Biomedical Engineering Reference
In-Depth Information
insult to injury, a second, separate effect also occurs for charged
redox species: the electric field in the double layer acts on these
molecules and modifies the surface concentrations from the purely
diffusive case considered in Section II.1, which further influences
the faradaic current. These two effects are collectively known as
Frumkin corrections
. Note that they are not specific to nanoelec-
trodes, and they also occur at the surface of macroscopic elec-
trodes. Recent theoretical work, however, suggests that thin-layer
cells may provide an ideal system for studying the Frumkin ef-
fect.
120
In addition to the above, the existence of a double layer with
characteristic dimension O
d
is expected to have further influence
on voltammetry for the case of electrodes with characteristic di-
mensions comparable to or smaller than O
d
. In our theoretical
treatment of the classical regime in Section II.1, we assumed that
mass transport of the redox species was entirely specified by diffu-
sion. Consider, however, a hemispherical electrode with radius
R
.
The concentration profile for steady-state, purely diffusive
transport from this electrode has the form
C
(
r
) = (
R
/
r
)
C
0
, where
C
0
is a constant. While it is not possible to rigorously define an
absolute length scale for this profile due to its power-law form,
qualitatively, most of the concentration change occurs on a size
scale of a few times
R
. More specifically, a value of 10
R
for the
size of the diffusion region is conventional. For a sufficiently small
electrode, however, the limit where O
d
becomes comparable to or
greater than 10
R
can be reached. In this case, the assumption of
purely diffusive transport clearly breaks down, since the electric
field from the double layer at the electrode permeates the diffusion
region.
This very regime was explored in a study by Chen and Ku-
cernak.
43
While studying the voltammetric response for the reduc-
tion of Ru(NH
3
)
6
3+
at carbon electrodes sized as low as 1 nm, they
observed that the limiting current decreased considerably in the
absence of supporting electrolyte compared to the case when 0.5M
KCl was present. This effect, however, was observed only at elec-
trodes having effective radii,
r
eff
, smaller than 20 nm. Similar ob-
servations were made regarding the suppression of rates of oxida-
tion of TMAFc
+
and IrCl
6
3-
and reduction of IrCl
6
2-
in the absence
of electrolyte at electrodes with radii less than 30 nm.
36, 41, 113
While a detailed understanding is yet lacking, these departures
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