When Si is anodically oxidized in a fluoride containing electrolyte, an oxide layer is grown. Simultaneously, the layer is etched by the fluoride containing electrolyte. The resulting stationary state exhibits a negative slope of the current-voltage characteristics in a certain range of applied voltage. We propose a physical model that reproduces this negative slope. In particular, our model assumes that the oxide layer consists of both partially and fully oxidized Si and that the etch rate depends on the effective degree of oxidation. Finally, we show that our simulations are in good agreement with measurements of the current-voltage characteristics, the oxide layer thickness, the dissolution valence, and the impedance spectra of the electrochemical system.
Using dynamic multifrequency analysis (DMFA), we investigated the oscillatory reaction dynamics that govern the anodic electrodissolution of p‐type silicon in fluoride‐containing electrolytes, in which the anodization of silicon is followed by the chemical etching of the oxide layer. By applying a constant voltage to the silicon electrode, stable oscillations are found in the presence of an external resistance. The dynamic impedance spectra acquired through DMFA were fitted to a suitable electrical equivalent circuit. In doing so, it was possible to investigate the temporal evolution of the kinetic parameters throughout the formation and dissolution of the silicon oxide.
Bistable microelectrodes with an S-shaped current-voltage characteristic have recently been shown to oscillate under current control, when connected in parallel. In other systems with equivalently coupled bistable components, such oscillatory instabilities have not been reported. In this paper, we derive a general criterion for when an ensemble of coupled bistable components may become oscillatorily unstable. Using a general model, we perform a stability analysis of the ensemble equilibria, in which the components always group in three or fewer clusters. Based thereon, we give a necessary condition for the occurrence of collective oscillations. Moreover, we demonstrate that stable oscillations may persist for an arbitrarily large number of components, even though, as we show, any equilibrium with two or more components on the middle, autocatalytic branch is unstable.
We investigate dynamics and bifurcations in a mathematical model that captures electrochemical experiments on arrays of microelectrodes. In isolation, each individual microelectrode is described by a one-dimensional unit with a bistable current-potential response. When an array of such electrodes is coupled by controlling the total electric current, the common electric potential of all electrodes oscillates in some interval of the current. These coupling-induced collective oscillations of bistable one-dimensional units are captured by the model. Moreover, any equilibrium is contained in a cluster subspace, where the electrodes take at most three distinct states. We systematically analyze the dynamics and bifurcations of the model equations: We consider the dynamics on cluster subspaces of successively increasing dimension and analyze the bifurcations occurring therein. Most importantly, the system exhibits an equivariant transcritical bifurcation of limit cycles. From this bifurcation, several limit cycles branch, one of which is stable for arbitrarily many bistable units.
Bistability, i.e. the coexistence of different states at identical external parameters, is a frequently encountered phenomenon in electrode reactions. If the reactions occur on individual, i.e. spatially separated, catalytically active areas that are all electrically connected, each individual active area can be considered as a bistable component, and the entire ensemble of all active areas as a many particle system of such interconnected bistable components. Examples range from (micro-)electrode arrays, where each of a usually moderate number of individual electrodes constitutes the bistable components, to insertion battery cathodes, where each of the billions of nano-particulate storage particles can be considered a bistable component. In this talk, we will demonstrate that despite of their completely different chemical nature, from a dynamic point of view, all these systems can be treated mathematically under a common framework. We will start by introducing the general mathematical description and compile key results that can be derived [1]: (1) All steady states are composed of one, two or three cluster states, i.e. each of the individual components takes on one of the three values which correspond to the three steady states of the individual bistable component at the corresponding common value of the external voltage. (2) Stable steady states possess at most one electrode on the intermediate, autocatalytic branch. As a consequence, the electrodes transition sequentially from one stable state to the other one as the current density is increased. This sequential activation is a generalization of the mosaic instability described in phase transition systems. (3) Ensembles of globally coupled bistable components might exhibit oscillations. We will derive necessary condition of when the many particle system might become oscillatory and illustrate this condition with an intuitive LC analogue. In the second part of the talk we will apply our general theory to two prominent examples. We will start with the CO oxidation on an array of Pt micro-electrodes, where experiments showed collective oscillations [2,3]. For this system, the necessary condition for a Hopf bifurcation to occur translates to the condition that components in two different groups (or, equivalently two of the three steady states of the bistable elements) have different slopes in either of the two dependences: the change of the coverage with potential at constant CO coverage, or in the change of the current with CO coverage at constant potential. While the first derivative is always positive, the current increases with coverage in the active state (where the coverage is low), but decreases in the passive (CO covered state) or the intermediate states which possess relatively high coverages. The existence of stable oscillations is indeed also observed in simulations. Besides oscillations, this system shows also period doubling cascades and chaos. Our second example involves Li insertion batteries. Here, the chemical potential of each storage nanoparticle exhibits a non-monotonic characteristic as a function of the degree of charging [4]. Hence, the two stable states of a bistable component can be identified with charged and discharged nanoparticle of the insertion material, and a battery under charging / discharging conditions can be classified as a bistable many component system. During charging / discharging with a constant current oscillations of the potential have been reported [5], whose origin has not yet unambiguously been identified. We present a simplified model of the dynamics of the Li ion batteries and again apply our criterion for a Hopf bifurcation to the model. In this case, the necessary condition is not fulfilled so that the origin of the observed oscillations remains in the dark. [1] M. Salman, Chr. Bick and K. Krischer, Phys. Rev. Research 2, 043125 (2020) [2] D. Alfonso Crespo-Yapur, Antoine Bonnefont, Rolf Schuster, Katharina Krischer, and Elena R. Savinova, ChemPhysChem 14, 1117 (2013). [3] S. Bozdech, Y. Biecher, E. R. Savinova, R. Schuster, K. Krischer, and A. Bonnefont, Chaos 28, 045113 (2018). [4] W. Dreyer, J. Jamnik, C. Guhlke, R. Huth, J. Moskon, M. Gaberscek, Nat Mater. 9, 448 (2010). [5] D. Li, Y. Sun, Zh. Zang, L. Gu, Z. Chen, H. Zhou, Joule 2, 1265 (2018)
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