Influence of Spherical Diffusion in Second Harmonic A.C. Polarography and Related Techniques SIR: The measurement of second harmonic currents and other "higherorder" current components-e.g., intermodulation and d.c. rectification components-arising from nonlinearity of the faradaic impedance under a.c. polarographic conditions has been alleged t o represent a potentially effective approach to studies of kinetics and mechanisms of electrode reactions (2-6, 7 , 9, 18-20, 22, 23). Unfortunately, application of these techniques has enjoyed only sporadic interest. Among factors seemingly dictating against widespread acceptance of these methods has been an over-abundance of apparently anomalous experimental results which show marked deviations from predictions of available theory and/or experimental results of other methods (3, 9, 12, 17, 18, 20). I n some cases, plausible qualitative explanations have been put forth to account for the anomalies (3, 18, 20). These explanations invoke mechanisms involving kinetic influence by processes (adsorption, coupled chemical reactions) not considered in the simple theory (3,9,18,22) which accounts only for rate control by linear diffusive mass transfer and/or heterogeneous charge transfer. To our knowledge, no one has suggested that the explanation for some of these anomalies might be found in the more subtle perturbations associated with curvature and motion of the surface of the dropping mercury electrode which was used in most of these experiments. Recently, it has been demonstrated theoretically that spherical diffusion can have a significant influence on fundamental harmonic ax. polarographic waves of systems in which the reduced form of the electroactive redox couple is in the amalgam state (11). In the present communication, theoretical and experimental evidence is presented to show that the same effect is associated with a x . polarographic waves of higherorder current components. It is shown that the effect of spherical diffusion serves as a reasonable basis for the explanation of a t least some of the anomalous results which have been reported. Because the present discussion is designed merely to demonstrate the existence of a contribution of spherical diffusion, specific consideration is confined to systems in which diffusion is the sole rate-determining step (the reversible case), However, extrapolation of the concepts to more complicated kinetic schemes is considered briefly.
THEORY
d=0,1,2 3
Theory for systems exhibiting rate control by diffusion to a stationary spherical electrode will be examined to demonstrate the theoretical basis for a significant spherical diffusion contribution, The stationary sphere electrode model yields equations which must be considered approximate for data obtained with the dropping mercury electrode because the contribution due to motion of the electrode surface relative to the solution (drop growth) is neglected as are the more insidious perturbations due to shielding (15, 16), depletion (1.9, 14),and streaming (I,%$). The equations presented may be considered exact for a well designed experiment involving a fixed-potential measurement with a hanging mercury drop. Other basic assumptions inherent in the theory and not.ation definitions have been given elsewhere (10, 11). Two basic cases are considered: (A) both forms of the redox couple soluble in the solution phase; (B) the reduced form soluble in the electrode, as in amalgam formation. The theory for second harmonic, third harmonic, and
j
=
e-i(sinwt)P P!
-
ro-1(D/w)1/2
sin (2wt
ir
- -) 4
(10)
where
[I
- exp (a2t)erfc ( ~ t ~ / ~(11) ) ]
and
It should be noted that Equation 10 differs from the corresponding planar diffusion result for the diffusion-controlled second harmonic wave by the factor F ( t ) (21,22). In other words, for planar diffusion (ro + a )
F(t)
e-i(sinwt)d X d!
(3)
The expression for the small amplitude ( A E 5 S/n mv.) second harmonic a x . polarographic current is obtained from the solution of Equation 1 for p = 2. Following the usual procedures and recognizing that for normal experimental conditions
(.;)
5
P
nF - ( E d . c . - Ri/z')
16 R2T2cosh3
intermodulation polarograms will be considered. Theoretical expressions will be presented without derivation as the course of derivation is essentially identical to that described in obtaining expressions for the fundamental harmonic (10, 11). A. Theory for the Second Harmonic A.C. Polarogram with Rate Control by Diffusion to a Stationary Spherical Electrode and Both Redox Forms Soluble in Solution. It has been shown previously (10, 11)) that for this case the infinite set of integral equations describing all current components flowing under a.c. polarographic conditions takes the form
-1
RT
~ 2 n 3 F 8 A C ~ * ( w D o ) 1 / 2 A E Zsinh F(t) I(2wt) =
.......
=
1
(13)
Thus, effects of spherical diffusion manifested in Equation 10 obviously are confined to the function, F ( t ) . This result exactly parallels the observed theoretical difference in the fundamental harmonic expressions for the diffusioncontrolled wave based on planar and spherical diffusion models (10, 11). I n fact, the F ( t ) function in Equation 10 is VOL 38, NO. 1 1 , OCTOBER 1966
1615
identical to the corresponding function arising in the fundamental harmonic wave equation (11) indicating that spherical diffusion contributions to the fundamental and second harmonic a.c. polarograms share a common origin. B. Theory for the Second Harmonic A.C. Polarogram with Rate Control by DifIusion to a Stationary Spherical Electrode and the Reduced Form Soluble in the Electrode. For this case the set of integral equations describing all current components differs from Equation 1 only by the definition of J R ( ~which ) takes the form (11) JR(t)
=
( D R ~ o - 'X ~) [erfc (D~'/'~o-'t'/2)- 21 (14)
D~1/270-1exp
Solution of the integral equation for p = 2 yields an expression for the small amplitude second harmonic current which is identical to Equation 10 except for the definition of F ( t ) . In this case one obtains (11)
[I
- exp (b2t)erfc (bt1/2)] (15)
where
b=
eiDoU2
rdl
- DR1/2
+ e?
(17)
where the function, F(t), is defined by Equation 11 for both forms soluble in solution and by Equation 15 for the reduced form soluble in the electrode. The planar diffusion analog of Equation 17 corresponds to a unity value of F ( t ) . Thus, the spherical diffusion contribution incorporated in the function F(t) alters the third harmonic polarogram in a manner which is identical to its influence on the lower order harmonics for 1616
ANALYTICAL CHEMISTRY
AE(sin w l t
+ sin wzt)
(18)
By employing Equation 18 in place of the usual applied potential expression, the intermodulation component may be obtained through the usual approach (10, 11). The small-amplitude current component a t the difference frequency is found in the solution of the integral equation for p = 2. One can show that this. component may be written
RESULTS AND DISCUSSION
The foregoing theoretical results together with previous work (11) indicate that all alternating current components arising from a diffusion-controlled process are influenced by spherical diffusion in precisely the same manner. The effect is incorporated in a time-dependent function, F ( t ) , which arises in the equations as a multiplying factor, that is, F ( t ) represents the ratio of theoretical predictions based on the stationary sphere model to those based on the stationary plane model. The form of F(t) depends only on whether the reduced form of the electroactive redox couple is soluble in the solution or electrode phase and not on the alternating current component in question. These observations indicate that the rationalization given for the influence of spherical diffusion on the reversible fundamental harmonic wave (11) is directly applicable to the higher-order current components considered here. As was pointed out (11), the spherical diffusion contribution originates in the d.c. process where spherical diffusion induces a deviation of the mean (d.c.) surface concentration components from the planar diffusion values. This deviation is represented in relative units by the function, F(t).
(16)
The remarks following Equations 9-11 in the previous section are also applicable to the present case. C. Theory for the Third Harmonic A.C. Polarogram with Rate Control by DifIusion to a Stationary Spherical Electrode : Cases A and B. Because of reported anomalies found in third harmonic a x . polarograms of Cd(I1) (8,18,19),it is of interest to examine the theory for the spherical diffusion contribution to this current component, which is defined by Equation 1 for p = 3 (small amplitudes). Solution of this equation yields an expression for the small-amplitude third harmonic alternating current which may be given in the form
x sin (3wt + r/4)
both Cases A and B. One can readily show that a similar result will arise in the theory for fourth and higher harmonic current components with diffusioncontrolled systems. D. Theory for the Intermodulation Polarogram with Rate Control by Diffusion to a Stationary Spherical Electrode: Cases A and B. In the usual form of intermodulation polarography (18, 20) two sinusoidal signals of different frequency are applied to the polarographic cell and the current component corresponding to the difference frequency is measured. If one assumes that the two applied sinusoidal signals are of equal amplitude, the applied potential function in intermodulation polarography may be written
L
' J
where F ( t ) is defined by Equation 11 for both forms soluble in solution and by Equation 15 for the reduced form soluble in the electrode. As in the cases of the current components discussed above, the expression for the intermodulation component given here differs from the corresponding planar diffusion expression (18) by the factor, F ( t ) . It should be noted that, as Paynter pointed out (18), Barker's r.f. polarography (2-4) may be considered a special r
form of intermodulation polarography and the basic features of the theoretical predictions for intermodulation and r.f. polarography are essentially identical. EXPERIMENTAL
Experimental data presented here were obtained with the aid of apparatus and measurement procedures which have been described in detail elsewhere (8). The usual precautions were exercised in selection of reagents and solution preparation (8).
It should be recognized that the predicted effect of spherical diffusion on the higher-order current components does not represent an influence on the degree of nonlinearity of the faradaic impedance with diffusion-controlled systems. The ratio of the magnitude of any higherorder current component to the fundamental harmonic current magnitude is the same for spherical diffusion and planar diffusion theory [the F ( t ) factor cancels in taking the ratio]. Thus, the relative nonlinearity of the faradaic impedance is unaltered by spherical diffusion. The effect of spherical diffusion on higher-order current components simply manifests an overall change in the magnitude of the faradaic impedance due to the influence of spherical diffusion on the magnitudes of the mean surface concentrations. Theoretical polarograms calculated on the basis of equations presented here exhibit a number of interesting features. As was pointed out previously (11), the F ( t ) function applicable to the case of both forms soluble in solution (Equation 11) deviates negligibly from unity except when Do and DR differ appreciably. For example, the theory predicts that spherical diffusion will alter peak magni-
-150
-130 -110
-90
-70
-50
-30 EDC
-10
10
- REV E+ (rnv.)
30
50
70
90
110
130
150
Figure 1. Calculated second harmonic a. c. polarogramt showing influence of spherical diffusion with both redox forms soluble in solution ro = 0.0528 Colcs. applicable to a diffusion-controlled procers where T = 25' C., n = 1.00, Co* = 1.00 X cm., A€ = 5.00 X 1O-'volt,o = 40s radions sec.-I, Do = 2.50 X 10- cm.* set.-? DR = 1.00 X 1 0-6 cm.l I = Coicd. from plonor diffusion model; any drop life 2 = Calcd. from stationary rphere model; drop life = 3 seconds 3 = Calcd. from stationary sphere model; drop life = 9 seconds. Abricisra = Ed.o. -E+' in mv. Ordinate = recond harmonic current In pa. X 1 O1 Curves ore reproductionsof CDC 3400 computer readout via digital Incremental plotter (8)
tudes of second harmonic or intermodulation polarograms by only 0.09% and 0.3% a t the first and second peaks, respectively, for such a system when = 1, T = 25' C., ro = 0.0528 cm., Do = 4.10 X 10-6cm. sec.-l, DR = 4.50 X 10-6 cm. sec.-1 (DR/Do = 1.10). This case involving a 10% difference in diffusion coefficients might be considered in the neighborhood of what one would typically encounter in a reversible process involving a metal ion-metal ion system. Of course, an experimentally significant spherical diffusion contribution is predicted for sufficiently large disparities in Do and DR as shown in Figure 1. For such a system, spherical diffusion introduces a time-dependence (mercury column height dependence) and an asymmetry into the higher harmonic polarograms; two effects which represent rather profound departures from predictions of planar diffusion theory. Calculations such as these suggest that applications of the classical planar diffusion theory to higher harmonic polarograms of systems characterized by solubility of both redox forms in solution is admissable except for the rare case where Do and DR differ excessively. A decidedly different situation is predicted for systems involving solubility of the reduced form in the electrode. Calculations for this case yield F(t) (Equa-
tion 15) values which deviate appreciably from unity for any realistic combination of n, Do, DR,t , ro, etc. ( 1 1 ) . As a result, the predicted polarograms of higher-order current components are characterized by a marked time-dependence and asymmetry as attested by Figures 2 and 3A. The asymmetry associated with the larger cathodic peak, the rather large time-dependence on the cathodic side of the wave and the overall enhancement in the magnitude of the wave illustrated in Figures 2 and 3A are predicted to be general characteristics of diffusion-controlled second harmonic and intermodulation polarograms of amalgam-forming systems. It is interesting to note that certain predictions of planar diffusion theory remain valid even when the contribution of spherical diffusion is predicted to be substantial. Calculations show that the peak separation in second harmonic, third harmonic , and intermodulation .polarograms is predicted to be essentially equal to the value predicted by planar diffusion theory (19, $1, 28). Likewise, the frequency-dependence of higher harmonic currents arising from diffusion-controlled systems is unaltered by spherical diffusion as is apparent from inspection of the above equations. Although the theory presented here has been limited to the simple case of diffusion-controlled or reversible sys-
tems, it carries certain obvious implications of importance for systems exhibiting more complicated kinetic schemes. The case of combined rate control by diffusion and heterogeneous charge transfer (the quasi-reversible case) represents a noteworthy example as the literature is abundant in theoretical and experimental studies of such systems, including investigations of higher-order current components (2-4, 6, 9, 18, 22). Relevant theory has been limited to the stationary plane electrode model. Predictions based on planar diffusion theory suggest that an asymmetry will be observed in polarograms of higher-order current components only when substantial charge transfer kinetic influence is combined with a charge transfer coefficient, a, which differs from 0.5 [except for very small charge transfer For example, an asymrates (W)]. metry in a second harmonic, intermodulation, or r.f. polarogram which is characterized by a larger cathodic peak is associated with a
