feedback occurs at longer times decreases with increasing rate
constant, causing the rapid decay in the current and the
appearance of the peak at shorter times.
Theoretical results and discussion
The normalised currentÈtime response of the tip UME
depends primarily on the tipÈsubstrate separation, K and c.
To limit the length of this paper, comprehensive theoretical
results are presented for the case of c \ 1, which both rep-
resents the most general case and was appropriate to the
experimental system of interest (vide infra).
At the larger tipÈsubstrate separations [Fig. 2(b)] the di†u-
sion time for the substrate-generated species to reach the tip
increases. Consequently, for a given value of K, the peak
current occurs at a longer time than for the closer tipÈ
substrate separation [Fig. 2(a)]. Moreover, as found for
feedback2,4h8 and TGÈSC5,6,8 measurements, the current
response becomes less sensitive to solution kinetics, the
greater the spacing between the tip and substrate. For the
SGÈTC mode in the present application, the e†ect of increas-
ing the inter-electrode spacing is to decrease the peak current
for a given value of K. These issues are illustrated further in
Fig. 3, which shows transients for K \ 10 at a range of values
of log(d/a) between [1.0 and [0.5. It is clear that the closer
the tipÈsubstrate separation, the larger the peak current, but
the shorter the peak time. From an experimental viewpoint,
close tipÈsubstrate separations lead to high sensitivity in terms
of tip current, but it is also evident that the tip response has
to be measured with the highest temporal resolution in this
situation.
The characteristic features of the tip current transients in
Fig. 2 and 3 are the peak current, peak time and post-half-
peak time. Comprehensive contour plots illustrating how
these features depend on the tipÈsubstrate separation and K
are given in Fig. 4. The magnitude of the normalised peak
current [Fig. 4(a)] increases as both K and the value of d/a are
decreased. In contrast, the peak time increases as the value of
d/a is increased and K is decreased [Fig. 4(b)]. In principle,
this contrasting variation in the peak current and peak time
with K and d/a should allow both to be determined from a
single transient measurement. The post-half-peak time has a
yet di†erent dependence on d/a and K, by increasing as K
decreases while showing only a minimal dependence on the
inter-electrode separation [Fig. 4(c)].
Calculated tip chronoamperometric characteristics for a
range of values of the normalised rate constant, at two tipÈ
substrate separations, log(d/a) \ [0.7 and [0.2 (typical of
relatively small and large separations) are shown in Fig. 2(a)
and (b), respectively. In the absence of following chemical
reactions (K \ 0), the tip current rises, after an initial lag
period, from zero to a steady-state value. The magnitude of
the steady-state current and the rise time depend on the tipÈ
substrate separation, as discussed in detail elsewhere.15 In
brief, the closer the tip and substrate, the shorter the inter-
electrode di†usion time. Consequently, in this limiting situ-
ation, the more rapid the current rise the sooner a steady state
is established. At steady-state, the concentration gradients
normal to the UME are steeper the closer the inter-electrode
spacing, resulting in an increasingly enhanced current with
decreasing electrode separation.
In contrast, for Ðnite values of K, the current does not reach
a steady state, rather it rises to a peak value and then
decreases at longer times. The value of both K, and the tipÈ
substrate separation, have a marked e†ect on the overall
shape of the tip current transient and the magnitude of the
peak current. For a given tipÈsubstrate separation, the larger
the value of K, the smaller the peak current, the shorter the
time to reach the peak, and the steeper the decay curve follow-
ing the peak. This is expected, since increasing the rate of the
following chemical reaction decreases the fraction of
substrate-generated O that initially reaches the tip in the Ðrst
pass, thereby decreasing the peak current. The extent to which
The implications of the above analysis are that it should be
possible to measure following chemical reaction rates via the
SGÈTC mode without any prior knowledge of the tipÈ
substrate separation. From a practical viewpoint, this opens
up the possibility of making such measurements with very
basic positioning apparatus. The range of measurable rate
constants will largely be governed by the timescale on which
tip currents can be recorded. For example, towards the fast
kinetic limit, a normalised peak current in excess of unity is
predicted for log K \ 2.5, with tipÈsubstrate separations
closer than log(d/a) \ [0.8. The normalised time at which the
peak occurs is, however, smaller than 0.01. For a tip UME
with a radius of 10 lm and a typical di†usion coefficient of
10~5 cm2 s~1, these Ðgures relate to a rate constant in excess
of 3000 s~1 and a corresponding peak time less than 10~3 s.
Under the experimental conditions of this study, with a large
Fig. 2 E†ect of K on the chronoamperometric characteristics for an
EC process in the SGÈTC mode at tipÈsubstrate separations of (a)
log(d/a) \ [0.7 and (b) log(d/a) \ [0.2. Normalised current data are
plotted as a function of normalised time, q. The dashed lines (i) show
the behaviour for K \ 0, while the solid lines are for log K values of:
(ii) [0.5; (iii) 0.0; (iv) 0.5; (v) 1.0 and (vi) 1.5.
Fig. 3 E†ect of tipÈsubstrate separation on the SGÈTC tip collector
chronoamperometric response. The theoretical data relate to K \ 10,
with a range of deÐned log(d/a) values.
J. Chem. Soc., Faraday T rans., 1998, V ol. 94
755