Figure 1.
Rolling-up of nanotubes from a graphene sheet
- Previous Article
- DCDS-S Home
- This Issue
-
Next Article
Rigidity of three-dimensional lattices and dimension reduction in heterogeneous nanowires
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 |
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 [
Keywords:
Carbon nanotubes,
Tersoff energy,
variational perspective,
new geometrical model,
stability,
Cauchy-Born rule.
Mathematics Subject Classification:
Primary: 82D25.
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. |
Figure 2.
Notation for bonds and bond angles
Figure 3.
Zigzag nanotube
Figure 4.
The construction of the function $\beta_z$
Figure 5.
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 $\ell=10$
Figure 6.
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 $\ell=10$
Figure 7.
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
Figure 8.
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
Figure 9.
Optimality of the configuration $(F^*_L,L)\in
\mathscr{F}_z$ (bottom point) for all given $L$ in a neighborhood of $L^*$
Figure 10.
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)
| [1] |
Shuyang Dai, Fengru Wang, Jerry Zhijian Yang, Cheng Yuan. On the Cauchy-Born approximation at finite temperature for alloys. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3131-3153. doi: 10.3934/dcdsb.2021176 |
| [2] |
Rabah Amir, Igor V. Evstigneev. A new perspective on the classical Cournot duopoly. Journal of Dynamics and Games, 2017, 4 (4) : 361-367. doi: 10.3934/jdg.2017019 |
| [3] |
Azeddine Elmajidi, Elhoussine Elmazoudi, Jamila Elalami, Noureddine Elalami. Dependent delay stability characterization for a polynomial T-S Carbon Dioxide model. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 143-159. doi: 10.3934/dcdss.2021035 |
| [4] |
Huiqiang Jiang. Energy minimizers of a thin film equation with born repulsion force. Communications on Pure and Applied Analysis, 2011, 10 (2) : 803-815. doi: 10.3934/cpaa.2011.10.803 |
| [5] |
Dinh-Liem Nguyen. The factorization method for the Drude-Born-Fedorov model for periodic chiral structures. Inverse Problems and Imaging, 2016, 10 (2) : 519-547. doi: 10.3934/ipi.2016010 |
| [6] |
Huaying Guo, Jin Liang. An optimal control model of carbon reduction and trading. Mathematical Control and Related Fields, 2016, 6 (4) : 535-550. doi: 10.3934/mcrf.2016015 |
| [7] |
Martin Frank, Benjamin Seibold. Optimal prediction for radiative transfer: A new perspective on moment closure. Kinetic and Related Models, 2011, 4 (3) : 717-733. doi: 10.3934/krm.2011.4.717 |
| [8] |
Neil K. Chada, Yuming Chen, Daniel Sanz-Alonso. Iterative ensemble Kalman methods: A unified perspective with some new variants. Foundations of Data Science, 2021, 3 (3) : 331-369. doi: 10.3934/fods.2021011 |
| [9] |
Changyan Di, Qingguo Zhou, Jun Shen, Li Li, Rui Zhou, Jiayin Lin. Innovation event model for STEM education: A constructivism perspective. STEM Education, 2021, 1 (1) : 60-74. doi: 10.3934/steme.2021005 |
| [10] |
Haifeng Hu, Kaijun Zhang. Stability of the stationary solution of the cauchy problem to a semiconductor full hydrodynamic model with recombination-generation rate. Kinetic and Related Models, 2015, 8 (1) : 117-151. doi: 10.3934/krm.2015.8.117 |
| [11] |
Anis Theljani, Ke Chen. An augmented lagrangian method for solving a new variational model based on gradients similarity measures and high order regulariation for multimodality registration. Inverse Problems and Imaging, 2019, 13 (2) : 309-335. doi: 10.3934/ipi.2019016 |
| [12] |
Teemu Lukkari, Mikko Parviainen. Stability of degenerate parabolic Cauchy problems. Communications on Pure and Applied Analysis, 2015, 14 (1) : 201-216. doi: 10.3934/cpaa.2015.14.201 |
| [13] |
Jingzhi Tie, Qing Zhang. An optimal mean-reversion trading rule under a Markov chain model. Mathematical Control and Related Fields, 2016, 6 (3) : 467-488. doi: 10.3934/mcrf.2016012 |
| [14] |
Frederik Riis Mikkelsen. A model based rule for selecting spiking thresholds in neuron models. Mathematical Biosciences & Engineering, 2016, 13 (3) : 569-578. doi: 10.3934/mbe.2016008 |
| [15] |
Kota Kumazaki. A mathematical model of carbon dioxide transport in concrete carbonation process. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 113-125. doi: 10.3934/dcdss.2014.7.113 |
| [16] |
Dayi He, Xiaoling Chen, Qi Huang. Influences of carbon emission abatement on firms' production policy based on newsboy model. Journal of Industrial and Management Optimization, 2017, 13 (1) : 251-265. doi: 10.3934/jimo.2016015 |
| [17] |
Chjan C. Lim, Da Zhu. Variational analysis of energy-enstrophy theories on the sphere. Conference Publications, 2005, 2005 (Special) : 611-620. doi: 10.3934/proc.2005.2005.611 |
| [18] |
L.R. Ritter, Akif Ibragimov, Jay R. Walton, Catherine J. McNeal. Stability analysis using an energy estimate approach of a reaction-diffusion model of atherogenesis. Conference Publications, 2009, 2009 (Special) : 630-639. doi: 10.3934/proc.2009.2009.630 |
| [19] |
Fanze Kong, Qi Wang. Stability, free energy and dynamics of multi-spikes in the minimal Keller-Segel model. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2499-2523. doi: 10.3934/dcds.2021200 |
| [20] |
Goro Akagi. Energy solutions of the Cauchy-Neumann problem for porous medium equations. Conference Publications, 2009, 2009 (Special) : 1-10. doi: 10.3934/proc.2009.2009.1 |
2020 Impact Factor: 2.425
Tools
Metrics
Other articles
by authors
[Back to Top]