Anal. Chem. 2005, 77, 6481-6486
General Theory for Migrational Voltammetry. Strong Influence of Diversity in Redox Species Diffusivities on Charge Reversal Electrode Processes. Wojciech Hyk* and Zbigniew Stojek
Department of Chemistry, Warsaw University, ul. Pasteura 1, 02-093 Warsaw, Poland
High sensitivity of the microelectrode response to the difference between the substrate and the product diffusion coefficients is predicted for the charge reversal processes. This effect is anticipated from the general theoretical model developed for the diffusional-migrational transport to microelectrodes. The model predicts the voltammetric wave heights for any type of electrode processes carried out in the presence of any number and concentration of nonelectroactive ions. It involves changes in diffusion coefficients of the redox species and assumes no homogeneous complications. Handy, analytical expressions for the limiting current and limiting potential can be derived for a system of a univalent product and univalent ions of supporting electrolyte. This case covers charge reversal processes of the following type: Sz f P( + ne (n + z ) sgn(n), |n| g 2). It has been shown that under migrational conditions the change in the ratio of the product and the substrate diffusivities (DP/DS) by as little as 10% results in significant changes in the voltammetric wave height. For 2-e charge reversal processes, a 10% increase in DP versus DS leads to a drop in the voltammetric wave height of 18.3% compared to that calculated for equal diffusion coefficients. The reversed change, i.e., the 10% decrease of the DP value with respect to DS, increases the voltammetric wave height by 30.5% compared to that obtained for equal diffusivities. The theoretical predictions were confronted with our recent experimental results obtained for the 2-e oxidation of sodium (6,8-diferrocenylmethylthio)octanoate, which process can be classified as the charge reversal reaction. The best fit was obtained for DP/ DS equal to 0.71. Recently, we have reported on the 2-e oxidation of the negatively charged diferrocene derivative (6,8-diferrocenylmethylthio)octanoate that may by classified as a charge reversal process.1 The charge, or sign, reversal processes are electrode * Corresponding author. E-mail:
[email protected]. Phone/fax: (+4822) 822 4889. (1) Hyk, W.; Nowicka, A.; Misterkiewicz, B.; Stojek, Z. J. Electroanal. Chem. 2005, 575, 321. 10.1021/ac051097g CCC: $30.25 Published on Web 08/25/2005
© 2005 American Chemical Society
processes that undergo with the reversal of the sign of electroactive species charge numbers in a single step. Thus, they require exchange of at least two electrons in one step. The main feature of this class of heterogeneous processes, being a direct consequence of the preservation of the electroneutrality principle, is a strong enhancement of the faradaic current under the conditions of supporting electrolyte deficit. The total elimination of supporting electrolyte from the system should result in a limitless increase of faradaic current as predicted by several models of migrational voltammetry for this type of process.2-5 In real physical systems, however, there are several unwanted effects that make this intriguing prediction impossible to observe. One of them is the unavoidable presence of an unspecified amount of ionic impurities (which act as supporting ions even if supporting electrolyte is not added), and the other one is the occurrence of a homogeneous, either chemical or redox, reaction between the product and the substrate. These effects drastically lower the enhancement of the voltammetric current even if the deliberately added supporting electrolyte is present at a very low level. As a consequence, in the absence of intentionally added supporting electrolyte instead of predicted ramp-shaped voltammograms, typical wave-shaped signals should be observed. The above considerations are only in part consistent with our experimental results. In fact, we were unable to record rampshaped voltammograms but the experimental voltammetric waves were significantly higher than the theoretical results predicted by the idealized models (i.e., models assuming the equality of redox-species diffusivities and the absence of homogeneous complications). Surprisingly, these differences are even greater if the experimental data are compared to the theoretical predictions based on the model that involves the comproportionation reaction.6 Interestingly, the discrepancies seen might be minimized if unequal diffusion coefficients of the electrode process (2) Amatore, C.; Fosset, B.; Bartelt, J.; Deakin, M. R.; Wightman, R. M. J. Electroanal. Chem. 1988, 256, 255. (3) Oldham, K. B. J. Electroanal. Chem. 1992, 337, 91. (4) Myland, J. C.; Oldham, K. B. J. Electroanal. Chem. 1993, 347, 49. (5) Ciszkowska, M.; Stojek, Z. J. Electroanal. Chem. 1999, 466, 129. (6) Amatore, C.; Bento, M. F.; Montenegro, M. I. Anal. Chem. 1995, 67, 2800.
Analytical Chemistry, Vol. 77, No. 19, October 1, 2005 6481
substrate and product are taken into account.7 A different transport mechanism for this class of electrode processes, other than that assumed in the theoretical models, may be considered as an alternative explanation of the unexpected results obtained. It is rather clear that the reversal of the redox-species charge sign during the electrode process may lead to significant changes in the conformation and solvation of the generated ion. These changes may affect directly the diffusion coefficient. Therefore, the interpretation of the voltammetric responses for such processes might be wrong if the assumption of the equality of the redox-species diffusivites is made. In this paper, we focus our attention on the influence of redoxspecies diffusivities on the microelectrode responses for charge reversal systems. To explore these new effects, a theoretical model that involves the changes in both diffusion coefficients and concentration of supporting electrolyte is developed. The model is a broad extension of our recent theory developed for the systems with no supporting electrolyte.8 The theoretical scheme presented in this paper can be applied to any electrode process carried out in the presence of any number of nonelectroactive ions. However, the handy analytical expressions for the limiting current and the electrostatic potential in the solution can only be derived for a univalent product and univalent ions of supporting electrolyte. This case covers charge reversal processes of the following type: Sz f P( + ne (n + z ) sgn(n), |n| g 2). THEORETICAL MODEL Model Cell. A two-electrode system with a hemispherical working microelectrode placed on an infinite insulating plane and semi-infinite transport field are assumed. Initially, the solution contains the electrode process substrate (S) of charge zS accompanied by the counterion and at least a trace of supporting ions. The model does not impose any limitation on the number and the charge of different electroinactive ions that form soluble electrolytes. It is assumed that the product (P) is initially absent in the solution. The electrode process SzSfPzP + ne proceeds in a single step and is not coupled to chemical reactions (where zP is the product charge number and n is the number of electrons transferred per molecule equal to zP - zS, positive for oxidation and negative for reduction). It is also assumed that the size of the microelectrode is not too small, so that the thickness of the double layer is an insignificant fraction of the transport layer thickness. Under such the conditions, the electroneutrality principle holds throughout the solution.9-12 No restrictions on the magnitudes of the diffusion coefficients of all species are imposed. Formulation of the Problem. In a system with no excess of inert electrolyte, both diffusion and migration contribute to the
total flux of the transported substance, fi, according to the NernstPlanck law
fi ∂Ci ziCiF ∂Φ + )∂r RT ∂r 2πr2D
(1) i
or using space variable transformation
∂Ci
+
∂(1/r)
ziCiF ∂Φ fi ) RT ∂(1/r) 2πDi
(2)
where Ci, zi, and Di denote ith species concentration, charge number, and diffusion coefficient, respectively, r is the radial distance, Φ is the local electrostatic potential existing in the solution, and F, R, and T have their usual meaning. Under the steady-state conditions, the fluxes become independent of time and for electroactive species they are, via Faraday’s law, proportional to the current (I) flowing through the interface due to the electrooxidation or electroreduction, while the fluxes of electroinactive species are equal to zero. The transport equations for the steady-state conditions, written for each solute, constitute the following system of differential equations
dcS I dΨ + zScS )-L dγ dγ I
(3)
dcP I dΨ + zPcP ) L dγ dγ θI
(4)
d
d
dcj dΨ + zjcj )0 dγ dγ
j ) 1,..., N
(5)
which were made dimensionless using the following substitutions
ci ) Ci/CbS
i ) S, P, N supporting ions
(6)
θ ) DP/DS
(7)
γ ) re/r
(8)
Ψ ) F/RT(Φ - Φb)
(9)
where N is the number of supporting ions plus the substrate counterion, DP and DS are diffusion coefficients of the product and the substrate, respectively, re is the microelectrode radius, ILd is the limiting current at hemispherical electrodes for the purely diffusional conditions, and superscript b is related to the bulk of the solution. By adding the relation for the electroneutrality principle N
zScS + zPcP +
∑z c ) 0 j j
any γ
(10)
j)1
(7) Leventis, N.; Oh, W. S.; Gao, X.; Rawashdeh, A.-M. M. Anal. Chem. 2003, 75, 4996. (8) Hyk, W.; Stojek, Z. Anal. Chem. 2002, 74, 4805. (9) Norton, J. D.; White, H. S.; Feldberg, S. W. J. Phys. Chem. B 1990, 94, 6772. (10) Oldham, K. B.; Feldberg, S. W. J. Phys. Chem. B 1999, 103, 1699. (11) Feldberg, S. W. Electrochem. Commun. 2000, 2, 453. (12) Oldham, K. B.; Bond, A. M. J. Electroanal. Chem. 2001, 508, 28.
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Analytical Chemistry, Vol. 77, No. 19, October 1, 2005
and the following boundary conditions
γ f 0:
cS f 1,
cP f 0,
the set of eqs 3-5 becomes solvable.
cj f cbj ,
Ψf0
(11)
Before solving, similarly to our derivation scheme presented earlier,8 the transport equations are rearranged with respect to Ψ, which is defined as a new independent variable.13 This yields the following form of the equation set
(
)
dcS I dΨ + zScS )-L dΨ dγ I
(
(13)
j ) 1,..., N
dγ
( )
ILd θ
N
cS ) e
∑
+
j)1
{
N
∑ j)1
{
2
- zS (zS cS + zP cP +
∑
2
zj cj)
-1
∑R )(e
λΨ
j
- 1) -
j)1
[zS(zS - zP)Rj + zj(zj - zP)cjb](e-zjΨ - 1) zj
}}
(19)
for λ * 0, i.e., for zSzP(θ - 1) * 0, where
Rj ) (zjθ(zj - zP)cbj )/(zSzP(θ - 1) - zj(zSθ - zP))
For zSzP(θ - 1) ) 0, i.e., for either equal diffusion coefficients of the substrate and the product, or uncharged substrate, or uncharged product, this relation has the form
I ILd
γ)
θ zP - zSθ
{
N
zS(zS - zP)(1 -
N
∑R )Ψ - ∑ j
j)1
j)1
{
1
[zS(zS zj
zP)Rj + zj(zj - zP)cbj ](e-zjΨ - 1) (15)
}}
(20)
j)1
zjθ(zj - zP)cbj
}
(e-zjΨ - eλΨ)
zSzP(θ - 1) - zj(zSθ - zP)
zj(zj - zS)cbj
(eλΨ - e-zjΨ)
zSzP(θ - 1) - zj(zSθ - zP) cj ) cbj e-zjΨ
{
N
1
N
2
(16)
cP )
∑
(14)
By multiplying eqs 12-14 by the reciprocal of eq 15, one eliminates the potential gradient in the left-hand sides of these equations and, finally, obtains the set of linear differential equations with respect to Ψ. They can be easily integrated8 and the following solutions are obtained
λΨ
{
zS (zS - zP)(1 zP - zSθ λ θ
j)1
Solving the Transport Equations. Before the transport equations are integrated, one will need to find the explicit form of the electrostatic potential gradient, dΨ/dγ. This task was obtained by summing eqs 3-5 multiplied by the corresponding charge of the species and using the electroneutrality condition. The result is as follows
)
γ)
N
)
)
I zP
ILd
d
dcj dΨ + zjcj )0 dΨ dγ
dΨ
I
(12)
dcP I dΨ + z P cP ) L dΨ dγ θId
(
to the bulk of the solution (Ψ ) 0 ) γ). The result is the following
j ) 1,..., N
}
(17)
(18)
where Rj is simplified to (zjθ(zP - zj)cbj )/(zj(zSθ - zP)). Equations 19 and 20 apply to any point in the solution under the steady-state conditions, including the most useful point, the one located at the solution-electrode interface (i.e., at r ) re). These equations, together with eqs 16-18, allow one to construct a voltammogram for any θ, cbj , zS, zP, and zj set. The algorithm consists of several steps to be performed:8 (1) selection of a small number either positive (oxidation) or negative (reduction) for the increment of the electrostatic potential at the electrode-solution interface, δΨ; (2) addition of the potential increment to the actual value of Ψ at γ ) 1 (Ψ1); (3) calculation of the dimensionless current (I/ILd) at γ ) 1 via eq 19 or 20; (4) calculation of the surface concentrations of the substrate and the product (cS,1 and cP,1seqs 16 and 17) using the actual value of Ψ1; (5) computation of the true electrode potential (Er - E0) using the appropriate expression (eqs 21-23) for the selected electrode process mechanism:
Er - E0 ) where
λ ) (zSzP(θ - 1))/(zP - zSθ)
Equations 16-18 are the general solutions of the equation set 3-5. They describe the potential dependencies of the concentrations of the substrate, the product, and N electroinactive ions, respectively. To find out how the electrostatic potential varies with the current and the distance from the electrode surface, one has to integrate relation 15 using the lower limit as that corresponding (13) Kharkats, Y. I. J. Electroanal. Chem. 1979, 105, 97.
( )
cP,1 RT ln nF cS,1
(21)
for reversible processes, where E0 refers to the standard potential of the redox couple,
{
[
]
βnF(Er - E0) I ) k c exp S,1 RT ILd cP,1 exp
[
]}
(1 - β)nF(Er - E0) RT
(22)
for quasi-reversible processes, where β is the electron-transfer Analytical Chemistry, Vol. 77, No. 19, October 1, 2005
6483
coefficient and k is the dimensionless standard heterogeneous rate constant (equal to ksre/DS), and
Er - E0 )
( )
RT I ln L βnF IdkcS,1
(23)
for totally irreversible processes; (6) evaluation of the potential imposed to the electrode (the total electrode potential, E) by adding the value calculated in step 5 to RTΨ1/F; (7) introduction of the dimensionless current from step 3 to the plot of the current against the total electrode potential from step 6; (8) verification whether cS,1 is very close to or less than zero. If so, the voltammogram is completed, otherwise, a new iteration (steps 2-8) is required. When the total concentration polarization of the substrate occurs at the electrode, the current and the electrostatic potential will approach their limits. The corresponding analytical expressions describing these parameters can only be derived for the processes that generate univalent product from the substrate of any charge accompanied by the univalent counterion and are carried out in the presence of at least a trace of supporting electrolyte consisting of univalent ions. This case covers electrooxidations producing univalent cations from an anionic substrate and electroreductions generating univalent anions from a cationic substrate, i.e., the charge reversal processes of Sz f P( + ne type (n + z ) sgn(n) and |n| g 2, N ) 2). By setting cS to 0 in eq 16 and by making some tedious rearrangements, one obtains the following expressions for the limiting electrostatic potential at the interface (limiting ohmic potential drop across the cell) and the limiting current
[
]
1 + 2θ(ξ + |z|) - |z| 1 + |z|θ ΨL ) -sgn(z) ln 1 + 2|z|θ - |z| 2θξ
(24) IL θ {1 + |z| + 2ξ[1 - exp(-sgn(z)ΨL)]} (25) ) ILd θ - 1 where ξ is the support ratio, defined as cbse/cbS (the ratio of the bulk concentrations of supporting electrolyte and the substrate). Equations 24 and 25 apply to the charge reversal processes of Sz f P( + ne type. They correspond to the steady-state conditions; therefore, they are, in fact, independent of the microelectrode geometry and the experimental technique used to produce them. Please note, that the absolute value of z is used in eqs 24 and 25, and “sgn” denotes the signum function. It is worth noting that eqs 24 and 25 are derived for the absence of the comproportionation reaction. For charge reversal electrode processes, however, instead of simultaneous transfer of n electrons in a single step, several independent one-electron transfers (seen as several waves on the voltammogram) can take place. This favors occurrence of the comproportionation reaction. On the other hand, if the difference between the half-wave potentials of neighboring waves is very small, then the efficiency of the comproportionation reaction is rather low and it cannot affect the predictions based on eqs 24 and 25 significantly. RESULTS AND DISCUSSION The graphical representation of eq 25 for the process S- f ( P ( 2e is illustrated by the three-dimensional plot in Figure 1. 6484
Analytical Chemistry, Vol. 77, No. 19, October 1, 2005
Figure 1. Theoretical steady-state limiting current, made dimensionless with respect to the diffusion-limited current, for the electrode processes of S- f P( ( 2e type, versus common logarithms of support ratio and the ratio of diffusion coefficients of the electrode process product and the substrate.
In this figure, the dimensionless limiting current is plotted against common logarithms of support ratio and DP/DS ratio. It is seen that the limiting response is extremely sensitive to the changes in the redox-species diffusivities, particularly for small values of support ratio. On the other hand, by increasing support ratio, i.e., by reducing migrational contribution to the total transport, the limiting current becomes independent of DP/DS ratio and approaches the value predicted for the purely diffusional conditions. To demonstrate the strong influence of the redox-species diffusion coefficients on the magnitude of the limiting current under the conditions of severe deficit of supporting electrolyte, we have compared the theoretical voltammograms calculated for equal diffusivities of the product and the substrate to those obtained for diffusion coefficients differing as little as by 10%. The comparison was made for the process S- f P( ( 2e at ξ ) 0.0001 for the three values of the DP/DS ratio (0.9, 1.0, and 1.1). The results are shown in Figure 2. The 10% drop in the DP value compared to DS increases the voltammetric wave height by 30.5% with respect to that obtained for equal diffusion coefficients. The 10% increase in the product diffusivity, in turn, lowers the voltammetric height by 18.3% compared to that calculated for equal diffusion coefficients. The corresponding concentration profiles of each species present in the system and calculated for the oxidation process are shown in Figure 3. As can be predicted, the smaller the DP/DS ratio, the larger is the concentration of the product in the layer adjacent to the electrode. The increased accumulation of the product is the direct cause for the larger enhancement of the oppositely charged substrate transport and, in consequence, the larger enhancement of the migration-driven faradaic current. In general, one can distinguish the following ranges of θ and ξ variables that define mathematical boundaries for the limiting
Figure 2. Theoretical voltammetric responses for the process Sf P( ( 2e calculated for the support ratio of 0.0001 and for DP/DS ratio of 0.9 (a), 1.0 (b), and 1.1 (c). The current is made dimensionless with respect to diffusion-limited current.
current given by eq 25 and illustrated in Figure 1:
1. DP , DS (θ f 0),
ξ
