Elastic properties of carbon nanotubes and nanoropes are investigated using
an empirical force-constant model. For single and multi-wall nanotubes the
elastic moduli are shown to be insensitive to details of the structure such as
the helicity, the tube radius and the number of layers. The tensile Young's
modulus and the torsion shear modulus calculated are comparable to that of the
diamond, while the the bulk modulus is smaller. Nanoropes composed of
single-wall nanotubes possess the ideal elastic properties of high tensile
elastic modulus, flexible, and light weight.Comment: 10 page

Lu, J. P.

English

arXiv

Elastic properties of carbon nanotubes and nanoropes are investigated using an empirical force-constant model. For single and multi-wall nanotubes the elastic moduli are shown to be insensitive to details of the structure such as the helicity, the tube radius and the number of layers. The tensile Young's modulus and the torsion shear modulus calculated are comparable to that of the diamond, while the the bulk modulus is smaller. Nanoropes composed of single-wall nanotubes possess the ideal elastic properties of high tensile elastic modulus, flexible, and light weight

Lu, Jian Ping

Carolina Digital Repository

arXiv:cond-mat/9704219v1  [cond-mat.mtrl-sci]  27 Apr 1997Elastic Properties of Carbon Nanotubes and NanoropesJian Ping LuDepartment of Physics and AstronomyUniversity of North Carolina at Chapel HillChapel Hill, North Carolina 27599jpl@physics.unc.edu(February 7, 2008)AbstractElastic properties of carbon nanotubes and nanoropes are investigated usingan empirical force-constant model. For single and multi-wall nanotubes theelastic moduli are shown to be insensitive to details of the structure such asthe helicity, the tube radius and the number of layers. The tensile Young’smodulus and the torsion shear modulus calculated are comparable to that ofthe diamond, while the the bulk modulus is smaller. Nanoropes composed ofsingle-wall nanotubes possess the ideal elastic properties of high tensile elasticmodulus, flexible, and light weight.PACS numbers: 61.46+w, 36.40+dTypeset using REVTEX1I. IntroductionThe discoveries of carbon nanotubes1 and the new efficient method of producing them2stimulate a great interest in these novel materials. The electronic3 and magnetic properties4of nanotubes depend sensitively on the structural details such as the tube radius and thehelicity. It has been speculated that nanotubes also posses novel mechanical properties.Recent measurements have inferred a Young’s modulus that is several times that of thediamond.5The mechanical properties of small single-wall nanotubes have been studied by severalgroups using molecular dynamics simulations.6,7 A Young’s modulus several times greaterthan that of the diamond was predicted. However, those calculations were restricted to smallsingle-wall nanotubes of few Å in radius. Most samples of nanotubes are either multi-wallor crystalline ropes of single-wall nanotubes.A practical method of investigating elastic properties is to use the empirical force-constant model. The phonon spectrum and elastic properties of the graphite has beensuccessfully calculated using such models.8 The similarity in local structure between thegraphite and the nanotubes ensure that a similar model is applicable for nanotubes. Theadvantage of such a model is that it can be easily applied to nanotubes of different size,helicity, and number of layers. One such model has been used to predict the phonon spec-trum of small single-wall nanotubes.9 Here we present results of applying a similar model tocalculate elastic properties of single and multi-wall nanotubes of various size and geometry,and that of crystalline nanoropes composed of single-wall nanotubes.II. The Force-constant ModelIn an empirical force-constant model, the atomic interactions near the equilibrium struc-ture are approximated by the sum of pair-wise harmonic potentials between atoms. In themost successful model for the graphite, interactions up to fourth-neighbor in-plane and out-of-plane interactions are included.8 The force constants are empirical determined by fittingto measured elastic constants and phonon frequencies.2The local structure of a nanotube layer can be constructed from the conformal mappingof the graphitic sheet on to a cylindrical surface. For a typical nanotube of few nm in radius,the curvature is small enough that one expects short-range atomic interactions to be veryclose to that in the graphite. Thus, we adopt the same parameters developed by Al-Jishi etal.8 for graphite for intra-plane interactions in all nanotubes.The different layers in a multi-wall nanotube are not well registered as they are in thesingle crystal graphite. Thus, one can not adopt the same set of parameters for the interlayerinteractions. Instead, we model the interlayer interactions in nanotube by the summation ofpair-wise van de Waals interactions, U(r) = 4ǫ ((σ/r)12 − (σ/r)6)). Such a model has beenused successfully to calculate the bulk properties of C60 solid.10 The van de Waals parameterσ = 3.4Å , ǫ = 12meV, were determined by fitting the interlayer distance and the elasticconstant c33 of the single crystal graphite.11III. Single-Layer NanotubesFollowing the notation of White et al.12, each single-layer nanotube is indexed by a pairof integers (n1, n2), corresponding to a lattice vector L = n1a1 + n2a2 on the graphiteplane, where a1, a2 are the graphite plane unit cell vectors. The structure of the nanotubeis obtained by the conformal mapping of a graphite strip onto a cylindrical surface. Thenanotube radius is given by R = ao√3(n21 + n22 + n1n2)/2π, where ao = 1.42Å is the C-Cbond length.In principle, force constants depend on the size of the nanotube as overlaps of π orbitalschange with the nanotube curvature.13 However, Such dependence is very weak. In thispaper, we neglect this effect and concentrate on the dependence of elastic properties on thegeometry and interlayer interactions.The elastic constants are calculated from the second derivatives of the energy densitywith respect to various strains.14 The tensile stiffness as measured by Young’s modulus isdefined as the stress/strain ratio when a material is axially strained. For most materials, theradial dimension is reduced when it is axially elongated. The ratio of the reduction in radial3dimension to the axial elongation defines the Poisson ratio ν. We first calculate the Poissonratio by minimizing the strain energy with respect to both the radial compression and theaxial extension. The Young’s modulus Y is then calculated from the second derivative ofthe strain energy density with respect to the axial strain at the fixed ν.Table.1 lists the bulk, Young’s and shear (referred to the torsional shear) moduli calcu-lated for selective examples of single-wall nanotubes. An important quantity in determin-ing values of elastic constants is the wall thickness h of nanotubes. Previous calculationshas taken h = 0.66Å for single layer nanotubes which leads to the unusual large Young’smodulus.7 For multi-wall nanotubes, all experiments indicate that the interlayer distance isthe same as that in the graphite, h = 3.4Å. Thus, it is reasonable to take the interlayerdistance h = 3.4Å as the wall thickness. We use the same values for all single multi-wallnanotubes. This enables us to compare results across nanotubes of different size and numberof layers. For comparison, elastic moduli of the graphite11 and that of the diamond14 arealso listed.Examine the numbers in Table.1 one concludes that: (1) Elastic moduli are insensitiveto the size and the helicity. (2) The Young’s and shear moduli of nanotubes are comparableto that of the diamond and that if in-plane graphite. (3) Single-wall nanotubes are stiff inboth the axial direction and the basal plane.VI. Multi-wall NanotubesThe interlayer distance in all experimentally observed multi-wall nanotubes is compa-rable to that in graphite. This puts a constrain on possible combinations of single-wallnanotubes to form multi-wall nanotubes. We have calculated elastic moduli for many differ-ent combinations. It is found that elastic properties are insensitive to different combinationsas long as the constrain – interlayer distance ≈ 3.4Å – is satisfied. Because of this insen-sitivity we use results for one series of multi-wall nanotubes to illustrate our main points.The series chosen is constructed from (5n, 5n), n = 1, 2, 3 · · · single-wall tubes. This is oneof the most likely structure for multi-wall tubes as its interlayer distance is very close to4that actually observed.15Table.2 lists the calculated elastic coefficients and the bulk, Young’s, and shear modulusfor this series of nanotubes up to 10 layers. The experimental values for the graphite and thediamond are also listed for comparison. One observes that the elastic moduli are essentiallyindependent of the number of layers. The same is true for all other multi-wall nanotubes wehave calculated. From Table.2 and its comparison with Table.1 one concludes: (1)The elasticmoduli vary little with the number of layers. (2)The interlayer van de Waals interactionscontribute less than 10% to the elastic moduli of multi-wall nanotubes.The Young’s modulus of multi-wall nanotubes was deduced recently by Treacy et al.5from the thermal vibrations of anchored tubes. Their values range from 0.4 to 4 TPa withthe average values of 1.6 TPa. These results are substantial larger than our calculatedvalues of 1 TPa. The discrepancy may be due to the large uncertainty in how to estimatethe Young’s modulus from their experiments. In their estimation the isotropic model wasassumed, our results clearly show that this is not true. More recent direct measurementof the multi-wall nanotubes using the AFM technique and the Euler’s buckling criteria hasyield a result of 1 TPa,16 in agreement with our calculations.V. Crystalline NanoropesThe discovery of a new efficient method of producing bulk quantity of single-wall nan-otubes has made it possible to make crystalline ropes of nanotubes.2 These nanoropes consistof 100 to 500 single-wall nanotubes of uniform size. Due to the weak inter-tube interactionsone expects these rope to be flexible in the basal plane, yet very stiff along the axial direction.We use same model described above to calculate the lattice constant and elastic moduliof these nanoropes. Table.3 summarize the bulk properties of nanoropes with nanotubesradius ranging from 1nm (the (5,5) tube) to 2nm (the (13,13) tube). Due to extremedisparity between the inter-tube and intra-tube interactions, we have neglected the couplingbetween the two interactions. Thus, the lattice constant a0 and the cohesive energy E0 aredetermined by inter-tube van de Waals interaction only. It is found that a0 and the cohesive5energy per atom scales with the tube radius R as a0 = 2R+3.2Å , E0 = 61.5(meV )/√R(Å ).For nanorope composed of the typical (10,10) tube, R = 6.78Å , a0 = 16.8Å , E0 = 23meV.The cohesive energy of the rope is comparable to that of the C60 solid (33mev).From Table.3 one observes that: (1) Nanorope is very anisotropic. (2) The basal planeis soft, while the axial direction is very stiff. (3) The C33 is about half that of the diamond.The weak inter-tube interaction ensures that the rope is flexible as individual tubes canrotate and slide with respect to each other easily. This is supported by the experimentalSEM images where long nanoropes are observed to be well bended and tangled.2VI. ConclusionsWe have investigated elastic properties of nanotubes and nanoropes using an empiricalforce constant model. The simplicity of the model enable us to explore the dependence ofelastic moduli on the nanotube geometry. It is shown that elastic properties are insensitiveto the radius, helicity, and the number of layers. The calculated Young’s modulus (∼ 1TPa) and shear modulus (∼ 0.5 TPa) are comparable to that of diamond for both singleand multi-wall individual nanotubes. Crystalline rope of nanotubes is very anisotropic inits elastic properties – soft on basal plane and stiff along the axial direction. The largeYoung’s modulus and flexibility of nanoropes make them ideal materials for nanometerscale engineering.Acknowledgments This work is supported by a grant from U.S. Department of Energy,and in part by a grant from The Petroleum Research Foundation.6REFERENCES1 S. Iijima, Nature 354, 56 (1991). T. W. Ebbesen and P. M. Ajayan, Nature 358, 220(1992).2 A. Thess et al., Science 273, 483 (1996).3 N. Hamada et al., Phys. Rev. Lett. 68, 1579 (1992); L. Wang et al., Phys. Rev. B 46,7175 (1992); Saito et at., ibid 46, 1804 (1992).4 J. P. Lu, Phys. Rev. Lett. 74, 1123 (1995).5 M. M. Treacy, T. W. Ebbesen, and J. M. Gibson, Nature 381, 678 (1996).6 D. H. Robertson, D.W. Brenner, and J.W. Mintmire, Phys. Rev. B. 45, 12592 (1992).7 B. I. Yakobson, C.J. Brabec, and J. Bernholc, Phys. Rev. Lett. 76, 2511 (1995).8 R. Al-Jishi and G. Dresselhaus, Phys. Rev. B. 26, 4514 (1982).9 R. Al-Jishi et. al., J. Chem. Phys. 209, 77 (1993).10 J. P. Lu et al., Phys. Rev. Lett. 68, 1551 (1992). X. P. Li et al., Phys. Rev. B 46, 4301(1992).11 O.L. Blakslee et al., J. Appl. Phys. 41, 3373 (1970). E.J. Seldin and C.W. Nezbeda, J.Appl. Phys. 41, 3389 (1970).12 C. T. White et al., Phys. Rev. B 47, 5485 (1993).13 J. P. Lu and W. Yang, Phys. Rev. B 49, 11421 (1994).14 J.B. Huntington, in Solid State Physics 7, ed. by F. Seitz and D. Turnbull, AcademicPress (1958).15 R.S. Ruoff et al., Nature 364, 514 (1993).16 H. Dai et al., Nature 384, 147 (1996).7TABLESTABLE I. Elastic moduli of selective single-wall nanotubes. (n1, n2) – index, R – radius innm. B,Y,M are bulk, Young’s and shear modulus in units of TPa (1013dy/cm2). ν is the Poissonratio. Experimental values for the graphite and the diamond are listed for comparison.(n1, n2) R B Y M ν(5,5) 0.34 0.191 0.971 0.436 0.280(6,4) 0.34 0.191 0.968 0.437 0.284(7,3) 0.35 0.191 0.968 0.454 0.284(8,2) 0.36 0.190 0.974 0.452 0.280(9,1) 0.37 0.191 0.968 0.465 0.284(10,0) 0.39 0.192 0.968 0.451 0.282(10,10) 0.68 0.191 0.972 0.457 0.278(50,50) 3.39 0.192 0.969 0.458 0.282(100,100) 6.78 0.192 0.969 0.462 0.282(200,200) 13.56 0.192 0.969 0.478 0.282Graphitea 0.0083 1.02 0.44 0.16Graphiteb 0.0083 0.0365 0.004 0.012Diamondc 0.442 1.063 0.5758 0.1041a Graphite along the basal plane.11b Graphite along the C axis.11c Diamond along the cube axis.148TABLE II. Elastic coefficients and moduli (in TPa) of multi-wall nanotubes constructed fromthe (5n, 5n), n = 1, 2, 3 · · · series of single-wall tubes. N – number of layers, R – radius of theout-most layer in nm. B,Y,M are bulk, Young’s and shear modulus (in TPa). Values for thegraphite and the diamond are listed for comparison.n R C11 C33 C44 C66 C13 Y M B1 0.34 0.397 1.05 0.189 0.134 0.147 0.97 0.436 0.1912 0.68 0.412 1.13 0.189 0.137 0.146 1.05 0.455 0.1943 1.02 0.413 1.15 0.189 0.138 0.146 1.08 0.464 0.1944 1.36 0.412 1.17 0.189 0.138 0.146 1.09 0.472 0.1945 1.70 0.411 1.18 0.189 0.139 0.146 1.10 0.481 0.1946 2.03 0.411 1.18 0.189 0.139 0.146 1.10 0.491 0.1947 2.37 0.410 1.18 0.189 0.139 0.146 1.11 0.502 0.1948 2.71 0.410 1.19 0.189 0.139 0.146 1.11 0.514 0.1949 3.05 0.410 1.19 0.190 0.139 0.146 1.11 0.527 0.19410 3.39 0.410 1.19 0.190 0.139 0.146 1.11 0.541 0.194Graphitea 1.06 0.036 0.004 0.440 0.015 1.02 0.008 0.440Diamond 1.07 1.07 0.575 0.575 0.125 1.06 0.442 0.575aYoung’s and shear moduli refer to the basal plane.9TABLE III. Lattice constant a0 (nm), cohesive energy per atom E0 (meV), and elastic moduli(in TPa) of crystalline nanorope made of single-wall (n, n) tubes. R(nm) is the radius of single-walltube.n R a0 E0 C11 C12 C335 0.33 0.99 33.5 0.066 0.022 0.7956 0.40 1.13 30.1 0.071 0.024 0.7367 0.47 1.26 28.2 0.078 0.024 0.6878 0.54 1.40 26.2 0.082 0.029 0.6419 0.61 1.54 24.7 0.085 0.029 0.60010 0.67 1.67 23.5 0.090 0.032 0.56311 0.74 1.81 22.5 0.098 0.035 0.53212 0.81 1.94 21.6 0.102 0.036 0.50213 0.88 2.08 20.7 0.106 0.036 0.47514 0.94 2.21 19.9 0.111 0.042 0.45215 1.01 2.35 19.3 0.118 0.043 0.43010

Elastic Properties of Carbon Nanotubes and Nanoropes

Abstract

Similar works

Full text

Available Versions

Carolina Digital Repository