Biomedical Engineering Reference
In-Depth Information
Equation (6) indicates that the relative importance of hetero-
geneous kinetics compared to mass transport is entirely determined
by the dimensionless ratio
0
Ak
Db
O
(7)
a quantity known as the dimensionless rate constant. In the limit of
very fast kinetics,
k
0
>>
Db/A
or, equivalently, O >> 1, Eq. (6) re-
duces to the fully reversible result obtained from the Nernst equa-
tion. Redox species at the electrode surface are then in local equi-
librium with the surface, and the heterogeneous kinetic parameters
do not affect the shape of the voltammogram. Once O becomes
sufficiently small, however, kinetics begin to have an appreciable
impact. As a rule of thumb, this occurs when O decreases below a
value of 10.
Very importantly, Eq. (6) indicates that the geometry influ-
ences the relative importance of kinetics through two closely relat-
ed, yet separate, mechanisms. First, it determines the area
A
and,
hence, the total electron transfer rate at a given surface concentra-
tion. Second, it influences the rate of mass transport of both R and
O, as expressed by the parameter
b
. This makes it clear why elec-
trodes with smaller dimensions are more sensitive to heterogene-
ous kinetics: for a given shape of electrode, scaling all the dimen-
sions simultaneously by a factor J causes
A
to change by a factor
J
2
and
b
by a factor J, leading to a net change of O by a factor J.
More simply put, shrinking all dimensions by a factor of 10 reduc-
es the electrode area by a factor of 100 and is equivalent to speed-
ing mass transport by a factor of 10.
The dual role played by geometry can, however, easily mis-
lead our intuition, as we now illustrate through two numerical ex-
amples. First, consider the simple case of a shrouded semi-
hemispherical electrode of radius
R
,
as shown in
Fig. 1a
. In this
case the area is simply given by
A
= 2S
R
2
, while
b
= 2S
R
, as dis-
cussed above. In this case Eq. (6) reduces to a more recognizable
expression,
Search WWH ::
Custom Search