# American Institute of Mathematical Sciences

February  2017, 10(1): 141-160. doi: 10.3934/dcdss.2017008

## Carbon-nanotube geometries: Analytical and numerical results

 1 Dipartimento di Ingegneria meccanica, energetica, gestionale, e dei trasporti (DIME), Università degli Studi di Genova, Piazzale Kennedy 1, I-16129 Genova, Italy 2 Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria 3 Faculty of Mathematics, Kyushu University, 744 Motooka, Nishiku, Fukuoka, 819-0395, Japan 4 Istituto di Matematica Applicata e Tecnologie Informatiche "E. Magenes" -CNR, v. Ferrata 1, I-27100 Pavia, Italy

Received  June 2015 Revised  October 2015 Published  December 2016

We investigate carbon-nanotubes under the perspective ofgeometry optimization. Nanotube geometries are assumed to correspondto atomic configurations whichlocally minimize Tersoff-type interactionenergies. In the specific cases of so-called zigzag and armchairtopologies, candidate optimal configurations are analytically identifiedand their local minimality is numerically checked. Inparticular, these optimal configurations do not correspond neither tothe classical Rolled-up model [5] nor to themore recent polyhedral model [3]. Eventually, theelastic response of the structure under uniaxial testing is numericallyinvestigated and the validity of the Cauchy-Born rule is confirmed.

Citation: Edoardo Mainini, Hideki Murakawa, Paolo Piovano, Ulisse Stefanelli. Carbon-nanotube geometries: Analytical and numerical results. Discrete and Continuous Dynamical Systems - S, 2017, 10 (1) : 141-160. doi: 10.3934/dcdss.2017008
##### References:
 [1] P. M. Agrawal, B. S. Sudalayandi, L. M. Raff and R. Komandur, 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), Comp. Mat. Sci., 41 (2008), 450-456.  doi: 10.1016/j.commatsci.2007.05.001. [2] M. E. Budyka, T. S. Zyubina, A. G. Ryabenko, S. H. Lin and A. M. Mebel, Bond lengths and diameters of armchair single-walled carbon nanotubes, Chem. Phys. Lett., 407 (2005), 266-271. [3] B. J. Cox and J. M. Hill, Exact and approximate geometric parameters for carbon nanotubes incorporating curvature, Carbon, 45 (2007), 1453-1462.  doi: 10.1016/j.carbon.2007.03.028. [4] M. S. Dresselhaus, G. Dresselhaus and R. Saito, Carbon fibers based on C60 ad their symmetry, Phys. Rev. B, 45 (1992), 6234-6242. [5] M. S. Dresselhaus, G. Dresselhaus and R. Saito, Physics of carbon nanotubes, Carbon Nanotubes, (1996), 27-35.  doi: 10.1016/B978-0-08-042682-2.50009-6. [6] W. E and D. Li, On the crystallization of 2D hexagonal lattices, Comm. Math. Phys., 286 (2009), 1099-1140.  doi: 10.1007/s00220-008-0586-2. [7] R. D. James, Objective structures, J. Mech. Phys. Solids, 54 (2006), 2354-2390.  doi: 10.1016/j.jmps.2006.05.008. [8] H. Jiang, P. Zhang, B. Liu, Y. Huans, P. H. Geubelle, H. Gao and K. C. Hwang, The effect of nanotube radius on the constitutive model for carbon nanotubes, Comp. Mat. Sci., 28 (2003), 429-442.  doi: 10.1016/j.commatsci.2003.08.004. [9] V. K. Jindal and A. N. Imtani, Bond lengths of armchair single-walled carbon nanotubes and their pressure dependence, Comp. Mat. Sci., 44 (2008), 156-162. [10] R. A. Jishi, M. S. Dresselhaus and G. Dresselhaus, Symmetry properties and chiral carbon nanotubes, Phys. Rev. B, 47 (1993), 166671-166674. [11] K. Kanamits and S. Saito, Geometries, electronic properties, and energetics of isolated single-walled carbon nanotubes, J. Phys. Soc. Japan, 71 (2002), 483-486.  doi: 10.1143/JPSJ.71.483. [12] A. Krishnan, E. Dujardin, T. W. Ebbesen, P. N. Yianilos and M. M. J. Treacy, Young's modulus of single-walled nanotubes, Phys. Rev. B, 58 (1998), 14013-14019.  doi: 10.1103/PhysRevB.58.14013. [13] J. Kurti, V. Zolyomi, M. Kertesz and G. Sun, The geometry and the radial breathing model of carbon nanotubes: Beyond the ideal behaviour, New J. Phys., 5 (2003), 1-21. [14] R. K. F. Lee, B. J. Cox and J. M. Hill, General rolled-up and polyhedral models for carbon nanotubes, Fullerenes, Nanotubes and Carbon Nanostructures, 19 (2011), 726-748.  doi: 10.1080/1536383X.2010.494786. [15] E. Mainini and U. Stefanelli, Crystallization in carbon nanostructures, Comm. Math. Phys., 328 (2014), 545-571.  doi: 10.1007/s00220-014-1981-5. [16] E. Mainini, H. Murakawa, P. Piovano and U. Stefanelli, Carbon-nanotube Geometries as Optimal Configurations preprint, 2016. [17] L. Shen and J. Li, Transversely isotropic elastic properties of single-walled carbon nanotubes, Phys. Rev. B, 69 (2004), 045414, Erratum Phys. Rev. B 81 (2010), 119902. doi: 10.1103/PhysRevB. 69. 045414. [18] L. Shen and J. Li, Equilibrium structure and strain energy of single-walled carbon nanotubes, Phys. Rev. B, 71 (2005), 165427.  doi: 10.1103/PhysRevB.71.165427. [19] F. H. Stillinger and T. A. Weber, Computer simulation of local order in condensed phases of silicon, Phys. Rev. B, 8 (1985), 5262-5271.  doi: 10.1103/PhysRevB.31.5262. [20] J. Tersoff, New empirical approach for the structure and energy of covalent systems, Phys. Rev. B, 37 (1988), 6991-7000.  doi: 10.1103/PhysRevB.37.6991. [21] M. M. J. Treacy, T. W. Ebbesen and J. M. Gibson, Exceptionally high Young's modulus observed for individual carbon nanotubes, Nature, 381 (1996), 678-680.  doi: 10.1038/381678a0. [22] M.-F. Yu, B. S. Files, S. Arepalli and R. S. Ruoff, Tensile Loading of Ropes of Single Wall Carbon Nanotubes and their Mechanical Properties, Phys. Rev. Lett., 84 (2000), 5552-5555.  doi: 10.1103/PhysRevLett.84.5552. [23] T. Zhang, Z. S. Yuan and L. H. Tan, Exact geometric relationships, symmetry breaking and structural stability for single-walled carbon nanotubes, Nano-Micro Lett., 3 (2011), 28-235.  doi: 10.1007/BF03353677. [24] X. Zhao, Y. Liu, S. Inoue, R. O. Jones and Y. Ando, Smallest carbon nanotibe is 3Å in diameter, Phys. Rev. Lett., 92 (2004), 125502.  doi: 10.1007/BF03353677.

show all references

##### References:
 [1] P. M. Agrawal, B. S. Sudalayandi, L. M. Raff and R. Komandur, 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), Comp. Mat. Sci., 41 (2008), 450-456.  doi: 10.1016/j.commatsci.2007.05.001. [2] M. E. Budyka, T. S. Zyubina, A. G. Ryabenko, S. H. Lin and A. M. Mebel, Bond lengths and diameters of armchair single-walled carbon nanotubes, Chem. Phys. Lett., 407 (2005), 266-271. [3] B. J. Cox and J. M. Hill, Exact and approximate geometric parameters for carbon nanotubes incorporating curvature, Carbon, 45 (2007), 1453-1462.  doi: 10.1016/j.carbon.2007.03.028. [4] M. S. Dresselhaus, G. Dresselhaus and R. Saito, Carbon fibers based on C60 ad their symmetry, Phys. Rev. B, 45 (1992), 6234-6242. [5] M. S. Dresselhaus, G. Dresselhaus and R. Saito, Physics of carbon nanotubes, Carbon Nanotubes, (1996), 27-35.  doi: 10.1016/B978-0-08-042682-2.50009-6. [6] W. E and D. Li, On the crystallization of 2D hexagonal lattices, Comm. Math. Phys., 286 (2009), 1099-1140.  doi: 10.1007/s00220-008-0586-2. [7] R. D. James, Objective structures, J. Mech. Phys. Solids, 54 (2006), 2354-2390.  doi: 10.1016/j.jmps.2006.05.008. [8] H. Jiang, P. Zhang, B. Liu, Y. Huans, P. H. Geubelle, H. Gao and K. C. Hwang, The effect of nanotube radius on the constitutive model for carbon nanotubes, Comp. Mat. Sci., 28 (2003), 429-442.  doi: 10.1016/j.commatsci.2003.08.004. [9] V. K. Jindal and A. N. Imtani, Bond lengths of armchair single-walled carbon nanotubes and their pressure dependence, Comp. Mat. Sci., 44 (2008), 156-162. [10] R. A. Jishi, M. S. Dresselhaus and G. Dresselhaus, Symmetry properties and chiral carbon nanotubes, Phys. Rev. B, 47 (1993), 166671-166674. [11] K. Kanamits and S. Saito, Geometries, electronic properties, and energetics of isolated single-walled carbon nanotubes, J. Phys. Soc. Japan, 71 (2002), 483-486.  doi: 10.1143/JPSJ.71.483. [12] A. Krishnan, E. Dujardin, T. W. Ebbesen, P. N. Yianilos and M. M. J. Treacy, Young's modulus of single-walled nanotubes, Phys. Rev. B, 58 (1998), 14013-14019.  doi: 10.1103/PhysRevB.58.14013. [13] J. Kurti, V. Zolyomi, M. Kertesz and G. Sun, The geometry and the radial breathing model of carbon nanotubes: Beyond the ideal behaviour, New J. Phys., 5 (2003), 1-21. [14] R. K. F. Lee, B. J. Cox and J. M. Hill, General rolled-up and polyhedral models for carbon nanotubes, Fullerenes, Nanotubes and Carbon Nanostructures, 19 (2011), 726-748.  doi: 10.1080/1536383X.2010.494786. [15] E. Mainini and U. Stefanelli, Crystallization in carbon nanostructures, Comm. Math. Phys., 328 (2014), 545-571.  doi: 10.1007/s00220-014-1981-5. [16] E. Mainini, H. Murakawa, P. Piovano and U. Stefanelli, Carbon-nanotube Geometries as Optimal Configurations preprint, 2016. [17] L. Shen and J. Li, Transversely isotropic elastic properties of single-walled carbon nanotubes, Phys. Rev. B, 69 (2004), 045414, Erratum Phys. Rev. B 81 (2010), 119902. doi: 10.1103/PhysRevB. 69. 045414. [18] L. Shen and J. Li, Equilibrium structure and strain energy of single-walled carbon nanotubes, Phys. Rev. B, 71 (2005), 165427.  doi: 10.1103/PhysRevB.71.165427. [19] F. H. Stillinger and T. A. Weber, Computer simulation of local order in condensed phases of silicon, Phys. Rev. B, 8 (1985), 5262-5271.  doi: 10.1103/PhysRevB.31.5262. [20] J. Tersoff, New empirical approach for the structure and energy of covalent systems, Phys. Rev. B, 37 (1988), 6991-7000.  doi: 10.1103/PhysRevB.37.6991. [21] M. M. J. Treacy, T. W. Ebbesen and J. M. Gibson, Exceptionally high Young's modulus observed for individual carbon nanotubes, Nature, 381 (1996), 678-680.  doi: 10.1038/381678a0. [22] M.-F. Yu, B. S. Files, S. Arepalli and R. S. Ruoff, Tensile Loading of Ropes of Single Wall Carbon Nanotubes and their Mechanical Properties, Phys. Rev. Lett., 84 (2000), 5552-5555.  doi: 10.1103/PhysRevLett.84.5552. [23] T. Zhang, Z. S. Yuan and L. H. Tan, Exact geometric relationships, symmetry breaking and structural stability for single-walled carbon nanotubes, Nano-Micro Lett., 3 (2011), 28-235.  doi: 10.1007/BF03353677. [24] X. Zhao, Y. Liu, S. Inoue, R. O. Jones and Y. Ando, Smallest carbon nanotibe is 3Å in diameter, Phys. Rev. Lett., 92 (2004), 125502.  doi: 10.1007/BF03353677.
Rolling-up of nanotubes from a graphene sheet
Notation for bonds and bond angles
Zigzag nanotube
The construction of the function $\beta_z$
The angle $\beta_z$ as a function of the angle $\alpha$ (above) and a zoom (below) with the points $(\alpha^{\rm ru}_z,\beta_z(\alpha^{\rm ru}_z))$ and $(\alpha^{\rm ch}_z,\beta_z(\alpha^{\rm ch}_z))$ for $\ell=10$
The angle $\beta_a$ as a function of the angle $\alpha$ (above) and a zoom (below) with the points $(\alpha^{\rm ru}_a,\beta_a(\alpha^{\rm ru}_a))$ and $(\alpha^{\rm ch}_a,\beta_a(\alpha^{\rm ch}_a))$ for $\ell=10$
The energy-per-particle $\widehat E_i$ in the zigzag (above) and in the armchair (below) family, as a function of the angle $\alpha$ for $\ell=10$, together with a zoom about the minimum
Comparison between energies of the optimal configurations and energies of their perturbations in the cases Z1, Z2, Z3 (left, from the top) and A1, A2, A3 (right, from the top). The marker corresponds to the optimal configuration $\mathcal{F}_i^*$ and value $\alpha$ represents the mean of all $\alpha$-angles in the configuration
Optimality of the configuration $(F^*_L,L)\in \mathscr{F}_z$ (bottom point) for all given $L$ in a neighborhood of $L^*$
Elastic response of the nanotube Z1 under uniaxial small (left) and large displacements (right). The function $L \mapsto E(F_L^*,L)$ (bottom) corresponds to the lower envelope of the random evaluations (top)
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