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The density effect of van der Waals forces on the elastic modules in graphite layers
Computational Materials Science 74 (2013) 138–142
Contents lists available at SciVerse ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
The density effect of van der Waals forces on the elastic modules in graphite layers Amir R. Golkarian, Mehrdad Jabbarzadeh ⇑ Department of Mechanical Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran
a r t i c l e
i n f o
Article history: Received 2 December 2012 Received in revised form 11 March 2013 Accepted 14 March 2013 Available online 12 April 2013 Keywords: Graphite layers Mechanical properties Van der Waals forces Non-linear spring
a b s t r a c t The influence of van der Waals forces as interlayer non-bonded interactions on the elastic modules in graphene structures based on finite element modeling is the aim of this paper. The graphite layers and interlayer interactions were simulated using fully nonlinear spring-like elements. Detailed investigations were done for four various densities of interlayer interactions about different number of layers. Results indicate that flake’s properties show different behavior with respect to the various densities of interlayer forces. It is observed that the effect of van der Waals forces depends on the chirality of layers. Increasing the number of layers just intensify the effect of van der Waals forces. The results are compared with obtained results from molecular dynamic and finite element simulations in the open literature. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction Graphene sheets are layers of carbon atoms that are arranged in hexagon forms frequently and regularly and lead to extraordinary mechanical, thermal and electrical properties. These significant properties motivate researchers to employ them as a reinforcement agent within other substances such as polymers. Consequently, prediction of mechanical properties of this category of nanostructures is a considerable task. The generation procedure of monolayer graphene sheets and single wall carbon nanotubes (SWCNTs) is a challenging task, as the most available types of these nanostructures in nature and the output of their generation procedures are mainly graphite layers and multi-walled carbon nanotubes (MWCNTs). Rafiee and Rafiee [1] tested the effects of different nanostructures such as graphene layers, SWCNTs and MWCNTs on the improvement of the mechanical properties of epoxy nano-composites by experiment. They reported superiority of graphene layers over other nanostructures. Therefore, simulation of graphite flakes has more importance than MWCNTs because, by achieving the graphite models, investigation of mechanical properties of nanocomposites reinforced by graphite layers would be available. Some experimental and molecular examinations have been performed in order to evaluate mechanical properties of graphite layers but, less continuum modeling have been presented in this case due to the complexity of modeling procedure and unknown behavior of van der Waals forces. The van der Waals forces are the agents ⇑ Corresponding author. Tel./fax: +98 5116625046. E-mail address: [email protected] (M. Jabbarzadeh). 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.03.026
of keeping graphite layers together and an atom can form them with more than one atom from other layers but it should be mentioned that these non-bonded interactions are significantly weaker than covalent bonds which are formed between two individual atoms. Li and Chou [2] simulated graphite layers by the molecular structural mechanics method. They modeled covalent bonds using linear beams and used non-linear truss rod elements in order to simulate interlayer van der Waals forces. For more simplification, they considered these non-covalent bonds only between the nearest neighboring atoms. They found negligible increase of the Young’s modules in graphite flakes by increasing the number of layers. Bao and Zhu [3] performed molecular dynamics (MD) simulation to evaluate the Young’s modules of graphene layers from one to five layers. They found negligible difference between the Young’s module of monolayer graphene sheet and graphite flakes with different number of layers. Cho and Luo [4] evaluated elastic constant of graphite nanoplatelets by molecular mechanic method. Tsai and Tu [5] characterized mechanical properties of zigzag graphite flakes using molecular dynamics simulation. They stated higher module for graphene layers than graphite flakes and suggested that using more number of graphene layers provides better reinforcement effect than graphite flakes. Behfar and Seifi [6] presented an analytical approach to estimate bending module of multi-layered graphene sheets. They derived a bending potential energy from the van der Waals interactions between the atoms in the neighboring layers of a double-layered graphene sheets. They found bending modules as a property for multi-layered graphene sheets and independent of length. In the case of MWCNTs, Kalamkarov et al. [7] simulated MWCNTs using
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A.R. Golkarian, M. Jabbarzadeh / Computational Materials Science 74 (2013) 138–142
analytical and numerical techniques. They used beam elements to simulate CNTs and employed non-linear springs in order to simulate van der Waals forces. For more convenience, they just considered the van der Waals forces between adjacent atoms in the relative distance less than 0.38 nm. They reported increasing effect of presence of van der Waals forces on the Young’s and shear modules of carbon nanotubes. Zhao and Min [8] investigated the dependency of elastic properties in graphene nanoribbons on size and chirality using tight-binding method and MD simulation. They found Young’s module of zigzag direction higher than armchair direction for the layers with same size. Georgantzinos and Giannopoulos [9] investigated mechanical properties of graphene structures using spring based finite element models. They used two groups of linear translational springs in order to simulate graphene sheets and non-linear springs for describing the van der Waals non-bonded interactions. They stated that modulus constants have decreased as flake’s thickness increased. Zhang and Wang [10] investigated the density effect of interlayer bonds on the mechanical properties in bilayer armchair graphene sheets by the molecular dynamics simulation. They reported strengthening effect on the shear module and reduction in the Young’s modules by increasing the density of interlayer bonds. As it can be seen from the literatures, different behaviors have been reported by researchers for elastic modules of graphite flakes due to the various used methods of simulations and unknown behavior of van der Waals forces. Some researchers have used MD simulation which is an expensive method with high amounts of computations and contains some limitations in length and time scales. Few studies have been carried out using structural mechanics approach which is a powerful method to simulate these nanostructures. Refs. [2,7] have used this method using beam elements to simulate covalent bonds which it needs to introduce arbitrary cross section and inertia for these elements which it would be inappropriate [11,12]. In addition, they just considered the nonbonded interactions between the nearest neighboring atoms. Ref. [9] has used spring elements to simulate covalent bonds, but they assumed linear behavior for their employed elements which is unacceptable due to the nonlinear nature of the interactions between atoms in these atomic structures [13,14]. In other hand, they considered the maximum possible amounts of van der Waals interactions, while it increases computations drastically and may not be true due to the existing discrepancies between some of their results and other reported results in the literatures. Therefore, the model used in this paper is based on three groups of fully nonlinear spring-like elements for both covalent bonds and van der Waals non-bonded interactions. On the other hand, in order to understand the effects of different possible densities of van der Waals interactions, the influences of four various densities on the Young and shear modulus of graphite flakes about one to four layered graphite flakes are investigated. The dependency of the effect of van der Waals forces on the chirality is examined, too.
2. Graphite layers description Each carbon atom in a graphene layer forms covalent bonds to three neighboring atoms in that layer. These carbon atoms are arranged in the form of hexagon cells and lead to the formation of graphene layers. The length of these covalent bonds is r0 = 0.1421 nm and the angle between each two adjacent bonds is hcc = 120° which leads to graphitic layers with thickness about t = 0.34 nm (Fig. 1). The atoms of each layer have just one 2p unhybridized orbital and can form p-bonds with the atoms in other layers and lead to the creation of graphite flakes. These weak van der Waals forces are the agent of keeping these layers together. The van der Waals forces are non-bonded interactions and have
an unknown behavior. Therefore, simulation of this category of nanostructures is a tedious task. These interactions have been simulated using Lennard-Jones (L-J) potential energy [13] which expresses each carbon atom can form non-bonded interaction with the atoms in other layers as long as the distance between them is less than 2.5r (cut-off distance) that r is the Lennard-Jones parameter and equals 0.34 nm. Whereas the separation distance between adjacent layers is about 0.34 nm [15] (Fig. 1), each carbon atom can form non-bonded interaction with about 60 atoms in adjacent layer. The real density of non-bonded interactions is unclear and different densities have been employed in the literatures. The model used in this study is a graphene sheet with the dimensions of 1.7 nm 1.6 nm with the separation distance between adjacent layers of 0.34 nm (Fig. 1) which by increasing the number of layers, same layers are repeated along the direction normal to the plane of the basic layer. 3. Molecular mechanic Based on molecular mechanic theory, the relative position of atoms in a nanostructure is regulated by a force field. The force field is expressed in the form of steric potential energy which is the sum of energies due to the bonded and non-bonded interactions between atoms. The following equation expresses these energies [16]:
V ¼ V r þ V h þ V / þ V x þ V v dW
ð1Þ
where Vr represents the bond stretching energy, Vh the bond angle bending energy, V/ the dihedral angle torsion energy, Vx the outof-plane torsion energy and VvdW the non-bonded energies of van der Waals interactions. The values of remaining energies are negligible against bond stretching and bond angle bending energies [17]. In the present model, the bond stretching and bond angle bending energies are used to simulate graphite layers. Morse potential energy [14] is used to describe these energies. Based on Morse function, the bond stretching energy is expressed as:
V r ¼ De f½1 ebðrr0 Þ 2 1g 10
ð2Þ 1
where De = 6.03105e , b = 26.26 nm , r0 = 0.1421 nm and r refers to the current distance between two atoms. In order to describe bond angle bending energy following form of Morse function is employed:
Vh ¼
1 kh ðDhÞ2 ½1 þ ksextic ðDhÞ4 2
ð3Þ
with kh = 0.9e18 N m/rad2, Dh = h h0, ho = 2.094 rad and ksextic = 0.754 rad4. The van der Waals interactions are the agent of keeping graphene layers together. These non-bonded interactions are characterized using the general Lennard-Jones potential [13] as the following equation:
12 6 r r V v dW ¼ 4e r r
ð4Þ
where r is the distance between interacting atoms and e and r are the L-J parameters which for carbon atoms the parameters are e = 0.38655 103 nN nm and r = 0.34 nm [18]. 4. Simulation method In this paper, in order to simulate graphite flakes, three types of non-linear translational spring-like elements are used (Fig. 1). Springs group ‘‘A’’ are responsible for stretching bond. The non-linear force–displacement behavior of these springs is derived from the first derivation of Eq. (2):
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Fig. 1. Finite element model.
FðDrÞ ¼ 2bDe ð1 ebDr ÞebDr
ð5Þ
where Dr is the deviation of bond length from the equilibrium distance (r0). The first derivation of Eq. (3) could be used for torsional springs, while for more convenience; non-linear translational springs which their non-linear force–displacement curve was developed in [19–21] are employed to simulate bond angles bending:
4 48 FðDRÞ ¼ 2 kh DR 1 þ 4 ksextic ðDRÞ4 r0 r0
ð6Þ
where DR ispthe ffiffiffi deviation of bond’s length from equilibrium distance (R0 ¼ 3 r0 Þ). These springs are categorized under the group ‘‘B’’ (Fig. 1). Springs group ‘‘C’’ are responsible for keeping graphene layers together as van der Waals forces (Fig. 1). The first derivation of Eq. (4) leads to the non-linear force–displacement relationship of this category of springs:
e r13 r7 FðrÞ ¼ 24 2 r r r
ð7Þ
Atomic mass of carbon atom (mc = 1.9943 1023 g) as point mass is attached to each node. The geometry of models is created in MATLAB software and the analysis is done in the commercial code ABAQUS 6.10. To define non-linear springs, CONN3D2 spring-like elements are used. 6. Numerical results and discussion The variation of Young and shear modules of graphite flakes respect to the variation of number of layers, density of van der Waals forces and chirality of layers was investigated. In the case of computing the Young modules, the atoms at the end of the graphite flakes along the length of the sheets were fully restricted (with respect to the chirality) and the tensile displacement was applied to the atoms of the free end in that direction (Fig. 2). The following equation was used to compute the Young module:
E¼
r F=wt ¼ e Dl=l
ð8Þ
it should be mentioned that in order to measure Dr, there are about 16 different initial lengths for springs group ‘‘C’’ in the graphite flakes which these differences are considered about the employed force–displacement relations in the present study.
where F is the sum of reaction forces along the length of the sheets, w is the sheets width, t is the sheets thickness, l is the initial sheet length and Dl is the applied displacement to the end of the sheets. In order to compute the shear module, shear displacement along the width of the sheet was applied to the free end of the sheets (Fig. 3) and the following equation was performed:
5. Finite element model
G¼
Four different numbers of layers which correspond to single up to fourfold layered armchair and zigzag graphite layers were examined. In each case, the effects of four different densities of van der Waals forces on the Young and shear modules of flakes were investigated. Based on L-J potential, each carbon atom can form nonbonded interactions with the atoms in other layers as long as the distance between them is less than 2.5r which is equal to 0.85 nm. Li and Chou [2] and Kalamkarov et al. [7] for more simplification, just considered the van der Waals forces between the atoms in the distance about 0.38 nm. Georgantzinos and Giannopoulos [9] considered the full distance of 0.85 nm as cut-off distance. In this paper, in order to compare the results, four different distances which contain 0.38, 0.42, 0.62 and 0.85 nm were employed as four different densities of interlayer interactions. The maximum numbers of interlayer interactions for each atom from one layer with the atoms from one another layer correspond to the cut-off distances of 0.38, 0.42, 0.62 and 0.85 nm are 4, 10, 31 and 60 interactions respectively.
where Dw is the applied shear displacement along the width of the sheet (with respect to the chirality). The variation of elastic modules for both armchair and zigzag layers versus the number of layers are illustrated in Figs. 4–7. According to the results, it can be deduced that different densities have different effects on the properties as reported by other
s F=wt ¼ c Dw=l
Fig. 2. Graphene sheet model under tensile displacement.
ð9Þ
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141
Fig. 3. Graphene sheet model under shear displacement.
Fig. 6. The variation of the Young modules of zigzag flakes versus the number of layers about different cut-off distances.
Fig. 4. The variation of the Young modules of armchair flakes versus the number of layers about different cut-off distances.
Fig. 7. The variation of the shear modules of zigzag flakes versus the number of layers about different cut-off distances.
Fig. 5. The variation of the shear modules of armchair flakes versus the number of layers about different cut-off distances.
researchers. Li and Chou [2], based on their linear model using linear beams as covalent bonds and non-linear truss rod elements as van der Waals forces, stated a negligible increase of the Young module of graphite flakes than graphene sheets by considering the value of 0.38 nm as cut-off distance. In the present study, for cut-off distance of 0.38 nm, same behavior was observed for the Young and shear modules of both armchair and zigzag graphite flakes and just about 1.5% increasing effect was observed. The effect of van der Waals forces for higher densities was completely
different. Georgantzinos and Giannopoulos [9] investigated the effect of van der Waals forces on the mechanical properties of armchair graphite flakes. They considered fully cut-off distance of 0.85 nm and employed linear spring for simulating covalent bonds and non-linear spring elements as interlayer interactions. They reported a decreasing effect due to the increasing the number of layers on the Young and shear modules. The same behavior was observed in the case of Young modules of armchair flakes (Fig. 4) but, different behavior for shear modules was observed (Fig. 5) which is in a good agreement with the reported results in [10]. Zhang and Wang [10] reported the strengthening effect on the shear modules and reduction in the Young’s modules by increasing the density of interlayer interactions. The same behavior was observed in the present FE model and decreasing effect about 7% for bilayer and 11% for fourfold layered flakes on the Young module and increasing effect about 14% for bilayer and 22% for fourfold layered flakes on the shear module were observed in the case of armchair layers (Figs. 4 and 5). According to the zigzag layers, weaker Young and shear module for the graphite flakes than graphene layers was observed (Figs. 6 and 7) that are in a good agreement with the results reported by Tsai and Tu [5]. They reported the Young
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module about 0.912 TPa and shear module about 0.358 TPa for the graphene sheets while these amounts have decreased to 0.795 TPa and 0.318 TPa in the case of graphite flakes. In the present study, the decreasing effect about 8% for bilayer and 13% for fourfold layered flakes on the Young module and 6% for bilayer and 7% for fourfold layered flakes on the shear module were observed (Figs. 6 and 7). Also, the Young’s modules about 1.0 TPa have been reported for graphene sheets via experimental works while the reported Young modulus in the case of graphite flakes have been decreased to 0.8 TPa [22–24] which prove the satisfactory performance of the present model. In regards to the results it is observed that the cut-off distance of 0.38 nm leads to increase the elastic modules of both armchair and zigzag flakes which are in a good agreement with the obtained results by literatures [2,3,7]. The cut-off distances larger than 0.38 nm lead to a decrease in the elastic modules of zigzag flakes and Young modules of armchair flakes. Different behavior is observed about the shear module of armchair flakes which by increasing the cut-off distance the shear module increases. By increasing the cut-off distance from 0.38 nm to 0.85 nm the effect of van der Waals forces increases. The results are in a good agreement with the results reported in the literatures [5,9,10]. Amongst the employed densities of van der Waals forces, the cut-off distance of 0.38 nm leads to the completely different results from the observed behaviors by experimental works and MD simulations [1,5,10]. The cut-off distance 0.42 nm, shows decreasing effect on the shear module of armchair layers which is not in agreement with the results in [10]. The obtained results from the cut-off distances 0.62 nm and 0.85 nm show good agreement with the reported results in the literatures [5,9,10]. Increasing the number of layers just intensifies the increasing or decreasing effect of van der Waals forces. For example, the Young module of tested armchair graphene sheet has decreased from 1.1463 to 1.0618 TPa for bilayer graphite flakes which by increasing the number of layers up to fourfold layered flakes, this amount has decreased to 1.0207 TPa. 7. Conclusion The density effect of van der Waals forces on the elastic modules of graphite flakes was investigated. The covalent and noncovalent bonds were simulated using fully non-linear spring-like elements. The effects of four different densities of interlayer interaction about different number of layers for both armchair and zigzag chiralities were examined and below results were obtained:
In the cut-off distance of 0.38 nm, all the elastic modules have increased which is in contrast with the reported behavior in recent literatures. Increasing the number of layers led to intensifying this behavior. In the case of 0.42 nm cut-off distance, all the elastic modules have decreased whereas the decrease of shear module of armchair flakes is in contrast with MD simulation results. Cut off distances larger than 0.42 nm led to decrease of the elastic modules of zigzag flakes and Young module of armchair flakes while the shear modules of armchair flakes have increased. Also, the increase of the number of layers just intensified the effect of van der Waals forces. Obtained results from 0.62 to 0.85 nm cut-off distances are comparable with MD results. The obtained results from the cut-off distances larger than 0.42 nm show good agreement with the reported results in the literatures.
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M.A. Rafiee, J. Rafiee, ACS Nano 3 (2009) 3884–3890. C. Li, T.-W. Chou, Int. J. Solid Struct 40 (2003) 2487–2499. W.X. Bao, C.C. Zhu, Physica B 352 (2004) 156–163. J. Cho, J.J. Luo, Compos. Sci. Technol. 67 (2007) 2399–2407. J. Tsai, J. Tu, Mater. Des. 31 (2010) 194–199. K. Behfar, P. Seifi, Thin Solid Films 496 (2006) 475–480. A.L. Kalamkarov, A.V. Georgiades, S.K. Rokkam, V.P. Veedu, M.N. GhasemiNejhad, Int. J. Solid Struct. 43 (2006) 6832–6854. H. Zhao, K. Min, Nano Lett. 9 (2009) 3012–3015. S.K. Georgantzinos, G.I. Giannopoulos, Mater. Des. 31 (2010) 4646–4654. Y. Zhang, C. Wang, Carbon 49 (2001) 4511–4517. T. Chang, H. Gao, J. Mech. Phys. Solids 51 (2003) 1059–1074. L. Nasdala, G. Ernst, Comput. Mater. Sci. 33 (2005) 443–458. J.E. Lennard-Jones, Proc. Roy. Soc. A 106 (1924) 441. K. Machida, Principles of Molecular Mechanics, Wiley ed., Wiley and Kodansha, 1999. B.T. Kelly, Physics of Graphite, Applied Science Press, London, 1981. A.K. Rappe, C.J. Casemit, J. Am. Chem. Soc. 114 (1992) 10024–10035. J.R. Xiao, B.A. Gama, Int. J. Solids Struct. 42 (2005) 3075–3092. L.A. Girifalco, R.A. Lad, J. Chem. Phys. 25 (4) (1956) 693–697. G.I. Giannopoulos, P.A. Kakavas, N.K. Anifantis, Comput. Mater. Sci. 41 (2008) 561–569. V. Parvaneh, M. Shariati, Comput. Mater. Sci. 50 (2011) 2051–2056. R. Rafiee, M. Heidarhaei, Compos. Struct. 94 (2012) 2460–2464. C. Lee, X. Wei, J.W. Kysar, J. Hone, Measurement of the elastic properties and intrinsic strength of monolayer graphene, Science 321 (5887) (2008) 385–388. C. Lee, X. Wei, J.W. Kysar, J. Hone, Science 321 (2008) 385. O.L. Blakslee, D.G. Proctor, E.J. Seldin, G.B. Spence, T. Weng, J. Appl. Phys. 41 (1970) 3373–3382.
Contents lists available at SciVerse ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
The density effect of van der Waals forces on the elastic modules in graphite layers Amir R. Golkarian, Mehrdad Jabbarzadeh ⇑ Department of Mechanical Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran
a r t i c l e
i n f o
Article history: Received 2 December 2012 Received in revised form 11 March 2013 Accepted 14 March 2013 Available online 12 April 2013 Keywords: Graphite layers Mechanical properties Van der Waals forces Non-linear spring
a b s t r a c t The influence of van der Waals forces as interlayer non-bonded interactions on the elastic modules in graphene structures based on finite element modeling is the aim of this paper. The graphite layers and interlayer interactions were simulated using fully nonlinear spring-like elements. Detailed investigations were done for four various densities of interlayer interactions about different number of layers. Results indicate that flake’s properties show different behavior with respect to the various densities of interlayer forces. It is observed that the effect of van der Waals forces depends on the chirality of layers. Increasing the number of layers just intensify the effect of van der Waals forces. The results are compared with obtained results from molecular dynamic and finite element simulations in the open literature. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction Graphene sheets are layers of carbon atoms that are arranged in hexagon forms frequently and regularly and lead to extraordinary mechanical, thermal and electrical properties. These significant properties motivate researchers to employ them as a reinforcement agent within other substances such as polymers. Consequently, prediction of mechanical properties of this category of nanostructures is a considerable task. The generation procedure of monolayer graphene sheets and single wall carbon nanotubes (SWCNTs) is a challenging task, as the most available types of these nanostructures in nature and the output of their generation procedures are mainly graphite layers and multi-walled carbon nanotubes (MWCNTs). Rafiee and Rafiee [1] tested the effects of different nanostructures such as graphene layers, SWCNTs and MWCNTs on the improvement of the mechanical properties of epoxy nano-composites by experiment. They reported superiority of graphene layers over other nanostructures. Therefore, simulation of graphite flakes has more importance than MWCNTs because, by achieving the graphite models, investigation of mechanical properties of nanocomposites reinforced by graphite layers would be available. Some experimental and molecular examinations have been performed in order to evaluate mechanical properties of graphite layers but, less continuum modeling have been presented in this case due to the complexity of modeling procedure and unknown behavior of van der Waals forces. The van der Waals forces are the agents ⇑ Corresponding author. Tel./fax: +98 5116625046. E-mail address: [email protected] (M. Jabbarzadeh). 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.03.026
of keeping graphite layers together and an atom can form them with more than one atom from other layers but it should be mentioned that these non-bonded interactions are significantly weaker than covalent bonds which are formed between two individual atoms. Li and Chou [2] simulated graphite layers by the molecular structural mechanics method. They modeled covalent bonds using linear beams and used non-linear truss rod elements in order to simulate interlayer van der Waals forces. For more simplification, they considered these non-covalent bonds only between the nearest neighboring atoms. They found negligible increase of the Young’s modules in graphite flakes by increasing the number of layers. Bao and Zhu [3] performed molecular dynamics (MD) simulation to evaluate the Young’s modules of graphene layers from one to five layers. They found negligible difference between the Young’s module of monolayer graphene sheet and graphite flakes with different number of layers. Cho and Luo [4] evaluated elastic constant of graphite nanoplatelets by molecular mechanic method. Tsai and Tu [5] characterized mechanical properties of zigzag graphite flakes using molecular dynamics simulation. They stated higher module for graphene layers than graphite flakes and suggested that using more number of graphene layers provides better reinforcement effect than graphite flakes. Behfar and Seifi [6] presented an analytical approach to estimate bending module of multi-layered graphene sheets. They derived a bending potential energy from the van der Waals interactions between the atoms in the neighboring layers of a double-layered graphene sheets. They found bending modules as a property for multi-layered graphene sheets and independent of length. In the case of MWCNTs, Kalamkarov et al. [7] simulated MWCNTs using
139
A.R. Golkarian, M. Jabbarzadeh / Computational Materials Science 74 (2013) 138–142
analytical and numerical techniques. They used beam elements to simulate CNTs and employed non-linear springs in order to simulate van der Waals forces. For more convenience, they just considered the van der Waals forces between adjacent atoms in the relative distance less than 0.38 nm. They reported increasing effect of presence of van der Waals forces on the Young’s and shear modules of carbon nanotubes. Zhao and Min [8] investigated the dependency of elastic properties in graphene nanoribbons on size and chirality using tight-binding method and MD simulation. They found Young’s module of zigzag direction higher than armchair direction for the layers with same size. Georgantzinos and Giannopoulos [9] investigated mechanical properties of graphene structures using spring based finite element models. They used two groups of linear translational springs in order to simulate graphene sheets and non-linear springs for describing the van der Waals non-bonded interactions. They stated that modulus constants have decreased as flake’s thickness increased. Zhang and Wang [10] investigated the density effect of interlayer bonds on the mechanical properties in bilayer armchair graphene sheets by the molecular dynamics simulation. They reported strengthening effect on the shear module and reduction in the Young’s modules by increasing the density of interlayer bonds. As it can be seen from the literatures, different behaviors have been reported by researchers for elastic modules of graphite flakes due to the various used methods of simulations and unknown behavior of van der Waals forces. Some researchers have used MD simulation which is an expensive method with high amounts of computations and contains some limitations in length and time scales. Few studies have been carried out using structural mechanics approach which is a powerful method to simulate these nanostructures. Refs. [2,7] have used this method using beam elements to simulate covalent bonds which it needs to introduce arbitrary cross section and inertia for these elements which it would be inappropriate [11,12]. In addition, they just considered the nonbonded interactions between the nearest neighboring atoms. Ref. [9] has used spring elements to simulate covalent bonds, but they assumed linear behavior for their employed elements which is unacceptable due to the nonlinear nature of the interactions between atoms in these atomic structures [13,14]. In other hand, they considered the maximum possible amounts of van der Waals interactions, while it increases computations drastically and may not be true due to the existing discrepancies between some of their results and other reported results in the literatures. Therefore, the model used in this paper is based on three groups of fully nonlinear spring-like elements for both covalent bonds and van der Waals non-bonded interactions. On the other hand, in order to understand the effects of different possible densities of van der Waals interactions, the influences of four various densities on the Young and shear modulus of graphite flakes about one to four layered graphite flakes are investigated. The dependency of the effect of van der Waals forces on the chirality is examined, too.
2. Graphite layers description Each carbon atom in a graphene layer forms covalent bonds to three neighboring atoms in that layer. These carbon atoms are arranged in the form of hexagon cells and lead to the formation of graphene layers. The length of these covalent bonds is r0 = 0.1421 nm and the angle between each two adjacent bonds is hcc = 120° which leads to graphitic layers with thickness about t = 0.34 nm (Fig. 1). The atoms of each layer have just one 2p unhybridized orbital and can form p-bonds with the atoms in other layers and lead to the creation of graphite flakes. These weak van der Waals forces are the agent of keeping these layers together. The van der Waals forces are non-bonded interactions and have
an unknown behavior. Therefore, simulation of this category of nanostructures is a tedious task. These interactions have been simulated using Lennard-Jones (L-J) potential energy [13] which expresses each carbon atom can form non-bonded interaction with the atoms in other layers as long as the distance between them is less than 2.5r (cut-off distance) that r is the Lennard-Jones parameter and equals 0.34 nm. Whereas the separation distance between adjacent layers is about 0.34 nm [15] (Fig. 1), each carbon atom can form non-bonded interaction with about 60 atoms in adjacent layer. The real density of non-bonded interactions is unclear and different densities have been employed in the literatures. The model used in this study is a graphene sheet with the dimensions of 1.7 nm 1.6 nm with the separation distance between adjacent layers of 0.34 nm (Fig. 1) which by increasing the number of layers, same layers are repeated along the direction normal to the plane of the basic layer. 3. Molecular mechanic Based on molecular mechanic theory, the relative position of atoms in a nanostructure is regulated by a force field. The force field is expressed in the form of steric potential energy which is the sum of energies due to the bonded and non-bonded interactions between atoms. The following equation expresses these energies [16]:
V ¼ V r þ V h þ V / þ V x þ V v dW
ð1Þ
where Vr represents the bond stretching energy, Vh the bond angle bending energy, V/ the dihedral angle torsion energy, Vx the outof-plane torsion energy and VvdW the non-bonded energies of van der Waals interactions. The values of remaining energies are negligible against bond stretching and bond angle bending energies [17]. In the present model, the bond stretching and bond angle bending energies are used to simulate graphite layers. Morse potential energy [14] is used to describe these energies. Based on Morse function, the bond stretching energy is expressed as:
V r ¼ De f½1 ebðrr0 Þ 2 1g 10
ð2Þ 1
where De = 6.03105e , b = 26.26 nm , r0 = 0.1421 nm and r refers to the current distance between two atoms. In order to describe bond angle bending energy following form of Morse function is employed:
Vh ¼
1 kh ðDhÞ2 ½1 þ ksextic ðDhÞ4 2
ð3Þ
with kh = 0.9e18 N m/rad2, Dh = h h0, ho = 2.094 rad and ksextic = 0.754 rad4. The van der Waals interactions are the agent of keeping graphene layers together. These non-bonded interactions are characterized using the general Lennard-Jones potential [13] as the following equation:
12 6 r r V v dW ¼ 4e r r
ð4Þ
where r is the distance between interacting atoms and e and r are the L-J parameters which for carbon atoms the parameters are e = 0.38655 103 nN nm and r = 0.34 nm [18]. 4. Simulation method In this paper, in order to simulate graphite flakes, three types of non-linear translational spring-like elements are used (Fig. 1). Springs group ‘‘A’’ are responsible for stretching bond. The non-linear force–displacement behavior of these springs is derived from the first derivation of Eq. (2):
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Fig. 1. Finite element model.
FðDrÞ ¼ 2bDe ð1 ebDr ÞebDr
ð5Þ
where Dr is the deviation of bond length from the equilibrium distance (r0). The first derivation of Eq. (3) could be used for torsional springs, while for more convenience; non-linear translational springs which their non-linear force–displacement curve was developed in [19–21] are employed to simulate bond angles bending:
4 48 FðDRÞ ¼ 2 kh DR 1 þ 4 ksextic ðDRÞ4 r0 r0
ð6Þ
where DR ispthe ffiffiffi deviation of bond’s length from equilibrium distance (R0 ¼ 3 r0 Þ). These springs are categorized under the group ‘‘B’’ (Fig. 1). Springs group ‘‘C’’ are responsible for keeping graphene layers together as van der Waals forces (Fig. 1). The first derivation of Eq. (4) leads to the non-linear force–displacement relationship of this category of springs:
e r13 r7 FðrÞ ¼ 24 2 r r r
ð7Þ
Atomic mass of carbon atom (mc = 1.9943 1023 g) as point mass is attached to each node. The geometry of models is created in MATLAB software and the analysis is done in the commercial code ABAQUS 6.10. To define non-linear springs, CONN3D2 spring-like elements are used. 6. Numerical results and discussion The variation of Young and shear modules of graphite flakes respect to the variation of number of layers, density of van der Waals forces and chirality of layers was investigated. In the case of computing the Young modules, the atoms at the end of the graphite flakes along the length of the sheets were fully restricted (with respect to the chirality) and the tensile displacement was applied to the atoms of the free end in that direction (Fig. 2). The following equation was used to compute the Young module:
E¼
r F=wt ¼ e Dl=l
ð8Þ
it should be mentioned that in order to measure Dr, there are about 16 different initial lengths for springs group ‘‘C’’ in the graphite flakes which these differences are considered about the employed force–displacement relations in the present study.
where F is the sum of reaction forces along the length of the sheets, w is the sheets width, t is the sheets thickness, l is the initial sheet length and Dl is the applied displacement to the end of the sheets. In order to compute the shear module, shear displacement along the width of the sheet was applied to the free end of the sheets (Fig. 3) and the following equation was performed:
5. Finite element model
G¼
Four different numbers of layers which correspond to single up to fourfold layered armchair and zigzag graphite layers were examined. In each case, the effects of four different densities of van der Waals forces on the Young and shear modules of flakes were investigated. Based on L-J potential, each carbon atom can form nonbonded interactions with the atoms in other layers as long as the distance between them is less than 2.5r which is equal to 0.85 nm. Li and Chou [2] and Kalamkarov et al. [7] for more simplification, just considered the van der Waals forces between the atoms in the distance about 0.38 nm. Georgantzinos and Giannopoulos [9] considered the full distance of 0.85 nm as cut-off distance. In this paper, in order to compare the results, four different distances which contain 0.38, 0.42, 0.62 and 0.85 nm were employed as four different densities of interlayer interactions. The maximum numbers of interlayer interactions for each atom from one layer with the atoms from one another layer correspond to the cut-off distances of 0.38, 0.42, 0.62 and 0.85 nm are 4, 10, 31 and 60 interactions respectively.
where Dw is the applied shear displacement along the width of the sheet (with respect to the chirality). The variation of elastic modules for both armchair and zigzag layers versus the number of layers are illustrated in Figs. 4–7. According to the results, it can be deduced that different densities have different effects on the properties as reported by other
s F=wt ¼ c Dw=l
Fig. 2. Graphene sheet model under tensile displacement.
ð9Þ
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141
Fig. 3. Graphene sheet model under shear displacement.
Fig. 6. The variation of the Young modules of zigzag flakes versus the number of layers about different cut-off distances.
Fig. 4. The variation of the Young modules of armchair flakes versus the number of layers about different cut-off distances.
Fig. 7. The variation of the shear modules of zigzag flakes versus the number of layers about different cut-off distances.
Fig. 5. The variation of the shear modules of armchair flakes versus the number of layers about different cut-off distances.
researchers. Li and Chou [2], based on their linear model using linear beams as covalent bonds and non-linear truss rod elements as van der Waals forces, stated a negligible increase of the Young module of graphite flakes than graphene sheets by considering the value of 0.38 nm as cut-off distance. In the present study, for cut-off distance of 0.38 nm, same behavior was observed for the Young and shear modules of both armchair and zigzag graphite flakes and just about 1.5% increasing effect was observed. The effect of van der Waals forces for higher densities was completely
different. Georgantzinos and Giannopoulos [9] investigated the effect of van der Waals forces on the mechanical properties of armchair graphite flakes. They considered fully cut-off distance of 0.85 nm and employed linear spring for simulating covalent bonds and non-linear spring elements as interlayer interactions. They reported a decreasing effect due to the increasing the number of layers on the Young and shear modules. The same behavior was observed in the case of Young modules of armchair flakes (Fig. 4) but, different behavior for shear modules was observed (Fig. 5) which is in a good agreement with the reported results in [10]. Zhang and Wang [10] reported the strengthening effect on the shear modules and reduction in the Young’s modules by increasing the density of interlayer interactions. The same behavior was observed in the present FE model and decreasing effect about 7% for bilayer and 11% for fourfold layered flakes on the Young module and increasing effect about 14% for bilayer and 22% for fourfold layered flakes on the shear module were observed in the case of armchair layers (Figs. 4 and 5). According to the zigzag layers, weaker Young and shear module for the graphite flakes than graphene layers was observed (Figs. 6 and 7) that are in a good agreement with the results reported by Tsai and Tu [5]. They reported the Young
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module about 0.912 TPa and shear module about 0.358 TPa for the graphene sheets while these amounts have decreased to 0.795 TPa and 0.318 TPa in the case of graphite flakes. In the present study, the decreasing effect about 8% for bilayer and 13% for fourfold layered flakes on the Young module and 6% for bilayer and 7% for fourfold layered flakes on the shear module were observed (Figs. 6 and 7). Also, the Young’s modules about 1.0 TPa have been reported for graphene sheets via experimental works while the reported Young modulus in the case of graphite flakes have been decreased to 0.8 TPa [22–24] which prove the satisfactory performance of the present model. In regards to the results it is observed that the cut-off distance of 0.38 nm leads to increase the elastic modules of both armchair and zigzag flakes which are in a good agreement with the obtained results by literatures [2,3,7]. The cut-off distances larger than 0.38 nm lead to a decrease in the elastic modules of zigzag flakes and Young modules of armchair flakes. Different behavior is observed about the shear module of armchair flakes which by increasing the cut-off distance the shear module increases. By increasing the cut-off distance from 0.38 nm to 0.85 nm the effect of van der Waals forces increases. The results are in a good agreement with the results reported in the literatures [5,9,10]. Amongst the employed densities of van der Waals forces, the cut-off distance of 0.38 nm leads to the completely different results from the observed behaviors by experimental works and MD simulations [1,5,10]. The cut-off distance 0.42 nm, shows decreasing effect on the shear module of armchair layers which is not in agreement with the results in [10]. The obtained results from the cut-off distances 0.62 nm and 0.85 nm show good agreement with the reported results in the literatures [5,9,10]. Increasing the number of layers just intensifies the increasing or decreasing effect of van der Waals forces. For example, the Young module of tested armchair graphene sheet has decreased from 1.1463 to 1.0618 TPa for bilayer graphite flakes which by increasing the number of layers up to fourfold layered flakes, this amount has decreased to 1.0207 TPa. 7. Conclusion The density effect of van der Waals forces on the elastic modules of graphite flakes was investigated. The covalent and noncovalent bonds were simulated using fully non-linear spring-like elements. The effects of four different densities of interlayer interaction about different number of layers for both armchair and zigzag chiralities were examined and below results were obtained:
In the cut-off distance of 0.38 nm, all the elastic modules have increased which is in contrast with the reported behavior in recent literatures. Increasing the number of layers led to intensifying this behavior. In the case of 0.42 nm cut-off distance, all the elastic modules have decreased whereas the decrease of shear module of armchair flakes is in contrast with MD simulation results. Cut off distances larger than 0.42 nm led to decrease of the elastic modules of zigzag flakes and Young module of armchair flakes while the shear modules of armchair flakes have increased. Also, the increase of the number of layers just intensified the effect of van der Waals forces. Obtained results from 0.62 to 0.85 nm cut-off distances are comparable with MD results. The obtained results from the cut-off distances larger than 0.42 nm show good agreement with the reported results in the literatures.
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