The We initially modelled the diffusion of the chemical species away from the electrode by treating the electrode as a disc source with a constant flux of the molecule across it. We then constructed the problem by assuming that the agar medium itself was not generating additional molecules, and therefore that the only source of said molecules was the disc electrode. This assumption, combined with the conservation of species, allows us to describe the concentration of the species within the agar with the partial differential equation known as the Laplace equation. The Laplace equation in this situation represents the idea that the number of molecules arriving at a point in space per unit time is equal to the number of molecules that leave that space per unit time.
The radially symmetric nature of the problem (the problem looks the same from any angle, as the electrode is a disc, not a rectangle or any other shape) allows us to use the 2D axisymmetric form of the Laplace equation, rather than the full 3D form, which makes solving the problem easier.

Due to the slow speed of diffusion, and the square root proportionality between time and diffusion distance, we assumed that the agar plate was arbitrarily deep, and that the concentration of the species tended to 0 as depth (z) increased. This can also be conceptualised as the diffusion “front” not reaching the bottom of the plate within the timescales that we are interested in. This assumption was also applied in the radial direction (ie the diffusion front does not reach the edge of the agar plate). This assumption was later confirmed through the use of Comsol modelling. (INSERT LINK?) We also constrained the problem to positive values of z, meaning that no molecules can leave the agar through its top surface. We also assumed that the chemical species analysed would not adsorb onto, or otherwise interact with, the agar medium.

The full mathematical solution for the problem, as well as the justification for the assumption that the molecules are only transported by diffusion, is given in the appendix (INSERT LINK FILE OR SMTH). However, we shall say here that the problem was formulated in cylindrical coordinates (again due to the radially symmetric nature of the problem), and non-dimensionalised both to simplify working and to generalise the problem to any desired scales (scales here meaning length scale of the problem and scale of diffusion, normalised for diffusion constant and flux through the electrode). The scalings are given below. The problem was evaluated analytically and by hand up until the last step. That last step involved the evaluation of a complicated infinite integral, which was non-trivial. However, because the integrand tends to 0 as the variable tends to infinity, use of numerical integration in this case was justified to give the solution for the dimensionless concentration profile. SCALING FORMULAE INSERT (IntegrandPlot1) The exact magnitude of concentration for a given species at a specific flux can be found through simply reinserting the diffusion coefficient for that species. Interestingly, the diffusion coefficients for the species of interest are all reasonably similar, and so give reasonably similar results. Despite this, a wider range of diffusion coefficients was tested (between 5e-10m^2s^-1 and 1e-9m^2s^-1) to account for any differences from literature value due to interactions with the agar medium, or differences due to molecular interactions within the solution.

Observing the scalings for the non-dimensionalised problem, it can be seen that the width of the concentration distribution is directly proportional to the electrode radius, and therefore scales linearly with it. The concentration at any given point is directly proportional to the flux through the electrode, and inversely proportional to the Diffusion coefficient. The half-maximal width of the concentration profile is 1.145 times the electrode diameter, which confirms the feasibility of generating spatially resolved regions of oxidising or reducing environment, with the resolution limited only by the electrode radius, which can be considered an engineering parameter. INSERT JUSTIFICATION FOR SPACING PLOT OF DIFFERENT D (SSforDs) SURF OF PROFILE RADIAL PLOT