Hostname: page-component-7dd5485656-s6l46 Total loading time: 0 Render date: 2025-10-31T16:18:10.173Z Has data issue: false hasContentIssue false
  • English
  • Français

Spatio-temporal organization in a morphochemical electrodeposition model: Hopf and Turing instabilities and their interplay

Published online by Cambridge University Press:  22 December 2014

DEBORAH LACITIGNOLA ,
BENEDETTO BOZZINI  and
IVONNE SGURA
DEBORAH LACITIGNOLA
Affiliation:
Dipartimento di Ingegneria Elettrica e dell'Informazione, Università di Cassino e del Lazio Meridionale, Via di Biasio, I-03043 Cassino, Italy email: d.lacitignola@unicas.it
BENEDETTO BOZZINI
Affiliation:
Dipartimento di Ingegneria dell'Innovazione, Università del Salento, via Monteroni, I-73100 Lecce, Italy email: benedetto.bozzini@unisalento.it
IVONNE SGURA
Affiliation:
Dipartimento di Matematica e Fisica “E. De Giorgi”, Università del Salento, via per Arnesano, I-73100 Lecce, Italy email: ivonne.sgura@unisalento.it
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we investigate from a theoretical point of view the 2D reaction-diffusion system for electrodeposition coupling morphology and surface chemistry, presented and experimentally validated in Bozzini et al. (2013J. Solid State Electr.17, 467–479). We analyse the mechanisms responsible for spatio-temporal organization. As a first step, spatially uniform dynamics is discussed and the occurrence of a supercritical Hopf bifurcation for the local kinetics is proved. In the spatial case, initiation of morphological patterns induced by diffusion is shown to occur in a suitable region of the parameter space. The intriguing interplay between Hopf and Turing instability is also considered, by investigating the spatio-temporal behaviour of the system in the neighbourhood of the codimension-two Turing--Hopf bifurcation point. An ADI (Alternating Direction Implicit) scheme based on high-order finite differences in space is applied to obtain numerical approximations of Turing patterns at the steady state and for the simulation of the oscillating Turing–Hopf dynamics.

Keywords

Reaction-diffusion modelsPattern formationTuring-Hopf instabilityADI methodFinite differences

Information

Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Issue 2 , April 2015 , pp. 143 - 173
Copyright
Copyright © Cambridge University Press 2014 

References

[1]Aragon, J. L., Barrio, R. A., Woolley, T. E., Baker, R. E. & Maini, P. K. (2012) Nonlinear effects on Turing patterns: Time oscillations and chaos. Phys. Rev. E 86, 026201, 1–4.CrossRefGoogle ScholarPubMed
[2]Banerjee, M. & Petrovskii S. (2011) Self-organised spatial patterns and chaos in a ratio-dependent predator-prey system. Theor. Ecol. 4, 37–53.CrossRefGoogle Scholar
[3]Baurmann, M., Gross, T. & Feudel, U. (2007) Instabilities in spatially extended predator-prey systems: Spatio-temporal patterns in the neighbourhood of Turing-Hopf bifurcations. J. Theor. Biol. 245, 220–229.CrossRefGoogle ScholarPubMed
[4]Bogoliubov, N. N. & Mitropolski, Y. A. (1961) Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach, New York.Google Scholar
[5]Boissonade, J., Dulos, E. & De, Kepper P. (1995) Onset and beyond Turing pattern formation. In: Kapral, R. & Showalter, K. (editors), Chemical Waves and Patterns, Kluwer Academic Publishers, Dordrecht.Google Scholar
[6]Borckmans, P., Dewel, G., De Wit, A., Dulos, E., Boissonade, J., Gauffre, F. & De Kepper, P. (2002) Diffusive instabilities and chemical reactions. Int. J. Bif. Chaos 12, 2307–2332.CrossRefGoogle Scholar
[7]Bozzini, B., Lacitignola, D. & Sgura, I. (2008) A reaction-diffusion model of spatial pattern formation in electrodeposition. J. Phys.: Conf. Ser. 96, 012051.Google Scholar
[8]Bozzini, B., D'Urzo, L., Lacitignola, D., Mele, C., Sgura, I. & Tondo, E. (2009) An investigation into the dynamics of Au electrodeposition based on the analysis of SERS spectral time series. Trans. Inst. Met. Fin. 87, 193–200.CrossRefGoogle Scholar
[9]Bozzini, B., Lacitignola, D. & Sgura, I. (2010) Morphological spatial patterns in a reaction diffusion model for metal growth. Math. Biosci. Eng. 7, 237–258.Google Scholar
[10]Bozzini, B., Sgura, I., Lacitignola, D., Mele, C., Marchitto, M. & Ciliberto, A. (2010) Prediction of morphological properties of smart-coatings for Cr replacement, based on mathematical modelling. Adv. Mat. Res. 138, 93–106.Google Scholar
[11]Bozzini, B., Lacitignola, D. & Sgura, I. (2011) Travelling waves in a reaction-diffusion model for electrodeposition. Math. Comput. Simul. 81, 1027–1044.CrossRefGoogle Scholar
[12]Bozzini, B., Lacitignola, D. & Sgura, I. (2011) 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.CrossRefGoogle Scholar
[13]Bozzini, B., Lacitignola, D., Mele, C. & Sgura, I. (2012) 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.Google Scholar
[14]Bozzini, B., Lacitignola, D., Mele, C. & Sgura, I. (2012) Morphogenesis in metal electrodeposition. Note di Matem. 32, 7–46.Google Scholar
[15]Bozzini, B., Lacitignola, D. & Sgura, I. (2013) Spatio-temporal organisation in Alloy electrodeposition: A morphochemical mathematical model and its experimental validation. J. Solid State Electrochem. 17, 467–479.CrossRefGoogle Scholar
[16]Mele, C., Catalano, M., Taurino, A. & Bozzini, B. (2013) Electrochemical fabrication of NPG-supported manganese oxide nanowires based on electrodeposition from eutectic urea/choline chloride ionic liquid. Electrochim. Acta 87, 918–924.CrossRefGoogle Scholar
[17]Bozzini, B., Gianoncelli, A., Mele, C. & Kiskinova, M. (2013) Electrochemical fabrication of nanoporous gold decorated with manganese oxide nanowires from eutectic urea/choline chloride ionic liquid. Part II - Electrodeposition of Au–Mn: A study based on soft X-ray microspectroscopy. Electrochim. Acta, 114, 889–896.CrossRefGoogle Scholar
[18]Castets, V., Dulos, E., Boissonade, J. & De Kepper, P. (1990) Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern. Phys. Rev. Lett. 64, 2953.CrossRefGoogle ScholarPubMed
[19]Crawford, J. D. (1991) Introduction to bifurcation theory. Rev. Mod. Phys. 63, 991–1037.CrossRefGoogle Scholar
[20]Cross, M. C. & Honenberg P. C. (1993) Pattern formation outside the equilibrium. Rev. Mod. Phys. 65, 851–1112.CrossRefGoogle Scholar
[21]Cross, M. & Greenside, H. (2009) Pattern Formation and Dynamics in Nonequilibrium Systems, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
[22]Dewel, G., Borckmans, P., De Wit, A., Rudovics, B., Perraud, J., Dulos, E., Boissonade, J. & Kepper, P. D. (1995) Pattern selection and localized structures in reaction-diffusion systems, Physica A 213, 181–198.CrossRefGoogle Scholar
[23]Dewel, G., De Wit, A., Metens, S., Verdasca, J. & Borckmans, P. (1996) Pattern selection in reaction-diffusion systems with competing bifurcations. Phys. Scr. T67, 51–57.CrossRefGoogle Scholar
[24]De Wit, A., Lima, D., Dewel, G. & Borckmans, P. (1996) Spatiotemporal dynamics near a codimension-two point. Phys. Rev. E 54, 261–271.CrossRefGoogle Scholar
[25]De Wit, A. (1999) Spatial patterns and spatiotemporal dynamics in chemical systems. Adv. Chem. Phys. 109, 435–513.Google Scholar
[26]Dobrovolska, Ts., Krastev, I., Zabinski, P., Kowalik, R. & Zielonka, A. (2011) Oscillations and self-organization phenomena during electrodeposition of silver-indium alloys. Experimental study. Arch. Metall. Mater. 56, 645–657.CrossRefGoogle Scholar
[27]Dobrovolska, Ts., Lopez-Sauri, D. A., Veleva, L. & Krastev, I. (2012) Oscillations and spatio-temporal structures during electrodeposition of AgCd alloys. Electrochim. Acta 79, 162–169.CrossRefGoogle Scholar
[28]Epstein, J. M. (1997) Nonlinear Dynamics, Mathematical Biology and Social Science, Addison-Wesley, Reading, MA.Google Scholar
[29]Gambino, G., Lombardo, M. C. & Sammartino, M. (2012) Turing instability and traveling fronts for a nonlinear reaction–diffusion system with cross-diffusion. Math. Comput. Simul. 82, 1112–1132.CrossRefGoogle Scholar
[30]Gambino, G., Lombardo, M. C. & Sammartino, M. (2013) Pattern formation driven by cross-diffusion in a 2D domain. Nonlinear Anal. Real World Appl. 14, 1755–1779.CrossRefGoogle Scholar
[31]Gross, T. & Feudel, U. (2006) Generalized models as a universal approach to the analysis of nonlinear dynamical systems. Phys. Rev. E 75, 1–14.Google Scholar
[32]Hoyle, R. (2006) Pattern Formation. An Introduction to Methods, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
[33]Just, W., Bose, M., Bose, S., Engel, H. & Scholl, E. (2001) Spatiotemporal dynamics near a supercritical Turing-Hopf bifurcation in a two-dimensional reaction-diffusion system. Phys. Rev. E 64, 1–12.CrossRefGoogle Scholar
[34]Kaminaga, A., Vanag, V. K. & Epstein, I. R. (2005) ‘Black spots’ in a surfactant-rich BZ-AOT microemulsion system. J. Chem. Phys. 122, 174706, 1–11.CrossRefGoogle Scholar
[35]Kapral, R. & Showalter, K. (1995) Chemical Waves and Patterns, Kluwer Academic Publishers, Dordrecht.CrossRefGoogle Scholar
[36]Koper, M. T. M. (1998) Non-linear phenomena in electrochemical systems. J. Chem. Soc. Faraday Trans. 94, 1369–1378.CrossRefGoogle Scholar
[37]Krastev, I. & Koper M. T. M. (1995) Pattern formation during the electrodeposition of a silver-antimony alloy. Physica A 213, 199–208.CrossRefGoogle Scholar
[38]Krastev, I. (2009) Self-structured silver alloy coatings and their properties. J. Eng. Process. Manage. 1, 104–112.Google Scholar
[39]Krastev, I., Dobrovolska Ts., Lacnjevac U. & Nineva S. (2012) Pattern formation during electrodeposition of indium-cobalt alloys. J. Solid State Electrochem. 16, 3449–3456.CrossRefGoogle Scholar
[40]Krischer, K., Mazouz, N. & Grauel, P. (2001) Fronts, waves, and stationary patterns in electrochemical systems. Angew. Chem. Int. Ed. 40, 850–869.3.0.CO;2-3>CrossRefGoogle ScholarPubMed
[41]Lacitignola, D., Bozzini, B. & Sgura, I. (2014) Spatio-temporal organization in a morphochemical electrodeposition model: Analysis and numerical simulation of spiral waves. Acta Appl. Math.,132, 377–389.CrossRefGoogle Scholar
[42]Lee, K. J., McCormick, W. D., Ouyang, Q. & Swinney, H. L. (1993) Pattern Formation by Interacting Chemical Fronts. Sci. New Ser. 261(5118), 192–194.Google ScholarPubMed
[43]Leppanen, T., Karttunen, M., Kaski, K. & Barrio, R. A. (2004) Turing systems as models of complex pattern formation. Braz. J. Phys. 34, 368–372.CrossRefGoogle Scholar
[44]Leppänen, T., Karttunen, M., Barrio, R. A. & Kaski, K. (2004) Morphological transitions and bistability in Turing systems. Phys. Rev. E 70, 066202.CrossRefGoogle ScholarPubMed
[45]Li, Y-J., Oslonovitch, J., Mazouz, N., Plenge, F., Krischer, K. & Ertl, G. (2001) Turing-type patterns on electrode surfaces. Science 291, 2395–2398.CrossRefGoogle ScholarPubMed
[46]Maini, P. K. & Othmer, H. G. (2001) Mathematical models for biological pattern formation. In: The IMA Volumes in Mathematics and its Applications - Frontiers in Applications of Mathematics, Vol. 121, Springer, New York.Google Scholar
[47]Maini, P. K. (2004) Using mathematical models to help understand biological pattern formation. C. R. Biol. 327, 225–234.CrossRefGoogle ScholarPubMed
[48]Malchow, H., Petrovskii, S. V. & Venturino, E. (2008) Spatiotemporal Patterns in Ecology and Epidemiology, Chapman & Hall, U.K.Google Scholar
[49]Meixner, M., De Wit, A., Bose, S. & Scholl, E. (1997) Generic spatiotemporal dynamics near codimension-two Turing Hopf bifurcations. Phys. Rev. E 55, 6690–6697.CrossRefGoogle Scholar
[50]Miguez, D. G., Alonso, S., Munuzuri, A. P. & Sagues, F. (2006) Experimental evidence of localized oscillations in the photosensitive chlorine dioxide-iodine-malonic acid reaction. Phys. Rev. Lett. 97, 178301, 1–4.CrossRefGoogle ScholarPubMed
[51]Murray, J. D. (2002) Mathematical Biology II; Spatial Models and Biomedical Applications, 3rd ed., Springer, New York.Google Scholar
[52]Orbán, M., Kurin-Csörgei, K., Zhabotinsky, A. M. & Epstein, I. R. (2001) A new chemical system for studying pattern formation: Bromate-Hypophosphite-Acetone - dual catalyst. Faraday Disc. 120, 11–19.CrossRefGoogle Scholar
[53]Ouyang, Q. & Swinney, H. L. (1991) Transition from a uniform state to hexagonal and striped Turing patterns. Nature 352, 610–612.CrossRefGoogle Scholar
[54]Perraud, J. J., Agladze, K., Dulos, E. & De Kepper, P. (1992) Stationary Turing patterns versus time-dependent structures in the chlorite-iodide-malonic acid reaction. Physica A 188, 1–16.CrossRefGoogle Scholar
[55]Ricard, M. R. & Michler, S. (2009) Turing Instabilities at Hopf bifurcation. J. Nonlinear Sci. 19, 467–496.CrossRefGoogle Scholar
[56]Rovinsky, A. & Menzinger M. (1992) Interaction of Turing and Hopf bifurcations in chemical systems. Phys. Rev. A 46, 6315–6322.CrossRefGoogle ScholarPubMed
[57]Rudovics, B., Dulos, E. & de Kepper, P. (1996) Standard and nonstandard Turing patterns and waves in the CIMA reaction. Phys. Scr. T67, 43–50.CrossRefGoogle Scholar
[58]Sagués, F. & Epstein, I. R. (2003) Nonlinear chemical dynamics. Dalton Trans. 1201–1217.CrossRefGoogle Scholar
[59]Saitou, M. & Fukuoka, Y. (2004) An experimental study on stripe pattern formation of Ag-Sb electrodeposits. J. Phys. Chem. B 108, 5380–5385.CrossRefGoogle Scholar
[60]Saitou, M. & Fukuoka, Y. (2004) A study on transient changes of surface morphologies in Ag and Sb coelectrodeposition. J. Electrochem. Soc. 151, C627–C632.CrossRefGoogle Scholar
[61]Saitou, M. & Fukuoka, Y. (2005) Stripe patterns in Ag-Sb co-electrodeposition. Electrochim. Acta 50, 5044–5049.CrossRefGoogle Scholar
[62]Sanders, J. A. & Verhulst, F. (1985) Averaging Methods in Nonlinear Dynamical Systems, Applied Mathematical Sciences, Vol. 59, Springer, New York.CrossRefGoogle Scholar
[63]Sgura, I., Bozzini, B. & Lacitignola, D. (2012) Numerical approximation of Turing patterns in electrodeposition by ADI methods. J. Comput. Appl. Math. 236, 4132–4147.CrossRefGoogle Scholar
[64]Sgura, I., Bozzini, B. & Lacitignola, D. (2012) Numerical approximation of oscillating Turing patterns in a reaction-diffusion model for electrochemical material growth. AIP Conf. Proc. 1493, 896–903.CrossRefGoogle Scholar
[65]Sherratt, J. A. (2012) Turing patterns in deserts. In: Cooper, S. B., Dawar, A. & Lowe, B. (editors), How the World Computes, Lecture Notes in Computer Science, Vol. 7318, Springer, New York, pp. 667–674.CrossRefGoogle Scholar
[66]Turing, A. M. (1952) The chemical bases of morphogenesis, Phil. Trans. R. Soc. London B 237, 37–72.Google Scholar
[67]Vanag, V. K. & Epstein I. R. (2008) Design and control of patterns in reaction-diffusion systems. Chaos 18, 026107, 1–11.CrossRefGoogle ScholarPubMed
[68]Vanag, V. K. & Epstein I. R. (2009) Pattern formation mechanisms in reaction-diffusion systems. Int. J. Dev. Biol. 53, 673–681.CrossRefGoogle ScholarPubMed
[69]Yang, L., Dolnik, M., Zhabotinsky, A. M. & Epstein, I. R. (2002) Spatial resonances and superposition patterns in a reaction-diffusion model with interacting Turing modes. Phys. Rev. Lett. 88, 208303, 1–4.CrossRefGoogle Scholar
[70]Yang, L. & Epstein, I. R. (2003) Oscillatory Turing patterns in reaction-diffusion systems with two coupled layers. Phys. Rev. Lett. 90, 178303, 1–4.CrossRefGoogle ScholarPubMed