Influence of inversion energy on elastic properties of single-walled carbon nanotubes

Materials Science and Engineering A 467 (2007) 78–88

Influence of inversion energy on elastic properties of single-walled carbon nanotubes H.W. Zhang ∗ , J.B. Wang, H.F. Ye Department of Engineering Mechanics, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116023, China Received 26 November 2006; received in revised form 15 February 2007; accepted 21 February 2007

Abstract An analytical molecular mechanics model, in which the inversion energy is considered besides the bond stretch and bond angle variation energies, is developed for the prediction of mechanical properties of single-walled carbon nanotubes (CNTs). The closed-form expressions for the surface Young’s modulus and the Poisson’s ratio of achiral and chiral carbon nanotubes are obtained based on the developed model. The present results show the influence of the tube diameter and the inversion energy on the mechanical properties of carbon nanotubes is significant when the tube diameter is small. With increasing the tube diameter, the surface Young’s modulus and the Poisson’s ratio approach, respectively, those of a graphite sheet and the influence of the inversion energy becomes weak gradually. © 2007 Elsevier B.V. All rights reserved. Keywords: Inversion energy; Molecular mechanics; Carbon nanotube; Energy method

1. Introduction Carbon nanotubes (CNTs) have attracted significant attentions since their first discovery by Iijima [1]. CNTs possess superior mechanical properties, such as high elastic modulus, strength and fracture strain and so on. Many studies including experiment, numerical simulation and analytical methods have been carried out to study unique mechanical properties of CNTs. Treacy et al. [2] measured the Young’s modulus of multi-walled carbon nanotubes (MWCNTs) with an average of 1.8 TPa. Yu et al. [3] reported that the Young’s moduli of single-walled carbon nanotubes (SWCNTs) are 0.32–1.47 TPa by the direct tensile loading tests. Based on the classical molecular dynamics, Liew et al. [4,5] studied the buckling behavior of CNTs and CNT bundles under axial compression. Hern´andez et al. [6] investigated the Young’s modulus of SWCNTs based on the tight binding theory and found that it is sizedependent. Li and Chou [7] developed a continuum mechanics model for mechanical properties of CNTs successfully by linking the molecular mechanics constants of force fields and frame sectional stiffness parameters. Arroyo and Belytschko [8] and Zhang et al. [9] extended the classical Cauchy-Born rule using an exponent map to study the mechanical properties of CNTs. Jiang et al. [10] established a hybrid continuum/atomistic model to study Stone-Wales transformation in SWCNTs. Guo et al. [11] and Wang et al. [12,13] developed a nanoscale continuum theory based on the higher order Cauchy-Born rule and predicted mechanical properties of SWCNTs. Besides experiment and numerical simulation methods, some analytical approaches have been developed recently. For example, Chang et al. [14,15] presented an analytical molecular mechanics model and obtained closed-form expressions for the elastic properties of SWCNTs under axial loading. Xiao et al. [16,17] extended the analytical model developed by Chang et al. [14] to torsion loading condition and predicted mechanical properties and nonlinear stress–strain relationships for defect-free CNTs by incorporating the modified Morse potential into the model and also studied the deformation of single- and multi-walled CNTs under ∗

Corresponding author. Tel.: +86 411 84706249; fax: +86 411 84708769. E-mail address: [email protected] (H.W. Zhang).

0921-5093/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2007.02.106

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radial pressure. Shen and Li [18] proposed an energy approach in the framework of molecular mechanics and obtained closedform expressions for the longitudinal Young’s modulus, major Poisson’s ratio, longitudinal shear, plane strain bulk, and in-plane shear moduli of CNTs. However, it is noticed from these analytical methods based on molecular mechanics [14–18] that only the bond stretch and the bond angle variation term energies are included in the calculation of system energy for predicting mechanical properties such as the Young’s modulus and the Poisson’s ratio but the inversion term energy is ignored. The inversion energy may have a significant contribution for the system energy when tube diameter is small. Besides, it is also noticed that most of the molecular mechanics based analytical studies [14,16,18] are focused on achiral CNTs, i.e., armchair and zigzag CNTs, other than chiral ones. In this paper, the energy approach based on molecular mechanics suggested by Shen and Li [18] is enhanced by incorporating the inversion energy into the system total energy. Based on the model proposed, the closed-form expressions for elastic properties of not only achiral but also chiral SWCNTs are obtained and the influence of inversion energy on elastic properties of SWCNTs is measured by comparing with the cases without the inversion energy. The present results may make it possible to enhance many of the current molecular mechanics based analytical approaches to consider the contribution of the inversion energy. 2. Molecular mechanics model From the viewpoint of the molecular mechanics based on the concept of molecular force field [19], the total potential energy of a molecular system can be expressed as a sum of several individual energy terms U = Uρ + Uθ + Uω + Uτ + Uvdw + Ues

(1)

where Uρ , Uθ , Uω and Uτ are the energies associated with bond stretching, angle variation, inversion and torsion; Uvdw and Ues are the energies associated with van der Waals and electrostatic interactions, respectively. For a SWCNT (see Fig. 1) subjected to an axial loading at small strain, van der Waals term Uvdw and electrostatic interaction Ues are negligible. It is assumed that only the nearest-neighbor interactions between atoms are countered for in the analysis. So only the first three terms Uρ , Uθ and Uω are considered in the present study. Under the assumption of small deformation, the harmonic approximation suffices to characterize the energy. So the total molecular potential energy U of a SWCNT can be expressed as U=

2 2 1 1 1 Cρ (dbi ) + Cθ (dθj ) + Cω (dωk )2 2 2 2 i

j

(2)

k

where Cρ , Cθ and Cω are the force constants associating with bond stretching, angle variation and bond inversion. These constants can be determined from empirical molecular potential or fitting to experimental data. As shown in Refs. [14,15,18], the properties of graphite are taken as a reference to obtain the force constants here. dbi , dθ j and dωk are the elongation of bond i, the variation of bond angle j and pyramidalization angle k, respectively. The definition of the pyramidalization angle given in Refs. [20,21] is adopted here. As shown in Fig. 2, it is assumed that the ␲ orbital axial vector is perfect and ␲-orbital and ␴-bonds around an atom all form are equal to angle θ ␴␲ . Then the pyramidalization angle is expressed as ω = θ␴␲ −

π 2

(3)

For SWCNT, the relationship between C–C bond angles θ i and θ ␴␲ can be derived as 3  i=1

cos−1

cos θi − cos2 θ␴␲ = 2π sin2 θ␴␲

(4)

The total system potential energy of a SWCNT with N atoms can be expressed as Π = U − W = NU0 − F dl

(5)

where U and W denote the molecular potential energy and the external force potential energy, respectively; U0 is the molecular potential energy of the representative atom; F the axial load; dl is the elongation of SWCNT. According to the minimum potential

Fig. 1. The calculation model of a SWCNT subjected to axial loading.

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Fig. 2. The pyramidalization angle.

energy principle δΠ = δ(U − W) = 0

(6)

the bond length, bond angles and the pyramidalization angle, which describe the local deformation around each atom in SWCNT, can be obtained through a set of linear equations derived from Eq. (6). 3. The study of elastic properties of SWCNTs As shown in Fig. 3, a SWCNT can be viewed as a cylinder rolled up from a graphite sheet. That is to say, we can visualize cutting  touches its the graphite sheet along lines OA, CB, OC and AB and rolling the tube so that the origination O of the roll-up vector C   tail C. The roll-up vector C can be described as C = na1 + ma2 , where a1 and a2 are unit vectors, and the integers (n, m) are the number of steps along the zigzag carbon bonds of the hexagonal lattice. T៝ is the translation vector. Thus, rectangle OABC in the graphite sheet denotes a periodicity of (n, m) chiral SWCNT, which is viewed as a representative segment in the present analysis. For a chiral SWCNT subjected to an axial loading, Fig. 4(a) shows the representative atom and its three nearest-neighbor bonds in unrolled planar of SWCNT, Fig. 4(b) shows the top view of the local structure along tube axial, Fig. 4(c) shows the space structure composed of bonds and bond angles around the representative atom. The simple relations among these bond angles θ 1 , θ 2 and θ 3 can be derived as cos θi = sin φj sin φk cos(αj − αk ) + cos φj cos φk

(i, j, k = 1, 2, 3, i = j = k)

(7)

where the subscripts i, j, k are even permutation in order and the summation convention is not available for them. The parameters such as φi , αi and θ i are defined in Fig. 4.

Fig. 3. A schematic illustration of a (n, m) chiral SWNT rolled up from a graphite sheet.

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Fig. 4. The representative atom and its nearest-neighbor bonds of chiral SWCNT (a) an unrolled state, (b) top view of the structure along tube axial, (c) a space state.

Due to the assumption of SWCNT formed by an ideal rolling from a graphite sheet [15,21], we can take φ1 , φ2 and φ3 for the undeformed chiral SWCNT as follows 2n + m φ1 = arccos √ , 2 2 n + m2 + mn

φ2 = φ1 +

4π , 3

φ3 = φ1 +

2π 3

(8)

and the angles between the projection of the bonds and the lateral axial, i.e., αi (i = 1, 2, 3), can be calculated by πm , 2(n2 + nm + m2 ) π(m + n) , α2 = − 2 2(n + nm + m2 ) πn α3 = 2 2(n + nm + m2 ) α1 =

(9)

Differentiating both sides of Eq. (7) leads to dθi = (Aij dφj + Aik dφk )

(10)

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where ⎧ sin φj cos φk − cos φj sin φk cos(αj − αk ) ⎪ ⎪ ⎨ Aij = sin θi cos φ − cos φk sin φj cos(αj − αk ) sin φ ⎪ j k ⎪ ⎩ Aik = sin θi

(11)

According to Eq. (4), we can obtain the relationship between the pyramidalization angle θ ␴␲ and bond angles for (n, m) chiral SWCNT and its differential expression can be obtained 3 Ii sin θi sin θ␴␲ dθi dω = dθ␴␲ = 3 i=1 = E1 dθ1 + E2 dθ2 + E3 dθ3 (12) i=1 Ii 2 cos θ␴␲ (1 − cos θi ) where 1

Ii = 



cos θi cos2 θ␴␲ 1− − sin2 θ␴␲ sin2 θ␴␲ Ii sin θi sin θ␴␲ Ei = 3 I i=1 i 2 cos θ␴␲ (1 − cos θi )

2 (13)

The length of the circumferential vector and the translation vector for a representative segment, i.e., the perimeter of CNT and the segment length l, can be written as  = mr1 sin φ1 − (n + m)r2 sin φ2 + nr3 sin φ3 |C|

(14)

and l = |T | =

[(2n + m)r1 cos φ1 + (−n + m)r2 cos φ2 − (n + 2m)r3 cos φ3 )] dR

(15)

Differentiating Eq. (15) leads to [(2n + m) cos φ1 dr1 − (n − m) cos φ2 dr2 − (n + 2m) cos φ3 dr3 − (2n + m)r1 sin φ1 dφ1 dl = d|T | =

+ (n − m)r2 sin φ2 dφ2 + (n + 2m)r3 sin φ3 dφ3 ] dR

where dR is the greatest common divisor of (2n + m, 2m + n). Based on Eq. (2), the total system potential energy of a (n, m) chiral SWNT can be expressed as  Π = N 41 Cρ (dr12 + dr22 + dr32 ) + 21 Cθ (dθ12 + dθ22 + dθ32 ) + 21 Cω dω2 − F dl

(16)

(17)

where N is the total atoms number in a periodicity of a (n, m) chiral SWNT with N = 4(n2 + nm + m2 )/dR . It is should be noticed that for SWCNT which is formed by rolling up a graphite sheet, the tip and the tail of the roll-up vector  should be conterminous no matter how SWCNT is deformed unless the carbon-carbon bond is broken. Thus, one can derive a C consistent equation for deformed SWCNT as follows [15] d[mr1 cos φ1 − (n + m)r2 cos φ2 + nr3 cos φ3 ] = 0

(18)

So dφ3 can be expressed by the other five variables as dφ3 = P1 dr1 + P2 dr2 + P3 dr3 + P4 dφ1 + P5 dφ2

(19)

where P1 =

m cos φ1 (n + m) cos φ2 n cos φ3 mr1 sin φ1 (n + m)r2 sin φ2 , P2 = − , P3 = , P4 = − , P5 = nr3 sin φ3 nr3 sin φ3 nr3 sin φ3 nr3 sin φ3 nr3 sin φ3

(20)

Substituting of Eq. (19) into Eq. (17) yields the following equation according to the minimum potential energy principle δΠ = N[(K11 dr1 + K12 dr2 + K13 dr3 + K14 dφ1 + K15 dφ2 )δdr1 + (K21 dr1 + K22 dr2 + K23 dr3 + K24 dφ1 + K25 dφ2 )δdr2 + (K31 dr1 + K32 dr2 + K33 dr3 + K34 dφ1 + K35 dφ2 )δdr3 + (K41 dr1 + K42 dr2 + K43 dr3 + K44 dφ1 + K45 dφ2 )δdφ1 + (K51 dr1 + K52 dr2 + K53 dr3 + K54 dφ1 + K55 dφ2 )δdφ2 − (M1 δdr1 + M2 δdr2 + M3 δdr3 +M4 δdφ1 +M5 δdφ2 )] = 0 (21)

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where K11 = Cρ /2 + H6 P12 , K12 = H6 P1 P2 = K21 , K13 = H6 P1 P3 = K31 , K14 = H3 P1 + H6 P1 P4 = K41 , K15 = H5 P1 + H6 P1 P5 = K51 , K22 = Cρ /2 + H6 P22 , K23 = H6 P2 P3 = K32 , K24 = H3 P2 + H6 P2 P4 = K42 , K25 = H5 P2 + H6 P2 P5 = K52 , K33 = Cρ /2 + H6 P32 , K34 = H3 P3 + H6 P3 P4 = K43 , K35 = H5 P3 + H6 P3 P5 = K53 ,

(22)

K44 = H1 + 2H3 P4 + H6 P42 , K45 = H2 + H3 P5 + H5 P4 + H6 P4 P5 = K54 , K55 = H4 + 2H5 P5 + H6 P52 F (C1 + C6 P1 ) F (C2 + C6 P2 ) F (C3 + C6 P3 ) F (C4 + C6 P4 ) F (C5 + C6 P5 ) M1 = , M2 = , M3 = , M4 = , M5 = NdR NdR NdR NdR NdR and H1 = B1 + 2Cω A21 A31 E2 E3 , H2 = B6 + Cω (E1 E2 A12 A21 + E1 E3 A12 A31 + E2 E3 A21 A32 ), H3 = B5 + Cω (E1 E2 A13 A21 + E1 E3 A13 A31 + E2 E3 A23 A31 ), H4 = B2 + 2Cω E1 E3 A12 A32 , H5 = B4 + Cω (E1 E2 A12 A23 + E1 E3 A13 A32 + E2 E3 A23 A32 ), H6 = B3 + 2Cω A13 A23 E1 E2 , C1 = (2n + m) cos φ1 , C2 = −(n − m) cos φ2 , C3 = −(n + 2m) cos φ3 , C4 = −(2n + m)r1 sin φ1 , C5 = (n − m)r2 sin φ2 , C6 = (n + 2m)r3 sin φ3 ,

(23)

B1 = (Cθ + E22 Cω )A221 + (Cθ + E32 Cω )A231 , B2 = (Cθ + E12 Cω )A212 + (Cθ + E32 Cω )A232 , B3 = (Cθ + E12 Cω )A213 + (Cθ + E22 Cω )A223 , B4 = (Cθ + E12 Cω )A12 A13 , B5 = (Cθ + E22 Cω )A21 A23 , B6 = (Cθ + E32 Cω )A31 A32 From Eq. (21), we can obtain the following equation ⎫ ⎧ ⎫ ⎡ ⎤⎧ dr1 ⎪ M1 ⎪ K11 K12 K13 K14 K15 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥⎪ ⎪ ⎪ ⎪ ⎪ dr2 ⎪ M2 ⎪ ⎪ ⎪ ⎢ K21 K22 K23 K24 K25 ⎥ ⎪ ⎪ ⎪ ⎨ ⎨ ⎬ ⎬ ⎢ ⎥ ⎢ K31 K32 K33 K34 K35 ⎥ dr3 = M3 ⎢ ⎥⎪ ⎪ ⎪ ⎪ ⎢ ⎥⎪ ⎪ ⎪ ⎪ ⎪ dφ1 ⎪ M4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ K41 K42 K43 K44 K45 ⎦ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎩ ⎭ ⎭ K51 K52 K53 K54 K55 dφ2 M5

(24)

Solving Eq. (24), we can obtain the solutions of dr1 , dr2 , dr3 , dφ1 and dφ2 as the function of F, and dφ3 can be calculated by Eq. (19). The axial strain and the circumferential strain of a (n, m) chiral SWCNT can be expressed as εz = =

d|T | dl d[(2n + m)r1 cos φ1 + (−n + m)r2 cos φ2 − (n + 2m)r3 cos φ3 ]  = = l |T | 3r0 (n2 + nm + m2 ) C1 dr1 + C2 dr2 + C3 dr3 + C4 dφ1 + C5 dφ2 + C6 dφ3  3r0 (n2 + nm + m2 )

(25)

and εθ =

 d|C| d[mr1 sin φ1 − (n + m)r2 sin φ2 + nr3 sin φ3 ] D1 dr1 + D2 dr2 + D3 dr3 + D4 dφ1 + D5 dφ2 + D6 dφ3 = = √ √ √ √ 2 2  |C| 3r0 n + m + mn 3r0 n2 + m2 + mn (26)

where D1 = m sin φ1 , D4 = mr1 cos φ1 ,

D2 = −(m + n) sin φ2 , D5 = −(m + n)r2 cos φ2 ,

D3 = n sin φ3 , D6 = nr3 cos φ3 ,

(27)

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When a SWCNT is viewed as a shell of cylinder with thickness t and radius R, its Young’s modulus and the Poisson’s ratio can be defined as F F E= = (28) ៝ Aεz |C|tεz vzθ = −

εθ εz

(29)

It is noted that the expression for the Young’s modulus of SWCNT contains the tube thickness t. However, as shown in Ref. [14], there have been some different values for tube thickness in existing studies. To avoid the decision of tube thickness, an alternative definition of the Young’s modulus, i.e., the surface Young’s modulus is adopted F (30) Es = Et = ៝ |C|εz which is independent of tube thickness t. As a result, the analytic expressions of the surface Young’s modulus and the Poisson’s ratio for a (n, m) chiral SWCNT can be expressed as √ 3F F (31) = Es =  C dr + C dr + C dr + C4 dφ1 + C5 dφ2 + C6 dφ3 εz |C| 1 1 2 2 3 3 and ν=−

√ εθ 3(D1 dr1 + D2 dr2 + D3 dr3 + D4 dφ1 + D5 dφ2 + D6 dφ3 ) =− εz C1 dr1 + C2 dr2 + C3 dr3 + C4 dφ1 + C5 dφ2 + C6 dφ3

(32)

where ri = 0.142 nm, i = 1, 2, 3. The surface Young’s modulus and the Poisson’s ratio for chiral SWCNTs without consideration of the influence of the inversion energy can also be obtained through steps similar to the above derivation with ignoring of the inversion energy in Eq. (2). Furthermore, we can also obtain the analytical expressions of the surface Young’s modulus and Poisson’s ratio for the limit cases such as (n, n) armchair and (n, 0) zigzag SWCNTs. As for (n, n) SWCNT, the analytical surface Young’s modulus and Poisson’s ratio can be expressed as √ 4 3Cρ Es = for (n, n) (33) 3Cρ a2 /4(Cθ + 2η21 Cθ + η22 Cω ) + 9 vzθ =

Cρ a2 /4(Cθ + 2η21 Cθ + η22 Cω ) − 1 Cρ a2 /4(Cθ + 2η21 Cθ + η22 Cω ) + 3

for (n, n)

(34)

where η1 = − η2 =

sin(α/2) cos(π/2n) 2 sin β

(35)

(cos β − cos2 θ␴␲ ) sin(α/2) cos(π/2n) + (sin α sin2 θ␴␲ )/2 sin 2θ␴␲ (1 − 2 cos β + cos2 (α/2))

The parameters a = b = 0.142 nm, α ≈ 2π/3 and β ≈ π − arccos[(1/2)(cos(π/2n)] can be found in Ref. [22]. As for (n, 0) zigzag SWCNT, the surface Young’s modulus and Poisson’s ratio can be simplified as √ 4 3Cρ Es = for (n, 0) 3Cρ a2 /(2Cθ + η 21 Cθ + η 22 Cω ) + 9 vzθ =

Cρ a2 /(2Cθ + η 21 Cθ + η 22 Cω ) − 1 Cρ a2 /(2Cθ + η 21 Cθ + η 22 Cω ) + 3

for (n, 0)

(36)

(37)

(38)

where η1 =

2 cos(π/2n) cos α cos(β/2)

(39)

η2 =

2 sin α(cos2 θ␴␲ − cos α) + 2 cos α sin(β/2) sin2 θ␴␲ cos(π/2n) sin 2θ␴␲ (1 − 2 cos α + cos2 (β/2))

(40)

The parameters a = b = 0.142 nm, α ≈ 2π/3 and β ≈ arccos[(1/4)−(3/4)cos(π/n)] can be found in Ref. [22].

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Similar to the case for (n, m) chiral SWCNT, the surface Young’s modulus and the Poisson’s ratio for armchair and zigzag SWCNTs without consideration of the influence of the inversion energy can also be obtained with ignoring of the inversion energy in Eq. (2). Force constants Cρ , Cθ and Cω will be calculated in the following. The experimental data of a graphite sheet are adopted to fit Eqs. (33), (34), (37) and (38) with n → ∞. It can be found for armchair and zigzag SWCNTs that for the case of n → ∞ Eqs. (35), (36), (39) and (40) are simplified as 1 η1 = − , 2

η2 = 0

(41)

η1 = −2,

η2 = 0

(42)

The surface Young’s modulus expressions Eqs. (33) and (37) and the Poisson’s expressions Eqs. (34) and (38) are simplified, respectively, as √ 4 3Cρ Es = (n → ∞) (43) Cρ a2 /2Cθ + 9 vzθ =

Cρ a2 /6Cθ − 1 Cρ a2 /6Cθ + 3

(n → ∞)

(44)

which are the same as those given in [15,18]. From Eqs. (43) and (44), it is found that the surface Young’s moduli and the Poisson’s ratios for armchair and zigzag SWCNTs both simplify as those of the graphite sheet as n → ∞. It also agrees with the face that the in-plane elastic constants of a graphite sheet are isotropic. Substituting Es = 0.36 TPa nm and νzθ = 0.16 for a graphite sheet into Eqs. (43) and (44) leads to Cρ = 742.31 nN/nm and Cθ = 1.416 nN nm. Force constant Cω = 4.65 nN nm [21] is also adopted here. 4. Results and discussions Substituting force constants Cρ = 742.31 nN/nm, Cθ = 1.416 nN nm and Cω = 4.65 nN nm into Eqs. (33), (34), (37) and (38), we can obtain the variations of the surface Young’s modulus and the Poisson’s ratio for (n, n) armchair and (n, 0) zigzag SWNTs with tube diameter, as shown in Figs. 5 and 6. The present results are labeled as the modified ones in Figs. 5 and 6. The results reported by Shen et al. [18] labeled as the unmodified ones are reproduced by ignoring the inversion energy in Eq. (2) and also shown in Figs. 5 and 6. Fig. 5 shows the variations of the surface Young’s modulus for armchair and zigzag SWCNTs with the tube diameter. From Fig. 5, it can be observed that the surface Young’s modulus for armchair and zigzag nanotubes increases with increasing tube diameter, and finally tends to the value 0.360 TPa nm of a graphite sheet. When the tube diameter less than 1 nm, the modulus depends strongly on the tube diameter and the dependence gradually becomes weak as the diameter increases. As for an armchair SWCNT and a zigzag one with the given tube diameter, the surface Young’s modulus for the former is slightly larger than that for the latter. By comparing the present results with Shen’s results [18], it can be found that the tendencies of the two results are similar, but the modified results

Fig. 5. The surface Young’s modulus for (n, n) armchair and (n, 0) zigzag SWCNTs.

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Fig. 6. The Poisson’s ratio for (n, n) armchair and (n, 0) zigzag SWCNTs.

are larger than those without consideration of the inversion energy. The influence of the inversion energy on the surface Young’s modulus is significant for armchair and zigzag SWCNTs with small diameter, and gradually becomes weak with increasing the tube diameter. Fig. 6 shows the variations of the Poisson’s ratio for armchair and zigzag SWCNTs with the tube diameter. The Poisson’s ratio decreases with increasing the tube diameter and the tendencies for armchair and zigzag SWCNTs are similar with each other. The curves show a strong dependence of the tube diameter that is less than 1 nm. As the diameter increases, the dependence becomes weak and the Poisson’s ratio gradually approaches the value 0.16 of a graphite sheet. The Poisson’s ratios for zigzag SWCNTs are slightly larger than those for armchair SWCNTs with their close tube diameters. Compared with the results of [18] without consideration of inversion energy, the present results are smaller but the trends predicted by different models are similar. Similar to the surface Young’s modulus, the influence of inversion energy is significant for armchair and zigzag SWCNTs with small tube diameter and the maximum difference reaches 9.3–10.1%. Substituting force constants Cρ = 742.31 nN/nm, Cθ = 1.416 nN nm and Cω = 4.65 nN nm into Eqs. (31) and (32), we can obtain the variations of the surface Young’s modulus and the Poisson’s ratio for (n, m) chiral SWCNTs with the variation of tube diameter as shown in Figs. 7 and 8. The results without consideration of the inversion energy labeled as the unmodified ones are also given by ignoring the inversion energy in Eq. (2). The results with consideration of the inversion energy are referenced as the modified ones. Fig. 7 shows the relationships between the surface Young’s modulus for chiral SWCNTs (including (2n, n), (4n, n), (6n, n)) and the tube diameter. Similar to the surface Young’s modulus for achiral SWCNTs, the moduli for chiral SWCNTs increase and approach that of a graphite sheet as the tube diameter increases. The influence of the diameter on the surface Young’s modulus is significant for chiral SWCNTs with small tube diameter. From Fig. 7, it can be observed that the modified results are slightly larger than the unmodified ones and the inversion energy has some effect on the Young’s modulus of chiral SWCNTs with small tube diameter.

Fig. 7. The surface Young’s modulus for chiral SWCNTs.

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Fig. 8. The Poisson’s ratio of chiral SWCNT.

Fig. 8 shows the relationships between the Poisson’s ratio for chiral SWCNTs (including (2n, n), (4n, n), (6n, n)) and the tube diameter. The variations of the Poisson’s ratio for chiral SWNTs are similar to those for achiral SWCNTs. Whether the inversion energy is considered or not, the Poisson’s ratio of chiral SWCNTs decreases with increasing the tube diameter and tends to the value 0.16 for a graphite sheet. The modified results are smaller than those without consideration of the inversion energy. From Fig. 8, it can be easily found that the influence of the inversion energy and the diameter on the Poisson’s ratio is obvious for chiral SWCNTs with small diameter. 5. Conclusions The present paper enhances the energy method [18] in the framework of molecular mechanics by incorporating the inversion energy in the total system potential energy besides the bond stretching and the angle variation energies. Based on the model, the analytical expressions of the surface Young’s modulus and the Poisson’s ratio for achiral and chiral SWCNTs are derived. The mechanical properties without consideration of the inversion energy for SWCNTs are also given and compared with those with taking the inversion energy into account. It can be found that the surface Young’s modulus increases while the Poisson’s ratio decreases with increasing tube diameter whether the inversion energy is included or not; when tube diameter is small, the influence of the inversion energy and the tube diameter on the mechanical properties is significant, but the influence becomes gradually weak with increasing the tube diameter. Acknowledgements The supports from the National Natural Science Foundation (10225212, 10421202, 10472022, 10640420176), the Program for Changjiang Scholars, NCET Program provided by the Ministry of Education and the National Key Basic Research Special Foundation of China (2005CB321704) are greatly acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

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