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Stable carbon configurations. (English) Zbl 1380.82052
Summary: Molecular Mechanics models molecules as configurations of particles interacting via classical potentials. The specific geometry of covalent bonding in carbon is described by the combination of an attractive-repulsive two-body interaction and a three-body bond-orientation part. We investigate the strict local minimality of specific carbon configurations under general assumptions on the interaction potentials. Carbyne, graphene, some fullerenes, and diamond are proved to be stable.

82D25 Statistical mechanical studies of crystals
82D80 Statistical mechanical studies of nanostructures and nanoparticles
Full Text: DOI
[1] Allinger, N.L.: Molecular Structure: Understanding Steric and Electronic Effects from Molecular Mechanics. Wiley, Amsterdam (2010)
[2] Yeung, AY; Friesecke, G; Schmidt, B, Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff-shape, Calc. Var. Partial Differ. Equ., 44, 81-100, (2012) · Zbl 1379.74002
[3] Baughman, RH, Dangerously seeking linear carbon, Science, 312, 1009-1110, (2006)
[4] Brenner, DW, Empirical potential for hydrocarbons for use in stimulating the chemical vapor deposition of diamond films, Phys. Rev. B, 42, 9458-9471, (1990)
[5] Brenner, DW; Shenderova, OA; Harrison, JA; Stuart, SJ; Ni, B; Sinnott, SB, A second-generation reactive empitical bond order (REBO) potential energy expression for hydrocarbons, J. Phys. Condens. Mater., 14, 783-802, (2002)
[6] Brook, BR; Bruccoleri, RE; Olafson, BD; States, DJ; Swaminathan, S; Karplus, M, CHARMM: a program for macromolecular energy, minimization, and dynamics calculations, J. Comput. Chem., 4, 187-217, (1983)
[7] Bundy, FP; Kasper, JS, Hexagonal diamond: a new form of carbon, J. Chem. Phys., 46, 3437, (1967)
[8] Butenko, Y; Siller, L; Hunt, MRC; Gogotsi, Y (ed.); Presser, V (ed.), Carbon onions, 279-302, (2014), New York
[9] Campbell, EK; Holz, M; Gerlic, D; Maier, JP, Laboratory confirmation of \(C_{60}^+\) as the carrier of two diffuse interstellar bands, Nature, 523, 322323, (2015)
[10] Chandraseker, K; Mukherjee, S; Paci, JT; Schatz, GC, An atomistic-continuum Cosserat rod model of carbon nanotubes, J. Mech. Phys. Solids, 57, 932-958, (2009)
[11] Clark, M; Cramer, RD; Opdenbosch, N, Validation of the general purpose tripos 5.2 force field, J. Comput. Chem., 10, 982-1012, (1989)
[12] Clayden, J., Greeves, N., Warren, S.G.: Organic Chemistry. Oxford University Press, Oxford (2012)
[13] Cox, BJ; Hill, JM, Exact and approximate geometric parameters for carbon nanotubes incorporating curvature, Carbon, 45, 1453-1462, (2007)
[14] David, WIF; Ibberson, RM; Matthewman, JC; Prassides, K; Dennis, TJS; Hare, JP; Kroto, HW; Taylor, R; Walton, DRM, Crystal structure and bonding of \(C_{60}\), Nature, 353, 147-149, (1991)
[15] Davoli, E., Piovano, P., Stefanelli, U.: Sharp \(N^{3/4}\) law for the minimizers of the edge-isoperimetric problem on the triangular lattice. Preprint http://cvgmt.sns.it/paper/2862/. Submitted 2015 · Zbl 1383.82063
[16] Davoli, E., Piovano, P., Stefanelli, U.: Wulff shape emergence in graphene. Math. Models Methods Appl. Sci. (2016). doi:10.1142/S0218202516500536 · Zbl 1355.82073
[17] Dresselhaus, MS; Dresselhaus, G; Saito, R, Carbon fibers based on \(C_{60}\) ad their symmetry, Phys. Rev. B, 45, 6234-6242, (1992)
[18] Dresselhaus, MS; Dresselhaus, G; Saito, R, Physics of carbon nanotubes, Carbon, 33, 883-891, (1995)
[19] Weinan, E; Li, D, On the crystallization of 2D hexagonal lattices, Commun. Math. Phys., 286, 1099-1140, (2009) · Zbl 1180.82191
[20] Farmer, B., Esedo\(\bar{\rm g}\)lu, S., Smereka, P.: Crystallization for a Brenner-like potential. Commun. Math. Phys. (2016). doi:10.1007/s00220-016-2732-6
[21] Flatley, LC; Theil, F, Face-centered cubic crystallization of atomistic configurations, Arch. Ration. Mech. Anal., 218, 363-416, (2015) · Zbl 1335.82006
[22] Friedrich, M., Piovano, P., Stefanelli, U.: The geometry of \(C_{60}\): a rigorous approach via molecular mechanics. SIAM J. Appl. Math. (2016, to appear) · Zbl 1353.82072
[23] Friesecke, G., Theil, F.: Molecular Geometry Optimization, Models. In: Engquist, B. (ed.) Encyclopedia of Applied and Computational Mathematics, pp. 951-957. Springer, New York (2015)
[24] Gajbhiye, SO; Singh, SP, Vibration characteristics of open- and capped-end single-walled carbon nanotubes using multi-scale analysis technique incorporating tersoff-Brenner potential, Acta Mech., 226, 3565-3586, (2015)
[25] van Gunsteren, W.F., Berendsen, H.J.C.: Groningen Molecular Simulation (GROMOS) Library Manual. BIOMOS b.v, Groningen (1987)
[26] Guo, H; Liu, R; Zeng, XC; Wu, X; Jiang, D-E (ed.); Chen, Z (ed.), Graphene-based architecture and assemblies, 153-182, (2013), Amsterdam
[27] Hanson, JC; Nordman, CE, The crystal and molecular structure of corannulene, \(C_{20}H_{10}\), Acta Cryst., B32, 1147-1153, (1976)
[28] Iijima, S, Helical microtubules of graphitic carbon, Nature, 354, 56-58, (1991)
[29] Itoh, M; Kotani, K; Naito, H; Sunada, T; Kawazoe, Y; Adschiri, T, New metallic carbon crystal, Phys. Rev. Lett., 102, 055703, (2009)
[30] Itzhaki, L; Altus, E; Basch, H; Hoz, S, Harder than diamond: determining the cross-sectional area and young’s modulus of molecular rods, Angew. Chem., 117, 7598, (2005)
[31] Itzhaki, L; Altus, E; Basch, H; Hoz, S, Harder than diamond: determining the cross-sectional area and young’s modulus of molecular rods, Angew. Chem. Int. Ed., 44, 7432-7435, (2005)
[32] Jiang, H; Zhang, P; Liu, B; Huans, Y; Geubelle, PH; Gao, H; Hwang, KC, The effect of nanotube radius on the constitutive model for carbon nanotubes, Comput. Math. Sci., 28, 429-442, (2003)
[33] Jishi, RA; Dresselhaus, MS; Dresselhaus, G, Symmetry properties and chiral carbon nanotubes, Phys. Rev. B, 47, 166671-166674, (1993)
[34] Kamatgalimov, AR; Kovalenko, VI, Deformation and thermodynamic instability of a \(C_{84}\) fullerene cage, Russ. J. Phys. Chem. A, 84, 4l721-4l726, (2010)
[35] Kroto, HW; Heath, JR; O’Brien, SC; Curl, RF; Smalley, RE, C 60: buckminsterfullerene, Nature, 318, 162-163, (1985)
[36] Kroto, HW, The stability of the fullerenes \(C_n\), with \(n=24, 28, 32, 36, 50, 60\) and 70, Nature, 329, 529-531, (1987)
[37] Lazzaroni, G., Stefanelli, U.: Chain-like ground states in three dimensions. (2016, in preparation)
[38] Lee, RKF; Cox, BJ; Hill, JM, General rolled-up and polyhedral models for carbon nanotubes, Fuller. Nanot. Car. N., 19, 726-748, (2011)
[39] Lewars, E.G.: Computational Chemistry, 2nd edn. Springer, New York (2011) · Zbl 1367.92005
[40] Lin, F; Sørensen, E; Kallin, C; Berlinsky, J; Sattler, D (ed.), \(C_{20}\), the smallest fullerene, (2010), New York
[41] Liu, M; Artyukhov, VI; Lee, H; Xu, F; Yakobson, BI, Carbyne from first principles: chain of \(C\) atoms, a nanorod or a nanorope?, ACS Nano, 7, 10075-10082, (2013)
[42] Mackay, AL; Terrones, H, Diamond from graphite, Nature, 35, 762, (1991)
[43] Mainini, E; Piovano, P; Stefanelli, U, Finite crystallization in the square lattice, Nonlinearity, 27, 717-737, (2014) · Zbl 1292.82043
[44] Mainini, E., Murakawa, H., Piovano, P., Stefanelli, U.: A numerical investigation on carbonnanotube geometries. Discr. Contin. Dyn. Syst. Ser. - S. (2016, to appear) · Zbl 1362.82059
[45] Mainini, E., Murakawa, H., Piovano, P., Stefanelli, U.: Carbon-nanotube geometries as optimal configurations. Submitted (2016) · Zbl 1394.82028
[46] Mainini, E; Stefanelli, U, Crystallization in carbon nanostructures, Commun. Math. Phys., 328, 545-571, (2014) · Zbl 1391.82058
[47] Mayo, SL; Olafson, BD; Goddard, WA, DREIDING: a generic force field for molecular simulations, J. Phys. Chem., 94, 8897-8909, (1990)
[48] Kass, D; Monneau, R, Atomic to continuum passage for nanotubes: a discrete Saint-Venant principle and error estimates, Arch. Ration. Mech. Anal., 213, 25-128, (2014) · Zbl 1292.82052
[49] Nasibulin, AG; etal., A novel hybrid carbon material, Nature Nanotechnol., 2, 156-161, (2007)
[50] Rappé, A.K., Casewit, C.L.: Molecular Mechanics Across Chemistry. University Science Books, Sausalito, CA (1997)
[51] Robertson, DH; Brenner, DW; Mintmire, JW, Energetics of nanoscale graphitic tubules, Phys. Rev. B, 45, 12592-12595, (1992)
[52] Schein, S; Friedrich, T, A geometric constraint, the head-to-tail exclusion rule, may be the basis for the isolated-pentagon rule for fullerenes with more than 60 vertices, Proc. Natl. Acad. Sci. USA, 105, 19142-19147, (2008)
[53] Schmidt, B, Ground states of the 2D sticky disc model: fine properties and \(N^{3/4}\) law for the deviation from the asymptotic Wulff-shape, J. Stat. Phys., 153, 727-738, (2013) · Zbl 1292.82027
[54] Stillinger, FH; Weber, TA, Computer simulation of local order in condensed phases of silicon, Phys. Rev. B, 8, 5262-5271, (1985)
[55] Sunada, T, Crystals that nature might miss creating, Notices Am. Math. Soc., 55, 208-215, (2008) · Zbl 1144.82071
[56] Tersoff, J, New empirical approach for the structure and energy of covalent systems, Phys. Rev. B, 37, 6991-7000, (1988)
[57] Theil, F, A proof of crystallization in two dimensions, Commun. Math. Phys., 262, 209-236, (2006) · Zbl 1113.82016
[58] Vázquez, S; Camps, P, Chemistry of pyramidalized alkenes, Tetrahedron, 61, 5147-5208, (2005)
[59] Wade, L.G.: Organic Chemistry, 8th edn. Pearson Prentice Hall, New York (2012)
[60] Weiner, PK; Kollman, PA, AMBER: assisted model building with energy refinement. A general program for modeling molecules and their interactions, J. Comput. Chem., 2, 287-303, (1981)
[61] Yakobson, BI; Campbell, MP; Brabec, CJ; Bernholc, J, High strain rate fracture and \(C\)-chain unraveling in carbon nanotubes, Comput. Mater. Sci., 8, 341-348, (1997)
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