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Atomistic simulation study of mechanical properties of periodic graphene nanobuds
Computational Materials Science 107 (2015) 163–169
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Atomistic simulation study of mechanical properties of periodic graphene nanobuds A. Fereidoon a, M. khorasani b, M. Darvish Ganji c,⇑, F. Memarian a,⇑ a
Department of Mechanical Engineering, Semnan University, Semnan, Iran Department of Mechanical Engineering, Semnan Branch, Islamic Azad University, Semnan, Iran c Young Researchers and Elite club, Central Tehran Branch, Islamic Azad University, Tehran, Iran b
a r t i c l e
i n f o
Article history: Received 14 January 2015 Received in revised form 27 April 2015 Accepted 8 May 2015
Keywords: Graphene Graphene nanobuds Mechanical properties Young’s modulus Molecular dynamic simulations
a b s t r a c t Among the graphene-based hybrid nanostructures, graphene nanobuds (GNBs); a hybrid of graphene/ fullerene architecture, are one of the most interesting nanostructured materials. In this study we have investigated the mechanical properties of graphene nanobud through molecular dynamic simulations. The effects of temperature, size of graphene sheet and also neck’s length on the mechanical properties of this novel material were investigated, for the first time. The calculated Young’s modulus ranges from 440 to 760 GPa and the failure stress changes from 100 to 200 GPa. The results show that the Young’s modulus changes a little with temperature increment while, with increasing the neck’s length the Young’s modulus decreases. We found that with both temperature and neck’s length increment, the failure stress decreases. Furthermore, when the size of graphene sheet was increased all the respected mechanical properties were increased too. These findings will augment the current understanding of the mechanical performance of GNBs, which will conduct the design of GNBs and shed lights on its various applications. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction Graphene is an allotrope of carbon in the form of a two-dimensional, atomic-scale, hexagonal lattice which consists of covalently bonded carbon atoms. Its intrinsic strength, predicted to exceed that of any other material. Many studies have shown that graphene Young’s modulus is around 1 TPa and Its fracture strength is above 100 GPa [1–9]. Also graphene is the basic structural element of other allotropes, including graphite, charcoal, carbon nanotubes and fullerenes. Fullerene C60, which is discovered in 1985 [10], is a molecule composed entirely of 60 sp2-hybridized carbon atoms which forms a hollow sphere with twenty hexagonal and twelve pentagonal rings [11]. Because of its exceptional structure, fullerene has the ability of attaching to different atoms or small molecules for modification or getting better properties [12]. Although fullerene has remarkable properties such as super conductivity, ferromagnetism, mechanical strength and thermal stability, because of its weak accessibility these properties are still unknown [13]. Two types of C–C bonds occurs in fullerene: one ⇑ Corresponding authors. Tel./fax: +98 1132320342 (M.D. Ganji). Tel./fax: +98 23 33383331 (F. Memarian). E-mail addresses: [email protected] (M.D. Ganji), [email protected] (F. Memarian). http://dx.doi.org/10.1016/j.commatsci.2015.05.004 0927-0256/Ó 2015 Elsevier B.V. All rights reserved.
type between hexagonal and pentagonal rings and the other between two hexagonal rings (Fig. 1(a)) [14]. The discoveries of these carbon nanostructures have played critical roles in the advancement of modern nanoscience and nanotechnology [15], due to their supernatural electrical [16], mechanical [5] and thermal [17] properties. Over the past decade, many efforts have been done to synthesize hybrid of carbon nanostructures with novel properties. The first hybrid nanostructure which is experimentally fabricated was carbon nanopeapod [18], in which carbon nanotube encapsulating fullerenes. Carbon nanobud (CNB) is another hybrid nanostructure which was constructed successfully in 2007. In this structure one or two C60 molecules covalently attached to the sidewall of single-walled carbon nanotubes (SWCNTs) [19]. It is expected that chemical interactions between C60 and SWCNT modified CNBs properties. In addition presence of C60 molecules weakening the tendency of adhesion among SWCNTs and prevent slipping of SWCNTs in composite materials. So it is predicted that CNBs with mentioned operation can increase mechanical properties of composite materials [11]. Another hybrid nanostructure which is predicted theoretically, is obtained from fullerene and graphene sheet. The combination of these two novel structures, will develop a 3D carbon-based network, which is named graphene nanobuds (GNBs) [20]. GNBs can reinforce composite materials similar to CNBs and it is forecasted
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(b) (a)
Deformation direction
(c)
Fig. 1. (a) Two types of C–C bonds occurs in fullerene, (b) octagonal and hexagonal rings at junction region, and (c) GNB under tension load along armchair direction.
that mechanical properties improve significantly in the presence of this unique nanostructure. Different properties of GNBs were investigated in recent years. Wang et al. [21] studied the magnetic properties of GNBs using spin-polarized density functional theory. Furthermore by using GCMC simulations the mechanism of argon adsorption onto GNBs were investigated [22]. Also structural and electronic properties of two prototype periodic GNBs by means of the first-principles calculations was considered by Wu et al. [20]. In GNBs, fullerenes are attached to a graphene monolayer. These junctions can formed differently: in first type C60 molecules are covalently bonded to a graphene monolayer [20] whereas in second type fragmented C60 are fused onto a defective graphene monolayer [20]. In both cases, the C60 molecules form a periodic lattice on the graphene monolayer. In this study we use the second type with different length of neck attached to defective graphene sheet. Effects of these junctions in electronic structures and thermal transport [23,24] have been studied, but how the mechanical properties can be affected by junctions is a novel field of interest that is investigated in this study. GNBs can reinforce composite materials similar to CNBs and it is forecasted that mechanical properties improve significantly in the presence of this unique nanostructure. Temperature is an important factor for producing new hybrid nanostructure and usually these structures are fabricated in high temperature [25]. Also carbon-based materials are useful in many electronic devices working in high temperature such as CPUs [26], transistors [27], diodes [28] and capacitors [29]. Furthermore from reported studies we believe that the elastic properties of single layer graphene sheet, such as Young’s moduli is temperature-dependent [30]. So elastic properties of the combination of graphene sheet and fullerene is predicted to be temperature dependent. Therefore understanding the behavior of GNBs in different temperatures is necessary for fabricating high quality
graphene based devices. Because of above mentioned reasons, in this work a series of GNBs are investigated in eight temperatures in order to consider effect of temperature on mechanical properties of GNBs entirely. Furthermore, additional attention is paid to the role of junctions, necks length and sheet size in the 3D network to gain further conception insight into the different mechanical properties. The revealed understanding of junction’s role at molecular-level can help control the performance of these structures [31]. 2. Atomistic modeling of graphene nanobud The powerful Large-scale Atomic/Molecular Massively Parallel Simulator (LAAMPS) code and Visual Molecular Dynamics (VMD) visualizer were performed for studying mechanical properties of GNBs. VMD software was used for animating and coloring molecules which supports more than 60 molecular file formats including LAMMPS output files [32]. The adaptive intermolecular reactive empirical bond order (AIREBO) potential was employed as a force field in LAMMPS package [33]. The AIREBO potential which was originated from Tersoff–Brenner potential allows the intermolecular and torsional interactions [34–36] in addition to the simulation of bond breaking and formation. So this potential can easily shows failure process in nanostructured systems. The AIREBO potential can be represented by a sum over pairwise interactions, including covalent bonding REBO interactions EREBO , LJ terms ELJ ij ij , and TORS [30]. torsion interactions Ekijl The full expressions of energy items are given in Eq. (1) [37].
" # X X TORS 1 XX REBO LJ E¼ E þ Eij þ Ekijl 2 i j–i ij k–i;jl–i;j;k
ð1Þ
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Also in order to maintain the temperature of the system the Nose–Hoover thermostat is used which provides good conversion of energy and reduce fluctuation in temperature [38]. Cut-off radius of AIREBO potential is set to be the original value 2 Å to eliminate the non-physical strain hardening of the stress– strain curve [39]. For this study the periodic boundary condition was applied and for reaching equilibration the isothermal isobaric ensemble (NPT) was performed. The pressure was fixed to be 1 bar and temperature varies from 1 K to 800 K for all simulation models. The non-equilibrium simulation mode was applied to allow GNB edges to move along the stretching direction at a constant strain rate (Fig. 1(c)). Time step in open literatures ranges from 0.1 to 1 fs for tensile test of graphene [40,41]. Researchers found that the failure point of graphene depends on the strain rate [41–43]. At lower strain rates, the system has more time to relax and reach equilibrium state, and hence the results would be more accurate. The used strain rate value varies between 0.0005 and 0.01 ps1 [41]. In the present study, strain rate and time step were assumed to be 0.01 ps1 and 1 fs, respectively. Before loading, the initial structure is relaxed for 20 ps to ensure that the system is in equilibrium especially because of applying different temperatures. To fully understand the effect of attaching fullerene on the mechanical properties of graphene sheet, we considered seven types of configurations in which length of neck in junction between fullerene and graphene sheet and size of graphene sheet, as illustrated in Fig. 2. The number of atoms for each system are listed in Table 1. For analyzing the mechanical behavior of GNBs under tensile loading, the stress–strain relations should be obtained. In order to calculate the stress–strain evolution during deformation, the per-atom stress tensor for each individual carbon atom is first calculated using Eq. (2) [44].
raij ¼
1 Va
1 a a a 1 X ab ab m vi vj þ f r 2 2 b¼1;n i j
! ð2Þ
where i and j are Cartesian indices and a and b are atomic indices. V, m, v, f and r are respectively the representative volume under stress, mass, velocity, force and distance between two atoms. By averaging the atomic stresses over all the atoms the total stress is then computed [44].
(a)
(e)
(b)
(f)
Table 1 Number of atoms for various GNB systems. Dimension (in Å)
23 23 28 28 36 36
# Neck 0
1
5
10
302 368 602
314 410 614
362 458 662
434 530 734
A nonlinear elastic response was normally suggested for graphene sheets and nanotubes according to open literature [5]. This non-linear stress–strain curve of pristine graphene is associated with the an-harmonic terms in the potential energy description [45]. Also numerical simulations confirmed that nonlinear elastic response is appropriate for these structures [3,46]. Since the structures of GNBs are similar to graphene sheets the same elastic behavior was predicted for GNBs. So for calculating the Young’s modulus the following equation was considered:
r¼
@U ¼ De2 þ Ee þ C @e
ð3Þ
where D is the third order elastic modulus, E is the Young’s modulus, and C is residual stress in GNBs. The values of failure stress, rF, and failure strain, eF, were defined at the point where the stress is reached to its maximum value and then spontaneous drop in the stress is observed. 3. Results and discussion At first, in order to validate the inter-atomic potential, the numerical approach and accuracy of calculations, mechanical properties for pristine graphene sheet is first determined, for which various experimental and theoretical reports in the open literatures are available [5]. A tensile load was applied on a square graphene sheet of 41.5 Å 41.5 Å, in armchair direction. The Young’s modulus was obtained by second order polynomial fit of the first section of stress–strain curve as shown in Fig. 3. The calculated Young’s modulus is 1.01 TPa, which agree well with the previous experimental report [5]. Also it is consistent with the nanoindentation measurements of 1.0 TPa [47] and ab initio predictions of 1.05 TPa [3]. In addition the obtained failure stress is 0.29 TPa with associated fracture strain of 0.44, while
(c)
(d)
(g)
Fig. 2. Different neck length and sheet size: (a) Neck-0, (b) Neck-1, (c) Neck-5, (d) Neck-10, (e) 23 23, (f) 28 28, and (g) 36 36.
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Fig. 3. (a) Stress–strain relations of GNBs under uniaxial tensile tests at 300 K, and (b) the second order polynomial section.
experimental [5] reports are 0.13 ± 0.1 TPa and 0.25. It can be deduced that our results overestimate the experimental one. The idealism of the computer simulation model compared with the real experimental samples may cause overestimating these values [48]. For example there are some defects in the real specimens but the simulated model is completely intact. It is also believed that defects generally degrade in-plane mechanical strength [49]. So the validity of the potential used and the simulation model is confirmed by obtained results. 3.1. Young’s modulus and stress strain evolution The armchair GNB is axially loaded at 300 K and the obtained stress–strain evolution is shown in Fig. 1(c). As it has been pointed out previously the Young’s modulus is obtained by second order polynomial fit of first section of stress–strain curve. The Young’s modulus of zero neck GNB with 25 25 Å graphene sheet is determined 623 GPa which is significantly lower than graphene one. As it was obvious from the result, the pristine graphene sheet exhibits
larger slope for stress–strain relations at lower strain than GNB. The lower slopes for GNBs could be explained as follows. After the structural optimization at the equilibrium regime, the covalent bonding between C60 molecules and graphene cause a distortion of the graphene surface. In this region the C60 molecule drags the carbon atoms of graphene near to the junction outward from their original flat surface. So their bonding is transformed from in-plane sp2-hybridization to off-plane sp3-hybridization. These off-plane sp3 bonds are easily rotate because these are not firmly constrained. Furthermore, as it was obvious from Fig. 1(b) at the junction region octagonal rings formed which have larger area than normal hexagonal rings and can be represented as a rupture point. When the junction is formed, the ring plane is bended and resulting in non-parallel p bond alignment, asymmetrical stress distribution and also cause dramatic angle distortion [45]. This angle distortion by changing bonds’ plane and hybridization can weaken the system significantly. In addition, destabilized nature of these bonds causes easier and earlier fracture of this hybrid structure which significantly reduce critical strength.
Fig. 4. Variation of Young’s modulus, failure stress and failure strain of GNBs at different temperatures.
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A. Fereidoon et al. / Computational Materials Science 107 (2015) 163–169 Table 2 Young’s modulus for 23 23 (Å) GNB at various temperatures.
Neck-0 Neck-1 Neck-5 Neck-10
1K
100 K
200 K
300 K
400 K
500 K
600 K
700 K
800 K
636.61 612.13 537.99 454.85
630.93 605.13 533.80 442.47
621.32 592.97 535.33 441.43
623.00 592.64 525.68 441.77
610.64 583.88 522.33 439.63
629.38 566.35 527.64 451.8
629.3 571.60 495.93 456.35
616.2 582.45 515.13 440.71
626.49 575.84 507.61 436.33
Table 3 Young’s modulus for 28 28 (Å) GNB at various temperatures.
Neck-0 Neck-1 Neck-5 Neck-10
1K
100 K
200 K
300 K
400 K
500 K
600 K
700 K
800 K
706.25 685.92 617.19 538.77
694.33 686.67 608.61 530.71
687.55 678.13 610.09 534.83
695.18 655.85 608.37 519.40
663.58 650.30 588.63 523.41
668.39 684.16 595.34 519.95
678.14 653.91 583.85 518.70
665.16 647.29 578.39 484.57
689.84 627.23 594.51 490.80
Table 4 Young’s modulus for 36 36 (Å) GNB at various temperatures.
Neck-0 Neck-1 Neck-5 Neck-10
1K
100 K
200 K
300 K
400 K
500 K
600 K
700 K
800 K
784.80 768.49 716.68 651.24
775.57 751.59 711.60 645.15
778.91 745.28 693.81 637.30
763.62 743.02 685.91 627.21
747.28 738.58 672.94 624.78
753.14 742.07 673.74 626.63
751.36 735.64 684.03 610.47
760.30 726.07 700.15 608.26
745.40 722.24 669.41 607.44
(a)
(c)
(b)
(d)
Fig. 5. Snapshots of failure process for neck-5 and 28 28 (Å) graphene sheet.
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Another important point in the stress–strain evolution, is the changing point where parabolic section alter to linear line part. At the beginning of tensile loading, sp2-hybridized bonds in graphene sheet are elongated gradually. With further loading, strain increased and junction region with its octagonal rings started to be stretched. In this stage when stress accumulated in junction region, bond breakage is observed to initiate at this place. So the stress–strain manner change at this level and stress increase dramatically. Therefore junction deteriorates the bonding and weaken the sheet, so obviously it is responsible for early fracture of GNBs, as discussed above. 3.2. Temperature dependence In order to investigate the effect of temperature on mechanical properties of GNBs, GNBs with different graphene size and neck length, subjected to uniaxial tension are simulated under temperature varying from 1 to 800 K. The manner of Young’s modulus, failure stress and strain against temperature for all GNB models are shown in Fig. 4 and the calculated results are listed in Tables 2–4. As shown in Fig. 4 in each graphene the Young’s modulus varies a little with temperature increment. With an increasing temperature, the axial Young moduli trend to decrease linearly. It is also seen from Fig. 4 that the effect of temperature increment on Young’s modulus of GNB with larger graphene sheet is stronger than that of with smaller sheet. The maximum reduction of Young’s modulus when the temperature ranges from 1 to 800 K for 23 23, 28 28 and 36 36 (Å) graphene sheet is about 1.6%, 2.3% and 5.02% respectively. Since size of junction region in all models is the same, therefore the number of off-plane sp3 bonds and octagonal rings are equal. So destabilized nature of these bonds is less effective in GNB with the largest graphene sheet. It means that larger base GNBs have larger Young’s modulus, failure stress and strain, so all in all the largest base GNBs is the stiffest one against fracture. Also it can be deduced from Fig. 4 that neck’s length increment cause significant reduction in Young’s modulus. Actually when height of neck increase, the structure become more unstable which make a meaningful reduction in all mechanical properties. Furthermore, Fig. 4 shows that in some cases the failure strian is constant in various temperatures. The reason is that the value which we considered for strain rate is 0.01 ps1 and maybe their difference in failure strain is less than 0.01 so it cannot be observed in graph while decreasing trend is completely obvious. Also, we found that with temperature increment the fracture happens in smaller strain. It is observed that bond length of carbon–carbon bond and the length of GNBs in the x direction change remarkably prior to complete failure. The length of GNB before bond breaking in the armchair direction increases 38.31%. To understand the failure process entirely, in Fig. 5, some representative snapshots of the GNB under armchair tensile deformation from the onset to the failure of the structures is presented. As it has been pointed out before, the junction obviously deteriorates the bonding and thus weakens the graphene sheet, which cause bond breakage in this region. Fig. 5 shows this procedure completely. At first stage just one carbon–carbon bond breaks which makes a defect in structure of GNB then defect grows rapidly and the system fails completely. 4. Conclusions In this paper we evaluated the mechanical properties of graphene nanobuds based on molecular dynamic simulations. We used Large-scale Atomic/Molecular Massively Parallel Simulator
code LAMMPS to investigate the effect of temperature, size of graphene sheet and neck’s length on mechanical properties of GNBs. The Young’s modulus was obtained from second order polynomial fit to first section of stress strain evolution. This diagram can also shows failure stress and strain. Three sizes of graphene sheet and four different neck’s length was considered. For validity, we compared our results with graphene ones. The calculated Young’s modulus of graphene is 1.01 TPa which is in good agreement with experimental value of 1.0 ± 0.1 TPa. Comparing Young’s modulus of GNBs at 300 K with the same size graphene shows that Young’s modulus of GNB is significantly less than graphene. This phenomenon occurs due to existence of nano-bud which makes defects in graphene lattice. The calculated Young’s modulus at 300 K for different models range from 440 to 763 GPa. We found that Young’s modulus decreases smoothly with temperature increment. Additionally with neck’s length and graphene size increment the Young’s modulus decreases and increases, respectively. So, for all graphene sheet size the neck-0 has larger Young’s modulus value and hence is mechanically stronger than the other counterparts. This behavior is observed for failure stress and strain too, while dependence of these two parameters on temperature is more than the Young’s modulus. It is noticeable that in higher temperature the failure strain is in lower range due to higher kinetic energy of atoms. Our MD simulation results provide a vital exploration of the mechanical properties of the graphene nanobuds structures, which will aid the design and also the applications of graphene-based hybrid nano-materials.
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169
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Atomistic simulation study of mechanical properties of periodic graphene nanobuds A. Fereidoon a, M. khorasani b, M. Darvish Ganji c,⇑, F. Memarian a,⇑ a
Department of Mechanical Engineering, Semnan University, Semnan, Iran Department of Mechanical Engineering, Semnan Branch, Islamic Azad University, Semnan, Iran c Young Researchers and Elite club, Central Tehran Branch, Islamic Azad University, Tehran, Iran b
a r t i c l e
i n f o
Article history: Received 14 January 2015 Received in revised form 27 April 2015 Accepted 8 May 2015
Keywords: Graphene Graphene nanobuds Mechanical properties Young’s modulus Molecular dynamic simulations
a b s t r a c t Among the graphene-based hybrid nanostructures, graphene nanobuds (GNBs); a hybrid of graphene/ fullerene architecture, are one of the most interesting nanostructured materials. In this study we have investigated the mechanical properties of graphene nanobud through molecular dynamic simulations. The effects of temperature, size of graphene sheet and also neck’s length on the mechanical properties of this novel material were investigated, for the first time. The calculated Young’s modulus ranges from 440 to 760 GPa and the failure stress changes from 100 to 200 GPa. The results show that the Young’s modulus changes a little with temperature increment while, with increasing the neck’s length the Young’s modulus decreases. We found that with both temperature and neck’s length increment, the failure stress decreases. Furthermore, when the size of graphene sheet was increased all the respected mechanical properties were increased too. These findings will augment the current understanding of the mechanical performance of GNBs, which will conduct the design of GNBs and shed lights on its various applications. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction Graphene is an allotrope of carbon in the form of a two-dimensional, atomic-scale, hexagonal lattice which consists of covalently bonded carbon atoms. Its intrinsic strength, predicted to exceed that of any other material. Many studies have shown that graphene Young’s modulus is around 1 TPa and Its fracture strength is above 100 GPa [1–9]. Also graphene is the basic structural element of other allotropes, including graphite, charcoal, carbon nanotubes and fullerenes. Fullerene C60, which is discovered in 1985 [10], is a molecule composed entirely of 60 sp2-hybridized carbon atoms which forms a hollow sphere with twenty hexagonal and twelve pentagonal rings [11]. Because of its exceptional structure, fullerene has the ability of attaching to different atoms or small molecules for modification or getting better properties [12]. Although fullerene has remarkable properties such as super conductivity, ferromagnetism, mechanical strength and thermal stability, because of its weak accessibility these properties are still unknown [13]. Two types of C–C bonds occurs in fullerene: one ⇑ Corresponding authors. Tel./fax: +98 1132320342 (M.D. Ganji). Tel./fax: +98 23 33383331 (F. Memarian). E-mail addresses: [email protected] (M.D. Ganji), [email protected] (F. Memarian). http://dx.doi.org/10.1016/j.commatsci.2015.05.004 0927-0256/Ó 2015 Elsevier B.V. All rights reserved.
type between hexagonal and pentagonal rings and the other between two hexagonal rings (Fig. 1(a)) [14]. The discoveries of these carbon nanostructures have played critical roles in the advancement of modern nanoscience and nanotechnology [15], due to their supernatural electrical [16], mechanical [5] and thermal [17] properties. Over the past decade, many efforts have been done to synthesize hybrid of carbon nanostructures with novel properties. The first hybrid nanostructure which is experimentally fabricated was carbon nanopeapod [18], in which carbon nanotube encapsulating fullerenes. Carbon nanobud (CNB) is another hybrid nanostructure which was constructed successfully in 2007. In this structure one or two C60 molecules covalently attached to the sidewall of single-walled carbon nanotubes (SWCNTs) [19]. It is expected that chemical interactions between C60 and SWCNT modified CNBs properties. In addition presence of C60 molecules weakening the tendency of adhesion among SWCNTs and prevent slipping of SWCNTs in composite materials. So it is predicted that CNBs with mentioned operation can increase mechanical properties of composite materials [11]. Another hybrid nanostructure which is predicted theoretically, is obtained from fullerene and graphene sheet. The combination of these two novel structures, will develop a 3D carbon-based network, which is named graphene nanobuds (GNBs) [20]. GNBs can reinforce composite materials similar to CNBs and it is forecasted
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A. Fereidoon et al. / Computational Materials Science 107 (2015) 163–169
(b) (a)
Deformation direction
(c)
Fig. 1. (a) Two types of C–C bonds occurs in fullerene, (b) octagonal and hexagonal rings at junction region, and (c) GNB under tension load along armchair direction.
that mechanical properties improve significantly in the presence of this unique nanostructure. Different properties of GNBs were investigated in recent years. Wang et al. [21] studied the magnetic properties of GNBs using spin-polarized density functional theory. Furthermore by using GCMC simulations the mechanism of argon adsorption onto GNBs were investigated [22]. Also structural and electronic properties of two prototype periodic GNBs by means of the first-principles calculations was considered by Wu et al. [20]. In GNBs, fullerenes are attached to a graphene monolayer. These junctions can formed differently: in first type C60 molecules are covalently bonded to a graphene monolayer [20] whereas in second type fragmented C60 are fused onto a defective graphene monolayer [20]. In both cases, the C60 molecules form a periodic lattice on the graphene monolayer. In this study we use the second type with different length of neck attached to defective graphene sheet. Effects of these junctions in electronic structures and thermal transport [23,24] have been studied, but how the mechanical properties can be affected by junctions is a novel field of interest that is investigated in this study. GNBs can reinforce composite materials similar to CNBs and it is forecasted that mechanical properties improve significantly in the presence of this unique nanostructure. Temperature is an important factor for producing new hybrid nanostructure and usually these structures are fabricated in high temperature [25]. Also carbon-based materials are useful in many electronic devices working in high temperature such as CPUs [26], transistors [27], diodes [28] and capacitors [29]. Furthermore from reported studies we believe that the elastic properties of single layer graphene sheet, such as Young’s moduli is temperature-dependent [30]. So elastic properties of the combination of graphene sheet and fullerene is predicted to be temperature dependent. Therefore understanding the behavior of GNBs in different temperatures is necessary for fabricating high quality
graphene based devices. Because of above mentioned reasons, in this work a series of GNBs are investigated in eight temperatures in order to consider effect of temperature on mechanical properties of GNBs entirely. Furthermore, additional attention is paid to the role of junctions, necks length and sheet size in the 3D network to gain further conception insight into the different mechanical properties. The revealed understanding of junction’s role at molecular-level can help control the performance of these structures [31]. 2. Atomistic modeling of graphene nanobud The powerful Large-scale Atomic/Molecular Massively Parallel Simulator (LAAMPS) code and Visual Molecular Dynamics (VMD) visualizer were performed for studying mechanical properties of GNBs. VMD software was used for animating and coloring molecules which supports more than 60 molecular file formats including LAMMPS output files [32]. The adaptive intermolecular reactive empirical bond order (AIREBO) potential was employed as a force field in LAMMPS package [33]. The AIREBO potential which was originated from Tersoff–Brenner potential allows the intermolecular and torsional interactions [34–36] in addition to the simulation of bond breaking and formation. So this potential can easily shows failure process in nanostructured systems. The AIREBO potential can be represented by a sum over pairwise interactions, including covalent bonding REBO interactions EREBO , LJ terms ELJ ij ij , and TORS [30]. torsion interactions Ekijl The full expressions of energy items are given in Eq. (1) [37].
" # X X TORS 1 XX REBO LJ E¼ E þ Eij þ Ekijl 2 i j–i ij k–i;jl–i;j;k
ð1Þ
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Also in order to maintain the temperature of the system the Nose–Hoover thermostat is used which provides good conversion of energy and reduce fluctuation in temperature [38]. Cut-off radius of AIREBO potential is set to be the original value 2 Å to eliminate the non-physical strain hardening of the stress– strain curve [39]. For this study the periodic boundary condition was applied and for reaching equilibration the isothermal isobaric ensemble (NPT) was performed. The pressure was fixed to be 1 bar and temperature varies from 1 K to 800 K for all simulation models. The non-equilibrium simulation mode was applied to allow GNB edges to move along the stretching direction at a constant strain rate (Fig. 1(c)). Time step in open literatures ranges from 0.1 to 1 fs for tensile test of graphene [40,41]. Researchers found that the failure point of graphene depends on the strain rate [41–43]. At lower strain rates, the system has more time to relax and reach equilibrium state, and hence the results would be more accurate. The used strain rate value varies between 0.0005 and 0.01 ps1 [41]. In the present study, strain rate and time step were assumed to be 0.01 ps1 and 1 fs, respectively. Before loading, the initial structure is relaxed for 20 ps to ensure that the system is in equilibrium especially because of applying different temperatures. To fully understand the effect of attaching fullerene on the mechanical properties of graphene sheet, we considered seven types of configurations in which length of neck in junction between fullerene and graphene sheet and size of graphene sheet, as illustrated in Fig. 2. The number of atoms for each system are listed in Table 1. For analyzing the mechanical behavior of GNBs under tensile loading, the stress–strain relations should be obtained. In order to calculate the stress–strain evolution during deformation, the per-atom stress tensor for each individual carbon atom is first calculated using Eq. (2) [44].
raij ¼
1 Va
1 a a a 1 X ab ab m vi vj þ f r 2 2 b¼1;n i j
! ð2Þ
where i and j are Cartesian indices and a and b are atomic indices. V, m, v, f and r are respectively the representative volume under stress, mass, velocity, force and distance between two atoms. By averaging the atomic stresses over all the atoms the total stress is then computed [44].
(a)
(e)
(b)
(f)
Table 1 Number of atoms for various GNB systems. Dimension (in Å)
23 23 28 28 36 36
# Neck 0
1
5
10
302 368 602
314 410 614
362 458 662
434 530 734
A nonlinear elastic response was normally suggested for graphene sheets and nanotubes according to open literature [5]. This non-linear stress–strain curve of pristine graphene is associated with the an-harmonic terms in the potential energy description [45]. Also numerical simulations confirmed that nonlinear elastic response is appropriate for these structures [3,46]. Since the structures of GNBs are similar to graphene sheets the same elastic behavior was predicted for GNBs. So for calculating the Young’s modulus the following equation was considered:
r¼
@U ¼ De2 þ Ee þ C @e
ð3Þ
where D is the third order elastic modulus, E is the Young’s modulus, and C is residual stress in GNBs. The values of failure stress, rF, and failure strain, eF, were defined at the point where the stress is reached to its maximum value and then spontaneous drop in the stress is observed. 3. Results and discussion At first, in order to validate the inter-atomic potential, the numerical approach and accuracy of calculations, mechanical properties for pristine graphene sheet is first determined, for which various experimental and theoretical reports in the open literatures are available [5]. A tensile load was applied on a square graphene sheet of 41.5 Å 41.5 Å, in armchair direction. The Young’s modulus was obtained by second order polynomial fit of the first section of stress–strain curve as shown in Fig. 3. The calculated Young’s modulus is 1.01 TPa, which agree well with the previous experimental report [5]. Also it is consistent with the nanoindentation measurements of 1.0 TPa [47] and ab initio predictions of 1.05 TPa [3]. In addition the obtained failure stress is 0.29 TPa with associated fracture strain of 0.44, while
(c)
(d)
(g)
Fig. 2. Different neck length and sheet size: (a) Neck-0, (b) Neck-1, (c) Neck-5, (d) Neck-10, (e) 23 23, (f) 28 28, and (g) 36 36.
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Fig. 3. (a) Stress–strain relations of GNBs under uniaxial tensile tests at 300 K, and (b) the second order polynomial section.
experimental [5] reports are 0.13 ± 0.1 TPa and 0.25. It can be deduced that our results overestimate the experimental one. The idealism of the computer simulation model compared with the real experimental samples may cause overestimating these values [48]. For example there are some defects in the real specimens but the simulated model is completely intact. It is also believed that defects generally degrade in-plane mechanical strength [49]. So the validity of the potential used and the simulation model is confirmed by obtained results. 3.1. Young’s modulus and stress strain evolution The armchair GNB is axially loaded at 300 K and the obtained stress–strain evolution is shown in Fig. 1(c). As it has been pointed out previously the Young’s modulus is obtained by second order polynomial fit of first section of stress–strain curve. The Young’s modulus of zero neck GNB with 25 25 Å graphene sheet is determined 623 GPa which is significantly lower than graphene one. As it was obvious from the result, the pristine graphene sheet exhibits
larger slope for stress–strain relations at lower strain than GNB. The lower slopes for GNBs could be explained as follows. After the structural optimization at the equilibrium regime, the covalent bonding between C60 molecules and graphene cause a distortion of the graphene surface. In this region the C60 molecule drags the carbon atoms of graphene near to the junction outward from their original flat surface. So their bonding is transformed from in-plane sp2-hybridization to off-plane sp3-hybridization. These off-plane sp3 bonds are easily rotate because these are not firmly constrained. Furthermore, as it was obvious from Fig. 1(b) at the junction region octagonal rings formed which have larger area than normal hexagonal rings and can be represented as a rupture point. When the junction is formed, the ring plane is bended and resulting in non-parallel p bond alignment, asymmetrical stress distribution and also cause dramatic angle distortion [45]. This angle distortion by changing bonds’ plane and hybridization can weaken the system significantly. In addition, destabilized nature of these bonds causes easier and earlier fracture of this hybrid structure which significantly reduce critical strength.
Fig. 4. Variation of Young’s modulus, failure stress and failure strain of GNBs at different temperatures.
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A. Fereidoon et al. / Computational Materials Science 107 (2015) 163–169 Table 2 Young’s modulus for 23 23 (Å) GNB at various temperatures.
Neck-0 Neck-1 Neck-5 Neck-10
1K
100 K
200 K
300 K
400 K
500 K
600 K
700 K
800 K
636.61 612.13 537.99 454.85
630.93 605.13 533.80 442.47
621.32 592.97 535.33 441.43
623.00 592.64 525.68 441.77
610.64 583.88 522.33 439.63
629.38 566.35 527.64 451.8
629.3 571.60 495.93 456.35
616.2 582.45 515.13 440.71
626.49 575.84 507.61 436.33
Table 3 Young’s modulus for 28 28 (Å) GNB at various temperatures.
Neck-0 Neck-1 Neck-5 Neck-10
1K
100 K
200 K
300 K
400 K
500 K
600 K
700 K
800 K
706.25 685.92 617.19 538.77
694.33 686.67 608.61 530.71
687.55 678.13 610.09 534.83
695.18 655.85 608.37 519.40
663.58 650.30 588.63 523.41
668.39 684.16 595.34 519.95
678.14 653.91 583.85 518.70
665.16 647.29 578.39 484.57
689.84 627.23 594.51 490.80
Table 4 Young’s modulus for 36 36 (Å) GNB at various temperatures.
Neck-0 Neck-1 Neck-5 Neck-10
1K
100 K
200 K
300 K
400 K
500 K
600 K
700 K
800 K
784.80 768.49 716.68 651.24
775.57 751.59 711.60 645.15
778.91 745.28 693.81 637.30
763.62 743.02 685.91 627.21
747.28 738.58 672.94 624.78
753.14 742.07 673.74 626.63
751.36 735.64 684.03 610.47
760.30 726.07 700.15 608.26
745.40 722.24 669.41 607.44
(a)
(c)
(b)
(d)
Fig. 5. Snapshots of failure process for neck-5 and 28 28 (Å) graphene sheet.
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Another important point in the stress–strain evolution, is the changing point where parabolic section alter to linear line part. At the beginning of tensile loading, sp2-hybridized bonds in graphene sheet are elongated gradually. With further loading, strain increased and junction region with its octagonal rings started to be stretched. In this stage when stress accumulated in junction region, bond breakage is observed to initiate at this place. So the stress–strain manner change at this level and stress increase dramatically. Therefore junction deteriorates the bonding and weaken the sheet, so obviously it is responsible for early fracture of GNBs, as discussed above. 3.2. Temperature dependence In order to investigate the effect of temperature on mechanical properties of GNBs, GNBs with different graphene size and neck length, subjected to uniaxial tension are simulated under temperature varying from 1 to 800 K. The manner of Young’s modulus, failure stress and strain against temperature for all GNB models are shown in Fig. 4 and the calculated results are listed in Tables 2–4. As shown in Fig. 4 in each graphene the Young’s modulus varies a little with temperature increment. With an increasing temperature, the axial Young moduli trend to decrease linearly. It is also seen from Fig. 4 that the effect of temperature increment on Young’s modulus of GNB with larger graphene sheet is stronger than that of with smaller sheet. The maximum reduction of Young’s modulus when the temperature ranges from 1 to 800 K for 23 23, 28 28 and 36 36 (Å) graphene sheet is about 1.6%, 2.3% and 5.02% respectively. Since size of junction region in all models is the same, therefore the number of off-plane sp3 bonds and octagonal rings are equal. So destabilized nature of these bonds is less effective in GNB with the largest graphene sheet. It means that larger base GNBs have larger Young’s modulus, failure stress and strain, so all in all the largest base GNBs is the stiffest one against fracture. Also it can be deduced from Fig. 4 that neck’s length increment cause significant reduction in Young’s modulus. Actually when height of neck increase, the structure become more unstable which make a meaningful reduction in all mechanical properties. Furthermore, Fig. 4 shows that in some cases the failure strian is constant in various temperatures. The reason is that the value which we considered for strain rate is 0.01 ps1 and maybe their difference in failure strain is less than 0.01 so it cannot be observed in graph while decreasing trend is completely obvious. Also, we found that with temperature increment the fracture happens in smaller strain. It is observed that bond length of carbon–carbon bond and the length of GNBs in the x direction change remarkably prior to complete failure. The length of GNB before bond breaking in the armchair direction increases 38.31%. To understand the failure process entirely, in Fig. 5, some representative snapshots of the GNB under armchair tensile deformation from the onset to the failure of the structures is presented. As it has been pointed out before, the junction obviously deteriorates the bonding and thus weakens the graphene sheet, which cause bond breakage in this region. Fig. 5 shows this procedure completely. At first stage just one carbon–carbon bond breaks which makes a defect in structure of GNB then defect grows rapidly and the system fails completely. 4. Conclusions In this paper we evaluated the mechanical properties of graphene nanobuds based on molecular dynamic simulations. We used Large-scale Atomic/Molecular Massively Parallel Simulator
code LAMMPS to investigate the effect of temperature, size of graphene sheet and neck’s length on mechanical properties of GNBs. The Young’s modulus was obtained from second order polynomial fit to first section of stress strain evolution. This diagram can also shows failure stress and strain. Three sizes of graphene sheet and four different neck’s length was considered. For validity, we compared our results with graphene ones. The calculated Young’s modulus of graphene is 1.01 TPa which is in good agreement with experimental value of 1.0 ± 0.1 TPa. Comparing Young’s modulus of GNBs at 300 K with the same size graphene shows that Young’s modulus of GNB is significantly less than graphene. This phenomenon occurs due to existence of nano-bud which makes defects in graphene lattice. The calculated Young’s modulus at 300 K for different models range from 440 to 763 GPa. We found that Young’s modulus decreases smoothly with temperature increment. Additionally with neck’s length and graphene size increment the Young’s modulus decreases and increases, respectively. So, for all graphene sheet size the neck-0 has larger Young’s modulus value and hence is mechanically stronger than the other counterparts. This behavior is observed for failure stress and strain too, while dependence of these two parameters on temperature is more than the Young’s modulus. It is noticeable that in higher temperature the failure strain is in lower range due to higher kinetic energy of atoms. Our MD simulation results provide a vital exploration of the mechanical properties of the graphene nanobuds structures, which will aid the design and also the applications of graphene-based hybrid nano-materials.
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