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Theoretical study of the EE reaction mechanism with comproportionation and different diffusivities of reactants
Electrochemistry Communications 12 (2010) 1378–1382
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Electrochemistry Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e l e c o m
Theoretical study of the EE reaction mechanism with comproportionation and different diffusivities of reactants Oleksiy V. Klymenko a,b, Irina Svir a,b,⁎, Christian Amatore a,⁎ a b
Ecole Normale Supérieure, Département de Chimie, UMR CNRS-ENS-UPMC 8640 “PASTEUR”, 24 rue Lhomond, 75231 Paris Cedex 05, France Kharkov National University of Radioelectronics, Mathematical & Computer Modelling Laboratory, 14 Lenin Avenue, 61166 Kharkov, Ukraine
a r t i c l e
i n f o
Article history: Received 5 July 2010 Received in revised form 20 July 2010 Accepted 20 July 2010 Available online 29 July 2010 Keywords: EE mechanisms Fast kinetics Comproportionation Different diffusion coefficients KISSA Simulation
a b s t r a c t In this article we investigate the origin of unexpected features appearing in voltammetry or double-step chronoamperometry of EE systems when the rate of comproportionation is extremely fast and the diffusion coefficients of the reactant and products highly differ. These features were noticed during the testing of our new software KISSA intended to solve any reaction mechanisms even when acute reaction fronts develop near the electrode surface or within the solution. To validate the principle of the new adaptive algorithm [C. Amatore, O. Klymenko, I. Svir, Electrochem. Commun. doi:10.1016/j.elecom.2010.06.009] implemented in KISSA we used analytical and numerical solutions for double-step chronoamperometry. This revealed that the peculiar current jumps stem from a rapid variation of the reaction front position when the starting material is still reducible at the electrode while the product of the second electron transfer is re-oxidized. The exact convergence between the predictions by these independent methods demonstrated that KISSA is perfectly accurate even under such extreme mechanistic conditions. © 2010 Elsevier B.V. All rights reserved.
1. Introduction In two preceding publications we introduced a novel method for the numerical simulation of electrochemical reaction mechanisms involving fast homogeneous steps leading to sharp reaction fronts either in the vicinity of the electrode surface or in the bulk of the solution [1,2]. One of the most representative examples of this type is the EE reaction mechanism with comproportionation [3–7]: 1
A + e ⇄B; E0 ; 2
ð1Þ 1
B + e⇄C; E0 bbE0 ;
ð2Þ
h i 1 2 A + C⇄2B; Keq = kf = kb = exp F E0 −E0 = RT NN1:
ð3Þ
It has been demonstrated before that the reaction scheme in Eqs. (1)–(3) leads to extremely rapid variations in the concentrations of the species occurring in the solution bulk which, nonetheless, do not affect the voltammetric waveshape when the diffusion coefficients of all species are equal [1,3]. However, when the diffusion coefficients of species A, B and C begin to significantly differ the
⁎ Corresponding authors. Ecole Normale Supérieure, Département de Chimie, UMR CNRS-ENS-UPMC 8640 “PASTEUR”, 24 rue Lhomond, 75231 Paris Cedex 05, France. Tel.: +33 144 32 33 88. E-mail addresses: [email protected] (I. Svir), [email protected] (C. Amatore). 1388-2481/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.elecom.2010.07.025
voltammogram may develop peculiar features such as rapid current variations on the reverse potential scan depicted in Fig. 1A. These voltammograms were obtained through numerical simulation using KISSA software [1,2]. In fact, this effect reveals itself differently depending on the ratios of the diffusion coefficients, rate of comproportionation in Eq. (3) and on the difference between the formal potentials of the two redox reactions (1) and (2). It should be noted that similar features have been noted previously for ECE-DISPtype reaction mechanisms [8] (where they were referred to as ‘current dips’). In order to ensure that this peculiarity is not an artefact introduced by the adaptive numerical algorithm employed for the simulations [1,2] and to predict the appearance and amplitude of the current jumps depending on the system parameters we conducted a thorough investigation of this situation for fast comproportionation. 2. Model Let us consider a simplified system involving an infinitely fast comproportionation reaction (3) since this will allow obtaining an analytical solution for a part of the problem. Moreover, instead of modelling a cyclic voltammetric response (which implies continuously changing boundary conditions at the electrode surface) we consider an ‘equivalent’ double potential step experiment which gives rise to the same current jump effect (see inset of Fig. 1B-a). At t b 0 the system contains species A exclusively (this corresponds to E → ∞). When at t = 0 the potential is instantaneously changed to a value E1 bb E20 both redox reactions (1) and (2) proceed under
O.V. Klymenko et al. / Electrochemistry Communications 12 (2010) 1378–1382
diffusion control (step I) so that both B and C are produced near the electrode. After a chosen duration the potential is jumped to a value E2 = E01 + E02 = 2 (see inset in Fig. 1B-a) and held there (step II) for the same duration as that of step I. During step II, owing to the condition E20 bb E10, C is oxidized while A is still reduced. Therefore, during step II the species C is consumed both at the electrode surface and through the homogeneous reaction (3) in the bulk of the solution. Our aim is to investigate the transition in the electrochemical current provoked during step II at the very moment when species C disappears completely and the reaction mechanism in Eqs. (1)–(3) reduces to the single reaction (1). The two steps I and II will be considered separately since they differ by boundary conditions applying at the electrode surface. We define the dimensionless variables and parameters: τ=
t x D D ; y = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; dB = B ; dC = C ; tmax DA DA DA tmax
a=
½A ½B ½C i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; b= ; c= ;Ψ = ½A0 ½A 0 ½A 0 nFS½A0 DA = tmax
ð4Þ
where tmax is the total duration of the two chronoamperometric steps and S is the electrode area. The time-dependent dimensionless position of the homogeneous reaction front is denoted μ(τ). 2.1. Step I Once the potential E1 has been applied to the electrode the reduction reactions (1) and (2) lead to the formation of species C at the electrode surface. Owing to the assumption of infinitely fast interaction between A and C these species can exist only in different parts of the space separated by an infinitely thin reaction front. Within each zone at both sides of the reaction front none of the species experiences homogeneous interactions and reaction (3) proceeds only at the point y = μ(τ). Since initially only species A is present in the solution (see above) and there is no homogeneous reaction the following initial conditions apply: τ = 0 : a = 1; b = 0; c = 0; μ = 0:
ð5Þ
In the space region 0 b y b μ(τ) the concentration distributions are governed by the equations: ð6aÞ
a≡0; 2
∂b ∂ b = dB 2 ; ∂τ ∂y
ð6bÞ
∂c ∂2 c = dC 2 : ∂τ ∂y
ð6cÞ
For the space region μ(τ) b y b ∞ the following equations are valid: ∂a ∂2 a ; = ∂τ ∂y2
ð7aÞ
Fig. 1. (A) CV responses simulated for the mechanism in Eqs. (1)–(3) with parameter values: [A]0 = 1 mM, DA = 10−5 cm2 s−1, DB = 10−6 cm2 s−1, DC = 10−7 cm2 s−1, E10 = 0.4 V, E20 = − 0.4V, v = 1V s − 1. kf values are given in M − 1s − 1. (B-a) Electrochemical current simulated using KISSA for dB = 0.1, dC = 0.01, and τI = τII = 0.5. Inset shows a scheme of the applied potentials in double-step chronoamperometry relative to formal potentials of A/B and B / C couples. Panels (B-b) and (B-c) compare the dimensionless current during step II in double-step chronoamperometry simulated using KISSA (solid curves) and computed numerically for infinitely fast kinetics (symbols) for parameter values: (B-b) dB = 0.5, dC = 0.25, and (B-c) dB = 0.2, dC = 0.04.
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∂b ∂2 b = dB 2 ; ∂τ ∂y
ð7bÞ
Space region μ(τ) b y b ∞: 2
ð7cÞ
c≡0: Eqs. (6a)–(7c) satisfy the boundary conditions: y=0:
a = 0;
ð8aÞ
b = 0;
ð8bÞ
dB y = μ ðτÞ :
∂b ∂c = −dC ; ∂y ∂y
ð8cÞ
a = 0;
ð8dÞ
bðμ−0; τÞ = bðμ + 0; τÞ;
ðcontinuity of bÞ
∂a ∂c = −dC ; ∂y ∂y ∂c ∂b −2dC = dB ∂y ∂y
ðequal influx of reagentsÞ
j
∂b − ∂y y = μ−0
j
a = 1; b = 0; c = 0:
ð8gÞ
∂½C ; ∂x x = 0
j
ð8hÞ ð8iÞ
ð9Þ
:
ð14aÞ
c = 0;
ð14bÞ
y = μ ðτÞ : a = 0;
ð14cÞ
c = 0;
ð14dÞ
∂a ∂c = −dC ; ∂y ∂y y=∞:
ðequal influx of reagentsÞ
ð14eÞ
a = 1;
ð14fÞ
c = 0:
ð14gÞ
Eq. (14e) provides a condition for the determination of the reaction front position μ(τ) during step II. The normalized electrochemical current follows the equation: Ψ = dC
∂c ∂y
j
:
ð15Þ
y=0
Once the reaction front position has reached the electrode surface (viz. when μ(τ) has reached zero) due to the complete depletion of C the above mathematical model simplifies: Eq. (13a) is valid for 0 b y b ∞ with boundary conditions Eqs. (14a) and (14f). At this stage the normalized current is determined exclusively by species A: ∂a ∂y
j
:
ð16Þ
y=0
ð10Þ 3. Analytical solution for step I
2.2. Step II During step II species C is consumed both at the electrode surface and in the solution bulk due to the reaction with A. Therefore the zone containing species C gradually shrinks and eventually disappears, at which point species A gets access to the electrode surface and the reaction mechanism reduces simply to reaction (1). Consider the concentrations a and c, which satisfy the following initial conditions at the beginning of step II (lower indices I and II denote the potential step to which the quantity refers): τ = τ0 : aII ð y; τ0 Þ = aI ð y; τ0 Þ; cII ð y; τ0 Þ = cI ð y; τ0 Þ; μII ðτ0 Þ = μI ðτ0 Þ: ð11Þ
Space region 0 b y b μ(τ): ð12aÞ 2
∂c ∂ c = dC 2 : ∂τ ∂y
a = 0;
y=0:
Ψ=−
y=0
a≡0;
Boundary conditions:
;
or in the normalized form ∂c ∂y
ð13bÞ
!
The condition in Eq. (8h) indicates that the rate of production of B at the reaction front (which is equal to the sum of its fluxes in the directions away from the reaction front) is linked to the influx of species C into the reaction zone by the stoichiometry of reaction (3). The electrochemical current during step I is defined by the flux of C as:
Ψ = dC
c≡0:
ð8eÞ
y=μ + 0
ðconservation of matterÞ
i = nFSDC
ð13aÞ
ð8fÞ
c = 0;
y=∞:
∂a ∂ a ; = ∂τ ∂y2
ð12bÞ
The problem in Eqs. (5)–(8i) can be solved analytically using the approach described in [9,10], which was done for a slightly more general situation in [11]. Using the notations adopted here the solution for step I can be presented as: in the space region 0 ≤ y ≤ μ(τ): ! H y bð y; τÞ = pffiffiffiffiffi erf pffiffiffiffiffiffiffiffi ; dB 2 dB τ
ð17aÞ
" !# ! H λ y cð y; τÞ = pffiffiffiffiffiffi erf pffiffiffiffiffiffi −erf pffiffiffiffiffiffiffiffi ; dC dC 2 dC τ
ð17bÞ
in the space region μ(τ) b y b ∞: y erfc pffiffiffi 2 τ ; að y; τÞ = 1− erfcðλÞ
ð18aÞ
O.V. Klymenko et al. / Electrochemistry Communications 12 (2010) 1378–1382
H bð y; τÞ = pffiffiffiffiffi dB
λ erf pffiffiffiffiffi dB
! !
! erfc
λ erfc pffiffiffiffiffi dB
2
y pffiffiffiffiffiffiffiffi ; dB τ
ð18bÞ
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species C has been depleted. Then the gradient of a that was prevailing on the right of the reaction front is assumed to apply at the electrode surface. 5. Simulation results
where H is a constant defined as
H=
h i exp λ2 d1 −1 C
erfcðλÞ
:
ð19Þ
Parameter λ defines the reaction front propagation speed through the relation pffiffiffi μ ðτÞ = 2λ τ;
ð20Þ
and satisfies the following transcendental equation !
λ 1 2 1 2 erfc pffiffiffiffiffi = exp λ − : dC dB dB
ð21Þ
The normalized electrochemical current during step I is: Ψ = dC
∂c ∂y
j
y=0
H = − pffiffiffiffiffiffi : πτ
ð22Þ
The simulated current response has the shape illustrated in Fig. 1Ba for the case of very different diffusion coefficients and evidences the same type of current jump as noted in Fig. 1A. Note that the reduction current during step I (τ b 0.5) instantly changes sign upon the potential step from E1 to E2 which ensures the oxidation of C. Later during step II, when species C is sufficiently exhausted for not compensating the reduction current due to species A, the current changes its sign once again. This happens within a very thin time frame leading to a current jump. Let us first focus on step I and compare the numerical solution by KISSA with the analytical current in Eq. (22) (see Fig. 2A). The excellent agreement between the numerical and analytical results indicates the correct treatment of the moving reaction front problem by the adaptive numerical method implemented in KISSA. Next, let us examine step II and compare the simulation results by KISSA with the numerical solution described in Section 4. This comparison is shown in Fig. 1B-b and B-c for two pairs of values of dB and dC giving rise to current jumps and evidences a strict agreement of the results. As noted in Fig. 1A, the occurrence of such current jumps
4. Numerical solution for step II Solution for step II cannot be derived in closed form and therefore was obtained numerically treating it as a moving boundary problem. For this purpose the front fixing approach was employed [12,13]. Thus, for the solution of the diffusion Eq. (12b) for the species C the following coordinate transformation was used: y = ξ μ ðτÞ;
ð23Þ
where ξ is a new spatial coordinate varying in a fixed interval [0, 1]. Eq. (12b) thus becomes: 2
μ ðτÞ
∂c ∂2 c dμ ∂c = dC 2 + ξ μ ðτÞ dτ ∂ξ ∂τ ∂ξ
ð24Þ
which satisfies homogeneous boundary conditions (c = 0) at both ξ = 0 and ξ = 1. Similarly, for the semi-infinite diffusion of A we use the following transformation: y = μ ðτÞ +
η ; 1−η
ð25Þ
where the new spatial variable η again belongs to the fixed interval [0, 1], and Eq. (13a) becomes: 2
∂a 4∂ a 3 ∂a 2 dμ ∂a −2ð1−ηÞ = ð1−ηÞ + ð1−ηÞ ; dτ ∂η ∂τ ∂η ∂η2
ð26Þ
satisfying the boundary conditions: η = 0:
a = 0;
ð27aÞ
η = 1:
a = 1:
ð27bÞ
Using implicit numerical methods of second order stable solutions can be achieved for any finite value of μ(τ). However, in simulations when μ(τ) reached a value of 0.001 the computation continued for species A only as described at the end of Section 2.2 assuming that all
Fig. 2. Comparison of simulation results by KISSA for double-step chronoamperometry (solid curves; see Fig. 1 legend for parameter values; kf = 5 × 1010 M − 1s − 1) and limiting solutions (symbols): (A) dimensionless currents during step I for different ratios of diffusion coefficients; (B) reaction front position as function of time (τI = τII = 0.5). Numbers at the curves correspond to: (1) dB = 1, dC = 1; (2) dB = 0.1, dC = 0.01; (3) dB = 0.2, dC = 0.04; (4) dB = 0.1, dC = 1; and (5) dB = 0.5, dC = 0.25.
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O.V. Klymenko et al. / Electrochemistry Communications 12 (2010) 1378–1382
demands large rate constants of comproportionation and small values of dC (Fig. 1B-b,B-c). Since in classical EE mechanisms dB and dC are expected to be close to unity, this explains why such features have never been reported experimentally to the best of our knowledge. In fact, Fig. 2B shows that the occurrence of such current jumps is directly related to the position of the reaction front during the second potential step. Note also that the simulations using KISSA (solid curves) strictly coincide with the analytical solution for τ ≤ 0.5 and with the numerically computed values for τ N 0.5 (symbols). The peculiar current jumps observed in Fig. 1A and B-a–B-c stem from the rapid variations of the reaction front created by the sudden vanishing of species C. Hence, the current “jump” observed in voltammetry or in chronoamperometry features not a real electron stoichiometry change but a change of the boundary which ultimately defines the concentration gradient near the electrode surface. This situation is then identical to that which gives rise to two electron waves in EE mechanisms [14]. 6. Conclusions Exploration of EE mechanisms under extreme conditions (huge comproportionation rate constants; highly different diffusion coefficients) by simulation thanks to the newly developed software KISSA revealed unexpected features in voltammetric or chronoamperometric currents. To find out whether these features had a sound mechanistic origin or resulted from a local loss of accuracy in our simulations we employed a combined analytical (first potential step) and numerical (second potential step) strategy radically independent of KISSA to scrutinize the same mechanistic situations under the conditions of double-step chronoamperometry. This confirmed the validity of simulations and predictions produced by means of KISSA and showed that current jumps resulted from the diffusion coefficient-dependent reaction front dynamics when species C has practically disappeared from the diffusion layer due to its combined oxidation at the electrode surface and its consumption in solution by
species A. Though the observation of such specific feature requires so drastic conditions that their experimental observability is doubtful, this work establishes that our software is perfectly accurate even under such extreme conditions. Acknowledgements In Paris this work was supported in part by CNRS (UMR8640 “PASTEUR” and LIA “XiamENS”), Ecole Normale Supérieure (ENS), Université Pierre et Marie Curie (UPMC), and by the French Ministry of Research. In Kharkov this work was supported by the Ministry of Education and Science of Ukraine. Dr. Klymenko thanks Mairie de la Ville de Paris for a ‘Research in Paris 2010’ award. Prof. Svir thanks CNRS for the Director of research position in UMR8640 during 2010. References [1] C. Amatore, O. Klymenko, I. Svir, Electrochem. Commun. doi:10.1016/j. elecom.2010.06.009. [2] C. Amatore, O. Klymenko, I. Svir, Electrochem. Commun. doi:10.1016/j. elecom.2010.06.008. [3] C.P. Andrieux, J.-M. Savéant, J. Electroanal. Chem. 28 (1970) 339. [4] Z. Rongfeng, D.H. Evans, J. Electroanal. Chem. 385 (1995) 201. [5] L.K. Bieniasz, J. Electroanal. Chem. 379 (1994) 71. [6] S.R. Belding, R. Baron, E.J.F. Dickinson, R.G. Compton, J. Phys. Chem. C 113 (2009) 16042. [7] E.O. Barnes, A.M. O'Mahony, S.R. Belding, R.G. Compton, J. Chem. Eng. Data 55 (2010) 2219. [8] C. Amatore, J. Pinson, J.-M. Savéant, A. Thiébault, J. Electroanal. Chem. 107 (1980) 75. [9] H.S. Carslaw, J.C. Jaeger, Conduction of heat in solids, 2nd ed., Clarendon Press, Oxford, 1959. [10] C.P. Andrieux, P. Hapiot, J.M. Savéant, J. Electroanal. Chem. 172 (1984) 49. [11] Y.-M. Tsou, F.C. Anson, J. Phys. Chem. 89 (1985) 3818. [12] A.A. Samarskii, P.N. Vabishchevich, Computational Heat Transfer, Editorial URSS, Moscow, 2003. [13] C. Amatore, O.V. Klymenko, A.I. Oleinick, I. Svir, Chem. Phys. Chem. 10 (2009) 1593. [14] C. Amatore, F. Bonhomme, J.-L. Bruneel, L. Servant, L. Thouin, J. Electroanal. Chem. 484 (2000) 1.
Contents lists available at ScienceDirect
Electrochemistry Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e l e c o m
Theoretical study of the EE reaction mechanism with comproportionation and different diffusivities of reactants Oleksiy V. Klymenko a,b, Irina Svir a,b,⁎, Christian Amatore a,⁎ a b
Ecole Normale Supérieure, Département de Chimie, UMR CNRS-ENS-UPMC 8640 “PASTEUR”, 24 rue Lhomond, 75231 Paris Cedex 05, France Kharkov National University of Radioelectronics, Mathematical & Computer Modelling Laboratory, 14 Lenin Avenue, 61166 Kharkov, Ukraine
a r t i c l e
i n f o
Article history: Received 5 July 2010 Received in revised form 20 July 2010 Accepted 20 July 2010 Available online 29 July 2010 Keywords: EE mechanisms Fast kinetics Comproportionation Different diffusion coefficients KISSA Simulation
a b s t r a c t In this article we investigate the origin of unexpected features appearing in voltammetry or double-step chronoamperometry of EE systems when the rate of comproportionation is extremely fast and the diffusion coefficients of the reactant and products highly differ. These features were noticed during the testing of our new software KISSA intended to solve any reaction mechanisms even when acute reaction fronts develop near the electrode surface or within the solution. To validate the principle of the new adaptive algorithm [C. Amatore, O. Klymenko, I. Svir, Electrochem. Commun. doi:10.1016/j.elecom.2010.06.009] implemented in KISSA we used analytical and numerical solutions for double-step chronoamperometry. This revealed that the peculiar current jumps stem from a rapid variation of the reaction front position when the starting material is still reducible at the electrode while the product of the second electron transfer is re-oxidized. The exact convergence between the predictions by these independent methods demonstrated that KISSA is perfectly accurate even under such extreme mechanistic conditions. © 2010 Elsevier B.V. All rights reserved.
1. Introduction In two preceding publications we introduced a novel method for the numerical simulation of electrochemical reaction mechanisms involving fast homogeneous steps leading to sharp reaction fronts either in the vicinity of the electrode surface or in the bulk of the solution [1,2]. One of the most representative examples of this type is the EE reaction mechanism with comproportionation [3–7]: 1
A + e ⇄B; E0 ; 2
ð1Þ 1
B + e⇄C; E0 bbE0 ;
ð2Þ
h i 1 2 A + C⇄2B; Keq = kf = kb = exp F E0 −E0 = RT NN1:
ð3Þ
It has been demonstrated before that the reaction scheme in Eqs. (1)–(3) leads to extremely rapid variations in the concentrations of the species occurring in the solution bulk which, nonetheless, do not affect the voltammetric waveshape when the diffusion coefficients of all species are equal [1,3]. However, when the diffusion coefficients of species A, B and C begin to significantly differ the
⁎ Corresponding authors. Ecole Normale Supérieure, Département de Chimie, UMR CNRS-ENS-UPMC 8640 “PASTEUR”, 24 rue Lhomond, 75231 Paris Cedex 05, France. Tel.: +33 144 32 33 88. E-mail addresses: [email protected] (I. Svir), [email protected] (C. Amatore). 1388-2481/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.elecom.2010.07.025
voltammogram may develop peculiar features such as rapid current variations on the reverse potential scan depicted in Fig. 1A. These voltammograms were obtained through numerical simulation using KISSA software [1,2]. In fact, this effect reveals itself differently depending on the ratios of the diffusion coefficients, rate of comproportionation in Eq. (3) and on the difference between the formal potentials of the two redox reactions (1) and (2). It should be noted that similar features have been noted previously for ECE-DISPtype reaction mechanisms [8] (where they were referred to as ‘current dips’). In order to ensure that this peculiarity is not an artefact introduced by the adaptive numerical algorithm employed for the simulations [1,2] and to predict the appearance and amplitude of the current jumps depending on the system parameters we conducted a thorough investigation of this situation for fast comproportionation. 2. Model Let us consider a simplified system involving an infinitely fast comproportionation reaction (3) since this will allow obtaining an analytical solution for a part of the problem. Moreover, instead of modelling a cyclic voltammetric response (which implies continuously changing boundary conditions at the electrode surface) we consider an ‘equivalent’ double potential step experiment which gives rise to the same current jump effect (see inset of Fig. 1B-a). At t b 0 the system contains species A exclusively (this corresponds to E → ∞). When at t = 0 the potential is instantaneously changed to a value E1 bb E20 both redox reactions (1) and (2) proceed under
O.V. Klymenko et al. / Electrochemistry Communications 12 (2010) 1378–1382
diffusion control (step I) so that both B and C are produced near the electrode. After a chosen duration the potential is jumped to a value E2 = E01 + E02 = 2 (see inset in Fig. 1B-a) and held there (step II) for the same duration as that of step I. During step II, owing to the condition E20 bb E10, C is oxidized while A is still reduced. Therefore, during step II the species C is consumed both at the electrode surface and through the homogeneous reaction (3) in the bulk of the solution. Our aim is to investigate the transition in the electrochemical current provoked during step II at the very moment when species C disappears completely and the reaction mechanism in Eqs. (1)–(3) reduces to the single reaction (1). The two steps I and II will be considered separately since they differ by boundary conditions applying at the electrode surface. We define the dimensionless variables and parameters: τ=
t x D D ; y = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; dB = B ; dC = C ; tmax DA DA DA tmax
a=
½A ½B ½C i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; b= ; c= ;Ψ = ½A0 ½A 0 ½A 0 nFS½A0 DA = tmax
ð4Þ
where tmax is the total duration of the two chronoamperometric steps and S is the electrode area. The time-dependent dimensionless position of the homogeneous reaction front is denoted μ(τ). 2.1. Step I Once the potential E1 has been applied to the electrode the reduction reactions (1) and (2) lead to the formation of species C at the electrode surface. Owing to the assumption of infinitely fast interaction between A and C these species can exist only in different parts of the space separated by an infinitely thin reaction front. Within each zone at both sides of the reaction front none of the species experiences homogeneous interactions and reaction (3) proceeds only at the point y = μ(τ). Since initially only species A is present in the solution (see above) and there is no homogeneous reaction the following initial conditions apply: τ = 0 : a = 1; b = 0; c = 0; μ = 0:
ð5Þ
In the space region 0 b y b μ(τ) the concentration distributions are governed by the equations: ð6aÞ
a≡0; 2
∂b ∂ b = dB 2 ; ∂τ ∂y
ð6bÞ
∂c ∂2 c = dC 2 : ∂τ ∂y
ð6cÞ
For the space region μ(τ) b y b ∞ the following equations are valid: ∂a ∂2 a ; = ∂τ ∂y2
ð7aÞ
Fig. 1. (A) CV responses simulated for the mechanism in Eqs. (1)–(3) with parameter values: [A]0 = 1 mM, DA = 10−5 cm2 s−1, DB = 10−6 cm2 s−1, DC = 10−7 cm2 s−1, E10 = 0.4 V, E20 = − 0.4V, v = 1V s − 1. kf values are given in M − 1s − 1. (B-a) Electrochemical current simulated using KISSA for dB = 0.1, dC = 0.01, and τI = τII = 0.5. Inset shows a scheme of the applied potentials in double-step chronoamperometry relative to formal potentials of A/B and B / C couples. Panels (B-b) and (B-c) compare the dimensionless current during step II in double-step chronoamperometry simulated using KISSA (solid curves) and computed numerically for infinitely fast kinetics (symbols) for parameter values: (B-b) dB = 0.5, dC = 0.25, and (B-c) dB = 0.2, dC = 0.04.
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O.V. Klymenko et al. / Electrochemistry Communications 12 (2010) 1378–1382
∂b ∂2 b = dB 2 ; ∂τ ∂y
ð7bÞ
Space region μ(τ) b y b ∞: 2
ð7cÞ
c≡0: Eqs. (6a)–(7c) satisfy the boundary conditions: y=0:
a = 0;
ð8aÞ
b = 0;
ð8bÞ
dB y = μ ðτÞ :
∂b ∂c = −dC ; ∂y ∂y
ð8cÞ
a = 0;
ð8dÞ
bðμ−0; τÞ = bðμ + 0; τÞ;
ðcontinuity of bÞ
∂a ∂c = −dC ; ∂y ∂y ∂c ∂b −2dC = dB ∂y ∂y
ðequal influx of reagentsÞ
j
∂b − ∂y y = μ−0
j
a = 1; b = 0; c = 0:
ð8gÞ
∂½C ; ∂x x = 0
j
ð8hÞ ð8iÞ
ð9Þ
:
ð14aÞ
c = 0;
ð14bÞ
y = μ ðτÞ : a = 0;
ð14cÞ
c = 0;
ð14dÞ
∂a ∂c = −dC ; ∂y ∂y y=∞:
ðequal influx of reagentsÞ
ð14eÞ
a = 1;
ð14fÞ
c = 0:
ð14gÞ
Eq. (14e) provides a condition for the determination of the reaction front position μ(τ) during step II. The normalized electrochemical current follows the equation: Ψ = dC
∂c ∂y
j
:
ð15Þ
y=0
Once the reaction front position has reached the electrode surface (viz. when μ(τ) has reached zero) due to the complete depletion of C the above mathematical model simplifies: Eq. (13a) is valid for 0 b y b ∞ with boundary conditions Eqs. (14a) and (14f). At this stage the normalized current is determined exclusively by species A: ∂a ∂y
j
:
ð16Þ
y=0
ð10Þ 3. Analytical solution for step I
2.2. Step II During step II species C is consumed both at the electrode surface and in the solution bulk due to the reaction with A. Therefore the zone containing species C gradually shrinks and eventually disappears, at which point species A gets access to the electrode surface and the reaction mechanism reduces simply to reaction (1). Consider the concentrations a and c, which satisfy the following initial conditions at the beginning of step II (lower indices I and II denote the potential step to which the quantity refers): τ = τ0 : aII ð y; τ0 Þ = aI ð y; τ0 Þ; cII ð y; τ0 Þ = cI ð y; τ0 Þ; μII ðτ0 Þ = μI ðτ0 Þ: ð11Þ
Space region 0 b y b μ(τ): ð12aÞ 2
∂c ∂ c = dC 2 : ∂τ ∂y
a = 0;
y=0:
Ψ=−
y=0
a≡0;
Boundary conditions:
;
or in the normalized form ∂c ∂y
ð13bÞ
!
The condition in Eq. (8h) indicates that the rate of production of B at the reaction front (which is equal to the sum of its fluxes in the directions away from the reaction front) is linked to the influx of species C into the reaction zone by the stoichiometry of reaction (3). The electrochemical current during step I is defined by the flux of C as:
Ψ = dC
c≡0:
ð8eÞ
y=μ + 0
ðconservation of matterÞ
i = nFSDC
ð13aÞ
ð8fÞ
c = 0;
y=∞:
∂a ∂ a ; = ∂τ ∂y2
ð12bÞ
The problem in Eqs. (5)–(8i) can be solved analytically using the approach described in [9,10], which was done for a slightly more general situation in [11]. Using the notations adopted here the solution for step I can be presented as: in the space region 0 ≤ y ≤ μ(τ): ! H y bð y; τÞ = pffiffiffiffiffi erf pffiffiffiffiffiffiffiffi ; dB 2 dB τ
ð17aÞ
" !# ! H λ y cð y; τÞ = pffiffiffiffiffiffi erf pffiffiffiffiffiffi −erf pffiffiffiffiffiffiffiffi ; dC dC 2 dC τ
ð17bÞ
in the space region μ(τ) b y b ∞: y erfc pffiffiffi 2 τ ; að y; τÞ = 1− erfcðλÞ
ð18aÞ
O.V. Klymenko et al. / Electrochemistry Communications 12 (2010) 1378–1382
H bð y; τÞ = pffiffiffiffiffi dB
λ erf pffiffiffiffiffi dB
! !
! erfc
λ erfc pffiffiffiffiffi dB
2
y pffiffiffiffiffiffiffiffi ; dB τ
ð18bÞ
1381
species C has been depleted. Then the gradient of a that was prevailing on the right of the reaction front is assumed to apply at the electrode surface. 5. Simulation results
where H is a constant defined as
H=
h i exp λ2 d1 −1 C
erfcðλÞ
:
ð19Þ
Parameter λ defines the reaction front propagation speed through the relation pffiffiffi μ ðτÞ = 2λ τ;
ð20Þ
and satisfies the following transcendental equation !
λ 1 2 1 2 erfc pffiffiffiffiffi = exp λ − : dC dB dB
ð21Þ
The normalized electrochemical current during step I is: Ψ = dC
∂c ∂y
j
y=0
H = − pffiffiffiffiffiffi : πτ
ð22Þ
The simulated current response has the shape illustrated in Fig. 1Ba for the case of very different diffusion coefficients and evidences the same type of current jump as noted in Fig. 1A. Note that the reduction current during step I (τ b 0.5) instantly changes sign upon the potential step from E1 to E2 which ensures the oxidation of C. Later during step II, when species C is sufficiently exhausted for not compensating the reduction current due to species A, the current changes its sign once again. This happens within a very thin time frame leading to a current jump. Let us first focus on step I and compare the numerical solution by KISSA with the analytical current in Eq. (22) (see Fig. 2A). The excellent agreement between the numerical and analytical results indicates the correct treatment of the moving reaction front problem by the adaptive numerical method implemented in KISSA. Next, let us examine step II and compare the simulation results by KISSA with the numerical solution described in Section 4. This comparison is shown in Fig. 1B-b and B-c for two pairs of values of dB and dC giving rise to current jumps and evidences a strict agreement of the results. As noted in Fig. 1A, the occurrence of such current jumps
4. Numerical solution for step II Solution for step II cannot be derived in closed form and therefore was obtained numerically treating it as a moving boundary problem. For this purpose the front fixing approach was employed [12,13]. Thus, for the solution of the diffusion Eq. (12b) for the species C the following coordinate transformation was used: y = ξ μ ðτÞ;
ð23Þ
where ξ is a new spatial coordinate varying in a fixed interval [0, 1]. Eq. (12b) thus becomes: 2
μ ðτÞ
∂c ∂2 c dμ ∂c = dC 2 + ξ μ ðτÞ dτ ∂ξ ∂τ ∂ξ
ð24Þ
which satisfies homogeneous boundary conditions (c = 0) at both ξ = 0 and ξ = 1. Similarly, for the semi-infinite diffusion of A we use the following transformation: y = μ ðτÞ +
η ; 1−η
ð25Þ
where the new spatial variable η again belongs to the fixed interval [0, 1], and Eq. (13a) becomes: 2
∂a 4∂ a 3 ∂a 2 dμ ∂a −2ð1−ηÞ = ð1−ηÞ + ð1−ηÞ ; dτ ∂η ∂τ ∂η ∂η2
ð26Þ
satisfying the boundary conditions: η = 0:
a = 0;
ð27aÞ
η = 1:
a = 1:
ð27bÞ
Using implicit numerical methods of second order stable solutions can be achieved for any finite value of μ(τ). However, in simulations when μ(τ) reached a value of 0.001 the computation continued for species A only as described at the end of Section 2.2 assuming that all
Fig. 2. Comparison of simulation results by KISSA for double-step chronoamperometry (solid curves; see Fig. 1 legend for parameter values; kf = 5 × 1010 M − 1s − 1) and limiting solutions (symbols): (A) dimensionless currents during step I for different ratios of diffusion coefficients; (B) reaction front position as function of time (τI = τII = 0.5). Numbers at the curves correspond to: (1) dB = 1, dC = 1; (2) dB = 0.1, dC = 0.01; (3) dB = 0.2, dC = 0.04; (4) dB = 0.1, dC = 1; and (5) dB = 0.5, dC = 0.25.
1382
O.V. Klymenko et al. / Electrochemistry Communications 12 (2010) 1378–1382
demands large rate constants of comproportionation and small values of dC (Fig. 1B-b,B-c). Since in classical EE mechanisms dB and dC are expected to be close to unity, this explains why such features have never been reported experimentally to the best of our knowledge. In fact, Fig. 2B shows that the occurrence of such current jumps is directly related to the position of the reaction front during the second potential step. Note also that the simulations using KISSA (solid curves) strictly coincide with the analytical solution for τ ≤ 0.5 and with the numerically computed values for τ N 0.5 (symbols). The peculiar current jumps observed in Fig. 1A and B-a–B-c stem from the rapid variations of the reaction front created by the sudden vanishing of species C. Hence, the current “jump” observed in voltammetry or in chronoamperometry features not a real electron stoichiometry change but a change of the boundary which ultimately defines the concentration gradient near the electrode surface. This situation is then identical to that which gives rise to two electron waves in EE mechanisms [14]. 6. Conclusions Exploration of EE mechanisms under extreme conditions (huge comproportionation rate constants; highly different diffusion coefficients) by simulation thanks to the newly developed software KISSA revealed unexpected features in voltammetric or chronoamperometric currents. To find out whether these features had a sound mechanistic origin or resulted from a local loss of accuracy in our simulations we employed a combined analytical (first potential step) and numerical (second potential step) strategy radically independent of KISSA to scrutinize the same mechanistic situations under the conditions of double-step chronoamperometry. This confirmed the validity of simulations and predictions produced by means of KISSA and showed that current jumps resulted from the diffusion coefficient-dependent reaction front dynamics when species C has practically disappeared from the diffusion layer due to its combined oxidation at the electrode surface and its consumption in solution by
species A. Though the observation of such specific feature requires so drastic conditions that their experimental observability is doubtful, this work establishes that our software is perfectly accurate even under such extreme conditions. Acknowledgements In Paris this work was supported in part by CNRS (UMR8640 “PASTEUR” and LIA “XiamENS”), Ecole Normale Supérieure (ENS), Université Pierre et Marie Curie (UPMC), and by the French Ministry of Research. In Kharkov this work was supported by the Ministry of Education and Science of Ukraine. Dr. Klymenko thanks Mairie de la Ville de Paris for a ‘Research in Paris 2010’ award. Prof. Svir thanks CNRS for the Director of research position in UMR8640 during 2010. References [1] C. Amatore, O. Klymenko, I. Svir, Electrochem. Commun. doi:10.1016/j. elecom.2010.06.009. [2] C. Amatore, O. Klymenko, I. Svir, Electrochem. Commun. doi:10.1016/j. elecom.2010.06.008. [3] C.P. Andrieux, J.-M. Savéant, J. Electroanal. Chem. 28 (1970) 339. [4] Z. Rongfeng, D.H. Evans, J. Electroanal. Chem. 385 (1995) 201. [5] L.K. Bieniasz, J. Electroanal. Chem. 379 (1994) 71. [6] S.R. Belding, R. Baron, E.J.F. Dickinson, R.G. Compton, J. Phys. Chem. C 113 (2009) 16042. [7] E.O. Barnes, A.M. O'Mahony, S.R. Belding, R.G. Compton, J. Chem. Eng. Data 55 (2010) 2219. [8] C. Amatore, J. Pinson, J.-M. Savéant, A. Thiébault, J. Electroanal. Chem. 107 (1980) 75. [9] H.S. Carslaw, J.C. Jaeger, Conduction of heat in solids, 2nd ed., Clarendon Press, Oxford, 1959. [10] C.P. Andrieux, P. Hapiot, J.M. Savéant, J. Electroanal. Chem. 172 (1984) 49. [11] Y.-M. Tsou, F.C. Anson, J. Phys. Chem. 89 (1985) 3818. [12] A.A. Samarskii, P.N. Vabishchevich, Computational Heat Transfer, Editorial URSS, Moscow, 2003. [13] C. Amatore, O.V. Klymenko, A.I. Oleinick, I. Svir, Chem. Phys. Chem. 10 (2009) 1593. [14] C. Amatore, F. Bonhomme, J.-L. Bruneel, L. Servant, L. Thouin, J. Electroanal. Chem. 484 (2000) 1.