Spiral waves on the sphere for an alloy electrodeposition model

Abstract This paper focuses on the emergence of spiral waves in a specific morpho-electrochemical reaction-diffusion model on a sphere. This study fits in the framework of the morphological control of material electrodeposited onto spherical particles that is crucial to the energetic efficiency of the recharge process as well as to the durability of energy storage devices. The spherical geometry for the electrode surface is of notable practical interest since spheres are the shape of choice for flow batteries and metal-air devices [35]. Motivated by this technological framework, in this paper we extend the results on pattern formation in [27] to include investigations on the spiral wave phenomenology. We show that spiral waves emerge because of the interplay between two specific model parameters: one regulating the oscillatory dynamics in the kinetics and the other one related to the domain size. We present systematic numerical simulations based on the finite element method LSFEM [45,50] accompanied by the computation of suitable indicators that allow to characterize and compare the spatio-temporal features. Interestingly, the model also supports a mechanism of spirals break up leading to a complex spatio-temporal phenomenology. The findings of our study have been validated with experimental results on Ag-In and Ag-Co electrodeposition.

[1]  I. Aranson,et al.  The world of the complex Ginzburg-Landau equation , 2001, cond-mat/0106115.

[2]  B. Bozzini,et al.  Parameter estimation for a morphochemical reaction-diffusion model of electrochemical pattern formation , 2018, Inverse Problems in Science and Engineering.

[3]  J. Keener,et al.  Spiral waves in the Belousov-Zhabotinskii reaction , 1986 .

[4]  Bifurcations of equilibria in a mathematical model for metal growth , 2018 .

[5]  Single and double rotor spiral waves on spherical surfaces , 1990 .

[6]  Richard Kollár,et al.  Coherent Structures Generated by Inhomogeneities in Oscillatory Media , 2007, SIAM J. Appl. Dyn. Syst..

[7]  T. Dobrovolska,et al.  Electrodeposition of gold–indium alloys , 2015 .

[8]  D. Lacitignola,et al.  Spatio-temporal organization in a morphochemical electrodeposition model: Hopf and Turing instabilities and their interplay , 2014, European Journal of Applied Mathematics.

[9]  L. Glass,et al.  Scroll waves in spherical shell geometries. , 2001, Chaos.

[10]  R A Barrio,et al.  Turing patterns on a sphere. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[11]  Markus Bär,et al.  Dependence of the spiral rotation frequency on the surface curvature of reaction-diffusion systems , 2003 .

[12]  M. Kiskinova,et al.  Depth-Dependent Scanning Photoelectron Microspectroscopy Unravels the Mechanism of Dynamic Pattern Formation in Alloy Electrodeposition , 2018, The Journal of Physical Chemistry C.

[13]  Faridon Amdjadi,et al.  REACTION-DIFFUSION EQUATIONS ON A SPHERE: MEANDERING OF SPIRAL WAVES , 1997 .

[14]  Peter Grindrod,et al.  The geometry and motion of reaction-diffusion waves on closed two-dimensional manifolds , 1987, Journal of mathematical biology.

[15]  D. Lacitignola,et al.  Intermetallics as key to spiral formation in In–Co electrodeposition. A study based on photoelectron microspectroscopy, mathematical modelling and numerical approximations , 2015 .

[16]  N. A. Saltykova Electrodeposition of platinum metals and alloys from chloride melts , 2003 .

[17]  Charles M. Elliott,et al.  Finite element methods for surface PDEs* , 2013, Acta Numerica.

[18]  J. Tsai Rotating spiral waves in λ - ω systems on circular domains , 2010 .

[19]  Anotida Madzvamuse,et al.  Preserving invariance properties of reaction–diffusion systems on stationary surfaces , 2016, 1609.02741.

[20]  Arnd Scheel,et al.  Bifurcation to spiral waves in reaction-diffusion systems , 1998 .

[21]  Zuntao Fu,et al.  The Hopf bifurcation and spiral wave solution of the complex Ginzburg–Landau equation , 2002 .

[22]  W. Baxter,et al.  Stationary and drifting spiral waves of excitation in isolated cardiac muscle , 1992, Nature.

[23]  X. Illa,et al.  Spontaneous formation of spiral-like patterns with distinct periodic physical properties by confined electrodeposition of Co-In disks , 2016, Scientific Reports.

[24]  D. Lacitignola,et al.  Cross-diffusion effects on a morphochemical model for electrodeposition , 2018 .

[25]  Bard Ermentrout,et al.  The Existence of Spiral Waves in an Oscillatory Reaction-Diffusion System , 1994, SIAM J. Appl. Math..

[26]  J. Keener A geometrical theory for spiral waves in excitable media , 1986 .

[27]  T. Dobrovolska,et al.  Pattern formation during electrodeposition of indium–cobalt alloys , 2012, Journal of Solid State Electrochemistry.

[28]  O. Faugeras,et al.  Standing and travelling waves in a spherical brain model: The Nunez model revisited , 2017, Physica D. Nonlinear phenomena.

[29]  L. Glass,et al.  Spiral wave dynamics in excitable media with spherical geometries. , 2006, Chaos.

[30]  Michio Yamada,et al.  Spiral wave behaviors in an excitable reaction-diffusion system on a sphere , 1998 .

[31]  D. Lacitignola,et al.  Spatio-temporal organization in alloy electrodeposition: a morphochemical mathematical model and its experimental validation , 2013, Journal of Solid State Electrochemistry.

[32]  Benedetto Bozzini,et al.  Characterization of the particulate anode of a laboratory flow Zn–air fuel cell , 2017, Journal of Applied Electrochemistry.

[33]  J. Ramos Dynamics of spiral waves in excitable media with local time-periodic modulation , 2002 .

[34]  Leon Glass,et al.  Topological constraints on spiral wave dynamics in spherical geometries with inhomogeneous excitability. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[35]  Spatio-Temporal Organization in a Morphochemical Electrodeposition Model: Analysis and Numerical Simulation of Spiral Waves , 2014 .

[36]  T. Dobrovolska,et al.  New examples of electrodeposited alloy systems with pattern formation , 2016 .

[37]  B. Roth Meandering of spiral waves in anisotropic cardiac tissue , 2001 .

[38]  Deborah Lacitignola,et al.  Turing pattern formation on the sphere for a morphochemical reaction-diffusion model for electrodeposition , 2017, Commun. Nonlinear Sci. Numer. Simul..

[39]  H. Henry Spiral wave drift in an electric field and scroll wave instabilities. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[40]  V. Davydov,et al.  Drift of spiral waves on nonuniformly curved surfaces , 2000 .

[41]  Jiang-Xing Chen,et al.  Controlling chaos by developing spiral wave from heterogeneity in excitable medium , 2009 .

[42]  A. Winfree When time breaks down , 1987 .

[43]  T. Dobrovolska,et al.  Self-organisation phenomena during electrodeposition of palladium–indium alloys , 2015 .