Electro-hydrodynamic instabilities near a cation-exchange microgranule in an electrolyte solution under an external electric field are studied numerically. Despite the smallness of the particle and practically zero Reynolds numbers, in the vicinity of the particle, several sophisticated flow regimes can be realized, including chaotic ones. The obtained results are analyzed from the viewpoint of hydrodynamic stability and bifurcation theory. It is shown that a steady-state uniform solution is a non-unique one; an extra solution with a characteristic microvortex, caused by non-linear coupling of the hydrodynamics and electrostatics, in the region of incoming ions is found. Implementation of one of these solutions is subject to the initial conditions. For sufficiently strong fields, the steady-state solutions lose their stability via the Hopf bifurcation and limit cycles are born: a system of waves grows and propagates from the left pole, θ = 180°, toward the angle θ = θ0 ≈ 60°. Further bifurcations for these solutions are different. With the increase in the amplitude of the external field, the first cycle undergoes multiple period doubling bifurcation, which leads to the chaotic behavior. The second cycle transforms into a homoclinic orbit with the eventual chaotic mode via Shilnikov’s bifurcation. Santiago’s instability [Chen et al., “Convective and absolute electrokinetic instability with conductivity gradients,” J. Fluid Mech. 524, 263 (2005)], the third kind of instability, was then highlighted: an electroneutral extended jet of high salt concentration is formed at the right pole (region of outgoing ions, θ = 0°). For a large enough electric field, this jet becomes unstable; the perturbations are regular for a small supercriticality, and they acquire a chaotic character for a large supercriticality. The loss of stability of the concentration jet significantly affects the hydrodynamics in this area. In particular, the Dukhin–Mishchuk vortex, anchored to the microgranule at θ ≈ 60°, under the influence of the jet oscillations loses its stationarity and separates from the microgranule, forming a chain of vortices moving off the granule. This phenomenon strongly reminds the Kármán vortices behind a sphere but has another physical mechanism to implement. Besides the fundamental importance of the results, the instabilities found in the present work can be a key factor limiting the robust performance of complex electrokinetic bio-analytical systems. On the other hand, these instabilities can be exploited for rapid mixing and flow control of nanoscale and microscale devices.

The influence of the texture of a hydrophobic surface on the electro-osmotic slip of the second kind and the electrokinetic instability near charge selective surfaces (permselective membranes, electrodes, or systems of microchannels and nanochannels) is investigated theoretically using a simple model based on the Rubinstein-Zaltzman approach. A simple formula is derived to evaluate the decrease in the instability threshold due to hydrophobicity. The study is complemented by numerical investigations both of linear and nonlinear instabilities near a hydrophobic membrane surface. Theory predicts a significant enhancement of the ion flux to the surface and shows a good qualitative agreement with the available experimental data.

A new kind of instability caused by Joule heating near charge-selective surfaces (permselective membranes, electrodes, or systems of micro- and nanochannels) is investigated theoretically using a model based on the Rubinstein-Zaltzman approach. A simple relation is derived for the marginal stability curves: Joule heating can either destabilize or stabilize the steady state, depending on the location of the space charge region relative to the gravity vector. For the destabilizing case, the short-wave Rubinstein-Zaltzman instability is replaced by a long-wave thermal instability. The physical mechanism of the thermal instability is found to be very different from Rayleigh-Bénard convection, and is based on a nonuniform distribution of the electrical conductivity in the electrolyte. The study is complemented by numerical investigations both of linear and nonlinear instabilities near a charge-selective surface. There is a good qualitative agreement with the analytics. A possible explanation of the discrepancy between the experimental data and our previous theoretical voltage-current characteristics is highlighted.

Numerical simulations are presented for the transient and steady-state response of a model electrodiffusive cell with a bipolar ion-selective membrane under electric current. The model uses a continuum Poisson-Nernst-Planck theory including source terms to account for the catalytic second Wien effect between ionogenic groups in the membranes and resolves the Debye layers at interfaces. The resulting electric field at the membrane junction is increased by as much as four orders of magnitude in comparison to the field external to the membrane. This leads to a significant amplification of the second Wien effect, creating an increased ionic flux due to the catalytic decomposition of water. The effect also induces an exaltation effect wherein the salt ion flux undergoes a concomitant increase as well. The interplay of effects results in a unique over-limiting current mechanism due to concentration polarization internal, rather than external, to the membranes. In addition to the case of two equal but oppositely charged membranes under the standard simplifying assumption of equal ionic diffusivities, two variations on this model are studied. Asymmetric diffusivities, representative of the actual mobility difference in dissociated water ions, and the effect of the membrane charge density ratio were also considered. The latter elucidates an overlimiting current shift mechanism for DNA adsorption on anion-selective membranes proposed by Slouka et al. [Langmuir 29, 8275 (2013)]. The former provides more realistic picture of multi-ion transport and demonstrates a surprising steady-state effect due to the asymmetry in the diffusivity of hydroxide and hydronium.

The instability of ultra-thin films of an electrolyte bordering a dielectric gas in an external tangential electric field is scrutinized. The solid wall is assumed to be either a conducting or charged dielectric surface. The problem has a steady one-dimensional solution. The theoretical results for a plug-like velocity profile are successfully compared with available experimental data. The linear stability of the steady-state flow is investigated analytically and numerically. Asymptotic long-wave expansion has a triple-zero singularity for a dielectric wall and a quadruple-zero singularity for a conducting wall, and four (for a conducting wall) or three (for a charged dielectric wall) different eigenfunctions. For infinitely small wave numbers, these eigenfunctions have a clear physical meaning: perturbations of the film thickness, of the surface charge, of the bulk conductivity, and of the bulk charge. The numerical analysis provides an important result: the appearance of a strong short-wave instability. At increasing Debye numbers, the short-wave instability region becomes isolated and eventually disappears. For infinitely large Weber numbers, the long-wave instability disappears, while the short-wave instability persists. The linear stability analysis is complemented by a nonlinear direct numerical simulation. The perturbations evolve into coherent structures; for a relatively small external electric field, these are large-amplitude surface solitary pulses, while for a sufficiently strong electric field, these are short-wave inner coherent structures, which do not disturb the surface.