We report stationary, nonequilibrium potential and adsorbate patterns with an intrinsic wavelength that were observed in an electrochemical system with a specific type of current/electrode-potential (I-phi(DL)) characteristic. The patterns emerge owing to the interplay of a self-enhancing step in the reaction dynamics and a long-range inhibition by migration currents rather than by diffusion. Theoretical analysis revealed that this self-structuring of the electrode occurs in all electrochemical systems with an S-shaped I-phi(DL) characteristic in wide and well-accessible parameter ranges. This unusual pattern-forming instability in electrochemical systems has all the characteristics of the mechanism proposed by Turing in 1952 in the framework of an early theory of morphogenesis. Our finding might account for structure formation in certain biological systems that have gradients in the electric potential and may open new paths for fabricating patterned electrodes.

A specific feature of pattern formation in electrochemical systems is the occurrence of accelerated fronts; they can be attributed to long-range spatial coupling. In this paper we demonstrate that different coupling functions can be realized by tuning easily accessible parameters: The range of the coupling crucially depends on the length scales of the system, and the strength of the coupling is proportional to the conductivity of the electrolyte. Simulations in the bistable regime are presented which illustrate how the front behavior changes qualitatively when length scales or conductivity are varied. ͓S1063-651X͑97͒12702-7͔PACS number͑s͒: 64.10.ϩh, 82.20.Mj, 82.45.ϩz, 47.54.ϩr SYSTEM AND MODELElectrochemical reactions take place at the electrodeelectrolyte interface, the reaction rate being decisively deter-*Present address:

Oscillatory behavior has been observed for almost all electrochemical reactions in a certain, although sometimes small, range of external parameters. Only in the past ten years has it been possible, however, to find a common explanation for the occurrence of these temporal self-organization phenomena of chemically completely different electrochemical reactions. The breakthrough was achieved because new methods and concepts, which had been developed in nonlinear dynamics to describe the spontaneous formation of order in various disciplines, could be applied. This development in turn was only possible because the underlying laws are universal at a certain abstract level. Oscillations are only one possible manifestation of nonlinear behavior. Examples of other features that are often closely associated with temporal instabilities are spatial structures and waves. Initiated by the theoretical progress and the development of new experimental techniques, spatial pattern formation in electrochemical systems has been targeted for investigations in the past few years. Based on these investigations, it can be predicted under which conditions temporal or spatial pattern formation can be expected. Furthermore, the possibility of predicting the occurrence of instabilities indicates that it might be feasible to exploit nonlinear effects to increase, for example, the yield of electrocatalytic reactions. Here we discuss physicochemical mechanisms that lead to pattern formation in electrochemical systems. At the same time, we stress the generic principles that are responsible for self-structuring processes in many chemical and biological systems.

There exist many experimental examples of spatial pattern formation in electrochemical systems. Using a recently proposed model, we present numerical simulations of the spatio-temporal behavior in the bistable and oscillatory regime. The simulations reveal that for some parameter regimes only spatially inhomogeneous solutions exist. The impact of the system parameters on the spatial coupling as well as on the existence of patterns is elaborated. The predictions coincide with experimental results for a simple electrochemical reaction, the reduction of peroxodisulfate.

Applying a constant voltage across an external resistor in series with an electroche variation of the experimental control mode from potentiostatic to galvanostatic. The model for electrochemical pattern formation presented for potentiostatic conditions in Ref. 1 is extended to account for this general case. The resistor introduces a global coupling into the model, interacting in a nontrivial manner with the nonlocal coupling through migration currents in the electrolyte which occur independently of the control mode. The complex interplay of these two coupling terms is investigated as a function of operating parameters, and is illustrated by simulations in the bistable regime of an electrochemical reaction.