The reduced 1D Poisson-Nernst-Planck (PNP) model of artificial nanopores in the presence of a permanent charge on the channel wall is studied. More specifically, we consider the limit where the channel length exceed much the Debye screening length an
d channels charge is sufficiently small. Ion transport is described by the nonequillibrium steady-state solution of the PNP system within a singular perturbation treatment. The quantities, 1/lambda -- the ratio of the Debye length to a characteristic length scale and epsilon -- the scaled intrinsic charge density, serve as the singular and the regular perturbation parameters, respectively. The role of the boundary conditions is discussed. A comparison between numerics and the analytical results of the singular perturbation theory is presented.
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 biologic
al ion channels [P. Yu. Apel, {it et al.}, Nucl. Instr. Meth. B {bf 184}, 337 (2001); Z. Siwy, {it et al.}, Europhys. Lett. {bf 60}, 349 (2002)]. Here, the ion current rectification is studied within a reduced 1D Poisson-Nernst-Planck (PNP) model of synthetic nanopores. A conical channel of a few $mathrm{nm}$ to a few hundred of nm in diameter, and of few $mu$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 non-equilibrium 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.
