Anal. Chem. 2008, 80, 7957–7963
Capacitive and Solution Resistance Effects on Voltammetric Responses of a Thin Redox Layer Attached to Disk Microelectrodes Christian Amatore,† Alexander Oleinick,†,‡ and Irina Svir*,†,‡ De´partement de Chimie, Ecole Normale Supe´rieure, UMR CNRS-ENS-UPMC 8640 “PASTEUR”, 24 rue Lhomond, 75231 Paris Cedex 05, France, and Mathematical and Computer Modelling Laboratory, Kharkov National University of Radioelectronics, 14 Lenin Avenue, 61166 Kharkov, Ukraine A rigorous theoretical analysis of cyclic voltammetry of surface-attached redox layers at disk microelectrodes is presented when effects enforced by the solution resistance and the electrode capacitance cannot be neglected. This allows a precise quantitative evaluation of the influence of each of the current components (faradaic, resistive, and capacitive) on the voltammetric shapes through numerical simulation. It is shown that the consideration of the solution resistance and capacitance effects is crucial for the correct treatment of experimental voltammograms at high-voltage scan rates when the resistance is not compensated. This article is a continuation of our previous work on the investigation of the effect of uncompensated solution resistance on steady-state and transient voltammograms at the disk microelectrode.1 In the first part,1 the steady state conditions made possible neglecting the capacitive components, though it was noted that this should play an important role in transient voltammetry at high scan rates. In this work we examine the interplay between capacitive and solution resistance effects when taking into account the heterogeneity of currents and resistance at a disk electrode to understand which role such effects can play and how this is reflected on the voltammetric response. This is performed here under the specific conditions of thin redox layers, i.e., when the redox species are confined within a thin layer attached to the surface of the disk microelectrode. Recently a similar system with a surface-attached layer on a planar macroelectrode undergoing reversible electron transfer under cyclic voltammetry was considered by Feldberg2 yielding an empirical expression linking the peak current and peak potential. Previous studies on the subject include also the seminal * Corresponding author. E-mail:
[email protected];
[email protected]. † Ecole Normale Supe´rieure. ‡ Kharkov National University of Radioelectronics. (1) Amatore, C.; Oleinick, A.; Svir, I. Theoretical Analysis of Microscopic Ohmic Drop Effects on Steady-State and Transient Voltammetry at the Disk Microelectrode: A Quasi-Conformal Mapping Modeling and Simulation. Anal. Chem. 2008, 80, 7947-7956. (2) Feldberg, S. W. Effect of Uncompensated Resistance on the Cyclic Voltammetric Response of an Electrochemically Reversible Surface-Attached Redox Couple: Uniform Current and Potential Across the Electrode Surface. J. Electroanal. Chem. In press, DOI: 10.1016/j.jelechem.2008.07.20. 10.1021/ac8012972 CCC: $40.75 2008 American Chemical Society Published on Web 10/01/2008
work on solution resistance at a disk electrode by Newman3 and a more detailed work by Oldham4 considering also the capacitive current component. However, these previous works considered the faradaic and capacitive components of the current as totally independent so that the overall current is given by their summation. Indeed, this assumption, which has been contested in the past by Save´ant,5 represents only a limiting situation when the solution resistance is negligible so that there is no ohmic contribution to the overall current. In reality the deviation of the current from the predicted one becomes increasingly important when the solution resistance increases. Moreover when the cell time constant is not negligible versus scan rate, the coupling of the faradaic, resistive, and capacitive currents is extremely strong even at microelectrodes. This coupling is even more intense when nonuniform distributions of the current density occur along the electrode surface as for the disk electrode. Thus the consideration of experimental voltammograms in order to extract quantitative and even qualitative information must take simultaneously into account all the above-mentioned coupled effects. In this paper we wish to present a comprehensive mathematical model which allows analyzing the effects of solution resistance and double layer capacitance (i.e., by obtaining each of the current components) on cyclic voltammograms at a disk microelectrode. On the other hand, the results of the numerical simulations presented in this paper establish that the usual practice of voltammogram correction by subtracting the capacitive current recorded using a blank solution (which is equivalent to considering faradaic and capacitive components independent) may lead to wrong interpretation of experimental data unless the cell resistance is negligible or fully compensated electronically.6 Though the analysis presented here is more complete and extended, this validates the previous caveat raised by Save´ant.5 (3) Newman, J. J. Electrochem. Soc. 1966, 113, 501–502. (4) Oldham, K. Electrochem. Commun. 2004, 6, 210–214. (5) Andrieux, C. P.; Garreau, D.; Hapiot, P.; Pinson, J.; Save´ant, J. M. J. Electroanal. Chem. 1988, 243, 321–335. (6) (a) Amatore, C.; Maisonhaute, E.; Simonneau, G. Electrochem. Commun. 2000, 2, 81–84. (b) Amatore, C.; Maisonhaute, E.; Simonneau, G. J. Electroanal. Chem. 2000, 486, 141–155.
Analytical Chemistry, Vol. 80, No. 21, November 1, 2008
7957
radial position on the disk surface1 and for the case of oxidation (including the solution resistance) may be written as
[
kf(r, t) ) k0 exp (1 - R)
[
]
nFE*(r, t) ) RgT
k0 exp (1 - R)
Figure 1. Sketch of the system under scrutiny. For sake of clarity the thickness (δ) of the redox film is considerably enlarged.
[
kb(r, t) ) k0 exp -R
MODEL Mathematical Model. A sketch of the modeled system is shown in Figure 1. The system under scrutiny consists of the microdisk electrode of radius rd covered with a film/thin layer of thickness δ containing an electroactive species A at the initial concentration c0 which undergoes an oxidation with the transfer of n electrons (A - ne- f B). We make the following assumptions which correspond to the common experimental practice: (i) the bulk solution outside the surface-attached layer contains an inert electrolyte but is free of electroactive particles which remain confined within the thin layer; (ii) the thickness of the layer is much smaller than the electrode radius, i.e. δ,rd; and (iii) the diffusion coefficients of the redox species A/B in the film are supposed to be equal and have a common value D; we do not need to specify whether or not D is a “true” or an “electron hopping” diffusion coefficient. The mathematical model can be written as
[
]
(1)
c)0
(2a)
∂c ∂2c 1 ∂c ∂2c )D 2 + + ∂t r ∂r ∂z2 ∂r with the initial (t ) 0): 0 e r e rd,
0 e z e δ,
and boundary conditions (t > 0) are z ) 0,
0 e r e rd,
r ) 0,
r ) rd,
D
∂c ) -kf(r, t)(c0 - c) + kb(r, t)c; ∂z (disk-electrode) (2b)
0 < z < δ,
0 < z < δ,
∂c ) 0; ∂r
∂c ) 0; ∂r
(side
(symmetry
axis)
[
z ) δ,
∂c ) 0; ∂z
(top
where k0 is the standard electrochemical rate constant; R is the transfer coefficient; F is the Faraday constant; Rg is the gas constant; T is the temperature; E0 is the formal potential of the A/B redox couple; di(r,t) dRe(r) is the ohmic drop applying at the distance r from the electrode center, where each of the factors (current and resistance) are associated with the infinitesimally small ring of the radius r. E*(r,t) is the effective potential across the interface between the metal conductor and its adjacent solution at the distance r from the electrode center: E*(r, t) ) E - E0 - di(r, t)dRe(r)
(4)
where E is the applied potential; di(r, t) is the overall current through the elementary ring; dRe(r) is the resistance of the current tube associated with the same ring. Note that though di(r,t) and dRe(r) are both differentials, their product is a true electrical potential drop applying at a distance r from the disk center at time t and therefore is not a differential. Since the thickness of the film (or thin layer) is much smaller than the disk radius, unless the film is extremely resistive vs the solution (which is generally not the case) we can neglect the film resistance difference vs that of the solution. Similarly, since the thin film thickness is infinitely small vs the disk radius in common experimental practice, the geometry of the solution cone defining its electrical resistance is essentially identical to that of a disk electrode placed into a semi-infinite electrolyte. Hence, all expressions related to the potential and elementary resistance obtained in ref 1 are applicable. We recall that the expression for the current derived ibidem reflects only the cylindrical symmetry of the system and therefore can be used here too:
di(r, t) dRe(r) ) -
π nFD ∂c 2 r - r2 2 γ ∂z √ d
(5)
E*(r, t) ) E - E0 +
π nFD ∂c 2 r - r2 , 2 γ ∂z √ d
(6)
boundary of the layer)
boundary of the layer) (2e)
where c is the concentration of B; t is the time, r, z are the cylindrical coordinates; kf(r,t) and kb(r,t) are the forward and backward electron transfer rate constants which depend on the 7958
]
nF (E - E0 - di(r, t)dRe(r)) (3b) RgT
(2c)
(2d)
0 e r e rd,
]
nFE*(r, t) ) RgT
k0 exp -R
]
nF (E - E0 - di(r, t)dRe(r)) (3a) RgT
Analytical Chemistry, Vol. 80, No. 21, November 1, 2008
where γ is the solution conductivity. Incorporating expressions in eqs 3-6 into the boundary condition (eq 2b) yields a nonlinear initial and boundary value problem for the diffusion eq 1 which must be solved numerically. The shape of the computational region (see Figure 1) allows performing simulations on a uniform rectangular grid in a bounded region (attached thin layer).
Modeling of Solution Resistance and Capacitive Currents. Here we consider the problem of the solution resistance and capacitive currents at a disk microelectrode. Using the classical formula, C)
Q V
(7)
where C is the capacitance, Q is the charge stored at the plates of a capacitor, and V is the voltage across the capacitor, one can write the corresponding expression for each infinitesimal ring at disk of radius r: sp * dQ(r, t) ) 2πr drC dl E (r, t)
(8)
where dQ(r, t) is the elementary charge of the ring with radius sp r and thickness dr; Cdl is the specific capacitance of the electrode interface per unit of surface area (viz., in F/cm2) of the double layer which is assumed to be independent of the potential and radial position; E*(r, t) is the effective potential with the inclusion of the solution resistance given by eq 4 with the potential applied to the electrode in the form E ) Emin + vt (Emin is the starting potential; v is the scan rate) and the overall current through the elementary ring represented by a sum of two components, di(r, t) ) dicap(r, t) + dif(r, t), with dicap(r, t) and dif(r, t) being the elementary capacitive and faradaic current components, respectively. For the sake of simplifying the expressions the following notation will be used dCdl(r) ) 2πr dr Cdlsp
(9)
Note that the charge stored at the electrode-solution interface at time t is equal to the integral of the capacitive current that has been flowing through the elementary ring (dicap(r, t)) at the radial position r over the duration of the experiment, i.e., dQ(r, t) )
∫ di t
0
(r, τ)dτ
cap
(10)
where τ is an auxiliary integration variable with the dimension of time. Differentiating the latter equation with respect to time one obtains
dicap(r, t) ) dCdl(r)
{
}
∂ [di(r, t)] ∂E*(r, t) ) dCdl(r) v - dRe(r) ∂t ∂t (11)
of the elementary ring on the disk surface (as well as solution and electrode-solution interface properties). Equation 12 reveals the complex interplay between faradaic and capacitive currents when the solution resistance and double layer capacitance are not negligible. It shows that the derivative of the overall current (with respect to time) flowing through an electrode surface element (being an infinitesimally thin ring for the disk electrode) is given by the difference between the increment of the capacitive current when the faradaic process is absent (dCdl(r)v) and that flowing (dicap(r, t)) in the system, related to the elementary time constant (dRe(r) dCdl(r)) of the surface element. Note that all the quantities involved in eq 12 are related to the elementary rings on the surface of the disk electrode. However if one were to write currents in terms of current densities and elementary capacitance using eq 9, the resulting equation would depend only on the radial position r and reveal no dependence on the spatial differential dr (the product dRe(r) dCdl(r) does not depend on dr; see discussion just after eq 12). It should be noted that eq 12 is valid for any electrode geometry including planar macroelectrodes (in which case the dependence on the spatial coordinate should be omitted and partial derivatives replaced with ordinary ones). While its differential form is more convenient for numerical calculations, we also give here the corresponding integral formulation:
di(r, t) )
E(t) - E0 1 dRe(r) dRe(r) dCdl(r)
t
0
(r, τ)dτ
cap
(13)
or
di(r, t2) - di(r, t1) )
v (t - t ) dRe(r) 2 1 1 dRe(r) dCdl(r)
∫
t2
t1
dicap(r, τ)dτ (14)
for the variation of the current over the time interval [t1, t2]. Dimensionless Variables. In order to impart generality to the results obtained here the following dimensionless variables are introduced:
τ)
Dt , rd2
R) θ)
r , rd
Z)
z , rd
nF ( E-E0), RgT * ) Cdl
and then rearranging eq 11 yields
∫ di
C) ν)
c , c0
Khet )
nFDc0 , γ
D Cdl, 4rd3γ
k0rd , D
Re* ) 4rdγRe,
f)
i i ) (15) ss 4nFDc0rd idisk
(12)
Note that all currents (overall, faradaic, and capacitive) are normalized as in the last expression in eq 15, i.e., with respect to the steady state diffusion-limited current iss disk at an inlaid disk electrode.7 Equation 12 after some rearrangement is rewritten
where dRe(r) dCdl(r) ) (π/2 γ)(rd2 - r2)1/2Cdlsp, which shows that the local time constant dRe(r) dCdl(r) depends only on the radius
(7) Bard, A.; Faulkner, L. Electrochemical Methods: Fundamentals and Applications, 2nd ed.; Wiley: New York, 2001.
∂ [di(r, t)] ∂ [dicap(r, t)] ∂ [dif(r, t)] dCdl(r)v - dicap(r, t) ) + ) ∂t ∂t ∂t dRe(r) dCdl(r)
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Figure 2. Simulated overall (capacitive + faradaic) current for different values of the heterogeneous rate constant: 100 (peaks O1R1); 10-5 (O2-R2); 10-7 (O3-R3); and 10-9 cm/s (O4-R4), which correspond to the oxidation (On) and reduction (Rn) paired processes at the disk microelectrode.
in terms of the dimensionless variables in the following way: dfcap(R, τ) ∂ [dfcap(R, τ)] ∂ [dff(R, τ)] + * ) µ(R) (16) + * ∂τ ∂τ dRe(R) dCdl(R) where
µ(R) )
rd2v ss Didisk Re(r)
,
dRe*(R)dCdl*(R) )
π D √1 - R2Cdlsp 2 rdγ
RESULTS AND DISCUSSION In order to represent the computational results in a comprehensive and illustrative way, we shall give them in both the dimensioned and dimensionless forms simultaneously, i.e., the bottom and left axes of the following figures correspond to the real physical values, while the top and right axes correspond to dimensionless quantities introduced above. The values of the dimensioned currents and potentials were evaluated throughout this section using the following set of parameters: disk radius rd ) 10 µm; layer thickness δ ) 0.5 µm; concentration of electroactive species in the thin layer at disk c0 ) 5 mM; diffusion coefficient D ) 10-5 cm2/s; specific capacitance of the electrode2 electrolyte interface Csp dl ) 10 µF/cm ; formal potential of the redox 0 couple E ) 0; transfer coefficient R ) 0.5; temperature T ) 298 K. Other parameters have the following values (unless otherwise stated): scan rate v ) 1 V/s; conductivity γ ) 5 × 10-3 A/V cm. When it was important to ensure that redox equilibrium was achieved at any E* value, a standard heterogeneous rate constant k0 ) 100 cm/s was imposed. We recognize that such value is unrealistically large but it allowed imposing absolute local equilibria while using kinetic formulations in eqs 3a and 3b. Figure 2 contains a series of cyclic voltammograms with relatively slow scan rate (v ) 1 V/s) illustrating the effect of the standard rate constant value. As expected, the reversible voltammogram (k0 ) 100 cm/s or Khet ) 104; O1-R1 peaks) demonstrates 7960
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Figure 3. Simulated overall (capacitive + faradaic) (a) and capacitive (b) currents for different values of the conductivity: 5 × 10-3 (1); 5 × 10-6 (2); and 5 × 10-7 (3) A/V cm.
a perfectly symmetrical response with respect to the potential axis, while in the irreversible case (k0 ) 10-5, 10-7, 10-9 cm/s or Khet ) 10-3, 10-5, 10-7; O{2, 3, 4}-R{2, 3, 4} peaks, respectively), the peaks significantly shift with the potential peak separation being the larger, the smaller the standard heterogeneous rate constant. Since under steady state conditions the time constant is relatively small, the capacitive component of the current results in an almost rectangular frame (which can be clearly seen in Figure 2) with a very fast exponential rise (which cannot be distinguished at the present potential scale) at starting (E ) -1 V) and switching (E ) 1 V) potentials. The results presented in Figure 3 describe the effect of solution resistance. As it was noted and fully explained in ref 1 for cyclic voltammetry in an “open” solution (without a surface-attached layer) at the inlaid disk microelectrode, the increase of solution resistance (decrease of conductivity) leads to a progressive shift of the current peak due to purely ohmic (i.e., linear) buildup of the current before the peak (see Figure 3a). Since in the previous work1 the capacitive currents were not considered, no conclusions were made about their behavior or coupling with the resistive effects. However, increasing the resistance proportionally increases the time constant which is evident from ever slower exponential current rise in Figure 3b, where only the capacitive component of the current is presented. Moreover, the interplay
Figure 4. Equivalent electric circuit.
between the faradaic and capacitive currents results in the repartition of the current densities which results in the capacitive current component behaving as depicted in Figure 3b. This behavior can be explained on the basis of eq 16. Indeed, the right-hand side (rhs) of the equation is constant for a fixed value of R, therefore when the faradaic component grows (due to the increase in the overpotential) the capacitive one should decrease to maintain a constant rhs and vice versa. This is exactly what is exemplified in Figure 3b (although one should keep in mind that this figure represents the overall current flowing across the disk but not its elementary part corresponding to a small current tube), the capacitive current smoothly decreases while the overall, and hence faradaic, current grows (see Figure 3a). After the peak, the overall current drops which leads to a significant increase of the capacitive component to compensate for this change. The physical origin of this phenomenon was originally described by Saveant5 but is recalled hereafter. Once the capacitive current has attained its steady state value under a linear potential sweep, the further increase in the overpotential forces a larger fraction of the current to flow through the faradaic circuit (see the scheme in Figure 4) which thus results in decreasing capacitive current. After the current peak, the faradaic component drops due to the significant depletion of the electroactive species within the film that leads to a spike in the capacitive current, which compensates for the double layer charge that would be accumulated in the absence of the faradaic process. It should be emphasized that such complex interplay between the faradaic and capacitive currents may lead to full deterioration of the information in an electrochemical response. Indeed, it is well-known that with increasing scan rate, the capacitive current eventually exceeds the faradaic component.6-8 However, when the solution resistance is significant, the faradaic current does not remain unchanged and due to the strong coupling described above it is severely altered and can no longer be described by the classical theory. This occurs because if the overall current in this situation may still be represented by the sum of capacitive and faradaic currents, both of these components considerably interfere, so that the capacitive current is no longer that obtained when the faradaic process is absent. Since the latter assumption of additivity of current components is widely used in experimental studies upon assuming that the two currents remain independent, one should be extremely cautious that the procedure may lead to erroneous interpretation of experimental data. This is exemplified in Figures 5 and 6. Figure 5 shows computed voltammograms (Figure 5a) and their capacitive components (Figure 5b) for several scan rates in the range of 10-200 kV/s. The conductivity value used for these computations was γ ) 5 × 10-3 A V/cm. It is clear that the increase in the scan rate drastically changes the shape of the (8) Amatore, C.; Jutand, A.; Pflu ¨ ger, F. J. Electroanal. Chem. 1987, 218, 361– 365.
Figure 5. Simulated overall (capacitive + faradaic) (a) and capacitive (b) currents for different values of the scan rate: 10 (1), 100 (2), and 200 (3) kV/s.
voltammogram (Figure 5a), but due to the interplay between the capacitive and resistance effects, the capacitive current (Figure 5b) substantially differs from the one expected from the measurements in a “dummy” cell (the latter is a simple exponential curve,7,8 see below). Figure 6 illustrates the effect in more detail where the cyclic voltammogram is given for v ) 100 kV/s and the same value of conductivity that was used for Figure 5. Figure 6a contains the overall current, its capacitive component and the capacitive current obtained from the dummy cell. Both capacitive currents are identical up to the moment when the faradaic current, and hence resistive effect, become significant. To give quantitative support to our above statement about the loss of information in the electrochemical signal under such conditions, the faradaic currents corresponding to a classical response (without taking into account the capacitive and resistance effects), the actual faradaic current (with account for the capacitive and resistance effects), and the one obtained by subtraction of the overall current and capacitive current from the dummy cell are given in Figure 6b. The inspection of the actual faradaic current reveals that the information contained in the classical response is corrupted by the decrease in the signal amplitude, widening the peaks creating a sluggish feature just after the current peak (the nature of this Analytical Chemistry, Vol. 80, No. 21, November 1, 2008
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Figure 7. Variations of the capacitive (a) or faradaic (b) normalized currents across the electrode surface illustrating the characteristic “wave” behavior discussed in the text under the conditions of Figure 6. The individual current variations correspond to different times equidistantly distributed across the range of potentials from 0.05 to 0.35 V. Figure 6. Cyclic voltammograms simulated at a scan rate of 100 kV/s (see other parameters in the text): (a): blue curve, overall current; green curve, capacitive component of the overall current; red curve, capacitive current in the absence of the faradaic process. (b) Blue curve, faradaic component of the overall current; red curve, faradaic component extracted from the overall current by subtraction of the red curve given in part a; light green curve, faradaic current computed without taking into account the capacitive and resistive effects.
sluggish behavior is fully documented in ref 1) and an increased peak separation. The same conclusion would apply to a semiinfinite solution of electroactive species, since in the present case the diffusion layers are small enough to be contained within the thin redox film so the latter behaves as an infinite solution. Nevertheless, it is interesting to note that though each peak is severely shifted, the formal potential deduced through the halfsum of peak potentials7 is not drastically affected (Figure 6b) when one uses the subtracting procedure assuming noninterference between capacitive and faradaic currents even when their interaction is severe.8 It should be noted also that the faradaic component obtained under the considered conditions has a peculiar feature, selfintersection. This should be attributed to the resistive effect since, as was pointed out in ref 1, the ohmic drop term (see eq 4) leads to a shift in the effective electrode potential with respect to the applied potential to smaller/bigger values depending on the direction of the scan. Henceforth, the current peak is shifted to 7962
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more positive potentials during the direct scan, while it shifted to more negative potentials during the reverse scan. The overall effect produces an intersection of the two branches of the voltammogram. Finally, Figure 7 describes the complexity of the system at hand by examining a close view of the electrochemical current simulations obtained at the rising part of the forward current wave shown in Figure 6a, as a function of the position of the electroactive electrode element vs the disk center. For convenience of the presentation, this is represented as the variation of the capacitive (Figure 7a) or faradaic (Figure 7b) normalized current (see eq 15) at different times (hence different potentials equidistantly distributed from 0.05 to 0.35 V) that flow according to the radial position R. It is observed that because of the different time constants and ohmic drops relative to each electrode element, the behaviors significantly differ. As was noted in our previous work,1 one observes a kind of wave propagation along the electrode surface, though here due to the time constant modulations across the electrode surface, the phenomenon is both smooth (in terms of the sharpness of the wave propagating across the electrode) and amplified in terms of the magnitude of the effect due to the coupling between capacitive and faradaic currents which was absent in ref 1. As was shown in ref 6, an effective way to preserve the electrochemical information in voltammetry in resistive solutions at extremely high scan rates is the use of electronic compensation
nonlinear algebraic equation systems. The typical computational grid was (NR) × (NZ) ) 150 × 150 with 4000 time steps per volt (or its dimensionless equivalent) that ensured the numerical error in peak current of less than 0.5% for all considered cases.
Figure 8. Cyclic voltammograms simulated under the same conditions as the one in Figure 6 except that a very high conductivity of the medium (γ ) 10 A/V cm) is considered, i.e., when the resistance is electronically compensated.6 Curve 1 is the overall current as recorded at the disk electrode. Curve 2 is the theoretical faradaic current predicted in the absence of ohmic or capacitive distortions. Curve 3 is the capacitive current in the absence of any faradaic reaction (viz., when c0 ) 0). Note that the faradaic current extracted from curve 1 after subtracting the capacitive current (curve 3) is also shown (though it is indistinguishable from curve 2).
of the ohmic drop. Results presented in Figure 8 provide a theoretical support for such an experimental approach. Indeed, Figure 8 contains the current-voltage response computed under the same conditions as the voltammogram depicted in Figure 6a, except for the conductivity value, which was drastically increased in this case to γ ) 10 A/V cm (i.e., 4 orders of magnitude higher). Such a drastic increase of the conductivity value is equivalent to the effect of an electronic compensation6 since it leads to the partial suppression of the ohmic drop effect. In this case, the overall current may indeed be accurately described as a sum of independent capacitive and faradaic components though the capacitive current is high vs the faradaic one. Because the reduction of the resistance simultaneously decreases the cell time constant, the extracted faradaic current fully coincides with the classical response (i.e., that predicted without taking into account the capacitive and resistance phenomena). COMPUTATION DETAILS The mathematical model (eqs 1-2) coupled with eqs 3 and 12 was numerically solved in the dimensionless cylindrical coordinates (R, Z) using a uniform computational grid since no singularities are encountered in the system under scrutiny. The singularities, related to the resistance and iRe product, were solved analytically by means of the appropriated quasi-conformal mapping9 as it was explained above in the text and in our previous work.1 We used the alternating direction implicit (ADI) method coupled with the Newton’s method for the solution of the resulting (9) (a) Amatore, C.; Oleinick, A. I.; Svir, I. J. Electroanal. Chem. 2006, 597, 69–76. (b) Oleinick, A.; Amatore, C.; Svir, I. Electrochem. Commun. 2004, 6, 588–594.
CONCLUSIONS Thanks to the widespread of ultramicroelectrodes, an increasing number of electrochemical measurements are presently performed under conditions where ohmic drop and/or capacitive currents are larger than those considered in classical theories of voltammetry. In this work it has been shown that the ensuing non-negligible resistive and capacitive components of the current need to be thoroughly evaluated in order to correctly analyze the voltammetric responses. Moreover, when the solution resistance is not negligible or compensated electronically, the two current components (viz., the faradaic and capacitive ones) strongly interfere when the time scale of the experiment is comparable to the cell time constant. This coupling is even more complex at microelectrodes, e.g., the disk microelectrode considered here, due to the nonuniform current densities resulting from different accessibility and different resistance at each electrode point. The latter results in a spatially inhomogeneous distribution of ohmic drop and time constants across the electrode surface which in turn leads to spatially distributed potentials across the electrochemical interface even when the electrode substrate is perfectly conducting. Hence, the overall current results from a time convolution of the responses of electrode surface elements rather than that assumed in classical views. On the other hand, it was shown that even when the capacitive currents are experimentally large as compared to faradaic ones, a small cell resistance allows first minimizing ohmic drop and second reducing the coupling between the faradaic and capacitive components of the current. Thus the results presented here substantiate the principle of background subtraction when the medium resistance is intrinsically small, as in in vivo experiments at small scan rates, or after electronic ohmic drop compensation at high scan rates. ACKNOWLEDGMENT The authors thank Dr. S.W. Feldberg for initiating our interest in this problem. In Paris, this work was supported in part by CNRS (UMR 8640 “PASTEUR” and LIA “XiamENS”), ENS, UPMC, and by the French Ministry of Research. I.S. thanks CNRS for the Research Director position in UMR 8640. A.O. thanks the Ministry of Education and Science of Ukraine and INTAS for the Young Scientist Fellowship 06-1000019-6451, which supports his postdoctoral research at ENS (Paris, France). NOTE ADDED AFTER ASAP PUBLICATION The paper was posted on the Web on 10/01/08. A minor correction (zero superscript to E) to eq 15 was made. The paper was reposted on 10/08/08. Received for review July 2, 2008. Accepted August 16, 2008. AC8012972
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