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Impurity-Induced Grain Boundary Strengthening in Polycrystalline Graphene Fanchao Meng, Dianyin Hu, and Jun Song J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b10731 • Publication Date (Web): 31 Dec 2015 Downloaded from http://pubs.acs.org on January 5, 2016

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The Journal of Physical Chemistry

Impurity-Induced Grain Boundary Strengthening in Polycrystalline Graphene Fanchao Meng1, Dianyin Hu2,3, and Jun Song1* 1. Department of Mining and Materials Engineering, McGill University, Montréal, Québec H3A 0C5, Canada 2. School of Energy and Power Engineering, Beihang University, Beijing 100191, China 3. Collaborative Innovation Center of Advanced Aero-Engine, Beijing 100191, China

*

Author to whom correspondence should be addressed. Email: [email protected] Tel.: +1 (514) 398-4592 Fax: +1 (514) 398-4492

1 ACS Paragon Plus Environment


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ABSTRACT First-principles density functional theory (DFT) calculations were performed to investigate the effects of boron (B) and nitrogen (N) doping on the fracture behaviors of a series of symmetric graphene grain boundaries (STGBs) under biaxial straining. Doping was found to generally enhance the fracture strength of STGBs, which was shown to be attributed to dopants mechanically lowering the local stretching energy and/or chemically strengthening the critical bond. We also showed that doping may also induce crack deflection to further improve the fracture resistance of graphene grain boundaries (GBs). Furthermore, we showed that the presence of multiple B and N dopants in the pentagon-heptagon ring(s) can contribute to promoting intergranular fracture in the presence of a small prestrain along the GBs. Our findings clarify the role of B and N dopants in GBs strengthening of polycrystalline graphene.


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I

INTRODUCTION The two-dimensional (2D) crystalline structure and extraordinary physical properties of

graphene1-7 make it unambiguously one of most attractive materials for researchers. Particularly the excellent mechanical properties of graphene, for example, high Young’s modulus ~1TPa and high intrinsic fracture strength ~130 GPa,3 promise exciting possibilities for nanocomposites with graphene additives,8-9 atomic-scale pressure barriers,10 graphene-based molecular filters11-12 and etc. The production of large-scale graphene sheet is made possible using chemical vapor deposition where graphene is grown on metal substrates.13-15 However, the large-scale fabricated graphene sheet is polycrystalline in nature,16-17 populated with interface defects, i.e., grain boundaries (GBs). At GBs, atoms are misaligned and of higher energy states compared to the pristine graphene lattice. Consequently, GBs are presumably more prone to fracture and play a vital role in determining the effective strength of a polycrystalline graphene. Numerous research studies have been performed to understand the GB failure in graphene. The failure of CVD-grown polycrystalline graphene was examined by Huang et. al.17 and Ruiz-Vargas et. al.18 through atomic force microscopy, showing that GBs can greatly degrade the mechanical strength of graphene membranes. Grantab et. al.19 showed that highangle symmetric tilt GBs (STGBs) with a high 5-7 defect density exhibit higher strength than low-angle STGBs, in sharp contrast to continuum predictions, and suggested the importance of local critical bonds. It was further discovered20-22 that the strength of a graphene grain boundary (GB) depends not only on the misorientation angles, but also on the detailed arrangement of pentagon-heptagon defects along the GB. The strength of graphene GBs was also shown to decrease substantially as the temperature rises.23 In order to attain the full potential of graphene in mechanical applications, it is necessary to limit or moderate the deteriorating effects of GBs.



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Besides the apparent strategy of reducing the GB density during growth of large-area graphene,24-25 it is crucial to define new strategies to strengthen GBs. Given that impurities are often introduced into graphene to modulate its electronic and electrocatalytic properties,26-28 it would be interesting to see if impurities can also interact with GBs to influence their mechanical properties. In this work, we focus on two common impurity atoms, B and N, and examined the effects of B and/or N doping on the fracture behaviors of four representative graphene STGBs. The exothermic doping sites along those STGBs for individual and multiple dopants are identified. Then, the graphene STGBs under different doping scenarios are deformed by a biaxial strain till failure. The correlation between fracture initiation strain with the local bond stretching energy and doping energetics is clarified, and the doping-induced crack deflection and propagation path alternation is demonstrated. The implications of B and N doping toward engineering the fracture behaviors of polycrystalline graphene are discussed.


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II

COMPUTATIONAL METHOD The fracture behaviors of a series of STGBs in graphene, with the misorientation angle θ

ranging from 9.5° to 21.8°, are investigated. The STGB is constructed by creating a V-shaped notch in an initially perfect graphene sheet to partition the sheet into two sides which are then rotated and stitched together to form the 5-7 defects (see Supporting Information for details), similar to the procedure outlined in Ref. 20. For the simulation, a rectangular cell containing a graphene monolayer is constructed with the cell dimension perpendicular to the monolayer set as 15 Å to eliminate the interlayer interactions. The graphene monolayer consists of two parallel STGBs. With STGBs essentially being arrays of dislocations that are of a long-range 1/r stress field,29-30 elastic interactions between GBs necessarily exist. As a consequence, the in-plane cell dimension perpendicular to the GB is set to be larger than 40 Å (i.e., between 41.26 Å and 45.32 Å depending on the orientation) as suggest in Ref. 31, in order to ensure minimal interactions between neighboring GBs and no size dependence of our results (see Supporting Information for details). Figure 1a shows a representative simulation supercell, constructed for STGBs with θ = 13.2°. The cell dimension along the GB direction also varies to accommodate STGBs of different misorientation angles, as illustrated in Figure 1b. One thing to note is that the Ʃ49 GB has a repeated unit of the dual 5-7 defects, unlike other STGBs where 5-7 defects are uniformly distributed along the GB line (see Figures 1b-c). The equilibrium lattice constant, a0 = 2.46 Å, is used in the above supercell construction, in agreement with previously reported values.32-33 B and/or N impurities are introduced to substitute C atoms at lattice sites of 5-7 defects (see Figure 1c) to study the influence of B and/or N doping on the facture behaviors of graphene GBs. The fracture simulations are conducted within the canonical (NVT) ensemble of ab-initio molecular dynamics based on spin polarized density-functional theory (DFT)34-35 using the



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Vienna Ab-initio Simulation Package (VASP).36 The exchange correlation functions are approximated by generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof (PBE),37 and the electron-ion interactions for elements C, B, and N are described using the projector augmented wave (PAW) method.38 A plane wave basis cut-off of 500 eV is used for all calculations. A biaxial strain is incrementally applied to deform the graphene sheet with the strain step being 0.25%, and the fracture initiation and crack propagation are accordingly monitored. In the following context the threshold strain at which fracture initiates, namely the fracture initiation strain, and the strain upon which the material fully fractures, namely the failure strain, are denoted as ε c and ε p , respectively. Prior to the biaxial deformation, the graphene sheet is relaxed using self-consistent calculations with the force tolerance being 0.03 eV/Å.39 During deformation, the atomic positions evolve within the ab initio molecular dynamics with a time step of 3 fs, and the temperature is maintained as 0.1 K using Nosé thermostat.36 In the case of a critical fracture event (i.e., crack initiation or propagation) occurring, 100 ion relaxation steps are performed for each loading step.


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Figure 1. (a) Atomic configuration of a representative STGB, i.e., Ʃ19 (θ = 13.2°), where the region enclosed by the black square indicates the periodic supercell used in our simulations. (b) The local atomic arrangements along STGBs Ʃ37 (θ = 9.5°), Ʃ19 (13.2°), Ʃ49 (16.4°), and Ʃ7 (21.8°). (c) The potential lattice sites for impurity doping for STGBs with (left) uniformly distributed 5-7 rings and (right) nonuniformly distributed 5-7 rings (i.e., Ʃ49).



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III

RESULTS AND DISCUSSION

3.1

Fracture behaviors of pristine graphene STGBs

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The fracture initiation strain, ε c , for the four graphene STGBs, i.e., Ʃ37 (θ = 9.5°), Ʃ19 (θ = 13.2°), Ʃ49 (θ = 16.4°), and Ʃ7 (θ = 21.8°), is plotted in Figure 2a, showing a nonmonotonic variation with respect to the GB angle θ. For those STGBs examined, the crack initiation was found to occur at the bond shared by the heptagon and hexagon, followed by crack propagation along the GB. The fracture process is illustrated by the charge density plots in Figure 2b, using the STGB with θ = 13.2° as a representative. In the context below, we refer to the bond at which fracture initiates as the critical bond. These observations are in close agreement with previous studies.19-22 The crack initiation can be attributed to high prestress in the critical bond (in the heptagon ring) prior to deformation.19, 40 To quantitatively assess the tendency of fracture for different STGBs, we adopt the concept of stretching energy (denoted as Fs) for the critical bond, which is defined by the generalized von Karman equation for a flexible solid membrane as41-42 Fs =

3 S ( rc − r0 ) 2 4

(1)

where S = Eh is the in-plane stretching modulus defined by the product of Young’s modulus E of pristine graphene monolayer and the interlayer distance h of graphite (3.34 Å). Young’s modulus E is calculated as 1.04 TPa being very close to previously reported values.3, 29, 43 rc represents the length of the critical bond before loading and r0 = 1.42 Å is the C-C bond length in pristine graphene monolayer. The values of Fs are plotted together with the fracture initiation strains for different STGBs in Figure 2a. A strong correlation between Fs and ε c is observed, i.e., a lower fracture initiation strain corresponds to a higher stretching energy of the critical bond.


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Figure 2. (a) Fracture initiation strain εc and stretching energy Fs for graphene STGBs of different GB angle θ. (b) A representative fracture initiation event along the Ʃ19 (θ = 13.2°) STGB, where the breakage at the critical bond (left) and subsequent propagation process (right) are illustrated by the charge density plots.

3.2

Segregation energetics of B and N at STGBs in graphene Before studying the influence of B and/or N doping on fracture behaviors of STGBs in

graphene, it is important to first evaluate the likelihood of B and/or N segregation at STGBs by examining the corresponding segregation energetics. The formation energy Ω X of a substitutional impurity (i.e., B or N) in graphene lattice is defined as 44-45 Ω X = E X − E0 − µ X + µC

(2)



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where E0 is the total energy of the pristine graphene supercell, E X is the total energy of the supercell after the introduction of one impurity atom X ( X = B or N), and µ X and µC are the chemical potentials for the impurity atom X and C , respectively, which can be obtained from α-boron bulk, N2 molecule, and perfect graphene sheet.44 The values of µC , µ B , and µ N are 9.22 eV, -6.68 eV, and -8.32 eV, respectively. The formation energies of B and N in perfect graphene are calculated to be 1.00 eV and 0.63 eV, respectively, closely matching the values preciously reported.44, 46-47 For the STGBs considered, the 5-7 rings are the regions where the local lattice symmetry is broken and were found to be potential locations for impurity segregation. The formation energies of B or N at different lattice sites (denoted by α , β , γ , δ , λ ,η and α * , β * , γ * , δ * , λ * ,η * , see Figure 1c) of the 5-7 ring(s) are shown in Table 1, indicating that the preferred sites as α and β for B, and λ and η for N (except for the case of Ʃ49 (θ = 16.4°) where λ * and η * are the more preferable), respectively. These preferred sites correspond to the sites where Ω X values are negative (except for the case of Ʃ748). Additionally, the simultaneous segregation of two B atoms or two N atoms (in the supercell) at STGBs is also considered, with the corresponding formation energy Ω X 2 ( X = B or N) defined in a similar way as eq 2 Ω X 2 = E X 2 − E0 − 2 µ X + 2 µC ,

(3)

where E X 2 is the total energy of the supercell after the introduction of two impurity atoms X ( X = B or N). For all the STGBs considered, the configurations corresponding to the lowest Ω B 2 and Ω N 2 are ββ and λλ ( λ *λ * for the Ʃ49 GB), respectively,49 as shown in Table 1.

Furthermore, in light of the segregation preferences discussed above, the simultaneous


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segregation of two B and two N atoms at ββ and λλ ( λ *λ * for the Ʃ49 GB) respectively is also considered, for which the formation energy is defined as Ω B 2 N 2 = E B 2 N 2 − E0 − 2 µ B − 2 µ N + 4 µ C ,

(4)

with EB 2 N 2 denoting the corresponding total energy of the supercell after the simultaneous segregation of two B and two N atoms. In addition, based on eqs 3-4, the separate contribution of two B or two N atoms to the formation energy in the case of their simultaneous segregation can be defined as Ω B 2 N 2 − Ω N 2 and Ω B 2 N 2 − Ω B 2 , respectively. These data are also listed in Table 1. In view of the energetics data shown in Table 1, we considered the following three categories of scenarios of impurity doping at STGBs in graphene: (i) a single substitutional impurity in each 5-7 ring(s),50 being either B atom at α or β , or N atom at λ or η ( λ * or η * for Ʃ49), (ii) two B atoms at ββ sites or two N atoms at λλ ( λ *λ * for Ʃ49) sites in each 5-7 ring, and (iii) coexistence of two B atoms at ββ sites and two N atoms at λλ ( λ *λ * for Ʃ49) sites in each 5-7 ring, to investigate the effects of doping on the fracture behaviors of STGBs in graphene. For simplicity, the doping configurations in the above three scenarios are denoted as Bϕ or Nψ , BB or NN, and B 2 N 2 respectively, with the subscripts ϕ and ψ indicate the doping

sites, being α or β for B and λ or η ( λ * or η * for Ʃ49) for N. It is important to put the above doping scenarios in the experimental context by discussing their feasibility. Gong et. al. reported that the conversion of regions in graphene into boron-, nitrogen- and carbon-containing domains can be achieved with full control of composition, spatial variation and dimension of the G/h-BNC/h-BN interfaces and domains sizes at the nanometer scale using lithography mask techniques.51 They also noted that the first step of 11 ACS Paragon Plus Environment


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such conversion is the substitution of N or B atoms at energetically favorable sites at lattice defects. With those doping sites we considered being exothermic and thus energetically favorable, we thus deem it possible to achieve preferential doping employing the fabrication method proposed by Gong et. al.51 Another important aspect worth clarifying is the possible competition between B/N substitution and B/N adatoms. From our preliminary calculations, we show that B/N adatoms at GBs are not energetically favorable. In addition, the effects of B/N adatoms on the fracture behaviors are expected to be limited in comparison to B/N substitutional doping (see Supporting Information for details). Therefore in the present study B/N adatoms were not studied. Table 1. Formation energies (eV) of B or N at different lattice sites (see Figure 1c) of the 5-7 ring (* indicates the additional set of lattice sites shown in Figure 1c for the particular case of Ʃ49 (θ = 16.4°)). The bolded numbers correspond to those doping cases examined in the present study Formation energy

ΩX

ΩX2

θ = 9.5°

θ = 13.2°

θ = 16.4°

N

B

N

B

N

B

N

B

N

-0.94

1.43

-0.70

1.36

-0.82

1.33

0.16

1.00

0.26

1.03

β

-0.53

0.96

-0.39

0.93

-0.79

1.00

0.34

0.65

0.13

0.80

γ

0.34

0.78

0.44

0.77

0.19

0.87

0.84

0.55

0.71

0.79

δ

0.91

-0.10

0.92

-0.06

0.63

0.13

1.17

-0.24

0.94

0.11

λ η

2.13

-0.61

2.25

-0.53

1.46

-0.20

2.20

-0.67

1.82

-0.21

2.28

-0.80

2.17

-0.67

1.52

-0.19

2.30

-0.80

1.59

-0.10

αβ

-0.00

-

0.37

-

0.03

-

1.65

-

1.80

-

ββ

-0.49

-

-0.25

-

-0.65

-

0.68

-

0.76

-

λλ

-

-0.54

-

-0.40

-

0.16

-

-0.71

-

0.15

λη

-

0.30

-

0.48

-

1.15

-

0.23

-

1.21

Lattice site

B

α

θ* = 16.4°

θ = 21.8°

Ω B2 N 2

ββλλ

-1.88

-1.72

-1.82

-1.24

Ω B2 N 2 − Ω N 2

ββλλ

-1.34

-1.32

-1.11

-1.39


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Ω B 2 N 2 − Ω B2

3.3

ββλλ

-1.39

-1.47

-1.17

-2.00

Fracture behaviors of doped graphene STGBs under Bϕ or Nψ doping With the residence of one dopant (i.e., Bϕ or Nψ ) per 5-7 ring(s), the fracture of

graphene STGBs still occurs at the bond shared by the heptagon and hexagon. This bond stays as a C-C bond except for the case of Bϕ doping where it becomes a C-B bond (see Figure 3c) (see Supporting Information for details). After crack is initiated, fracture propagates along the GB as the one shown in Figure 2b for all Bϕ or Nψ doped STGBs. The corresponding values of ε c and Fs are plotted in Figures 3a and 3b respectively, together with the ones for pristine graphene STGBs. Note that in the calculation of Fs for the particular case of Bβ doping (see eq 1), rc is the length of the critical C-B bond prior to loading while r0 represents the corresponding length of a C-B bond (1.483 Å) in the case of one B atom embedded in a large pristine graphene monolayer. From Figure 3a we see that Nψ doping leads to very tiny increase in ε c , in agreement with its effect on Fs where overall a fairly small decrease in Fs is observed. The minor influence of Nψ doping on GB fracture is largely expected considering the appreciable distance between the site of Nψ doping and the critical bond. Meanwhile, we note that Bϕ doping has a profound influence on ε c . From Figure 3a we see that except for Ʃ7 (θ = 21.8°), B doping, particularly Bα , significantly enhances the fracture resistance of graphene STGBs. On the other hand, we see from Figure 3b that Bϕ doping elevates Fs. These effects of Bϕ doping are in sharp contrast with the ε c -Fs correlations observed for the cases of pristine (see Figure 2) and Nψ -doped graphene STGBs (see Figure 3a) where a smaller Fs value signals higher fracture resistance, i.e., larger



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value of ε c . To understand this discrepancy, we note that different from the pristine and Nψ doped graphene STGBs, Bϕ doping renders the critical bond to be C-B. Hence, besides the local mechanics, the chemistry of the critical bond is necessarily modified because of Bϕ doping. Figure 3b shows the formation energies of Bϕ doping at different graphene STGBs. We see that the formation energy exhibits negative values at Ʃ37 (θ = 9.5°), Ʃ19 (θ = 13.2°), and Ʃ49 (θ = 16.4°). The negative formation energy indicates that the resultant C-B bond is stronger than the original C-C bond. Consequently, the data in Figure 3b suggest that Bϕ doping would increase

ε c for Ʃ37, Ʃ19, and Ʃ49, directly in accordance with the trend observed in Figure 3a. Additionally, the data in Figure 3b suggests that for these three STGBs Bα doping would lead to larger increment in ε c than Bβ doping given the lower formation energy, again in agreement with what’s observed in Figure 3a. For the particular case of Ʃ7, Bβ doping exhibits rather small (albeit positive) formation energies, indicating limited influence of Bβ doping on the critical bond. This is consistent with the fact that ε c stays almost unchanged under Bβ doping.


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Figure 3. (a) Fracture initiation strain εc and (b) (top) stretching energy FS under or doping, and (bottom) formation energy ΩB under doping (at sites and ) for graphene STGBs of different GB angle θ. The open black symbols in (a) and (b) respectively represent corresponding εc and Fs values for graphene STGBs in absence of dopants. (c) Illustration of the fracture initiation event under doping using Ʃ19 (θ = 13.2°) as a representative, showing that the critical bond is C-B in nature.

3.4

Fracture behaviors of doped graphene STGBs under BB or NN doping When the graphene STGBs are subjected to BB or NN doping, the critical bond remains

the bond shared by the heptagon and hexagon, being C-C in nature under NN doping and C-B in nature under BB doping (except for Ʃ7 where the critical bond becomes C-C, see below). Figure 4a and Figure 4b respectively show the corresponding ε c and Fs values for different graphene STGBs. Similar to the scenarios of single dopant per 5-7 ring(s), NN doping shows negligible effect on ε c and Fs while BB doping greatly enhances ε c (except for the case of Ʃ7). The negligible effect of NN doping on ε c and Fs again can be attributed to the N dopants being a large distance away from the critical bond. On the other hand, BB doping is shown to



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significantly increase ε c for Ʃ37, Ʃ19, and Ʃ49. This can be explained by the combinative effects of considerable lower Fs values and negative formation energies (see Figure 4b) at those STGBs. For the case of Ʃ7 under BB doping, fracture initiates at C-C bond shared by heptagon and pentagon (see Supporting Information for details). Like the scenario of Bϕ doping, the fracture initiation is not really affected for Ʃ7 under BB doping. This is expected given the invariance of Fs and the critical bond being C-C (the strength of which is thus not directly affected by BB doping).

Figure 4. (a) Fracture initiation strain εc and (b) (top) stretching energy Fs under BB or NN doping, and (bottom) formation energy Ω under BB doping for graphene STGBs of different GB angle θ. The open black symbols in (a) and (b) respectively represent corresponding εc and Fs values for the graphene STGBs in absence of dopants. Note that for Ʃ7, formation energy is zero because the critical bond is C-C.

For the subsequent fracture, the crack propagates along the GB in the case of BB doping (see Supporting Information for details). However, crack deflection is observed for the case of NN doping, as illustrated in Figure 5a. This is likely a result of the strong bonding within the NNdoped pentagon as indicated by the charge density plot in Figure 5a. Crack deflection increases the fracture surface area and grows the crack under the mixed mode (tension and shear) rather than simply mode-I tension,52 and thus serves to enhance the fracture toughness. To quantify the


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toughness gain due to the crack deflection, we introduce a toughening ratio α Γ , employing the Faber and Evans model and treating the NN-doped pentagon as a disk-shape solid inclusion53

αΓ =

Γ − Γ0 = 0.56V p Γ0 R

(5)

where Γ and Γ 0 denote the fracture energy with and without crack deflection, VP is the volume fraction of the NN-doped pentagon, and R is a geometry parameter defined as the ratio between the radius and thickness of the NN-doped pentagon. R is treated as a constant for all STGBs considered. Here VP can be calculated as Vp =

2π r 2 A

(6)

where r is taken the maximal inner radius of the NN-doped pentagon and A is the in-plane area of the corresponding GB. α Γ gives a quantitative indication of the toughening resulting from the crack deflection. In Figure 5b, we plot the α Γ together with ∆ε p , being the NN-doping induced increment in the failure strain ε p , for the STGBs considered. A good correlation between α Γ and ∆ε p is observed. Nonetheless, some deviation is observed at θ = 16.4° (Ʃ49), where ∆ε p exhibits a higher value than the level suggested by α Γ . This is likely due to the fact that for Ʃ49 the location of crack deflection (the pentagon containing λ *λ * ) is rather distant away from the fracture initiation site (see Supporting Information for details).



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Figure 5. (a) Typical fracture behaviors of graphene STGBs under NN doping using Ʃ19 (θ = 13.2°) as a representative, where (left) the fracture initiation at the critical bond and (right) full propagation with crack deflection at the NN-doped pentagon are illustrated. (b) The predicted toughening ratio Γ (see eq 5) and simulated failure strain increment ∆ for different STGBs under NN doping.

3.5

Fracture behaviors of doped graphene STGBs under B 2 N 2 doping In the scenario of B 2 N 2 doping, the critical bond is always the bond shared by the

heptagon and hexagon, being C-B for all STGBs. The fracture initiation strain ε c is further enhanced by doping, as shown in Figure 6a. This large increase in ε c can be well explained by the sizable reduction in Fs augmented by the bond (i.e., the C-B bond) strengthening by BB doping (see Figure 6b). Note that the bond strengthening comes from BB doping, and thus in Figure 6b Ω B 2 N 2 − Ω N 2 (see Table 1), which represents the effective formation energy of BB 18 ACS Paragon Plus Environment

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doping (under the scenario of B 2 N 2 doping), is plotted. Comparing Figure 6 with Figure 4, one particular distinction we note is that the fracture initiation of Ʃ7 is greatly influenced by B 2 N 2 doping but little affected by separate BB or NN doping. This suggests mutual interactions between BB and NN dopant pairs.

Figure 6. (a) Fracture initiation strain εc and (b) (top) stretching energy FS and (bottom) effective formation energy (Ω Ω ) of BB doping for graphene STGBs doped with four dopants per 5-7 ring of different GB angle θ.

For the subsequent crack growth under B 2 N 2 doping, crack propagates along the GB with or without crack deflection. This suggests that BB doping weaken the effects of NN doping on the 5 ring, again evidencing mutual interplay between BB and NN dopant pairs (see Supporting Information for details). Though the GB remains the “weak” path in the graphene sheet, the significant GB strengthening from B 2 N 2 doping may contribute to promoting other crack paths. Indeed from our study we find that with the aid of a small prestrain along the GB, the fracture initiation and propagation in B 2 N 2 doped graphene STGBs can be altered,54 as illustrated in Figure 7a using Ʃ19 (θ = 13.2°) as a representative. The threshold prestrain required to induce the change of fracture path is shown in Figure 7b. The results in Figure 7 suggest that B and/or N doping can possibly promote transgranular type of fracture in polycrystalline graphene.



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Figure 7. (a) The new fracture initiation and propagation behaviors of graphene STGBs illustrated by GB with θ = 13.2° stimulated by 2% prestrain along the GB followed by biaxial strain. (b) Critical prestrain needed to induce the new fracture path for different graphene STGBs.


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IV

CONCLUSIONS In summary, the effects of B and/or N doping on the fracture behaviors of a series of

symmetric grain boundaries (STGBs) under biaxial straining were investigated using firstprinciple calculations. The preferential substitutional doping site(s) of single and multiple dopants were identified. The fracture initiation strain at a STGB was shown to be prescribed by the characteristics of the critical bond at which fracture first occurs. The stretching energy of the critical bond was defined on base of the generalized von Karman equation. For graphene STGBs in absence of dopants, the fracture initiation strain exhibits a non-monotonic trend as the GB angle varies and was shown to directly correlate with the stretching energy. For B and/or N doped graphene STGBs, doping was shown to enhance the fracture strength mechanically by lowering the stretching energy and/or chemically by strengthen the critical bond. In particular, the chemical strengthening is mainly attributed to B doping, which modifies the critical bond from C-C to C-B. In addition, we showed that the presence of two N dopants in the pentagon unit can result in crack deflection. By treating the dual N-doped pentagon as a disk-shape solid inclusion, the additional resistance due to crack deflection was shown to be well predicted by the classical Faber and Evans model. Furthermore, we showed that the significant GB strengthening resulted from multiple B and N dopants in the pentagon-heptagon ring(s) can contribute to promoting intergranular fracture in the presence of a small prestrain along the GB. Our findings provide important information toward GB strengthening of polycrystalline graphene.



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ACKNOWLEDGEMENTS We greatly acknowledge the financial support from McGill Engineering Doctoral Award and National Sciences and Engineering Research Council (NSERC) Discovery grant (grant # RGPIN 418469-2012). We also thank Supercomputer Consortium Laval UQAM McGill and Eastern Quebec for providing computing power and Mr. Bin Ouyang for helpful discussions.


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SUPPORTING INFORMATION AVAILABLE Supporting Information. Additional details of computational methods and supporting results. This information is available free of charge via the Internet at http://pubs.acs.org.



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