The Painlevé equation of the second kind for the binary ionic transport in diffusion boundary layers near ion-exchange membranes at overlimiting current

In ion-exchange membrane systems such as electrodialysis, the overlimiting current phenomenon still remains a difficult topic to study due to the complicated solution method for the Nernst–Planck–Poisson model. In this study, the Nernst–Planck–Poisson equations were prepared to simulate the steady-state binary ionic transport in the diffusion boundary layer near an ion-exchange membrane. The system of differential equations was converted into the Painlevé equation of the second kind in such a way that the converted model domain explicitly shows the transition point from the space-charge region to the electroneutral region in the diffusion boundary layer even before the differential equation is solved. Based on this property, mathematical expressions were proposed to estimate the limiting current density and the width of the space-charge region in the diffusion boundary layer near an ideally perm-selective ion-exchange membrane. The so-called Airy solution of the Painlevé equation of the second kind was used to describe the ionic transport in the space-charge region. It was also found that the Airy function of the second kind with its derivative describes the behavior of the electric double layer developed from the ion-exchange membrane surface. In addition, a relatively simple numerical method, including a stability criterion, was used to solve the Painlevé equation of the second kind to simulate the ionic transport in the diffusion boundary layer near an ion-exchange membrane.