Comparing Quantum, Molecular and Continuum Models for Graphene at Large Deformations

Comparing quantum, molecular and continuum models for graphene at large deformations

Aningi Mokhalingam\dagger 1, Reza Ghaffari\S 2, Roger A. Sauer\S 3 and Shakti S. Gupta\dagger 4

\S Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany \dagger Department of Mechanical Engineering, IIT Kanpur, Kanpur - 208016, India

Abstract In this paper, the validity and accuracy of three interatomic potentials, commonly used to study carbon nanostructures in molecular dynamics, and the continuum shell model of Ghaffari and Sauer [1] are investigated. The mechanical behavior of single-layered graphene sheets (SLGSs) near zero Kelvin is studied for this comparison. The validity of the molecular and continuum models is assessed by direct comparison with density functional theory (DFT) data available in the literature. The molecular simulations are carried out employing the MM3, Tersoff and REBO+LJ potentials. The continuum formulation uses an anisotropic hyperelastic material model in the framework of the geometrically exact Kirchhoff-Love shell theory and isogeometric finite elements. For the comparison, the nonlinear response of a square graphene sheetunder uniaxial stretching, biaxial stretching and pure bending is studied. Results from the continuum model are in good agreement with those of DFT. The results from the MM3 potential agree well with the DFT results up to the instability point, whereas those from the REBO+LJ and Tersoff potentials agree with the DFT results only within the range of small deformations. In contrast to the other potentials, the Tersoff potential yields auxetic response in SLGSs under uniaxial stretch. Additionally, the transverse vibration frequencies of a pre-stretched graphene sheet and a carbon nanocone are obtained using the continuum model and molecular simulations with the MM3 potential. The variations of the frequencies obtained from these two approaches agree within an accuracy of about 95\%. Keywords: Anisotropic hyperelasticity; carbon nanocone; continuum mechanics; graphene; Kirchhoff-Love shells; molecular dynamics.

1 Introduction

Graphene is an atom-thick two-dimensional (2D) hexagonal lattice of covalently bonded carbon atoms, which can be exfoliated from bulk graphite [2]. Due to its excellent mechanical [3], electrical and thermal properties [4], it has many industrial applications in the fields of nanocom- posites [5], nano-electromechanical systems [6] and electronic devices [7]. Structural pentagonal and heptagonal defects in graphene lead to the formation of other carbon nanostructures such arXiv:1908.05090v1 [physics.comp-ph] 14 Aug 2019 as fullerenes and carbon nanocones (CNCs) [8,9]. CNCs have potential applications in the scanning probe microscopy [10], field emission electron source [11] and molecular pumps [12]. In such applications, the physical properties are investigated by using either experimental or computational techniques based on ab-initio [13], molecular [14] (i.e. molecular dynamics (MD),

1Email: [email protected] 2Email: [email protected] 3Email: [email protected] 4Corresponding author, email: [email protected]

1 molecular mechanics (MM) and molecular structural mechanics) and continuum methods. In Table1, the elastic moduli of a single layered graphene sheet (SLGS) along the armchair, E\mathrm{A}\mathrm{C}, and zigzag, E\mathrm{Z}\mathrm{Z}, directions obtained from these methods are reported. As seen, some authors report different stiffness in the armchair and zigzag directions. This table also reveals ambiguity in the literature regarding isotopic behaviour of SLGS even in the small deformation regime. Subsequently in this paper, relevant discussion on this aspect from our findings are reported. In MD and MM simulations, the accuracy of the results depends on the potential defining

Table 1: Elastic properties of graphene reported in the literature. NA = Not available.

Ref. Year Method/potential E\mathrm{A}\mathrm{C} [TPa] E\mathrm{Z}\mathrm{Z} [TPa] Thickness [nm] [15] 2000 Ab-initio 1.100 1.100 0.340 [16] 2006 Molecular structural mechanics 1.096--1.125 1.106--1.201 0.340 (Tersoff-Brenner) [17] 2006 DFT 1.250 1.250 0.340 [18] 2007 Ab-initio 1.050 1.050 0.334 [3] 2008 Experimental (nano indentation) 1 \pm 0.100 1 \pm 0.100 0.335 [19] 2009 DFT 0.964 0.964 0.340 [20] 2009 Molecular structural mechanics 1.040 1.042 0.340 [21] 2009 Quantum molecular dynamics 1.100 0.600 0.335 [22] 2009 Orthogonal tight-binding and MD 1.01 \pm 0.030 1.01 \pm 0.030 0.335 [23] 2009 Truss-type analytical models with AMBER 1.378 1.303 NA with MORSE 1.379 1.957 NA [24] 2010 MD (AIREBO) 0.890 0.830 NA [25] 2010 Molecular structural mechanics 0.721 0.737 0.340 [14] 2010 MD (Tersoff) 1.130 1.050 0.335 [26] 2010 Molecular mechanics (MM3) 3.380 3.400 0.100 [27] 2012 MD (AIREBO) 1.097 0.961 0.335 [28] 2013 Molecular structural mechanics and MD 1.070 1.070 0.335 [29] 2014 Molecular structural mechanics 1.100 1.100 0.34 (Modified MORSE) [30] 2017 Space frame approach with AMBER 0.780 0.819 NA with MORSE 0.890 0.938 NA [31] 2018 Multiscale model (MM3) 0.927 0.927 0.335 atomic interactions. For carbonaceous structures, popular potentials are: MM3 [32], Tersoff [33], the first and second generation reactive empirical bond order (REBO)[34, 35], adap- tive intermolecular reactive empirical bond order (AIREBO) [36], and ReaxFF [37]. Recently, Lebedeva et al. [38] investigated the applicability of these potentials (except MM3) up to 3\% uniaxial elongation and reported that the considered potentials fail to reproduce precisely the experimental and ab-initio in-plane and out-of-plane deformations of a SLGS. Employing the MM3 potential in MM simulations, Gupta and Batra [26] reported significant increase of the transverse vibration frequencies of SLGS under pre-stretch. Similarly, Liao et al. [39] studied the influence of temperature, cone height, and cone angles on the mechanical behavior ofCNCs under uniaxial tensile and compression employing the Tersoff potential. Some equivalent continuum structures have been proposed in the linear regime to study vibrations of SLGSs [20, 25, 40--44] and CNCs [45--47]. Using a continuum plate model, Jiang et al. [41] investigated the effect of size, shape and boundary conditions on vibration of SLGSsand multi-layered graphene sheets (MLGSs). Apart from the elastic properties, the vibrational behavior of SLGSs has also been studied using molecular structural mechanics [20, 25, 48]. Using MM simulations based on the universal force field (UFF) model, Chowdhury et al.[42] reported that the natural frequencies of SLGSs are insensitive to the chirality. Singh and Patel [49] used a multiscale method to study the effect of pre-tension on the nonlinear static and dynamic response of SLGSs. They have reported that the pre-tension significantly increases natural fre-

2 quencies. Singh and Patel [50] also studied the nonlinear elastic response of SLGSs and reported that SLGSs show softening behavior at small strains and hardening behavior at large bending. Fakhrabadi et al. [45] studied the vibrational properties of CNCs of different heights and cone angles using MM simulations and reported that the transverse vibrational frequencies reduce with increase in cone angle and height. Hu et al. [46] modelled CNCs as tapered beams using the Euler-Bernoulli and Timoshenko beam theory to study transverse vibrations. In this paper, the nonlinear mechanical response of a square graphene sheet under uniaxial and biaxial in-plane stretch is studied using MM/MD, DFT and a hyperelastic continuum model based on DFT data [51]. The REBO+LJ and Tersoff potentials are used in the MD simulations and the MM3 potential is used in the MM simulations. The validity of the interatomic potentials is investigated in the linear and nonlinear deformation regime by comparing the results from MM/MD simulations with those obtained from DFT simulations [51]. The frequencies of the transverse vibrations of the SLGS and a CNC under stretch, obtained from MM simulations, are compared with those obtained from the continuum model. The remainder of this paper is arranged as follows: Section2 describes the molecular simulation methods and the interatomic potentials used in the present study. A brief introduction to the continuum model is given in Section3. Numerical results are then presented in Section4 followed by conclusions in Section5.

2 Molecular simulations

This section presents details about molecular simulations and the mathematical expressions of the interatomic potentials considered. Molecular simulations are carried out by solving the Newtonian equations of motion for the atoms, \bfitF I = mI \bfitr\"I , (1) where \bfitr I and mI are the position and mass of the atom I, and \bfitF I is the interatomic force on \bigl( N\bigr) atom I. \bfitF I can be determined from the potential function U r as \partial U \bfitF I = - , (2) \partial \bfitr I N where r := \{ \bfitr 1, \bfitr 2, \bfitr 3, ..., \bfitr N \} are the positions of all the atoms. At this juncture, we differentiate between the two techniques used in the current work, namely, molecular mechanics (MM) and molecular dynamics (MD). In MM simulation, systems are modeled at 0 K, which means that the average velocity fluc- tuations of the atoms are zero. They are thus static problems that can be solved by finding their minimum energy configurations. Here, this is done using the Broyden-Fletcher-Goldfarb- Shanno (BFGS) method [52]. Before applying any loads, the system under consideration (here SLGS/CNC) should itself be brought into the minimum energy configuration. We do this initial relaxation step by bringing the root mean square gradient of U to 0.001 Kcal/mol-A.\r Subse- quently, we apply the desired boundary conditions and compute the eigenvalues (frequencies) and eigenvectors (mode shapes) of the mass-weighted Hessian of the SLGS/CNCs at various stretch values using the Vibrate subroutine of Tinker [53]. In MD simulations, the specification of a finite temperature leads to the fluctuating velocityin the system. MD systems thus require transient solution approaches. Further, they need to be thermally equilibrated in order to reach quasi-static states at macro-scales. However, first the relaxed or minimum energy configuration of the system should be obtained. This is donehere with the Polak-Ribiere's conjugate gradient method [54] in a quasi-static approach with abso- lute zero temperature and zero velocity. After this step, the system is thermally equilibrated at

3 constant volume and temperature of 0.1 K for 50 ps with a timestep of 1 fs. The temperature is maintained at 0.1 K employing a Nose-Hoover thermostat [55] with three chains during deformations. The MD simulations are performed with the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [56]. In molecular simulations at approximately 0 K, the virial stress is defined by [57, 58]

Np ij 1 \sum i j \sigma\= I := fIJ rIJ , (3) VI J=1 where VI is the volume occupied by atom I, and i and j denote the Cartesian components along i the x, y and z directions, respectively. Np is the number of neighboring atoms of atom I, fIJ is the force on atom I due to atom J, and rIJ is the distance between atom I and J. For 2D materials, such as graphene, it is natural to introduce the stress as force per length. Given (3), this stress follows as ij VI ij \sigma I := \sigma\=I , (4) AI where AI is the sheet area attributed to atom I. Through definition Eq. (4), the usage of a thickness for SLGSs is avoided. The interatomic potentials are described in the following subsections.

2.1 MM3 Potential

The MM3 potential consists of the first and higher order expansions of bond stretching, angle bending, and torsion. This potential also incorporates the cross terms5 among the mentioned contributions and between angle bending and out-of-plane bending [32]. The expression of the MM3 potential is given by [32] \sum \sum \bigl( \bigr) \sum \sum U\mathrm{M}\mathrm{M}3 = U\mathrm{s} + U\theta + U\phi + U\mathrm{s}\theta + U\theta \theta\prime + U\phi \mathrm{s} + U\mathrm{v}\mathrm{d}\mathrm{W} , (5) I J I K where U\mathrm{s}, U\theta and U\phi are the energy contributions corresponding to changes in bond length, bond angle and dihedral angle, respectively. U\mathrm{s}\theta , U\theta \theta\prime and U\phi \mathrm{s} account for energies of the cross-term interactions between bond stretch and angle bending, angle bending and out-of-plane-bending, and bond stretch and dihedral angle, respectively. U\mathrm{v}\mathrm{d}\mathrm{W} defines van der Waals (vdW) attraction 6 and steric repulsion in the form (r\mathrm{c}/rIJ ) and exp( 12- rIJ /r\mathrm{c}), where rc is the cut-off distance. Further details of these terms are given in Appendix A.1.

2.2 REBO+LJ Potential

The REBO+LJ potential consists of two parts. The covalent bonds between carbon atoms are modeled using the REBO potential, which is widely used for the formation and breaking of bonds in carbonaceous structures. The REBO part of the potential is [36]

\mathrm{R}\mathrm{E}\mathrm{B}\mathrm{O} \sum \sum \bigl[ \bigr] U = E\mathrm{R}(rIJ ) + bIJ E\mathrm{A}(rIJ ) , (6) I J=I+1 where rIJ is the distance between a pair of atoms and bIJ is an empirical bond-order term. E\mathrm{R} and E\mathrm{A} are, respectively, the repulsive and attractive terms and are given in Appendix A.2.

5The cross term between the bending and torsion is not considered.

4 The vdW attraction and steric repulsion are modeled by the standard 12-6 Lennard-Jones (LJ) potential [59] \Biggl[ \Biggr] \biggl( \sigma \biggr) 12 \biggl( \sigma \biggr)6 U \mathrm{L}\mathrm{J} = 4\epsilon - , (7) rIJ rIJ where \sigma and \epsilon are the LJ parameters. The vdW energy is only included when the covalent bond energy from the REBO potential become zero, i.e., after breakage of the covalent bond.

2.3 Tersoff Potential

The Tersoff potential is a pair-like potential in which the strength of a bond depends onthe local environment, i.e., an atom with fewer neighboring atoms forms a stronger bond than the atom with more neighboring atoms. The Tersoff potential is [33] 1 \sum \sum U \mathrm{T}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{o}ff = f (r )\bigl[f (r ) + b f (r )\bigr] , IJ 2 \mathrm{c} IJ \mathrm{R} IJ IJ \mathrm{A} IJ (8) I J\not =I where rIJ is the distance between the atom pairs I and J, bIJ is the bond order term, f\mathrm{R} and f\mathrm{A} are the repulsive and attractive terms, respectively, and f\mathrm{c} is a smooth cutoff function. Details of the potential are given in Appendix A.3.

3 Continuum model

As a homogenized structure, the SLGS and CNC are modeled based on the shell formulation of Duong et al. [60] and the anisotropic hyperelastic material model of Ghaffari and Sauer [1]. Ghaffari and Sauer [1] formulated the strain energy density, per unit area of the initial configuration, based on a set of invariants \scrJi , i.e. [1, 61] \mathrm{d}\mathrm{i}\mathrm{l} \mathrm{d}\mathrm{e}\mathrm{v} W (\scrJ1 , \scrJ2 , \scrJ3 ) = W\mathrm{m} (\scrJ1 ) + W\mathrm{m} (\scrJ2 , \scrJ3 ; \scrJ1 ) + W\mathrm{b}(\kappa1 , \kappa2 ; \scrJ1 ) , (9) \mathrm{d}\mathrm{i}\mathrm{l} \mathrm{d}\mathrm{e}\mathrm{v} where W\mathrm{m} and W\mathrm{m} are the pure dilatational and deviatoric parts of the membrane strain energy density, respectively, and W\mathrm{b} is the bending strain energy density. These terms are defined as \mathrm{d}\mathrm{i}\mathrm{l} \bigl[ \bigr] W\mathrm{m} := \varepsilon 1 - (1 +\alpha \^ \scrJ1 ) exp( \alpha- \^ \scrJ1 ) , (10) \mathrm{d}\mathrm{e}\mathrm{v} W\mathrm{m} := 2 \mu (\scrJ1 ) \scrJ2 + \eta (\scrJ1 ) \scrJ3 , c W := J \mathrm{b} \bigl( \kappa2 + \kappa2 \bigr) , (11) \mathrm{b} 2 1 2 where \mu and \eta are defined as \beta\^ \scrJ \mu (\scrJ1 ) := \mu 0 - \mu 1 e 1 , (12) 2 \eta (\scrJ1 ) := \eta0 - \eta 1 \scrJ1 . \^ The material constants \varepsilon ,\alpha \^ , \mu 0, \mu 1, \beta , \eta 0, \eta 1 and c\mathrm{b} are given in Tables2 and3. J is the surface area change, and \kappa1 and \kappa2 are the principal surface curvatures [64]. \scrJ1 and \scrJ2 capture isotropic dilatation and shear deformation, respectively, while \scrJ3 captures anisotropic shear deformation. \scrJi are given by

\scrJ1 := ln J = ln(\lambda 1\lambda2 ) , \biggl( 2 2 \biggr) 1 \lambda 1 \lambda 2 \scrJ2 := 2 + 2 - 2 , (13) 4 \lambda 2 \lambda 1 \biggl( \biggr)3 1 \lambda 1 \lambda 2 \scrJ3 := - cos(6\theta ) . 8 \lambda 2 \lambda 1

5 Table 2: Membrane behavior: Hyperelastic material constants determined by fitting to DFT calculations based on generalized gradient approximation (GGA) according to Kumar and Parks [62] and Shirazian et al. [51].

\alpha\^ \varepsilon [N/m] \mu 0 [N/m] \mu 1 [N/m] \beta\^ \eta0 [N/m] \eta1 [N/m] Kumar and Parks [62] 1.53 93.84 172.18 27.03 5.16 94.65 4393.26 Shirazian et al. [51] 1.435 103.9 182.6 34.94 4.665 83.46 3932

Table 3: Bending stiffness according to various atomistic models. FGBP = first generation Brenner potential; SGBP = second generation Brenner potential; QM = quantum mechanics.

FGBP Lu et al. [63] SGBP Lu et al. [63] QM Kudin et al. [13] c\mathrm{b} [nN\cdotnm] 0.133 0.225 0.238

\lambda 1 and \lambda2 are the principal surface stretches with \lambda 1 \geq \lambda 2. \theta is the maximum stretch angle relative to the armchair direction and defined as

\theta := arccos (\bfitY 1 \cdot \bfitx\^ ) , (14) where \bfitY 1 is the direction of the maximum stretch, see Ghaffari et al. [61] and Ghaffari and Sauer [1] for details. The material has isotropic behavior under pure dilatation, and anisotropic behavior only appears under large shear deformation. Material model (9) is implemented in the nonlinear finite shell element formulation of Duong et al.[60]. The discretized weak form can be written as [60] M u\" + f\mathrm{i}\mathrm{n}\mathrm{t}(u) = f\mathrm{e}\mathrm{x}\mathrm{t}(u) , (15) where M is the mass matrix (see Ghaffari and Sauer [65] for the mass matrix of graphene), u is the displacement vector and f\mathrm{i}\mathrm{n}\mathrm{t} and f\mathrm{e}\mathrm{x}\mathrm{t} are the internal and external force vectors, respec- e e tively. They are assembled from the elemental contributions f\mathrm{i}\mathrm{n}\mathrm{t} and f\mathrm{e}\mathrm{x}\mathrm{t}, respectively, using e e the standard finite element assembly procedure. f\mathrm{i}\mathrm{n}\mathrm{t} and f\mathrm{e}\mathrm{x}\mathrm{t} are defined by \int \int f e := \tau \alpha \beta N\mathrm{T} \bfita dA + M \alpha \beta N\mathrm{T} \bfitn dA, \mathrm{i}\mathrm{n}\mathrm{t} ,\alpha \beta 0 ;\alpha \beta (16) e e \Omega0 \Omega0 \int e \mathrm{T} f\mathrm{e}\mathrm{x}\mathrm{t} = N \bfitt ds , (17) e \partial t\Omega e e for the special case of zero body forces and zero boundary moments. Here \Omega 0 and \Omega denote element domains in the reference and current configuration, respectively, and along \partial \bfitt \Omega ,the boundary traction \bfitt is applied. \bfita\alpha are the covariant tangent vectors and \bfitn is the normal vector to the shell surface. N, N,\alpha and N;\alpha \beta denote the shape function matrix and its first parametric derivative and second covariant derivative [60], and (\bullet )\mathrm{T} is the transpose operator. The contra- variant components of the surface Kirchhoff stress tensor and the moment tensor are given by [60] \partial W \tau \alpha \beta = , (18) \partial\alpha a \beta

\alpha \beta \partial W M0 = , (19) \partial\alpha b \beta where a\alpha \beta and b\alpha \beta are the surface metric and curvature tensor components. The Cauchy and Kirchhoff stress components are connected by 1 \sigma \alpha \beta = \tau \alpha \beta . (20) J

6 Using a Taylor expansion about u\widehat such that u = u\widehat + \Delta u, the linearized equations of motion become \" M \Delta u\" + K \Delta u = (- f\mathrm{i}\mathrm{n}\mathrm{t} + M u\widehat - f\mathrm{e}\mathrm{x}\mathrm{t}) , (21) where K := \partial (f\mathrm{i}\mathrm{n}\mathrm{t} - f\mathrm{e}\mathrm{x}\mathrm{t})/\partial u is the tangent stiffness matrix (see Ghaffari and Sauer[65] and Duong et al. [60] for details). For harmonic vibrations we have

\Delta u := \Deltau \= e - i\omega t , (22) where \Delta u\= are \omega are the mode shape and frequency of the structure, respectively. Using (22), Eq. (21) can be transformed to the standard eigenvalue problem

K \Delta u\= = \omega 2 M \Delta u\= . (23)

At each load increment, we solve Eq. (15) iteratively using the Newton-Raphson method. Subse- quently eigenvalue problem (23) is solved at each load increment in order to obtain the variation of the frequencies with the loading.

4 Numerical results

This section presents numerical results pertaining to the deformation and vibrations of a square SLGS and a CNC. The three molecular models of Sec.2 are compared with DFT data from Shirazian et al. [51] and the DFT-based continuum model of Sec.3. First, the minimum energy configuration of the SLGS is given in Subsection 4.1. Then the procedure for the in-plane stretch of the SLGS is described in Subsection 4.2.1, and the variation of strain energy and stresses is studied in Subsections 4.2.2 and 4.2.3. The out-of-plane bending of SLGS is then discussed in Subsection 4.3, followed by the modal analysis of SLGS and CNC in Subsection 4.4. The transverse vibration frequencies of SLGS are calculated up to the instability point, where the ellipticity of the elasticity tensor is lost [62]. A pointwise summary of the main findings is finally given in Subsection 4.5.

4.1 Minimum energy configuration of SLGS

A SLGS with dimensions 10 nm \times 10 nm and an initial bond length of 0.142 nm is considered. The distribution function of the bond length for the relaxed configuration employing the three potentials is shown in Fig.1. In all cases, the equilibrium (i.e. mean) bond length is slightly different than the experimental value 0.1422 nm[66]. The difference may be attributed to the finite size of the specimen and different functional forms of the potentials and the constants used therein. In all the cases almost 90\% of bonds are stretched to their respective mean length. Further, the sharp bond length distribution of the MM3 potential indicates that the SLGS is in a homogeneous state of deformation after relaxation. Further, from Fig.1, we note that after the minimization the SLGS shrinks more when the MM3 potential is employed compared to REBO+LJ. On the other hand, the SLGS dilates after relaxation when the Tersoff potential is employed.

4.2 In-plane stretching of SLGS

This section examines the elastic response of the square SLGS under uniaxial and biaxial stretch.

7 100 MM3 90 REBO+LJ Tersoff 80

0 1.3 1.35 1.4 1.45 1.5 1.55 1.6

Figure 1: The distribution function of the bond length after relaxation.

(a) (b) (c) Figure 2: Pre-stressed SLGS: Stretch along the (a) zigzag and (b) armchair direction, and (c) under pure dilatation.

4.2.1 Procedure

The MM simulations (with MM3) of the SLGS are performed with and without periodic boundary conditions (PBCs) to investigate the size effect. In the absence of PBCs, the position inthe current configurationx ( , y) of the edge atoms is given by

\biggl[ x\biggr] \biggl[\lambda 0 \biggr]X \biggl[ \biggr] = 1 , (24) y 0 \lambda 2 Y where X and Y are the initial position of the edge atoms in the relaxed configuration. For uniaxial stretch, \lambda 1 = \lambda and \lambda2 = 1 or \lambda 1 = 1 and \lambda 2 = \lambda (i.e. the lateral direction is kept fixed), while for pure dilatation, \lambda1 = \lambda 2 = \lambda (see Fig.2). When PBCs are used, the periodic simulation box is deformed along X and/or Y with a stretch increment of 0.1 A.\r At each increment, the edge atoms are kept fixed and the system is relaxed to compute the potential energy and virial stress. In the MD simulations (with REBO+LJ and Tersoff), PBCs are employed and(24) is applied on the thermally equilibrated SLGS with a constant strain rate of 0.001/ps. In the continuum simulations, (24) is applied quasi-statically without using PBCs.

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(c) (f) Figure 3: Comparison of the total potential energy of a square SLGS under uniaxial stretch along (a) armchair, and (b) zigzag directions, and (c) in pure dilatation. The corresponding error E(W ) according to Eq. (25) is shown in (d), (e) and (f).

9 4.2.2 Strain energy

Figure3 shows the potential energy variation versus the uniaxial and biaxial stretch computed from molecular simulations, the DFT-based continuum model and DFT [51]. The relative error in the strain energy W is shown in Fig.3d,3e and3f. The relative error in the quantity X with respect to the DFT results is defined as X - X E(X) = \mathrm{D}\mathrm{F}\mathrm{T} , (25) max(X\mathrm{D}\mathrm{F}\mathrm{T}) where X results from the continuum and/or molecular simulations and X\mathrm{D}\mathrm{F}\mathrm{T} is the DFT re- sult. In MM simulations, it is found that the variation in the energy and stress (discussed subsequently) are almost identical with and without PBCs. These results confirm that the size effect on the elastic response is negligible when the diagonal length of the SLGS is over10.0nm as reported by Zhao et al. [22]. The strain energy results from the MM3 potential agree with the DFT results up to the stretch \lambda 1 \approx 1.15 for uniaxial loading and \scrJ1 \approx 0.3 for biaxial loading within \approx 5\% error. The results from the REBO+LJ and Tersoff potentials agree within an accuracy of \approx 80\%. The continuum results are in excellent agreement with those from DFT simulations for the whole range under study (within \approx 0.05\% error). This is due to the fact, that the continuum model hasbeen calibrated directly from DFT data [51].

4.2.3 Stresses

Next, the stresses from the different approaches are compared. For uniaxial stretch, the SLGS is stretched either along the armchair or zigzag direction. Three different stresses are examined in the following. 1. Stress along the stretch direction. The variation in \sigma 11 -- the stress along the stretch direction -- and the corresponding errorare shown in Fig.4 for stretch along the armchair direction, and Fig.5, for stretch along the zigzag direction. Fig.4 shows that the molecular simulation results are in good agreement with the DFT results up to \lambda1 = 1.1 within an error of \approx 5\%. Beyond this stretch, \sigma11 computed

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(a) (b) Figure 4: Variation of (a) stress \sigma11 and (b) error E(\sigma11 ) according to Eq. (25) as a function of stretch \lambda1 in the armchair direction. from the MM3 potential shows gradual hardening. \sigma11 computed from the other two potentials

10 follow the DFT results up to \lambda 1 = 1.19 and then there is a sudden rise, which is discusssed subsequently.

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(a) (b) Figure 5: Variation of (a) stress \sigma 11 and (b) error E(\sigma 11) according to Eq. (25) as a function of stretch \lambda 1 in the zigzag direction.

Similarly Fig.5 shows that the molecular simulation results follow the DFT results up to \lambda 1 = 1.13. Beyond this stretch, the MM3 results exhibit much higher stresses than the other cases. This may be due to the absence of cutoff function in MM3 as is present in the other two potentials, which initiates bond breaking in the structure. The results from the other two potentials continue to follow the DFT results up to \lambda 1 = 1.25.

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0 0 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.3 1.4 1.5 1.6 1.7 1.8

(a) (b) Figure 6: Distribution function of the bond length for stretch along the (a) armchair and (b) zigzag directions at the stretch \lambda 1 = 1.25 for the REBO+LJ potential.

As Fig.4 shows a sudden rise in \sigma 11 is noticed in the results with the REBO+LJ and Tersoff potentials in the armchair direction. In order to explain this behavior, Figs.6a and6b show the distribution of the bond lengths at \lambda1 = 1.25 for stretch along the armchair and zigzag directions, respectively, employing the REBO+LJ potential. It is found that around 36\% of bonds are stretched to a bond length of more than 0.17 nm when the stretch is along the armchair direction, which activates the cutoff function (31). Due to the discontinuity of the second

11 derivative of the cutoff function a sudden rise in the stress is recorded6. In the case of the Tersoff potential, the anomalous response in \sigma 11 in the armchair direction at \lambda 1 = 1.19 is due to elongations of bonds beyond the cutoff radius (36) of 0.18 nm. 2. Stress perpendicular to the stretch direction. The variation in the lateral stress \sigma22 -- the stress in the perpendicular direction to the stretch -- and its error according to Eq.(25) as a function of \lambda1 are shown in Figs.7 and8. The figures show that the results from the MM3 potential agree well with the DFT results upto \lambda 1 = 1.15 and \lambda 1 = 1.1 in the armchair and zigzag directions, respectively. As Fig.7 shows for stretch in the armchair direction, the lateral stresses from the REBO+LJ potential agree well with the DFT results within the small deformation regime (upto \lambda 1 = 1.025) and then show gradual softening. However, for stretch in the zigzag direction (see Fig.8), \sigma22 from REBO+LJ matches well with \sigma 22 from DFT up to \lambda1 = 1.25. Contrary to the results from the MM3 and REBO+LJ potentials, the Tersoff potential produces negative lateral stress for uniaxial stretch in both directions.

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-100 -2

-4 -150 1 1.05 1.1 1.15 1.2 1.25 1 1.05 1.1 1.15 1.2 1.25

(a) (b) Figure 7: Variation of (a) stress \sigma 22 and (b) error E(\sigma 22) according to Eq. (25) as a function of stretch \lambda 1 in the armchair direction.

To explain this exceptional behavior of the Tersoff potential, we have performed simulations without the constraints on the lateral edge atoms and calculated the Poisson's ratio \nu = \varepsilon- \mathrm{l}\mathrm{a}\mathrm{t}/\varepsilon\mathrm{l}\mathrm{o}\mathrm{n} , where \varepsilon \mathrm{l}\mathrm{a}\mathrm{t} and \varepsilon \mathrm{l}\mathrm{o}\mathrm{n} are the lateral and longitudinal strains, respectively. The strains are computed by taking the ratio of the change in periodic box dimensions with the initial box dimensions. For stretch along the zigzag and armchair directions, the SLGS exhibits a negative Poisson's ratio for all the stretch ratios according to the Tersoff potential. This has also been reported in the literature [38, 67, 68]. Due to this behavior, the Tersoff potential stands in sharp contrast to all the other methods. It is noticed that the stress response computed from the continuum model for both the loading cases are in good agreement with those from DFT. Both the molecular and continuum stresses are equal at smaller stretches, and it is evident that the SLGS shows anisotropy at higher stretches. Additionally, the anisotropy predicted from both the approaches is different, i.e., molecular simulations predict that the armchair direction is stiffer whereas DFT and continuum predict otherwise. To elucidate the anisotropy of the SLGS, we have plotted the two dominating energy terms

6REBO+LJ is sensitive to the choice of cutoff distances. Other values have shown to lead to bond breaking (and hence a sudden stress drop) at much lower strains.

12 12 150

10 100

50 6

4 0

2 -50

-100 -2

-4 -150 1 1.05 1.1 1.15 1.2 1.25 1 1.05 1.1 1.15 1.2 1.25

(a) (b) Figure 8: Variation of (a) stress \sigma 22 and (b) error E(\sigma 22) according to Eq. (25) as a function of stretch \lambda 1 in the zigzag direction. of the MM3 potential in Fig.9. The figure shows that the energy contribution from bond- stretching (term Us in Eq. (5)) is the same for stretch in both directions. The contribution

0 1 1.05 1.1 1.15 1.2 1.25 1.3

Figure 9: Comparison of the dominant energy terms of the MM3 potential when the square SLGS is stretched along the armchair (AC) and zigzag directions (ZZ). of angle-bending on the other hand (term U\theta ) to the total energy for both cases is equal up to \lambda 1 = 1.17, but deviates beyond this stretch. It increases more strongly when the SLGS is stretched in the armchair direction. The remaining energy contributions in Eq. (5) are not significant for this deformation state. Thus, we conclude that U\theta is responsible for the anisotory in SLGS at large deformations. 3. Surface tension. Finally, the variation in the surface tension \gamma = (\sigma11 +\sigma 22)/2, under pure dilatation and its corresponding percentage error is shown in Fig. 10. As seen, the results from molecular simulation employing the MM3 potential agree with the results from DFT up to \scrJ1 = 0.28 and then show gradual hardening, whereas the results from the REBO+LJ and Tersoff potentials agree only up to \scrJ1 = 0.08 and then deviate. Additionally, the results from the later two potentials exhibit a sharp rise at \scrJ1 = 0.39 and 0.43, respectively. At those values of \scrJ1 , \approx 96\% of bonds are \mathrm{m}\mathrm{i}\mathrm{n} stretched to more than 0.17 nm and 0.18 nm (rIJ of respective potentials) using the REBO+LJ

13 and Tersoff potentials, respectively.

80 100

50 50

30 0

0 -50 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5

(a) (b) Figure 10: Variation of (a) surface tension \gamma and (b) error E(\gamma ) according to Eq. (25) as a function of dilatation \scrJ1 .

4.3 Out-of-plane bending of SLGS

In molecular simulations, the bending stiffness of the SLGS can be calculated in two different ways. They both assume linear elastic bending behavior. In the first approach, the SLGS is considered as a thin, linearly elastic, homogeneous and isotropic plate with thickness h, whose bending stiffness is given by[69]

Eh3 c = , (26) \mathrm{b} 12(1 - \nu 2) where E and \nu are the Young's modulus and Poisson's ratio, respectively. To avoid introducing a thickness of the SLGS in the first approach, the first bending frequency of a square SLGSis computed instead in our work. In MM simulations, the Vibrate module in Tinker is used to find the natural frequencies. In MD, on the other hand, the equilibrated SLGS is deformed into the mode shape corresponding to the first natural frequency and then allowed to vibrate freely in the NVE setup. The time history of the atoms closest to the center of the SLGS is then evaluated to determine the frequencies of the transverse vibration using fast Fourier transform (FFT). The obtained frequencies are then compared with those determined from an equivalent plate model. The formula for the transverse frequencies of a linearly elastic, homogeneous and isotropic square plate of side a is given by [70]

2 \sqrt{} \lambda mn c\mathrm{b} fmn = 2 , (27) 2\pi a \rho \mathrm{s} where m and n are the half wave numbers along the x and y directions, c\mathrm{b} and \rho\mathrm{s} are the bending stiffness and area mass density (mass of total carbon atoms per area), respectively. 2 For m = 1 and n = 1, the constant \lambda 11 is 35.99 [70]. The bending stiffness determined from the molecular simulations according to Eqs. (26) and (27) (see Table4) is higher than 1.49 eV, which is the value from DFT [13, 51]. The discrepancy may be associated with the presence of pre-stress in the molecular simulations, which also causes higher frequencies at zero stretch. This is discussed further below.

14 Table 4: Bending stiffnessc ( \mathrm{b}) obtained from an equivalent plate model. Here K, \nu and h are the basal plane stiffness, Poisson ratio and the effective thickness of a graphene sheet, respectively. Here, first three rows of the table are calculated from Eq. 26 and remaining are from Eq. 27.

Potential/model K (N/m) \nu h (nm) c\mathrm{b} (eV) MM3 [26] 340.0 0.210 0.100 1.86 REBO-I [71] 235.0 0.412 0.062 0.56 REBO-II [71] 243.0 0.397 0.057 0.49 MM3 (Current work) - - - 2.11 REBO+LJ (Current work) - - - 2.27 Tersoff (Current work) - - - 2.08

In the second approach, the bending energy of the SLGS is obtained by computing the potential energy of relaxed carbon nanotubes of different radii with respect to the ground state energy of SLGS. The potential energy is fitted by a quadratic curve. The second derivative of this curve then corresponds to the bending stiffness. The obtained values are listed in Table5. We note that the second approach is problematic, as relaxed CNTs usually change radius and therefore also contain in-plane strain energy that is usually not accounted for in the stiffness calculation. As a consequence the bending stiffness may be overestimated.

Table 5: Bending stiffness c\mathrm{b} in [eV] from an equivalent CNT model.

Potential/model c\mathrm{b} (armchair) c\mathrm{b} (zigzag) MM3 (Current work) 3.271 3.146 REBO+LJ (Current work) 2.184 2.235 Tersoff (Current work) 2.078 2.010

The variation of the bending energy with the curvature for armchair and zigzag carbon nanotubes is shown in Fig. 11a and 11b. The bending energy from the DFT and DFT-equivalent 2 continuum model in (11) is given by W\mathrm{b} = c\mathrm{b}\kappa1 /2 , where c\mathrm{b} is the bending stiffness and \kappa1 is\surd the curvature\surd radius. The curvature radius of a CNT is the reciprocal of the radius 2 2 R = ( 3acc/2\pi ) n + m + nm , where acc is the equilibrium bond length of C-C, and n and m are the chirality indices. The bending energy calculated from the MM3 potential differs more

0.5 0.7 DFT DFT 0.45 REBO+LJ REBO+LJ Tersoff 0.6 Tersoff 0.4 MM3 MM3 MM3 (Gupta et al. [73]) 0.35 0.5

0.3 0.4 0.25 0.3 0.2

0.15 0.2

0.1 0.1 0.05

0 0 0 1 2 3 4 0 1 2 3 4 5

(a) (b) Figure 11: Variation of the bending energy with the bending curvature for (a) armchair and (b) zigzag CNTs. than the other two potentials considered. The difference may be attributed to the presence of

15 higher order and cross-interaction terms of the potential. In their study, Gupta et al. [72] computed the bending stiffness of different radii and chiralities of CNTs by equating the frequencies from MM simulations employing the MM3 potential with that of a shell model. The bending energy computed from these calculations is shown in Fig. 11(b).

4.4 Modal analysis

In the following we discuss the effect of incremental prestretch in a SLGS and a CNC ontheir first few modes of vibrations.

4.4.1 Square SLGS

After having established the validity and accuracy of the MM3 potential for stretches up to \approx 1.1 in the previous section, we now study the transverse modes of vibrations of a SLGS using this potential. The vibration response is then compared with that obtained from the DFT-based continuum model at different stretch states. In the continuum model, the stretch is applied up to the loss of ellipticity of the elasticity tensor. This limiting value of stretch is also used in the molecular simulations. The mode shapes and frequency variation of the transverse vibrations

(a)

(b) (c)

Figure 12: Mode shapes of a relaxed square graphene sheet determined by a MM simulation: Shape corresponding to frequency (a) \omega 11, (b) \omega 12, and (c) \omega22 . Here the two indices in \omega represent the number of half waves along the local Cartesian coordinates, and the dotted red lines show the nodal lines. of the first three modes of the SLGS with increasing stretch/dilatation are shown inFig. 12 and Fig. 13, respectively. The molecular simulations are performed without applying PBCs. As Fig. 13 shows, the frequency vs. stretch curves obtained from the two approaches agree within \approx 95\%. It is observed that for all the three loading cases, the frequencies increase monotonically with the stretch. The rate of increase in the frequency is higher at small stretches than at higher stretches. Singh and Patel [49] have reported a similar behaviour for a rectangular SLGS under uniaxial stretch. It is also observed that at zero strain, the frequencies from the molecular simulations are \approx 15\% higher than those computed from the continuum model. This difference may be due to residual stresses in the relaxed configuration of SLGS, which are not accounted for in the present continuum model. For uniaxial stretch along the armchair direction, the frequencies from the molecular simulations and those from the continuum model agree up to \lambda 1 = 1.15 after which the MM simulation results show a stiffer behavior. This is consistent with the variation in the stress with stretch shown in Figs.4 and5. At higher stretches, the variation in the frequency is higher when the SLGS is stretched along the armchair direction

16 0.8 0.8

0.7 0.7

0.6 0.6

0.5 0.5

0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1

0 0 1 1.05 1.1 1.15 1.2 1 1.05 1.1 1.15 1.2

(a) (b) 1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0 0.005 0.01 0.015 0.02 0.025 0.03

(c) Figure 13: Frequencies of a 10 nm \times 10 nm, all-edge-clamped graphene sheet under (a) uniaxial stretch along the armchair direction, (b) uniaxial stretch along the zigzag direction and (c) pure dilatation. than in the zigzag direction. This mild anisotropic response is attributed to the angle-bending energy, which contributes significantly to the total potential energy when the SLGS is stretched along the armchair direction. Contrary to the molecular simulation results, the continuum model shows the zigzag direction to be stiffer7.

4.4.2 Carbon nanocone

The modal analysis of a pre-stretched simply-supported CNC is carried out using MM simulations employing the MM3 potential. The modal frequencies obtained from the MM simulations are compared with those obtained from the continuum model at each stretch. For this study, a truncated cone is selected with initial apex angle \alpha = 14.2\circ , height H = 12.4 nm, and tip radius R = 1.017 nm. Before computing the modal frequencies in the MM simulations, the minimum energy configuration of the structure is found at each stretch level. The stretch is obtained by fixing all the degrees-of-freedom of the edge atoms of the larger radius andincre- mentally moving the atoms of the smaller radius by 0.1 A\r in the axial direction. We note that

7I.e. the mode shapes with a higher number of sinusoidal waves along the stiffer direction have a higher frequencies than the one with higher waves along the softer direction.

17 0.6

0.55

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1 1 1.02 1.04 1.06 1.08

(a) (b)

Figure 14: Variation in the total potential energy (a) and frequencies (b) of a CNC under uniaxial stretch. Here, \langle \lambda1 \rangle denotes the average stretch along the CNC axis. this leads to an inhomogeneous stretch state in the CNC, due to its tapered geometry. The following results are therefore taken at the average stretch \langle \lambda1 \rangle = h/H, where h is the current height. The variations in the potential energy and the modal frequencies computed from the MM3 potential and the continuum model are shown in Fig. 14. The corresponding mode shapes computed from the MM3 potential are shown in Fig. 15. For moderate values of the stretch,

(a) (b) (c) (d) Figure 15: Mode shapes of a relaxed simply-supported carbon nanocone determined by MM simulations: Shape corresponding to frequency (a) \omega21 , (b) \omega 31, (c) \omega01 and (d) \omega 22. Here the two indices in \omega represent the number of circumferential waves and axial half waves in the order. the potential energy from both the approaches are in good agreement. As in the case of SLGS, the modal frequencies from both the approaches differ slightly in the unstretched configuration, possibly due to the absence of residual stresses in the continuum model. The frequency of the lower order modes \omega 0,1 and \omega2 ,1 obtained from the MM3 potential and the continuum model are in good agreement. Based on continuum shell theory, we conjecture that this agreement is due to the dominant membrane/stretching deformation in the lower modes for which the present continuum model is calibrated. This is also the reason for a good match in the strain energy since for moderate value of \langle \lambda1 \rangle only the membrane deformations are dominating. At

18 higher modes bending deformation dominates and hence there is a disagreement between the frequencies computed from the two approaches. Finally, we note that the (0,1) mode frequency remains unaffected for the range of \langle\lambda 1\rangle considered here.

4.5 Summary of main findings

Examining the elastic response of SLGS, we find that compared to DFT: 1) The MM3 potential results agree up to a stretch of 1.1 (10\% strain). 2) The REBO+LJ potential results only agree within the small deformation regime. 3) The Tersoff potential fails to capture the behavior correctly. 4) The continuum model results are in good agreement. 5) The Tersoff potential predicts negative Poisson ratio for SLGS. Examining the vibrations of SLGS and CNC, we find that: 6) The transverse vibrational frequencies increase monotonically with the pre-stretch. 7) The frequencies from atomic simulations differ from the continuum model at zero stretch. 8) The bending frequency \omega01 of CNC is nearly unaffected by the stretch.

5 Conclusions

The nonlinear mechanical response of square SLGS under large uniaxial stretch and pure dilatation is investigated with continuum and atomistic simulations employing the MM3, REBO+LJ and Tersoff potentials. The obtained results are compared with DFT results available inthe literature for the two stretch states. The stress results obtained from the continuum model are in good agreement with DFT results. From the atomistic models, we find that the MM3 is most accurate among the potentials considered in this study. We also studied the transverse vibrational frequencies of a square SLGS and a CNC with the continuum model and molecular simulations employing the MM3 potential. The variation in frequencies at different stretches from the two approaches is compared and are found to be in good agreement within an accuracy of about 95\%. In future work, the present study should be extended to finite temperatures.

Acknowledgement

Financial support from the German Research Foundation (DFG) through grant GSC 111 is gratefully acknowledged. The authors also thank Farzad Shirazian for his comments on the atomistic simulations.

A Description of the molecular potentials

In this section, the mathematical expressions of the used potentials are given.

19 A.1 MM3 Potential

The terms in Eq. (5) are given for atoms I, J, K and L by

\biggl[ \biggl( 7 \biggr) \biggr] U = 71.94K (r - r\mathrm{e} )2 1 - 2.55(r - r\mathrm{e} ) + 2.55(r - r\mathrm{e} )2 , \mathrm{s} \mathrm{s} IJ IJ IJ IJ 12 IJ IJ \biggl[ \mathrm{e} 2 \mathrm{e} 5- \mathrm{e} 2 U\theta = 0.021914K\theta (\theta IJK - \theta IJK ) 1 - 0.014(\theta IJK - \theta IJK ) + 5.6(10) (\theta IJK - \theta IJK ) \biggr] - 7 \mathrm{e} 3 - 10 \mathrm{e} 4 - 7.0(10) (\theta IJK - \theta IJK ) + 9.0(10) (\theta IJK - \theta IJK ) , V V V U = 1 (1 + cos \phi ) + 2 (1 - cos 2\phi ) + 3 (1 + cos 3\phi ) , \phi 2 IJKL 2 IJKL 2 IJKL \mathrm{e} \mathrm{e} \mathrm{e} U\mathrm{s}\theta = 2.51118K\mathrm{s}\theta [(rIJ - rIJ ) + (rJK - rJK )] (\theta IJK - \theta IJK ) , K U = 11.995 \phi \mathrm{s} (r - r\mathrm{e} )(1 + cos(3\phi )) , \phi \mathrm{s} 2 IJ IJ IJKL \mathrm{e} \mathrm{e} U\theta \theta\prime = 0- .021914K\theta \theta\prime (\theta IJK - \theta IJK )(\theta IKL - \theta IKL) , \Biggl[ \biggl( \biggr)6 \biggl( \biggr)\Biggr] r\mathrm{v} 5 rIJ U\mathrm{v}\mathrm{d}\mathrm{W} = \epsilon\mathrm{e} 2- .25 + 1.84(10) exp 12- .0 , rIJ r\mathrm{v} (28) where rIJ , \theta IJK and \phi IJKL are the bond length, the angle between the bonds and the torsion angle, respectively. The parameters with superscript e define the equilibrium values at which the total potential energy is minimum. K\mathrm{s}, K\theta , V1, V2, V3, \epsilon\mathrm{e} , r\mathrm{v}, K\mathrm{s}\theta , K\phi s and K\theta \theta\prime are the potential parameters and are listed in Table6.

Table 6: The potential parameters of MM3 [32].

Parameter value K\mathrm{s} 4.49 mdyne/A\r 2 K\theta 0.67 mdyne-A/rad\r V1 0.185 Kcal/mol V2 0.170 Kcal/mol V3 0.520 Kcal/mol \epsilon\mathrm{e} 0.027 Kcal/mol r\mathrm{v} 2.04 A\r K\mathrm{s}\theta 0.130 mdyne/rad K\phi s 0.059 mdyne/rad \r 2 K\theta \theta\prime 0.240 mdyne-A/rad

A.2 REBO+LJ Potential

The repulsive (E\mathrm{R}) and attractive (E\mathrm{A}) terms in Eq. (6) are

\biggl( Q \biggr) IJ - \alpha IJ rIJ E\mathrm{R}(rIJ ) = wij(rIJ ) 1 + AIJ e , (29) rIJ

3 n \sum n - \beta rIJ E\mathrm{A}(rIJ ) = w- IJ (rIJ ) BIJ e IJ , (30) n=1 n n where QIJ , AIJ , \alpha IJ , BIJ and \betaIJ are the potential parameters that depend on the type of atoms I and J and are given in Table7. wIJ is the bond-weighting factor that depends on

20 \mathrm{m}\mathrm{i}\mathrm{n} \mathrm{m}\mathrm{a}\mathrm{x} the cutoff distances (rIJ and rIJ ) and varies from zero to one. The REBO interactions are \mathrm{m}\mathrm{i}\mathrm{n} \mathrm{m}\mathrm{a}\mathrm{x} gradually turned off, when the bond length is in the range rIJ < rIJ < rIJ . This is achieved through the bond-weighting factor \prime wIJ (rIJ ) = S (t\mathrm{c}(rIJ )) , (31) \prime where the switching and scaling functions S and tc 1 S\prime (t) = \Theta ( t- ) + \Theta ( t- )\Theta (1 - t) (1 + cos(\pi)) t , (32) 2 \mathrm{m}\mathrm{i}\mathrm{n} rIJ - rIJ t\mathrm{c}(rIJ ) = \mathrm{m}\mathrm{a}\mathrm{x} \mathrm{m}\mathrm{i}\mathrm{n} . (33) rIJ - rIJ Here \Theta (t) is the Heaviside step function.

Table 7: The REBO+LJ potential parameters for carbon [36].

Parameter value Q 0.313460 A\r \alpha 4.746539 A\r 1- A 10953.544 eV B1 12388.792 eV B2 17.567065 eV B3 30.714932 eV \beta 1 4.720452 A\r 1- \beta 2 1.433213 A\r 1- \beta 3 1.382691 A\r 1- \epsilon 0.002840 eV \sigma 3.4 A\r \mathrm{m}\mathrm{i}\mathrm{n} rIJ 1.7 A\r \mathrm{m}\mathrm{a}\mathrm{x} rIJ 2.0 A\r

A.3 Tersoff Potential

The terms in the Tersoff potential of Eq. (8) are given by

- \lambda IJ rIJ f\mathrm{R}(rIJ ) = AIJ e , (34)

\mu- IJ rIJ f\mathrm{A}(rIJ ) = B- IJ e , (35) and \left\{ \mathrm{m}\mathrm{i}\mathrm{n} 1 rIJ < rIJ , \biggl[ \biggl( \mathrm{m}\mathrm{i}\mathrm{n} \biggr) \biggr] 1 (rIJ - rIJ )\pi \mathrm{m}\mathrm{i}\mathrm{n} \mathrm{m}\mathrm{a}\mathrm{x} f\mathrm{c}(r) := 1 + cos \mathrm{m}\mathrm{a}\mathrm{x} \mathrm{m}\mathrm{i}\mathrm{n} rIJ < rIJ < rIJ , (36) 2 rIJ - rIJ \mathrm{m}\mathrm{a}\mathrm{x} 0 rIJ > rIJ . where AIJ , BIJ , \lambdaIJ and \mu IJ are the parameters that depend on paired atom types. The cutoff \mathrm{m}\mathrm{i}\mathrm{n} \mathrm{m}\mathrm{a}\mathrm{x} function (f\mathrm{c}) describes a gradual decrease of the bond strength between rIJ and rIJ . In Eq. (8), bIJ is the bond order term. It depends on neighboring atoms and it is defined by

n n 1- \bigl( I I \bigr) 2 nI (37) bIJ := \chi ij 1 + \beta I \zeta IJ , \sum \zetaIJ = f\mathrm{c}(rIK ) \omega IK g(\theta IJK ) , (38) K\not =I,J 2 2 cI cI g(\theta IJK ) = 1 + 2 - 2 2 , (39) dI dI + (hI - cos(\theta IJK )) n \mathrm{m}\mathrm{i}\mathrm{n} \mathrm{m}\mathrm{a}\mathrm{x} where \theta IJK is the angle between bond IJ and IK. A, B, \lambda, \mu , \beta , n, c, d, h, rIJ , rIJ , \chi and \omega are the potential parameters that for carbon are given in Table8.

21 Table 8: The Tersoff potential parameters for carbon [33].

Parameter Value A 1.3936 \times 103 eV B 3.467 \times 102 eV \lambda 3.4879 A\r 1- \mu 2.2119 A\r 1- \beta 1.5724 \times 10 7- n 7.2751 \times 10 1- c 3.8049 \times 104 d 4.384 \times 100 h - 5.7058 \times 10 - 1 \mathrm{m}\mathrm{i}\mathrm{n} rIJ 1.8 A\r \mathrm{m}\mathrm{a}\mathrm{x} rIJ 2.1 A\r \chi 1 \omega 1

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