COMSOL 6.4 - Voltammetry at a Microdisk Electrode

Cyclic voltammetry is a common electroanalytical technique. Since the 1980s, it has been common in voltammetry to use a microdisk electrode as the working electrode (Ref. 1). This is a disk electrode with a radius on the order of micrometers, embedded in an insulator whose surface is flush with the electrode.

Figure 1: Schematic of the simulation geometry for a microdisk electrode.

These very small electrodes have advantageous mass transport properties that can maximize the measured current density, and so enable the study of electrochemical behavior that would not be observable by conventional voltammetry as performed on a large macroelectrode. (See the model Cyclic Voltammetry at a Macroelectrode in 1D.)

This example demonstrates the use of a common approximation in which an electrode with microscale dimensions is assumed to have stationary (equilibrium) diffusion properties on the time scale of a voltammetry study. This simplifies the analysis because a time-dependent model is not required. Instead, a Parametric Sweep is used to assemble a voltammogram under a quasistatic approximation.

Model Definition

The model contains a 2D axisymmetric domain surrounded by a concentric region in which Infinite Elements are used to extend the bulk solution in the model to ‘infinity’. The approximation that the bulk solution is infinitely distant is suitable if the electrochemical cell is several orders of magnitude larger than the electrode.

The x-axis is divided by a point at the electrode radius, re, which equals 10 μm. At r < re, this axis represents the working electrode (microdisk) where the electrochemical reaction takes place. At r > re, this axis represents the surrounding insulator in-plane with the disk electrode.

DOMAIN EQUATIONS

We assume the presence of a large quantity of supporting electrolyte. This is inert salt that is added in electroanalytical experiments to increase the conductivity of the electrolyte without otherwise interfering with the reaction chemistry. Under these conditions, the resistance of the solution is sufficiently low that the electric field is negligible, and we can assume that the electrolyte potential ϕl = 0 (Ref. 2).

The Electroanalysis interface implements chemical species transport equations for the reactant and product species of the redox couple subject to this assumption. The domain equation is the diffusion equation (also known as Fick’s 2nd law), which describes the chemical transport of the electroactive species A and B. At steady-state, this reduces to:

Here Di represents the diffusion coefficient and ci the concentration of a species.

BOUNDARY EQUATIONS

At the bulk boundary (r → ∞), we assume a uniform concentration equal to the bulk concentration for the reactant. The product has zero concentration here, as in bulk.

At the insulating (inert) surface, the normal flux of both species A and B equals zero, since this surface is impermeable and neither species reacts there.

At the electrode boundary, the reactant species A oxidizes (loses one electron) to form the product B. By convention, electrochemical reactions are written in the reductive direction:

The stoichiometric coefficient is –1 for B, the “reactant” in the reductive direction, and +1 for A, the “product” in the reductive direction. This formulation is consistent even in examples such as this model where at certain applied potentials, the reaction proceeds favorably to convert A to B. The number of electrons transferred, n, equals one.

The current density for this reaction is given by the electroanalytical Butler–Volmer equation for an oxidation:

in which iloc is the local current density, k0 is the heterogeneous rate constant of the reaction, αc is the cathodic transfer coefficient, and η is the overpotential at the working electrode. Additionally, F, R, T, represent Faraday’s constant, the ideal gas constant and the temperature, respectively.

According to Faraday’s laws of electrolysis, the flux of the reactant and product species are proportional to the current density drawn:

Here n is the normal vector of the boundary, Ji the molar flux, and the product -n · Ji the molar flux across the boundary. Additionally, νi represents the stoichiometric factor and n the number of electrons transferred.

This is expressed in the Electrode Surface boundary condition.

The total current, Iel, recorded at the disk electrode can be extracted by integrating the local current density across the electrode surface. For this purpose, the Electroanalysis interface defines an electrode current variable according to

Note that it is not sufficient to simply multiply by the area of the electrode, because the current density may be nonuniform.

STATIONARY STUDY

In contrast to macroelectrode voltammetry, a voltammogram recorded at a microdisk does not exhibit hysteresis. Diffusion is so fast on the time scale of the experiment that a stationary approximation is suitable. A quasistatic approximation applies when:

where v is the voltammetric scan rate (SI unit: V/s). The two terms in this inequality are respectively the diffusive and voltammetric time scales of the system.

Within the Stationary study, a parametric sweep is used to study the range of applied potentials achieved in the voltammogram.

Results and Discussion

The stationary concentration profile around a microdisk electrode (Figure 2) has a distinct shape. At large distances from the electrode, the concentration profile is roughly hemispherical, but close to the disk edge the flux is elevated. For fast kinetics the concentrations on the electrode surface are roughly equilibrated and so are uniform. This leads to unequal flux over the surface of the electrode — it is nonuniformly accessible.