Advertisement

Archive for Rational Mechanics and Analysis

, Volume 231, Issue 1, pp 465–517 | Cite as

Characterization of Optimal Carbon Nanotubes Under Stretching and Validation of the Cauchy–Born Rule

  • Manuel FriedrichEmail author
  • Edoardo Mainini
  • Paolo Piovano
  • Ulisse Stefanelli
Open Access
Article

Abstract

Carbon nanotubes are modeled as point configurations and investigated by minimizing configurational energies including two- and three-body interactions. Optimal configurations are identified with local minima and their fine geometry is fully characterized in terms of lower-dimensional problems. Under moderate tension, we prove the existence of periodic local minimizers, which indeed validates the so-called Cauchy–Born rule in this setting.

Notes

Acknowledgements

Open access funding provided by Austrian Science Fund (FWF). M.F. acknowledges support from the Alexander von Humboldt Stiftung. E.M. acknowledges support from the Austrian Science Fund (FWF) project M 1733-N20. P. P. acknowledges support from the Austrian Science Fund (FWF) project P 29681, and from the Vienna Science and Technology Fund (WWTF), the City of Vienna, and the Berndorf Private Foundation through Project MA16-005. U.S. acknowledges support from the Austrian Science Fund (FWF) projects P 27052, I 2375, and F 65 and from the Vienna Science and Technology Fund (WWTF) through project MA14-009. The authors would like to acknowledge the kind hospitality of the Erwin Schrödinger International Institute for Mathematics and Physics, where part of this research was developed under the frame of the thematic program Nonlinear Flows.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Agrawal, P.M., Sudalayandi, B.S., Raff, L.M., Komandur, R.: Molecular dynamics (MD) simulations of the dependence of C-C bond lengths and bond angles on the tensile strain in single-wall carbon nanotubes (SWCNT). Comput. Mater. Sci. 41, 450–456 (2008)CrossRefGoogle Scholar
  2. 2.
    Allinger, N.L.: Molecular Structure: Understanding Steric and Electronic Effects from Molecular Mechanics. Wiley, New York (2010)CrossRefGoogle Scholar
  3. 3.
    Arroyo, M., Belytschko, T.: Continuum mechanics modeling and simulation of carbon nanotubes. Meccanica 40, 455–469 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bajaj, C., Favata, A., Podio-Guidugli, P.: On a nanoscopically-informed shell theory of single-wall carbon nanotubes. Eur. J. Mech. A Solids 42, 137–157 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Braides, A., Lew, A., Ortiz, M.: Effective cohesive behavior of layers of interatomic planes. Arch. Ration. Mech. Anal. 180, 151–182 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brenner, D.W.: Empirical potential for hydrocarbons for use in stimulating the chemical vapor deposition of diamond films. Phys. Rev. B 42, 9458–9471 (1990)ADSCrossRefGoogle Scholar
  7. 7.
    Brook, B.R., Bruccoleri, R.E., Olafson, B.D., States, D.J., Swaminathan, S., Karplus, M.: CHARMM: a program for macromolecular energy, minimization, and dynamics calculations. J. Comput. Chem. 4, 187–217 (1983)CrossRefGoogle Scholar
  8. 8.
    Budyka, M.E., Zyubina, T.S., Ryabenko, A.G., Lin, S.H., Mebel, A.M.: Bond lengths and diameters of armchair single-walled carbon nanotubes. Chem. Phys. Lett. 407, 266–271 (2005)ADSCrossRefGoogle Scholar
  9. 9.
    Cao, G.X., Chen, X.: The effects of chirality and boundary conditions on the mechanical properties of single-wall carbon nanotubes. Int. J. Solid. Struct. 44, 5447–5465 (2007)CrossRefzbMATHGoogle Scholar
  10. 10.
    Charlier, J.-C., Lambin, Ph: Electronic structure of carbon nanotubes with chiral symmetry. Phys. Rev. B 57, R15037 (1998)ADSCrossRefGoogle Scholar
  11. 11.
    Clark, M., Cramer III, R.D., Van Opdenbosch, N.: Validation of the general purpose tripos 5.2 force field. J. Comput. Chem. 10, 982–1012 (1989)CrossRefGoogle Scholar
  12. 12.
    Clayden, J., Greeves, N., Warren, S.G.: Organic Chemistry. Oxford University Press, Oxford (2012)Google Scholar
  13. 13.
    Conti, S., Dolzmann, G., Kirchheim, B., Müller, S.: Sufficient conditions for the validity of the Cauchy-Born rule close to \(SO(n)\). J. Eur. Math. Soc. (JEMS) 8, 515–530 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cox, B.J., Hill, J.M.: Exact and approximate geometric parameters for carbon nanotubes incorporating curvature. Carbon 45, 1453–1462 (2007)CrossRefGoogle Scholar
  15. 15.
    Cox, B.J., Hill, J.M.: Geometric structure of ultra-small carbon nanotubes. Carbon 46, 711–713 (2008)CrossRefGoogle Scholar
  16. 16.
    Davoli, E., Piovano, P., Stefanelli, U.: Wulff shape emergence in graphene. Math. Models Methods Appl. Sci. 26(12), 2277–2310 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Demczyk, B.G., et al.: Direct mechanical measurement of the tensile strength and elastic modulus of multiwalled carbon nanotubes. Mater. Sci. Eng. A 334, 173–178 (2002)CrossRefGoogle Scholar
  18. 18.
    Dresselhaus, M.S., Dresselhaus, G., Saito, R.: Carbon fibers based on \(\text{C}_{60}\) ad their symmetry. Phys. Rev. B 45(11), 6234–6242 (1992)ADSCrossRefGoogle Scholar
  19. 19.
    Dresselhaus, M.S., Dresselhaus, G., Saito, R.: Physics of carbon nanotubes. Carbon 33, 883–891 (1995)CrossRefGoogle Scholar
  20. 20.
    Li, W.E.D.: On the crystallization of 2D hexagonal lattices. Commun. Math. Phys. 286(3), 1099–1140 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ming, W.E.P.: Cauchy-Born rule and the stability of crystalline solids: dynamic problems. Acta Math. Appl. Sin. Engl. Ser. 23, 529–550 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ming, W.E.P.: Cauchy-Born rule and the stability of crystalline solids: static problems. Arch. Ration. Mech. Anal. 183, 241–297 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jorio, A., Dresselhaus, G., Dresselhaus, M.S. (Eds.).: Carbon Nanotubes Advanced Topics in the Synthesis, Structure, Properties and Applications, Topics in Applied Physics, vol. 111. Springer, New York, 2011Google Scholar
  24. 24.
    El Kass, D., Monneau, R.: Atomic to continuum passage for nanotubes: a discrete Saint-Venant principle and error estimates. Arch. Ration. Mech. Anal. 213, 25–128 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ericksen, J.L.: On the Cauchy-Born rule. Math. Mech. Solids 13, 199–220 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Ericksen, J.L.: The Cauchy and Born hypotheses for crystals. In: Phase Transformations and Material Instabilities in Solids (Madison, Wis., 1983), vol. 52, pp. 61–77, Publ. Math. Res. Center Univ. Wisconsin. Academic Press, Orlando, 1984Google Scholar
  27. 27.
    Farmer, B., Esedoḡlu, S., Smereka, P.: Crystallization for a Brenner-like potential. Commun. Math. Phys. 349, 1029–1061 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Favata, A., Podio-Guidugli, P.: A new CNT-oriented shell theory. Eur. J. Mech. A/Solids 35, 75–96 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Favata, A., Micheletti, A., Podio-Guidugli, P.: A nonlinear theory of prestressed elastic stick-and-spring structures. J. Eng. Sci. 80, 4–20 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Favata, A., Podio-Guidugli, P.: A shell theory for carbon nanotube of arbitrary chirality. In: Shell and Membrane Theories in Mechanics and Biology. Advanced Structured Materials, vol. 45, pp. 155–167. Springer, Cham, 2015Google Scholar
  31. 31.
    Favata, A., Micheletti, A., Podio-Guidugli, P., Pugno, N.M.: Geometry and self-stress of single-wall carbon nanotubes and graphene via a discrete model based on a 2nd-generation REBO potential. J. Elast. 125, 1–37 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Friedrich, M., Piovano, P., Stefanelli, U.: The geometry of \(C_{60}\): a rigorous approach via molecular mechanics. SIAM J. Appl. Math. 76, 2009–2029 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Friedrich, M., Schmidt, B.: An atomistic-to-continuum analysis of crystal cleavage in a two-dimensional model problem. J. Nonlinear Sci. 24, 145–183 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Friedrich, M., Schmidt, B.: An analysis of crystal cleavage in the passage from atomistic models to continuum theory. Arch. Ration. Mech. Anal. 217, 263–308 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Friesecke, G., James, R.D., Müller, S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Commun. Pure Appl. Math. 55, 1461–1506 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Friesecke, G., Theil, F.: Validity and failure of the Cauchy-Born hypothesis in a two-dimensional mass-spring lattice. J. Nonlinear Sci. 12, 445–478 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Geim, A.K., Novoselov, K.S.: The rise of graphene. Nat. Mater. 6, 183–191 (2007)ADSCrossRefGoogle Scholar
  38. 38.
    van Gunsteren, W.F., Berendsen, H.J.C.: Groningen Molecular Simulation (GROMOS) Library Manual. BIOMOS b.v, Groningen (1987)Google Scholar
  39. 39.
    Gupta, A., Sakthivela, T., Seal, S.: Recent development in 2D materials beyond graphene. Progr. Mat. Sci. 73, 44–126 (2015)CrossRefGoogle Scholar
  40. 40.
    Han, F., Azdoud, Y., Lubineau, G.: Computational modeling of elastic properties of carbon nanotube/polymer composites with interphase regions. Part I: micro-structural characterization and geometric modeling. Comput. Mater. Sci. 81, 641–651 (2014)CrossRefGoogle Scholar
  41. 41.
    Iijima, S.: Helical microtubules of graphitic carbon. Nature 354, 56–58 (1991)ADSCrossRefGoogle Scholar
  42. 42.
    James, R.D.: Objective structures. J. Mech. Phys. Solids 54, 2354–2390 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Jiang, H., Zhang, P., Liu, B., Huans, Y., Geubelle, P.H., Gao, H., Hwang, K.C.: The effect of nanotube radius on the constitutive model for carbon nanotubes. Comput. Mater. Sci. 28, 429–442 (2003)CrossRefGoogle Scholar
  44. 44.
    Jindal, V.K., Imtani, A.N.: Bond lengths of armchair single-walled carbon nanotubes and their pressure dependence. Comput. Mater. Sci. 44, 156–162 (2008)CrossRefGoogle Scholar
  45. 45.
    Jishi, R.A., Dresselhaus, M.S., Dresselhaus, G.: Symmetry properties and chiral carbon nanotubes. Phys. Rev. B 47, 166671–166674 (1993)CrossRefGoogle Scholar
  46. 46.
    Kanamitsu, K., Saito, S.: Geometries, electronic properties, and energetics of isolated single-walled carbon nanotubes. J. Phys. Soc. Jpn. 71(2), 483–486 (2002)ADSCrossRefGoogle Scholar
  47. 47.
    Krishnan, A., Dujardin, E., Ebbesen, T.W., Yianilos, P.N., Treacy, M.M.J.: Young's modulus of single-walled nanotubes. Phys. Rev. B 58, 14013–14019 (1998)ADSCrossRefGoogle Scholar
  48. 48.
    Kroto, H.W., Heath, J.R., O'Brien, S.C., Curl, R.F., Smalley, R.E.: C 60: buckminsterfullerene. Nature 318, 162–163 (1985)ADSCrossRefGoogle Scholar
  49. 49.
    Kroto, H.W.: The stability of the fullerenes \(C_n\), with \(n=24, 28, 32, 36, 50, 60\) and \(70\). Nature 329, 529–531 (1987)ADSCrossRefGoogle Scholar
  50. 50.
    Kurti, J., Zolyomi, V., Kertesz, M., Sun, G.: The geometry and the radial breathing model of carbon nanotubes: Beyond the ideal behaviour. New J. Phys. 5, 1–21 (2003)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Lazzaroni, G., Stefanelli, U.: Chain-like ground states in three dimensions. In preparation, 2017Google Scholar
  52. 52.
    Lee, R.K.F., Cox, B.J., Hill, J.M.: General rolled-up and polyhedral models for carbon nanotubes. Fuller. Nanotub. Carbon Nanostruct. 19, 726–748 (2011)ADSCrossRefGoogle Scholar
  53. 53.
    Lewars, E.G.: Computational Chemistry, 2nd edn. Springer, New York (2011)CrossRefzbMATHGoogle Scholar
  54. 54.
    Li, X., Yang, W., Liu, B.: Bending induced rippling and twisting of multiwalled carbon nanotubes. Phys. Rev. Lett. 98, 205502–205505 (2007)ADSCrossRefGoogle Scholar
  55. 55.
    Mainini, E., Murakawa, H., Piovano, P., Stefanelli, U.: Carbon-nanotube geometries: analytical and numerical results. Discrete Contin. Dyn. Syst. Ser. S 10, 141–160 (2017)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Mainini, E., Murakawa, H., Piovano, P., Stefanelli, U.: Carbon-nanotube geometries as optimal configurations. Multiscale Model. Simul. 15(4), 1448–1471 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Mainini, E., Stefanelli, U.: Crystallization in carbon nanostructures. Commun. Math. Phys. 328(2), 545–571 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Mannix, A.J., Kiraly, B., Hersma, M.C., Guisiger, N.P.: Synthesis and chemistry of elemental 2D materials. Nat. Rev. Chem. 1, 14 (2017)CrossRefGoogle Scholar
  59. 59.
    Mas-Ballesté, R., Gómez-Navarro, C., Gómez-Herrero, J., Zamora, F.: 2D materials: to graphene and beyond. Nanoscale 3, 20 (2011)ADSCrossRefGoogle Scholar
  60. 60.
    Mayo, S.L., Olafson, B.D., Goddard, W.A.: DREIDING: a generic force field for molecular simulations. J. Phys. Chem. 94, 8897–8909 (1990)CrossRefGoogle Scholar
  61. 61.
    Morris, J.E., Iniewski, K.: Graphene, carbon nanotubes, and nanostructures: techniques and applications. CRC Press, Boca Raton (2013)Google Scholar
  62. 62.
    Novoselov, K.S., et al.: Two-dimensional gas of massless Dirac fermions in graphene. Nature 438, 197–200 (2005)ADSCrossRefGoogle Scholar
  63. 63.
    Poncharal, P., Wang, Z.L., Ugarte, D., de Heer, W.A.: Electrostatic deflections and electro-mechanical resonances of carbon nanotubes. Science 283, 1513–1516 (1999)ADSCrossRefGoogle Scholar
  64. 64.
    Rappé, A.K., Casewit, C.L.: Molecular Mechanics Across Chemistry. University Science Books, Sausalito, CA (1997)Google Scholar
  65. 65.
    Rochefort, A., et al.: Electrical and mechanical properties of distorted carbon nanotubes. Phys. Rev. B 60, 13824–13830 (1999)ADSCrossRefGoogle Scholar
  66. 66.
    Ru, C.Q.: Axially compressed buckling of a doublewalled carbon nanotube embedded in an elastic medium. J. Mech. Phys. Solids 49, 1265–1279 (2001)ADSCrossRefzbMATHGoogle Scholar
  67. 67.
    Schmidt, B.: On the derivation of linear elasticity from atomistic models. Netw. Heterog. Media 4, 789–812 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    Stefanelli, U.: Stable carbon configurations. Boll. Unione Mat. Ital 9(10), 335–354 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    Stillinger, F.H., Weber, T.A.: Computer simulation of local order in condensed phases of silicon. Phys. Rev. B 8, 5262–5271 (1985)ADSCrossRefGoogle Scholar
  70. 70.
    Tersoff, J.: New empirical approach for the structure and energy of covalent systems. Phys. Rev. B 37, 6991–7000 (1988)ADSCrossRefGoogle Scholar
  71. 71.
    Treacy, M.M.J., Ebbesen, T.W., Gibson, J.M.: Exceptionally high Young's modulus observed for individual carbon nanotubes. Nature 381, 678–680 (1996)ADSCrossRefGoogle Scholar
  72. 72.
    Tuukkanen, S., et al.: Stretching of solution processed carbon nanotube and graphene nanocomposite films on rubber substrates. Synth. Met. 191, 28–35 (2014)CrossRefGoogle Scholar
  73. 73.
    Wang, X., Wang, X., Xiao, J.: A non-linear analysis of the bending modulus of carbon nanotubes with rippling deformations. Compos. Struct. 69, 315–321 (2005)CrossRefGoogle Scholar
  74. 74.
    Warner, J.H., Young, N.P., Kirkland, A.I., Briggs, G.A.D.: Resolving strain in carbon nanotubes at the atomic level. Nat. Mater. 10, 958–962 (2011)ADSCrossRefGoogle Scholar
  75. 75.
    Weiner, P.K., Kollman, P.A.: AMBER: Assisted model building with energy refinement. A general program for modeling molecules and their interactions. J. Comput. Chem. 2, 287–303 (1981)CrossRefGoogle Scholar
  76. 76.
    Yakobson, B.I., Brabec, C.J., Bernholc, J.: Nanomechanics of carbon tubes: instabilities beyond linear response. Phys. Rev. Lett. 76, 2511–2514 (1996)ADSCrossRefGoogle Scholar
  77. 77.
    Yu, M.-F., Files, B.S., Arepalli, S., Ruoff, R.S.: Tensile loading of ropes of single wall carbon nanotubes and their mechanical properties. Phys. Rev. Lett. 84, 5552–5555 (2000)ADSCrossRefGoogle Scholar
  78. 78.
    Zanzotto, G.: On the material symmetry group of elastic crystals and the Born rule. Arch. Ration. Mech. Anal. 121, 1–36 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  79. 79.
    Zhang, D.-B., Dumitric̆a, T.: Elasticity of ideal single-walled carbon nanotubes via symmetry-adapted tight-binding objective modeling. Appl. Phys. Lett. 93, 031919 (2008)ADSCrossRefGoogle Scholar
  80. 80.
    Zhao, X., Liu, Y., Inoue, S., Jones, R.O., Ando, Y.: Smallest carbon nanotube is \(3\)Å in diameter. Phys. Rev. Lett. 92(12), 125502 (2004)ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Manuel Friedrich
    • 1
    Email author
  • Edoardo Mainini
    • 2
  • Paolo Piovano
    • 3
  • Ulisse Stefanelli
    • 3
    • 4
  1. 1.Applied Mathematics MünsterUniversity of MünsterMünsterGermany
  2. 2.Dipartimento di Ingegneria Meccanica, Energetica, Gestionale e dei Trasporti (DIME)Università degli Studi di GenovaGenovaItaly
  3. 3.Faculty of MathematicsUniversity of ViennaViennaAustria
  4. 4.Istituto di Matematica Applicata e Tecnologie Informatiche “E. Magenes” - CNRPaviaItaly

Personalised recommendations