Instabilities, bifurcations, and transition to chaos in electrophoresis of charge-selective microparticle

Abstract

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.