Carbon Nanotubes as Thin ShellsThin shell models for predicting the mechanical behavior of carbon nanotubes (CNTs) have been developed with excellent results, however, the small size of CNTs leads to some notable issues. Because CNTs have dimensions on the order of nanometers, continuum models are not strictly applicable, and certain assumptions must be made in order to utilize them. The motivation for using a continuum approach is that it is much simpler and computationally less demanding than atomistic approaches.

Introduction

Carbon NanotubesIn 1991, carbon nanotubes (CNTs), cylindrical allotropes of carbon, were discovered using carbon arc discharge [1]. This new form of carbon exhibits excellent physical, electrical, and thermal properties and has been the subject of a growing amount of research. Made from a single sheet of graphene (a sheet of hexagonally patterned carbon, 1 atom thick) rolled into a cylinder, CNTs have diameters as small as 0.5 nm and aspect ratios as high as 132,000,000:1 [2]. There are generally two classes of CNTs: single wall nanotubes (SWNTs), which are comprised of a single cylinder of graphene, and multiwall nanotubes (MWNTs), which include several concentric SWNTs stacked within each other. Figure 1 shows a schematic of the two types of CNTs.

Figure 1: Single wall and Multiwall nanotubes [3]

There are three general classes of CNT patterns: armchair, zigzag, and chiral. Figure 2 shows a schematic of the three patterns of CNTs. This pattern plays an especially important role in determining the electrical properties of CNTs.

Figure 2: (A) armchair, (B) zigzag, and (C) chiral CNTs [4]

While the procedure for producing CNTs is very different from that of graphene, one can think of CNTs as being cut from a sheet of graphene then rolled into a cylinder. The angle of the cut, called the chiral angle, determines the pattern of the nanotube as shown in Figure 3.

Figure 3: Chiral angles [5]

The chiral vector can be described by

C⃗h=n a⃗1+m a⃗2

where the integers (m ,n ) are the number of steps along the zigzag carbon bonds of the hexagonal lattice in their respective directions [5]. A chiral angle of 0o, or equivalently a rollup vector of (n ,0 ), results in a zigzag pattern. A chiral angle of 30o and rollup vector of (n ,n ) results in an armchair pattern, while all other angles result in what are called chiral patterns [5].

Experimental MethodsCarbon nanotubes have excellent mechanical properties that vary depending on tube dimensions, number of defects, whether they are SWNTs or MWNTs, and sometimes chirality. Many studies have been done trying to measure the mechanical properties of CNTs and, because of the challenges associated with the size of CNTs, there is fairly wide scatter among the data. Young’s modulus is generally reported to be around 1 TPa, and strength has been reported from 13 to 150 GPa with most estimates being around 45 GPa [5-16]. Poisson’s ratio of a CNT has been estimated to be 0.16 [17].

Because CNTs are so small, it is difficult to directly measure their mechanical properties in conventional ways but many different strategies have been used to make reliable estimates. The first measurement was made by Treacy et al. [6] by observing, in a TEM, the amplitude of thermal vibrations of cantilevered CNTs. This resulted in an average Young’s modulus of 1.8 TPa, but with significantly scattered data. Since then, the same method was improved upon by Krishnan et al.[8], getting an average Young’s modulus of 1.25 TPa with much less scatter. Both of these methods assumed a solid homogenous cylindrical cantilever with length L and inner and outer radii a and b respectively.

Another technique was to bend the tube with an AFM and measure the deflection of the cantilevered CNT. This technique resulted in average Young’s moduli of 1.28 TPa in low defect MWNTs [9] and 0.81 TPa in high defect SWNT [10]. The strength can also be measured in this way, but failure is due to buckling rather than breaking of atomic bonds. Strength was estimated at 28.5 GPa using this method but it should be noted that this value is likely lower than the tensile strength [9]. Another group measured the strain on the outer wall of a bent nanotube to calculate a strength of 150 GPa [13].

Yu et al. directly performed tensile tests on MWNTs in a SEM to find a Young’s modulus from 0.27 to 0.95 TPa and a strength ranging from 11 to 63 GPa [11]. Another group immersed CNTs in a matrix material in order to subject it to tensile loads [14]. They found an estimated strength of 45 GPa. A SWNTs rope was exposed to a sideways pull to calculate at strength of at least 45 GPa [15]. With a combination of tensile and bending tests, another group estimated the strength to be 150 GPa [16].

CNTs have also shown extraordinary resilience, enduring significant deformations and changes to their shape without permanent atomic rearrangements [12]. They have been found to develop kinks or flatten into ribbons, and subsequently return to their original shape without damage.

Continuum ModelYakobson, Brabec, and Bernholc [18] are generally credited for creating the framework for a thin shell method of analyzing SWNTs. This method is referred to as the YBB method and it has shown to be very good at predicting CNT behavior, especially under axial compression, bending, and torsion. They verified their model using molecular dynamics (MD) simulations.

Axial LoadingIn axial loading, buckling can be predicted by a continuum approach with appropriate choice of parameters. Since the properties of a nanotube are essentially isotropic, they can be approximated by a uniform shell with only two elastic parameters: flexural rigidity, D, and in-plane stiffness, C. The energy of a shell is given by the surface integral of the quadratic form of local deformation [19] according to

E=12∫∫{D [(κ x+κ y )2−2 (1−v ) (κ xκ y−κ xy2) ]+ C

1−υ2[( εx+ε y )2−2 (1−v ) (ε xε y−ε xy2 )]}dS

where κ is the curvature variation, ε is the in-plane strain, and x and y are local coordinates. The values of D and C can be determined by comparison between ab initio and semiempirical studies of nanotube energetics at small strains [18]. Using data from [20], Yakobson et al. found that C=59 eV/atom and D=0.85 eV. Using the standard relations D=Y h3/12(1−υ2) and C=Yh, Jakobson et al. found an elastic modulus of 5.5 TPa and a wall thickness of 0.066 nm [18].

The YBB model assumes a perturbation in the form of Fourier harmonics, with M azimuthal lobes and N half waves along the tube as shown in Figure 4.

Figure 4: Example of buckling pattern. Here (M,N) is (4,4). [18]

This can be expressed as a series of sine and cosines of arguments 2My /d and Nπx /L. At a critical strain, the energy variation drops to zero for this shape disturbance and the cylinder becomes unstable, reducing its energy by assuming an (M ,N )pattern. For a nanotube with a diameter of 1 nm and length greater than 10 nm, the critical strain occurs at M=N=1 and the tube buckles sideways as a whole, preserving its circular cross section. The critical strain is similar to that of a simple rod, as shown by

ε c=12

(πd /L )2

For shorter tubes the lowest critical strain corresponds to M=2 and the critical strain shows little dependence on length and can be estimated by

ε c=4 √D /C d−1

which for a nanotube becomes

ε c=(0.077nm )d−1

For a nanotube of length 6 nm, the lowest critical strain occurs for M=2 and N=2 or 3 and are very close to the value obtained by the MD simulation. Figure 5 shows the strain energy (normalized by its second derivative) as a function of strain.

Figure 5: Strain energy, normalized by second derivative of strain energy, with respect to strain of a MD simulation of a SWNT of length 6 nm with 1nm diameter and armchair chirality under compressive axial

load [18]

Figure 6 shows the corresponding buckling shape changes that occur at points b, c, d, and e under axial compression.

Figure 6: Buckling shapes resulting from MD simulation of a SWNT of length 6 nm with 1nm diameter and armchair chirality under

compressive axial load [18]

BendingIn bending, only one side of the cylinder is compressed and can buckle. It can be assumed that it buckles when the local strain is close to the critical strain of axial compression. The local strain is given by ε=Kd /2 where K is the local curvature. Accordingly, the local curvature is given by

K= (0.155nm)d−2

which is within 4% of the MD simulations of SWCNTs. The strain energy curve and its derivative (both normalized to the second derivative) is shown in Figure 7.

Figure 7: Strain energy, normalized by second derivative of strain energy, with respect to bend angle of a MD simulation of a SWNT 8 nm

long, 2 nm in diameter, with chirality (13, 0) under bending load [18]

and the shape of the buckling due to bending is given in Figure 8.

Figure 8: Buckling shapes resulting from MD simulation of a SWNT 8 nm long, 2 nm in diameter, with chirality (13, 0) under bending load [18]

TorsionThe continuum analysis of torsion is similar to that of axial loading and bending but it involves skew harmonics of the form Nπx /L±2My /d . For overall beam buckling (M=1 ), the critical azimuthal angle between the tube ends, ϕc, is given by

ϕc=2 (1+υ )π

and tube flattening (M=2 ) occurs at

ϕc=(0.055nm3 /2 ) Ld−5 /2

For L≤136d5 /2 tube flattening should occur first. This was true for all of the MD simulations undertaken by Yakobson et al. but the values were significantly higher than predicted by the thin shell calculations. The circular ends of the nanotube were assumed to remain circular in the MD simulation, which is justifiable since most nanotubes are capped with a hemisphere, which is essentially half of a fullerene, which greatly increases the radial stiffness.

Figure 9 shows the strain energy curve and its derivative (both normalized to the second derivative) as calculated in the MD simulation.

Figure 9: Strain energy, normalized by second derivative of strain energy, with respect to torsion angle of a MD simulation of a SWNT 23

nm long, 1 nm in diameter, with chirality (13, 0) under torsion [18]

Figure 10 shows the shapes formed by flattening and by sideways buckling.

Figure 10: Buckling shapes resulting from MD simulation of a SWNT 23 nm long, 1 nm in diameter, with chirality (13, 0) under torsion [18]

Yakobson’s ParadoxWhile the YBB method is excellent at predicting many different behaviors of nanotubes, especially in compression, bending, and torsion, it also predicts an elastic modulus of about 5 TPa, which is significantly higher than the generally accepted 1 TPa and thickness of 0.066 nm, which is much less than the diameter of a carbon atom. This anomaly is known as Yakobson’s paradox and is thought to be due to the ill-defined wall thickness [21].

Molecular dynamics approaches require one to assume the thickness in order to determine the elastic modulus. This is problematic since the thickness of an atomic monolayer is ambiguous. Table 1 shows some of the assumed thicknesses used by different groups and the resulting moduli.

Table 1: Calculated thickness and elastic modulus of CNTs [18, 21-33]

One approach is to use the interlayer atomic spacing of graphite (0.34 nm) as the wall thickness of CNTs [21, 23, 25, 26]. Another approach is to model the CNT as a thin shell and fitting the atomistic results to get a thickness ranging from 0.06 nm to 0.09 nm [18, 28-33]. The range of thicknesses arrived at using this method can be attributed to the fact that the thickness calculated depends on the method of loading [21]. For uniaxial tension, the thickness is around 0.06 nm [18, 30, 33] while for equibiaxial stretching it is about 0.09 nm [28]. Note that the groups that chose a thickness of 0.34 nm calculate an elastic modulus of about 1 TPa while the groups that use a thickness between 0.06 and 0.09 nm determined the elastic modulus to be five times higher.

ConclusionsThe YBB thin shell model accurately and repeatably predicts the buckling behavior of carbon nanotubes with a very low computational cost. The results match up well with both experimental results and atomistic predictions. The thin shell model, however, provides ambiguous results when determining the mechanical properties. Since the elastic modulus of a hollow tube depends on the thickness of the wall, and the thickness of a monatomic sheet does not have an exact and meaningful value, the elastic modulus has a wide range of predicted values based on how one chooses to define the wall thickness.

Works Cited

1. Iijima, S., Helical microtubules of graphitic carbon. nature, 1991. 354(6348): p. 56-58.2. Wang, X., et al., Fabrication of ultralong and electrically uniform single-walled carbon nanotubes

on clean substrates. Nano Letters, 2009. 9(9): p. 3137-3141.3. Pearson, C., The Role of Chemistry in History, 2007.4. Baughman, R.H., A.A. Zakhidov, and W.A. de Heer, Carbon nanotubes--the route toward

applications. Science, 2002. 297(5582): p. 787-792.5. Thostenson, E.T., Z. Ren, and T.-W. Chou, Advances in the science and technology of carbon

nanotubes and their composites: a review. Composites science and technology, 2001. 61(13): p. 1899-1912.

6. Treacy, M.M.J., T.W. Ebbesen, and J.M. Gibson, Exceptionally high Young's modulus observed for individual carbon nanotubes. Nature, 1996. 381(6584): p. 678-680.

7. Poncharal, P., et al., Electrostatic deflections and electromechanical resonances of carbon nanotubes. Science, 1999. 283(5407): p. 1513-1516.

8. Krishnan, A., et al., Young’s modulus of single-walled nanotubes. Physical Review B, 1998. 58(20): p. 14013.

9. Wong, E.W., P.E. Sheehan, and C.M. Lieber, Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes. Science, 1997. 277(5334): p. 1971-1975.

10. Salvetat, J.-P., et al., Elastic and shear moduli of single-walled carbon nanotube ropes. Physical Review Letters, 1999. 82(5): p. 944.

11. Yu, M.-F., et al., Strength and breaking mechanism of multiwalled carbon nanotubes under tensile load. Science, 2000. 287(5453): p. 637-640.

12. Dresselhaus, M.S., et al., Carbon nanotubes. 2000: Springer.13. Falvo, M., et al., Bending and buckling of carbon nanotubes under large strain. Nature, 1997.

389(6651): p. 582-584.14. Wagner, H., et al., Stress-induced fragmentation of multiwall carbon nanotubes in a polymer

matrix. Applied Physics Letters, 1998. 72(2): p. 188-190.15. Walters, D., et al., Elastic strain of freely suspended single-wall carbon nanotube ropes. Applied

Physics Letters, 1999. 74(25): p. 3803-3805.16. Demczyk, B., et al., Direct mechanical measurement of the tensile strength and elastic modulus

of multiwalled carbon nanotubes. Materials Science and Engineering: A, 2002. 334(1): p. 173-178.

17. Muster, J., et al., Scanning force microscopy characterization of individual carbon nanotubes on electrode arrays. Journal of Vacuum Science & Technology B: Microelectronics and Nanometer Structures, 1998. 16(5): p. 2796-2801.

18. Yakobson, B.I., C.J. Brabec, and J. Bernholc, Nanomechanics of carbon tubes: Instabilities beyond linear response. Physical Review Letters, 1996. 76(14): p. 2511-2514.

19. Landau, L. and E. Lifshitz, Elasticity theory. 1975.20. Robertson, D., D. Brenner, and J. Mintmire, Energetics of nanoscale graphitic tubules. Physical

Review B, 1992. 45(21): p. 12592.21. Huang, Y., J. Wu, and K. Hwang, Thickness of graphene and single-wall carbon nanotubes.

Physical review B, 2006. 74(24): p. 245413.22. Lu, J.P., Elastic properties of carbon nanotubes and nanoropes. Physical Review Letters, 1997.

79(7): p. 1297-1300.23. Hernandez, E., et al., Elastic properties of C and BxCyNz composite nanotubes. Physical Review

Letters, 1998. 80(20): p. 4502-4505.

24. Odegard, G.M., et al., Equivalent-continuum modeling of nano-structured materials. Composites Science and Technology, 2002. 62(14): p. 1869-1880.

25. Li, C. and T.-W. Chou, A structural mechanics approach for the analysis of carbon nanotubes. International Journal of Solids and Structures, 2003. 40(10): p. 2487-2499.

26. Jin, Y. and F. Yuan, Simulation of elastic properties of single-walled carbon nanotubes. Composites Science and Technology, 2003. 63(11): p. 1507-1515.

27. Tserpes, K. and P. Papanikos, Finite element modeling of single-walled carbon nanotubes. Composites Part B: Engineering, 2005. 36(5): p. 468-477.

28. Kudin, K.N., G.E. Scuseria, and B.I. Yakobson, C_ {2} F, BN, and C nanoshell elasticity from ab initio computations. Physical Review B, 2001. 64(23): p. 235406.

29. Tu, Z.-c. and Z.-c. Ou-Yang, Single-walled and multiwalled carbon nanotubes viewed as elastic tubes with the effective Young’s moduli dependent on layer number. Physical Review B, 2002. 65(23): p. 233407.

30. Vodenitcharova, T. and L. Zhang, Effective wall thickness of a single-walled carbon nanotube. Physical Review B, 2003. 68(16): p. 165401.

31. Pantano, A., D. M Parks, and M.C. Boyce, Mechanics of deformation of single-and multi-wall carbon nanotubes. Journal of the Mechanics and Physics of Solids, 2004. 52(4): p. 789-821.

32. Goupalov, S., Optical transitions in carbon nanotubes. Physical Review B, 2005. 72(19): p. 195403.

33. Wang, L.F., et al., Size dependence of the thin-shell model for carbon nanotubes. Physical Review Letters, 2005. 95(10).