Hemispherical diffusion, semi-infinite

Electrode geometry — Figure 3. Concentration profiles at an array of inlaid disks at different time in response to an electrochemical perturbation. a Semi-infinite planar diffusion at short times b hemispherical diffusion at intermediate times c semi-infinite linear diffusion due to overlap of concentration profiles at long times... 

Unlike macroelectrodes which operate under transient, semi-infinite linear diffusion conditions at all times, UMEs can operate in three diffusion regimes as shown in the Figure for an inlaid disk UME following a potential step to a diffusion-limited potential (i.e., the Cottrell experiment). At short times, where the diffusion-layer thickness is small compared to the diameter of the inlaid disc (left), the current follows the - Cottrell equation and semi-infinite linear diffusion applies. At long times, where the diffusion-layer thickness is large compared to the diameter of the inlaid disk (right), hemispherical diffusion dominates and the current approaches a steady-state value. 

The characteristic diffusion time for any UME geometry where the transition from semi-infinite linear diffusion (transient) to hemispherical or spherical diffusion (steady state) occurs may be given as... 

FIG. 3 Steady-state isoconcentration contours projected in the x-z plane corresponding to semi-infinite diffusion from hemispherical (solid lines) and disk-shaped (dashed lines) pore openings. Contours are plotted for Cs/2, Cs/4, and Cs/8, where CB is the concentration at the surface of the pore. The disk-shape pore drawn in the figure has a radius a the radius of the corresponding hemispherical pore opening (not shown), r0, is equal to lalrr. 

Fig. 3. Representations of the diffusive fields at (a) a semi-infinite planar electrode, (b) a hemispherical electrode, and (c) a finite disc electrode.

Let us consider again the reaction O + R in an experiment involving a step of any magnitude, but in contrast to the limitations of the previous section, let us allow the experiment to proceed beyond the regime where semi-infinite linear diffusion applies. For the moment let us also restrict the electrode geometry to a sphere or hemisphere of radius tq. Species O is present in the bulk, but R is absent. We begin each experiment at a potential at which no current flows and at = 0, we change E instantaneously to a value anywhere on the reduction wave. 

II.5.10 Semi-infinite Hemispherical Diffusion for Faradaic Processes... 

These equations can be solved for semi-infinite external diffusion, where both Red and Ox forms are in the solution outside the sphere (diffusion to a spherical or hemispherical hanging mercury electrode, metallic solid spherical electrode), or they may diffuse inside the sphere (amalgam formation at mercury electrode, intercalation of Li into particles, hydrogen absorption into spherical hydrogenabsorbing particles). 

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