Nonlinear electrochemical relaxation around conductors

We analyze the simplest problem of electrochemical relaxation in more than one dimension—the response of an uncharged, ideally polarizable metallic sphere (or cylinder) in a symmetric, binary electrolyte to a uniform electric field. In order to go beyond the circuit approximation for thin double layers, our analysis is based on the Poisson-Nernst-Planck (PNP) equations of dilute solution theory. Unlike most previous studies, however, we focus on the nonlinear regime, where the applied voltage across the conductor is larger than the thermal voltage. In such strong electric fields, the classical model predicts that the double layer adsorbs enough ions to produce bulk concentration gradients and surface conduction. Our analysis begins with a general derivation of surface conservation laws in the thin double-layer limit, which provide effective boundary conditions on the quasineutral bulk. We solve the resulting nonlinear partial differential equations numerically for strong fields and also perform a time-dependent asymptotic analysis for weaker fields, where bulk diffusion and surface conduction arise as first-order corrections. We also derive various dimensionless parameters comparing surface to bulk transport processes, which generalize the Bikerman-Dukhin number. Our results have basic relevance for double-layer charging dynamics and nonlinear electrokinetics in the ubiquitous PNP approximation.

Figure 1

(a) Schematic diagram of metallic colloidal sphere in a binary electrolyte subjected to an applied electric field, which has the same relaxation as a metallic hemisphere on a flat insulting surface, shown in (c). We also consider the analogous two-dimensional problems of a metallic cylinder (b) or a half cylinder on an insulating plane (d).

Figure 2

Schematic diagram of normal and tangential fluxes involved in the surface conservation law (32). The shaded circle represents the metal particle; the dashed circle represents the outer “edge” of the double layer.

Figure 3

Numerical solutions for the concentration $c$ (top) and excess electric potential $\psi$ (bottom) for $E=10$, $ϵ=0.01$, $\delta =1$. Notice the large gradients in the concentration profile near the surface of the sphere.

Figure 4

Bulk concentration $\widehat{c}$ and electric potential $\widehat{\varphi }$ at the surface of the sphere for varying values of the applied electric field. In these figures, $ϵ=0.01$ and $\delta =1$. Notice that for $E=15$, the poles ($\theta =0$ and $\theta =\pi$) are about to be depleted of ions (i.e., $\widehat{c}\approx 0$).

Figure 5

Surface charge density $ϵq$ (top) and excess surface concentration of neutral salt $ϵw$ (bottom) and for varying values of the applied electric field. In these figures, $ϵ=0.01$ and $\delta =1$. Notice that for large applied fields, $ϵw=O(1∕E)$ and $ϵq=O(1∕E)$ so that the surface conduction terms in (60, 61) are $O\left(1\right)$.

Figure 6

Tangential surface fluxes for the surface charge density $\mid {\mathbf{J}}_{t,q}\mid$ (top) and excess surface concentration of neutral salt $\mid {\mathbf{J}}_{t,w}\mid$ (bottom) for varying values of the applied electric field. In these figures, $ϵ=0.01$ and $\delta =1$. Notice that for large applied fields, the surface fluxes are $O\left(1\right)$ quantities and make a non-negligible leading-order contribution in (60, 61).

Figure 7

Comparison of the magnitudes of $\mid {\mathbf{J}}^{\left(e\right)}\mid$ (solid lines) and $\mid {\mathbf{J}}^{\left(d\right)}\mid$ (dashed lines) for the excess surface fluxes of $ϵq$ (top) and $ϵw$ (bottom) for an applied electric field value of $E=15$. In these figures, $ϵ=0.01$ and $\delta =1$. Notice that in both cases, the surface diffusion is on the order of $1∕\widehat{c}E$ times the surface conduction.

Figure 8

Tangential component of bulk electric field $\mid E_t\mid$ at surface of sphere for varying values of the applied electric field. In this figure, $ϵ=0.01$ and $\delta =1$.

Figure 9

Normal flux of current $c\partial \varphi ∕\partial n$ (top) and neutral salt $\partial c∕\partial n$ (bottom) into the double layer for varying values of the applied electric field. In these figures, $ϵ=0.01$ and $\delta =1$.

Figure 10

Diffusion currents drive transport of neutral salt near the surface of the sphere. In this figure, $E=10$, $ϵ=0.01$, and $\delta =1$. Notice that streamlines of neutral salt are closed; current lines start on the surface of the sphere near the equator and end closer to the poles.

Figure 11

Comparison of the magnitudes of bulk electromigration $\mid {\mathbf{F}}_{\rho }^{\left(e\right)}\mid$ (solid lines) and diffusion $\mid {\mathbf{F}}_c^{\left(d\right)}\mid$ (dashed lines) fluxes as a function of distance from the surface of the sphere at $\theta =0,\pi ∕4$, and $\pi ∕2$ (top). The figure on the bottom zooms in on the diffusion flux. In these figure, $E=10$, $ϵ=0.01$, and $\delta =1$. Notice that the magnitude of the electromigration dominates diffusion and that the electromigration term itself becomes dominated by the contribution from the applied electric field a short distance from the surface of the sphere.

Figure 12

Tangential surface fluxes for the cation $\mid {\mathbf{J}}_{t,+}\mid$ (top) and anion $\mid {\mathbf{J}}_{t,-}\mid$ (bottom) for varying values of the applied electric field. In these figures, $ϵ=0.01$ and $\delta =1$. Notice that for large applied fields, the surface fluxes are $O\left(1\right)$ quantities (in the appropriate hemisphere) which leads to a non-negligible leading-order contribution in (47).

Figure 13

Exponential relaxation time constants for the charge density $\rho$ and the accumulated surface charge density $q$ at weak applied fields as a function of $ϵ$ and $\delta$. The top panel shows the relaxation time constant for the charge density at the surface of the sphere, ${\tau }_{\rho }(r=1)$. The bottom panel shows the relaxation time constant for the accumulated surface charge density, ${\tau }_q$.

Figure 14

Five asymptotically distinct regions of space-time that govern the dynamic response of a metal colloid sphere to an applied electric field. Note the nested spatial boundary layers at the $RC$ time $[t=O(ϵ\left)\right]$.

Figure 15

Time evolution of the dominant coefficients in the Legendre polynomial expansion of the bulk electric potential in the weakly nonlinear regime at the $RC$ time. In this figure $v=0$, $E=5$, and $\delta =0.1$. Notice that the dipolar term dominates the solution.

Figure 16

Time evolution of the dominant coefficients in the Legendre polynomial expansion of the bulk electric potential in the weakly nonlinear regime at the $RC$ time when the sphere has a nonzero applied voltage. In this figure $v=3$, $E=5$, and $\delta =0.1$. Note that both symmetric and antisymmetric terms make non-negligible contributions.

Figure 17

Time evolution of the dominant coefficients in the Legendre polynomial expansion of the bulk electric potential in the weakly nonlinear regime at the $RC$ time for low (top) and high (bottom) Stern capacitance values. In these figures $v=0$ and $E=5$. Note that double-layer charging is retarded when $\delta$ is small but that the slowdown in double-layer charging is suppressed for larger $\delta$ values.