Dislocation Shielding of a Nanocrack in Graphene: Atomistic

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Dislocation Shielding of a Nanocrack in Graphene: Atomistic Simulations and Continuum Modeling Fanchao Meng, Cheng Chen, and Jun Song J. Phys. Chem. Lett., Just Accepted Manuscript • Publication Date (Web): 20 Sep 2015 Downloaded from http://pubs.acs.org on September 28, 2015

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The Journal of Physical Chemistry Letters

Dislocation Shielding of a Nanocrack in Graphene: Atomistic Simulations and Continuum Modeling Fanchao Meng, Cheng Chen, and Jun Song* Department of Mining and Materials Engineering, McGill University, Montréal, Québec H3A 0C5, Canada

*

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 Combining atomistic simulations and continuum modeling, we studied dislocation shielding of a nanocrack in monolayer graphene under mode-I loading. Different crackdislocation configurations were constructed and the shielding effects on the threshold stress intensity for crack propagation were examined. Excellent agreement between simulation results and linear-elastic fracture mechanics (LEFM) predictions was achieved. As the separation between the crack-tip and dislocation, i.e., rR , varies (with respect to the crack size a ), the shielding effect exhibits two different dependences on rR , scaling as 1 / rR1/ 2 for rR a ≪ 1 (neartip) while 1 rR for rR a ≫ 1 (far-field), respectively. Particularly, the far-field 1 rR scaling was shown to be a direct manifestation of the stress field of dislocation in graphene. Our work presents a systematic study of nanoscale crack-dislocation interactions in graphene, providing valuable information on defect engineering of graphene.

Table of Contents Image:

Keywords: Dislocation shielding; Fracture; Molecular dynamics; Defect engineering; Graphene.


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Graphene, an allotrope of carbon, has the structure of a planar monolayer of carbon atoms densely packed into a 2D hexagonal crystal lattice.1 It promises many next-generation applications due to its novel mechanical,2-3 thermal,4-5 and electronic properties.6-7 In particular, graphene possesses superior mechanical properties, e.g., a Young’s modulus as high as ~1 TPa and an intrinsic fracture strength ~130 GPa,2 showing great potential for applications requiring high stiffness and strength such as in pressure barriers8 and sensors.9 With the unavoidable introduction of various topological defects (e.g., dislocations10 and grain boundaries11) and crack-like flaws (e.g., slits and holes12-13) during the fabrication process, large-scale graphene sheet often experiences appreciable structural modifications and subsequently changes in its properties from the pristine state. Those defects bring rich possibilities to engineer graphene for new avenues in applications, such as molecule capacitors14 and filters.15-16 In addition, with the associated lattice distortion, they may interplay with external strain field to enable strain engineering to graphene, e.g., tuning catalytic properties17 and chemical reactivity.18 On the other hand, the defects may degrade the mechanical properties of graphene to result in premature failure,13,

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challenging the stability and durability of graphene devices.

Despite many efforts directed towards understanding the fracture and failure in graphene,13, 19-22 the effects of topological defects on the fracture behaviors of graphene with existing cracks remain largely elusive. Among the topological defects, dislocation is particularly important because of its long-range stress field.23 When in the vicinity of the crack, dislocations necessarily modify the stress states at the crack tip and subsequently the critical stress intensity factor for crack propagation. Thus, concise knowledge of crack-dislocation interactions is key to understanding structural evolution and ripping of graphene, and thus targeted defect engineering of graphene.



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In this work, we studied the shielding effects of dislocations on a nanocrack in graphene using both atomistic simulations and continuum modeling. The threshold stress intensity and fracture behaviors of cracks of different geometries, along either armchair (AC) or zigzag (ZZ) direction, with or without the influence of dislocations, were examined by large-scale molecular dynamics (MD) simulations. The simulation results were then compared with the predictions from continuum modeling. In the end, the spatial variation of the shielding effect and its relevance to the dislocation stress field were discussed. To examine the effects of dislocation shielding, we constructed several crack-dislocation configurations, as illustrated in Figure 1. The crack-dislocation configuration consists of an internal crack along either the AC or ZZ direction, accompanied by two or four dislocations symmetrically residing on both sides of the crack. Two crack heights, i.e., h1AC = 2a0 and

h2AC = 3a0 for AC cracks, and h1ZZ = 4 3a0 3 and h2ZZ = 2 3a0 for ZZ cracks, respectively, (see Figure 1) are considered, where a0 = 2.42 Å is the lattice constant of the pristine graphene.24-25 Two types of dislocation configurations, being (1,0) pair (Figures 1a-b) and (1,1) quartet (Figures 1c-d) for AC cracks, and (1,1) pair (Figures 1e-f) and (1,0) quartet (Figures 1g-h) for ZZ cracks, were considered, where the notations (1,0) and (1,1) represent edge dislocations with

r r Burger’s vectors b(1,0) = a0 and b(1,1) = 3a0 , respectively. For simplicity, below we refer to different crack-dislocation combinations as αβ / ( m, n) where αβ = AC or ZZ and ( m, n) = (1,0) or (1,1) indicate the types of crack and dislocation, respectively. The geometrical relationship between the crack and dislocations is further illustrated in Figure 1i, where the relative location of a dislocation with respect to the crack can be described by the length parameter, rR and rL ,


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and angle parameters, θ R and θ L , while the relative orientation of the dislocation can be described by an angle α .

Figure 1. Sample crack-dislocation configurations of armchair cracks of heights (a) h1AC and (b) h2AC, with the (1,0) dislocation pair, and (c) h1AC and (d) h2AC, with the (1,1) dislocation quartet, and zigzag cracks of heights (e) h1ZZ and (f) h2ZZ, with the (1,1) dislocation pair, and (g) h1ZZ and (h) h2ZZ, with the (1,0) dislocation quartet. (i) schematically illustrates a configuration of an internal crack of size 2a with a pair of dislocations, each of Burgers vector . The relative position of dislocations with respect to the



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crack are described by geometrical parameters , , , , which are measured from crack tips to the pentagon tip (of the dislocation) where the highest compressive stress locates. and are the glide and climb components of the Burgers vector, respectively.

The simulation supercell was deformed by an in-plane uniaxial tensile stress along the direction perpendicular to the crack (i.e., Y direction in Figure 1) to drive crack propagation. During the deformation, the stress versus strain response of the graphene sheet was monitored. Figure 2 shows the typical stress-strain curves for AC and ZZ cracks. We see the stress increases monotonically until the onset of crack propagation that is signalled by an abrupt drop in stress. The linear fashion of the stress-strain response prior to the onset of fracture and the subsequent catastrophic failure indicate that the fracture of graphene is brittle in nature, consistent with previous studies.26-27 Moreover, from the inset figures of Figures 2a-b that illustrate atomic configurations before, at, and after crack initiation, we see that for both AC and ZZ cracks (with atoms colored according to atomic stress σ yy ), the bond at the very crack tip experiences the highest tensile stress immediately prior to crack initiation. This bond will break upon further loading, followed by crack propagation. The ZZ crack exhibits a clean unidirectional cleavage, while crack deflection and serrated crack path are observed for the AC crack. Nonetheless, for both cases the propagation is along the ZZ edge. The preference of cracking along the ZZ edge can be attributed to the fact that the fracture energy of the ZZ edge is lower than that of the AC edge (see fracture energy values in Table 1). In addition, we see that during the crack propagation in AC crack, the newly-formed ZZ edge can experience edge reconstruction to form alternating 5-7 rings along the fracture surface. This edge reconstruction was also reported in previous studies.28-29 Using the stress value (denoted as σ c ) immediately prior to the drop, we can determine the threshold mode I stress intensity factor K Ι corresponding to the onset of crack propagation, 6 ACS Paragon Plus Environment

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through the relation K Ι = Yσ c π a where Y is a dimensionless parameter30 ( Y ≈ 1 in our model systems). Alternatively, K Ι can also be obtained via the means of J-integral,31 which was verified to yield essentially the same value as the previous means.

Figure 2. Typical stress-strain curves for graphene with (a) armchair crack and (b) zigzag crack. The inset figures show local atomic configurations at the crack tip before (I), at (II), and after (III) crack initiations. Atoms are colored according to the local atomic stress . Particularly for the armchair crack, surface reconstruction of the newly-forned zigzag edge into 5-7 rings (the grey-shaded region in IV) was observed.



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Denoting the threshold stress intensity factors for a crack influenced by dislocations and a reference crack in dislocation-free graphene as K Ιeff and K Ι0 , respectively, the effect of dislocation shielding can be quantified as ∆K I / K I0 = ( K Ieff − K I0 ) / K I0 . Figure 3 shows the simulated ∆K I / K I0 as a function of rR for different crack-dislocation configurations. From Figure 3, we note that ∆K I / K I0 decreases monotonically as rR increases and eventually decays to near zero values for large rR values. Another interesting observation from Figure 3 is that the evolution of ∆K I / K I0 is independent of the crack height (or bluntness). In addition, simulations were also performed to examine the influence of the crack shape and our results indicate that dislocation shielding is also insensitive to the crack shape (see Supporting Information for details). This suggests that the stress concentration at the crack tip only depends on the local atomic sharpness rather than the global geometry of the crack.


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Figure 3. Simulated and LEFM predicted ∆K Ι / K Ι0 as functions of the crack tip-dislocation distance rR for armchair cracks influenced by (a) a (1,0) dislocation pair and (b) a (1,1) dislocation quartet, and zigzag cracks influenced by (c) a (1,1) dislocation pair and (d) a (1,0) dislocation quartet.

The dislocation shielding effect is a direct manifestation of the interplay between stress fields of dislocation and crack. Thus, to understand and subsequently be able to predict the effects of shielding shown in Figure 2, below we analyze dislocation shielding in the framework of continuum mechanics. In light of the inherent brittleness of graphene32 (also see Figures 2a-b), linear-elastic fracture mechanics (LEFM) is employed in the continuum analysis.33 Considering a pair of dislocations symmetrically positioned with respect to an internal crack as illustrated in Figure 1i, the continuum prediction of the shielding of dislocations on the crack under plane stress condition can be expressed as 0 Ι

∆K I K =

1

Eby2

4 πa

Γ

f1 cos α −

1 4 πa

Ebx2 f 2 sin α , Γ

(1)

where E is Young’s modulus, bx and by are the magnitudes of Burger’s vector of the glide and climb dislocation components, Γ is the work of fracture, and α is the angle between Burger’s

r r vector b and its climb component b y . The values of the above parameters are listed in Table 1 (see Supporting Information for details). f1 and f 2 are two geometrical functions defined as follows a f1 = [2 cos θ * + sin θ L sin(θ * + θ L ) + sin θ R sin(θ * + θ R )] , r%

(2)

a f 2 = [sin θ L cos(θ * + θ L ) + sin θ R cos(θ * + θ L )] , r%

(3)

r% = ( rL rR )1/2 and θ * = 1/ 2(θ L + θ R ) .

(4)

with



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where the length parameter, rL and rR , and angle parameters, θ L and θ R , are illustrated in Figure 1i. The predicted evolutions of ∆K I / K I0 are plotted in Figure 3 together with the simulated data, showing an excellent agreement. This verifies that the shielding effect is a direct consequence of the interactions between the stress fields of the crack and dislocations. Table 1. The calculated Young’s modulus and fracture energy in this study

E (TPa )

Cal.a Ref.b 0.955 1.01±0.03 a. This work. b. Ref. 27.

Γ ZZ ( J m2 ) Cal.a Ref.c 10.49 10.04

Γ AC ( J m 2 ) Cal.a Ref.c 11.27 10.04 c. Ref. 34.

A further examination of eqs 1-4 reveals that the dependence of ∆K I / K I0 on rR can be further simplified under two extreme conditions, i.e., rR a ≪ 1 and rR a ≫ 1. For simplicity in the follows, we refer to these two conditions (and the associated solutions) as near-tip and farfield conditions (solutions). Under the near-tip ( rR a ≪ 1) condition, the geometrical functions f1 and f 2 can be reduced to f1 ≈ f1NT = f 2 ≈ f 2NT

a 1 3 (2 cos θ R + sin θ R sin θ R ) , 2rR 2 2

(5)

a 3 = (sin θ R cos θ R ) , 2rR 2

which then yields 0 I

∆K I / K ≈ ∆K

NT I

0 I

/K =

1

Eby2

4 πa

Γ

NT

f1 cos α −

1 4 πa

Ebx2 NT f 2 sin α . Γ

(6)

On the other hand, under the far-field ( rR a ≫ 1) condition, f1 and f 2 can be simplified as (see Supporting Information for details)


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f1 ≈ f1FF = f2 ≈ f

FF 2

2a (cos θ R + sin θ R sin 2θ R ) , rR

(7)

2a = sin θ R cos 2θ R , rR

from which we have ∆K I / K I0 ≈ ∆K IFF / K I0 =

1

Eby2

4 πa

Γ

f1FF cos α −

1 4 πa

Ebx2 FF f 2 sin α . Γ

(8)

It is interesting to note that the near-tip and far-field solutions show distinct dependence on rR , scaling as rR−1 2 and rR−1 , respectively. Examining the differences between these two solutions and MD simulated results, i.e., δ NT and δ FF defined below

δ NT = ∆K ΙNT K Ι0 − ( ∆K Ι K Ι0 ) MD ,

(9)

δ FF = ∆K ΙFF K Ι0 − ( ∆K Ι K Ι0 ) MD ,

(10)

we see from Figure 4 that the near-tip solution gives a better prediction for rR < 50 Å, but deviates from the simulated results for larger rR , while the opposite is observed for the far-field solution. In particular, we note δ FF converges to zero for sufficiently large rR , suggesting that the shielding effect scale as rR−1 (same as the far field solution, see eq 7) when the dislocation and crack tip are far apart from each other. With shielding being a direct manifestation of dislocation stress field, this scaling confirms the 1 rR decay of the nonlocal stress fields of dislocations in graphene, in agreement with previous computational35 and experimental23 studies. This was further illustrated in the plot of σ YY (see Supporting Information for details) shown in Figure S4.



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Figure 4. The deviations between the simulated shielding effect and near-tip solution (filled symbols) and far-field solution (open symbols) as functions of the crack-tip/dislocation distance rR for armchair cracks influenced by (a) a (1,0) dislocation pair and (b) a (1,1) dislocation quartet, and zigzag cracks influenced by (c) a (1,1) dislocation pair and (d) a (1,0) dislocation quartet.

In summary, dislocation shielding of a nanocrack, along either armchair or zigzag direction, in graphene was studied using both atomistic simulations and continuum linear-elastic fracture mechanics (LEFM). Our results show that fracture occurs in a brittle fashion and the simulated shielding effect is well predicted by LEFM. We also found that dislocation shielding is independent of the crack bluntness, indicating the stress concentration at the crack tip depends on the local atomic sharpness rather than the global geometry of the crack. Depending on the separation, i.e., rR , between the dislocation and crack tip, the shielding effect can be represented by near-tip and far-field solutions that scale as rR−1 2 and rR−1 , respectively. In particular, the rR−1


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scaling directly confirms the 1 rR decay of the stress fields of dislocations in graphene. The present study elucidates the micromechanics underlying crack-dislocation interactions in graphene, providing essential information for predictive optimization of mechanical properties and controllable structural modification of graphene through topological defect engineering.



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COMPUTATIONAL METHOD Dislocations were constructed by inserting semi-infinite graphene strips of graphene into the otherwise pristine graphene lattice as outlined in Ref. 36 (see Supporting Information for details), while cracks were created by simply removing atoms. The supercell used in our simulation is periodic along all directions, and rectangular in shape with in-plane dimensions of 600~800 Å in width and 700~1000 Å in height depending on the details of crack-dislocation configurations (benchmark calculations were performed to ensure ignorable size dependence due to dislocation/crack interactions across the periodic boundaries). An interlayer separation of 15 Å is set in the direction perpendicular to the graphene sheet to eliminate the interlayer interactions. In the simulations, the fracture behaviors as rR varies were investigated (see Supporting Information for details on the variation of rR ). Meanwhile, the reference fracture behaviors of the dislocation-free supercells were also examined. MD simulations using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS)37 were employed to investigate the fracture and dislocation shielding in graphene. The interatomic interactions are described by the adaptive intermolecular reactive empirical bond order (AIREBO) potential,38 which was widely adopted to simulate mechanical behaviors in graphene.27,

39-41

The cut-off distance of the AIREBO potential has been modified from the

original value of 1.7 Å to 2.0 Å to avoid the non-physical post-hardening behavior as suggested by previous studies.27, 39-41 In the fracture simulation, the atoms evolved (with a timestep of 1 fs) within a NPT ensemble42-43 to maintain a constant temperature of T = 1 K and zero normal pressure along X direction (see Figure 1). The system was initially relaxed for 100 ps to attain equilibrium, following which the supercell was deformed by an in-plane tensile loading with a strain rate of 0.001 ps-1 along Y direction (see Figure 1) to drive crack propagation.


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



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SUPPORTING INFORMATION AVAILABLE Details of computational method and additional results. This information is available free of charge via the Internet at http://pubs.acs.org.


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