Relationships between the elastic and fracture

Home

Search

Collections

Journals

About

Contact us

My IOPscience

Relationships between the elastic and fracture properties of boronitrene and molybdenum disulfide and those of graphene

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2017 Nanotechnology 28 064002 (http://iopscience.iop.org/0957-4484/28/6/064002) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 61.129.42.30 This content was downloaded on 12/01/2017 at 09:25 Please note that terms and conditions apply.

You may also be interested in: 2D microscopic model of graphene fracture properties Peter Hess Elastic, plastic, and fracture mechanisms in graphene materials Colin Daniels, Andrew Horning, Anthony Phillips et al. Fracture mechanics of monolayer molybdenum disulfide Xiaonan Wang, Alireza Tabarraei and Douglas E Spearot The defect-induced fracture behaviors of hexagonal boron-nitride monolayer nanosheets under uniaxial tension Xiong Qi-lin, Li Zhen-huan and Tian Xiao-geng Mechanical properties of hybrid graphene and hexagonal boron nitride sheets as revealed by molecular dynamic simulations Shijun Zhao and Jianming Xue Failure criterion for graphene in biaxial loading—a molecular dynamics study Hessam Yazdani and Kianoosh Hatami Atomic vacancies significantly degrade the mechanical properties of phosphorene Zhen-Dong Sha, Qing-Xiang Pei, Ying-Yan Zhang et al. The fracture toughness of graphene during the tearing process Ying Wang and Zishun Liu Mechanical properties of borophene films: a reactive molecular dynamics investigation Minh Quy Le, Bohayra Mortazavi and Timon Rabczuk

Nanotechnology Nanotechnology 28 (2017) 064002 (9pp)

doi:10.1088/1361-6528/aa52e4

Relationships between the elastic and fracture properties of boronitrene and molybdenum disulfide and those of graphene Peter Hess Institute of Physical Chemistry, University of Heidelberg, Im Neuenheimer Feld 253, D-69120 Heidelberg, Germany E-mail: [email protected] Received 5 October 2016, revised 8 December 2016 Accepted for publication 9 December 2016 Published 9 January 2017 Abstract

A consistent set of 2D elastic and fracture properties of hexagonal boron nitride (h-BN) monolayers (boronitrene) and molybdenum disulfide (MoS2) nanosheets is derived. Reported literature values for Young’s moduli and fracture strengths, based on experiments and DFT calculations, were used to estimate the line or edge energy with a local 2D bond-breaking model. Consistent information was obtained for intrinsic fracture properties. The basic mechanical properties of boronitrene are roughly 25% lower than the corresponding graphene values. This is consistent with the tensile bond force model, and the lower ionic-covalent bonding energy of sp2-hybridized B–N bonds in comparison with sp2-hybridized carbon bonds. While the intrinsic stiffness and strength of MoS2 correlate with the strength of its constituent chemical bonds, DFT calculations of the line or edge energy scale with roughly two times the Mo–S bonding energy, whereas the 2D bond-breaking model yields a correlation similar to that found for h-BN. Additional failure properties such as the fracture toughness and strain energy release rate were determined. Together with the intrinsic strengths a Griffith plot of the effective strength of defective h-BN and MoS2 versus the square root of half the defect size of single defects such as (multi)vacancies and micro-cracks exhibits a slope similar to that of the graphene plot. Keywords: boronitrene, MoS2, 2D mechanical properties, 2D bond-breaking model, 2D Giffith relation (Some figures may appear in colour only in the online journal) 1. Introduction

covalent bonding of h-BN caused by a significant charge transfer of 0.47 electron charge from B to N [4]. Among the transition metal dichalcogenide (TMD) nanosheets, MoS2 is the most widely studied, owing to its semiconducting properties with direct band gap. MoS2 has a central hexagonal plane of molybdenum (Mo) sandwiched between two hexagonal sulfur (S) planes [5]. The Mo−S bonds also have a significant ionic character with negatively charged S atoms around positively charged Mo atoms. There is an excess charge of 0.205 electron on each S atom and a depletion of 0.410 electron on each Mo atom [6]. The Mo−S bond length is r0=0.242 nm and as the thickness of a single

The one-atom-thick sp2-bonded hexagonal boron nitride (hBN) monolayer (boronitrene) is together with graphene one of the thinnest known materials with remarkable mechanical properties [1, 2]. The isostructural and isoelectronic h-BN monolayer has distinct similarities to graphene with comparable bond lengths of r0=0.145 nm and r0=0.142 nm and layer thicknesses of h=0.332 nm and h=0.334 nm for boronitrene and graphene, respectively [3]. Nevertheless, the two materials show characteristic differences in their mechanical behavior, primarily due to the mixed ionic– 0957-4484/17/064002+09$33.00

1

© 2017 IOP Publishing Ltd Printed in the UK

Nanotechnology 28 (2017) 064002

P Hess

sheet we take the effective layer thickness of single MoS2 sheets in the layered bulk material of h=0.615 nm [7–9]. It is important to note that also a much smaller layer thickness of only h≈0.321 has been suggested by several authors [10]. Preparing single free-standing h-BN monolayers seems to be much more difficult than preparing graphene layers owing to the partially ionic character of the B−N bonds. Since the B and N atoms are superimposed in succession in condensed structures, the formation of double layers or multilayered structures is preferred, owing to stronger interlayer interaction [11]. Nonetheless, the preparation of freestanding monolayers and growth on suitable substrates by low pressure chemical vapor deposition (CVD) has been achieved [12, 13]. However, to date no nanoindentation measurements on free-standing monolayers have been reported. Only for ultrathin h-BN films with at least two to five atomic layers have the 2D Young’s modulus and 2D intrinsic strength been determined experimentally by nanoindentation [14]. Besides the modulus and strength of multilayered h-BN sheets, the fracture toughness has been measured for sheets containing ten monolayers [15]. While the partially ionic character of B−N bond leads to a stabilization of double or multilayered structures, making the preparation of monolayers difficult [11], for single sheets of MoS2 the Young’s modulus and intrinsic strength have been measured by nanoindentation employing punctual loads exerting biaxial stress [7, 16]. Monolayers of MoS2 are easily available by mechanical exfoliation from the mineral molybdenite, unlike monolayers of h-BN, which must be manufactured synthetically. The fracture behavior of boronitrene is described by two directions, namely failure in the zigzag and armchair directions. In boronitrene zigzag cracks are not symmetric, since in one of the two edges boron atoms occupy the edge tips, whereas in the other edge nitrogen atoms are located at the outermost position. In contrast, edges with armchair chirality are symmetric [17]. In this work always the direction of the fractured or cleaved edge is considered and not the direction of load or tension. For mode I fracture by uniaxial tension zigzag edges normally possess the lower strength, line energy, fracture toughness, and critical strain energy release rate. Owing to the complex failure process of the triplelayered atomic structure of MoS2, the fracture mechanism has been rarely studied. Similar to the situation in boronitrene, the edges of zigzag cracks are not symmetric. At the outermost position, one edge tip is composed of pairs of S atoms, while the other edge tip is made up of Mo atoms. In contrast to this, armchair edges are symmetric, exhibiting two different tip structures, either a pair of S atoms or one Mo atom. Molecular dynamics (MD) simulations indicate that under mixed mode I and II loading both zigzag and armchair cracks prefer to extend mostly along a zigzag path, suggesting that the zigzag direction is the preferred crack propagation path similar to graphene [18] (see figure 1). At present, the most accurate information on the mechanical properties of intrinsic h-BN monolayers originates from DFT calculations and the information on defective layers comes mainly from MD simulations [19]. While the

Figure 1. Schematic presentation of the tri-layer system of MoS2, showing the zigzag and armchair directions of failure.

two theoretical approaches yield good agreement for the Young’s modulus, in many cases substantially higher intrinsic strength values have been obtained by MD simulations, which are sometimes comparable to or even surpass the critical fracture stress of graphene. Since graphene defines the ultimate limits of mechanical stiffness and strength of singlelayer 2D materials these MD results must be considered with care. The theoretical treatment of mechanical properties is not as well developed for MoS2 as for graphene and boronitrene owing to the complex three-dimensional structure of MoS2 with two elements of different electronegativity. A survey of the complementary lattice dynamical and mechanical properties of graphene and MoS2 has been published recently [5]. DFT calculations have been reported by several groups based on biaxial and uniaxial loads, as will be discussed later [20, 21]. As in the case of defective 2D solids, most theoretical work is based on MD simulations [22–26]. Nanoindenter experiments are analyzed in terms of 2D mechanical properties. However, monolayers are not 2D in the strict mathematical sense, because they possess a subnanometer thickness. Therefore, these quasi-two-dimensional layers have a volume, which, of course, is extremely small in comparison with conventional volumes. Nevertheless, it is possible to formally assign to each monolayer a 3D mechanical property by dividing the 2D quantity by the corresponding layer thickness. This allows a formal comparison with the well-known properties of normal bulk materials. One should keep in mind, however, that the third dimension is not variable but fixed at a subnanometer thickness. Here the description by 2D mechanical properties is preferred because it is independent of the layer thickness. In cases where the thickness assumed by the authors is not specified, h=0.332 nm (boronitrene) and h=0.615 nm (MoS2) were used to transform formal 3D to quasi 2D properties. While point and line defects play an important role in potential technological applications of large-area samples grown, for example, by CVD, a systematic theoretical investigation of the deteriorating effects of different kinds of defects is difficult. Optimized mechanical properties are a prerequisite to achieve the necessary durability and performance of these layers in low-dimensional devices. Several MD simulations of the influence of (multi)vacancies, Stone– Wales defects, and microcracks on selected mechanical properties of boronitrene have been published fairly recently 2

Nanotechnology 28 (2017) 064002

P Hess

and microcracks in h-BN and MoS2 is compared with the behavior of graphene to verify the existence of a 2D relationship analog to the well-known 3D Griffith plot.

Table 1. Collection of the mean intrinsic values of the Young’s modulus (h-BN: [36–41], MoS2: [7, 10, 20, 55, 59–61]), fracture strength (h-BN: [39–41, 43], MoS2: [20, 33, 55, 56]), line (edge) energy (h-BN: [45, 46], MoS2: [10, 62]), fracture toughness (h-BN: [15], MoS2: [18, 63]), and strain energy release rate (values were calculated from the line energies) of boronitrene and MoS2 with those of graphene [44]. Results derived by the bond-breaking model from the mean stiffness and strength values are assigned by stars.

E2D (N m−1)

s 2D m (N m−1)

γ1D (nJ m−1)

KIC (N m−1/2)

Graphene

340

37

Boronitrene

270

28

MoS2

131

9.8

1.6 2.3* 1.2 1.7* 1.2 0.71*

1.1×10−3 1.2×10−3* 8.2×10−4 9.6×10−4* 8.3×10−4 4.3×10−4*

2. Two-dimensional continuum models 2.1. Two-dimensional bond-breaking model

The 2D bond rupture or cleavage of an isolated perfect monolayer into two parts can be described by a 2D bondbreaking model [35] derived from the 3D cleavage model for localized bond rupture [34] 1D 2D 1/2 . s 2D m = [(g E ) (4r0 )]

(1 )

Here s 2D m is the 2D intrinsic strength or critical fracture stress (in N m−1), E2D the 2D Young’s modulus (in N m−1), γ1D the breaking force (in N) or line (edge) energy (in J m−1), and r0 the nanoscopic length scale represented by the equilibrium bond length of the covalent bonds of boronitrene (r0=0.145 nm) and MoS2 (r0=0.242 nm).

[27–29]. The observed behavior will be compared with the extensive data already available for defective graphene to reveal the analogies and differences in their mechanical behavior. According to the structural complexity several new types of point defects have been recently defined and characterized by atomic resolution imaging and first-principles calculations in MoS2 [30]. Since the monosulfur vacancy (VS) has the lowest formation energy, it is the most common defect. Besides this widespread defect and the disulfur vacancy (V2S) four less common intrinsic point defects could be observed. These are a vacancy complex consisting of a Mo atom and three S atoms within one plane (VMo3S) and Mo with nearby three disulfur pairs (VMo6S). Furthermore, two antiside defects exist, where a Mo atom substitutes for two S atoms or two S atoms substitute for Mo [30]. In a joint experiment-theory investigation besides the two antiside defects MoS2 and S2Mo the defects MoS and SMO and VMo have been studied [31]. Our understanding of how these point defects and grain boundary effects influence the mechanical properties is still at the beginning [32, 33]. In this situation of incomplete knowledge of relevant mechanical data, there is a strong rationale for analytical continuum models suitable for estimating missing mechanical failure properties. Here a 2D bond-breaking model, based on Morse-type interaction, is used to connect literature results on the intrinsic 2D elastic modulus and 2D fracture strength with the intrinsic line (edge) energy [34, 35]. The line energy is a fundamental, however, very difficult to determine mechanical property that is employed to estimate the formal intrinsic 2D fracture toughness and critical strain energy release rate. Furthermore, an analytical 2D fracture model derived from the 3D Griffith relationship is employed to correlate the real strength of defective graphene, boronitrene, and MoS2 monolayers with the square root of half the defect size of point and line defects. For these complex defect structures essentially MD simulations have been performed and continuum mechanics makes important contributions to elucidate the fracture mechanics of such distorted networks. The reduction of strength by defects such as isolated vacancies

In general, the 2D bond-breaking model gives access to any of the three mechanical quantities 2D elastic modulus, 2D intrinsic strength, and breaking force (line or edge energy), when the other two are known from experiment or theory. Therefore, this analytical model is a valuable tool for evaluating missing intrinsic material properties of brittle 2D solids. Predominantly the very difficult to determine line energy of perfect 2D crystals is of interest in this respect. 2.2. Two-dimensional Griffith model of defective monolayers

Based on the fundamental connection between mechanical properties of quasi 2D and formal 3D solids via the thickness of the covalently bonded monolayer, a 2D version of the Griffith relation can be easily obtained. The resulting relationship provides a useful tool for studying 2D solids containing structural defects [35] 1D 2D 1/2 , s 2D cr = [(2g E ) (pa )]

(2 )

1D where s 2D the line or cr is the critical 2D fracture strength, γ 2D edge energy, E the 2D Young’s modulus, and a the halflength of a central crack in the monolayer or the depth of an edge crack. Owing to the reduction of dimensionality, 2D cracks are spatially limited to the plane. This drastically reduces the number of possible crack orientations and configurations with respect to an applied load such as uniaxial tension. The ultimate or critical fracture strength is achieved by a tensile load perpendicular to the crack extending along the direction with the lowest cleavage energy. This means a realistic description of monolayer failure can be achieved with the 2D Griffith concept. An important question is the validity of the Griffith relationship at short length scales around and below 1 nm, as already studied by several groups for graphene [35].

3

Nanotechnology 28 (2017) 064002

P Hess

discussed below. This correspondence of breaking forces is consistent with the assumption that the bond-breaking model describes the bare dissociation of the corresponding bonds, leaving edges along the crack path without appreciable reconstruction of the edge tip. Under normal conditions chemical reconstruction stabilizes the reactive edges formed by fracture and leads to a reduction of the line or edge energy. Note that calculation of the line energy of pristine boronitrene is more difficult than for graphene even without taking edge reconstruction into account, owing to the lack of inversion symmetry and the binary composition. Since h-BN is composed of heterogeneous atoms the nontrivial angle range increases from 30° in graphene to 60° in h-BN. Several theoretical estimates of the line energy are available for reconstructed nanoribbons with a typical width