Importance of Correct Prediction of Initial Concentrations in

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Importance of Correct Prediction of Initial Concentrations in Voltammetric Scans: Contrasting Roles of Thermodynamics, Kinetics, and Natural Convection Christian Amatore,*,† Oleksiy V. Klymenko,†,‡ and Irina Svir*,†,‡ †

Département de Chimie, Ecole Normale Supérieure 24, rue Lhomond, 75005 Paris, France Mathematical and Computer Modelling laboratory, Kharkov National University of Radio Electronics, 14 Lenin Avenue, Kharkov, 61166, Ukraine

‡

ABSTRACT: In order to successfully model an electrochemical reaction mechanism one must ensure that all the equations, including initial conditions, satisfy the pertinent thermodynamic and kinetic relationships. Failure to do so may lead to invalid results even if they are mathematically correct. This fact has been previously emphasized (Luo, W.; Feldberg, S. W.; Rudolph, M. J. Electroanal. Chem. 1994, 368, 109−113; Rudolph, M. Digital Simulation in Electrochemistry. In Physical Electrochemistry; Rubenstein, I., Ed.; Marcel Dekker: New York, 1995; Chapter 3) and existing computer software for electrochemical simulations, such as DigiSim (Rudolph, M.; Reddy, D. P.; Feldberg, S. W. Anal. Chem. 1994, 66, 589A; http://www.basinc.com/products/ec/digisim/), offer the option of enforcing the so-called “pre-equilibration” which evaluates thermodynamic concentrations of all species prior to beginning a voltammetric scan. Although this approach allows setting consistent thermodynamic values it may result in a nonrealistic initial concentrations set because it corresponds to the whole solution status at infinite time for infinite kinetic constants. However, the perturbation created by the working electrode poised at its rest potential is necessarily limited by the size of the electrode, reaction kinetics, and duration of the rest period. Furthermore, natural convection limits even more the importance of the perturbation. This is analyzed theoretically through comparison of simulation results by DigiSim and KISSA-1D software for certain common electrochemical mechanisms in order to illustrate the importance of correct prediction of initial concentrations.

P

that a planar macroelectrode with surface area S is immersed in an electrochemical cell of volume Vsoln containing an electroactive species A at a concentration [A]bulk. We also assume that the rest potential is set to the top of the wave of the redox couple to ensure diffusion-limited conversion of the initial species. After a relatively short Cottrellian transient (lasting typically for 10−15 s) during which the diffusion layer (δdiff ∝ (πDt)1/2) generated since the effective voltammetric scan started increases up to when it reaches the end, δ, of the stagnant layer so that the current tends toward a quasi-steadystate value:5−9

rescribing changes in equilibrium concentration values enforced by setting a working electrode at a rest potential, Erest, throughout a whole solution implies that the electrochemical cell remained at its rest potential for a sufficiently long time to achieve exhaustive electrolysis. However, in typical electrochemical experiments, rest times are sufficiently short so that the concentrations of all species remain unchanged in most of the solution bulk and are altered according to the applied rest potential only within a relatively thin stagnant volume adjacent to the electrode as opposed to the bulk solution where concentrations are uniform due to its spontaneous mixing by natural convection.1−9 These effects must be taken care of in a general electrochemical simulator as shown below and implemented in the new software KISSA-1D and KISSA-2D for simulation of one- and two-dimensional problems.10−13 Evidently, in the following we assume that the chemical composition of the system prior to poising the working electrode at its rest potential is correctly defined in agreement with the relevant thermodynamics and kinetics.14a

i(t ) = nFSD

(1)

where D is the diffusion coefficient, [A]soln is the bulk concentration of species A at time t, and NA is the quantity of species A in the solution (in moles), n is the number of transferred electrons.

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THEORY Let us estimate the time duration necessary to establish equilibrium concentrations in the whole solution volume. Suppose © 2012 American Chemical Society

[A]soln d[A]soln dN = −nF A ≈ −nFVsoln δ dt dt

Received: December 1, 2011 Accepted: February 13, 2012 Published: February 13, 2012 2792

dx.doi.org/10.1021/ac203188b | Anal. Chem. 2012, 84, 2792−2798

Analytical Chemistry

Article

species present in the solution from t = 0 until the end of rest period, t = trest, yields the true prescan concentration profiles. This approach does not introduce any bias with respect to the change of bulk concentrations as long as the product kapptrest remains much smaller than unity. For the sake of simplicity its results will only be documented hereafter considering a planar electrode.

Rearranging the latter expressions we arrive at the following differential equation describing the evolution of the bulk concentration of electroactive species: d[A]soln SD =− dt = −kapp dt [A]soln Vsoln δ

(2)

where kapp = SD/Vsolnδ is the apparent rate of electrochemical conversion. Taking typical values as S = 1 mm2, D = 10−5 cm2 s−1, Vsoln = 20 mL, and δ = 200 μm we obtain the value kapp = 2.5 × 10−7 s−1. This means that in order to bring the concentration from its initial value [A]bulk to within 1% of the target equilibrium value (zero in this case) the working electrode would have to be held at the rest potential for over 200 days even when this is set on the plateau of a redox wave. This simple calculus illustrates that, unless equilibrium concentration values are very close to their initial (bulk) values, the pure thermodynamic pre-equilibration approach cannot reflect the true situation because the perturbation created by the working electrode during its rest period is necessarily limited to the stagnant volume adjacent to it. Therefore, concentrations prevailing at the end of the rest period need to be evaluated not by brutally enforced thermodynamics but through solving a system of partial differential equations describing mass transport and reactions of all species while the electrode is poised at its rest potential: ∂cj ∂t

=

1 ∂ ⎛ app λ ∂cj ⎞ ⎜D j x ⎟ + Rj ∂x ⎠ x λ ∂x ⎝

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Although one generally sets the rest potential of any voltammetric scan far from the peak potential there are several circumstances in which this is meaningless or not feasible. For example, when tracking voltammetrically the progress of an electrolysis the potential needs to be set at E1/2 in order to assess the relative proportions of the redox couple species. For this reason it was of interest to compare the effect of Erest for a moderately long rest time (e.g., trest = 100 s) on the ensuing cyclic voltammograms (CVs) when enforcing a standard application of thermodynamic pre-equilibration or when taking into account kinetics and natural convection. When a sufficiently high scan rate (e.g., v = 1 V s−1) is used the time-dependent change of the diffusion layer during the CV scan is small vis-à-vis the size of the natural convection layer. However, because of the rest potential period, natural convection affects the CV wave depending on the value of the rest potential and on the fact that pre-equilibration is not considered (Figure 1a) or enforced (Figure 1b). This is illustrated for three values of Erest: E0 −0.5 V, E0, and E0 +0.5 V; each scan starts at a potential coinciding with the respective rest potential. Without enforcing pre-equilibration (i.e., trest = 0 s, Figure 1a) CVs begin with awkward chronoamperometric features corresponding to the sudden electrolysis of bulk concentrations of A and B. Conversely, such chronoamperometric features disappear when enforcing correct initial concentrations (i.e., trest = 100 s, Figure 1b), and the initial currents have the expected sign depending on Erest. At smaller scan rates (viz., Figure 1, parts c and d, for v = 0.05 V s−1) the same general effect is observed, though now the extent of the diffusion layer created during each forward voltammetric scan reaches the end of the stagnant layer much before the potential inversion time. Hence, in Figure 1d all voltammetric traces result superimposed except for the forward scan starting at Erest = 0 V, in agreement with usual experimental observations under such circumstances. This simple example illustrates the importance of correctly describing the effects of the pre-electrolysis performed while the working electrode is poised at its prescan rest potential. Several commercial simulation softwares, such as DigiSim, include already an option for preequilibration, though to the best of our knowledge none consider that initial concentrations may drastically differ from their thermodynamic predicted values because of limitations due to kinetics and natural convection. In the following we wish to illustrate the performance of the present approach by comparing our results (KISSA-1D) to those of DigiSim when the pre-equilibration setting is selected (note that this option may be unselected at will in DigiSim or KISSA-1D).14

(3)

where cj is the concentration of the jth species, t is the time, x is the coordinate normal to the electrode surface, λ is the geometrical parameter (i.e., λ = 0 for planar electrode, λ = 1 for the cylinder, and λ = 2 for the sphere), Rj is the reaction term13, and Djapp is the apparent diffusion coefficient of the jth species. The latter respects the presence of spontaneous convection in the solution through its coordinate dependence:5 app

Dj

= Dj0[1 + γ(x /δ)4 ]

SIMULATION RESULTS FOR SIMPLE ET

(4)

Dj0

where is the real diffusion coefficient of the jth species and γ = π4/26. Note that this model (eqs 3 and 4) has been fully validated experimentally at planar electrodes.5−9 In nonextremely viscous solutions δ ranges from 100 to 250 μm, and its value may be derived from the long time limit, ilim, of the chronoamperometric current for a simple electron-transfer (ET) mechanism under otherwise identical experimental conditions: δ = nFSD[A]bulk/ilim, as follows from eq 1. Hence, the evaluation of apparent diffusion coefficients in eq 4 does not require any adjustment.5−9 Note also that δ is formally independent of the exact electrode shape and dimension provided it is embedded in a sufficiently large flat inactive wall. For totally spherical or cylindrical electrodes the matter is more delicate since eq 4 assumes that microscopic hydrodynamic movements of the solution are limited by an insulating substrate in which the electrode is embedded.5 Though, whenever simulations at spherical and cylindrical electrodes are used as fast approximations for disk and band electrodes, respectively, this assumption seems reasonable. Solving eqs 3 and 4 with the pertinent boundary conditions13 and electrode potential fixed at Erest for all 2793



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Figure 1. Voltammograms for simple ET (A − e ⇄ B) simulated by KISSA-1D considering natural convection for different rest potentials at v = 1 V s−1 (a and b) or v = 0.05 V s−1 (c and d). trest = 0 s (a and c) or 100 s (b and d). D = 10−5 cm2 s−1, δ = 150 μm, and S = 1 mm2. [A]bulk = 0.2 mM and [B]bulk = 0.8 mM in order to simulate a situation where the progress of a bulk electrolysis is assessed voltammetrically. The beginning and direction of each scan are indicated by crosses with arrows. Note that in panel d the CVs for Erest = −0.5 and 0.5 V are superimposed over most of the potential excursion range.

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Defining KET = exp[−(F/RT)(E − E0)], the normalized equilibrium concentrations of reactants (e.g., as those set as initial ones by DigiSim in the pre-equilibration mode) are

COMPARISON OF SIMULATION RESULTS BY DIGISIM AND KISSA-1D FOR THE EC MECHANISM Consider the EC reaction scheme: E0

(5)

aeq =

KET 1 + KET + K

(10)

K = k+/k−

(6)

beq =

1 1 + KET + K

(11)

A − e ⇄ B, B ⇄ C,

At equilibrium the following relations hold: ceq =

RT [B] E − E0 = ln F [A]

(7)

K[B] = [C]

(8)

1 + KET + K

(12)

The results presented in Figure 2 correspond to different Erest values and the following parameters: [A]bulk = 1 mM, [B]bulk = 0 mM, D = 10−5 cm2 s−1, E0 = 0 V, v = 1 V s−1, k+ = 100 s−1, K = 100 (i.e., E0* = −0.118 V); additionally, for KISSA-1D, δ = 150 μm, trest = 100 s, whereas DigiSim is operated in its preequilibration mode. It is observed that when the rest potential lies to the left from the apparent formal potential E0* the results obtained by the two approaches practically coincide. However, if Erest is set at E0* or lies to its right drastic differences are observed between the two sets of predicted CVs. Indeed, as follows from eq 10

so that the apparent formal potential of the redox wave is given by RT E 0* = E 0 − ln K F

K

(9)

when k+ and k− are large enough versus the scan rate. 2794



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the concentration of species A imposed in DigiSim as the initial one in the whole solution in this case is much less than unity, whereas when evaluated by KISSA-1D it remains unchanged in the bulk solution and gradually decreases in the stagnant layer to reach the value in eq 10 only at the very electrode surface.

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E⃗CE⃖ MECHANISM (ELECTRON-TRANSFER CATALYSIS, ETC).15−19 Consider the following reaction scheme: E10

(13)

irreversible

(14)

A − e ⇄ B, k

B → C,

C + e ⇄ D,

E20 > E10

(15)

in which the direct reaction A → D is hampered kinetically though being thermodynamically extremely favorable (note that when E10 and E20 do not differ by more than several volts this implies that the reaction B → C is largely exergonic which justifies that for simplicity we consider it as irreversible here). The plots in Figure 3 compare CVs simulated by KISSA-1D or DigiSim operated in the pre-equilibration mode14b for different Erest values and the following parameters: [A]bulk = 1 mM, [B]bulk = [C]bulk = [D]bulk = 0 mM, D = 10−5 cm2 s−1, E10 = −0.1 V, E20 = 0.1 V, v = 1 V s−1, k = 100 s−1, S = 1 mm2. Additionally for KISSA-1D: δ = 150 μm, trest = 100 s. Note that for such an electron-transfer-catalyzed mechanism, assuming complete thermodynamic equilibrium irrespective of kinetics imposes that, whatever the electrode rest potential, the whole bulk solution of A is converted into a solution of D. Under the conditions of natural convection employed in the KISSA1D simulations the scan rate was chosen sufficiently large so that the diffusion layer created during each scan remained well within the stagnant layer. Thus, when pre-equilibration is not enforced, the results obtained by KISSA-1D (i.e., with trest = 0 s) and DigiSim coincide (see Figure 3, part Aa, Ab, and Ac). When Erest is negative enough (Figure 3Aa), both simulated CVs evidence a clear voltammetric peak for the A/B redox couple at E = −0.112 V, though its current peak intensity is drastically reduced due to the ETC operating during the voltammetric scans.15−19 However, the A/B waves simulated by KISSA-1D with trest = 100 s and DigiSim when the preequilibration mode is enforced drastically differ (compare Figure 3Ba). The peak at E = −0.112 V totally disappears in the DigiSim prediction, whereas it remains present when simulated with KISSA-1D. This happens because, in agreement with numerous experimental observations,15−19 in KISSA-1D the effects of the rest potential conditions prevailing at the electrode surface are attenuated by natural convection, whereas in DigiSim the pre-equilibration mode enforces them throughout the whole solution. Thus, concentration values at the electrode surface after a 100 s long pre-equilibration at Erest = −0.5 V as computed by KISSA-1D are as follows: [A]0 = 0.999992 mM, [B]0 = 1.732 × 10−10 M, [C]0 = 5.844 × 10−19 M, [D]0 = 8.043 × 10−9 M, although a strict enforcement of thermodynamic requirements at the rest potential would lead to [A]0 = 4.174 × 10−17 M, [B]0 = 7.246 × 10−24 M, [C]0 = 7.240 × 10−14 M, [D]0 ≈ 1 mM (as computed by DigiSim with the equilibrium constant of reaction 14 equal to 1010).

Figure 2. Comparison of simulated voltammograms by KISSA-1D and DigiSim (ref 14b) for an EC mechanism when the electrode is initially poised at different rest potentials during trest = 100 s. (a) Erest = −0.5 V ≪ E0* = −0.118 V (i.e., KET = 2.83 × 108, so aeq ≈ 1, beq = 3.534 × 10−9, ceq = 3.534 × 10−7 from eqs 10−12); (b) Erest = 0 V ∼ E0* = −0.118 V (i.e., KET = 1, so aeq = 0.0098, beq = 0.0098, ceq = 0.9804 from eqs 10−12); (c) Erest = +0.5 V ≫ E0* = −0.118 V (i.e., KET = 3.53 × 10−9, so aeq = 3.499 × 10−11, beq = 0.0099, ceq = 0.9901 from eqs 10−12). In all cases, v = 1 V s−1 and S = 1 mm2. 2795



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Figure 3. Comparison of CVs predicted by KISSA-1D and DigiSim for an E⃗ CE⃖ mechanism (eqs 13−15) with (right panel) or without (left panel) pre-equilibration at different rest potentials: (Aa and Ba) Erest = −0.5 V; (Ab and Bb) Erest = 0 V; (Ac and Bc) Erest = 0.5 V. Black crosses with arrows indicate the beginning and direction of voltage scan. In all cases v = 1 V s−1, S = 1 mm2. trest = 0 s (left panel) or 100 s (right panel) for KISSA-1D. Note that in panel Bb the scan begins and ends at E = 0 V, where the curve almost meets itself.

A similar situation is observed when Erest = 0 (Figure 3, parts Ab and Bb). When thermodynamic conditions are enforced throughout the whole solution (as does DigiSim when its preequilibration setting is enabled)14b the A/B peak is absent since

species A and B almost completely disappear when imposing thermodynamic values while the concentrations of C and D become [C] = 0.02001 mM, [D] = 0.97999 mM (see Figure 4b). Conversely, when evaluated with KISSA-1D, it is seen that, 2796



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Figure 4. Initial concentration profiles achieved at Erest = 0 for the E⃗ CE⃖ mechanism (eqs 13−15) as simulated (a) by KISSA-1D for trest = 100 s or (b) by DigiSim when the pre-equilibration mode is selected (ref 14b).

“MicroNanoChem”) and CNRS for financial support of this project in ENS (UMR 8640). Dr. Klymenko thanks the Mairie de la Ville de Paris for a one-year fellowship at ENS (UMR 8640).

though drastically decreased in the very vicinity of the electrode surface, species A is still present in large amounts in the stagnant layer after the rest period due to the compensation of the ETC role by natural convection (Figure 4a). Finally, resting the potential at Erest = 0.5 V leads to a pure C reduction curve in the case of pre-equilibration in DigiSim (Figure 3Bc) but still shows an oxidation peak for A on the backward scan when simulated using KISSA-1D.

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CONCLUSIONS Imposing equilibrium concentration values throughout the solution volume at the end of the rest potential period, i.e., upon assuming an infinite rest time period and neglecting the influence of natural convection, may lead to extremely significant deviations of the simulated results from reality. This occurs because even at long rest times the concentration profiles may reach their steady-state shapes only within the stagnant solution layer imposed by spontaneous natural convection. These steady-state concentration profiles result from compensation of the concentration changes imposed on the one hand by the working electrode at its surface during the rest time period and on the other hand by the natural convection concentration supply at the other extremity of the stagnant layer which tends to continuously restore bulk concentrations. The results presented in this work illustrate the importance of setting correct initial conditions by taking into account both thermodynamic conditions and natural convection in order to obtain realistic solutions.14 This is especially critical in the case of electrontransfer-catalyzed reactions or of ECE-type mechanisms when the second redox couple is thermodynamically favored at the rest potential.

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REFERENCES

(1) Luo, W.; Feldberg, S. W.; Rudolph, M. J. Electroanal. Chem. 1994, 368, 109−113. (2) Rudolph, M. Digital Simulation in Electrochemistry. In Physical Electrochemistry; Rubenstein, I., Ed.; Marcel Dekker: New York, 1995; Chapter 3. (3) Rudolph, M.; Reddy, D. P.; Feldberg, S. W. Anal. Chem. 1994, 66, 589A. (4) DigiSim homepage. http://www.basinc.com/products/ec/ digisim/. Accessed on February, 2012. (5) Amatore, C.; Szunerits, S.; Thouin, L.; Warkocz, J. S. J. Electroanal. Chem. 2001, 500, 62−70. (6) Baltes, N.; Thouin, L.; Amatore, C.; Heinze, J. Angew. Chem., Int. Ed. 2004, 43, 1431−1435. (7) Amatore, C.; Knobloch, K.; Thouin, L. J. Electroanal. Chem. 2007, 601, 17−28. (8) Amatore, C.; Pebay, C.; Thouin, L.; Wang, A. Electrochem. Commun. 2009, 11, 1269−1272. (9) Amatore, C.; Pebay, C.; Thouin, L.; Wang, A.; Warkocz, J. C. Anal. Chem. 2010, 82, 6933−6939. (10) Amatore, C.; Klymenko, O.; Svir, I. Electrochem. Commun. 2010, 12, 1170−1173. (11) Amatore, C.; Klymenko, O.; Svir, I. Electrochem. Commun. 2010, 12, 1165−1169. (12) Klymenko, O.; Oleinick, A.; Svir, I.; Amatore, C. Russ. J. Electrochem. 2012, in press. (13) Klymenko, O.; Svir, I.; Oleinick, A.; Amatore, C. ChemPhysChem 2012, 13, 845−859. (14) (a) As pointed out by one of the reviewers, it is critical not to confuse the concentrations of reactants initially introduced in a solution and the chemical composition of this solution prior to any electrochemical experiments. Indeed, complexation, acido−basic, etc., reactions may equilibrate the solution components as a function of the relevant thermodynamics and kinetics, thus changing the solution composition vs that following arithmetically from the amounts of its individual components used during its preparation. For example, this is the case in the most seminal experiments of organic electrochemistry (Heyrovsky) during which the introduction of an aldehyde or a ketone in an aqueous electrolyte results in its hydration, hence giving rise to CE mechanisms because the electroactive species is the aldehyde or a

AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work has been supported in parts by the CNRS (UMR 8640), Ecole Normale Supérieure (ENS, Paris), University Pierre and Marie Curie (UPMC), and the French Ministry of Research. The authors thank ANR (Chaire d’Excellence project 2797



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ketone though it may be present only at very small concentrations after chemical equilibration of the solution. In the present work, we assume that one knows the exact composition of the solution under scrutiny prior to setting the working electrode potential to its rest value. (b) For acute mechanistic situations, depending on the selected rest potential, commercial softwares which allow imposing thermodynamic conditions throughout the whole solution, i.e., without taking into account the effect of natural convection, may lead to erroneous predictions whenever their “pre-equilibration” option is enabled. However, this is not an entirely negative issue provided that this option may be disabled (as, e.g., offered by DigiSim). Indeed, comparing simulated voltammograms obtained with and without enabling this option may be useful for users and provide direct evidence that great precaution should be taken in selecting the experimental rest potential (S. W. Feldberg, private communication, January 2012) in their simulations or in their experiments. (15) Feldberg, S. W.; Jeftic, L. J. Phys. Chem. 1972, 76, 2439−2446. (16) Amatore, C.; Savéant, J.-M.; Thiébault, A. J. Electroanal. Chem. 1979, 103, 303−320. (17) Bezems, G. J.; Rieger, P.; Visco, S. J. Chem. Soc., Chem. Commun. 1981, 6, 265−266. (18) Darchen, A.; Mahé, C.; Patin, H. J. Chem. Soc., Chem. Commun. 1982, 14, 243−245. (19) Zizelman, P. M.; Amatore, C.; Kochi., J. K. J. Am. Chem. Soc. 1984, 106, 3771−3784.

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Importance of Correct Prediction of Initial Concentrations in

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