Far-from-equilibrium phenomena at local sites of reaction

Abstract

We investigate some far-from-equilibrium effects of localized nonlinear reaction sites, such as electrode surfaces or catalytic membranes, on an otherwise stable reacting-diffusing bulk medium. We obtain an integral equation as the formal solution of the partial differential equations describing reaction and diffusion in the total system, in which the nonlinearities of the local reaction sites are fully retained; the kernel of that integral equation is a propagator function which describes the linearized bulk dynamics in the absence of the localized reactions. Analytical solutions to the integral equations are found for several mechanisms which demonstrate the existence of a variety of typically far-from-equilibrium phenomena. The equations for steady states are derived and used to show the existence of multiple steady states for a model system. A determinantal condition for the stability of steady states is obtained and applied to another model system consisting of a localization of the Prigogine-Lefever mechanism with no bulk reactions. The system is found to have oscillatory instability whose frequency depends on transport processes in the bulk as well as on the parameters of the localized reactions. Waves and dissipative space structures on a planar localized reaction site are shown to obey ordinary integral equations which are solved, in the case of waves, for two model cases. The waves are shown to exist only in a region about the plane of the local reaction. Propagation occurs due to bulk transport processes coupling both local and bulk reaction in the vicinity of the reactive surface. Waves are shown to propagate along the membrane or surface only for certain ranges of values of the wavevector. A threshold wave phenomenon is shown to exist for one case, such that over the entire range of allowed wavevectors waves exist only beyond a minimal amplitude about a stable steady state. Symmetry-breaking instabilities in a system with two equivalent localized sites are considered and we show that these can occur only for intermediate values of separation of the local sites.