Anal. Chem. 1988, 60,62-66
62
Discrete Convolution of Staircase Voltammograms Jan C. Myland* and Keith B. Oldham Trent University, Peterborough, Ontario, Canada
Mgllalstalrcase voltanmetry generates a sequence of current data at equally spaced potentlais. Such data is of llmited value because it depends, In a complex fashion, on measurement thw and step slze. An exact technlque ls described by which reversible stalrcase data may be transformed Into a wave-shsped response resembling a neopolarogram. DHferentlath leads to a readout analogous to a dmerentlal pulse polarogram. Experlments confirm predtctions for both displays. Analytical advantages of this technlque are suggested by Its rapidity and Its excellent dlscrhnlnatlon against charglng current.
Because the potential E has the constant value Ej= Ei -
jAE during the interval 0' - 1)At < t < jAt, m has the constant value mj during the same interval, with m , = 3 [ l - t a n h { m ( E nF j-Eh) 2
1 An applied potential in the form of a staircase ( I ) , as illustrated in Figure l, has two decided advantages over the classical ramp signal used in linear sweep voltammetry and cyclic voltammetry. First, if the current readout is made as late as possible in the "tread" of the staircase, the (nonfaradaic current)/ (faradaic current) ratio is markedly reduced. Second, because it can be generated digitally, the potential signal can be applied very accurately and reproducibly. However, even for reversible reactions at planar electrodes, the current-versus-potential readout from a staircase is unattractively asymmetric and depends in a complex fashion on such parameters as AE, the step height (I), and +, the ratio of +At, the readout delay, to At, the step length (2). As AI3 0 and 1, the staircase voltammogram approaches a linear-scan voltammogram, but one needs to be very close to these limits in order to reproduce linear-scan voltammetry within experimental error (3). Stefani and Seeber ( 4 ) have 0 restriction may be removed. Subshown how the AE sequent to the submission of this article two papers on staircase voltammetry have been published (5, 6). In this article we shall present a procedure for converting a staircase voltammogram into a signal-versus-potentialgraph that has the form of a classical polarogram (or its derivative) and that relates closely to surface concentration data. The form of the graphs is totally independent of the AE,At, and parameters. Our theoretical treatment is restricted to the reversible reaction
-
+
-
-
+
O(so1n)
+ ne
-
R(so1n or amal)
(1)
occuring at a planar electrode, with R initially absent.
Eh = Eo' + (RT/2nF) In (DR/Do)
(2)
All symbols have their usual significances and are, as well, defined in Appendix A. Under the prescribed conditions, it is known (7) that the faradaic semiintegral m is a function only of the electrode potential E, the functional form also expressing the surface concentration excursions, thus
+ tanh
nFmti
- h))]
(4)
2RT
where md is an abbreviation for nAFcobDo'J2and h = (Ei-
Eh)/a. The semiintegral m suffers a discontinuity at times t = jAt, where j = 1,2 , 3 , .... Consider the effect of the discontinuity at the instant t = (J- 1)At on the ensuing current. Then, by the laws of the fractional calculus (8)the semiderivative i of m after t = (J- 1)At is given by
( J - 1 ) A t < t < JAt Here iJ-'(t) is the functional form that the current had during (J- 2)At < t < (J- 1)At (and would have continued to have subsequent to ( J - 1 ) A t had not a discontinuity occurred at that instant). Figure 2 will clarify this. Now, because mJ mJdl is a constant, its semiderivative is easily expressed, so that
( J - 1 ) A t < t < JAt The term iJ-'(t) in eq 6 may likewise be expressed in terms of iJ-*(t) and a term to account for the perturbation in current that occurred at the instant t = ( J - 2)At; thus iJ(t) = iJ-dt)
THEORY The signal applied to the working electrode has the form depicted in Figure 1. The staircase commences at a potential Ei sufficiently positive that the faradaic current is negligible; in practice, it will suffice if there are about 5RT/(nFAE) steps prior to Eh, the half-wave potential, given by
=
+
mJ-l
- mJ-2
[ ~ (- tJAt
+ 2At)l1/'
J+ [ ~ (mtJAt - + At)]'/' mJ-l
(7)
and this procedure may be repeated indefinitely until one arrives at
(J- 1 ) A t < t < J A t Unlike eq 6 and 7, there is no leading current term in eq 8. This is because io is negligible. In practical staircase voltammetry, the current is not measured continuously during the interval following a step but (usually) only at a single instant t = (J - 1 + @ ) A tas
0 1987 American Chemical Society 0003-2700/8S/0360-0062$01.50/0
ANALYTICAL CHEMISTRY, VOL. 60, NO. 1, JANUARY 1, 1988 current ,measurement
Table I. Weighting Factors for I$ = 1
i
1.000 000 000 0 0.292 893218 8 0.215 542 949 6 0.178 486 1400 0.155 687 255 2 0.139 857 870 4 0.128045 036 4 0.1187947890 0.111 297 207 8 0.105 060 6718 0.099 767 207 6
3 4 5 6 7 8 9 10
J'A t
0 1 2
3 4 5 6 7 8
Figure 2. Nomenclature of the experimental current and its semi-
integral. illustrated in Figure 1. If we denote this measured datum by iJ, then from eq 8 mj-mi-l
(9)
+ 4 - j)'/'
If this is written as a difference of two summations and the summation index is redefined in the second sum, eq 9 may be developed into ZJ
=
mJ (T4At)V2 J-1
i
0.070 859576 6 0.057 942 507 4 0.050 216 974 1 0.044 935465 7 0.041 032 485 4 0.037 996 7754 0.035 548396 8 0.033 519 507 4 0.031 802 5530 0.014 232 705 5 0.010 064 942 4
20
30 40 50 60 70 80 90 100 500 1000
4
i
5
wj
Table 11. Weighting Factors for Other Values of 6
Figure 1. Diagram illustrating the applied staircase waveform. -
(TAt)ll2j=1 ( J
i
wj
0 1 2
L
iJ = -
63
9 10 20 30 40 50 60 70 80 90 100
lo4
0.1
0.5
0.9
0.010 00 0.009 90 0.009 83 0.009 77 0.009 73 0.009 68 0.009 64 0.009 61 0.009 57 0.009 54 0.009 51 0.009 28 0.009 10 0.008 96 0.008 84 0.008 73 0.008 63 0.00854 0.008 45 0.008 37
0.316 23 0.220 88 0.18062 0.156 77 0.14049 0.12844 0.11906 0.11148 0.10520 0.099 87 0.095 28 0.069 17 0.057 01 0.049 60 0.044 49 0.040 69 0.037 73 0.035 33 0.033 33 0.031 64
0.707 11 0.298 86 0.21833 0.180 14 0.15680 0.14067 0.12868 0.11930 0.11171 0.105 41 0,10007 0.070 97 0.058 00 0.050 26 0.044 96 0.04105 0.03801 0.03556 0.03353 0.031 81
0.948 68 0.295 75 0.21663 0.179 09 0.15609 0.140 14 0.12826 0.11897 0.11144 0.105 18 0.099 87 0.070 90 0.057 96 0.050 23 0.044 94 0.041 04 0.038 06 0.035 55 0.033 52 0.031 81
This equation expresses the semiintegral as a weighted sum, w& wliJ-l w2iJ-p ... wJ-lil,of the currents. Accurate values of a selection of the weights in eq 15 are presented as Table I. They were generated as the 4 = 1 instance of the general recursive algorithm
+
+
+ +
i k=l
Equation 10 provides a method of calculating iJ from mJ, mJ+ mJ-2,.... However, we seek a procedure by which mJ might be determined from iJ,iJ+ iJ-2,.... To demonstrate how the latter procedure may be implemented, let us consider the #I = 1 example and write eq 10 in the rounded numerical form
i., = mJ - 0.293mJ-, - 0.130mJ-2 - 0.077rnJ-, - 0.053mJ-4 - ... (11))
j = 1, 2, 3, ... (16) Some weights for nonunity values of the fraction 4 are included in Table 11. Notice that the weights rapidly become &independent and approach l / ( r j 1 / 2as) j becomes large. Once the weights appropriate to a particular fractional value of 4 are known, it is a simple matter to calculate exact values of the semiintegral by J- 1
Similarly
mj = (nAt)1/2E
(TAt)1/2ij-,= mJ-l - 0.293mJ-2 - 0.130mJ_, - 0.077mJ-4 (TAt)'/'i~-2=
- 0.293mJ-, - 0.130mJ-, - ...
312~4
(TAt)'/2i~-3 = mj-3 - 0.293mJ-, - ...
... (12) (13) (14)
etc. If the above equations are suitably weighted and added, one obtains
(iJ + 0.293iJ-1 + 0.216iJ-2 + 0.178iJ-3 + 0.156ij-4
+ ...)
= mj (15)
j=O
J Wlij-J
= (TAt)lI2CWj-liJ J=1
(17)
from the measured current samples. The analogy between summation 17 and classical convolution leads us to refer to the procedure of obtaining mJ from iJ,iJ+ iJ-2,... as "discrete convolution". Notice that both w and i depend on 4, but that their convolution removes this dependence, m being independent of the readout fraction, 9. It should be emphasized that the above derivation in no way depends on the step size AE being small. The only penalty paid for using a large AE is that one then has fewer
64
ANALYTICAL CHEMISTRY, VOL. 60, NO. 1, JANUARY 1, 1988
Table 111. Experimental Conditions expt
analyte
potential span, mV
1 2 3 4 5
Pb(I1) Pb(I1) Pb(I1) Pb(I1) Pb(I1) Pb(I1) + Cd(I1)
-2OO+-500 -200--500 -200--500 -2OO+-500 -200 -500 -200 -800
6
-
+
AE,mV 10
10 10 10 2 2
At, s
@
0.2 0.2
0.2495 0.4995 0.7495 0.9995 0.9975 0.9950
0.2
0.2 0.04 0.02
points on an mJ versus E j graph. When an experimental staircase voltammogram is discretely convolved to generate a neopolarogram (an m versus E plot) a degree of smoothing is introduced. If more noise removal is required, a technique such as Fourier smoothing (9) or Savitzky-Golay smoothing (10-12) may be applied. Alternatively, these techniques may be used to differentiate the neopolarogram, to produce a derivative curve (akin to a differential pulse polarogram) that is more easily quantified. We make use of Savitzky-Golay differentiation in the Experimental Section of this article. The Savitzky-Golay method permits one to differentiate the neopolarogram accurately, even though the semiintegral is known only at a set of evenly spaced potentials. The equation of the derivative neopolarogram, found by differentiating eq 3, is
e = -dm -dt
dm-- -u- dm _-AE -_ At
dE
I
1
1
200
I
I
300
I
400
500
-E/mV Figure 3. Staircase voltammograms for the first five experiments: 0 = experiment 1, X = experiment 2, A = experiment 3, 0 = experiment 4, and 0 = experiment 5. See Table I11 for experimental
conditions.
-, u, N
A
3
a
=L \
- E,)) (18)
This equation describes a peaked curve, the peak height e,, equaling nFvmdf (4RT) and the peak potential being E,. The width of the peak at half height is
(&)
i
