Buckling and Collapse of Embedded Carbon Nanotubes

O. Lourie; D. M. Cox; H. D. Wagner

doi:10.1103/PhysRevLett.81.1638

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Buckling and Collapse of Embedded Carbon Nanotubes

O. Lourie¹, D. M. Cox^1,2, and H. D. Wagner^1,*

¹Department of Materials & Interfaces, The Weizmann Institute of Science, Rehovot 76100, Israel
²Exxon Research and Engineering, Route 22 East, Annandale, New Jersey 08801

^*Author to whom correspondence should be addressed. Email address: cpwagner@wis.weizmann.ac.il

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Phys. Rev. Lett. 81, 1638 – Published 24 August, 1998

DOI: https://doi.org/10.1103/PhysRevLett.81.1638

Abstract

Experimental observations of various deformation and fracture modes under compression of single multiwalled carbon nanotubes, obtained as a result of embedment within a polymeric film, are reported. Based on a combination of experimental measurements and the theory of elastic stability, the compressive strengths of thin- and thick-walled nanotubes are found to be about 2 orders of magnitude higher than the compressive strength of any known fiber.

References (20)

S. Iijima, Nature (London) 354, 56 (1991).
S. Iijima, C. Brabec, A. Maiti, and J. Bernholc, J. Chem. Phys. 104, 2089 (1996).
M. M. J. Treacy, T. W. Ebbesen, and J. M. Gibson, Nature (London) 381, 678 (1996).
G. Overney, W. Zhong, and D. Tomanek, Z. Phys. D 27, 93 (1993).
D. H. Robertson, D. W. Brener, and J. W. Mintmire, Phys. Rev. B 45, 12 592 (1992).
N. Chopra, L. Benedict, V. Crespi, M. Cohen, S. Louie, and A. Zettl, Nature (London) 377, 135 (1995).
J. M. Molina, S. S. Savinsky, and N. V. Khokhriakov, J. Chem. Phys. 104, 4652 (1996).
B. I. Yakobson, C. J. Brabec, and J. Bernholc, Phys. Rev. Lett. 76, 2511 (1996).
P. M. Ajayan, O. Stephan, C. Colliex, and D. Trauth, Science 265, 1212 (1994).
P. Calvert, Nature (London) 357, 365 (1992).
T. W. Ebbesen, Annu. Rev. Mater. Sci. 24, 235 (1994).
P. M. Ajayan, Condens. Matter News 4, 9 (1995).
A. Kelly and N. H. MacMillan, Strong Solids (Clarendon Press, Oxford,1986), 3rd ed., p. 6.
J. Prescott, Applied Elasticity (Dover Publications, New York,1946), p. 100.
R. Feynman, R. Leyton, and M. Sands, The Feynman Lectures in Physics (Addison-Wesley, Reading, MA,1964), Vol. 2.
S. Timoshenko, Theory of Elastic Stability (McGraw-Hill, New York,1936), Chaps. 2 and 9.
Y. Lanir and Y. C. B. Fung, J. Compos. Mater. 6, 387 (1972).
H. Allen and P. Bulson, Background to Buckling (McGraw-Hill, London,1980), Chap. 7; A. H. Cottrell, The Mechanical Properties of Matter (John Wiley & Sons, New York,1964), Chap. 5.
The morphological singularities under stress predicted by Yakobson et al. [[8]] were calculated for free nanotubes, and for tube structures that are intermediate between the geometries for buckling and collapse observed here. Indeed, assuming that $h$ is equal to the length of a C-C bond (0.14 nm), with $D = 1 nm$ and $L = 6 nm$ [[8]], one has $h / r \approx 0.29$ and $(L / r)_{\min} \approx 12$ , which is transitional between the values of the same parameters for buckling [for which $h / r \approx 0.72$ and $(L / r)_{\min} \approx 16.4$ ] and collapse [for which $h / r \approx 0.08$ and $(L / r)_{\min} \approx 11$ ] observed here. The results presented here and those of Yakobson et al. may therefore be fully compatible.
For thin-walled tubes we have used Euler's expression as an approximation, since it is strictly valid for full rods only. It may be adapted to the case of hollow rods by using $I = (p / 4) ({4 h r}^{3} {+ 4 h}^{3} {r - 6 h}^{2} r^{2} {- h}^{4})$ for the area moment of inertia, and $A = π ({2 r h - h}^{2})$ for the tube cross section. Using $ξ = h / r$ , the resulting expression for the Euler stress is $σ = E (π r / l)^{2} f (ξ)$ , where the function $f (ξ) = (4 ξ + 4 ξ^{3} - 6 ξ^{2} - ξ^{4}) / (2 ξ - ξ^{2})$ decreases monotonically to 1 as $ξ$ tends to 1.

[c1] S. Iijima, Nature (London) 354, 56 (1991).

[c2] S. Iijima, C. Brabec, A. Maiti, and J. Bernholc, J. Chem. Phys. 104, 2089 (1996).

[c3] M. M. J. Treacy, T. W. Ebbesen, and J. M. Gibson, Nature (London) 381, 678 (1996).

[c4] G. Overney, W. Zhong, and D. Tomanek, Z. Phys. D 27, 93 (1993).

[c5] D. H. Robertson, D. W. Brener, and J. W. Mintmire, Phys. Rev. B 45, 12 592 (1992).

[c6] N. Chopra, L. Benedict, V. Crespi, M. Cohen, S. Louie, and A. Zettl, Nature (London) 377, 135 (1995).

[c7] J. M. Molina, S. S. Savinsky, and N. V. Khokhriakov, J. Chem. Phys. 104, 4652 (1996).

[c8] B. I. Yakobson, C. J. Brabec, and J. Bernholc, Phys. Rev. Lett. 76, 2511 (1996).

[c9] P. M. Ajayan, O. Stephan, C. Colliex, and D. Trauth, Science 265, 1212 (1994).

[c10] P. Calvert, Nature (London) 357, 365 (1992).

[c11] T. W. Ebbesen, Annu. Rev. Mater. Sci. 24, 235 (1994).

[c12] P. M. Ajayan, Condens. Matter News 4, 9 (1995).

[c13] A. Kelly and N. H. MacMillan, Strong Solids (Clarendon Press, Oxford,1986), 3rd ed., p. 6.

[c14] J. Prescott, Applied Elasticity (Dover Publications, New York,1946), p. 100.

[c15] R. Feynman, R. Leyton, and M. Sands, The Feynman Lectures in Physics (Addison-Wesley, Reading, MA,1964), Vol. 2.

[c16] S. Timoshenko, Theory of Elastic Stability (McGraw-Hill, New York,1936), Chaps. 2 and 9.

[c17] Y. Lanir and Y. C. B. Fung, J. Compos. Mater. 6, 387 (1972).

[c18] H. Allen and P. Bulson, Background to Buckling (McGraw-Hill, London,1980), Chap. 7; A. H. Cottrell, The Mechanical Properties of Matter (John Wiley & Sons, New York,1964), Chap. 5.

[c19] The morphological singularities under stress predicted by Yakobson et al. [[8]] were calculated for free nanotubes, and for tube structures that are intermediate between the geometries for buckling and collapse observed here. Indeed, assuming that $h$ is equal to the length of a C-C bond (0.14 nm), with $D = 1 nm$ and $L = 6 nm$ [[8]], one has $h / r \approx 0.29$ and $(L / r)_{\min} \approx 12$ , which is transitional between the values of the same parameters for buckling [for which $h / r \approx 0.72$ and $(L / r)_{\min} \approx 16.4$ ] and collapse [for which $h / r \approx 0.08$ and $(L / r)_{\min} \approx 11$ ] observed here. The results presented here and those of Yakobson et al. may therefore be fully compatible.

[c20] For thin-walled tubes we have used Euler's expression as an approximation, since it is strictly valid for full rods only. It may be adapted to the case of hollow rods by using $I = (p / 4) ({4 h r}^{3} {+ 4 h}^{3} {r - 6 h}^{2} r^{2} {- h}^{4})$ for the area moment of inertia, and $A = π ({2 r h - h}^{2})$ for the tube cross section. Using $ξ = h / r$ , the resulting expression for the Euler stress is $σ = E (π r / l)^{2} f (ξ)$ , where the function $f (ξ) = (4 ξ + 4 ξ^{3} - 6 ξ^{2} - ξ^{4}) / (2 ξ - ξ^{2})$ decreases monotonically to 1 as $ξ$ tends to 1.

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Buckling and Collapse of Embedded Carbon Nanotubes

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