Rectification in synthetic conical nanopores: A one-dimensional Poisson-Nernst-Planck model

Ion transport in biological and synthetic nanochannels is characterized by phenomena such as ion current fluctuations and rectification. Recently, it has been demonstrated that nanofabricated synthetic pores can mimic transport properties of biological ion channels [P. Yu. Apel et al., Nucl. Instrum Methods Phys. Res. B 184, 337 (2001); Z. Siwy et al., Europhys. Lett. 60, 349 (2002)]. Here, the ion current rectification is studied within a reduced one-dimensional (1D) Poisson-Nernst-Planck (PNP) model of synthetic nanopores. A conical channel of a few nm to a few hundred nm in diameter, and of a few $\mu \text{m}$ long is considered in the limit where the channel length considerably exceeds the Debye screening length. The rigid channel wall is assumed to be weakly charged. A one-dimensional reduction of the three-dimensional problem in terms of corresponding entropic effects is put forward. The ion transport is described by the nonequilibrium steady-state solution of the 1D Poisson-Nernst-Planck system within a singular perturbation treatment. An analytic formula for the approximate rectification current in the lowest order perturbation theory is derived. A detailed comparison between numerical results and the singular perturbation theory is presented. The crucial importance of the asymmetry in the potential jumps at the pore ends on the rectification effect is demonstrated. This so constructed 1D theory is shown to describe well the experimental data in the regime of small-to-moderate electric currents.

Figure 1

(Color online) A sketch of the profile of the conical nanopore. The wall inside the pore exhibits a surface charge density $\sigma$. The local radius of the pore $R\left(z\right)$ is given by Eq. (5). The drawing does not reflect the actual physical $\left(z:r\right)$ proportions.

Figure 2

(Color online) Concentration profile $c_{{\text{K}}^{+}}\left(z\right)$ at equilibrium ($c_L=c_R=0.1M$, $\Phi \left(z=0\right)=\Phi \left(z=L\right)=0$). Calculations are done for the short pore (a) and the long pore (b). Solid line and squares: $\sigma =-0.02\text{e}/{\text{nm}}^2$; dashed line and circles: $\sigma =-0.1\text{e}/{\text{nm}}^2$. Symbols in all cases denote the results of the numerical solution, and lines represent the results of the perturbation theory. The insets depict a closer look into the left and the right boundary layers, respectively.

Figure 3

(Color online) Current voltage $\left(I-U\right)$ characteristics of the short pore (a) and the long pore (b) for $U$ from $-0.1\text{V}$ to 0.1 V. Solid line and squares: $\sigma =-0.02\text{e}/{\text{nm}}^2$; dashed line and circles: $\sigma =-0.1\text{e}/{\text{nm}}^2$. Symbols denote the numerical solution, and the lines represent the results of first order perturbation theory (in $ϵ$).

Figure 4

(Color online) Current voltage $\left(I-U\right)$ characteristics of the short pore (a) and the long pore (b) for $U$ from $-0.5\text{V}$ to 1 V. Solid line and squares: $\sigma =-0.02\text{e}/{\text{nm}}^2$; dashed line and circles: $\sigma =-0.1\text{e}/{\text{nm}}^2$. Symbols in all cases stand for numerical solution, and lines represent first order perturbation theory (in $ϵ$).

Figure 5

(Color online) The potassium and chloride currents in the short pore for $\sigma =-0.02\text{e}/{\text{nm}}^2$. The solid line represents the fourth order (in $ϵ$) perturbation calculation for potassium and the dashed line for the chloride current. Symbols in all cases present the numerical solution (circles: potassium current; squares: chloride).

Figure 6

(Color online) In panel (a) we depict the potential profile $\Phi \left(z\right)$ at equilibrium ($c_L=c_R=0.1M$, $\Phi \left(0\right)=\Phi \left(L\right)=0$) for $\sigma =-0.02\text{e}/{\text{nm}}^2$. In panel (b) we depict the $I-U$ dependence and in panel (c) we show the rectification measure $\alpha$. The calculations are done for the long pore (dashed line and circles) and the short pore (solid line and squares). The symbols in all cases stand for numerical solution; the lines represent the perturbation theory (in $ϵ$): up to second order (a), and in (b) and (c) up to fourth order.

Figure 7

(Color online) The current voltage $\left(I-U\right)$ dependence in the experiment (symbols) and the 1D-PNP theory (lines). Only the numerical results are given because the perturbation approach fails for the large surface charges used in the experiment. We depict results for one value of the surface charge density $\sigma =-1.0\text{e}/{\text{nm}}^2$ and for $c_L=c_R=0.1M$, which lies within the range of experimentally measured values. We use two different pore geometries: panel (a) $R\left(z=0\right)=3\text{nm}$, $R\left(L\right)=265\text{nm}$, $L=12000\text{nm}$; panel (b) $R\left(z=0\right)=3.5\text{nm}$, $R\left(L\right)=340\text{nm}$, $L=12000\text{nm}$.