Extreme nonequilibrium electrophoresis of an ion-selective microgranule

We investigate the electrophoresis of an ion-selective microgranule in an electrolyte solution. A semianalytical analysis of a small length of the electric double layer as well as overlimiting and extreme overlimiting currents is complemented by the direct numerical study of the full nonstationary Nernst-Planck-Poisson-Stokes system, with the corresponding boundary and initial conditions. Our results are in reasonably good agreement with the available experimental data. Moreover, they can be used to modify Dukhin's formula for the electrophoretic velocity. A steady-state solution is observed for moderate electric fields. Three boundary layers, nested inside each other, are formed in this solution as follows: an electric double layer, a space-charge region, and a thin diffusion layer. Only the electric double layer is present in the area of the outgoing ion flux. This flux generates a jet of high electric conductivity. Increasing the external field makes this jet narrower, but its conductivity increases. At the point on the granule surface where the ion flux vanishes, a separation of the diffusion boundary layer occurs. For sufficiently strong fields, the steady-state solution loses stability. Instability arises in the diffusion layer region but manifests itself in other regions. In particular, it generates electrokinetic microvortices. Two kinds of microvortices are found: large steady Dukhin-Mishchuk vortices and electrokinetic vortices that propagate from the pole of the particle towards the Dukhin-Mishchuk vortices. At small supercriticality, the oscillations of the unknowns are periodic, but increasing the external field makes the flow chaotic in the Feigenbaum scenario fashion.

Figure 1

Mobility vs $E_{\infty }$ for different $p$.

Figure 2

Schematic of the flow near the granule under the external electric field $E_{\infty }$ and the fluid flow velocity $u_{\infty }$ at infinity, directed along the $x$ axis. The electrophoretic velocity of the particle is in the opposite direction. In the spherical polar system $x=rcos\theta$ and $y=rsin\theta ,u=u_{\theta }$ is the tangential velocity at the particle surface and $v=u_r$ is the normal velocity at the particle surface. At the comoving reference frame, the far-field velocity condition is $u\to -u_{\infty }sin\theta$ and $v\to u_{\infty }cos\theta$.

Figure 3

Thin inner boundary layers, nested inside each other: the EDL, I; the SCR, II; and the diffusion layer, III. Also shown is the electroneutral outer region IV with $K=2$. Points 1 and $1^{\prime }$ separate regions of the incoming and outgoing fluxes. Inset A shows the typical structure of the incoming flux region. Inset B shows the typical structure of the outgoing flux region.

Figure 4

(a) Electric conductivity $K(r,\theta )$ for a moderate external electric field $E_{\infty }=5$. The dashed line stands for the edge of the diffusion layer. (b) Cross sections of $K\left(r\right)$ at $\theta =0^{\circ }$ and $\theta ={180}^{\circ }$. The behavior in the EDL, i.e., the depleted and enriched regions, is shown in the insets.

Figure 5

(a) Electric conductivity $K(r,\theta )$ for $E_{\infty }=10$. The dashed line stands for the edge of the diffusion layer. The values have been clipped from the actual maximal value down to $K=5$ in order to achieve contrast with the diffusion layer. (b) Cross sections of $K\left(r\right)$ at $\theta =0^{\circ }$ and $\theta ={180}^{\circ }$. The behavior in the EDL, i.e., the depleted and enriched regions, is shown in the insets.

Figure 6

(a) Electric conductivity $K(r,\theta )$ for $E_{\infty }=20$. The dashed line stands for the edge of the diffusion layer. The values have been clipped from the actual maximal value down to $K=5$ in order to achieve contrast with the diffusion layer. (b) Cross sections of $K\left(r\right)$ at $\theta =0^{\circ }$ and $\theta ={180}^{\circ }$. The behavior in the EDL, i.e., the depleted and enriched regions, is shown in the insets.

Figure 7

(a) Charge density $\rho (r,\theta )$ for a moderate external electric field $E_{\infty }=5$. (b) Cross sections of $\rho \left(r\right)$ at $\theta =0^{\circ }$ and $\theta ={180}^{\circ }$ near the granule surface.

Figure 8

(a) Charge density $\rho (r,\theta )$ for $E_{\infty }=10$. (b) Cross sections of $\rho \left(r\right)$ at $\theta =0^{\circ }$ and $\theta ={180}^{\circ }$ near the granule surface.

Figure 9

(a) Charge density $\rho (r,\theta )$ for $E_{\infty }=20$. (b) Cross sections of $\rho \left(r\right)$ at $\theta =0^{\circ }$ and $\theta ={180}^{\circ }$ near the granule surface.

Figure 10

Theoretical stream function distribution for (a) $E_{\infty }=5$, (b) $E_{\infty }=10$, and (c) $E_{\infty }=20$. (d) Electrokinetic flow from [38] around a 1-mm ion-selective particle at ${\tilde{E}}_{\infty }=100$ V/cm, which corresponds to dimensionless $E_{\infty }=200$ and Debye number $\nu \approx 8.68\times {10}^{-5}$.

Figure 11

(a) Charge distribution $\rho =c^{+}-c^{-}$ and (b) electric conductivity $K=c^{+}+c^{-}$, with $E_{\infty }=30$ for regions I and II of the standing and traveling waves, respectively. Arrows show the direction of propagation.

Figure 12

(a) Blow-up of region I of Fig. 11. (b) Spikes for a flat membrane [27, 42]. The normal to the membrane coordinate is strongly compressed in comparison to the direction along the membrane.

Figure 13

(a) Cross section of $\rho \left(r\right)$ at $\theta ={180}^{\circ }$ and at different time instants. (b) Cross section of $K\left(r\right)$ at $\theta ={180}^{\circ }$ and at different time instants. The wave is bouncing back from the edge of the diffusion layer defined by $\delta$.

Figure 14

Stream-function distribution for the unstable regime $E_{\infty }=30$. Here I stands for the Rubinstein-Zaltzman microvortices and II stands for the Dukhin-Mishchuk vortices. The arrows show the propagation of the Rubinstein-Zaltzman microvortices.

Figure 15

Electric current normalized to its maximum value along the microgranule surface $0<\theta <{180}^{\circ }$, for different strengths of the external field $E_{\infty }$. The inset shows the variation of the ion flux in time for the unstable case, which shows a traveling wave. In this inset, all subsequent curves are for $t_0+n\mathrm{\Delta }t,n=1,2,3$.

Figure 16

Time records of the electrophoretic velocity $u_{\infty }$ at different strengths of the electric field $E_{\infty }$.

Figure 17

Schematic bifurcation diagrams. (a) Solutions of Eq. (18) have an imperfect pitchfork bifurcation, which is structurally unstable [48]. (b) Plot of Eq. (17); for a small but finite $\nu$ this kind of bifurcation is destroyed. The solid line corresponds to the solution that makes physical sense.

Figure 18

Separation angle of the diffusion boundary layer ${\theta }_0$ versus $\chi$ at $\varkappa =0.26$. The solid line stands for the semianalytical solution and the triangles stand for the numerics.

Figure 19

Normalized drop of the electrical potential $F$ in the region $0<y<y_m$ as a function of $\chi$ at $\theta ={180}^{\circ },F=F_m$, at $\varkappa =0.26$. The solid line stands for the semianalytical solution and the triangles stand for the numerics. The inset shows $F_m$ along the surface of the granule at $\chi =1$.

Figure 20

Length of the SCR, $Y_m$, at different points of the granule surface. The solid line stands for the semianalytical solution and the triangles joined by the dashed line stand for the numerics.

Figure 21

Ion flux $J$ on the surface versus the angle $\theta$ for different $\chi$. The dashed line stands for the DNS results and the solid line stands for the analytics.

Figure 22

Electrophoretic velocity according to different theoretical approaches as a function of $\chi$. Curves 1 and 2 are the predictions of our semianalytical theory: 1 stands for $\varkappa =0.1$ and 2 stands for $\varkappa =1$. Curve 3 corresponds to the DNS with $\varkappa =0.26$ and $\nu =0.002$. The dashed line corresponds to Dukin's relation (45).

Figure 23

Typical spectrum of the linear stability problem for the semianalytical solution in the complex $\lambda$ plane for $\chi =2$. The top inset shows a conjugate pair of unstable eigenvalues in the vicinity of the origin. The other insets show the real part of eigenfunctions at different points of the spectrum as a function of $\theta$.

Figure 24

Maximum growth rate mode ${\lambda }_R$ as a function of $\chi$. Here ${\chi }_{*}^a$ and ${\chi }_{*}^N$ are analytical and numerical critical values, respectively. Regions I and II correspond to the numerical steady and unsteady regimes, respectively.

Figure 25

The solid line stands for the results of our DNS and triangles stand for the experiment [31]. The dimensionless parameter $\chi$ is (a) $\chi =0.035$, (b) $\chi =0.088$, and (c) $\chi =0.151$. In these plots the angle is counted in the reverse direction to match the experiments.

Figure 26

Experimental dimensionless velocity $u_{\infty }$ as a function of $E_{\infty }$. Here I, II, and III are the region of electrophoresis of the first kind, the transition region, and the region of electrophoresis of the second kind, respectively. Lines 1, 2, and 3 stand for $u_{\infty }\sim {\text{const}}_1{E}_{\infty },u_{\infty }\sim {\text{const}}_2{E}_{\infty }^{4/3}$, and $u_{\infty }\sim {\text{const}}_2{E}_{\infty }^2$, respectively.

Figure 27

Comparison of the theoretical and experimental electrophoretic velocities for the universal variables $U_{\infty }$ vs $\chi$. The markers for the experimental points are given in Table 2; the solid line stands for the results of our DNS at $\nu =0.002$ and the dashed line stands for the semianalytical results. Inset A shows the behavior of $u_{\infty }$ for small $E_{\infty }$. The solid line stands the DNS and the dashed line stands for the analytical solution. Inset B shows the behavior of $u_{\infty }$ for large $E_{\infty }$.