R0-fll92 82S THE INFLUENCE OF UNCONPENSATED SOLUTION RESISTANCE ON
THE DETERHINATION A. . __»''i
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The Influence of Uncompensated Solution Resistance on
the Determination of Standard Electrochemical Rate Constants
Using Cyclic Voltammetry, and Some Comparisons with AC Voltammetry
David F. Milner and Michael J. Weaver
Department of Chemistry, Purdue University,
West Lafayette, Indiana 47907, U.S.A.
88 3 22 0 91
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abstract
A digital simulation analysis Is presented of the deleterious effects
of uncompensated solution resistance, upon the evaluation of standard
rate constants, using cyclic voltammetry. The results are expressed
In terms of systematic deviations of "apparent measured" rate constants,
> evaluated in the conventional manner without regard for R^gi from
the corresponding actual values, k®b(true), as a function of and other
experimental parameters. Attention Is focused on the effects of altering
the electrode area and the double-layer capacitance upon the extent of the
deviations between k®b(app) and k®b(true), and on comparisons with
corresponding simulated results obtained from phase-selective ac Impedance
data. The extent to which k®b(spp) < small R^^ values was
found to be similar for the cyclic and ac voltammetrlc techniques. The
latter method is, however, regarded as being preferable under most
circumstances in view of the greater of minimizing, as well as
evaluating, R^^ for ac impedance measurements. The influence of solution
resistance on k®b measurements using microelectrodes and without IR
compensation is also considered.
In recent years cyclic voltammetry (CV) has been utilized to determine
standard electrochemical rate constants, k®, for a great number of redox
couples under widely varying conditions. The rate constants have been most
commonly derived from the cyclic voltammograms by using the Nicholson-Shain
1 2
type of analysis, although other methods have also been employed.
Particularly given the common and sometimes indiscriminant use of CV for
this purpose, it is Important to ascertain clearly the range of system
properties and measurement conditions over which the observed rate
parameters are indeed valid. This is of particular concern when relatively
rapid rate constants are required to be evaluated, since the effect of even
small positive amounts of uncompensated resistance, can easily be
1 a
misinterpreted as slow electrode kinetics under these conditions. In
addition to the deleterious influence of solution resistance, measurement
nonidealities associated with the effects of potentiostat bandwidth and
double-layer charging current also need to be considered.
Digital simulations of cyclic voltammograms of varying complexity have
been performed for as long as cyclic voltammetry has been employed for
s 13
evaluating k for quaslreversible systems. ' These simulations have
primarily been of the "idealized" form in which no instrumental or
electrochemical artifacts, such as arising from R^g, the double-layer
capacitance or finite amplifier bandwidths, are considered to affect
the cell response. Although significant attention has been given to the
development of algorithms which account for the influence of upon the
cyclic voltammograms,^ surprisingly little effort has been directed towards
providing analyses which enable the experimentalist to readily deduce the
consequences for the reliable measurement of k®. A significant difficulty
■ ■»*
Is that the solution resistance distorts cyclic voltanunograms in a manner
qualitatively similar to that of sluggish electrode kinetics.This can
produce considerable uncertainty in evaluating the latter in the presence
s 2-1
of the former, especially for moderate or large k values (> 10 cm s ).
While it is usual to employ positive-feedback compensation so to minimize
Rusi is extremely difficult to reduce to zero without severely
distorting the potential-time ramp applied to the cell.^
By means of digital simulations we have recently examined the combined
influence of uncompensated solution resistance and other measurement
nonidealities on the determination of k^ by means of ac impedance
techniques.^ For this purpose it was found useful to distinguish between
"apparent" observed rate constants, k®^(app), that are obtained for a given
instrument and measurement conditions by means of the conventional analysis
assuming such nonidealities to be absent, and the desired actual ("true")
value of the rate constant, k®^(true). The extent of the systematic
deviations between k®jj(app) and k®|j(true) was examined as a function of R^^
and the magnitude of k®|j(true) in order to provide means by which the
reliability of the RqJjCspp) values could be diagnosed, and to enable the
extent of the corrections to k®^(app) arising from the presence of
measurement nonidealities to be determined.^
The present communication summarizes the results of a comparable
digital simulation study aimed at assessing the reliability of k^^(app)
values obtained by using cyclic voltammetry. Of particular interest is the
extent to which the presence of uncompensated solution resistance provides
an effective upper limit to the reliable evaluation of k®^(true) for a
given set of measurement conditions, and how this limit for cyclic
I
voltammetry compares to that for corresponding ac Impedance measurements.
Simulation parameters are chosen so to correspond to a range of conditions
appropriate to typical aqueous and nonaqueous media. The effect of varying
the electrode area is considered, including conditions appropriate for
microelectrodes.
SIMULATION PROCEDURES
The digital simulations of cyclic voltammograms performed in this
study were of two types. For those instances where the effect of was
4a
not considered, the method of Nicholson was used, while those where this
influence was included were finite difference simulations similar to those
3
described by Feldberg.
The values of the various system parameters employed in the
simulations were selected to be representative of those commonly
encountered for redox couples in nonaqueous solvents as well as aqueous
media. The ranges of R,_ values chosen (< 50 ohm) are based on the minimum
values that we have generally been able to obtain in studies utilizing
common commercial potentiostats, such as the PAR 173/179 system (vide
infra). The reactant (and product) diffusion coefficient was taken to be 1
-5 2 *1
X 10 cm s , the electrochemical transfer coefficient was assumed to
equal 0.5, the number of electrons transferred, n, equals one, and the
reactant concentration was taken to be lm{l unless otherwise noted. The
voltammetric sweep rate was set at 20 V s'^ unless specified otherwise.
The general procedure involved simulating cyclic voltammograms for a
suitable range of system input parameters, including k®^j(true) and R^^
values, and extracting "apparent observed" rate constants, k®^(app), from
the simulated voltammetrlc curves In the same fashion as is conventional
for experimental data. For simplicity, this involved evaluating
from the difference between the cathodic and anodic peak potentials,
using AEp'f plots in the manner prescribed by Nicholson. Where
necessary, simulations were performed under "ideal" conditions (i.e. for
^us’ ^dl ~ generate more complete AE^-t data and thereby avoid any
interpolation errors encountered by using Table I of ref. la. Strict
adherence was also paid to the potential scan limits specified for the
validity of the AEp-* relation in ref. la.
All simulation programs were written in FORTRAN-77 and executed on a
DEC LSI 11-73 microcomputer under the RSXll-M operating system.
RESULTS AND DISCUSSION
Origins of Distortion of Cyclic Voltammograms from Uncompensated Solution
Resistance
Before examining systematically the extent to which the presence of
Ryg can lead to systematic differences between h®^(app) as determined by
cyclic voltammograms and the required k®^(true) values, it is useful to
clarify the origins of the underlying distortions in the current-potential
profiles. Under most conditions, the solution resistance will be
Incompletely compensated by the potentiostat positive-feedback circuitry so
that R^^g will be positive. The presence of R^^ will yield a net ohmic
potential drop, iR^g, through the solution in proportion to the current, i,
flowing. Since the current peaks for the negative- and positive-going
potential sweeps have opposite signs, the measured cathodic-anodic peak
separation, AE_, will clearly be larger for positive R,,^ than the "ideal"
5
case, where R^g * 0.
An additional, less obvious, source of distortion arises from the fact
that when R^g ^ 0 only the time derivative of the overall cell potential,
dEceii/dt, will be held constant during the voltammetrlc scan, rather than
the corresponding derivative of the double-layer potential, dE^^/dt, as is
required for the exact applicability of the usual CV treatment. As a
consequence, dE^j^/dt will vary with so to distort the measured AEp
still further in the presence of R^g. This is illustrated in Fig. 1, which
shows simulated i - and (dE^j^/dt) - traces for a negative-going
potential sweep, with (dE^^gj^j^/dt) - i/ - 100 V s*^. The electrode area. A,
is taken as 0.2 cm , R^g as 50 ohms, and as 20 /iF cm . The magnitude
of dE^j^/dt is such that it is less negative when i is decreasing (Fig. lb).
This "acceleration" of the effective sweep rate in the vicinity of the
current peak contributes to larger AEp values than in the absence of
uncompensated solution resistance.
The nonfaradaic current, i^£, resulting from nonzero values will
also contribute to this distortion. This is because 1^^ enhances the total
current throughout the voltammogram, and therefore further enlarges AEp
than when R^g - 0. This capacitive contribution to AEp, A(AEp), can be
determined approximately from A(AEp) - 2 where v is the applied
potential scan rate (‘^^cell^**^^ ’ However, the precise influence of
upon AEp will be more complex, not only because is usually potential
dependent, but also since the scan rate v in this relation should really be
identified with (dE^j^/dt) rather than with (‘^^cell/*^^^ ■
It is well known that the presence of can lead to large systematic
errors in the determination of k®^, since increasing or decreasing k®^
both lead to greater AE^ values under a given set of experimental
la
conditions. A common procedure is to "fit" the experimental current-
potential curves to corresponding simulated curves obtained for a set of
s 2 c 7
input parameters and trial k^^ values until the best match is obtained. ’
Although this procedure is in a sense preferable to the examination of AEp
values alone, it is not always obvious that obtaining a "good fit" of the
simulated to the experimental data is due to the correctness of the choice
of k®^ rather than to an incorrect choice of other system variables,
especially
The crux of the problem is that the diagnosis of AEp values greater
than the "reversible" limit (i.e. for k®^ *), (59/n)mV at 25*C, as
arising purely from the presence of "finite" (i.e. measurable) k®^ values
rather than at least partly from R^^ > 0 is far from straightforward. For
example. Fig. 2 shows a pair of simulated cyclic voltammograms, the points
S *" 1
corresponding to k^^ - 0.04 cm s and R^^ - 0, and the solid trace
referring to k®^^ - « and R^^ — 50 ohm. (The latter value is chosen here
since it approximates the magnitude of R^^ that can be anticipated in many
O
nonaqueous media. ) The close similarity in the two curves, in addition to
the almost identical AEp values, clearly make the evaluation of k®^ values
even as small as 0.04 cm s‘^ fraught with difficulty unless R^^ is known
accurately, or preferably diminished substantially below 50 ohm. Although
it is common to examine the dependence of AEp on the sweep rate, i/, in
WHUiUyililWIMJMJBUIJUBlWMJBTJWIWWJWWWUWIBWIlItlU^^ wwiKnjwwruwiwvivuini^
order to evaluate this procedure provides little or no diagnosis for
the dominant presence of finite electrode kinetics rather than since
both show a similar dependence
la
This is illustrated in Fig. 3,
which shows plots of dE^ against 1 / for the same conditions as in Fig. 2.
Both the resistance-dominated (solid curve) and kinetics-dominated
I
(squares)plots are almost indistinguishable. Clearly, then, substantial
systematic errors in the evaluation of k®^, so that k®^j(app) < k®^(true),
can occur for analyses of this type unless is known accurately.
In principle, a distinction between resistive- and kinetically-
dominated dEp behavior can be obtained by varying the bulk reactant
O
concentration, C^, while holding all other parameters fixed. As shown in
Fig. A, if the sweep-rate dependent dE is dominated by R , then dE
r wa |ii
should increase roughly linearily with whereas dEp will be independent
of if the dEp response is dominated by sluggish kinetics, at least if
these are first order so that k®^ is independent of This approach,
however, is often not a practical one. In circumstances where either
positive-feedback iR compensation and/or a Luggin probe is used to minimize
Rus It is usually very difficult to maintain precisely the same R^^ values
on successive measurements using solutions of varying reactant
concentration. Moreover, the presence of nonfaradaic current leads to some
influence of R^^ upon dEp even when -» 0 (Fig. 4). Consequently, it is
difficult to use this strategy to diagnose the influence of R^^ upon the
cyclic voltammograms except under relatively favorable circumstances.
It is therefore apparent that an accurate knowledge of R^^ is a
prerequisite for the reliable determination of k®^ by means of cyclic
voltammetry. As noted above, we have previously utilized digital
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simulations to examine deviations between k®|j(true) and k®^(app) as a
function of R for ac impedance measurements, where k^. (app) is determined
from the conventional analysis that assumes that R
0.” Such
relationships enable estimates of k®j^(true) to be obtained from measured
k®b(®PP) values if R^^ is known, as well as providing an upper limit to the
k®b(®PP) values for which meaningful kinetic information can be extracted.
Examples of such relationships obtained for cyclic voltammetry are
shown in Figs. 5 and 6 in the form of plots (solid curves) of log k®^(app)
versus R^^ for various k®jj(true) values. The k®^(app) values were obtained
from the AEp values for simulated cyclic voltammograms by using the
la
Nicholson analysis and ignoring the influence of The solid traces
A, B, and C in Figs. 5 and 6 refer to k®^(true) values of ®, 1.0, and 0.1
cm s , respectively. Complete coincidence between these k^^Ctrue) values
and the corresponding k®|j(app) values in these plots is only observed when
R^g - 0. A sweep rate of 20 V s'^ is used in these simulations, although
similar results were obtained at least over the range 1 < »/ < 100 V s'^.
The remaining conditions in Fig. 5 and 6 are also identical, with - 20
■ 2 9
ftF cm , except that the electrode area is taken to be 0.2 and 0.02 cm ,
respectively, in these two figures. Comparable results to those in Figs. 5
and 6 were also obtained if smaller C^j^ values, in the range 2-20 fiF cm’^,
were employed. Figure 7 shows such a set of results, obtained for the same
O
conditions as in Fig. 6, but with C^jj^ - 2 /iF cm .
From Fig. 5 we can see that for an electrode area of 0.2 cm a
significant distinction between a completely reversible reaction [i.e.
where k®^(true) - *] and that for which k®jj(true) - 1 cm s‘^ is only
possible for very small R values (< 3 ohm). In other words, the knh(app)
9
values are independent of k®jj(true) under these conditions, being virtually
indistinguishable for larger (Fig. 5), so that no meaningful kinetic
data can be extracted if k^jj^Ctrue) > 1 cm s . Also evident from Fig. 5 is
that if is larger than about 10 ohms, no value for k®^(app) could be
obtained that is larger than about 0.1 cm s"^, a frequently reported value
(e.g. ref. 9). Furthermore, the avoidance of substantial discrepancies
between k®^(app) and k®^(true) when the latter approaches 1 cm s'^ requires
that only extremely small uncompensated resistances be present. For
example, in order to evaluate a kQjj(true) value of 1 cm s to 50% accuracy
S “1
[i.e. to obtain kQjj(app) > 0.5 cm s under these conditions] it is
required that R^^^ < 2 ohm.
Comparing Fig. 5 and 6 shows that decreasing the electrode area
Improves somewhat this unfavorable situation. Thus for an electrode area
2
of 0.02 cm (Fig. 6) there is a clear experimental distinction between
k®b(app) values corresponding to k||j(true) values of ® and 1 cm s’^ (curves
A and B) even for moderate R^^ values (ca. 20 ohms), indicating that some
kinetic information would be contained in experimental data gathered under
these circumstances. Nevertheless, the k®^(app) values in curve B fall
c
markedly below kQjj(true) under these conditions, so that the corrections
necessary to extract the latter from the former are still substantial. The
s -1
errors involved in evaluating smaller k^jj(true) values, around 0.1 cm s ,
are relatively small since then k®^(app) - k®|j(true) even for R^g < 40 ohm
(Fig. 6). This diminished influence of R^g with decreasing electrode area
is, however, slightly misleading since the effective solution resistance
will increase under these conditions, yielding probable increases in •
10
1^-Jk*
luence ol
>oiutlon
ance on Rat
impedancf
Given that ac voltanunetry, particularly employing phase-selective
impedance measurements, provides the most common means of evaluating
other than by using cyclic voltammetry, it is of interest to compare
quantitatively the extent to which the utility of these two techniques is
impaired by the presence of uncompensated resistance. There is ample
reason, however, to expect the nature and extent of the influences of
upon these two techniques to be significantly different. In the ac
voltammetric experiment using a stationary electrode, the principal effect
of is to force the phase of the ac potential waveform across the double
layer to differ from that applied by the potentiostat, yielding errors in
the apparent phase angle of the current. While the presence of R^^ also
forces the magnitude of the ac potential waveform across the double layer
to differ from that controlled by the potentiostat, this is of little
consequence since it is generally only the phase angle, rather than the
magnitude, of the current which is of relevance to the evaluation of
standard rate constants.
The error introduced into the evaluation of by ac voltammetry
therefore depends principally on the combined influence of R^^ and the
double-layer capacitance, and is diminished with decreasing electrode area
to the extent that the total capacitance is also decreased. The effects of
Rus cyclic voltammetry have distinctly different origins, as discussed
above. Although the extent of the error in evaluating k®^j with cyclic
voltammetry also decreases with the electrode area, this arises primarily
as a result of the decreased total current flowing through the cell.
11
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As a consequence, we elected to compare simulated values of kob(aPP) :
obtained from phase-selective ac impedance as well as cyclic voltammetry by |
I
using conventional data analyses for the following three conditions: (a) J
electrode area A - 0.2 cm^, - 20fiF cm*^; (b) A - 0,02 cm^, - 20nF j
cm ; (c) A “ 0.02 cm , - 2pF cm The first two cases span the range ]
of areas commonly encountered with these two techniques, while the second
and third cases cover the typical range of capacitance values. The '
frequency range taken for the ac voltammetric data was between 100 and 500 I
I
Hz, the analysis utilizing the frequency dependence of the quadrature to |
I
in-phase current ratio, Iq/Ij (see ref. 6b for simulation details). The I
resulting plots of log versus obtained with ac voltammetry for j
cases a, b, and c are shown as the dashed curves in Figs. 5, 6, and 7,
respectively, to be compared with the solid curves in each of these figures
which (as already noted) show corresponding results obtained with cyclic
voltammetry. (As noted above, the three log curves in each of
these figures labeled A, B, and C, refer to log k®^j(true) values of <*>, 1.0
and 0.1 cm s*^, respectively).
2
For the large (0.2 cm ) electrode (Fig. 5), the degree to which
s s
kob(®PP) falls below k^^j(true) for a given R^^ value is more marked for ac
voltammetry than for cyclic voltammetry. In other words, under the
specific conditions prescribed by these simulations the degree of error
induced in k®jj(app) by ignoring solution resistance effects, and therefore
the extent of the corrections required to extract k®j^(true) from k®jj(app),
is smaller for cyclic voltammetry than for ac voltammetry. However, for
the smaller (0.02 cm ) electrode having the same double-layer capacitance
-2 s
per unit area (20^F cm ), the log kQ^(app) - R^^ curves obtained for the
two techniques are more comparable (Fig. 6). Moreover, the substitution of
a smaller double-layer capacitance (Fig. 7) yields slightly superior log
kob(app) • Ryg curves for ac as compared with cyclic voltammetry (Fig. 7).
These examples therefore suggest that roughly comparable errors are
introduced into cyclic and ac voltammetrlc measurements of under
"typical" conditions in the presence of at least small amounts of
uncompensated resistance. However, consideration of other factors lead to
the latter technique being clearly favored for this purpose. Most
Importantly, it is crucial to minimize the value of the uncompensated
resistance. This can be achieved relatively readily in the ac experiment
since a small amplitude waveform having relatively low frequencies (> 2000
Hz) is employed. In contrast, in the cyclic voltammetric experiment the
abrupt change in the time derivative of the potential which occurs when the
scan direction changes is equivalent to the sudden injection of high-
frequency "noise", which will result in a dampened oscillation ("ringing")
of the current. The avoidance of severe distortion in the i-E profile for
the return scan therefore normally requires the presence of significant
uncompensated resistance (> few ohms).
In addition, the accurate estimation of for a given level of
electronic resistance compensation in the ac impedance experiment is
relatively straightforward, either by evaluating the quadrature and in-
phase currents at potentials well separated from the ac voltammetric wave
or by ac measurements in conjunction with a dummy-cell arrangement.^®’^
(The same procedures could, of course, be employed to estimate R for a
us
given set of experimental conditions, including the resistance compensation
setting, used in a cyclic voltammetric experiment.) An alternative
approach to evaluate is to select a redox couple which under the
measurement conditions employed is known to exhibit k®|j(true) -► «, so that
the measured response is necessarily dominated by Although this
method has obvious merits, the selection of such a redox couple requires
11 12
careful consideration. ’ In our experience, using a variety of
potentiostats of commercial and in-house design, it is difficult to either
measure or minimize even for ac impedance measurements to much less
than 5 ohms or so in typical nonaqueous media. The corresponding minimum
Ryg values attainable in cyclic voltammetric experiments are often
substantially larger.
Another advantage of the phase-selective impedance approach is that a
distinction between apparent electrochemical irreversibility brought about
by the presence of uncompensated resistance rather than by finite electrode
kinetics can be made by examining the dependence of the (Ip/Ij) ratio upon
the ac frequency, If the former factor is predominant, then the slope
of the (Iq/Ij)-w plot will increase with increasing w, rather than be
independent of w as will be the case when electrode kinetics controls this
response.This diagnostic situation can be contrasted with the
inability, noted above, of achieving a ready distinction between the
dominant presence of R^^ and finite electrode kinetics by varying the sweep
rate in cyclic voltammetry.
Some Con siderations for Microelectrodes
The virtues of employing electrodes of especially small dimensions,
having radii down to (or below) 1 micron (so-called "microelectrodes" or
"ultramicroelectrodes") for electrochemical measurements have recently been
explored extensively, including preliminary applications to electrode
14
kinetics.^® The advantage of such a marked diminution in electrode area
can readily be seen by recalling that while the faradaic current decreases
with the square of the electrode radius the effective solution resistance
10a
only increases in inverse proportion to the radius. Clearly, then, in
the absence of IR compensation the deleterious influence of solution
resistance will be dramatically reduced by employing micro- rather than
conventional electrodes. This advantage is, however, offset somewhat by
the IR compensation that is much more readily applied with larger
electrodes.
Figure 8 shows illustrative plots of log k®^(app) versus log k®j^(true)
obtained from simulated cyclic voltammograms for several conditions that
are typically encountered with microelectrodes. All curves refer to a
-1 -2
sweep rate of 1000 V s and a double-layer capacitance of 20 /iF cm .
(This sweep rate was chosen since comparable values are typically employed
with microelectrodes and it is sufficiently rapid so to avoid influences
from spherical diffusion even at the small electrodes considered here.)
The solid traces A-C refer to an electrode diameter of 1 #ira, having R^^
values of 1 X 10^, 7 x 10^, and A x 10^ ohm, respectively. These R^^
values correspond to solution specific resistances, p, of 20, 140, and 800
ohm cm, respectively, obtained from the relation R^ - p/4r where r is the
radius of a disk electrode, since in the absence of iR compensation R^^
will equal the solution resistance R^. These values were chosen since they
are appropriate for concentrated aqueous electrolytes,^^ and O.liJ
Q
electrolytes in acetonitrile and dichloroethane, respectively. While
there is reasonable agreement between k®^(app) and k®^(true) for curves A
and B at least up to k^^ - 10 cm s , indicating the virtual absence of R^^
15
effects up to this point, for curve C substantial deviations are observed
even for markedly smaller rate constants. This illustrates the need to
consider resistive effects when evaluating rapid values even with
extremely small microelectrodes when relatively high resistance media are
employed.
The solid curves E-F in Fig. 8 are the corresponding plots obtained
for the same electrolyte (i.e. same p values) and other conditions as in
curves A-C, but for an electrode diameter of 5 pm. This size is more
typically employed at the present time in microelectrode experiments
utilizing cyclic voltammetry.^^ While reasonable concordance between
k®b(app) and k®^(true) is seen for k®^^ < 10 cm s*^ in the least resistive
media (curve D), the results obtained for conditions corresponding to
typical nonaqueous media (curves E and F) are substantially inferior.
Indeed, the 5 pm electrode in dichloroethane shows that k®jj(app) «
k®b(true) at least for k®^(true) > 1 cm s*^; i.e. the AEp values are
determined almost entirely by solution resistance effects under these
conditions.
Figure 8 also contains a comparative log k®^(app) - log k®^(true)
trace obtained for the conditions of curve E, but utilizing ac rather than
cyclic voltammetry in the fashion prescribed above (dashed trace). As for
the behavior noted above for the larger size electrodes, the influence of
solution resistance upon the ac and cyclic voltammetric response is seen to
be similar. Comparable behavior was also obtained for the other conditions
considered in Fig. 8.
The foregoing considerations demonstrate that the deleterious
influence of uncompensated solution resistance imposes a severe and
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sometimes unexpected limitation on the magnitude of standard rate constants
that can be evaluated by means of cyclic voltammetry. In order to extract
meaningful values using this technique, it is clearly imperative to
obtain reliable estimates of for the measurement conditions employed,
and to demonstrate that is sufficiently small so that
approximates k®^(true). The observation of k®^(app) values that approach
the limiting value dictated by the known uncompensated resistance; i.e.
those corresponding to k®^(true) -► *, provides a clear signal that the
desired electrode kinetics are not measurable under the conditions
employed. Given a reliable knowledge of R^g, desired k^^^Ctrue) values can
still be evaluated in the intermediate case of partial kinetic-resistive
control from the k^^^Capp) values by using the appropriate k®^(app)-
kQ^(true) relation extracted from digital simulations.
Although the extent of the systematic errors in k^^^Capp) values
obtained by cyclic and ac voltammetry are surprisingly comparable for
typical R^g values, the latter technique would seem to exhibit clear
advantages for the evaluation of fast electrode kinetics under most
conditions. This stems both from the ability to better minimize and
evaluate R^^ with ac impedance measurements, and from the diagnosis of
dominant resistive effects from the observation of nonlinear (Ip/Ij) -
plots.
As has already been well documented,^® the use of cyclic voltammetry
with microelectrodes offers real advantages for evaluating standard rate
constants. However, even under these conditions the influence of cell
resistance can provide a serious Impediment to the evaluation of k®^ values
greatly in excess of 1 cm s*^, especially for high resistance media and in
the absence of IR compensation. It therefore would appear that values
evaluated by any of these approaches, especially for moderately fast
s *1
reactions > 0.1 cm s ) should be regarded with some skepticism in the
absence of due consideration of solution resistance effects by the
experimenter.
Acknowledgment
This work is supported by the Office of Naval Research.
18
References
1. (a) R. S. Nicholson, Anal. Chem. , 12 (1965), 1351; (b) R. S.
Nicholson and I. Shain, Anal. Chem., 2^ (1964), 706.
2. (a) J. M. Saveant and D. Tessier, J. Electroanal. Chem., S5 (1975),
57; (b) J. M. Saveant and D. Tessier, J. Electroanal. Chem., 21
(1977), 225; (c) K. B. Oldham, J. Electroanal. Chem., 21 (1976), 371;
(d) P. E. Whitson, H. W. VandenBoren, and D. H. Evans, Anal. Chem., 4^
(1973), 1298; (e) B. Aalstad and V. D. Parker, J. Electroanal. Chem.,
122 (1981), 183; (f) B. Speiser, Anal. Chem., 57 (1985), 1390.
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Advances", A. J. Bard, ed., M. Dekker, New York, Vol. 3, 1969, p. 199.
4. (a) R. S. Nicholson, Anal. Chem., 21 (1965), 667; (b) W. T. DeVries
and E. VanDalen, J. Electroanal. Chem., (1965), 183.
5. (a) D. Garreau and J.-M. Saveant, J. Electroanal. Chem., Si (1978),
63; (b) D. F. Milner, Ph.D. thesis, Purdue University, 1987.
6. (a) D. F. Milner and M. J. Weaver, J. Electroanal. Chem., in press;
(b) D. F. Milner and M. J. Weaver, J. Electroanal. Chem., 191 (1985),
411.
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8. V. D. Parker, "Electroanalytical Chemistry - A Series of Advances", A.
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Chem., 20. (1986), 2911.
19
11. T. Gennett and M. J. Weaver, Anal. Chem., ^ (1984), 1444.
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13. J. Nevnnan, J. Electrochem. Soc., 113 (1966), 501.
14. J. F. Chambers, J. Phys. Chem., 6Z (1958), 1136.
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Fife. 1
Comparison between a simulated current-potential curve for a linear-
sweep voltammogram (A) and the corresponding time derivative of the
potential across the double layer, dE^^/dt (B). Simulation conditions are:
2
reactant concentration - ImM; electrode area A - 0.2 cm , diffusion
coefficient D — 1 x 10'^ cm s*^; sweep rate i/ - 100 V s*^, uncompensated
.2
resistance - 50 0, double-layer capacitance, — 20 /iF cm , true
standard rate constant, k®^(true) - ®.
.2
Illustrative comparison of the simulated effects of uncompensated
solution resistance and finite electrode kinetics on cyclic voltammograms.
Solid trace is for - ImH, j/ - 20 V s*^, - 0, D - 1 x 10'^ cm s‘^,
2 s .1
A - 0.2 cm , - 0, and k^^Ctrue) - 0.04 cm s Squares are obtained
for the same conditions, but for R^^^ - 50 fl and k®jj(true) - ®.
F-U. ?
Illustrative comparison of the simulated effects of uncompensated
solution resistance (squares) and finite electrode kinetics (solid trace)
on cyclic voltammetric potential peak separations, dEp, as a function of
sweep rate, u. Simulation conditions as in Fig. 2.
21
ru. ^
Dependence of cyclic voltammetric potential peak separation, AEpi upon
reactant concentration in the presence of uncompensated solution
resistance, values are A, 50 fl; B, 20 0; C, 5 n. Other
s 19
simulation conditions are: (true) - «o, i/ - 20 V s , - 20 /iF cm ,
D - 1 X 10"^ cm^ s'^, A - 0.2 cm^.
I1&, ?
Plots of log k®^(app), where k®^(app) is the "apparent" rate constant
extracted from simulated cyclic voltammograms (solid traces) or ac
voltammograms (dashed traces) assuming R^^ — 0, against R^^ for various
values of the actual ("true") rate constant, k®^(true). Curves A, B, and C
are for kQ^j(true) - «, 1, and 0.1 cm s* , respectively. Other simulation
conditions are - ImJI, - 20 ptF cm*^, A - 0.2 cm^, D - 1 x 10’^ cm^
s The cyclic voltammetric sweep rate is 20 V s'^, and the ac impedance
data are for frequencies between 100 and 500 Hz (see text for further
details).
Fts. ^
As for Fig. 5, but for an electrode area, A, of 0.02 cm^.
E1&, 7
As for Fig. 6, but for a double-layer capacitance, of 2 /iF cm*^.
Fig. 8
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Illustrative relationships between k®^(app) and k®^(true) for some
typical conditions encountered with cyclic voltammetry using
microelectrodes. Curves A-C and D-F are for 1.0 and 5 fim diameter
electrodes, respectively. Key to solution resistance conditions: A,
Rys - 10^ n (p - 20 n cm); B, - 7 x 10^ 0 (p - 140 0 cm) ; C, -
4 X 10^ O (p - 800 n cm): D, R,^ - 2 x 10^ 0 (p - 20 0 cm); E, R.^ - 10^ n
(p - 140 n cm); F, R^^ - 7 x 10^ 0 (p - 800 n cm). Other simulation
conditions are - ImM, u - 1000 V s*^, - 20 pF cm'^, D - 1 x 10'^ cm^
s The dashed curve is from a simulation corresponding to curve E, but
obtained for ac impedance data taken between ac frequencies of 100-500 Hz.
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