(297f) Rectification and Rectification Inversion of Ion Currents in Conical Nanopores

As shown in some recent experiments, ion transport in perm-selective media can exhibit rectification behavior like semiconductor diodes. Such rectification can be a result of asymmetrical geometry (Siwy et al, Europhys Lett 2002), concentration gradient (Cheng and Guo, Nano Lett 2007) and non-uniform surface charge (Daiguji, Oka and Shirono, Nano Lett 2005). Here, we focus on mechanism behind rectification induced by conical geometry and explain why the rectification can reverse with change in bulk concentration, as observed by some recent numerical simulations (Momotenko and Girault, JACS 2011) and experiments (Gilad Yossifon--unpublished data).  The conic geometry allows us to carry out rigorous cross-section averaging to produce a one-dimensional Nernst-Planck equation for ion transport of a symmetric electrolyte. The averaged transport equation is complemented by Donnan type boundary conditions due to Boltzmann equilibrium between the boundary ions with the bulk. Using an expansion in a small parameter, corresponding to the ratio of the effective surface charge concentration over the bulk ionic strength, we show that the selectivity jump along the conic pore produces non-equilibrium intra-pore enrichment and depletion under positive/reverse bias and is hence responsible for the rectification phenomenon. We also offer a simple expression for the rectification factor, as a function of geometric parameters, surface charge, voltage and bulk ionic strength,  that satisfactorily collapse literature data and data from our collaborators at high bulk ionic strengths.  However, at very low bulk concentrations, we find rectification inversion can happen with the increase of applied voltage due to external depletion, which is also asymmetric because of the different diffusion lengths outside the two entrances. By matching the intrapore current with the limiting current due to external depletion (Yossifon et al, PRE 2009), we obtain a simple estimate of the crossover voltage for rectification inversion, which agrees with our numerical simulation. Conic pore rectification is hence due to asymmetric enrichment/depletion both within and outside the nanopore.