Theory of electrochemical pattern formation

Filament dynamics in confined chemical gardens and in filiform corrosion

Once rescaled by the preferred wavenumber ω* of the curvature power spectrum, both filiform corrosion and chemical garden filaments display similar dynamics.

Spatially distributed current oscillations with electrochemical reactions in microfluidic flow cells

The formation of spatiotemporal patterns is investigated by using a chemical reaction on the surface of a high-aspect-ratio metal electrode positioned in a flow channel. A partial differential equation model is formulated for nickel dissolution in sulfuric acid in a microfluidic flow channel. The model simulations predict oscillatory patterns that are spatially distributed on the electrode surface; the downstream portion of the metal surface exhibits large-amplitude, nonlinear oscillations of dissolution rates, whereas the upstream portion displays small-amplitude, harmonic oscillations with a phase delay. The features of the dynamical response can be interpreted by the dependence of local dynamics on the widely varying surface conditions and the presence of strong coupling. The patterns can be observed for both contiguous and segmented metal surfaces. The existence of spatially distributed current oscillations is confirmed in experiments with Ni electrodissolution in a microfluidic device. The results show the impact of a widely heterogeneous environment on the types of patterns of chemical reaction rates.

Dynamics of reaction-diffusion patterns controlled by asymmetric nonlocal coupling as a limiting case of differential advection

A one-component bistable reaction-diffusion system with asymmetric nonlocal coupling is derived as a limiting case of a two-component activator-inhibitor reaction-diffusion model with differential advection. The effects of asymmetric nonlocal couplings in such a bistable reaction-diffusion system are then compared to the previously studied case of a system with symmetric nonlocal coupling. We carry out a linear stability analysis of the spatially homogeneous steady states of the model and numerical simulations of the model to show how the asymmetric nonlocal coupling controls and alters the steady states and the front dynamics in the system. In a second step, a third fast reaction-diffusion equation is included which induces the formation of more complex patterns. A linear stability analysis predicts traveling waves for asymmetric nonlocal coupling, in contrast to a stationary Turing patterns for a system with symmetric nonlocal coupling. These findings are verified by direct numerical integration of the full equations with nonlocal coupling.

Synchronization of electrochemical oscillators with differential coupling

Experiments are presented to describe the effect of capacitive coupling of two electrochemical oscillators during Ni dissolution in sulfuric acid solution. Equivalent circuit analysis shows that the coupling between the oscillators occurs through the difference between the differentials of the electrode potentials. The differential nature of the coupling introduces strong negative nonisochronicity (i.e., phase shear, strong dependence of the period on the amplitude) in the coupling mechanism with smooth oscillators (under conditions just above a Hopf bifurcation point). Because of the negative nonisochronicity, asymmetrically coupled oscillators exhibit anomalous phase synchronization in the form of frequency difference enhancement. At strong coupling bistability is observed between in-phase and antiphase synchronized states. With relaxation oscillators, in contrast to the resistive coupling where antiphase synchronization can occur, the typical system response with weak coupling is out-of-phase synchronization. When the capacitance is applied on the individual resistors attached to the electrodes the oscillators exhibit weak positive nonisochronicity; this is in contrast with the strong negative nonisochronicity obtained with cross coupling. The proposed coupling configurations reveal the importance of the nonisochronicity level of oscillations for the experimentally observed synchronization patterns and also provide efficient ways of tuning the nonisochronicity level of the oscillations. This latter feature can be exploited to design synchronization features with a combination of resistive (difference) and capacitive (differential) coupling.

The formation mechanism of defects, spiral wave in the network of neurons

A regular network of neurons is constructed by using the Morris-Lecar (ML) neuron with the ion channels being considered, and the potential mechnism of the formation of a spiral wave is investigated in detail. Several spiral waves are initiated by blocking the target wave with artificial defects and/or partial blocking (poisoning) in ion channels. Furthermore, possible conditions for spiral wave formation and the effect of partial channel blocking are discussed completely. Our results are summarized as follows. 1) The emergence of a target wave depends on the transmembrane currents with diversity, which mapped from the external forcing current and this kind of diversity is associated with spatial heterogeneity in the media. 2) Distinct spiral wave could be induced to occupy the network when the target wave is broken by partially blocking the ion channels of a fraction of neurons (local poisoned area), and these generated spiral waves are similar with the spiral waves induced by artificial defects. It is confirmed that partial channel blocking of some neurons in the network could play a similar role in breaking a target wave as do artificial defects; 3) Channel noise and additive Gaussian white noise are also considered, and it is confirmed that spiral waves are also induced in the network in the presence of noise. According to the results mentioned above, we conclude that appropriate poisoning in ion channels of neurons in the network acts as ‘defects’ on the evolution of the spatiotemporal pattern, and accounts for the emergence of a spiral wave in the network of neurons. These results could be helpful to understand the potential cause of the formation and development of spiral waves in the cortex of a neuronal system.

Kinetic insights over a PEMFC operating on stationary and oscillatory states

Kinetic investigations in the oscillatory state have been carried out in order to shed light on the interplay between the complex kinetics exhibited by a proton exchange membrane fuel cell fed with poisoned H(2) (108 ppm of CO) and the other in serie process. The apparent activation energy (E(a)) in the stationary state was investigated in order to clarify the E(a) observed in the oscillatory state. The apparent activation energy in the stationary state, under potentiostatic control, rendered (a) E(a) ≈ 50-60 kJ mol(-1) over 0.8 V < E < 0.6 V and (b) E(a) ≈ 10 kJ mol(-1) at E = 0.3 V. The former is related to the H(2) adsorption in the vacancies of the surface poisoned by CO and the latter is correlated to the process of proton conductivity in the membrane. The dependence of the period-one oscillations on the temperature yielded a genuine Arrhenius dependence with two E(a) values: (a) E(a) around 70 kJ mol(-1), at high temperatures, and (b) E(a) around 10-15 kJ mol(-1), at lower temperatures. The latter E(a) indicates the presence of protonic mass transport coupled to the essential oscillatory mechanism. These insights point in the right direction to predict spatial couplings between anode and cathode as having the highest strength as well as to speculate the most likely candidates to promote spatial inhomogeneities.

Synchronized Current Oscillations of Formic Acid Electro-oxidation in a Microchip-based Dual-Electrode Flow Cell

We investigate the oscillatory electro-oxidation of formic acid on platinum in a microchip-based dual-electrode cell with microfluidic flow control. The main dynamical features of current oscillations on single Pt electrode that had been observed in macro-cells are reproduced in the microfabricated electrochemical cell. In dual-electrode configuration nearly in-phase synchronized current oscillations occur when the reference/counter electrodes are placed far away from the microelectrodes. The synchronization disappears with close reference/counter electrode placements. We show that the cause for synchronization is weak albeit important, bidirectional electrical coupling between the electrodes; therefore the unidirectional mass transfer interactions are negligible. The experimental design enables the investigation of the dynamical behavior in micro-electrode arrays with well-defined control of flow of the electrolyte in a manner where the size and spacing of the electrodes can be easily varied.

Confined Spatio-Temporal Chaos During Metal Electrodissolution: Simulations

The electrodissolution of metal electrodes exhibits often sustained oscillations of the current density. We study the spatio-temporal dynamics of a metal disk electrode embedded in an insulator in a cylindrical electrochemical cell with a ring-shaped CE in the oscillatory region. The chosen cell geometry introduces a strong radial parameter dependence into the system. For low conductivity the dynamics is spatio-temporally chaotic. A small aspect ratio of the cell supports 2-dimensional space-time chaos on the entire disk electrode. For larger aspect ratios and thus larger parameter gradients, however, we observe a ‘self-confinement’ of the turbulent dynamics to the central part of the disk electrode, the outer ring of the electrode exhibiting angularly uniform oscillations. It can be expected that the state of confined chaos is not restricted to electrochemical systems but can be established also in other dynamical systems by introducing an appropriate parameter gradient.

Spontaneous oscillations and waves during chemical vapor deposition of InN

We report observations of self-sustaining spatiotemporal chemical oscillations during metal-organic chemical vapor deposition of InN onto GaN. Under constant supply of vapor precursors trimethylindium and NH3, the condensed-phase cycles between crystalline islands of InN and elemental In droplets. Propagating fronts between regions of InN and In occur with linear, circular, and spiral geometries. The results are described by a model in which the nitrogen activity produced by surface-catalyzed NH3 decomposition varies with the exposed surface areas of GaN, InN, and In.

Oscillatory electrodeposition of metal films at liquid/liquid interfaces induced by the large surface energy of growing deposits

Electrodeposition of zinc (Zn) at an aqueous ZnSO4/n-butylacetate (BuAc) interface (liquid/liquid (LL) interface) showed a potential oscillation in the region of the current density exceeding the diffusion-limited one, accompanied by formation of two-dimensional Zn film with a concentric pattern at the LL interface. In-situ optical microscopic inspections revealed that the oscillatory growth of the Zn film synchronized with meniscus oscillation of the LL interface. The vigorous growth of the deposits occurs only when the shape of the meniscus becomes hollow on the negative potential side of the potential oscillation. On the other hand, on the positive side, the meniscus becomes almost flat and the deposits formed in the preceding stage are thickened. A mechanism is proposed to explain the oscillatory Zn electrodeposition coupled with the meniscus oscillation, on the basis of the fact that the interfacial tension at the growing metal/aqueous solution interface is extremely large.

Mixed-mode oscillations and cluster patterns in an electrochemical relaxation oscillator under galvanostatic control

We studied the dynamics of a prototypical electrochemical model, the electro-oxidation of hydrogen in the presence of poisons, under galvanostatic conditions. The lumped system exhibits relaxation oscillations, which develop mixed-mode oscillations (MMOs) for low preset currents. A fast-slow analysis of the homogeneous dynamics reveals that the MMOs arise from a fast oscillating subsystem and a one-dimensional slow manifold. In the spatially extended system, the galvanostatic constraint imposes a synchronizing global coupling that drives the system into cluster patterns. The properties of the cluster patterns (CPs) result from an intricate interplay of the nature of the local oscillators, the global constraint, and a nonlocal coupling through the electrolyte. In particular, we find that the global constraint suppresses small-amplitude oscillations of MMOs and prevents domains oscillating out of phase from occupying equal regions in phase space. The nonlocal coupling causes each individual clustered region to oscillate on a different limit cycle. Typically multistability of CPs is found. Coexisting patterns possess different oscillation periods and a different total fraction in space that occupies the in-phase or out-of-phase state, respectively.

Edge effects in an electrochemical reaction: HCOOH oxidation on a Pt ribbon

The use of a ribbon-shaped Pt electrode gives rise to edge effects of the interfacial potential, as is predicted from the potential theory in the form of the corresponding reaction-migration equation. They are studied in the bistable region of formic acid oxidation. Essentially, the edges tend to be more passive than the bulk of the electrode, which also causes a passivation (activation) transition to originate from the edges (center) of the ribbon. The experimental results are in agreement with simulations of the reaction-migration system.

Two-dimensional electrochemical turbulence during the electrodissolution of metal disk electrodes: Model calculations

We present numerical studies of the spatio-temporal dynamics of disk electrodes with local limit cycle oscillations. The simulations are done with a realistic 3-D geometry of the electrochemical cell and disk-shaped working electrodes (WE). Spatio-temporal chaos is shown to exist from a critical electrode size onwards. It is analyzed by Karhunen-Loève decomposition and Hilbert transform. The former shows that the chaos becomes more complex with increasing system size, the latter allows features that generate the spatio-temporal complexity to be identified, namely, spatially extended 1-D phase defects and topological defects.

Creating bound states in excitable media by means of nonlocal coupling

We consider pulses of excitation in reaction-diffusion systems subjected to nonlocal coupling. This coupling represents long-range connections between the elements of the medium; the connection strength decays exponentially with the distance. Without coupling, pulses interact only repulsively and bound states with two or more pulses propagating at the same velocity are impossible. Upon switching on nonlocal coupling, pulses begin to interact attractively and form bound states. First we present numerical results on the emergence of bound states in the excitable Oregonator model for the photosensitive Belousov-Zhabotinsky reaction with nonlocal coupling. Then we show that the appearance of bound states is provided solely by the exponential decay of nonlocal coupling and thus can be found in a wide class of excitable systems, regardless of the particular kinetics. The theoretical explanation of the emergence of bound states is based on the bifurcation analysis of the profile equations that describe the spatial shape of pulses. The central object is a codimension-4 homoclinic orbit which exists for zero coupling strength. The emergence of bound states is described by the bifurcation to 2-homoclinic solutions from the codimension-4 homoclinic orbit upon switching on nonlocal coupling. We stress that the high codimension of the bifurcation to bound states is generic, provided that the coupling range is sufficiently large.

Pattern formation in stiff oscillatory media with nonlocal coupling: a numerical study of the hydrogen oxidation reaction on Pt electrodes in the presence of poisons

The impact of the strength of negative (desynchronizing) global coupling (NGC) on the spatiotemporal dynamics of an electrochemical relaxation oscillator is studied numerically with a prototypical model, the electro-oxidation of hydrogen in the presence of poisons. The results are compared with recent experiments. The NGC has a destabilizing effect on the homogeneous oscillations. Both, in theory and in experiments, the basic patterns found with increasing global coupling strength are modulated oscillations, target patterns (including an asymmetric variant), and modulated pulses, the average spatial inhomogeneity during an oscillation increasing with the intensity of the NGC. It is suggested that this scenario is typical for strong relaxation oscillations, and a comparison with an electrochemical oscillator exhibiting harmonic oscillations points to the fact that the critical coupling strength, upon which the complete synchronization is destroyed, is larger for relaxation oscillations than for harmonic oscillations. In addition, the numerical simulations predicted two- and three-phase cluster patterns at high coupling strength. Also in experiments cluster patterns were observed, however only in parameter regions of the local dynamics which were different from the one investigated in this study.

Macroscopically Uniform Nanoperiod Alloy Multilayers Formed by Coupling of Electrodeposition with Current Oscillations

The electrodeposition from an acidic solution containing Cu(2+), Sn(2+), and a cationic surfactant gave a negative differential resistance (NDR) and a current oscillation in a narrow potential region of about 20 mV lying slightly more negative than the onset potential for Sn-Cu alloy deposition. Scanning Auger microscopic inspection has indicated that alloy films deposited during the oscillation have a clear alternate multilayer structure composed of two alloy layers of different compositions. The multilayer had the period of thickness of 40-90 nm and was uniform over a macroscopically wide area of about 1 mm x 1 mm. Detailed investigations have revealed that the NDR arises from adsorption of a cationic surfactant (acting as an inhibitor for diffusion of metal ions) on the alloy surface, and the oscillation comes from coupling of the NDR with the ohmic drop in the electrolyte.

Metal Latticeworks Formed by Self-Organization in Oscillatory Electrodeposition

Strikingly well-ordered Sn latticeworks, standing perpendicular to the substrate, are formed spontaneously in oscillatory electrodeposition. Cooperation of various processes, such as needle formation by autocatalytic crystal growth, cuboid formation under a reaction-limited condition, and autocatalytic oxidation at closest-packed surfaces, enabled the self-organization of the latticeworks. The mechanism is generally applicable to deposition of other metals such as Zn, Pb, and Cu. The present work has opened a promising, unique field toward the formation of highly ordered 3-D micro- or nanostructures at solid surfaces.