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Atomistic finite element model for axial buckling of single-walled carbon nanotubes
Physica E 43 (2010) 58–69
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Physica E journal homepage: www.elsevier.com/locate/physe
Atomistic finite element model for axial buckling of single-walled carbon nanotubes R. Ansari n, S. Rouhi Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran
a r t i c l e in f o
a b s t r a c t
Article history: Received 16 March 2010 Accepted 17 June 2010 Available online 23 June 2010
An atomistic finite element model is developed to study the buckling behavior of single-walled carbon nanotubes with different boundary conditions. By treating nanotubes as space-frame structures, in which the discrete nature of nanotubes is preserved, they are modeled using three-dimensional elastic beam elements for the bonds and point mass elements for the atoms. The elastic moduli of the beam elements are determined via a linkage between molecular mechanics and structural mechanics. Based on this model, the critical compressive forces of single-walled carbon nanotubes with different boundary conditions, geometries as well as chiralities are obtained and then compared. It is indicated that at low aspect ratios, the critical buckling load of nanotubes decreases considerably with increasing aspect ratios, whereas at higher aspect ratios, buckling load slightly decreases as the aspect ratio increases. It is also indicated that increasing aspect ratio at a given radius results in the convergence of buckling envelops associated with armchair and zigzag nanotubes. & 2010 Elsevier B.V. All rights reserved.
1. Introduction Carbon nanotubes (CNTs), discovered in 1991 by Iijima [1], have been extensively studied by many research workers. This is largely due to their extraordinary mechanical, thermal and electrical properties over other existing materials known to man. Due to difficulties encountered in experiments at the nanoscale, theoretical modeling of nanostructured materials is of great interest. Theoretical analyses in the literature come under two main categories: atomistic approaches [2–13] and continuum mechanics approaches [14–34]. The former includes the molecular dynamics (MD), tight-binding molecular dynamics (TBMD) and density functional theory (DFT). These approaches are often computationally expensive, especially for large-scale CNTs a with high number of walls, so that alternative continuum models were proposed. By establishing a linkage between structural mechanics and molecular mechanics, Odegard et al. [35] and Li and Chou [36] developed a molecular structural mechanics approach for modeling carbon nanotubes. Based on their approach, Tserpes and Papanikos [38] proposed a FE model for single-walled carbon nanotubes. They determined the elastic moduli of beam element using a linkage between molecular and continuum mechanics. They also found the dependence of elastic modulus on diameter and chirality of the nanotubes.
n
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The buckling of carbon nanotubes is a matter of great technical interest to many research workers. Yakobson et al. [2] compared the results of molecular dynamics (MD) and continuum shell model and deduced that the continuum model provides a remarkably accurate ‘‘roadmap’’ of nanotube behavior beyond Hooke’s law if model parameters are properly chosen. Ru [15–17] used a classical continuum shell model allowing for the van der Waals (vdW) interlayer interactions to study the buckling of single- and multi-walled carbon nanotubes under axial compression and external pressure. Wang et al. [24] Wang and Yang [25] studied the torsional and bending buckling of multi-walled carbon nanotubes using solid shell elements. Li and Chou [37] employed the molecular structural mechanics approach to model the elastic buckling of single- and multi-walled carbon nanotubes under axial compression and bending loading. Chang et al. [20] obtained analytical solutions for the critical buckling strain of single-walled carbon nanotubes based on a molecular mechanics model. They reported that zigzag tubes are more stable than armchair tubes of the same diameter. He et al. [26] derived explicit formulas for the van der Waals (vdW) interaction between any two layers of a multi-walled carbon nanotube (CNT), which are modeled by elastic shells. Based on their results, vdW interaction will lead to a higher critical buckling load in multi-walled CNTs. Zhang and Shen [10] used MD simulations to analyze the buckling and postbuckling of single-walled carbon nanotubes in thermal environments. They reported that the influence of the vdW interactions on the postbuckling behavior of SWCNTs under axial compression can be neglected, while the
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vdW forces affect the postbuckling equilibrium paths of SWCNTs under torsion and external pressure. They [11] also presented the thermal buckling of initially compressed single-walled carbon nanotubes subjected to a uniform temperature rise using the MD simulations. Yao et al. [30] carried out extensive FEM simulations to investigate the buckling behaviors of SWCNTs, DWCNTs and MWCNTs under bending deformation. They concluded that their results agree with atomistic techniques and can drastically reduce the computational cost of atomistic models, especially for largescale CNTs and systems with a number of CNTs. Guo et al. [31] employed the atomic-scale finite element method to study bending buckling of single-walled carbon nanotubes (SWNTs). Murmu and Pradhan [34] implemented the nonlocal elasticity and Timoshenko beam theory to investigate the stability response of SWCNT embedded in an elastic medium. The stability characteristics of armchair and zigzag nanotubes under compression have been studied in this article. An atomistic modeling approach has been implemented to achieve this goal. The critical compressive forces for simply supported, clamped and cantilever SWCNTs of different geometries and chiralities are calculated.
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Fig. 2. Schematic of (a) zigzag and (b) armchair nanotubes.
2. Atomic structure of single-walled carbon nanotubes (SWCNTs) Carbon nanotubes can be considered as a graphene sheet of hexagonal lattice that is rolled up into a hollow seamless cylinder. Fig. 1 shows the geometrical parameters needed to describe the structures of a carbon nanotube. The atomic structure of an SWNT is often defined using the ! chiral vector C h given by ! ! ! ð1Þ C h ¼ n a 1 þm a 2 ! ! where a 1 and a 2 are base vectors and n and m are integers that characterize the type of CNT under study. Armchair nanotubes are defined by (m,m) and zigzag nanotubes by (n,0). Other combinations of n and m are referred to as chiral nanotubes. Schematic of nanotubes with different chiralities, including armchair and zigzag nanotubes, is illustrated in Fig. 2. The direction of the chiral vector is termed as chiral angle y, which is given by ð2n þ mÞ
y ¼ cos1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
ðm2 þ mnþ n2 Þ
ð2Þ
Substituting n¼ m and n ¼0 for armchair and zigzag nanotubes gives the chiral angles of 301 and 01, respectively. The radius of
Fig. 3. Rolling up a planar hexagonal lattice into a nanotube.
nanotube is obtained as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Length of Ch a ðm2 þ mn þn2 Þ R¼ ¼ ð3Þ 2p 2p pffiffiffi where a ¼ a0 3, in which a0 is the equilibrium bond length of the atoms in the graphite sheet. Another important geometrical ! parameter of a SWNT is the translation vector T depicted in Fig. 1, which is directed along the SWNT axis and perpendicular to the ! ! chiral vector C h . In the graphene plane, T is given by ! T ¼
2m þn 2n þ m ! ! a 1 a2 gcdð2m þ n,2n þ mÞ gcdð2m þ n,2n þmÞ
ð4Þ
where gcd stands for the greatest common divisor of the arguments. The magnitude of the translation vector corresponds to the length of the SWNT unit cell, which is marked in grey in Fig. 1, and is given by the expression pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3a0 m2 þmn þn2 ð5Þ L¼ gcdð2mþ n,2n þmÞ To form the configuration of a nanotube of radius R from a planar hexagonal lattice indicated in Fig. 3, the following mapping is used: x1 ¼ X1 ,
Fig. 1. Schematic of a graphene sheet and definitions of geometrical parameters for describing a CNT.
x2 ¼ R sinðX2 =RÞ,
x3 ¼ R cosðX2 =RÞR
ð6Þ
in which (X1,X2) represent the coordinates of an atom on the planar sheet and (x1,x2,x3) are the corresponding images in the nanotube configuration.
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regarded as large molecules consisting of carbon atoms whose positions are regulated by a force field due to electron–nucleus and nucleus–nucleus interactions. The force field can be expressed in the form of steric potential energy, which depends
3. Atomistic finite element modeling of SWCNTs The geometric resemblance between nanoscopic fullerenes and macroscopic frame structures has led to the idea of molecular structural mechanics [36]. Likewise, SWCNTs can be treated as space-frame structures whose behavior can be described by the classical structural mechanics methods. To this end, a threedimensional finite element model comprising beam elements for the bonds and concentrated masses for the carbon atoms is employed. The elastic properties of these beams are determined by the correlation between molecular mechanics and structural mechanics. This linkage was first proposed by Odegard et al. [35] and then by Li and Chou [36], which is briefly described herein. In the context of molecular mechanics, carbon nanotubes can be
Table 1 Selected radii for nanotubes of different chiralities. Armchair and shell Chirality (5,5) 3.39 ˚ Radius ðAÞ Zigzag Chirality ˚ Radius ðAÞ
(9,0) 3.52299
(11,11) 7.458
(15,15) 10.17
(30,30) 20.34
(44,44) 29.832
(19,0) 7.43743
(26,0) 10.1775
(51,0) 19.9636
(77,0) 30.1412
N
N Stretching
ΔL
L
M
M α
α
Bending (variation of bond) Δ T
T
Dihedral angle torsion
Van der Waals Out of plane torsion Fig. 4. Equivalence of molecular mechanics and structural mechanics for interatomic interactions.
Fig. 5. Schematic of armchair and zigzag nanotubes with (a) simply supported, (b) clamped–free and (c) clamped boundary conditions.
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solely on the relative positions of the nuclei constituting the carbon molecule. The general expression for the total steric potential energy of an SWNT at small strains, when neglecting the electrostatic interactions, can be obtained by the sum of energies
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the due to valence of bonded interactions or bonded and nonbonded interactions. [39] X X X X X Utot ¼ Ur þ Uy þ U| þ Uo þ Uvdw ð7Þ
Table 2 Selected lengths for nanotubes of different chiralities. (5,5)
(9,0)
(11,11)
(19,0)
(15,15)
(26,0)
(30,30)
(51,0)
(44,44)
(77,0)
3.68927 7.37854 9.83805 13.5273 17.2166 20.9059 23.3654 27.0546 30.7439 34.4332 36.8927 40.582 44.2712 47.9605 50.42 54.1093 57.7985 61.4878 63.9473 67.6366
2.84 7.1 11.36 13.49 17.75 22.01 24.14 28.4 32.66 34.79 39.05 43.31 45.44 49.7 51.83 56.09 60.35 62.48 66.74 71
7.37854 14.7571 22.1356 29.5141 36.8927 44.2712 51.6498 60.258 67.6366 75.0151 82.3937 89.7722 97.1507 104.529 111.908 119.286 126.665 134.043 141.422 148.8
7.1 15.62 22.01 30.53 36.92 45.44 51.83 60.35 66.74 75.26 81.65 90.17 96.56 105.08 111.47 119.99 126.38 134.9 141.29 147.68
11.0678 20.9059 30.7439 40.582 50.42 60.258 70.0961 81.1639 90.0019 100.84 110.678 120.516 130.354 140.192 150.03 159.868 169.706 179.544 190.612 200.45
11.36 22.01 30.53 41.18 51.83 60.35 71 81.65 90.17 102.95 109.34 119.99 130.64 139.16 149.81 160.46 171.11 179.63 190.28 200.93
20.9059 40.582 60.258 79.9341 100.84 120.516 140.192 161.098 180.744 200.45 220.126 239.802 259.479 280.384 300.06 319.737 340.642 360.319 379.995 399.671
19.88 41.18 60.35 81.65 100.82 119.99 141.29 160.46 179.63 200.93 220.1 239.27 260.57 279.74 301.04 320.21 339.38 360.68 379.85 399.02
29.5141 60.258 89.7722 120.516 150.03 179.544 210.288 239.802 270.546 300.06 329.575 360.319 389.833 420.577 450.091 479.605 510.349 539.863 570.607 600.121
30.53 60.35 90.17 119.99 149.81 179.63 209.45 239.27 271.22 301.04 330.86 360.68 390.5 420.32 450.14 479.96 509.78 539.6 569.42 599.24
Fig. 6. Critical compressive force of simply supported nanotubes of radii R ¼ 3:39 A˚ (armchair) and R ¼ 3:52299 A˚ (zigzag) versus nanotube aspect ratio.
Fig. 7. Critical compressive force of clamped nanotubes of radii R ¼ 3:39 A˚ (armchair) and R ¼ 3:52299 A˚ (zigzag) versus nanotube aspect ratio.
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where Ur,Uy,U|,Uo and Uvdw are energies associated with bond stretching, bond angle bending, dihedral angle torsion, out-ofplane torsion and the van der Waals forces due to non-covalent interactions, respectively. Fig. 4 presents a schematic of the interatomic interactions and their structural mechanics equivalences.
A number of potential functions have been proposed to describe the interatomic interactions of carbon atoms [39–42]. For covalent systems, the main contributions to the total steric energy come from the first four terms of Eq. (7). Assuming that the covalent interactions between carbon atoms can be represented by simple harmonic terms [43,44], the vibrational
Fig. 8. Critical compressive force of clamped–free nanotubes of radii R ¼ 3:39 A˚ (armchair) and R ¼ 3:52299 A˚ (zigzag) versus nanotube aspect ratio.
Fig. 9. Critical compressive force of simply supported nanotubes of radii R ¼ 7:458 A˚ (armchair) and R ¼ 7:43743 A˚ (zigzag) versus nanotube aspect ratio.
Fig. 10. Critical compressive force of clamped nanotubes of radii R ¼ 7:458 A˚ (armchair) and R ¼ 7:43743 A˚ (zigzag) versus nanotube aspect ratio.
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potential energies resulting from interactions between covalently bonded carbon atoms can be represented by the following expressions: Ur ¼ ð1=2Þkr ðrr0 Þ2 ¼ ð1=2Þkr ðDrÞ2
ð8Þ
1 1 k ðyy0 Þ2 ¼ ky ðDyÞ2 2 y 2
ð9Þ
Uy ¼
Ut ¼ U| þ Uo ¼ ð1=2Þkt ðD|Þ2
63
ð10Þ
where kr, ky and kt denote the force constants associated with the stretching, bending and torsion of bonds, respectively; Dr, Dy and D| correspond to the deviation of bond length, bond angle and dihedral angle from the equilibrium position,respectively. The elements representing the bond are assumed to be elastic beams with Young’s modulus E, length L, cross-sectional area A and
Fig. 11. Critical compressive force of clamped–free nanotubes of radii R ¼ 7:458 A˚ (armchair) and R ¼ 7:43743 A˚ (zigzag) versus nanotube aspect ratio.
Fig. 12. Critical compressive force of simply supported nanotubes of radii R ¼ 10:17 A˚ (armchair) and R ¼ 10:1775 A˚ (zigzag) versus nanotube aspect ratio.
Fig. 13. Critical compressive force of clamped nanotubes of radii R ¼ 10:17 A˚ (armchair) and R ¼ 10:1775 A˚ (zigzag) versus nanotube aspect ratio.
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moment of inertia I. The strain energy under pure tension N is given by UA ¼
1 2
Z
L 0
N2 1 N2 1 EA ðDLÞ2 dL ¼ ¼ 2 EA 2 L EA
ð11Þ
The strain energy of the beam element under pure bending moment M is UM ¼
1 2
Z 0
L
M2 2EI 2 1 EI ð2aÞ2 dL ¼ a ¼ L 2L EI
Fig. 14. Critical compressive force of clamped–free nanotubes of radii R ¼ 10:17 A˚ (armchair) and R ¼ 10:1775 A˚ (zigzag) versus nanotube aspect ratio.
Fig. 15. Critical compressive force of simply supported nanotubes of radii R ¼ 20:34 A˚ (armchair) and R ¼ 19:9636 A˚ (zigzag) versus nanotube aspect ratio.
Fig. 16. Critical compressive force of clamped nanotubes of radii R ¼ 20:34 A˚ (armchair) and R ¼ 19:9636 A˚ (zigzag) versus nanotube aspect ratio.
ð12Þ
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where a is the rotational angle of beam ends. The strain energy of the beam element under pure twisting moment T is also given by UT ¼
1 2
Z
L 0
T2 1 T 2L 1 GJ ¼ ðDbÞ2 dL ¼ 2 GJ 2 L GJ
ð13Þ
65
where J,G and Db are the polar moment of inertia, shear modulus and relative rotations of beam ends, respectively. The potential energy terms given by Eqs. (8)–(10) and Eqs. (11)–(13) in two different systems (molecular and structural) are equivalent. Hence, the structural mechanics parameters EA,EI and GJ can be
Fig. 17. Critical compressive force of clamped–free nanotubes of radii R ¼ 20:34 A˚ (armchair) and R ¼ 19:9636 A˚ (zigzag) versus nanotube aspect ratio.
Fig. 18. Critical compressive force of simply supported nanotubes of radii R ¼ 29:832 A˚ (armchair) and R ¼ 30:1412 A˚ (zigzag) versus nanotube aspect ratio.
Fig. 19. Critical compressive force of clamped nanotubes of radii R ¼ 29:832 A˚ (armchair) and R ¼ 30:1412 A˚ (zigzag) versus nanotube aspect ratio.
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directly related to their molecular mechanics counterparts kr,ky and kt, respectively,
EA ¼ kr , L
EI ¼ ky , L
GJ ¼ kt L
ð14Þ
The elastic properties of the isotropic beam elements of diameter d and length L can be obtained as follows: sffiffiffiffiffiffiffi ky d¼4 , kr
E¼
k2r L , 4pky
G¼
k2r kt L 8pk2y
Fig. 20. Critical compressive force of clamped–free nanotubes of radii R ¼ 29:832 A˚ (armchair) and R ¼ 30:1412 A˚ (zigzag) versus nanotube aspect ratio.
Fig. 21. Critical compressive force of an armchair nanotube of radius R ¼ 3:39 A˚ different boundary conditions versus nanotube aspect ratios.
Fig. 22. Critical compressive force of a zigzag nanotube of radius R ¼ 3:52299 A˚ with different boundary conditions versus nanotube aspect ratios.
ð15Þ
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Here, the values of kr, ky and kt and L are taken to be 6.52 10 7 N/nm, 8.76 10 10 and 2.78 10 10 Nnm/rad2, respectively [42,45–46]. The covalent bond distance of the carbon atoms in the hexagonal lattice is also 0.142 nm. Introducing these values into Eq. (15), one can arrive at 2 2 ˚ E ¼ 5:488 108 N=A˚ and G ¼ 8:701 109 N=A˚ for d ¼ 1:466 A, the beam elements. These parameters are fed to the FE model as inputs.
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4. Numerical results By treating SWCNTs as space-frame structures, the atomistic finite element model comprised of beam elements is developed through the ANSYS commercial FE code. For the modeling of the bonds, the three-dimensional elastic BEAM4 element is used. It has six degrees of freedom at each node, including translations in the x, y, and z directions and rotations about the x-, y-, and z-axes.
Fig. 23. First ten mode shapes of an armchair nanotube under clamped–free boundary condition.
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The computed element properties such as the cross-sectional area, area moment of inertia (Iyy ¼Izz) and torsional moment of 2
inertia (Ixx) are considered to be 1:68794 A˚ , 0:453456 A˚
4
and
4
0:22682 A˚ , respectively. A schematic of the armchair and zigzag SWCNTs with clamped (bridge), clamped–free (cantilever) and simply supported boundary conditions is presented in Fig. 5. The geometry of an SWCNT can be defined in terms of the nanotube chirality prescribed by unit vector indices(n,m), or its radius R, accordingly, and nanotube length L. Since the dimensions
of armchair SWCNTs do not coincide necessarily with those of zigzag ones, the closest possible dimensions are selected for comparison. The selected dimensions are listed in Tables 1 and 2. Based on the modeling procedure described above, buckling behaviors of armchair and zigzag nanotubes have been studied here. The axial load applied is uniformly distributed among the atoms at one end of the nanotubes. The critical buckling compressions of chiral nanotubes, i.e. armchair and zigzag ones, for a number of nanotube radii are graphically compared in Figs. 6–20. From the figures, it is observed that compressive
Fig. 24. First ten mode shapes of a zigzag nanotube under clamped–free boundary condition.
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buckling force decreases when nanotube aspect ratio increases. At low aspect ratios, on increase of aspect ratio will result in a significant decrease in the critical buckling load. On the other hand, at high aspect ratios, increasing aspect ratio will result in a slight decrease in the critical compressive force. Also it can be seen that the buckling envelopes associated with armchair and zigzag SWCNTs coincide with each other as nanotube aspect ratio increases, so that the value of critical compressive force for sufficiently long nanotubes is independent of chirality of the tube. According to Figs. 6–20, the dependence of critical buckling stress on chirality is influenced by the boundary conditions of the nanotube under study. For instance, the buckling curves associated with clamped–free boundary conditions come to convergence at higher aspect ratios as compared with other two selected boundary conditions. In other words, the effect of chirality on clamped–free nanotubes is shown to be more prominent. Through comparison of figures corresponding to nanotubes of small and large radii, it can be deduced that increasing the radius will result in decreasing the discrepancy between the critical buckling loads of armchair and zigzag nanotubes. To study the influence of nanotube end conditions on the critical compressive force, the buckling envelops of SWCNTs with simply supported, clamped and cantilever boundary conditions are depicted in Figs. 21 and 22. It can be observed that the critical compressive force of fully clamped boundary conditions is more than simply supported ones and the buckling curve associated with simply supported boundary conditions is higher than that of clamped–free nanotubes. It can be further observed that these buckling envelops tend to converge as the L/R ratio increases. This means that the influence of nanotube end conditions diminishes for long SWCNTs, as would be expected. Represented graphically in Figs. 23 and 24 are the first ten buckling deformations of cantilever SWCNTs of radius 10:17 A˚ and length 81:1639 A˚ for the armchair-type structure and of radius 10:1775 A˚ and length 81:65 A˚ for the zigzag-type structure. Interestingly, the buckling deformations associated with the zigzag model are very similar to those associated with the armchair counterpart.
5. Concluding remarks The elastic buckling analysis of zigzag and armchair singlewalled carbon nanotubes subjected to axial compression was studied in this paper. The atomistic finite element model of nanotubes based on the space frame was developed. Comparisons between the critical buckling forces of two models were made for a wide range of nanotube geometries as well as different boundary conditions. It was indicated that at low aspect ratios, the critical buckling load of nanotubes decreases considerably with increasing aspect ratios, whereas at higher aspect ratios, buckling load slightly decreases as aspect ratio increases. It was also indicated that increasing aspect ratio at a given radius results in the convergence of buckling envelops associated with armchair and zigzag SWCNTs. For nanotubes of large radii, the discrepancy between armchair and zigzag nanotubes decreases. This means that the magnitude of critical buckling stress for sufficiently long nanotubes of large radii is independent of chirality of the tube.
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The influence of nanotube end conditions on the critical buckling forces was shown to diminish for SWCNTs of large aspect ratios. The dependence of the critical buckling stresses on the chirality of nanotubes was also shown to be sensitive to nanotube end conditions.
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Physica E journal homepage: www.elsevier.com/locate/physe
Atomistic finite element model for axial buckling of single-walled carbon nanotubes R. Ansari n, S. Rouhi Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran
a r t i c l e in f o
a b s t r a c t
Article history: Received 16 March 2010 Accepted 17 June 2010 Available online 23 June 2010
An atomistic finite element model is developed to study the buckling behavior of single-walled carbon nanotubes with different boundary conditions. By treating nanotubes as space-frame structures, in which the discrete nature of nanotubes is preserved, they are modeled using three-dimensional elastic beam elements for the bonds and point mass elements for the atoms. The elastic moduli of the beam elements are determined via a linkage between molecular mechanics and structural mechanics. Based on this model, the critical compressive forces of single-walled carbon nanotubes with different boundary conditions, geometries as well as chiralities are obtained and then compared. It is indicated that at low aspect ratios, the critical buckling load of nanotubes decreases considerably with increasing aspect ratios, whereas at higher aspect ratios, buckling load slightly decreases as the aspect ratio increases. It is also indicated that increasing aspect ratio at a given radius results in the convergence of buckling envelops associated with armchair and zigzag nanotubes. & 2010 Elsevier B.V. All rights reserved.
1. Introduction Carbon nanotubes (CNTs), discovered in 1991 by Iijima [1], have been extensively studied by many research workers. This is largely due to their extraordinary mechanical, thermal and electrical properties over other existing materials known to man. Due to difficulties encountered in experiments at the nanoscale, theoretical modeling of nanostructured materials is of great interest. Theoretical analyses in the literature come under two main categories: atomistic approaches [2–13] and continuum mechanics approaches [14–34]. The former includes the molecular dynamics (MD), tight-binding molecular dynamics (TBMD) and density functional theory (DFT). These approaches are often computationally expensive, especially for large-scale CNTs a with high number of walls, so that alternative continuum models were proposed. By establishing a linkage between structural mechanics and molecular mechanics, Odegard et al. [35] and Li and Chou [36] developed a molecular structural mechanics approach for modeling carbon nanotubes. Based on their approach, Tserpes and Papanikos [38] proposed a FE model for single-walled carbon nanotubes. They determined the elastic moduli of beam element using a linkage between molecular and continuum mechanics. They also found the dependence of elastic modulus on diameter and chirality of the nanotubes.
n
Corresponding author. Tel./fax: + 98 131 6690276. E-mail address: [email protected] (R. Ansari).
1386-9477/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2010.06.023
The buckling of carbon nanotubes is a matter of great technical interest to many research workers. Yakobson et al. [2] compared the results of molecular dynamics (MD) and continuum shell model and deduced that the continuum model provides a remarkably accurate ‘‘roadmap’’ of nanotube behavior beyond Hooke’s law if model parameters are properly chosen. Ru [15–17] used a classical continuum shell model allowing for the van der Waals (vdW) interlayer interactions to study the buckling of single- and multi-walled carbon nanotubes under axial compression and external pressure. Wang et al. [24] Wang and Yang [25] studied the torsional and bending buckling of multi-walled carbon nanotubes using solid shell elements. Li and Chou [37] employed the molecular structural mechanics approach to model the elastic buckling of single- and multi-walled carbon nanotubes under axial compression and bending loading. Chang et al. [20] obtained analytical solutions for the critical buckling strain of single-walled carbon nanotubes based on a molecular mechanics model. They reported that zigzag tubes are more stable than armchair tubes of the same diameter. He et al. [26] derived explicit formulas for the van der Waals (vdW) interaction between any two layers of a multi-walled carbon nanotube (CNT), which are modeled by elastic shells. Based on their results, vdW interaction will lead to a higher critical buckling load in multi-walled CNTs. Zhang and Shen [10] used MD simulations to analyze the buckling and postbuckling of single-walled carbon nanotubes in thermal environments. They reported that the influence of the vdW interactions on the postbuckling behavior of SWCNTs under axial compression can be neglected, while the
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vdW forces affect the postbuckling equilibrium paths of SWCNTs under torsion and external pressure. They [11] also presented the thermal buckling of initially compressed single-walled carbon nanotubes subjected to a uniform temperature rise using the MD simulations. Yao et al. [30] carried out extensive FEM simulations to investigate the buckling behaviors of SWCNTs, DWCNTs and MWCNTs under bending deformation. They concluded that their results agree with atomistic techniques and can drastically reduce the computational cost of atomistic models, especially for largescale CNTs and systems with a number of CNTs. Guo et al. [31] employed the atomic-scale finite element method to study bending buckling of single-walled carbon nanotubes (SWNTs). Murmu and Pradhan [34] implemented the nonlocal elasticity and Timoshenko beam theory to investigate the stability response of SWCNT embedded in an elastic medium. The stability characteristics of armchair and zigzag nanotubes under compression have been studied in this article. An atomistic modeling approach has been implemented to achieve this goal. The critical compressive forces for simply supported, clamped and cantilever SWCNTs of different geometries and chiralities are calculated.
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Fig. 2. Schematic of (a) zigzag and (b) armchair nanotubes.
2. Atomic structure of single-walled carbon nanotubes (SWCNTs) Carbon nanotubes can be considered as a graphene sheet of hexagonal lattice that is rolled up into a hollow seamless cylinder. Fig. 1 shows the geometrical parameters needed to describe the structures of a carbon nanotube. The atomic structure of an SWNT is often defined using the ! chiral vector C h given by ! ! ! ð1Þ C h ¼ n a 1 þm a 2 ! ! where a 1 and a 2 are base vectors and n and m are integers that characterize the type of CNT under study. Armchair nanotubes are defined by (m,m) and zigzag nanotubes by (n,0). Other combinations of n and m are referred to as chiral nanotubes. Schematic of nanotubes with different chiralities, including armchair and zigzag nanotubes, is illustrated in Fig. 2. The direction of the chiral vector is termed as chiral angle y, which is given by ð2n þ mÞ
y ¼ cos1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
ðm2 þ mnþ n2 Þ
ð2Þ
Substituting n¼ m and n ¼0 for armchair and zigzag nanotubes gives the chiral angles of 301 and 01, respectively. The radius of
Fig. 3. Rolling up a planar hexagonal lattice into a nanotube.
nanotube is obtained as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Length of Ch a ðm2 þ mn þn2 Þ R¼ ¼ ð3Þ 2p 2p pffiffiffi where a ¼ a0 3, in which a0 is the equilibrium bond length of the atoms in the graphite sheet. Another important geometrical ! parameter of a SWNT is the translation vector T depicted in Fig. 1, which is directed along the SWNT axis and perpendicular to the ! ! chiral vector C h . In the graphene plane, T is given by ! T ¼
2m þn 2n þ m ! ! a 1 a2 gcdð2m þ n,2n þ mÞ gcdð2m þ n,2n þmÞ
ð4Þ
where gcd stands for the greatest common divisor of the arguments. The magnitude of the translation vector corresponds to the length of the SWNT unit cell, which is marked in grey in Fig. 1, and is given by the expression pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3a0 m2 þmn þn2 ð5Þ L¼ gcdð2mþ n,2n þmÞ To form the configuration of a nanotube of radius R from a planar hexagonal lattice indicated in Fig. 3, the following mapping is used: x1 ¼ X1 ,
Fig. 1. Schematic of a graphene sheet and definitions of geometrical parameters for describing a CNT.
x2 ¼ R sinðX2 =RÞ,
x3 ¼ R cosðX2 =RÞR
ð6Þ
in which (X1,X2) represent the coordinates of an atom on the planar sheet and (x1,x2,x3) are the corresponding images in the nanotube configuration.
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regarded as large molecules consisting of carbon atoms whose positions are regulated by a force field due to electron–nucleus and nucleus–nucleus interactions. The force field can be expressed in the form of steric potential energy, which depends
3. Atomistic finite element modeling of SWCNTs The geometric resemblance between nanoscopic fullerenes and macroscopic frame structures has led to the idea of molecular structural mechanics [36]. Likewise, SWCNTs can be treated as space-frame structures whose behavior can be described by the classical structural mechanics methods. To this end, a threedimensional finite element model comprising beam elements for the bonds and concentrated masses for the carbon atoms is employed. The elastic properties of these beams are determined by the correlation between molecular mechanics and structural mechanics. This linkage was first proposed by Odegard et al. [35] and then by Li and Chou [36], which is briefly described herein. In the context of molecular mechanics, carbon nanotubes can be
Table 1 Selected radii for nanotubes of different chiralities. Armchair and shell Chirality (5,5) 3.39 ˚ Radius ðAÞ Zigzag Chirality ˚ Radius ðAÞ
(9,0) 3.52299
(11,11) 7.458
(15,15) 10.17
(30,30) 20.34
(44,44) 29.832
(19,0) 7.43743
(26,0) 10.1775
(51,0) 19.9636
(77,0) 30.1412
N
N Stretching
ΔL
L
M
M α
α
Bending (variation of bond) Δ T
T
Dihedral angle torsion
Van der Waals Out of plane torsion Fig. 4. Equivalence of molecular mechanics and structural mechanics for interatomic interactions.
Fig. 5. Schematic of armchair and zigzag nanotubes with (a) simply supported, (b) clamped–free and (c) clamped boundary conditions.
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solely on the relative positions of the nuclei constituting the carbon molecule. The general expression for the total steric potential energy of an SWNT at small strains, when neglecting the electrostatic interactions, can be obtained by the sum of energies
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the due to valence of bonded interactions or bonded and nonbonded interactions. [39] X X X X X Utot ¼ Ur þ Uy þ U| þ Uo þ Uvdw ð7Þ
Table 2 Selected lengths for nanotubes of different chiralities. (5,5)
(9,0)
(11,11)
(19,0)
(15,15)
(26,0)
(30,30)
(51,0)
(44,44)
(77,0)
3.68927 7.37854 9.83805 13.5273 17.2166 20.9059 23.3654 27.0546 30.7439 34.4332 36.8927 40.582 44.2712 47.9605 50.42 54.1093 57.7985 61.4878 63.9473 67.6366
2.84 7.1 11.36 13.49 17.75 22.01 24.14 28.4 32.66 34.79 39.05 43.31 45.44 49.7 51.83 56.09 60.35 62.48 66.74 71
7.37854 14.7571 22.1356 29.5141 36.8927 44.2712 51.6498 60.258 67.6366 75.0151 82.3937 89.7722 97.1507 104.529 111.908 119.286 126.665 134.043 141.422 148.8
7.1 15.62 22.01 30.53 36.92 45.44 51.83 60.35 66.74 75.26 81.65 90.17 96.56 105.08 111.47 119.99 126.38 134.9 141.29 147.68
11.0678 20.9059 30.7439 40.582 50.42 60.258 70.0961 81.1639 90.0019 100.84 110.678 120.516 130.354 140.192 150.03 159.868 169.706 179.544 190.612 200.45
11.36 22.01 30.53 41.18 51.83 60.35 71 81.65 90.17 102.95 109.34 119.99 130.64 139.16 149.81 160.46 171.11 179.63 190.28 200.93
20.9059 40.582 60.258 79.9341 100.84 120.516 140.192 161.098 180.744 200.45 220.126 239.802 259.479 280.384 300.06 319.737 340.642 360.319 379.995 399.671
19.88 41.18 60.35 81.65 100.82 119.99 141.29 160.46 179.63 200.93 220.1 239.27 260.57 279.74 301.04 320.21 339.38 360.68 379.85 399.02
29.5141 60.258 89.7722 120.516 150.03 179.544 210.288 239.802 270.546 300.06 329.575 360.319 389.833 420.577 450.091 479.605 510.349 539.863 570.607 600.121
30.53 60.35 90.17 119.99 149.81 179.63 209.45 239.27 271.22 301.04 330.86 360.68 390.5 420.32 450.14 479.96 509.78 539.6 569.42 599.24
Fig. 6. Critical compressive force of simply supported nanotubes of radii R ¼ 3:39 A˚ (armchair) and R ¼ 3:52299 A˚ (zigzag) versus nanotube aspect ratio.
Fig. 7. Critical compressive force of clamped nanotubes of radii R ¼ 3:39 A˚ (armchair) and R ¼ 3:52299 A˚ (zigzag) versus nanotube aspect ratio.
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where Ur,Uy,U|,Uo and Uvdw are energies associated with bond stretching, bond angle bending, dihedral angle torsion, out-ofplane torsion and the van der Waals forces due to non-covalent interactions, respectively. Fig. 4 presents a schematic of the interatomic interactions and their structural mechanics equivalences.
A number of potential functions have been proposed to describe the interatomic interactions of carbon atoms [39–42]. For covalent systems, the main contributions to the total steric energy come from the first four terms of Eq. (7). Assuming that the covalent interactions between carbon atoms can be represented by simple harmonic terms [43,44], the vibrational
Fig. 8. Critical compressive force of clamped–free nanotubes of radii R ¼ 3:39 A˚ (armchair) and R ¼ 3:52299 A˚ (zigzag) versus nanotube aspect ratio.
Fig. 9. Critical compressive force of simply supported nanotubes of radii R ¼ 7:458 A˚ (armchair) and R ¼ 7:43743 A˚ (zigzag) versus nanotube aspect ratio.
Fig. 10. Critical compressive force of clamped nanotubes of radii R ¼ 7:458 A˚ (armchair) and R ¼ 7:43743 A˚ (zigzag) versus nanotube aspect ratio.
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potential energies resulting from interactions between covalently bonded carbon atoms can be represented by the following expressions: Ur ¼ ð1=2Þkr ðrr0 Þ2 ¼ ð1=2Þkr ðDrÞ2
ð8Þ
1 1 k ðyy0 Þ2 ¼ ky ðDyÞ2 2 y 2
ð9Þ
Uy ¼
Ut ¼ U| þ Uo ¼ ð1=2Þkt ðD|Þ2
63
ð10Þ
where kr, ky and kt denote the force constants associated with the stretching, bending and torsion of bonds, respectively; Dr, Dy and D| correspond to the deviation of bond length, bond angle and dihedral angle from the equilibrium position,respectively. The elements representing the bond are assumed to be elastic beams with Young’s modulus E, length L, cross-sectional area A and
Fig. 11. Critical compressive force of clamped–free nanotubes of radii R ¼ 7:458 A˚ (armchair) and R ¼ 7:43743 A˚ (zigzag) versus nanotube aspect ratio.
Fig. 12. Critical compressive force of simply supported nanotubes of radii R ¼ 10:17 A˚ (armchair) and R ¼ 10:1775 A˚ (zigzag) versus nanotube aspect ratio.
Fig. 13. Critical compressive force of clamped nanotubes of radii R ¼ 10:17 A˚ (armchair) and R ¼ 10:1775 A˚ (zigzag) versus nanotube aspect ratio.
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moment of inertia I. The strain energy under pure tension N is given by UA ¼
1 2
Z
L 0
N2 1 N2 1 EA ðDLÞ2 dL ¼ ¼ 2 EA 2 L EA
ð11Þ
The strain energy of the beam element under pure bending moment M is UM ¼
1 2
Z 0
L
M2 2EI 2 1 EI ð2aÞ2 dL ¼ a ¼ L 2L EI
Fig. 14. Critical compressive force of clamped–free nanotubes of radii R ¼ 10:17 A˚ (armchair) and R ¼ 10:1775 A˚ (zigzag) versus nanotube aspect ratio.
Fig. 15. Critical compressive force of simply supported nanotubes of radii R ¼ 20:34 A˚ (armchair) and R ¼ 19:9636 A˚ (zigzag) versus nanotube aspect ratio.
Fig. 16. Critical compressive force of clamped nanotubes of radii R ¼ 20:34 A˚ (armchair) and R ¼ 19:9636 A˚ (zigzag) versus nanotube aspect ratio.
ð12Þ
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where a is the rotational angle of beam ends. The strain energy of the beam element under pure twisting moment T is also given by UT ¼
1 2
Z
L 0
T2 1 T 2L 1 GJ ¼ ðDbÞ2 dL ¼ 2 GJ 2 L GJ
ð13Þ
65
where J,G and Db are the polar moment of inertia, shear modulus and relative rotations of beam ends, respectively. The potential energy terms given by Eqs. (8)–(10) and Eqs. (11)–(13) in two different systems (molecular and structural) are equivalent. Hence, the structural mechanics parameters EA,EI and GJ can be
Fig. 17. Critical compressive force of clamped–free nanotubes of radii R ¼ 20:34 A˚ (armchair) and R ¼ 19:9636 A˚ (zigzag) versus nanotube aspect ratio.
Fig. 18. Critical compressive force of simply supported nanotubes of radii R ¼ 29:832 A˚ (armchair) and R ¼ 30:1412 A˚ (zigzag) versus nanotube aspect ratio.
Fig. 19. Critical compressive force of clamped nanotubes of radii R ¼ 29:832 A˚ (armchair) and R ¼ 30:1412 A˚ (zigzag) versus nanotube aspect ratio.
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directly related to their molecular mechanics counterparts kr,ky and kt, respectively,
EA ¼ kr , L
EI ¼ ky , L
GJ ¼ kt L
ð14Þ
The elastic properties of the isotropic beam elements of diameter d and length L can be obtained as follows: sffiffiffiffiffiffiffi ky d¼4 , kr
E¼
k2r L , 4pky
G¼
k2r kt L 8pk2y
Fig. 20. Critical compressive force of clamped–free nanotubes of radii R ¼ 29:832 A˚ (armchair) and R ¼ 30:1412 A˚ (zigzag) versus nanotube aspect ratio.
Fig. 21. Critical compressive force of an armchair nanotube of radius R ¼ 3:39 A˚ different boundary conditions versus nanotube aspect ratios.
Fig. 22. Critical compressive force of a zigzag nanotube of radius R ¼ 3:52299 A˚ with different boundary conditions versus nanotube aspect ratios.
ð15Þ
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Here, the values of kr, ky and kt and L are taken to be 6.52 10 7 N/nm, 8.76 10 10 and 2.78 10 10 Nnm/rad2, respectively [42,45–46]. The covalent bond distance of the carbon atoms in the hexagonal lattice is also 0.142 nm. Introducing these values into Eq. (15), one can arrive at 2 2 ˚ E ¼ 5:488 108 N=A˚ and G ¼ 8:701 109 N=A˚ for d ¼ 1:466 A, the beam elements. These parameters are fed to the FE model as inputs.
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4. Numerical results By treating SWCNTs as space-frame structures, the atomistic finite element model comprised of beam elements is developed through the ANSYS commercial FE code. For the modeling of the bonds, the three-dimensional elastic BEAM4 element is used. It has six degrees of freedom at each node, including translations in the x, y, and z directions and rotations about the x-, y-, and z-axes.
Fig. 23. First ten mode shapes of an armchair nanotube under clamped–free boundary condition.
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The computed element properties such as the cross-sectional area, area moment of inertia (Iyy ¼Izz) and torsional moment of 2
inertia (Ixx) are considered to be 1:68794 A˚ , 0:453456 A˚
4
and
4
0:22682 A˚ , respectively. A schematic of the armchair and zigzag SWCNTs with clamped (bridge), clamped–free (cantilever) and simply supported boundary conditions is presented in Fig. 5. The geometry of an SWCNT can be defined in terms of the nanotube chirality prescribed by unit vector indices(n,m), or its radius R, accordingly, and nanotube length L. Since the dimensions
of armchair SWCNTs do not coincide necessarily with those of zigzag ones, the closest possible dimensions are selected for comparison. The selected dimensions are listed in Tables 1 and 2. Based on the modeling procedure described above, buckling behaviors of armchair and zigzag nanotubes have been studied here. The axial load applied is uniformly distributed among the atoms at one end of the nanotubes. The critical buckling compressions of chiral nanotubes, i.e. armchair and zigzag ones, for a number of nanotube radii are graphically compared in Figs. 6–20. From the figures, it is observed that compressive
Fig. 24. First ten mode shapes of a zigzag nanotube under clamped–free boundary condition.
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buckling force decreases when nanotube aspect ratio increases. At low aspect ratios, on increase of aspect ratio will result in a significant decrease in the critical buckling load. On the other hand, at high aspect ratios, increasing aspect ratio will result in a slight decrease in the critical compressive force. Also it can be seen that the buckling envelopes associated with armchair and zigzag SWCNTs coincide with each other as nanotube aspect ratio increases, so that the value of critical compressive force for sufficiently long nanotubes is independent of chirality of the tube. According to Figs. 6–20, the dependence of critical buckling stress on chirality is influenced by the boundary conditions of the nanotube under study. For instance, the buckling curves associated with clamped–free boundary conditions come to convergence at higher aspect ratios as compared with other two selected boundary conditions. In other words, the effect of chirality on clamped–free nanotubes is shown to be more prominent. Through comparison of figures corresponding to nanotubes of small and large radii, it can be deduced that increasing the radius will result in decreasing the discrepancy between the critical buckling loads of armchair and zigzag nanotubes. To study the influence of nanotube end conditions on the critical compressive force, the buckling envelops of SWCNTs with simply supported, clamped and cantilever boundary conditions are depicted in Figs. 21 and 22. It can be observed that the critical compressive force of fully clamped boundary conditions is more than simply supported ones and the buckling curve associated with simply supported boundary conditions is higher than that of clamped–free nanotubes. It can be further observed that these buckling envelops tend to converge as the L/R ratio increases. This means that the influence of nanotube end conditions diminishes for long SWCNTs, as would be expected. Represented graphically in Figs. 23 and 24 are the first ten buckling deformations of cantilever SWCNTs of radius 10:17 A˚ and length 81:1639 A˚ for the armchair-type structure and of radius 10:1775 A˚ and length 81:65 A˚ for the zigzag-type structure. Interestingly, the buckling deformations associated with the zigzag model are very similar to those associated with the armchair counterpart.
5. Concluding remarks The elastic buckling analysis of zigzag and armchair singlewalled carbon nanotubes subjected to axial compression was studied in this paper. The atomistic finite element model of nanotubes based on the space frame was developed. Comparisons between the critical buckling forces of two models were made for a wide range of nanotube geometries as well as different boundary conditions. It was indicated that at low aspect ratios, the critical buckling load of nanotubes decreases considerably with increasing aspect ratios, whereas at higher aspect ratios, buckling load slightly decreases as aspect ratio increases. It was also indicated that increasing aspect ratio at a given radius results in the convergence of buckling envelops associated with armchair and zigzag SWCNTs. For nanotubes of large radii, the discrepancy between armchair and zigzag nanotubes decreases. This means that the magnitude of critical buckling stress for sufficiently long nanotubes of large radii is independent of chirality of the tube.
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The influence of nanotube end conditions on the critical buckling forces was shown to diminish for SWCNTs of large aspect ratios. The dependence of the critical buckling stresses on the chirality of nanotubes was also shown to be sensitive to nanotube end conditions.
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