The bond force constant and bulk modulus of small fullerenes using density functional theory and finite element analysis

  • A. TapiaEmail author
  • C. Villanueva
  • R. Peón-Escalante
  • R. Quintal
  • J. Medina
  • F. Peñuñuri
  • F. Avilés
Original Paper


Dedicated bond force constant and bulk modulus of C n fullerenes (n = 20, 28, 36, 50, 60) are computed using density functional theory (DFT). DFT predicts bond force constants of 611, 648, 675, 686, and 691 N/m, for C20, C28, C36, C50, and C60, respectively, indicating that the bond force constant increases for larger fullerenes. The bulk modulus predicted by DFT increases with decreased fullerene diameter, from 0.874 TPa for C60 to 1.830 TPa for C20. The bond force constants predicted by DFT are then used as an input for finite element analysis (FEA) of the fullerenes, considered as spatial frames in structural models where the bond stiffness is represented by the DFT-computed bond force constant. In agreement with DFT, FEA predicts that smaller fullerenes are stiffer, and underestimates the bulk modulus with respect to DFT. The difference between the FEA and DFT predictions of the bulk modulus decreases as the size of the fullerene increases, from 20.9 % difference for C20 to only 4 % difference for C60. Thus, it is concluded that knowing the appropriate bond force constant, FEA can be used as a plausible approximation to model the elastic behavior of small fullerenes.


Bond force constant Density functional theory Elastic properties Finite element analysis Fullerenes 



This research was supported by FOMIX-CONACYT under grant No. 170297 directed by Dr. J. A. Tapia (UADY).

Supplementary material

894_2015_2649_MOESM1_ESM.docx (38 kb)
ESM 1 (DOCX 38 kb)


  1. 1.
    Osawa E (1970) Superaromaticity. Kagaku (Chem) 25:854Google Scholar
  2. 2.
    Bochvar DA, Gal’pern EG (1973) Energy spectra of certain condensed cyclopentadienyl and cycloheptatrienyl systems. Proc Acad Sci USSR 209:239Google Scholar
  3. 3.
    Stankevich IV, Nikerov MV, Bochvar DA (1984) The structural chemistry of crystalline carbon: geometry, stability, and electronic spectrum. Russ Chem Rev 53:640CrossRefGoogle Scholar
  4. 4.
    Kroto HW, Heath JR, O’Brien SC, Curl RE, Smalley RE (1985) C60: buckminsterfullerene. Nature 318:162CrossRefGoogle Scholar
  5. 5.
    Grossman JC, Mitas L, Raghavachari K (1995) Structure and stability of molecular carbon: importance of electron correlation. Phys Rev Lett 75:3870CrossRefGoogle Scholar
  6. 6.
    Lin F, Sørensen ES, Kallin C, Berlinsky AJ (2007) Strong correlation effects in the fullerene C20 studied using a one-band Hubbard model. Phys Rev B 76:033414Google Scholar
  7. 7.
    Zhang C-Y, Wu H-S, Jiao H (2007) Aromatic C20F20 cage and its endohedral complexes X@C20F20(X = H−, F−, Cl−, Br−, H, He). J Mol Model 13:499Google Scholar
  8. 8.
    Tang SW, Sun LL, Feng JD, Sun H, Wang RS, Chang YF (2009) DFT investigation on the structures and stabilities of exohedral derivatives for D3 C32 fullerene: C32Xn (X = H and Cl). Eur Phys J D53:197Google Scholar
  9. 9.
    Fowler PW, Heine T, Rogers KM, Sandall JPB, Seifert G, Zerbetto F (1999) C36, a hexavalent building block for fullerene compounds and solids. Chem Phys Lett 300:369CrossRefGoogle Scholar
  10. 10.
    Guo T, Diener MD, Chai Y, Alford MJ, Haufler RE, McClure SM et al (1992) Uranium stabilization of C28: a tetravalent fullerene. Science 257(5077):1661CrossRefGoogle Scholar
  11. 11.
    Prinzbach H, Weller A, Landenberger P, Wahl F, Worth J, Scott LT et al (2000) Gas-phase production and photoelectron spectroscopy of the smallest fullerene, C20. Nature 407:60Google Scholar
  12. 12.
    Wang Z, Ke X, Zhu Z, Zhu F, Ruan M, Chen H et al (2001) A new carbon solid made of the world’s smallest caged fullerene C20. Phys Lett A 280:351Google Scholar
  13. 13.
    Iqbal Z, Zhang Y, Grebel H, Vijayalakshmi S, Lahamer A, Benedek G et al (2003) Evidence for a solid phase of dodecahedral C20. Eur Phys J B 31:509CrossRefGoogle Scholar
  14. 14.
    Kietzmann H, Rochow R, Ganteför G, Eberhardt W, Vietze K, Seifert G et al (1998) Electronic structure of small fullerenes: evidence for the high stability of C32. Phys Rev Lett 81:5378Google Scholar
  15. 15.
    Krätschmer W, Lamb L, Fostiropoulos D, Huffman D (1990) Solid C60: a new form of carbon. Nature 347:354CrossRefGoogle Scholar
  16. 16.
    Domene MC, Fowler PW, Mitchell D, Seifert G, Zerbetto F (1997) Energetics of C20 and C22 fullerene and near-fullerene carbon cages. J Phys Chem A 101:8339Google Scholar
  17. 17.
    Lin F, Sørensen ES, Kallin C, Berlinsky AJ (2010) C20, the smallest fullerene. In: Sattler KD (ed) Handbook of nanophysics. CRC, Boca RatonGoogle Scholar
  18. 18.
    Dresselhaus MS, Dresselhaus G, Eklund PC (1996) Science of fullerenes and carbon nanotubes. Academic Press, ElsevierGoogle Scholar
  19. 19.
    Chae S-R, Hotze EM, Wiesner MR (2014) Possible applications of fullerene nanomaterials in water treatment and reuse. In: Street A, Sustich R, Duncan J, Savage N (eds) Nanotechnology applications for clean water, 2nd edn. Elsevier, Netherlands, pp 329–338, Chap. 21Google Scholar
  20. 20.
    Afreen S, Muthoosamy K, Manickam S, Hashim U (2015) Functionalized fullerene (C60) as a potential nanomediator in the fabrication of highly sensitive biosensors. Biosens Bioelectron 63:354CrossRefGoogle Scholar
  21. 21.
    Ogasawara T, Ishida Y, Kasai T (2009) Mechanical properties of carbon fiber/fullerene-dispersed epoxy composites. Compos Sci Technol 69:2002Google Scholar
  22. 22.
    Kay ER, Leigh DA, Zerbetto F (2007) Synthetic molecular motors and mechanical machines. Angew Chem Int 46:72CrossRefGoogle Scholar
  23. 23.
    Kang JW, Hwang HJ (2004) Fullerene nano ball bearings: an atomistic study. Nanotechnology 15:614CrossRefGoogle Scholar
  24. 24.
    Du AJ, Pan ZY, Ho YK, Huang Z, Zhang ZX (2002) Memory effect in the deposition of C20fullerenes on a diamond surface. Phys Rev B 66:035405CrossRefGoogle Scholar
  25. 25.
    Zhang ZX, Pan ZY, Wang YX, Li ZJ, Wei Q (2003) Simulations of the nanomechanical properties of compressed small fullerenes. Mod Phys Lett B 17:877CrossRefGoogle Scholar
  26. 26.
    Lu X, Chen Z (2005) Curved Pi-Conjugation, aromaticity, and the related chemistry of small fullerenes (C60) and single-walled carbon nanotubes. Chem Rev 105:3643Google Scholar
  27. 27.
    Plestenjak B, Pisanski T, Graovac A (1996) Generating fullerenes at random. J Chem Inf Model 36:825CrossRefGoogle Scholar
  28. 28.
    Fowler PW, Manolopoulos DE (2006) An atlas of fullerenes. Dover, MineolaGoogle Scholar
  29. 29.
    Ehlich R, Landenberger P, Prinzbach H (2001) Coalescence of C20 fullerenes. J Chem Phys 115:5830Google Scholar
  30. 30.
    Oku T, Suganuma K (2001) High-resolution electron microscopy and structural optimization of C36, B36N36 and Fe@B36N36 clusters. Diam Relat Mater 10(3–7):1205Google Scholar
  31. 31.
    Diudea MV, Nagy CL (2012) C20-related structures: diamond D5. Diam Relat Mater 23:105Google Scholar
  32. 32.
    Wen YW, Liu X, Duan X, Chen R, Shan B (2013) First-principles study of the structural, energetic and electronic properties of C20-carbon nanobuds. Model Simul Mater Sci Eng 21:035006Google Scholar
  33. 33.
    Ruoff RS, Ruoff AL (1991) The bulk modulus of C60 molecules and crystals: a molecular mechanics approach. Appl Phys Lett 59:1553Google Scholar
  34. 34.
    Woo SJ, Lee SH, Kim E, Lee KH, Lee YH (1992) Bulk modulus of the C60 molecule via the tight binding method. Phys Lett A 162:501Google Scholar
  35. 35.
    Enyashin AN, Ivanovskii AL (2009) Structural, electronic and elastic properties of ultra-light diamond-like crystalline allotropes of carbon-functionalized fullerenes C28. Chem Phys Lett 473:108Google Scholar
  36. 36.
    Kovalev OO, Kuzkin VA (2011) Analytical expression for bulk moduli and frequencies of volumetrical vibrations of fullerenes C20 and C60. Nanosyst: Phys Chem Math 2:65Google Scholar
  37. 37.
    Ahangari MG, Fereidoon A, Ganji MD, Sharifi N (2013) Density functional theory based molecular dynamics simulation study on the bulk modulus of multi-shell fullerenes. Physica B 423:1Google Scholar
  38. 38.
    Peón-Escalante R, Villanueva C, Quintal R, Avilés F, Tapia A (2014) The bond force constant and bulk modulus of C60. Comput Mater Sci 83:120Google Scholar
  39. 39.
    Ortíz-Saavedra J, Aguilera-Granja F, Dorantes-Dávila J, Morán-López JL (1993) The s and p character of the electronic structure of C20, C60 and C70. Solid State Commun 85:767Google Scholar
  40. 40.
    Adams GB, O’Keeffe M, Ruoff RS (1994) Van der Waals surface areas and volumes of fullerenes. J Phys Chem 98:9465CrossRefGoogle Scholar
  41. 41.
    Zubov VI, Tretiakov NP, Teixeira-Rabelo JN, Sanchez-Ortiz JF (1994) Calculations of the thermal expansion, cohesive energy and thermodynamic stability of a Van der Waals crystal-fullerene C60. Phys Lett A 194:223Google Scholar
  42. 42.
    Li AY, Li QS (1998) A new method for electronic structure of the fullerene C20. J Mol Struct THEOCHEM 432:115Google Scholar
  43. 43.
    Wang ZY, Su KH, Fan HQ, Hu LD, Wang X, Li YL et al (2007) Mechanical and electronic properties of C60 under structure distortion studied with density functional theory. Comput Mater Sci 40:537Google Scholar
  44. 44.
    Bai H, Du R, Qiao W, Huang Y (2010) Structures, stabilities and electronic properties of C50 dimers. J Mol Struct THEOCHEM 961(1–3):42Google Scholar
  45. 45.
    Bai H, Qiao W, Zhu Y, Huang Y (2012) Theoretical study on one-dimensional C50 polymers. Diam Relat Mater 26:20Google Scholar
  46. 46.
    Fan Z-Q, Zhang Z-H, Deng X-Q, Tang G-P, Zhu HL, Ren Y et al (2014) Structural, electronic, and transport properties of Ih-symmetry-breaking C60 isomers. Comput Mater Sci 91:15Google Scholar
  47. 47.
    Belytschko T (2009) A review of extended/generalized finite element methods for material modeling. Model Simul Mater Sci Eng 17:043001CrossRefGoogle Scholar
  48. 48.
    Computational chemistry list LTD (1999) Columbus, Ohio, (Accessed in April 2014)
  49. 49.
    Hohenberg P, Kohn W (1964) Inhomogeneous electron gas. Phys Rev 136:B864CrossRefGoogle Scholar
  50. 50.
    Perdew JP, Burke K, Ernzerhof M (1996) Generalized gradient approximation made simple. Phys Rev Lett 77:3865CrossRefGoogle Scholar
  51. 51.
    Ordejón P, Artacho E, Soler JM (1996) Self-consistent order-n density functional calculations for very large systems. Phys Rev B 53, R10441Google Scholar
  52. 52.
    Soler JM, Artacho E, Gale JD, García A, Junquera J, Ordejón P, Sánchez-Portal D (2002) The SIESTA method for ab initio order-N materials simulation. J Phys Condens Matter 14:2745Google Scholar
  53. 53.
    Troullier N, Martins JL (1991) Efficient pseudopotentials for plane-wave calculations. Phys Rev B 43:1993Google Scholar
  54. 54.
    Anglada E, Soler JM, Junquera J, Artacho E (2002) Systematic generation of finite-range atomic basis sets for linear-scaling calculations. Phys Rev B 66:205101Google Scholar
  55. 55.
    Gobre VV, Tkatchenko A (2013) Scaling laws for van der Waals interactions in nanostructured materials. Nat Commun 4:2341CrossRefGoogle Scholar
  56. 56.
    Cornell WD, Cieplak P, Bayly CI, Gould IR, Merz KM Jr, Ferguson DM et al (1995) A second generation force field for the simulation of proteins, nucleic acids, and organic molecules. J Am Chem Soc 117:5179CrossRefGoogle Scholar
  57. 57.
    Arulmozhiraja S, Kolandaivel P (1997) Force constants and chemical hardness. Mol Phys 92:353CrossRefGoogle Scholar
  58. 58.
    de Berg M, Cheong O, van Kreveled M, Overmars M (2008) Computational geometry: algorithms and applications. Springer, BerlinCrossRefGoogle Scholar
  59. 59.
    Ansys 13.0, 2010. Swanson analysis systems, HoustonGoogle Scholar
  60. 60.
    Li C, Chou T-W (2003) A structural mechanics approach for the analysis of carbon nanotubes. Int J Solids Struct 40:2487Google Scholar
  61. 61.
    Kalamkarov AAL, Georgiades AV, Rokkam SK, Veedu VP, Ghasemi Nejhad MN (2006) Analytical and numerical techniques to predict carbon nanotubes properties. Int J Solids Struct 43:6832CrossRefGoogle Scholar
  62. 62.
    Tserpes KI, Papanikos P (2005) Finite element modeling of single-walled carbon nanotubes. Compos Part B 36:468Google Scholar
  63. 63.
    Giannopoulos GI, Kakavas PA, Anifantis NK (2008) Evaluation of the effective mechanical properties of single walled carbon nanotubes using a spring based finite element approach. Comput Mater Sci 41:561CrossRefGoogle Scholar
  64. 64.
    Saada AS (1974) Elasticity: Theory and Applications. Robert E. Krieger publishing Co., Malabar, FloridaGoogle Scholar
  65. 65.
    Kaur N, Gupta S, Jindal VK, Dharamvir K (2010) Pressure induced transformations in condensed and molecular phases of C60. Carbon 48:744CrossRefGoogle Scholar
  66. 66.
    Weeks DE, Harter WG (1988) Vibrational frequencies and normal modes of buckminsterfullerene. Chem Phys Lett 144:366CrossRefGoogle Scholar
  67. 67.
    Crawford BL, Miller FA (1949) The planar vibrations of benzene. J Chem Phys 17:249CrossRefGoogle Scholar
  68. 68.
    Rappé AK, Casewit CJ, Colwell KS, Goddard WA III, Skid WM (1992) UFF, a full periodic table force field for molecular mechanics and molecular dynamics simulations. J Am Chem Soc 114:10024CrossRefGoogle Scholar
  69. 69.
    Giannopoulos GI, Georgantzinos SK, Kakavas PA, Anifantis NK (2013) Radial stiffness and natural frequencies of fullerenes via a structural mechanics spring-based method. Fullerenes Nanotubes Carbon Nanostruct 21:248Google Scholar
  70. 70.
    Medina J, Avilés F, Tapia A (2015) The bond force constant of graphene and benzene calculated by density functional theory. Mol Phys. doi: 10.1080/00268976.2014.986241 Google Scholar
  71. 71.
    Amer MS, Maguire JF (2009) On the compressibility of C60 individual molecules. Chem Phys Lett 476:232Google Scholar
  72. 72.
    Horikawa T, Kinoshita T, Suito K, Onodera A (2000) Compressibility measurement of C60 using synchrotron radiation. Solid State Commun 114:121Google Scholar
  73. 73.
    Levin VM, Blank VD, Prokhorov VM, Soifer JM, Kobelev NP (2000) Elastic modules of solid C60: measurement and relationship with nanostructure. J Phys Chem Solids 6:1017CrossRefGoogle Scholar
  74. 74.
    Popov M, Mordkovich V, Perfilov S, Kirichenko A, Kulnitskiy B, Perezhogin I, Blank V (2014) Synthesis of ultrahard fullerite with a catalytic 3D polymerization reaction of C60. Carbon 76:250CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • A. Tapia
    • 1
    Email author
  • C. Villanueva
    • 1
  • R. Peón-Escalante
    • 1
  • R. Quintal
    • 1
  • J. Medina
    • 1
    • 2
  • F. Peñuñuri
    • 1
  • F. Avilés
    • 1
    • 2
  1. 1.Facultad de IngenieríaUniversidad Autónoma de YucatánMéridaMexico
  2. 2.Centro de Investigación Científica de Yucatán, A.CUnidad de MaterialesMéridaMexico

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