Bulk electroconvective instability at high Péclet numbers

Bulk electroconvection pertains to flow induced by the action of a mean electric field upon the residual space charge in the macroscopic regions of a locally quasielectroneutral strong electrolyte. For a long time, controversy has existed in the literature as to whether quiescent electric conduction from such an electrolyte into a uniform charge-selective solid, such as a metal electrode or ion exchange membrane, is stable with respect to bulk electroconvection. While it was recently claimed that bulk electroconvective instability could not occur, this claim pertained to an aqueous, low-molecular-weight electrolyte characterized by an order-unity electroconvection Péclet number. In this paper, we show that the bulk electroconvection model transforms into the leaky dielectric model in the limit of infinitely large Péclet number. For the leaky dielectric model, conduction of the above-mentioned type is unstable, and so it is in the bulk electroconvection model for sufficiently large Péclet numbers. Such instability is sensitive to the ratio of the diffusivity of the cations to the anions. For infinite Péclet number, the case with equal ionic diffusivities is a bifurcation point separating stable and unstable regimes at the low-current limit. Further, for a cation-selective solid, when the Péclet number is finite and the anions are much more diffusive than the cations, an unreported bulk electroconvective instability is possible at low current. At higher currents and large Péclet numbers, we found that the system is unstable for all cation-to-anion diffusivity ratios, but passes from a monotonic instability to an oscillatory one as this ratio passes through unity.

Figure 1

Sketch of the problem’s geometry.

Figure 2

Imaginary part of the dominant eigenvalue $S_1$ as a function of wave number as computed from the low-current HM model.

Figure 3

Dominant eigenvalue $S_2$ as a function of wave number as computed from the low-current MHM model.

Figure 4

Wave number dependence of the dominant eigenvalue for the BE formulation at low current, $S_3^0$. When $L=-100$, the system has a band of wave numbers where the growth rates are positive and the system is unstable, corresponding to $D<1$. When $L=100$, the system is stable, corresponding to $D>1$.

Figure 5

Marginal stability curve for the BE formulation when $D<1$; the minimum corresponds to critical values $L_c\equiv \mathrm{Pe}{\alpha }^2(D-1)∕(D+1)=-68$, $k_c=4.74$.

Figure 6

Real and imaginary parts of the eigenvalues $S_4^1$ and $S_4^2$ as a function of Péclet number for $k=4$. The two real eigenvalues merge around ${\mathrm{Pe}}_2^{*}=900$ and form a complex conjugate pair. As the Péclet number increases to infinity, the eigenvalues computed from the BE model converge to those computed from the HM model.

Figure 7

Critical Péclet number as a function of $D$ at low current. The system is unstable when Pe exceeds this critical value. When $D>1$ the low-current analysis predicts that the system is always stable. Note that the $y$ axis uses ${\mathrm{Pe}}_1=\mathrm{Pe}{\alpha }^2$ from Eq. (71).

Figure 8

Critical voltage as a function of $D$ at high Péclet number at $k=4$. The solid curve is computed from the BE formulation; the two dashed curves are computed from the low-current formulation for $D<1$ and the MHM model for $D>1$.

Figure 9

Two largest eigenvalues when $V=4$, $k=4$, and $\mathrm{Pe}={10}^4$. The solid line denotes the real part of the two most dominant eigenvalues and the dashed line the magnitude of the imaginary part. A transition to oscillating instability occurs as the system passes through $D=1$.

Figure 10

Marginal stability curve at finite current, plotting the control parameter $L$ vs wave number. The voltages are $V=0.01$, 0.1,1,2,4 and $D=0.1$.

Figure 11

Dependence of the critical value of Pe on voltage for $D=0$.

Figure 12

Marginally stable Péclet number versus voltage for different values of $D>1$ and $k=4$. The dashed lines show the voltage computed from the MHM model (infinite Pe). The solid curves are computed from the BE model.