Approximate time-dependent current-voltage relations for currents exceeding the diffusion limit

The time-dependent behavior of one-dimensional ion transport into a permselective medium is theoretically modeled in this work for currents exceeding the diffusion limit. Leveraging the findings of Yariv [E. Yariv, Phys. Rev. E 80, 051201 (2009)], we derive three separate expressions for the potential drop for short, intermediate, and long times. We show that the potential drop correlates to the time evolution of the space-charge layer adjacent to the permselective interface. Our approximate models show remarkable correspondence to numerical simulations.

Figure 1

The current-voltage, $i-V$, response comprises three regions: linear Ohmic resistance, $R$; limiting current, $i_{\lim }$; and overlimiting current, $i_{\mathrm{OLC}}$.

Figure 2

Schematic of current-voltage relation surpassing the limiting currents. The solid blue line is given by Eq. (1) while dashed lines with increasing $\varepsilon$ are determined by Eq. (41).

Figure 3

Time evolution of the potential drop similar to what was shown in Ref. [42] but with different simulation parameters (as given in Fig. 6).

Figure 4

Steady-state profiles for (a) the positive concentration distribution, $c^{+}$, and (b) the space-charge density, $q$ [Eq. (14)], for various currents. Simulation parameters are $\varepsilon ={10}^{-4},N=25$. For presentation purposes in (a) we show concentration in the range of 0–1 but note that there is a boundary layer near $x=0$ such that $c^{+}(x=0)=N$. In (b) which is a log-log plot, we show that the space charge is primarily dominated by the positive charge carrier such that $q(x=0)\approx \frac{1}{2}{c}^{+}(x=0)=N/2$.

Figure 5

A schematic of the 1D geometry within the domain $x\in [0,1]$. The domain is divided into four regions: CBL, SCL, TL, and DL. The order of magnitude of the length of each region is given;see main text for more details (Sec. 2d).

Figure 6

Time-dependent profiles for (a) the positive concentration distribution, $c^{+}$, and (b) space-charge density, $q$, for different times. Simulation parameters are $\varepsilon ={10}^{-4},N=25,i=2.4$.

Figure 7

Time evolution of the voltage, $V\left(t\right)$. Simulation parameters as in Fig. 6.