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Mechanical responses of pristine and defective C3N nanosheets studied by molecular dynamics simulations
Computational Materials Science 147 (2018) 316–321
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Mechanical responses of pristine and defective C3N nanosheets studied by molecular dynamics simulations A.H.N. Shirazi b, R. Abadi b, M. Izadifar b, N. Alajlan a, T. Rabczuk a,b,⇑ a b
Department of Computer Engineering, College of Computer and Information Sciences, King Saud University, Riyadh, Saudi Arabia Institute of Structural Mechanics, Bauhaus-Universität Weimar, Weimar, Germany
a r t i c l e
i n f o
Article history: Received 27 November 2017 Received in revised form 25 January 2018 Accepted 28 January 2018
Keywords: C3N Nanosheet Molecular dynamics Defects Cracks
a b s t r a c t The purpose of this study is to investigate the mechanical properties of a new two-dimensional graphene like material, crystalline carbon nitride with the stoichiometry of C3N. The extraordinary properties of C3N make it an outstanding candidate for a wide variety of applications. In this work, the mechanical properties of C3N nanosheets have been studied not only in the defect-free form, but also with critical defects such as line cracks and notches using molecular dynamics simulations. Different crack lengths and notch diameters were considered to predict the mechanical response at different temperatures under the uniaxial tensile loading. Our simulation results show that larger cracks and notches reduce the strength of the nanosheets. Moreover, it was shown the temperature rise has a weakening effect on the tensile strength of C3N. Our study can provide useful information with respect to the thermomechanical properties of pristine and defective graphene like C3N 2D material. Ó 2018 Elsevier B.V. All rights reserved.
1. Introduction Single layered materials, so-called two dimensional (2D) materials have captured the interest of researchers of different fields since the isolation of graphene from graphite reported in 2004 [1–5]. Graphene is among the lightest and strongest materials with unique mechanical and thermal properties and therefore has been used in numerous applications [6–8]. Graphene can withstand a tensile strength of 130.5 GPa which is hundred times of the strength of high strength metals [9]. In-plane mechanical properties of graphene are much better than those for diamond [10]. Graphene has shown the highest measured thermal conductivity among all the materials ever known [11]. Due to the exceptionally high strength, thermal conductivity and light weight, graphene has been widely employed in the fabrication of advanced polymer composites with enhanced mechanical and thermal conduction properties [12–23]. Graphene has an extremely high ratio of surface area to mass [24] which makes it ideal for energy conversion and storage [25]. Furthermore, it is suitable in super-capacitors as conductive plates [26]. Such a super-capacitor can store more energy per unit volume than all other types of capacitors. The graphene’s fascinating properties can be tailored for the special appli-
⇑ Corresponding author at: Department of Computer Engineering, College of Computer and Information Sciences, King Saud University, Riyadh, Saudi Arabia. E-mail address: [email protected] (T. Rabczuk). https://doi.org/10.1016/j.commatsci.2018.01.058 0927-0256/Ó 2018 Elsevier B.V. All rights reserved.
cations such as nanoelectronics, mechanical robust components, optoelectronics, thermal management systems, energy storage and energy conversion [27]. The surface of the graphene can be modified by the foreign atoms such as nitrogen and oxygen which can be used to modify the properties [28,29]. However, graphene as a semimetal has a zero band gap which makes it difficult to be used in electronic circuits as a transistor [30]. Semiconductors can carry the electrical charge when excited by an external thermal or electrical field. The zero band gap in graphene can be modified by the chemical doping or fabrication of 2D heterostructures in which another 2D materials are used [31–36]. Recently a new 2D material with the stoichiometry of C3N has been fabricated by polymerization of diaminophenazine. The nitrogen atoms are uniformly distributed in the structure of the pristine C3N which is analogous to graphene [37]. The 2D polyaniline C3N shows the characteristics of a semiconductor with a small band gap [38]. Hence, it can solve the limitation of graphene which has zero band gap. It has been shown that monolayer C3N has a band gap of 1.09 eV and is an indirect semiconductor [39]. The surface of the pristine C3N can be engineered through functionalization with nonmetallic and semimetallic elements. The adsorption of foreign adatoms can tune further properties of C3N [40]. Ab inito molecular dynamics simulation show that C3N can be stable at a temperature up to 4000 K. Density functional theory (DFT) calculations showed C3N as an extremely stiff material with a Young’s modulus of 1090 GPa [38].
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The properties of the newly synthesized graphene like C3N have not been entirely explored. There exists only limited research on the mechanical characteristics of C3N and to the best of our knowledge the effects of defects and temperature on the mechanical properties have not been explored so far. As a common fact, in the commercially available materials there exist always defects in various forms. These defects can drastically reduce the ultimate tensile strength of the material [41–44]. Therefore, it is critical to study the mechanical properties of the materials in presence of defects such as cracks and notches [45–47]. The temperature effect on the properties of the C3N is also a very worthy issue to be explored since it withstands high temperatures up to 4000 K. In the current study, we investigate the mechanical properties of the C3N with defects such as cracks and notches. We study the propagation of the crack and notch for different crack lengths and different notch diameters at different temperatures up to 900 K. The comprehensive insight provided by this study can be very useful for the design of nano-devices employing the semiconducting C3N nanosheets. 2. Molecular dynamics modelling Classical molecular dynamics simulations were used to predict the mechanical properties of C3N nanosheets with various cracks and notches. The molecular dynamics simulations were performed with the open-source software of LAMMPS (Large-scale Atomic/ Molecular Massively Parallel Simulator) [48]. The optimized Tersoff potential was used to define the carbon atoms interactions [49]. The Tersoff potential parameters for the nitrogen-carbon atoms interactions were taken from the work of Kinarci et al. [50]. The post processing of the results were performed through the open-source visualization software called OVITO [51]. The calculated stresses, strains, and the atomistic positions through the uniaxial tension were recorded. The top-view of the single layer C3N is illustrated in Fig. 1. Each nitrogen atom is surrounded by three carbon atoms in a hexagonal network through strong polar covalent bonds. This structure is very similar to graphene and one can imagine the nitrogen atoms are doped in the graphene structure and occupy the place of carbon atoms in pristine graphene in a regular layout. Similarly to graphene, monolayer C3N has two major orientations of armchair and zigzag. The mechanical properties of pristine C3N nanosheets under uniaxial tension loading were calculated. The simulations were performed for a nanosheet which contains 33,600 atoms. The boundary conditions are periodic along the inplane directions. First, the simulation box was relaxed via NoséHoover barostat and thermostat (NPT) method to obtain a stress free condition for the model. Then, a strain rate of 108 s1 with
time step size of 0.25 fs was applied in the MD simulations. In order to avoid the void formation during the simulation process, the atomic positions were remapped according to the size of the simulation box. The stress in values were calculated based on Virial theorem at each time step [52]. The calculated stresses for all atoms were averaged over the time steps of the simulation. To provide the uniaxial loading condition, the periodic simulation box along the structure width was adjusted using the NPT method to ensure negligible stress in this direction [53,54]. 3. Results and discussion In the current study, the molecular model was verified by comparison with molecular dynamics simulation results of Mortazavi [38]. At a temperature of 300 K, the nanosheet showed an ultimate tensile strength of 128 GPa in armchair direction and 125 GPa in zigzag direction. These results agree well to those reported in [38]. A pristine nanosheet with dimensions of 300 300 Å was used to study the influence of the temperatures (200, 300, 500, 700, and 900 K). Since the applied boundary conditions were periodic and we constructed a large super-cell, the acquired results are convincingly size independent. The models were stretched in the zigzag orientation and the resulting stress-strain curves of the pristine C3N nanosheet is depicted in Fig. 2. It is worthy to note that in the stress calculations, we considered the volume of the structure using a thickness of 3.2 Å [38] for the single-layer C3N. At higher temperatures, the ultimate tensile strength decreases which in turn also decreases the modulus of elasticity as the temperature rises. At 300 K, the ultimate stress is 128 GPa at a strain of 0.17. The tensile stress is 82 GPa when the temperature is increased to 900 K which is about 36% lower than the ultimate stress at 300 K. The maximum tensile stress occurs at 200 K at a strain of 0.18 while the strain at the highest simulated temperature is 0.11, which is about 38% less than the maximum strain at the 200 K. At the higher temperatures, the nanosheet can be less elongated as compared to that at the room temperatures in which there
Fig. 2. The stress-strain response of the pristine C3N nanosheet under the uniaxial tension at the temperatures of 200, 300, 500, 700, and 900 K.
Table 1 Young’s Modulus (E) of the pristine nanosheet at the 200, 300, 500, 700, and 900 K. Fig. 1. Atomic layout of C3N with a honeycomb structure including both carbon and nitrogen atoms. Each nitrogen atom is surrounded by three carbon atoms. The unit cell is shown in the structure of C3N that contains two nitrogen atoms and six carbon atoms.
T (K)
200
300
500
700
900
E (GPa)
953
939
911
885
867
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Fig. 3. (a) The tensile strength of the nanosheet in the presence of crack with different lengths which are studied at a range of temperatures from 200 K to 900 K, (b) Engineering strain at maximum tensile strength of the C3N nanosheet with various cracks at different temperatures.
Fig. 4. Stress intensity factor for the crack of length L/12 at temperatures of 200, 300, 500, 700, and 900 K.
exist less molecular motions. At the beginning parts of the curves, there is a linear relation between stress and strain. From the slope of this zone, the elasticity modulus can be calculated, see Table 1.
Table 1 also shows the lower Young’s modulus at higher temperatures. The modulus of elasticity at atomic scale is directly related to the chemical bonds which are weakened at higher temperatures. After the linear region, there is the nonlinear part until the nanosheet fractures. At fracture, the stresses suddenly drop. Crack propagation depends on the orientation of the crack in the material [55]. We have studied the mode I crack in which the applied loading is perpendicular to the crack orientation. To build models with cracks, we applied disconnected interatomic bonds for the atoms in the two side of the crack. With these pre-crack, the molecular dynamics simulation were done in order to explore the crack propagation in the C3N nanosheet. Different initial crack lengths were considered, i.e. L/12, L/6, L/3, and L/2, where L is the dimension of the square C3N nanosheet. In Fig. 3a, the maximum stress that the nanosheet can tolerate under tension loadings are illustrated for different crack lengths at different temperatures. The results are compared with the crack free model, which is defined as 0L. The maximum stress of the pristine model was discussed in the previous section and here the amount of the maximum stress at different temperatures is illustrated. The tensile strength decreases dramatically when cracks exist. The tensile strength decreases with increasing crack length. The tempera-
Fig. 5. The process of the crack propagation (a–e) in a C3N nanosheet with the length of L/6 at 300 K.
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ture rise has a weakening influence on the ultimate tensile strength of the nanosheet as well. Fig. 3b depicts the strains at the maximum tensile stress. After the failure of the C3N nanosheet, the atomic bonds are broken and bigger deformations occur. The results for the pre-cracked models have been compared with the results of the pristine model. As expected, the defect free C3N nanosheet can withstand more tensile stress. According to Linear Elastic Fracture Mechanics (LEFM), the critical stress intensity factor for a plate with periodic centred cracks under mode I fracture can be obtained as [56]
K IC ¼ rf
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pa 2h tan 2h
ð1Þ
where rf is the fracture stress, 2h is the width of the nanosheet, and 2a is the initial crack length. The above mentioned formula has been used for a periodic structure of the C3N nanosheet. For crack propagation, we have calculated the critical stress intensity factor for the crack length of L/12 at different temperatures. As can be seen from Fig. 4, the stress intensity factor only slightly decreases as the temperature increases. The simulations were conducted for five
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samples with uncorrelated initial conditions in order to define the error-bars in the acquired results. The crack propagation with a sample of initial crack length equal L/6 at 300 K is illustrated in Fig. 5. Stress concentrations of 130 GPa occur close to the crack tip. The strain at the end of process (e), where crack propagates the whole length of the crack, is 0.09. We next focus on the influence of the notch on the mechanical properties of C3N. Different notch diameters of L/12, L/6, L/3, and L/2 were studied, which are similar to the values for the crack lengths. In Fig. 6a, the maximum tensile stress for the different diameters are depicted at different temperatures. They are compared with the values for the pristine model. The ultimate tensile strength decreases as the notch diameter increases. As the temperature increases, the tensile strength decreases as well. At 300 K, the maximum tensile strength for the notch-free model is 128 GPa while its value is only 44 GPa for a notch of length L/2 at 900 K. In other words, the ultimate tensile stress decreases by an amount of 65%. The same decreasing trend is observed for all models when the temperature rises from 300 K to 900 K. At 900 K for the pristine
Fig. 6. (a) The ultimate tensile strength of the nanosheet in presence of the notch defect with different notch diameters at different temperatures of 200, 300, 500, 700, and 900 K. (b) Engineering strain at maximum tensile strength in presence of notch with different diameters.
Fig. 7. The process of the C3N nanosheet fracture under the uniaxial tensile loading for a notch with the length of L/6 at 500 K.
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helpful for the use of ultra stiff and thermally stable C3N for various applications such as those in nanocomposites and nanoelectronics.
Acknowledgement The authors extend their appreciation to the Distinguished Scientist Fellowship Program (DSFP) at King Saud University for funding this work.
References
Fig. 8. Comparison for the maximum strength of samples with two cracks and two notches in.
model, the tensile strength is 82 GPa while for a notch of L/2 diameter its value decreases by 64% to 29 GPa. Fig. 6b shows the maximum engineering strain and stress at fracture. The maximum strain decreases when the temperature rises and the notches become bigger. Fig. 7 illustrates the fracture process for a C3N nanosheet containing a notch of L/6 diameter. The simulations were performed at 500 K until the nanosheet ruptures along the orientation perpendicular to the loading direction. The linear zone of the stressstrain curve (a) is shown at the initiation step where the engineering strain is 0.02 and the C3N nanosheet is still within the elastic range. The increase in engineering strain is 78% in the whole process showing the nanosheet experiences large deformations. The stresses at the top and bottom of the notch increase dramatically. However they are not that high on the left and right side of the notch. A comparison of the results of the pre-cracked and pre-notched nanosheets is plotted in Fig. 8. Two different crack lengths are compared with two different notch diameters. The temperature rise were studied as well. The stresses of the C3N nanosheet with the notch defect are larger in both cases. In another words, the C3N nanosheet can withstand higher tensile stresses for notch-like defects as compared to crack-like defect. 4. Conclusions We studied the mechanical properties of C3N nanosheets and analyzed the effects of cracks, notches and temperature. It was shown that the existence of cracks and notches have strong weakening effects on the ultimate tensile strength of the material. The simulations for different crack lengths and different notch diameters were performed at five different temperatures from 200 K to 900 K. For the pristine structure, the temperature rise decreases the tensile strength. It was also shown that by increasing the crack length both the ultimate tensile strength and its corresponding engineering strain decreases. As expected, the temperature rise decreases the tensile strength since at higher temperatures atomic bonds become weaker. The critical stress intensity factor ðK IC Þ was also calculated for a crack model at different temperatures. It was shown that the temperature raise only slightly decreases the critical stress intensity factor. The simulations were conducted at the same temperature conditions for the C3N nanosheet with different notch diameters. Similar to the pre-cracked models, the notches also reduce the strength of the nanosheet. Finally, the results of the crack models were compared to the results of the notch models, which confirm that the nanosheets with notch defects can withstand higher tensile stresses. Our study can be
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Mechanical responses of pristine and defective C3N nanosheets studied by molecular dynamics simulations A.H.N. Shirazi b, R. Abadi b, M. Izadifar b, N. Alajlan a, T. Rabczuk a,b,⇑ a b
Department of Computer Engineering, College of Computer and Information Sciences, King Saud University, Riyadh, Saudi Arabia Institute of Structural Mechanics, Bauhaus-Universität Weimar, Weimar, Germany
a r t i c l e
i n f o
Article history: Received 27 November 2017 Received in revised form 25 January 2018 Accepted 28 January 2018
Keywords: C3N Nanosheet Molecular dynamics Defects Cracks
a b s t r a c t The purpose of this study is to investigate the mechanical properties of a new two-dimensional graphene like material, crystalline carbon nitride with the stoichiometry of C3N. The extraordinary properties of C3N make it an outstanding candidate for a wide variety of applications. In this work, the mechanical properties of C3N nanosheets have been studied not only in the defect-free form, but also with critical defects such as line cracks and notches using molecular dynamics simulations. Different crack lengths and notch diameters were considered to predict the mechanical response at different temperatures under the uniaxial tensile loading. Our simulation results show that larger cracks and notches reduce the strength of the nanosheets. Moreover, it was shown the temperature rise has a weakening effect on the tensile strength of C3N. Our study can provide useful information with respect to the thermomechanical properties of pristine and defective graphene like C3N 2D material. Ó 2018 Elsevier B.V. All rights reserved.
1. Introduction Single layered materials, so-called two dimensional (2D) materials have captured the interest of researchers of different fields since the isolation of graphene from graphite reported in 2004 [1–5]. Graphene is among the lightest and strongest materials with unique mechanical and thermal properties and therefore has been used in numerous applications [6–8]. Graphene can withstand a tensile strength of 130.5 GPa which is hundred times of the strength of high strength metals [9]. In-plane mechanical properties of graphene are much better than those for diamond [10]. Graphene has shown the highest measured thermal conductivity among all the materials ever known [11]. Due to the exceptionally high strength, thermal conductivity and light weight, graphene has been widely employed in the fabrication of advanced polymer composites with enhanced mechanical and thermal conduction properties [12–23]. Graphene has an extremely high ratio of surface area to mass [24] which makes it ideal for energy conversion and storage [25]. Furthermore, it is suitable in super-capacitors as conductive plates [26]. Such a super-capacitor can store more energy per unit volume than all other types of capacitors. The graphene’s fascinating properties can be tailored for the special appli-
⇑ Corresponding author at: Department of Computer Engineering, College of Computer and Information Sciences, King Saud University, Riyadh, Saudi Arabia. E-mail address: [email protected] (T. Rabczuk). https://doi.org/10.1016/j.commatsci.2018.01.058 0927-0256/Ó 2018 Elsevier B.V. All rights reserved.
cations such as nanoelectronics, mechanical robust components, optoelectronics, thermal management systems, energy storage and energy conversion [27]. The surface of the graphene can be modified by the foreign atoms such as nitrogen and oxygen which can be used to modify the properties [28,29]. However, graphene as a semimetal has a zero band gap which makes it difficult to be used in electronic circuits as a transistor [30]. Semiconductors can carry the electrical charge when excited by an external thermal or electrical field. The zero band gap in graphene can be modified by the chemical doping or fabrication of 2D heterostructures in which another 2D materials are used [31–36]. Recently a new 2D material with the stoichiometry of C3N has been fabricated by polymerization of diaminophenazine. The nitrogen atoms are uniformly distributed in the structure of the pristine C3N which is analogous to graphene [37]. The 2D polyaniline C3N shows the characteristics of a semiconductor with a small band gap [38]. Hence, it can solve the limitation of graphene which has zero band gap. It has been shown that monolayer C3N has a band gap of 1.09 eV and is an indirect semiconductor [39]. The surface of the pristine C3N can be engineered through functionalization with nonmetallic and semimetallic elements. The adsorption of foreign adatoms can tune further properties of C3N [40]. Ab inito molecular dynamics simulation show that C3N can be stable at a temperature up to 4000 K. Density functional theory (DFT) calculations showed C3N as an extremely stiff material with a Young’s modulus of 1090 GPa [38].
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The properties of the newly synthesized graphene like C3N have not been entirely explored. There exists only limited research on the mechanical characteristics of C3N and to the best of our knowledge the effects of defects and temperature on the mechanical properties have not been explored so far. As a common fact, in the commercially available materials there exist always defects in various forms. These defects can drastically reduce the ultimate tensile strength of the material [41–44]. Therefore, it is critical to study the mechanical properties of the materials in presence of defects such as cracks and notches [45–47]. The temperature effect on the properties of the C3N is also a very worthy issue to be explored since it withstands high temperatures up to 4000 K. In the current study, we investigate the mechanical properties of the C3N with defects such as cracks and notches. We study the propagation of the crack and notch for different crack lengths and different notch diameters at different temperatures up to 900 K. The comprehensive insight provided by this study can be very useful for the design of nano-devices employing the semiconducting C3N nanosheets. 2. Molecular dynamics modelling Classical molecular dynamics simulations were used to predict the mechanical properties of C3N nanosheets with various cracks and notches. The molecular dynamics simulations were performed with the open-source software of LAMMPS (Large-scale Atomic/ Molecular Massively Parallel Simulator) [48]. The optimized Tersoff potential was used to define the carbon atoms interactions [49]. The Tersoff potential parameters for the nitrogen-carbon atoms interactions were taken from the work of Kinarci et al. [50]. The post processing of the results were performed through the open-source visualization software called OVITO [51]. The calculated stresses, strains, and the atomistic positions through the uniaxial tension were recorded. The top-view of the single layer C3N is illustrated in Fig. 1. Each nitrogen atom is surrounded by three carbon atoms in a hexagonal network through strong polar covalent bonds. This structure is very similar to graphene and one can imagine the nitrogen atoms are doped in the graphene structure and occupy the place of carbon atoms in pristine graphene in a regular layout. Similarly to graphene, monolayer C3N has two major orientations of armchair and zigzag. The mechanical properties of pristine C3N nanosheets under uniaxial tension loading were calculated. The simulations were performed for a nanosheet which contains 33,600 atoms. The boundary conditions are periodic along the inplane directions. First, the simulation box was relaxed via NoséHoover barostat and thermostat (NPT) method to obtain a stress free condition for the model. Then, a strain rate of 108 s1 with
time step size of 0.25 fs was applied in the MD simulations. In order to avoid the void formation during the simulation process, the atomic positions were remapped according to the size of the simulation box. The stress in values were calculated based on Virial theorem at each time step [52]. The calculated stresses for all atoms were averaged over the time steps of the simulation. To provide the uniaxial loading condition, the periodic simulation box along the structure width was adjusted using the NPT method to ensure negligible stress in this direction [53,54]. 3. Results and discussion In the current study, the molecular model was verified by comparison with molecular dynamics simulation results of Mortazavi [38]. At a temperature of 300 K, the nanosheet showed an ultimate tensile strength of 128 GPa in armchair direction and 125 GPa in zigzag direction. These results agree well to those reported in [38]. A pristine nanosheet with dimensions of 300 300 Å was used to study the influence of the temperatures (200, 300, 500, 700, and 900 K). Since the applied boundary conditions were periodic and we constructed a large super-cell, the acquired results are convincingly size independent. The models were stretched in the zigzag orientation and the resulting stress-strain curves of the pristine C3N nanosheet is depicted in Fig. 2. It is worthy to note that in the stress calculations, we considered the volume of the structure using a thickness of 3.2 Å [38] for the single-layer C3N. At higher temperatures, the ultimate tensile strength decreases which in turn also decreases the modulus of elasticity as the temperature rises. At 300 K, the ultimate stress is 128 GPa at a strain of 0.17. The tensile stress is 82 GPa when the temperature is increased to 900 K which is about 36% lower than the ultimate stress at 300 K. The maximum tensile stress occurs at 200 K at a strain of 0.18 while the strain at the highest simulated temperature is 0.11, which is about 38% less than the maximum strain at the 200 K. At the higher temperatures, the nanosheet can be less elongated as compared to that at the room temperatures in which there
Fig. 2. The stress-strain response of the pristine C3N nanosheet under the uniaxial tension at the temperatures of 200, 300, 500, 700, and 900 K.
Table 1 Young’s Modulus (E) of the pristine nanosheet at the 200, 300, 500, 700, and 900 K. Fig. 1. Atomic layout of C3N with a honeycomb structure including both carbon and nitrogen atoms. Each nitrogen atom is surrounded by three carbon atoms. The unit cell is shown in the structure of C3N that contains two nitrogen atoms and six carbon atoms.
T (K)
200
300
500
700
900
E (GPa)
953
939
911
885
867
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Fig. 3. (a) The tensile strength of the nanosheet in the presence of crack with different lengths which are studied at a range of temperatures from 200 K to 900 K, (b) Engineering strain at maximum tensile strength of the C3N nanosheet with various cracks at different temperatures.
Fig. 4. Stress intensity factor for the crack of length L/12 at temperatures of 200, 300, 500, 700, and 900 K.
exist less molecular motions. At the beginning parts of the curves, there is a linear relation between stress and strain. From the slope of this zone, the elasticity modulus can be calculated, see Table 1.
Table 1 also shows the lower Young’s modulus at higher temperatures. The modulus of elasticity at atomic scale is directly related to the chemical bonds which are weakened at higher temperatures. After the linear region, there is the nonlinear part until the nanosheet fractures. At fracture, the stresses suddenly drop. Crack propagation depends on the orientation of the crack in the material [55]. We have studied the mode I crack in which the applied loading is perpendicular to the crack orientation. To build models with cracks, we applied disconnected interatomic bonds for the atoms in the two side of the crack. With these pre-crack, the molecular dynamics simulation were done in order to explore the crack propagation in the C3N nanosheet. Different initial crack lengths were considered, i.e. L/12, L/6, L/3, and L/2, where L is the dimension of the square C3N nanosheet. In Fig. 3a, the maximum stress that the nanosheet can tolerate under tension loadings are illustrated for different crack lengths at different temperatures. The results are compared with the crack free model, which is defined as 0L. The maximum stress of the pristine model was discussed in the previous section and here the amount of the maximum stress at different temperatures is illustrated. The tensile strength decreases dramatically when cracks exist. The tensile strength decreases with increasing crack length. The tempera-
Fig. 5. The process of the crack propagation (a–e) in a C3N nanosheet with the length of L/6 at 300 K.
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ture rise has a weakening influence on the ultimate tensile strength of the nanosheet as well. Fig. 3b depicts the strains at the maximum tensile stress. After the failure of the C3N nanosheet, the atomic bonds are broken and bigger deformations occur. The results for the pre-cracked models have been compared with the results of the pristine model. As expected, the defect free C3N nanosheet can withstand more tensile stress. According to Linear Elastic Fracture Mechanics (LEFM), the critical stress intensity factor for a plate with periodic centred cracks under mode I fracture can be obtained as [56]
K IC ¼ rf
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pa 2h tan 2h
ð1Þ
where rf is the fracture stress, 2h is the width of the nanosheet, and 2a is the initial crack length. The above mentioned formula has been used for a periodic structure of the C3N nanosheet. For crack propagation, we have calculated the critical stress intensity factor for the crack length of L/12 at different temperatures. As can be seen from Fig. 4, the stress intensity factor only slightly decreases as the temperature increases. The simulations were conducted for five
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samples with uncorrelated initial conditions in order to define the error-bars in the acquired results. The crack propagation with a sample of initial crack length equal L/6 at 300 K is illustrated in Fig. 5. Stress concentrations of 130 GPa occur close to the crack tip. The strain at the end of process (e), where crack propagates the whole length of the crack, is 0.09. We next focus on the influence of the notch on the mechanical properties of C3N. Different notch diameters of L/12, L/6, L/3, and L/2 were studied, which are similar to the values for the crack lengths. In Fig. 6a, the maximum tensile stress for the different diameters are depicted at different temperatures. They are compared with the values for the pristine model. The ultimate tensile strength decreases as the notch diameter increases. As the temperature increases, the tensile strength decreases as well. At 300 K, the maximum tensile strength for the notch-free model is 128 GPa while its value is only 44 GPa for a notch of length L/2 at 900 K. In other words, the ultimate tensile stress decreases by an amount of 65%. The same decreasing trend is observed for all models when the temperature rises from 300 K to 900 K. At 900 K for the pristine
Fig. 6. (a) The ultimate tensile strength of the nanosheet in presence of the notch defect with different notch diameters at different temperatures of 200, 300, 500, 700, and 900 K. (b) Engineering strain at maximum tensile strength in presence of notch with different diameters.
Fig. 7. The process of the C3N nanosheet fracture under the uniaxial tensile loading for a notch with the length of L/6 at 500 K.
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helpful for the use of ultra stiff and thermally stable C3N for various applications such as those in nanocomposites and nanoelectronics.
Acknowledgement The authors extend their appreciation to the Distinguished Scientist Fellowship Program (DSFP) at King Saud University for funding this work.
References
Fig. 8. Comparison for the maximum strength of samples with two cracks and two notches in.
model, the tensile strength is 82 GPa while for a notch of L/2 diameter its value decreases by 64% to 29 GPa. Fig. 6b shows the maximum engineering strain and stress at fracture. The maximum strain decreases when the temperature rises and the notches become bigger. Fig. 7 illustrates the fracture process for a C3N nanosheet containing a notch of L/6 diameter. The simulations were performed at 500 K until the nanosheet ruptures along the orientation perpendicular to the loading direction. The linear zone of the stressstrain curve (a) is shown at the initiation step where the engineering strain is 0.02 and the C3N nanosheet is still within the elastic range. The increase in engineering strain is 78% in the whole process showing the nanosheet experiences large deformations. The stresses at the top and bottom of the notch increase dramatically. However they are not that high on the left and right side of the notch. A comparison of the results of the pre-cracked and pre-notched nanosheets is plotted in Fig. 8. Two different crack lengths are compared with two different notch diameters. The temperature rise were studied as well. The stresses of the C3N nanosheet with the notch defect are larger in both cases. In another words, the C3N nanosheet can withstand higher tensile stresses for notch-like defects as compared to crack-like defect. 4. Conclusions We studied the mechanical properties of C3N nanosheets and analyzed the effects of cracks, notches and temperature. It was shown that the existence of cracks and notches have strong weakening effects on the ultimate tensile strength of the material. The simulations for different crack lengths and different notch diameters were performed at five different temperatures from 200 K to 900 K. For the pristine structure, the temperature rise decreases the tensile strength. It was also shown that by increasing the crack length both the ultimate tensile strength and its corresponding engineering strain decreases. As expected, the temperature rise decreases the tensile strength since at higher temperatures atomic bonds become weaker. The critical stress intensity factor ðK IC Þ was also calculated for a crack model at different temperatures. It was shown that the temperature raise only slightly decreases the critical stress intensity factor. The simulations were conducted at the same temperature conditions for the C3N nanosheet with different notch diameters. Similar to the pre-cracked models, the notches also reduce the strength of the nanosheet. Finally, the results of the crack models were compared to the results of the notch models, which confirm that the nanosheets with notch defects can withstand higher tensile stresses. Our study can be
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