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

DO WE NEED A HOW? MAYBE A QUICK COMSOL GUIDE??

This model was constructed in order to test assumptions used in the analytical solution, specifically the assumptions that the diffusion “front” would not reach the edges of the plate within the relevant timescale. To do this, we used a time-dependent study with the 2D axisymmetric geometry pictured below, where the disc on the top surface is the source electrode. INSERT SIMPLE 2D AXISYMMETRIC FIG
Results and Conclusions
As can be seen from the figure below, after 20 minutes, the concentration front was nowhere near the bottom of the plate, let alone the sides of the plate. Note that the edge of the domain shown is not the edge of the plate; the domain shown has a radius of 1cm, and the true edge of the plate would be much further from the electrode than this. The model corroborates the analytical solution, showing a very steep concentration gradient within (INSERT)mm of the edge of the electrode, again making a good argument for the achievement of spatial resolution in a patterning application such as ours. INSERT 20MINS CONC PROFILE (2D SIMPLE) From this model, we can conclude that the assumptions in the analytical solution regarding the diffusion “front” are valid, and that it is valid to treat the dish in the analytical solution as both arbitrarily deep and wide. Additionally, it is reasonable to think that localised oxidised or reduced volumes can be induced in the agar plate in a period of 20 minutes by the use of electrodes, which was confirmed experimentally.

When a manufacturer produces a PCB with pads on the underside, such as the one we ordered, the pads are plated, and then the underside of the PCB lacquered. This means that the actual surface of the pad is slightly recessed from the surface of the rest of the PCB. INSERT IMAGE OF RECESSED BOIIIIIIIIIIIIIIIIIIII EXTRA RECESS LIKE THE GREAT RECESSION Therefore, we wanted to test how a small recession such as this would affect the concentration profile around the electrode. We again used 2D axisymmetric geometry, shown below, where the electrode disc is the top surface of the recessed cylinder.
Results and Conclusions INSERT GRAPHIC
The simulation shows that even very small recessions drastically reduce the diffusive flux away from the surface. This is because when the surface is recessed, even by a few um, all flux of the species must be perpendicular to the surface. This constraint drastically limits the space into which the molecules can diffuse, as there is a concentration gradient only in the z-direction, as opposed to in both the r- and z-directions in the flush case. This in turn limits the diffusive flux into the surrounding environment, and results in an increased concentration at the electrode surface due to limited diffusion out of the recession. INSERT GRAPHIC- FLUX LINE COMPARISONS
It should be noted that the decreased diffusive flux into the wider environment is not necessarily a bad thing, as although it limits the volume a particular electrode can affect, it increases the sharpness of the concentration profile (and hence gives a sharper on/off transition for induced cells at the boundary), and increases the concentration directly underneath the electrode, resulting in greater fold induction of any electro-inducible organisms in this region. Indeed, the magnitude of recession of electrodes within their mounting could be approached as an engineering specification, with a smaller recession resulting in wider but less well-defined fields of influence, and a larger recession resulting in smaller, sharper and more highly induced fields of influence.