  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.
