From nanotubes to nanoholes: scaling of selectivity in uniformly charged nanopores through the Dukhin number for 1:1 electrolytes

TL;DR

This study investigates how the selectivity of uniformly charged nanopores scales with system parameters, identifying the Dukhin number as a key scaling parameter in different pore length regimes, supported by modeling results.

Contribution

It introduces the Dukhin number and a modified version as effective scaling parameters for nanopore selectivity, linking double layer behavior to pore geometry and charge.

Findings

Dukhin number effectively scales selectivity in the nanotube limit.

Modified Dukhin number applies to the nanohole limit.

Transition between surface and volume conduction characterized by inflection point.

Abstract

Scaling of the behavior of a nanodevice means that the device function (selectivity) is a unique smooth and monotonic function of a scaling parameter that is an appropriate combination of the system's parameters. For the uniformly charged cylindrical nanopore studied here these parameters are the electrolyte concentration, $c$, voltage, $U$, the radius and the length of the nanopore, $R$ and $H$, and the surface charge density on the nanopore's surface, $\sigma$. Due to the non-linear dependence of selectivites on these parameters, scaling can only be applied in certain limits. We show that the Dukhin number, $\mathrm{Du}=|\sigma|/eRc\sim |\sigma|\lambda_{\mathrm{D}}^{2}/eR$ ($\lambda_{\mathrm{D}}$ is the Debye length), is an appropriate scaling parameter in the nanotube limit ($H\rightarrow\infty$). Decreasing the length of the nanopore, namely, approaching the nanohole limit ($H\rightarrow 0$), an alternative scaling parameter has been obtained that contains the pore length and is called the modified Dukhin number: $\mathrm{mDu}\sim \mathrm{Du}\, H/\lambda_{\mathrm{D}}\sim |\sigma|\lambda_{\mathrm{D}}H/eR$. We found that the reason of non-linearity is that the double layers accumulating at the pore wall in the radial dimension correlate with the double layers accumulating at the entrances of the pore near the membrane on the two sides. Our modeling study using the Local Equilibrium Monte Carlo method and the Poisson-Nernst-Planck theory provides concentration, flux, and selectivity profiles, that show whether the surface or the volume conduction dominates in a given region of the nanopore for a given combination of the variables. We propose that the inflection point of the scaling curve may be used to characterize the transition point between the surface and volume conductions.