Prediction of stiffness and strength of single-walled carbon nanotubes by molecular-mechanics based finite element approach

Materials Science and Engineering A 390 (2005) 366–371

Prediction of stiffness and strength of single-walled carbon nanotubes by molecular-mechanics based finite element approach Xuekun Sun∗ , Wenming Zhao Department of Mechanical, Aerospace and Manufacturing Engineering, Syracuse University, Syracuse, NY 13244, USA Received 15 January 2004; received in revised form 5 August 2004

Abstract Molecular-mechanics based finite element approach was used to predict the tensile stiffness and strength of single-walled carbon nanotubes. Different types of nanotubes, such as Arm-Chair, Zig-Zag, and chiral type, were discussed in detail. Nanotube stiffness was predicted to be independent of both the nanotube diameter and the nanotube helicity, but Poisson ratio was dependent of the nanotube diameter. In addition to the stiffness, nanotube strength was also analyzed by molecular-mechanics based finite element approach. Modified Morse potential function was selected to model the breakage of C C chemical bond with the separation energy of 7.7 eV. Nanotube strength was predicted at 77–101 GPa with the fracture strain around 0.3. The nanotube strength was found to be moderately dependent of the nanotube helicity, but independent of the nanotube diameter. © 2004 Elsevier B.V. All rights reserved. Keywords: Carbon nanotube; Molecular-mechanics; Finite element analysis; Nanocomposites; Stiffness; Strength

1. Introduction Carbon nanotubes (CNTs), since reported first time by Iijima [1] in 1991, have attracted great attentions worldwide because of their excellent properties, such as extremely high stiffness and strength. They have been widely considered as a great potential to be used as reinforcing elements in advanced composite materials. The mechanical properties of CNTs were widely investigated both experimentally and theoretically. The experimental techniques reported to determine the mechanical properties of CNTs include high-resolution transmission electron microscopy (HRTEM), micro-Raman spectroscopy (MRS) and atomic force microscopy (AFM). The first measurement of Young’s modulus of multi-wall nanotubes (MWNTs) came from Treacy et al. [2]. HRTEM was used to measure the thermal vibration amplitudes of MWNTs, and Young’s modulus derived varied from 0.40 to 4.15 TPa. Later, this technique was also employed for single-wall nanotubes (SWNTs) at ∗

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room temperature, while the Young’s modulus were determined at 0.9–1.9 TPa [3]. By using MRS, Lourie and Wagner [4] measured Young’s modulus of SWNTs and MWNTs. Averagely, it was 3.0 TPa for SWNTs and 2.4 TPa for MWNTs. It was more difficult to test the nanotube strength than the stiffness, so there are few papers published on the strength testing. Yu et al. [5] used AFM technique under in-site scanning electronic microscope (SEM) to test both the stiffness and the strength of MWNTs. The nanotube tensile stiffness was found at 0.27–0.95 TPa. The strength of the outmost layer of MWNTs varied from 11 to 63 GPa, and the failure strain was around 0.12. His experimental results have been widely used for the validation of theoretical works by other authors. Theoretically, molecular dynamics (MD) and molecular structural mechanics (MSM) are two main approaches to predict the stiffness and strength of CNTs. Yao and Lordi [6], by using MD simulation, predicted the tensile stiffness of SWNTs at 1.0 TPa that increased with the decreasing nanotube diameter. Ozaki et al. [7] predicted that the stiffness of SWNTs was sensitive only to the nanotube helicity instead of the diameter. In contrast, Liu [8] found that the nanotube stiffness was independent of both the helicity and the diam-

X. Sun, W. Zhao / Materials Science and Engineering A 390 (2005) 366–371

eter. Jin and Yuan [9] took MD simulation of SWNTs and predicted the mechanical properties along longitudinal and cross-sectional direction. The tensile stiffness of SWNTs was predicted at 1.23 TPa. At the same time, they also concluded that the tensile stiffness of SWNTs was independent of the nanotube diameter. As a recently developed analytical approach, the molecular structural mechanics (MSM) has attracted great attention and been used widely in the nano technology because of its accuracy and efficiency. Compared with MD simulation, MSM simulation was accepted as a faster tool with the fair accuracy for the prediction of the mechanical properties of advanced composites at molecular or atomic level. Through MSM simulation, Odegard et al. [10] calculated the tensile stiffness of SWCTs, in which truss elements were used to model the tensile C C bonds and the C C C bending potential energy was simply modeled by an equivalent truss element. Li and Chou [11] introduced a beam-like model in which the beam-element properties were determined from both the C C tensile potential energy and the C C C bending potential energy. It was more attractive that Wang et al. [12] successfully developed a specific finite element approach, named molecular-mechanics based finite element approach, in which all the potential energies (inter- and intramolecules) were considered. This approach has been used to model the response of polymer networks under external forces. The investigation of the strength of CNTs has not been reported as much as that of their stiffness. Currently, Yu’s experimental data was widely accepted although the strength variation was significant (from 11 to 63 GPa). A few theoretical works have been reported to analyze the material fracture under molecular/atomic scale [13,14]. Yakobson et al. [15] predicted a large fracture strain of 0.3 and a high strength of 300 GPa for SWNTs by using MD simulation. The strength of SWNTs was concluded to be independent of the nanotube helicity. Through MD simulation, Belytschko et al. [16] also found that the nanotube strength was moderately dependent of the nanotube helicity, and a small breaking strain of 0.1–0.13 was predicted. In this paper, molecular-mechanics based finite element approach was used to analyze the tensile stiffness and strength of SWNTs. The tensile stiffness was predicted at around 0.4 TPa, which was independent of both the nanotube diameter and the nanotube helicity. The Poission ratio varied from 0.31 to 0.35 that decreased with the increasing nanotube diameter. In order to predict the strength of SWNTs, a modified Morse potential function with a separation energy was selected to model the breakage of C C tensile bond. For a separation energy of 7.7 eV, the strength of SWNTs was predicted at 80–101 GPa, which was found to be moderately dependent of the nanotube helicity. The nanotube strength increased with the increasing nanotube helicity, so the Arm-Chair nanotubes had the biggest strength.

367

Fig. 1. The wedge of the graphite lattice with rolling scheme of [n1 , n2 ] nanotube.

2. Definition of single-walled carbon nanotubes White et al. in Ref. [17] described how to roll out a nanotube from a graphite sheet lattice. Since the periodic symmetry along circumferential direction, only one-twelfth wedge is shown in Fig. 1. If we introduced the set of primitive lattice vectors R1 and R2 depicted in Fig. 1, then vector R can be expressed as R = n1 R1 + n2 R2 . Each tubule can be rolled out along the tubule axis with the sheet width from the point of [0, 0] to point [n1 , n2 ], where n1 and n2 were a pair of two integers. Therefore, the corresponding nanotube was named as [n1 , n2 ] nanotube, for example: [10, 0], [10, 10] nanotube. Accordingly, the radius r and the helical angle θ of [n1 , n2 ] nanotube can be calculated as follows: √
b 3n2 2 2 r= 3.0(n1 + n1 n2 + n2 ), tg(θ) = 2π 2.0n1 + n2 (1) θ was the angle between vector R and R1 . Accordingly, the nanotubes of θ = 30◦ were named Arm-Chair nanotubes, for example [10, 0], [15, 0] nanotube. The nanotubes of θ = 0◦ were named as Zig-Zag nanotubes, such as [5, 5], [10, 10] nanotube. And the chiral nanotubes were defined for 0◦ < θ < 30◦ .

3. Molecular-mechanics based finite element approach Wang et al. [12] developed a nano-scale finite element approach for polymer networks, named molecular-mechanics based finite element approach. It was based upon the atomic force fields inside materials. Two kinds of elements, chemical bond element and Lennard–Jones element, were established to model the intra- and inter-molecular potential energies, respectively. The chemical bond element that represented the intra-molecular potential energies of stretching, bending and torsional energy, is shown in Fig. 2. It was two-node elas-

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Fig. 3. Coordinate system of chemical bond element.

in which Ur , Uθ , Uφ and UL–J were the stretching, bending, torsional and Lennard–Jones potential energy, respectively. For the chemical bond element with the consideration of stretching and bending potential energies, the 3D element stiffness matrix can be derived from the force equilibrium as in Ref. [12]. Fig. 3 shows the coordinate systems (CS) of i–j element, in which oxyz was the element CS, and o1 x1 y1 z1 , o2 x2 y2 z2 were the local nodal CS at nodes i and j, respectively. Local nodal CS at each node was defined from its corresponding bending plane with z1 (or z2 ) axis vertical to the bending plane. The finite element formulae for i–j element can be written as:

Fig. 2. (a) Chemical bonds and (b) chemical bond element.

tic rod element with an elastic joint at each end. The element stiffness was derived from the corresponding force field and static force equilibrium of the element. Meanwhile, the Lennard–Jones bond, which was the inter-molecular potential energy, was modeled as a spring element with the nonlinear stiffness derived from Lennard–Jones potential energy. Finally, the total potential energy in the nano-scale finite element model can be written as:     U= Ur + Uθ + Uφ + UL–J ,

{F៝ e } = [Ke ] · {

ue } while

1 1 Uθ = Kθ (θ − θ0 )2 , Kr (r − r0 )2 , 2 2 1 Uφ = Kφ (cos(3φ) + 1), 2    σ 12  σ 6 UL–J = 4ε − r r Ur =



Kr   0   0    0    0   e   0 K =  −Kr   0    0    0   0  0

(3)

{F៝ e } = {Fx , Fy , Fz , Mx , My , Mz }T , {

ue } = {u, v, w, θx , θy , θz }T (2)

0

0

c1 L2 c2 L2

c2 L2 c3 L2

0 − aL2

0

0

0

− aL2

0

− aL3

0

0

0

0

a1 L

− aL3 a2 L

a3

0

−a2

0

0

0

− Lc12 − Lc22

− Lc22 − Lc32

0

0

0

0

0

− bL2 b1 L

− bL3 b2 L

0

0

0

0

0

0 a2 L a3 L

There were six degrees-of-freedom (DOF) at each node, three transitional DOF and three rotational DOF. The element stiffness matrix was derived as:  0 −Kr 0 0 0 0 0 a1 b1  0 − Lc12 − Lc22 0 − bL2 L L   c3 b3 a2 b2  c2 0 − L2 − L2 0 − L L L   0 0 0 0 0 0 0    a3 a2 −a2 0 0 0 0  L L  a1 a2 a1 0 −L −L 0 0 0   (4)  0 Kr 0 0 0 0 0   c1 c2 b2 − aL1 0 0 − bL1   L L2 L2  c3 b3 a2 c2 b2  −L 0 0 −L  L L2 L2  0 0 0 0 0 0 0    b3 b2 0 0 0 b −b 3 2  L L b1 b2 0 0 − L − L 0 −b2 b1

X. Sun, W. Zhao / Materials Science and Engineering A 390 (2005) 366–371

369

Fig. 4. Tensile force vs. strain curve (C C bond from modified Morse potential function).

in which a1 = 2Kθi · n21 ,

a2 = 2Kθi · n1 · k1 ,

a3 = 2Kθi · k12 ,

b1 = 2Kθj · n22 ,

b2 = 2Kθj · n2 · k2 , c2 = a2 + b2 ,

b3 = 2Kθj · k22 ,

c3 = a3 + b3 ,

c1 = a1 + b1 , Fig. 5. (a) Arm-Chair, (b) Zig-Zag and (c) chiral nanotubes.

n1 = cos(៝y, y៝ 1 ),

k1 = cos(៝z, y៝ 1 ), n2 = cos(៝y, y៝ 2 ), k2 = cos(៝z, y៝ 2 )

4. Force-field constants and modified Morse potential function In this paper, the tensile stiffness and strength of singlewall carbon nanotubes were modeled by the molecularmechanics based finite element approach. The force-field constants in Eq. (2) were taken from the MM3 force field of Allinger et al. [18–20]. Due to the nature of SWNTs and loading condition, only stretching and bending potential energies were considered in this study. The parameters used to calculate the tensile stiffness in this paper were as follows: Kr = 3.26 × 10

−7

Kθ = 4.38 × 10

−10

nJ/bond nm , 2

nJ/angle rad2 ,

r0 = 0.14 nm

(5)

Morse potential function has been accepted widely to characterize the tensile breakage of chemical bonds. A modified Morse potential function was used by Belytschko et al. [16] to simulate the failure of SWNTs, in which the expression of the stretching potential energy Ur in Eq. (2) was replaced by: 2

Ur = De {[1 − e−β(r−r0 ) ] − 1}

(6)

De was the separation (dissociation) energy of chemical bond. For C C bond, 7.7 eV (177.5 kcal/mol) of De was used in this paper as in Ref. [11]. Fig. 4 shows the tensile force versus strain curves of C C bond from the modified Morse potential function of Eq. (5). The initial slope of all curves were same as Kr in Eq. (4) so that for each De , β value can be calculated by:  Kr β= (7) 2De

5. Stiffness and strength of single-walled carbon nanotubes Molecular-mechanics based finite element approach was used to simulate the response of SWNTs under the tensile load along nanotube’s axis direction. Three groups of nanotubes, Arm-Chair, Zig-Zag, and chiral nanotubes (see Fig. 5), were analyzed for their tensile stiffness and strength. Modified Morse potential function was employed to model the breakage of C C chemical bond. The thickness of SWNTs was selected as 0.34 nm which was equal to the graphite layer spacing. The tensile stiffness of Arm-Chair and Zig-Zag SWNTs are shown in Fig. 6. The results for chiral nanotubes were in between them. So, the tensile stiffness of SWNTs was independent of the nanotube helicity. Fig. 6 also shows that

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X. Sun, W. Zhao / Materials Science and Engineering A 390 (2005) 366–371

Fig. 6. The variation of the tensile stiffness with the nanotube diameter (Arm-Chair and Zig-Zag nanotubes).

the tensile stiffness did not increase too much with increasing nanotube diameters. For Arm-Chair nanotubes with the diameters from 0.617 to 2.216 nm, the tensile stiffness increased from 0.39 to 0.399 TPa. For Zig-Zag nanotubes with the diameters from 0.668 to 2.67 nm, the tensile stiffness increased from 0.386 to 0.402 TPa. Most likely, we can conclude that the tensile stiffness of SWNTs was independent of the diameter as well, specifically when the diameter was bigger than 1.4 nm. The Poisson ratio ν12 was also calculated when the nanotubes undertook the tensile load (see Fig. 7). 1-direction was the longitudinal direction of nanotubes, and 2-direction was the radial one. The Poisson ratio ν12 decreased with the increasing nanotube diameter. They varied from 0.31 to 0.35. In addition to the tensile stiffness, the tensile strength of SWNTs was predicted as well in this paper. Modified Morse potential energy of Eq. (6) was employed with the separation energy De of 7.7 eV. Fig. 8 shows the variation of the tensile strength with the nanotube diameter for Arm-Chair

Fig. 8. The variation of the tensile strength with the nanotube diameter (ArmChair nanotubes).

Fig. 9. The variation of the tensile strength with the nanotube helicity.

nanotubes. It was obvious that the tensile strength was independent of the nanotube diameter. Same conclusion can be received for Zig-Zag nanotubes and the chiral nanotubes as well. Fig. 9 shows the variation of the tensile strength of SWNTs with the nanotube helicity. The tensile strength increased with the increasing nanotube helicity. And the Arm-Chair nanotubes owed the biggest tensile strength. The tensile strength of SWNTs varied from 77 to 101 GPa with the fracture strain around 0.3.

6. Conclusions

Fig. 7. The variation of Poisson ratio with the nanotube diameter (Arm-Chair nanotubes).

By using molecular-mechanics based finite element approach, the tensile stiffness and strength of single-walled carbon nanotubes were predicted. Modified Morse potential function was used to model the breakage of C C chemical bond, and the effect of the nanotube diameter and the nanotube helicity was discussed. Following can be concluded for carbon nanotubes:

X. Sun, W. Zhao / Materials Science and Engineering A 390 (2005) 366–371

1. The tensile stiffness was around 0.4 TPa, and the tensile strength varied from 77 to 101 GPa. 2. The tensile stiffness was independent of both the nanotube diameter and the nanotube helicity. 3. The Poisson ratio varied from 0.31 to 0.35, which decreased with the increasing nanotube diameter. 4. The tensile strength was independent of the nanotube diameter, but moderately dependent of the nanotube helicity. It increased with the increasing helicity. Arm-Chair nanotubes had the biggest tensile strength.

References [1] S. Iijima, Nature (1991) 354. [2] M.M.J. Treacy, T.W. Ebbesen, J.M. Gibson, Nature 381 (1996) 678–683. [3] A. Krishnan, E. Dujardin, T.W. Ebbesen, P.N. Yianilos, M.M.J. Treacy, Phys. Rev. B 58 (20) (1998) 14013–14019. [4] O. Lourie, H.D. Wagner, J. Mater. Res. 13 (9) (1998) 2418–2422. [5] M.F. Yu, O. Lourie, M.J. Dyer, K. Moloni, T.F. Kelly, R.S. Ruoff, Science 287 (2000) 637–640. [6] N. Yao, V. Lordi, J. Appl. Phys. 84 (4) (1998) 1939–1943.

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[7] T. Ozaki, Y. Iwasa, T. Mitani, Phys. Rev. Lett. 84 (8) (2000) 1712–1715. [8] J.P. Liu, Phys. Rev. Lett. 79 (7) (1997) 1297–1300. [9] Y. Jin, F.G. Yuan, Proceedings of the 43rd AIAA SDM Conference, AIAA-2002-1430, Denver, CO, April 22–25, 2002. [10] G.M. Odegard, T.S. Gates, L.M. Nichoson, K.E. Wise, NASA/TM2001-210863. [11] C. Li, T.W. Chou, Proceedings of the 43rd AIAA SDM Conference, AIAA-2002-1520, Denver, CO, April 22–25, 2002. [12] Y. Wang, C. Sun, X. Sun, J. Hinkley, G.M. Odegard, T. Gates, Proceedings of the 43rd AIAA SDM Conference, AIAA-2002-1318, Denver, CO, April 22–25, 2002. [13] A. Omeltchenko, J. Yu, R.K. Kalia, P. Vashishta, Phys. Rev. Lett. 78 (1997) 2148–2151. [14] O.A. Shenderova, D.W. Brenner, A. Omeltchenko, X. Su, L.H. Yang, Phys. Rev. B 61 (2000) 3877–3888. [15] B.I. Yakobson, M.P. Campbell, C.J. Brabec, J. Bernholc, Comput. Mater. Sci. 8 (4) (1997) 341–348. [16] T. Belytschko, S.P. Xiao, G.C. Schatz, R. Ruoff, Phys. Rev. B 65 (25) (2002), Art No. 235430. [17] C.T. White, D.H. Robertson, J.W. Mintmire, Phys. Rev. B 47 (9) (1993) 5485–5488. [18] N.L. Allinger, Y.H. Yuh, J.H. Lii, J. Am. Chem. Soc. 111 (1989) 8551–8566. [19] J.H. Lii, N.L. Allinger, J. Am. Chem. Soc. 111 (1989) 8566–8575. [20] J.H. Lii, N.L. Allinger, J. Am. Chem. Soc. 111 (1989) 8576–8582.