  The scaling behavior for the rectification of bipolar nanopores is studied
using the Nernst-Planck equation coupled to the Local Equilibrium Monte Carlo
method. The bipolar nanopore's wall carries $\sigma$ and $-\sigma$ surface
charge densities in its two half regions axially. Scaling means that the device
function (rectification) depends on the system parameters (pore length, $H$,
pore radius, $R$, concentration, $c$, voltage, $U$, and surface charge density,
$\sigma$) via a single scaling parameter that is a smooth analytical function
of the system parameters. Here, we suggest using a modified Dukhin number,
$\mathrm{mDu}=|\sigma|l_{\mathrm{B}}^{*}\lambda_{\mathrm{D}}HU/(RU_{0})$, where
$l_{\mathrm{B}}^{*}=8\pi l_{\mathrm{B}}$, $l_{\mathrm{B}}$ is the Bjerrum
length, $\lambda_{\mathrm{D}}$ is the Debye length, and $U_{0}$ is a reference
voltage. We show how scaling depends on $H$, $U$, and $\sigma$ and through what
mechanisms these parameters influence the pore's behavior.
