Scaling of selectivity in uniformly charged nanopores through a modified Dukhin number for 1:1 electrolytes

Abstract: We show that a modified version of the Dukhin number is an appropriate scaling parameter for the ionic selectivity of uniformly charged nanopores. The modified Dukhin number is an unambiguous function of the variables $\sigma$ (surface charge), $R$ (pore radius), and $c$ (salt concentration), and defined as $\mathrm{mDu}=|\sigma|/e(R/\lambda)$, where $\lambda$ is the screening length of the electrolyte carrying the $c$ dependence ($\lambda\sim c^{-1/2}$). Scaling means that the device function (selectivity) is a smooth and (in this case) monotonic function of mDu. The original Dukhin number defined as $\mathrm{Du}=|\sigma|/eRc$ ($c^{-1}$ dependence) was introduced to indicate whether the surface or the volume conduction is dominant in the pore. The modified version satisfies scaling and characterizes selectivity in the intermediate regime, where both surface and bulk conductions are present and the pore is neither perfectly selective, nor perfectly non-selective. Our modeling study using the Local Equilibrium Monte Carlo method and the Poisson-Nernst-Planck theory provides the radial flux profiles from which the radial selectivity profile can be computed. These profiles show in which region of the nanopore the surface or the volume conduction dominates for a given combination of the variables $\sigma$, $R$, and $c$. We show that the inflection point of the scaling curve may be used to characterize the transition point between the surface and volume conductions.