Anal. Chem. 1996, 68, 4377-4388
Equivalence between Microelectrodes of Different Shapes: Between Myth and Reality Christian Amatore* and Bruno Fosset
De´ partement de Chimie, Ecole Normale Supe´ rieure, URA CNRS 1679, 24 rue Lhomond, 75231 Paris Cedex 05, France
Diffusion and diffusion coupled with homogeneous and/ or heterogeneous kinetics are investigated at microelectrodes under conditions of steady-state (disk or hemisphere) or quasi-steady-state (hemicylinder and band) diffusion, in order to search for the possible existence of equivalence relationships between these electrodes of different shapes. The goal of this work is to answer the following question: Is it possible to extract sound kinetic and thermodynamic information from experimental voltammograms obtained at a disk or a band microelectrode, using theories or simulations performed respectively at a would-be equivalent hemispherical or a hemicylindrical microelectrode? It is shown that no true equivalence exists for either pair (band/hemicylinder or disk/hemisphere). Despite the fact that there is no true equivalence, there are time domains where the pair components may become equivalent. This occurs at short times for both sets when planar diffusion prevails and in the quasisteady-state regime for the band/hemicylinder pair, provided that no slow homogeneous kinetic steps are considered and that different rate constants are considered at each electrode (see text). Conversely, no equivalence exists between disk and hemisphere electrodes in the steady-state domain except for the already noted Nernstian electron transfer mechanism. Under all other kinetic situations, the disk and hemisphere electrodes are not equivalent. The fundamental physicochemical reasons for this major difference between the two microelectrodes pairs are discussed. Because of the inherent angular isotropy of spheres or cylinders, Fick’s laws at spherical and cylindrical electrodes can be formulated as a function of a single space variable, viz., the distance r from the center or axis of the electrode, respectively. This simplifies greatly the formulation and, therefore, the analytical solution of Fick’s laws. This is also of interest for numerical solutions. Indeed, if the evaluation of a given numerical solution in a one-dimensional space requires N memories and T CPU, the same evaluation in the “physical” three-dimensional space would require about N3 memories and T3 CPU for an identical accuracy. Most of the theories of transport at ultramicroelectrodes take advantage of this simplification and, hence, have been developed for electrodes having such idealistic geometries.1 The same is true for most numerical procedures. This contrasts with the fact that the electrodes most widely used in actual experiments are (1) For reviews, see: (a) Amatore, C. In Physical Electrochemistry; Rubinstein, I., Ed.; M. Dekker: New York, 1995; Chapter 4, pp 136-208. (b) Wightman, R. M.; Wipf, D. O. In Electroanalytical Chemistry; Bard, A. J., Ed.; M. Dekker: New York, 1989; Vol. 15, pp 267-353. S0003-2700(96)00421-0 CCC: $12.00
© 1996 American Chemical Society
disk- and band-shaped not only because of easier manufacturing but also, and mainly, because such geometries allow controlled cleaning of their surface through simple grinding/polishing procedures.1b,2 Yet, because of the large, or even infinite, current densities at the edge(s) of such electrodes, analytical treatment of mass transport and kinetics become difficult.1 This is even more true for simulations, which then require specific and delicate procedures,3 whose complexity adds to the inherent excessive time consumption and memory occupation since calculations can no longer be performed in a one-dimensional space (vide supra). This inherent dichotomy between “easier” theory and “easier” experiments has stimulated the quest for “equivalence” relationships, i.e., of relationships by which kinetic information contained in an experimental voltammogram obtained, e.g., at a disk or at a band electrode could be extracted on the basis of analytical theories or of simulations performed respectively at a hemispherical or a hemicylindrical one. This was clearly true one decade and half ago, viz., during the historical times of ultramicroelectrodes,1 because theories at band and disk electrodes were not readily available, not to mention precise simulations which would have exceeded the power of desktop computers available at the time.3 Today, owing to the availability of several theories for mass transport coupled with simple elementary kinetic steps,1,2 and of powerful algorithms1a,4 associated with a day-to-day increasing power of desktop computers, the need for such equivalence relationships is expected to have decreased or even to have disappeared. However, one should observe (and already observes) exactly the opposite. On the one hand, the variety of kinetic situations that are encountered grows exponentially, due to the increasing penetration of microelectrodes in laboratories with different kinetic preoccupations. On the other hand, fast and efficient software are now available for evaluation of a nearly infinite variety of kinetic combinations at spherical or cylindrical electrodes.5 It is then expected that this situation will encourage experimentalists to perform mechanistic interpretations and kinetic measurements on the basis of experiments obtained at disk (or band) electrodes through the use of theories or simulations performed at spherical or cylindrical ones. This gives a renewed topicality to “equivalence (2) Microelectrodes: Theory and Applications; Montenegro, M. I., Queiros, M. A., Daschbach, J. L., Eds.; NATO ASI Series E, 197; Kluwer Academic Press: Dordrecht, 1991. (3) See, e.g.: Varco Shea, T.; Bard, A. J. Anal. Chem. 1987, 59, 2102. (4) Speiser, B. In Electroanalytical Chemistry; Bard, A. J., Rubinstein, I., Eds.; M. Dekker: New York, 1996; Vol. 19, pp 1-108. (5) See, e.g.: DigiSim from Bioanalytical Systems, Inc., based on the Fast Implicit Finite Difference method introduced by M. Rudolph, which proposes simulations at spherical and cylindrical electrodes virtually for all possible electrochemical mechanisms. For a description of the simulator, see: Rudolph, M.; Reddy, D. P.; Feldberg, S.W. Anal. Chem. 1994, 66, 589A and references therein.
Analytical Chemistry, Vol. 68, No. 24, December 15, 1996 4377
relationships”. However, and despite the prevalent feeling that their existence should not be questioned, they have not yet been always properly established, except in restricted cases where their need is no more of interest now.7 This prompted us to re-examine in detail whether such equivalence relationships actually exist or are only a convenient myth and to investigate the physicochemical roots of such equivalencies or nonequivalencies. We wish to present here the results of these investigations. THEORY Common intuition tells that a disk electrode does not differ significantly from a hemispherical one; the same is true for a band/hemicylinder pair. This feeling has its mathematical roots in the fact that the expressions of the two Fick’s laws tend to be identical for each pair of electrodes, provided one considers large distances from their surfaces, i.e., provided that the diffusion layers extending near these electrodes are extremely thick. This becomes obvious upon considering the limits of the gradient or of the Laplacian at a band (eqs 1 and 2) or a disk (eqs 3 and 4) electrode at large values of r:
||
δC f δr grad C 1 δC r δΩ (∇2C)band )
r98 ∞
band
(1) hemicylinder
∂2C 1 ∂C 1 ∂2C + + 98 (∇2C)hemicylinder ) ∂r2 r ∂r r2 ∂Ω2 r98∞
||
δC f δr grad C 1 δC r δφ (∇2C)disk )
|| δC δr 0
f 98 grad C
∂2C 1 ∂C (2) + ∂r2 r ∂r
|| δC δr 0
f 98 grad C r98 ∞
disk
(3)
hemisphere
(
)
1 ∂ ∂2C 2 ∂C ∂C + 2 sin φ 98 2 + ∂r r ∂r r sin φ ∂φ ∂φ r98∞ (∇2C)hemisphere )
∂2C 2 ∂C (4) + ∂r2 r ∂r
Note that these expressions are valid in any cross-section plane containing the symmetry axis of the electrode and are written using the cylindrical or spherical coordinates defined in Figure 1. At short time scales, i.e., when the smallest dimension of the electrode is much larger than the diffusion layer thickness (viz., when (Dt)1/2 , r0 or w/2, r0 being the radius of the disk and w the width of the band), the shape of the electrode is irrelevant. Planar diffusion occurs and edge currents are negligible, so that the electrode shapes do not affect mass transport or homogeneous or heterogeneous kinetics. Current densities are thus independent of the electrode shapes. All electrodes are then equivalent since voltammetric currents obtained at two electrodes are identical to within a scaling factor equal to the relative surface (6) See, e.g.: Amatore, C. A.; Fosset, B.; Deakin, M. R.; Wightman. R. M. J. Electroanal. Chem. 1987, 225, 33. (7) Amatore, C. A.; Deakin, M. R.; Wightman. R. M. J. Electroanal. Chem. 1986, 207, 23.
4378 Analytical Chemistry, Vol. 68, No. 24, December 15, 1996
Figure 1. Definition of the cylindrical (band/hemicylinder) or spherical (disk/hemisphere) coordinates used in the physical space to describe a point located in any plane orthogonal to the insulating plane in which the electrode is embedded and passing through the center (disk, hemisphere) or orthogonal to the axis (band, hemicylinder) of the electrode. Definitions of Γ and θ for each electrode shape (Table 1) refer to the appropriate set of coordinates.
areas. The equivalence relationships always exist and are straightforward in these short time scales. They reflect only the identity of the surface areas, viz., r0hemicylinder ) w/π, r0hemisphere ) 2-1/2r0disk.1 So, the need to investigate for fruitful equivalencies arises only when the diffusion layer thickness is not much smaller than the smallest dimension of the electrode (Dt not , r02 or w2/4), a situation which corresponds to a transition from transient electrochemistry toward steady or quasi steady states or to steady(disk or hemisphere) or quasi-steady-state (band or hemicylinder) regimes.1 As will be shown in the Discussion, there is no possible equivalence between either pair during the transition between transient regimes and steady-state or quasi-steady-state behaviors, or when slow homogeneous steps are involved. In the following, we wish, therefore, to focus only steady-state or quasi-steady-state regimes (Dt . r02 or w2/4) where equivalence may exist. Even under such delimited conditions, there can never be true equivalence between the disk and hemisphere electrodes except for a simple Nernstian electron transfer. For the band hemisphere pair in the quasi-steady-state regime (Dt . w2/4), there are several kinetic situations of frequent experimental interest where certain scaling of kinetic parameters will produce equivalence. This is detailed hereafter on the basis of several kinetic cases which represents canons of most frequently encountered experimental situations. Case I. Nernstian Electron Transfers or Current Plateaus in the Absence of Homogeneous Kinetics. Despite the existence of mathematical identities at infinite values of r, the above set of equations shows also that the mass transport at a disk, or a band, differs significantly from that at a hemisphere, or at a hemicylinder respectively, at short distances of the electrode surfaces (viz., when r ≈ r0 or w/2). However, when steady-state or quasi-steady-state regimes are considered, this discrepancy is only apparent: it exists only in “our” physical space. Indeed, we have shown in previous works based on conformal mapping techniques1a,6-9 that the four different geometries described above are projections in different spaces of a unique one existing in a fictitious (Γ,θ) space (see Figure 2 and Table 1 for the relationships between the “true” space coordinates and Γ and θ). Irrespective of the electrode shape, the system obeys the following boundary conditions in the (Γ,θ) space: (8) Deakin, M. R.; Wightman. R. M.; Amatore, C. A. J. Electroanal. Chem. 1986, 215, 49. (9) Amatore, C. A.; Fosset, B. J. Electroanal. Chem. 1992, 328, 21.
Table 1. Definition of the Physical Space Coordinatesa as a Function of Γ and θ, and of the Equivalent Diffusion Layer δ, According to the Geometry of the Ultramicroelectrode (See Figure 2) electrode
dimension(s)
hemisphere
r0 (radius)
disk
r0 (radius)
hemicylinder
r0 (radius) l (length)b
band
w (width) l (length)b
space transform Γ ) 1-r0/r θ ) φ/(2π) or θ ) φ/π x ) r0 (1-θ2)1/2/cos(πΓ/2) z ) r0 θ tan(πΓ/2) Γ ) ln(r/r0) θ ) Ω/(2π) or θ ) Ω/π withc cyl ) ln[(2/r0)(Dt)1/2] x ) (w/2) cosh(Γ) cos(πθ/2) y ) (w/2) sinh(Γ) sin(πθ/2) withc band ) ln[(8/w)(Dt)1/2]
σ ) A/δd
diffusion layer (δ)
D(Γ,θ)/D
ζ(θ) Γ)4
2πr0
r0
r0
(1 -
4r0
πr0/4
r0(πθ/2)
1 [tan(Γπ/2)]2 + θ2
πl/cyl
r0cyl
cyl r0
exp(-2cylΓ)
πl/band
(w/π)band
band (w/2) sin(πθ/2)
1 [sinh(Γband)]2 + [sin(πθ/2)]2
a See Figure 1 for the definition of the physical coordinates. b The length, being millimetric, is considered much larger than any diffusion layers. These expressions of are valid only when (Dt)1/2 . r0 or w. See ref 1a for expressions of when this requirement is not fulfilled. d A, electrode surface area.
c
constant as it is for a Nernstian condition, the concentrations are necessarily independent of θ. They are thus simply given by integration of ∂2C/∂Γ2 ) 0, i.e.,
C ) Cel + Γ(Cbulk - Cel)
(8)
Since the changes of variables in Table 1 conserve the fluxes,1a,6-9 eq 8 imposes that the current i at either electrode considered in Table 1 is given by a unique expression:
∫ (∂C ∂ν )
i ) nFD Figure 2. Projections of the fictitious (Γ,θ) space (a) into the physical spaces (b-e) corresponding to four common electrode geometries: hemispherical (b), disk (c), hemicylindrical (d), or band (e) electrodes. In each space, the electrode is represented by the black solid area whose limit corresponds to the projection of the line at Γ ) 0 represented in (a). To illustrate the effect of the space distortion along the projections, the projections of the lines of equations Γ ) 0,0.1,0.2, ..., 0.8, and 0.9 in (a) are represented in each physical space in (b-e). Note that these lines correspond also to the lines of isoconcentration defined by (C - Cel)/(Cbulk - Cel) ) 0, 0.1, 0.2, ..., 0.8, 0.9, and 1.0 for steady-state or quasi-steady-state diffusion when the electrode surface concentration is constant (see text). Note also that the line at Γ ) 1.0 (or at C ) Cbulk), which closes the conformal space in (a), cannot be represented in (b-e) because it corresponds to an infinite distance from the electrode center or axis.
f[Cel(θ),(∂C/∂Γ)el(θ)] ) 0, for Γ ) 0
(5)
C 98 Cbulk, ∀θ
(6)
∂C ) 0, for θ ) 0 (electrode edge) or 1 ∂θ
(7)
Γ981
where Cel(θ) and (∂C/∂Γ)el are the values of the concentration and of its derivative at the electrode surface (viz., at a given θ and at Γ ) 0), f an implicit or explicit function of these variables, and Cbulk the bulk concentration. This implies that, in the absence of homogeneous chemical reaction (vide infra when homogeneous steps are considered), a dependence of the system on θ, viz., on the electrode shape, can be introduced only through the function f involved in the electrode boundary condition at the electrode surface (eq 5). When f is independent of θ, e.g., when the concentration at the electrode is
A
0
electrode
∫ (∂C ∂Γ)
dA ) nFDσ
1
0
Γ)0
dθ )
nFDσ(Cbulk - Cel) (9)
where A is the electrode surface area in the true space, ν the coordinate normal to the electrode surface in the true space (viz., r for the hemisphere or hemicylinder, z for the disk, and y for the band, see Figure 1), and σ is a geometric parameter with dimensions of length (see Table 1). σ depends on the change of variables performed to project the true space onto the (Γ,θ) space, that is, on the particular electrode shape.1a,6-9 Noting δ ) A/σ (see Table 1) affords the more classical forms in eq 10:
i)
nFAD(Cbulk - Cel) ) δ 1 nFADCbulk ) δ 1 + exp[nF(E - E°)/RT] ilim 1 + exp[nF(E - E°)/RT]
(10)
where E° is the standard potential of the redox couple considered. δ plays the role of an equivalent diffusion layer. Therefore, eq 10 establishes that all four electrodes considered here are equivalent and produce voltammograms that differ only because of the current scaling factor σ ) A/δ, which controls the limiting current value, ilim ) nFADCbulk/δ. It ensues that, as established before, an equivalence relationship between these electrodes exists under the present conditions and requires only the identity of their ilim values, viz., of their σ parameters. Analytical Chemistry, Vol. 68, No. 24, December 15, 1996
4379
If the equivalence is sought in terms of equivalent dimensions, this general equivalence relationship must be examined separately for the disk/hemisphere and band/hemicylinder couples because each pair leads to a different current-time dependence, viz., to steady-state (disk/hemisphere) or quasi-steady-state (band/hemicylinder) currents. Then the relationships are straightforward: r0hemisphere ) (2/π)r0disk for the disk/hemisphere pair10 and r0hemicylinder ) wband/4 for the band/hemicylinder one.6 These relationships have been established previously along more complicated analytical procedures.6,10 However, they are shown here to derive from the simple fact that all four electrodes considered are avatars of a single fictitious one existing in the (Γ,θ) space (Figure 2). Note also that these results remain true even when the electron transfers are not Nernstian, provided only current plateaus are considered. Indeed, limiting currents correspond to a zero concentration of the electroactive bulk species at the electrode surfaces and, therefore, to a constant concentration boundary condition (viz., Cel ) 0 replacing eq 5). It results that the above formalism, and therefore its conclusions, apply also to the determination of current plateaus unless the overall electron consumption depends on slow homogeneous kinetics (vide infra). Case II. Irreversible Electron Transfer in the Absence of Homogeneous Kinetics. When the electron transfer kinetics are irreversible, the boundary condition at the electrode surface couples the local electrode concentration, Cel, and the local current density, di/dA, at each point of the electrode:
di ∂C ) nFkel(E)Cel ) nFD dA ∂ν
( )
ν)0
(11)
For example, although they involve different ζ(θ) functions (Table 1), a band and a hemicylindrical electrode may afford the same quasi-steady-state irreversible voltammograms, as has been established in a previous work of this group.6 This requires that r0hemicylinder ) wband/4 because of the identity of their σ parameters (vide supra), but also that a different heterogeneous rate constant is considered at each electrode. Indeed, the voltammograms coincide only when [kel(E)]hemicylinder ) (4/π)[kel(E)]band, which compensates for their different ζ(θ) functions.1a,6 Thus, in the case of irreversible charge transfers, equivalence between a band and a hemicylinder requires the simultaneous fulfillment of two relationships, one of a geometric nature, and the other of a kinetic nature.1a,6 We wish now to show that a similar set of relationships cannot exist for the disk/hemisphere pair, which is the fundamental reason why these electrodes produce different voltammograms for slow charge transfer kinetics.10,11 For this, let us consider an irreversible electron transfer,
A + ne 98 B
( )
kel(E) C ζ(θ) ) θ,Γ)0 D el
( )
(12)
where the function ζ(θ) (see Table 1) depends on the change of variables performed, i.e., on the electrode shape.1a,6-9 When ζ(θ) depends on θ (Table 1), eq 12 imposes that the local flux or the concentration at the electrode surface, or presumably both, depend on θ. This shows that the function f in eq 5 now depends on θ. Therefore, within the (Γ,θ) space, the concentration profiles at the band and disk electrodes are expected to depend on θ, with a specific dependence associated with each specific ζ(θ) function, viz., each specific shape. The isoconcentration lines created at each electrode can no longer be projected rigorously on each other since even in the (Γ,θ) space (compare Figure 2) they depend on the real electrode shape. This suggests that there is no strict equivalence between the electrodes in a “mathematical” sense as in case I. However, this does not establish that there is no equivalence at all, i.e., that two electrodes of different shape may give rise to experimentally indistinguishable current-potential curves although their isoconcentration lines differ mathematically in the (Γ,θ) space. Indeed, although the above reasoning is mathematically correct, it does apply only to the local current densities and not necessarily to the overall current since this is an integrated variable. (10) Oldham, K. B.; Zoski, C. G. J. Electroanal. Chem. 1988, 256, 11.
4380
Analytical Chemistry, Vol. 68, No. 24, December 15, 1996
(13)
under steady-state conditions (vide supra). In the (Γ,θ) space, the steady-state second Fick’s law for the hemisphere is
∂2C )0 ∂Γ
(14)
and is associated with the boundary conditions in eqs 6 and 7 and the electrode condition (eq 5):
(∂C ∂Γ)
where kel(E) is the potential-dependent heterogeneous rate constant. When transposed to the (Γ,θ) space, this relationship becomes
kel(E) ∂C ∂ν C ) ∂Γ θ,Γ)0 D el ∂Γ
(E°, k°, R)
θ,Γ)0
) Λhemisphere eRξCel
(15)
where ξ ) nF(E° - E)/RT, and Λhemisphere ) k°r0hemisphere/D. Integration of eq 14 gives readily
C ) Cel + Γ(Cbulk - Cel)
(16)
where Cel is obtained through derivation of eq 16 vs Γ and application of eq 15:
Cbulk - Cel ) Λhemisphere eRξCel
(17)
The current ih at the hemisphere is then finally given by the classical equation12
ih ) nFDσh
∫ (∂C ∂Γ) 1
0
θ,Γ)0
dθ ) Cbulk - Cel ) Cbulk
Λhemisphere eRξ 1 + Λhemisphere eRξ
(18)
where σh ) 2πr0 (Table 1). (11) Note that, in ref 10, Oldham and Zoski have already established that irreversible voltammograms at a disk and at a hemisphere electrode do not coincide. We wish, nevertheless, to examine this case here in order to illustrate how the approach through the (Γ,θ) space allows the origin of this nonequivalence to be identified. On the other hand, this will serve to establish all the other results presented here for the disk/hemisphere pair. (12) Bard, A. J.; Faulkner, L. R. Electrochemical Methods; Wiley: New York, 1980.
In the (Γ,θ) space, the steady-state second Fick’s law for the disk is9
[
]
2 cos (πΓ/2) π
2
[
]
∂ ∂2C ∂C + (1 - θ)2 )0 ∂θ ∂Γ2 ∂θ
θ,Γ)0
( )
π ) (k° eRξ/D)ζ(θ)Cel ) Λdisk eRξ θ Cel 2
Cav
0
0
(21)
1
0
θ,Γ)0
1
∂Γ
∂Cav ∂Γ
) Cbulk - Cav,el )
∫ C θ dθ )Θ ∫ C dθ 1
π Θ(ξ) ) 2
(22)
el
0
(28)
0
1
el
For the band/hemicylinder pair, the analogous requirement (see ζ(θ) in Table 1) was shown to be true under quasi-steady-state conditions:6
This establishes that Cav obeys
Cav ) Cav,el + Γ(Cbulk - Cav,el)
∫C 1
(23) Θband(ξ) )
0
el
sin(πθ/2) dθ
∫C 1
0
el
dθ
∫ sin(πθ/2) dθ 2 = ) π ∫ dθ 1
0
1
0
where Cav,el ) Cav(Γ ) 0) is obtained as follows. Integration of the electrode boundary condition (eq 20) over θ yields
∫C
π Λdisk eRξ 2
1
0
elθ
dθ )
∫
() (∫ ) ( ) ∂C ∂Γ
1
0
∂
θ,Γ)0
dθ )
1
0
C dθ
∂Γ
)
θ,Γ)0
∂Cav ∂Γ
Γ)0
(24)
d Owing to eq 23, and since Cel g 0, eq 24 shows that
∫ C θ dθ ) Θ(ξ)∫ C dθ ) Λ
π Cbulk - Cav,el ) Λdisk eRξ 2 Λdisk eRξ
1
el
0
1
0
el
(27)
Equation 27 is reminiscent of the condition found previously for the equivalence between a band and a hemicylindrical electrode,6 in the sense that it shows that equivalence between a disk and a hemispherical electrode requires not only a geometric relationship (i.e., between their radii) but also a kinetic relationship (i.e., between the rate constants considered at each electrode). However, since the individual k° values are necessarily constant, eq 27 requires that Θ(ξ) is independent of the potential, i.e.,
0
)0
Γ)0
where σd ) 4r0 (Table 1). Identity of the disk (eq 26) and hemisphere (eq 18) currents imposes, therefore, that σd ) σh, viz., that r0hemisphere ) (2/π)r0disk (Table 1), and that ΛdiskΘ(ξ) ) Λhemisphere, viz., that
2
2
( )
Λdisk eRξΘ(ξ) (26) Cbulk 1 + Λdisk eRξΘ(ξ)
Thus, integrating eq 19 over θ, and taking advantage of the boundary conditions in eq 7, shows that
∂ Cav
dθ )
k°hemisphere ) [(π/2)Θ(ξ)]k°disk
0
1
1
(20)
where Λdisk ) k°r0disk/D. Let us introduce the average concentration, Cav(Γ), defined by
∫ C(Γ,θ) dθ ) ∫ C(Γ,θ) dθ (Γ) ) ∫ dθ
∫ (∂C ∂Γ)
(19)
and is associated with the boundary conditions in eqs 6 and 7 and the electrode condition,
(∂C ∂Γ )
id ) nFDσd
disk
eRξΘ(ξ)Cav,el (25)
where Θ, such as 0 e Θ e (π/2),13 depends on Cav,el and is, therefore, a function of ξ. The current id is then (13) This results from application of the chord theorem in algebra to the integral in the first member of eq 24. Thus, the area of a product f(θ)g(θ), over the interval [θ1,θ2], over which f(θ) keeps a constant sign (viz., either f(θ) g 0 or f(θ) e 0), is equal to the area of g(θ) over the same interval times a scaling factor Θ, such as Θ ∈ [f(θ1),f(θ2)]. This is used here with f(θ) ≡ (π/2)θ and g(θ) ≡ Cel(θ), so Θ ∈ [0,π/2].
(29)
This permitted us to introduce a notion of kinetic equivalence between the two electrodes: k°hemicylinder ) (4/π)k°band. For the disk, we could not evaluate analytically the function Cel(θ) and, therefore, Θ(ξ) at finite potentials. However, it can be shown by considering asymptotic limits that eq 28 cannot be true. When the potential is set near the foot of the wave, Cel ≈ Cbulk. So, in eq 28, Cel can be replaced by Cbulk under the two integrals, which affords limξf-∞Θ(ξ) ≈ π/4. Conversely, when the electrode potential approaches the wave plateau, Cel ≈ 0. The situation is then equivalent to that observed for a Nernstian electron transfer (viz., a constant concentration boundary condition). This imposes that the current density at the electrode surface is nearly constant in the (Γ,θ) space (see case I). From eq 20, it results that Cel ≈ (∂C/∂G)θ,Γ)0[exp(-Rξ)]/[Λdiskζ(θ)] ∝ θ-1. Introduction of this value into eq 28 shows that limξf+∞Θ(ξ) ≈ 0. Θ(ξ) being by definition (eq 28) a continuous function of ξ, it is also a continuous function of i/ilim, Θ[ξ(i/ilim)]. This function has two different values, at i/ilim ) 0 and at i/ilim ) 1 (viz., 0 for ξ f +∞). Although the analytical variations of Θ(ξ) could not be obtained, this is sufficient to establish analytically that Θ(ξ) cannot be constant. Note that the limit Θ(ξ) f π/4 (viz., near the foot of the wave, i/ilim f 0) corresponds to the constant concentration approximation at the disk surface, while the limit Θ(ξ) f 0 (viz., near the Analytical Chemistry, Vol. 68, No. 24, December 15, 1996
4381
plateau of the wave, i/ilim f 1) corresponds to the constant current density approximation. Therefore, at a disk electrode, the boundary condition at the electrode surface shifts progressively from one asymptotic boundary condition to the other while the wave is described. The complete variations of Θ(ξ) can be obtained by simulation9 and, hence, from direct numerical evaluation of the pertinent integrals in eq 28. Alternatively, it can be determined indirectly, using an empirical equation reported for irreversible electron transfer voltammograms at a disk.10 Indeed, rewriting eq 26 shows that
∫ C θ dθ ) ∫ C dθ 1
π Θdisk(ξ) ) 2
el
0
1
0
el
1
(
R[nF(E°-E)/RT]+ln(k°r0/D)disk
e
)
i i -i lim
(30)
disk
where i/ilim at the disk electrode may be obtained through an empirical formula proposed by Oldham and Zoski (eq 24 in ref 10). Both approaches, viz., direct computation of the integrals in eq 28 by numerical simulation, or indirect through eq 30, gave almost identical results in the potential range of interest. The resulting variations of Θ(i/ilim) presented in Figure 3a confirm that the function Θ(ξ) defined in eq 28 for the disk electrode is not constant, and therefore that the disk and the hemisphere electrode cannot be equivalent.11 Note that both methods verify that limξf-∞Θ(ξ) ≈ π/4. However, if simulations show effectively that limξf∞Θ(ξ) ≈ 0, the indirect method does not. This is reflects only an insufficient accuracy of the empirical i/ilim equation when i/ilim approaches unity.10 Indeed, eq 30 magnifies any slight error on i/ilim when this ratio approaches unity. Figure 3a shows also that, in the experimentally most useful section of the voltammogram, viz., between E1/4 and E3/4, Θ(ξ) is nearly constant and can be approximated at better than 5% by its average value over the interval: Θ(ξ) ≈ Θav ) 0.726. This suggests that, in this range of potential values, the disk current can be approximated with a reasonable accuracy by that of a hemispherical electrode with r0hemisphere ) (2/π)r0disk and k°hemisphere ) (πΘav/2)k°disk ≈ 1.14k°disk, despite the fact that a true equivalence between the two electrodes does not exist. This prediction is verified by the comparison of the voltammograms simulated at each electrode using the above approximate relationships (Figure 3b). It is seen that, although there is not a strict coincidence, the two voltammograms do not differ much in the zone of main experimental interest (viz., for E ∈ [E1/4,E3/4]). They deviate much more in the region closer to the plateau, reflecting the larger curvature of the function Θ(ξ) in this range (Figure 3a). In the (Γ,θ) space, this corresponds to a shift from an almost constant concentration boundary condition at the disk surface (Θ(ξ) ≈ Θav ) 0.726) to a constant current density condition (Θ(ξ) ≈ 0) when the current plateau is approached. The fact that Θ(ξ) remains almost constant except near the wave plateau (Figure 3a) arises from the fact that the slow charge transfer condition (eq 11) acts as a positive feedback, provided that Cel/Cbulk differs significantly from zero. This can be substantiated using the following intuitive argument. Let us consider that, at some point of the electrode surface, the concentration Cel is much larger than Cav,el. Because of eq 11, the local current density is then also necessarily much larger than the average electrode 4382 Analytical Chemistry, Vol. 68, No. 24, December 15, 1996
Figure 3. Irreversible electron transfer at a disk electrode (case II). (a) Variations of the function Θ(ξ) defined in eq 28 as a function of i/ilim, for steady-state conditions (solid curve, based on simulations; dashed curve, based on eq 30 and an empirical fitting equation given in ref 10). Note that the function Θ(ξ) may be easily obtained as a function of ξ by combining this curve with that giving i/ilim vs ξ ) nF(E° - E)/RT represented in (b). (b) Simulated steady-state irreversible voltammogram at a disk electrode (solid line) or analytical steadystate irreversible voltammogram (eq 18) at a hemispherical electrode (dashed line). The voltammogram at the hemispherical electrode is evaluated for r0hemisphere ) (2/π)r0disk and k°hemisphere ) (Θavπ/2)k°disk ≈ 1.14k°disk (see text). (c) Simulated concentration profiles at the disk electrode in the (Γ,θ) space at E ) E1/2 at the edge (θ ) 0, solid line) or at the center (θ ) 1, dashed line) of the electrode in the (Γ,θ) space. Simulations were performed under steady-state conditions (Tc ) RT/ (nFv) ) 104r02/D) for a slow charge transfer mechanism with Λ ) k°r0/D ) 0.01.
current density. This implies that the local rate of electrolysis is also larger than the average one. It then follows that Cel should be smaller than Cav,el, in contradiction with the initial hypothesis. The reverse hypothesis (viz., Cel much smaller than Cav,el) is similarly shown to be incoherent. Thus, Cel cannot differ significantly from Cav,el, and therefore Θ(ξ) ≈ π/4 ) 0.785. Conversely, when Cel/Cbulk approaches zero, the current density tends to be constant over the electrode surface. Because of eq 11, this forces Cel to vary considerably over the electrode surface (in relative values only, since it remains always close to zero) and, hence, Θ(ξ) to deviate significantly from π/4. In other words, in
the (Γ,θ) space, the electrode boundary conditions shift from a nearly constant concentration condition (i.e., Θ(ξ) ≈ π/4, valid over most part of the wave) to a nearly constant current density (i.e., Θ(ξ) f 0) when the wave plateau is approached. Case III. Extremely Fast Homogeneous Followup Kinetics Taking Place near the Electrode Surface. When the electrode electron transfer step is associated with homogeneous follow-up reactions, three different situations may be encountered. When the kinetics are extremely slow (viz., λ ) khomo(Cbulk)m-1 (a2/D) , 1, a being the radius or the width of the electrode and m the reaction order of the species considered), they cannot affect the diffusion of molecules (pure diffusion regime). Therefore, the above conclusions remain valid. When the kinetics are slow but sufficiently large to affect diffusion (viz., λ ≈ 1), the concentration profiles obey a mixed kinetic-diffusional regime all over the diffusion layer (mixed kinetic/diffusion regime). As will be made clear in case VI (vide infra), the concentration profiles in the (Γ,θ) space are specific to each electrode shape, and there is no possible equivalence. Therefore, the search for a possible equivalence bears some meaning only in the third limiting situation, i.e., when the homogeneous rate constants are extremely large (viz., λ . 1, pure kinetic regime). Then, the reactive species exists only within a restricted space domain, the kinetic layer, whose width µ is infinitely smaller than the diffusion layer (viz., µ , a).14 Outside the kinetic layer, all the species diffuse or are present at very small concentrations. The kinetic layer plays the role of a fake boundary B whose role is to impose a different relationship between a concentration and its gradient on either side of the boundary.14 The transient intermediate species (R) of interest is produced at the electrode surface and exists only within the kinetic layer. In the kinetic layer, since µ , a, the curvature radii of the isoconcentration surfaces are necessarily infinite compared to the local width of the kinetic layer. Therefore, diffusion reaction occurs under a planar regime in the kinetic layer. The concentration, [R], of the species obeys then the simplified diffusion reaction equation:14
∂2[R] ∂ν2
) (khomo/D)[R]m
(31)
where ν is the spatial coordinate normal to the electrode surface (vide supra). Multiplying both sides of eq 31 by 2(∂[R]/∂ν) and integrating over ν from 0 to µ, noting that 2(∂[R]/∂ν)(∂2[R]/∂ν2) ) ∂[(∂[R]/∂ν)2]/∂ν, affords
[( ) ] [ ( ) ∂[R] ∂ν
2
)
ν)0
i.e.,
∂[R] ∂ν
2khomo
] [
D(m + 1) ν)0
)-
([R]ν)0)(m+1), 2khomo
]
D(m + 1)
1/2
species O is controlled by eq 11, which imposes the current density. The current is then independent of eq 32, which serves only to control [R]ν)0, the surface concentration of the transient intermediate.14 The existence of any equivalence is then controlled by the factors already discussed in case II. So, let us focus hereafter on the only case where an analysis is required, viz., when a Nernst law applies. Owing to the flux conservation (first Fick’s laws) and considering Nersnt’s law, eq 32 is converted into eq 33:
( ) [ ∂[O] ∂ν
)
(14) Amatore, C. In Organic Electrochemistry; Lund, H., Baizer, M. M., Eds.; M. Dekker: New York, 1991; Chapter 2, pp 108-109.
]
1/2
e(m+1)ξ/2([O]ν)0)(m+1)/2 (33)
D(m + 1)
ν)0
where ξ keeps its previous definition, ξ ) nF(E° - E)/RT, with n being the number of electrons transferred in the direction required to convert the stable species O into the transient intermediate R. It is seen that the structure of eq 33 is identical to that of a fictitious electrode boundary condition analogous to eq 11. Its transposition into the (Γ,θ) space involves, therefore, again a dependence on the electrode shape through the function ζ(θ):
( ) ∂[O] ∂Γ
[
)
2khomo
]
1/2
D(m + 1)
θ,Γ)0
e(m+1)ξ/2ζ(θ)([O]el)(m+1)/2 (34)
Outside of the kinetic layer, the stable species diffuses, and its average concentration (defined as in eq 21) is given by eq 23. The current due to this couple is then such as
∫ ( ∂Γ )
i ) nFDσ
∂[O]
1
0
θ,Γ)0
dθ )
( ) ∂[O]av ∂Γ
)
Γ)0
[O]bulk -
∫ [O] 1
el
dθ (35)
(m+1)/2 ζ(θ) el)
dθ (36)
0
i.e., using eq 34,
[O]bulk -
[
∫ [O] 1
el
0
2khomo
dθ )
]
1/2
D(m + 1)
∫ ([O]
e(m+1)ξ/2
1
0
At the hemispherical or hemicylindrical electrodes, ζ(θ) does not depend on θ (viz., ζ(θ) ) ζ0 as given in Table 1) and [O]el is independent of θ. The two integrals in eq 36 can be replaced by their values as a function of the constant [O]el value, showing that this latter variable is the solution of
[
([R]ν)0)(m+1)/2 (32)
since by definition of µ, [R] and ∂[R]/∂ν tend to zero when ν f µ (note that ∂[R]/∂ν is necessarily negative since R is produced by the electrode). This flux is opposed to that of O its redox partner species, and the two surface concentrations are coupled through a Nernst law or a Butler-Volmer kind of relationship. When an irreversible Butler-Volmer law applies, the situation is akin to case II: the gradient at ν ) 0 of the stable redox partner
2khomo
[O]el + ζ0
2khomo
]
D(m + 1)
1/2
e(m+1)ξ/2([O]el)(m+1)/2 ) [O]bulk (37)
the current being then given by
i ) nFDσ([O]bulk - [O]el)
(38)
Equation 35 shows that the equivalence between two electrodes implies the identity of their σ factors as well as the identity of their average electrode concentrations. As for the irreversible Analytical Chemistry, Vol. 68, No. 24, December 15, 1996
4383
electron transfer case, this latter condition requires the existence of a kinetic relationship, viz., of a relationship between the rate constants considered at each electrode. This kinetic condition must be independent of the potential, since the true rate constants are potential independent. This may occur only if Ξ(ξ) defined in eq 39 is a constant, i.e.,
∫ [O] Ξ(ξ) ) [∫ [O] 1
K are respectively 0 and -(kf + kb)[K]. On the other hand, because I is not electroactive at the electrode surface, one obtains immediately
(∂[S] ∂ν )
)
ν)0
(∂[K] ∂ν )
)-
ν)0
(∂[R] ∂ν )
ν)0
(43)
(m+1)/2
ζ(θ) dθ
el
0
1
0
el
dθ](m+1)/2
) Ξ0
(39)
Nernst’s equation relating the electrode concentrations of O and R is replaced by the following fictitious boundary condition:
[K]ν)0 + (kf/kb)[S]ν)0 ) (1 + kf/kb)[O]ν)0 ) If so, the kinetic relationship is simply given by the identity of the khomoΞ0 values at each electrode. The situation is then formally identical to that met with in case II. For the band/hemicylinder pair, following the same reasoning developed in our previous work dealing with irreversible charge transfer,6 it follows readily that, in both cases, the electrode surface concentration tends to be even (viz., independent of θ), so that
{
cylr0 1 Ξ(ξ) ) 0 ζ(θ) dθ ) w band π
∫
(hemicylinder) (band)
(40)
Since the identity of the σ factors of the band and the hemicylindrical electrode (Table 1) imposes the equality of their parameters, one has necessarily r0hemicylinder ) w/4. Thus, the kinetic equivalence (eq 40) requires only that khomohemicylinder ) (4/π)khomoband, i.e., a condition identical to that previously obtained between the heterogeneous rate constants.6 It is noteworthy that this relationship is independent of the reaction order of the species, i.e., of the reaction considered.14 We were not able to determine analytically the function Ξ(ξ) relative to the disk. However, the present problem is formally identical to that met with in the case of a heterogeneous rate constant (case II). There is even a strict identity between the two problems when m ) 1. Using the same reasoning as for case II, it follows that, at the disk electrode, Ξ(ξ) tends toward π/4 at the foot of the wave (limit at ξ f -∞), while it tends toward zero near the plateau (limit at ξ f +∞). This shows that, for a disk, Ξ(ξ) necessarily depends on the potential and, therefore, that there is no equivalence between a disk and a hemispherical electrode when fast homogeneous kinetics are involved (vide infra, case VI, for slow kinetics). Case IV. Extremely Fast Homogeneous Antecedent Kinetics. Let us consider a typical CE sequence, where I is not electroactive and is in rapid equilibrium with the electroactive species O (viz., λ ) (kf + kb)[I]bulk(a2/D) . 1, where a is the radius or the width of the electrode). Let us consider the situation where O/R is a Nernstian redox couple (vide infra for the converse):
IhO O + ne h R
(kf,kb) (E°)
(41) (42)
Let us introduce the fictitious concentrations [S] ) [I] + [O] and [K] ) [O] - (kf/kb)[I]. Linear combination of the kinetic terms which correspond to the real species I and O shows that the fictitious kinetic terms which apply on the fictitious species S and 4384
Analytical Chemistry, Vol. 68, No. 24, December 15, 1996
[R]ν)0 exp[-ξ + ln(1 + kf/kb)] (44)
In addition, application to the fictitious species K of the formalism used above for the transient intermediate R in case III (with m ) 1 here), yields
( ) ∂[K] ∂ν
ν)0
)-
(
)
kf + kb D
1/2
[K]ν)0
(45)
Thus, the situation is mathematically identical to those investigated above for a heterogeneous reaction (case II) or a fast homogeneous follow-up reaction (case III), although it features now fast homogeneous antecedent reversible kinetics. Because of this formal identity, all the results obtained in cases II and III immediately transpose to the present case. It is then readily deduced that there is no equivalence between the disk and the hemisphere, while there is equivalence between the band and the hemicylinder (see Table 2). Case V. Extremely Fast Homogeneous Kinetics Taking Place Far from the Electrode Surface. This situation is encountered, for example, when one species is generated at the electrode and titrates another species present in the bulk solution. It is commonly found in electrochemistry; compare, for example, conproportionation reactions, redox catalysis, ECL, etc.12 When the rate constant of the homogeneous step is extremely fast (i.e., λ ) khomoCbulk(a2/D) . 1, for a bimolecular reaction), the reactants tend to titrate each other whenever they are present in the same solution domain. It results that only one species may “survive” in a given solution domain; this is the one with the largest local concentration, the other being completely depleted.15 This allows the diffusion layer to be separated into two main zones: one is adjacent to the electrode and contains only the species generated at the electrode surface, and the other extends from the bulk solution and contains only the coreactant present into the bulk solution.15 In each zone, the pertinent species diffuses only since the coreactant species cannot enter the zone, thus making the rate of the chemical step negligible. These two zones are separated by a very thin strip of solution, B, in which both species coexist at comparable and extremely small concentrations. This has been established rigorously in a previous work dealing with conproportionation reactions15 and is immediately transposable to any other comparable situation, such as ECL, redox catalysis, etc.12 This zone acts as a virtual electrode standing in the midst of the solution, being a “sink” for each reactant and a source for the product(s) of the reaction. The (15) Amatore, C.; Bento, M. F.; Montenegro, M. I. Anal. Chem. 1995, 67, 2800.
Table 2. Equivalence between the Usual Electrodes Considered in This Work under Steady-State or Quasi-Steady-State Condtionsa examples
disk/hemisphere (steady-state)
band/hemicylinder (quasi-steady-state)
O + ne h R (fast) O + ne f R (slow, kel)
r0hs ) (2/π)r0dk no equivalence
I h O (fast, kf, kb) O + ne h R (fast) mR + Zxsf P (fast, k) (CE, EC, ECE, DISP1, DIM, etc.) any combination of cases (III + IV)-A with case II
no equivalence
r0hc ) w/4 r0hc ) w/4 kelhc ) (4/π)kelbd r0hc ) w/4 khc ) (4/π)kbd
cases case I case II cases (III + IV)-A
cases (III + IV)-B
no equivalence r0hs ) (2/π)r0dk
case V-B case V-C
O + ne h R (fast) mR + ZstofP (fast, k) P + n′e h Q (fast) (EC, ECE, DISP1, etc.) or O + 2ne h R (fast) R + O f 2P (fast, k) P + ne h R (fast) (conproportionations) or O + ne h R (fast) mR + Zsto f O (fast, k) (redox catalysis) cases V-A when species P when P + n′e f Q (slow, kel) any combination of case V-A with case II
case VI
any combinations involving slow homogeneous kinetics (ka2/D ≈ 1, a ) w or r0)
no equivalence
case V-A
no equivalence no equivalence
r0hc ) w/4 khc ) (4/π)kbd kelhc ) (4/π)kelbd r0hc ) w/4
r0hc ) w/4 r0hc ) w/4 kelhc ) (4/π)kelbd no equivalence
a The four electrodes are grouped into two classes owing to their steady-state or quasi-steady-state behavior. The superscripts hs, dk, hc, and bd indicate that the variable considered pertains to the hemisphere, the disk, the hemicylinder, or the band electrode, respectively. r0 or w are the electrode radius and width, respectively. kel represents the potential-dependent heterogeneous rate constant and k the homogeneous ones. The subscripts xs and sto indicate that the coreactant species Z are considered in excess or in stoichiometric or less amounts, respectively. b These examples are not exhaustive and should be considered only as illustrative of classes of mechanisms.
reaction product(s) diffuse(s) freely in each of the two adjacent zones. All the other species which are not involved in the reaction diffuse freely across the boundary B. The only role of the boundary B is to exchange the fluxes of the two coreactants and products on each side of the boundary and to maintain at zero the concentration of each reactant on the surface B:15
( )
∂CP D ∂ν P
( )
∂CP -D B+ ∂ν P
( )
∂C)η D B∂ν P
CB -
)
-
+ CB +
)0
( )
∂C+ ) -η D B∂ν B+ (46) P
the concentrations of interest are C- ) [(ΓB - Γ)/ΓB]Cel for Γ e ΓB, and C ) 0
for Γ g ΓB (48) + C+ ) [(Γ - ΓB)/(1 - ΓB)]Cbulk for Γ g ΓB, and C+ ) 0
for Γ e ΓB (49)
+
(47)
where ν is the coordinate normal to the boundary at the location considered, C+ and C- are the concentrations of the two reactants, CP is that of (or of one of) the product(s) formed with the stoichiometric order ηP; superscripts +, -, or P represents each side of the boundary or the product(s), respectively. Note that when the product(s) P is (are) not electroactive, it (they) do(es) not need to be considered. Thus in the following we consider that they are electroactive. In our physical space, the boundary B is a surface. In the (Γ,θ) space, it is projected into a line of general equation B(Γ,θ) ) 0.1a In all the (Γ,θ) space, all species diffuse only, except at the boundary B(Γ,θ) ) 0. When all electrode boundary conditions are θ independent, the solution is obvious in the (Γ,θ) space. The boundary equation is Γ ) ΓB,15 where ΓB is a constant, and
P CP ) CPel + φΓ for Γ e ΓB, and CP ) Cbulk + ψΓ
for Γ g ΓB (50) Z CZ ) CZel + (Cbulk - CZel)Γ
(51)
where ΓB, φ and ψ are constant parameters, the subscripts el or bulk refer to θ-independent values at the electrode surface or in the bulk, respectively, and the superscript Z stands for any species not involved in the reaction. The values of ΓB, φ, and ψ are easily obtained by application of the flux conditions in eq 46. Of interest + here is only the value of ΓB ) D-C-el/(D-Cel + D+Cbulk ), which shows that the boundary located at Γ ) ΓB moves into the solution when Cel changes, e.g., because the potential of the electrode varies. The above set of equations (48-51) is independent of the electrode shape, which means that the situation, although somewhat more complicated, is akin to that considered above for a simple Nernstian electron transfer (case I). Therefore, there is a true equivalence between the four electrodes, the equivalence Analytical Chemistry, Vol. 68, No. 24, December 15, 1996
4385
relationship being only geometric and requiring only then the equality of their σ values (Table 1). Conversely, when the boundary conditions, or some of them, depend on θ (e.g., when one of the products P or a reactant undergoes irreversible electron transfers at the electrode surface), electrode concentrations become a priori dependent on θ through the dependence of ζ(θ) (compare eq 12). This shows that the above set of eqs 48-51 can no longer be a solution of the diffusion problem. For the disk and band electrodes, the concentration profiles in the (Γ,θ) space become θ-dependent because of the boundary equation (compare case II). The situation is then formally identical to case II, and for the same reasons, no equivalence may exist between the disk and hemispherical electrodes. For the band/hemicylinder case, for the same reason as above (i.e., because the electrode concentrations are nearly independent of θ for slow charge transfer kinetics),6 there is still equivalence under these conditions. Equivalence involves then a geometric relationship (viz., r0hemicylinder ) wband/4) and a kinetic one imposed by the slow charge transfer kinetics (viz., [kel(E)]hemicylinder ) (4/π)[kel(E)]band).6 The presence of the extremely fast homogeneous bimolecular step imposes by itself no specific relationship. Case VI. Slow Homogeneous Kinetics. Irrespective of the antecedent or follow-up nature of the chemical step, or of the position where it occurs (viz., in a domain adjacent to the electrode surfacescases III and IVsor far in the solutionscase V), when at least one rate constant khomo is such that its dimensionless parameter λ ) khomo(Cbulk)m-1(a2/D) is comparable to unity, the kinetic layer relative to this step (at the electrode or within the solution) extends necessarily over a large fraction of the diffusion layer. Then, no simplification can be performed on the second Fick’s laws. In the (Γ,θ) space, the simplification introduced for the treatment of diffusion is compensated by the fact that a pseudodiffusion coefficient,1a,6-9 D(Γ,θ), always a function of Γ and possibly also of θ (disk or band), must be considered. It must be stressed here that D(Γ,θ) depends on the electrode shape (see Table 1) because it depends on the change of variables performed. In the absence of homogeneous kinetics, or in the presence of extremely fast ones (cases III-V), this does not introduce any difficulty, since D(Γ,θ)(∇Γ,θ)2C ) 0 is equivalent to (∇Γ,θ)2C ) 0. This is the reason why the existence of this pseudodiffusion coefficient has been omitted up to now. However, when a slow homogeneous step is considered, the whole diffusion operator must be considered, leading to the following expression after division by D(Γ,θ):
(∇Γ,θ)2C - β2[khomo/D(Γ,θ)]Cm ) (∇Γ,θ)2C - κ(Γ,θ)Cm ) 0 (52)
where β, a constant parameter with dimensions of a length, is introduced because of the dimensionless formulation of eq 52.1a,6-9 Equation 52 shows that the global effect of the (Γ,θ)-dependent diffusion coefficient amounts in fact to consider a fictitious rate constant, κ(Γ,θ) ) β2khomo/D(Γ,θ), that depends on Γ, possibly on θ (disk or band), and therefore on the electrode shape. This establishes that, even when the electrode boundary conditions is of the Nernstian kind (viz., constant concentration on the electrode surface), a dependence on the electrode shape is nevertheless introduced through that of the fictitious rate constant κ(Γ,θ). This 4386
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establishes unconditionally that no equivalence may exist between electrodes of different shapes under these conditions. DISCUSSION We have examined above paradigms of the most typical kinetic situations which can be encountered in classical electrochemical experiments.12 The only other typical case which has not been considered here deals with migration involvement and has been extensively investigated in previous works.15-17 From this overview, it appears that true equivalence may exist between the four classical electrodes considered here under a few circumstances (case I, case V Nernstian kind), that in other special ones (cases II-IV, case V non-Nernstian kind) equivalence may exist only between band and hemicylinder electrodes but not between disk and hemispherical ones, and finally that in the most general and useful situation (case VI) there is absolutely no equivalence possible at all. Equivalence Relationships. When there is equivalence, this requires generally a double set of conditions, one of geometrical nature (between the electrode dimensions) and the other of kinetic nature (between the rate constants). Thus, different values of rate constants have to be considered at each equivalent electrode. This is particularly important for kinetic measurements, when one wants to extract rate constants from voltammograms obtained, e.g., at band electrodes by using theories or simulations developed at a hemicylindrical electrode. Even when this is allowed because there is equivalence between the two electrodes under the conditions considered, the rate constant value obtained at the hemicylindrical equivalent must be corrected afterward. It is noteworthy that, in these cases, the true rate constants derive from those determined at the equivalent hemicylindrical electrode by application of the same scaling factor (Table 2), irrespective of the homogeneous or heterogeneous nature or of the reaction order of the kinetic step. All these geometric and kinetic relationships are recalled in Table 2 when they exist. They were developed for steady-state (disk/hemisphere) or quasi-steady-state (band/hemicylinder) conditions. We have not searched for possible equivalencies during the transitions between planar diffusion (viz., when (Dt)1/2 , r0 or w) and steady- or quasi-steady-state (viz., when (Dt)1/2 . r0 or w), i.e., for equivalencies that would apply at any value of Dt/a2 for a ) r0 or w. Indeed, it is obvious that no equivalence may exist under such circumstances, whatever the mechanism considered. This is established as follows. Under steady- or quasi steady state, a minimum requirement (but not always sufficient) for equivalence, if any, is that the two electrodes considered must have the same σ values: r0hemisphere ) (2/π)r0disk for the hemisphere/ disk pair and r0hemicylinder ) w/4 for the hemicylinder/band one (Table 2). On the other hand, equivalence at short time scales, i.e., when planar diffusion occurs, imposes only the equality of the surface area of the two electrodes. For the hemisphere/disk pair, this imposes that r0hemisphere ) 2-1/2r0disk, while one obtains r0hemicylinder ) w/(2π) for the hemicylinder/band pair. These relationships cannot be reconciled with those that result from the identity of the σ values. The two sets of relationships are exclusive and cannot be verified simultaneously for the planar and steady (or quasi-steady)-state limits. This result is sufficient to establish (16) Amatore, C.; Deakin, M. R.; Wightman. R. M. J. Electroanal. Chem. 1987, 220, 49. (17) Amatore, C.; Fosset, B.; Bartelt, J.; Deakin, M. R.; Wightman. R. M. J. Electroanal. Chem. 1988, 256, 255.
that, whatever the voltammetric variable considered, its variations with Dt/a2 (a ) r0 or w) at one electrode cannot be derived using a unique analytical relationship from those determined at another electrode, since the would-be analytical relationship must necessarily be different in the short and long time domains. Thus, no equivalence can be found between any couple of electrodes of different shapes during the transition between planar diffusion and steady or quasi steady state.6 Experimental Significance of Nonequivalence. What precedes concerns “true equivalence”. This is defined here by the fact that two equivalent electrodes give rise to voltammograms that are exactly superimposable, or at least whose differences are so very small that they cannot be detected experimentally (as, for example, for the band/hemicylinder pair in cases II-V).5 Empirical equivalence relationships could be proposed under some circumstances where a true equivalence does not exist (Table 2), but these would be necessarily ad hoc and fuzzy because they are limited to a range of potential values, to a range of rate constants, to certain particular kinetic schemes, etc. Although the existence of such “locally approximated” ad hoc equivalencies may prove extremely useful for some experimental purposes, we did not consider them here because they do not reflect real physicochemical phenomena but rather particular and local ad hoc data processing approximations (compare, e.g., Figure 3b). To stress this point, let us consider the following typical situation. For example, for the disk/hemisphere couple in case II, eqs 18 and 26 coincide mathematically if, at the hemisphere, the rate constant variation with the potential is such as [kel(E)]hemisphere ) (π/2)Θ(i/ilim)[kel(E)]disk. Alternatively, a voltammogram computed at a hemisphere could be used to generate a posteriori a voltammogram for a disk of same σ by introducing a “shape-overpotential” such as Edisk ) Ehemisphere + (RT/RnF) ln[Θ(i/ilim)]. Thus, considering that the function Θ(i/ilim) may be evaluated once for all (Figure 3a), one could think of generating voltammograms at disk electrodes for a broad range of rate constants on the basis of the above shape-overpotential correction and using only theories or simulations performed at hemispherical electrodes. The fact that this is nonsense is readily shown by considering a situation in which two successive irreversible electron transfers occur (e.g., EirrevEirrev mechanism). Then, each electrochemical step should impose its own shape-overpotential correction, which is impossible except when the two E1/2s coincide. Similarly, considering an EirrevCfastE situation (combination of cases II and III) would lead to the same impossibility, since the shape-overpotential correction for the extremely fast chemical step would impose now Edisk ) Ehemisphere + [2RT/(m+1)nF]ln[Ξ(E - E1/2)] in addition to the above one. Both corrections are obviously impossible to use simultaneously, except maybe in the very particular case when m ) 0 (e.g., Michaelis-Menten-like kinetics) and R ) 1/2. These simple and ubiquitous cases illustrate adequately the impossibility of relying blindly on such ad hoc “equivalencies” when no real equivalence exists (Table 2). It can be concluded that, except under a few circumstances (Table 2), interpretation of data obtained at a disk or a band cannot be performed by using theories or simulations derived at a hemisphere or at a hemicylinder, respectively. Theoretical predictions must then be obtained by considering the true shape of the electrode. In these cases, it must be emphasized that deriving
the numerical or analytical solutions in the (Γ,θ) conformal space is considerably simplified with respect to deriving it in the real space. The (Γ,θ) conformal space suppresses or at least smoothes the problems connected with the infinite extension of diffusion layers (band and hemicylinder) and with discontinuities (disk and band) at the edge(s) of the electrode (compare Figure 2). This greatly reduces the size and complexity of simulation grids.6-9,15-18 What precedes shows that nonequivalence between two electrodes imposes that they produce different voltammograms for the same system, at least from a strict theoretical point of view. However, since in this work we have been focusing only on the physicochemical origin of equivalencies and nonequivalencies, it is hard to predict how far the two voltammograms differ. Furthermore, this is also a matter of theoretical accuracy versus experimental precision. Indeed, experimentally, the degree of adequacy of using, e.g., simulations at a hemisphere to extract data obtained at a disk depends also on the precision of the data and on the sought degree of accuracy of the procedure. This is apparent in Figure 3b. Whenever the experimental scatter of data points is much larger than the difference between the two curves, either curve may be used, provided the right kinetic parameter is considered. However, this amounts to interchange precision and accuracy, since one plays then on a poor precision to avoid the need for a good accuracy. To the best of our knowledge, this has been investigated thoroughly only for the slow charge transfer case (case II)6,10 and would result in an infinite, sterile, and tedious task if this should be extended to any possible mechanism. However, based on the identities between the boundary conditions and between the equivalence requirements (see, e.g., eqs 28, 30, and 39), it is expectable that, for most mechanisms, the voltammetric differences will result of the same order as those shown in Figure 3b. Indirect validation of this intuitive statement is given by a former work, where ECE and DISP1 mechanisms have been compared at disk and hemispherical electrodes.18 It was thus shown that the hemispherical approximation was sufficient to understand the trends and fundamental kinetic features observed experimentally.18 Why Band/Hemicylinder Pairs Are More Often Equivalent than Disk/Hemisphere Ones. To conclude this discussion, we wish to elaborate on the origin of the difference between the disk/hemisphere pair and the band/hemicylinder one. Indeed, we have established here or previously6 that, under some specific circumstances (Table 2), equivalence exists only between band and hemicylinder electrodes but not between disk and hemispherical ones.10 This conclusion may seem puzzling since the “physical” relation existing between the shapes of the electrode appears globally similar within one pair. In some cases (Table 2), an equivalence exists for the band/ hemicylinder pair but not for the disk/hemisphere one. This arises because, in the first case, the following ratio of integrals,
∫ C ζ(θ) dθ Θ(ξ) or Ξ(ξ) ∝ [∫ C dθ] 1
χ
el
0
1
0
χ
(53)
el
(where χ is a constant; compare, e.g., eqs 28, 29, and 39) is nearly independent of the potential, while in the second case it depends (18) Ciolkowski, E. L.; Maness, K. M.; Cahill, P. S.; Wightman, R. M.; Evans, D. H.; Fosset, B.; Amatore, C. Anal. Chem. 1994, 66, 3611.
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largely on it (see, e.g., Figure 3a). The physical origin of this important difference is that at the band, the concentration tends to be constant over the electrode surface, so that
Θ(ξ) or Ξ(ξ) ∝
∫ ζ(θ) dθ ) ∫ ζ(θ) dθ ) constant C [∫ dθ] 1
Celχ
1
0
χ
el
1
0
χ
0
(54)
while this is not the case at the disk. This important difference is related to that existing between the diffusion layers. The thickness of the diffusion layer at the band or hemicylinder electrode extends to infinity when time increases (because the term in Table 1 tends to infinity), thus leading to a quasi-steadystate behavior. This contrasts with the steady-state diffusion layer at a disk or hemisphere whose dimension is comparable to its radius. Therefore, at the band electrode, the effect of heterogeneous kinetics, or of extremely fast homogeneous kinetics occurring near the electrode surface, is “smoothed” during the long diffusional pathway, while this cannot occur for the disk. Indeed, in the presence of heterogeneous kinetics or extremely fast homogeneous kinetics taking place near the electrode surface, the concentration profiles are affected over distances from the electrode surface that are comparable to the electrode width or radius. Thus, in the disk case, the kinetically perturbed region (i.e., where concentrations are not only a function of the distance to the center or to the axis of the electrode) fills an important fraction of the diffusion layer (compare, e.g., Figure 3c), while it represents only a very small fraction of it at the band electrode. This becomes readily apparent when one considers the (Γ,θ) space in Figure 2. Owing to the definitions of Γ and θ for each case (Table 1), the diffusion layer thickness (viz., 0 e Γ e 1) is independent of the electrode shape in the (Γ,θ) space. This implies that, in this space, a region which extends from the band electrode surface over distances comparable to the electrode width corresponds a very thin strip adjacent to the electrode surface. Since the (Γ,θ) space is constructed so that all gradients remain finite, the concentration variations that occur across this thin strip are necessarily negligible. This imposes that the variation of the
concentration at the electrode surface is also negligible, irrespective of the presence of heterogeneous or extremely fast homogeneous kinetics taking near the electrode surface.19 Thus, although the average electrode concentration depends on these kinetics, the situation remains formally identical to case I for what concerns the diffusional concentration drop in the diffusion layer, since most of this drop occurs in a range where the system is θ independent. This is the main root of the equivalence between band or hemicylinder electrodes. Conversely, for the disk, the kinetically perturbed region extends over a significant fraction of the diffusion layer, even in the (Γ,θ) space (see, e.g., Figure 3c), since the diffusion layer is also comparable to the electrode radius. Concentration profiles remain θ dependent over the most part of the diffusion layer, thus affecting strongly the distribution of the concentrations on the electrode surface (Figure 3c). In some respect, for the disk the situation is always similar to case VI, although the true kinetic steps which are at the origin of this dependence on θ may be limited to the electrode surface or to its close vicinity (cases IIIV). This explains why a disk and a hemisphere cannot be equivalent under most circumstances involving kinetics (Table 2). In other words, the disk and the hemisphere cannot be equivalent under most electrochemical circumstances because they obey a true steady-state behavior (i.e., finite diffusion layer of dimension comparable to the electrode radius), while the band and the hemicylinder remain equivalent except in case VI because they obey quasi-steady-state behavior (i.e., steady state within a time-dependent diffusion layer of dimension considerably exceeding the electrode width or radius).1a ACKNOWLEDGMENT This work has been supported in part by CNRS (URA 1679) and by ENS and MESR. C.A. acknowledges discussions with Steve Feldberg, which helped greatly to improve the presentation of some key features of this paper. Received for review April 26, 1996. Accepted September 26, 1996.X AC960421S
(19) This is, indeed, the principle of the analytical demonstration that is given in ref 6.
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Abstract published in Advance ACS Abstracts, November 1, 1996.
