Mechanical properties of zigzag-shaped carbon nanotubes: the roles of the geometric parameters

Abstract

In our previous work, we predicted that zigzag-shaped carbon nanotubes (Z-CNTs) show significant electromechanical properties where under uniaxial tensile strain, a semiconductor-to-metal or metal-to-semiconductor transition can be observed [Phys. Chem. Chem. Phys., 2013, 15, 17134–17141]. Thus, investigating the mechanical properties of Z-CNTs is of key importance for exploring their applications in mechanical and electromechanical devices. Here, a series of Z-CNTs of different geometric parameters were constructed and optimized using the density functional method. Mechanical properties, such as intrinsic strength, Young's modulus and elastic constant, were studied to find their relationships to geometric parameters. Generally, all of the intrinsic strength, Young's modulus and elastic constant increase with increasing tubular radius, but decrease with increasing the pitch. In particular, fitting formulae were given to describe the relationships between the intrinsic strength/Young's modulus and the tubular radius, which show exponential relationships. Explanations were also given for dependence of mechanical properties on the geometric parameters of the Z-CNTs.

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Introduction

In the family of carbon nanotubes (CNTs), there is one kind of unique curved structure consisting of pentagons and heptagons.^1–3 Generally, there are three typical kinds of curved CNTs, i.e. toroidal CNTs,^4–6 in-plane kinked CNTs^7–9 and out-of-plane carbon nanocoils.^10–14 The existence of pentagons and heptagons brings different properties from the pristine CNTs composed of hexagons, which greatly extends the applications of CNT-based materials.³

A zigzag-shaped carbon nanotube (Z-CNT), an interesting structure which belongs to the kinked CNTs, has periodic in-plane kinks, and has a sawtooth like morphology. It can be fabricated by different methods. Via catalytic decomposition of ethylene, Gao et al.¹⁵ synthesized the Z-CNTs in 2000. They pointed out that the detailed shape of a tube is determined by the fractions and creation rates of pentagons, hexagons, and heptagons, which was kinetically controlled growth process. During creation of pentagon–heptagon pairs, if they are oriented in the same plane, a Z-CNT will be formed; otherwise, a helical CNT will be formed if pentagon–heptagon pairs rotate along the tube body. On the other hand, through changing the direction of applied electric field during dc plasma enhanced chemical vapor deposition (CVD), AuBuchon et al.^8,16 obtained the Z-CNTs with a bending angle of ∼90° and the same diameter before and after bending. Besides, by controlling gas flow orientation, the Z-CNTs can be formed during aligning the CNTs with the aid of CNT–lattice interaction.¹⁷

Due to the unique sawtooth-like geometry, the Z-CNTs are promising for mechanical and electromechanical devices. It's well-known that the CNT shows excellent mechanical properties with a Young's modulus (E) up to ∼1.0 TPa and an intrinsic strength (τ) of ∼130 GPa.^18,19 However, like the carbon nanocoils,^20,21 kinks formed by pentagons and heptagons in Z-CNTs will reduce the mechanical strength compared with pristine CNTs. It was found that for a (5, 5)² Z-CNT, the Young's modulus is 561 GPa, ∼60% of that of the (5, 5) CNT.⁹ As the tubular radius increases, the Young's modulus increases monotonously. Similarly, the intrinsic strength of the (5, 5)² Z-CNT is 68.7 GPa, ∼70% of that of the (5, 5) CNT.⁹ It also increases with the tubular radius continuously. Therefore, compared with the straight CNTs, the Z-CNTs seem more flexible. As for the electronic properties of the Z-CNTs, the pentagons and heptagons would either keep the metallicity of the pristine CNTs or transform the metallic CNTs into semiconducting.^9,22 As indicated by Wu et al.'s calculations,²² pentagons and heptagons make the step-kinked (5, 0) CNT semiconducting with a band gap of ∼0.76 eV, totally break the metallicity of perfect (5, 0) CNT.²³ While, both the perfect and step-kinked (6, 0) CNTs are metallic. Our previous DFT simulations obtained consistent results that the metallic (5, 5), (7, 7), and (8, 8) CNTs transfer to semiconducting Z-CNT counterparts due to incorporation of pentagons and heptagons, but the (6, 6) Z-CNT keeps metallic.⁹ In particular, the Z-CNTs show significant electromechanical properties. No matter metallic or semiconducting the Z-CNT is, under uniaxial tensile strain, it will undergo a semiconductor-to-metal or metal-to-semiconductor transition.⁹

Since the Z-CNTs are promising for mechanical and electromechanical devices, to gain a deep insight of mechanical properties of the Z-CNTs, here we constructed a series of Z-CNTs with different geometric parameters to study of the roles of geometric parameters on the mechanical properties, focusing on the effects of tubular radius and pitch. Moreover, empirical formulae were fitted to describe the relationships between the intrinsic strength/Young's modulus and the tubular radius.

Structural models and computational methods

Structural models

Previously, we proposed a simple way to construct atomistic structural models of the curved CNTs with different geometric parameters.³ It can be used to build the structural models of carbon nanocoils, carbon nanotori and Z-CNTs.^9,21,24–26 As shown in Fig. 1a, by introducing pentagon and heptagon pairs into CNTs, one-dimensional (1D) periodic Z-CNTs were built. The Z-CNTs can possess different geometric parameters, as defined by a symbol (n, n)^m. The chiral vector (n, n) indicates that Z-CNTs were built from (n, n) CNTs. It is also used to describe the tubular radius (r) by:where with δ indicating the C–C bond length. Fig. 1b is a side view to illustrate Z-CNTs with different tubular radii. The superscript m means the number of hexagons between two neighbour pentagon and heptagon along the periodic direction. It can denote the pitch (λ) of a Z-CNT, since λ is proportional to m. Fig. 1c is a side view to present Z-CNTs with different pitches, i.e. the 1D lattice parameters of the Z-CNT unitcells.

Fig. 1 Schematic diagrams of the atomic structural models for (n, n)^m Z-CNTs. (a) is a structural model for a (5, 5)¹ Z-CNT; (b) is a side view to show Z-CNTs with different tubular radii, taking the (n, n)¹ Z-CNTs with n = 5–8 as examples; and (c) is a side view to show Z-CNTs with different pitches, taking the (5, 5)^m Z-CNTs with m = 1–4 as examples. The pentagons and heptagons are highlighted in colour in (a) and in thick sticks in (b) and (c).

Computational methods

Employing the plane-wave pseudopotential technique as implemented in the Vienna ab initio simulation package (VASP)²⁷ with the PW91 functional²⁸ for the exchange–correlation interaction and the PAW pseudopotential^29,30 for the ion–electron interaction, all the structural models were fully relaxed. To avoid the interaction between the Z-CNT and its periodic images, a vacuum thickness of 12 Å was adopted. Besides, a value of 400 eV was chosen for kinetic energy cutoff to ensure well convergence of the stresses. For the periodic structures of different sizes, the Monkhorst–Pack grids³¹ with a separation of 0.03 Å⁻¹ were used. All geometric structures were fully relaxed using the conjugate gradient algorithm until the force on each atom was smaller than 0.01 eV Å⁻¹.

To investigate the mechanical properties, each Z-CNT was carefully relaxed to achieve the equilibrium state within 1D periodic boundary condition. Starting from the 1D equilibrium lattice length, i.e. the pitch λ, we gradually elongated the Z-CNT until reaching the fracture strain and obtain the strain–stress curves, as shown in Fig. S1.† Here, the stress means the effective stress, as defined in eqn (2). Through the strain–stress curves, Young's modulus E and intrinsic strength τ can be obtained, where τ is determined by the dropping point of the stress–strain curves and E was calculated in the linear part of the strain–stress curve by the following the formula:³²

where V is volume of the supercell, V₀ is the effective volume of the 1D structural unit of the Z-CNT in its equilibrium configuration, σ is the output stress in VASP calculations, respectively. Thus (V₀/V) × σ is the effective stress. ε is the strain, which is calculated by the division of displacement to pitch. For a Z-CNT, the effective volume is calculated by V₀ = 2πr × λ × Δd, where Δd = 3.4 Å is the van der Waals (vdW) thickness of tube wall.²¹ The tubular radii for the (n, n) CNTs with n = 5, 6, 7 and 8 are 3.40, 4.08, 4.76 and 5.44 Å, separately, which agrees with the results in our previous work.²¹ Based on the Young's modulus, the elastic constant k of each Z-CNT can be calculated according to the following equation:where A = 2πr × Δd is the cross-sectional area through which the force is applied.

Results and discussion

Pitch and binding energy

After optimization, the pitch for each (n, n)^m Z-CNT with n ranging from 5 to 8 and m ranging from 1 to 6 was obtained and listed in Table 1. It can be noticed that there is a linear relationship between λ and m:

λ_m+1 = λ_m + αm, m = 1, 2, 3,…, n (4)

Table 1 Pitch λ (in the unit of Å) of (n, n)^m Z-CNTs

m	(n, n)
	(5, 5)	(6, 6)	(7, 7)	(8, 8)
1	8.80	8.68	8.64	8.63
2	13.07	12.91	12.84	12.81
3	17.35	17.15	17.04	17.00
4	21.62	21.38	21.22	21.19
5	26.00	25.57	25.42	25.36
6	30.29	29.80	29.60	29.53
α	4.30	4.22	4.19	4.18

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The slope α is around 4.26, which is the lattice parameter of a hexagonal graphene unit cell sheet along the armchair direction. Because increase of m by one means adding a hexagon almost along the periodic orientation. On the other hand, as the chiral index n increases from 5 to 8, the λ decreases and trends to a limit of the perfect CNT counterpart due to lowering of curvature.

Based on the equilibrium structures of the Z-CNTs, their binding energies per atom were computed to evaluate the thermodynamic stability, as presented in Fig. 2. It can be noticed that the binding energies of the Z-CNTs are much lower than that of C₆₀ fullerene, indicating they are more stable than C₆₀ fullerene. While, due to incorporation of pentagons and heptagons, the Z-CNTs are slightly less stable than the CNTs. However, as n and m increase, the Z-CNTs become more stable due to reduction of curvature and pentagon–heptagon rate. Increase of n suggests a large tubular radius, and the stability of Z-CNT will converge to that of graphene. Increase of m means a large tubular length, and the stability of Z-CNT will converge to that of perfect CNT, as indicated by m = ∞ in Fig. 2.

Fig. 2 Binding energies of the (n, n)^m Z-CNTs compared to the pristine (n, n) CNTs (the case of m trending to infinity) and C₆₀ fullerene (the blue square).

Intrinsic strength

To study the mechanical properties of Z-CNTs, the strain–stress curves were investigated. Starting from the equilibrium structures, the (n, n)^m Z-CNTs with n ranging from 5 to 8 and m ranging from 1 to 6 were uniaxially elongated with a tensile strain step of 1% until the fracture strain ε_c was reached, which is define by the critical strain where the strain–stress curve starts to drop. The ε_c is ∼15% for all the (n, n)^m Z-CNTs studied here, which is lower than that (∼20%) of their straight CNT counterparts due to incorporation of pentagons and heptagons, as shown in Fig. S1.† This is reasonable since existence of Stone–Wales defects will lower the intrinsic strength and fracture strain of the CNTs.^33,34 The stress corresponding to the fracture strain is the intrinsic strength τ, as summarized in Table 2. τ of both the CNTs and graphene are consistent will previous reports,^34–36 indicating the reasonability of our method. Clearly, τ of the Z-CNTs is lower than that of the CNT counterparts, demonstrating incorporation of pentagons and heptagons will reduce the strength of the perfect CNTs. This result agrees well with previous reports that existence of Stone–Wales type pentagon–heptagon defects will degrade the intrinsic strengths of the CNTs,³⁴ where τ drops from 135 to 125 GPa and from 124 to 115 GPa for the (5, 5) armchair CNT and (10, 0) zigzag CNT, respectively.

Table 2 Intrinsic strength τ of the (n, n)^m Z-CNTs with comparison to those of corresponding (n, n) CNTs and that of graphene

m	(n, n)
	(5, 5)	(6, 6)	(7, 7)	(8, 8)	Graphene
1	78.5	81.5	85.4	89.1	111.8
2	68.7	72.4	76.6	81.4
3	66.5	68.1	71.0	74.7
4	61.3	62.1	65.3	68.9
5	57.9	58.9	61.3	62.9
6	54.6	55.4	57.9	59.4
CNT	96.0	100.7	103.6	105.2

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On the other hand, τ of the Z-CNTs closely depends on the geometric parameters. As n increases, τ of the Z-CNTs increase accordingly and gradually converges to that of graphene (111.8 GPa). Since n represents the tubular radius r, the relationship between τ and r was plotted in Fig. 3, which can be expressed by a fitting formula:

τ = A + Be^−(C/r) (5)

where A, B and C are three fitting parameters. The detailed fitting process can be found in ESI.† Previously, we obtained the r–τ relationship of τ = 34.7 + 77.1e^−(2.8/r) for (n, n)² Z-CNTs with A = 34.7 GPa, B = 77.1 GPa and C = 2.8.⁹ Fig. S2† presents the values and standard errors of parameters A, B and C for all the (n, n)^m Z-CNTs studied here.

Fig. 3 Relationships between τ and r for (n, n)^m Z-CNTs with n ranging from 5 to 8 and m ranging from 1 to 6, where the τ at 1/r = 0 stands for graphene.

A simple picture can be given to explain the relationship between τ and r for the Z-CNTs. As r increases, curvature of the Z-CNT decreases, which will further release the stress and shorten C–C bond lengths. As a result, shortening of C–C bond lengths to the value of that of graphene means an enhancement of C–C bond strength, contributing to the increase of τ. Taking (n, n)² Z-CNTs as examples, as n increases from 5 to 8, the average C–C bond strength shortens from 1.430 to 1.425 Å.⁹ Besides, change of bond lengths and bond angles at the defective sites means a distorted sp² hybridization, which can also affect the electronic structures of the Z-CNTs.⁹

Besides, it can be also noticed from Fig. 3 that as m increases, τ of the Z-CNTs decreases inversely. For example, as m increase from 1 to 6, τ of the (5, 5)^m Z-CNTs drops from 78.5 to 54.6 GPa, as listed in Table 2. To explain the relationship between τ and m, we can make such an assumption. From Fig. 2, we know that as m trends to infinity, the binding energy of (n, n)^m Z-CNTs will converge to that of the (n, n) CNT. Thus, we can take the τ of (n, n) CNT as the upper limit of (n, n)^m Z-CNTs, corresponding to the value of m = 0. From eqn (2), the effective stress is proportional to V₀/V. We can further expand it by:

where S is the cross section area of the supercell for a (n, n)^m Z-CNT. As m increases, the height of the Z-CNT increases significantly, which leads to increase of S. Therefore, τ of (n, n)^m Z-CNTs will decrease with m.

Young's modulus

Using eqn (2), the Young's modulus E of a Z-CNT can be also calculated from the strain–stress curve in its linear part. Here, E of (n, n)^m Z-CNTs were calculated and compared with those of the perfect (n, n) CNTs and graphene, as listed in Table 3. Again, the calculated E of (n, n) CNTs (n = 5–8) and graphene is about 1.0 TPa, in good agreement with previous experimental measurements^18,19 and theoretical predictions.^36,37 Similar to the case of τ, pentagons and heptagons will degrade the Young's modulus of the perfect CNT, since E of each (n, n)^m Z-CNT is lower than that of its (n, n) CNT counterpart. This result is consistent with the case of carbon nanocoils, which incorporate similar pentagon–heptagon pairs.²¹

Table 3 Young's modulus E (in a unit of GPa) of the (n, n)^m Z-CNTs compared with those of the pristine (n, n) CNTs and graphene

m	(n, n)
	(5, 5)	(6, 6)	(7, 7)	(8, 8)	Graphene
1	590	609	646	687	1037
2	561	580	613	651
3	514	531	580	610
4	477	499	536	568
5	435	458	495	526
6	403	426	458	487
CNT	908	922	935	957

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To show the dependence of E on the geometric parameters of Z-CNTs, we plotted the relationship between E and r in Fig. 4. It can be noticed that E of the Z-CNTs increase with r and gradually converges to its upper limit, i.e. E of graphene (1037 GPa). Generally, the relationship between E and r can be also expressed by a fitting formula similar to that of the τ–r:

E = A′ + B′e^−(C′/r) (7)

where A′, B′ and C′ are three fitting parameters. Also, the detailed fitting process was presented in ESI.† The values and standard errors of parameters A′, B′ and C′ for all the (n, n)^m Z-CNTs studied here were shown in Fig. S3.† Previously, we obtained the E–r relationship of E = 470 + 567e^−(6.4/r) for (n, n)² Z-CNTs with A′ = 470 GPa, B′ = 567 GPa and C′ = 6.4.⁹ Since E is proportional to the effective stress as shown in eqn (2), the explanation for the relationship between τ and r can be also used to elucidate the dependence of E on r. Therefore, release of stress and shortening of C–C bond lengths during increasing r are the main factor for the increased E.⁹

Fig. 4 Relationships between E and r for (n, n)^m Z-CNTs with n ranging from 5 to 8 and m ranging from 1 to 6, where the E at 1/r = 0 stands for graphene.

Different from the tendency of E–r, E of the (n, n)^m Z-CNTs decreases inversely as m increases. For example, E of the (5, 5)^m Z-CNTs drops from 590 to 403 GPa as m increases from 1 to 6, as presented in Table 3. Similar to the case of τ, decrease of E as m increases can be attributed to the lowering of V₀/V.

Elastic constant

Based on the calculated Young's modulus, we further calculated the elastic constant k of the (n, n)^m Z-CNTs with n ranging from 5 to 8 and m ranging from 1 to 6, as presented in Table 4. Also, k of the (n, n)^m Z-CNTs increases with enlarging r. From Tables 1 and 3, it can be noticed that increase of tubular radius r will lead to a shorter pitch λ and a larger Young's modulus E. Thus, according to eqn (3), we get lower k as r increases. However, as m increases, k of the (n, n)^m Z-CNTs will decrease inversely due to increase of λ and decrease of E.

Table 4 Elastic constant k (in a unit of N m⁻¹) of the (n, n)^m Z-CNTs with n ranging from 5 to 8 and m ranging from 1 to 6

m	(n, n)
	(5, 5)	(6, 6)	(7, 7)	(8, 8)
1	487	612	760	925
2	312	392	485	591
3	215	270	346	417
4	160	203	257	312
5	122	156	198	241
6	97	125	157	192

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We further compared k between the (n, n)^m Z-CNTs and (n, n) CNTs at the same pitch. Since k decrease with m, we only considered the largest k for each (n, n)^m Z-CNT studied here, i.e. m = 1. As a result, k of each (n, n)¹ Z-CNT is 65%, 66%, 69%, and 72% of that of each (n, n) CNT for n = 5, 6, 7, and 8, respectively. Thus, at the same pitch, a (n, n)^m Z-CNT is more flexible than its (n, n) CNT counterpart.

Conclusions

Z-CNTs with different tubular radii and pitches were constructed and studied using first-principles calculations. Dependence of the mechanical properties on the geometric parameters of the Z-CNTs was detailed discussed, including the relationship between the intrinsic strength and the tubular radius/pitch, the relationship between the Young's modulus and the tubular radius/pitch, and the relationship between the elastic constant and the tubular radius/pitch. Especially, fitting formulae were obtained to describe the relationship between the intrinsic strength/Young's modulus and the tubular radius, where both the intrinsic strength and the Young's modulus are exponentially inverse to the tubular radius. Increase of the intrinsic strength and the Young's modulus with the tubular radius is ascribed to release of stress and decrease of pentagon–heptagon rate. Decrease of the effective stress leads to lowering the intrinsic strength and the Young's modulus during increasing the pitch. Compared with the perfect CNT counterparts, the Z-CNTs show lower elastic constant, indicating good flexibility.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (11504040), and Fundamental Research Funds for the Central Universities of China (DUT14RC(3)114).

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6ra01260d

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