3.2. Mechanical Properties of Single-Walled C3NNTs
The Young’s modulus of each SWC
3NNT was determined as follows. After plotting the stress–strain (
σ–
ε) curves, a second-order polynomial was fitted to the linear part of the curves, and the Young’s modulus was calculated using Equation (1) [
39,
40]:
where
D,
E, and
C are the third-order elastic modulus, the Young’s modulus, and the residual stress of the nanotubes, respectively. The stress–strain curve of a (6,6) armchair C
3NNT at 300 K displayed in
Figure 3a,b shows a zoom of the linear part used for the second-order polynomial fit.
According to
Figure 3a, the highest stress obtained is observed at a strain of about 40%. Then, the stress drops to a considerably lower amplitude, therefore the coordinates of this point correspond to the values of failure stress and failure strain. We repeated these steps for all SWC
3NNTs and then for SWCNTs to compare and validate our results. The plots of Young’s modulus, failure stress, and failure strain are displayed in
Figure 4. This figure shows that the variation of all the studied properties as a function of the radius of the nanotubes is rather limited. Indeed, as seen in
Figure 4a, by increasing the radius of the nanotubes, the Young’s modulus of armchair SWC
3NNTs increases by about 24 GPa, passing from 951.6 GPa in structure (4,4) to 975.5 GPa in structure (10,10), and then drops to 970.8 GPa in structure (12,12). Similarly, an upward behavior is observed in the Young’s modulus of armchair CNTs, except for the thickest ones where the values tend to stabilize, rising from 983.3 GPa in structure (4,4) to 1085.4 GPa in structure (12,12).
We observed the same behavior by comparing the Young’s modulus of zigzag and armchair SWC
3NNTs. The Young’s modulus of zigzag SWC
3NNTs increases slightly, from 903.8 GPa in (8,0) to 935.2 GPa in (20,0), similar to the upward trend of armchair SWC
3NNTs. A general comparison between the modulus of SWC
3NNTs and SWCNTs reveals that the values obtained for SWC
3NNTS are much lower than for the corresponding SWCNTs at any chirality or radius considered. The higher mechanical behavior of CNTs compared to C
3NNTs is caused by the length of the C-C bond, which is about 1.445 Å in the ideal structure of CNTs, while the length of the C-N bond in C
3Ns is about 1.468 Å. The shorter bond length between the elements of the nanostructures leads to higher mechanical properties as demonstrated by Ghorbanzadeh et al. [
16]. As mentioned earlier, we modeled CNTs in the present paper to have the possibility of validating our simulation and obtaining results for C
3NNTs. The Young’s modulus that we calculated for SWCNTs is around 1000 GPa, which is close to what has already been reported in previous theoretical articles, ranging from 0.5 to 1.5 TPa [
41,
42,
43,
44,
45], as well as in experimental reports [
46,
47,
48]. For instance, Treacy et al. reported a value of around 2 TPa for the elastic modulus of individual SWCNTs of different diameters and lengths [
46], and Yu et al. reported an average modulus of 1002 GPa for 8 SWCNT ropes using atomic force microscopy (AFM) [
48]. These results demonstrate the accuracy of our simulation and therefore also of our results for C
3NNTs.
Furthermore, from
Figure 4b,c, we can see that the failure stress and failure strain of zigzag SWC
3NNTs and SWCNTs increase with the radius of the nanotubes, while those of armchair SWC
3NNTs and SWCNTs decrease under the same conditions. The highest failure stress and failure strain, equal to 282.6 GPa and 0.44%, respectively, were found for armchair (6,6) and (4,4) C
3NNTs, respectively. In an MD-case study, Shirazi et al. reported the ultimate tensile strength of armchair and zigzag C
3N sheets at 300 K, equal to 128 GPa and 125 GPa, which are both lower than what we obtained for our strongest structures (250.21 GPa for armchair and 138.29 for zigzag) [
32]. The results calculated for the mechanical properties of armchair and zigzag SWC
3NNTs and SWCNTs are listed in
Table 2 and
Table 3.
To take the effect of temperature into account, (10,10) and (18,0) single-walled C
3NNTs and the CNTs with the closest dimensions were modeled and subjected to a uniaxial tensile loading while the temperature increased from 300 to 900 K. The calculated results are shown in
Figure 5.
All the aforementioned mechanical properties decreased with temperature, whether for CNTs or for C
3NNTs. Thus, the highest values for all tested samples were obtained at 300 K and the lowest values at 900 K. The (10,10) armchair C
3NNT had a higher elastic modulus at any temperature compared to the (18,0) zigzag structure, and the C
3NNTs showed a lower modulus than the CNTs. The Young’s modulus of (18,0) and (10,10) C
3NNTs were respectively 2% and 5% lower at 900 K than at 300 K. Likewise, the failure stress and failure strain of all the samples decreased when the temperature increased. However, although the failure stress of zigzag and armchair C
3NNTs was lower than those of the corresponding CNTs at most temperatures, these structures generally failed at a higher strain rate than those of CNTs. The same observations were reported by Shirazi et al. [
32]. They noted a total decrease of 36% for the stress at failure of C
3N sheets at 900 K compared to 200 K, in accordance with our own finding.
3.3. Mechanical Properties of Double-Walled C3NNTs
At this point, to probe the effect of adding walls to C
3NNTs on their mechanical properties, we modeled and tested four zigzag and six armchair double-walled C
3NNTs (DWC
3NNTs) as well as the corresponding DWCNTs. It should be mentioned that there are some limitations to the modeling of multi-walled nanotubes, in particular regarding the interlayer distances. If the interlayer distance exceeds a certain value, no van der Waals interaction occurs between layers and the structure does not form. On the other hand, if the distance were less than a specific value, the structure would collapse because of instability. Therefore, to have an armchair and zigzag structures with the closest dimensions enabling reasonable comparisons to be made, we selected the distances according to the schematic views displayed in
Figure 6. Thus, we could model stable structures with close and comparable dimensions.
After testing all the modeled samples under uniaxial tensile loading at a constant temperature of 300 K, the mechanical properties were plotted in
Figure 7. As seen in
Figure 7a, by increasing the interlayer distance, the Young’s modulus of all samples first increased and then constantly decreased to slightly lower values. Unlike single-walled nanotubes, the Young’s modulus of zigzag DWC
3NNTs and SWCNTs was higher than that of armchair DWC
3NNTs of similar radius, while DWC
3NNTs had a lower elastic modulus compared to DWCNTs, as already observed for single-walled nanotubes. The results of double-walled armchair and zigzag C
3NNTs and CNTs are collected in
Table 4 and
Table 5, respectively. The highest values obtained for DWC
3NNTs occurred in structures (8,0),(16,0) and (4,4),(8,8), 1448.7 GPa and 1418.6 GPa, respectively, and this property was about 10% lower in the weakest structure with respect to the strongest one, whatever the chirality.
Besides, making comparisons between the results of single and double-walled C
3NNTs reveals that adding a wall to these nanotubes increased the elastic modulus so that DWC
3NNTs have a considerably higher elastic modulus than SWC
3NNTs. In addition, and just like SWCNTs, DWCNTs showed a higher Young’s modulus than DWC
3NNTs regardless of the chirality. The Young’s modulus of DWCNTs was higher than that of SWCNTs, just like for C
3NNTs. The highest Young’s modulus of DWCNTs found in the zigzag structure (8,0),(16,0), 1553.3 GPa, is close to that reported earlier [
49,
50], further supporting the accuracy of our simulation and the results calculated for DWC
3NNTs.
Finally, considering
Figure 7b,c, we found that failure stress and failure strain of zigzag DWC
3NNTs were nearly two times lower than those of armchair DWC
3NNTs having a close radius. No chirality showed a significant trend by increasing the radius, and a similar trend was observed for DWCNTs. We obtained the highest failure stress and failure strain of DWC
3NNTs in the (4,4),(8,8) armchair: 374.4 GPa and 0.435%, respectively.
3.4. Mechanical Properties of Triple-Walled C3NNTs (TWC3NNTs)
In this section, we added one more wall to DWC
3NNTs to compare the mechanical behavior of TWC
3NNTs to that of SWC
3NNTs and DWC
3NNTs. For that purpose, we modeled and tested one zigzag and one armchair TWC
3NNT of structures (8,0),(14,0),(20,0) and (4,4),(8,8),(12,12), respectively, and the corresponding TWCNTs. A schematic view of the modeled structures is displayed in
Figure 8, and the obtained results are presented in
Table 6.
By examining this table, we found a significant growth in the Young’s modulus of TWC3NNTs compared to double- or single-walled nanotubes, whether CNTs or C3NNTs, and regardless of chirality. The Young’s modulus of armchair TWC3NNT (1850.4 GPa) was higher than that of zigzag TWC3NNT (1760.7 GPa), unlike what we had obtained for DWC3NNTs. In addition, the Young’s modulus of TWC3NNTs was lower than that of TWCNTs with the same structure, as for single-walled C3NNTs and CNTs. Besides, the failure stresses of armchair TWC3NNT and TWCNTs were not only higher than for zigzag TWC3NNTs and TWCNTs, but also higher than for double- and single-walled nanotubes.
Moreover, we compared the mechanical properties of single-walled (4,4) and (8,0) C
3NNTs with double- and triple-walled C
3NNTs made up of these two basic structures in
Figure 9. From this figure, it can be observed that the addition of walls to the single-walled nanotubes had a remarkable impact on the Young’s modulus of both C
3NNTs and CNTs, regardless of the chirality. By increasing the radius following the addition of one then two more walls to the SWC
3NNTs, the modulus of (4,4) single-wall armchair C
3NNT increased by 32% and 48%, respectively, in structures (4,4),(8,8) and (4,4),(8,8),(12,12), respectively. Similarly, the modulus of (8,0),(14,0) and (8,0),(14,0),(20,0) zigzag structures were respectively 33% and 46% higher than that of the (8,0) SWC
3NNT. The same behavior is observed in TWCNTs compared to their corresponding double and single-walled counterparts.
3.5. Mechanical Properties of C3N Nanobuds
Nanobuds are 3D nanostructures that form when a fullerene or nanocage is randomly attached to the outer surface of nanotubes or graphenic structures in the synthesis process. The specific features of nanocages and fullerenes, including their porous shells and nanometric thickness, make them an appropriate choice for developing novel 3D nanostructures [
51]. In the present work, we attached randomly one, two, three, and four C
60 fullerene molecules to the outer surface of zigzag and armchair SWC
3NNTs to form C
3N nanobuds. One armchair and one zigzag C
3NNTs with the closest dimension of structures (10,10) and (18,0) were modeled and tested. The plots of the mechanical properties of armchair and zigzag SWC
3NNTs after attaching one, two, three, and four C
60 to their surface are given in
Figure 10.
Similar to what we had seen above for SWC
3NNTs, the mechanical properties of armchair nanobuds were higher than zigzag nanobuds, and all the studied properties decreased constantly as the number of C
60 increased. This could be due to first, the increase in the effective surface area of the nanobuds with the number of attached fullerenes, and second to the associated increase in stress concentration, a higher number of attached fullerenes implying a higher stress concentration. In addition, the properties of C
3N nanobuds were lower than for simple SWC
3NNTs, either in zigzag or armchair structures. The highest elastic modulus was calculated for structures (10,10)-1C
60: 874.5GPa, and (18,0)-1C
60: 865.8 GPa, i.e., was 10% and 7% lower than (10,10) and (18,0) SWC
3NNTs, respectively. With four C
60 attached to armchair and zigzag nanobuds, the modulus was reduced by almost 20% compared to those with only one C
60. The same kind of results have been reported by other studies on other types of nanobuds. Mashhadzadeh et al. in their DFT-based research, reported that the Young’s modulus of graphene-like BeO reduced considerably by increasing the number of attached nanocages [
20]. Ghorbanzadeh et al. used DFT calculations to compare the mechanical properties of simple CNTs with CNT nanobuds. They found a reduction in Young’s modulus of armchair and zigzag CNTs after attaching a C
60 molecule to their surface. The values obtained for the mechanical properties of C
3N nanobuds are presented in
Table 7.
Figure 11 shows a snapshot of the failure process of a (18,0)-1C
60 SWC
3NNT. It can be seen that, as expected, the failure started around the region where the fullerene was placed on the nanotube surface. This is due to the higher stress concentration existing around this region, which facilitates the formation and propagation of cracks.
3.6. Mechanical Properties of Defective C3NNTs
In the end, we examined the effect of point defects on the mechanical properties of SWC
3NNTs. (10,10) armchair and (18,0) C
3NNTs were selected and we modeled the defective samples with one and two vacancies as well as Stone–Wales defects. All the designed samples are presented in
Figure 12, where the two types of Stone–Wales defects can be seen. Type one (STW-1) forms when a horizontal C-N (or C-C) bond rotates 90 degrees, and type 2 (STW-2) forms once a skewed C-N (or C-C) bond rotates 90 degrees.
After having implemented tensile tests at a constant temperature of 300 K and at constant strain rate ε of 10
8 s
−1, the results are presented as bar graph in
Figure 13. According to this figure, creating defects on the surface of armchair and zigzag C
3NNTs resulted in a reduction of all mechanical properties compared to pristine SWC
3NNTs. The Young’s modulus of different defected zigzag and armchair C
3NNTs are close to each other (with higher values for most types of armchair), with very little differences and no significant trend. However, the failure stress and failure strain of defective armchair C
3NNTs are considerably higher than the corresponding zigzag C
3NNTs. The results obtained for each property are collected in
Table 8 for each type of defect.
This table shows that the highest reduction in all properties occurred when two atoms were removed from the surface of zigzag and armchair SWC
3NNTs. In contrast, defects of types STW-1 and STW-2 resulted in the lowest reduction in properties compared to pristine SWC
3NNTs. The lowest Young’s modulus, failure stress, and failure strain, 900.3 GPa, 122.68 GPa, and 0.182%, respectively, were about 7%, 9%, and 15% lower than for the corresponding pristine SWC
3NNTs. These values corresponded to two-atom vacancy C
3NNTs, including armchair structure defect type c (2-C), zigzag structure defect type d (2-C), and zigzag structure defect type e (1-C, 1-N), respectively. The same kind of results have been reported by previous researches. Shirazi et al. provided the same results for C
3N nanosheets with crack-type defects [
32]. They showed that increasing the crack length could significantly decrease the mechanical response of the defective sheets. However, Sadeghzadeh et al. observed different results for their C
3N sheet depending on the vacancy concentration [
31]. They demonstrated that defective C
3N had higher elastic modulus and failure strain than the defect-free sheets, which is different from the findings of Shirazi’s and ours. In another article, an adverse effect of point defects on the mechanical properties of graphene-like ZnO structures was reported by Ghorbanzade et al. [
15]. In a MD-based study, Albooye et al. also reported that increasing the number of missing atoms reduced the Young’s modulus of defective BNNTs so that the highest modulus was obtained in pristine structures, and the lowest modulus was observed in those containing three-atom vacancies [
26]. In another MD studies, Gupta et al. investigated Young’s modulus, failure stress, and failure strain of hybrid single-layer graphene; they reported a reduction in the behavior of all properties with respect to non-defective graphene monolayers by imposing Stone–Wales and nanopore defects [
52].
Furthermore, a snapshot of the failure process of a (10,10) armchair SWC
3NNT including a two-atom vacancy defect is presented in
Figure 14. Similar to what happens in nanobuds, the failure of defective C
3NNT begins from the defective region due to its higher stress concentration, and then the cracks propagate until the complete rupture of the structure.