Biomedical Engineering Reference
In-Depth Information
1. Theory
Because heterogeneous kinetics feature prominently in the experi-
ments reviewed below, we begin by briefly reviewing the relevant
theory at the level of Butler-Volmer kinetics. While this topic will
already be familiar to many readers, we aim through our exposi-
tion to bring particular attention to the role played by the electrode
geometry in determining the predicted voltammogram shape.
We consider a simple outer-sphere heterogeneous electron-
transfer reaction at an electrode surface, O +
e
-
'
R. O and R
are present in the bulk solution at concentrations
C
O
b
and
C
R
b
, re-
spectively. We further assume the presence of a large excess of
inert salt, such that mass transport in this simple system will be
determined by the diffusion of the species O and R to or from the
surface. Finally, we also assume for notational simplicity that the
diffusion coefficients for O and R have an identical value,
D
. Quite
generally for any electrode of finite size immersed in a volume of
much larger size in all three dimensions (a condition which nearly
always applies to nanoelectrodes immersed in a macroscopic vol-
ume), a true steady-state limiting current exists.
The total rate of
mass transport of O from the bulk solution to the electrode,
J
O
, can
then be written as
17
J
DC
(
b
C b
e
)
(1)
O
O
O
Here,
C
O
e
is the concentration of O at the electrode surface (to be
determined further into the calculation), while
b
is a parameter
with units of length that characterizes the electrode geometry. The
parameter
b
can be thought of consisting of two components: a
numerical constant that depends only on the
shape
of the electrode,
multiplied by a characteristic length that characterizes the
size
of
the electrode. Determining the value of
b
is equivalent to solving
the diffusion equation for this particular geometry. A well-known
example is a shrouded semi-hemispherical electrode of radius
R
,
for which
b
= 2S
R
. The corresponding expression for the rate of
mass transport of reduced species R from the electrode surface to
the bulk can be written as
b
R
R
J
D
C
C
(2)
R
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