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Electrodes are modified with polymer films to grant novel permeability. Often, redox probes partition from solution into film and are electrolyzed at the electrode. This creates a flux of probe into the polymer film and a flux of electrolyzed probe out of the polymer film. Transport of the probe through the film is governed by diffusion and migration, mathematically described from the Nernst-Planck equation as J_{i}=-D_{i}((∂C_{i}(x,t))/(∂x))-((z_{i}F)/(RT))D_{i}C_{i}(x,t)((∂Φ(x,t))/(∂x)) where x is the distance from the electrode, t is time, C_{i}(x,t) is space and time dependant concentration of the probe i, z_{i} is the charge of the probe i, F is Faraday's constant, R is the gas constant, T is absolute temperature, J_{i} is the flux of the probe i, D_{i} is the diffusion constant of the probe i and Φ(x,t) is the space and time dependant potential.
In most natural systems, charge accumulation is not appreciably noticed, the system behaves in such a way that a charged ion is neutralized by a counterion. This is called electroneutrality and is mathematically represented by Laplace's condition on the potential, ((∂²Φ)/(∂x²))=0. In some systems, it is not clear if counterions are readily available to neutralize an ion. In such a system, there may not be electroneutrality, giving Poisson's equation to replace Laplace's condition as ((∂²Φ)/(∂x²))=-(F/ɛ)∑_{i}z_{i}C_{i}(x,t) where ɛ is the relative permittivity. The addition of Poisson's condition makes the system nonsolvable. In addition, the magnitude of F/ɛ creates difficulty simulating the system using standard techniques. The first system investigated determines the concentration and potential profiles over the polymer membrane of a fuel cell without electroneutrality.
In some systems, the probes can not easily diffuse around each other, certain polymer film environments prevent such a swap of location as diffusion is commonly thought to occur. A more generalized form of the Nernst-Planck equation describes spatially varying diffusion coefficient as J=-D(x,t)((∂C(x,t))/(∂x))-((zF)/(RT))D(x,t)C(x,t)((∂Φ(x,t))/(∂x)). D(x,t) is space and time dependent diffusion, usually thought of with a physical diffusion term and an ion hopping term. The second system this thesis investigates is a modified electrode system where electron hopping is responsible for a majority of the probe transport within the film.
Lastly, the beginnings of a method are presented to easily determine the physical diffusion rate of a probe within a modified electrode system based on known system parameters.