The Molecular Mechanics Model of Carbon Allotropes
Mohsen Motamedi
1
(
faculty of engineering, university of shahreza, Isfahan, Iran
)
AMIRHOSSEIN NAGHDI
2
(
Student of mechanical enginnering, university of shahreza,isfahan, iran
)
Keywords: Carbon nanotubes, Natural Frequencies, Graphene, Buckyballs, Young&rsquo, s Modulus,
Abstract :
Due to its valency, carbon can form too many allotropes. A number of well-known forms of carbon include graphene, carbon nanotubes, capped carbon nanotubes, buckyballs, and nanocones. The remarkable mechanical properties of these carbons have attracted researchers. Numerous studies have been conducted on carbon nanotubes or graphene. In the present study, however, we applied the molecular mechanic method in order to model five forms of carbon with a uniform approach and draw a detailed comparison between the allotropes of carbon. Furthermore, we obtained Young’s modulus and natural frequencies for every form of carbon, which can be useful for researchers. The results show that increasing the diameter of the carbon nanotube will decrease its strength (decreases the Young’s modulus). Also, the capped carbon nanotube is stronger than the non-capped nanotube. This is because of the end bonds of the carbon nanotube. Also, the results show that Buckyball has extraordinary properties. Its strength is three times more than that of the carbon nanotube with the same diameter.
The Molecular Mechanics Model of Carbon Allotropes
Abstract:
Due to its valency, carbon can form too many allotropes. A number of well-known forms of carbon include graphene, carbon nanotubes, capped carbon nanotubes, buckyballs, and nanocones. The remarkable mechanical properties of these carbons have attracted researchers. Numerous studies have been conducted on carbon nanotubes or graphene. In the present study, however, we applied the molecular mechanics method in order to model five forms of carbon with a uniform approach and draw a detailed comparison between the allotropes of carbon. Furthermore, we obtained Young’s modulus and natural frequencies for every form of carbon, which can be useful for researchers. The results show that increasing the diameter of the carbon nanotube will decrease its strength (decreases the Young’s modulus). Also, the capped carbon nanotube is stronger than the non-capped nanotube. This is because of the end bonds of the carbon nanotube. Also, the results show that Buckyball has extraordinary properties. Its strength is three times more than that of the carbon nanotube with the same diameter.
keywords: Graphene, Carbon Nanotubes, Buckyballs, Young’s Modulus, Natural Frequencies
1. Introduction
Carbon can form numerous allotropes because of its valency. Two well-known forms of carbon are diamond and graphite. These years, many allotropes and forms of carbon have been detected and discovered, such as buckyballs and graphene sheets. Larger-scale structures of carbon encompass nanotubes, nanobuds, and nanocones. Other unusual forms of carbon exist at very high temperatures or extreme pressures.
Single-walled carbon nanotubes (SWCNTs) are tubes with a diameter of 4–5 A˚ [1-3]. The ideal characteristics of SWCNTs originate from their long macro-morphology (high aspect ratio, length/diameter), remarkable mechanical properties (Young’s modulus = 1–1.8 TPa), transport conductivity, and thermal conductivity (3000W/m K) [3]. SWCNTs have been considered as the building blocks of various nanoscale electronic and mechanical devices due to their significant structural and mechanical properties [4-6].
Ideally, graphite comprises of infinite layers of sp2-hybridized carbon atoms. Within a layer (called a graphene sheet), each carbon atom bonds to three other carbon atoms, forming a planar array of fused hexagons. Graphite is also a good electrical and thermal conductor in the plane directions. The thermal conductivity values are 
∼ 15–20 W
and 
∼0.05–0.1 W 
[7-10]. It must be noted that these values depend on the sample history and temperature.
Buckyballs and fullerenes (including C60 and C70) [11–12] form when the dangling bonds at the edges of a real (finite) graphene layer are connected to each other. The chemistry and physical properties of fullerenes have received much attention in this decade, with several functionalized derivatives being reported [12,13,14]. Owing to their many interesting physical properties [18,19], solar cells [15] and biological applications [16,17] and their derivatives have been recently reviewed.
Several studies have been conducted on the mechanical properties of the allotropes of carbon. in the experimental works, Blakslee et al. [20] reported that the Young’s modulus about 1.06 ± 0.02 TPa for graphite. Frank et al. [21] measured the Young’s modulus of five layers graphene sheets about 0.5 TPa. Gomez-Navarro et al. [22] applied the tip-induced deformation method and estimated the Young’s modulus of a graphene (with 1 nm thickness) at 0.25 ± 0.15 TPa. Lee et al. [23] employed nano indentation in the center of a monolayer graphene sheet with an atomic force microscope; in addition, they measured the Young’s modulus at 1 ±0.1 TPa by assuming that the thickness of graphene was 0.335 nm. Various ab initio (molecular dynamic) calculations on graphene found that the Young’s modulus values of graphene was 1.11 TPa [24] or 1.24 ±0.01 TPa [25] by assuming that the thickness of graphene was 0.34 nm. MD simulation has also been utilized to gain the mechanical properties of graphene. For example, the Young’s modulus of graphene was 1.272 TPa [26] with the modified Brenner potential. Also, Memarian et al. [27] obtained the Young's modulus of graphene by molecular mechanics method about 1078.86 and 982.01 GPa in zigzag and armchair directions, respectively.
Many researches have been done on CNTs. The results shows that CNTs have wide ranges of 270 - 5500 GPa for Young’s modulus and 240-2300 GPa for shear modulus [28-37, 39]. Nevertheless, the majority of the results are distributed close to a Young’s modulus of 1000 GPa or 1 TPa and a shear modulus of 400 GPa.
In the present study, we applied the molecular mechanics model in order to model graphene, carbon nanotube (CNT), buckyballs, and nanocones to obtain the mechanical properties and vibrational properties of these allotropes of carbon. It’s the first time which all of carbon’s allotropes simulated by same method to obtained their mechanical properties. Also, the properties of buckyballs and nanocones were obtained for the first time by molecular mechanics method.
2. Modelling
A SWNT can be visualized as a hollow cylinder and formed by rolling over a graphite sheet. It can be uniquely characterized by a vector C in terms of a set of two integers (n, m) corresponding to graphite vectors 
and 
(Figure 1) [38].
(1)
Figure 1. A nanotube (n,m) is formed by rolling a graphite sheet [38].
Thus, the SWNT is constructed by rolling up the sheet in such a way that the two end-points of the vector are superimposed. This tube is denoted as the (n, m) tube with the diameter given by
(2)
2.1 Molecular mechanic method
CNTs are frame-like structure which their bonds can be treated like as the beam members and carbon atoms as the joints. If the electrostatic interactions are neglected, the total potential energy (Utotal) characterizing the force field can be obtained as the sum of energies due to valence (or bonded) and non-bonded interactions, which is given by [39]:
(3)
Here, 
and 
correspond with the energy associated with bond stretch interactions, bond angle bending, torsion, and van der Waals forces (non-covalent). Figure 2 [39] illustrates various inter-atomic interactions at the molecular level.
Figure 2. Equivalence of molecular mechanics and structural mechanics for covalent and non-covalent interactions between carbon atoms (a) Molecular mechanics model and (b) structural mechanics model [39].
Force fields can describe the interaction between individual carbon atoms. Kalamkarov et al. [39] used simple harmonic functions to represent covalent interactions between carbon atoms. The energies associated with each covalent component of Eq. (4) can be mathematically described as [40]:
(4)
To establish the linkage between the force constants in molecular mechanics and the beam element stiffness in structural mechanics, Table 1 was employed [41] .
Table 1. beam element properties
Structure mechanics | Molecular mechanics | Energy form | |
| Stretching energy | ||
|
| Bending energy | |
|
| Torsional energy | |
In this reference, the parameters are bond stretching increment (
), axial stretching deformation (
), bond angle change (
), total rotation angle (
), angle of bond twisting (
), and torsion angle (
).
Therefore, the force constants 
can be directly represented by:
(5) 
The values of the force constants
, based on the experience of dealing with graphite sheets, are selected as follows: 
, 
, 
[41].
In this study, the BEAM element was selected to simulate the carbon bonds using ABAQUS package. Figure 3 shows how the hexagonal lattice of the CNT can be simulated as the structure of a space frame.
Figure 3. BEAM element to simulate carbon bonds.
3. Results and Discussion
3.1. Carbon nanotubes
Carbon nanotubes (CNTs) are tube-like nanostructures which have unusual properties. These tubes are useful for nanotechnology, electronics, optics, and other fields of material science.
To estimate the Young’s modulus of nano particles in this research, the following relationships from Beer [49] were utilized:
, 
(6)
where, F is the load applied to the beam, L is the initial length of CNT, A is the cross-sectional area of the beam, and 
is the calculated displacement of the beam.
Studying the vibrational behavior of carbon-based nano-particles is critical for various industrial applications, including oscillators and nanocomposites [42].To be able to investigate the vibrational behavior of carbon nanotubes (CNTs), three major boundary conditions were considered. The first boundary condition was the cantilevered beam. The second one was the bridged CNT, and the third one was without any boundary conditions. Table 3 indicates the results for nanotubes with chiral (5,0). Figure 4 and Figure 5 show vibrational behavior and displacement behavior of CNTs.
Figure 4. Vibrational behavior of zigzag nanotubes (5,0)
Figure 5.displacement of zigzag (5,0) CNT due to applied load
The Young’s modulus of CNTs with chiral (10,0) and (5,0) was 965 (GPa) and 1393 (GPa), respectively (Table 2). In previous work on nanotubes, The results shows that CNTs have wide ranges of 270 - 5500 GPa for Young’s modulus [28-37, 39]. Nevertheless, the majority of the results are distributed close to a Young’s modulus of 1000±200 GPa which have good agreement with this work.
3.2 Capped tube
To be able to investigate the vibration behavior of the capped tube, three major boundary conditions were considered. The first boundary condition was the cantilevered beam. The second one was the bridged CNT, and the third one was without any boundary conditions. the natural frequencies of capped CNTs were obtained, and the mode shapes can be observed in Figure 6. Also Table 3 presents the results for the capped tube.
Figure 6. Vibrational behavior of capped tube.
The Young’s modulus of capped tube with chiral (10,0) was 1173 (GPa) (Table 2). Schematic of stretched capped CNT shows in Figure 7. Comparing these results with other studies, shows good agreements. Lu [43] reported a Young’s modulus of∼1 TPa, a shear modulus of∼0.5 TPa for nanotubes. A molecular dynamics model was used by Yao et al. [44] who obtained Young’s modulus about 1 TPa .
Figure 7.displacement contour due to applied load to capped tube
3.3 Graphene
Graphene is an allotrope of carbon in the form of two-dimensional and hexagonal lattice sheet.
To be able to investigate the vibration behavior of the graphene sheet, three major boundary conditions were considered. The first boundary condition was the cantilevered condition. The second one was the bridged condition, and the third one was without any boundary conditions. Table 4 demonstrates the results. In addition, the mode shapes of graphene can be seen in Figure 8.
Figure 8. Vibrational behavior modes of graphene sheet
To estimate the Young’s modulus of the graphene sheet in the this study, the graphene were pulled (Figure 9) and was employed, where 
is the measured stress, and 
is the calculated strain of graphene sheet.
Figure 9.displacement due to applied load to graphene sheet
The Young’s modulus of a graphene sheet with 24.422 nm width and 25 nm length was 1091 (GPa) (Table 2). The Young's modulus has been reported in experimental results between 0.5 to 2 TPa [45]. But most researches have reported Young’s modulus close to 1 TPa [46]. So, the results are in good agreement with the previous researches.
3.4 Buckyballs
Buckyball is a spherical shape of carbon allotropes (fullerene molecule C60). This structure is made of twenty hexagons and twelve pentagons (such as a soccer ball).
The vibrational behavior and natural frequency of Buckyball were investigated with no boundary conditions. The mode shapes of buckyball are shown in Figure 10. It shall be noticed that there is not any report for buckyballs natural frequencies.
Figure 10. The modes shapes of buckyball
To estimate the Young’s modulus of buckyball in this work, the relationship ( ) was used, where is the measured stress and is the calculated strain of buckyball (Figure 11).
Figure 11.displacement due to applied load to buckyball
The Young’s modulus of the buckyball with 60 atoms was 3628 (GPa) (Table 2).
There are a little work on mechanical properties of buckyballs. Tomoharu [47] indicated that the application of a large amount of stress, almost 2.5 GPa, did not damage the fullerene structures also, Jeremy et al. [48] studied on hardness of C70 and showed that its hardness was more than graphite hardness. So, it can be concluded that the obtained Young’s modulus of buckyball has good agreement with mentioned works.
3.5 Nano-cone
Carbon nanocones are conical structures which are made of carbon and have at least one dimension of the order one micrometer or smaller.
To be able to investigate the vibration behavior of the nanocone, two major boundary conditions were considered. The first boundary condition was the cantilevered condition, and the second one was with no boundary conditions. The natural frequencies of nanocones are shown in Table 4. Moreover, the mode shapes are depicted in Figure 12. For comparing results, Narjabadifam and coworkers [50], found natural frequency of nanocones using molecular dynamics method. Their frequencies are the same order of current study which shows good agreement. Also, Ansari et al. [51] founding vibration properties of nanocones using molecular mechanics method and compared their results by molecular dynamic method.
Figure 12. vibrational bihavior for nano-cone
The Young’s modulus of the nano cone with 20 nm height (Figure 13) was 3590 (GPa) (Table 2). It shall be noticed that there was not observed any simulation or experimental studied on nano-cones in literatures. But because of shape of nano-cones, it can be concluded that it’s strength is more than graphene sheet.
Figure 13.displacement due to applied load to nano-cone
Table 2.Young’s modules of Carbon’s allotropes
| Diameter[nm] | Length[nm] | Young’s Modulus[GPa] |
SWCNT (5,0) | 3.887 | 25 | 1393 |
SWCNT (10,0) | 7.774 | 25 | 965 |
Capped SWCNT (10,0) | 7.774 | 25 | 1173 |
Graphene sheet | - | 25*24.422 | 1091 |
Buckyball | 7.11294 | - | 3628 |
Nano cone | - | 20 | 3590 |
.
Mode ID | Zigzag(5,0) | Zigzag(10,0) | Bucky ball | ||||
cantilevered | bridged | free | cantilevered | bridged | none | free | |
1 | 127 | 3350 | 130 | 604 | 7849 | 266 | 476 |
2 | 127.1 | 3351 | 163 | 604.1 | 7850 | 525 | 764 |
3 | 3125 | 12320 | 184 | 4423 | 17450 | 1360 | 1090 |
4 | 3543 | 17140 | 4209 | 9393 | 21840 | 1740 | 1440 |
5 | 3544 | 17150 | 4211 | 9400 | 21890 | 1939 | 10580 |
6 | 8149 | 32550 | 12610 | 12330 | 31880 | 3590 | 10590 |
7 | 19630 | 47250 | 22360 | 16950 | 31890 | 14750 | 10595 |
Table 3. Natural frequencies (GHz) for SWCNT with different structure and boundary conditions
.
Mode ID | Graphene sheet | Nano-cone | Capped tube | |||||
cantilevered | bridged | free | cantilevered | free | cantilevered | bridged | free | |
1 | 146 | 5494 | 381 | 6311 | 127 | 485 | 1853 | 70.2 |
2 | 1035 | 8355 | 464 | 6324 | 429 | 492 | 1862 | 73.2 |
3 | 5590 | 22860 | 772 | 8019 | 1330 | 1236 | 3178 | 658 |
4 | 9278 | 39840 | 951 | 8023 | 5350 | 3063 | 8237 | 1045 |
5 | 12180 | 49640 | 1010 | 15470 | 5360 | 3457 | 12590 | 1285 |
6 | 13170 | 73770 | 2737 | 15480 | 5855 | 3468 | 12660 | 1286 |
7 | 36390 | 84520 | 5818 | 16980 | 8023 | 12380 | 12910 | 2066 |
Table 4. Natural frequencies (GHz) for other carbon’s allotropes
Based on Table 2, it can be concluded that:
- Increasing the diameter of the carbon nanotube will decrease its strength (decreases the Young’s modulus), which can be concluded from the formula 
as well. From this formula, we can find that increasing the Area (A) leads to decreasing the Young’s modulus (E). The nanotube with a higher diameter has a higher section area (A) and, thus, a lower Young’s modulus.
- The capped carbon nanotube is stronger than the non-capped nanotube. This is because of the end bonds of the carbon nanotube.
- Buckyball has extraordinary properties. Its strength is three times more than carbon nanotube with the same diameter.
Based on Table 3 and Table 4, it can be concluded that:
- For carbon nanotubes, the highest frequency accrues at the bridged boundary condition. Based on the Euler Bernoulli theory, the fundamental frequency can be expressed by [52]:
, where k represents the stiffness, and m is the mass of the structure. The bridged boundary condition leads to increasing the stiffness of the structure, and then frequency will increase as well.
For this reason, the cantilevered carbon nanotubes have higher frequency than the free-carbon nanotubes.
4. Conclusion
The interesting mechanical properties of allotropes of carbon have attracted researchers thus far. Although many studies have been performed on carbon nanotube or graphene, but in the present study we applied the molecular mechanics method to model five forms of carbon allotropes in order to draw a detailed comparison between the allotropes of carbon. The Young’s modulus and natural frequencies were obtained for every form of carbon, which can be useful for researchers. We indicated that the capped carbon nanotube is stronger than the non-capped one. Furthermore, we found that the strength of buckyball is three times more than that of the carbon nanotube with the same diameter.
References
[1] B. Xue, X. Shao, W. Cai, Computational Materials Science, 43, 5319 (2008).
[2] Y. Lin, W. Cai, X. Shao, Journal of Molecular Structure, 767, 87 (2006).
[3] M. Motamedi, M. Eskandari, M. Yeganeh, Materials and Design, 34, 603 (2012).
[4] Z. Qin, Q. Qin, X. Feng, Physics Letters A, 372, 6661 (2008).
[5] M. Meyyappan, “Carbon nanotubes : science and applications”, NASA Ames Research Center, 2005.
[6] S. Melchor, J.A. Dobado, Journal of chemical information and computer sciences, 44,1639 (2004).
[7] D.D.L. Chung , Journal of Materials Science, 37, 1475 (2002).
[8] I.L. Spain, A.R. Ubbelohde, D.A. Young, Philos. Trans. Royal Soc. A, 262, 345 (1967).
[9] C.N. Hooker, A.R. Ubbelohde, D.A. Young, Proc. Royal Soc. A, 284, 17 (1965).
[10] R. Taylor , K.E. Gilchrist, L.J. Poston, Carbon, 6, 537 (1968).
[11] H.W. Kroto, J.R. Heath, S.C. O’Brien, R.F. Curl, R.E. Smalley, Nature, 318, 162 (1985).
[12] F. Wudl, J. Mater. Chem., 12, 1959 (2002).
[13] C. Thilgen, A. Herrmann, F. Diederich, Angew. Chem. Int. Ed. Engl., 36, 2268 (1997).
[14] F. Wudl, Acc. Chem. Res., 25, 157 (1992).
[15] J.L. Segura, N. Martin, D.M. Guldi, Chem. Soc., 34, 31 (2005).
[16] A.W. Jensen, S.R. Wilson, D.I. Schuster, Bioorg. Med. Chem., 4, 767 (1996).
[17] S. Bosi, T. Da Ros, G. Spalluto, M. Prato, Eur. J. Med. Chem., 38, 913 (2003).
[18] C.S. Foote, Topic Curr. Chem., 169, 347 (1994).
[19] M.S. Dresselhaus, G. Dresselhaus, P.C. Eklund, J. Mater. Res., 8, 2054 (1993).
[20] O.L. Blakslee, D.G. Proctor, E.J. Seldin, G.B. Spence, T. Weng, J. Appl. Phys., 41(8), 3373 (1970).
[21] I.W. Frank, D.M. Tanenbaum, A.M. van der Zande, P.L. McEuen, J. Vac. Sci. Technol. B, 25 (6), 2558 (2007).
[22] C. Gomez-Navarro, M. Burghard and K. Kern, Nano Lett., 8(7), 2045 (2008).
[23] C. Lee, X. Wei and J. W. Kysar, Hone, measurement of the elastic properties and intrinsic strength of monolayer graphene, Science 2008, 321(18): 385-388.
[24] G.V. Lier, C.V. Alsenoy, V.V. Doren, P. Geerlings, Chem. Phys. Lett., 326, 181 (2000).
[25] E. Konstantinova, S.O. Dantas and P.M.V. Barone, Phys. B, 74, 035417 (2006).
[26] S. Gupta, K. Dharamvir, V. K. Jindal, Phys. ReV. B, 72, 165428 (2005).
[27] F. Memarian, A. Fereidoon, M. Darvish Ganji, Superlattices and Microstructures, 85, 348 (2015).
[28] C. Li, T. Chou, International Journal of Solids and Structures, 40, 2487 (2003).
[29] C. Li, T. Chou, Composites Science and Technology, 63, 1517 (2003).
[30] M.M. Zaeri, S. Ziaei-Rad, A. Vahedi, F. Karimzadeh, Carbon, 48, 3916 (2010).
[31] C.L. Zhang, H. S. Shen, Applied Physics Letters, 89, 081904 (2006).
[32] M. Rossi, M. Meo, Composites Science and Technology, 69,1394 (2009).
[33] A. Sears, R.C. Batra, Physical Review B, 69, 235406 (2004).
[34] F. Li , H.M. Cheng, S. Bai, G. Su, M.S. Dresselhaus, Applied Physics Letters, 77, 3161 (2000).
[35] E.T. Thostenson, C. Li, T.W. Chou, Composites Science and Technology, 65, 491 (2005).
[36] A. Fereidoon, M. Ghorbanzadeh Ahangari, M.D. Ganji, M. Jahanshahi, Computational Materials Science, 53 (1), 377 (2012).
[37] D. Qian, G.J. Wagner, W.K. Liu, M.F.Yu, R.S. Ruoff, Applied Mechanics Reviews, 55, 495 (2002).
[38] M. Meyyappan, “Carbon nanotubes : science and applications”, NASA Ames Research Center,2005.
[39] A.L. Kalamkarov, A.V. Georgiades, S.K. Rokkam, V.P. Veedu, M.N. Ghasemi-Nejhad, Int. J. Solid Struct., 43, 6832 (2006).
[40] A.F. Ávila, A.C. Eduardo, A.S. Neto, Computers and Structures, 89, 878 (2011).
[41] C. Li, T. Chou, International Journal of Solids and Structures, 40, 2487 (2003).
[42] R.F. Gibson, E.O. Ayorinde, Y.F. Wen, Compos. Sci. Tech., 67, 1 (2007).
[43] J.P. Lu, Phys. Rev. Lett. 79 (7), 1297(1997).
[44] N. Yao, V. Lordi, J. Appl. Phys. 84 (4) 1939 (1998)
[45] R. Daniel, Condensed Matter. Physics, 501686 (2012).
[46] B.T. Kelly, “Physics of Graphite”, Applied Science, London, UK, 1981.
[47] T. Tokunaga, K. Kaneko,K. Sato, Z. Horita, Scripta Materialia, 58, 735 (2008)
[48] J.R. Patterson, S.A. Catledge, Y.K. Vohra, PHYSICAL REVIEW LETTERS, 8, 25 (2000)
[49] F.P. Beer, R. Johnson, “Mechanics of Materials”, McGraw-Hill Education, 7th edition, 2014.
[50] A. Narjabadifam, F. Vakili-Tahami, M. Zehsaz, Computational Materials Science, 137, 55 (2017)
[51] R. Ansari, A. Momen, S. Rouhi, S. Ajori, Shock and Vibration, 8 (2014)
[52] J.Y. Anand, S.C. Satish, S.P. Harsha, Sensors and Actuators A: Physical, 168(2), 275 (2011).