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Carbon-nanotube geometries as optimal configurations. (English) Zbl 1394.82028
Carbon nanotubes show remarkable strength and, depending on the topology, high electrical conductivity or semiconductivity. Such exceptional properties make carbon nanotubes interesting for the development of innovative technologies. Carbon nanotubes are presently used in the production of composite materials, coatings, microelectronics, energy-storage devices and biotechnologies. The fine geometry of nanotubes is presently still debated, and different models are available. The most nanotube models are geometrical in nature, they reside on sets of geometrical postulates. Here, an approach to investigate carbon nanotube geometries within the frame of molecular mechanics is proposed instead of a variational approach. Atom configurations are modeled as a collection of particle positions, to which a configurational energy is associated. Specific carbon nanotubes geometries are singled out by focusing on their stability. This concept is central with respect to the process of molecular structuring, since among the many theoretically possible geometries, only those showing some suitable stability can be expected to be realized. This concept is formalized here by interpreting stability as minimality of the configurational energy with respect to small perturbations. Some analytical justification of this fact is provided. In particular, stability with respect to a large class of small perturbations including tractions and diameter dilations, among many others, is proved.

82D80 Statistical mechanical studies of nanostructures and nanoparticles
82D25 Statistical mechanical studies of crystals
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[1] N. L. Allinger, Molecular Structure: Understanding Steric and Electronic Effects from Molecular Mechanics, Wiley, New York, 2010.
[2] D. S. Bethune, C. H. Kiang, M. S. De Vries, G. Gorman, R. Savoy, et al., Cobalt catalysed growth of carbon nanotubes with single-atomic-layer walls, Nature, 363 (1993), pp. 605–607.
[3] B. R. Brook, R. E. Bruccoleri, B. D. Olafson, D. J. States, S. Swaminathan, and M. Karplus, CHARMM: A program for macromolecular energy, minimization, and dynamics calculations, J. Comput. Chem., 4 (1983), pp. 187–217
[4] J. Clayden, N. Greeves, and S. G. Warren, Organic Chemistry, Oxford University Press, Oxford, 2012.
[5] M. Clark, R. D. Cramer III, and N. Van Opdenbosch, Validation of the general purpose tripos \(5.2\) force field, J. Comput. Chem., 10 (1989), pp. 982–1012.
[6] B. J. Cox and J. M. Hill, Exact and approximate geometric parameters for carbon nanotubes incorporating curvature, Carbon, 45 (2007), pp. 1453–1462.
[7] B. J. Cox and J. M. Hill, Geometric structure of ultra-small carbon nanotubes, Carbon, 46 (2008), pp. 711–713.
[8] M. F. De Volder, S. H. Tawfick, R. H. Baughman, and A. J. Hart, Carbon nanotubes: Present and future commercial applications, Science, 339 (2013), pp. 535–539.
[9] M. S. Dresselhaus, G. Dresselhaus, and R. Saito, Carbon fibers based on C\(_{60}\) ad their symmetry, Phys. Rev. B, 45 (1992), pp. 6234–6242.
[10] M. S. Dresselhaus, G. Dresselhaus, and R. Saito, Physics of Carbon Nanotubes, Carbon, 33 (1995), pp. 883–891.
[11] W. E and D. Li, On the crystallization of 2D hexagonal lattices, Comm. Math. Phys., 286 (2009), pp. 1099–1140. · Zbl 1180.82191
[12] W. E and P. Ming, Cauchy–Born rule and the stability of crystalline solids: Static problems, Arch. Ration. Mech. Anal., 183 (2007), pp. 241–297. · Zbl 1106.74019
[13] D. El Kass and R. Monneau, Atomic to continuum passage for nanotubes. A discrete Saint-Venant principle and error estimates, Arch. Ration. Mech. Anal., 213 (2014), pp. 25–128. · Zbl 1292.82052
[14] R. S. Elliott, N. Triantafyllidis, and J. A. Shaw, Stability of crystalline solids-I: Continuum and atomic lattice considerations, J. Mech. Phys. Solids, 54 (2006), pp. 161–192. · Zbl 1120.74392
[15] R. S. Elliott, J. A. Shaw, and N. Triantafyllidis, Stability of crystalline solids-II: Application to temperature-induced martensitic phase transformations in a bi-atomic crystal, J. Mech. Phys. Solids, 54 (2006), pp. 193–232. · Zbl 1120.74391
[16] B. Farmer. S. Esedoḡlu, and P. Smereka, Crystallization for a Brenner-like potential, Comm. Math. Phys., 349 (2017), pp. 1029–1061. · Zbl 1381.74044
[17] M. Friedrich, E. Mainini, P. Piovano, and U. Stefanelli, Characterization of Optimal Carbon Nanotubes Under Stretching and Validation of the Cauchy–Born Rule, arXiv:1706.01494, submitted, 2017.
[18] M. Friedrich, P. Piovano, and U. Stefanelli, The geometry of \(C_{60}\): A rigorous approach via Molecular Mechanics, SIAM J. Appl. Math., 76 (2016), pp. 2009–2029. · Zbl 1353.82072
[19] W. F. van Gunsteren and H. J. C. Berendsen, Groningen Molecular Simulation (GROMOS) Library Manual, BIOMOS b.v., Groningen, 1987.
[20] P. J. F. Harris, Carbon Nanotube Science Synthesis, Properties and Applications, Cambridge University Press, Cambridge, 2009.
[21] S. Iijima, Helical microtubules of graphitic carbon, Nature, 354 (1991), pp. 56–58.
[22] S. Ijima and T. Ichihashi, Single-shell carbon nanotubes of \(1\)-nm diameter, Nature, 363 (1993), pp. 603–605.
[23] R. D. James, Objective structures, J. Mech. Phys. Solids, 54 (2006), pp. 2354–2390. · Zbl 1120.74312
[24] A. Jorio, G. Dresselhaus, and M. S. Dresselhaus, eds., Carbon nanotubes advanced topics in the synthesis, structure, properties and applications, Topics in Applied Physics, Vol. 111. Springer, New York, 2011. · Zbl 1175.20001
[25] K. Kanamits and S. Saito, Geometries, electronic properties, and energetics of isolated single-walled carbon nanotubes, J. Phys. Soc. Japan, 71 (2002), pp. 483–486.
[26] 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), pp. 14013–14019.
[27] 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), pp. 726–748.
[28] E. G. Lewars, Computational Chemistry, 2nd ed., Springer, New York, 2011. · Zbl 1367.92005
[29] E. Mainini and U. Stefanelli, Crystallization in carbon nanostructures, Comm. Math. Phys., 328 (2014), pp. 545–571. · Zbl 1391.82058
[30] E. Mainini, H. Murakawa, P. Piovano, and U. Stefanelli, Carbon-nanotube geometries: Analytical and numerical results, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), pp. 141–160. · Zbl 1362.82059
[31] S. L. Mayo, B. D. Olafson, and W. A. Goddard, DREIDING: A generic force field for molecular simulations, J. Phys. Chem., 94 (1990), pp. 8897–8909.
[32] M. Monthioux and V. L. Kuznetsov, Who should be given the credit for the discovery of carbon nanotubes?, Carbon, 44 (2006), pp. 1621–1623.
[33] T. W. Odom, J. L. Huang, P. Kim, and C. M. Lieber, Atomic structure and electronic properties of single-walled carbon nanotubes, Nature, 391 (1998), pp. 62–64.
[34] L. V. Radushkevich and V. M. Lukyanovich, O strukture ugleroda, obrazujucegosja pri termiceskom razlozenii okisi ugleroda na zeleznom kontakte, Zurn. Fisic. Chim., 26 (1952), pp. 88–95.
[35] A. K. Rappé and C. L. Casewit, Molecular Mechanics Across Chemistry, University Science Books, Sausalito, CA, 1997.
[36] D. Sfyris, Phonon, Cauchy–Born and homogenized stability criteria for a free-standing monolayer graphene at the continuum level, Eur. J. Mech. A Solids, 55 (2016), pp. 134–148. · Zbl 1406.74160
[37] L. Shen and J. Li, Transversely isotropic elastic properties of single-walled carbon nanotubes, Phys. Rev. B, 69 (2004), 045414.
[38] L. Shen and J. Li, Erratum: Transversely isotropic elastic properties of single-walled carbon nanotubes, Phys. Rev. B, 81 (2010), 119902.
[39] L. Shen and J. Li, Equilibrium structure and strain energy of single-walled carbon nanotubes, Phys. Rev. B, 71 (2005), 165427.
[40] U. Stefanelli, Stable carbon configurations, Boll. Unione Mat. Ital. (9), 10 (2017), pp. 335–354. · Zbl 1380.82052
[41] F. H. Stillinger and T. A. Weber, Computer simulation of local order in condensed phases of silicon, Phys. Rev. B, 8 (1985), pp. 5262–5271.
[42] J. Tersoff, New empirical approach for the structure and energy of covalent systems, Phys. Rev. B, 37 (1988), pp. 6991–7000.
[43] M. M. J. Treacy, T. W. Ebbesen, and J. M. Gibson, Exceptionally high Young’s modulus observed for individual carbon nanotubes, Nature, 381 (1996), pp. 678–680.
[44] X. Wang, Q. Li, J. Xie, Z. Jin, J. Wang, Y. Li, K. Jiang, and S. Fan, Shoushan. Fabrication of ultralong and electrically uniform single-walled carbon nanotubes on clean substrates, Nano Lett., 9 (2009), pp. 3137–3141.
[45] P. K. Weiner and P. A. Kollman, AMBER: Assisted model building with energy refinement. A general program for modeling molecules and their interactions, J. Comput. Chem., 2 (1981), pp. 287–303.
[46] 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), pp. 5552–5555.
[47] 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), pp. 28–235.
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