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Fracture mechanics of monolayer molybdenum disulfide
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 Nanotechnology 26 175703 (http://iopscience.iop.org/0957-4484/26/17/175703) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 129.173.72.87 This content was downloaded on 23/06/2015 at 13:46
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Nanotechnology Nanotechnology 26 (2015) 175703 (6pp)
doi:10.1088/0957-4484/26/17/175703
Fracture mechanics of monolayer molybdenum disulfide Xiaonan Wang1, Alireza Tabarraei1 and Douglas E Spearot2 1
Department of Mechanical Engineering, University of North Carolina Charlotte, 9201 University City Blvd., Charlotte, NC 28223, USA 2 Department of Mechanical Engineering, University of Arkansas, 1 University of Arkansas, Fayetteville, AR 72701, USA E-mail: [email protected] Received 5 December 2014 Accepted for publication 20 January 2015 Published 2 April 2015 Abstract
Molecular dynamics (MD) modeling is used to study the fracture toughness and crack propagation path of monolayer molybdenum disulfide (MoS2) sheets under mixed modes I and II loading. Sheets with both initial armchair and zigzag cracks are studied. The MD simulations predict that crack edge chirality, tip configuration and the loading phase angle influence the fracture toughness and crack propagation path of monolayer MoS2 sheets. Furthermore, under all loading conditions, both armchair and zigzag cracks prefer to extend along a zigzag path, which is in agreement with the crack propagation path in graphene. A remarkable out-of-plane buckling can occur during mixed mode loading which can lead to the development of buckling cracks in addition to the propagation of the initial cracks. Keywords: molybdenum disulfide, two-dimensional materials, molecular dyanmics, fracture (Some figures may appear in colour only in the online journal) 1. Introduction
mechanical failure in the form of cracks, it is essential to understand the failure properties of monolayer MoS2. In contrast to three-dimensional materials whose fracture properties have been widely studied, research on the fracture properties of two-dimensional (2D) materials is scarce. Most of the studies on the fracture properties of 2D materials are limited to graphene [11–15]; fracture properties of other 2D materials have been rarely investigated. The failure mechanisms of MoS2 are more complex than those of graphene. This is mainly due to the more complicated atomic structure of MoS2. MoS2 has a binary system (composed of two elements) and in contrast to graphene, which has a truly planar structure, a single sheet of MoS2 has a triple-layered atomic structure. These differences in the atomic structure necessitate separate investigations of the failure mechanism of monolayer MoS2.
Monolayer molybdenum disulfide (MoS2) is a recently exfoliated two-dimensional material in which two atomic layers of close-packed sulfur (S) encompass a close-packed layer of molybdenum (Mo) (see figure 1). Monolayer MoS2 displays excellent electrical and optical properties which makes it appealing for a wide range of applications, such as flexible optoelectronic devices, photodetectors, integrated logic circuits, field effect transistors and sensors [1–6]. Its semiconducting nature [7] allows it to overcome the zerobandgap of graphene, while still sharing many of graphene’s advantages for electronic applications [8, 9]. Besides fantastic physical properties, its remarkable mechanical properties such as high Young’s modulus and high flexibility [10] make it a promising candidate as a filler in nanocomposite materials. To ensure that nanodevices and nanomaterials designed based on monolayer MoS2 preserve their structural integrity during fabrication and service time, it is necessary to gain a fundamental understanding of the mechanical properties of monolayer MoS2. In particular, to predict and prevent 0957-4484/15/175703+06$33.00
2. Computational methods In this paper, we use molecular dynamics (MD) simulations to investigate the fracture properties of monolayer MoS2 1
© 2015 IOP Publishing Ltd Printed in the UK
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Nanotechnology 26 (2015) 175703
Figure 2. A pre-cracked molybdenum disulfide sheet. The
asymptotic crack tip displacement fields are applied to the boundary atoms shown in red.
Figure 1. Atomic structure of MoS2. Mo atoms are shown in blue while S atoms are shown in yellow.
under mixed opening mode (mode I) and in-plane shear (mode II) loading. In a monolayer MoS2 sheet, the interatomic interactions consists of pair-wise (Mo–Mo, S–S and Mo–S) and angle bending (S–Mo–S and Mo–S–Mo). These interatomic interactions are modeled using a recently developed reactive empirical bond order (REBO) potential [16]. This potential allows bond breaking and formation during mechanical deformation and has been used to model large deformation of bulk MoS2 [17–20]. All the MD simulations are conducted using the LAMMPS package [21] and a Nosé– Hoover thermostat [22] is used to maintain the temperature at 300 K. Our MD model is a square domain with side lengths of 110 Å , consisting of about 4700 atoms, as shown in figure 2. The structures of zigzag and armchair cracks studied in this paper are shown in figure 3. Initial armchair and zigzag cracks are generated by removing three or four layers of atoms, respectively. Due to the lattice structure of MoS2, the edges of zigzag cracks (ZZ) are not symmetric; one edge is composed of pairs of S atoms at its outermost atomic layer while the other surface is made of Mo atoms. On the other hand, while the surfaces of armchair cracks are symmetric, armchair cracks can have two different tip structures: a pair of S atoms (AC–S) or one Mo atom (AC–Mo) at their tips. The two tips of armchair cracks are demonstrated in figure 3. The effects of the crack tip structure on its toughness are studied through MD simulations. The MD simulations are performed by applying crack tip asymptotic displacement fields to the outermost layers of atoms shown in figure 2. It is assumed that the far field behavior is linear and isotropic since the crack tip strain field reduces rapidly and MoS2 behavior is anisotropic only under large strains [10, 23]. The crack tip asymptotic displacement
Figure 3. Schematic of three crack types: (a) ZZ crack, (b) AC crack with a pair of S atoms at the crack tip, and (c) AC crack with one Mo atom at the crack tip.
fields for a linear isotropic material under mixed modes I and II fracture are given by ux =
2
r ⎡ app θ⎛ θ⎞ ⎢ KI cos ⎝⎜ κ − 1 + 2 sin2 ⎠⎟ 2π ⎣ 2 2 ⎤ ⎛ ⎞ θ θ + KIIapp sin ⎜ κ + 1 + 2 cos2 ⎟ ⎥ , 2⎝ 2 ⎠⎦
1+v E
(1)
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Nanotechnology 26 (2015) 175703
Figure 4. Crack propagation paths of zigzag cracks at different loading phase angles. Atoms with a broken bond are shown in red. (a) Crack path on the undeformed configurations, (b) top view of the deformed configuration, and (c) out-of-plane deformations of the sheets.
uy =
⎡ app θ⎛ θ⎞ ⎢ KI sin ⎝⎜ κ + 1 − 2 cos2 ⎠⎟ ⎣ 2 2 θ⎛ θ ⎞⎤ − KIIapp cos ⎜ κ − 1 − 2 sin2 ⎟ ⎥ , ⎝ 2 2 ⎠⎦
1+v E
simulations can be sensitive to loading rate, and due to the lack of experimental data on the fracture properties of monolayer MoS2, the results presented in this paper mainly provide qualitative insights regarding fracture properties and trends. Nevertheless, we have run select MD simulations for 500 000 time steps relaxation and observe no qualitative differences in the presented results. The potential used in this paper consists of two components: the first component considers the interaction between covalently bonded atoms and the second component describes the van der Waals interactions. The covalent component of the potential employs two cutoff distances Rmin and Rmax to allow a smooth transition of cutoff function from one to zero. To obtain the critical stress intensity factors, the bonds lengths are checked at the end of each load increment. If the length of any bonds at the crack tip is larger than the Rmax (equal to 3.05 Å for Mo–S interactions), the bonds are considered app broken, and the Keff corresponding to that load increment is cr considered as the critical stress intensity factor (Keff ).
r 2π
(2)
where ux and uy are the displacement components in the x and y directions, E is the Young’s modulus which equals 200 GPa, [10] ν = 0.29 is the Poisson’s ratio, [23] r and θ are the polar coordinates shown in figure 2, κ is the Kolosov constant which for a plane stress condition is 3 − v . KIapp and 1+v KIIapp are the applied stress intensity factors for modes I and II loadings. The effective stress intensity factor is app Keff = (KIapp )2 + (KIIapp )2 , and the loading phase angle describing the ratio of the modes I and II loading is defined as K app I ϕ = tan−1( K app ), hence a loading phase angle of II 0° corresponds to a pure mode I loading and a loading phase angle of 90° corresponds to a pure mode II loading. The boundary conditions are applied in increments of app ΔKeff = 0.01 MPa m . The velocity-Verlet scheme with a time step of 1 fs is utilized for the purpose of time integration. After each loading increment, the position of the boundary atoms are kept fixed while the position of internal atoms are relaxed for 60 000 time steps, this corresponds to a strain rate of 5 × 10−5 ps−1. Since the results obtained in MD
3. Results and discussion Some snapshots of the crack propagation paths of zigzag cracks under mixed mode loadings are shown in figure 4. Under a pure mode I loading (ϕ = 0 ) zigzag cracks extend 3
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Nanotechnology 26 (2015) 175703
Figure 5. Crack propagation paths of armchair cracks (AC–S) at different loading phase angles. Atoms with a broken bond are shown in red. (a) Crack path on the undeformed configurations, (b) top view of the deformed configuration, and (c) out-of-plane deformations of the sheets.
Figure 6. Out-of-plane deformation of MoS2 sheets as a function of loading phase angle.
cr Figure 7. Critical stress intensity factors Keff of armchair and zigzag
cr cr are represented with solid lines while K BP are cracks. KCP represented with dashed lines.
along a self-similar path. When the phase angle is not zero, zigzag cracks extend via a kink forming at an angle with the initial crack. The crack trajectories of armchair cracks shown in figure 5 indicate that even under pure mode I loading, armchair cracks propagate via a kink. Investigation of the crack propagation paths of armchair and zigzag cracks shows that both cracks propagate mostly along a zigzag direction;
suggesting that the zigzag direction has a smaller surface energy than the armchair direction. It can be seen in figures 4(c) and 5(c) that when loading phase angle is large, i.e., when mode II loading is dominant, the MoS2 sheet undergoes out-of-plane buckling. The 4
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Nanotechnology 26 (2015) 175703
cr Table 1. Critical stress intensity factors Keff (MPa m ) of cracks at different loading phase angles ϕ. The numbers in brackets correspond to the stress intensity factor at which a buckling crack nucleates.
ϕ cr Keff (MPa m )
ZZ ZZ (negative) AC–S AC–Mo
0°
15°
30°
45°
60°
75°
90°
1.3 1.3 1.8 1.5
1.3 1.7 1.8 1.3
1.3 1.4 1.3 1.3
1.5(4.6) 1.8(4.0) 1.8(4.0) 1.5(5.7)
1.2(2.9) 1.5(2.8) 1.5(3.2) 1.3(4.5)
1.8(2.5) 1.6(2.7) 1.6(2.0) 1.2(2.5)
1.3(2.3) 1.2(2.8) 1.8(2.2) 1.6(2.6)
buckling is due to the compressive stress generated by mode II component of loading. Since the thickness of the MoS2 sheet is very small, the compressive stress induces out-ofplane deformation, a phenomenon reminiscent of buckling of thin cracked plates under tensile and shear loading [24–27]. The magnitude of out-of-plane buckling of armchair and cri zigzag cracks at Keff is shown in figure 6, which clearly shows that out-of-plane deformation is negligible when the mode I component is dominant and becomes more severe when the phase angle increases. Furthermore, these curves show that out-of-plane deformations of armchair cracks with a pair of sulfur atoms at their tips (AC–S) are always larger than that of the armchair cracks with one molybdenum atom (AC– Mo) at their tips. Figures 4 and 5 also indicate that if the out-of-plane buckling is excessive, buckling cracks nucleate at the surface of the initial crack. In this paper, for each loading phase, two critical stress intensity factors are reported. One corresponds to the critical stress intensity factor at which the initial crack cr propagates (KCP ), and the other one corresponds to the critical stress intensity factor at which a bond breaks due to the cr buckling (KBP ) of a crack surface. The critical effective stress intensity factors of the armchair and zigzag cracks are presented in figure 7 and table 1. Due to the asymmetry of the zigzag crack edges, the loading phase angle is varied from − 90° to 90°. Plots in figure 7 show cr that the KCP of the AC–S cracks is larger than that of AC–Mo atom at its tip. This is mainly due to different interatomic interactions between Mo–Mo and S–S atoms in MoS2. In the REBO potential, the cut-off distance of Mo–Mo pairwise interactions is 3.8 Å whereas the cut-off distance of S–S interactions is 3.0 Å . In an armchair crack with one Mo atom at its tip, the distance between the four S atoms at the immediate vicinity of the tip (see figure 3) is larger than S–S cut-off distance, hence they don’t interact with each other. On the other hand, in armchair cracks with two S atoms at the crack tip, the distance between the two Mo atoms located in the immediate vicinity of the crack tip is less than their cut-off distance, hence they interact with each other. Due to this extra interatomic interaction, more energy is required to deform the crack tip neighborhood which in turn leads to an increase in the fracture toughness. cr Figure 7 also indicates that in general KCP of zigzag cracks is larger when the crack surface with Mo atoms at its outermost layer is under compression, i.e., loading phase cr angle is less than zero. Furthermore, comparing KCP of
armchair and zigzag cracks shows that under pure mode I loading armchair cracks are tougher than zigzag cracks. Based on figure 7, no buckling crack is generated when the phase angle is less than 45°, i.e., no buckling cracks are observed when mode I loading is dominant. By increasing the phase angle, the magnitude of the out-of-plane deformation cr increases and buckling cracks nucleate. The magnitude of KBP of both armchair and zigzag cracks significantly reduces when the loading phase angles approaches pure mode II loading. cr Moreover, KBP of AC–S cracks are always smaller than that of AC–Mo cracks. This is in agreement with the out-of-plane deformation curves presented in figure 6. Furthermore, the plots of figure 7 indicate that for the loading rate presented in this paper, buckling cracks always nucleate after the propagation of the initial cracks. However, from supplemental MD simulations, we have noticed that by increasing the loading rate, the buckling cracks might nucleate before the propagation of the initial crack.
4. Conclusion and summary In summary, we have employed MD simulations to study the failure mechanism of armchair and zigzag cracks in monolayer MoS2 sheets under mixed fracture modes I and II loading. MD simulations predict that similar to graphene, both armchair and zigzag cracks prefer to propagate along a zigzag surface. The critical stress intensity factors of MoS2 depends on the loading phase angle, crack edge chirality and cr crack tip structure and is in the range of KCP = 1–1.8 (MPa m ). A main difference between the fracture in MoS2 sheets and graphene is the buckling fracture. Although under mode II loading both graphene and MoS2 sheets undergo outof-plane deformation, buckling cracks have not been observed in graphene. This might be due to a higher strength of graphene which limits the amplitude of buckling. The amplitude of buckling in MoS2 depends on the phase angle; no buckling is observed if the phase angle is less than 45°, and buckling cracks develop only if the phase angle is larger than 45°.
References [1] Zeng Z, Yin Z, Huang X, Li H, He Q, Lu G, Boey F and Zhang H 2011 Single-layer semiconducting nanosheets: high-yield preparation and device fabrication Angew. Chem., Int. Ed. 50 11093–7 5
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Nanotechnology 26 (2015) 175703
[14] Le M and Batra R 2013 Single-edge crack growth in graphene sheets under tension Comput. Mater. Sci. 69 381–8 [15] Hwangbo Y et al 2014 Fracture characteristics of monolayer cvd-graphene Sci. Rep. 4 4439 [16] Liang T, Phillpot S and Sinnott S 2009 Parametrization of a reactive many-body potential for Mo-S systems Phys. Rev. B 79 245110 [17] Stewart J and Spearot D 2013 Atomistic simulations of nanoindentation on the basal plane of crystalline molybdenum disulfide (MoS2) Modelling Simul. Mater. Sci. Eng. 21 045003 [18] Bucholz E and Sinnott S 2013 Structural effects on mechanical response of MoS2 nanostructures during compression J. Appl. Phys. 114 034308 [19] Dang K and Spearot D 2014 Effect of point and grain boundary defects on the mechanical behavior of monolayer MoS2 under tension via atomistic simulations J. Appl. Phys. 116 013508 [20] Dang K, Simpson J and Spearot D 2014 Phase transformation in monolayer molybdenum disulphide MoS2 under tension predicted by molecular dynamics simulations Scr. Mater. 76 41–4 [21] http://lammps.sandia.gov [22] Nosé S 1984 A unified formulation of the constant temperature molecular dynamics methods J. Chem. Phys. 81 511 [23] Cooper R, Lee C, Marianetti C, Wei X, Hone J and Kysar J 2013 Nonlinear elastic behavior of two-dimensional molybdenum disulfide Phys. Rev. B 87 035423 [24] Markstrom K and Storakers B 1980 Buckling of cracked members under tension Int. J. Solid Mech. 16 217–29 [25] Sih G and Lee Y 1986 Tensile and compressive buckling of plates weakened by cracks Theor. Appl. Fract. Mech. 6 129–38 [26] Brighenti R 2005 Buckling of cracked thin-plates under tension or compression Thin-Walled Struct. 43 209–24 [27] Brighenti R 2010 Influence of a central straight crack on the buckling behaviour of thin plates under tension, compression or shear loading Int. J. Mech. Mater. Des. 6 73–87
[2] Radisavljevic B, Radenovic A, Brivio J, Giacometti V and Kis A 2011 Single-layer MoS2 transistors Nat. Nanotechnology 6 147–50 [3] Mak K, He K, Shan J and Heinz T 2012 Control of valley polarization in monolayer MoS2 by optical helicity Nat. Nanotechnology 7 494–8 [4] Jariwala D, Sangwan V, Lauhon L, Marks T and Hersam M 2014 Emerging device applications for semiconducting twodimensional transition metal dichalcogenides ACS Nano 8 1102–20 [5] Perkins F, Friedman A, Cobas E, Campbell P, Jernigan G and Jonker B 2013 Chemical vapor sensing with monolayer MoS2 ACS Nano 13 668–73 [6] Wang Q, Kalantar-Zadeh K, Kis A, Coleman J and Strano M 2012 Electronics and optoelectronics of two-dimensional transition metal dichalcogenides Nat. Nanotechnology 7 699–712 [7] Mak F, Lee C, Hone J, Shan J and Heinz T 2010 Atomically thin MoS2: a new direct-gap semiconductor Phys. Rev. Lett. 105 136805 [8] Osada M and Sasaki T 2012 Two-dimensional dielectric nanosheets: novel nanoeletronics from nanocrystal builidng blocks Adv. Mater. 24 210–5528 [9] Balendhran S, Walia S, Nili H, Ou J, Zhuiykov S, Kaner R, Sriram S, Bhaskaran M and Kalantar-Zadeh K 2013 Twodimensional molybdenum trioxide and dichalcogenides Adv. Funct. Mater. 23 3952–70 [10] Li T 2012 Ideal strength and phonon instability in single-layer MoS2 Phys. Rev. B 85 235407 [11] Xu M, Tabarraei A, Paci J and Belytschko T 2012 Coupled quantum/continuum mechanics study of graphene fracture Int. J. Fract. 173 163–73 [12] Zhang B, Mei L and Xiao H 2012 Nanofracture in graphene under complex mechanical stresses Appl. Phys. Lett. 101 121915 [13] Dewapriya M, Rajapakse R and Phani A 2014 Atomistic and continuum modelling of temperature-dependent fracture of graphene Int. J. Fract. 187 199–212
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Fracture mechanics of monolayer molybdenum disulfide
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 Nanotechnology 26 175703 (http://iopscience.iop.org/0957-4484/26/17/175703) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 129.173.72.87 This content was downloaded on 23/06/2015 at 13:46
Please note that terms and conditions apply.
Nanotechnology Nanotechnology 26 (2015) 175703 (6pp)
doi:10.1088/0957-4484/26/17/175703
Fracture mechanics of monolayer molybdenum disulfide Xiaonan Wang1, Alireza Tabarraei1 and Douglas E Spearot2 1
Department of Mechanical Engineering, University of North Carolina Charlotte, 9201 University City Blvd., Charlotte, NC 28223, USA 2 Department of Mechanical Engineering, University of Arkansas, 1 University of Arkansas, Fayetteville, AR 72701, USA E-mail: [email protected] Received 5 December 2014 Accepted for publication 20 January 2015 Published 2 April 2015 Abstract
Molecular dynamics (MD) modeling is used to study the fracture toughness and crack propagation path of monolayer molybdenum disulfide (MoS2) sheets under mixed modes I and II loading. Sheets with both initial armchair and zigzag cracks are studied. The MD simulations predict that crack edge chirality, tip configuration and the loading phase angle influence the fracture toughness and crack propagation path of monolayer MoS2 sheets. Furthermore, under all loading conditions, both armchair and zigzag cracks prefer to extend along a zigzag path, which is in agreement with the crack propagation path in graphene. A remarkable out-of-plane buckling can occur during mixed mode loading which can lead to the development of buckling cracks in addition to the propagation of the initial cracks. Keywords: molybdenum disulfide, two-dimensional materials, molecular dyanmics, fracture (Some figures may appear in colour only in the online journal) 1. Introduction
mechanical failure in the form of cracks, it is essential to understand the failure properties of monolayer MoS2. In contrast to three-dimensional materials whose fracture properties have been widely studied, research on the fracture properties of two-dimensional (2D) materials is scarce. Most of the studies on the fracture properties of 2D materials are limited to graphene [11–15]; fracture properties of other 2D materials have been rarely investigated. The failure mechanisms of MoS2 are more complex than those of graphene. This is mainly due to the more complicated atomic structure of MoS2. MoS2 has a binary system (composed of two elements) and in contrast to graphene, which has a truly planar structure, a single sheet of MoS2 has a triple-layered atomic structure. These differences in the atomic structure necessitate separate investigations of the failure mechanism of monolayer MoS2.
Monolayer molybdenum disulfide (MoS2) is a recently exfoliated two-dimensional material in which two atomic layers of close-packed sulfur (S) encompass a close-packed layer of molybdenum (Mo) (see figure 1). Monolayer MoS2 displays excellent electrical and optical properties which makes it appealing for a wide range of applications, such as flexible optoelectronic devices, photodetectors, integrated logic circuits, field effect transistors and sensors [1–6]. Its semiconducting nature [7] allows it to overcome the zerobandgap of graphene, while still sharing many of graphene’s advantages for electronic applications [8, 9]. Besides fantastic physical properties, its remarkable mechanical properties such as high Young’s modulus and high flexibility [10] make it a promising candidate as a filler in nanocomposite materials. To ensure that nanodevices and nanomaterials designed based on monolayer MoS2 preserve their structural integrity during fabrication and service time, it is necessary to gain a fundamental understanding of the mechanical properties of monolayer MoS2. In particular, to predict and prevent 0957-4484/15/175703+06$33.00
2. Computational methods In this paper, we use molecular dynamics (MD) simulations to investigate the fracture properties of monolayer MoS2 1
© 2015 IOP Publishing Ltd Printed in the UK
X Wang et al
Nanotechnology 26 (2015) 175703
Figure 2. A pre-cracked molybdenum disulfide sheet. The
asymptotic crack tip displacement fields are applied to the boundary atoms shown in red.
Figure 1. Atomic structure of MoS2. Mo atoms are shown in blue while S atoms are shown in yellow.
under mixed opening mode (mode I) and in-plane shear (mode II) loading. In a monolayer MoS2 sheet, the interatomic interactions consists of pair-wise (Mo–Mo, S–S and Mo–S) and angle bending (S–Mo–S and Mo–S–Mo). These interatomic interactions are modeled using a recently developed reactive empirical bond order (REBO) potential [16]. This potential allows bond breaking and formation during mechanical deformation and has been used to model large deformation of bulk MoS2 [17–20]. All the MD simulations are conducted using the LAMMPS package [21] and a Nosé– Hoover thermostat [22] is used to maintain the temperature at 300 K. Our MD model is a square domain with side lengths of 110 Å , consisting of about 4700 atoms, as shown in figure 2. The structures of zigzag and armchair cracks studied in this paper are shown in figure 3. Initial armchair and zigzag cracks are generated by removing three or four layers of atoms, respectively. Due to the lattice structure of MoS2, the edges of zigzag cracks (ZZ) are not symmetric; one edge is composed of pairs of S atoms at its outermost atomic layer while the other surface is made of Mo atoms. On the other hand, while the surfaces of armchair cracks are symmetric, armchair cracks can have two different tip structures: a pair of S atoms (AC–S) or one Mo atom (AC–Mo) at their tips. The two tips of armchair cracks are demonstrated in figure 3. The effects of the crack tip structure on its toughness are studied through MD simulations. The MD simulations are performed by applying crack tip asymptotic displacement fields to the outermost layers of atoms shown in figure 2. It is assumed that the far field behavior is linear and isotropic since the crack tip strain field reduces rapidly and MoS2 behavior is anisotropic only under large strains [10, 23]. The crack tip asymptotic displacement
Figure 3. Schematic of three crack types: (a) ZZ crack, (b) AC crack with a pair of S atoms at the crack tip, and (c) AC crack with one Mo atom at the crack tip.
fields for a linear isotropic material under mixed modes I and II fracture are given by ux =
2
r ⎡ app θ⎛ θ⎞ ⎢ KI cos ⎝⎜ κ − 1 + 2 sin2 ⎠⎟ 2π ⎣ 2 2 ⎤ ⎛ ⎞ θ θ + KIIapp sin ⎜ κ + 1 + 2 cos2 ⎟ ⎥ , 2⎝ 2 ⎠⎦
1+v E
(1)
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Nanotechnology 26 (2015) 175703
Figure 4. Crack propagation paths of zigzag cracks at different loading phase angles. Atoms with a broken bond are shown in red. (a) Crack path on the undeformed configurations, (b) top view of the deformed configuration, and (c) out-of-plane deformations of the sheets.
uy =
⎡ app θ⎛ θ⎞ ⎢ KI sin ⎝⎜ κ + 1 − 2 cos2 ⎠⎟ ⎣ 2 2 θ⎛ θ ⎞⎤ − KIIapp cos ⎜ κ − 1 − 2 sin2 ⎟ ⎥ , ⎝ 2 2 ⎠⎦
1+v E
simulations can be sensitive to loading rate, and due to the lack of experimental data on the fracture properties of monolayer MoS2, the results presented in this paper mainly provide qualitative insights regarding fracture properties and trends. Nevertheless, we have run select MD simulations for 500 000 time steps relaxation and observe no qualitative differences in the presented results. The potential used in this paper consists of two components: the first component considers the interaction between covalently bonded atoms and the second component describes the van der Waals interactions. The covalent component of the potential employs two cutoff distances Rmin and Rmax to allow a smooth transition of cutoff function from one to zero. To obtain the critical stress intensity factors, the bonds lengths are checked at the end of each load increment. If the length of any bonds at the crack tip is larger than the Rmax (equal to 3.05 Å for Mo–S interactions), the bonds are considered app broken, and the Keff corresponding to that load increment is cr considered as the critical stress intensity factor (Keff ).
r 2π
(2)
where ux and uy are the displacement components in the x and y directions, E is the Young’s modulus which equals 200 GPa, [10] ν = 0.29 is the Poisson’s ratio, [23] r and θ are the polar coordinates shown in figure 2, κ is the Kolosov constant which for a plane stress condition is 3 − v . KIapp and 1+v KIIapp are the applied stress intensity factors for modes I and II loadings. The effective stress intensity factor is app Keff = (KIapp )2 + (KIIapp )2 , and the loading phase angle describing the ratio of the modes I and II loading is defined as K app I ϕ = tan−1( K app ), hence a loading phase angle of II 0° corresponds to a pure mode I loading and a loading phase angle of 90° corresponds to a pure mode II loading. The boundary conditions are applied in increments of app ΔKeff = 0.01 MPa m . The velocity-Verlet scheme with a time step of 1 fs is utilized for the purpose of time integration. After each loading increment, the position of the boundary atoms are kept fixed while the position of internal atoms are relaxed for 60 000 time steps, this corresponds to a strain rate of 5 × 10−5 ps−1. Since the results obtained in MD
3. Results and discussion Some snapshots of the crack propagation paths of zigzag cracks under mixed mode loadings are shown in figure 4. Under a pure mode I loading (ϕ = 0 ) zigzag cracks extend 3
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Nanotechnology 26 (2015) 175703
Figure 5. Crack propagation paths of armchair cracks (AC–S) at different loading phase angles. Atoms with a broken bond are shown in red. (a) Crack path on the undeformed configurations, (b) top view of the deformed configuration, and (c) out-of-plane deformations of the sheets.
Figure 6. Out-of-plane deformation of MoS2 sheets as a function of loading phase angle.
cr Figure 7. Critical stress intensity factors Keff of armchair and zigzag
cr cr are represented with solid lines while K BP are cracks. KCP represented with dashed lines.
along a self-similar path. When the phase angle is not zero, zigzag cracks extend via a kink forming at an angle with the initial crack. The crack trajectories of armchair cracks shown in figure 5 indicate that even under pure mode I loading, armchair cracks propagate via a kink. Investigation of the crack propagation paths of armchair and zigzag cracks shows that both cracks propagate mostly along a zigzag direction;
suggesting that the zigzag direction has a smaller surface energy than the armchair direction. It can be seen in figures 4(c) and 5(c) that when loading phase angle is large, i.e., when mode II loading is dominant, the MoS2 sheet undergoes out-of-plane buckling. The 4
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cr Table 1. Critical stress intensity factors Keff (MPa m ) of cracks at different loading phase angles ϕ. The numbers in brackets correspond to the stress intensity factor at which a buckling crack nucleates.
ϕ cr Keff (MPa m )
ZZ ZZ (negative) AC–S AC–Mo
0°
15°
30°
45°
60°
75°
90°
1.3 1.3 1.8 1.5
1.3 1.7 1.8 1.3
1.3 1.4 1.3 1.3
1.5(4.6) 1.8(4.0) 1.8(4.0) 1.5(5.7)
1.2(2.9) 1.5(2.8) 1.5(3.2) 1.3(4.5)
1.8(2.5) 1.6(2.7) 1.6(2.0) 1.2(2.5)
1.3(2.3) 1.2(2.8) 1.8(2.2) 1.6(2.6)
buckling is due to the compressive stress generated by mode II component of loading. Since the thickness of the MoS2 sheet is very small, the compressive stress induces out-ofplane deformation, a phenomenon reminiscent of buckling of thin cracked plates under tensile and shear loading [24–27]. The magnitude of out-of-plane buckling of armchair and cri zigzag cracks at Keff is shown in figure 6, which clearly shows that out-of-plane deformation is negligible when the mode I component is dominant and becomes more severe when the phase angle increases. Furthermore, these curves show that out-of-plane deformations of armchair cracks with a pair of sulfur atoms at their tips (AC–S) are always larger than that of the armchair cracks with one molybdenum atom (AC– Mo) at their tips. Figures 4 and 5 also indicate that if the out-of-plane buckling is excessive, buckling cracks nucleate at the surface of the initial crack. In this paper, for each loading phase, two critical stress intensity factors are reported. One corresponds to the critical stress intensity factor at which the initial crack cr propagates (KCP ), and the other one corresponds to the critical stress intensity factor at which a bond breaks due to the cr buckling (KBP ) of a crack surface. The critical effective stress intensity factors of the armchair and zigzag cracks are presented in figure 7 and table 1. Due to the asymmetry of the zigzag crack edges, the loading phase angle is varied from − 90° to 90°. Plots in figure 7 show cr that the KCP of the AC–S cracks is larger than that of AC–Mo atom at its tip. This is mainly due to different interatomic interactions between Mo–Mo and S–S atoms in MoS2. In the REBO potential, the cut-off distance of Mo–Mo pairwise interactions is 3.8 Å whereas the cut-off distance of S–S interactions is 3.0 Å . In an armchair crack with one Mo atom at its tip, the distance between the four S atoms at the immediate vicinity of the tip (see figure 3) is larger than S–S cut-off distance, hence they don’t interact with each other. On the other hand, in armchair cracks with two S atoms at the crack tip, the distance between the two Mo atoms located in the immediate vicinity of the crack tip is less than their cut-off distance, hence they interact with each other. Due to this extra interatomic interaction, more energy is required to deform the crack tip neighborhood which in turn leads to an increase in the fracture toughness. cr Figure 7 also indicates that in general KCP of zigzag cracks is larger when the crack surface with Mo atoms at its outermost layer is under compression, i.e., loading phase cr angle is less than zero. Furthermore, comparing KCP of
armchair and zigzag cracks shows that under pure mode I loading armchair cracks are tougher than zigzag cracks. Based on figure 7, no buckling crack is generated when the phase angle is less than 45°, i.e., no buckling cracks are observed when mode I loading is dominant. By increasing the phase angle, the magnitude of the out-of-plane deformation cr increases and buckling cracks nucleate. The magnitude of KBP of both armchair and zigzag cracks significantly reduces when the loading phase angles approaches pure mode II loading. cr Moreover, KBP of AC–S cracks are always smaller than that of AC–Mo cracks. This is in agreement with the out-of-plane deformation curves presented in figure 6. Furthermore, the plots of figure 7 indicate that for the loading rate presented in this paper, buckling cracks always nucleate after the propagation of the initial cracks. However, from supplemental MD simulations, we have noticed that by increasing the loading rate, the buckling cracks might nucleate before the propagation of the initial crack.
4. Conclusion and summary In summary, we have employed MD simulations to study the failure mechanism of armchair and zigzag cracks in monolayer MoS2 sheets under mixed fracture modes I and II loading. MD simulations predict that similar to graphene, both armchair and zigzag cracks prefer to propagate along a zigzag surface. The critical stress intensity factors of MoS2 depends on the loading phase angle, crack edge chirality and cr crack tip structure and is in the range of KCP = 1–1.8 (MPa m ). A main difference between the fracture in MoS2 sheets and graphene is the buckling fracture. Although under mode II loading both graphene and MoS2 sheets undergo outof-plane deformation, buckling cracks have not been observed in graphene. This might be due to a higher strength of graphene which limits the amplitude of buckling. The amplitude of buckling in MoS2 depends on the phase angle; no buckling is observed if the phase angle is less than 45°, and buckling cracks develop only if the phase angle is larger than 45°.
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