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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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
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: 0-471-89384-6
A.M. Zhabotinsky, in Oscillatory Processes in Biological and Chemical Systems, ed. G.M. Frank (Science Publications, Moscow, 1967), p. 252
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)
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
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
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
T.S. Briggs, W.C. Rauscher, An oscillating iodine clock. J. Chem. Educ. 50, 496 (1973). https://doi.org/10.1021/ed050p496
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
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
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
A.T. Winfree, The Geometry of Biological Time, 2nd edn. (Springer, New York, 2001)
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Pfeiffer, P. (2019). Chemical Oscillations and Spiral Waves. In: Tsuji, K., Müller, S.C. (eds) Spirals and Vortices. The Frontiers Collection. Springer, Cham. https://doi.org/10.1007/978-3-030-05798-5_8
Download citation
DOI: https://doi.org/10.1007/978-3-030-05798-5_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-05797-8
Online ISBN: 978-3-030-05798-5
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)