Discussion Forum

I am writing in hopes of soliciting some advice about a difficulty I’ve had using the Electrochemistry Module to build 3-D simulations of steady state diffusion at microelectrodes as a function of the electrode potential (steady state voltammograms).

I have been successful in building 2-D axisymmetric models of microdisk electrodes starting from this application file: www.comsol.com/model/voltammetry-at-a-microdisk-electrode-12877, which uses the electroanalysis physics interface. I’ve been able to simulate problems using different 2-D geometries/conditions and obtain realistic results: The current–potential curve has the expected sigmoidal shape centered on the reaction formal potential. The diffusion limited current agrees roughly with the analytical expression for a disk microelectrode (i = 4nFDCr). Concentration plots in COMSOL show the expected behavior: at potentials far past the reaction formal potential, the reactant concentration (in this case, the reduced species) at the electrode surface is nearly zero, and the product concentration is nearly equal to the bulk reactant concentration, reflecting the flux balance in the problem.

I ran into difficulties when trying to transition from a 2-D axisymmetric model to a 3-D model. My approach was to construct a 3-D model of the microdisk and then apply the same boundary conditions and study settings found in the 2-D axisymmetric application file (in this case, a parametric sweep through different electrode potentials). As in the application file, the problem consists of a disk microelectrode (radius either 5 or 10 microns) in a solution of a reduced species (diffusion coefficient 1 x 10^−9 m^2/s). The reduced species undergoes reversible one electron oxidation at the microelectrode, controlled using the “electrode reaction” boundary condition. I did not implement the infinite elements domain used in the application file; instead I applied a concentration condition to the outer boundary of the simulation.

My attempted 3-D model does not provide realistic results at all potentials. At potentials near or negative of the reaction formal potential, the current–potential curve shows the expected behavior. However, at potentials far positive of the formal potential (i.e. when the oxidation reaction should be running at its diffusion limited rate), the model makes wildly inaccurate predictions about the current (e.g. mA currents when nA currents are expected). The anomalous behavior kicks in at around +0.1 V vs. the reaction formal potential.

From examining the concentration plots, I spotted the likely cause: at these more extreme potentials, “hot spots” of anomalous concentration appear on the electrode surface. For example, handfuls of mesh elements show product concentrations in excess of the reactant bulk concentration (violating the flux balance), or elements showing negative reactant concentration.

My first instinct was that the mesh was not fine enough to get a realistic result in a 3-D model. However, even with element sizes much finer than those used in the 2-D model (boundary size node settings: maximum element size 50 nm, minimum element size 1 nm), I found similar anomalous currents. My next tactic was to apply a dense boundary layer mesh (64 boundary layers, 1.025 stretching factor, 0.25 thickness adjustment) to the electrode surface. This improved the result, but the voltammogram still shows unrealistic behavior (except now I see anomalous spikes in the 10 nA range instead of mA). Changing the basic geometry of the model (e.g. modelling the entire disk electrode as the end of a cylinder vs. modelling only a portion of the disk as a circle inscribed on a plane) did not seem to improve things.

I’ve attached a file here showing the expected results obtained from the 2-D simulation and then the anomalous results in the 3-D simulation. I am grateful for any tips the community has. I will say that I’m a beginner at COMSOL; it’s possible that I’m missing something very basic about how the solver or meshing works. I know that multiple authors have done 3-D COMSOL simulations of microdisk electrodes, although most of that work predates the electrochemistry module.