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Chemical Oscillations and Spiral Waves

  • Patricia PfeifferEmail author
Chapter
Part of the The Frontiers Collection book series (FRONTCOLL)

Abstract

Pattern formation is one of nature’s most fascinating phenomena. Starting with the evolution of life: cells and compartments start to differentiate such that they are able to undertake different tasks leading to life of complex organisms. Additionally, cells are able to release messenger substances, which may lead to an aggregation of cells as in the slime mold Dictyostelium discoideum . In this chapter, the formation of wave patterns, especially of spirals in non-equilibrium systems, is described. Starting with the revision of important aspects contributing to the historical development of synergetics, oscillating chemical reactions, such as the Belousov–Zhabotinsky reaction are described. Some theoretical aspects of reaction-diffusion systems and wave propagation in excitable media are outlined. The development and propagation of waves and thus, of spirals is described in such systems. At the end, the Belousov–Zhabotinsky reaction embedded in a compartmentalized system, namely an emulsion, is studied. Under the chosen conditions target patterns or spirals with segmented wave fronts evolve. These segmented waves (dashes) develop from a smooth one due to an instability. However, instead of forming a spiral turbulence, these dashes remain in an ordered configuration and form beautiful patterns.

References

  1. 1.
    K. Showalter, J.J. Tyson, Luther’s 1906 discovery and analysis of chemical waves. J. Chem. Educ. 64, 742–744 (1987).  https://doi.org/10.1021/ed064p742
  2. 2.
    B.P. Belousov, in Oscillations and Traveling Waves in Chemical Systems, ed. by R.J. Field, M. Burger (Wiley, New York, 1984), pp. 605–614. ISBN: 0-471-89384-6Google Scholar
  3. 3.
    A.M. Zhabotinsky, in Oscillatory Processes in Biological and Chemical Systems, ed. G.M. Frank (Science Publications, Moscow, 1967), p. 252Google Scholar
  4. 4.
    R.J. Field, R.M. Noyes, Oscillations in chemical systems. V. Quantitative explanation of band migration in the Belousov–Zhabotinskii reaction. J. Am. Chem. Soc. 96, 2001–2006 (1974)CrossRefGoogle Scholar
  5. 5.
    A.L. Hodgkin, A.F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952).  https://doi.org/10.1113/jphysiol.1952.sp004764
  6. 6.
    H. Haken, P. Plath, W. Ebeling, Y. Romanovsky, Beiträge zur Geschichte der Synergetik-Allgemeine Prinzipien der Selbstorganisation in Natur und Gesellschaft (Springer Spektrum, Wiesbaden, 2016).  https://doi.org/10.1007/978-3-658-12952-1
  7. 7.
    R.J. Field, E. Körös, R.M. Noyes, Oscillations in chemical systems. II. Thorough analysis of temporal oscillation in the bromate-cerium-malonic acid system. J. Am. Chem. Soc. 94, 8649–8664 (1972).  https://doi.org/10.1021/ja00780a001
  8. 8.
    T.S. Briggs, W.C. Rauscher, An oscillating iodine clock. J. Chem. Educ. 50, 496 (1973).  https://doi.org/10.1021/ed050p496
  9. 9.
    A.M. Turing, The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B 237, 37–72 (1952).  https://doi.org/10.1098/rstb.1952.0012
  10. 10.
    V. Castets, E. Dulos, J. Boissonade, P. De Kepper, Experimental evidence of a sustained standing turing-type nonequilibrium chemical pattern. Phys. Rev. Lett. 64, 2953–2956 (1990).  https://doi.org/10.1103/PhysRevLett.64.2953
  11. 11.
    I. Prigogine, R. Lefever, Symmetry breaking instabilities in dissipative systems. II. J. Chem. Phys. 48, 1695–1700 (1968).  https://doi.org/10.1063/1.1668896
  12. 12.
    A.T. Winfree, The Geometry of Biological Time, 2nd edn. (Springer, New York, 2001)CrossRefzbMATHGoogle Scholar
  13. 13.
    A.N. Zaikin, A.M. Zhabotinsky, Concentration wave propagation in two-dimensional liquid-phase self-oscillating system. Nature 225, 535–537 (1970).  https://doi.org/10.1038/225535b0
  14. 14.
    J.J. Tyson, J.P. Keener, Singular perturbation theory of traveling waves in excitable media (a review). Physica D 32, 327–361 (1988).  https://doi.org/10.1016/0167-2789(88)90062-0
  15. 15.
    S.C. Müller, T. Plesser, B. Hess, Two-dimensional spectrophotometry of spiral wave propagation in the Belousov–Zhabotinskii reaction: I. Experiments and digital data representation. Physica D 24, 71–86 (1987).  https://doi.org/10.1016/0167-2789(87)90067-4
  16. 16.
    A.M. Pertsov, M. Wellner, J. Jalife, Eikonal relation in highly dispersive excitable media. Phys. Rev. Lett. 78, 2656–2659 (1997).  https://doi.org/10.1103/PhysRevLett.78.2656
  17. 17.
    C. Luengviriya, U. Storb, M.J.B. Hauser, S.C. Müller, An elegant method to study an isolated spiral wave in a thin layer of a batch Belousov–Zhabotinsky reaction under oxygen-free conditions. Phys. Chem. Chem. Phys. 8, 1425–1429 (2006).  https://doi.org/10.1039/B517918A
  18. 18.
    R.J. Field, R.M. Noyes, Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction. J. Chem. Phys. 60, 1877–1884 (1974).  https://doi.org/10.1063/1.1681288
  19. 19.
    A.F. Taylor, B.R. Johnson, S.K. Scott, Effect of oxygen on wave propagation in the ferroin-catalysed Belousov–Zhabotinsky reaction. J. Chem. Soc. Faraday Trans. 94, 1029–1033 (1998).  https://doi.org/10.1039/a708600h
  20. 20.
    H.-F. Eicke, J. Naudts, Non-linear field effects due to activation-energy controlled charge transport in microemulsions. Chem. Phys. Lett. 142, 106–109 (1987).  https://doi.org/10.1016/0009-2614(87)87260-3
  21. 21.
    V.K. Vanag, I.R. Epstein, Patterns of nanodroplets: the Belousov–Zhabotinsky–Aerosol OT-microemulsion system, in Self-Organized Morphology in Nanostructured Materials, ed. by K. Al-Shamery, J. Parisi. Springer Series in Materials Science, vol. 99 (Springer, Berlin, 2008), pp. 89–113.  https://doi.org/10.1007/978-3-540-72675-3_5, ISBN: 978-3-540-72674-6
  22. 22.
    L.J. Schwartz, C.L. DeCiantis, S. Chapman, B.K. Kelley, J.P. Hornak, Motions of water, decane, and Bis(2-ethylhexyl)sulfosuccinate sodium salt in reverse micelle solutions. Langmuir 15, 5461–5466 (1999).  https://doi.org/10.1021/la9812119
  23. 23.
    V.K. Vanag, I.R. Epstein, Pattern formation in a tunable medium: the Belousov–Zhabotinsky reaction in an aerosol OT microemulsion. Phys. Rev. Lett. 87, 228301 (2001).  https://doi.org/10.1103/PhysRevLett.87.228301
  24. 24.
    V.K. Vanag, Waves and patterns in reaction-diffusion systems. Belousov–Zhabotinsky reaction in water-in-oil microemulsions. Phys.-Uspekhi 47, 923–941 (2004).  https://doi.org/10.1070/PU2004v047n09ABEH001742
  25. 25.
    Y. Feldman, N. Kozlovich, I. Nir, N. Garti, V. Archipov, Z. Idiyatullin, Y. Zuev, V. Fedotov, Mechanism of transport of charge carriers in the sodium Bis(2-ethylhexyl) sulfosuccinate-water-decane microemulsion near the percolation temperature threshold. J. Phys. Chem. 100, 3745–3748 (1996).  https://doi.org/10.1021/jp9525595
  26. 26.
    V.S. Zykov, A.S. Mikhailov, S.C. Müller, Wave instabilities in excitable media with fast inhibitor diffusion. Phys. Rev. Lett. 81, 2811–2814 (1998).  https://doi.org/10.1103/PhysRevLett.81.2811
  27. 27.
    D. Horváth, V. Petrov, S.K. Scott, K. Showalter, Instabilities in propagating reaction-diffusion fronts. J. Chem. Phys. 98, 6332–6343 (1993).  https://doi.org/10.1063/1.465062
  28. 28.
    M. Markus, G. Kloss, I. Kusch, Disordered waves in a homogeneous, motionless excitable medium. Nature 371, 402–404 (1994).  https://doi.org/10.1038/371402a0
  29. 29.
    P. Dähmlow, V.K. Vanag, S.C. Müller, Effect of solvents on the pattern formation in a Belousov–Zhabotinsky reaction embedded into a microemulsion. Phys. Rev. E 89, 010902 (2014).  https://doi.org/10.1103/PhysRevE.89.010902
  30. 30.
    Z. Nagy-Ungvarai, A.M. Pertsov, B. Hess, S.C. Müller, Lateral instabilities of a wave front in the Ce-catalyzed Belousov–Zhabotinsky reaction. Physica D 61, 205–212 (1992).  https://doi.org/10.1016/0167-2789(92)90163-H

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of PhysicsOtto von Guericke University MagdeburgMagdeburgGermany

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