Spiral waves on the sphere for an alloy electrodeposition model. (English) Zbl 1510.78044
Summary: 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 [the third author et al., “Depth-dependent scanning photoelectron microspectroscopy unravels the mechanism of dynamic pattern formation in alloy electrodeposition”, J. Phys. Chem. C 122, No. 2, 15996–6007 (2018; doi:10.1021/acs.jpcc.8b01267)]. Motivated by this technological framework, in this paper we extend the results on pattern formation in [J. Maselko, “Symmetrical double rotor spial waves on spherical surfaces”, J. Chem. Soc. Faraday Trans. 94, 2343–2345 (1998)] 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 [the first author et al., Commun. Nonlinear Sci. Numer. Simul. 48, 484–508 (2017; Zbl 1510.92034); M. Frittelli et al., IMA J. Numer. Anal. 39, No. 1, 235–270 (2019; Zbl 1483.65155)] 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.
MSC:
| 78A57 | Electrochemistry |
| 78A55 | Technical applications of optics and electromagnetic theory |
| 35B36 | Pattern formations in context of PDEs |
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
| 78M10 | Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory |
Keywords:
reaction-diffusion; spiral waves; lumped surface finite element method (LSFEM); electrodepositionReferences:
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