Introduction
Successful realization of graphene with outstanding properties [
1] have attracted scientists to focus more on novel two dimensional (2D) materials in various research communities. Wide range of experimental and theoretical studies confirmed the remarkable properties of 2D materials in comparison to their bulk counterparts [
2–
4]. Hexagonal boron-nitride (h-BN) bulk materials were investigated experimentally by Lynch and Drickamer back to 1960s [
5]. In recent years, h-BN nanosheets [
6] have been fabricated with outstanding semiconducting, mechanical strength, and heat conduction properties [
7–
9]. h-BN nanostructures have been discussed in wide range of research studies due to their extraordinary mechanical, electrical and chemical characteristics. They have proven to show high thermal stability up to 1000°C [
10] and thermal conductivity of 500 W·m
−1·K
−1 [
11,
12]. Their prospect applications are in aerospace, energy storage/conversion, medical and many others industries [
13,
14]. Crystal structure of h-BN is strongly stable due to its covalent bonding nature and is very close to that of the graphene. However, h-BN presents a wide band gap of 5–6 eV and acts like an insulator, in a sharp contrast with the zero band gap and highly electrical conductivity of graphene [
15,
16].
The materials durability during the service strongly depends on their tensile strengths under extreme loading conditions. However, materials may have primary defects which can reduce the prospect performance. Especially in lower scales, deficiency of precise control during growth or processing, some possible defects may form spontaneously, which can deteriorate the application performance of materials and suppress the mechanical and thermal conduction properties and affect the electronic properties [
17]. Defect assessment thus plays a critical role in material’s commercial applications as micro/nano electro mechanical (MEMS/NEMS) devices. Generally, material properties have to be investigated in the presence of various kind of defects like; impurity atoms, point vacancy, Stone-Wales, bivacancy, cracks, grain boundaries, and notches as they may substantially affect the properties [
18–
25]. Experimental advances [
26,
27] demonstrated the importance of ongoing studies to explore the defect related deterioration of material properties for novel structures. Since the complexities and high cost of experimental methods in lower scales, computational approaches have become a complementary and versatile alternative. A wide range of approaches from quantum mechanics (QM) and molecular dynamics (MD) to continuum mechanics (CM) including multiscale modeling have been developed and employed for this purpose [
28–
30]. Katzir et al. [
31] were the first to carry out the study of point defects, impurities and vacancies in h-BN by electron paramagnetic resonance (EPR). Jiménez et al. [
32] identified the formation of point defects in boron nitride by X-ray absorption near edge spectroscopy (XANES). Hirano et al. [
33] synthesized the amorphous boron nitride by pressure pyrolysis of borazine with high yield below 700°C at 100 MPa. Amorphous boron nitride powder was prepared by mechanically milling hexagonal boron nitride and used as a starting material for studying boron nitride phase transformation behavior. Fabricated amorphous boron nitride (ABN) powder was significantly reactive with humidity in the air [
34]. Mortazavi and Ahzi [
35] conducted classical MD simulations to investigate the effects of point vacancy, Stone-Wales and bivacancy defects on the thermal conductivity and tensile responses of single-layer graphene sheets. They found out decrement in the mechanical properties of graphene by increase of defects concentrations. They confirmed that defects can substantially suppress the thermal conduction, considerably larger than the mechanical properties. Ding et al. [
36] investigated vacancy defects between interface of graphene and h-BN using density functional theory (DFT). They found out vacancies have significant effect on the mechanical and electronic properties of the hybrid graphene/h-BN. Güryel et al. [
37] studied structural defects and chemical functionalization influence on mechanical properties of graphene employing finite element method. They demonstrated reduction of the elastic limit in the presence of structural defects. Han et al
. [
38] and Abadi et al. [
39] performed MD simulations to evaluate the effects of temperature, strain rate, and grain size on the mechanical properties of pristine h-BN nanosheets. Their results reveal decreasing trends in the ultimate tensile strength and failure strain by increasing the defect content and temperature. In this study, we conducted extensive MD simulations to explore the mechanical deterioration and failure mechanism of the h-BN with/without defects. Various initial and well-known defects like cracks, notches, and Stone-Wales point defects were considered in the h-BN nanosheets. The mechanical properties of the pristine and defective h-BN in various temperatures were estimated by applying uniaxial tension loading.
MD simulations
MD simulations in this work were conducted employing the large-scale atomic/molecular massively parallel simulator (LAMMPS) [
40]. The interactions between boron and nitrogen atoms were represented by modified Tersoff potential proposed by Matsunaga et al. [
41] which have been successfully employed in the previous studies for the mechanical properties of h-BN nanosheets [
9,
39]. Once the system was fully relaxed by Nosé-Hoover barostat and thermostat method (NPT) [
42–
45], the mechanical properties of the pristine and defected h-BN in various temperatures were estimated by applying the uniaxial tension loading [
46–
48]. Periodic boundary conditions were applied in the plane directions, so the simulated systems are representative of nanosheets and not the nanoribbons. During the uniaxial loading, the size of the simulation box along the loading direction was increased by a constant engineering strain rate of 2 × 10
8 s
−1, in which we used a time step of 0.25 fs, which is small enough to simulate the thermal and mechanical properties by the MD simulations [
49–
51]. To guarantee the uniaxial stress condition, the size of the box in the perpendicular direction of loading was controlled by the NPT method to reach the negligible stress values in the transverse direction of loading [
46–
48,
52,
53]. The stress tensors were computed based on the Virial theorem [
54]. The atomistic models were visualized using the OVITO [
55] package. Worthy to note that the methods which do not require any discretization have been presented in very recent works on the basis of deep machine learning [
56–
58].
In Fig. 1, the atomic lattice of the h-BN monolayer is illustrated, which shows the honeycomb structure as that of the graphene. In this work we studied the h-BN monolayers with different degree of defects, from a little density to fully amorphized structures. In Fig. 2, fully amorphized h-BN monolayers after the relaxation at the room temperatures are illustrated. In this work, we considered Stone-Wales defects on the h-BN nanosheet mechanical properties. Stone-Wales defect does not contain removed or added atoms. In this defect, by rotating one of B-N bonds by 90° with respect to the midpoint of B-N bond, four hexagons transform into two pentagons and two heptagons. We found that in order to have stabilized defective h-BN structures, system should be initially energy-minimized using the conjugate gradients method. The defects concentrations in Stone-Wales defects are introduced by the number of formed non-hexagon lattices with respect to the total number of hexagons in the pristine structure.
Results and discussions
First of all, in order to have reference structural parameters for comparison with defected lattice properties, we investigate the mechanical properties of pristine nanosheet with dimensions of 300 × 300 and the thickness of 3.4 Å within a temperature range of 200, 300, 400, 500, 700, and 900 K. Figure 3 revealed stress-strain curves of all considered configurations. The ultimate tensile stress/strain decreased by increasing temperature and subsequently results in elastic module decrement as we demonstrated in Table 1. High temperature values increase phonon frequencies which cause easier stretch of the atomic bonds [
59–
61]. According to our results pristine h-BN nanosheets at 300 K can keep its load bearing ability within a considerable strain value of 0.38, which is higher in comparison to the pristine graphene with the value of 0.27 and 0.20 [
62]. They also exhibited more strong tensile capacity than 2D phagraphene and graphene-like C
3N with maximum strain values of 0.13 [
63] and 0.18 [
64,
65] at the same direction and temperature, respectively.
Influence of crack and notch on the ultimate tensile strength and failure mechanisms of the h-BN lattice was investigated by considering four different sizes. First, cracks with length of L/3, L/6, L/9, and L/12 lengths were modeled, in which L is the size of the sheets width. To investigate crack propagation, cracks were considered to in plane tension loading along the perpendicular to the loading direction. Several MD simulations in various temperatures range of 200, 300, 600, and 900 K were performed. Figure 4 represent failure mechanisms and crack propagation of the h-BN nanosheet with an initial crack length of L/9 at 300 K. Maximum stress concentration of 95 GPa occurs at the strain value of 0.18 which indicates propagation crack leading to the final and complete rupture of the sheet.
Maximum bearing stress of h-BN nanosheet in the presence of cracks with various lengths and temperatures are summarized in Fig. 5(a). Crack free pristine lattice is considered as 0L to compare the results. As expected, ultimate tensile strength decreases by increasing the crack length, and takes 30% of the pristine lattice strength for the crack length of L/3. Accordingly, temperature rise cause to worsening of the ultimate tensile strength of the nanosheet. Strain values at the maximum tensile stress were exhibited in Fig. 5(b). As it clear from these results, maximum bearing strains of h-BN nanosheet decreases by increasing of the crack length and temperature as well.
Initial notch defect was investigated to demonstrate failure mechanisms and defect propagation under various diameter of L/3, L/6, L/9, L/12 and temperature range of 200, 300, 600, 900 K. Figure 6 depicts the failure mechanisms and crack propagation of the sample h-BN nanosheet with initial notch diameter of L/9 at room temperature. In Fig. 6(d), defect start to propagate in the transverse direction of loading at the strain value of 0.16 and experience the complete failure strain value at Fig. 6(f) with value of 0.17. The maximum stress concentration of 89 GPa occurs at the failure strain value of 0.17.
To elaborate the ultimate tensile strength of pristine and h-BN nanosheets with notches, maximum stresses and equivalent strains were reported in Figs. 7(a) and 7(b), respectively. These results illustrate the decrement of the maximum stress/strain due to increase of notch diameter and temperature as well. Strain at maximum stress of the defected lattice with smaller notch diameters of L/6 and L/9 at room temperature are less than strain of the lattice that the system with a larger notch diameter of L/3. This observation is because of the folding of the lattice in perpendicular direction and exhibiting higher tensile strain under stretch.
The elastic modulus and tensile strength of the h-BN nanosheet as a function of crack or notch sizes at different temperatures are compared in Fig. 8. In all calculated cases, elastic modulus and tensile strength decrease by increasing the size of crack/notch or temperature. Our results show that h-BN nanosheet even with largest defect size could exhibit a remarkably high elastic modulus of 480 and 475 GPa and tensile strengths of 62 and 71 GPa at 300 K for crack and notch, respectively. In comparison to the amorphized graphene with the 500 GPa elastic modulus and 60 GPa tensile strength [
66], we predict h-BN structures bearing considerable loading even in presence of high defects. In addition, we found that h-BN nanosheets maintain their excellent mechanical properties even at high defects and temperature levels.
In Fig. 9, stress-strain responses of defect-free and highly defective h-BN monolayers are compared. Initial defects were imposed in various concentration range of 10%, 40%, and 70%. As it can be observed, a linear relation exists at low strain level which is followed by a nonlinear response up to the tensile strength point and leading to a sudden decline which corresponds to the nanosheet failure. Stone-Wales defects reduced the ultimate tensile strength by increasing defect concentrations. As it clears from results, imposing 70% defect concentration reduced ultimate tensile strength to the value of 20% less than the defect-free nanosheet. Since the number of atoms in Stone-Wales defects is as same as the defect-free h-BN nanosheet, minimum effect on ultimate tensile strength could be observed in comparison to the crack and notch defects.
Ultimate tensile stress/strain and associate elastic modulus were shown in Fig. 10. To better understand the defect dependent variations, we normalized the values by related pristine amounts in room temperature (P-300K). The elastic modulus and tensile strength decrease by increasing defects concentration. Our results show that h-BN nanosheet with high defect concentration show the elastic modulus of 200 GPa and tensile strengths of 135 GPa at 300 K. Obviously, imposing 70% defect concentration reduced elastic modulus to the 30% of the defect-free nanosheet value. It could be concluded, more than 10% concentration of Stone-Wales defects have the most serious reduction effect on the elastic modulus of h-BN nanosheets in comparison to the crack and notch defective nanosheets.
Conclusions
We conducted extensive MD simulations to investigate and explore the mechanical properties and failure mechanism of h-BN nanosheets under critical conditions. The pristine h-BN at room temperature exhibits remarkable strain at failure of 0.38 and ultimate tensile stress of 170 GPa. To provide a comprehensive vision concerning mechanical aspects of h-BN nanosheets, we modeled most critical types of defects in engineering materials, cracks, notches, and point vacancies with various sizes and concentrations under the different loading temperatures. At 300 K, ultimate tensile stresses of 95, 89, and 130 GP at corresponding strain levels of 0.18, 0.17, and 0.35 were estimated for the systems with the largest crack, notch, and Sone-Wales defects, respectively. MD simulations showed that the decrement of the maximum tensile stress/strain due to increase of defect size and temperature. Consequently, our results demonstrate that h-BN nanosheet, even with large crack and notch, could exhibit remarkably high elastic modulus of 480 and 475 GPa at 300 K, respectively. However, Stone-Wales defects yield the most serious elastic modulus reduction in comparison to crack or notch. Our MD results reveal outstanding mechanical properties of h-BN nanosheets with and without defects, which are attractive for the designing of nanodevices with h-BN as a building block.
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