Journal of Mathematical Chemistry

, Volume 54, Issue 3, pp 765–776 | Cite as

Pentagonal chains and annuli as models for designing nanostructures from cages

  • Vladimir R. Rosenfeld
  • Andrey A. Dobrynin
  • Josep M. Oliva
  • Juanjo Rué
Original Paper

Abstract

Carbon is the most versatile of chemical elements in combining with itself or other elements to form chains, rings, sheets, cages, and periodic 3D structures. One of the perspective trends for creating new molecules of nanotechnological interest deals with constructs which may be formed by chemically linking of cage molecules. The growing interest in fullerene polyhedra and other molecules with pentagonal rings raises also a question about geometrically consistent in \({\mathbb {E}}^{3}\) nanoarchitectures which may be obtained by aggregating many such molecules. Simple examples are chains and rings assembled from pyramidal (car)borane subunits. Adequate geometrical models of such objects are a chain and an annulus built from regular pentagons wherein any two adjacent pentagons share an edge. Among arising combinatorial problems may be both analytical and constructive enumeration of such chains and annuli drawn in plane with no two edges crossing each other. This may also employ several mathematical disciplines, such as geometry, (spectral) graph theory, semigroup theory, theory of fractals, and others. We discuss some practical approaches for solving the mentioned mathematical problem.

Keywords

Cage molecule (Car)borane Nanoarchitecture Pentagonal chains and annuli Constructive and analytical enumeration Quasicrystal 

References

  1. 1.
    V.R. Rosenfeld, D.J. Klein, Enumeration of substitutional isomers with restrictive mutual positions of ligands. I. Overall counts. J. Math. Chem. 51(1), 21–37 (2013)CrossRefGoogle Scholar
  2. 2.
    V.R. Rosenfeld, D.J. Klein, Enumeration of substitutional isomers with restrictive mutual positions of ligands. II. Counts with restrictions on subsymmetry. J. Math. Chem. 51(1), 239–264 (2013)CrossRefGoogle Scholar
  3. 3.
    V.R. Rosenfeld, J.M. Oliva, D.J. Klein, Chain carborane molecules. Monatsh. Chem. 143(3), 361–364 (2012)CrossRefGoogle Scholar
  4. 4.
    V.R. Rosenfeld, D.J. Klein, J.M. Oliva, Enumeration of polycarborane isomers: especially dicarboranes. J. Math. Chem. 50(7), 2012–2022 (2012)CrossRefGoogle Scholar
  5. 5.
    J.M. Oliva, J. Rué, D. Hnyk, J.D. Kennedy, V.R. Rosenfeld, Borane polyhedra as building blocks for unknown but potentially isolatable new molecules—extensions based on computations of the known \({\rm B}_{\rm 18}{\rm H}_{\rm 22}\) isomers. Croat. Chem. Acta 86(4), 485–494 (2013). (86)Google Scholar
  6. 6.
    C.J. Cramer, Essentials of Computational Chemistry: Theories and Models (Wiley, Chichester, England, 2004)Google Scholar
  7. 7.
    D. Hnyk, M.L. McKee (Eds.), Boron: The Fifth Element. Challenges and Advances in Computational Chemistry and Physics, vol. 20 (Springer, Chicago, Harvard, Vancouver, 2015)Google Scholar
  8. 8.
    R.N. Grimes, Carboranes (Academic Press, London, 2011)Google Scholar
  9. 9.
    M. Liśkiewicz, M. Ogihara, S. Toda, The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes. Theor. Comput. Sci. 304(1–3), 129–156 (2003)CrossRefGoogle Scholar
  10. 10.
    A.J. Guttmann, Self-avoiding walks and polygons—an overview. Asia Pac. Math. Newsl. 2(4), 1–10 (2012)Google Scholar
  11. 11.
    A. De Luca, S. Varricchio, Factorial languages whose growth function is quadratically upper bounded. Inform. Process. Lett. 30, 283–288 (1989)CrossRefGoogle Scholar
  12. 12.
    A. De Luca, S. Varricchio, Some combinatorial properties of the Thue–Morse sequence and a problem in semigroups. Theor. Comput. Sci. 63, 333–348 (1989)CrossRefGoogle Scholar
  13. 13.
    A. De Luca, S. Varricchio, A combinatorial theorem on \(p\)-power-free words and an application to semigroups. RAIRO Inform. Théor. Appl. 24(3), 205–228 (1990). AGoogle Scholar
  14. 14.
  15. 15.
    V. I. Arnold, Huygens and Barrow, Newton and Hooke—pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals, Birkhäuser Verlag, Basel, Boston, Berlin, 1990. The 1st edition in: Series The Modern Mathematics for Students, Nauka, Moscow, 1989 (in Russian)Google Scholar
  16. 16.
    F. Harary, R.C. Read, The enumeration of tree-like polyhexes. Proc. Edinb. Math. Soc. (Series 2) 17(1), 1–13 (1970)CrossRefGoogle Scholar
  17. 17.
    S.B. Elk, An algorithm to identify and count coplanar isomeric molecules formed by the linear fusion of cyclopentane modules. J. Chem. Inf. Comput. Sci. 27(2), 67–69 (1987)CrossRefGoogle Scholar
  18. 18.
    J. Brunvoll, B.N. Cyvin, S.J. Cyvin, Enumeration and classification of coronoid hydrocarbons. J. Chem. Inf. Comput. Sci. 27(1), 14–21 (1987)CrossRefGoogle Scholar
  19. 19.
    S.J. Cyvin, J. Brunvoll, B.N. Cyvin, Enumeration of unbranched catacondensed polygons systems: general solution for two kinds of polygons. J. Chem. Inf. Comput. Sci. 37(3), 460–465 (1997)CrossRefGoogle Scholar
  20. 20.
    B.N. Cyvin, S.J. Cyvin, J. Brunvoll, A.A. Dobrynin, Enumeration of unbranched catacondensed systems of congruent polygons. Vychisl. Sist. 155, 3–14 (1996)Google Scholar
  21. 21.
    G. Brinkmann, A.A. Dobrynin, A. Krause, Fast generation of polycyclic chains with arbitrary ring sizes. MATCH Commun. Math. Comput. Chem. 41, 137–144 (2000)Google Scholar
  22. 22.
    F. Harary, A.J. Schwenk, The number of caterpillars. Discrete Math. 6, 359–365 (1973)CrossRefGoogle Scholar
  23. 23.
    J. O’Rourke, Art of Gallery Theorems and Algorithms (Oxford University Press, New York, Oxford, 1987)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Vladimir R. Rosenfeld
    • 1
  • Andrey A. Dobrynin
    • 2
  • Josep M. Oliva
    • 3
  • Juanjo Rué
    • 4
  1. 1.Department of Computer Science and MathematicsAriel UniversityArielIsrael
  2. 2.Sobolev Institute of MathematicsSiberian Branch of Russian Academy of SciencesNovosibirskRussia
  3. 3.Instituto de Química-FísicaCSICMadridSpain
  4. 4.Institut für MathematikFreie Universität BerlinBerlinGermany

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