Analytical Chemistry Chemical Cytometry Quantitates Superoxide

trode 28 by a finite element using Comsol Multiphysics software.
The thickness of the convection-free layer, δconv, was determined
before each experiment by chronoamperometry as described
previously. 18
RESULTS AND DISCUSSION
An important property of disk ultramicroelectrodes is that their
diffusion layers develop with time until reaching a steady-state
limit imposed by hemispherical-type diffusion. However, natural
convection may interfere with the mass transport as soon as the
thickness of the expanding diffusion layer becomes comparable
to δconv. In such a case, a steady-state regime is still achieved
but is then controlled by the respective contributions of
diffusion and natural convection. Depending on the electrode
radius, r0, and the thickness of the convection-free layer, δconv,
two situations may be encountered, whether the diffusion at
the electrode surface is planar or not. Indeed, at short time
scales, the thickness of the diffusion layer is considerably
smaller than the electrode radius. The electrodes then behave
as electrodes of infinite dimensions, and planar diffusion
operates. In that particular situation, the Cottrell equation
applies with
i planar )( nFADc°
√πDt
for electrodes of surface area A. Using the Nernst formulation,
eq 7 is similar to that given in hydrodynamic electrochemical
methods: 10
i )( nFADc°
δ
where δ ) (πDt) 1/2 . Therefore, natural convection interferes
significantly with the mass transport as soon as δconv ≈ (πDt) 1/2 .
Whenever this condition is not met, the diffusion layer may
develop, possibly reaching a hemispherical behavior until being
eventually limited by δconv.
However, since diffusion and natural convection occur together
in steady-state regimes, their contributions in mass transport
remain difficult to comprehend. To solve this problem, one needs
first to investigate the concentration profiles established under
the pure diffusional steady-state regime, i.e., without any influence
of natural convection. In such a case, solution of eq 1 shows that
most of the concentration gradients operate over a distance
comparable to the electrode radius (Figure 1A). Concentration
along the z axis (Figure 1B) varies according to
c 2 z
) arctan( c° π r0) The diffusion layer presents a hemispherical shape, and the
steady-state current is given by:
(7)
(8)
(9)
i hemisph )(4nFr 0 Dc° (10)
(28) Amatore, C.; Fosset, B. J. Electroanal. Chem. 1992, 328, 21.
Figure 1. Steady-state concentration profile simulated at a disk
electrode without considering the influence of natural convection: (A)
2D concentration profile with isoconcentration lines ranging from c/c°
) 0.1 to c/c° ) 0.9. (B) Concentration profile along the vertical axis
of symmetry.
Using the Nernst formulation again, comparison between eqs 8
and 10 leads to an equivalent diffusion layer thickness, δ ) πr0/
4. This thickness differs slightly from δz, which may be obtained
by extrapolating the concentration gradient at z ) 0, r ) 0 along
the z axis. Indeed, when z f 0, the concentration profile at r
) 0 tends to (Figure 1B)
c 2 z
)
c° π r0 (11)
which gives δz ) πr0/2. The difference in the δ and δz values
results from the nonradial distribution of diffusion fields at disk
electrodes. Concentration gradients are higher at the electrode
edges than at the center (Figure 1A). Since δ is evaluated from
the integration of concentration gradients over the whole electrode
surface (eq 5), it is necessarily smaller than δz. In the following,
the variation of these two parameters will provide an accurate
estimation of the influence of natural convection, either from
the current (i.e., through δ) or from the alteration of concentration
profiles in the z direction (i.e., through δz), where natural
convection prevails.
Analytical Chemistry, Vol. 82, No. 16, August 15, 2010
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