Diffusion Controlled

In a static solution where mass transport due to convection is negligible and migration has been suppressed, movement of electroactive species from the bulk of the solution to the electrode surface is controlled by diffusion. Diffusion is the movement of species down a concentration gradient. The rate that a species moves due to diffusion is proportional to the concentration gradient, as described by Fick’s first law of diffusion for a 1D system, equation 1.13,2, 3

(1.13)

where is the concentration gradient at a specific point in the x direction at a given time, where the x axis is perpendicular to the electrode surface. Figure 1.5 illustrates the time-dependant concentration profile due to diffusion at an electrode surface under conditions where at x = 0, c = 0 and where t3>t2>t1; the diffusion layer

thickness is defined as the distance which the concentration changes from 0 to cb. As

the concentration gradient decreases, the flux of the species towards the electrode surface decreases. The time-dependant expansion of the diffusion layer depends upon the experimental conditions employed.

Figure 1.5 The concentration profile within the diffusion layer for a system where the rate of ET is limited by diffusion to the electrode surface.

When Fick’s laws are considered in more than one dimension the WE geometry plays a significant effect upon the concentration gradient. Several geometries important to work within this thesis are summarised below.

Figure 1.6(a) shows a 2D representation of a disc macroelectrode, where a flat electrode, assumed to be uniformly active, is embedded within a co-planar insulating surface. When the reaction is driven at the electrode surface the local concentration of the mediator is decreased, creating a difference in concentration from the bulk concentration. Assuming diffusive mass transport control, the ET reaction will proceed at a rate greater than the rate of diffusion to the WE surface. With time the diffusion layer grows. For a macroelectrode the expanding concentration gradient creates a predominantly linear diffusion profile as the contribution due to edge diffusion is small in comparison to the linear diffusion to the WE surface. Over time the expansion of the diffusion layer results in a reduced flux to the electrode surface. This is observed in a CV as a peak current as mass transport cannot support the increasing rate of ET at the electrode surface. For CA measurements, stepping from a potential where no reaction occurs to a potential where ET is diffusion-controlled, the current response can be described by the Cottrell equation (equation 1.14),2, 3

(1.14)

where it is the current response with respect to time, and cb is the bulk concentration

Figure 1.6 Schematic of the diffusion profiles (black lines) caused by (a) the cross section through a macro disc electrode and (b) the cross section of a disc ultra micro electrode.

If the dimensions of the electrode are reduced to the micrometer scale, then diffusion to the edges of the electrode becomes significant. A disc or hemispherical electrode of diameter less than 100 µm is often referred to as an ultramicroelectrode (UME).12-14 The developed diffusion profile to a UME disc electrode imbedded co- planar within an insulator is shown in figure 1.6 (b). At short times, after a potential has been applied to the electrode which is suitable to initiate diffusion controlled electrolysis, the UME initially behaves as a macroelectrode, where an initial linear profile is established. With time the contribution of edge diffusion to the linear diffusion regime results in a hemispherical concentration profile. The diffusion controlled flux is thus greater for an UME than for a macroelectrode. The initial diffusion regime can be predicted using the Cottrell equation for semi-infinite linear diffusion (equation 1.14) and holds until the diffusional edge effects significantly contribute to the flux to the electrode surface.15 _{When edge diffusion becomes}

significant the flux to the UME surface deviates from that predicted by the Cottrell equation and reaches a steady state where the diffusion layer thickness becomes

constant.16 The current response for both regimes of a UME can be predicted by the Shoup-Szabo equation (equation 1.15 to equation 1.17),17

(1.15)

(1.16) (1.17)

where r is the radius of the disc. Figure 1.7 shows the effect of edge diffusion upon the time-dependant current response at a UME as predicted by the Shoup-Szabo equation, compared to linear diffusion only, predicted by the Cottrell equation.

Figure 1.7 Comparison of the Cottrell predicted response (-) to the Shoup-Szabo equation (-) for a disc electrode of r = 12.5 µm, D = 6 x 10-6 _{cm s}-1_{, and c}

b = 1 x10-3 mol dm-3.

The above cases consider diffusion to an individual electrode within an infinite insulating plane. However if multiple electrodes are present the diffusion profiles to the electrodes can become perturbed. The developing diffusion profiles for two

UMEs positioned so that the diffusion profiles will overlap on a typical timescale is shown in figure 1.8.18-20 Initially the diffuse layer expands at each UME as described previously for an individual UME (profiles 1-3). As the diffuse layer expands with time the presence of the diffusion field associated with the neighbouring UME perturbs the regular hemispherical profile; an effect referred to as diffusional overlap. This is shown in profiles 4 and 5 of figure 1.8. As time progresses overlap results in an essentially linear profile.21, 22 As the thickness of the diffusion layer is dependant upon the diffusion coefficient of the electroactive analyte and geometry of the UME employed the critical inter-electrode spacing for diffusional overlap is dependant on the experimental conditions utilised.20

Figure 1.8 The expansion of the diffuse layer with time to two UMEs, profiles 1-6 represent the movement of cb away from the electrode surface with time. Profile 1 represents the initial linear

diffusion, profiles 2 and 3 show the establishment of hemispherical diffusion to the individual UMEs, profiles 4 and 5 show diffusional overlap occurring, and profile 6 shows the development of a diffusion profile similar to the profile that would be obtained if the central insulating region was active.

Expanding the two electrode example further to an array of UMEs where complete diffusional overlap is possible on the timescale of the electrochemical experiment, a measured current response is the same as that for an entirely active

diffusional overlap does not occur (e.g. UMEs are spaced sufficiently far apart) then the array will maintain the characteristic electrochemical response of an UME (e.g. a CV with a limiting current for a mass transport controlled reaction), however the overall current response will be the sum of the currents from each individual UME in the array. Simulations of uniform21 and random22 UME arrays have been used to visualise the diffusional profiles to arrays. A significant advantage of using the array geometry is the reduction in the active electrode area compared to an electrode of the same geometric area (i.e. including the insulator area), resulting in the capacitance of the array system being smaller.23 The reduced non-Faradaic background signal provides an enhancement in the signal-to-noise levels allowing for lower concentrations to be detected than for a conventional all active electrode material.24