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A problem in nonlinear ion transport
Abstract
We regard ion motion as the superposition of diffusion (random movement), migration (motion under the influence of an electric field), and convection (hydrodynamic streaming). We consider an experiment in which two species of ions are transported in a medium by diffusion and migration. The ions are either "oxidized" or "reduced". The medium extends from $\xi$ = 0, at which there is an electrode, to $\xi = \infty$. Initially, only the oxidized species is present. At $\tau$ = 0 the voltage at the electrode is perturbed causing the oxidized species at the electrode to gain electrons, thereby becoming reduced. The motion of ions gives rise to an electric current $i(\tau$). We consider the case when the perturbation is so sharp that the concentration of the oxidized species drops immediately to zero. For this experiment, examining the current as a function of time is called chronoamperometry. Experimental evidence suggests that in chronoamperometry $i(\tau) \propto \tau\sp{{-}1\over2}$, in which case we say that the current response is Cottrellian. We model chronoamperometry by an initial-boundary value problem for the concentration (of the reduced species) and the charge. By means of a similarity substitution, we reduce this problem to a two-point boundary value problem with constraint. For the reduced model, we show that a unique solution exists and the current response is Cottrellian. To understand the quantitative nature of the solution, we treat the problem with three numerical algorithms. For two of the algorithms, we apply a Sinc-Galerkin method to obtain numerical results. The first algorithm is a direct implementation of the Sinc method to the model. We approximate the solution by a linear combination of Sinc functions, substitute this approximation into the differential equation and proceed as in a Galerkin method. The second algorithm is a "shooting method". The third is an iteration scheme which incorporates the Sinc-Galerkin method. Comparisons of the methods are provided and demonstrate a coincidence in results for the three schemes.
Subject Area
Mathematics|Chemistry|Electromagnetism
Recommended Citation
Pfabe, Kristin Anne, "A problem in nonlinear ion transport" (1995). ETD collection for University of Nebraska-Lincoln. AAI9536622.
https://digitalcommons.unl.edu/dissertations/AAI9536622