Skip to main content
SpringerLink
Log in

Analysis of quantum effects of fine scaling on the axial buckling of MWCNTs based on the density functional theory and molecular mechanics method

  • Published:
Applied Physics A Aims and scope Submit manuscript

Abstract

In this paper, quantum and molecular mechanics are used to study the quantum effects of fine scaling on the buckling strength of multi-walled carbon nanotubes under axial loading, as well as the effects of changes in length, diameter, chirality, wall number and length-to-diameter ratio of the structure. To this end, the total potential energy of the system is calculated with the consideration of both bond stretching and bond angular variations. The density functional theory along with the generalized gradient approximation function is used to obtain the relevant elastic constants of the nanotubes. An excellent agreement is found between the present numerical results and those found in the literature which confirms the validity as well as the accuracy of the present closed-form solution. The results show that in the effective longitudinal range for quantum effects of fine scaling, any change that leads to a change in the size of the structure has significant effects on the buckling strength of the structure. By increasing the diameter due to the increase in the number of walls or chirality and increasing the length of the structure, the critical buckling strength experiences a decreasing trend and this decrease is highly dependent on the increasing of the diameter due to the increase in the number of walls. In addition, zigzag multi-walled carbon nanotubes are more resistant than armchair multi-walled nanotubes, and the critical buckling strain of multi-walled carbon nanotubes with different chiralities is in the range of zigzag and armchair nanotubes. In other words, it can be said that quantum effects of fine scaling cause more buckling strengthening of the structure against external axial loads and with each longitudinal change that reduces the quantum effects of fine scaling, the strength of the structure decreases sharply.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22

Similar content being viewed by others

References

  1. S. Iijima, Helical microtubules of graphitic carbon. Nature 354(6348), 56–58 (1991)

    Article  ADS  Google Scholar 

  2. H. Abe, T. Shimizu, A. Ando, H. Tokumoto, Electric transport and mechanical strength measurements of carbon nanotubes in scanning electron microscope. Phys. E 24(1–2), 42–45 (2004)

    Article  Google Scholar 

  3. B. Hornbostel, P. Pötschke, J. Kotz, S. Roth, Mechanical properties of triple composites of polycarbonate, single-walled carbon nanotubes and carbon fibres. Phys. E 40(7), 2434–2439 (2008)

    Article  Google Scholar 

  4. B. WenXing, Z. ChangChun, C. WanZhao, Simulation of Young’s modulus of single-walled carbon nanotubes by molecular dynamics. Phys. B 352(1–4), 156–163 (2004)

    Article  ADS  Google Scholar 

  5. P. Aghdasi, R. Ansari, Structural and mechanical properties of Sb and SbX (X=H, F, Cl and Br) monolayers. Solid State Commun. 311, 113849 (2020)

    Article  Google Scholar 

  6. P. Aghdasi, R. Ansari, S. Rouhi, M. Goli, On the elastic and plastic properties of the bismuthene adsorbed by H, F, Cl and Br atoms. Superlattices Microstruct. 135, 106242 (2019)

    Article  Google Scholar 

  7. P. Aghdasi, R. Ansari, S. Rouhi, M. Goli, H.A. Gilakjani, Investigating the effects of H and F adsorption on the elastic and plastic properties of arsenene nanosheets. Phys. B: Condens. Matter 574, 411672 (2019)

    Article  Google Scholar 

  8. P. Aghdasi, R. Ansari, S. Rouhi, S. Yousefi, A DFT-based finite element approach for studying elastic properties, buckling and vibration of the arsenene. J. Mol. Gr. Modell. 101, 107725 (2020)

    Article  Google Scholar 

  9. P. Aghdasi, R. Ansari, S. Rouhi, S. Yousefi, M. Goli, H.R. Soleimani, Investigating elastic and plastic characteristics of monolayer phosphorene under atomic adsorption by the density functional theory. Phys. B: Condens. Matter 600, 412603 (2021)

    Article  Google Scholar 

  10. P. Aghdasi, R. Ansari, S. Yousefi, M. Goli, Structural and mechanical properties of pristine and adsorbed puckered arsenene nanostructures: A DFT study. Superlattices Microstruct. 139, 106414 (2020)

    Article  Google Scholar 

  11. P. Aghdasi, S. Yousefi, R. Ansari, Structural and mechanical properties of antimonene monolayers doped with transition metals: a DFT-based study. J. Mol. Model. 27(1), 15 (2021)

    Article  Google Scholar 

  12. M. Goli, R. Ansari, S. Rouhi, P. Aghdasi, S.M. Mozvashi, Influence of F and H adsorption on the elasto-plastic properties of silicene: a DFT investigation. Phys. E: Low-dimens. Syst. Nanostruct. 119, 113984 (2020)

    Article  Google Scholar 

  13. S. Yousefi, R. Ansari, P. Aghdasi, S.M. Mozvashi, Structural and mechanical properties characterization of arsenene nanosheets under doping effect of transition metals: a DFT study. Phys. E: Low-dimens. Syst. Nanostruct. 124, 114349 (2020)

    Article  Google Scholar 

  14. M.J. Treacy, T.W. Ebbesen, J.M. Gibson, Exceptionally high young’s modulus observed for individual carbon nanotubes. Nature 381(6584), 678–68 (1996)

    Article  ADS  Google Scholar 

  15. E.W. Wong, P.E. Sheehan, C.M. Lieber, Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes. Science 277(5334), 1971–1975 (1997)

    Article  Google Scholar 

  16. M.-F. Yu, O. Lourie, M.J. Dyer, K. Moloni, T.F. Kelly, R.S. Ruoff, Strength and breaking mechanism of multiwalled carbon nanotubes under tensile load. Science 287(5453), 637–640 (2000)

    Article  ADS  Google Scholar 

  17. B. Yakobson, Mechanical relaxation and “intramolecular plasticity” in carbon nanotubes. Appl. Phys. Lett. 72(8), 918–920 (1998)

    Article  ADS  Google Scholar 

  18. T. Xiao, X. Xu, K. Liao, Characterization of nonlinear elasticity and elastic instability in single-walled carbon nanotubes. J. Appl. Phys. 95(12), 8145–8148 (2004)

    Article  ADS  Google Scholar 

  19. H.-S. Shen, C.-L. Zhang, Postbuckling of double-walled carbon nanotubes with temperature dependent properties and initial defects under combined axial and radial mechanical loads. Int. J. Solids Struct. 44(5), 1461–1487 (2007)

    Article  MATH  Google Scholar 

  20. M. Dresselhaus, P. Eklund, Phonons in carbon nanotubes. Adv. Phys. 49(6), 705–814 (2000)

    Article  ADS  Google Scholar 

  21. R.F. Gibson, E.O. Ayorinde, Y.-F. Wen, Vibrations of carbon nanotubes and their composites: a review. Compos. Sci. Technol. 67(1), 1–28 (2007)

    Article  Google Scholar 

  22. R. Ansari, M. Mirnezhad, S. Sahmani, An accurate molecular mechanics model for computation of size-dependent elastic properties of armchair and zigzag single-walled carbon nanotubes. Meccanica 48(6), 1355–1367 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. B.I. Yakobson, C. Brabec, J. Bernholc, Nanomechanics of carbon tubes: instabilities beyond linear response. Phys. Rev. Lett. 76(14), 2511 (1996)

    Article  ADS  Google Scholar 

  24. C.-L. Zhang, H.-S. Shen, Buckling and postbuckling analysis of single-walled carbon nanotubes in thermal environments via molecular dynamics simulation. Carbon 44(13), 2608–2616 (2006)

    Article  Google Scholar 

  25. S.-C. Fang, W.-J. Chang, Y.-H. Wang, Computation of chirality-and size-dependent surface Young’s moduli for single-walled carbon nanotubes. Phys. Lett. A 371(5–6), 499–503 (2007)

    Article  ADS  Google Scholar 

  26. C. Li, T.-W. Chou, Modeling of elastic buckling of carbon nanotubes by molecular structural mechanics approach. Mech. Mater. 36(11), 1047–1055 (2004)

    Article  Google Scholar 

  27. R. Ansari, R. Rajabiehfard, B. Arash, Nonlocal finite element model for vibrations of embedded multi-layered graphene sheets. Comput. Mater. Sci. 49(4), 831–838 (2010)

    Article  Google Scholar 

  28. R. Ansari, S. Rouhi, Atomistic finite element model for axial buckling of single-walled carbon nanotubes. Phys. E 43(1), 58–69 (2010)

    Article  Google Scholar 

  29. R. Ansari, S. Sahmani, B. Arash, Nonlocal plate model for free vibrations of single-layered graphene sheets. Phys. Lett. A 375(1), 53–62 (2010)

    Article  ADS  Google Scholar 

  30. G.M. Odegard, T.S. Gates, L.M. Nicholson, K.E. Wise, Equivalent-continuum modeling of nano-structured materials. Compos. Sci. Technol. 62(14), 1869–1880 (2002)

    Article  Google Scholar 

  31. T. Vodenitcharova, L. Zhang, Effective wall thickness of a single-walled carbon nanotube. Phys. Rev. B 68(16), 165401 (2003)

    Article  ADS  Google Scholar 

  32. C. Ru, Effective bending stiffness of carbon nanotubes. Phys. Rev. B 62(15), 9973 (2000)

    Article  ADS  Google Scholar 

  33. T. Chang, H. Gao, Size-dependent elastic properties of a single-walled carbon nanotube via a molecular mechanics model. J. Mech. Phys. Solids 51(6), 1059–1074 (2003)

    Article  ADS  MATH  Google Scholar 

  34. L. Shen, J. Li, Transversely isotropic elastic properties of single-walled carbon nanotubes. Phys. Rev. B 69(4), 045414 (2004)

    Article  ADS  Google Scholar 

  35. J. Xiao, B. Gama, J. Gillespie Jr., An analytical molecular structural mechanics model for the mechanical properties of carbon nanotubes. Int. J. Solids Struct. 42(11–12), 3075–3092 (2005)

    Article  MATH  Google Scholar 

  36. H.-S. Shen, Postbuckling prediction of double-walled carbon nanotubes under hydrostatic pressure. Int. J. Solids Struct. 41(9–10), 2643–2657 (2004)

    Article  MATH  Google Scholar 

  37. P. Poncharal, Z. Wang, D. Ugarte, W.A. De Heer, Electrostatic deflections and electromechanical resonances of carbon nanotubes. Science 283(5407), 1513–1516 (1999)

    Article  ADS  Google Scholar 

  38. C. Wang, C. Ru, A. Mioduchowski, Axially compressed buckling of pressured multiwall carbon nanotubes. Int. J. Solids Struct. 40(15), 3893–3911 (2003)

    Article  MATH  Google Scholar 

  39. S. Seifoori, F. Abbaspour, E. Zamani, Molecular dynamics simulation of impact behavior in multi-walled carbon nanotubes. Superlattices and Microstruct. 140, 106447 (2020)

    Article  Google Scholar 

  40. R. Rahmani, M. Antonov, Axial and torsional buckling analysis of single- and multi-walled carbon nanotubes: finite element comparison between armchair and zigzag types. SN Appl. Sci. 1(9), 1134 (2019)

    Article  Google Scholar 

  41. J. Ding, L. Cheng, Ultra-high three-point bending fatigue fracture characteristics of CFRP modified by MWCNTs and fatigue life data analysis. Compos. Struct. 259, 113468 (2021)

    Article  Google Scholar 

  42. L. Li, S. Chen, Z. Wei, X. Qi, M. Xia, Y. Wang, Experimental and DFT study of thiol-stabilized Pt/CNTs catalysts. Phys. Chem. Chem. Phys. 14(48), 16581–16587 (2012)

    Article  Google Scholar 

  43. S. Beigy, J. Akbarian, “Study of mechanical properties of modified and unmodified double-walled carbon nanotubes,” (in en). Medbiotech J. 03(04), 150–154 (2019)

    Google Scholar 

  44. S. Baroni et al., "Quantum ESPRESSO: open-source package for research in electronic structure, simulation, and optimization," Code available from https://www/quantum-espresso.org2005

  45. A. Szabo, N.S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory (Courier Corporation, Chelmsford, 2012).

    Google Scholar 

  46. L. Hedin, New method for calculating the one-particle green’s function with application to the electron-gas problem. Phys. Rev. 139(3A), A796 (1965)

    Article  ADS  Google Scholar 

  47. J.P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple. Phys. Rev. Lett. 77(18), 3865 (1996)

    Article  ADS  Google Scholar 

  48. J.P. Perdew, K. Burke, Y. Wang, Generalized gradient approximation for the exchange-correlation hole of a many-electron system. Physical Review B 54(23), 16533 (1996)

    Article  ADS  Google Scholar 

  49. M. Mirnezhad, R. Ansari, S.R. Falahatgar, P. Aghdasi, Torsional buckling analysis of MWCNTs considering quantum effects of fine scaling based on DFT and molecular mechanics method. J. Mol. Gr. Modell. 104, 107843 (2021)

    Article  Google Scholar 

  50. N.L. Allinger, Conformational analysis. 130. MM2. A hydrocarbon force field utilizing V1 and V2 torsional terms. J. Am. Chem. Soc. 99(25), 8127–8134 (1977)

    Article  Google Scholar 

  51. U. Burkert, Molecular mechanics. ACS monograph 17, 114–115 (1982)

    Google Scholar 

  52. A.R. Leach, Molecular Modelling: Principles and Applications 2001 (Prentice Hall, Harlow, 1996), p. 2

    Google Scholar 

  53. R. Ansari, M. Mirnezhad, H. Rouhi, M. Bazdid-Vahdati, Prediction of torsional buckling behaviour of single-walled SiC nanotubes based on molecular mechanics. Eng. Comput. 32(6), 1837–1866 (2015)

    Article  Google Scholar 

  54. C. White, D. Robertson, J. Mintmire, Helical and rotational symmetries of nanoscale graphitic tubules. Phys. Rev. B 47(9), 5485 (1993)

    Article  ADS  Google Scholar 

  55. Y. Zhang, V. Tan, C. Wang, Effect of strain rate on the buckling behavior of single-and double-walled carbon nanotubes. Carbon 45(3), 514–523 (2007)

    Article  Google Scholar 

  56. T. Chang, G. Li, X. Guo, Elastic axial buckling of carbon nanotubes via a molecular mechanics model. Carbon 43(2), 287–294 (2005)

    Article  MathSciNet  Google Scholar 

  57. K. Liew, X. He, S. Kitipornchai, Buckling characteristics of embedded multi-walled carbon nanotubes. Proc. R. Soc. A: Math., Phys Eng. Sci. 461(2064), 3785–3805 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  58. K. Liew, C. Wong, X. He, M. Tan, S. Meguid, Nanomechanics of single and multiwalled carbon nanotubes. Phys. Review B 69(11), 115429 (2004)

    Article  ADS  Google Scholar 

  59. D. Kulathunga, K. Ang, J. Reddy, Accurate modeling of buckling of single-and double-walled carbon nanotubes based on shell theories. J. Phys.: Condens. Matter 21(43), 435301 (2009)

    Google Scholar 

  60. T. Chang, W. Guo, X. Guo, Buckling of multiwalled carbon nanotubes under axial compression and bending via a molecular mechanics model. Phys. Rev. B 72(6), 064101 (2005)

    Article  ADS  Google Scholar 

  61. M. Meo, M. Rossi, Prediction of Young’s modulus of single wall carbon nanotubes by molecular-mechanics based finite element modelling. Compos. Sci. Technol. 66(11–12), 1597–1605 (2006)

    Article  Google Scholar 

  62. J.S. Bunch et al., Electromechanical resonators from graphene sheets. Science 315(5811), 490–493 (2007)

    Article  ADS  Google Scholar 

  63. I. Frank, D.M. Tanenbaum, A.M. van der Zande, P.L. McEuen, Mechanical properties of suspended graphene sheets. J. Vac. Sci. Technol. B: Microelectron. Nanometer Struct. Process., Measurement, Phenomena 25(6), 2558–2561 (2007)

    Article  ADS  Google Scholar 

  64. S. Kirtania, D. Chakraborty, Finite element based characterization of carbon nanotubes. J. Reinf. Plast. Compos. 26(15), 1557–1570 (2007)

    Article  ADS  Google Scholar 

  65. R. Khare et al., Coupled quantum mechanical/molecular mechanical modeling of the fracture of defective carbon nanotubes and graphene sheets. Phys. Rev. B 75(7), 075412 (2007)

    Article  ADS  Google Scholar 

  66. M.M. Shokrieh, R. Rafiee, Prediction of Young’s modulus of graphene sheets and carbon nanotubes using nanoscale continuum mechanics approach. Mater. Des. 31(2), 790–795 (2010)

    Article  Google Scholar 

  67. K. Min, N.R. Aluru, Mechanical properties of graphene under shear deformation. Appl. Phys. Lett. 98(1), 013113 (2011)

    Article  ADS  Google Scholar 

  68. Q. Lu, R. Huang, Nonlinear mechanical properties of graphene nanoribbons. Mater Res Soc 12, 1284 (2011)

    Google Scholar 

  69. X. Lu, Z. Hu, Mechanical property evaluation of single-walled carbon nanotubes by finite element modeling. Compos. B Eng. 43(4), 1902–1913 (2012)

    Article  MathSciNet  Google Scholar 

  70. S. Iijima, C. Brabec, A. Maiti, J. Bernholc, Structural flexibility of carbon nanotubes. J. Chem. Phys. 104(5), 2089–2092 (1996)

    Article  ADS  Google Scholar 

  71. J.P. Lu, Elastic properties of carbon nanotubes and nanoropes. Phys. Rev. Lett. 79(7), 1297 (1997)

    Article  ADS  Google Scholar 

  72. A. Krishnan, E. Dujardin, T. Ebbesen, P. Yianilos, M. Treacy, Young’s modulus of single-walled nanotubes. Phys. Rev. B 58(20), 14013 (1998)

    Article  ADS  Google Scholar 

  73. O. Lourie, H. Wagner, Evaluation of Young’s modulus of carbon nanotubes by micro-Raman spectroscopy. J. Mater. Res. 13(9), 2418–2422 (1998)

    Article  ADS  Google Scholar 

  74. G. Gao, T. Cagin, W.A. Goddard III., Energetics, structure, mechanical and vibrational properties of single-walled carbon nanotubes. Nanotechnology 9(3), 184 (1998)

    Article  ADS  Google Scholar 

  75. J.-P. Salvetat et al., Mechanical properties of carbon nanotubes. Appl. Phys. A 69(3), 255–260 (1999)

    Article  ADS  Google Scholar 

  76. J.P. Salvetat et al., Elastic modulus of ordered and disordered multiwalled carbon nanotubes. Adv. Mater. 11(2), 161–165 (1999)

    Article  Google Scholar 

  77. J.-P. Salvetat et al., Elastic and shear moduli of single-walled carbon nanotube ropes. Phys. Rev. Lett. 82(5), 944 (1999)

    Article  ADS  Google Scholar 

  78. M.-F. Yu, B.S. Files, S. Arepalli, R.S. Ruoff, Tensile loading of ropes of single wall carbon nanotubes and their mechanical properties. Phys. Rev. Lett. 84(24), 5552 (2000)

    Article  ADS  Google Scholar 

  79. T.W. Tombler et al., Reversible electromechanical characteristics of carbon nanotubes underlocal-probe manipulation. Nature 405(6788), 769–772 (2000)

    Article  ADS  Google Scholar 

  80. C. Li, T.-W. Chou, A structural mechanics approach for the analysis of carbon nanotubes. Int. J. Solids Struct. 40(10), 2487–2499 (2003)

    Article  MATH  Google Scholar 

  81. W. Duan, Q. Wang, K.M. Liew, X. He, Molecular mechanics modeling of carbon nanotube fracture. Carbon 45(9), 1769–1776 (2007)

    Article  Google Scholar 

  82. W. Ding, L. Calabri, K. Kohlhaas, X. Chen, D. Dikin, R. Ruoff, Modulus, fracture strength, and brittle vs. plastic response of the outer shell of arc-grown multi-walled carbon nanotubes. Exp. Mech. 47(1), 25–36 (2007)

    Article  Google Scholar 

  83. J.-N. Lu, H.-B. Chen, Analysis of single-walled carbon nanotubes using a chemical bond element model. Chin. J. Chem. Phys. 21(4), 353 (2008)

    Article  Google Scholar 

  84. A.F. Ávila, G.S.R. Lacerda, Molecular mechanics applied to single-walled carbon nanotubes. Mater Res 11(3), 325–333 (2008)

    Article  Google Scholar 

  85. M. Huang, Studies of mechanically deformed single wall carbon nanotubes and graphene by optical spectroscopy.( Citeseer, 2009)

  86. M. Rossi, M. Meo, On the estimation of mechanical properties of single-walled carbon nanotubes by using a molecular-mechanics based FE approach. Compos. Sci. Technol. 69(9), 1394–1398 (2009)

    Article  Google Scholar 

  87. R. Faccio, P.A. Denis, H. Pardo, C. Goyenola, A.W. Mombrú, Mechanical properties of graphene nanoribbons. J. Phys.: Condens. Matter 21(28), 285304 (2009)

    Google Scholar 

  88. E. Mohammadpour, M. Awang, M.Z. Abdullah, Predicting the Young’s modulus of single-walled carbon nanotubes using finite element modeling. J. Appl. Sci. 11(9), 1653–1657 (2011)

    Article  ADS  Google Scholar 

  89. M.R. Falvo et al., Bending and buckling of carbon nanotubes under large strain. Nature 389(6651), 582–584 (1997)

    Article  ADS  Google Scholar 

  90. C. Ru, Elastic buckling of single-walled carbon nanotube ropes under high pressure. Phys Rev B 62(15), 10405 (2000)

    Article  ADS  Google Scholar 

  91. B.G. Demczyk et al., Direct mechanical measurement of the tensile strength and elastic modulus of multiwalled carbon nanotubes. Mater. Sci. Eng., A 334(1–2), 173–178 (2002)

    Article  Google Scholar 

  92. D. Bozovic, M. Bockrath, J.H. Hafner, C.M. Lieber, H. Park, M. Tinkham, Plastic deformations in mechanically strained single-walled carbon nanotubes. Phys Rev B 67(3), 033407 (2003)

    Article  ADS  Google Scholar 

  93. G. Guhados, W. Wan, X. Sun, J.L. Hutter, Simultaneous measurement of Young’s and shear moduli of multiwalled carbon nanotubes using atomic force microscopy. J. Appl. Phys. 101(3), 033514 (2007)

    Article  ADS  Google Scholar 

  94. X.L. Wei, Y. Liu, Q. Chen, M.S. Wang, L.M. Peng, The very-low shear modulus of multi-walled carbon nanotubes determined simultaneously with the axial young’s modulus via in situ experiments. Adv. Func. Mater. 18(10), 1555–1562 (2008)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mechanical Engineering, University of Guilan, UniversityCampus 2, Rasht, Iran

    M. Mirnezhad

  2. Faculty of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran

    R. Ansari, S. R. Falahatgar & P. Aghdasi

Authors
  1. M. Mirnezhad
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Ansari
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. R. Falahatgar
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. P. Aghdasi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Ansari.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mirnezhad, M., Ansari, R., Falahatgar, S.R. et al. Analysis of quantum effects of fine scaling on the axial buckling of MWCNTs based on the density functional theory and molecular mechanics method. Appl. Phys. A 127, 248 (2021). https://doi.org/10.1007/s00339-021-04380-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00339-021-04380-5

Keywords

Search

Navigation