Anal. Chem. 1995, 67, 2800-281 1
Migrational Effects on Second Waves of EE Mechanisms under Steady State or Quasi Steady State Regimes Christian Amatore,*lt M. Fatima Bento,* and M. Irene Montenegro*,* Ecole Normale Superieure, Departement de Chimie, URA CNRS 1679,24 rue Lhomond, 75231 Paris Cedex 05,France, and Departamento de Qulmica, Universidade do Minho, Largo do Paqo, 4719 Braga Codex, Portugal
A + 2ne * A2"-
An analytical theory is developed to predict the current
plateaus at second waves of EE mechanismswhen the rate constant of the reproportionation reaction is extremely large and migration contributes to the transport of molecules because of a reduced concentration of the s u p porting electrolyte. By comparison to those we previously developed for single waves, the present analytical solutions establish that the effect of migration at the second wave may be considerably magnified with respect to the effect on the f b t wave even when the diffusion coeflicients of all species are equal. Moreover, the current plateaus of second waves differ significantlyfrom those that would be evaluated upon considering a direct reduction or oxidation of the substrate without involvement of the reproportionation reaction. This difference arises because this reaction segregates the diffusion layer into two adjacent regions with extremely different compositions. The theory is tested on the two cases (z = 0, n = 1, dicyano(fluoren-9-ylidene)methane;z = 2 , n = 1, methylviologen dication) which are predicted to give the largest effects. The results are found to be in remarkable agreement with the experimental measurements.
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Does 1 1always make 2 in electrochemistry?' Stepwise o n e electron transfer reductions or oxidations are ubiquitous in the electrochemistry of organic or organometallic substrates. Except under special circumstances, e.g., when chemical homogeneous reactions are interposed between the two steps or when important reorganization of electronic structures occurs, the second electron transfer step is more dficult than the first one. This leads to the observation of a set of two waves, which are generally explained by the following sequence (n = 1, reduction; n = -1, oxidation):
PI(first wave)
A+ne=-A'"-
A'"- + n e ~t ~
2 " -
(second wave)
(1)
(2)
However, it was recognized very early that the sequence in eqs 1 and 2, together with their combination in eq 3, is not a realistic ' Ecole
Normale Superieure.
' Universidade do Minho.
(1) This provocative question is adapted from the title of a work by Rongfeng and Evans, kindly communicated to us before its publication (Rongfeng, Z.; Evans, D. H. j. Electround. Chem., in press).
2800 Analytical Chemistry, Vol. 67, No. 77, September 7, 7995
(second wave)
(3)
representation of the systems at hand, even when the various species considered are perfectly chemically stable. Indeed, because the species formed at the second wave is a much stronger reductant (n = 1) or oxidant (n = -1) than that formed at the first wave, an extremely exergonic homogeneous electron transfer has to be considered in addition:
Based on electron transfer theories, this homogeneous electron transfer is expected to proceed with a rate constant k close to the diffusion limit, Le., approaching 109-1010 M-l s-l.? Arate constant with such a magnitude implies that species A and Azn- cannot coexist in the solution under most electrochemical circumstances? Indeed, for a millimolar solution, freezing reaction 4 would require the use of scan rates in the range of 1 x lo6 V s-l in cyclic voltammetry, potential pulse durations much below a tenth of a microsecond in chronoamperometry, or electrodes with radii of a few nanometers in steady state voltammetry at ultramicroelectrodes. However, it has been established that the effect of reaction 4 on the shape of the two successive redox waves is almost negligible whenever the diffusion coefficients of the three species A, An--,and AZn-are not very different! The second wave current plateau is then equal to that of the first wave, reflecting that the overall electron stoichiometry at the second wave (eqs 1 and 2 or eq 3) is twice that at the first wave (eq 1). In other words, one has 1 1 = 2. This is no longer true when the diffusion coefficients of the three species differ ~ignificantly.~ The effect of reaction 4 is then unraveled, and the plateau current of the second wave may differ significantly from that of the first, despite the overall electron stoichiometry at the second wave being still
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(2) For a discussion of this aspect, see: Amatore, C. In Organic Electrochemistry; Lund. H., Baizer, M. M.. Eds.; M. Dekker: New York, 1991; Chapter 4, pp 32-35,45-53. (3) (a) For a millimolar solution, a rate constant in the order of the diffusion limit corresponds to a half-life of -30 ns. This corresponds in cyclic voltammetry to a time scale in the range of half a million volts per second (cf.: Amatore, C.; Jutand, A; Huger, F.J. Electround. Chem. 1987,218, 361) or in steady state voltammetry to a disk electrode with a radius of - 5 nm using D = 5 x lo-'' cm2 s-'. Both limits are achieved today only on an exceptional basis. (b) In this work, only rate constants within this order of magnitude are considered. Therefore. they cannot be determined on the basis of the theory presented here that is valid within a much larger time scale or for electrodes of much larger radii (see text). (4) For a discussion, see the work mentioned in ref 1.
0003-2700/95/0367-2800$9.00/0 0 1995 American Chemical Society
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0.05 pm alumina (Buehler) on a polishing machine (Buehler, Ecomet 3). They were washed with water and dried before use. Prior to each experiment, the electrodes were polished with 0.05 pm alumina, rinsed with water, polished with a wet polishing cloth, rinsed again, and dried. All solutions were prepared with the highest grade chemicals available in high-purity N,N-dimethylformamide @MF) or acetonitrile, previously dried with molecular sieves, 4 A. The supporting electrolytes used were Bu4NC104, Bu4NBF4, Me4NBF4,and Bu4NPF6. The f i s t was available commercially in high purity; BmNBF4 or Me4NBF4 were prepared by mixing aqueous solutions of NaBF4 and BudNHS04 or Me4NHS04, respectively; Bu4NPFs was prepared by mixing NH4PF6 and Bu4NHS04. The product in each case was recrystallized in methanol/water and dried under vacuum. N,M-Dimethyl-4,4'-bipy1idinium hexatluorophosphate (MV') was prepared by mixing aqueous solutions of the corresponding chloride salt and ammonium hexatluorophosphate. The resulting precipitate was washed with water and dried under vacuum. The dicyano (fluoren-9-ylidene)methaneg @CN) was prepared by the Knoevenagel condensation from 4fluorenone and malononitrile.ll Experiments were conducted at room temperature, and all solutions were degassed by nitrogen bubbling prior to experiments and maintained under a nitrogen atmosphere. To check for the possible effects of mechanical vibrations, some experiments were duplicated using a vibration-free table. Within the accuracy of their reproducibility, current measurements performed under these conditions were identical to those obtained without a vibration-free table. Correction for Variation of Dfision Coefficients with Ionic Strength. To correct limiting currents from variations of diffusion coefficients with ionic strength, we resorted to empirical formulas that were obtained as follows.12 The limiting current of the first wave, @", of the compound investigated was measured for a few large and constant excesses of the supporting electrolyte (whose bulk concentration is noted Ce?lk) as a function of the substrate concentration, @ (y = Cesbu'k/Co>> 1; see Figures 4b and 5b). For each substrate concentration and for each given excess y , the ionic strength I was evaluated and iIiml/@plotted as a function of P2(compare Figures 4b and 5b). This afforded regression lines whose equations, iIiml/(@) = e ( K - P), were then used to correct empirically for each y and @ pair in Figures 4c,d and 5c,d, the two limiting currents from the variations of diffusion coefficients as a function of ionic strength. This was performed as follows:12
twice that at the first wave. One then has apparently 1 1 # 2. This has been recently reviewed and verified by simulation as well as experimentally by Evans and R ~ n g f e n g . ~ In the following, we wish to establish that the effect of reaction 4 can also be unraveled when all species have identical diffusion coefficients, provided that migration is made effective by lowering the supporting electrolyte concentration with respect to that of the substrate. White and colleagues have already pointed out this effect experimentally using steady state voltammetry at ultramicroelectrodes.jI6 Indeed, they showed convincingly that their results did not follow Oldham's "general theory" for migration at second waves of EE mechanism^.^ They ascribed these deviations to the effect of reaction 4 and established this clearly by means of digital simulation.j,6 However, digital simulations could not handle very large rate constants for reaction 4. When k increases, species A and Azn- can coexist only within a thin strip of solution whose thickness tends toward zero when k tends toward infinity.* Since the location of this thin kinetic layer into the solution is a priori unknown (vide infra), one cannot rely on the use of special simulation grids, e.g., locally contracted grids, to bypass this inherent problem. As a result, simulations must be restricted to a range of rate constants that is much smaller than the expected ne.^,^ We wish to show that this inherent problem can be solved easily for extremely large rate constants, that is, when the thickness of the kinetic layer where A and Azfl- coexist tends toward zero. Moreover, under these circumstances, one can establish analytical equations for the current plateaus of steady state voltammetric waves. The predictions of this analytical approach are tested experimentally using two systems. The first one, reduction of dicyano(fluoren-9-ylidene)methane,is used to test the theory in the case when the initial substrate is neutral and undergoes two successive chemically reversible one-electron transfers? The second one, reduction of methylviologen dication,6 is used to test a case where the effects are enhanced because the substrate is charged twice positively. EXPERIMENTAL SECTION
Voltammograms were obtained using a Hi-Tek type PPRl wave form generator and a current follower based on a RS 071 operational amplifier. Data acquisitions were performed with an ACER computer and further recorded on a Roland plotter, type DXY-980A. All experiments were carried out in a two-electrode cell placed in a Faraday cage. The reference/counter electrode was a saturated calomel electrode (SCE), and all potentials are quoted versus SCE. The working electrodes were platinum microdisks with diameters of 4.7, 5.4, 26.7, and 28.3 ym or gold microdisks with diameters of 7, 12, and 29 ym, made from cross sections of adequate metal wires (Goodfellow) sealed into soft glass.1° Electrode diameters were calibrated by measuring the limiting voltammetric current for a known concentration of ferrocene in acetonitrile/Bu4NC104.'0 The electrodes were polished first with fine emery paper and then with a polishing cloth with 0.3 and (5) Norton, J. D.; Benson, W. E.; White, H. S.; Pendley, B. D.; Abruiia, H. D. Anal. Chem. 1991,63, 1909. (6) Norton, J. D.; White, H. S. J. Electroanal. Chem. 1992,325, 341. (7) Myland, J. C.; Oldham, IC B. J. Electround. Chem. 1993,347, 49. (8) For a millimolar solution, the width of this thin solution strip will approach a few tens of nanometers. (9) Pardini, V. L.; Roullier, L.; Utley, J. H. P.; Weber, A/. Chem. Soc., Perkin Trans. 2 1981,1520. (10) Baur, J. E.; Wightman, R M. /. Electroanal. Chem. 1991,305. 73.
'~
where the subscripts ref and measd indicate respectively the values taken for reference i d the measured ratio. The following K values were used: DCN, K = 1.76 M112 (Figure 4b; e = 0.145 M-II2; correlation coefficient, 0.979, n = 16); W +K,= 2.11 M1I2 (Figure 5b; e = 0.107 M-I/*; correlation coefficient, 0.977, n = 10). For DCN this was used only for the first wave. For the (11)McElvain, S. M.; Clemens, D. H. In Organic Syntheses; Rabjohn, N., Ed.; Wiley & Sons: New York, 1963; Vol. 4, p 463. (12) Amatore. C.; Deakin, M. R; Wightman. R M./. Electroanal. Chem. 1987, 225, 49. 1
Analytical Chemistry, Vol. 67, No. 17, September
I,
1995
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second wave, (ilimp)/(iliml)was plotted, taking advantage of the fact that the current plateau of the first wave is independent of y (compare Figure 4c) because z = O.I2 This allowed us to take implicitly into account any variations of D with the ionic strength without resorting to any explicit evaluation of the cause($. Indeed, these variations are difficult to predict in organic solvents. Ionic strength affects obviously mobility of ions because of variations in electrostatic forces applied to ionic species, but it also affects diffusion coefficients (particularly for neutral molecules) because of changes in viscosity due to solvent structuration by inert ions. This latter effect is perfectly illustrated by the fact that a linear correlation was observed between ionic strength and the reciprocal of the viscosity of TBAB (0-0.2 M) solutions in D M F P i 2 = -2.37 2.95/viscosity; correlation coefficient, 0.993, n = 7 (data not shown; l i n M;viscosities, in cP, were determined with an Ostwald viscosimeter at room temperature).
negligible, eqs 8-10 tend toward their steady state limit, Le.,
as soon as k becomes infinite. Noting d t ) ,the closed surface of domain U ( t ) ,application of Green's theorem to eqs 11-13 shows that when k tends toward infinity,
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RESULTS
A. Theory. (i) Formulation of the Problem for Intinite Values of 12. Before we start to address the problem of coupling migration and dfision, we want to establish the basis and validity of the analytical formulation that will be developed to handle rate constants k with extremely large values. To simplify the formulation of the mathematical expressions, the sequence of reactions 1, 2, and 4 is recast under the following form: A
+ ne
B + ne
--L
--L
B
@, (first wave)
C A
(second wave)
+ C - 2B (lz)
(5)
(6)
(7)
the charges of A, B, and C being respectively z, z - n and z - 2n. At any electrode, at any time, and irrespective of the electrochemical method used, the equations governing the time and space concentrations of each of the three species are:13 a[Al/at = -divoA) - k[Al [Cl
(8)
+ 2k[Al [Cl
(9)
a[Bl/at = - d i v o B )
a[Cl/at = -diva,) - R[Al[Cl
(10)
When k tends toward infinity, the last term in the above three equations tends toward infinity, except when either [AI or IC1 is extremely small. Let us consider a domain of solution, cii(t), where at a given time t [A] and [C] have finite values. In this domain, the partial derivative versus time must remain finite because of the continuity of the concentration with the time (note that this is always true except maybe in pulse methods, when %l(t) is almost adjacent to the electrode surface, i.e., immediately after a potential pulse). In this domain, the divergence of the flux of each species and the (infinite) kinetic term must then tend toward the same limit. Thus, even under nonsteady state conditions, in any domain O(t) where neither [A] nor [C] is (13) Bard, A. J.: Faulkner, L. R Electrochemical Methods; J. Wiley & Sons: New York, 1980: Chapter 4, p 119f.
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Analytical Chemistry, Vol. 67, No. 77,September 1, 7995
All fluxes in the solution must remain finite, with the exception of particular points (viz., the edges of electrodes), so the last term in eq 14 must remain finite, provided that such points do not belong to the surface p , the limit at r = p plays the role of a virtual electrode with a potential set on the plateau of the A/B wave. Conversely, at p, the concentration of species C is maintained at zero and its flux at p- is equal in magnitude and opposed in sign to that of species B. This shows that, within the space r < p , the limit at r = p plays the role of a bulk solution boundary for the redox couple B/C. The corresponding virtual bulk solution contains only species B at a concentration [Bl,, that is precisely the concentration of B imposed by the virtual electrode located at p in the subspace r > p. Note that these results are independent of the true electrode potential, provided that species C is generated at its surface. What precedes establishes that when the rate constant of the reaction 4 tends toward infinity,the effect of this reproportionation reaction is to segregate the diffusion layer into two virtual adjacent diffusion 1 a ~ e r s . Within l~ the transformed space described by r, these virtual diffusion layers are equivalent to two thin-layer cells containing only species B and C (0 < < p ) or A and B (u .= 1). A cross-talk between these virtual thin-layer cells occurs through their common boundary at r = p and is ensured via the common species B:
This formulation is independent of the electrode geometry, provided that this geometry can lead to the establishment of a steady state or a quasi steady state regime, since it does not depend on the exact relationship between the coordinate r and those of the physical space. In particular, this remains true if the steady state regime occurs because of natural or forced convection, provided that the Nemst layer approximation can be applied.
In this latter case, r = d / 6 , where d is the distance from the electrode and 6 the thickness of the Nemst layer. To conclude this section, let us point out that we have not considered the fate of any species other than A, B, and C. Indeed, because of the steady state or quasi steady state approximation, in the space r, the flux of any such species is also constant. It is thus equal to zero at any r value (except maybe at r = p because of the equation's discontinuity at this boundary), provided that the species is not electroactive or involved in chemical reactions. This is the case in particular for all inert ions, including the possible counterion(s) of species A or those of the supporting electrolyte. (ii) Formulation of the Local Electrical Field, E. In the following we will assume that the electroneutrality law applies. As it has been established recently by White and Smith,2O this is, however, an inconsistent approximation. Indeed, because of the Laplace equation, considering that the solution remains electroneutral implies that the divergence of the electrical field is zero. For example, in the one-dimensional space r defined above, this would correspond to a constant electrical field, a conclusion that is inconsistent with the variations of the electrical field deduced from Ficks laws by assuming the electroneutrality law. In practice, this apparent incoherence is easily bypassed by assuming that the electroneutrality law is not exactly satisfied but only closely approached. As shown by White and Smith?@ the concentration of the residual charge at any point in the solution is always much less than the concentrations of the ionic species which produce this residual charge, provided that the electrode remains of finite size and the concentrations considered are not too high. For micrometric electrodes, millimolar solutions, and classical electrochemical solvents, the electroneutrality law may then be considered as being approached within an adequate accuracy.2@The following theory is developed for such conditions only.21 To proceed further, let us consider that the bulk solution contains initially besides species A (whose charge is z) at concentration a series of N inert ions S, with charges z, and concentrations 0 = yJ@, this including the counterion(s) of species A Under steady state or quasi steady state regime and within the transformed space r, the fluxes of all species S, are such that
e,
Addition of eqs 29-32 affords the following (Z indicates a
(19) An identical situation exists at paired band electrodes operated in ECL mode;
(20) Smith, C. P.; White, H. Anal. Chem. 1993,65, 3343. (21) The electroneutrality law may also break down when n and z are such that either the signs of z and z - n or those of L - n and z - 2n are opposed. With n = 1 or -1, this situation is impossible. Under such circumstances,
Cf.: Amatore, C.; Fosset, B.; Maness, K M.; Wightman, R M. Anal. Chem. 1993,65, 2311.
use of the electroneutrality law may lead to the prediction of infinity currents (compare, e.g., z = 1 and n = 2 in the second column of Table 3).20."
2804 Analytical Chemistry, Vol. 67, No. 17, September 1, 1995
summation over the N inert species Si)
Table 2. Equivalent Diffusion Layers (S) for Common Ultramicroelectroder under Steady State or Quasi Steady State Regimes ( P= #IFADCO/S)~*
electrode
whose integration gives
JA + JB+J~+ Icls,= 0
(34)
since this quantity is equal to zero at the electrode surface. Multiplication of each eq 29-32 by the charge of the corresponding species and addition of the resulting expressions affords
dimensions
sphere/hemisphere YO (radius) disk ro (radius) cylinder/hemicyEnder ro (radius) 1 (length). band w (width) 1 (length).
area
diffusion layer (6)
4nroz or 2nro2 YO nro2 nr0/4 Zzrd or xrol YO In [2(Dt)1/2/r~l
wl
(w/n)In[8(Dt)1/2/wl
The length, being millimetric, is considered much larger than any diffusion layers at these electrodes (1 >> 6).
= he, where ze is positive and the & sign used depends of the charge of the ion Si. This restriction allows the factorization of zj2 = z,2 in eq 41, and the elimination of the ensuing sum over [Si] by combination with eq 39. This shows that in this most frequent experimental case (particularly that with z, = l), evaluation of the local electrical field requires only the knowledge of the local concentrations of the electroactive species A, B, and C: zj
Because of the Nemst-Einstein relation, the flux of each species M is given by13 = -D(a[M]/X - zM[M]FE/R~
JM
(37)
with
where E is the local electrical field. Introduction of this expression into eq 34 affords
+ [Bl + [Cl + x[Sjl)/ar= -(FE/RT)(z[Al +
NAI
(2 -
n) [Bl
+ (Z - 2n) IC1 + x ~ j [ S j l >(38)
The expression on the right-hand side of eq 38 tends toward zero when the electroneutrality law is approached. Integration of the left-hand side of eq 38 then yields
(iii) Migrational Effect at Second Waves of EE Mechanisms. To simplify the presentation of the following, let us introduce a = [AI/@, 6 = [Bl/@, c = [Cl/@, and II, = i/(nFA@D/G) where 6 is an apparent diffusion layer corresponding to the particular electrode geometry considered, defined in Table 2 for sphere, disk,cylinder, or band electrodes.1s With these notations, owing to the results established in sections i and ii, the formulation of the problem simpMes to the following:
O~r-=p(a=O), which is the conservation of matter equation. On the other hand, owing to the relationship in eq 37, eq 36 affords a{z[Al IzJA
+ (Z - 4[Bl + (2 - 2n) [CI +
x
z.[s.i}/ar I I =
+ (2 - n)JB + (2 - 2n)Jclr=o/D - (FE/RT){2[AI + (Z
- n)'[B]
+ (Z - 2n)'[C] + x~:[Sj]}
(40)
da/dr = q db/dr = v
When the electroneutrality law is approached, the left-hand side of eq 40 tends toward zero, showing that
FE/RT
+ (Z - n)JB + (Z - 2n)JcIr=o/[D{2[AI + (Z - n)'[B] + (Z - ~ P Z ) ~ + [ CC~:[Sjl}l ] (41)
[zJA
To proceed further, let us consider that the inert ions, including the counterion(s) of species, have identical absolute charges, i.e.,
- n(z)aly/o
(46)
- n(z - n)bv/a
(47)
where the definition of u as a function of a, b, and c follows from eq 43. Owing to the fact that the most frequent experimental cases involve inert counterions with an absolute charge of unity (Ze = 1) and that 1:l supporting electrolytes are generally used, the following section will be developed for this situation, since it simplifies its formulation and the further use of the resulting analytical expressions. It should, however, be noted that this is not a restrictive hypothesis for the theory, since it can be adapted Analytical Chemistty, Vol. 67, No. 17, September 1, 1995
2805
easily for a situation where z, is not unity. For z, = 1,taking into account the above notations and the fact that the concentration eq 43 simplifies to of the counterion(s) of species A is then IzIe,
o=
(2 - l ) a + [ ( z -
- llb + [(z - 2 n y
- llc
+
(1 + IzI + 2 ~ )(48) where y is the molar excess of the 1:l supporting electrolyte versus the species A in the initial bulk solution. When the potential of the electrode is set on the plateau of the second wave, the above differential equations are associated to the following boundary conditions:
r=o, b 0 = o
[(z -
= 1 + (z/n){[(~ + IZ
- nl(l+
- (7
+ P ) ” ~ I-
+ 2y) ln([A”2 + (A + Q)’”I/
/ZI
[rl/’+ (7 + Q)’’~I>} (60) where
(49)
Equation 48 simplifies into each subdomain of r values owing to the fact that either a or c is zero. Thus,
o s r < , ~ ( ~ =o o= [) (, ~ - ~ ) ~ - i i b + [(z - 2 ~ - )l ]~+~ (1+ / z I + 27) ,U < r s 1 ( c = o ) , O = (2 - i ) a +
reversible wave.22 Therefore, the value of rp in each subsystem < r 5 1, or 0 5 I‘ < p) is readily obtained by simple transposition of the results and procedures given in ref 22. For p < r 5 1, the above system is strictly identical to that solved in ref 22. One has then, by simple transposition of the pertinent variables,
(52)
n y - l l b + (1 + 121 + 2y) (53)
For evaluation of rpocy
