Page 145 - Analytical Electrochemistry 2d Ed - Jospeh Wang

130 PRACTICAL CONSIDERATIONS

For disk, spherical, and hemispherical geometries, the general expression for the
radial component in equation (4-14) is given by

i radial ˆ arnFDC 4-15†

where r is the electrode radius and a is a function of the electrode geometry. For
disks, spheres, and hemispheres the a values are equal to 4, 4p, and 2p, respectively.
Such radial diffusion leads to a larger ¯ux at the perimeter of the electrode than at
the center, and hence to a nonuniform current density.
The extent to which the planar or radial component dominates depends on the
relative dimensions of the electrode and the diffusion layer, as expressed by the
2
dimensionless parameter Dt=r where t is the electrolysis time and r is the smallest
0
0
2
dimension of the electrode (84). For large (>1) values of Dt=r (i.e., diffusion layer
0
thickness that exceeds the size of the electrode), the current approaches steady state,
and sigmoidal voltammograms are observed. In contrast, planar diffusion dominates
2
at small values of Dt=r , and a peak-shaped behavior is observed. Hence, depending
0
on the time scale of the experiment (i.e., the scan rate), the same electrode may
exhibit peak-shaped or sigmoidal voltammograms (e.g., Figure 4-24). Similarly, in
chronoamperometric experiments, a modi®ed Cottrell equation predicts that a
steady-state current is reached rapidly after the potential step (e.g., within
10 ms and 1.3 s for 1 mm and 10 mm diameter disks, respectively). The change
from semi-in®nite planar diffusion to semi-in®nite hemispherical diffusion, asso-
ciated with the decrease in the electrode dimension, is illustrated in Figure 4-25,
which displays computed concentration pro®les for a given time after the start of a
chronoamperometric experiment at a disk with different radii.

4-5.4.2 Con®gurations of Microelectrodes Electrodes of different materi-
als have been miniaturized in many geometrical shapes (Figure 4-26), with the
common characteristic that the electrode dimension is signi®cantly smaller than the
diffusion layer at the surface (for ordinary voltammetric time scales, e.g., 1±10 s).
The most commonly used is a circular conductor (of around 10 mm diameter),
embedded in an insulating plane (the so-called microdisk electrode) (a). Other
common geometries include the microring (b), microcylinder (c), microhemisphere
(d), or microband (e) electrodes. Cylinder and band (line) microelectrodes, which
can be several millimeters long, yield larger (and hence more easily measured)
currents, while maintaining an enhanced diffusional ¯ux. Band electrodes of
nanoscopic dimensions can be fabricated by sealing (``sandwiching'') ultrathin
carbon and metal ®lms between insulating supports and polishing one end of the
sandwich, or via photolithographic deposition of a thin metal ®lm on an insulating
substrate. The fabrication of most microelectrode geometries (with the exception of
microcylinders) is technically demanding. Special attention should be given to
proper sealing (between the active surface and insulating sheath) to assure good
performance and to minimize stray capacitances. Fine metal (Pt, Au, Ir) wires,
carbon ®bers, or thin metal ®lms are commonly used for these preparations.
Molecular (nanometer)-sized electrodes being developed in several laboratories