1610
Anal. Chem. 1988, 60, 1610-1621
(12) SleVln, W. &aph/te Furnace AAS: A Source Book; Perkin-Elmer Corp., RldgefleM, CT, 1984. (13) Mlzblek, A. W.; Wlllls, R. J. Opt. Lett. 1981, 6 , 528. (14) Molnar. C. J.; Reeves, R. D.; Wlnefordner. J. D.; Glenn, M. T.; Ahlstrom, J. R.; Savory, J. Appl. Spectrosc. 1972, 26, 606. (15) Goforth, D. Ph.D. Dissertation, University of Florlda, Gainesville, FL, 1986. (18) Omenetto, N.; Wlnefordner, J. D. Appl. Spectrosc. 1972, 26, 555. (17) Omenetto, N.; Wlnefordner, J. D. f r o g . Anal. At. Spectrosc. 1979, 72, 1. (18) Travis, J. C.; Turk, G. C.; DeVoe, J. R.; Schenck, P.K.; Van DIjk, C. A. frog. Anal. At. Spectrosc. 1904. 7 , 199.
(19) Omenetto, N.; Smith, B. W.; Hart, L. P. Fresenlus’ Z . Anal. Chem. lB86. 683. ..., 324. . .-. (20) Magnusson, I.; Sjostrom, S.; Lejon, M.; Rubinsztein-Dunlop, H. Spectrochlm. Acta. Part B 1887. 428. 713. (21) Bolshov, M. A.; Zybin, A. V.; Koloshnikov, V. G.; Koshelev, K. N. Spectrochlm. Acta Part B 1977, 328, 279. (22) Omenetto, N.; Berthoud, T.; Cavalli, P.; Rossi, G. Appl. Spectrosc. 1985, 39, 500.
.
RECEIVED for review December 9, 1987. Accepted April 1, 1988. Research supported by NIH-lROlGM38434-01.
Fourier Spectrum of a Voltammetric Wave Jan C. Myland, Keith B. Oldham,* and Zhu Guoyi’ Department of Chemistry, Trent University, Peterborough, Ontario, Canada K9J 7B8
A voltammetric wave may be parameterlred, exactly or approxlnately, by three quantnies reflecting the height, position, and steepness of the voltammugram. Expresslons are derived, in terms of these three parameters, for the amplCtudes and phase angles of all the Fourier components of a sultabty modifled voltammetric wave. Modlflcatlon Is necessary to remove the dlscontlnuitles Introduced on periodlratlon. A continuous slgnal Is fhst treated, to wMch the more practlcal discrete case Is then shown to be equlvalent. I t Is demonstrated that almost all of the lnformatlon content of the slgnal resides In the fundamental skwsold ahd Its first hamronk. The properUes of these two sinusoids can be used to regenerate the wave or, more usefully, to “Fourler fit” the data; that Is, to generate the characteristic parameters of the wave without subjectlvity or graphical representatlon. Thus thls method of data analysis lends itself to automated vottammetrk assay vla a Fourler fitting algorlthm, such as the one presented here. Experiments are reported that verlfy the procedure for a varlety of electrochemlcal systems.
In electroanalytical chemistry, signals arising from low analyte concentrations are inevitably contaminated by noise. A recent publication (1) from this laboratory discussed a method of diminishing noise in a voltammetric signal by the sequence (a) Fourier transformation of the signal, (b) attenuation of the high-frequency components of the transform, and (c) inverse Fourier transformation. In order that step b in this procedure be performed intelligently, it is necessary that the amplitudes of frequencies present in the transform of the noise-free voltammogram be known. An object of the present article is to provide this information for voltammetric waves. We also show how to perform “Fourier fitting” of experimental data, that is parameterize a voltammetric wave in terms of the half-wave potential, effective electron number, and plateau height from its Fourier transforms. If one wants simply to determine these parameters, then in fact steps b and c above become redundant.
ANATOMY OF VOLTAMMETRIC WAVES Electrochemical techniques whose outputs give rise to voltammetric waves, when plotted versus potential or time, include hydrodynamicvoltammetry (21, classical polarography
(3), tast polarography (4),normal pulse polarography ( 5 ) , neopolarography (6) (or convolution voltammetry (7)), discretely convolved staircase voltammetry (8),and steady-state voltammetry a t inlaid disk microelectrodes (9). The output may be a steady-state current (2,9),an averaged current (3), a sample current ( 4 , 5 ) ,or a current processed by semiintegration (6, 7) or by some other convolution (8); In what follows, any such output will be represented by f. Voltammetric waves are engendered by a potential ramp that starts at an initial potential Ei and proceeds (continuously or discontinuously, depending on the technique) at a constant rate v to a final value Ef. If time from the start of the experiment is denoted by t and t f is the duration of the experiment, then the various potentials and times are interrelated by
Ei - -- Ef - v = - -Ei - E
- Ei - E I / ~
(1) tl/2 Here tll2 is the time to reach the half-wave pptential El,*, which is the potential at which the output f has a value midway between ita limiting values at very positive and very negative potentials. Figure 1shows the characteristic shape of a voltammetric wave. All such waves approach a constant value at extreme polarization; this value is determined solely by diffusion and is termed the diffusion-limited value fd. It is convenient to subdivide the voltammetric wave into three regions: the ”foot”, where f is close to zero; the-“escarpment”, where f rises steeply; and the “plateau”,where f is close to fd. Of course, there are no sharp divisions between these regions, but for our present purpose we may quantify “close to” to mean ”within 2% of”, so that the regions are demarkated. foot: 0 5 f 5 0.02fd
tf
t
Thermodynamics and/or electrode kinetics, in addition to diffusion, limit the value off in the escarpment region, and so influence the shape of the voltammetric wave (10, 11). Reversible waves are controlled by thermodynamics and diffusion but not by kinetics. Their shapes are independent of the particular electrochemical technique. In the case of a transient reduction, the Heyrovsky-Ilkovic equation (12)
*
Present address: Changchun I n s t i t u t e of Applied Chemistry, Acadernica Sinica, Changchun, Jilin, People’s Republic of China.
0003-2700/88/0380-1810$01.50/00 1988 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 00, NO. 15, AUGUST 1, 1988
1611
Table I. Parameters for Equation 12 That Describe, Exactly or Approximately, the Shapes of Voltammetric Waves electrode reaction
technique
reversible
any transient any steady state classical polarography tast polarography normal pulse polarography neopolarography steady-state voltammetry at inlaid disk microelectrodes
reversible
irreversible irreversible irreversible irreversible
irreversible
U
n n 1.04an 1.09an 1.22an 1.18an 0.94an
Elj z
E, + (RT/2nF) In ( D R / D o ) E, + (RT/nF) In (DRlDo) E, + (RT/2anF) In (5.34k,2r/D0) E, + (RTf2anF) In (1.82k:r/Do) E, + (RT/2anF) In (5.36k:r/Do) E, + (RT/PanF)In (1.524k,2RT/anFuDo) E, + (RT/2anF) In (0.536k,2?/DO2)
for an irreversible reduction, whereas under similar conditions neopolarograms have a shape described by (17)
I
where
/
I /
/ F / / /
/
/
01 /
t-
0 Figure 1. Characterlstlc shape of a voltammetric wave.
applies. For steady-state experiments the "114" in eq 3 should be replaced by "1/2" (9). All symbols are defined in Appendix A. Irreversible waves are controlled by electrode kinetics and diffusion. Their shapes do depend on the particular electrochemical technique adopted. For tast (instantaneous) polamgraphy,the problem of the expanding electrode was first solved by Koutecky (13). His expression for the irreversible wave may be written as two alternative infinite series
(14) where X = k , ( ~ / D o ) l /expi-anF(E ~
- E,)/RT')
(5)
The leading coefficients, y1 and So, of these series both equal unity and others are calculable via the recursion formulas
and
where B(,) signities the complete beta function (15). The same variable X and the same coefficients, y j and Sj, occur in the formulas
for the wave shape in classical (averaged) polarography. Normal pulse polarography obeys the equation (16)
f = a l / 2 f d ~exp(i2) erfc(x)
(9)
All of eq 4,8,9,and 10 are based on the Butler-Volmer model of electrode kinetics (18). Usually irreverisble voltammetric waves are less steep than reversible waves of equal electron number, though this is not necessarily so (11). Unlike the reversible wave, which possesses inversion symmetry about the half-wave point, each of the irreversible curves described by eq 4,8,9, or 10, is asymmetric in that the escarpment has a steeper slope above the half-wave point than at the corresponding point below. Nevertheless, the asymmetry is mild enough to escape casual notice, especially in the present context of noisy voltammetric waves. Accordingly we propose to replace each of eq 4,8,9,and 10 by an approximate equation of the form
where v and Ell2 are empirical parameters chosen to obtain the best match between (12) and the exact equation. Values of these parameters are already in the literature (19-21) for classical polarography and tast polarography, and they have recently been determined for normal pulse polarography and neopolarography (22) as well as for steady-state voltammetry at inlaid disk microelectrodes (9). AU these values are included in Table I, which also includes entries for the reversible case because eq 3 is itself a special case of (12). Thermodynamics, kinetics, and diffusion all conspire to control the shapes of quasi-reversible voltammetric waves. These waves are intermediate between reversible and irreversible. One therefore finds their shapes to be closely approximated by eq 12, with suitably chosen v and ElI2values. We shall not further pursue quasi-reversibility. When eq 12 is combined with the definition given in ( 2 ) of the escarpment region, one finds that this region spans a potential range of ( 4 R T I v F ) In ( 7 ) ,equal to 2 0 0 / u mV. In evaluating this voltage, we assumed a temperature of 25.0 OC; a similar assumption enters all subsequent voltage evaluations. There is, of course, no limit in eq 12 to the potential range encompassed by the voltammogram but in practice data are available only over the restricted range Ei 1 E 1 Ef.We shall impose the following stipulations on this range: (i) The entire escarpment region must be embraced; this means
ANALYTICAL CHEMISTRY, VOL. 60, NO. 15, AUGUST 1, 1988
1612
tf
Ef
E,
Figure 3. “Periodization”of the difference function, illustrated for curve I of Figure 2.
I
1 / 1 1 /
i
i
I
t-
0
I I
I
0
i
t
tf
to the initial and final signals fi and f p With use of eq 12, this can be recast as
”[
f = 2 1+
tanh {$(ti
- tl,&I -
Flgwe 2. Illustration of the significance of the a parameter. Curves I, 11, and 111 have a = x13, 0, and - x / 3 , respectively. The dashed line connects the end points of curve I.
(ii) Inequality 13 implies that u(Ei- Ef)cannot be less than (4RT/F) In (7),but we shall impose a more severe restriction, namely
For a reason that will be explained in the next ssction, it is advantageousto Fourier transform the difference f - f instead off itself. We shall represent the f - f difference by f.
u(Ei -Ef) 1 (8RT/F) In (7) = 400 mV
FOURIER SERIES OF A VOLTAMMETRIC WAVE In its usual modern sense, Fourier transformation means transforming a finite set of data points. To understand this discrete case, it is advantageous first to consider the Fourier transformation of a continuous periodic function or, as the operation is more usually described, to find the Fourier series that corresponds to such a function. The Fourier series encodes information on the shape of the function as an infinite set of cosine and sine coefficients co, sl, cl, s2,c2, s3, ... (also known as “real” and “imaginary” amplitudes). We wish to find t_he Fourier teries corresponding to the difference function f - f, where f is given by eq 12 and the linear function f is given by (21). This difference function f exists only in the time range 0 It Itf and is not periodic. However, the Fourier operation treats f as if it were periodic. That is, replicas off are assumed to exist in the time ranges tf I t I2tf, -tf 5 t I0, 2tf It I3tf, -2tf I t I -tf, 3tf 5 t I4tf, etc. as illustrated in Figure 3. When “periodized” in this way, the f function contains nope of the discontinuities that would have been present had f alone been periodized. It is to avoid such discontinuities that f is subtracted. The cosine and sine Fourier coefficients off are given by the definite integrals
(14)
on the total potential range scanned by the voltammogram. In words, this means that the total range must be at least twice the width of the escarpment region. (iii) We further stipulate (15) which in words means that the half-wave potential lies somewhere within the middle one-third of the overall potential range. If we insist on meeting (14) and (15), inequality 13 becomes redundant and will be dispensed with. It is convenient to define two parameters, as follows: u = r ( t f - 2t,jz)/tf = ~ ( 2 E 1 /-2 Ei
- E f ) / ( E i- E,)
b = - 2a2RT - 2n2RT -- 507 mV UFUtf u ( E -~ E J F u ( E -~ E,)
(16) (17)
The a parameter specifies the asymmetry of the voltammetric wave. It is zero if the half-wave potential occurs in the center of the potential range scanned, so that the foot and the plateau are of equal widths, as in curve I1 of Figure 2; it is positive if El12is encountered earlier than tf/2, so that the wave displays more plateau than foot, as in curve I; it is negative for the situation shown as curve I11 in Figure 2. Stipulation 15 limits the asymmetry to the range -1.047
-x N
- Iu 3
x
I-
3
N
1.047
(18)
The b parameter reflects the total width of the voltammetric scan; it is large for a narrow scan and small for a wide scan. Stipulation 14 places the limit
on the permissible values of the b parameter. Figures 1 and 2 each show a diagonal line connecting the first and last points of the voltammogram. The equation of this straight line is
7 = Ti + t(ff - f i ) / tf
(20)
where f i and ff are the end points of the line, usually equal
The f function may be regenerated unchanged by the Fourier inversion procedure
or it may be processed in various ways if c k and s k are suitably modified between the transformationand inversion operations. Exact expressions for c k and sk, as functions of the parameters fd, a, and b, are derived in Appendix B. Evident from the exact equations B.26 and B.27 is that the values of c k and s k are affected by the values of each of the three parameters. The effect of a may be largely decoupled
ANALYTICAL CHEMISTRY, VOL. 60, NO. 15, AUGUST 1, 1988
E,
E, I
I
I
1613
We summarize the conclusions of this section and its attendant appendices as follows. After subtraction of a linear function joining the ends of the wave, the Fourier spectrum of a voltammetric wave is fully described by the cosine and sine coefficients Ck
'v
(-)k(fdb/~)csch ( k b ) sin (ka)
k = 1, 2, 3, ..., K (33)
Sk
(-)k(fdb/r) csch (kb)
COS
(ka)
k = 1, 2, 3, ..., K (34)
even though the individual approximations in (33) and (34) are occasionally invalid. The co coefficient is given by approximation 28, but note that it is also found from (33) by treating k as continuous and taking the k 0 limit. More useful than c k and s k are the equivalent parameters Pk and @k, given by definitions 25 and 26 and approximations 29 and 30, the latter pair being always valid.
-
Flguro 4. Comparison of an exact voltammetric wave (full line) with that calculated as explalned in the text (dotted line). Worst case values of the a and b parameters (a = d 3 , b = s2/[4 In (7)])have been assumed.
by forming the square root of the sum of squares of the coefficients (ck2
+ 8k2)lI2= P k
k = 1, 2, 3, ...
(25)
and the inverse tangent of their quotient Arctan (Ck/Sk) =
@k
k = 1, 2, 3, ...
(26)
We shall refer to the quantities defined in the last two equations as the amplitude and phase angle of the kth Fourier component. In terms of these quantities, the inversion formula (eq 24) for regenerating the wave-minus-straight-linefunction may be rewritten as
where Po,the amplitude of the zeroth-order Fourier component, equals cg. As Appendix C demonstrates, the simple formula
PO = CO
m
afd/r
(28)
is a valid approximation, as are
pk
N
(fdb/a) csch ( k b ) @k
ka
k = 1, 2, 3, ..., K (29)
k = I, 2, 3, ..., K
(30)
where
K = int(7/b)
(31)
provided that stipulations 18 and 19 are met. Moreover, the summation 27 converges fast enough that f is accurately reproduced by terms up to k = K. That is
Though Appendix C validates approximations 28,29, and 30, a more impressive confirmation is provided by Figure 4. The solid line in this diagram is an exact voltammetric wave corresponding to the worst case permitted by stipulations 18 and 19. The dotted line in the figure was computed by (a) calculating th_eamplitudes and phase angles of the difference function f = f - f by eq 28-30, (b) performing an inversion by means of (32), and (c) adding the straight line f to the recovered f to regenerate the wave. The near-perfect agreement of the two curves shows that the approximations are more than adequate.
DISCRETE FOURIER TRANSFORMATION Here we consider the application of Fourier transforma;tip to a finite !et (:ever less than 18) of voltammetric data fo, f l , fit ...,f j , ...,fN-1, f N with equal time intervals At between points. The transformation treats the N + 1 data as repetitive with period NAt and therefore the continuous function f = f - f that occurred in the precfding section is to be replaced by a set of discrete points f j = f j - fo - jf, - f o ) / N .The end points, fo and f, of the subtractive _linear_array are set equal to the first and last signal values, f o and f N , of the voltammogram [or, perhaps, to slightly different values that are more typical of the experimental values at the start of the foot and the end of the plateau]. To delineate the shape of the voltammetric wave adequately, we shall stipulate that at least nine points must lie on the escarpment. This is equivalent to specifyingthat the voltage increment between successive points obeys the inequality
uAt =
AE 5 (24 mV)/u
(35)
and that the total number of points is determined by
N I21/b
(36)
There is a slight advantage in choosing N to be a prime number. This ensures that trigonometric arguments equal to multiples of 7r/2 are never encountered in eq 37 and 38. The integrals that were used to evaluate the Fourier coefficients in the indiscrete case are now replaced by summations that may be regarded as trapezoidal approximations to the Fourier integrals. To emphasize that the discrete Fourier coefficients, which are more often referred to as the Fourier transforms, depend on the number of data points, we shall use the notations ck(N and sk(N. The formulas for the transformation are
and
("1 ')
k = 1, 2, 3, ...,int - (38) Unlike the ck, s k coefficients, there is only a finite set of distinct ck(N) and Sk(M transforms, N in all, because of aliasing (23).
1614
ANALYTICAL CHEMISTRY, VOL. 60, NO. 15, AUGUST 1, 1988
The inversion formula for the discrete case
Table 11. Comparison of Exact Fourier Amplitudes and Phase Angles with Those Calculated by the Approximations C.2 and C.3, for Typical Values, B = 0.524 and b = 0.951, of the Parameters pk/ f d
k 1 2
3
closely parallels eq 24. The term ( - ) j C N / 2 " / 2 is absent if N is odd. In principle, one could determine the transforms of a digitized voltammetric wave directly from summations 37 and 38. It is easier, however, to utilize the relationships
4 5 6 7 8 9
10 11 12
m
Ck(M
= ck
+ m=l
+ CmN+k
CmN-k
k = 0 , 1, 2,
(3
..., int -
13 14 15
(40)
*k
exact
approx
exact
approx
0.27500 0.09241 0.03504 0.01349 0.00521 0.00202 0.00078 0.00030 0.00011 0.00005 0.00002 0.00001 0.00000 0.00000 0.00000
0.27495 0.09243 0.03503 0.01350 0.00521 0.00201 0.00078 0.00030 0.00012 0.00004 0.00002 0.00001 0.00000 0.00000
0.524 1.048 1.572 2.097 2.619 3.146 3.666 4.193 4.723 5.212 5.846 6.109 7.026 6.299 7.702
0.524 1.048 1.572 2.096 2.620 3.144 3.668 4.192 4.716 5.240 5.764 6.288 6.812 7.336 7.860
0.00000
All these results require that stipulations 18 and 19 be met. Restatements of these stipulations are k = 1, 2, 3, ..., int( N 1 -1 ) (41) (23)between the discrete and indiscrete coefficients. Appendix D is devoted to evaluating these summations, using the continuous c and s coefficients given in eq 28, 33, and 34. In analogy with eq 25 and 26, we define
Pk"
=
([Ck'"]'
+ [sk(']2)1/2
k = 1, 2, 3,
(42)
and @k(m
= Arctan
(Ck(m/Sk(M)
k = 1, 2, 3, ...
(43)
because these are more useful than the transforms themselves. Equation 40 shows that k may assume values up to N / 2 . However, we shall be more restrictive and take interest only in k values that do not exceed K , as given by (31). Under these circumstances it is demonstrated in Appendix D that the error in Pk(m
bfd
- csch ( k b ) 7r
k = 1, 2, 3,
...,K
N
k = 1, 2, 3, ..., K
ka
(45)
to within an angle of radians, and that the zeroth-order Fourier cosine transform is c0(N N co
N
afd/7r
(46)
In summary, we conclude that, provided N is not less than 21/b and k is not more than K, the discrete Fourier amplitudes P k ( M and phase angles * k ( N ) , as well as the zeroth Fourier coefficient cJN, are equal to their continuous counterparts Pk, + k , and c,, to within a precision compatible with experimental reproducibility. These conclusions were based on results from the previous section of this article which themselves were validated only for k values not exceeding K = int(7/b). Hence we restate our final results as
PJN = co('") Pk(m
= ([Ck'']2
1 '1
afd/i?
N 2 21/b
+ [Sk(m]2)1/2
= Arctan
(ck(m/sk(m]
(47)
bfd
- csch (kb) (48) i?
E
ka
-1 a -
