ELSEVIER Physica D 110 (1997) 154-168 PHYSICA Electro-osmotic oscillations Konstantin Gedalin Department of Physics, Ben-Gurion University, POB 653, Beer-Sheva 84105, Israel Received 18 December 1996; received in revised form 7 April 1997; accepted 25 April 1997 Communicated by Y. Kuramoto Abstract In this paper we reproduce the continuum model of electro-osmotic oscillations at a non-charged porous membrane and study their onset with a focus on the singular nature of this transmission (singular Hopf bifurcation), resulting in a rapid transformation of the oscillations, harmonic at the onset, into the jerky ones of relaxation type. PACS: 66.10. Cb; 82.40 Bj; 82.65 Fr; 87.22 Fg 1. Introduction Electro-osmotic oscillations have been the object of interest by physicists, chemists and mathematicians ever since Teorell [24-27] observed this phenomenon in a laboratory set-up devised to the mimic nerve excitation and comprising as a basic element a porous, electrolyte solution saturated glass filter. Electro-osmotic oscillations may likely represent a common source of oscillations in biological and synthetic membranes in set-up largely different from that of Teorell. Before going into a detailed discussion of these oscillations, we wish to point out that they represent by no means the only type of oscillations encountered in purely diffusional (inertia-and source-less) ionic systems. Thus, to mention but a few examples, Forgacs [6] observed oscillations of membrane potential upon passing a current through a cation exchange electrodialysis membrane separating two identical solutions of silver nitrate. The characteristic frequency of these oscillations was much higher than that typically recorded for electro-osmotic oscillations in Teorell's set-up. This pioneering work by Forgacs, followed by that of Seno and Yamabe [22] who observed what would be nova-days called an autosoliton train in an experimental set-up similar to that of Forgacs, triggered an extensive amount of research on electric oscillations and low frequency noise at electrodialysis membranes [9,10,16,23]. The physical mechanism behind these phenomena remains largely unclear, although recently some evidence appeared [17] that they might have to do with electroconvection [4,8,21], a mechanism closely related to electro-osmosis. In order to outline the essence of the phenomenon of electro-osmotic oscillations, let us consider the following imaginary experiment, reproducing in general lines the classical one by Teorell. Let two vessels, 1 and 2, contain an electrolyte solution (e.g., KC1) at two different concentrations C1 < C2 and be connected by a porous filter (capillary), filled by the same solution. Let the vessel 1 (with concentration C]) be 0167-2789/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved PII S0167-2789(97)00075-4