Bäcklund flux quantization in a model of electrodiffusion based on Painlevé. II. (Q2880351)

The authors consider a model of steady one-dimensional two-ion electrodiffusion across a liquid junction. It involves three coupled first-order nonlinear ordinary differential equations and has the second-order Painlevé II equation at its core. In its simplest form, the model deals with one-dimensional transport across an infinite slab occupying \(0\leq x\leq \delta\), for two types of ions carrying equal and opposite charges. With the concentrations of the ionic species and the induced electric field within the slab denoted by \(c_{\pm}(x)\) and \(E(x)\), respectively, the governing system of coupled ordinary differential equations (ODSs) obtained is NEWLINE\[NEWLINEc'_{+}(x)=(\bar{z}e/k_{\beta}T)E(x)c_{+}(x)-\Phi_{+}/D_{+},NEWLINE\]NEWLINE NEWLINE\[NEWLINEc'_{-}(x)=-(\bar{z}e/k_{\beta}T)E(x)c_{-}(x)-\Phi_{-}/D_{-},NEWLINE\]NEWLINE NEWLINE\[NEWLINEE'(x)=(4\pi\bar{z}e/\epsilon)[c_{+}(x)-c_{-}(x)],NEWLINE\]NEWLINE for \(0<x<\delta\). Here, \(\Phi_{+}\) and \(\Phi_{-}\) denote the steady (constant) fluxes of the two species in the \(x\) direction across the slab, \(\bar{z}\) their common valence, and \(D_{\pm}\) the corresponding diffusion contains, while \(k_{\beta}\) denotes Boltzmann's constant, \(e\) the electronic charge and \(T\) the ambient absolute temperature within the solution in the slab. The authors note that solutions are grouped by Bäcklund transformations into infinite sequences, partially labeled by two Bäcklund invariants. Each sequence is characterized by evenly-spaced quantized fluxes of the two ionic species, and hence evenly-spaced quantization of the electric current density. Finite subsequences of exact solutions are identified, with positive ionic concentrations and quantized fluxes, starting from a solution with zero electric field found by Planck and suggesting an interpretation as a ground state plus excited states of the system. Positivity of ionic concentrations is established whenever Planck's charge-neutral boundary conditions apply. Exact solutions are obtained for the electric field and ionic concentrations in well-stirred reservoirs outside each face of the junction, enabling the formulation of more realistic boundary conditions. In an approximate form these lead to radiation boundary conditions for Painlevé II. Illustrations of numerical solutions are presented, and the problem of establishing compatibility of boundary conditions with the structure of flux-quantizing sequences is discussed.