#### DOI

10.17077/etd.8pvk7ew0

#### Document Type

Dissertation

#### Date of Degree

Summer 2010

#### Degree Name

PhD (Doctor of Philosophy)

#### Degree In

Applied Mathematical and Computational Sciences

#### First Advisor

Johna Leddy

#### Second Advisor

Gerhard Strohmer

#### Third Advisor

Bruce Ayati

#### First Committee Member

Bruce Ayati

#### Second Committee Member

Colleen Mitchell

#### Third Committee Member

Tong Li

#### Abstract

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.

#### Keywords

Electroneutrality, Fuel Cell, Membrane, Modeling, Modified Electrode, Nafion

#### Pages

xi, 101 pages

#### Bibliography

Includes bibliographical references (pages 99-101).

#### Copyright

Copyright 2010 Stephanie Ann Schmidt

#### Recommended Citation

Schmidt, Stephanie Ann. "Mathematical models of ion transport through nafion membranes in modified electrodes and fuel cells without electroneutrality." PhD (Doctor of Philosophy) thesis, University of Iowa, 2010.

https://doi.org/10.17077/etd.8pvk7ew0