Direct numerical simulation of electrokinetic instability and transition to chaotic motion

Abstract

A new type of instability -electrokinetic instability -and an unusual transition to chaotic motion near a chargeselective surface (semi-selective electric membrane, electrode or system of micro-/nanochannels) was studied by numerical integration of the Nernst-Planck-Poisson-Stokes system and a weakly nonlinear analysis near the threshold of instability. A special finite-difference method was used for the space discretization along with a semi-implicit 3 1 3 -step Runge-Kutta scheme for the integration in time. The linear stability problem was resolved by a spectral method.Two kinds of initial conditions were considered: (a) white noise initial conditions to mimic "room disturbances" and subsequent natural evolution of the solution; (b) an artificial monochromatic ion distribution with a fixed wave number to simulate regular wave patterns. The results were studied from the viewpoint of hydrodynamic stability and bifurcation theory. The threshold of electroconvective movement was found by the linear spectral stability theory, the results of which were confirmed by numerical simulation of the entire system. The following regimes, which replace each other as the potential drop between the selective surfaces increases, were obtained: one-dimensional steady solution; two-dimensional steady electroconvective vortices (stationary point in a proper phase space); unsteady vortices aperiodically changing their parameters (homoclinic contour); periodic motion (limit cycle); and chaotic motion.The transition to chaotic motion did not include Hopf bifurcation. Numerical resolution of the thin concentration polarization layer showed spike-like charge profiles along the surface, which could be, depending on the regime, either steady or aperiodically coalescent.The numerical investigation confirmed the experimentally observed absence of regular (near-sinusoidal) oscillations for the overlimiting regimes. There is a qualitative agreement of the experimental and the theoretical values of the threshold of instability, the dominant size of the observed coherent structures, and the experimental and theoretical volt-current characteristics.

Two-dimensional numerical simulation of microscale thermoelectrokinetic instability

“…Electroconvective vortices observed in [3] differ from those obtained during our calculations: they are short-wave and their centres are situated on the border of electric double layer, and our vortices are long-wave and situated in the centre of the channel. (Fig.…”

“…Further magnification of external electric field leads to reappear linear dependence of the electric current from the voltage. Transition from limited to overlimiting mode is caused by appearance of liquid movement owing to instability development, theoretically predicted in [2] and numerically confirmed in [3]. However, it must be said, that numerical and experimental data used to match only qualitatively, but not quantitatively, e.g.…”

“…The designed algorithm is based on one suggested by Nikitin in [5] and adopted for electrokinetic problems in [3]. A finite-difference method with second-order accuracy is applied for the spatial discretization.…”

“…1). For two-dimensional direct numerical simulation of the problem, the special algorithm based on one suggested in [5] and developed in [3] was designed. …”

Experimental Verification of Overlimiting Current by Surface Conduction and Electro-Osmotic Flow in Microchannels

“…Noticeable progress has been made in understanding electroconvection during ion concentration polarization (ICP) due to electro-osmotic flows near the membrane or nanochannel interface driving salt depletion [5,[7][8][9][10]. Due to the complexity of direct numerical simulation of the Poisson equation (for electric potential), Nernst-Planck equations (for ion concentrations), and Navier-Stokes equations (for fluid flows) in multidimensional geometries [9,11,12], as well as inherent limitations of the classical dilute solution model [13], it is crucial to directly observe particle motions and flow fields in precisely controlled micro-or nanofluidic geometries [14,15].…”