Journal of Statistical Physics

, Volume 101, Issue 1–2, pp 495–508 | Cite as

Reaction–Diffusion Patterns in Confined Chemical Systems

  • P. De Kepper
  • E. Dulos
  • J. Boissonade
  • A. De Wit
  • G. Dewel
  • P. Borckmans
Article

Abstract

We review on the effects of the feed mode on pattern selection observed in chemical systems operated in open spatial reactors. In two-side-fed reactors, strong parameter ramps naturally confine patterns in a stratum. The reactor thickness acts both as a genuine bifurcation parameter and on the pattern dimensionality. Depending on that thickness, standard 2D hexagon and stripe Turing patterns or more complex 3D planforms are observed. In thin one-side-fed reactors, patterning process must escape the imposed fixed boundary conditions either by devices introducing mixed boundary conditions or by an intrinsic phenomenon dubbed “spatial bistability.” We show that in most cases, for a comprehensive understanding of experimental observations, the full 3D aspects have to be taken into account.

reaction–diffusion chemical pattern Turing structure nonlinear dynamics boundary conditions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    A. M. Turing, Philos. Trans. R. S. London, B 327:37 (1952).Google Scholar
  2. 2.
    J. F. G. Auchmuty and G. Nicolis, Bull. Math. Biol. 37:323 (1975).Google Scholar
  3. 3.
    G. Nicolis and I. Prigogine, Self Organization in Nonequilibrium Systems (Wiley, New York, 1977).Google Scholar
  4. 4.
    V. Castets, E. Dulos, J. Boissonade, and P. De Kepper, Phys. Rev. Lett. 64:2953 (1990).Google Scholar
  5. 5.
    D. Walgraef, G. Dewel, and P. Borckmans, Adv. Chem. Phys. 49:311 (1982); A. De Wit, P Borckmans, G. Dewel, and D. Walgraef, Physica D 61:289 (1992).Google Scholar
  6. 6.
    P. Borckmans, G. Dewel, A. De Wit, and D. Walgraef, in Chemical Waves and Patterns, Chap. 10, Understanding Chemical Reactivity, Vol. 10, R. Kapral and K. Showalter, eds. (Kluwer Academic Publisher, 1994).Google Scholar
  7. 7.
    A. De Wit, Adv. Chem. Phys. 109:435 (1999).Google Scholar
  8. 8.
    A. De Wit, P. Borckmans, and G. Dewel, Proc. Nat. Acad. Sci. (USA) 94:12765 (1997).Google Scholar
  9. 9.
    T. Ohta, M. Mimura, and P. Kobayashi, Physica D 34:115 (1989).Google Scholar
  10. 10.
    A. Hagberg and E. Meron, Phys. Rev. E 48:705 (1993); A. Hagberg and E. Meron, Non Linearity 7:805 (1994); A. Hagberg and E. Meron, Chaos 4:477 (1994); A. Hagberg, E. Meron, I. Rubinstein, and B. Zaltman, Phys. Rev. E 55:4450 (1997).Google Scholar
  11. 11.
    Q. Ouyang, Z. Noszticzius, and H. L. Swinney, J. Chem. Phys. 96:6773 (1992).Google Scholar
  12. 12.
    P. De Kepper, J. J. Perraud, B. Rudovics, and E. Dulos, Int. J. Bif. Chaos 4:1215 (1994); J. J. Perraud, A. De Wit, E. Dulos, P. De Kepper, G. Dewel, and P. Borckmans, Phys. Rev. Lett. 71:1272 (1993).Google Scholar
  13. 13.
    H. Kidachi, Prog. Theor. Phys. 63:1152 (1980).Google Scholar
  14. 14.
    A. Rovinsky and M. Menzinger, Phys. Rev. A 46:6315 (1992).Google Scholar
  15. 15.
    A. De Wit, D. Lima, G. Dewel, and P. Borckmans, Phys. Rev. E 54:261 (1996).Google Scholar
  16. 16.
    I. Lengyel, G. Rabai, and I. R. Epstein, J. Am. Chem. Soc. 112:4606 (1990); I. Lengyel, G. Rabai, and I. R. Epstein, J. Am. Chem. Soc. 112:9104 (1990); I. Lengyel, J. Li, K. Kustin, and I. R. Epstein, J. Am. Chem. Soc. 118:3708 (1996).Google Scholar
  17. 17.
    I. Lengyel and I. R. Epstein, Science 251:650 (1990); I. Lengyel and I. R. Epstein, Proc. Nat. Sci. (USA) 89:3977 (1992).Google Scholar
  18. 18.
    J. E. Pearson and W. J. Bruno, Chaos 2:513 (1992).Google Scholar
  19. 19.
    J. Boissonade, J. Phys. France 49:541–546 (1988).Google Scholar
  20. 20.
    M. Herschkowitz-Kaufman and G. Nicolis, J. Chem. Phys. 56:1890 (1972).Google Scholar
  21. 21.
    S. Kadar, I. Lengeyl, and I. R. Epstein, J. Phys. Chem. 99:4054 (1995).Google Scholar
  22. 22.
    S. Setayeshgar and M. C. Cross, Phys. Rev. E 58:4485 (1998).Google Scholar
  23. 23.
    V. Dufiet and J. Boissonade, Phys. Rev. E 53:4883 (1996).Google Scholar
  24. 24.
    P. Borckmans, A. De Wit, and G. Dewel, Physica A 188:137 (1992).Google Scholar
  25. 25.
    Q. Ouyang and H. L. Swinney, Nature 352:610 (1991).Google Scholar
  26. 26.
    B. Rudovics, E. Dulos, and P. De Kepper, Physica Scripta T 67:43 (1996).Google Scholar
  27. 27.
    V. Dufiet and J. Boissonade, Physica A 188:158 (1992).Google Scholar
  28. 28.
    G. Dewel, P. Borckmans, and D. Walgraef, J. Phys. C 12:L491 (1979); D. Walgraef, G. Dewel, and P. Borckmans, Phys. Rev. A 21:397 (1980); D. Walgraef, G. Dewel, and P. Borckmans, Adv. Chem. Phys. 49:311 (1982).Google Scholar
  29. 29.
    L. M. Pismen, J. Chem. Phys. 72:1900 (1980).Google Scholar
  30. 30.
    E. Dulos, P. Davies, B. Rudovics, and P. De Kepper, Physica D 98:53 (1996).Google Scholar
  31. 31.
    B. Rudovics, Ph.D. Thesis (Bordeaux, 1995).Google Scholar
  32. 32.
    Q. Ouyang, G. H. Gunaratne, and H. L. Swinney, Chaos 3:707 (1993).Google Scholar
  33. 33.
    M. G. M. Gomes, Phys. Rev. E 60:3741 (1999).Google Scholar
  34. 34.
    E. Dulos, A. Hunding, J. Boissonade, and P. De Kepper, in Transport and Structure: Their Competitive Roles in Biophysics and Chemistry, Lecture Notes in Physics, Vol. 352, S. C. Mueller, J. Paosi, and W. Zimmermann, eds., to appear.Google Scholar
  35. 35.
    B. Rudovics, E. Barillot, P. W. Davies, E. Dulos, J. Boissonade, and P. De Kepper, J. Phys. Chem. 103:1790 (1999).Google Scholar
  36. 36.
    P. W. Davies, P. Blanchedeau, E. Dulos, and P. De Kepper, J. Chem. Phys. 102:8236 (1998).Google Scholar
  37. 37.
    P. Blanchedeau, J. Boissonade, and P. De Kepper, Physica D, submitted.Google Scholar
  38. 38.
    K. J. Lee, D. McCormick, Q. Ouyang, and H. L. Swinney, Science 261:192 (1993); K. J. Lee and H. L. Swinney, Phys. Rev. E 51:1899 (1995).Google Scholar
  39. 39.
    P. Blanchedeau and J. Boissonade, Phys. Rev. Lett. 81:5007 (1998).Google Scholar
  40. 40.
    M. Bachir, P. Borckmans, and G. Dewel, Phys. Rev. E 59:86223 (1999).Google Scholar

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • P. De Kepper
    • 1
  • E. Dulos
    • 1
  • J. Boissonade
    • 1
  • A. De Wit
    • 2
  • G. Dewel
    • 2
  • P. Borckmans
    • 2
  1. 1.Centre de Recherche Paul PascalC.N.R.S. and Université Bordeaux IPessacFrance
  2. 2.Service de Chimie-Physique and Center for Nonlinear Phenomena and Complex SystemsUniversité Libre de BruxellesBruxellesBelgium

Personalised recommendations