Since single vacancy and multi-vacancy defects are always presented in CNTs due to post-processing, including ion^[33] and electron irradiation, ^[34] and so on. Moreover, these vacancies can transform into other defects by saturating the dangling bonds.^{[33– 36]} It is demonstrated that vacancy defects have a great effect on the axial mechanical characteristics of individual SWCNTs.^[24] Herein, we mainly concentrate on the responses of defective SWCNT (10, 10) and SWCNT (17, 0) to radial mechanical load. The one type of defect studied is a divacancy defect, as depicted in Figs. 1(c) and 1(d), which will be referred to as non-reconstructed defects.^[24] The other type of defect studied is 5-8-5 defect, which is produced by saturating the dangling bonds with the adjacent carbon atoms. These configurations are presented in Fig. 1(e) for SWCNT (10, 10) and in Fig. 1(f) for SWCNT (17, 0). As is well known, the defect– defect interaction can influence the structural phase transition. Therefore, the lengths of all SWCNTs are set to be about 25 Å , thus the defect density (0.04 Å ^{− 1}) of SWCNT is low enough and this interaction does not significantly disturb the onset of radial collapse.

Fig. 1.

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	Fig. 1. Intrinsic SWCNTs (10, 10) and SWCNTs (17, 0) structure ((a) and (b)), and divacancy defect ((c) and (d)), and 5-8-5 defect ((e) and (f)).

MD and MM simulations, which are implemented using a commercial software package called Materials Studio (MS) developed by Accelrys Software, Inc, are conducted to investigate the radial collapse of a series of SWCNTs with divacancy and 5-8-5 defects. In all of our MM and MD simulations, we have used COMPASS force-field, parametrized, tested and validated first ab initio force-field, which enables an accurate prediction of various gas-phase and condensed-phase properties of most of the common organic and inorganic materials.^{[37– 40]} Moreover, it has been successfully used to study the mechanical properties and radial collapses of CNTs under hydrostatic pressure.^{[27– 32, 41, 42]}

In the present paper, periodic boundary conditions are used in all systems, including intrinsic SWCNTs (10, 10) and (17, 0) and SWCNTs with divacancy and 5-8-5 defects with the same diameter and length of 25 Å . First, MM simulations are performed to find the thermal stable morphology and achieve a conformation with minimum potential energy for all SWCNTs. Second, MM simulations are carried out to optimize all SWCNT systems at atmospheric pressure, and the systems are subsequently subjected to step-wise monotonically hydrostatic pressure increments to investigate the radial collapses of all systems with FORCITE mode. After each pressure increment the unit cell volume is allowed to equilibrate for at least 30 ps at each step in the NPT ensemble, and a fixed time step size of 1 fs is used in all cases. An Andersen thermostat method is employed to control the system at a temperature of 1 K to avoid the thermal effect, and the interactions are determined within a cutoff distance of 9.5 Å . The reduced volume ratio (V/V₀) gives information about the structural transition, which is obtained by measuring the unit cell volume after equilibration at each hydrostatic pressure.

As is well known, there is at least one defect in high-quality CNTs per 4 μ m.^[44] Single vacancy and multi-vacancy are some of the most abundant and important point defects. It has been found that vacancy can deteriorate the deformation response of CNTs and further their mechanical characteristics and can also be underlying locations where an irreversible deformation response is started. However, defects formed by saturating dangling bonds of vacancies defects can give rise to many advantageous effects. For example, it may partly alleviate the deterioration in their mechanical characteristics and even enhance mechanical properties of SWCNT.^[45]

In this section, we discuss the effects of divacancy and 5-8-5 defects on radial mechanical properties of SWCNT (10, 10) and SWCNT (17, 0). Figures 1(c)– 1(f) demonstrates the structures of SWCNT (10, 10) and SWCNT (17, 0) with divacancy and 5-8-5 defects, and their loading and unloading curves are shown in Fig. 2. It can be found that the divacancy and 5-8-5 defects have little influence on the radial elasticity of SWCNT (10, 10). All intrinsic SWCNT (10, 10), SWCNT (10, 10) with vacancy defect, SWCNT (10, 10) with 5-8-5 defect can return to their origin states. However, a 5-8-5 defect has a significant influence on the radial elasticity of SWCNT (17, 0), it make inelastic SWCNT (17, 0) change into elastic SWCNT (17, 0), which indicates that the 5-8-5 defect formed by saturating dangling bonds of divacancy defect can partly enhance the mechanical properties of SWCNT.

In order to further study the effects of divacancy and 5-8-5 defects on the radial collapse of SWCNT, we list the values of P_c of the SWCNTs in Table 3. It is found that the values of P_c of the intrinsic SWCNT (10, 10), SWCNT (10, 10) with divacancy, and 5-8-5 defects are 5.5, 4.1, and 4.9 GPa, respectively, while the values of P_c of the intrinsic SWCNT (17, 0), SWCNT (17, 0) with divacancy, and 5-8-5 defects are 0.4, 0.35, and 2.4 GPa, respectively. It is demonstrated that the divacancy defect can make the value of P_c of SWCNT (10, 10) decrease by 25.0% and make the value of P_c of SWCNT (17, 0) decrease by 12.5%, while 5-8-5 defect can make the value of P_c of SWCNT (10, 10) decrease by 11.0%. It is shown that divacancy defect can deteriorate the radial mechanical properties of SWCNTs while the 5-8-5 defect can alleviate the deterioration in their radial mechanical properties. To our surprise, the 5-8-5 defect can make the value of P_c of SWCNT (17, 0) increase by 500%. In other words, the 5-8-5 defect formed by saturating dangling bonds of divacancy defect can greatly enhance the radial mechanical properties of SWCNT (17, 0).

Based on the above discussion, the 5-8-5 defect in SWCNTs can enhance the mechanical properties of SWCNTs, which can be desired to improve the mechanical characteristics of SWCNT-based composite and also provide the mechanical stability for electric nano-circuit formed by SWCNTs with covalent inter-tube junctions.

Chirality, divacancy, and 5-8-5 defects have significant influence on the radial collapse of SWCNTs. The SWCNT capacity of resistance to collapse is directly related to its structure. It is recognized that the elementary unit of SWCNT is a six-member ring composed of six C− C bonds. The bending capacity of six C− C bonds along the bending direction should determine radial mechanical characteristics of SWCNT. Herein, a model is established to represent the bending capacity of six C− C bonds and further explain the SWCNT capacity of resistance to collapse.^[31]

The bending directions of six C− C bonds in SWCNT (10, 10) and SWCNT (17, 0) upon loading are shown in Fig. 3. We define L as the projection of six C− C bond length along bending direction. L represents the bending capacity of six C− C bonds. In other words, the larger the L, the more difficultly the six C− C bond bends and the larger the value of P_c which the SWCNT is corresponding to will be.^[31] It is easy to calculate the L₁ (L of a six-member ring in SWCNT (10, 10)) = 5.728 Å > L₂ (L of a six-member ring in SWCNT (17, 0)) = , 4.960 Å , which is demonstrated because the SWCNT (17, 0) is easier to bend and collapse than SWCNT (10, 10).

Fig. 3.

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	Fig. 3. Bending directions of six C− C bonds in (a) SWCNT (10, 10) and (b) SWCNT (17, 0) upon loading.

Now, the model is applied to the SWCNT with divacancy or 5-8-5 defects. Figure 4 shows the atomic networks of four six-member rings, corresponding to divacancy and 5-8-5 defects in SWCNT (10, 10) and SWCNT (17, 0). A divacancy defect is formed by deleting two carbon atoms of four six-membered rings, as shown in Figs. 4(c) and 4(d). The 5-8-5 defect can be formed by saturating dangling bonds of divacancy defect, as shown in Figs. 4(e) and 4(f). We can obtain the structures (b), (d), and (f) by rotating structures (a), (c), and (e) 30° towards the right. As shown in Table 3, it is calculated that L_a (L of four six-member rings in SWCNT (10, 10)) = 15.09 Å , L_b (L of four six-member rings in SWCNT (17, 0)) = 10.39 Å , L_c (L of corresponding divacancy defect in SWCNT (10, 10)) = 10.73 Å , L_d (L of corresponding divacancy defect in SWCNT (17, 0)) = 6.93 Å , and L_a > L_c, L_b > L_d, which indicates that the SWCNT with divacancy defect is easier to collapse.

Fig. 4.

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	Fig. 4. The atomic networks and bending directions of four six-member rings ((a) and (b)), corresponding divacancy defects ((c) and (d)) and corresponding 5-8-5 defects ((e) and (f) in SWCNT (10, 10) and SWCNT (17, 0).

It has been demonstrated that a variety of different polygons, such as hexagon, pentagon, and octagon are composed of sp₂-hybridized carbon atoms.^[46] Therefore, the current model should be applicable to SWCNTs with the 5-8-5 defect. Figure 5 shows the bond lengths and bond angles of the 5-8-5 defect structure which are marked by b₁, b₂, b₃, b₄, b₅, and α , β , γ , φ , respectively. Therefore, the L of the 5-8-5 defect along bending direction can be calculated as follows.

Fig. 5.

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	Fig. 5. Lengths and angles of 5-8-5 defect structure.

In Fig. 4(e), L_e (L of corresponding 5-8-5 defect in SWCNT (10, 10)) = 2b₁cos(2π /3 − α /2) + 2b₁sin(α /2 + π /6) + 2b₂cos(4π /3 − β − α /2) + 2b₂cos(β + α /2 − 7π /6) + 2b₃cosπ /6 + 2b₅cosπ /3 + 2b₄cos(φ − π /3) + 2b₄cos(φ − 2π /3) = 13.77 Å .

In Fig. 4(f), L_f (L of corresponding 5-8-5 defect in SWCNT (17, 0)) = 2b₃ + 4b₄cos(α /2) + 4b₂sin(γ − π /2) + 4b4sin(φ − π /2) = 14.39 Å .

According to the above calculations, we can obtain L_a > L_e > L_c and L_f > L_b > L_d, andP_{c(intrinsic SWCNT (10, 10))} (5.5 GPa) > P_{c(SWCNT (10, 10) with 5-8-5)} (4.9 GPa) > P_{c(SWCNT (10, 10) with divacancy)} (4.1 GPa)andP_{c(SWCNT (17, 0) with 5− 8− 5)} (2.4 GPa) > P_{c(intrinsic SWCNT (17, 0))} (0.4 GPa) > P_{c(SWCNT (17, 0) with divacancy)} (0.35 GPa).This indicates that the results are in good agreement with the law that the larger the value of L, the bigger the value of P_c of the corresponding SWCNT is, and therefore is the effect law of the familiar 5-7-7-5 defect on the radial collapse of SWCNT.^[31] Based on the above discussion, it is demonstrated that our model is very applicable to the understanding of the influences of chirality and defect on radial mechanical properties of SWCNT. Therefore, the established model is very useful to guide the application of CNTs with detects in high-load composites.