Skip to main content
SpringerLink
Log in

Spatio-Temporal Organization in a Morphochemical Electrodeposition Model: Analysis and Numerical Simulation of Spiral Waves

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

In this paper we derive Hopf instability conditions for the morphochemical mathematical model for alloy electrodeposition introduced and experimentally validated in Bozzini et al. (J. Solid State Electrochem., 17:467–479, 2013). Using normal form theory we show that in the neighborhood of the Hopf bifurcation, essential features of the system dynamics are captured by a specific Complex Ginzburg-Landau Equation (CGLE). The derived CGLE yields analytical results on the existence and stability of spiral waves. Moreover, the arising of spiral instability is discussed in terms of the relevant system parameters and the related phenomenology is investigated numerically. To face with the numerical approximation of the spiral structures and of their longtime oscillating behavior we apply an Alternating Direction Implicit (ADI) method based on high order finite differences in space.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Aranson, I.S., Kramer, L., Weber, A.: Theory of interaction and bound states of spiral waves in oscillatory media. Phys. Rev. E 47, 3231–3241 (1993)

    Article  Google Scholar 

  2. Aranson, I.S., Kramer, L.: The world of the complex Ginzburg-Landau equation. Rev. Mod. Phys. 74, 100–143 (2002)

    Article  MathSciNet  Google Scholar 

  3. Berenstein, I., Muñuzuri, A.P., Yang, L., Dolnik, M., Zhabotinsky, A.M., Epstein, I.R.: Breathing spiral waves in the chlorine dioxide-iodine-malonic acid reaction-diffusion system. Phys. Rev. E 78, 025101 (2008), 3 pp.

    Article  Google Scholar 

  4. Bozzini, B., Lacitignola, D., Sgura, I.: Morphological spatial patterns in a reaction diffusion model for metal growth. Math. Biosci. Eng. 7, 237–258 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bozzini, B., Lacitignola, D., Sgura, I.: Travelling waves in a reaction-diffusion model for electrodeposition. Math. Comput. Simul. 81, 1027–1044 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bozzini, B., Lacitignola, D., Sgura, I.: Frequency as the greenest additive for metal plating: mathematical and experimental study of forcing voltage effects on electrochemical growth dynamics. Int. J. Electrochem. Sci. 6, 4553–4571 (2011)

    Google Scholar 

  7. Bozzini, B., Lacitignola, D., Mele, C., Sgura, I.: Coupling of morphology and chemistry leads to morphogenesis in electrochemical metal growth: a review of the reaction-diffusion approach. Acta Appl. Math. 122, 53–68 (2012)

    MATH  MathSciNet  Google Scholar 

  8. Bozzini, B., Lacitignola, D., Sgura, I.: Spatio-temporal organisation in alloy electrodeposition: a morphochemical mathematical model and its experimental validation. J. Solid State Electrochem. 17, 467–479 (2013)

    Article  Google Scholar 

  9. Dobrovolska, T.S., Veleva, L., Krastev, I., Zielonka, A.: Composition and structure of silver-indium alloy coatings electrodeposited from cyanide electrolytes. J. Electrochem. Soc. 152, C137–C142 (2005)

    Article  Google Scholar 

  10. Dobrovolska, T.S., Krastev, I., Zabinski, P., Kowalik, R., Zielonka, A.: Oscillations and self-organization phenomena during electrodeposition of silver-indium alloys. Experimental study. Arch. Metall. Mater. 56, 645–657 (2011)

    Google Scholar 

  11. Gambino, G., Lombardo, M.C., Sammartino, M.: Turing instability and traveling fronts for a nonlinear reaction-diffusion system with cross-diffusion. Math. Comput. Simul. 82, 1112–1132 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gambino, G., Lombardo, M.C., Sammartino, M.: Pattern formation driven by cross-diffusion in a 2D domain. Nonlinear Anal., Real World Appl. 14, 1755–1779 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  13. Huber, G., Alstrom, P., Bohr, T.: Nucleation and transients at the onset of vortex turbulence. Phys. Rev. Lett. 69, 2380–2383 (1992)

    Article  Google Scholar 

  14. Ipsen, M., Hynne, F., Sorensen, P.G.: Amplitude equations and chemical reaction-diffusion systems. Int. J. Bifurc. Chaos 7, 1539–1554 (1997)

    Article  MATH  Google Scholar 

  15. Ipsen, M., Hynne, F., Sorensen, P.G.: Systematic derivation of amplitude equations and normal forms for dynamical systems. Chaos 8, 834–852 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ipsen, M., Hynne, F., Sorensen, P.G.: Amplitude equations for reaction-diffusion systems with a Hopf bifurcation and slow real modes. Physica D 136, 66–92 (1999)

    Article  MathSciNet  Google Scholar 

  17. Krastev, I., Koper, M.T.M.: Pattern formation during the electrodeposition of a silver-antimony alloy. Physica A 213, 199–208 (1995)

    Article  Google Scholar 

  18. Lacitignola, D., Bozzini, B., Sgura, I.: Spatio-temporal organization in a morphochemical electrodeposition model: Hopf and Turing instabilities and their interplay (submitted)

  19. Raub, E., Schall, A.: Silber-Indium-Legierungen. Z. Met.kd. 30, 149–151 (1938)

    Google Scholar 

  20. Reichenbach, T., Mobilia, M., Frey, E.: Self-organization of mobile populations in cyclic competition. J. Theor. Biol. 254, 368–383 (2008)

    Article  MathSciNet  Google Scholar 

  21. Van Saarloos, W.: Front propagation into unstable states. Phys. Rep. 386, 29–222 (2003)

    Article  MATH  Google Scholar 

  22. Sgura, I., Bozzini, B., Lacitignola, D.: Numerical approximation of Turing patterns in electrodeposition by ADI methods. J. Comput. Appl. Math. 236, 4132–4147 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  23. Weber, A., Kramer, L., Aranson, I.S., Aranson, L.: Stability limits of traveling waves in the transition to spatio-temporal chaos in the complex Ginzburg-Landau equation. Physica D 61, 279–283 (1992)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank the anonymous Referees for their helpful comments and remarks.

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria Elettrica e dell’Informazione, Università di Cassino e del Lazio Meridionale, Via di Biasio, 43, 03043, Cassino, Italy

    Deborah Lacitignola

  2. Dipartimento di Ingegneria dell’Innovazione, Università del Salento, Via per Monteroni, 73100, Lecce, Italy

    Benedetto Bozzini

  3. Dipartimento di Matematica e Fisica “E. De Giorgi”, Università del Salento, Via per Arnesano, 73100, Lecce, Italy

    Ivonne Sgura

Authors
  1. Deborah Lacitignola
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Benedetto Bozzini
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ivonne Sgura
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Deborah Lacitignola.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lacitignola, D., Bozzini, B. & Sgura, I. Spatio-Temporal Organization in a Morphochemical Electrodeposition Model: Analysis and Numerical Simulation of Spiral Waves. Acta Appl Math 132, 377–389 (2014). https://doi.org/10.1007/s10440-014-9910-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-014-9910-3

Keywords

Search

Navigation