Azam Salmankhani
email: azamsalmankhani@gmail.com

Zohre Karami
email: zohrekarami.2013@yahoo.com

Hamed Amin
email: amin.hamed.m@gmail.com

Maryam Zarghami Mashhadzadeh

Mohammad Reza Dehaghani

Vanessa Saeb

Alain Fierro

Mohammad Reza Celzard

Saeb

Amin Hamed Mashhadzadeh
email: amin.hamedmashhadzadeh@yahoo.com

Maryam Zarghami Dehaghani
email: maryam_zz67@yahoo.com

Mohammad Reza Saeb

Vanessa Fierro
email: vanessa.fierro@univ-lorraine.fr

Alain Celzard
email: alain.celzard@univ-lorraine.fr

A theoretical scenario for the mechanical failure of boron carbide nanotubes

Keywords: BC 3 nanotubes, Molecular Dynamics, Mechanical Properties, Nanobud, Defect

Introduction

Carbon nanotubes (CNTs) are best known for their excellent mechanical strength, high thermal and electrical conductivity and good sensing capability, making them excellent candidates for the development of technical materials and systems (see Figure 1S in the Supplementary Information, SI) [START_REF] Saeed | Carbon nanotubes-properties and applications: a review[END_REF][START_REF] Rao | Carbon Nanotubes and Related Nanomaterials: Critical Advances and Challenges for Synthesis toward Mainstream Commercial Applications[END_REF]. Chemical doping with heteroelements is a promising technique for improving their mechanical, electronic, optical and chemical properties [START_REF] Talla | Effect of boron and nitrogen co-doping on CNT's electrical properties: Density functional theory[END_REF][START_REF] Chen | Optical properties of BC3 nanotubes[END_REF][START_REF] Miyamoto | Electronic properties of tubule forms of hexagonal ${\mathrm{BC}}_{3}[END_REF][START_REF] Jana | Effect of chemical doping of boron and nitrogen on the electronic, optical, and electrochemical properties of carbon nanotubes[END_REF][START_REF] Wei | Electrical transport in pure and boron-doped carbon nanotubes[END_REF][START_REF] Zhang | Theoretical Studies on the Electronic and Optical Properties of Honeycomb BC(3) monolayer: A Promising Candidate for Metal-free Photocatalysts[END_REF].

A particular type of boron-doped CNTs can be obtained when a quarter of the carbon in the graphene is replaced by boron atoms, resulting in a new structure called BC 3 graphene. Like CNTs, BC 3 nanotubes (BC 3 NTs) are formed when a BC 3 sheet is wound along its chiral vector [START_REF] Zhao | Electronic and Photonic Properties of Doped Carbon Nanotubes[END_REF][START_REF] Borowiak-Palen | Formation of novel nanostructures using carbon nanotubes as a frame[END_REF][START_REF] Tomanek | Calculation of electronic and structural properties of BC3[END_REF]. For BC 3 compounds, only the B-C and C-C bonds are considered stable, while the B-B bonds are neglected because they are less stable. BC 3 nanosheets have good temperature resistance and are of high crystalline quality [START_REF] Tanaka | Novel macroscopic BC3 honeycomb sheet[END_REF].

The mechanical properties and failure mechanism of the nanostructures are of great importance, particularly at high temperatures [START_REF] Dadrasi | Mechanical properties of silicon-germanium nanotubes: A molecular dynamics study[END_REF][START_REF] Memarian | Molecular dynamic study of mechanical properties of single/double wall SiCNTs: Consideration temperature, diameter and interlayer distance[END_REF]. Although BC 3 NTs have been extensively studied from an electronic or chemical perspective, less attention has been paid to understanding the mechanical failure of BC 3 NTs. In a recent work, we have modeled and simulated the mechanical properties of either defect-free or defective C 3 N nanotubes (C 3 NNTs) and their nanobuds, and analyzed and compared the results with the corresponding modeled CNTs as a reference/control geometry for validation [START_REF] Salmankhani | Mechanical Properties of C3N Nanotubes from Molecular Dynamics Simulation Studies[END_REF].

In this framework, we have developed BC 3 nanotubes by replacing a quarter of the CNTs atoms with boron atoms in order to explore the effect of chirality, tube diameter, number of walls and temperature on the elastic modulus, failure stress and failure strain of flawless and defective BC 3 NTs, as well as BC 3 nanobuds. Since no research has reported on the mechanical properties of BC 3 NTs, the results of our previous work on corresponding CNTs with the same chirality and under the same loading conditions have been considered as a reference to theoretically validate our new results obtained by Molecular Dynamics (MD) simulation [START_REF] Salmankhani | Mechanical Properties of C3N Nanotubes from Molecular Dynamics Simulation Studies[END_REF].

Computational design

The molecular dynamics modeling in the present study was performed using the large-scale atomic/molecular massively parallel simulator (LAMMPS) package to predict the mechanical properties of BC 3 NTs in either armchair or zigzag chirality. To model C-C bonds, we selected the modified Tersoff potential presented by Lindsay and Broido [START_REF] Lindsay | Optimized Tersoff and Brenner empirical potential parameters for lattice dynamics and phonon thermal transport in carbon nanotubes and graphene[END_REF].

Moreover, to define the interaction of carbon and boron atoms, we used the Tersoff potential parameters developed by Kinaci et al. [START_REF] Kınacı | Thermal conductivity of BN-C nanostructures[END_REF]. Armchair and zigzag BC 3 NTs with an approximate length of 5 nm were modeled and the results obtained were compared with those of the corresponding carbon nanotubes (CNTs) as reference materials from our previous article [START_REF] Salmankhani | Mechanical Properties of C3N Nanotubes from Molecular Dynamics Simulation Studies[END_REF]. To calculate the mechanical properties, the stress-free structures at the simulated temperature were first performed with the isothermal-isobaric (NPT) ensemble at 300 K (pressure of 1 bar) with the time span of 100 ps. Next, the simulation box was stretched along the loading axial direction (z-direction) with the strain rate of 10 9 s -1 . To measure the effects of temperature, single-walled nanotubes in either zigzag or armchair chirality were modeled and tested at temperatures varying from 300K to 900K. Besides, the effects of wall addition, fullerene attachment, and different types of defects on the elastic modulus, failure stress and failure strain of BC 3 nanotubes were determined under the same loading conditions. We have provided a brief flowchart of the total modeling process used in the present study in Figure 1.

This flowchart follows the same procedure as that used in our previous article [START_REF] Salmankhani | Mechanical Properties of C3N Nanotubes from Molecular Dynamics Simulation Studies[END_REF].

Results and discussion

Mechanical properties of single-walled BC 3 NTs

To study the mechanical properties, we first modeled zigzag and armchair single-walled BC 3 nanotubes (SWBC 3 NTs) and their corresponding CNTs using molecular dynamics (MD) simulation. A schematic of the modeled SWBC 3 NTs and their stoichiometry are shown in Figure S2 and Table 1, both derived from the reference materials [START_REF] Salmankhani | Mechanical Properties of C3N Nanotubes from Molecular Dynamics Simulation Studies[END_REF]. The values of radius corresponding to each chirality are tabulated in Table S1. In Figure S2 and Table 1, DWBC 3 NTs and TWBC 3 NTs refer to double-walled and triple-walled nanotubes, respectively. Next, all samples were subjected to uniaxial tensile loading with a strain rate of 0.001 ps -1 when the temperature was set at 300K, and the stress-strain ( -ε) curves were extracted to calculate Young's modulus, failure stress, and failure strain. A schematic view of a (8,0) zigzag BC 3 NT under uniaxial tensile loading is shown in Figure S3. To obtain the Young's modulus, a second-order polynomial was fitted to the linear part 107 of the stress-strain curves using the following equation [START_REF] Albooyeh | Effect of point defects and low-density carbon-doped on mechanical properties of BNNTs: A molecular dynamics study[END_REF]. 108 [START_REF] Saeed | Carbon nanotubes-properties and applications: a review[END_REF] where E, D, and C are the Young's modulus, third-order elastic modulus and residual stress 109 of the nanotubes, respectively. A stress-strain curve plotted for a [START_REF] Jana | Effect of chemical doping of boron and nitrogen on the electronic, optical, and electrochemical properties of carbon nanotubes[END_REF][START_REF] Jana | Effect of chemical doping of boron and nitrogen on the electronic, optical, and electrochemical properties of carbon nanotubes[END_REF] armchair BC 3 NT is 110 presented in Figure 2a, and Figure 2b shows a zoom of the linear part of the plot used for the 111 second-order polynomial fit. As seen in Figure 2a, the highest stress obtained is observed at a 112 strain rate of about 40%. Then, the stress drops to a considerably lower amplitude. Therefore, 113 the coordinates of this point correspond to the values of failure stress and failure strain. to the maximum value of 780.5 GPa in structure [START_REF] Borowiak-Palen | Formation of novel nanostructures using carbon nanotubes as a frame[END_REF][START_REF] Borowiak-Palen | Formation of novel nanostructures using carbon nanotubes as a frame[END_REF] and then decreased to 776.7 GPa in structure [START_REF] Tanaka | Novel macroscopic BC3 honeycomb sheet[END_REF][START_REF] Tanaka | Novel macroscopic BC3 honeycomb sheet[END_REF]. Similarly, an upward behavior was observed in zigzag SWBC 3 NTs, i.e., the Young's modulus increased from the lowest value of 718.4 GPa in structure (8,0) to the highest value of 751.7GPa in structure (16,0), and then decreased slightly to 741.5 GPa in structure [START_REF] Hsieh | Theoretical variations in the Young's modulus of single-walled carbon nanotubes with tube radius and temperature: A molecular dynamics study[END_REF]0). By increasing the radius, the elastic modulus of the armchair and zigzag SWBC 3 NTs increased by approximately 7% and 4%, respectively. The same behavior was observed for the Young's modulus of armchair SWCNTs while zigzag SWCNTs remained almost unchanged by increasing the radius of nanotubes [START_REF] Salmankhani | Mechanical Properties of C3N Nanotubes from Molecular Dynamics Simulation Studies[END_REF]. The results previously obtained [START_REF] Salmankhani | Mechanical Properties of C3N Nanotubes from Molecular Dynamics Simulation Studies[END_REF] with armchair and zigzag SWCNTs as reference materials were in the range of 1 TPa, which is close to other data [START_REF] Van Lier | Ab initio study of the elastic properties of single-walled carbon nanotubes and graphene[END_REF][START_REF] Hsieh | Theoretical variations in the Young's modulus of single-walled carbon nanotubes with tube radius and temperature: A molecular dynamics study[END_REF][START_REF] Mohammadpour | Predicting the Young's Modulus of Single-Walled Carbon Nanotubes using Finite Element Modeling[END_REF], thereby demonstrating the accuracy of the used model and the obtained results for SWBC 3 NTs. In their experimental study, Yu et al. reported the average modulus obtained for 8 SWCNT ropes using atomic force microscopy, equal to 1002

GPa [START_REF] Yu | Tensile Loading of Ropes of Single Wall Carbon Nanotubes and their Mechanical Properties[END_REF]. Furthermore, Treacy et al. calculated an elastic modulus in the range of 2TPa for individual SWCNTs of different lengths and radii [START_REF] Treacy | Exceptionally high Young's modulus observed for individual carbon nanotubes[END_REF]. Moreover, the failure stress plots in Figure 3b reveal that armchair SWBC 3 NTs have higher stress at break than zigzag SWBC 3 NTs (almost three times as high), and the same behavior was observed in armchair SWCNTs compared to zigzag ones as derived from reference samples [START_REF] Salmankhani | Mechanical Properties of C3N Nanotubes from Molecular Dynamics Simulation Studies[END_REF]. The results obtained for armchair and zigzag SWBC 3 NTs and their corresponding SWCNTs are gathered in Tables S2 andS3, respectively. In addition, by increasing the radius of the nanotubes, the failure stress of the armchair SWBC 3 NTs slowly decreased, while on the other hand, the failure stress of zigzag SWBC 3 NTs increased (in the limits of certain fluctuations). The highest failure stress of the armchair and zigzag SWBC 3 NTs obtained was observed in structures [START_REF] Chen | Optical properties of BC3 nanotubes[END_REF][START_REF] Chen | Optical properties of BC3 nanotubes[END_REF] and (20,0) with values of 283.71 GPa and 107.6 GPa, respectively. Figure 3c shows the failure strain of the SWBC 3 NTs and their corresponding reference SWCNTs (from [START_REF] Salmankhani | Mechanical Properties of C3N Nanotubes from Molecular Dynamics Simulation Studies[END_REF]). According To this figure, and as already noted for elastic modulus and failure stress, the failure strain of zigzag and armchair SWBC 3 NTs was higher than or very close to that of SWCNTs with close radius, and the failure strain of the armchair SWBC 3 NTs was higher than that of the zigzags.

While the failure strain of the armchair SWBC 3 NTs and SWCNTs was almost unchanged, this property decreased significantly from the highest value of 0.295% in zigzag structure (8,0) to the lowest value of 0.205% in structure (16,0), followed by a small increase to 0.218% in structure (20,0). Overall, armchair SWBC 3 NTs exhibited higher mechanical properties than zigzag SWBC 3 NTs.

To understand the effects of temperature on the mentioned properties, we selected a zigzag and an armchair SWBC 3 NT with the closest radius, of structures (18,0) and (10,10), respectively, and examined their properties while the temperature was varied from 300K to 900K as shown in Figure 4. We also used the results of our previous article for comparison with reference CNTs subjected to the same range of temperature [START_REF] Salmankhani | Mechanical Properties of C3N Nanotubes from Molecular Dynamics Simulation Studies[END_REF]. This figure shows that temperature had a negative effect on the mechanical properties of both SWBC 3 NTs and SWCNTs. The Young's modulus of the armchair SWBC 3 NTs indeed constantly decreased from 780.5 GPa at 300K to 711.2 GPa at 900K (i.e., by 8%) while the modulus of zigzag SWBC 3 NTs was reduced from 746.4 GPa at 300K to 668.4 GPa at 900K (i.e., by 10%).

Besides, Figures 4b and4c reveal that, although the failure stress and failure strain of the armchair SWBC 3 NTs and SWCNTs were higher than for the corresponding zigzag structures at lower temperatures, these properties decreased sharply in armchair structures, reaching a lower value at higher temperatures compared to zigzag SWBC 3 NTs or SWCNTs. The minimum failure stress and failure strain were found in armchair SWBC 3 NTs at 900K with values of 76.85 GPa and 0.14%, respectively. The same kind of results were observed by Zahedi et al. for BC 3 single-layers [START_REF] Khademi-Zahedi | Mechanical properties of graphene-like BC3; a molecular dynamics study[END_REF]. They found that the Young's modulus of BC 3 single-layers decreased by about 10% and 3% in armchair and zigzag directions at 900K compared to 200K, respectively.

Mechanical properties of double-walled BC 3 NTs

In order to determine the effects of wall addition on the mechanical properties of SWBC 3 NTs, six armchair and four zigzag double-walled (DWBC 3 NTs) were modeled and tested, and the results were compared with those of the corresponding reference CNTs from [START_REF] Salmankhani | Mechanical Properties of C3N Nanotubes from Molecular Dynamics Simulation Studies[END_REF]. A schematic view of the designed DWBC 3 NTs is displayed in Figure S4, and the results of Young's modulus, failure stress and failure strain are presented in Figure 5. All tests were implemented at a constant temperature of 300K while the samples were subjected to uniaxial tensile loading. It should be mentioned that there are some limitations for modeling multi-walled nanotubes regarding interlayer distance. If the distance between layers exceeds a critical value, see Figure S5 for upper and lower limits of interlayer distance, no van der Waals interaction can take place between the layers and therefore no stable structure is formed. Otherwise, when the interlayer distance is below a certain value, the structure is unstable and collapses. Consequently, the distances shown in Figure S4 were selected to model stable armchair and zigzag structures with similar dimensions.

Figure 5a shows that by increasing the radius of the nanotubes, which is the consequence of the increase of the interlayer distance, the Young's modulus progressively decreased in both DWBC 3 NTs and reference DWCNTs samples from [START_REF] Salmankhani | Mechanical Properties of C3N Nanotubes from Molecular Dynamics Simulation Studies[END_REF]. The elastic modulus of two of the zigzag DWBC 3 NTs including structures (8,0),(14,0) and (8,0),(16,0) was higher than those for armchair DWBC 3 NTs of a similar radius, including structures (4,4), [START_REF] Wei | Electrical transport in pure and boron-doped carbon nanotubes[END_REF][START_REF] Wei | Electrical transport in pure and boron-doped carbon nanotubes[END_REF] and [START_REF] Chen | Optical properties of BC3 nanotubes[END_REF][START_REF] Chen | Optical properties of BC3 nanotubes[END_REF], [START_REF] Zhang | Theoretical Studies on the Electronic and Optical Properties of Honeycomb BC(3) monolayer: A Promising Candidate for Metal-free Photocatalysts[END_REF][START_REF] Zhang | Theoretical Studies on the Electronic and Optical Properties of Honeycomb BC(3) monolayer: A Promising Candidate for Metal-free Photocatalysts[END_REF], and two others had a lower modulus than the corresponding armchairs DWBC 3 NTs. The maximum Young's modulus of the armchair and zigzag DWBC 3 NTs were 1122.9 GPa and 1094.4 GPa, obtained for structures (8,0),(16,0) and (4,4), [START_REF] Zhang | Theoretical Studies on the Electronic and Optical Properties of Honeycomb BC(3) monolayer: A Promising Candidate for Metal-free Photocatalysts[END_REF][START_REF] Zhang | Theoretical Studies on the Electronic and Optical Properties of Honeycomb BC(3) monolayer: A Promising Candidate for Metal-free Photocatalysts[END_REF], respectively, and this property was about 8% and 5% lower in the thicker zigzag and armchair DWBC 3 NTs compared to the thinner ones. Comparison of the Young's modulus of DWBC 3 NTs and DWCNTs reveals the higher modulus of DWCNTs. The highest Young's modulus of DWCNTs was 1553.3 GPa, calculated for the structure (8,0), [START_REF] Lindsay | Optimized Tersoff and Brenner empirical potential parameters for lattice dynamics and phonon thermal transport in carbon nanotubes and graphene[END_REF]0). This value is within the range of previous reports for DWCNTs, thus supporting the accuracy of the results of this study [START_REF] Badehian | Transport and mechanical properties of double-walled carbon nanotubes as a function of interwall distance[END_REF][START_REF] Doh | Prediction of the mechanical behavior of double walled-CNTs using a molecular mechanics-based finite element method: Effects of chirality[END_REF].

Unlike Young's modulus, the failure stress of armchair DWBC 3 NT and DWCNT did not change significantly, as seen in Figure 5b. This property generally decreased in both structures by increasing the radius of double-walled nanotube despite some ups and downs.

For the zigzag structures, it can be observed that as the radius increased, the failure stress of the zigzag DWCNTs remained virtually unchanged, while the failure stress of DWBC Besides, the examination of Figure 5c reveals that the failure strain of the armchair DWCNTs and DWBC 3 NTs did not change significantly with the increase in the radius of the nanotubes. However, like the failure stress, the failure strain of zigzag DWBC 3 NTs decreased significantly (10%) as the radius of the nanotubes increases. The failure strain of the armchair DWBC 3 NT was nearly twice as high as that obtained for the corresponding zigzag DWBC 3 NT, and the highest failure strain of DWBC 3 NT was 0.416% and 0.293%, calculated for structures (4,4), [START_REF] Zhang | Theoretical Studies on the Electronic and Optical Properties of Honeycomb BC(3) monolayer: A Promising Candidate for Metal-free Photocatalysts[END_REF][START_REF] Zhang | Theoretical Studies on the Electronic and Optical Properties of Honeycomb BC(3) monolayer: A Promising Candidate for Metal-free Photocatalysts[END_REF] and (8,0),(16,0), respectively, similar to the failure stress.

The results obtained for the armchair and zigzag DWBC3NT and DWCNT are summarized in Tables S4 andS5.

Effect of adding walls on the Young's modulus of BC 3 NTs

In order to measure the mechanical properties of BC 3 NTs with respect to the number of walls, a third wall was added to the DWBC 3 NT and a zigzag and an armchair TWBC 3 NT were modeled, while the results of the corresponding reference TWCNT of similar radius were used (from [START_REF] Salmankhani | Mechanical Properties of C3N Nanotubes from Molecular Dynamics Simulation Studies[END_REF]). In other words, the main objective of this section is to examine how the addition of the second and third walls to a selected zigzag and armchair single-walled BC 3 NT could affect its Young's modulus. To this end, (4,4)(8,8) [START_REF] Tanaka | Novel macroscopic BC3 honeycomb sheet[END_REF][START_REF] Tanaka | Novel macroscopic BC3 honeycomb sheet[END_REF] armchair and (8,0)(14,0)(20,0) zigzag TWBC 3 NTs were modeled and tested under the same loading conditions at a constant temperature of 300K. A schematic view of the designed structures is shown in Figure S5, and the corresponding results are given in Table S6.

According to Figure 6, comparisons were made between the Young's modulus of BC 3 NTs having one, two, and three walls. In this figure, the Young's modulus of both zigzag and armchair SWBC 3 NT increased significantly as the number of walls increased from one to two and then to three. The addition of one wall to a (4,4) SWBC 3 NT improved its Young's modulus by about 33% (from 724.8 GPa to 1094.4 GPa), and the addition of another wall further increased the Young's modulus of the (4,4), [START_REF] Zhang | Theoretical Studies on the Electronic and Optical Properties of Honeycomb BC(3) monolayer: A Promising Candidate for Metal-free Photocatalysts[END_REF][START_REF] Zhang | Theoretical Studies on the Electronic and Optical Properties of Honeycomb BC(3) monolayer: A Promising Candidate for Metal-free Photocatalysts[END_REF] armchair structure by about 27%, reaching 1505.9 GPa in the structure (4,4), [START_REF] Zhang | Theoretical Studies on the Electronic and Optical Properties of Honeycomb BC(3) monolayer: A Promising Candidate for Metal-free Photocatalysts[END_REF][START_REF] Zhang | Theoretical Studies on the Electronic and Optical Properties of Honeycomb BC(3) monolayer: A Promising Candidate for Metal-free Photocatalysts[END_REF], [START_REF] Tanaka | Novel macroscopic BC3 honeycomb sheet[END_REF][START_REF] Tanaka | Novel macroscopic BC3 honeycomb sheet[END_REF], so that its total increase was about 60%. Similarly, the modulus of the zigzag (8,0) SWBC 3 NT first increased by 34% to reach 1103.5 GPa in the (8,0),(14,0) structure, followed by a further improvement of 21% to 1404.4 GPa in the (8,0),(14,0),(20,0) structure. The same behavior was observed for TWCNTs confirms the reasonableness of the results presented. The results obtained demonstrate that increasing the number of walls has a synergistic effect on the Young's modulus of the considered BC 3 NT, so that it could have the same effect on other BC 3 NTs of different dimensions by adding more walls. Figure 6. Young's modulus as a function of dimensions for different chiral single-, double-and triple-walled BC 3 NT.

Mechanical properties of BC 3 nanobuds

Recently, 3D nanostructures have received more attention. Nanobuds are a class of 3D nanostructures that form once a fullerene or nanocage is attached to the outer surface of a nanotube or graphene sheet. Research shows remarkable properties and applications of nanobuds due to their porous shells and nanometric thickness [START_REF] Qi | Multi-shelled hollow micro-/nanostructures[END_REF][START_REF] Ashrafian | Greatly enhanced adsorption of platinum on periodic graphene nanobuds: A first-principles study[END_REF][START_REF] Nezam Abadee | Removing phenol contaminants from wastewater using graphene nanobuds: DFT and reactive MD simulation investigations[END_REF]. In the present study, C 60 fullerene was selected to model BC 3 nanobuds by attaching one, two, three and four C 60 to the outer surface of SWBC 3 NT. An armchair and a zigzag single-walled BC 3 NTs with the closest dimension, having (10,10) and (18,0) structures, respectively, were selected to model nanobuds. These structures were selected because they have the closest dimensions when we compared the dimensions of all the armchair and zigzag single-walled structures. Thus, we were able to provide results that could be better compared. All samples were tested under the same loading condition at a constant temperature of 300K. The graphs of the Young's modulus, failure stress, and failure strain obtained in Figure 7 show that all properties consistently decreased as the number of the attached fullerenes increased, whether in armchair or zigzag chirality. This could be associated with the concentration of stresses that exist around the regions where bonds form between the fullerene and the surface of the nanotube. The higher the number of fullerenes, the higher the stress concentration and the more the mechanical properties are degraded. Similar results were reported in previous articles concerning other types of nanobuds.

Thus, Ghorbanzadeh et al. [START_REF] Ghorbanzadeh Ahangari | Mechanical and electronic properties of carbon nanobuds: First-principles study[END_REF] used DFT calculations to compare the mechanical properties of simple CNTs and CNT nanobuds. They found a tendency for the Young's modulus of zigzag and armchair CNTs to decrease after attaching one C 60 molecule. Mashhadzadeh et al. [START_REF] Hamed Mashhadzadeh | Theoretical studies on the mechanical and electronic properties of 2D and 3D structures of Beryllium-Oxide graphene and graphene nanobud[END_REF] reported in their DFT-based research that the Young's modulus of BeO nanobuds decreased considerably by increasing the number of attached nanocages. All values obtained for the mechanical properties of BC 3 nanobuds are shown in Table S7.

An overview of the failure process of a (18,0)-2C 60 nanobud under continuous tensile loading is provided in Figure 8. As can be seen, the failure started in the regions close to the attached fullerene molecules, due to higher concentration of stresses in these areas, which facilitates and accelerates crack formation and propagation.

Conclusions

In the present article, molecular dynamics (MD) simulation was used to study the mechanical properties of BC 3 nanotubes. We examined the Young's modulus, failure stress and failure strain of BC 3 NT, BC 3 nanobuds and BC 3 NT with different types of defects as a function of factors such as chirality, size, number of walls and temperature. We also used the results of CNTs of similar chirality and dimensions, obtained from a previous work, to compare and validate the present results. Our results demonstrated that the addition of walls to single-walled BC 3 NT increased the Young's modulus, failure stress, and failure strain remarkably, regardless of chirality, so that triple-walled BC 3 NT had markedly better properties than double-walled nanotubes, and double-walled nanotubes showed better properties compared to single-walled ones, again regardless of chirality. The highest Young's modulus obtained was observed for TWBC 3 NT and the lowest modulus was calculated for SWBC 3 NTs. In addition, armchair structures had higher mechanical properties than zigzag structures in most of the nanotubes studied, and BC 3 NT had lower properties compared to CNTs. The highest Young's modulus of the armchair and zigzag SWBC 3 NT was calculated to be 780.5 GPa and 751.7 GPa for (10,10) and (16,0) structures, respectively.

The modeling of double and triple-walled BC 3 NTs based on structures (4,4) and (8,0), with the closest radius, revealed that the Young's modulus of the armchair (4,4), [START_REF] Zhang | Theoretical Studies on the Electronic and Optical Properties of Honeycomb BC(3) monolayer: A Promising Candidate for Metal-free Photocatalysts[END_REF][START_REF] Zhang | Theoretical Studies on the Electronic and Optical Properties of Honeycomb BC(3) monolayer: A Promising Candidate for Metal-free Photocatalysts[END_REF], [START_REF] Tanaka | Novel macroscopic BC3 honeycomb sheet[END_REF][START_REF] Tanaka | Novel macroscopic BC3 honeycomb sheet[END_REF] TWBC 3 NT and (4,4),(8,8) DWBC 3 NT was nearly 60% and 33% higher than (4,4) SWBC 3 NTs, respectively. Similarly, the Young's modulus of the zigzag (8,0),(14,0),(20,0) TWBC 3 NT and (8,0),(14,0) DWBC 3 NT was 55% and 34% higher than that of (8,0) SWBC 3 NT, respectively.

Furthermore, after modeling BC 3 nanobuds by attaching 1 to 4 C 60 molecules to the outer surface of SWBC 3 NT, we observed that all mechanical properties were slightly reduced by increasing the number of attached fullerenes. Thus, the lowest Young's modulus, failure stress and failure strain of each chirality were obtained in nanobuds with four attached fullerenes. The lowest Young's modulus of the armchair and zigzag BC 3 nanobuds was calculated at 556.6 GPa and 531.4 GPa for structures (10,10)-4C 60 and (18,0)-4C 60 , respectively, and were 28% and 30% lower than the corresponding pristine SWBC 3 NTs, respectively. In the end, we modeled and tested SWBC 3 NTs with embedded one-atom and two-atom vacancies as well as Stone-Wales defects and found that Stone-Wales defects imposed the smallest reduction in Young's modulus and failure stress, regardless of chirality, while two-atom vacancies resulted in the largest reduction in these properties. With respect to failure strain, the greatest reduction was observed for two-atom vacancies and Stone-Wales defects in the case of armchair and zigzag structures, respectively. We calculated the highest Young's modulus, failure stress, and failure strain of the defective BC 3 NTs in armchair structures with defect types j (STW-2) and h (STW-1), and the corresponding values were 769.1 GPa, 260.32 GPa, and 0.384%, respectively. Overall, the outstanding results of this study could provide a new prospect for the wider use of 1D carbon-based materials in a variety of applications.

Figure 1 .

 1 Figure 1. Flowchart of the overall simulation steps including modeling, energy minimization and calculation of properties.

  114

Figure 2 .

 2 Figure 2. (a) Stress-strain curve for a (6,6) armchair BC 3 NT, and (b) zoom on the second-order polynomial fit.

Figure 3 .

 3 Figure 3. (a) Young's modulus, (b) failure stress and (c) failure strain of armchair and zigzag structures of SWBC 3 NTs under uniaxial tensile tests at 300K and their corresponding reference SWCNTs materials (from [15]).

Figure 4 .

 4 Figure 4. (a) Young's modulus, (b) failure stress and (c) failure strain of (10,10) and (18,0) SWBC 3 NTs and SWCNTs reference materials (from [15]) under uniaxial tensile tests as a function of temperature.

Figure 5 .

 5 Figure 5. (a) Young's modulus, (b) failure stress and (c) failure strain of armchair and zigzag DWBC 3 NTs 300K and their corresponding reference DWCNT from [15].

Figure 7 .

 7 Figure 7. (a) Young's modulus, (b) failure stress, and (c) failure strain of BC 3 nanobuds under uniaxial tensile tests at 300K.

Figure 8 .

 8 Figure 8. A snapshots of the failure process of a (18,0)-2C 60 BC 3 nanobud.

3. 5 .

 5 Figure 9. (A) Embedded defects in armchair BC 3 NTs: (a) one carbon atom vacancy, (b) one boron atom vacancy, (c) and (d) two carbon atoms vacancy, (e) and (f) one-carbon, one-boron vacancy, (g) and (i) Stone-wales type 1, )h( and ) j) Stone-wales type 2. (B) Same as (A) but for zigzag BC 3 NTs.

Figure 10 .Figure 11 .

 1011 Figure 10. (a) Young's modulus, (b) failure stress, and (c) failure strain of defective (10,10) BC 3 NT and (18,0) BC 3 NT.

Table 1 .

 1 Stoichiometry and chirality of SWBC 3 NTs, DWBC 3 NTs, TWBC 3 NTs.

	Chirality	Number of	Chirality	Number of
		Atoms		atoms
	(4,4)	328	(4,4),(7,7)	902
	(6,6)	492	(4,4),(8,8)	984
	(8,8)	656	(4,4),(9,9)	1066
	(10,10)	820	(4,4),(10,10)	1148
	(12,12)	984	(4,4),(11,11)	1230
	(8,0)	384	(4,4),(12,12)	1312
	(10,0)	480	(8,0),(14,0)	1056
	(12,0)	576	(8,0),(16,0)	1152
	(14,0)	672	(8,0),(18,0)	1248
	(16,0)	768	(8,0),(20,0)	1344
	(18,0)	864	(4,4),(8,8),(12,12)	1968
	(20,0)	960	(8,0),(14,0),(20,0)	2016
	106

Data availability

The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.

structures we found a greater reduction in the failure strain of BC 3 NT with Stone-Wales defects compared to those containing atom vacancies. The highest Young's modulus, failure stress, and failure strain calculated for defective BC 3 NTs of armchair structure with defect types j (STW-2) and h (STW-1) were 769.1 GPa, 260.32 GPa and 0.384%, respectively. In contrast, the lowest Young's modulus and failure stress of all studied defective BC 3 NT were found for zigzag structures with defect type e (one-carbon, one-boron): 711.2 GPa and 87.16 GPa, respectively. Zigzag BC 3 NT with defect type j had the lowest failure strain of 0.19%.

All the values obtained for the mechanical properties of defective BC 3 NT are shown in Table S8.

Previous research has also provided the same kind of results. Thus, Zahedi et al.

investigated the effects of crack-and notch-type defects on the mechanical properties of BC 3 single layers and reported a greater reduction in tensile strength properties as crack or notch dimensions increased [START_REF] Khademi-Zahedi | Mechanical properties of graphene-like BC3; a molecular dynamics study[END_REF]. In their paper on molecular dynamics, Albooye et al. [START_REF] Albooyeh | Effect of point defects and low-density carbon-doped on mechanical properties of BNNTs: A molecular dynamics study[END_REF] demonstrated that the Young's modulus of defective BNNT was reduced by increasing the number of missing atoms, with the highest modulus being found in pristine structures while the lowest modulus was reported for structures with three-atom vacancies. In another MD-based research, Gupta et al. [START_REF] Kumar Gupta | Probing the compound effect of spatially varying intrinsic defects and doping on mechanical properties of hybrid graphene monolayers[END_REF] studied the Young's modulus, failure stress, and failure strain of hybrid graphene single-layers and showed a downward trend in all the properties mentioned by creating Stone-Wales and nanopore defects compared to pristine single-layer graphene. Ghorbanzadeh et al. [START_REF] Ahangari | Effect of various defects on mechanical and electronic properties of zinc-oxide graphene-like structure: A DFT study[END_REF] investigated the effects of point defects on the properties of ZnO graphene-like surfaces in a DFT study, and also reported the negative effect of point defects on the mechanical properties of this material. Figure 11 presents a snapshot of the failure process of a (10,10) armchair BC 3 NT with two atom vacancies (defect type e). As expected, the failure occurred in the defective areas due to the higher stress concentration, and the process continued until the destruction of the nanostructure was complete.