Effects of Free Edges and Vacancy Defects on the Mechanical Properties of Graphene

Transcription

1 Proceedings of the 14th IEEE International Conference on Nanotechnology Toronto, Canada, August 18-21, 214 Effects of Free Edges and Vacancy Defects on the Mechanical Properties of Graphene M. A. N. Dewapriya and R. K. N. D. Rajapakse School of Engineering Science, Simon Fraser University, Burnaby, BC V5A1S6 Abstract Defects are unavoidable during synthesizing and fabrication of graphene based nanoelecromechanical systems. This paper presents a comprehensive molecular dynamics simulation study on the mechanical properties of finite graphene with vacancy defects. We characterize the strength and stiffness of graphene using the concept of surface stress in three-dimensional crystals. Temperature and strain rate dependent atomistic model is also presented to evaluate the strength of defective graphene. Free edges have a significant impact on the stiffness; the strength, however, is less affected. The vacancies exceedingly degrade the strength and the stiffness of graphene. These findings provide a remarkable insight into the strength and the stiffness of defective graphene, which is critical in designing experimental and instrumental applications. Index Terms Graphene fracture, vacancy defects, molecular dynamics, nanomechanics, effects of free edges. I. INTRODUCTION The extraordinary electromechanical properties of graphene have drawn remarkable attention from scientists and engineers. Graphene based nanoelectromechanical systems (NEMS), such as resonators [1], have demonstrated intriguing applications in various engineering disciplines from telecommunication [2] to biomedicine [3]. However, as in many crystalline materials, defects are unavoidable during synthesizing and fabrication of graphene based NEMS [3]. Defects, such as vacancies (missing atoms), drastically reduce the strength and stiffness of graphene that critically influence the performance of NEMS [4]. On the other hand, edges and interfaces present in a finite, narrow sheet change the thermo-mechanical properties and even influence the stability of graphene [5]. Classical continuum mechanics break down at the nanoscale [6]. The modified continuum models such as nonlocal elasticity [7] are also not applicable to systems made of graphene, which is a single atomic layer. These nanoscale systems can be analyzed by using first-principle methods. Such simulations are computationally very expensive (often impractical) when applied to systems with several thousands of atoms. Graphene based systems can be effectively modelled using atomistic methods such as molecular dynamics (MD) to compromise between the accuracy and the computational cost. This paper presents a comprehensive MD simulation study that investigates the effects of vacancies on the mechanical properties of finite graphene. We also show that the strength and the stiffness of defective graphene can be characterized by using the concept of surface stress in threedimensional crystals. Temperature and strain rate dependent atomistic model is also presented to evaluate the strength of defective graphene. II. MOLECULAR DYNAMICS SIMULATIONS We performed MD simulations using LAMMPS package [8] with adaptive intermolecular reactive empirical bond order (AIREBO) potential field [9]. A. AIREBO Potential Field The AIREBO potential consists of three sub-potentials, which are the reactive empirical bond order (REBO), Lennard-Jones, and torsional potentials. The REBO potential gives the energy stored in atomic bonds; the Lennard-Jones potential considers the non-bonded interactions between the atoms, and the torsional potential includes the energy from torsional interactions between the atoms. According to the REBO potential [1], the energy stored in a bond between atom i and atom j can be expressed as E ij REBO = f r ij ( )! V R A ij + b ij V ij " # $, (1) where V ij R and V ij A are the repulsive and the attractive potentials, respectively; b ij is the bond order term, which modifies the attractive potential depending on the local bonding environment; r ij is the distance between the atoms i and j; f(r ij ) is the cut-off function. The cut-off function in REBO potential [1], given in (2), limits the interatomic interactions to the nearest neighbors. ( 1, r ij < R (1) " f (r ij )= 1+ cos π ( r ij )% ) $ R(1) ' # $ ( R (2) R (1) )&', R(1) < r ij < R (2), R (2) < r + ij, (2) /$ IEEE 8

2 where R (1) and R (2) are the cut-off radii, which are 1.7 Å and 2 Å, respectively. The values of cut-off radii are defined based on the first and the second nearest neighboring distances of the relevant hydrocarbon. The cut-off function, however, causes a non-physical strain hardening in carbon nanostructures [11]. Therefore, modified cut-off radii, ranging from 1.9 Å to 2.2 Å, have been used to eliminate this non-physical strain hardening [12]. In this study, we used a truncated cut-off function f t (r ij ), given in (3) [13], to eliminate this strain hardening.! # 1, r ij < R f t (r ij )=", r ij > R $# where the value of R is 2 Å. Similar cut-off functions have been used in [12] and [14] to simulate the fracture of graphene. B. Simulation Parameters Length of graphene sheets was 1 nm; periodic boundary conditions were used along the longitudinal direction while the transverse edges were kept free. Width of the sheets were changed from ~1 nm to 25 nm. Fig. 1 shows armchair and zigzag graphene sheets. The sheets were allowed to relax over ps before applying strain; the time step was.5 fs. During the relaxation period, the pressure component along the transverse direction was kept at zero using NPT ensemble implemented in LAMMPS. The NPT ensemble controls the temperature by using Nośe-Hoover thermostat, which induces a non-physical thermal expansion in graphene [5]. This thermal expansion was eliminated by introducing an initial random out-of-plane displacement perturbation (~.5 Å) to the carbon atoms. The simulation temperature was K. Strain was applied by pulling the sheet along the longitudinal direction at a strain rate of 1 9 s -1. Stress perpendicular to the pulling direction was kept at zero to simulate uniaxial tensile test. Fig. 1. armchair and zigzag graphene sheets. The size of the sheets is nm 1 nm. The arrows indicated the direction of the applied strain. C. Calculation of Stress (3) Stress in MD simulations has been interpreted using either the Cauchy stress [5,12] or the virial stress [15]. The Cauchy stress is computationally efficient than the virial stress. However, the Cauchy stress induces a non-physical initial stress (at zero strain) at higher temperatures, whereas the virial stress gives the initial stress as zero [15]. The Cauchy stress is the gradient of the potential energy per unit volume vs strain curve; the virial stress [16], σ ij, is defined as σ ij = 1 V # 1 % $ % 2 N β=1 & R β α ( i R i )F αβ j m α v α α i v j ( '(, (4) α where i and j are the directional indices (x, y, and z); α is a number assigned to an atom; β is a number assigned to neighbouring atoms of atom α which varies from 1 to N; R i β is the position of atom β along the direction i; F j αβ is the force along the direction j on atom α due to atom β; m α and v α are the mass and the velocity of atom α, respectively; V is the total volume. The definition of volume in the virial stress, however, is ambiguous; the virial stress is quite similar to the Cauchy stress when instantaneous volume is used in the virial calculation [15]. In this work, we used the instantaneous volume to calculate the virial stress. Thickness of graphene was assumed 3.4 Å, which is the interlayer spacing of graphene in graphite. Five MD simulations, with different randomly distributed vacancies, were performed for each vacancy concentration and a given width. The strength and the stiffness are less sensitive (<5%) to the distribution of vacancies in the sheet. Therefore, the average strength and stiffness of these five simulations were used for the analysis. III. RESULTS AND DISCUSSION A. Effects of Free Edges and Defects The stress-strain curve of graphene is nonlinear as shown in Fig. 2. Therefore, we obtained the stiffness by considering the stress-strain curve up to.3 strain, where the curve is linear. Fig. 2 shows that the free edges do not have a significant effect on the tensile strength of graphene, which is indicted by the insignificant change in the tensile strength as the width increases. However, the width has a great influence on the stiffness. This width effect is not significant beyond 6 nm. Figure 2 shows the influence of vacancy defects on the stress-strain curve of a 12 nm wide graphene sheet, where the effect of width does not prevail. The figure shows that vacancies greatly reduce the strength of graphene. The stiffness is also significantly affected. Figure 3 shows that the stiffness gradually decreases with the increase of vacancy concentration. At all the considered vacancy concentrations, the stiffness reduces by ~% as the width decreases up to ~1 nm. However, the 9

3 edge effects become insignificant at widths larger than ~5 nm as the number of atoms at the edges is negligible compared to those in the bulk. Stress (GPa) Stress (GPa) 2 1 width = 1 nm w = 2 nm w = 3 nm w = 6 nm.5.1 Strain 2 1 pristine single vac..5% vac. 1%. 2% Strain Fig. 2. Stress-strain curves of graphene with various widths and vacancy concentrations. B. Continuum Modeling of Edge Effect When a finite graphene sheet of width w is subjected to an axial strain ε, the potential energy per unit length can be expressed using the concept of surface stress in a threedimensional crystal as [5] U(ε, w) = U + 2τε + E s ε 2 + E b ε 2 w 2, (5) σ (ε, w) = 2τ w +! 2E s w + E $ # b &ε. (6) " % Therefore, the effective elastic moduli (E eff ) of a finite sheet can be written as E eff = 2E s w + E b. (7) We obtained E s (GPa nm) and E b (GPa) by regression analysis, and the corresponding values are given in Table 1. The best-fit curves, in the form of (7), are plotted in Fig. 3, and these curves capture the effects of free edges quite well. Figure 3 shows that free edges do not have a significant effect on the strength as observed in Fig. 2; however, the vacancies drastically reduce the strength. Even a single vacancy reduces the strength by ~15%, whereas the stiffness is not affected. In the case of single vacancy, the vacancy percentage decreases with increasing width due to the increase in the number of atoms, thereby the widthstrength relationship is quite different compared to the other curves in Fig. 3. Similar to (7), the strength σ ult can be expressed as σ ult =2σ s,ult w + σ b,ult, (8) where σ s,ult (GPa nm) and σ b,ult (GPa) are the representative ultimate tensile strengths of the surface and the bulk, respectively; the values are given in Table 1. Table 1 shows that σ s,ult of zigzag sheets are positive, except in the case of single vacancy, which indicates that the strength increases as the width decreases. However, σ s,ult of zigzag sheets are not significant compared to σ b,ult ; therefore, the increase in strength is not significant. Stiffness (GPa) pristine single vac..5% vac. 1% 2% Width (nm) where U is the potential energy at zero strain; τ is the edge stress which arises from the difference of the energies in the edge and interior atoms; E b and E s are the bulk and the edge elastic moduli, respectively. The stress in the sheet is given by 91

4 Strength (GPa) Width (nm) Fig. 3. Variations in the stiffness and the ultimate tensile strength of armchair graphene with width and vacancy concentration. The both and have the same legend. The curves in Fig. 3 and represent (7) and (8), respectively. TABLE I SURFACE AND BULK PROPERTIES OF GRAPHENE Vacancy concentration σ s,ult σ b,ult Es E b % (ac) % (ac) % (ac) % (ac) Single vac. (ac) % (zz) % (zz) % (zz) % (zz) Single vac. (zz) C. Kinetic Modeling of Strength We recently used the Arrhenius equation and the Bailey s criterion to model the temperature and strain rate dependent fracture strength of defective graphene [17]; an overview of this model is presented below. This model, however, does not take into account the effects of free edges. The Bailey s criterion of durability [18] provides a basis to estimate the lifetime of materials at various temperatures [19]. The criterion is expressed as t f dt = 1, (9) τ ( T,t) where t f is the time (t) taken to the fracture; τ(t,t) is the durability function at temperature T, which is generally determined by experiments [18]. The Arrhenius equation, however, is a good approximation to the durability function [12]. The Arrhenius equation [2] expresses the temperature dependent rate of a chemical reaction (k) as k = A exp[δe/(k B T)], where A is a constant that depends on the chemical bonding; ΔE is the activation energy barrier; k B is the Boltzmann constant. When a mechanical force F is applied to a molecule, the activation energy barrier reduces by an amount of FΔx, where Δx is the change in the atomic coordinates due to F [21]. We defined a durability function for graphene in the form of Arrhenius equation as τ ( T,t) = τ ( ) n exp " U β γσ t % $ ', # k B T & (1) where τ is the vibration period of atoms that is 5 fs for carbon in graphene [5]; n is the number of bonds in the sheet; U is the interatomic bond dissociation energy that is 4.95 ev for a carbon-carbon bond [1]; β represents the reduction of average bond dissociation energy due to presence of vacancies; we defined β, using MD simulations at K, as!# β = " $# 1, α =.165α + k, α >, (11) where α is the vacancy percentage. Even the presence of a single vacancy reduces the strength drastically; this strength reduction is considered by the constant k. The values of k are 1.13 and 1.21 for armchair and zigzag sheets, respectively [17]. γ = vq, where v is the activation volume, which is 8.25 Å 3 ; the value of v is close to the representative volume of a carbon atom in graphene, which is 8.6 Å 3. q is a directional constant that takes into account the different bond orientation along the armchair and zigzag directions [17]; q is 1 for armchair sheets and it is 91.7/18.9 (.82) for zigzag sheets, where 91.7 and 18.9 are the tensile strengths, in GPa, of armchair and zigzag sheets at K, respectively. σ(t) is the stress at time t, which we expressed in terms of the strain rate ε as σ ( t) = a( εt ) + b( εt ) 2, (12) where a and b are the second and the third order elastic moduli, respectively; the values of a and b were obtained from regression analysis of the stress-strain curves given by MD simulations at K, where, a and b are 1.11 TPa and TPa for armchair sheet, the corresponding values for zigzag sheet are.91 TPa and -1. TPa. We calculated the failure time t f by numerically solving (9). We obtained the fracture stress σ(t f ) by substituting the t f into (12). Fig. 4 shows that the fracture strength given by the proposed model agrees quite well with the MD simulations results. The proposed model is computationally quite efficient than molecular dynamics simulations. 911

5 Strength (GPa) Strength (GPa) % model 2% MD Temperature (K) 2% model 2% MD pristine model pristine MD single vac. model single vac. MD 1% model 1% MD Temperature (K) pristine model pristine MD single vac. model single vac. MD 1% model 1% MD Fig. 4. Comparison of the model predicted strength of armchair and zigzag graphene with molecular dynamics simulations. IV. CONCLUSIONS In summary, we used molecular dynamics simulations to study the influence of free edges and vacancy concentration on the strength and stiffness of graphene. Results reveal that vacancy defects have a profound impact on the strength and stiffness. We also present an atomistic model to assess the temperature and strain rate dependent fracture strength of defective graphene. The model is computationally very efficient and quite accurate compared to the molecular dynamics simulations. ACKNOWLEDGMENT This work was financially supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. Computing resources were provided by WestGrid and Compute/Calcul Canada. REFERENCES [1] C. Chen, S. Rosenblatt, K. I. Bolotin, W. Kalb, P. Kim, I. Kymissis, et al., Performance of monolayer graphene nanomechanical resonators with electrical readout, Nature Nanotechnology, vol. 4, pp , Dec 29. [2] C. Y. Chen, S. Lee, V. V. Deshpande, G. H. Lee, M. Lekas, K. Shepard, et al., Graphene mechanical oscillators with tunable frequency, Nature Nanotechnology, vol. 8, pp , Dec 213. [3] K. S. Novoselov, V. I. Fal'ko, L. Colombo, P. R. Gellert, M. G. Schwab, and K. Kim, A roadmap for graphene, Nature, vol. 4, pp , Oct 212. [4] A. Zandiatashbar, G. H. Lee, S. J. An, S. Lee, N. Mathew, M. Terrones, et al., Effect of defects on the intrinsic strength and stiffness of graphene, Nature Communications, vol. 5, Jan 214. [5] M. A. N. Dewapriya, A. S. Phani, and R. Rajapakse, Influence of temperature and free edges on the mechanical properties of graphene, Modelling Simul. Mater. Sci. Eng., vol. 21, Sep 213. [6] L. Tapaszto, T. Dumitrica, S. J. Kim, P. Nemes-Incze, C. Hwang, and L. P. Biro, Breakdown of continuum mechanics for nanometrewavelength rippling of graphene, Nature Physics, vol. 8, pp , Oct 212. [7] F. Khademolhosseini, A. S. Phani, A. Nojeh, and R. K. N. D. Rajapakse, Nonlocal Continuum Modeling and Molecular Dynamics Simulation of Torsional Vibration of Carbon Nanotubes, IEEE Trans. Nano., vol. 11, pp , Jan 212. [8] S. Plimpton, Fast parallel algorithms for short-range moleculardynamics, J. Comp. Phys., vol. 117, pp. 1-19, Mar [9] S. J. Stuart, A. B. Tutein, and J. A. Harrison, A reactive potential for hydrocarbons with intermolecular interactions, J. Chem. Phys., vol. 112, pp , Apr 2. [1] D. W. Brenner, Empirical potential for hydrocarbons for use in simulating the chemical vapor-deposition of diamond films, Phys. Rev. B, vol. 42, pp , 19. [11] O. A. Shenderova, D. W. Brenner, A. Omeltchenko, X. Su, and L. H. Yang, Atomistic modeling of the fracture of polycrystalline diamond, Phys. Rev. B, vol. 61, pp , Feb 2. [12] H. Zhao and N. R. Aluru, Temperature and strain-rate dependent fracture strength of graphene, J. Appl. Phys., vol. 18, 64321, Sep 21. [13] M. A. N. Dewapriya, Molecular dynamics study of effects of geometric defects on the mechanical properties of graphene, M.A.Sc. Thesis, Dept. of Mech. Eng., Univ. of British Columbia, Canada, Apr 212. [14] B. Zhang, L. Mei, and H. F. Xiao, Nanofracture in graphene under complex mechanical stresses, Appl. Phys. Lett., vol. 11, , Sep 212. [15] M. A. N. Dewapriya, R. K. N. D. Rajapakse, and A. S. Phani, 214, Atomistic and continuum modelling of temperature-dependent fracture of graphene, Int. J. Fract., vol. 187, pp , Feb 214. [16] D. H. Tsai, The virial theorem and stress calculation in molecular dynamics, J. Chem. Phys.,, pp , [17] M. A. N. Dewapriya and R. K. N. D. Rajapakse, Molecular Dynamics Simulations and Continuum Modeling of Temperature and Strain Rate Dependent Fracture Strength of Graphene with Vacancy Defects, J. Appl. Mech., vol. 81, 811, Jun 214. [18] J. Bailey, An Attempt to Correlate Some Tensile Strength Measurements on Glass: III, Glass Ind., vol. 2, pp , [19] A. D. Freed, and A. I. Leonov, The Bailey Criterion: Statistical Derivation and Applications to Interpretations of Durability Tests and Chemical Kinetics, Z. Angew. Math. Phys., vol. 53, pp.1 166, Jan 22. [2] S. Arrhenius, On the Reaction Rate of the Inversion of the Non- Refined Sugar Upon Souring, Z. Phys. Chem., 4, pp , [21] T. L. Kuo, S. Garcia-Manyes, J. Y. Li, I. Barel, H. Lu, B. J. Berne, et al., "Probing static disorder in Arrhenius kinetics by single-molecule force spectroscopy," PNAS, vol. 17, pp , Jun