko, Rodney Ruoff, and Horacio Espinosa. This work was supported Scientific Research Defense University Research Initiative on Nanotech- by a number of centers, including the Center for Cancer Nanobiotechnology nology and Multidisciplinary University Research Initiative programs.

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6892 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0702187104 Schatz IOP PUBLISHING NANOTECHNOLOGY Nanotechnology 18 (2007) 155708 (6pp) doi:10.1088/0957-4484/18/15/155708 Size effect in the tensile fracture of single-walled carbon nanotubes with defects

M Yang, V Koutsos and M Zaiser

School of Engineering and Electronics, and Centre for Materials Science and Engineering, The University of Edinburgh, Institute for Materials and Processes, Edinburgh EH9 3JL, UK

E-mail: [email protected]

Received 2 January 2007, in ﬁnal form 20 February 2007 Published 16 March 2007 Online at stacks.iop.org/Nano/18/155708 Abstract Molecular simulation is used to determine the fracture strength of single-walled carbon nanotubes (SWNT) containing different concentrations of randomly distributed point defects. The results are analysed using Weibull statistics, and the dependence of the statistical distribution of fracture strengths on defect concentration is established. Arguments from extreme order statistics are then used to formulate a relationship between the length of SWNT and their fracture strength. The results of this investigation help to explain the large differences between SWNT fracture strengths measured in experiments (13Ð52 GPa) and those obtained from theoretical calculations assuming defect-free nanotubes (∼185 GPa).

1. Introduction mechanics based on Brenner’s potential is generally considered accurate in simulating the mechanical properties of carbon Most of the studies on carbon nanotubes regard them as ‘ideal’ nanotubes during quasi-static deformation and at not too high defect-free structures. However, since the discovery of carbon temperatures [21, 22]. (Given the high binding energies, nanotubes [1], people have found out that defects on carbon thermal energies are negligibly small at ambient temperature nanotubes are very common and have significant influence where most deformation experiments have been carried out.) on their properties, especially mechanical properties [2Ð6]. This potential is characterized by the quantum-mechanical Several kinds of defects may occur on carbon nanotubes. concept of bond order formalism, and is particularly useful In the present investigation, we focus on three kinds in modelling graphite and carbon nanotube structures when of localized (point-like) defects: functionalization defects changes in bonding may occur. The fracture strength of caused by chemical functionalization [7Ð9], StoneÐWales defected carbon nanotubes observed in simulations with type topological defects (5-7-7-5 defects) [3, 10, 11]and Brenner’s potential shows reasonable agreement with results structural defects (vacancies) [12] as shown in figure 1.We obtained from quantum mechanics computations [23]. use molecular mechanics to calculate the fracture strength of comparatively ‘short’ SWNT containing such defects in varying concentrations ranging from 0 to 7.5% for vacancies 2. Methodology and functionalization defects, and from 0 to 2.5% for StoneÐ Wales defects (for these, we use lower defect concentrations To simulate SWNT under tensile load, we carry out a sequence since one topological defect changes the structure of four of small elongation steps. In every step, we move the atoms at adjacent hexagons). We then use statistical arguments both ends of a carbon nanotube by a small amount (0.01 Aperû to address the size dependence of fracture strength and step) in the axial direction from the centre of the SWNT as extrapolate our results to nanotube lengths used in experiments. shown in figure 2. Then we relax the deformed SWNT by Computation techniques including quantum mechanics, energy minimization, using Brenner’s code to evaluate the molecular mechanics and continuum shell modelling have SWNT potential energy. A rigid restraint is applied which fixes been used to evaluate mechanical properties of carbon the atoms at the ends to prevent the SWNT from restoring its nanotubes [13Ð20]. Among these techniques, molecular initial structure (lowest-energy structure at zero load).

[9] Peng H, Alemany L B, Margrave J L and [22] Brenner D W, Shenderova O A, Harrison J A, Stuart S J, Khabashesku V N 2003 Sidewall carboxylic acid Ni B and Sinnott S B 2002 A second-generation reactive functionalization of single-walled carbon nanotubes J. Am. empirical bond order (REBO) potential energy expression Chem. Soc. 125 15174Ð82 for hydrocarbons J. Phys.: Condens. Matter 14 783Ð802 [10] Zhao Q, Nardelli M B and Bernholc J 2002 Ultimate strength [23] Mielke S L, Troya D Z, Zhang S, Li J, Xiao S, Car R, Ruo R S, of carbon nanotubes: A theoretical study Phys. Rev. B Schatz G C and Belytschko T 2004 The role of vacancy 65 144105 defects and holes in the fracture of carbon nanotubes Chem. [11] Grujicic M, Cao G and Singh R 2003 The effect of topological Phys. Lett. 390 413Ð20 defects and oxygen adsorption on the electronic transport [24] Sammalkorpi M, Krasheninnikov A, Kuronen A, properties of single-walled carbon-nanotubes Appl. Surf. Sci. Nordlund K and Kaski K 2004 Mechanical properties of 211 166Ð83 carbon nanotubes with vacancies and related defects Phys. [12] Lu A J and Pan B C 2004 Nature of single vacancy in achiral Rev. B 70 245416 carbon nanotubes Phys.Rev.Lett.92 105504 [25] Shen L X and Li J 2004 Transversely isotropic elastic [13] Zhou G, Duan W and Gu B 2001 First-principles study on properties of single-walled carbon nanotubes Phys. Rev. B morphology and mechanical properties of single-walled 69 045414 carbon nanotube Chem. Phys. Lett. 333 344Ð9 [26] Bhattacharya B and Lu Q 2006 The asymptotic properties of [14] Belytschko T, Xiao S P, Schatz G C and Ruoff R S 2002 random strength and compliance of single-walled carbon Atomistic simulations of nanotube fracture Phys. Rev. B nanotubes using atomistic simulation J. Stat. Mech. P06021 65 235430 [27] Lu Q and Bhattacharya B 2005 Effect of randomly occurring [15] Guo Y and Guo W 2003 Mechanical and electrostatic StoneÐWales defects on mechanical properties of carbon properties of carbon nanotubes under tensile loading and nanotubes using atomistic simulation Nanotechnology electric ﬁeld J. Phys. D: Appl. Phys. 36 805Ð11 16 555Ð66 [16] Buehler M J, Yong K and Gao H 2004 Deformation [28] Barber A, Kaplan-Ashiri I, Cohen S R, Tenne R and mechanisms of very long single-wall carbon nanotubes Wagner H D 2005 Stochastic strength of nanotubes: subject to compressive loading J. Eng. Mater. Technol. An appraisal of available data Compos. Sci. Technol. 126 245Ð9 65 2380Ð4 [17] Natsuki T and Endo M 2004 Stress simulation of carbon [29] Lu C, Danzer R and Fischer F D 2002 Fracture statistics of nanotubes in tension and compression Carbon 42 2147Ð51 brittle materials: Weibull or normal distribution Phys. Rev. E [18] Natsuki T, Tantrakarn K and Endo M 2004 Prediction of elastic 65 067102 properties for single-walled carbon nanotubes Carbon [30] Trustrum K and Jayatilaka A 1979 On estimating the Weibull 42 39Ð45 modulus for a brittle material J. Mater. Sci. 14 1080Ð4 [19] Wang Y, Wang X and Ni X 2004 Atomistic simulation of the [31] Yu M F, Files B S, Arepalli S and Ruoff R S 2000 Tensile torsion deformation of carbon nanotubes Model. Simul. loading of ropes of single wall carbon nanotubes and their Mater. Sci. Eng. 12 1099Ð107 mechanical properties Phys.Rev.Lett.84 5552Ð5 [20] Chang T, Li G and Guo X 2005 Elastic axial buckling of [32] Galambos J 1978 The Asymptotic Theory of Extreme Order carbon nanotubes via a molecular mechanics model Carbon Statistics (New York: Wiley) p 352 43 287Ð94 [33] Zhang S L, Mielke S L, Khare R, Troya D, Ruoff R S, [21] Brenner D W 1990 Empirical potential for hydrocarbons for Schatz G C and Belytschko T 2005 Mechanics of defects in use in simulating the chemical vapour-deposition of carbon nanotubes: Atomistic and multiscale simulations diamond ﬁlms Phys. Rev. B 42 9458Ð71 Phys. Rev. B 71 115403

6 PHYSICAL REVIEW B 75, 125414 ͑2007͒

Studies of nanotube-based resonant oscillators through multiscale modeling and simulation

Shaoping Xiao and Wenyi Hou Department of Mechanical and Industrial Engineering and Center for Computer-Aided Design, The University of Iowa, Iowa City, Iowa 52242, USA ͑Received 30 October 2006; published 12 March 2007͒ We propose a multiscale method to study nanotube-based resonant oscillators. In the multiscale model, nanotubes are modeled via molecular dynamics, while the metal paddle is modeled as a rigid body. The molecular and continuum models are attached to each other through the interfaces on which carbon atoms are located. We employ the concepts of “virtual” atoms and bonds to effectively couple the molecular and continuum models. Using the proposed multiscale method, we investigate both linear and nonlinear characteristics of resonant oscillators. Effects of vacancy and temperature on mechanisms of oscillators are discussed.

DOI: 10.1103/PhysRevB.75.125414 PACS number͑s͒: 85.35.Kt, 61.46.Fg

I. INTRODUCTION position between 0° and almost 180°. All of the fabricated resonant oscillators described above had a paddle with a vol- Since carbon nanotubes have extraordinary mechanical ume of about 0.04 ␮m3 and a nanotube with a length of and electrical properties,1 they have been utilized as essential around 500 nm. Therefore, a molecular-dynamics model of components in the design of novel nanoscale materials and such an oscillator may contain up to trillions of atoms and devices. Tremendous molecular-dynamics simulations have become infeasible for current computer resources. been conducted to study the physical phenomena of Since molecular dynamics has limitations in simulating nanotube-based composites and devices. Xiao and Hou2 large nanosystems, multiscale methods are attractive to sci- studied the mechanics of nanocomposites in which defected entists and engineers. Recently developed multiscale model- carbon nanotubes were embedded through molecular- ing techniques have shown promise in treating phenomena at both nano- and larger scales. Multiscale methods can be di- dynamics simulations. Srivastava3 employed molecular dy- vided into two classes: hierarchical multiscale methods and namics to discuss and test a phenomenological model for the concurrent multiscale methods. Most hierarchical models rotational dynamics of a single laser-powered molecular contain a continuum approximation based on the properties motor that powered carbon nanotube-based gears. Molecular ͑ ͒ 4 of a subscale model, such as a molecular-dynamics MD dynamics also assist researchers to investigate the model. The intrinsic properties of the material are deter- temperature-related energy dissipation of nanoscale devices. mined at the atomic level and embedded in the continuum 5–7 Recently, a new nanoscale device in which an individual model according to a homogenization procedure.9,10 How- carbon nanotube serves as a torsional spring and mechanical ever, the effects of defects cannot be considered with nano- support has been successfully fabricated. However, numeri- scale continuum approximation. Concurrent multiscale cal modeling and studies of this device have not been re- methods11–13 employ an appropriate model in which different ported. methodologies are employed in each spatial scale simulta- Williams et al.5,6 reported the fabrication of nanoscale neously. The typical concurrent multiscale methods include mechanical devices incorporating multiwalled carbon nano- the macroatomistic ab initio dynamics ͑MAAD͒ method11 tubes ͑MWNTs͒ as the torsional spring elements. They uti- and the bridging domain coupling method.12,13 They mainly lized electron-beam lithography to pattern a device element coupled a continuum model ͑ﬁnite element methods͒ with a directly onto an individual MWNT on a silicon dioxide sub- molecular model ͑molecular dynamics͒. Consequently, large strate. Consequently, the device consisted of a suspended le- models can be simulated without losing physical phenom- ver, i.e., the “paddle,” connected by an MWNT as a torsion enon details at the nanoscale. beam. Papadakis et al.7 used similar techniques to synthesize In this paper, we develop a multiscale method that couples asymmetric oscillators that were also called resonant oscilla- a continuum model and a molecular model to study the me- tors. The metal paddles in their experiments were not cen- chanical behavior of nanotube-based resonant oscillators. In tered on the MWNTs, and the MWNTs were strained prima- the proposed multiscale model, the nanotube is modeled with rily in torsion. Once the paddle was given an electrostatic molecular dynamics, while the metal paddle is modeled as a force, the oscillation was observed and the measured reso- continuum. The edge-to-edge coupling technique13 is em- nance frequencies were in the range of 1–9 MHz. Such os- ployed in this multiscale method to efﬁciently attach the con- cillators can be used as sensors and clocks for high- tinuum model and the molecular model. Without losing ac- frequency electronics. For example, with these nanoscale curacy, the metal paddle is treated as a rigid body since it has resonant oscillators, the clocks can be achieved with a single- only small deformation during the torsional oscillation. stage device.7 More recently, Meyer et al.8 built a torsional The outline of this paper is as follows. A multiscale mod- pendulum with an individual single-walled carbon nanotube eling of nanotube-based resonant oscillators is proposed in ͑SWNT͒, which was also used as a torsional spring and me- Sec. II. The coupling of molecular dynamics and rigid body chanical support for the metal paddle. They reported that this kinetics is introduced. In Sec. III, we discuss oscillation SWNT-based pendulum could be reproducibly turned to any mechanisms of resonant oscillators that are linear oscillator

1098-0121/2007/75͑12͒/125414͑9͒ 125414-1 ©2007 The American Physical Society STUDIES OF NANOTUBE-BASED RESONANT… PHYSICAL REVIEW B 75, 125414 ͑2007͒ had lower energy dissipate rates, i.e., higher quality factors. oscillators can also be simulated via the proposed multiscale The issue of energy dissipation should be considered method. Such modeling and simulation will be our future when designing nanoelectromechanical systems utilizing research. nanotube-based resonant oscillators. It should be noted that it was possible to simulate half of the resonant oscillator systems since the systems simulated ACKNOWLEDGMENTS in this paper were symmetric. However, if the metal paddle does not attach to the center of the carbon nanotube, the We acknowledge support from the Army Research Ofﬁce whole system must be modeled. On the other hand, we only ͑Contract No. W911NF-06-C-0140͒ and the National Sci- consider SWNTs in this paper. The MWNT-based resonant ence Foundation ͑Grant No. 0630153͒.

1 V. N. Popov, Mater. Sci. Eng., R. 43,61͑2004͒. 10 S. Xiao and W. Yang, Comput. Mater. Sci. 37, 374 ͑2006͒. 2 S. Xiao and W. Hou, Phys. Rev. B 73, 115406 ͑2006͒. 11 F. Abraham, J. Broughton, N. Bernstein, and E. Kaxiras, Euro- 3 D. Srivastava, Nanotechnology 8, 186 ͑1997͒. phys. Lett. 44, 783 ͑1998͒. 4 S. Xiao, D. Andersen, R. Han, and W. Hou, J. Comput. Theor. 12 S. Xiao and T. Belytschko, Comput. Methods Appl. Mech. Eng. Nanosci. 3, 142 ͑2006͒. 193, 1645 ͑2004͒. 5 P. Williams, S. Papadakis, A. Patel, M. Falvo, S. Washburn, and 13 T. Belytschko and S. Xiao, Int. J. Multiscale Comp. Eng. 1,115 R. Superfine, Phys. Rev. Lett. 89, 255502 ͑2002͒. ͑2003͒. 6 P. Williams, S. Papadakis, A. Patel, M. Falvo, S. Washburn, and 14 T. Belytschko, S. P. Xiao, G. C. Schatz, and R. S. Ruoff, Phys. R. Superfine, Appl. Phys. Lett. 82, 805 ͑2003͒. Rev. B 65, 235430 ͑2002͒. 7 S. J. Papadakis, A. R. Hall, P. A. Williams, L. Vicci, M. R. Falvo, 15 W. C. Young, Roark’s Formulas for Stress and Strain ͑McGraw- R. Superfine, and S. Washburn, Phys. Rev. Lett. 93, 146101 Hill, New York, 1989͒. ͑2004͒. 16 W. Hou and S. Xiao, J. Nanosci. Nanotechnol. ͑unpublished͒. 8 J. Meyer, M. Paillet, and S. Roth, Science 309, 1539 ͑2005͒. 17 S. Mielke, D. Troya, S. Zhang, J. Li, S. Xiao, R. Car, R. Ruoff, G. 9 E. Tadmor, R. Phillips, and M. Ortiz, Int. J. Solids Struct. 37, 379 Schatz, and T. Belytschko, Chem. Phys. Lett. 390, 413 ͑2004͒. ͑2000͒. 18 W. G. Hoover, Phys. Rev. A 31, 1695 ͑1985͒.

125414-9 PHYSICAL REVIEW B 75, 075412 ͑2007͒

Coupled quantum mechanical/molecular mechanical modeling of the fracture of defective carbon nanotubes and graphene sheets

Roopam Khare,1 Steven L. Mielke,2 Jeffrey T. Paci,2 Sulin Zhang,1 Roberto Ballarini,3 George C. Schatz,2,* and Ted Belytschko1,† 1Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208-3111, USA 2Department of Chemistry, Northwestern University, Evanston, Illinois 60208-3113, USA 3Department of Civil Engineering, University of Minnesota, Minneapolis, Minnesota 55455, USA ͑Received 12 July 2006; revised manuscript received 29 November 2006; published 9 February 2007͒ Coupled quantum mechanical/molecular mechanical ͑QM/MM͒ calculations were used to study the effects of large defects and cracks on the mechanical properties of carbon nanotubes and graphene sheets. The semi-empirical method PM3 was used to treat the QM subdomains and a Tersoff-Brenner potential was used for the molecular mechanics; some of the QM calculations were also done using density functional theory ͑DFT͒. Scaling of the Tersoff-Brenner potential so that the modulus and overall stress-strain behavior of the QM and MM models matched quite closely was essential for obtaining meaningful coupled calculations of the mechanical properties. The numerical results show that at the nanoscale, the weakening effects of holes, slits, and cracks vary only moderately with the shape of the defect, and instead depend primarily on the cross section of the defect perpendicular to the loading direction and the structure near the fracture initiation point. The fracture stresses for defective graphene sheets are in surprisingly good agreement with the Grifﬁth formula for defects as small as 10 Å, which calls into question the notion of nanoscale ﬂaw tolerance. The energy release rate at the point of crack extension in graphene was calculated by the J-integral method and exceeds twice the surface energy density by 10% for the QM͑DFT͒/MM results, which indicates a modest lattice trapping effect.

DOI: 10.1103/PhysRevB.75.075412 PACS number͑s͒: 68.65.Ϫk, 62.25.ϩg, 61.50.Ah, 81.07.De

I. INTRODUCTION ing of the role of defects in CNTs necessitates electronic structure calculations. Due to their favorable mechanical properties, carbon One way to treat a system of large molecules is to adopt ͑ ͒ nanotubes CNTs and exfoliated graphene sheets have at- linear scaling QM methods, which reduce the order of com- tracted considerable interest as nanoreinforcements for poly- putation to O͑N͒, where N is the total number of orbitals. mer composites. Electronic structure calculations1–7 of the Another approach to treating large systems is to couple a QM mechanical behavior of pristine CNTs predict fracture method to an MM method so that the important regions of strengths in the range of 75–135 GPa and ultimate strains of the system are treated quantum mechanically and MM inter- as much as 30%. However, manufacture of mass-quantities of perfect CNTs or graphene sheets may prove challenging; actions are used elsewhere. In fracture studies, only part of in practice, mechanical properties will often be limited by the system, such as the vicinity of defects, requires an accu- the presence of defects—in many cases of substantial size. rate treatment of bond breaking; for such systems QM meth- Oxidative puriﬁcation treatments8–10 are commonly used in ods can be used for these regions and MM methods can be the production of CNTs and we have suggested6,11 that these applied to the rest of the system. Although the MM potential can result in pitting, which provides a plausible explanation does not need to be able to model bond fracture accurately, it for the deviations between the CNT strength measurements must still predict stiffnesses and strengths that are consistent of Yu et al.12 and theoretical predictions. One route for the with the QM results; in the following we will present a exfoliation of graphene sheets involves thermal exfoliation simple scaling scheme to improve the compatibility of the of graphite oxide.13 During this process approximately 30% QM and MM mechanical properties. of the carbon atoms are lost as CO2; thus, the resulting sheets Here we describe a set of coupled quantum mechanical/ are expected to be highly defected. A detailed understanding molecular mechanical ͑QM/MM͒ calculations of the fracture of the consequences of such defects may be crucial to the of CNTs with large defects and the fracture of graphene effective utilization of these materials. sheets. We also consider the effect of lattice trapping18–21 by Previous electronic structure calculations of defected calculating the energy release rate of a crack in a QM/MM CNTs were limited to small defects because of the high com- model of a graphene sheet using the J-integral22 method. 6,14 putational cost of QM calculations. Therefore, the studies The rest of the paper is organized as follows. Section II of larger defects were restricted to MM calculations. How- gives an overview of the coupling method and its implemen- 15,16 17 ever, the modiﬁed second generation Tersoff-Brenner tation. In Sec. III the results of fracture in the presence of ͑ ͒ MTB-G2 potential employed in those calculations tends to defects such as one- and two-atom vacancy defects as well as systematically underestimate the strength of both pristine and nanosized holes and slits are presented. Section IV describes defected CNTs as compared to QM calculations, and has the results for lattice trapping in graphene sheets. Conclu- been shown3 to predict qualitatively different mechanisms sions are given in Sec. V. for the fracture of defected CNTs. Thus, a better understand-

1098-0121/2007/75͑7͒/075412͑12͒ 075412-1 ©2007 The American Physical Society COUPLED QUANTUM MECHANICAL/MOLECULAR… PHYSICAL REVIEW B 75, 075412 ͑2007͒ nanoscale ͑5–20 nm͒ defects: for any defect, the strength is accuracy of the coupled QM/MM calculations for the failure below the theoretical strength, as would be expected. stress is probably only a few percent and the results are more Crack-like defects, which we called slits, were con- useful for comparing defects of various sizes than in obtain- structed by removing four rows of carbon atoms in the zig- ing quantitative values of failure stresses. zag CNTs and by removing three rows of carbon atoms in We introduced a simple scaling scheme to improve the the armchair CNTs. The resulting dangling bonds were compatibility of the MM and QM models. If the MM inter- capped with hydrogen atoms. This is to be contrasted with action potential is used unscaled, mismatches between the crack models based on omitting bonds between adjacent at- stiffness and strength of the MM and QM subdomains result oms in MM calculations, which are pervasive in the litera- in highly spurious behavior. Comparisons of the QM/MM ture. Such defects cannot exist in electronic structure models calculations with MM calculations using the scaled potential because interactions between nearby atom pairs cannot sim- show good agreement for the failure stresses and stress-strain ply be neglected at moderate distances. Thus, the ability of curves. In most cases, the differences were less than 15%, and the qualitative pattern of dependence on defect size such schemes to accurately model crack behavior is ques- agreed well. It should be noted that this good agreement is tionable. Crack-like defects can be formed by displacing a only achieved for the scaled MM potential. The details of the lattice according to the asymptotic near-field of elastic frac- fracture processes predicted by the QM/MM method and the ture mechanics, as we reported in Sec. IV. However, such MM method differ significantly. For example, QM/MM cal- cracks will not exist in a solid in a stress-free state. culations show significantly more elongation of the bond at To ascertain the magnitude of lattice trapping in graphene, the crack tip. we computed the energy release rate using a discrete The results provide further credence to the hypothesis6 J-integral and compared it to twice the surface energy den- that large defects such as holes are the reason behind the low sity, 2␥. These results indicate a modest amount of lattice CNT fracture strengths observed in some experiments.12 Al- trapping; the energy release rate calculated by DFT for a though the QM/MM models predict failure stresses that are graphene sheet at fracture exceeds 2␥ by 10%. about 40% higher than unscaled MM results modeled The coupled QM/MM calculations were performed with previously,6,14 they are still in the range observed in the Yu et the ONIOM methodology. We checked the accuracy of the al. experiments.12 method by performing a series of calculations for small defects with QM fragments of increasing size. The resulting ACKNOWLEDGMENTS stress-strain curves agreed closely over most of the range We thank Diego Troya for helpful conversations. We even for relatively small QM fragments. The fracture stresses gratefully acknowledge grant support from the NASA Uni- and strains also appear to converge, but are more sensitive to versity Research, Engineering and Technology Institute on the QM fragment size and even for the largest two fragments Bio Inspired Materials ͑BIMat͒ under award No. NCC-1- studied, the fracture stresses and strains differed from the 02037, from the Army Research Office under Grant No. pure QM results by 4% and 8%, respectively, for a ͓10,0͔ W911NF-05–1–0049, and from the National Science Foun- CNT with a two-atom vacancy defect. Thus, the absolute dation.

*Electronic address: [email protected] 5, 163 ͑2005͒. †Electronic address: [email protected] 11 S. L. Mielke, T. Belytschko, and G. C. Schatz, Annu. Rev. Phys. 1 T. Ozaki, Y. Iwasa, and T. Mitani, Phys. Rev. Lett. 84, 1712 Chem. 58, 185 ͑2007͒. ͑2000͒. 12 M.-F. Yu, O. Lourie, M. J. Dyer, K. Moloni, T. F. Kelly, and R. S. 2 T. Dumitrica, T. Belytschko, and B. I. Yakobson, J. Chem. Phys. Ruoff, Science 287, 637 ͑2000͒. 118, 9485 ͑2003͒. 13 H. C. Schniepp, J.-L. Li, M. J. McAllister, H. Sai, M. Herrera- 3 D. Troya, S. L. Mielke, and G. C. Schatz, Chem. Phys. Lett. 382, Alonso, D. H. Adamson, R. K. Prud’homme, R. Car, D. A. 133 ͑2003͒. Saville, and I. A. Aksay, J. Phys. Chem. A 110, 8535 ͑2006͒. 4 S. Ogata and Y. Shibutani, Phys. Rev. B 68, 165409 ͑2003͒. 14 S. Zhang, S. L. Mielke, R. Khare, D. Troya, R. S. Ruoff, G. C. 5 G. Dereli and C. Ozdogan, Phys. Rev. B 67, 035416 ͑2003͒. Schatz, and T. Belytschko, Phys. Rev. B 71, 115403 ͑2005͒. 6 S. L. Mielke, D. Troya, S. Zhang, J.-L. Li, S. Xiao, R. Car, R. S. 15 O. A. Shenderova, D. W. Brenner, A. Omeltchenko, X. Su, and L. Ruoff, G. C. Schatz, and T. Belytschko, Chem. Phys. Lett. 390, H. Yang, Phys. Rev. B 61, 3877 ͑2000͒. 413 ͑2004͒. 16 T. Belytschko, S. P. Xiao, G. C. Schatz, and R. S. Ruoff, Phys. 7 T. Dumitrica, M. Hua, and B. I. Yakobson, Proc. Natl. Acad. Sci. Rev. B 65, 235430 ͑2002͒. U.S.A. 103, 6105 ͑2006͒. 17 D. W. Brenner, O. A. Shenderova, J. A. Harrison, S. J. Stuart, B. 8 A. G. Rinzler, J. Liu, H. Dai, P. Nikolaev, C. B. Huffman, F. J. Ni, and S. B. Sinnott, J. Phys.: Condens. Matter 14, 783 ͑2002͒. Rodriguez-Macias, P. J. Boul, A. H. Lu, D. Heymann, D. T. 18 R. Thomson, C. Hsieh, and V. Rana, J. Appl. Phys. 42, 3154 Colbert et al., Appl. Phys. A: Mater. Sci. Process. 67,29͑1998͒. ͑1971͒. 9 R. C. Haddon, J. Sippel, A. G. Rinzler, and F. Papadimitrakopou- 19 J. R. Rice, J. Mech. Phys. Solids 26,61͑1978͒. los, MRS Bull. 29, 252 ͑2004͒. 20 W. A. Curtin, J. Mater. Res. 5, 1549 ͑1990͒. 10 Y.-Q. Xu, H. Peng, R. H. Hauge, and R. E. Smalley, Nano Lett. 21 N. Bernstein and D. W. Hess, Phys. Rev. Lett. 91, 025501 ͑2003͒.

075412-11 Experimental Mechanics (2007) 47: 25–36 DOI 10.1007/s11340-006-9344-6

Modulus, Fracture Strength, and Brittle vs. Plastic Response of the Outer Shell of Arc-grown Multi-walled Carbon Nanotubes

W. Ding & L. Calabri & K.M. Kohlhaas & X. Chen & D.A. Dikin & R.S. Ruoff

Received: 27 March 2006 /Accepted: 25 May 2006 / Published online: 1 August 2006 # Society for Experimental Mechanics 2006

Abstract The fracture strengths and elastic moduli of Introduction arc-grown multi-walled carbon nanotubes (MWCNTs) were measured by tensile loading inside of a scanning The experimental discovery of multi-walled carbon electron microscope (SEM). Eighteen tensile tests nanotubes (MWCNTs) in 1991 [1] and single-walled were performed on 14 MWCNTs with three of them carbon nanotubes (SWCNTs) in 1993, [2, 3]has being tested multiple times (3Â,2Â, and 2Â, respec- spawned considerable interest in carbon nanotubes. tively). All the MWCNTs fractured in the Bsword-in- Theoretical studies have predicted that defect-free sheath’’ mode. The diameters of the MWCNTs were CNTs should have a Young’s modulus of ~1 TPa, [4– measured in a transmission electron microscope (TEM), 6] similar to the in-plane modulus of graphite (if a shell and the outer diameter with an assumed 0.34 nm shell thickness of 0.34 nm is used). Among numerous thickness was used to convert measured load- examples, MWCNTs have been studied in nanoscale displacement data to stress and strain values. An devices, [7–9] and as filler in composites [10–12]. The unusual yielding before fracture was observed in two mechanical properties of CNTs are clearly of both tensile loading experiments. The 18 outer shell fracture intrinsic and practical importance. strength values ranged from 10 to 66 GPa, and the 18 Several techniques have been developed for explor- Young’s modulus values, obtained from a linear fit of ing the mechanical properties of individual CNTs. For the stress–strain data, ranged from 620 to 1,200 GPa, example, the elastic moduli of cantilevered MWCNTs with a mean of 940 GPa. The possible influence of stress were obtained from mechanical resonance observed concentration at the clamps is discussed. inside a TEM [13–16]. The elastic moduli of MWCNTs deposited across pores on an alumina nanopore Keywords Nanotube . Fracture . Yield . Modulus . membrane were obtained from bending tests carried Strength out with an AFM, [17] and the same approach was used to obtain the elastic and shear moduli of SWCNT ropes [18]. The fracture strengths and elastic moduli of MWCNTs [19] and SWCNT ropes [20] have been W. Ding I K.M. Kohlhaas I X. Chen I D.A. Dikin obtained by tensile testing inside a SEM. R.S. Ruoff (*) We present a detailed study of the mechanical Department of Mechanical Engineering, properties of arc-grown MWCNTs by tensile loading Northwestern University, inside an SEM. Individual MWCNTs were tensile Evanston, IL 60208, USA e-mail: [email protected] loaded until fracture by using a piezo-actuated nano- manipulator [21]. The fracture strengths, failure strains L. Calabri and elastic moduli of the MWCNTs were obtained CNR-INFM-National Research on nanoStructures from the measured geometry, the load at failure, and and bioSystems at Surfaces (S3), via Campi 213/a, the load-displacement (which with the measured 4100 Modena, Italy geometry is converted to stress–strain) data. No

SEM 36 Exp Mech (2007) 47: 25–36

single-walled carbon nanotubes. Chem Phys Lett 386(4–6): the electronic band structure, elasstic constants, and equa- 239–243. tion of state for graphite. Phys Rev B 55(17):11202–11211. 28. Bom D, Andrews R, Jacques D, Anthony J, Chen BL, Meier 36. Ding W, Dikin DA, Chen X, Wang X, Li X, Piner R, Ruoff MS, Selegue JP (2002) Thermogravimetric analysis of the RS, Zussman E (2005) Mechanics of hydrogenated amor- oxidation of multiwalled carbon nanotubes: evidence for the phous carbon deposits from electron beam induced deposi- role of defect sites in carbon nanotube chemistry. Nano Lett tion of a paraffin precursor. J Appl Physi 98:014905. 2(6):615–619. 37. Li CY, Ruoff RS, Chou TW (2005) Modeling of carbon 29. Belytschko T, Xiao SP, Schatz GC, Ruoff RS (2002) nanotube clamping in tensile tests. Compos Sci Technol Atomistic simulations of nanotube fracture. Phys Rev B 65(15–16):2407–2415. 65(23):235430. 38. Pugno NM, Ruoff RS (2004) Quantized fracture mechanics. 30. Mielke SL, Troya D, Zhang S, Li JL, Xiao SP, Car R, Ruoff Philos Mag 84(27):2829–2845. RS, Schatz GC, Belytschko T (2004) The role of vacancy 39. Nardelli MB, Yakobson BI, Bernholc J (1998) Brittle and defects and holes in the fracture of carbon nanotubes. Chem ductile behavior in carbon nanotubes. Phys Rev Lett Phys Lett 390(4–6):413 – 420. 81(21):4656 – 4659. 31. Zhang SL, Mielke SL, Khare R, Troya D, Ruoff RS, Schatz 40. Yakobson BI (1998) Mechanical relaxation and Bintramolecular GC, Belytschko T (2005) Mechanics of defects in carbon plasticity’’ in Carbon Nanotubes. Appl Phys Lett 72(8):918 – nanotubes: atomistic and multiscale simulations. Phys Rev B 920. 71(11):115403. 41. Yu MF, Yakobson BI, Ruoff RS (2000) Controlled sliding 32. Sader JE, Larson I, Mulvaney P, White LR (1995) Method and pullout of nested shells in individual multiwalled carbon for the calibration of atomic-force microscope cantilevers. nanotubes. J Phys Chem B 104:8764–8767. Review of Scientific Instruments 66(7):3789–3798. 42. Zussman E, Chen X, Ding W, Calabri L, Dikin DA, 33. Chen XQ, Zhang SL, Wagner GJ, Ding WQ, Ruoff RS Quintana JP, Ruoff RS (2005) Mechanical and structural (2004) Mechanical resonance of quartz microfibers and characterization of electrospun PAN-derived carbon nano- boundary condition effects. J Appl Physi 95(9):4823 – 4828. fibers. Carbon 43:2175–2185. 34. Ding W, Calabri L, Chen X, Kohlhaas KM, Ruoff RS (2006) 43. Lu S, Guo Z, Ding W, Ruoff RS (2006) Analysis of a Mechanics of crystalline boron nanowires. Compos Sci microelectromechanical system testing stage for tensile Technol 66:1109–1121. loading of nanostructures. Review of Scientific Instruments 35. Boettger JC (1997) All-electron full-potential calculation of 77(5):1 – 4.

SEM INSTITUTE OF PHYSICS PUBLISHING NANOTECHNOLOGY Nanotechnology 18 (2007) 044026 (3pp) doi:10.1088/0957-4484/18/4/044026 Optimized adhesives for strong, lightweight, damage-resistant, nanocomposite materials: new insights from natural materials

PKHansma1, P J Turner1 and R S Ruoff2

1 Physics Department, Broida Hall, University of California, Santa Barbara, CA 93106, USA 2 Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3111, USA

E-mail: [email protected], [email protected] and [email protected]

Received 25 July 2006, in final form 15 November 2006 Published 21 December 2006 Online at stacks.iop.org/Nano/18/044026 Abstract From our investigations of natural composite materials such as abalone shell and bone we have learned the following. (1) Nature is frugal with resources: it uses just a few per cent glue, by weight, to glue together composite materials. (2) Nature does not avoid voids. (3) Nature makes optimized glues with sacrificial bonds and hidden length. We discuss how optimized adhesives combined with high specific stiffness/strength structures such as carbon nanotubes or graphene sheets could yield remarkably strong, lightweight, and damage-resistant materials.

1. Introduction Sacrificial bonds and hidden length in structural molecules and composites have been found to greatly increase the fracture The abalone shell, a composite of calcium carbonate plates toughness of biomaterials by providing a reversible, molecular- sandwiched between organic material, is 3000 times more scale energy-dissipation mechanism. This mechanism relies fracture resistant than a single crystal of the pure mineral [1]. on the energy, of order 100 eV, needed to reduce entropy The organic component, comprising just a few per cent of and increase enthalpy as molecular segments are stretched the composite by weight, working together with the structural after being released by the breaking of weak bonds, called geometry [2, 3], is thought to hold the key to nacre’s fracture sacrificial bonds. This energy is relatively large compared to toughness [4]. the energy needed to break the polymer backbone, of order a In addition to the details of the structural geometry, one few eV. In many biological cases, the breaking of sacrificial of the main mechanisms that lets a few per cent of ‘glue’ bonds has been found to be reversible, thereby additionally make such an enormous difference in fracture toughness is the providing a ‘self-healing’ property to the material. Due to the sacrificial bond and hidden length mechanism [4, 5]. By ‘glue’ nanoscopic nature of this mechanism, single molecule force in this context we mean polymer adhesive molecules that hold spectroscopy [6Ð8] using an atomic force microscope has been together the hard elements in a composite structure. In the a useful tool to investigate this mechanism [4, 5]. case of biological structures these glues tend to be composed Bone consists of mineralized collagen fibrils and a non- of proteins, proteoglycans, and glycoproteins. The study of fibrillar organic matrix, which acts as a ‘glue’ that holds the such glues is just in its infancy, but so far it appears that one mineralized fibrils together [9]. Here again the glue is just a of the key factors is the presence of charged side groups on the few per cent by weight. This glue may resist the separation of biological polymer adhesive molecules that can form sacrificial mineralized collagen fibrils. As in the case of the abalone shell, bonds with other charged groups on the polymers and on the in addition to the details of the structural geometry, one of the hard elements—sometimes with the help of ions in solution. main mechanisms that lets a few per cent of glue make such For example, Ca++ ions can help form sacrificial bonds an enormous difference in fracture toughness is the sacrificial between negatively charged side groups such as phosphate bond and hidden length mechanism [4, 10]. This mechanism groups. is also used in spider silk [11] and some diatoms, which are

0957-4484/07/044026+03$30.00 1 © 2007 IOP Publishing Ltd Printed in the UK Nanotechnology 18 (2007) 044026 PKHansmaet al needs to consider the statistical nature of defects that results NSF CMS-0625085 ‘Fracture mechanics of nanowires and in a decrease of strength with length or size for conventional nanostructures’. materials; this can perhaps be mitigated, and perhaps even eliminated, by future developments in fabrication techniques References for the nanotubes. One must however appreciate the challenge [1] Jackson A P, Vincent J F V and Turner R M 1988 Proc. R. Soc. of synthesizing carbon nanotubes that are completely defect B 234 415 free, and there is no evidence that such defect free CNTs have [2] Abdala A A, Milius D L, Adamson D H, Aksay I A and been produced to date. If one accepts that the CNTs will have Prudhomme R K 2004 Am. Chem. Soc. 227 U525 (Abstracts some defects that reduce their strength, then a real optimized of papers) [3] Evans A G, Suo Z, Wang R Z, Aksay I A, He M Y and adhesive must be chosen as a compromise between the benefits Hutchinson J W 2001 J. Mater. Res. 16 2475 of longer strong elements, such as a lower percentage of the [4] Smith B L, Schaffer T E, Viani M, Thompson J B, weaker glue, with the disadvantage that longer strong elements Frederick N A, Kindt J, Belcher A, Stucky G D, Morse D E may themselves be weaker! It is thus of interest to consider and Hansma P K 1999 Nature 399 761 the types of defects that have the largest impact on strength of [5] Fantner G E et al 2006 Biophys. J. 90 1411 [6] Rief M, Gautel M, Oesterhelt F, Fernandez J M and Gaub H E CNTs—namely point defects or clusters of missing atoms (in 1997 Science 276 1109 short, holes) [27Ð29, 32, 33]. Indeed, even a few missing atoms [7] Rief M, Fernandez J M and Gaub H E 1998 Phys. Rev. Lett. and thus small holes in the tube structure can cause a significant 81 4764 reduction in strength, as is discussed in detail elsewhere [27]. [8] Rief M, Gautel M, Schemmel A and Gaub H E 1998 Biophys. In this regard, it is possible that graphene sheets, if extracted J. 75 3008 [9] Fantner G E et al 2005 Nat. Mater. 4 612 from ‘high quality’ graphite, might contain both smaller, and [10] Fantner G E et al 2006 Biophys. J. BIOFAST 90 1411Ð8 also fewer, critical defects. On the other hand, synthetic [11] Becker N, Oroudjev E, Mutz S, Cleveland J P, Hansma P K, approaches which yield essentially perfect (defect-free) carbon Hayashi C Y, Makarov D E and Hansma H G 2003 Nat. nanotubes may yet be found. Mater. 2 278 [12] Gebeshuber I C, Kindt J H, Thompson J B, Del Amo Y, Stachelberger H, Brzezinski M, Stucky G D, Morse D E 3. Summary and Hansma P K 2004 J. Microsc.-Oxford 214 101 [13] Gebeshuber I C, Kindt J H, Thompson J B, Del Amo Y, The concept of an optimized adhesive is based on using only Stachelberger H, Brzezinski M A, Stucky G D, Morse D E and Hansma P K 2003 J. Microsc.-Oxford 212 292 enough adhesive to fully transfer the desired load to the strong [14] Gebeshuber I C, Kindt J H, Thompson J B, DelAmo Y, elements in a composite material. This full transfer will Stachelberger H, Brzezinski M, Stucky G D, Morse D E and require a fixed number of adhesive molecules—independent Hansma P K 2000 Biophys. J. 78 10a of the length of the strong element. Thus, the longer the [15] Reilly D T and Burstein A H 1975 J. Biomech. 8 393 strong elements, the less the fractional weight of the glue in [16] Cezayirlioglu H, Bahniuk E, Davy D T and Heiple K G 1985 J. Biomech. 18 61 the composite material. Fractional weights as small as a few [17] Braidotti P, Branca F P and Stagni L 1997 J. Biomech. 30 155 per cent are found in natural materials such as abalone shells [18] Crist B 1995 Annu. Rev. Mater. Sci. 25 295 and bone. [19] Qian D, Liu W K and Ruoff R S 2003 Compos. Sci. Technol. Making synthetic composite materials with just a few 63 1561 per cent of glue remains a challenge for the future. Nature gives [20] Zhang S L, Mielke S L, Khare R, Troya D, Ruoff R S, Schatz G C and Belytschko T 2005 Phys. Rev. B 71 115403 us some hints to help. For example, it is not important to avoid [21] Ruoff R S, Calabri L, Ding W and Pugno N M 2005 Rev. Adv. microscopic voids. In fact it is better to have microscopic voids Mater. Sci. 10 110 than to fill them with excess glue molecules because these can [22] Ruoff R S and Lorents D C 1995 Carbon 33 925 actually weaken the material. Moreover, it is useful to use [23] Ruoff R S, Tersoff J, Lorents D C, Subramoney S and entropic elasticity or friction to dissipate energy in the glue Chan B 1993 Nature 364 514 [24] Ramanathan T, Liu H and Brinson L C 2005 J. Polym. Sci. B molecules rather than in breaking strong bonds if we want a 43 2269 tough material. Finally, it is useful to focus on having the only [25] Zhang Y C and Wang X 2005 Int. J. Solids Struct. 42 5399 bonds that are broken during energy dissipation be weak bonds [26] Moulton S E, Minett A I, Murphy R, Ryan K P, McCarthy D, that can be reformed, such as Coulomb or van der Waals or Coleman J N, Blau W J and Wallace G G 2005 Carbon hydrogen bonds if we want a self-healing material. 43 1879 [27] Pugno N M and Ruoff R S 2004 Phil. Mag. 84 2829 [28] Mielke S L, Troya D, Zhang S, Li J L, Xiao S P, Car R, Acknowledgments Ruoff R S, Schatz G C and Belytschko T 2004 Chem. Phys. Lett. 390 413 [29] Belytschko T, Xiao S P, Schatz G C and Ruoff R S 2002 Phys. We thank Leonid Pechenik, Dan Morse, Galen Stucky, Herb Rev. B 65 235430 Waite, Jonathan Adams, Philipp Thurner, James Weaver [30] Dumitrica T, Hua M and Yakobson B I 2006 Proc. Natl Acad. and Doug Rehn for useful conversations. We acknowledge Sci. USA 103 6105 support from the NASA University Research, Engineering [31] Ruoff R S 2006 Proc. Natl Acad. Sci. USA 103 6779 [32] Yu M F, Lourie O, Dyer M J, Moloni K, Kelly T F and and Technology Institute on Bio-Inspired Materials under Ruoff R S 2000 Science 287 637 award No NCC-1-02037 and the NIH under award RO1 [33] Yu M F, Files B S, Arepalli S and Ruoff R S 2000 Phys. Rev. GM 065354-05. RSR further acknowledges support from Lett. 84 5552

3 PHYSICAL REVIEW B 75, 014301 ͑2007͒

Parallel replica dynamics for driven systems: Derivation and application to strained nanotubes

Blas Pedro Uberuaga,1 Steven J. Stuart,2 and Arthur F. Voter3 1Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 2Department of Chemistry, Clemson University, Clemson, South Carolina 29634, USA 3Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA ͑Received 3 April 2006; revised manuscript received 26 July 2006; published 3 January 2007͒ We show that parallel replica dynamics can be extended to driven systems ͑e.g., systems with time- dependent boundary conditions͒. Each processor simulates a replica at a driving rate that is M times faster than the desired rate, where M is the number of processors. As in regular parallel replica dynamics, when a transition to a new state is detected on any processor, the times are summed and every processor is restarted in the new state. The state-to-state dynamics are shown to be correct if the processors run at the same speed and the system is driven slowly enough ͑on each processor͒ so that the escape rates do not depend on the time history of the drive. We demonstrate the algorithm by stretching a carbon nanotube with a preexisting vacancy, noting a signiﬁcant dependence of the nature of nanotube yield on the strain rate. In particular, we are able to achieve strain rates slow enough such that the time scale for vacancy diffusion is faster than that for mechanical yield at a temperature of 2000 K. We thus observe vacancy-induced morphological changes in the nanotube structure, providing some insight into previously unexplained experimental features.

DOI: 10.1103/PhysRevB.75.014301 PACS number͑s͒: 02.70.Ns, 62.20.Fe, 82.20.Db, 61.46.Fg

I. INTRODUCTION molecule.3 The time scale for such an experiment typically exceeds that accessible to molecular dynamics because of the The molecular-dynamics ͑MD͒ simulation method has limitations in how fast the experiment can be done. For this proven marvelously powerful in a large variety of studies in type of system, it has not been obvious that AMD methods physics, chemistry, materials science, and biochemistry. It can be directly applied, as the state-to-state rate constants are provides a view of system behavior with full atomic detail, in general changing with time. requiring only a form for the interatomic forces and the as- In this paper, we show that parallel replica dynamics can sumption that the dynamics are well described by classical in fact be generalized to treat this type of driven-system case. mechanics. An important limitation, however, is the fact that We derive the relevant equations in the presence of time- MD simulation times are limited to at most microseconds— dependent rate constants and discuss the additional require- and often much less—while technologically important and ments on the system and on the implementation for the ap- scientifically interesting processes typically take place over proach to be valid. We then demonstrate the method by longer times. For many systems, the evolution on these stretching ͑to yield͒ a carbon nanotube with a preexisting longer time scales consists of extended periods of uninterest- vacancy at strain rates inaccessible to direct MD. For differ- ing excursions within a single state of the system ͑e.g., ther- ent strain rates, we see significant differences in both how mal vibrations within a local energy basin͒, punctuated by the nanotube yields and in the details of atomic motion lead- occasional fluctuation-based events that take the system to a ing up to yield. We also see the formation of features that are new state. For this type of system, the recently developed the result of morphological changes caused by the motion of accelerated molecular-dynamics ͑AMD͒ methods1 offer the preexisting vacancy. Such features have been previously ways to reach longer times than MD. In the AMD approach, observed in experiment in which nanotubes suspended on a the key concept is to modify the dynamics in some way that substrate were irradiated with electrons,4 though their origin encourages the system to escape from each state more was unknown. quickly, but without corrupting the relative probabilities of the different possible escape paths. Detailed information II. PARALLEL REPLICA DYNAMICS about the dynamics within each state is sacrificed in exchange for attaining accelerated state-to-state dynamics that The complete derivation of parallel replica dynamics has 2 are faithful to the underlying rate constants. The most accu- been given previously; we summarize it here. The parallel rate of these AMD techniques is the parallel replica dynam- replica method assumes only that, once a correlation time ics method,2 in which time is parallelized by running replica has passed since the system entered the present state, the trajectories on a number of processors, each independently probability distribution function for the time to the next es- searching dynamically for an escape path from the current cape is exponential, state of the system. p ͑t͒dt = k exp͑− kt͒dt. ͑1͒ An important class of systems for which the time-scale single problem often arises is one in which the boundary conditions This applies to a typical MD simulation performed on a are changing in time. A typical situation might be a time- single processor. Here, k is the total escape rate to leave that dependent tensile load applied to either a structural material, state, i.e., the sum of the rates for all the individual escape a nanoscale system such as a nanotube, or a biological paths. This exponential condition holds for any chaotic, er-

8 J. A. Harrison, S. J. Stuart, D. H. Robertson, and C. T. White, J. 413 ͑2004͒. Phys. Chem. B 101, 9682 ͑1997͒. 17 A. J. Stone and D. J. Wales, Chem. Phys. Lett. 128, 501 ͑1986͒. 9 K. M. Liew, C. H. Wong, X. Q. He, M. J. Tan, and S. A. Meguid, 18 P. M. Ajayan, V. Ravikumar, and J.-C. Charlier, Phys. Rev. Lett. Phys. Rev. B 69, 115429 ͑2004͒. 81, 1437 ͑1998͒. 10 Y. Wang, X. X. Wang, X. G. Ni, and H. A. Wu, Comput. Mater. 19 M. B. Nardelli, B. I. Yakobson, and J. Bernholc, Phys. Rev. Lett. Sci. 32, 141 ͑2005͒. 81, 4656 ͑1998͒. 11 H. J. Liu and K. J. Cho, Appl. Phys. Lett. 85, 807 ͑2004͒. 20 O. A. Shenderova, D. W. Brenner, A. Omeltchenko, X. Su, and L. 12 E. W. Wong, P. E. Sheehan, and C. M. Lieber, Science 277, 1971 H. Yang, Phys. Rev. B 61, 3877 ͑2000͒. ͑1997͒. 21 T. Belytschko, S. P. Xiao, G. C. Schatz, and R. Ruoff, Phys. Rev. 13 M. M. J. Treacy, T. W. Ebbesen, and J. M. Gibson, Nature ͑Lon- B 65, 235430 ͑2002͒. don͒ 381, 678 ͑1996͒. 22 S. Zhang, S. L. Mielke, R. Khare, D. Troya, R. S. Ruoff, G. C. 14 M. Huhtala, A. V. Krasheninnikov, J. Aittoniemi, S. J. Stuart, K. Schatz, and T. Belytschko, Phys. Rev. B 71, 115403 ͑2005͒. Nordlund, and K. Kaski, Phys. Rev. B 70, 045404 ͑2004͒. 23 J. Y. Huang, S. Chen, Z. Q. Wang, K. Kempa, Y. M. Wang, S. H. 15 M. Sammalkorpi, A. Krasheninnikov, A. Kuronen, K. Nordlund, Jo, G. Chen, M. S. Dresselhaus, and Z. F. Ren, Nature ͑London͒ and K. Kaski, Phys. Rev. B 70, 245416 ͑2004͒. 439, 281 ͑2006͒. 16 S. L. Mielke, D. Troya, S. Zhang, J.-L. Li, S. Xiao, R. Car, R. S. 24 C. Wei, K. Cho, and D. Srivastava, Phys. Rev. B 67, 115407 Ruoff, G. C. Schatz, and T. Belytschkob, Chem. Phys. Lett. 390, ͑2003͒, and references therein.

014301-9 PHYSICAL REVIEW B 74, 245411 ͑2006͒

Vacancy defects and the formation of local haeckelite structures in graphene from tight-binding molecular dynamics

Gun-Do Lee,1 C. Z. Wang,3 Euijoon Yoon,1 Nong-Moon Hwang,2 and K. M. Ho3 1School of Materials Science and Engineering and Inter-university Semiconductor Research Center (ISRC), Seoul National University, Seoul 151-742, Korea 2National Research Laboratory of Charged Nanoparticles, School of Materials Science and Engineering, Seoul National University, Seoul 151-742, Korea 3Ames Laboratory and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA ͑Received 8 June 2006; revised manuscript received 20 October 2006; published 11 December 2006͒ The dynamics of multivacancy defects in a graphene layer is investigated by tight-binding molecular dynamics simulations and by ﬁrst principles calculation. The simulations show that four single vacancies in the graphene layer ﬁrst coalesce into two double vacancies, each consisting of a pentagon-heptagon-pentagon ͑5-8-5͒ defective structure. While one of the 5-8-5 defects further reconstructs into a 555-777 defect, which is composed of three pentagonal rings and three heptagonal rings, another 5-8-5 defect diffuses toward the reconstructed 555-777 defect. During the 5-8-5 defect diffusion process, three interesting mechanisms, i.e., “dimer diffusion,” “chain diffusion,” and “single atom diffusion,” are observed. Finally, the four single vacancies reconstruct into two adjacent 555-777 defects, which is a local haeckelite structure.

DOI: 10.1103/PhysRevB.74.245411 PACS number͑s͒: 61.72.Ji, 81.05.Uw, 61.80.Az, 71.15.Pd

I. INTRODUCTION of multivacancies, the dynamical behavior of vacancies, and the effects of vacancies on the structure and stability of the Vacancy is one of the most common defects in crystalline graphene layer, atomistic simulation studies of multivacan- solids and affects profoundly the physical properties of the cies in a graphene layer will be very useful and highly de- solids. Low concentration of vacancy defects present in sirable. graphite1 during defective growth or as part of the thermal Since classical molecular dynamics simulations are not equilibrium process. These defects are much more prevalent reliable for studying such complex systems as vacancy de- in electron or ion irradiated materials and are believed to be fects in graphite or carbon nanotube and ab initio molecular the predominant defects on irradiated graphite surfaces.2 dynamics is too expensive for long time dynamical simula- Since graphite is commonly used as a substrate in various tion of large systems, we chose to perform the quantum mo- microscopy techniques,3 characterization of surface defect in lecular dynamics simulation for these systems using the graphite at the atomic scale is an important research drive. tight-binding molecular dynamics ͑TBMD͒ method. Re- Point defects induced by irradiation damage in graphite is cently, we have modified the environment-dependent tight- also a subject of great scientific and technological interest binding ͑TB͒ carbon potential by Tang et al.16 We have also because of the application of graphite as moderators in ther- used this modified TB potential to investigate the structure 4 mal nuclear reactors. Furthermore, the emerging field of car- and energetics of vacancy and adatoms in a graphene layer 5 bon nanoscience shares a lot of useful information from and carbon nanotube and performed TBMD simulations to graphitic systems including defect structure and energetics. study the dynamics of vacancy in a graphene layer. In this Various carbon nanostructures such as carbon nanotube paper, TBMD simulation studies of the structure and dynam- branched junctions6–8 have been produced through genera- ics of multivacancies in a graphene layer will be reported. In tion and recombination of vacancy defects in single-walled particular, the collective behavior of the vacancies and the carbon nanotubes. diffusion of a double vacancy observed from the TBMD Vacancy in graphitic systems has attracted considerable experimental and theoretical studies for many years. Various simulations will be discussed in detail. advanced experimental techniques such as scanning tunnel- II. CALCULATIONAL METHOD ing microscope ͑STM͒,2 positron annihilation spectroscopy,9 and transmission electron microscope10 have been used to investigate the structure and properties of vacancies in graph- The TBMD simulations are performed using the recently ite and carbon nanotubes. At the same time, a number of modified environment dependent tight-binding ͑EDTB͒ car- theoretical calculations have also been performed to study bon potential. We note that the original EDTB carbon poten- the vacancy in graphite.11–15 However, most of the previous tial by Tang et al. is not sufficient to describe the angle theoretical studies focused only on the structure of single dependence of the bonds because the effective interatomic vacancy. Much less is known for the dynamics of the va- distance is scaled only by the coordination number which is cancy as well as the structure and dynamics of multivacan- not angular sensitive.16 In our modified potential, the angle cies. In order to gain more information about the structures dependence of bonds is taken into account by redefining the

1098-0121/2006/74͑24͒/245411͑5͒ 245411-1 ©2006 The American Physical Society VACANCY DEFECTS AND THE FORMATION OF LOCAL… PHYSICAL REVIEW B 74, 245411 ͑2006͒ elite structure in a graphite or carbon nanotube after the ir- ACKNOWLEDGMENTS radiation. The authors would like to acknowledge the support from IV. SUMMARY KISTI under the 7th Strategic Supercomputing Applications We have performed TBMD simulation for a graphene Support Program. The use of the computing system of the layer with four scattered single vacancies. In the simulation, Supercomputing Center is also appreciated. This work is the single vacancies ﬁrst coalesce into double vacancies of supported by the Korea Research Foundation Grant funded 5-8-5 defect structure and then further reconstruct into a lo- by the Korea Government ͑MOEHRD͒͑KRF-2005-041- cal haeckelite structure consisting of two neighboring 555- D00406͒. This work was supported by the BK21 Program 777 defects. During the reconstruction process, we also ob- through the Ministry of Education, Korea. Ames Laboratory served three interesting mechanisms for the diffusion of a is operated for the U.S. Department of Energy by Iowa State 5-8-5 vacancy. These mechanisms are “dimer diffusion,” University under Contract No. W-7405-Eng-82. This work “chain diffusion,” and “single atom diffusion” mechanisms. was also supported by the Director for Energy Research, The simulated STM image of local haeckelite structure Ofﬁce of Basic Energy Sciences including a grant of com- shows that the carbon atoms in pentagonal rings contribute to puter time at the National Energy Research Supercomputing the bright spots. Center ͑NERSC͒ in Berkeley.

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245411-5 ARTICLE IN PRESS

International Journal of Mechanical Sciences 48 (2006) 1464–1470 www.elsevier.com/locate/ijmecsci

Stone–Wales transformation: Precursor of fracture in carbon nanotubes

J. Songa, H. Jiangc, D.-L. Shib, X.-Q. Fengb, Y. Huanga,Ã, M.-F. Yua, K.-C. Hwangb

aDepartment of Mechanical Science and Engineering, University of Illinois, 1206 W. Green Street, Urbana, IL 61801, USA bDepartment of Engineering Mechanics, Tsinghua University, Beijing 100084, China cDepartment of Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZ 85287-6106, USA

Received 14 October 2005; received in revised form 3 February 2006; accepted 29 March 2006 Available online 22 August 2006

Abstract

The fracture strain of carbon nanotubes (CNTs) obtained by molecular dynamics is about 30%, which is much higher than the experimental results (10–13%). The present study shows that this difference results mainly from defects in CNTs. As the tensile strain reaches a few percent, defects are nucleated in the form of Stone–Wales transformation (901 rotation of a bond). A bond in the vicinity of rotated bond breaks as the tensile strain reaches about 13%, which agrees well with the experimental results. Therefore, the Stone–Wales transformation is the precursor of CNT fracture. r 2006 Elsevier Ltd. All rights reserved.

Keywords: Stone–Wales transformation; Bond breakage; Fracture; Carbon nanotube

1. Introduction van der Waals interactions between CNT walls are weak, this failure is essentially the same as that for single-wall Carbon nanotubes (CNTs) possess superior properties CNTs. Yu et al. [17] also measured the failure strain of and have many potential applications such as nanoelec- single-wall CNT bundles, and found even lower failure tronics, nanoscale electromechanical systems (NEMS), and strains around 5–6%. Belytschko et al. [18] and Dumitrica nanocomposites. The mass density of CNTs is only one- et al. [19] showed that this discrepancy between the sixth of that for steel, but their Young’s modulus is six atomistic and experimental studies of failure strains can times higher than steel and is of the order 1 TPa; the be attributed to the nonphysical cutoff function introduced strength of CNTs is of the order 50 GPa, which is two in Brenner’s [12] interatomic potential. They used a orders of magnitude higher than that of steel (e.g., see the modified Morse potential to fit the Brenner potential for review articles, [1–5]). strain up to 10%, and predicted the failure strain between The atomistic studies have shown that CNTs have large 10% and 16%, which agrees with most of Yu et al.’s [16] tensile fracture strain around 30% [6–8] or even higher [9]. experimental results. The continuum theory of Zhang et al. [10] and Jiang et al. Belytschko et al. [18] introduced a weak bond in the [11] based on interatomic potentials for carbon [12,13] also CNT to serve as the site for bond breakage in their predicts the fracture strain in the same range [14,15]. atomistic simulations of CNT fracture. The strength of the However, the experimental studies of Yu et al. [16] for weak bond was 10% lower than others. The weak bond multiwall CNTs found the tensile failure strain of CNTs broke first upon loading, and bond breakage rapidly between 10% and 13%, which is much lower than the propagated to neighbor bonds, leading to brittle fracture of aforementioned atomistic simulations or atomistic-based the CNT. It is unclear, however, whether the fracture strain continuum studies. Most multiwall CNTs fail in a sheath- of CNTs depends on this imperfection. It is also unclear like pattern with only the outer nanotube failing. Since the how fracture starts in a perfect CNT without any preexisting defects. ÃCorresponding author. Tel.: +1 217 265 5072; fax: +1 217 244 6534. The purpose of this paper is to study the failure of single- E-mail address: [email protected] (Y. Huang). wall CNTs without introducing any initial imperfections.

0020-7403/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2006.03.019 ARTICLE IN PRESS J. Song et al. / International Journal of Mechanical Sciences 48 (2006) 1464–1470 1469

zigzag (10,0) much smaller than the fracture strain (around 30%) reported in molecular dynamics studies, but it agrees well with the experimental results. Therefore, the Stone–Wales 5 bond breakage transformation is the precursor of CNT fracture. In other words, bond breakage does not occur directly in a perfect CNT which requires 30% strain. Instead, it follows Stone– Stone-Wales Wales transformation and occurs at only 13% strain. 0 Acknowledgments E (ev) ∆ 9.8% YH acknowledges the support from NSF (Grants 00- 13.3% -5 99909, 01-03257, and 03-28162 via the Nano-CEMMS bond breakage Center at UIUC), Ofﬁce of Naval Research (Grant N00014-01-1-0205, Program Manager Dr. Y.D.S. Raja- pakse), and NSFC. KCH. and XQF acknowledge support -10 from NSFC and the Ministry of Education, China. 0 5 10 15 20 25 30 ε% References

Fig. 6. The energy difference DE ¼ EEperfect versus the tensile strain e for the zigzag (10,0) carbon nanotube, where Eperfect is the energy for the [1] Ruoff RS, Lorents DC. Mechanical and thermal properties of carbon system without any defects, and E is the energy for system with Stone– nanotubes. Carbon 1995;33:925–30. Wales transformation and/or bond breakage. [2] Govindjee S, Sackman JL. On the use of continuum mechanics to estimate the properties of nanotubes. Solid State Communications 1999;110:227–30. [3] Yakobson BI, Avouris P. Mechanical properties of carbon nanotubes. In: Dresselhaus MS, Dresselhaus G, Avouris P, editors. Carbon nanotubes. Topics in applied physics, vol. 80. Berlin– Heidelberg, Germany: Springer; 2001 p. 287–329. [4] Qian D, Wagner GJ, Liu WK, Yu MF, Ruoff RS. Mechanics of carbon nanotubes. Applied Mechanics Reviews 2002;55:495–533. (a) [5] Huang Y, Wang ZL. Mechanics of carbon nanotubes. In: Karihaloo B, Ritchie R, Milne I, editors. Comprehensive structural integrity handbook, Gerberwich W, Yang W, editors., Interfacial and nanoscale fracture, vol. 8. Amsterdam: Elsevier; 2003. p. 551–579 [chapter 8.16]. [6] Dereli G, Ozdogan C. Structural stability and energetics of single- walled carbon nanotubes under uniaxial strain. Physical Review B 2003;67:0354161–6. [7] Ogata S, Shibutani Y. Ideal tensile strength and band gap of single- (b) walled carbon nanotubes. Physical Review B 2003;68:1654091–4. Fig. 7. Schematic diagram of bond breakage following Stone–Wales [8] Mielke SL, Troua D, Zhang S, Li JL, Xiao S, Car R, et al. The role of transformation; (a) breakage of the rotated bond; (b) bond breakage in the vacancy defect and holes in the fracture for carbon nanotubes. vicinity of rotated bond. Chemical Physics Letters 2004;390:413–20. [9] Yakobson BI, Campbell MP, Brabec CJ, Bernholc J. High strain rate fracture and C-chain unraveling in carbon nanotubes. Computational shown in Fig. 7(b) gives the lowest strain for bond Material Science 1997;8:341–8. breakage in the vicinity of rotated bond. The critical strain [10] Zhang P, Huang Y, Geubelle PH, Klein PA, Hwang KC. The elastic (9.8%) for Stone–Wales transformation in zigzag (10,0) modulus of single-wall carbon nanotubes: a continuum analysis incorporating interatomic potentials. International Journal of Solids CNT is much higher than its counterpart (5.5%) for (5,5) and Structure 2002;39:3893–906. armchair CNT, but the strains for bond breakage are [11] Jiang H, Zhang P, Liu B, Huang Y, Geubelle PH, Gao H, et al. The rather close (12.7% versus 13.3%), and both agree well effect of nanotube radius on the constitutive model for carbon with the fracture strain reported in Yu et al.’s [16] nanotubes. Computational Material Science 2003;28:429–42. experiments. [12] Brenner DW. Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond ﬁlms. Physical Review B 1990;42:9458–71. 4. Concluding remarks [13] Brenner DW, Shenderova OA, Harrison JA, Stuart SJ, Ni B, Sinnott SB. A second-generation reactive empirical bond order (rebo) We have used a hybrid atomistic/continuum model to potential energy expression for hydrocarbons. Journal of Physics study the Stone–Wales transformation and bond breakage Condensed Matter 2002;14:783–802. [14] Zhang P, Huang Y, Gao H, Hwang KC. Fracture nucleation in in carbon nanotubes (CNTs) subject to tension. It is shown single-wall carbon nanotubes under tension: a continuum analysis that bond breakage occurs after the Stone–Wales trans- incorporating interatomic potentials. Journal of Applied Mechanics formation, and the failure strain is about 13%, which is 2002;69:454–8. Computational Materials Science 38 (2006) 271–281 www.elsevier.com/locate/commatsci

A comparison of diﬀerent methods of Young’s modulus determination for single-wall carbon nanotubes (SWCNT) using molecular dynamics (MD) simulations

Paras M. Agrawal a, Bala S. Sudalayandi a, Lionel M. Raﬀ b, Ranga Komanduri a,*

a School of Mechanical and Aerospace Engineering, Oklahoma State University, 218 Engineering North, Stillwater, OK 74078, USA b Department of Chemistry, Oklahoma State University, Stillwater, OK 74078, USA

Received 23 December 2005; received in revised form 9 February 2006; accepted 28 February 2006

Abstract

The computed values of Young’s modulus (Y) of single-wall carbon nanotubes given by four common methods based on (i) the determination of stress for a fixed value of strain, (ii) the determination of strain energy for a fixed value of strain, (iii) the longitudinal vibrations, and (iv) the transverse vibrations, and a new method (v) based on the determination of strain for a fixed value of stress have been compared to check the consistency of different methods. The computed values of Y are found to be in agreement with each other with the exception that results of the transverse vibration method differ from those given by other methods when the aspect ratio, namely, the ratio of length to the radius of the tube, is small; the results of the transverse vibration method for the tubes of small diameter are also found to differ from those given by other methods when the commonly used value of thickness of the tube, 3.4 A˚ , is assumed. The solutions of these problems are discussed in terms of an appropriate consideration for the value of thickness and diameter dependent end- correction in the length of the tube. Effect of defects in the form of vacancies, van der Waals (VDW) interactions, chirality, and diameter of the carbon nanotubes on Y has also been investigated. Y is found to be sensitive to the number of vacancies. Y is found to decrease by 1% when VDW interactions between carbon atoms are ignored. Y is also found to be lower for an armchair tube compared to a zigzag tube of the same diameter. As regards the dependence of Y on diameter, we found that as the diameter increases from 7A˚ to 25 A˚ , Young’s modulus drops by 4% and 8%, respectively, for armchair and zigzag tubes. These results are discussed and compared with other experimental and computed results reported in the literature. Ó 2006 Elsevier B.V. All rights reserved.

PACS: 81.07.De; 62.20.Dc; 02.70.Ns

Keywords: Carbon nanotubes; Young’s modulus; MD simulations

1. Introduction rapid progress towards the synthesis of large quantity and high quality CNTs in recent years, carbon nanotubes Since the discovery of carbon nanotubes (CNTs) in 1991 have been used as nanofillers to enhance the mechanical by Iijima [1], much attention has been focused on the strength of polymeric matrices [3–6]. Among the mechani- exploration and application of their extraordinary physical cal properties of nanotubes, the axial Young’s modulus (Y) properties. Qian et al. [2] have shown that an addition of shows an extraordinary feature in that its value is excep- just 1 wt.% CNTs results in a 25% increase in the tensile tionally high. strength of polystyrene-based composite film. With the The determination of Young’s modulus for CNTs has been a subject of considerable interest [7]. The computation * Corresponding author. Tel.: +1 405 744 5900; fax: +1 405 744 7873. of Y of CNTs may be classified into two categories. One is E-mail address: [email protected] (R. Komanduri). molecular dynamics (MD) simulation using a potential

Dr. G. Hazelrigg, and Dr. J. Cao of the Division of Design, [26] E. Hernańdez, C. Goze, P. Bernier, A. Rubio, Appl. Phys. A: Mater. Manufacturing, and Industrial Innovation, Dr. B. M. Kra- Sci. Process. 68 (1999) 287. mer, Engineering Centers Division, and Dr. J. Larsen [27] C. Goze, L. Vaccarini, L. Henrard, P. Bernier, E. Hernańdez, A. Rubio, Synth. Met. 103 (1999) 2500. Basse, Tribology and Surface Engineering program for [28] L. Vaccarini, C. Goze, L. Henrard, E. Hernańdez, P. Bernier, A. their interest and support of this work. This project was Rubio, Carbon 38 (2000) 1681. also sponsored by a DEPSCoR grant on the Multiscale [29] T. Ozaki, Y. Iwasa, T. Mitani, Phys. Rev. Lett. 84 (2000) 1712. Modeling and Simulation of Material Processing [30] Z. Xin, Z. Jianjun, O. Zhong-can, Phys. Rev. B 62 (2000) 13692. (F49620-03-1-0281). The authors thank Dr. Craig S. Hart- [31] Z. Peralta-Inga, S. Boyd, J.S. Murray, C.J. O’Connor, P. Politzer, Struct. 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Mechanical properties of ultrananocrystalline diamond prepared in a nitrogen-rich plasma: A theoretical study

Jeffrey T. Paci,1,* Ted Belytschko,2 and George C. Schatz1,† 1Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3113, USA 2Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3111, USA ͑Received 27 April 2006; revised manuscript received 5 September 2006; published 13 November 2006͒ We examine the mechanical properties of ultrananocrystalline diamond ͑UNCD͒ produced by plasma- enhanced chemical vapor deposition, with a focus on thin ﬁlms created with high levels of nitrogen in the plasma. A model with several of the attributes of the corresponding experimental UNCD is developed and its properties explored. Simulations are performed using semiempirical quantum mechanics and density functional theory. Our results predict a Young’s modulus of 0.69 TPa, failure strain of 0.13, and a tensile fracture stress of 61 GPa which are 66%, 100%, and 61%, respectively, of those predicted for UNCD produced in the absence of nitrogen. As in the case of UNCD produced without nitrogen in the plasma deposition, the fracture stress ͑␴ ͒ f =61 GPa is very large compared to that observed experimentally; these indicate that the experimental specimens contain large defects and some estimates are made of the size of these defects using the Grifﬁth formula with the surface energy computed here. The effect of nitrogen on the mechanical properties of atom-wide UNCD grain boundaries is also investigated. Throughout, the accuracy of the various simulation methods is compared and evaluated.

DOI: 10.1103/PhysRevB.74.184112 PACS number͑s͒: 62.20.Mk, 62.25.ϩg, 81.05.Uw

I. INTRODUCTION essentially butt up against each other, high-level plasma- nitrogen films are better envisioned as diamond grains em- Plasma-enhanced chemical vapor deposition techniques bedded in large amounts of GB material. can be used to make thin diamond films composed of ex- 2 ͑ ͒ The GB’s are composed of a mix of sp - and tremely small 3–5 nm diamond grains and atom-wide sp3-hybridized carbon.13 Near-edge x-ray-absorption fine- ͑ϳ ͒ 1–4 grain boundaries 0.2–0.4 nm wide . The material, structure spectroscopy suggests that 13.5% of the carbon in ͑ ͒ which is called ultrananocrystalline diamond UNCD , has films produced with 20% plasma nitrogen is sp2 ͑ very impressive mechanical properties hardness, fracture hybridized.13 Assuming that all of the sp2 carbon is confined ͒ 5–8 stress, smoothness . Adding nitrogen to the plasma used to to the GB’s, this 13.5% value suggests that 0.135/0.287 or make these films has a dramatic effect on their electrical sp2 9–14 47% of GB carbon atoms are hybridized. conductivity. Because of these properties, UNCD is an Some nitrogen is incorporated into UNCD films produced excellent candidate for use in the production of microelec- 15,16 with nitrogen in the plasma. High-resolution secondary-ion- tromechanical and nanoelectromechanical systems. mass spectroscopy indicates that the level of nitrogen in the Low-level plasma-nitrogen films have morphologies films is a maximum when they are grown with 18% plasma which are similar to those produced in the absence of nitro- nitrogen.13 At this nitrogen level, the nitrogen concentration gen. However, the structure of UNCD changes significantly in the resulting films is 2.2ϫ1020 cm−3 ͑1–2 nitrogen atoms as the level of nitrogen in the plasma is increased above per 1000 carbon atoms͒. The nitrogen is thought to be 10 ͑ ͒ 5%. Grain boundaries GB’s become significantly wider, present at the GB’s rather than being within the SCD ͑ ͒ and the average single-crystal diamond SCD grain size in- grains.18 creases. For example, films produced using 20% plasma ni- The presence of SCD grains cemented to each other by trogen have GB’s which are ϳ2 nm wide and average grain ͑ 13 wide GB layers which have widths which are on the same sizes of 16 nm. length scale as the grain radii͒ makes nitrogen-rich plasma Increased plasma-nitrogen levels also result in an in- UNCD films a kind of composite material. These films are creased GB volume fraction. This fraction can be estimated 17 different than most composites in that the distinct materials using are, in this case, both made of carbon atoms, whereas in most 3⌬͑d − ⌬͒2 others, each component has a higher level of chemical dis- VGB = , ͑1͒ tinctness. Nevertheless, this UNCD is composed of grains d3 which are SCD like and of GB’s which exhibit, because of where ⌬ is the GB thickness and d the average grain diam- their high level of sp2 hybridization, some properties which eter. For UNCD produced without nitrogen in the plasma, are not exhibited by other diamond thin films. ⌬ϳ0.3 nm and dϳ4 nm, which suggests a VGB=0.19, and There appears to be excellent connectivity between the 20% plasma-nitrogen films have a ⌬ϳ2 nm and dϳ16 nm, two phases, resulting in a material that is extremely stiff and so VGB=0.29. strong. For example, UNCD has a Young’s modulus19 E 19 ␴ The differences in volume fraction and average grain size =0.850 TPa and a tensile fracture stress f =2–3 GPa. suggest that whereas for films produced in the absence of Both of these values approach those of natural diamond20–22 ͑ ϳ ␴ ϳ ͒ plasma nitrogen one can envision a material in which grains E 1 TPa and f 4 GPa .

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184112-9 Composites: Part B 37 (2006) 662–669 www.elsevier.com/locate/compositesb

A progressive fracture model for carbon nanotubes

K.I. Tserpes a, P. Papanikos b,*, S.A. Tsirkas c

a Laboratory of Structural Mechanics, Department of Rural and Surveying Engineering, National Technical University of Athens, Zografou Campus, 9 Iroon Polytechniou St, 15780 Athens, Greece b Department of Product and Systems Design Engineering, University of the Aegean, Ermoupolis, Syros 84100, Greece c Institute of Structures and Advanced Materials (ISTRAM), Patron-Athinon 57 St, Patras 26441, Greece

Received 19 January 2005; accepted 1 February 2006 Available online 19 April 2006

Abstract

An atomistic-based progressive fracture model for simulating the mechanical performance of carbon nanotubes by taking into account initial topological and vacancy defects is proposed. The concept of the model is based on the assumption that carbon nanotubes, when loaded, behave like space-frame structures. The finite element method is used to analyze the nanotube structure and the modified Morse interatomic potential to simulate the non-linear force field of the C–C bonds. The model has been applied to defected single- walled zigzag, armchair and chiral nanotubes subjected to axial tension. The defects considered were: 10% weakening of a single bond and one missing atom at the middle of the nanotube. The predicted fracture evolution, failure stresses and failure strains of the nanotubes correlate very well with molecular mechanics simulations from the literature. 2006 Elsevier Ltd. All rights reserved.

Keywords: A. Nanostructures; B. Fracture; C. Finite Element Analysis (FEA); Carbon nanotubes

1. Introduction and interlaminar properties of currently used advanced composites as well as their alignment perpendicular to Carbon nanotubes (CNTs), due to their extraordinary cracks in order to slow down the crack growth by bridging mechanical properties, have stimulated great interest and up the crack faces [3,4]. extensive research with regard to the measurement of their The effective use of CNTs in structural applications exact mechanical properties and search for potential struc- depends on their mechanical performance as stand alone tural applications ever since their discovery by Iijima [1]. units. Experimental observations have revealed that topo- Specific characteristics, such as the exceptionally high stiff- logical defects, such as the Stone–Wales defect and vacancy ness and strength, which are in the range of TPa, the defects, are commonly present in CNTs [5]. Defects extreme resilience, the ability to sustain large elastic strain degrade the mechanical performance of CNTs, since they as well as the high aspect ratio and low density make CNTs alter not only their inelastic properties but also the elastic, the ideal reinforcing material for a new class of superstrong such as the Young’s modulus and Poisson’s ratio. The lon- nano-composites [2]. Besides the use of CNTs as conven- gitudinal and transverse stiffnesses as well as the flexural tional carbon fibers for reinforcing polymer matrix, several rigidity in tension, torsion and bending are, consequently, potential applications have been lately explored. Amongst being altered. For example, Chandra et al. [6] have shown them, is the use of CNTs for improving the out-of-plane that the presence of Stone–Wales defect reduces the stiffness of the defected area by about 30–50% resulting in reduction of the nanotube Young’s modulus. Mechanical properties and deformation of CNTs have * Corresponding author. Tel.: +30 210 3616344; fax: +30 22810 97109. been extensively studied during the last few years both E-mail address: [email protected] (P. Papanikos). experimentally and theoretically. From the viewpoint of

[4] Qian D, Dickey EC, Andrews R, Rantell T. Load transfer and [11] Tserpes KI, Papanikos P. Finite element modeling of single-walled deformation mechanisms in carbon nanotube-polystyrene compos- carbon nanotubes. Compos Part B Eng 2005;36:468–77. ites. Appl Phys Lett 2000;76(20):2868–70. [12] Brenner DW. Empirical potential for hydrocarbons for use in [5] Ebbesen TW, Takada T. Topological and sp3 defect structures in simulating the chemical vapor deposition of diamond films. Phys nanotubes. Carbon 1995;33(7):937–78. Rev B 1990;42:9458. [6] Chandra N, Namilae S, Shet C. Local elastic properties of carbon [13] Xiao JR, Gama BA, Gillespie Jr JW. An analytical molecular nanotubes in the presence of Stone–Wales defects. Phys Rev B structural mechanics model for the mechanical properties of carbon 2004;69:094101. nanotubes. Int J Solids Struct 2005;42:3075–92. [7] Belytschko T, Xiao SP, Schatz GC, Ruoff RS. Atomistic simulations [14] Sun X, Zhao W. Prediction of stiffness and strength of single-walled of nanotube fracture. Phys Rev B 2002;65:235430. carbon canotubes by molecular-mechanics based finite element [8] Mielke SL, Troya D, Zhang S, Li J-L, Xiao S, Car R, et al. The role approach. Mater Sci Eng 2005;390:366–71. of vacancy defects and holes in the fracture of carbon nanotubes. [15] Chen C, Fleck NA. Size effects in the constrained deformation of Chem Phys Lett 2004;390:413–20. metallic foams. J Mech Phys Solids 2002;50:955–77. [9] Liew KM, He XQ, Wong CH. On the study of elastic and plastic [16] Buryachenko VA. Effective elastic moduli of triply periodic partic- properties of multi-walled carbon nanotubes under axial tension ulate matrix composites with imperfect unit cells. Int J Solids Struct using molecular dynamics simulations. Acta Materiala 2004;52: 2005;42:4811–32. 2521–7. [17] Yu MF, Lourie O, Dyer MJ, Moloni K, Kelly TF, Ruoff RS. [10] Li C, Chou T-W. A structural mechanics approach for the analysis of Strength and breaking mechanism of multiwalled carbon nanotubes carbon nanotubes. Int J Solids Struct 2003;40:2487–99. under tensile load. Science 2000;287:637. INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER J. Phys.: Condens. Matter 18 (2006) S1971–S1990 doi:10.1088/0953-8984/18/33/S14

On the strength of the carbon nanotube-based space elevator cable: from nanomechanics to megamechanics

Nicola MPugno

Department of Structural Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Italy

E-mail: [email protected]

Received 31 January 2006, in ﬁnal form 16 March 2006 Published 4 August 2006 Online at stacks.iop.org/JPhysCM/18/S1971

Abstract In this paper various deterministic and statistical models, based on new quantized theories proposed by the author, are presented for estimating the strength of a real, and thus defective, space elevator cable. The cable, ∼100 000 km in length, is composed of carbon nanotubes, ∼100 nm long: thus, its design involves nanomechanics and megamechanics. The predicted strengths are extensively compared with the experimental and atomistic simulation results for carbon nanotubes available in the literature. All these approaches unequivocally suggest that the megacable strength will be reduced by a factor at least of ∼70% with respect to the theoretical nanotube strength, today (erroneously) assumed in the cable design. The reason is the unavoidable presence of defects in so huge a cable. Preliminary in-silicon tensile experiments confirm the same finding. The deduced strength reduction is sufficient to place in doubt the effective realization of the space elevator, that if built as designed today will certainly break (in the author’s opinion). The mechanics of the cable is also revised and possible damage sources discussed. (Some figures in this article are in colour only in the electronic version) Invited paper presented at Nanoscience and Nanotechnology 2005

1. Introduction

Aspace elevator (ﬁgure 1)basically consists of a cable attached to the Earth’s surface for carrying payloads into space (Artsutanov 1960). If the cable is long enough, i.e., around 150 000 km (reducible by a counterweight), the centrifugal forces exceed the gravity of the cable, that will work under tension (Pearson 1975). The elevator would stay ﬁxed geosynchronously. Once sent far enough, climbers would be accelerated by the Earth’s

0953-8984/06/331971+20$30.00 © 2006 IOP Publishing Ltd Printed in the UK S1971 On thestrength of the carbon nanotube-based space elevator cable S1989 to be reduced by a factor of at least ∼70% with respect to the theoretical strength of a carbon nanotube, assumed in the current design. Such a reduction is sufﬁcient to cast doubt on the effective realization of the space elevator. It is the author’s opinion that the cable, if realized as designed today (see Edwards and Westling 2003), will break.

Acknowledgments

The author thanks Professors Carpinteri, Ruoff and Troger for discussion and especially Dorothy Hesson for the English grammar supervision.

References

Artsutanov Y 1960 V Kosmos na Elektrovoze Komsomol-skaya Pravda (July) (contents described in Lvov 1967 Science 158 946–947) Barber A H, Kaplan-Ashiri I, Cohen S R, Tenne R and Wagner H D 2005 Stochastic strength of nanotubes: an appraisal of available data Compos. Sci. Technol. at press Belytschko T, Xiao S P and Ruoff R 2002 Effects of defects on the strength of nanotubes: experimental–computational comparisons LosAlamosNational Laboratory preprint physics/0205090 Brown E T and Hoek E 1978 Trends in relationship between measured in situ stresses and depth Int. J. Rock Mech. Min. Sci. Geomech. 15 211–5 Carpinteri A 1982 Notch sensitivity in fracture testing of aggregative materials Eng. Fract. Mech. 16 467–81 Carpinteri A and Pugno N 2002 A fractal comminution approach to evaluate the drilling energy dissipation Int. J. Numer. Anal. Methods Geomech. 26 499–513 Carpinteri A and Pugno N 2004 Evolutionary fractal theory of erosion and experimental assessment on MIR space station Wear 257 408–13 Carpinteri A and Pugno N 2005 Are the scaling laws on strength of solids related to mechanics or to geometry? Nat. Mater. 4 421–3 Edwards B C 2000 Design and deployment of a space elevator Acta Astronaut. 10 735–44 Edwards B C 2003 The Space Elevator NIAC Phases I and II Final reports Edwards B C and Westling E A 2003 The Space Elevator: A Revolutionary Earth-to-Space Transportation System Spageo Inc. Gombers J and Johnson P 2005 Dynamic triggering of earthquakes Nature 437 830 GriffithAA1921 The phenomenon of rupture and flow in solids Phil. Trans. R. Soc. A 221 163–98 Iijima S 1991 Helical microtubules of graphitic carbon Nature 354 56–8 Mielke et al 2004 The role of vacancy defects and holes in the fracture of carbon nanotubes Chem. Phys. Lett. 390 413–20 Neuber H 1958 Theory of Notch Stresses (Berlin: Springer) Novozhilov V 1969 On a necessary and sufficient criterion for brittle strength Prikl. Mat. Mekh. 33 212–22 Pearson J 1975 The orbital tower: a spacecraft launcher using the Earth’s rotational energy Acta Astronaut. 2 785–99 Pugno N 2002 AQuantized Griffith’s criterion Italian Group of Fracture Mtg on Fracture Nanomechanics (Vigevano, Italy, Sept. 2002) Pugno N 2004a New quantized failure criteria: application to nanotubes and nanowires Preprint cond-mat/0411556 Pugno N 2004b Preprint cond-mat/0504520 Pugno N 2006a Int. J. Fract. at press Pugno N 2006b Dynamic quantized fracture mechanics Int. J. Fracture at press Pugno N 2006c A universal scaling law for nanoindentation, but not only Preprint cond-mat/0601370 (submitted) Pugno N and Ruoff R 2004 Quantized fracture mechanics Phil. Mag. 84 2829–45 Pugno N and Ruoff R 2006 Nanoscale Weibull statistics J. Appl. Phys. 99 1–4 (Preprint cond-mat/0504518) Qian D, Liu W K and Ruoff R S 2003 Load transfer mechanism in carbon nanotube ropes Compos. Sci. Technol. 63 1561–9 Steindl A and Troger H 2005 Is the sky-hook configuration stable? Nonlinear Dyn. 40 419–31 Wang Q and Varadan V K 2005 Stability analysis of carbon nanotubes via continuum models Smart Mater. Struct. 14 281–6 Weibull W 1939 AStatistical Theory of the Strength of Materials. Ingeni¬orsvetenskapsakademiens Handlingar 151 Yakobson B I, Brabec C J and Bernholc J 1996 Nanomechanicsofcarbon tubes: instabilities beyond linear range Phys. Rev. Lett. 76 2511–4 J. Phys. Chem. B 2006, 110, 13037-13044 13037

Ozonization at the Vacancy Defect Site of the Single-Walled Carbon Nanotube

Lei Vincent Liu, Wei Quan Tian,† and Yan Alexander Wang* Department of Chemistry, UniVersity of British Columbia, VancouVer, BC V6T 1Z1, Canada ReceiVed: October 19, 2005; In Final Form: April 26, 2006

The ozonization at the vacancy defect site of the single-walled carbon nanotube has been studied by static quantum mechanics and atom-centered density matrix propagation based ab initio molecular dynamics within a two-layered ONIOM approach. Among five different reaction pathways at the vacancy defect, the reaction involving the unsaturated active carbon atom is the most probable pathway, where ozone undergoes fast dissociation at the active carbon atom at 300 K. Complementary to the experiments, our work provides a microscopic understanding of the ozonization at the vacancy defect site of the single-walled carbon nanotube.

1. Introduction with Stone-Wales defects.19-21 To the best of our knowledge, Single-walled carbon nanotubes (SWCNTs) have been in- there has been no reported theoretical studies on the chemical tensively studied during the past decade since the discoveries reaction of ozone with the vacancy defect sites of the SWCNTs. of Iijima in the early 1990s.1 A lot of potential applications of Vacancy defects can either occur as native defects or be induced 22,23 the SWCNTs have been proposed due to their unique proper- by ion or electron irradiation of the SWCNTs. The structures, ties: high Young’s modulus, high thermal conductivity, and mechanical and electronic properties, and potential applications high aspect ratio structure, etc. Over the years, applications of of the SWCNTs with vacancy defects have been recently 23-30 the SWCNTs have been successfully realized as chemical predicted theoretically. Direct observations of the vacancy sensors,2 hydrogen storage materials,3 and vacuum electronic defects on graphite and double-walled carbon nanotubes have devices.4 Access to the interior of the SWCNTs is essential for been reported recently by Iijima and co-workers using in situ 31,32 most of these applications. high-resolution transmission electron microscopy technique. Unfortunately, most of the SWCNTs are synthesized with In our previous studies, we have shown that the vacancy defect closed hemispherical fullerenelike end-caps, which prevent introduces localized electronic states at the defect site, which internal adsorption of chemical reagents.5 It is thus often leads to regioselectivity on the sidewall and further facilitates 25-27 necessary to open the capped ends via chemical means,6-14 the selective functionalization of the SWCNTs. Such which take advantage of the higher reactivity of the end-caps. reactivity of the vacancy defect was utilized to cut the SWCNTs 33 Such a higher reactivity is a result of the fact that the in a well-controlled oxidative way by Smalley and co-workers, pyramidalization angles of any hemispherical fullerenelike end- recently. caps are bigger than those of the sidewalls of the nanotubes. In this work, we present our theoretical studies of the Gas-phase ozone oxidation is one of the well-developed end- ozonization at the single vacancy defect site on the sidewall of opening methods.9-13 The oxidation process removes the caps the (5,5) SWCNT to further understand the chemical properties and introduces or enlarges vacancy defects on the sidewall, of the vacancy defect site. We anticipate a similar reactivity of producing two kinds of functional groups, esters and quinones, the vacancy defects on the outmost layer of the multiwalled at the ends or at the defective sites of the sidewall.9-13 After carbon nanotubues (CNTs), because the large interlayer distance high-temperature thermal treatment, these two functional groups makes the chemical effect of the inner CNTs on the outmost 11-13 CNT unlikely. Moreover, the chirality of the SWCNT should will decompose and emit a large amount of CO and CO2. Olefin ozonolysis can be understood through the standard have a more important effect on the reactivity of the vacancy Criegee’s mechanism15 (Scheme 1). As an 18-valence-electron, defect than the diameter of the SWCNT, since the chirality of 1,3-dipolar molecule, ozone (2) reacts with an olefin (1) via the SWCNT can readily modify the structure of the vacancy 24 the 1,3-dipolar cycloaddition (1,3-DC) to the π bond of the defect. It is our hope that our studies will make the first step olefin and forms the primary ozonide (3), which has an unstable toward the comprehensive understanding of the chemistry of five-membered ring. The C-C single bond and one of the O-O the vacancy defect site of CNTs of various forms. bonds of the primary ozonide then break to produce a carbonyl 2. Computational Methods compound (5) and a carbonyl oxide (6) in a zwitterion form. These two compounds (5 and 6) will recombine to form an We first built a fragment of the (5,5) SWCNT of 120 carbon ozonide (7). Recently, a similar Criegee’s mechanism atoms with 20 end-capping hydrogen atoms, C120H20. The has been proposed in the reactions of ozone with C60 and the hydrogen atoms are used to saturate the carbon atoms with SWCNTs.16-18 dangling bonds at the two ends. We then removed one carbon Several theoretical studies have been carried out to understand atom from the middle of the sidewall of C120H20, producing an the ozonization of the perfect SWCNTs17,18 and the SWCNTs ideal single vacancy, which contains three carbon atoms with dangling bonds (Figure 1a). Due to the large system size, we * Corresponding author. E-mail: [email protected]. first employed the semiempirical AM1 method34 to optimize † Current address: Department of Material Sciences, Faculty of Engi- neering Sciences, Kyushu University, 6-1 Kasugakoen, Kasuga, Fukuoka, the geometry. We further refined the AM1 optimized geometry 816-8580, Japan. with density functional theory (DFT) at the B3LYP/6-31G(d) 10.1021/jp055999x CCC: $33.50 © 2006 American Chemical Society Published on Web 06/14/2006 13044 J. Phys. Chem. B, Vol. 110, No. 26, 2006 Liu et al.

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ORIGINAL ARTICLE

Dynamic quantized fracture mechanics

N. M. Pugno

Received: 6 May 2005 / Accepted: 7 August 2006 © Springer Science+Business Media B.V. 2006

Abstract A new quantum action-based theory, ond range impact failures of 2024-T3 aircraft alu- dynamic quantized fracture mechanics (DQFM), is minum alloy. Since LEFM has been successfully presented that modifies continuum-based dynamic applied also at the geophysics size-scale, it is con- fracture mechanics (DFM). The crack propaga- ceivable that DQFM theory can treat objects that tion is assumed as quantized in both space and span at least 15 orders of magnitude in size. time. The static limit case corresponds to quantized fracture mechanics (QFM), that we have recently Keywords Dynamic fracture · Quantized developed to predict the strength of nanostruc- fracture · Finite fracture · Nanostructures · tures. DQFM predicts the well-known forbidden Nanotubes · Strength · Impacts strength and crack speed bands—observed in atomistic simulations—which are unexplained by continuum-based approaches. In contrast to DFM and linear elastic fracture mechanics (LEFM), that are 1 Introduction shown to be limiting cases of DQFM and which can treat only large (with respect to the “frac- Two classic treatments of linear elastic fracture ture quantum”) and sharp cracks under moderate mechanics (LEFM) are Griffith’s criterion (1920), loading speed, DQFM has no restrictions on treat- an energy-based method, and a method based on ing defect size and shape, or loading rate. Simple the stress-intensity factor developed by examples are discussed: (i) strengths predicted by Westergaard (1939). These have been shown to DQFM for static loads are compared with experi- be equivalent, as in the correlation between (sta- mental and numerical results on carbon nanotubes tic) energy release rate and stress-intensity factors containing nanoscale defects; (ii) the dynamic frac- formulated by Irwin (1957). An extension towards ture initiation toughness predicted by DQFM is dynamic fracture mechanics (DFM) was proposed compared with experimental results on microsec- by Mott (1948), which included in Griffith’s energy balance the contribution of the kinetic energy. International Conference on Fracture XI — Symposium Dynamic stress-intensity factors were then also 34, on Physics and Scaling in Fracture. proposed, as well as the dynamic generalization of N. M. Pugno (B) Irwin’s correlation, see the Freund’s book (1990). Department of Structural Engineering, Politecnico di Since LEFM and DFM can be applied only to large Torino, Corso Duca degli Abruzzi 24, 10129 Torino, and sharp cracks under moderate loading rates, we Italy e-mail: [email protected] choose to modify them by accounting for the dis- 168 N. M. Pugno

References Murakami H. (1986) Stress intensity factors handbook. Publ. Pergamon, Oxford, UK Belytschko T (2004) The role of vacancy defects and holes in Neuber H (1958) Theory of notch stresses. Springer, Berlin the fracture of carbon nanotubes. Chem Phys Lett 390: Novozhilov V (1969) On a necessary and sufficient criterion 413–420 for brittle strength. Prik Mat Mek, 33: 212–222 Belytschko T, Xiao SP, Ruoff R, (2002) Effects of defects Ogata S, Shibutani Y (2003) Ideal tensile strength and band on strength of nanotubes: experimental-computational gap of single-walled carbon nanotube. Phys Rev B 68: comparison. Los Alamos National Laboratory, Preprint 165409–1/4 Archive, Physics pp 1–6 Orowan E (1948) Fracture and strength of solids. Rep Pro- Carpinteri A, Pugno N (2005) Are the scaling laws on gress Phys XII: 185 strength of solids related to mechanics or to geometry? Owen DM, Zhuang SZ, Rosakis AJ, Ravichandran G (1998) Nat Mat 4: 421–423. Experimental determination of dynamic crack initiation Freund LB (1990) Dynamic fracture mechanics. Cambridge and propagation fracture toughness in thin aluminum University Press sheets. Int J Fract 90: 153–174 Gao H, Ji B, Jaeger IL, Arzt E, Fratzl P (2003) Materials Pechenik L, Levine H, Kessler D, (2002) Steady-state mode become insensitive to flaws at nanoscale: lesson from I cracks in a viscoelastic triangular lattice. J Mech Phys nature. Proce Natl Acad Sci USA 100: 5597–5600 Sol 50: 583–613 Griffith AA (1920) The phenomenon of rupture and flow in Petrov Yu V, Sitnikova WV (2004) Dynamic cracking resis- solids. Phil Trans Roy Soc A221: 163–198 tance of structural materials predicted from impact frac- Heizler SI, Kessler DA (2002) Mode-I fracture in a non- ture on aircraft alloy. Tech Phys 49: 57–60 linear lattice with viscoelastic forces. Phys Rev E 66: Petrov Yu V (1996) Quantum analogy in the mechanics of 016126–1/10. fracture solids. Phys Solid State 38: 1846–1850 Hellan K (1985) An Introduction to fracture mechanics. Pugno N, Ruoff R (2004) Quantized fracture mechanics. Phil McGraw-Hill Book Company Mag 84(27): 2829–2845 Hirai Y et al(2003) Molecular dynamics studies on mechani- Pugno N (2006) On the strength of the nanotube-based space cal properties of carbon nano tubes with pinhole defects. elevator cable: from nanomechanics to megamechanics. Jpn J Appl Phys 42: 4120–4123 J Phys-Condens Mat 18: S1971–S1990 Holland D, Marder M (1999) Cracks and atoms. Adv Mat Slepyan L (1981) Dynamics of brittle fracture in lattice. Dok- 11: 793–806 lady Soviet Phys 26: 538–540 Irwin GR (1957) Analysis of stresses and strains near the Tada H, Paris PC, Irwin GR (1985) The stress intensity factor end of a crack traversing a plate. Trans ASME J Appl handbook 2nd edn. Paris Productions Incorporated Mech E24: 361–364 Taylor D, Cornetti P,Pugno N (2005) The fracture mechanics Kessler D, Levine H (2003) Does the continuum theory of of finite crack extensions. Eng Frac Mech 72: 1021–1028 dynamic fracture work? Phys Rev E 68:036118–1/4 Westergaard HM (1939) Bearing pressures and cracks. Marder M (1991) New dynamical equations for cracks. Phys J Appl Mech 6: 49–53 Rev Lett 66: 2484–2487 Yu M-F, Lourie O, Dyer MJ, Moloni K, Kelly TF, Ruoff, RS Marder M, Gross S (1995) Origin of crack tip instabilities. (2000) Strength and breaking mechanism of multiwalled J Mech Phys Sol 43: 1–48 carbon nanotubes under tensile load. Science 287: 637– Marder M, Liu X (1993) Instability in lattice fracture. Phys 640 Rev Lett 71: 2417–2420 Zhang S, Mielke SL, Khare R, Troya D, Ruoff RS, Schatz Mielke SL, Troya D, Zhang S, Li J-L, Xiao S, Car R, Ruoff GC, Belytschko T (2004) Mechanics of defects in carbon RS, Schatz GC, nanotubes: atomistic and multiscale simulations. Phys Mott NF, (1948) Brittle fracture in mild steel plates. Engi- Rev B 71: 115403-1/12 neering 165: 16–18 Morozov NF, Petrov Yu V, Utkin AA (1990) Dokl Akad Nauk SSSR 313(2): 276 [Sov Phys Dokl 35:646] ournal of Statistical Mechanics: Theory and Experiment JAn IOP and SISSA journal

The asymptotic properties of random strength and compliance of single-walled carbon nanotubes using Stat J. atomistic simulation

1 2 Baidurya Bhattacharya and Qiang Lu .Mec 1 Department of Civil Engineering, Indian Institute of Technology, Kharagpur, WB 721302, India 2 Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, USA

E-mail: [email protected] and [email protected] h.

Received 13 February 2006 Accepted 31 May 2006 P06021 (2006) Published 30 June 2006

Online at stacks.iop.org/JSTAT/2006/P06021 doi:10.1088/1742-5468/2006/06/P06021

Abstract. Mechanical response of deformable bodies is often concerned with either thesum or the extreme of an underlying random process. This paper investigates the asymptotic statistical properties of ultimate strength (σu)and compliance (C)ofsingle-walled nanotubes (SWNTs) containing random defects using the technique of atomistic simulation (AS). The defects considered are of the Stone–Wales (SW) kind and a Matern hard-core random field applied on a finite cylindrical surface is used to describe the spatial distribution of the SW defects. Ananotube can be viewed as consisting of nominally identical segments of equal length possessing a stationary distribution of ultimate strength, σu.Under a weak dependence condition among the segmentstrengths(that decay to zero with increasing distance between the segments), consistent with the non-local nature of atomic interactions, formalized here in the form of strong mixing, the asymptotic properties of σu (as the extreme of the strong mixing sequence) and C (as the sum of a related strong mixing sequence) are studied with increasing tube length, l.Theextremal index, measuring the stochastic dependence in the strength field, is estimated. We simulate a set of displacement controlled tensile loading up to fracture of (6, 6) SWNTs with length between 49 and 492 A.˚ With increasing l,thedistribution of σu is found to shift to the left and become narrower and appears to fit the Weibull distribution rather well; the compliance of the tube increases with increasing l and becomes asymptotically normal. The compliance

c 2006 IOP Publishing Ltd and SISSA 1742-5468/06/P06021+21$30.00 Random strength and compliance of single-walled nanotubes of SW defects per unit tube surface area was kept constant. Seven values of l spanning an order of magnitude were considered (from 49 to 490 A)˚ and the loading was adjusted such that the strain rate was the same for each tube length. The strength distribution was found to shift to the left and become narrower with increasing l,andalsoappeared to ﬁt the Weibull distribution rather well. The distribution of C as the scaled sum of the reciprocal of the strong mixing strength sequence was studied with increasing tube length as well. The compliance of the tube increased with increasing length and became asymptotically normal. Finally, the compliance and strength of the tube were found to

be asymptotically uncorrelated. These results appeared to validate the strong mixing property of the strength ﬁeld. These ﬁndings can be used in future studies to better Stat J. model the random mechanical behaviour of nanotubes and nanotube based devices. Acknowledgment

Asmallsetofpreliminary results from this work was presented at the Ninth International Conference on Structural Safety and Reliability held in Rome, Italy, in June 2005. .Mec

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Deep levels in the band gap of the carbon nanotube with vacancy-related defects ͒ Gunn Kima Department of Physics, North Carolina State University, Raleigh, North Carolina 27695 and School of Physics, Seoul National University, Seoul 151-747, Korea ͒ Byoung Wook Jeong and Jisoon Ihmb School of Physics, Seoul National University, Seoul 151-747, Korea ͑Received 8 October 2005; accepted 4 April 2006; published online 9 May 2006͒ We study the modification in the electronic structure of the carbon nanotube induced by vacancy-related defects using the first-principles calculation. Three defect configurations which are likely to occur in semiconducting carbon nanotubes are considered. A vacancy-adatom complex is found to bring about a pair of localized states deep inside the energy gap. A pentagon-octagon-pentagon topological defect produced by the divacancy is structurally stable and gives rise to an unoccupied localized state in the gap. We also discuss the character of partially occupied localized state produced by a substitutional impurity plus a monovacancy. © 2006 American Institute of Physics. ͓DOI: 10.1063/1.2202112͔

As excellent conductors1,2 with long mean free path and 1͑a͒. If a single atom is removed from the nanotube, two of remarkably high mobility, carbon nanotubes ͑CNTs͒ have di- three carbon atoms around the vacancy rebond ͑bond length verse potential applications in nanoelectronics. Though the of 1.57 Å͒ by the Jahn-Teller distortion. The third atom has properties of the atomically perfect CNT have been under- the dangling bond even after the relaxation.12–14 The mono- stood relatively well, experimental and theoretical reports on vacancy alone does not reproduce the experimental spectra the defects such as vacancy and impurity substitution are of a pair of deep levels.5,6 The adatom may be mobile be- limited. Recently, Gómez-Navarro et al. showed that even a cause it has a lower diffusion barrier ͑ϳ0.47 eV͒͑Ref. 15͒ low concentration of vacancies in single-walled carbon nano- than that for vacancy migration ͑ϳ1.7 eV͒.16 Here the bond tubes ͑SWNTs͒ can produce a large decrease in their electri- length between the adsorbed C atom and the atom on the cal conductance.3 Through a sequence of high resolution tube is 1.34 Å. It is a typical C–C double bond and indicates transmission electron microscopy images recorded in situ on that the adatom has two unpaired electrons. The binding en- one of the SWNTs; on the other hand, Hashimoto et al. ergy of the adsorbed C atom in the model system is about showed that the adatoms appear mostly in the vicinity of the 5.6 eV. It is comparable with the bonding energy of the C–C vacancies.4 In the scanning tunneling microscopy ͑STM͒ ex- double bond ͑ϳ6.3 eV͒. Thus we conclude that our model periment on a semiconducting CNT, localized gap states structure is energetically stable. were observed in semiconducting SWNTs.5,6 In particular, a When a vacancy-adatom complex like our model struc- pair of gap states was found far from the band gap edge ture is introduced, two unoccupied ﬂat levels appear within forming deep levels in the STM measurement. the semiconducting gap, as plotted in Fig. 1͑b͒. Since they In this letter, we report the electronic structure of CNTs exist far from the band edges of the ͑17,0͒ CNT, they can be with various vacancy-related defects. The ﬁrst-principles called deep levels. In many cases, a deep level in the semi- pseudopotential calculations are carried out based on the conductor acts as a recombination center and decreases density functional theory7 within the local density approximation8 with spin polarization for the exchange- correlation functional. The ionic potential is described with the norm-conserving Troullier-Martins pseudopotential.9 Wave functions are expanded in a double-␨ basis set with an 10,11 energy cutoff of 80 Ry implemented in the SIESTA code. As a model system, we choose the ͑17,0͒ zigzag CNT of ϳ13.4 Å in diameter and ϳ0.5 eV in the band gap. The supercell size in the lateral direction is 25 Å to avoid the interaction between neighboring CNTs and that in the axial direction is 22 Å. The atomic position is relaxed until the forces on the atoms are reduced to within 0.02 eV/Å. First, we have tested many different geometries with vacancies to reproduce experimentally observed two unoccupied deep levels in a semiconducting nanotube.5,6 A simple model of a vacancy-adatom complex is presented in Fig.

FIG. 1. ͑a͒ Side view of the schematic ball-and-stick model and ͑b͒ band ͒ a Electronic mail: [email protected] structure for the vacancy-adatom complex system of the ͑17,0͒ CNT. The ͒ b Electronic mail: [email protected] Fermi level is set to zero. Two unoccupied ﬂatbands occur in the band gap.

0003-6951/2006/88͑19͒/193107/3/$23.0088, 193107-1 © 2006 American Institute of Physics Downloaded 19 Nov 2007 to 140.109.112.41. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp 193107-3 Kim, Jeong, and Ihm Appl. Phys. Lett. 88, 193107 ͑2006͒

band gap of the semiconducting CNT. In the realistic case of the random distribution of defects, there would be slightly broadened density of states in the gap produced by nearly degenerate localized gap states. The broadening will be enhanced as the defect density increases. These states can behave as recombination centers of electron and hole carriers. As in bulk semiconductors such as silicon and gallium ars- enide, these levels can signiﬁcantly affect the transport and optical properties of the semiconducting CNTs. This work is supported by the SRC program ͑Center for Nanotubes and Nanostructured Composites͒ of MOST/ KOSEF, the KRF ͑Grant No. KRF-2005-070-C00041͒, and the MOST through the NSTP ͑Grant No. M1-0213-04-001͒.

1T. Durkop, S. A. Getty, E. Cobas, and M. S. Fuhrer, Nano Lett. 4,35 ͑2004͒. 2A. Javey, J. Guo, Q. Wang, M. Lundstrom, and H. Dai, Nature ͑London͒ 424, 654 ͑2003͒. 3C. Gómez-Navarro, P. J. De Pablo, J. Gómez-Herrero, B. Biel, F. J. Garcia-Vidal, A. Rubio, and F. Floresi, Nat. Mater. 4, 534 ͑2005͒. 4A. Hashimoto, K. Suenaga, A. Gloter, K. Urita, and S. Iijima, Nature FIG. 4. ͑Color online͒ Localized states of the semiconducting ͑17,0͒ CNT ͑London͒ 430, 870 ͑2004͒. with a monovacancy plus a substitutional impurity. Side views of the 5H. Kim, J. Lee, S. Lee, Y. Kuk, J.-Y. Park, and S.-J. Kahng, Phys. Rev. B isodensity surface of the gap states due to a monovacancy with a boron and 71, 235402 ͑2005͒. nitrogen substitutional impurity are shown in ͑a͒ and ͑b͒, respectively, and 6S. Lee, G. Kim, H. Kim, B.-Y. Choi, J. Lee, B. W. Jeong, J. Ihm, Y. Kuk, top views of them in ͑c͒ and ͑d͒. The values for the red and blue isodensity and S.-J. Kahng, Phys. Rev. Lett. 95, 166402 ͑2005͒. 3 7 surfaces are ±0.02e/a0, where the sign is that of the wave function. W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 ͑1965͒. 8D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 ͑1980͒. 9N. Troullier and J. L. Martins, Phys. Rev. B 43, 1993 ͑1991͒. junctions with complex topological defects reveal common 10 19 P. Ordejón, E. Artacho, and J. M. Soler, Phys. Rev. B 53, R10441 ͑1996͒. features in electronic properties of the localized states. 11D. Sánchez-Portal, P. Ordejón, E. Artacho, and J. M. Soler, Int. J. Quan- However, the wave functions may not have the mirror sym- tum Chem. 65, 453 ͑1997͒. metry in general. Vacancy-related defects produce localized 12S. L. Mielke, D. Troya, S. Zhang, J.-L. Li, S. Xiao, R. Car, R. S. Ruoff, G. levels near the Fermi level even in the metallic CNT.18 The C. Schatz, and T. Belytschko, Chem. Phys. Lett. 390,413͑2004͒. 13 difference from the semiconducting CNT is that, since there Y. Ma, P. O. Lehtinen, A. S. Foster, and R. M. Nieminen, New J. Phys. 6, ͑ ͒ exists a ﬁnite density of states at all energies in metallic 68 2004 . 14J. Rossato, R. J. Baierle, A. Fazzio, and R. Mota, Nano Lett. 5,197 CNTs, the localized states are actually in resonance with the ͑2005͒. extended states with ﬁnite broadening. 15P. O. Lehtinen, A. S. Foster, A. Ayuela, A. Krasheninnikov, K. Nordlund, In summary, relatively deep localized gap states caused and R. M. Nieminen, Phys. Rev. Lett. 91, 017202 ͑2003͒. by various vacancy-related defects are found in the semicon- 16A. A. El-Barbary, R. H. Telling, C. P. Ewels, M. I. Heggie, and P. R. ducting SWNT in ab initio pseudopotential calculations. Be- Briddon, Phys. Rev. B 68, 144107 ͑2003͒. 17 ͑ ͒ cause of the periodic boundary condition in the supercell A. Hangleiter, Phys. Rev. B 35,9149 1987 . 18H. J. Choi, J. Ihm, S. G. Louie, and M. L. Cohen, Phys. Rev. Lett. 84, method, our systems correspond to the case where defects 2917 ͑2000͒. are arranged in order. Band structures show weak band dis- 19Y.-W. Son, S. B. Lee, C.-K. Lee, and J. Ihm, Phys. Rev. B 71, 205422 persions due to weakly interacting gap states in the intrinsic ͑2005͒.

Downloaded 19 Nov 2007 to 140.109.112.41. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp Symmetry-, time-, and temperature-dependent strength of carbon nanotubes

Traian Dumitrica, Ming Hua, and Boris I. Yakobson†

Departments of Mechanical Engineering & Materials Science and Chemistry, Rice University, Houston, TX 77005

Communicated by Robert F. Curl, Rice University, Houston, TX, February 3, 2006 (received for review December 6, 2005) Although the strength of carbon nanotubes has been of great actually dominant remained unanswered. In this article, we interest, their ideal value has remained elusive both experimen- present a simultaneous study of the two mechanisms, combining tally and theoretically. Here, we present a comprehensive analysis molecular dynamics (MD) simulations, careful quantum me- of underlying atomic mechanisms and evaluate the yield strain for chanical evaluation of the energy characteristics for the key arbitrary nanotubes at realistic conditions. For this purpose, we configurations, and reaction rate theory (22) for the probability combine detailed quantum mechanical computations of failure of yield event. This approach allows one to calculate breaking- nucleation and transition-state barriers with the probabilistic ap- strain values for nanotubes of different symmetry and diameter proach of the rate theory. The numerical results are then summa- at different temperatures and load rates. Such a comprehensive rized in a concise set of equations for the breaking strain. We reveal view, inaccessible with direct MD, results in a strength map a competition between two alternative routes of brittle bond relating the load level, its duration, temperature, and chirality of breaking and plastic relaxation, determine the domains of their the sample. The results of these calculations indicate that both dominance, and map the nanotube strength as a function of chiral ductile-type bond flip and the brittle bond-breaking mechanisms symmetry, tensile test time, and temperature. coexist and either can play the dominant role in failure in a particular test. mechanics ͉ plasticity ͉ isomerization ͉ rate theory Results and Discussion he highest strengths of solids are obtained from specimens of It is important to remember that even for a well defined flawless CHEMISTRY Tutmost uniformity and perfection. Even a single defect can nanotube the tensile failure process depends considerably on a cause drastic loss of strength. Thin solid filaments (whiskers) number of parameters, such as sample type (diameter d and have long been viewed as material structures that can sustain the chiral symmetry, i.e., the angle 0° Յ ␹ Յ 30° between the roll-up greatest mechanical tension (1, 2). Small cross sections permit vector and the zigzag roll-up vector), applied strain ␧,test little room for defects in their bulk, and the only heterogeneity duration t (or similarly, the strain rate d␧͞dt Ϸ ␧͞t), and is caused by inevitable presence of the surface and the interfacet temperature T. Searching such a multidimensional parameter edges. Discovery (3) of carbon nanotubes (CNTs) offered, at space in direct MD is impractical, even with the least taxing ENGINEERING least in principle, the next level of perfection, when in a classical interatomic force field. However, MD remains a good cylindrical network all atoms are equivalently tied to the neigh- tool for performing a preliminary hands-off search to identify bors, and no ‘‘weak spot’’ is apparent. This intrinsic uniformity, the primary failure modes, which should then further be ex- together with the known strength of carbon bonds, must lead to plored in detail. We have performed such simulations with extreme resistance to mechanical tension, as has been antici- quantum [nonorthogonal tight-binding approximation (23)] pated all along (4, 5). On the other hand, establishing the quantitative level of breaking strain and identifying the details of MD. We specifically considered different lattice temperatures atomic-scale rearrangements responsible for initial yield turned and different applied tension (fixed degree of elongation). out to be elusive both experimentally and theoretically. Different CNT types were considered with the test duration in In recent years, much progress has been made in elucidating subnanosecond range, well shorter than any experimental test. the atomic mechanisms of CNT failure. In experiment, refined Fig. 1 illustrates, through selected representative configurations, loading techniques often based on atomic force microscopy and the two main possibilities that emerge in the course of extensive combined with electron microscopic imaging allowed one to simulations. Under high tension, the load is transferred differ- measure the breaking-strain level and observe the overall failure ently to the bonds according to their orientations relative to the patterns (6–10). The reported experimental values of breaking axis (color-coded in Fig. 1a). Further, the type of first lattice strain ranged within 2–19% because of variability of the samples transformation depends qualitatively on temperature. At low T, and measurement conditions (6–8). In theory, bond rotation thermal fluctuations appear insignificant and the yield event is [that is a concerted movement of two atoms, known in chemistry purely ‘‘mechanical.’’ In this mode, one of the highly elongated as Stone-Wales isomerization (SW) (11)] has been recognized as bonds (marked blue in Fig. 1, for this ‘‘cold’’ mechanism) breaks a key step in mechanical relaxation (12–14). It leads to the lowest and the crack-like configurations emerge (Fig. 1b). As discussed energy defect, a cluster of two pentagons and heptagons, 5͞7͞ later, careful minimization with constraint (maintaining the 7͞5. In the lattice of hexagons (the nanotube body) it represents tensile strain) shows (18) these states to be metastable, shallow a dislocation dipole, which explains its formation under high energy minima, corresponding to the distinguishable n broken tension. This particular relaxation step is most favorable ther- bonds (n ϭ 1, 2, and 6 in the examples of Fig. 1 b–d), a nucleating modynamically, but because of the high barrier of SW (15–17) it brittle crack. requires thermal activation. In contrast, another mechanism recently analyzed (18) needs no thermal activation but occurs at higher tension as a sequence of direct brittle bond-breaking Conﬂict of interest statement: No conﬂicts declared. steps, when a series of ‘‘lattice-trapped’’ states (19) can be Freely available online through the PNAS open access option. identified. Further work has also begun (20, 21) in computing Abbreviations: CNT, carbon nanotube; SW, Stone-Wales isomerization; MD, molecular how the defects can reduce the CNT strength. dynamics; DFT, density functional theory. Despite these insights, the questions at what strain an ideal †To whom correspondence should be addressed. E-mail: [email protected]. tube begins to yield and which primary atomic rearrangement is © 2006 by The National Academy of Sciences of the USA www.pnas.org͞cgi͞doi͞10.1073͞pnas.0600945103 PNAS ͉ April 18, 2006 ͉ vol. 103 ͉ no. 16 ͉ 6105–6109 1. Kelly, A. & Macmillan, N. H. (1986) Strong Solids (Clarendon, Oxford). 23. Goringe, C. M., Bowler, D. R. & Hernandez, E. (1997) Rep. Prog. Phys. 60, 2. Colbert, D. T. & Smalley, R. E. (2002) in Perspectives on Fullerene Nanotech- 1447–1512. nology, ed. Osawa, E. (Kluwer, Dordrecht, The Netherlands), pp. 3–10. 24. Kaxiras, E. & Duesbery, M. S. (1993) Phys. Rev. Lett. 70, 3752–3755. 3. Iijima, S. (1991) Nature 354, 56–58. 25. Yakobson, B. I., Campbell, M. P., Brabec, C. J. & Bernholc, J. (1997) Comput. 4. Calvert, P. (1992) Nature 357, 365–366. Mater. Sci. 8, 341–348. 5. Service, R. F. (1998) Science 281, 940–942. 26. Wei, C., Cho, K. & Srivastava, D. (2003) Phys. Rev. B 67, 115407. 6. 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Dumitrica et al. PNAS ͉ April 18, 2006 ͉ vol. 103 ͉ no. 16 ͉ 6109 COMMUNICATION www.rsc.org/chemcomm | ChemComm

Perforated organometallic nanotubes prepared from a Rh N-heterocyclic carbene using a porous alumina membrane

Sathyajith Ravindran,a G. T. Senthil Andavan,b Chunglin Tsai,a Cengiz S. Ozkan*a and T. Keith Hollis*b

Received (in Berkeley, CA, USA) 28th October 2005, Accepted 3rd March 2006 First published as an Advance Article on the web 15th March 2006 DOI: 10.1039/b515332h

Fabrication of perforated organometallic nanotubes using a di- 3 N NaOH followed by centrifugation. The aqueous supernatant rhodium bis(N-heterocyclic carbene) complex by a simple was discarded and a yellow solid was isolated. This material was nanoporous template wetting technique is described along with rinsed with water and dried at room temperature.{ characterization data from scanning electron microscopy (SEM), transmission electron microscopy (TEM), energy dispersive X-ray spectroscopy (EDS), proton NMR and Mass spectroscopy.

Many groups world-wide are involved in the development of 1,2 nanotubes. The ubiquitous carbon nanotubes receive much Preliminary identification of the morphology was conducted by 3 attention. The fabrication of nanotubes from polymers via a optical microscopy after dissolving the template but before re- 4 templating methodology has been reported recently. Nanotubes suspension and washing (Fig. 1). Bunches of aligned strands or have been demonstrated to have potential application for fibers were noted as may be seen in Fig. 1. The alignment of the 5 hydrogen storage and fuel cell technology. Despite these large nanotubes is a simple consequence of the structure of the porous scale synthetic efforts very few examples of organometallic alumina membrane. Individual nanotubes appeared to be about 6 nanotubes have been reported. We report the preparation of 20 mminlength. nanotubes from an organometallic material based on the porous The structure and constitution of these nanotubes were alumina membrane templating methodology. Furthermore, the addressed first. Samples for scanning electron microscopy (SEM) organometallic nanotubes contain semi-regular perforations were prepared and immediately imaged for morphological (pores) along the main axis of the tube. This highly desirable information and for quantitative length and diameter measure- property opens the possibility of controlled drug delivery with ments (Fig. 2). No gold coating was used for the SEM sample 7 biocompatible metals and many other applications. preparation and yet the imaging was free from any charge build- The construction of nanotubes, especially single-walled carbon up. This fact is indirect evidence that the nanotubes may be nanotubes (SWNTs), has focused on solid walled materials. In conductive. Fig. 2 shows images of these tubes with pores at part this is driven by the applications that have been envisaged and regular intervals. The regularity is so astonishing that it causes the available methodologies. Much has been reported concerning doubt that these are due to the surface profile of the alumina defects (holes) in the walls of SWCNTs and their spherical counter template. Many groups aroundtheworldhaveusedporous 8,9 part C60. Solid-walled nanotubes are the microscopic counter- alumina membranes for the fabrication of nanomaterials but there parts to rigid PVC pipe. An alternative macroscopic model is are no similar reports in the literature that indicate perforation perforated pipe, which has regularly spaced pores that serve many features on the nanotubes such as those observed in our case. functions. Therefore, we suggest that the pores are due to the nature of the The orange, crystalline di-Rh bis(N-heterocyclic) carbene (NHC) complex 1 depicted below was recently reported by two of us.10 The potential for crosslinking the Rh centers into a polymeric array made them interesting potential starting materials for nanotube synthesis. The nanotubes were fabricated by wetting a porous alumina membrane template with a toluene solution of 1 andallowingthesolutiontopermeatethemembraneuntilitwas saturated.4a,11 The solvent was allowed to evaporate under ambient conditions. The excess precipitate was removed from the surface of the membrane, and the alumina was dissolved with aDept of Mechanical Engineering, University of California, Riverside, CA 92521, USA. E-mail: [email protected]; Fax: 951 827 2899; Tel: 951 827 5016 bDepartment of Chemistry, University of California, Riverside, CA 92521, USA. E-mail: [email protected]; Fax: 951 827 4713; Fig. 1 Optical images at 10006 magnification illustrating the nature of Tel: 951 827 3024 and alignment of the fibers obtained initially.

1616 | Chem. Commun., 2006, 1616–1618 This journal is ß The Royal Society of Chemistry 2006 in air. The excess solid was removed from the top of the template with a by liquid secondary ion mass spectroscopy (LSIMS). An identical 1 10 toluene dampened Kimwipe . The template was dissolved with 3N NaOH observation was made for the starting material 1. (2 mL), the solid was centrifuged, and the supernatant was decanted. The These observations make even more intriguing the question of nanotubes were washed twice by suspension in distilled H2Ofollowedby the nature of the molecular forces or non-covalent interactions centrifugation and decantation. They were suspended in water and drop- holding the perforated organometallic nanotube together. There cast onto a Si substrate and dried before analysis. Optical microscopy images were collected from a Nikon Eclipse E600. A Leo Supra 55 was are several possible explanations. It could be intermolecular van used to collect the SEM images and a Philips CM300 was used to collect der Waals attractions between butyl groups with or without the TEM images and the EDS data. High resolution proton NMR data concomitant p-stacking of the aromatic rings.13 Alternatively, the were recoded on a Varian Inova 500 MHz instrument. porous alumina membrane may catalyze a ligand exchange process whereby Rh–I–Rh and/or Rh–cyclooctadiene–Rh bridges 1 For a recent review see: G. Cao Nanostructures & Nanomaterials: 14,15 Synthesis, Properties & Applications; Imperial College Press, London, are formed. The ready dissolution of the nanotubes in 2004. chloroform and the spectroscopic data (vide supra)seemto 2 For a recent example of an inorganic nanotube see: J. Roggenbuck and preclude the formation of m-oxo (Rh–O–Rh)or m-hydroxo (Rh– M. Tiemann, J. Am. Chem. Soc., 2005, 127, 1096; S. M. Liu, L. M. Gan, OH–Rh) bridges during treatment with NaOH.16,17 The suggested L. H. Liu, W. D. Zhang and H. C. Zeng, Chem. Mater., 2002, 14, 1391. 3 For a recent report see: J. Charlier, A. De Vita, X. Blase and R. Car, explanations account for the ready solubility in chloroform and Science (Washington, D.C.), 1997, 275,646. account for recovery of 1. 4(a) M. Steinhart, J. H. Wendorff, A. Greiner, R. B. Wehrspohn, In conclusion, we have fabricated nanotubes from a di-Rh K. Nielsch, J. Schilling, J. Choi and U. Gosele, Science (Washington, N-heterocyclic carbene complex by a facile technique. These D.C.), 2002, 296, 1997; (b)X.ZhangandS.K.Manohar,J. Am. Chem. Soc,2005,127, 14156. organometallic nanotubes seem to have good electrical conductive 5 C.Wang,M.Waje,X.Wang,J.M.Tang,R.C.HaddonandY.S.Yan, properties and show promise as nanowires. In addition, the Nano Lett., 2004, 4, 345. presence of perforations (nanopores) at regular intervals in the 6 X.-S. Wang, M. A. Winnik and I. Manners, Angew. Chem., Int. Ed., 2004, 43,3703;X.-S.Wang,H.Wang,N.Coombs,M.A.Winnikand nanotubes opens the possibility of numerous other applications. In I. Manners, J. Am. Chem. Soc., 2005, 127, 8924. the case of our nanotubes, the perforations around their sidewalls 7 A. Bianco, K. Kostarelos, C. D. Partidos and M. Prato, Chem. would make it easier for material to diffuse in and out of the tubes. Commun., 2005, 571. Work is underway to ascertain the chemical structure of the tubes, 8 For a leading reference see: R. H. Telling, C. P. Ewels, A. A. El-Barbary and M. I. Heggie, Nat. Mater., 2003, 2, 333–337; S. L. Mielke, D. Troya, to determine the molecular or non-covalent forces holding the S.Zhang,J.-L.Li,S.Xiao,R.Car,R.S.Ruoff,G.C.Schatzand tubes together, to insert objects into the tubes and to gain insight T. Belytschko, Chem. Phys. Lett., 2004, 390, 413. into the source of the perforations (nanopores), which will enhance 9Y.Rubin,Chem.-Eur. J., 1997, 3, 1009; Y. Rubin, T. Jarrosson, our ability to control them. G.-W.Wang,M.D.Bartberger,K.N.Houk,G.Schick,M.Saunders andR.J.Cross,Angew. Chem., Int. Ed., 2001, 40, 1543; G. Schick, We thank Professor Wenbin Lin (UNC-Chapel Hill) for helpful T. Jarrosson and Y. Rubin, Angew. Chem., Int. Ed., 1999, 38, 2360; discussions, and Dr. Dan Borchardt for assistance with NMR R. Stackow, G. Schick, T. Jarrosson, Y. Rubin and C. S. Foote, J. Phys. spectra. We (T.K.H.) gratefully acknowledge support from the Chem. B, 2000, 104, 7914. National Science Foundation (CHE- 0317089) and (C.S.O.) the 10 G. T. S. Andavan, E. B. Bauer, C. S. Letko, T. K. Hollis and F. S. Tham, J. Organomet. Chem., 2005, 690, 5938. Center on Functional Engineered Nano Architectonics funded by 11 J. C. Hulteen and C. R. Martin, J. Mater. Chem., 1997, 7, 1075. the Microelectronics Advanced Research Corporation (MARCO- 12 T. L. Amyes, S. T. Diver, J. P. Richard, F. M. Rivas and K. Toth, A01010-59708) and (C.S.O.) the Center for Nanoscience J. Am. Chem. Soc., 2004, 126, 4366. Innovation for Defense (H94003-04-2-0404). 13 C. A. Hunter, Chem. Soc. Rev., 1994, 23, 101. 14 For a recent example of a Rh(NHC) complex forming Rh–I–Rh dimers see: R. J. Rubio, G. T. S. Andavan, E. B. Bauer, T. K. Hollis, J. Cho, Notes and references F. S. Tham and B. Donnadieu, J. Organomet. Chem., 2005, 690, 5353. 15 For a recent example of a bis(Ni(NHC)) complex with a bridging COD { Experimental: The 25 mm alumina disc templates with pore size of see: T. Schaub and U. Radius, Chem.-Eur. J., 2005, 11, 5024. 200 nm were procured from Whatman Inc. The alumina template was 16 For an example of crystallographically characterized Rh–O–Rh see: sonicated for 10 min in deionised water to clean the pores and allowed to R. S. Hay-Motherwell, G. Wilkinson, B. Hussainbates and dry. Air and moisture stable di-Rh 1 (9.98 mg) was dissolved in toluene M. B. Hursthouse, Polyhedron, 1990, 9, 2071. (5 mL). An aliquot of the solution was applied to the top of a template by 17 For an example of crystallographically characterized Rh–OH–Rh see: microsyringe and the solvent allowed to evaporate overnight under a lamp O.Gevert,J.WolfandH.Werner,Organometallics, 1996, 15,2806.

1618 | Chem. Commun., 2006, 1616–1618 This journal is ß The Royal Society of Chemistry 2006 Computational Materials Science 35 (2006) 432–441 www.elsevier.com/locate/commatsci

Analysis of localized failure of single-wall carbon nanotubes

Jia Lu *, Liang Zhang

Department of Mechanical and Industrial Engineering, Center for Computer Aided Design, The University of Iowa, Iowa City, IA 52242-1527, USA

Received 24 September 2004; accepted 25 February 2005

Abstract

A method is developed to determine the conditions for the onset of localized failure of carbon nanotubes. Examples of failure modes include ductile necking under tension or localized crushing under compression. A nanoscale continuum theory for carbon nanotube is adapted. The onset of localized failure is identiﬁed by the singularity point of the acoustic tensor derived from continuum energy function based on Tersoﬀ–Brenner potential. The analysis predicts 35–44% of breaking strains for tension and 18–25% compressive strain for plastic collapse. The results are in agreement with molecular dynamics simulations and experimental estima- tions reported in the literature. Ó 2005 Elsevier B.V. All rights reserved.

Keywords: Carbon nanotube; Strain localization; Material failure; Acoustic tensor

1. Introduction resulting in sideway buckling at as early as 5% of compressive strain. Large diameter tubes may buckle earlier Carbon nanotubes (CNT) have been found to have by way of wall rippling. Torsion of a tube above a crit- exceptional mechanical and electronic properties. Appli- ical value results in ribbon-like flatten shapes. When cations of CNTs as building blocks for high-strength bended beyond a buckling point, multiple kinks develop material or nanodevice have been proposed and investi- on the compressive side of the wall. These simulations gated in many studies. The mechanical characteristics of agree with the experimental findings by Iijima et al. nanotube reinforced material or nanodevices with nano- [13]. Arroyo and Belytschko [3,4] replicated these buck- tube as building blocks depend on the intrinsic mechan- ling patterns using finite element formulation based on ical properties of CNTs. These include not only the crystal elasticity derived from bond-order potentials, elastic modulus of CNTs, which describes the elastic and predicted a family of buckling patterns and post- property at small strains, but also the stability and fail- buckling motions. ure of CNTs at large strains. In these deformations, the CNT remains in the elastic Early studies show that CNTs can be subject to very limit, and restores its original geometry once the load is large strain without causing failure or defect formation. removed. Another category of instability, which we Yakobson et al. [30,31] investigated the buckling behav- characterize as localized failure in a sense to be made ior of CNTs under large deformation using molecular clear later, has also been observed and reported in the dynamics (MD). It has been showed that under axial literature. Yakobson et al. [31] and recently Marques compression, nanotubes exhibit structural instabilities et al. [17], studied the failure of CNT under tension at large strains. The initial response appears homogeneous; the hexagon bonds extend as the tube is stretched. At a * Corresponding author. Tel.: +1 319 3356405; fax: +1 319 3355669. certain critical level, one of few C–C bonds break almost E-mail address: [email protected] (J. Lu). instantaneously, and the resulting ‘‘hole’’ in a tube wall

Appendix A. Comparison with buckling analysis References

This appendix provides a brief comparison with the [1] M. Arroyo, T. Belytschko, An atomistic-based finite deformation buckling analysis in [33] and the current work for the membrane for single layer crystalline films, Journal of the Mechanics and Physics of Solids 50 (9) (2002) 1941–1977. special case of armchair tube under uniaxial tension. [2] M. Arroyo, T. Belytschko, A finite deformation membrane based Starting from a relation between elasticity tensors A on inter-atomic potentials for the transverse mechanics of and D nanotubes, Mechanics of Materials 35 (3–6) (2003) 193–215. ¼ S d þ F F D ; ðA:1Þ [3] M. Arroyo, T. Belytschko, Finite crystal elasticity of carbon Aiajb ab ij im jq maqb nanotubes based on the exponential Cauchy–Born rule, Physical where S is the second Piola–Kirchhoff stress and Review B 69 (2004) 115–415. dS [4] M. Arroyo, T. Belytschko, Finite element methods for the non- D ¼ 2 dC is the referential elasticity tensor, the acoustic tensor can be written as linear mechanics of crystalline sheets and nanotubes, International Journal for Numerical Methods in Engineering 59 (2004) 419– Qij ¼ SabN aN bdij þ F imF jqDmaqbN aN b. ðA:2Þ 456. [5] T. Belytschko, S.P. Xiao, G.C. Shatz, R.S. Ruoff, Atomistic For the uniaxial tension specified in (29), the non- simulations of nanotube fracture, Physical Review B 65 (2002) zero terms are axial stress S11, the axial stretch kZ and 235–430. [6] Q.W. Brenner, Empirical potential for hydrocarbons for use in hoop stretch kH. Setting N = EZ, the nonzero entries of Q are simulating the chemical vapor deposition of diamond films, Physical Review B 42 (1990) 9458–9471. 2 [7] H. Gao, Y. Huang, F.A. Abraham, Continuum and atomistic Q ¼ SZZ þ k DZZZZ ; 11 Z studies of intersonic crack propagation, Journal of the Mechanics 2 Q22 ¼ SZZ þ kHDHZHZ ; ðA:3Þ and Physics of Solids 49 (2001) 2113–2132. Q ¼ k k D . [8] H. Jiang, X.Q. Feng, Y. Huang, K.C. Hwang, P.D. Wu, Defect 12 Z H ZZHZ nucleation in carbon nanotubes under tension and torsion: Stone– The condition detQ = 0 reduces to Wales transformation, Computer Methods in Applied Mechanics ! ! and Engineering 193 (2004) 3419–3429. [9] H. Jiang, P. Zhang, B. Liu, Y. Huang, P.H. Geubelle, H. Gao, SZZ SZZ DZZZZ þ 2 DHZHZ þ 2 DZZHZ ¼ 0. ðA:4Þ K.C. Hwang, The effect of nanotube radius on the constitutive kZ kH model for carbon nanotubes, Computational Materials Science 28 (2003) 429–442. Assuming that Bðcos hij; cos hikÞ in T–B potential is con- [10] R. Hill, Acceleration waves in solids, Journal of Mechanics and stant and that g = 0, as in Ref. [33], the tangent modulus Physics of Solids 10 (1962) 1–16. takes the form [11] R. Hill, On the theory of plane strain in finitely deformed compressible materials, Mathematical Proceedings of the Cam- 1 X3 V 00 V 0 bridge Philosophical Society 86 (1979) 161–178. D ¼ i i A A A A ; ðA:5Þ [12] L. Van Hove, Sur lextension de la condition de legendre du calcul A r2 r3 i i i i 0 i¼1 i i des variations aux integrals multiples a plusieurs fonctions inconnues. Koninklijke Nederlandse Akademie van Wetens- where Vi = V(ri)=VR(ri) BVA(ri). In this case, chappen. Afdeling Natuurkunde. Proceedings of the Section of DZZHZ = DHZZZ = DZHZZ = DZZZH = 0 holds regard- Sciences 50 (1947) 18–23 less of chirality angle. The jump condition reduces to [13] S. Iijima, C. Brabec, A. Matti, J. Bernholc, Structural flexibility of carbon nanotubes, Journal of Chemical Physics 104 (5) (1996) 2 2 2089–2092. SZZ þ kZ DZZZZ ¼ 0orSZZ þ kHDHZHZ ¼ 0; ðA:6Þ [14] P. Klein, H. Gao, Crack nucleation and growth as strain whichever reaches first. In tension, actual computation localization in a virtual-bond continuum, Engineering Fracture 2 Mechanics 61 (1998) 21–48. shows that SZZ þ kHDHZHZ 6¼ 0 within the range of stretch considered, therefore the critical state is charac- [15] J. Li, K.J. Van Vliet, T. Zhu, S. Yip, S. Suresh, Atomistic 2 mechanisms governing elastic limit and incipient plasticity in terized by SZZ þ kZ DZZZZ ¼ 0. crystals, Nature 418 (2002) 303–310. The criterion deduced by Huangs group (Eq. (32) of [16] O. Lourie, D.M. Cox, H.D. Wagner, Buckling and collapse of [33]) states that embedded carbon nanotubes, Physical Review Letters 81 (8) ! ! (1998) 1638–1641. SZZ SZZ mpR 2 [17] M.A.L. Marques, H.E. Troiani, M. Miki-Yoshida, M. Jose- DZZZZ þ DHHHH þ ¼ D ; k2 k2 L HHZZ Yacaman, A. Rubio, On the breaking of carbon nanotubes under Z H tension, Nano Letters 4 (5) (2004) 811–815. ðA:7Þ [18] S.L. Mielke, D. Troya, S. Zhang, J.L. Li, S. Xiao, R. Car, R.S. Ruoff, G.C. Schatz, T. Belytschko, The role of vacancy defects where R is the tube radius and L is the tube length. Now, and holes in the fracture of carbon nanotubes, Chemical Physics if R !1, to make the right-hand side finite, the first Letters 390 (2004) 413–420. factor should be diminishingly small, which yields the [19] M.B. Nardelli, B.I. Yakobson, J. Bernholc, Brittle and ductile same condition. In [35], Huang et al. further showed behavior in carbon nanotubes, Physical Review Letters 81 (1998) 4656. that for axisymmetric buckling, the critical strain for [20] A. Needleman, Analysis of plastic flow localization in metals, an armchair is almost insensitive to tube geometry. Applied Mechanics Review 45 (1992) S3–S18. PHYSICAL REVIEW B 73, 115406 ͑2006͒

Fracture of vacancy-defected carbon nanotubes and their embedded nanocomposites

Shaoping Xiao and Wenyi Hou Department of Mechanical and Industrial Engineering, and Center for Computer-Aided Design, The University of Iowa, Iowa City, Iowa 52242, USA ͑Received 11 December 2005; revised manuscript received 17 January 2006; published 8 March 2006͒

In this paper, we investigate effects of vacancy defects on fracture of carbon nanotubes and carbon nanotube/ aluminum composites. Our studies show that even a one-atom vacancy defect can dramatically reduce the failure stresses and strains of carbon nanotubes. Consequently, nanocomposites, in which vacancy-defected nanotubes are embedded, exhibit different characteristics from those in which pristine nanotubes are embedded. It has been found that defected nanotubes with a small volume fraction cannot reinforce but instead weaken nanocomposite materials. Although a large volume fraction of defected nanotubes can slightly increase the failure stresses of nanocomposites, the failure strains of nanocomposites are always decreased.

DOI: 10.1103/PhysRevB.73.115406 PACS number͑s͒: 61.46.Fg, 81.05.Ni

I. INTRODUCTION experiments11 with an n-atom defect model ͑vacancy due to n atoms missing͒. However, since the bonds along the hypo- It is known that carbon nanotubes ͑CNTs͒ have large ten- thetical crack were not reconstructed, the physical plausibil- sile modulus1,2 and high thermal conductivity.3,4 The Young’s ity of these defects remains in question. In research per- moduli of CNTs are around 1 Tpa and their thermal conduc- formed in collaboration with Belytschko, Car, Ruoff, Schatz, tivity can be 6600 W/m K. On the other hand, CNTs are etc., the role of vacancy defects and holes in the fracture of expected to have high strength. Previous theoretical analyses CNTs13 was studied. Both quantum mechanical and molecu- and numerical simulations predicted failure strengths of up lar mechanics calculations indicated that the holes due to 5–7 to 300 Gpa for CNTs. Consequently, they have been pro- one- and two-atom vacancy defects could reduce failure posed as ideal fibers for the manufacture of the next genera- stresses by as much as ϳ26%. In their studies, nanotubes tion of composite materials with mechanical and thermal 8–10 were assumed at zero temperature. management applications. However, low failure stresses, Since high strengths of CNTs were predicted, it was as- which are in the range of 21–63 Gpa, were observed in the sumed that the toughness of their embedded composite ma- experiments.11 Such observation conflicted with theoretical terials could be significantly increased.19–21 Generally, three and numerical analyses outcomes. types of materials can be used as the matrix: polymers,21,22 Some researchers have pointed out the significant effects ceramics,23,24 and metal.25,26. Because polymers have low of defects on nanotube fracture.12,13 Defects in CNTs can arise from various causes. Chemical defects consist of atoms/ density and are easy to shape, they are the first choice as the groups covalently attached to the carbon lattice of the tubes matrix of fiber reinforced composites. As a structural mate- such as oxidized carbon sites or chemical vapor rial, ceramics present many advantages over polymers, such deposition.14,15 Topological defects correspond to the pres- as high rigidity and hardness, even at high temperature, and ence of rings other than hexagons, mainly studied as low sensitivity to corrosion. However, they are brittle. Such pentagon/heptagon pairs.16,17 Incomplete bonding defects weakness can be made up by reinforcing the composites with like vacancies may be caused through impact with high- CNTs. Lately, there has been more interest in using metal as energy electrons in the transmission electron microscopy en- the matrix material for composites. It has been found that the vironment, see Banhart,18 or defects in the original outer fracture toughness of a metal matrix composite with nano- nanotube shell. tubes can be increased by up to 200%.27 Chemical defects usually occur during functionalizing In this paper, we will first study the failure mechanism of CNTs so that chemical bonds can be formed between CNTs vacancy-defected CNTs using molecular dynamics. As a dif- and the matrix material in nanocomposites. Consequently, ference from previous research work, vacancy defects are the mechanical properties of nanocomposites can be signifi- modeled by taking out atoms and then reconstructing bonds. cantly enhanced19,20 because of the strong interfacial load Various temperatures will be considered to investigate tem- transfer. However, we do not think that functionalization will perature effects on the fracture of CNTs. We also study size have significant effects on the nanotube fracture itself. effects of vacancy defects on a nanotube fracture at room 5/7/7/5 dislocation, also called Stone-Wales dislocation, re- temperature, i.e., T=300 K. Then, nanotube-embedded alu- sults in high failure strengths5,7 in comparison with the ex- minum ͑CNT/Al͒ composites are considered to investigate perimental results. Incomplete bonding, especially vacancy, effects of vacancy defects on fracture of nanocomposites. In will form nanoscale cracks or holes that can have large varia- this paper, only nonbonded interatomic interaction, i.e., van tions in size. Such an initial mechanism can dramatically der Waals energy, is considered at the CNT/Al interface, reduce the strength of CNTs. Belytschko et al.12 obtained since no chemical reactions were observed during processing reasonable results that can account for the fractures in the of CNT/Al composites in the experimentation.27

1098-0121/2006/73͑11͒/115406͑7͒/$23.00115406-1 ©2006 The American Physical Society FRACTURE OF VACANCY-DEFECTED CARBON¼ PHYSICAL REVIEW B 73, 115406 ͑2006͒ the vacancy. We also found that temperature effects on the a large volume fraction defected nanotubes can increase the strength of vacancy-defected nanotubes are not as signiﬁcant failure stress, i.e., strength, of nanocomposites, the defected as those on the strength of pristine nanotubes. In addition, we nanotubes reduce the failure strain of CNT/Al composites. employed CNT/Al nanocomposites to demonstrate whether We should point out that only the weak CNT/matrix interface vacancy-defected nanotubes can reinforce composite materi- is considered in this paper. Strong load transfer at the CNT/ als or not. Due to experimental observation, no chemical matrix interface may be achievable through functionalization bonds exist at the CNT/Al interface. Several interesting phe- of nanotubes. This will be our future research topic. nomena were observed for defected CNT/Al composites other than pristine CNT/Al composites. At ﬁrst, a critical volume fraction should be reached for vacancy-defected ACKNOWLEDGMENT nanotubes to reinforce the aluminum matrix. For example, 14% is the critical volume fraction if ͑5,5͒ nanotubes with a The authors acknowledge startup fund support from the two-atom vacancy defect are expected to increase the College of Engineering and the Center for Computer-Aided strength of CNT/Al nanocomposites. Second, although with Design ͑CCAD͒ at the University of Iowa.

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115406-7 PHYSICAL REVIEW B 73, 045435 ͑2006͒

Interwall interaction and elastic properties of carbon nanotubes

Elena Bichoutskaia* University Chemical Laboratory, Lensﬁeld Road, Cambridge CB2 1EW, United Kingdom

Malcolm I. Heggie Department of Chemistry, University of Sussex, Falmer, Brighton BN1 9QJ, United Kingdom

Andrey M. Popov and Yurii E. Lozovik Institute of Spectroscopy, Troitsk, Moscow region, 142190, Russia ͑Received 4 September 2005; revised manuscript received 9 November 2005; published 30 January 2006͒

The physical properties of a wide range of nonchiral single-walled carbon nanotubes ͑SWNT͒ and double- walled carbon nanotubes ͑DWNT͒ with nonchiral commensurate walls are studied. Equilibrium structures of SWNT and DWNT, as well as the interwall interaction energies of DWNT, are computed using a local density approximation within density functional theory with periodic boundary conditions and Gaussian-type orbitals. Based on ab initio structural characteristics, elastic properties of SWNT and DWNT are calculated. Relative motion of the walls of DWNT with different radii and chiralities is explored using ab initio results for the interwall interaction energies. Relative positions of nonchiral commensurate walls of DWNT which correspond to extrema of the interwall interaction energy are derived. For DWNT with incompatible rotational symmetries of the walls, the possibility of orientational melting is predicted. Ab initio values of barriers to relative rotation and sliding of the walls of DWNT are used to calculate threshold forces. For nonreversible telescopic extension of the walls, maximum overlap of the walls for which threshold forces are greater than capillary forces is estimated. A method for selecting pairs of nonchiral commensurate walls in multiwalled carbon nanotubes ͑MWNT͒ is proposed.

DOI: 10.1103/PhysRevB.73.045435 PACS number͑s͒: 68.35.Fx, 61.48ϩc

I. INTRODUCTION unit cells of constituent SWNT. Lack of commensurability Carbon nanotubes are promising material for potential between the neighboring nested SWNT implies a dramatic weakening of the corrugation in the interwall interaction components of future nanoelectronic and nanomechanical 14 devices.1–4 Weak interwall interactions within MWNT pro- potential. Barriers to the relative motion of the commensurate walls of sufficiently long DWNT are proportional to the vide perfect bearing for possible novel nanodevices based on ⌬ ⌬ ⌬ relative sliding, rotation, or screwlike motion of the nanotube length: U= UcNc, where Uc is the barrier per unit cell and N is the number of unit cells in the nanotube. walls.5–10 Theoretical study of the interwall interaction and c Conceivably, there is a possibility of fabrication of DWNT relative motion of the walls in carbon nanotubes holds the with commensurate walls with a custom-ordered value of key to success of these applications. barriers to relative motion of the walls. Therefore, these Neglecting their structure at the ends, which can be either DWNT can be considered as potential components in nan- open or closed, carbon nanotubes are single or multiple lay- odevices for which a precise control of motion of the walls is ers of a cylinder rolled up from graphene sheets. Only one required. In contrast to the commensurate case, barriers to parameter is needed to fully determine the structure of the relative motion of incommensurate walls of DWNT do not middle section of a SWNT: the chirality index ͑n, m͒ which increase with the nanotube length, but fluctuate near the av- 5,6,15 corresponds to a two-dimensional lattice vector c=na1 erage value. Such incommensurate systems, even if they +ma2, where a1 and a2 are equivalent lattice vectors of contain thousands of carbon atoms, have barriers to the rela- graphene.11 A segment defined by the vector c becomes the tive motion of the walls comparable to those of a single unit circumference of cylindrical surface of a nanotube wall cell. These systems hold promise for application in mechani- which can be well modeled by an infinite tube, where peri- cal elements, providing perfect bearings for possible odic boundary conditions are applicable.12 Two types of nanodevices.7–9,16 SWNT characterized by the chirality index of ͑n, n͒ and ͑n, The most commonly used convention employs the term 0͒ have a simple translational symmetry, and these are re- “commensurate walls” for the walls which are commensurate ferred to as armchair and zigzag nanotubes forming different with their structures obtained by graphene plane mapping on pattern of hexagons in circumference. DWNT consist of two a cylindrical surface with the bond lengths kept constant coaxially arranged SWNT with the interwall distance close ͑see, for example, Refs. 14, 15, and 17–20͒. Otherwise, the to the graphite interlayer distance of 3.335 Å.13 walls are defined as incommensurate. However, the bond The walls of DWNT are commensurate if the ratio of the lengths of the walls of nanotubes slightly differ from those in lengths of their unit cells is a rational fraction. In this case, a graphite, and for this reason the lengths of unit cells of iso- DWNT is a quasi-one-dimensional crystal with the length of lated commensurate walls are also slightly different. Inter- unit cell equal to the lowest common factor of the lengths of wall interactions in DWNT lead to the contraction ͑or expan-

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045435-9 INSTITUTE OF PHYSICS PUBLISHING NANOTECHNOLOGY Nanotechnology 17 (2006) 864–870 doi:10.1088/0957-4484/17/3/042 Equilibrium conﬁguration and continuum elastic properties of ﬁnite sized graphene

CDReddy1,2,SRajendran1 and K M Liew1,2

1 School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, 639798, Singapore 2 Nanyang Centre for Supercomputing and Visualisation, Nanyang Technological University, 50 Nanyang Avenue, 639798, Singapore

E-mail: [email protected]

Received 15 July 2005, in final form 12 October 2005 Published 16 January 2006 Online at stacks.iop.org/Nano/17/864 Abstract This paper presents a continuum mechanics approach to modelling the elastic deformation of finite graphene sheets based on Brenner’s potential. The potential energy of the graphene sheet is minimized for determining the equilibrium configuration. The four edges of the initially rectangular graphene sheet become curved at the equilibrium configuration. The curving of the sides is attributed to smaller coordination number for the atoms at the edges compared to that of the interior atoms. Considering two graphene models, with only two or all four edges constrained to be straight, the continuum Young’s moduli of graphene are computed applying the Cauchy–Born rule. The computed elastic constants of the graphene sheet are found to conform to orthotropic material behaviour. The computed constants differ considerably depending on whether a minimized or unminimized configuration is used for computation.

1. Introduction very high elastic modulus and tensile strength (approximately 10–100 times more than the hardest steel [5]) and low weight, Since the discovery of carbon nanotubes by Iijima et al [1] carbon nanotubes have found potential applications in the in 1991, the prospects of this new material have motivated areas of space as well as material reinforcement in composite widespread research towards several potential applications. technologies [6]. Carbon nanotubes (CNTs) behave like Graphene is a term that refers to a single layer of carbon atoms asemiconductor or metal, depending on the orientation of which are densely packed into a hexagonal ring structure, carbon atoms, and find wide range of potential applications and is widely used to describe the properties of carbon based in the field of electronic industry. CNTs have very high materials including graphite, large fullerenes, nanotubes etc. current carrying capacities (approximately 1000 times as that The discovery of the single-walled carbon nanotube (SWNT) of copper wires [7]) and have high thermal stability up to wasfirst reported in 1993 [2, 3]. The diameters of SWNTs 2800 ◦Cinvacuum, so they find wide applications in the range from 0.4 to 2–3 nm and the lengths are usually of electrical industry. micrometre order. Usually SWNTs come in the form of Many researchers have reported analysis of carbon bundles which are hexagonally arranged to form a crystal-like nanotubes by theoretical modelling. There are mainly two structure [4]. bonding potentials used in theoretical modelling, namely, Extremely small size, outstanding physical properties and direct bonding potential and indirect bond potential (interlayer unique atomic arrangement of carbon nanotubes are some of potential) to compute the mechanical properties. The bonding the attractive features that have triggered intensive research potentials can be further categorized in terms of three models, in a wide variety of fields, i.e.,chemistry, physics, material i.e., force field model, bond order model and semi-empirical science, medicine, and engineering. Many researchers have model. The molecular mechanics force field (MM2, MM3) reported in the literature theoretical and experimental results was introduced by Allinger and co-workers [8, 9]. A generic showing as high an elastic modulus as 1 TPa, that exceeds force field was proposed by Mayo et al [10]. The bond order those of any previously existing materials. Because of the model was proposed by Abell [11], and extension to the carbon

1.09 [1] Iijima S 1991 Nature 354 56 [2] Iijima S and Ichihashi T 1993 Nature 363 603 1.08 [3] Bethune D S, Klang C H, de Vries M S, Gorman G, Savoy R, Vazquez J and Beyers R 1993 Nature 363 605 1.07 [4] Thess A et al 1996 Science 273 483 [5] Erik T T, Zhifeng R and Tsu-Wei C 2001 Compos. Sci.

1.06 Technol. 61 1899 [6] Smitherman D V J 2000 Space Elevators An Advanced

1.05 EarthÐSpace Infracture for the New Millennium NASA/cp-2000-210429

Young's modulus, Y (TPa) [7] Collins P G and Avouris P 2000 Sci. Am. 283 62 1.04 [8] Allinger N L 1977 J. Am. Chem. Soc. 99 8127 [9] Allinger N L, Yuh Y H and Lii J H 1989 J. Am. Chem. Soc. 1.03 111 8551 [10] Mayo S L, Olafson B D and Goddard W A 1990 J. Phys. Chem. 1.02 94 8897 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Poisson's ratio, ν [11] Abell G C 1985 Phys. Rev. B 31 6184 [12] Tersoff J 1988 Phys. Rev.Lett. 61 2879 Figure 8. Young’s modulus versus Poisson’s ratio of graphene. [13] Brenner D W 1990 Phys. Rev. B 42 9458 [14] Brenner D W, Harrison J A, White C T and Colton R J 1991 Thin Solid Films 206 220 [15] Robertson D H, Brenner D W and White C T 1992 J. Phys. vertical direction (to simulate lateral contraction) by assuming Chem. 96 6133 avariablevalue of Poisson’s ratio. This becomes necessary [16] Harrison J A, White C T, Colton R J and Brenner D W 1992 Surf. Sci. 271 57 as the lateral contraction is not automatic because potential [17] Tupper K J and Brenner D W 1994 Thin Solid Films 253 185 minimization is not done to obtain equilibrium configuration. [18] Robertson D H, Brenner D W and Mintmire J W 1992 Phys. Figure 8 shows the plot of Young’s modulus against the Rev. B 45 12592 assumed Poisson’s ratio values. The minimum point gives us [19] Brenner D W, Shenderova O A, Harrison J A, Stuart S J, Ni Band Sinnott S B 2002 J. Phys.: Condens. Matter the second set of values (1 TPa, 0.25). 14 783 Thefirst set of values is expected to be more accurate [20] Pettifor D G and Oleinik I I 2000 Phys. Rev.Lett. 84 4124 because the values have been computed allowing for local [21] Girifalco L A, Hodak M and Lee R S 2000 Phys. Rev. B adjustments of atomic positions to reach the equilibrium 62 13104 configurations. [22] Kudin K N, Scuseria G E and Yakobson B I 2001 Phys. Rev. B 64 235406 [23] Lier G V, Alsenoy C V, Doren V V and Greelings P 2000 5. Summary and conclusions Chem. Phys. Lett. 326 181 [24] Arroyo M and Belytschko T 2004 Phys. Rev. B 69 115415 [25] Novoselov K S, Giem A K, Morozov S V, Jiang D, Zhang Y, The elastic constants of finite graphene sheet have been Dubonos S V, Grigorieva I V and Firsov A A 2004 Science determined by modelling it as a continuum. It has been 306 666 observed that the equilibrium adjustments of atoms have much [26] Tersoff J 1988 Phys. Rev. B 37 6991 influence on the computed elastic constants. Computations [27] Yakobson B I, Brabec C J and Bernholc J 1996 Phys. Rev.Lett. 76 2511 considering equilibrium adjustments lead to Young’s modulus [28] Zhang P, Jiang H, Huang Y, Geubelle P H and and Poisson’s ratio values around 0.7 TPa and 0.4, respectively, Hwang K C 2002 Int. J. Solids Struct. 39 3893 whereas computations ignoring equilibrium adjustments yield [29] Friesecke G and Theil F 2002 J. Nonlinear Sci. 12 445 1TPaand0.25, respectively. Nevertheless, papers reporting [30] Lu J P 1997 Phys. Rev. Lett. 79 1297 the elastic constant values often do not mention whether [31] Reddy C D, Rajendran S and Liew K M 2005 Int. J. Nanosci. 4 631 equilibrium adjustments have been considered in their [32] Shen L and Li J 2004 Phys. Rev. B 69 045414 computations. This may be one of the reasons for the wide [33] Yu M F, Files B S, Arepalli S and Ruoff R S 2000 Phys. Rev. scatter in the elastic properties found in the literature. Lett. 84 5552 Thevariation of bond lengths caused by equilibrium [34] Sammalkorpi M, Krasheninnikov A, Kuronen A, Nordlund K and Kaski K 2004 Phys. Rev. B 70 245416 adjustments of atoms during the potential minimization [35] Yoon J, Ru C Q and Mioduchowski A 2005 Trans. ASME J. process has also been studied. The elastic constants Y1, Y2, Appl. Mech. 72 10 ν12, ν21 and G12 have been computed for two graphene models, [36] Chang T and Gao H 2003 J. Mech. Phys. Solids 51 1059 namely, with two straight edges and four straight edges. The [37] Mielke S L, Troya D, Zhang S, Li J, Xiao S, Car R, Ruoff R S, Schatz G C and Belytschko T 2004 Chem. Phys. Lett. study of inter-relationship between these constants seems to 390 413 suggest that graphene behaves like an orthotropic material. [38] Ogata S and Shibutani Y 2003 Phys. Rev. B 68 165409

870 JOURNAL OF APPLIED PHYSICS 99, 024301 ͑2006͒

Nanoscale Weibull statistics ͒ N. M. Pugnoa Department of Structural Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino 10129, Italy R. S. Ruoff Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208-3111 ͑Received 15 April 2005; accepted 29 November 2005; published online 17 January 2006͒ In this paper a modification of the classical Weibull statistics is developed for nanoscale applications. It is called nanoscale Weibull statistics. A comparison between nanoscale and classical Weibull statistics applied to experimental results on fracture strength of carbon nanotubes clearly shows the effectiveness of the proposed modification. A Weibull’s modulus of ϳ3 is deduced for nanotubes. The approach can treat ͑also͒ a small number of structural defects, as required for nearly defect-free structures ͑e.g., nanotubes͒ as well as a quantized crack propagation ͑e.g., as a consequence of the discrete nature of matter͒, allowing to remove the paradoxes caused by the presence of stress intensifications. © 2006 American Institute of Physics. ͓DOI: 10.1063/1.2158491͔

I. INTRODUCTION ␴͑P͒ m P ͑␴͒ =1−expͭ− ͵ ͫ ͬ dVͮ, ͑1a͒ f 1 ␴ Weibull statistics for strength ͑or time to failure, fatigue V 0V life, etc.͒ of solids and deterministic linear elastic fracture or equivalently, mechanics2 ͑LEFM͒ do not apply properly at the nanoscale. Weibull statistics assumes that the number of critical flaws is ␴ m P ͑␴͒ =1−expͫ− V*ͩ ͪ ͬ, ͑1b͒ proportional to the volume or to the surface area of the struc- f ␴ 0V ture, whereas single-crystal nanostructures are anticipated to ␴ ͑ be either defect-free or to have a small number of ͑critical͒ where 0V and m are Weibull’s scale with anomalous physi- defects. Recently LEFM, which assumes infinite ideal cal dimension͒ and shape ͑dimensionless͒ parameters, re- strength of solids, as well as large ͑with respect to the so- spectively, and V* is an “equivalent” volume that refers to a 5 called “plastic zone”͒ and perfectly sharp cracks, has been reference ͑e.g., the maximum͒ stress ␴ in the specimen, de- modified and a theory, quantized fracture mechanics3 ͑QFM͒, fined by comparing Eqs. ͑1a͒ and ͑1b͒. If the specimen is has been presented that quantizes the crack advancement. under uniform tension ␴͑P͒ϵ␴ and V* ϵV. QFM is intended for treating defects of any size and shape The surface-flaw-based Weibull distribution simply re- ͑e.g., atomic vacancies and nanoholes͒. In this paper we places the volume V in Eqs. ͑1͒ with the surface area S of the ͑ ␴ ␴ ͒ present a modification of the Weibull statistics for describing specimen and 0V with a new constant 0S , the strength of solids ͑also͒ at the nanoscale. We apply this ␴͑P͒ m statistical treatment to the largest collection of carbon nano- P ͑␴͒ =1−expͭ− ͵ ͫ ͬ dSͮ, ͑2a͒ f 4 ␴ tube strengths available. The Weibull modulus for nano- S 0S tubes is obtained as ϳ3; furthermore, the statistical data analysis suggests that a small number of defects were critical ␴ m P ͑␴͒ =1−expͫ− S*ͩ ͪ ͬ. ͑2b͒ for such nanotubes. An application to different types of whis- f ␴ 0S kers is also discussed. The proposed approach, coupled with ␴ ␴ quantized fracture mechanics, can treat stress distribution Note that 0V or 0S have the anomalous physical di- also if dominant stress intensifications are present, thus re- mensions of a stress times a volume or a surface raised to ͑ ͒ ͑ ͒ moving the classical paradoxes related to the nonconver- 1/m, so that the exponents in Eqs. 1 and 2 are evidently gence of the Weibull integrals. dimensionless. ͑␴ ͒ The cumulative probability Pf i can be obtained experimentally as6 i − 1/2 II. CLASSICAL WEIBULL STATISTICS ͑␴ ͒ ͑ ͒ Pf i = , 3 Classical Weibull statistics1 assumes the probability of N failure Pf for a specimen of volume V under uniaxial stress where N is the total number of tests and the observed ␴͑P͒͑a function of the considered point P in the volume V͒ ␴ ␴ strengths 1 ,..., N are ranked in ascending order. as The volume- and surface-based approaches become identical for the case of fracture of the external wall of nano- ͒ a Electronic mail: [email protected] tubes under ͑nearly͒ uniform tension, such as for the 19

Downloaded 19 Nov 2007 to 140.109.112.41. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp 024301-4 N. M. Pugno and R. S. Ruoff J. Appl. Phys. 99, 024301 ͑2006͒

ACKNOWLEDGMENTS 9A. Carpinteri and N. M. Pugno, Nat. Mater. 4, 421 ͑2005͒. 10Y. T. Zhu, W. R. Blumenthal, S. T. Taylor, and T. C. Lowe, J. Am. Ceram. The authors would like to thank S. Puzzi and A. Carpin- Soc. 80, 1447 ͑1997͒. teri for the helpful scientiﬁc discussions. One of the authors 11S. B. Batdorf, in Fracture Mechanics of Ceramics, edited by R. C. Bradt, ͑ ͒ ͑R.S.R.͒ appreciates support from the NASA University Re- D. P. H. Hasselman, and F. F. Lange Plenum, New York, 1978 ,Vol.3,p. 1. search, Engineering and Technology Institute on Bio Inspired 12E. J. Gumbel, Statistics of Extremes ͑Columbia University Press, New Materials ͑BIMat͒ under Award No. NCC-1-02037. York, 1958͒. 13A. Carpinteri, Int. J. Solids Struct. 31,291͑1994͒. 14 1W. Weibull, J. Appl. Mech. 18,293͑1951͒. A. Carpinteri, Mech. Mater. 18,89͑1994͒. 15 2A. A. Grifﬁth, Philos. Trans. R. Soc. London, Ser. A 221, 163 ͑1920͒. A. Carpinteri and N. M. Pugno, Int. J. Fract. 128, 253 ͑2004͒. 16 3N. M. Pugno and R. S. Ruoff, Philos. Mag. 84,2829͑2004͒. P. D. Bayer and R. E. Copper, J. Mater. Sci. 2, 233 ͑1967͒. 17 4M.-F. Yu, O. Lourie, M. J. Dyer, K. Moloni, T. F. Kelly, and R. S. Ruoff, P. D. Bayer and R. E. Copper, J. Mater. Sci. 2, 347 ͑1967͒. Science 287,637͑2000͒. 18S. S. Brenner, Fiber Composite Materials ͑American Society for Metals, 5J. Bagdahn and W. N. Sharpe, J. Microelectromech. Syst. 12, 302 ͑2003͒. Metals Park, OH, 1965͒. 6C. A. Johnson, Fract. Mech. Ceram. 5,365͑1983͒. 19A. P. Levitt, Whisker Technology ͑Wiley-Interscience, New York, 1970͒. 7M.-F. Yu, M. J. Dyer, G. D. Skidmore, H. W. Rohrs, X. K. Lu, K. D. 20H. Iwanaga and C. Kawai, J. Am. Ceram. Soc. 81,773͑1998͒; S. Ogata Ausman, J. R. Von Ehr, and R. S. Ruoff, Nanotechnology 10, 244 ͑1999͒. and Y. Shibutani, Phys. Rev. B 68, 165409 ͑2003͒. 8S. L. Mielke, D. Troya, S. Zhang, J.-L. Li, S. Xiao, R. Car, R. S. Ruoff, G. 21S. Ogata, N. Hirosaki, C. Kocer, and Y. Shibutani, Acta Mater. 52,233 C. Schatz, and T. Belytschko, Chem. Phys. Lett. 390,413͑2004͒. ͑2004͒.

Downloaded 19 Nov 2007 to 140.109.112.41. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp PHYSICAL REVIEW B 72, 245409 ͑2005͒

Theoretical study of the stability of defects in single-walled carbon nanotubes as a function of their distance from the nanotube end

Feng Ding* Department of Physics, Göteborg University, SE-412 96, Göteborg, Sweden ͑Received 1 June 2005; revised manuscript received 20 October 2005; published 7 December 2005͒

Point defects, including atom vacancies, adatom, and Stone-Wale defects, close to a ͑5,5͒ single-walled carbon nanotube ͑SWNT͒ open end were studied by density functional theory ͑DFT͒, semiempirical PM3 methods, and the empirical Brenner potential. It is found that defect stability increases as they become closer to the SWNT open end. Based on these results, a model for removing defects in a growing SWNT is proposed, where the defects diffuse to the SWNT end. Furthermore, the calculations show that the semiempirical PM3 method compares well with DFT results, and is accurate enough for studying defect formation in SWNTs. In contrast, the empirical Brenner potential yields large errors and is sometimes not even qualitatively correct.

DOI: 10.1103/PhysRevB.72.245409 PACS number͑s͒: 61.46.ϩw, 61.72.Ji, 68.65.Ϫk, 31.15.Ew

I. INTRODUCTION of a SWNT on a catalyst particle surface, various defects ͑ ͒ As the most important nanomaterial, carbon nanotubes e.g., pentagons, heptagons, adatoms, and vacancies are 16–18 ͑CNTs͒, including single-walled carbon nanotubes ͑SWNTs͒ readily formed. A scooter mechanism has been proposed 10 and multiwalled carbon nanotubes ͑MWNTs͒, have been to explain how a catalyst atom fixes these defects. The studied extensively since their discovery in 19911 and 1993.2 catalyst atom stays on the open end of the SWNT and, when Compared with other nanomaterials, such as semiconducting a defect is formed, the catalyst atom “scoots” to the defect nanowires3 and nanobelts,4 CNTs distinguish themselves by position and fixes it. If all defects are fixed by the scooter their extremely small diameters ͓e.g., the diameter of the mechanism, the SWNT maintains an open end during the smallest SWNT is 0.4 nm ͑Ref. 5͔͒, ultralong lengths ͑e.g., nucleation process. Such a mechanism can explain the heal- isolated SWNTs up to several centimeters6 and decimeter ing of defects on the open end of a growing SWNT only. If long SWNT bundles7 have been produced in experiments͒, some defects are not healed at the CNT end during nucle- high chemical stability, outstanding mechanical properties ation, they will not be healed by the scooter mechanism and, ͑such as tensile strengths of up to 150 GPa, which are more in the absence of a second healing mechanism, will remain in than 300 times that of steel͒,8 and excellent electronic the CNT wall. That is, if the scooter mechanism were the properties.9 All of these favorable physical, chemical, and only way to heal defects, the working efficiency of the cata- mechanical properties make CNTs the most likely candidate lyst atom must be extremely high to grow pristine SWNTs. for a diverse range of applications, such as very small elec- For example, the defect density on a high quality SWNT tronic devices, ultralarge scale integrated circuits, high den- may be as low as one defect per micron, and if we assume sity memories, chemical sensors, superhard and superstrong one defect is formed for every 10 added carbon atoms on the materials, hydrogen storage, and nanomachines. tube end then the catalyst atom must fix about 99.99% of the Although much progress has been made to produce CNTs defects. This must be done in a very short time scale since in a controlled way, it is still not sufficiently well controlled the growth rate of SWNTs is very high ͑e.g., it has been to allow for industrial applications. The growth mechanism reported that SWNT growth rates are as large as 20 ␮m/s in is not well understood. One important problem that is related CCVD experiments at 1173 K,19 which means each defect is to the growth mechanism is how CNT defects are fixed dur- exposed to the catalyst for just 10−5 s͒. Such high catalytic ing the nucleation process, especially when CNTs grow at efficiency is not probable. Hence in order to understand the relatively low temperatures.10,11 For example, typical tem- growth mechanism of high quality CNTs, one should con- peratures for catalytic chemical vapor deposition ͑CCVD͒ sider more mechanisms for healing the defects, especially for CNT growth are 800–1500 K,12 which is several times healing the defects in CNT walls. lower than the melting point of graphite, about 4100 K. Cer- In this contribution we consider a mechanism where de- tainly, thermal annealing at these low temperatures cannot fix fects in SWNT walls diffuse to the open end of the growing a high density of defects in condensed carbon structures to SWNT, where they can be healed by, for example, the form the perfect graphitic layers that are needed to nucleate scooter mechanism. The position dependence stability of CNT walls. In the experimental production of CNTs, a lot of various point defects close to the SWNT open end was stud- amorphous carbon is often formed as the main byproduct.13 ied by density functional theory ͑DFT͒, the semiempirical In contrast, carbon atoms near catalyst particles form high PM3 method,20 and the empirical Brenner potential.21 quality graphitic layers,14,15 which implies that the catalyst plays a crucial role in healing defects in nucleated carbon II. METHOD OF THE THEORETICAL STUDY structures at low temperatures. Recent molecular dynamics ͑MD͒ simulations also show There are three kinds of point defects in CNTs: vacancy, that, when free carbon atoms incorporate into the open end add-atom ͑AA͒, and Stone-Wale ͑SW͒ defects,22 which can

CCVD.47 In addition, CNTs produced in high temperature also found that the PM3 method, which is computationally CCVD often have a higher quality than those produced by cheaper than the DFT method, is sufﬁciently accurate to low temperature CCVD.48 study defect formation energies and structures. On the other hand, the Brenner potential is only qualitatively correct. IV. CONCLUSION ACKNOWLEDGMENTS In conclusion, studies based on DFT, PM3, and the Bren- ner potential show that the stability of point defects in ͑5,5͒ The author is grateful to Professor Arne Rosén and Dr. SWNTs increases as they become nearer to the open end of Kim Bolton for valuable discussions, as well as for time the nanotube. Based on these results, we propose that defects allocated on the Swedish National Supercomputing facilities in SWNTs can be healed by their diffusing to the nanotube and for ﬁnancial support from the Swedish Foundation for end, where they are removed by the catalyst particle. It is Strategic Research and the Swedish Research Council.

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245409-6 COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 65 (2005) 2407–2415 www.elsevier.com/locate/compscitech

Modeling of carbon nanotube clamping in tensile tests

Chunyu Li a, Rodney S. Ruoﬀ b, Tsu-Wei Chou a,*

a Department of Mechanical Engineering, University of Delaware, 126 Spencer Laboratory, Newark, DE 19716, USA b Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, USA

Received 21 June 2005; accepted 22 June 2005 Available online 18 August 2005

Abstract

In this paper, the stress distributions in carbon nanotube clamps such as those formed by the electron beam induced deposition (EBID) technique are analyzed and the contributing factors, including nanotube position, stiffness of clamp material, and thickness of the clamping pad between the AFM tip and the nanotube are examined for the case of tensile loading of the nanotube. The nanotube is modeled at the atomistic scale by the molecular structural mechanics approach and is assumed to be defect free. The clamp material is analyzed by the continuum finite element method. The nanotube and the clamp are assumed to be bonded perfectly to each other. This bonding condition sets the upper limit of clamping capacity. The simulation results indicate that the location and intensity of stress concentration are sensitive to the nanotube orientation. Misaligned nanotubes are likely to break near the edge of the clamp. The clamp material with a lower stiffness (for the stiffness range studied) and a thicker clamping pad between the nanotube and the AFM tip reduce the magnitude of stress concentrations in the clamp. Ó 2005 Elsevier Ltd. All rights reserved.

Keywords: Carbon nanotube; Clamping; Nanocomposites; Multiscale modeling; Tensile testing

1. Introduction ultra-precision clamping of specimens [4]. One of the interesting challenges in this regard is the clamping of The term ÔclampingÕ here refers to the secure fastening carbon nanotubes for tensile tests. of an already positioned specimen. The sound clamp- Since their discovery, carbon nanotubes offer tremen- ing of a specimen is critical to producing reliable test re- dous opportunity for nanotechnology applications. sults of mechanical properties, such as elastic modulus, Examples of their potential applications include as rein- tensile strength, hardness and fracture toughness. At forcements for composites [5,6], components of nanoelec- the macroscopic scale, traditional methods for specimen tromechanical systems [7–9], and storage for hydrogen fastening utilize mechanical, pneumatic or hydraulic fuel [10]. Intensive research has been focused on the clamping [1]. At the microscopic scale, electrostatic and mechanical and physical properties of carbon nanotubes. magnetic clamping techniques, for instance, are com- Theoretical and experimental studies have confirmed that monly used for precision machining operations [2,3]. carbon nanotubes possess extraordinary axial stiffness More recently, the development of nanotechnology re- and tensile strength [11–15]. In these experiments, the quires the characterization of properties of nanostruc- appropriate clamping of carbon nanotubes is critical. tured materials and, hence, the research of nanoscale There are several suitable methods for attaching a nanotube to a substrate. These include using adhesives * Corresponding author. Tel.: +1 302 831 1550; fax: +1 302 831 such as acrylates, methacrylates or epoxies, employing 3619. electrostatic forces, exploiting the chemical affinity be- E-mail address: [email protected] (T.-W. Chou). tween the substrate and the nanotube, and synthesizing

0266-3538/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2005.06.019 C. Li et al. / Composites Science and Technology 65 (2005) 2407–2415 2415 tensile testing. In this paper, we analyze the stress distri- [9] Li CY, Chou TW. Strain and pressure sensing using single-walled bution of nanotube clamps formed by the EBID tech- carbon nanotubes. Nanotechnology 2004;15:1493–6. nique and examine the contributing factors, including [10] Darkrim FL, Malbrunot P, Tartaglia GP. Review of hydrogen storage by adsorption in carbon nanotubes. Int J Hydrogen nanotube position, stiffness of clamp material, and Energy 2002;27:193–202. thickness of the clamping pad between the AFM tip [11] Treacy MMJ, Ebbesen TW, Gibson TM. Exceptionally High and the nanotube. The numerical analysis is performed YoungÕs modulus observed for individual carbon nanotubes. by a multiscale modeling technique. The nanotube is Nature 1996;381:680–7. modeled at the atomistic scale by the molecular struc- [12] Wong EW, Sheehan PE, Lieber CM. Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes. tural mechanics approach, and the clamp material is Science 1997;277:1971–5. analyzed by the continuum finite element method. The [13] Salvetat JP, Bonard JM, Thomson NH, et al. Mechanical nanotube and the clamp material are assumed to be per- properties of carbon nanotubes. J Appl Phys A 1999;69: fectly bonded. The simulation results indicate that the 255–60. location and intensity of stress concentration are sensi- [14] Yu MF, Lourie O, Dyer MJ, Moloni K, Kelly TF, Ruoff RS. Strength and breaking mechanism of multi-walled carbon nano- tive to the nanotube orientation. The clamp material with tubes under tensile load. Science 2000;287:637–40. a relatively lower stiffness reduces the magnitude of stress [15] Cuenot S, Demoustier-Champagne S, Nysten B. Elastic modulus concentration. Also, the increase of the thickness of the of polypyrrole nanotubes. Phys Rev Lett 2000;85:1690–3. clamping pad between a nanotube and the AFM tip re- [16] Dai HJ, Hafner JH, Rinzler AG, Colbert DT, Smalley RE. duces the tensile stress concentration. The misalignment Nanotubes as nanoprobes in scanning probe microscopy. Nature 1996;384:147–50. between the applied tensile force and the nanotube orien- [17] Madsen DN, Molhave K, Mateiu R, Rasmussen AM, Brorson M, tation results in bending and torsional loads on the Jacobsen CJH, et al. Soldering of nanotubes onto microelec- nanotube and, hence, additional interfacial shear stress trodes. Nano Lett 2003;3:47–9. concentrations in the clamp material. Also, the misalign- [18] Liu K, Avouris P, Bucchignano J, Martel R, Sun S, Michl J. ment is in part responsible for having caused failure of Simple fabrication scheme for sub-10 nm electrode gaps using electron-beam lithography. Appl Phys Lett 2002;80:865–7. carbon nanotubes near the edge of the clamp. [19] Storm AJ, Chen JH, Ling XS, Zandbergen HW, Dekker C. Fabrication of solid-state nanopores with single-nanometre precision. Nature Mater 2003;2:537–40. Acknowledgements [20] Silvis-Cividjian N, Hagen CW, Kruit P, VanderStam MAJ, Groen HB. Direct fabrication of nanowires in an electron microscope. Appl Phys Lett 2003;82:3514–6. This work is supported by the National Science [21] Dong LX, Arai F, Fukuda T. Electron-beam-induced deposition Foundation (NIRT Program, Grant No. 0304506, with carbon nanotube emitters. Appl Phys Lett 2002;81:1919–21. Dr. Ken P. Chong, Program Director) and the Army [22] Mailly F, Giani A, Bonnot R, Temple-Boyer P, Pascal-Delannoy Research Office (Grant No. DAAD 19-02-1-0264, F, Foucaran A, et al. Anemometer with hot platinum thin film. Dr. Bruce LaMattina, Program Director). Sensors Actuat A 2001;94:32–8. [23] Baba M, Sano T, Iguchi N, Iida K, Sakamoto T, Kawaura H. DNA size separation using artificially nanostructured matrix. Appl Phys Lett 2003;83:1468–70. References [24] Li CY, Chou TW. Multiscale modeling of carbon nanotube reinforced polymer composites. J Nanosci Nanotechnol [1] Raabe J. Hydro power: the design, use, and function of 2003;3:423–30. hydromechanical, hydraulic, and electrical equipment. VDI-Ver- [25] Li CY, Chou TW. A structural mechanics approach for lag; 1985. the analysis of carbon nanotubes. Int J Solids Struct [2] Kalkowski G, Risse S, Harnisch G, Guyenot V. Electrostatic 2003;40:2487–99. chucks for lithography applications. Microelectron Eng [26] Li CY, Chou TW. Elastic properties of single-walled carbon 2001;57:219–22. nanotubes in transverse directions. Phys Rev B 2004;69:073401. [3] De Stefani JD. Magnetic Chucks Attract New User. Tool Product [27] Li CY, Chou TW. Vibrational behaviors of multi-walled carbon 1994;60:61–4. nanotube-based nanomechancial resonators. Appl Phys Lett [4] Ding W, Dikin DA, Chen X, Wang X, Li X, Piner RD, et al. 2004;84(1):121–3 (SCI 759BH). Clamping nano-structures using electron beam induced deposi- [28] Li CY, Chou TW. Quantized molecular structural mechanics tion. J Appl Phys 2005;98:014905. modeling for specific heat of single-walled carbon nanotubes. [5] Thostenson ET, Ren Z, Chou TW. Advances in the science and Phys Rev B 2005;71:075409. technology of carbon nanotubes and their composites: a review. [29] Li CY, Chou TW. Axial and radial thermal expansion of single- Compos Sci Technol 2001;61:1899–912. walled carbon nanotubes. Phys Rev B 2005;71:235414. [6] Thostenson ET, Li CY, Chou TW. Nanocomposites in context. [30] Tersoff J. Structural properties of sp3-bonded hydrogenated Compos Sci Technol 2005;65:491–516. amorphous carbon. Phys Rev B 1991;44:12039–42. [7] Li CY, Chou TW. Single-walled carbon nanotubes as ultrahigh [31] Mielke SL, Troya D, Zhang S, Li JL, Xiao SP, Car R, et al. The frequency nanomechanical resonators. Phys Rev B role of vacancy defects and holes in the fracture of carbon 2003;68:073405. nanotubes. Chem Phys Lett 2004;390:413–20. [8] Li CY, Chou TW. Mass detection using carbon nanotube-based [32] Pugno NM, Ruoff RS. Quantized fracture mechanics. Philos Mag nanomechanical resonators. Appl Phys Lett 2004;84:5246–8. 2004;84:2829–45. COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 65 (2005) 2380–2384 www.elsevier.com/locate/compscitech

Stochastic strength of nanotubes: An appraisal of available data

A.H. Barber, I. Kaplan-Ashiri, S.R. Cohen, R. Tenne, H.D. Wagner *

Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot 76100, Israel

Received 14 July 2005; accepted 14 July 2005 Available online 2 September 2005

Abstract

This paper summarizes and discusses the limited statistically significant, currently available, experimental data for the tensile strength of individual nanotubes of any sort. Only three such data sets currently exist: two for multi-wall carbon nanotubes and one for multi-wall WS2 nanotubes. It is shown here that Weibull–Poisson statistics accurately fits all strength data sets and thus seems to apply at the nano-scale as well as it does at the micro- and macro-scales. The significance and trends of the Weibull shape and scale parameters, and their relation to the specific structural features of the different nanotubes, are discussed in each case. More recent fracture analyses are also discussed and, in that context, the role of defects in quasi-perfect structures in relation to the theoretical strength is examined. Ó 2005 Elsevier Ltd. All rights reserved.

Keywords: Nanotubes; Nanoscale; Weibull; Biocomposites

1. Introduction F(rf) is often simply termed the probability of failure (denoted by Pf), a is the scale parameter (it has dimen- The strength of any material is inherently governed sions of stress and is smaller for longer or thicker fibers), by the statistical distribution of defects in its microstruc- and b is the shape parameter (no dimensions). And in- ture, and by specimen shape and volume. In fibrous deed, most fibers used in composite materials (carbon/ materials, the comparatively small fiber diameter repre- graphite, Kevlar, glass) do follow quite accurately the sents an upper bound for critical defect size. It is com- Weibull–Poisson statistical model. monly assumed that, in strong micro-scale fibers such Going down in scale by 2–3 orders of magnitude as carbon and glass, the severity of the flaws follows a (from a 10-lm diameter fiber down to a 10–100 nm Poisson distribution and the strength of a fiber is then nanotube), questions may be asked as to the validity of determined by the most severe flaw, so that the fiber fails this classical model: is the defect population still describ- when the weakest point in the fiber fails. This behavior able by the random Poisson model? Would not the crit- may conveniently be modeled by a Weibull two-param- icality of individual defects in nanometer-scale objects be eter distribution. If rf is the failure strength of the nano- expected to be more severe than in micron- or macro- tube, the cumulative distribution function F(rf) for the sized specimens, because of the closeness in scale between two-parameter Weibull distribution is given by defects and structures? Can the stochastic strength still r b be fitted to a Weibull model and is this model still appro- F ðr Þ¼1 exp f . ð1Þ priate at the nanoscale? What is the role of single defects f a in a nanotube with an almost perfect structure? As a matter of fact, the mechanical strength of * Corresponding author. Tel.: +972 89342594; fax: +972 89344137. carbon nanotubes is found to be noticeably higher – E-mail address: [email protected] (H.D. Wagner). by one to two orders of magnitude – than that of

0266-3538/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2005.07.021 2384 A.H. Barber et al. / Composites Science and Technology 65 (2005) 2380–2384 at the same time, causes more variability in strength. [3] Sammalkorpi M, Krasheninnikov A, Kuronen A, Nordlund K, Such a strength-enhancing mechanism by wall-to-wall Kaski K. Phys Rev B 2004;70:245416. interactions may possibly be observed in future research [4] Mielke SL, Troya D, Zhang S, Li J-L, Xiao S, Car R, et al. Chem Phys Lett 2004;390:413. with other multi-tubular structures. The novel QFM [5] Pugno NM, Ruoff RS. Phil Mag 2004;84:2829. scheme shows promises to model the strength of nano- [6] Gao H, Ji B, Jager IL, Arzt E, Fratzl P. Proc Natl Acad Sci tubes as well, despite the presence of a free parameter. 2003;100:5597. [7] Carpinteri A, Pugno NM. Nature Mat 2005;4:421. [8] Li F, Cheng HM, Bai S, Su G. Appl Phys Lett 2000;77:3161. Acknowledgements [9] Wagner HD, Lourie O, Feldman Y, Tenne R. Appl Phys Lett 1998;72:188. [10] Zhu HW, Xu CL, Wu DH, Wei BQ, Vajtai R, Ajayan PM. This project was supported by the (CNT) Thematic Science 2002;296:884. European network on ‘‘Carbon Nanotubes for Future [11] Walters DA, Ericson LM, Casavant MJ, Liu J, Colbert DT, Industrial Composites’’ (EU), the NOESIS European Smith KA, et al. Appl Phys Lett 1999;74:3803. project on ‘‘Aerospace Nanotube Hybrid Composite [12] Pan ZW, Xie SS, Lu L, Chang BH, Sun LF, Zhou WY, et al. Structures with Sensing and Actuating Capabilities’’, Appl Phys Lett 1999;74:3152. [13] Barber AH, Andrews R, Schadler LS, Wagner HD [submitted]. the Minerva Foundation, the G.M.J. Schmidt Minerva [14] Yu M-F, Lourie O, Dyer MJ, Moloni K, Kelly TF, Ruoff RS. Centre of Supramolecular Architectures, and the Israeli Science 2000;287:637. Academy of Science. H.D. Wagner is the recipient of the [15] Kaplan-Ashiri I, Cohen SR, Gartsman K, Ivanovskaya V, Heine Livio Norzi Professorial Chair. T, Seifert G, et al. [submitted]. [16] Yu M-F, Files BS, Arepalli S, Ruoff RS. Phys Rev Lett 2000;84:5552. [17] Mann NR, Schafer RE, Singpurwalla ND. Methods for statistical References analysis of reliability and life data. New York: Wiley; 1974. p. 216. [1] Griffith AA. Phil Trans Roy Soc 1920;A221:163. [18] Wagner HD, Aronhime J, Marom G. Proc Roy Soc Lond A [2] Weibull W. J Appl Mech 1951;18:293. 1990;428:493. Meccanica (2005) 40: 455–469 © Springer 2005 DOI 10.1007/s11012-005-2133-y

Continuum Mechanics Modeling and Simulation of Carbon Nanotubes

MARINO ARROYO1,∗ and TED BELYTSCHKO2 1Departament de Matematica` Aplicada III, Laboratori de Calcul` Numeric` (LaCaN), Universitat Politecnica` de Catalunya, E-08034 Barcelona, Spain 2Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, USA

(Accepted: 27 July 2005)

Abstract. The understanding of the mechanics of atomistic systems greatly benefits from continuum mechanics. One appealing approach aims at deductively constructing continuum theories starting from models of the interatomic interactions. This viewpoint has become extremely popular with the quasicontinuum method. The application of these ideas to carbon nanotubes presents a peculiarity with respect to usual crystalline materials: their structure relies on a two-dimensional curved lattice. This renders the cornerstone of crystal elasticity, the Cauchy–Born rule, insufficient to describe the effect of curvature. We discuss the application of a theory which corrects this deficiency to the mechanics of carbon nanotubes (CNTs). We review recent developments of this theory, which include the study of the convergence characteristics of the proposed continuum models to the parent atomistic models, as well as large scale simulations based on this theory. The latter have unveiled the complex nonlinear elastic response of thick multiwalled carbon nanotubes (MWCNTs), with an anomalous elastic regime following an almost absent harmonic range.

Key words: Continuum mechanics, Carbon nanotubes, Finite elasticity, Atomistic models, Nanotube- based devices.

1. Introduction

Carbon nanotubes (CNTs) can be viewed as graphene sheets (a two-dimensional hexagonal lattice of carbon atoms) wrapped into cylinders one atom thick of speciﬁc chirality and diameter. Although irradiation or the synthesis process can produce defects into the crystal structure, these nanostructures are remarkably perfect. Since their discovery in 1991 [18], CNTs have attracted much attention due to their unique structure and mechanical, chemical and electronic properties [11]. These properties have made of CNTs the central element in an array of nanostructured materials, as well as nanosensors and devices. Mechanics plays a central role in nanotube-based nanotechnology. On the one hand, the exceptional mechanical properties (a nomi- nal Young’s modulus of 1 TPa, ideal strength of over 100 MPa) has prompted intense research in nanostructured materials such as nanotube-based nanocomposites. On the other hand, the strong function of other properties such as electrical conductance [37]

∗e-mail: [email protected] Modeling and Simulation of Carbon Nanotubes 469

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Published by Maney Publishing (c) IOM Communications Ltd eevd1 uy20;acpe 5Jl 2005 July 15 10.1179/174328405X63999 accepted DOI Institute 2005; the July of 10 behalf Received on Maney by Published ß omo ilctosaeal ofcltt haigat shearing facilitate the to in able imperfections are metals, specimen dislocations of the of case of form volume the the In as decreases. greater become defects * UK 3QZ, CB2 Cambridge, Cambridge of University Street, Metallurgy, Pembroke and Science Materials of Department are they as sharply increases crystals smaller. made of strength The appropriate make strength to Theoretical scenario. this lecture in feature this steels many how in show to are attempt comparisons dreams. our shall there beyond strength I literature possess because popular Before which and materials scientific strength of cost. modern the review of in to affordable to promises important meaning easy is an is it the that material, novel one has this chunks, describing large and in made manufacture be can a that over viable commercially applications. be of range would broad isotropic material an a cost. strong, maintaining reasonable Such a while at properties make of shape, combination attractive arbitrary to of difficult components is materialswhichcanbeusedtomanufacturelarge it In limitations. have particular, mechan- unfortunately other strengthening many of and isms These bonds. carbon– covalent of carbon stretching involves based the deformation if of Carbon strong mode incredibly only perfection. become principle in from can materials benefit whereas microstructure crystals the of single scale the strength- reducing be by can extremelyened creating metals of Polycrystalline ways materials. many of strong think to possible is It Introduction rapid or processing mechanical for apply. and need in produce the large Keywords: are to without alloying that cheap items made expensive is making be of for and used use can cooling be the which can which without dimensions, iron, steel three of bulk alloy all strong in of a combination is have now extraordinary this are We an All elements. crystals in Hatfield toughness. 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Garcia Carlos hard especial thank on an work is for original Caballero the It Garcia Francisca the transition. acknowledge providing to nanotube pleasure of for to illustration schematic Endo graphene the M. the and graphene to the of grateful images is author The Acknowledgements 0Cluae ant n atniesattemperatures start martensite and bainite Calculated 10 si lastecs,teermi ayparameters many remain there case, the always is As h ujc fs uhrsac…BTIN BUT research… much so CHUNKS! of are LARGE essential subject which the has nanomaterials of the structural all and of achieved produce has objectives bainite to effect, hard In cheap sections. the large very very in properties residual is uniform range long It have stresses. not does after treatment steel the heat Therefore, does cooling. and rapid uniform), require processing, it not mechanical of require has not all ever, does hardest (almost the ductility is considerable temperatures, low very at ttmn 5: Statement e2i2nwt aito ncro concentration carbon in variation with Fe–2Ni–2Mn a Fe–0 . 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The mechanical properties of single-crystal and ultrananocrystalline diamond: A theoretical study

Jeﬀrey T. Paci a,*, Ted Belytschko b, George C. Schatz a

a Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3113, USA b Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3111, USA

Received 1 July 2005; in ﬁnal form 1 August 2005

Abstract

We examine the mechanical properties of single-crystal and ultrananocrystalline diamond (UNCD) by simulating their fracture using semiempirical quantum mechanics and density functional theory. Our results predict a failure strain of 0.13 and a fracture stress of 100 GPa for UNCD, which are 37% and 43%, respectively, that of single-crystal diamond. The YoungÕs modulus of UNCD is E = 1.05 TPa which is only slightly smaller than that of single-crystal diamond (E = 1.09 TPa). The UNCD fracture stress value (rf = 100 GPa) is very large compared to that observed experimentally (rf < 5 GPa). We use Griﬃth theory to show that this difference is due to defects in UNCD. 2005 Elsevier B.V. All rights reserved.

1. Introduction We are interested in examining the strain-to-fracture behavior of these UNCD films using semiempirical Thin diamond films composed of extremely small (3– quantum mechanics and density functional theory. 5 nm) diamond grains and atom-wide grain boundaries Strain is applied to the diamond cluster, and the posi- (0.2 nm wide) can now be produced using plasma- tions of the atoms are adjusted to minimize its energy, enhanced chemical vapor deposition techniques [1–4]. subject to the constraint provided by the strain. The material, which is called ultrananocrystalline dia- The structure of UNCD films makes a complete mond (UNCD), is very strong with mechanical proper- examination of their fracture properties difficult. In ties (Vickers hardness, HV = 88 GPa [5], and YoungÕs addition to the grain boundaries which form between modulus, E = 950 GPa [6]) similar to those of single- adjacent 3–5 nm diamond grains, these films contain crystal diamond (HV = 100 GPa [7], and E = 1050 GPa other types of defects. Groups of diamond grains tend [8,9]). The films are also smooth, with a mean surface to form structures which are 100 nm in size. Interfaces roughness of 20–40 nm [10,11]. Nitrogen-doping of form between adjacent groups of diamond grains, pro- the plasma can be used to make these films conduct sig- ducing grain boundaries with structures which are not nificant amounts of electric current [12–17]. These prop- well understood. erties combine to make UNCD an excellent candidate In this work, our primary focus will be on the for use in the development of microelectromechanical mechanical properties of atom-wide grain boundaries systems [5,18]. which form between adjacent 3–5 nm diamond grains. References to the term Ôgrain boundaryÕ in the remainder of the Letter are meant to refer to these grain bound- * Corresponding author. Fax: +1 847 467 4996. aries. The effect of defects larger than these grain bound- E-mail address: [email protected] (J.T. Paci). ary structures will, however, also be discussed.

0009-2614/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.08.019 358 J.T. Paci et al. / Chemical Physics Letters 414 (2005) 351–358 energy of the clusters as they were strained to failure. In [8] H.J. McSkimin, P. Andreatch Jr., J. Appl. Phys. 43 (1972) 2944. the second, MSINDO was used to calculate structures as [9] M.H. Grimsditch, A.K. Ramdas, Phys. Rev. B 11 (1975) 3139. a function of strain, and PBE used to calculate energies [10] C. Zuiker, A.R. Krauss, D.M. Gruen, X. Pan, J.C. Li, R. Csencsits, A. Erdemir, C. Bindal, G. Fenske, Thin Solid Films 270 of the structures. In the third approach, PBE was used, (1995) 154. as MSINDO was in the first approach, to calculate both [11] A. Erdemir, G.R. Fenske, A.R. Krauss, D.M. Gruen, T. energies and structures as a function of strain. The third McCauley, R.T. Csencsits, Surf. Coat. Technol. 120–121 (1999) approach is the most accurate but is very computation- 565. ally time-consuming. The second approach is only mod- [12] B. Fausett, M.C. Granger, M.L. Hupert, J. Wang, G.M. Swain, D.M. Gruen, Electroanalysis 12 (2000) 7. estly more computationally expensive than the first but [13] S. Bhattacharyya, O. Auciello, J. Birrell, J.A. Carlisle, L.A. predicts YoungÕs modulus, failure strain and fracture Curtiss, A.N. Goyette, D.M. Gruen, A.R. Krauss, J. Schlueter, A. stress values which are within 8%, 6% and 6%, respec- Sumant, P. Zapol, Appl. Phys. Lett. 79 (2001) 1441. tively, of the full-strain PBE values for single-crystal dia- [14] T.D. Corrigan, D.M. Gruen, A.R. Krauss, P. Zapol, R.P.H. mond, and 9%, 23% and 16%, respectively, for UNCD. Chang, Diam. Relat. Mater. 11 (2002) 43. [15] J. Birrell, J.A. Carlisle, O. Auciello, D.M. Gruen, J.M. Gibson, MSINDO alone was much less accurate. Appl. Phys. Lett. 81 (2002) 2235. Experimentally observed fracture stress values for [16] J. Birrell, J.E. Gerbi, O. Auciello, J.M. Gibson, D.M. Gruen, J.A. single-crystal diamond and UNCD are approximately Carlisle, J. Appl. Phys. 93 (2003) 5606. 1/50th of the corresponding theoretical values. We have [17] J.E. Gerbi, O. Auciello, J. Birrell, D.M. Gruen, B.W. Alphenaar, shown that simple grain boundaries cannot be responsi- J.A. Carlisle, Appl. Phys. Lett. 83 (2003) 2001. [18] O. Auciello, J. Birrell, J.A. Carlisle, J.E. Gerbi, X. Xiao, B. Peng, ble for this decrease in strength and that Griffith theory, H.D. Espinosa, J. Phys. Condens. Matter 16 (2004) R539. based on the calculated surface energy, suggests defects [19] O.A. Shenderova, D.W. Brenner, A. Omeltchenko, X. Su, L.H. on the scale of 500 nm; defects of this scale have been Yang, Phys. Rev. B 61 (2000) 3877. observed experimentally. [20] B. Ahlswede, K. Jug, J. Comput. Chem. 20 (1999) 563, 572. [21] D. Sańchez-Portal, P. Ordejoń, E. Artacho, J.M. Soler, Int. J. Quantum Chem. 65 (1997) 453. [22] I.N. 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Stability of carbon nanotubes under electron irradiation: Role of tube diameter and chirality

A. V. Krasheninnikov,1,2 F. Banhart,3 J. X. Li,3 A. S. Foster,1 and R. M. Nieminen1 1Laboratory of Physics, Helsinki University of Technology, P.O. Box 1100, Helsinki 02015, Finland 2Accelerator Laboratory, P.O. Box 43, FIN-00014 University of Helsinki, Finland 3Institut für Physikalische Chemie, Universität Mainz, D-55099 Mainz, Germany ͑Received 21 June 2005; revised manuscript received 14 July 2005; published 22 September 2005͒

As recent experiments demonstrate, the inner shells of multiwalled carbon nanotubes are more sensitive to electron irradiation than the outer shells. To understand the origin of such counterintuitive behavior, we employ a density-functional-theory based tight-binding method and calculate the displacement threshold energies for carbon atoms in single-walled nanotubes with different diameters and chiralities. We show that the displacement energy and the defect production rate strongly depend on the diameter of the nanotube and its chirality, with the displacement energy being lower, but saturating towards the value for graphite when the tube diameter increases. This implies that the threshold electron energies to produce damage in nanotubes with diameters smaller than 1 nm are less than the commonly accepted value for graphitic nanoparticles. We also calculate the displacement energies for carbon atoms near defects and show that if a single vacancy is formed, it will likely be transformed to a double vacancy, as the nanotube atomic network with double vacancies has no energetically unfavorable undercoordinated atoms.

DOI: 10.1103/PhysRevB.72.125428 PACS number͑s͒: 61.46.ϩw, 61.80.Az, 68.37.Lp

I. INTRODUCTION displacements. Uniform irradiation of SWNTs ͑Ref. 9͒ re- Electron irradiation of carbon nanotubes has proved to be sulted in surface reconstructions and drastic dimensional an outstanding example of nanoengineering. In modern elec- changes, as a corollary of which the apparent diameter of the tron microscopes with field emission guns, the electron beam nanotubes decreased from 1.4 to 0.4 nm. Experimental and can be focused onto areas of several Å. This allows one to theoretical studies also demonstrated that the electron beam selectively modify nanostructures on an atomic scale by dis- creates defects nonuniformly: carbon atoms are most rapidly placing or removing atoms from pre-defined regions. Carbon removed from surfaces lying normal to the beam nanotubes are particularly suited for beam-induced nanoma- direction.6,10 nipulation since the graphitic network has a high tendency to Very recently, it has been shown experimentally13 that re-arrange after the formation of point defects. For example, MWNTs subjected to electron irradiation are destroyed from welding1 and coalescence2 of carbon nanotubes by the elec- inside and that it is the inner shell which collapses first. tron beam have been demonstrated, which opens a different Concurrently with these experiments, qualitatively similar way for making a network of connected nanotubes for mate- results12 were obtained for double-walled nanotubes at dif- rial reinforcement and use in nanoelectronics. The mechani- ferent temperatures. cal properties of nanotube bundles,3 and in principle macro- It has been demonstrated13 that a lower stability of the scopic nanotube samples,4 can be improved due to intertube inner shells is due to a combination of two effects: a higher links produced by electron irradiation. defect production rate in the inner shells and fast migration The irradiation-induced structural transformations in of carbon interstitials via the inner hollow in the axial direc- single-walled and multiwalled nanotubes ͑SWNTs/MWNTs͒ tion. The defect production rate is related to the tube diam- are due to the defects, mostly vacancies and interstitials, cre- eter, as the displacement threshold is lower for thin nano- ated by the impacts of energetic electrons followed by satu- tubes with highly curved atomic network. ration of highly reactive dangling bonds at undercoordinated In this work, we theoretically study the relationship be- carbon atoms. Thus knowing the defect production mecha- tween the tube diameter and displacement rate in more de- nism and how the defect production rate is related to the tail. We also address the role of chirality. Finally, to better beam characteristics is extremely important for controlling understand the evolution of nanotubes under high-dose irra- the transformations. Moreover, as carbon nanotubes are rou- diation, we calculate the displacement energies for carbon tinely characterized nowadays in the transmission electron atoms near defects which can exist in nanotubes or appear microscope ͑TEM͒, the complete understanding of the inter- due to interaction with the electron beam. action of energetic electrons with nanotubes should also minimize the amount of damage when it is an undesirable II. EVOLUTION OF MWNTs UNDER HIGH-DOSE side effect. ELECTRON IRRADIATION: TEM EXPERIMENTS Several TEM studies on irradiation-induced defects in carbon nanotubes have already been carried out.5–13 Early Similar to recent experiments,13 MWNTs were exposed to experiments5 showed that SWNTs exposed to focused elec- an intense focused electron beam in a TEM ͑FEI Tecnai tron irradiation were locally deformed and developed neck- F-30͒ with field emission electron gun operating at 300 kV. like features due to removal of carbon atoms by knock-on To prevent the agglomeration of interstitial atoms,8 the speci-

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Biological structures mitigate catastrophic fracture through various strategies

R. BALLARINI1,∗,R.KAYACAN2, F.-J. ULM3, T. BELYTSCHKO4 and A.H. HEUER1 1Case Western Reserve University, Civil Engineering, 10900 Euclid Avenue, Cleveland, OH 44106-7201, USA 2Suleyman Demirel University 3Massachusetts Institute of Technology 4Northwestern University ∗Author for correspondence. (E-mail: [email protected])

Received 4 July 2005; accepted in revised form 10 October 2005

Abstract. Gao et al. (PNAS, 100, 5597–5600 (2003)) have argued that load-bearing mineralized hard tissues, including bones, shells, and teeth, are nanocomposites, in which the mineral phase has nanoscale dimensions that ensure optimum strength and flaw tolerance. In particular, it has been claimed that the thickness of these brittle building blocks, being smaller than a critical size, h∗, of the order of tens of nanometers, renders them insensitive to the presence of crack-like flaws and enables them to achieve near-theoretical strength, which is why Nature employs nanoscale features in mineralized biological composites. We find this point of view, which Gao et al. and others have quoted in subsequent publications and presentations, unpersuasive and present several counterexamples which show that biological structures, as a result of being comprised of relatively fragile constituents that fracture at stress levels several orders of magnitude smaller than the theoretical strength, adopt various strategies to develop mechanical responses that enable them to mitigate catastrophic failure. Nanoscale structural features are not a result of an innate resistance to very high stresses.

Key words: Biological structures, crack bridging, ﬂaw-intolerance, ﬂaw-tolerance, nanoscale structures, toughening.

1. Cracks always weaken brittle solids

It has been realized since the seminal work of Griffith in (1921) that cracks or flaws always weaken brittle solids, and that increasing strength with decreasing specimen dimensions is not a question of flaw tolerance but of the decreasing probability that a “strength-defining” (Griffith) flaw is present in the area or volume being loaded. The argument of Gao et al. that nanoscale components in mineralized hard tissue could (and for toughness and strength of the overall structure are required to) attain (near) theoretical strength and flaw tolerance must therefore be rigorously examined. We will first present a classical elementary atomistic model that shows that a brittle structure containing a crack tens of nanometers long cannot achieve its theoretical strength, contrary to the argument of Gao et al.; this atomistic model is relevant to all brittle solids containing nanoscale defects, including mineralized hard tissues. We conclude that it is unlikely that if Nature’s design were optimized, it would be Biological structures mitigate catastrophic fracture 197

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Journal of the Mechanics and Physics of Solids 53 (2005) 1929–1950 www.elsevier.com/locate/jmps

Predicting the elastic properties of single-walled carbon nanotubes

H.W. ZhangÃ, J.B. Wang, X. Guo

State Key Laboratory of Structural Analysis and Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, P.R. China

Accepted 4 May 2005

Abstract

Advances in the prediction of the mechanical properties of single-walled carbon nanotubes (SWNTs) are reviewed in this paper. Based on the classical Cauchy-Born rule, a new computational method for the prediction of Young’s modulus of SWNTs is investigated. Compared with the existing approaches, the developed method circumvents the difﬁculties of high computational efforts by taking into consideration of the microstructure of nanotube and the atomic potential of hydrocarbons. Numerical results of Young’s modulus and its variation with respect to the deformation gradient tensor are given and discussed. The results obtained are in good agreement with those obtained by laboratory experiments and other numerical methods. r 2005 Elsevier Ltd. All rights reserved.

Keywords: Carbon nanotube; Cauchy-Born rule; Atomic potential function; Young’s modulus

1. Introduction

Since Iijima (1991) found multi-walled carbon nanotubes (MWNTs) in the soot produced by the arc discharge technique, carbon nanotubes (CNTs) have attracted much attention due to their superior mechanical, thermal and electrical properties

ÃCorresponding author. Tel.: +86 411 84706249; fax: +86 411 84708769. E-mail addresses: [email protected] (H.W. Zhang), [email protected] (X. Guo).

H.W. Zhang et al. / J. Mech. Phys. Solids 53 (2005) 1929–1950 1949

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Qiang Lu and Baidurya Bhattacharya1

Department of Civil and Environmental Engineering, University of Delaware, Newark, DE 19716, USA

E-mail: [email protected]

Received 13 October 2004, in final form 26 January 2005 Published 25 February 2005 Online at stacks.iop.org/Nano/16/555 Abstract The remarkable mechanical properties of carbon nanotubes (CNTs) have generated a lot of interest in recentyears. While CNTs are found to have ultra-high stiffness and strength, an enormous scatter is also observed in available laboratory results. This randomness is partly due to the presence of nanoscale defects, heterogeneities etc, and this paper studies the effects of randomly distributed Stone–Wales (SW or 5–7–7–5) defects on the mechanical properties of single-walled nanotubes (SWNTs) using the technique of atomistic simulation (AS). A Matern hard-core random field applied on a finite cylindrical surface is used to describe the spatial distribution of the Stone–Wales defects. We simulate a set of displacement controlled tensile loadings up to fracture of SWNTs with (6, 6) armchair and (10, 0) zigzag configurations and aspect ratio around six. A modified Morse potential is adopted to model the interatomic forces. We find that fracture invariably initiates from a defect if one is present; for a defect-free tube the crack initiates at quite random locations. The force–displacement curve typically behaves almost linearly up to about half way, although there is no obvious yield point. Three mechanical properties—stiffness, ultimate strength and ultimate strain—are calculated from the simulated force and displacement time histories. The randomness in mechanical behaviour resulting only from initial velocity distribution was found to be insignificant at room temperature. The mean values of stiffness, ultimate strength and ultimate strain of the tube decrease as the average number of defects increases, although the coefficients of variation do not show such a monotonic trend. The introduction of an additional defect has the most pronounced effect on the randomness in mechanical properties when the tube is originally defect free. We also find that, for a given mean number of defects in the tube, the zigzag configuration has less strength and less ultimate strain on average, but more uncertainty in its stiffness and ultimate strain, compared with the armchair tube. (Some figures in this article are in colour only in the electronic version)

1 Author to whom any correspondence should be addressed.

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J. load Science 287 637–40 Quantum Chem. 96 239–46

565 PHYSICAL REVIEW B 71, 115403 ͑2005͒

Mechanics of defects in carbon nanotubes: Atomistic and multiscale simulations

Sulin Zhang,1,* Steven L. Mielke,2 Roopam Khare,1 Diego Troya,2,† Rodney S. Ruoff,1 George C. Schatz,2 and Ted Belytschko1,‡ 1Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3111, USA 2Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3113, USA ͑Received 11 August 2004; revised manuscript received 3 November 2004; published 3 March 2005͒

Molecular mechanics ͑MM͒ calculations together with coupling methods bridging MM and finite crystal elasticity are employed to simulate the fracture of defected carbon nanotubes ͑CNTs͒ and to compare with the available experimental results. The modified second generation Brenner potential ͑MTB-G2͒ is adopted in the calculations. Our MM calculations show fair agreement with quantum mechanical ͑QM͒ benchmarks, and indicate that one- and two-atom vacancies reduce the fracture strength of CNTs by 20% –33% ͑whereas the QM calculations predict 14% –27%͒, but these fracture strengths are still much higher than the experimental data. We then demonstrate that this experimental and theoretical discrepancy can be attributed to the presence of large-scale defects, such as those that may arise from oxidative purification processes. Simulations on multiwalled CNTs and tubes twisted prior to tensile loading show negligible effects on the fracture strength, which indicates that these are not the causes of low experimental values. The effects of chirality and tube diameter on fracture strengths are also investigated.

DOI: 10.1103/PhysRevB.71.115403 PACS number͑s͒: 61.50.Ah, 62.25.ϩg, 68.65.Ϫk, 81.07.De

I. INTRODUCTION predict such weakening, which suggests that these potentials Predicting the strength of carbon nanotubes ͑CNTs͒ is an must be used with caution when treating defected CNTs. interesting challenge from both the scientific and engineering Irradiation with energetic ions or electrons can knock carbon viewpoints. From a scientific viewpoint, a CNT ostensibly atoms out of the hexagonal lattice, producing single-atom or offers a clean model for the study of fracture, since the frac- multiatom vacancies in CNTs.13–15 Density functional theory ture of a single molecule should involve only chemical bond ͑DFT͒ calculations showed that vacancy defects can form breaking at the atomistic scale without other complications links between adjacent graphite layers,16 providing a mecha- such as grain boundaries. From an engineering viewpoint, a nism for improved intershell or intertube mechanical thorough understanding of CNT fracture is needed for the coupling.17,18 In a recent study on the fracture of CNTs,7 it design of CNT-reinforced composites. So far, comparisons of was argued that large defects could be introduced in experimental data and theoretical calculations have mani- MWCNTs by oxidative purification processes.19,20 fested large discrepancies. According to the experimental Due to the small size of CNTs, fracture experiments are measurements of Yu et al.,1 the fracture strengths of 19 mul- extremely challenging, and measurements of the tensile fail- tiwalled CNTs ͑MWCNTs͒ ranged from 11 to 63 GPa with a ure strength of individual tubes are fairly limited.1,21 QM mean value of 27.8 GPa ͑see Fig. 1͒. However, recent quan- calculations2–7 have therefore been used to elucidate the frac- tum mechanical ͑QM͒ calculations2–7 for pristine tubes agree ture of CNTs; however, the computational cost limits QM reasonably well with each other and indicate that the fracture studies to CNTs with relatively small dimensions. Molecular of nanotubes is brittle at room temperature with a fracture stress in the range of 75–135 GPa depending on tube chirality. It is thus of interest to examine whether plausible defects or other possible effects stemming from the differences between the experiments and the numerical models could explain these discrepancies. The cause of defects and their effects on the physical properties of CNTs have attracted considerable attention. One of the most intensively studied defects is the 5-7-7-5 dislocation formed by a Stone-Wales ͑SW͒ transformation.8 It has been shown by QM calculations that the SW transformation is energetically favored above a tensile strain of about ϳ5% –6% for armchair tubes9,10 and ϳ12% for zigzag tubes.10 Aggregation of SW defects has been hypoth- esized to lead to crack initiation;11 however, QM analysis6 indicates that aggregations of SW defects do not markedly reduce the fracture strength of CNTs—at least at moderate temperatures where brittle failure mechanisms prevail. It was FIG. 1. Distribution of fracture stresses in the experiment of Yu also noted6 that empirical bond-order potentials12 incorrectly et al. ͑Ref. 1͒.

TABLE IV. Size effects on the fracture strength of DWCNTs. resemblance to cracks͒ lower the fracture strength more sig- The outer tube contains a two-atom vacancy ͑symmetric͒, while the nificantly, falling in the upper range of the experimental ob- inner tube is defect free. Douter is the diameter of the outer tubes, servations. Slits and holes with a comparable cross section ⌬ ⌬ and Router and Rinner are the radius decrements just prior to frac- were observed to weaken tubes to a similar degree. The ef- ture as compared to the zero-strain state of the outer and inner fect of tube chirality on fracture was explored; fracture ␴ ⌬ tubes, respectively. The values for cr and Router in parentheses strength increased monotonically with increasing chiral correspond to the case when the inner shells are absent. angle and armchair tubes were most resistant to the weakening effects of holes. ͑ ͒ ␴ ͑ ͒ ⌬ ͑ ͒ ⌬ ͑ ͒ CNTs Douter Å cr GPa Router Å Rinner Å In addition to the MM calculations, calculations using a coupling method that bridges MM and finite crystal elasticity ͓50, 0͔/͓24, 24͔ 39.15 68.7͑67.8͒ 0.07͑0.12͒ 0.03 were presented. This coupling method enables the study of ͓71, 0͔/͓36, 36͔ 55.59 68.8͑68.3͒ 0.10͑0.13͒ 0.07 large-diameter SWCNTs and MWCNTs. Our simulations ͓116, 0͔/͓62, 62͔ 90.83 69.3͑68.8͒ 0.17͑0.24͒ 0.14 show that the presence of inner tubes only slightly increases ͓220, 0͔/͓122, 122͔ 172.26 69.6͑69.1͒ 0.32͑0.42͒ 0.26 the fracture strength of the CNTs considered, indicating small intershell mechanical coupling. Simulations of ͓388, 0͔/͓219, 219͔ 303.80 70.1͑69.4͒ 0.50͑0.78͒ 0.43 DWCNTs with two-atom vacancy defects in the outer shell show that the fracture strength is size dependent, but the variation is only a few GPa for the range of tube diameters V. CONCLUDING REMARKS considered. Twisting the tube prior to loading and other load Motivated by discrepancies between theoretical and ex- imperfections were observed to negligibly affect the fracture perimental fracture strengths of CNTs, we studied the effects strength of MWCNTs, but reduced the fracture strength of of vacancy defects, holes, and slits on fracture strength using SWCNTs by as much as ϳ4% at a twisting angle of 15°. MM and coupled MM/CM techniques. Where possible, these Therefore, imperfections in the loading are not a likely results are compared to available quantum mechanical calcu- source of the low experimental fracture strengths. lations and fair agreement is observed. The MM calculations show that one- and two-atom vacancy defects weaken CNTs ACKNOWLEDGMENTS by 20% –33%, whereas QM calculations have shown We thank Professor Shaoping Xiao and Dr. Marino Ar- 14% –27% reductions in strengths of these defects. The royo for helpful discussions. We gratefully acknowledge the computed fracture strengths are slightly greater than the grant support from the NASA University Research, Engi- highest experimental values of Yu et al. and substantially neering and Technology Institute on Bio Inspired Materials greater than the average measured fracture strength. Holes ͑BIMat͒ under award No. NCC-1-02037 ͑Jeff Jordan, Pro- ͑which may readily be introduced by oxidative purification gram Manager͒. R.S.R. also appreciates support from the Of- processes͒ and slits ͑which are less likely to be experimen- fice of Naval Research grant ͑RSR: No. N000140210870, tally relevant, but which have formal interest due to their Program manager Mark Spector͒.

*Electronic address: [email protected] 5346 ͑1998͒. †Current address: Department of Chemistry, Virginia Tech, 107 11 B. I. Yakobson, Appl. Phys. Lett. 72, 918 ͑1998͒. Davidson Hall, Blacksburg, VA, 24061-0212, USA. 12 D. W. Brenner, O. A. Shenderova, J. A. Harrison, S. J. Stuart, B. ‡Electronic address: [email protected] Ni, and S. B. Sinnott, J. Phys.: Condens. Matter 14, 783 ͑2002͒. 1 M.-F. Yu, O. Lourie, M. J. Dyer, K. Moloni, T. F. Kelly, and R. S. 13 P. M. Ajayan, V. Ravikumar, and J.-C. Charlier, Phys. Rev. Lett. Ruoff, Science 287, 637 ͑2000͒. 81, 1437 ͑1998͒. 2 T. Ozaki, Y. Iwasa, and T. Mitani, Phys. Rev. Lett. 84, 1712 14 A. V. Krasheninnikov, K. Nordlund, M. Sirvio, E. Salonen, and J. ͑2000͒. Keinonen, Phys. Rev. B 63, 245405 ͑2001͒. 3 G. Dereli and C. Ozdogan, Phys. Rev. B 67, 035416 ͑2003͒. 15 A. V. Krasheninnikov, K. Nordlund, and J. Keinonen, Phys. Rev. 4 T. Dumitrica, T. Belytschko, and B. I. Yakobson, J. Chem. Phys. B 65, 165423 ͑2002͒. 118, 9485 ͑2003͒; 119, 1281͑E͒͑2003͒. 16 R. H. Telling, C. P. Ewels, A. A. El-Barbary, and M. I. Heggie, 5 S. Ogata and Y. Shibutani, Phys. Rev. B 68, 165409 ͑2003͒. Nat. Mater. 2, 333 ͑2003͒. 6 D. Troya, S. L. Mielke, and G. C. Schatz, Chem. Phys. Lett. 382, 17 A. Kis, G. Csanyi, J.-P. Salvetat, T.-N. Lee, E. Couteau, A. J. 133 ͑2003͒. Kulik, W. Benoit, J. Brugger, and L. Forro, Nat. Mater. 3, 153 7 S. L. Mielke, D. Troya, S. Zhang, J.-L. Li, S. Xiao, R. Car, R. S. ͑2004͒. Ruoff, G. C. Schatz, and T. Belytschko, Chem. Phys. Lett. 390, 18 M. Huhtala, A. V. Krasheninnikov, J. Aittoniemi, S. J. Stuart, K. 413 ͑2004͒. Nordlund, and K. Kaski, Phys. Rev. B 70, 045404 ͑2004͒. 8 A. J. Stone and D. J. Wales, Chem. Phys. Lett. 128, 501 ͑1986͒. 19 D. T. Colbert, J. Zhang, S. M. McClure, P. Nikolaev, Z. Chen, J. 9 M. B. Nardelli, B. I. Yakobson, and J. Bernholc, Phys. Rev. B 57, H. Hafner, D. W. Owens, P. G. Kotula, C. B. Carter, J. H. R4277 ͑1998͒. Weaver, A. G. Rinzler, and R. E. Smalley, Science 266, 1218 10 P. Zhang, P. E. Lammert, and V. H. Crespi, Phys. Rev. Lett. 81, ͑1994͒.

115403-11 Chemical Physics Letters 403 (2005) 16–21 www.elsevier.com/locate/cplett

Fracture paths and ultrananocrystalline diamond

Jeﬀrey T. Paci a, Lipeng Sun a, Ted Belytschko b, George C. Schatz a,*

Received 23 November 2004; in ﬁnal form 15 December 2004 Available online 11 January 2005

Abstract

We use the simulated fracture of ultrananocrystalline diamond (UNCD) to illustrate how different fracture paths can result in different predictions of system properties. At zero temperature, the system is unable to explore the potential energy surface far from the fracture path being investigated. This can result in misleading predictions for the mechanical properties of UNCD. In non-zero temperature simulations, the system can explore more of the potential energy surface, but these are computationally intense simulations. We show how lower bounds to the energy path during fracture can be determined in pure and nitrogen-doped UNCD without doing finite temperature simulations. 2004 Elsevier B.V. All rights reserved.

1. Introduction mechanical simulations we describe in this work are performed at T = 0. Strain is applied to the diamond struc- Using plasma-enhanced chemical vapor deposition ture, and the atoms adjust their positions in an attempt techniques, it is now possible to make thin diamond to minimize the energy of the cluster, subject to the con- films containing extremely small (3–5 nm) diamond straint provided by the strain. In this minimization, only grains and atom-wide grain boundaries (0.2 nm wide) the local potential energy surface (PES) is explored. A [1–4]. This material, called ultrananocrystalline dia- more global exploration of the PES is possible when mond (UNCD), has very impressive mechanical proper- thermal effects are included. In the latter case atoms ties (97 GPa hardness and 967 GPa YoungÕs modulus, are vibrating about their lattice positions as strain is ap- which are similar to single-crystal diamond) [5,6]. These plied. Such an exploration can dramatically effect the films can be doped with nitrogen to produce diamond reaction path taken on the way to diamond fracture. films with significant electrical conduction [7–13]. These These thermal effects are important, as physically realis- properties combined with the fact that the films are tic fracture paths tend to be those which require the min- smooth (mean surface roughness = 20–40 nm) [14,15], imum amount of energy necessary for a given process makes them an excellent candidate for use in the devel- [18]. However, the sampling of thermal paths is a very opment of microelectromechanical systems [16,17]. time-consuming process when electronic structure calcu- We are interested in examining the strain-to-fracture lations are used to determine energies. behavior of these UNCD films using semiempirical A partial solution to this problem is to examine more quantum mechanics. A complete examination of dia- than one, or several fracture paths at T = 0. In this mond fracture would necessarily involve some consider- scheme the various paths explore different regions of ation of the effect of temperature, T. The quantum the PES. The examination of many such paths should allow for a thorough exploration of the PES. In real * Corresponding author. Fax: +1 847 491 7713. systems thermal effects would allow the system to find E-mail address: [email protected] (G.C. Schatz). its way between some of the paths. As a result, an

Table 2 Stresses at selected strains, v along the various fracture paths for UNCD doped with 16 nitrogen atoms v = 0.03 v = 0.06 v = 0.09 v = 0.12 v = 0.15 v = 0.18 r(GS) 41 77 – 107 69 5 r(AGBS) 41 72 77 – – – r(ATGBS) 41 67 – 8 – – r(ABGBS) 32 125 – – – – The units of stress are GPa. same is true for the pure UNCD crystal although the [8] Q. Chen, D.M. Gruen, A.R. Krauss, T.D. Corrigan, M. Witek, two grain boundaries in that case are not sufficiently dif- G.M. Swain, J. Electrochem. Soc. 148 (2001) E44. ferent to produce a significant difference in asymptotic [9] S. Bhattacharyya, O. Auciello, J. Birrell, J.A. Carlisle, L.A. Curtiss, A.N. Goyette, D.M. Gruen, A.R. Krauss, J. Schlueter, energy, as shown in Fig. 2. A. Sumant, P. Zapol, Appl. Phys. Lett. 79 (2001) 1441. [10] T.D. Corrigan, D.M. Gruen, A.R. Krauss, P. Zapol, R.P.H. Chang, Diam. Relat. Mater. 11 (2002) 43. 4. Conclusions [11] J. Birrell, J.A. Carlisle, O. Auciello, D.M. Gruen, J.M. Gibson, Appl. Phys. Lett. 81 (2002) 2235. [12] J. Birrell, J.E. Gerbi, O. Auciello, J.M. Gibson, D.M. Gruen, Presented here is a theoretical investigation of differ- J.A. Carlisle, J. Appl. Phys. 93 (2003) 5606. ent paths leading to UNCD fracture. Both pure and [13] J.E. Gerbi, O. Auciello, J. Birrell, D.M. Gruen, B.W. Alphenaar, nitrogen-doped UNCD were investigated. Because the J.A. Carlisle, Appl. Phys. Lett. 83 (2003) 2001. diamond cluster size required for meaningful simula- [14] C. Zuiker, A.R. Krauss, D.M. Gruen, X. Pan, J.C. Li, R. tions means thermal effects are difficult to fully incorpo- Csencsits, A. Erdemir, C. Bindal, G. Fenske, Thin Solid Films 270 (1995) 154. rate in quantum-mechanical studies, an investigation of [15] A. Erdemir, G.R. Fenske, A.R. Krauss, D.M. Gruen, T. multiple fracture paths was made. Results from the dif- McCauley, R.T. Csencsits, Surf. Coat. Technol. 120–121 (1999) ferent fracture paths were examined, and a better 565. approximation to the real fracture energy path pro- [16] A.R. Krauss, O. Auciello, D.M. Gruen, A. Jayatissa, A. Sumant, vided. This makes it possible to locate lower bounds J. Tucek, D.C. Mancini, N. Moldovan, A. Erdemir, D. Ersoy, M.N. Gardos, H.G. Busmann, E.M. Meyer, M.Q. Ding, Diam. to the energy path during fracture without doing ther- Relat. Mater. 10 (2001) 1952. mal sampling or simulated annealing. [17] O. Auciello, J. Birrell, J.A. Carlisle, J.E. Gerbi, X. Xiao, B. Peng, H.D. Espinosa, J. Phys.: Condens. Matter 16 (2004) R539. [18] R.D. Levine, R.B. Bernstein, Molecular Reaction Dynamics Chemical Reactivity, Oxford University Press, New York, Acknowledgments 1987. [19] B. Ahlswede, K. Jug, J. Comput. Chem. 20 (1999) 563, p. 572. We gratefully acknowledge grant support from the [20] K. Jug, G. Geudtner, T. Homann, J. Comput. Chem. 21 (2000) National Science Foundation (Grant CMS 974. 500304472). We thank Peter Zapol and Michael Stern- [21] T. Bredow, G. Geudtner, K. Jug, J. Comput. Chem. 22 (2001) 861. berg for useful input, and Steven L. Mielke for helpful [22] I.N. Levine, Quantum Chemistry, fourth ed., Prentice Hall, New comments on the manuscript. Jersey, 1991. [23] T. Bredow, G. Geudtner, K. Jug, J. Comput. Chem. 22 (2001) 89. [24] J.D. Head, M.C. Zerner, Chem. Phys. Lett. 122 (1985) 264. References [25] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in FORTRAN 77, second ed., Cambridge [1] D.M. Gruen, S. Liu, A.R. Krauss, J. Luo, X. Pan, Appl. Phys. University Press, Cambridge, 1996. Lett. 64 (1994) 1502. [26] P. Zapol, M. Sternberg, L.A. Curtiss, T. Frauenheim, D.M. [2] L.C. Qin, D. Zhou, A.R. Krauss, D.M. Gruen, Nanostruct. Gruen, Phys. Rev. B 65 (2001) 045403. Mater. 10 (1998) 649. [27] P. Keblinski, D. Wolf, S.R. Phillpot, H. Gleiter, J. Mater. Res. 13 [3] D.M. Gruen, Ann. Rev. Mater. Sci. 29 (1999) 211. (1998) 2077. [4] X. Xiao, J. Birrell, J.E. Gerbi, O. Auciello, J.A. Carlisle, J. Appl. [28] P. Keblinski, D. Wolf, F. Cleri, S.R. Phillpot, H. Gleiter, Mater. Phys. 96 (2004) 2232. Res. Soc. Bull. 23 (1998) 36. [5] H.D. Espinosa, B.C. Prorok, B. Peng, K.H. Kim, N. Moldovan, [29] F. Cleri, P. Keblinski, L. Colombo, D. Wolf, S.R. Phillpot, O. Auciello, J.A. Carlisle, D.M. Gruen, D.C. Mancini, Exp. Europhys. Lett. 46 (1999) 671. Mech. 43 (2003) 256. [30] O.A. Shenderova, D.W. Brenner, A. Omeltchenko, X. Su, L.H. [6] H.D. Espinosa, B. Peng, B.C. Prorok, N. Moldovan, O. Auciello, Yang, Phys. Rev. B 61 (2000) 3877. J.A. Carlisle, D.M. Gruen, D.C. Mancini, J. Appl. Phys. 94 [31] J.T. Paci, T. Belytschko, G.C. Schatz (to be published). (2003) 6076. [32] S.L. Mielke, D. Troya, S. Zhang, J.-L. Li, S. Xiao, R. Car, R.S. [7] B. Fausett, M.C. Granger, M.L. Hupert, J. Wang, G.M. Swain, Ruoff, G.C. Schatz, T. Belytschko, Chem. Phys. Lett. 390 (2004) D.M. Gruen, Electroanalysis 12 (2000) 7. 413. PHYSICAL REVIEW B 70, 245416 (2004)

Mechanical properties of carbon nanotubes with vacancies and related defects

M. Sammalkorpi,1,* A. Krasheninnikov,2 A. Kuronen,1 K. Nordlund,2 and K. Kaski1 1Laboratory of Computational Engineering, Helsinki University of Technology, P.O. Box 9203, 02015 HUT, Finland 2Accelerator Laboratory, University of Helsinki, P.O. Box 43, FIN 00014 University of Helsinki, Finland (Received 15 March 2004; revised manuscript received 24 May 2004; published 14 December 2004)

Although as-grown carbon nanotubes have relatively few defects, defects can appear at the puriﬁcation stage or be deliberately introduced by irradiation with energetic particles or by chemical treatment when aiming at the desired functionality. The defects, especially vacancies, give also rise to a deleterious effect—deterioration of axial mechanical properties of nanotubes. By employing molecular dynamics simulations and continuum theory we study how the Young’s modulus and tensile strength of nanotubes with vacancy-related defects depend on the concentration of defects and defect characteristics. We derive an analytical expression, with coefﬁcients parametrized from atomistic computer simulations, which relates the Young’s modulus and defect density in carbon nanotubes. We further show that the tensile strength and critical strain of single-walled nanotubes decrease by nearly a factor of 2 if an unreconstructed vacancy is present. However, this deterioration in the mechanical characteristics is partly alleviated by the ability of nanotubes to heal vacancies in the atomic network by saturating dangling bonds.

DOI: 10.1103/PhysRevB.70.245416 PACS number(s): 81.07.De, 61.80.Jh, 62.25.ϩg

I. INTRODUCTION Very recent experiments24 on electron irradiation of car- Carbon nanotubes (CNTs) have extremely high axial bon nanotube bundles followed by mechanical testing of the Young’s modulus of about 1 TPa (Refs.1–6) and tensile bundle bending modulus (which is proportional to the strength approaching 60 GPa.1,7 These exceptional mechani- Young’s modulus) indicate that small dose irradiation gives cal properties along with low weight of CNTs and recent rise to a very large improvement in the mechanical properties improvements in their synthesis and purification techniques of irradiated bundles. This result was understood in terms of make CNTs ideal candidates for reinforcement of various irradiation-induced inter-tube links which provided load materials, e.g., polymers.8–10 High stiffness and the tensile transfer and correspondingly enhanced the shear modulus in- strength of CNTs should also provide the mechanical stabil- side the bundle. However, high-dose irradiation resulted in deterioration of mechanical characteristics due to accumula- ity for electric nano-circuits formed by CNTs with covalent tion of the irradiation-induced damage, and specifically va- inter-tube junctions.11 cancies, in the nanotube atomic network. These outstanding mechanical characteristics hold for In addition to linking the nanotubes by covalent bonds, nearly perfect CNTs. However, if CNTs have defects in the irradiation has experimentally been demonstrated to give rise atomic network, one can expect that due to their quasi-one- to complete welding11 and coalescence26 of nanotubes thus dimensional atomic structure even a small number of defects opening new ways for electron/ion beam-assisted engineer- 12 will result in some degradation of their characteristics. The ing of nano-circuits. The driving force for these structural defects can appear at the stage of CNT growth and transformations was found to be the formation of vacancies purification,13,14 or later on during device or composite pro- with chemically reactive dangling bonds followed by anneal- duction. Moreover, defects in CNTs can deliberately be cre- ing of the damage in which the bonds were saturated. How- ated by chemical treatment15 or by irradiation11,16–18 to ever, even spatially-localized irradiation will create defects achieve the desired functionality. not only in the junction region, but also in the rest of the As an example of this, defects are expected to increase system due to, e.g., sputtered carbon atoms. This will inevi- CNT adhesion to a polymer matrix,18,19 which should result tably result in deterioration of the mechanical stability of the in improvements of the composite mechanical characteris- system. tics. Likewise, defects may enhance the overall characteris- Therefore, to understand the role of defects in mechanical tics of bundles of single-walled nanotubes (SWNTs) and strength and to fully exploit the advantages potentially pro- multi-walled nanotubes (MWNTs). In these structures the in- vided by the irradiation techniques, one should know how teractions between intact nanotubes are governed by weak vacancy-related defects influence the mechanical character- van der Waals forces, so that the axial mechanical load is istics of CNTs. carried only by the SWNTs at the rope perimeter20 or by the Although continuum methods27–29 work well for perfect outermost shell in MWNTs. Thus, creating strong defect- materials, they cannot directly be applied to nanotubes with mediated covalent bonds between SWNTs in defects, as these methods assume the material to be perfect. bundles16,17,21–24 or between shells of MWNTs25 by, for ex- However, a combination of these methods and atomistic ample, irradiation should provide load transfer to the inner simulations can be used for evaluating elastic properties of tubes (shells). On the other hand, irradiation will create not nanotubes with defects, while only atomistic methods can be only covalent bonds between the tubes but also defects in the employed for simulating the plastic behavior. In this paper, atomic network. by employing atomistic computer simulations and analytical

R4277 (1998). 45 P. Zhang, P. E. Lammert, and V. H. Crespi, Phys. Rev. Lett. 81, 38 H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, A. 5346 (1998). DiNola, and J. R. Haak, J. Chem. Phys. 81, 3684 (1984). 46 G. G. Samsonidze, G. G. Samsonidze, and B. I. Yakobson, Phys. 39 T. Belytschko, S. P. Xiao, G. C. Schatz, and R. S. Ruoff, Phys. Rev. Lett. 88, 065501 (2002). Rev. B 65, 235430 (2002). 47 M. B. N. Q. Zhao and J. Bernholc, Phys. Rev. B 65, 144105 40 B. I. Yakobson, M. P. Campbell, C. J. Brabec, and J. Bernholc, (2002). Comput. Mater. Sci. 8, 341 (1997). 48 F. Banhart, Rep. Prog. Phys. 62, 1181 (1999). 41 T. Xiao and K. Liao, Phys. Rev. B 66, 153407 (2002). 49 A. A. El-Barbary, R. H. Telling, C. P. Ewels, M. I. Heggie, and P. 42 C. Wei, K. Cho, and D. Srivastava, Appl. Phys. Lett. 82, 2512 R. Briddon, Phys. Rev. B 68, 144107 (2003). (2003). 50 R. W. Herzberg, Deformation and Fracture Mechanics of Enigi- 43 C. Wei, K. Cho, and D. Srivastava, Phys. Rev. B 67, 115407 neering Materials (Wiley, New York, 1996). (2003). 51 S. L. Mielke, D. Troya, S. Zhang, J. Li, S. Xiao, R. Car, R. S. 44 T. Dumitrica, T. Belytschko, and B. I. Yakobson, J. Chem. Phys. Ruoff, G. C. Schatz, and T. Belytschko, Chem. Phys. Lett. 390, 118, 9485 (2003). 413 (2004).

245416-8 Downloaded By: [Academia Sinica] At: 06:30 19 November 2007 V P alr utocra h hsclyuraoal eola.W hoet modify to choose We load. assumption zero classical unreasonable strength, the physically material the the with exceeds at or combined deri- occur equals is stress must length; 1939) maximum failure the (Westergaard crack when tip occurs crack infinitesimal failure the that for at field also stress the ved from in Conversely failure. singularity at be a load to infinite elasticity, an shown predict (LEFM), incorrectly the mechanics 1957), (Irwin fracture these equivalent on both elastic length, linear based infinitesimal continuum-based of For crack of method a paradox. methods containing a well-known element structural a and elastic present linear these method, a 1939); energy-based (Westergaard factor an stress-intensity (1920), criterion Griffith’s hilosophical ol. z y h hnmnno rcuei fgetitrs.Tocasctetet are treatments classic Two interest. great of is fracture of phenomenon The uhrfrcrepnec.Eal r-ruoff@northwestern.edu Email: correspondence. for [email protected] Author Email: correspondence. for Author ldn IadtaigII,adtesaiiyo h rcuepoaain,are propagations, fracture the of stability the I, (opening way. and Modes simple III), in fracture a different in tearing strength The treated and restric- ideal shape. no and II finite has size QFM sliding a defect LEFM, treating predicts to contrast on QFM In tions prediction. length Orowan’s mechanics crack with frac- fracture agreement vanishing elastic non-linear with linear For second-order with to (NLFM). first-order and to (LEFM), mechanics agreeing holes, ture self-consistent, and spring is cracks two-dimensional QFM of with fracture networks. graphene on and simulations mechanics-based nanotubes statistical carbon mechanics/dynamics and molecular of with fracture also of and simulation films; thin polysilicon and whiskers, IL Evanston, University, Northwestern Engineering, Mechanical of Department tessta r elpeitdb F:srntspeitdb F r com- are QFM ‘quantized’ nanotubes, by at carbon predicted occurs on strengths loading results QFM: of experimental by type with predicted and pared well criterion Fracture geometry are remarkable. given Griffith’s that are a in stresses with implications strain- differentials systems the and The tiny differences; of finite proposed. stress- with also mechanics; substituted are are fracture analogs continuum-based QFM based modifies that sented 4 N 84, eateto tutrlEgneig oienc iTrn,CroDc degli Duca Corso Torino, di Politecnico Engineering, Structural of Department hlspia Magazine Philosophical e nrybsdter,qatzdfatr ehnc QM,i pre- is (QFM), mechanics fracture quantized theory, energy-based new A o. 7 2829–2845 27, M agazine [ eevd2 a 04adacpe 0My2004 May 30 accepted and 2004 May 28 Received unie rcuemechanics fracture Quantized SN17–45pitIS 4864 online 1478–6443 print/ISSN 1478–6435 ISSN ,21S O:10.1080/14786430412331280382 DOI: and http://www.tandf.co.uk/journals buz 4 02,Italy 10129, 24, Abruzzi ioaM Pugno M. 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N. of cracks short with nanotubes of strength the 2 for length predictions corresponding 2844 oarmt tesfield stress remote a to eoti h euto qain()if (3a) equation (7) applying equation hand, of other result the On the hole). obtain the we without element the of strength r neetn:()for (i) interesting: are radius dimensionless with osiuielw n pligeuto 3)b nerto,w bantersl of strength result the the between obtain ratio we the integration, gives by that elastic (3b) (7), linear equation equation isotropic applying Assuming and predictions). our laws affect constitutive not does stress shearing (the iyadcascltninlciein htasmstecniumhptei,i.e., hypothesis, continuum the assumes that criterion, tensional a classical and city wanaga rwin l lumberg elytschko elytschko elytschko, eale azant agdahn reager hasiotis arpinteri ! rory riffith, irsch irai, ellan ashimoto urakami ielke H h tesna oeo radius of hole a near stress The ) for 0); addad .D,adS and D., P. , .R,1957, R., G. , . D M., , Y. . uy19,DeTeredrEatzttudDeBdrnsedrFestigkeitslehre, der Bedurfnisse Die und Elastizitat der Theorie Die 1898, July R., , 6 n B and etcrf eenDeshrIngenieure Duetscher Verein Zeitschrift 95 65 xeietlcmuainlcomparison, experimental-computational Archive aaeTheories Damage .L,T L., S. , .P,adC and P., Z. , . 1985, K., , 15. , . n P and M., , l . n K and H., , . S J., , 9128. , 235430. , . n K and I., , tal. et .A,1920, A., A. . 1986, H., , . 1999, T., , S . 1997, A., , . D M., , . X T., , . X T., , elinger elytschko 2 . n X and T., auskardt , a 2003, , ! harpe roya ,4 Physics / oevcnyi nieltodmninllattice), two-dimensional ideal an in vacancy (one 1/2 a iao nrdcint rcueMechanics Fracture to Introduction ,6 rolovitz owling iao aris rn.ASME Trans. nauss . Z D., , .L,W L., R. , awai edolin .N,adJ and N., W. , # .P,adR and P., S. , r a tutrlMechanics Structural p.J pl Phys. appl. J. Jpn. tesItniyFcosHandbook Factors Intensity Stress oeua Simulation Molecular 1–6. , ,8 hl rn.Ry Soc. Roy. Trans. Phil. .P,S P., S. , ¼ ¼ iao .H,K H., R. , . 2004, T., , .C,1967, C., P. , Ofr nvriyPress). University (Oxford a . 1998, C., , 2 2 .G,2003, G., W. , ! hang . 2003, S., , .F,T F., N. , s(ish19;seCritr 1997) Carpinteri see 1898; (Kirsch is ! ! r eotdi al n oprdwt F and QFM with compared and 2 table in reported are . 1991, L., , .J,1986, J., D. , 1 1 0, * þ ¼ ang . L S., , chatz ant , / r r uoff } 2 2 hm hs Lett. Phys. Chem. .ap.Mech appl. J. 2 2 adaan / A5. .G,adG and Z.-G., , C a .A.Crm Soc. Ceram. Am. J. n.J utsaeCmu.Engng Comput. Multiscale J. Int. opper . n R and A., , i n h da tegho h lmn ie,the (i.e., element the of strength ideal the and , ! n.J rc.Mech. Fract. J. Int. References .L,X J.-L., , þ . 02 ffcso eet nsrnt fnanotubes: of strength on defects of Effects 2002, R., , tblt fStructure of Stability cne at (center .C,adR and C., G. , , iclrholes Circular .Mc.Py.Solids Phys. Mech. J. ;(i for (ii) 1; 2 2 42 : . 2003, O., , hs Rev. Phys. , ! ! nfidApproach Unified A ! .H,adS and H., T. , 4120. , 23 1 1 , . A221 143. , þ þ .Assuming 0. iao itchie ., o lmsNtoa Laboratory National Alamos Los 3 3 . C S., , E24 x r r 163. , elbart 4 4 , 4 4 , y .Mcolcrmc.Syst. Microelectromech. J. B, NwYr:McGraw-Hill). York: (New 390 361. , .O,1995, O., R. , ¼ !1 o 2 cos ar , uoff Ofr:Pergamon). (Oxford: 4 37 , 81 413. , )i nifiiepaesubjected plate infinite an in 0) mith r . R R., , 3 5500. , .. 1991, W.M., , 2 2 773. , : 247. , # .S,2003, S., R. , v , Elastic fteeeetwt hole, a with element the of , o 2 cos .N,1980, N., K. , ¼ Lno:Camn&Hall). & Chapman (London: 51 / uoff # 1551. , C , # ¼ ! .ap.Phys. appl. J. Inelastic , .S,S S., R. , , ,trelmtcases limit three 0, / cs felasti- of (case 1/3 1 115. , .Ce.Phys. Chem. J. hs Rev. Phys. , chatz n.J Fract J. Int. / rcueand Fracture , C 12 , , 78 ! Preprint 302. , .C., G. , 3083. , 0.71. B, ., , ARTICLE IN PRESS

Computational Materials Science xxx (2007) xxx–xxx www.elsevier.com/locate/commatsci

Density functional calculations of response of single-walled armchair carbon nanotubes to axial tension

Ali Ebrahimi a,*, Hossein Ehteshami b, Marzie Mohammadi a

a Department of Chemistry, University of Sistan and Balouchestan, Zahedan, Iran b Department of Mechanical Engineering, University of Sistan and Balouchestan, Zahedan, Iran

Received 24 December 2006; received in revised form 25 March 2007; accepted 11 May 2007

Abstract

The response of single-walled armchair carbon nanotubes (SWACNTs) to axial tension was studied using density functional calculations. A new response causing an abrupt change in nanotube structure at specific strains was detected. Atom rearrangement results in a lower energy than expected. The geometry of armchair nanotube plays an important role in the observed response, with the effect of curvature being important. There is a meaningful relationship between rearrangement strain and nanotube diameter. Rearrangement can be explained using the Poisson effect, which increases with the lateral displacement and is inversely proportional to nanotube index number. Ó 2007 Elsevier B.V. All rights reserved.

PACS: 61.46.Fg; 31.15.Ew

Keywords: Armchair nanotube; Rearrangement; Tension; Response mechanism; Circumferential pressure; DFT

1. Introduction axial strains and very flexible against non-axial strains [2]. CNTs combine high stiffness with resilience and the Strength of materials emerges from strength of chemical ability to buckle and collapse in a reversible manner [3– bonds. Not only the strength of chemical bonds plays an 5]. Despite the very large strain, any bond breaking or important role in mechanical ability of materials but switching did not occur in nanotubes. Excellent mechanical arrangement of bonds is also important. For instance, response to severe deformation and strain is the main cause though the strength of C–C single (or double) bond is of attracting much experimental [5–14] and theoretical approximately equal in different substances, known allo- [1,3,4,15–27] investigations to them since their discovery tropes of carbon (namely diamond, graphite, and fullerene) in 1991. It was demonstrated that carbon nanotubes have have entirely different physical properties. Carbon nano- exceptional mechanical properties. Their excellent flexibil- tubes (CNTs) are very important members of fullerenes. ity during bending has been observed by experiment and In a cylindrical network of CNTs all atoms are equivalently simulated by theory [3–5]. Loading techniques, often based tied to the neighbors, and no ‘‘weak spot’’ is apparent. This on atomic force microscopy that combined with electron intrinsic uniformity, together with the known strength of microscopic imaging, allow one to measure the breaking- carbon bonds, must lead to extreme resistance to mechan- strain level and Young modulus, and to observe the overall ical tension [1]. The strong in-plane graphitic C–C bonds failure patterns experimentally [1]. At first, the elastic mod- make free defect CNTs that are strong and stiff against ulus in multi-walled carbon nanotubes (MWCNTs) has experimentally been determined by Treacy and co-workers [5]. It has been determined equal to 1.8 ± 0.9 TPa by * Corresponding author. Fax: +98 541 2446565. measuring thermal vibrations using TEM. Wong et al. E-mail address: [email protected] (A. Ebrahimi). [10] obtained the lower value 1.28 ± 0.59 TPa using the

Please cite this article in press as: A. Ebrahimi et al., Comput. Mater. Sci. (2007), doi:10.1016/j.commatsci.2007.05.006 ARTICLE IN PRESS

6 A. Ebrahimi et al. / Computational Materials Science xxx (2007) xxx–xxx

concept. The effect of tube diameter has also been examined. The results show that the rearrangement strain defers to higher values by the increase in the diameter of CNT. Qualitatively, we introduced a formula for circumferential pressure that can successfully predict the delay in rearrangement strain. References Fig. 4. (a) The situation of transverse tension in graphene. (b) A hexagon of graphene sheet and deformed structure undergoes transverse tension; L [1] T. Dumitrica, M. Hua, B.I. Yakobson, PNAS 103 (2006) 6105–6109. is the magnitude of lateral displacement. [2] D. Srivastava, C. Wei, K. Cho, Appl. Mech. Rev. 56 (2003) 215–230. [3] S. Iijima, C. Brabec, A. Maiti, J. Bernholc, J. Chem. Phys. 104 (1996) 2089–2092. armchair tubes directly exert a structural pressure on cir- [4] B.I. Yakobson, C.J. Brabec, J. Benholc, Phys. Rev. Lett. 76 (1996) cumferential bonds during tension due to Poisson effect. 2511–2514. At specific strains, structural pressure increases enough to [5] M.M.J. Treacy, T.W. Ebbesen, J.M. Gibson, Nature 381 (1996) 678– overcome resistant forces and rearrange the structure. 680. [6] J.P. Salvetat, J.M. Bonard, N.H. Thomson, A.J. Kulik, L. Forro, W. Physically, the pathway that the structure moves along Benoit, et al., Appl. Phys. A 62 (1999) 255–260. depends on strain rate, temperature, etc., which by the [7] J.P. Salvetat, G.A.D. Brriggs, J.M. Bonard, R.R. Bacsa, A.J. Kulik, way is not the subject of this work. It is deduced from T. Stockli, et al., Phys. Rev. Lett. 82 (1999) 944–974. Table 1 that the structural effect is in the reverse proportion [8] D.A. Walters, L.M. Ericson, M.J. Casavant, L. Liu, D.T. Colbert, to the nanotube index. On the other hand, rearrangement K.A. Smith, et al., Appl. Phys. Lett. 74 (1999) 3803–3805. [9] M.R. Falvo, G.J. Clary, R.M. Taylor II, V. Chi, F.P. Brooks Jr., S. strain defer to the higher values by the increase of nano- Washburn, et al., Nature 74 (1997) 582–584. tube diameter. It is well-known that in armchair nanotubes [10] E.W. Wong, P.E. Sheehan, C.M. Lieber, Science 277 (1997) 1971– the Poisson ratio decreases by the increase of diameter [37]. 1975. Another evident of reverse proportionality of tube index [11] M.-F. Yu, O. Lourie, K. Moloni, T. Kelly, R.S. Ruoff, Science 287 with structural effect is obtained by comparison between (2000) 637–640. the results of Mielke et al. [20] and Ogata and Shibutani [12] M.-F. Yu, B.S. Files, S. Arepalli, R.S. Ruoff, Phys. Rev. Lett. 84 (2000) 5552–5555. [38]. As can be seen in Fig. 4b, the deformed hexagon [13] H.E. Troiani, M. Miki-Yoshida, G.A. Camacho-Bragado, M.A.L. undergoes the transverse tension; L corresponds to lateral Marques, A. Rubio, J.A. Ascencio, et al., Nano. Lett. 3 (2003) 751– displacement. Obviously, the escalation of strain increases 755. the lateral displacement. As a criterion for a comparison [14] M.A.L. Marques, H.E. Troiani, M. Miki-Yoshida, M. Jose-Yac- aman, A. Rubio, Nano. Lett. 4 (2004) 811–815. between the rate of this effect and the diameter of nano- [15] P. Zhang, P.E. Lammert, V.H. Crespi, Phys. Rev. Lett. 81 (1998) tube, we established a simple formula: 5346–5349. L [16] E. Hernandez, C. Goze, P. Bernier, A. Rubio, Phys. Rev. Lett. 80 P / ð1Þ (1998) 4502–5405. n [17] G.G. Samsonidze, B.I. Yakobson, Phys. Rev. Lett. 88 (2002) 065501- where P is Poisson effect, L is lateral displacement and n is 1-4. nanotube index number. Our formula implies that (3,3) [18] Q.Z. Zhao, M.B. Nardelli, J. Bernholc, Phys. Rev. B 65 (2002) 144105-1-6. armchair nanotube is the first one to reach to adequate [19] G.G. Samsonidze, B.I. Yakobson, Comput. Mater. Sci. 23 (2002) 62– strain. (4,4), (5,5), (6,6) and (8,8) armchair nanotubes are 72. the latter ones, respectively. As can be seen in Table 1 [20] S.L. Mielke, D. Troya, S. Zhang, J.L. Li, S.P. Xiao, R. Car, et al., and Fig. 1, there is a good agreement between qualitative Chem. Phys. Lett. 390 (2004) 413–420. prediction of formula and calculations. [21] T. Belytschko, S.P. Xiao, G.C. Schatz, R. Ruoff, Phys. Rev. B 65 (2002) 235430-1-8. [22] M.B. Nardelli, B.I. Yakobson, J. Bernholc, Phys. Rev. Lett. 81 (1998) 4656–4659. 4. Conclusions [23] L.G. Zhou, S.Q. Shi, Appl. Phys. Lett. 83 (2003) 1222–1224. [24] B.I. Yakobson, Appl. Phys. Lett. 72 (1998) 918–920. A new response mechanism was detected along tension [25] T. Dumitrica, T. Belytschko, B.I. Yakobson, J. Chem. Phys. 118 of single-walled armchair carbon nanotubes by density (2003) 9485–9488. functional calculation using B3LYP method. Structural [26] D. Troya, S.L. Mielke, G.C. Schatz, Chem. Phys. Lett. 382 (2003) 133–141. changes are not topological, but help nanotube to bear [27] T. Natsuki, M. Endo, Carbon 42 (2004) 2147–2151. more strain. In this mechanism, an important factor is [28] B.G. Demczyk, Y.M. Wang, J. Cumings, M. Hetman, W. Han, A. the changes of bond lengths that are precisely aligned with Zettl, R.O. Ritchie, Mater. Sci. Eng. A 334 (2002) 173–178. circumference. These bonds are physically in correlation [29] D. Qian, W.K. Liu, M. Yu, R.S. Ruoff, Appl. Mech. Rev. 55 (2002) with longitudinal bonds. This behavior can obviously be 495–533. [30] M.S. Dresselhaus, G. Dresselhaus, A. Jorio, Annu. Rev. Mater. Res. seen at a specific strain. In relationship with Poisson effect, 34 (2004) 247–278. a structural effect is also attended which is called circumfer- [31] J. Bernholc, D. Brenner, M.B. Nardelli, V. Meunier, C. Roland, ential pressure. This mechanism can be explained by this Annu. Rev. Mater. Res. 32 (2002) 347–375.

Please cite this article in press as: A. Ebrahimi et al., Comput. Mater. Sci. (2007), doi:10.1016/j.commatsci.2007.05.006 DOI: 10.1002/adma.200700776 Ultrastrong, Stiff, and Lightweight Carbon-Nanotube Fibers**

By Xiefei Zhang, Qingwen Li, Terry G. Holesinger, Paul N. Arendt, Jianyu Huang, P. Douglas Kirven,

COMMUNICATION Timothy G. Clapp, Raymond F. DePaula, Xiazhou Liao, Yonghao Zhao, Lianxi Zheng, Dean E. Peterson, and Yuntian Zhu*

From the stone ages to modern history, new materials have assessed by a material’s specific strength and specific stiffness, often been the enablers of revolutionary technologies.[1] For a which are defined as the strength or stiffness (Young’s modu- wide variety of envisioned applications in space exploration, lus) of a material divided by its density.[9] The combination of energy-efficient aircraft, and armor, materials must be signifi- high strength, high stiffness, and low density affords CNTs cantly stronger, stiffer, and lighter than what is currently with extremely high values for specific strength and specific available. Carbon nanotubes (CNTs) have extremely high stiffness. The most effective way to utilize these properties is strength,[2–5] very high stiffness,[6,7] low density, good chemical to assemble CNTs into fibers. However, despite extensive stability, and high thermal and electrical conductivities.[8] worldwide efforts to date, the specific strength and specific These superior properties make CNTs very attractive for stiffness of CNT fibers that have been reported by various re- many structural applications and technologies. Here we report search groups are much lower than currently available com- CNT fibers that are many times stronger and stiffer per mercial fibers.[10–22] In early studies, researchers attempted to weight than the best existing engineering fibers and over reinforce polymer fibers with short CNTs, but the reinforce- twenty times better than other reported CNT fibers. Addi- ment was limited by several issues, including poor dispersion, tionally, our CNT fibers are nonbrittle and tough, making poor alignment, poor load transfer, and a low CNT volume them far superior to existing materials for preventing cata- fraction.[10–15] Recently, pure CNT fibers (also called yarns) strophic failure. These new CNT fibers will not only make were reported with and without twisting.[16–22] For example, tens of thousands of products stronger, lighter, safer, and Zhang et al.[20] demonstrated that spinning from aligned CNT more energy efficient, but they will also bring to fruition many arrays could significantly improve the strength of CNT fibers envisioned technologies that have been to date unavailable by twisting them. However, to date no breakthrough has been because of material restrictions. reported in the specific strength and specific stiffness of CNT Strong, stiff, and lightweight are critical property require- fibers. ments for materials that are used in the construction of space Here we report CNT fibers with values for specific strength shuttles, airplanes, and space structures. These properties are and specific stiffness that are much higher than values reported for any current engineering fibers as well as previously reported CNT fibers. As shown in Figure 1, the specific strength – 200 [*] Dr. Y. T. Zhu, Dr. X. F. Zhang,[+] Dr. Q. W. Li,[+] Dr. T. G. Holesinger, Dr. P. N. Arendt, R. F. Depaula, Dr. Y. H. Zhao, Dr. L. X. Zheng, Dr. D. E. Peterson

Los Alamos National Laboratory 6 150 Los Alamos, NM 87545 (USA) E-mail: [email protected] Dr. J. Y. Huang Sandia National Laboratory 100 Albuquerque, NM 87123 (USA) P. D. Kirven Sigma-K Corporation 50 T1000 511 Clayton Rd, Durham, NC 27703 (USA) Prof. T. G. Clapp M70J Textile Engineering 0 North Carolina State University, Raleigh, NC 27606 (USA) 0 20 40 60 80 100 120 140 Dr. X. Z. Liao 8 School of Aerospace, Mechanical, and Mechatronic Engineering Specific Stiffness, E/ρ (10 cm) University of Sydney Sydney, NSW 2006 (Australia) Figure 1. Comparison of the specific strength and specific stiffness (stiff- [+] These authors contributed equally to this work. ness is defined as Young’s modulus) of our CNT fibers (filled circles) to [**] This project is supported by the Laboratory Directed Research & De- other existing engineering fibers (unfilled circles) [9], the strongest and velopment program of the Los Alamos National Lab. Supporting in- stiffest carbon fibers (filled squares) [23], and CNT fibers reported pre- formation is available online from Wiley InterScience or from the viously (filled diamonds) [20–22]. For more information on these data author. points, see Figure S1 in the Supporting Information.

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