Asymmetry-induced electric current rectification in permselective systems

For a symmetric ion permselective system, in terms of geometry and bulk concentrations, the system response is also symmetric under opposite electric field polarity. In this work we derive an analytical solution for the concentration distribution, electric potential, and current-voltage response for a four-layered system comprised of two microchambers connected by two permselective regions of varying properties. It is shown that any additional asymmetry in the system, in terms of the geometry, bulk concentration, or surface charge property of the permselective regions, results in current rectification. Our work is divided into two parts: when both permselective regions have the same surface charge sign and the case of opposite signs. For the same sign case we are able to show that the system behaves as a dialytic battery while accounting for field-focusing effects. For the case of opposite signs (i.e., bipolar membrane), our system exhibits the behavior of a bipolar diode where the magnitude of the rectification can be of order ${10}^2-{10}^3$.

Figure 1

Schematic describing a two-dimensional four-layered system comprised of two microchambers (regions 1 and 4) connected by two permselective mediums (regions 2 and 3). The geometry of each region is described in the schematic while the surface charge properties (or the counterion concentrations) of the permselective regions are defined in Secs. 3a and 4a. Black lines prescribe zero flux BC $\left({\mathit{j}}_{\pm }·\mathit{n}=0,\partial \varphi /\partial n=0\right)$ while the magenta lines at the two opposite ends $x=0,{\mathrm{\Delta }}_4$ prescribe Dirichlet BCs and are given in the schematic.

Figure 2

The (a) $I-V$ curves and (b) $\mathit{RF}$ for a 1D system. All lengths scales are unity $\left(L_{1,4}=d_{2,3}=1\right)$ and $N_{2,3}={10}^2$. The bulk at $x={\mathrm{\Delta }}_4$ is kept constant ${\widehat{c}}_4=1$, while the bulk concentration ${\widehat{c}}_1$ is varied. The simulations were conducted for the following values: $L_{1,4}=d_{2,3}=1,N_{2,3}={10}^2,{\widehat{c}}_1=2,{\widehat{c}}_4=1,\varepsilon ={10}^{-4}$.

Figure 3

The (a) $I-V$ curves and (b) $\mathit{RF}$ for a 2D system where the height $h_2$ was varied. All the length scales are unity $\left(L_{1,4}=d_{2,3}=1\right)$, while the remaining heights are $H_{1,4}=0.4,h_3={10}^{-2}$ and $N_{2,3}={10}^2$. The bulk at $x={\mathrm{\Delta }}_4$ is kept constant ${\widehat{c}}_4=1$. The bulk concentration ${\widehat{c}}_1=1$ unless otherwise specified. The simulations were conducted for the following values: $L_{1,4}=d_{2,3}=1,H_{1,4}=0.4,h_3={10}^{-2},N_{2,3}={10}^2,{\widehat{c}}_4=1,{\widehat{c}}_1=2,\varepsilon ={10}^{-3}$. For the case of symmetric bulk concentrations $V_0=0$ while for the asymmetric case used here $V_0=-ln2$.

Figure 4

The $I-V$ curves for a 1D system. All lengths scales are unity $\left(L_{1,4}=d_{2,3}=1\right)$ and $N_2=-N_3={10}^2$. For the sake of comparison we have added the positive branch of the $I-V$ for an ideal permselective system with $N_{2,3}={10}^2$. The simulations were conducted for the following values: $L_{1,4}=d_{2,3}=1,N_2=-N_3={10}^2,{\widehat{c}}_4=1,{\widehat{c}}_1=1,\varepsilon ={10}^{-4}$.

Figure 5

The (a) $I-V$ curves and (b) $\mathit{RF}$ for a 2D system where the geometry is varied. The nominal geometry is such that the length of each region is unity $\left(L_{1,4}=d_{2,3}=1\right)$ and the heights are $H_{1,4}=0.4,h_{2,3}={10}^{-2}$. For all curves a value of $N_2=-N_3={10}^2$ was used. The simulations (red squares) were conducted for the following values: $L_{1,4}=d_{2,3}=1,H_{1,4}=0.4,h_{2,3}={10}^{-2},N_2=-N_3={10}^2,{\widehat{c}}_{1,4}=1,\varepsilon ={10}^{-3}$.