Comparative Fracture Toughness of Multilayer Graphenes and

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Comparative Fracture Toughness of MultiLayer Graphenes and Boronitrenes Xianlong Wei, Si Xiao, Faxin Li, Daiming Tang, Qing Chen, Yoshio Bando, and Dmitri Golberg Nano Lett., Just Accepted Manuscript • Publication Date (Web): 02 Jan 2015 Downloaded from http://pubs.acs.org on January 3, 2015

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26x8mm (300 x 300 DPI)

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Comparative Fracture Toughness of Multilayer Graphenes and Boronitrenes Xianlong Wei,1,* Si Xiao,2 Faxin Li,3,* Dai-Ming Tang,4,5 Qing Chen,1 Yoshio Bando,5 Dmitri Golberg 5,* 1

Key Laboratory for the Physics and Chemistry of Nanodevices, Department of Electronics, Peking University, Beijing 100871, P. R. China

2

3

Department of Engineering Mechanics, Tsinghua University, Beijing 100084, P. R. China

State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, P. R. China 4

International Centre for Young Scientists, National Institute for Materials Science (NIMS), Namiki 1-1, Tsukuba, Ibaraki, 305-0044, Japan

5

International Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), Namiki 1-1, Tsukuba, Ibaraki, 305-0044, Japan

1 ACS Paragon Plus Environment

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ABSTRACT: We report the comparative in-situ fracture toughness testing on single-edge V/Unotched multilayer graphenes and boronitrenes in a high-resolution transmission electron microscope (HRTEM). The nanostructures of notch tips and fracture edges of the tested specimens are unambiguously resolved using HRTEM. By analyzing the notch tip stresses using finite element method, the fracture toughness of multilayer graphenes and boronitrenes is determined to be 12.0±3.9 and 5.5±0.7 MPa m , respectively, taking into account the notch tip blunting effects.

KEYWORDS: graphene, boronitrene, fracture toughness, two-dimensional nanosheets, transmission electron microscope

Graphene and its boron nitride (BN) analogue, boronitrene, both attract broad attentions due to their exceptional physical properties and great potentials in a rich variety of applications, especially in electronic nanodevices. Graphene is a zero bandgap semi-metal with ultrahigh carrier mobility while boronitrene is a wide bandgap insulator.1 By using graphene as a channel or electrode material and boronitrene as the gate dielectric or underlying support, highperformance field-effect transistors have been fabricated.2, 3 The distinct electrical properties together with the quite similar lattice structures of graphene and boronitrene make it feasible to construct graphene/boronitrene heterostructures with fascinating electrical properties in both planar and lateral configurations.4-6 Benefiting from high flexibility, graphenes, boronitrenes and their heterostructures are particularly valuable for flexible electronics,7, 8 which strongly relies on the mechanical properties of graphene and boronitrene. Moreover, graphene and boronitrene


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have been demonstrated to be the key components in nanoelectromechanical systems and reinforced composites.9-11 For these applications, it is very important to measure the basic mechanical properties of graphene and boronitrene, such as Young’s modulus, tensile strength and fracture toughness. Under indentation of a freestanding monolayer graphene with an atomic force microscope (AFM) tip the intrinsic strength and Young’s modulus of the mechanically exfoliated graphene were measured to be as high as 130 GPa and 1 TPa, respectively.12 As compared to an intrinsic strength of 86.5 GPa and Young’s modulus of 1.006 TPa calculated for graphene (assuming a monolayer thickness of 0.34 nm), comparable intrinsic strength of 78.8 GPa and Young’s modulus of 0.836 TPa were predicted for boronitrene (assuming a monolayer thickness of 0.33 nm).13 Thus graphene and boronitrene are consequently believed to be the two strongest materials on Earth if uniform bond breaking in their single-crystalline forms is considered. However, for flexible (but brittle) atomically thin graphenes and boronitrenes, it has been rare to witness uniform bond breaking in practice, especially for large-area atomic films. The relevant engineering strength of large-area two-dimensional (2D) nanosheets is usually governed by the local bond breaking at the pre-existing defects, which serve as the stress concentrators. The capability to resist the local bond breaking near defects is described by fracture toughness which is another important mechanical quantity of solids. Due to the insurmountable difficulties in specimen fabrication and mechanical testing, previous studies on the fracture of graphenes and boronitrenes have mostly been relied on sole computational simulations.14-18 Up to date, as compared to well established Young’s modulus and intrinsic strength,12,19 the fracture toughness of 2D-atomic films has seldom been addressed in experiments except the recent pioneering work of Zhang et al.20 These authors conducted in situ tensile testing of center-cracked graphenes in a


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scanning electron microscope (SEM) and obtained their fracture toughness based on the classical Griffith fracture theory. However, the geometrical structures of crack tips were not resolved in their work due to the spatial resolution limitations of SEM, and thus the effects of crack blunting on the measured fracture toughness were not evaluated. Since the fracture of a material is quite sensitive to the geometrical structures of crack tips it is particularly timely-warrant to develop a method to determine fracture toughness of 2D materials taking into account the crack blunting effects. Herein, we report the comparative in situ fracture toughness testing in a high-resolution transmission electron microscope (HRTEM) on multilayer graphene and boronitrene samples having V/U-shaped pre-made single-edge notches. Taking the advantages of HRTEM, the curvature radius, the open angle of the notch tip, and notch dimensions were unambiguously determined. By analyzing the notch tip stresses just before fracture and taking into account the local notch tip structures using finite element method, the apparent stress intensity factor (SIF) for V/U-notched samples was calculated and the fracture toughness of multilayer graphene and boronitrene was determined. The experimental fracture testing on both graphene and boronitrene nanosheets, as well as the direct visualization of their fracture edges using HRTEM, enables us to correlate the measured fracture toughness with the particular nanostructures of fracture edges and to perform the first comparative study of the fracture properties of the these two important 2D nanomaterials.


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Figure 1. (a) A photograph of the front end of AFM-TEM holder. (b) Schematic drawing of fracture toughness testing of multilayer graphene and boronitrene nanosheets. (c-d) Schematic drawings of V (c) and U (d) shaped pre-notched nanosheets tested in our work. The dimensions of nanosheets (L, W, t), dimensions of notches (d, s) and the width of force applying segment (l) are marked. (e) The local structure of a notch tip, as described by curvature radius (R) and open angle (θ).

Our experiments were carried out in situ in HRTEM equipped with a side-entry AFM-TEM holder (Nanofactory Instruments AB), which combines a force-measuring system and a highprecision piezoelectric tube-driven nanomanipulating system (Figure 1a). The graphene and boronitrene nanosheets tested in our experiments were synthesized by arc-discharge method21 and an induction heating method,22 respectively. A graphene or boronitrene sample for fracture testing was fabricated by firstly edge clamping a rectangular graphene or boronitrene nanosheet between the tip of a force-sensing cantilever (AFM tip) and the movable W tip and then cutting a


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notch at the middle of the clamped edge by a 300 keV focused electron probe (Figure 1b). The details of sample fabrication are shown in Supporting Information. To fracture the nanosheets, the W tip was retracted at a constant speed of ~2-4 nm/s with the tension direction perpendicular to the crack and the tensile force was simultaneously recorded through the deflection of the force-sensing cantilever with a resolution of a few nanonewtons until the nanosheets were split off. The whole testing process was monitored and video-recorded in real time under TEM observations. As shown in Figure 1c and d, the nanosheets with a notch of V or U shapes were adopted for testing. In addition to the nanosheet dimensions (e.g. length (L), width (w) and thickness (t)) and notch dimensions (e.g. notch depth (d) and open width (s)), the curvature radius (R) and open angle (θ) of the notch tip were determined under HRTEM for each nanosheet for the complete characterization of the local notch tip nanostructures. Figure 2 shows the fracture testing of a single-edge V-notched multilayer graphene nanosheet. The nanosheet was clamped between the AFM and W tips at its upper edge with the clamped segment width (l) of ~11 nm (Figure 2a). The rectangular section of interest between the two tips was 170 nm long and 54 nm wide. HRTEM image taken from the clamped edge indicates that the nanosheet exhibits five well-defined lattice fringes (Figure 2b). Since the nanosheet was cut down from a flattened nanotube,23 it is actually 10 layers thick with a thickness of 3.4 nm. A Vshaped notch with a depth of 23 nm, an open angle of 70°, and a curvature radius of notch tip of 4.0 nm was made at the middle of the clamped edge (Figure 2c). The nanosheet was subjected to tension and finally fractured from the tip of the V-notch into two pieces with the fracture edges perpendicular to the tensile direction (Figure 2d). The tensile force increased linearly to the critical value (FC) of 1210 nN and then abruptly dropped to zero (Figure 2e), indicating that the nanosheet fractured in a brittle way, in agreement with the previous observations.20 A real-time


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video of the fracture process is presented in Supporting Information (Video S1). Taking into account local structures of the notch tip and the dimensions of the notch and nanosheet as determined by HRTEM, the stress field of the nanosheet just before the moment of fracture was simulated using finite element method (details of the simulation are shown in Supporting Information), as shown in Figure 2f. The maximum stress at the front of the notch tip was finalized to be 196.1 GPa.


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Figure 2. (a) TEM image of a multilayer graphene nanosheet (C_#3 in Table S2) clamped between the AFM and W tips. (b) HRTEM image of the framed area in (a). (c) TEM image of the nanosheet in (a) after a V-shaped notch was fabricated at its upper edge. As outlined in the figure, the notch tip has a curvature radius of 4.0 nm and an open angle of 70°. (d) TEM image of the nanosheet after it was fractured from the notch tip. (e) Force curve recorded during the nanosheet fracture testing. Due to the pre-existing tensile stress before the fracture testing, the force curve does not pass through the origin of the coordinate plane. (f) Stress distribution in the nanosheet just before the moment of fracture. The upper figure is a magnification of the framed area in the lower one.

The fracture testing of a U-notched multilayer boronitrene nanosheet is shown in Figure 3. The nanosheet was 150 nm long and 60 nm wide with the clamped segment length of 12 nm (Figure 3a). HRTEM image of the clamping edge of the nanosheet shows that it has 10 layers exhibiting a thickness of 3.5 nm (Figure 3b). The well-defined lattice fringes and the fast Fourier transform (FFT) pattern of the nanosheet indicate that it is highly crystalline. The U-shape notch with a depth of 25 nm, an open width of 29 nm was produced at its upper edge (Figure 3c). The notch tip had a curvature radius of 2.8 nm (Figure 3d). Similar to the fracture of the graphene nanosheet in Figure 2, the boronitrene nanosheet fractured into two pieces from the notch tip in a brittle mode with the fracture edges perpendicular to the tensile direction (Figure 3e). The tensile force increased approximately linearly until an abrupt drop to zero. The maximum tensile force was 850 nN (Figure 3f). Figure 3g depicts the stress distribution of the nanosheet just before the moment of fracture, exhibiting a maximum stress of 154.6 GPa at the notch tip. A real-time


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video of the fracture process of the boronitrene nanosheet is presented in Supporting Information (Video S2).

Figure 3. (a) TEM image of a multilayer boronitrene nanosheet (BN_#3 in Table S2) clamped between the AFM and W tips. (b) High-resolution TEM image of the framed area in (a). The


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inset is a FFT pattern of the high-resolution TEM image. (c) TEM image of the nanosheet after a U-shaped notch was fabricated at its upper edge and (d) corresponding HRTEM image of the notch. As outlined in (d), the notch tip has a radius of curvature of 2.8 nm. (e) TEM image of the nanosheet after fracture testing, and (f) the force curve recorded during the fracture testing. (g) Stress distribution in the nanosheet just before the moment of fracture. The upper figure is a magnification of the framed area in the lower one.

During conventional fracture toughness ( K IC ) testing, the specimens are machined to standardized configurations such as the single-edge-notched beam in particular dimensions, and the fracture toughness can be straightforwardly calculated using a simple formula from a SIF handbook once the critical tensile force is measured. However, for the present graphene or boronitrene nanosheets, it is rather difficult to tailor the testing samples into standardized configurations. In this work, thanks to the in-situ TEM imaging, both the geometry and dimensions of the nanoscale samples and notches were accurately measured, and the stress fields in the testing samples were simulated by using finite element method. This enables us to determine K IC through apparent stress intensity factor and taking into account the effects of crack blunting. For the V-notched sample, the apparent SIF K IV value can be expressed by the stress field at the notch tip and its geometrical parameters (the curvature radius R and the open angle θ ) as:24  K IV = σ max 2π r01− λ1 

 (1 + µ1 ) χ d 1 + χ c1 q   / 1 +   (1 + λ1 ) + χ b1 (1 − λ1 ) 4(q − 1)  

(1)


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, where σ max is the maximum principle stress at the front of the notch tip, r0 and q are functions of R and θ as: q = 2 − θ / 180° , and r0 = R ( q − 1) / q . λ1 , µ1 , χ b1 , χ c1 , χ d 1 are solely dependent on θ and can be determined by looking in the Table S1 shown in Supporting Information. For the U-notched sample, the apparent SIF K IU can be written as: 25 K IU = σ max π R / 4

(2)

For V/U-notched samples, the critical apparent SIF K CV /U value has the following approximate relationship with the fracture toughness K IC :25

K CV /U π R = 1+ K IC 4 lch

(3)

, where l ch is the characteristic length of the material defined as lch = ( K IC / f t ) 2 with ft being intrinsic tensile strength,25 which is 86.5 GPa for graphene and 78.8 GPa for boronitrene.13 So one can get fracture toughness as:

K IC = (( K CV /U ) 2 −

π R 2 1/2 ft ) 4

(4)

It can be seen from Equation 4 that the larger the curvature radius of the notch tip becomes, the more significant difference between K CV /U and K IC is apparent. In this way, the notch tip blunting effect is taken into account. Therefore, by calculating the maximum stress at the front of the notch tip using finite element method and measuring the curvature radius, and open angle of the notch tip, the sample fracture toughness can be straightforwardly obtained by using the Equations 1, 2 and 4. The fracture toughness of the graphene nanosheet in Figure 2 and the boronitrene nanosheet in Figure 3 was determined to be 14.1 and 6.3 MPa m , respectively.


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In total, we were able to measure three graphene and three boronitrene nanosheets. A summary from all the measured samples is shown in Table S2 in Supporting Information. The fracture toughness (KIC) of the measured by us multilayer nanosheets, together with theoretically estimated KIC values and previously reported KIC values for monolayer graphene and/or boronitrene, are presented in Figure 4. The measured by us multilayer graphene nanosheets exhibit an average KIC of 12.0 MPa m with a standard deviation of 3.9 MPa m . For boronitrene samples, we obtain an average KIC of 5.5 MPa m with a standard deviation of 0.7

MPa m . To the best of our knowledge, this is the first experimental measurements of the fracture toughness of boronitrene. From Table S2, it can be seen that the characteristic length of multilayer graphene nanosheets has an average value of about 21 nm, while that of the multilayer boronitrene nanosheets is only about 4.8 nm. This indicates that the fracture toughness of multilayer graphene nanosheets is not so sensitive to the notch tip radius as for the multilayer boronitrene ones, i.e., the notch tip blunting effect in graphene nanosheets is less pronounced.

Figure 4. A column diagram showing the fracture toughness (KIC) of multilayer graphene and boronitrene nanosheets measured in this work, together with theoretically estimated KIC for


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monolayer graphene and boronitrene, and previously reported KIC for mono/bi-layer graphene. The color code is shown in the upper right frame. The theoretical KIC value was estimated as K IC = 2γ E with the γ reported in Ref. 26 and E reported in Ref. 13.

It can be seen from Figure 4 that the measured by us KIC values of multilayer graphene and boronitrene nanosheets are larger than the theoretically estimated KIC values for their monolayer counterparts, both for the fractures along armchair or zigzag edges. The theoretical KIC value was estimated by using a formula: K IC = 2γ E with E being the Young’s modulus and γ being the surface energy, or edge energy in case of 2D materials. Monolayer graphene and boronitrene were reported to have the Young’s modulus of 1.006 TPa and 0.836 TPa, respectively.13 The edge energies of monolayer graphene and boronitrene were calculated to be 1.00 and 0.76 eV/Å for armchair edges, and 1.17 and 1.24 eV/Å for zigzag edges.26 Assuming a monolayer thickness of 0.34 nm for graphene and 0.33 nm for boronitrene, the edge energy of graphene and boronitrene can be respectively rewritten as 4.70 and 3.68 J/m2 for armchair edges and 5.50 and 6.01 J/m2 for zigzag edges. According to the reported Young’s modulus and edge energy, KIC values of monolayer graphene and boronitrene are estimated to be 3.08 and 2.48 MPa m for the fracture along armchair edges and 3.33 and 3.17 MPa m for zigzag edges, thus much smaller numbers than the ones determined by us. In contrary to quite similar KIC values for monolayer graphene and boronitrene based on the estimations, the obtained by us values of KIC for multilayer graphene nanosheets is about two times larger than those for multilayer boronitrene nanosheets. The determined herein KIC value of multilayer graphene nanosheets is


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also larger than that of mono/bi-layer graphene measured by Zhang et al., who obtained the KIC value of 4.0±0.6 MPa m for graphene fracture along a smooth edge.20 To get deeper insights into the fracture of the present graphene and boronitrene nanosheets and to further understand the determined KIC values, we took closer observations of the fracture edge nanostructures by HRTEM, as illustrated in Figure 5a and 5b. It can be seen that, instead of atomically smooth fracture edges (as it should be in the case of ideally brittle fracture along armchair or zigzag edges) rough fracture edges are natural for the present graphene and boronitrene nanosheets. These are thought to be caused by crack meandering and/or branching. These factors significantly increase the energy required for a crack to propagate as compared to the fracture along atomically smooth edges. This explains why determined by us KIC values of multilayer graphenes and boronitrenes are considerably larger than those of monolayer graphene theoretically estimated for the fractures along armchair or zigzag edges and previously measured for the smooth fracture edges. The fracture edges of graphene nanosheets are found to be much rougher than those of boronitrene nanosheets (Figure 5a), implying more pronounced additional energy dissipation in them. This is in a good agreement with the presently measured higher fracture toughness of graphene nanosheets compared to boronitrene ones.


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Figure 5. (a-b) HRTEM images of typical fracture edges of multilayer graphene (a) and boronitrene (b) nanosheets as outlined by white lines. (c) Schematic diagram showing the crack meandering of graphene or boronitrene monolayer as a result of the fracture preference along the armchair and zigzag directions. The blue solid arrows indicate the fracture edge of a monolayer along the armchair or zigzag directions. The red dashed arrow indicates the practical fracture direction of a tested multilayer nanosheet.

We now discuss the mechanisms responsible for the rough fracture edges of the tested herein nanosheets and their higher fracture toughness. As shown in Figure S1, the monolayers in the tested multilayer graphene stacks are disordered having a rotational mismatch with respect to each other. The fracture of monolayer graphenes was observed to predominantly propagate in the armchair or zigzag directions.27 So the crack propagation path of a whole multilayer nanosheet pile is rarely aligned along the armchair or zigzag directions peculiar to an individual monolayer. To make the fracture of a monolayer propagate averagely along the fracture direction of the whole nanosheet, it is expected to fracture alternately along armchair or zigzag directions


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resulting in crack meandering as shown in Figure 5c. Moreover, since each monolayer has a different alignment relative to the fracture direction of the whole nanosheet, each monolayer has a different fracture path. Therefore, the fracture edge of a multilayer graphene nanosheet with disordered layer stacking is expected to be rough along both the in-plane and thickness directions. However, the crack meandering in Figure 5c due to disordered layer stacking is found to increase the fracture toughness value by less than two times only (see Supporting Information), thus this cannot fully account for the obtained by us fracture toughness values that are 2-4 times larger than the theoretically estimated and previously measured values for the smooth fracture edges. Previous molecular dynamics simulations have shown that there is crack branching during the fracture of a rectangular monolayer graphene when it is tensile loaded along one of its edges,27 just as graphene nanosheets were tensile loaded in our experiments. So, the crack branching is expected to occur during fracture of multilayer nanosheets as well, when they are tensile loaded along one of their edges, especially for the tested by us nanosheets with disordered layer stacking. Since crack branching can easily multiply the length of crack path by several times when crack propagates in a complex way, it can significantly increase the fracture toughness of the measured multilayer graphene nanosheets. It should be noted that, graphene films were uniformly tensile loaded through two opposite edges in the work of Zhang et al.20 Previous molecular dynamics simulations have shown that graphene tensile loaded in this way exhibit smooth fracture edges without crack branching.27 This may be one of the reasons for the different values of fracture toughness measured by these authors and by us. Moreover, structural defects in the nanosheets introduced during their synthesis and fracture testing under high-energy electron beam irradiation in TEM may also contribute to the roughness of the fracture edges and to the measured higher fracture toughness. For example, the direction of crack propagation was


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found to change when a crack propagates across a grain boundary.20,27 Such crack meandering due to grain boundary is expected to increase fracture toughness as well. The rough fracture edges of boronitrene samples and their higher fracture toughness are attributed to the similar reasons discussed above. However, as shown in Figure S2 (also see the inset of Figure 3b), the tested boronitrene nanosheets exhibited more orderly stacked piles compared to graphene nanosheets, probably caused by the enhanced interlayer interactions in them due to the partial ionic character of B-N bonds.28 The more ordered layer stacking in boronitrene samples may be responsible for their smoother fracture edges and lower fracture toughness. From Table S2, it can be calculated that the relative standard error of the measured fracture toughness K IC values for graphene and boronitrene nanosheets are 18.9% and 7.6%, respectively. Considering the difficulties and corresponding uncertainties associated with the fabrication and fracture testing of the nanoscale notched specimens, such errors are acceptable. A minor scatter of the measured fracture toughness values indicates that the utilized combined in-situ fracture testing in TEM and finite element analysis provide a feasible and reliable method for determining the fracture toughness of a 2D nanomaterial. Furthermore, the local structures of notch tips unambiguously resolved under HRTEM in our work and their effect on the fracture toughness values, explicitly taken into account herein, make a significant step forward compared to the previous work of Zhang et al.20 In summary, single-edge V/U-notched multilayer graphenes and boronitrenes were in situ fractured in HRTEM. Taking advantages of this method, the local nanostructures of notch tips and fracture edges of the tested specimens were unambiguously resolved. By analyzing the notch tip stress using finite element method and calculating the apparent SIF for V/U notched samples, and by taking into account the notch tip blunting effects, the fracture toughness of multilayer


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graphene and boronitrene nanosheets was determined to be 12.0±3.9 and 5.5±0.7 MPa m , respectively. Importantly, this is the first time that the fracture toughness of boronitrene was quantified in the experiments. Our results suggest that fracture toughness of multilayer graphene and boronitrene nanosheets depends on the orderliness of layer stacking revealing higher fracture toughness for more disordered piles. Our work provides a novel method for determining the fracture toughness of 2D nanomaterials with the notch tip blunting effects taken into account. The determined herein fracture toughness of graphenes and boronitrenes is envisaged to pave the way toward a smart mechanical design of high-end nanodevices based on graphene, boronitrene and their heterostructures.

Supporting Information. Details of specimen fabrication and stress field simulation, the values of the parameters in Equation 1, summary of all measured samples, structural and compositional information of graphene and boronitrene nanosheets, a theoretical discussion about the effects of crack meandering on the fracture toughness of graphene and real-time videos of fracture testing. This material is available free of charge via the Internet at http://pubs.acs.org.

Corresponding Authors * [email protected], [email protected], [email protected]

ACKNOWLEDGMENT The work in PKU was supported by the National Basic Research Program of China (Grant No. 2013CB933604), the National Natural Science Foundation of China (Grant No. 61371001,


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61321001 and 11304003), and the Foundation for the Author of National Excellent Doctoral Dissertation of China (Grant No. 201241), and Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20130001110030). The work in NIMS was supported by the International Center for Materials Nanoarchitectonics (MANA) of the National Institute for Materials Science (NIMS), Japan. The authors thank Prof. Jun Lou in Rice University and Prof. Bin Liu in Tsinghua University for valuable discussions.

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(8) Hsu, A.; Wang, H.; Shin, Y. C.; Mailly, B.; Zhang, X.; Yu, L.; Shi, Y.; Lee, Y. H.; Dubey, M.; Kim, K. K.; Kong, J.; Palacios, T. P. IEEE 2013, 101, 1638. (9) Bunch, J. S.; van der Zande, A. M.; Verbridge, S. S.; Frank, L. W.; Tanenbaum, D. M.; Parpia, J. M.; Craighead, H. G.; McEune, P. L. Science 2007, 315, 490. (10) Stankovich, S.; Dikin, D. A.; Dommett, G. H. B.; Kohlhaas, K. M.; Zimney, E. J.; Stach, E. A.; Piner, R. D.; Nguyen, S. T.; Ruoff, R. S. Nature 2006, 442, 282. (11) Liu, F.; Mo, X. S.; Gan, H. B.; Guo, T. Y.; Wang, X. B.; Chen, B.; Chen, J.; Deng, S. Z.; Xu, N. S.; Sekiguchi, T.; Golberg, D.; Bando, Y. Sci. Rep. 2014, 4, 4211. (12) Lee, C.; Wei, X.; Kysar, J. W.; Hone, J. Science 2008, 321, 385. (13) Andrew, R. C.; Mapasha, R. E.; Ukpong, A. M.; Chetty, N. Phys. Rev. B 2012, 85, 125428. (14) Omeltchenko, A.; Yu, J.; Kalia, R. K.; Vashishta, P. Phys. Rev. Lett. 1997, 78, 2148. (15) Xu, M.; Tabarraei, A.; Paci, J. T.; Oswald, J.; Belytschko, T. Int. J. Fract. 2012, 173, 163. (16) Cao, A. J.; Qu, J. M. Appl. Phys. Lett. 2013, 102, 071902. (17) Le, M.-Q.; Batra, R. C. Comp. Mater. Sci. 2013, 69, 381. (18) Terdalkar, S. S.; Huang, S.; Yuan, H.; Rencis, J. J.; Zhu, T.; Zhang, S. Chem. Phys. Lett. 2010, 494, 218. (19) Song, L.; Ci, L.; Lu, H.; Sorokin, P. B.; Jin, C. H.; Ni, J.; Kvashnin, A. G.; Kvashnin, D. G.; Lou, J.; Yakobson, B. I.; Ajayan, P. M. Nano Lett. 2010, 10, 3209.


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(20) Zhang, P.; Ma, L.; Fan, F.; Zeng, Z.; Peng, C.; Loya, P. E.; Liu, Z.; Gong, Y.; Zhang, J.; Zhang, X.; Ajayan, P. M.; Zhu, T.; Lou, J. Nat. Commun. 2014, 5, 3782. (21) Ando, Y.; Iijima, S. Jpn. J. Appl. Phys. 1993, 32, L107. (22) Zhi, C. Y.; Bando, Y.; Tang, C. C.; Golberg, D. Solid State Commun. 2005, 135, 67. (23) Chopra, N. G.; Benedict, L. X.; Crespi, V. H.; Cohen, M. L.; Louie, S. G.; Zettl, A. Nature 1995, 377, 135. (24) Filippi, S.; Lazzarni, P.; Tovo, R. Int. J. Solids Struct. 2002, 39, 4543. (25) Gomez, F. J.; Guinea, G. V.; Elices, M. Int. J. Fract. 2006, 141, 99. (26) Huang, B.; Lee, H.; Gu, B.-L.; Liu, F.; Duan, W. H. Nano Res. 2012, 5, 62. (27) Kim, K.; Artyukhov, V. I.; Regan, W.; Liu, Y.; Crommie, M. F.; Yakobson, B. I.; Zettl, A. Nano Lett. 2012, 12, 293. (28) Celik-Aktas, A.; Zuo, J. M.; Stubbins, J. F.; Tang, C. C.; Bando, Y. Acta Cryst. 2005 , A61 , 533.


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Table of Contents Graphic:


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Comparative Fracture Toughness of Multilayer Graphenes and

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