Chirality independence in critical buckling forces of super carbon nanotubes

Solid State Communications 148 (2008) 63–68

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Chirality independence in critical buckling forces of super carbon nanotubes Ying Li a , XinMing Qiu a , Fan Yang a , Xi-Shu Wang a , Yajun Yin a,b,∗ , Qinshan Fan a,b a

Department of Engineering Mechanics, Tsinghua University, 100084, Beijing, PR China

b

Division of Mechanics, Nanjing University of Technology, 210000, Nanjing, PR China

article

info

Article history: Received 28 April 2008 Received in revised form 16 June 2008 Accepted 8 July 2008 by F. Peeters Available online 19 July 2008 PACS: 61.46.Fg 62.20.mq 46.70.Lk Keywords: A. Nanostructures A. Super carbon nanotube D. Mechanical properties D. Elastic buckling

a b s t r a c t The elastic buckling of super carbon nanotubes (STs) under compression and bending were studied by the molecular structure mechanics method. Under compression, the elastic buckling mode of the ST is similar to that of a cantilever column. However, the bending buckling mode of ST could transform from shell wall buckling to beam buckling as the length of ST increases. Therefore, the classical Euler formula could give good prediction both to the critical compression force for ST and to the critical bending force for very long STs. A phenomenological formula for the critical bending force of ST was given in this work, which could well predict the critical bending buckling of STs and bridge over the transformation of the bending buckling mode. Two types of chirality independent phenomena were found: first, the critical compression force carried by the unit carbon nanotube inside STs is independent of the chirality of STs; second, the critical bending force of ST’s is independent of the chirality of STs. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction Hierarchical structures exist widespread in biological systems, such as bone, tooth, nacre (shell) et al., and the attachment pads of geckos [1,2]. Such hierarchical structures could supply the extraordinary mechanical properties of its composites (e.g. bone, nacre) [1–4] and strengthen the adhesion abilities of geckos [5]. It is also found that such hierarchical materials become insensitive to flaws at the nanoscale [6]. Therefore, hierarchical structures, indeed, play important roles in biological systems for their selfhealing capacity and their multi-functionality [1,2]. The lessons drawn from hierarchical biological materials could obviously help us to design new engineering materials. Recently, a kind of hierarchical carbon nanotubes, named super carbon nanotubes (STs), were proposed as an attempt to assemble a large number of single wall carbon nanotubes (SWCNTs) [7,8]. As shown in Fig. 1, the lowest order ST, ST(0) , is the SWCNT. Using the ST(0) as a building block, the first order ST, ST(1) , could be obtained. Then the high order STs, ST(k) , could be built up by its previous order ST, ST(k−1) . Thus hierarchical carbon nanotubes can be generated. As hierarchical biological materials, hierarchical STs could also have exceptional properties that may not exist in SWCNTs.

∗ Corresponding author at: Department of Engineering Mechanics, Tsinghua University, 100084, Beijing, PR China. Tel.: +86 10 62795536; fax: +86 10 62781824. E-mail address: [email protected] (Y. Yin). 0038-1098/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2008.07.025

Fig. 1. The scheme of the hierarchical ST structure.

As recently as in the last two years, there have been a series of articles about STs. In the study of Coluci et al. [7], the tight binding total energy and density of states calculations showed the metallic and semiconducting behaviors of the ST structures. Based on the geometry conservation law, Yin et al. [8] have proved that the Y-branched junctions with angles of 1200 in STs satisfied both the minimum energy and symmetry geometry. Pugno [9] also theoretically evaluated the strength and stiffness of STs-reinforced composites. Based on different numerical simulation methods, such as the equivalent shell model [10], the Euler beam model [11], and the molecular dynamics (MD) model [12,13], the mechanical properties of the STs were calculated. The following results were revealed: (a) STs have super flexibility [10–13] and ultra-low

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Y. Li et al. / Solid State Communications 148 (2008) 63–68

density (about 22.8 kg/m3 ), which is only 1% of the bulk density of graphite (2200 kg/m3 ) and 0.3% of that of steel (7860 kg/m3 ). (b) STs have remarkable specific strength as high as 52% [13]. However, the specific strength of SWCNT is only about 9% [13]. Here the specific strength of materials is defined as σf /E with σf and E the tensile strength and the Young’s modulus of materials respectively. (c) STs showed comparable tension force-bearing capacity as SWCNT and higher shear modulus than nanoropes [14]. Besides, the potential fabrication technologies for STs and STbased nano devices have also been studied [15,16]. In short, STs have shown great potential as reinforcement phases in composite materials, due to their large regular scale, super flexibility, ultrahigh specific strength and ultra-low density. An important problem in the applications of STs as nano devices is their stability. Although there has been much work about the buckling of carbon nanotubes (CNTs) [17–20], no work has been done on the buckling of STs. Here, elastic buckling of the first order STs, ST(1) , will be studied by the molecular structure mechanics (MSM) method [21]. 2. Method and model 2.1. The molecular structure mechanics (MSM) method In this article, the MSM method proposed by Li and Chow [21] is employed. The key idea of the MSM method is to use beam elements in characterizing the covalent bond. Numerical results of MSM had shown good agreement with the experimental and molecule dynamics (MD) results, not only in tensile, bending, torsional simulations, elastic buckling and vibration simulations of SWCNTs [20–23], but also in the mechanical properties and vibration simulations of the first order STs [14,15]. The ST structure is assumed to be a space frame, and each covalent bond is simplified as a beam with circular cross section in MSM simulation. The deformation of the ST is easily obtained by structure mechanics theory, with given tensile resistance Eb Ab , flexural rigidity Eb Ib and torsional stiffness Gb Jb of the beam. These parameters are obtained from the local potential energy of the covalent bond, according to the energy equivalence principle [21, 22]: Eb Ab = kr ,

Eb Ib = kθ ,

Gb Jb = kτ

(1)

where kr , kθ and kτ denote the bond stretching, bond bending and torsional resistance force constants, respectively. Therefore, the beam element is supposed to have the following geometrical and mechanical parameters [22]:

s r =2

kθ kr

,

Eb =

k2r b 4π kθ

,

k2r kτ b

Gb =

(2)

8π k2θ

where r is the radius of the beam cross section, Eb and Gb the Young’s modulus and shear modulus, respectively, b the length of the covalent bond. b is usually taken as 1.421 Å. The values for the constants are selected from Refs. [21,22] as follows: −1

kr = 938 kcal mol

−1

kθ = 126 kcal mol

−1

kτ = 40 kcal mol

−2

Å

= 6.52 × 10

−7

−2

rad

−2

rad

−1

N nm

= 8.76 × 10

−10

= 2.78 × 10

−10

(3) −2

(4)

.

(5)

N nm rad

−2

N nm rad

Therefore, the parameters of the beam element may be obtained as follows: r = 0.735 Å,

Eb = 5.49 TPa,

Gb = 0.871 TPa.

(6)

From the MSM method, the tensile resistance EA, flexural rigidity EI and torsional stiffness GJ of ST(1) could be easily determined by the uniaxial tension, bending and torsional

simulations, respectively [14]. For example, the transversal loading F is applied at the free end of a cantilever ST. The corresponding transversal displacement δ at the free end should be [24]:

δ=

FL3 3EI

.

(7)

Therefore, in the bending simulation, the flexural rigidity EI of ST could be obtained from Eq. (7), EI = FL3 /(3δ). The values of EI for some STs are listed in Table 1. Similar to the calculation of EI, the GJ could also be obtained when the torsion moment is applied on the free end of the cantilever ST [14]. According to mechanics of materials theory, GJ = TL/θ , where the T and θ is the torsion moment and the torsional angle at the free end of the cantilever ST, respectively. The values of GJ for some STs are also listed in Table 1. The above mechanical properties (EA, EI and GJ) of ST(1) could be easily determined by the MSM method. If the ST(1) is considered as a ‘‘super bond’’, a similar MSM method could also be used to determine the mechanical properties of ST(2) according to the Eqs. (1) and (2). Then the MSM method could be applied to super nanotube ST(k) of arbitrary generation k, provided that the three elastic parameters (Eq. (1)) of the ‘‘super-bonds’’ have been computed for ST(k−1) . Finally, such a ‘‘hierarchical’’ MSM method will be very useful for exploring the properties of the ST(k) with arbitrary level number k. The eigenvalue buckling analysis predicts the theoretical buckling force of an ideal linear elastic structure. In textbook an eigenvalue buckling analysis of a column will match the classical Euler solution [25]. In a practical structure, the global stiffness matrix includes global elastic stiffness matrix Ke and global geometric stiffness matrix Kg . If Ke is known, Kg could be obtained by following conventional procedures in the structure analysis [20]. The external load P is assumed to be P = λP ∗

(8)

where λ denotes a constant multiplier and P is the relative magnitudes of the applied force. Then the global geometric stiffness matrix Kg could be obtained as follows: ∗

Kg = λKg∗

(9)

∗

where Kg represents the geometric stiffness matrix for the applied load P ∗ . This equation is based on the fact that the geometric stiffness matrix (Kg∗ ) is proportional to the internal forces at the onset of the loading step (P ∗ ). The global elastic stiffness matrix can be considered as unchanged for a wide range of displacement a. Thus there is relation [20]:

(Ke + λKg∗ )a = λP ∗ .

(10)

At the bifurcation point, the stiffness of the structure vanishes. Therefore, the determinant of the structure stiffness matrix must be zero, hence

|Ke + λKg∗ | = 0.

(11)

By solving the eigenvalue Eq. (11), the lowest value of λ, i.e. λcr , could be obtained. The critical buckling load is given by ∗ Pcr = λcr Pcr .

(12)

Since the external load is not specified in the above deviation, compression, bending, even torsion and other types of loads could be applied. Above procedure could be easily implemented in commercial finite element software, such as ANSYS. In this article, the structure of STs is generated by MATLAB, and the mechanical properties and buckling loads of STs are simulated by the MSM and ANSYS.

Y. Li et al. / Solid State Communications 148 (2008) 63–68

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Table 1 The bending rigidity EI and torsion stiffness GJ of STs with different arm tube aspect ratio α

α

EI (nN nm2 ) GJ (nN nm2 )

[6, 0]@(6, 0)

[6, 0]@(6, 0)

[6, 0]@(6, 0)

[8, 0]@(6, 0)

[4, 0]@(6, 0)

[4, 4]@(6, 0)

8.768 1954 1308

6.954 1855 1382

5.140 1733 1452

5.140 4367 –

5.140 419 –

5.140 3491 –

Fig. 2. The [6, 0]@(6, 0) ST (a) front view and (b) side view. In the [6, 0]@(6, 0) ST, the (6, 0) arm tubes (box) are connected by the Y-junctions (circle). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

2.2. Super carbon nanotube and parameter definitions Similar to the notation of CNT chirality, (m, n), the chirality of first order STs is tagged with [M , N ]. Therefore, [M , N ]@(m, n) represents a [M , N ] ST constructed by (m, n) CNT [13]. As shown in Fig. 2, the (m, n) CNT arm tube has length l and diameter d, and the diameter D of [M , N ]@(m, n) STs could be easily determined by p D = l 3(M 2 + N 2 + MN )/π [13]. In the ST(1) structure, the CNT arm tubes along the axial direction of ST are straight and connected by two adjacent Y junctions. Therefore, the length l of the CNT arm tube could be defined as the distance between the central points of two adjacent Y junctions along the axial direction of the ST (see Fig. 2). The aspect ratio of CNT arm tubes is defined as α = l/d. Similarly, the aspect ratio of ST is defined as α˜ = L/D with L the length of the ST. For a certain [M , N ]@(m, n) ST, the diameter D changes with l, which could be represented by the aspect ratio α of CNT arm tube. When the D is fixed, the L will linearly increase as α˜ increases. Therefore, the effect of α and α˜ on the critical buckling force of STs will be considered here. In present work, the (6, 0) SWCNT is used in constructing STs with different chiralities, i.e. [8, 0], [6, 0], [4, 0], [5, 5], [4, 4] and [3, 3], respectively. In order to check the effect of the arm tube chirality, the (4, 4) SWCNT is also used to build the [6, 0] and [4, 4] STs for comparison. As shown in Fig. 2(a), the [6, 0]@(6, 0) ST could be considered as the covalent bond in (6, 0) SWCNT substituted by the (6, 0) arm tube (red box), which is connected by the corresponding CNT Y-junction (red circle). Now look at the Fig. 2(b). The red filled discs represent the cross section of the [6, 0]@(6, 0) ST. Two buckling loads will be considered here: (a) compression and (b) bending, as shown in Fig. 3(a) and (b), respectively. For the STs, one end is fixed and the other end is applied with the compression force or bending force. According to procedure given above, the critical compression force or bending force of STs could be obtained.

Fig. 3. The cantilever [6, 0]@(6, 0) ST (a) under compression (b) under bending.

3. Compression buckling of STs As shown in Fig. 4(a), the buckling mode of slender ST (e.g. α˜ = 3.610) under compression, is similar to that of a cantilever column. If the ST is relatively shorter, such as α˜ < 3, its deformation is restricted to the loading region (see the localized deformation near the free end of ST in Fig. 4(b)). The buckling modes in Fig. 4(a) and (b) belong to ‘‘global buckling’’ and ‘‘local buckling’’, respectively. Under compression, the global buckling mode will appear in slender ST, while local buckling will appear in stubby ST’s. The Fig. 5a shows the effect of aspect ratio α˜ on the critical compression force of [6, 0]@(6, 0) STs with different arm tube aspect ratio α . The critical compression forces for cantilever STs obviously decease as the α˜ increases. For a fixed α˜ , the critical compression force decreases when the α increases. Therefore, both the α and α˜ have the similar effects on the critical compression force. If the ST is assumed to be a continuum cylindrical shell, the critical compression force could be calculated according to classical Euler formula for cantilever columns [25]: c Pcr =

π 2 EI

(13) 4L2 where EI is the flexural rigidity of the ST. According to the flexural rigidities EI listed in Table 1, the critical compression force could be calculated from Eq. (13) and plotted in Fig. 5a (see the dash lines). It could be seen that the critical compression forces calculated from Eq. (13) are in good agreement with the MSM results (Eqs. (10)– (12)) for slender STs (such as α˜ > 3) that exhibit global buckling. As the classical Euler formula is based on the global buckling presumption, Eq. (13) cannot give proper results for stubby STs that exhibit local buckling deformation. For zigzag STs constructed by the (6, 0) SWCNT with α = 5.140, the critical compression forces are plotted in Fig. 5b. Besides, the critical compression forces predicted by Eq. (13) for corresponding cantilever columns are also plotted in the Fig. 5b. As shown in

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Y. Li et al. / Solid State Communications 148 (2008) 63–68

Fig. 5b. The α˜ effect on the critical compression force of [8,0]@(6, 0), [6, 0]@(6, 0), [4, 0]@(6, 0) STs with arm tube aspect ratio α = 5.140.

Fig. 4. The buckling mode of [6, 0]@(6, 0) ST with arm tube aspect ratio α = 5.140 under compression (a) ST aspect ratio α˜ = 3.610 (b) ST aspect ratio α˜ = 1.796.

Fig. 5c. The α˜ effect on the critical compression force carried by unit CNT. The STs are all constructed by (6, 0) arm tube with aspect ratio α = 5.140. The unit CNT is also referenced as the individual CNT arm tube at the end of ST.

Fig. 5a. The α˜ effect on the critical compression force of [6, 0]@(6, 0) STs with arm tube aspect ratio α = 8.768, 6.954 and 5.140.

Fig. 5a, Eq. (13) could give proper critical compression force for zigzag STs with α˜ > 3. For the [N , 0] STs with the same arm tube ((6, 0) SWCNT with α = 5.140), the critical compression force could increase as N increases. If we consider the compression of the individual (or unit) CNT arm tube at the end of the ST (i.e. the red disc in Fig. 2(b)), we may calculate the critical compression force carried by the unit CNT inside the ST with different chirality (see Fig. 5c). It is interesting to note that for the slender STs with the same arm tube (α˜ > 3), the buckling force carried by the unit CNT is almost unaffected by the chirality of STs. Such a phenomenon was not found in the nanoropes. In nanoropes, the critical buckling force carried by unit CNT is dependent on the number of CNT, which could be induced by the inter-tube van der Waals interactions [26]. To reveal the effect of arm tube chirality on the critical compression force of ST, the critical compression forces of [6, 0]@(6, 0), [4, 4]@(6, 0), [4, 4]@(4, 4) and [6, 0]@(4, 4) STs are plotted in Fig. 5d. The aspect ratio of arm tube is α = 5.140 for (6, 0) CNT

and α = 4.977 for (4, 4) CNT. They are assumed to be comparable by ignoring 3% difference. From the Fig. 5d, the differences between the critical compression forces of [6, 0]@(6, 0) ST and [6, 0]@(4, 4) ST are quit small. Similar phenomenon also exists between the [4, 4]@(6, 0) and [4, 4]@(4, 4) STs. Considering the aspect ratio α of (6, 0) arm tube is slightly larger than that of (4, 4) arm tube, we may conclude that the effect of arm tube chirality on the critical compression force is detected to be small and even neglectable in present work. The critical compression forces of STs shown in Figs. 5a and 5b are in the range of 0–14 nN, which is smaller than that of SWCNTs (0–40 nN) [20]. The critical compression force of the unit CNT (0–2 nN) is also much smaller than the critical compression forces of SWCNT. Although the classical Euler formula (Eq. (13)) cannot give a proper prediction of the critical compression forces of SWCNTs [20], it does give the proper prediction of the critical compression forces of STs. Therefore, STs behave like continuum shells instead of SWCNTs. The critical compression forces of STs are much smaller than that of CNT bundles (about several times or tens times of the critical compression forces of SWCNTs [26]), which will restrict the application of STs to compression.

Y. Li et al. / Solid State Communications 148 (2008) 63–68

Fig. 5d. The α˜ effect on the critical compression force of [6, 0]@(6, 0) and [4, 4]@ (6, 0) STs with arm tube aspect ratio α = 5.140, [6, 0]@(4, 4) and [4, 4]@(4, 4) STs with aspect ratio α = 4.977.

67

Fig. 7a. The α˜ effect on the critical bending force of [6, 0]@(6, 0) STs with arm tube aspect ratio α = 8.768, 6.954 and 5.140.

Fig. 7b. The α˜ effect on the critical bending force of [8, 0]@(6, 0), [6, 0]@(6, 0), [4, 0]@(6, 0) STs with arm tube aspect ratio α = 5.140.

force also decreases when α or α˜ increases. Although the cantilever beam model is again not suitable, the classical Euler formula [25] is still given here for comparison with the MSM result:

√ Fig. 6. The buckling mode of [6, 0]@(6, 0) ST with arm tube aspect ratio α = 5.140 under bending (a) ST aspect ratio α˜ = 5.424 (b) ST aspect ratio α˜ = 8.144.

4. Bending buckling of STs Fig. 6 shows the buckling mode of the ST under bending, which is quit different with that of the cantilever beam under bending. Local buckling is obvious even the aspect ratio α˜ of ST is quit large, i.e. α˜ = 5.424. Such a local buckling mode is quite similar to that of a thin shell under bending. Only for very large α˜ , e.g. α˜ = 8.144, will the global buckling mode appear. Such a phenomenon is similar to the buckling of SWCNTs under compression — The compression buckling mode of a short SWCNT is similar to that of the shell wall; while the compression buckling mode of a long SWCNT is similar to that of column [27]. Because local buckling in bending ST is dominant, the classical Euler formula is not valid again, unless α˜ is large enough. Fig. 7a shows the effect of aspect ratio α˜ on the critical bending force of [6, 0]@(6, 0) STs with a different arm tube aspect ratio α . Similar to the critical compression force, the critical bending

b Pcr

= 4.013

EIGJ

(14) L2 where the GJ is the torsional stiffness of the ST. The critical bending force in Eq. (14) is plotted in Fig. 7a. It is clear that the classical Euler formula could not predict the critical bending force of ST accurately unless α˜ > 7. A phenomenological formula for the critical bending force of ST is proposed here: b Pcr = P0 · α˜ β

(15)

where P0 and β are the parameters fitted for the critical bending force. The values of P0 and β are listed in Table 2. In Fig. 7b, the critical bending forces of different chirality STs constructed from (6, 0) CNT arm tube with α = 5.140 are plotted and compared with Eq. (15) fitted from [6, 0]@(6, 0) ST. It is noted that the critical bending forces are all near the fitting curve. Therefore, it is reasonable to believe that the critical bending forces of STs are independent of the chirality of STs. Moreover, Eq. (15) also bridges over the transformation of bending buckling mode, from the shell wall buckling, to beam buckling.

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Y. Li et al. / Solid State Communications 148 (2008) 63–68

of STs. Both the critical compression force carried by unit CNT inside ST and critical bending forces of STs are independent of the ST chirality. Currently, only the Y-junctions of arm tube SWCNT (6, 0) and (4, 4) were generated in our research. In the construction of STs, it is important to find the real positions of all carbon atoms at the Yjunctions. Clearly, it will be more difficult to do this for arm tubes with larger diameters, such as (18, 0) or (8, 8). However, we think the conclusions given in present work will be still valid for STs with larger diameter arm tubes, i.e. [6, 0]@(18, 0) or [4, 4]@(8, 8). Acknowledgements

Fig. 7c. The α˜ effect on the critical bending force of [6, 0]@(6, 0) and [4, 4]@(6, 0) STs with arm tube aspect ratio α = 5.140, [6, 0]@(4, 4) and [4, 4]@(4, 4) STs with arm tube aspect ratio α = 4.977. Table 2 b The parameters P0 , β fitted for critical bending force Pcr = P0 · α˜ β of [6, 0]@(6, 0) STs with different arm tube aspect ratio α

α

P0 (nN)

β

8.768 6.954 5.140

4.87534 ± 0.26444 9.40355 ± 0.29331 16.39526 ± 0.35998

−1.16134 ± 0.04657 −1.26184 ± 0.03271 −1.25382 ± 0.02401

In order to study the effect of arm tube chirality, the critical bending forces of [6, 0]@(6, 0), [6, 0]@(4, 4), [4, 4]@(6, 0) and [4, 4]@(4, 4) STs are plotted in Fig. 7c. The arm tube chirality also has little effect on the critical bending force of ST, which is similar to the case of compression buckling. The critical bending forces of STs range in 0–14 nN and are near to that of SWCNTs (0–10 nN) [20]. 5. Conclusions Under compression, the ST buckles like a column. Under bending, the buckling mode of ST transforms from shell wall buckling to beam buckling as the length of ST increases. Therefore, the classical Euler formula is valid for compression buckling of STs and bending buckling of very long STs (α˜ > 7). A phenomenological formula for the critical bending force of ST exists, which could well predict both the critical bending force of STs and bridge over the bending buckling mode transformation. There are two types of chirality independence in the critical forces

Supports by the Chinese NSFC (Grant No. 10572076) are gratefully acknowledged. Y. Li and X.S. Wang would like to thank NSFC (Grant No. 10772091) and National Basic Research Program of China through Grant No. 2007CB936803. X.M. Qiu would like to thank the Chinese NSFC (Grant No. 10502027) and the National Basic Research Program of China through Grant No. 2006CB601202. Q. Fan would like to acknowledge the financial support from Nanjing University of Technology. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]

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