Electroconvective instability near an ion-selective surface: A mesoscopic lattice Boltzmann study

Direct numerical simulations of electroconvection instability near an ion-selective surface are conducted using a mesoscopic lattice Boltzmann method (LBM). An electrohydrodynamic model of ion transport and fluid flow is presented. We numerically solve the Poisson-Nernst-Planck equations for the electric field and the Navier-Stokes equations for the flow field. The results cover Ohmic, limiting, and overlimiting current regimes, and they are in good agreement with the asymptotic analytical solution for the relationship between current and voltage. The influences of different ion transport mechanisms on the voltage-current relationship are discussed. The results reveal that the electroconvection mechanism is as important as other ion transport mechanisms in electrohydrodynamic flow. By comparing the contribution of different regions in the numerical domain, we find that the flow in the extended space charge layer is dominated by electroconvection. We also study the influences of multiple driving parameters, and the electrohydrodynamic coupling constant plays a dominant role in triggering convective instability. The flow pattern and ion concentration distribution are described in detail. Moreover, the route of flow from steady state to periodic oscillation and then to chaos is discussed.

Figure 1

Sketch of a typical current-voltage relationship at an ion-selective surface.

Figure 2

Model of electroconvection near the ion-selective surface (not to scale). A pair of symmetric counter-rotating vortexes form with diffusion, transition, extended space-charge, and electric double (cathodic Debye) layers. The numerical domain is periodic in the $x$-direction.

Figure 3

Spatial discretization of the D2Q9 and D2Q5 lattices.

Figure 4

The current-voltage ($I\text{−}V$) curve when a bias voltage is applied at the ion-selective surface for case A. Three independent regimes are shown: Ohmic, limiting, and overlimiting. The dashed line is derived by Yariv's asymptotic overlimiting current approximation. The inset shows the relationship between the maximum velocity and voltage, and the abscissa of the inset is consistent with the main plot.

Figure 5

Ion distribution curve between the ion-surface and bulk electrolyte when there is no flow. The cases of 1, 3, and 6 ${\varphi }_0$ refer to the Ohmic regime; the others are for the limiting regime. The scatters in the figures are only used to distinguish different cases. (a) Concentrations of cations (solid line) and anions (dashed line); (b) charge density ${\rho }_e$.

Figure 6

Anion concentration contour for (a) $26{\varphi }_0$, (b) $30{\varphi }_0$, and (c) $50{\varphi }_0$. We divide the results into two parts. The purple represents the velocity field and the green is the anion concentration. The darker regions correspond to larger velocity and concentration. The dimensionless magnitude of the velocity is marked on the right.

Figure 7

Decomposed cation flux: diffusion, electric migration, and convection. The figure shows six cases at different voltages; green lines with diamonds represent electric migration, blue lines with triangles represent diffusion, and red lines represent convection. The diffusion limiting current is obvious because it is nearly constant when the voltage is higher than the Ohmic regime. The electric migration carries more ions at higher voltage.

Figure 8

Average variables along the $x$ direction. (a) Average work done by electric force; the subplot is average kinetic energy. (b) $x$-average charge density.

Figure 9

Relationship between current-voltage and velocity-voltage; the legends correspond to Table 1. (a) Current under different voltages; solid lines are our numerical results, and dashed lines represent Yariv's asymptotic solution. (b) Maximum velocity under different voltages.

Figure 10

Typical snapshots of the lateral electric field and charge density for $V=40{\varphi }_0$. The lateral electric field intensity is on the left and the charge density is on the right. The order from top to bottom is consistent with Table 1. The label in the figure is the dimensionless magnitude of the variables.

Figure 11

Average concentrations of cations and anions for the voltage $V=40{\varphi }_0$.

Figure 12

Time-dependent evolution of the maximum velocity of case D. (a) Evolution of the maximum velocity with time. (b) Spatial phase diagram reconstructed by a delay time; the dotted line represents the moving trace in phase space. (c) Fourier spectrum of the maximum velocity.

Figure 13

Illustration of the method to determine the instability criterion: (a) time history of the peak velocity in the exponential growth stage, and (b) linear fitting of the growth rates for cases A–E. The intersections of the lines with the $x$ axis present the voltage corresponding to neutral stability.

Figure 14

The current and maximum velocity at different aspect ratios of numerical domains. The subfigures are typical snapshots of the ion concentration field for different aspect ratios of the numerical domain.