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Analytical solutions of integral equations for modelling of reversible electrode processes under voltammetric conditions
Journal of Electroanalytical Chemistry 619–620 (2008) 164–168
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Journal of Electroanalytical Chemistry journal homepage: www.elsevier.com/locate/jelechem
Analytical solutions of integral equations for modelling of reversible electrode processes under voltammetric conditions Valentin Mircˇeski a,*, Zˇivorad Tomovski b a b
Institute of Chemistry, Faculty of Natural Sciences and Mathematics, Ss Cyril and Methodius University, P.O. Box 162, 1000 Skopje, Republic of Macedonia Institute of Mathematics, Faculty of Natural Sciences and Mathematics, Ss Cyril and Methodius University, P.O. Box 162, 1000 Skopje, Republic of Macedonia
a r t i c l e
i n f o
Article history: Received 22 January 2008 Received in revised form 31 March 2008 Accepted 7 April 2008 Available online 10 April 2008 Keywords: Mathematical modelling Integral equations Voltammetry Laplace transform
a b s t r a c t Mathematical procedure for analytical solutions of integral equations referring to a voltammetric experiment at a stationary planar electrode of reversible electrode processes is described. The procedure is Rt based upon Laplace transforms and definition of an auxiliary function qðtÞ ¼ 0 IðsÞds, where I(t) is the faradaic current–time function. The auxiliary function q(t) represents the charge consumed in the course of the voltammetric experiment. The methodology is exemplified by considering simple charge transfer reaction, an EC0i catalytic mechanism, and an electrode mechanism involving partial adsorption of both components of the redox couple. All solutions are derived in terms of integrals that need to be further numerically evaluated. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Modelling of a voltammetric experiment can be accomplished applying two different strategies. The first, and nowadays probably most popular one, is simulation of the experiment following completely numerical procedure based on the method of finite differences [1–4]. The second approach requires rigorous algebraic manipulation of differential equations describing the mass transfer regime, in order to provide solutions that relate the critical parameters of the voltammetric experiment, i.e., faradaic current, time and potential. The latter approach, in a general sense, is mathematically more elegant and more accurate. However, an analytical solution in a closed form can be obtained only for rather simple experimental conditions, which is exemplified by the classic works of Cottrell [5], Ilkovicˇ [6], Karaoglanoff [7] and others. For other experimental conditions, e.g., a common voltammetric experiment at a stationary electrode, the rigorous mathematical modelling must be finalized by a numerical algorithm that provides the final link between the current and the electrode potential. For solving differential equations describing mass transfer regime Laplace transform has been frequently used [8], as it provides a powerful means for simplification of the mathematical complexity. Application of Laplace transform ends with solutions in a form of integral equations, in which the current–time function I(t) appears as an argument of the integration. For instance, let us consider a simple electrode reaction of a dissolved redox couple at a * Corresponding author. E-mail address: [email protected] (V. Mircˇeski). 0022-0728/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jelechem.2008.04.001
stationary planar electrode in the absence of concurrent homogeneous chemical reactions: n
OxðaqÞ þ ne ¼ RedðaqÞ
ð1Þ
Assuming that the mass transfer can be described by a linear semi-infinite diffusion model, the solutions that relate the surface concentrations of electroactive species with the faradaic current are well known: Z t 1 1 pffiffiffiffi IðsÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds pðt sÞ nFA D 0 Z t 1 1 pffiffiffiffi cRed ð0; tÞ ¼ IðsÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds pðt sÞ nFA D 0 cOx ð0; tÞ ¼ cOx
ð2Þ ð3Þ
where A is the electrode surface area, D is the common diffusivity, cOx is the bulk concentration, and all other symbols have their usual meaning. Eqs. (2) and (3) are derived by solving Fick’s second law, and they are generally valid irrespective of the voltammetric method applied. They are derived under provision the reductive current has a positive sign. Moreover, at the beginning of the voltammetric experiment (t = 0), the surface concentration of Ox specie equals its bulk concentration (cOx ð0; 0Þ ¼ cOx ), while Red was initially absent from the solution. The identical integral appearing in both Eqs. (2) and (3) is known as a semi-integral of the current, or Riemann– Liouville fractional integral of half order [9]. The integral represents convolution operation of the current function I(t) and the time function p1ffiffiffi . It is important to emphasize that Oldham [10–13] develpt oped a valuable methodology for modelling of a variety of voltammetric experiment by virtue of semi-integration, or its
V. Mircˇeski, Zˇ. Tomovski / Journal of Electroanalytical Chemistry 619–620 (2008) 164–168
converse operation, semi-differentiation [9]. Combining the surface concentrations with the Nernst or Butler–Volmer equations, one obtains an integral equation that provides a link between the current I(t) and the potential function E(t), thereby predicting the outcome of the voltammetric experiment. Assuming that reaction (1) is electrochemically reversible, the final integral equation reads: cOx 1 pffiffiffiffi ¼ 1 þ expðuðtÞÞ nFA D
Z
t
0
1 IðsÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds pðt sÞ
ð4Þ
where is relative dimensionless electrode potential function, is the standard potential of the redox couple, and E(t) is the potential function depending on the respective voltammetric technique. Integral Eq. (4) can be solved numerically by the algorithm proposed in the seminal work of Nicholson and Shain [14], or by related methods [15–18]. It is an Abel type integral equation, which can be also analytically solved, as demonstrated by Gokhshtein [19] and Matsuda and Ayabe [20], where the solution is given in a form of an integral. Olmstead and Nicholson proposed a general numerical procedure based on the step-function method [21] that can be applied to solve Eq. (4) as well as related Voltera type integral equations arising in the theory of voltammetry [22]. The latter method gained a wide application as it provides simple solutions in a form of recurrent formula that can be easily adopted for each voltammetric technique. According to the step-function method, the total time interval of the voltammetric experiment is divided into definite number of time increments. Over each time increment, the current function is regarded as being a constant, and only an integral of the time function (i.e., f ðtÞ ¼ p1ffiffiffi in Eq. (4)) pt has to be evaluated. In our recent work, we have proposed modification of the step-function method in order to cope with complex experimental cases, exemplified by an electrode mechanism which involves adsorption of the redox couple and, at the same time, a concurrent chemical reaction [23]. In such complex case, the analytical form of the time function f(t) cannot be obtained, and its values, together with its integral can only be numerically evaluated. In this procedure, the solution can be obtained in an implicit form, requiring double application of the numerical step-function method, which sacrifices the accuracy of the estimated current values and increases the computational time. On the contrary, in the present communication we make an attempt to develop a general methodology for analytical solution of integral equations in order to avoid the numerical approach as much as possible. Using Laplace transform, a mathematical procedure for analytical solution of integral equation (4) in a time domain is presented. In addition, we present for the first time analytical solutions for a reversible diffusion controlled electrode reaction coupled with an irreversible regenerative chemical reaction (EC0i catalytic mechanism) and a reversible mechanism in which both components of the redox couple adsorb, in the absence of a concurrent chemical reaction. In the present procedure the solutions are given in an explicit form, enabling the current values to be calculated with much higher accuracy compared to the previous implicit numerical approach [23].
2. Results and discussion 2.1. Reversible electrode reaction at a planar electrode Eq. (4) is known as a singular integral equation of Abel type, with a kernel kðtÞ ¼ p1ffiffiffi , where k(0) = 1. Let us define the following pt auxiliary functions: gðtÞ ¼
cOx 1 þ expðuðtÞÞ
ð5Þ
165
and qðtÞ ¼
Z
t
IðsÞds
ð6Þ
0
Obviously, the function q(t) has a physical meaning, representing the charge consumed in the course of the voltammetric experiment. In addition, KðpÞ; GðpÞ; IðpÞ; and Q ðpÞ are Laplace transforms of k(t), g(t), I(t) and q(t), respectively, where p is dummy Laplace variable. The Laplace transforms of the charge and current functions are related as follows: Q ðpÞ ¼
1 IðpÞ p
ð7Þ
In addition, KðpÞ ¼ p1ffiffip. Applying the convolution theorem to (4) one gets 1 GðpÞ ¼ bIðpÞ pffiffiffi p
ð8Þ
where b ¼ nFA1pffiffiDffi. Taking into account Eq. (7), the latter equation is rearranged to GðpÞ Q ðpÞ ¼ pffiffiffi b p
ð9Þ
Applying inverse Laplace transform to Eq. (9), the function q(t) is found as follows: Z t 1 gðt sÞ pffiffiffi ds ð10Þ qðtÞ ¼ pffiffiffi b p 0 s Substituting for g(t) and b in (10), one gets pffiffiffiffi Z c nFA D t ds pffiffiffi qðtÞ ¼ Ox pffiffiffi p sð1 þ expðuðt sÞÞÞ 0
ð11Þ
From (6) follows that the current is the first derivation of the charge function q(t), i.e., IðtÞ ¼
d qðtÞ dt
ð12Þ
Taking into account the formula for derivation of the convolution integral Z t Z t d d ð13Þ f ðsÞgðt sÞds ¼ f ðsÞ ðgðt sÞÞds þ f ðtÞgð0Þ dt 0 dt 0 the final solution reads ! pffiffiffiffi Z t d euðtsÞ dt ðuðt sÞÞ c nFA D 1 1 p ffiffi IðtÞ ¼ Ox pffiffiffi ds þ pffiffiffi p t 1 þ euð0Þ sð1 þ euðtsÞ Þ2 0
ð14Þ
Provided the potential function /(t) is analytically given, Eq. (14) represents a general solution valid for any voltammetric technique. For instance, in linear sweep voltammetry (LSV), the potential function has the simples form, i.e., ð15Þ where Ei is the initial potential and v is the scan rate. Note that Eq. (15) applies for a reductive potential scan. Hence, the exact solution for LSV is ! pffiffiffiffi Z t c nFA D 1 eðui hðtsÞÞ h 1 1 pffiffiffi p ffiffi IðtÞ ¼ Ox pffiffiffi ds þ ð16Þ p s ð1 þ eðui hðtsÞÞ Þ2 t 1 þ eui 0 where . Unfortunately, the integral in (16) cannot be solved in a closed form, and it has to be numerically evaluated. The latter equation can be easily transformed into dimensionless form by defining the dimensionless current as pffiffiffiffi pffiffiffiffiffiffiffiffi. The solid line in Fig. 1 shows a dimensionless LS W ¼ nFAcI RT vDnF Ox voltammogram calculated on the basis of Eq. (16). The numerical
V. Mircˇeski, Zˇ. Tomovski / Journal of Electroanalytical Chemistry 619–620 (2008) 164–168
166
0.8
0.5
0.6
0.4
0.4
0.3
0.2
0.2
0 -0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
E − E° / V
0.1 -0.2
0 -0.3
-0.2
-0.1
0
0.1
0.2
0.3
E − E° / V
-0.4
Fig. 1. Comparison of the LS voltammograms for reaction (1) calculated with the aid of Eq. (16) (line) and numerical step-function method (circles) [21]. For the stepfunction method, the potential increment was dE = 0.1 mV. The dimensionless cupffiffiffiffi pffiffiffiffiffiffiffiffi. rrent is defined as W ¼ nFAcI RT vDnF Ox
integration has been carried out with the standard operations for numerical integration of the software package MATHCAD [24]. For numerical integration, MATHCAD uses the Romberg method as the default, where the interval of integration is divided into equally spaced subintervals [25]. The features of the voltammogram agree well with those simulated with other methods [21]. Fig. 1 also shows a comparison between voltammograms calculated with Eq. (16) (solid line) and simulated by the standard step-function numerical method [21,22] (circles). For numerical simulations, the step potential was dE = 0.1 mV. The agreement between the two methods is obvious. The average relative difference between the dimensionless current values is 0.025%. The general applicability of Eq. (16) to more complex voltammetric techniques is illustrated by considering the case of squarewave voltammetry. It has been recently proposed that the potential form of the ramped SWV can be represented by the following Fourier series [26]:
EðtÞ ¼ Ei vt þ
K 4Esw X sin½ð2n 1Þx t 2n 1 p n¼1
with K ¼ 1;
ð17Þ
where x is the angular frequency and Esw is the amplitude of the square-wave. For K P 20, Eq. (17) gives an excellent approximation to the real ramped square-wave voltage. Note that in this version of the technique, square-wave is superimposed on a dc ramp, rather than on a staircase voltage as frequently used in the standard Osteryoung-type SW voltammetry [27]. Thus, the general solution (14) can be easily adopted for ramped SWV recognising that K d nF 4xEsw X uðt sÞ ¼ h þ cos½xð2n 1Þðt sÞ dt RT p n¼1
ð18Þ
Fig. 2 shows square-wave voltammogram calculated with this approach. Its properties are in excellent agreement with theoretical voltammograms simulated with other methods [27].
Fig. 2. Dimensionless square-wave voltammogram for reaction (2) calculated by combining Eqs. (14), (17), and (18). The dimensionless current is defined as I pffiffiffiffi W¼ . The conditions of the calculation are: frequency of the square-wave nFAcOx
Df
f = 10 Hz, amplitude Esw = 50 mV, scan rate of the dc ramp v = 0.1 V s1, and the angular frequency is defined as x ¼ 2pf . Other conditions are the same as for Fig. 1.
2.2. EC0i mechanism (reversible catalytic electrode mechanism) An EC0i electrode mechanism is represented by the following reaction scheme: ð19Þ The electrode reaction is assumed to be reversible, whereas the chemical reaction is irreversible, attributed with a first order rate constant k in units of s1. The integral equation describing reaction scheme (19) at a stationary planar electrode reads [14] Z t cOx 1 ekðtsÞ pffiffiffiffi ¼ IðsÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds ð20Þ uðtÞ 1þe pðt sÞ nFA D 0 kt
, where k(0) = 1. The In this case, kernel function is kðtÞ ¼ epffiffiffi pt other auxiliary functions are defined as in the previous section. Applying convolution theorem to (20) yields: b GðpÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi IðpÞ pþk
ð21Þ
Considering (7), Eq. (21) is transformed into to the following form: pffiffiffiffiffiffiffiffiffiffiffi pþk Q ðpÞ ¼ GðpÞ ð22Þ bp The latter equation can be easily rearranged to ! 1 1 k pffiffiffiffiffiffiffiffiffiffiffi GðpÞ þ pffiffiffiffiffiffiffiffiffiffiffi GðpÞ Q ðpÞ ¼ b pþk p pþk
ð23Þ
The inverse Laplace transform of (23) is pffiffiffiffi Z qðtÞ ¼ cOx nFA D
t 0
pffiffiffi eks ds pffiffiffiffiffi þ k uðtsÞ ps 1 þ e
Z 0
t
! pffiffiffiffiffi erfð ksÞ ds 1 þ euðtsÞ
ð24Þ
V. Mircˇeski, Zˇ. Tomovski / Journal of Electroanalytical Chemistry 619–620 (2008) 164–168
Taking into account (12) and (13), the final solution for the current function is ! pffiffiffiffi Z t eks euðtsÞ d ðuðt sÞÞ ek t 1 dt pffiffiffiffiffi p ffiffiffiffiffi ffi IðtÞ ¼ cOx nFA D ds þ ps p t 1 þ euð0Þ ð1 þ euðtsÞ Þ2 0 Z t pffiffiffiffiffiffiffi pffiffiffiffiffi euðtsÞ d ðuðt sÞÞ dt erfð ksÞ ds þ cOx nFA D k ð1 þ euðtsÞ Þ2 0 pffiffiffiffiffi 1 þerfð k t Þ ð25Þ 1 þ euð0Þ
1.8
Under conditions of LSV, the solution reads ! pffiffiffiffi Z t eks eðui hðtsÞÞ h ekt 1 pffiffiffiffiffi p ffiffiffiffiffi ffi IðtÞ ¼ cOx nFA D ds þ ps ½1 þ eðui hðtsÞÞ 2 p t 1 þ eui 0 Z t pffiffiffiffiffiffiffi pffiffiffiffiffi eðui hðtsÞÞ h erfð ksÞ ds þ cOx nFA D k ½1 þ eðui hðtsÞÞ 2 0 pffiffiffiffiffi 1 þerfð kt Þ 1 þ eui
0.8
1.6
167
5
1.4 1.2 1 4
0.6 3
0.4 0.2
2 1
ð26Þ
It can be readily shown that for k = 0 the solution (26) is identical with the solution for a simple charge transfer reaction given by Eq. (16). Fig. 3 depicts several LS voltammograms calculated with the aid of Eq. (26) for various rate constants. By increasing the rate of the regenerative chemical reaction the voltammogram transforms into a sigmoid curve, as expected for a steady-state voltammetric experiment [14].
0 -0.25 -0.2 -0.15 -0.1 -0.05
0
0.05 0.1 0.15 0.2 0.25
E − E° / V Fig. 3. Dimensionless LS voltammograms for catalytic EC0i reaction mechanism (reaction 19) calculated on the basis of Eq. (26) for catalytic rate constant k = 1 (1); 3.16 (2); 10 (3); 31.62 (4); and 100 s1 (5). The scan rate is v = 1 V s1. Other conditions are the same as for Fig. 1.
2.3. Electrode mechanism with adsorption of both reactant and product
OxðaqÞ ¼ OxðadsÞ
Here, a concern is to a complex electrode mechanism at a stationary planar electrode in which both reactant and product adsorb on the electrode surface obeying a linear adsorption isotherm law. The adsorption equilibria are attributed with a common adsorption constant b in units of cm (reaction scheme 27).
RedðaqÞ ¼ RedðadsÞ
n
OxðaqÞ þ ne ¼ RedðaqÞ n
ð27Þ
n
The integral equation describing the electrode mechanism is [27] pffiffi Z t 2 pffiffiffiffiffiffiffiffiffiffi cOx cOx ea t erfcða tÞ a 2 p ffiffiffi ffi ¼ IðsÞea ðtsÞ erfcða t sÞds 1 þ euðtÞ 1 þ euðtÞ nFA D 0 ð28Þ
1 0.6
4 3
0.8
0.5
2
0.4
0.6
1
0.3 0.4 0.2 0.2 0.1
0 -0.3
-0.2
-0.1
0
E − E° / V
0.1
0.2
0.3
0 -0.3
-0.2
-0.1
0
0.1
0.2
0.3
E − E° / V
Fig. 4. (A) Comparison of the LS voltammograms calculated with the aid of Eq. (16) (solid line), referring to simple reaction of a dissolved redox couple (reaction 1), and Eq. (32) (circles), referring to adsorption complicated electrode mechanism (reaction 27) for very week adsorption characterized with log(b cm1) = 3.5. (B) Effect of the adsorption constant on LS voltammograms calculated with the application of Eq. (32) for log(b cm1) = 3.5 (1); 2.5 (2); 2 (3); and 1 (4). The other conditions of the calculation for both (A) and (B) are: v = 1 V s1, D = 5 106 cm2 s1, T = 298 K, and n = 1.
V. Mircˇeski, Zˇ. Tomovski / Journal of Electroanalytical Chemistry 619–620 (2008) 164–168
168
pffiffi with a kernel kðtÞ ¼ expða2 tÞerfcða tÞ, where a is an auxiliary pffiffiffi adsorption parameter, a ¼ bD. The functions g(t), q(t), GðpÞ; IðpÞ; and Q ðpÞ are defined as in the previous cases. In addition, pffi 2
c ea t erfcða t Þ
and let FðpÞ to be the Lawe define the function f ðtÞ ¼ Ox 1þeuðtÞ 1 pffiffi place transform of f(t). The function KðpÞ ¼ pffiffipðaþ is the Laplace pÞ transform of k(t) [8]. Applying Laplace transform to (28) yields IðpÞ GðpÞ FðpÞ ¼ ab pffiffiffi pffiffiffi pða þ pÞ
ð29Þ
any voltammetric technique, providing the potential-time function of the particular technique is analytically given. To the best of our knowledge, we provide for the first time analytical solutions of integral equations of EC0i catalytic mechanism and mechanism complicated by adsorption of the redox couple under conditions of LS voltammetry. Though all solutions are derived without involving any previous approximation, they involve an integral than needs finally to be numerically evaluated.
Taking into account Eq. (7) and rearranging one gets Q ðpÞ ¼
1 1 ðGðpÞ FðpÞÞ þ pffiffiffi ðGðpÞ FðpÞÞ ba b p
The inverse Laplace transform of (30) is pffi 2 Rt cOx c ea t erfcða t Þ 1 ffi qðtÞ ¼ ba Ox 1þeuðtÞ þ 1b 0 p1ffiffiffi ps 1þeuðtÞ Rt
p1ffiffiffiffi 0 ps
1b
2 cOx ea ðtsÞ
pffiffiffiffiffi erfcða tsÞ
1þeuðtsÞ
cOx 1þeuðtsÞ
ð30Þ
ds ð31Þ
ds
Acknowledgements V. Mircˇeski acknowledges gratefully the financial support of A.v. Humboldt-Stiftung and the Ministry of Education and Science of the Republic of Macedonia. Zˇ. Tomovski acknowledges with gratitude the financial support of the ICP, University of Stuttgart, Germany, as well as the support of Dr. Rudolf Hilfer.
The first derivation of function q(t) yields the current, as follows:
0 1 pffiffi pffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 2 2 2 aea t ða p t a p t erfða t Þ ea t Þ ð1 þ euðtÞ Þ ea t erfcða tÞeuðtÞ u0 ðtÞ cOx @ euðtÞ u0 ðtÞ pffiffiffiffi pt A IðtÞ ¼ ba ð1 þ euðtÞ Þ2 ð1 þ euðtÞ Þ2 Z
c þ Ox b
t 0
1 euðtsÞ u0 ðt sÞ 1 1 pffiffiffiffiffi ds þ pffiffiffiffiffiffi ps ð1 þ euðtsÞ Þ2 p t ð1 þ euð0Þ Þ
Z 0
t
1 Jðt sÞ 1 1 pffiffiffiffiffi ds pffiffiffiffiffiffi ps ð1 þ euðtsÞ Þ2 p t 1 þ euð0Þ
where
! ð32Þ
References
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffiffiffiffi ða pðt sÞea ðtsÞ erfcða t sÞ aÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ euðtsÞ Þ Jðt sÞ ¼ pðt sÞ pffiffiffiffiffiffiffiffiffiffi 2 ea ðtsÞ erfcða t sÞeuðtsÞ u0 ðt sÞ 2
and u0 ðtÞ ¼ dtd uðtÞ. derived from
The solution for (32) by
LSV can be easily recognising that . When b ? 0, the solution (32) converges to (14), i.e., the electrode mechanism with adsorption of the redox couple simplifies to the simple charge transfer reaction of a dissolved redox couple, which supports the correctness of the solution (32). This is confirmed by the results shown in Fig. 4A, where the current functions for LSV calculated with formula (16) and (32) are compared, the latter referring to very week adsorption, i.e., log(b) = 3.5. In addition, Fig. 4B displays the effect of the increasing adsorption strength on the shape of LS voltammograms calculated by application of Eq. (32). The peak current increases, whereas the diffusion tail of the voltammogram descends by enhancing the strength of adsorption, being in agreement with the expected properties of the considered electrode mechanism. 3. Conclusion
We have demonstrated that integral equations describing reversible electrode processes at a stationary planar electrode under voltammetric conditions can be effectively solved by applying Laplace transforms. A rigorous mathematical procedure is developed upon definition of an auxiliary charge function Rt qðtÞ ¼ 0 IðsÞds and its first derivation that yields the final current–time function. The presented procedure is applicable to
[1] S.W. Feldberg, Digital simulation of electrochemical surface boundary phenomena: multiple electron transfer and adsorption, in: J.S. Mattson, H.B. Mark, H.C. MacDonald (Eds.), Electrochemistry: Calculations, Simulation and Instrumentation, vol. 2, Marcel Dekker, New York, 1972, p. 185. [2] B. Speiser, Numerical simulation of electroanalytical experiments: recent advances in methodology, in: A.J. Bard (Ed.), Electroanalytical Chemistry, vol. 19, Marcel Dekker, New York, 1996, p. 1. [3] L.K. Bieniasz, Computer Chem. 21 (1997) 1. [4] L.K. Bieniasz, Electrochem. Commun. 3 (2001) 149. [5] F.G. Cottrell, Z. Physik. Chem. 42 (1902) 385. [6] D. Ilkovicˇ, Collect. Czech. Chem. Commun. 6 (1934) 498. [7] Z. Karaoglanoff, Z. Elecktrochem. 12 (1906) 5. [8] M.R. Spiegel, Laplace Transforms, Schaum Publications Co., New York, 1965. [9] K.B. Oldham, J. Spainer, The Fractional Calculus, Academic Press, New York, 1974. [10] K.B. Oldham, J. Electroanal. Chem. 430 (1997) 1. [11] P.J. Mahon, K.B. Oldham, J. Electroanal. Chem. 445 (1998) 179. [12] P.J. Mahon, K.B. Oldham, J. Electroanal. Chem. 464 (1999) 1. [13] P.J. Mahon, K.B. Oldham, Electrochim. Acta 46 (2001) 953. [14] R.S. Nicholson, I. Shain, Anal. Chem. 36 (1964) 706. [15] J.C. Myland, K.B. Oldham, J. Electroanal. Chem. 153 (1983) 43. [16] A.C. Ramamurthy, S.K. Rangarajan, Electrochim. Acta 26 (1981) 111. [17] L.B. Bieniasz, J. Electroanal. Chem. 347 (1993) 15. [18] L.B. Bieniasz, Computer Chem. 16 (1992) 311. [19] Y.P. Gokhshtein, Dokl. Akad. Nauk. SSSR 126 (1959) 598. [20] H. Matsuda, Y. Ayabe, Z. Electrochem. 59 (1955) 494. [21] M.L. Olmstead, R.S. Nicholson, J. Electroanal. Chem. 16 (1968) 145. [22] R.S. Nicholson, M.L. Olmstead, in: J.S. Mattson, H.B. Mark, H.C. Macdonald (Eds.), Electrochemistry: Calculations Simulation and Instrumentation, vol. 2, Marcel Dekker, New York, 1972, pp. 120–137. [23] V. Mircˇeski, J. Electroanal. Chem. 545 (2003) 29. [24] MATHCAD, MathSoft Inc., Cambridge, MA, 1998. [25] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes, Cambridge University, New York, 1992. [26] J. Zhang, Si.-X. Guo, A.M. Bond, M.J. Honeychurch, K.B. Oldham, J. Phys. Chem. B 109 (2005) 8935. [27] V. Mircˇeski, Š. Komorsky-Lovric´, M. Lovric´, Square-wave voltammetry, Theory and Application, Springer, 2007.
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Journal of Electroanalytical Chemistry journal homepage: www.elsevier.com/locate/jelechem
Analytical solutions of integral equations for modelling of reversible electrode processes under voltammetric conditions Valentin Mircˇeski a,*, Zˇivorad Tomovski b a b
Institute of Chemistry, Faculty of Natural Sciences and Mathematics, Ss Cyril and Methodius University, P.O. Box 162, 1000 Skopje, Republic of Macedonia Institute of Mathematics, Faculty of Natural Sciences and Mathematics, Ss Cyril and Methodius University, P.O. Box 162, 1000 Skopje, Republic of Macedonia
a r t i c l e
i n f o
Article history: Received 22 January 2008 Received in revised form 31 March 2008 Accepted 7 April 2008 Available online 10 April 2008 Keywords: Mathematical modelling Integral equations Voltammetry Laplace transform
a b s t r a c t Mathematical procedure for analytical solutions of integral equations referring to a voltammetric experiment at a stationary planar electrode of reversible electrode processes is described. The procedure is Rt based upon Laplace transforms and definition of an auxiliary function qðtÞ ¼ 0 IðsÞds, where I(t) is the faradaic current–time function. The auxiliary function q(t) represents the charge consumed in the course of the voltammetric experiment. The methodology is exemplified by considering simple charge transfer reaction, an EC0i catalytic mechanism, and an electrode mechanism involving partial adsorption of both components of the redox couple. All solutions are derived in terms of integrals that need to be further numerically evaluated. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Modelling of a voltammetric experiment can be accomplished applying two different strategies. The first, and nowadays probably most popular one, is simulation of the experiment following completely numerical procedure based on the method of finite differences [1–4]. The second approach requires rigorous algebraic manipulation of differential equations describing the mass transfer regime, in order to provide solutions that relate the critical parameters of the voltammetric experiment, i.e., faradaic current, time and potential. The latter approach, in a general sense, is mathematically more elegant and more accurate. However, an analytical solution in a closed form can be obtained only for rather simple experimental conditions, which is exemplified by the classic works of Cottrell [5], Ilkovicˇ [6], Karaoglanoff [7] and others. For other experimental conditions, e.g., a common voltammetric experiment at a stationary electrode, the rigorous mathematical modelling must be finalized by a numerical algorithm that provides the final link between the current and the electrode potential. For solving differential equations describing mass transfer regime Laplace transform has been frequently used [8], as it provides a powerful means for simplification of the mathematical complexity. Application of Laplace transform ends with solutions in a form of integral equations, in which the current–time function I(t) appears as an argument of the integration. For instance, let us consider a simple electrode reaction of a dissolved redox couple at a * Corresponding author. E-mail address: [email protected] (V. Mircˇeski). 0022-0728/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jelechem.2008.04.001
stationary planar electrode in the absence of concurrent homogeneous chemical reactions: n
OxðaqÞ þ ne ¼ RedðaqÞ
ð1Þ
Assuming that the mass transfer can be described by a linear semi-infinite diffusion model, the solutions that relate the surface concentrations of electroactive species with the faradaic current are well known: Z t 1 1 pffiffiffiffi IðsÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds pðt sÞ nFA D 0 Z t 1 1 pffiffiffiffi cRed ð0; tÞ ¼ IðsÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds pðt sÞ nFA D 0 cOx ð0; tÞ ¼ cOx
ð2Þ ð3Þ
where A is the electrode surface area, D is the common diffusivity, cOx is the bulk concentration, and all other symbols have their usual meaning. Eqs. (2) and (3) are derived by solving Fick’s second law, and they are generally valid irrespective of the voltammetric method applied. They are derived under provision the reductive current has a positive sign. Moreover, at the beginning of the voltammetric experiment (t = 0), the surface concentration of Ox specie equals its bulk concentration (cOx ð0; 0Þ ¼ cOx ), while Red was initially absent from the solution. The identical integral appearing in both Eqs. (2) and (3) is known as a semi-integral of the current, or Riemann– Liouville fractional integral of half order [9]. The integral represents convolution operation of the current function I(t) and the time function p1ffiffiffi . It is important to emphasize that Oldham [10–13] develpt oped a valuable methodology for modelling of a variety of voltammetric experiment by virtue of semi-integration, or its
V. Mircˇeski, Zˇ. Tomovski / Journal of Electroanalytical Chemistry 619–620 (2008) 164–168
converse operation, semi-differentiation [9]. Combining the surface concentrations with the Nernst or Butler–Volmer equations, one obtains an integral equation that provides a link between the current I(t) and the potential function E(t), thereby predicting the outcome of the voltammetric experiment. Assuming that reaction (1) is electrochemically reversible, the final integral equation reads: cOx 1 pffiffiffiffi ¼ 1 þ expðuðtÞÞ nFA D
Z
t
0
1 IðsÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds pðt sÞ
ð4Þ
where is relative dimensionless electrode potential function, is the standard potential of the redox couple, and E(t) is the potential function depending on the respective voltammetric technique. Integral Eq. (4) can be solved numerically by the algorithm proposed in the seminal work of Nicholson and Shain [14], or by related methods [15–18]. It is an Abel type integral equation, which can be also analytically solved, as demonstrated by Gokhshtein [19] and Matsuda and Ayabe [20], where the solution is given in a form of an integral. Olmstead and Nicholson proposed a general numerical procedure based on the step-function method [21] that can be applied to solve Eq. (4) as well as related Voltera type integral equations arising in the theory of voltammetry [22]. The latter method gained a wide application as it provides simple solutions in a form of recurrent formula that can be easily adopted for each voltammetric technique. According to the step-function method, the total time interval of the voltammetric experiment is divided into definite number of time increments. Over each time increment, the current function is regarded as being a constant, and only an integral of the time function (i.e., f ðtÞ ¼ p1ffiffiffi in Eq. (4)) pt has to be evaluated. In our recent work, we have proposed modification of the step-function method in order to cope with complex experimental cases, exemplified by an electrode mechanism which involves adsorption of the redox couple and, at the same time, a concurrent chemical reaction [23]. In such complex case, the analytical form of the time function f(t) cannot be obtained, and its values, together with its integral can only be numerically evaluated. In this procedure, the solution can be obtained in an implicit form, requiring double application of the numerical step-function method, which sacrifices the accuracy of the estimated current values and increases the computational time. On the contrary, in the present communication we make an attempt to develop a general methodology for analytical solution of integral equations in order to avoid the numerical approach as much as possible. Using Laplace transform, a mathematical procedure for analytical solution of integral equation (4) in a time domain is presented. In addition, we present for the first time analytical solutions for a reversible diffusion controlled electrode reaction coupled with an irreversible regenerative chemical reaction (EC0i catalytic mechanism) and a reversible mechanism in which both components of the redox couple adsorb, in the absence of a concurrent chemical reaction. In the present procedure the solutions are given in an explicit form, enabling the current values to be calculated with much higher accuracy compared to the previous implicit numerical approach [23].
2. Results and discussion 2.1. Reversible electrode reaction at a planar electrode Eq. (4) is known as a singular integral equation of Abel type, with a kernel kðtÞ ¼ p1ffiffiffi , where k(0) = 1. Let us define the following pt auxiliary functions: gðtÞ ¼
cOx 1 þ expðuðtÞÞ
ð5Þ
165
and qðtÞ ¼
Z
t
IðsÞds
ð6Þ
0
Obviously, the function q(t) has a physical meaning, representing the charge consumed in the course of the voltammetric experiment. In addition, KðpÞ; GðpÞ; IðpÞ; and Q ðpÞ are Laplace transforms of k(t), g(t), I(t) and q(t), respectively, where p is dummy Laplace variable. The Laplace transforms of the charge and current functions are related as follows: Q ðpÞ ¼
1 IðpÞ p
ð7Þ
In addition, KðpÞ ¼ p1ffiffip. Applying the convolution theorem to (4) one gets 1 GðpÞ ¼ bIðpÞ pffiffiffi p
ð8Þ
where b ¼ nFA1pffiffiDffi. Taking into account Eq. (7), the latter equation is rearranged to GðpÞ Q ðpÞ ¼ pffiffiffi b p
ð9Þ
Applying inverse Laplace transform to Eq. (9), the function q(t) is found as follows: Z t 1 gðt sÞ pffiffiffi ds ð10Þ qðtÞ ¼ pffiffiffi b p 0 s Substituting for g(t) and b in (10), one gets pffiffiffiffi Z c nFA D t ds pffiffiffi qðtÞ ¼ Ox pffiffiffi p sð1 þ expðuðt sÞÞÞ 0
ð11Þ
From (6) follows that the current is the first derivation of the charge function q(t), i.e., IðtÞ ¼
d qðtÞ dt
ð12Þ
Taking into account the formula for derivation of the convolution integral Z t Z t d d ð13Þ f ðsÞgðt sÞds ¼ f ðsÞ ðgðt sÞÞds þ f ðtÞgð0Þ dt 0 dt 0 the final solution reads ! pffiffiffiffi Z t d euðtsÞ dt ðuðt sÞÞ c nFA D 1 1 p ffiffi IðtÞ ¼ Ox pffiffiffi ds þ pffiffiffi p t 1 þ euð0Þ sð1 þ euðtsÞ Þ2 0
ð14Þ
Provided the potential function /(t) is analytically given, Eq. (14) represents a general solution valid for any voltammetric technique. For instance, in linear sweep voltammetry (LSV), the potential function has the simples form, i.e., ð15Þ where Ei is the initial potential and v is the scan rate. Note that Eq. (15) applies for a reductive potential scan. Hence, the exact solution for LSV is ! pffiffiffiffi Z t c nFA D 1 eðui hðtsÞÞ h 1 1 pffiffiffi p ffiffi IðtÞ ¼ Ox pffiffiffi ds þ ð16Þ p s ð1 þ eðui hðtsÞÞ Þ2 t 1 þ eui 0 where . Unfortunately, the integral in (16) cannot be solved in a closed form, and it has to be numerically evaluated. The latter equation can be easily transformed into dimensionless form by defining the dimensionless current as pffiffiffiffi pffiffiffiffiffiffiffiffi. The solid line in Fig. 1 shows a dimensionless LS W ¼ nFAcI RT vDnF Ox voltammogram calculated on the basis of Eq. (16). The numerical
V. Mircˇeski, Zˇ. Tomovski / Journal of Electroanalytical Chemistry 619–620 (2008) 164–168
166
0.8
0.5
0.6
0.4
0.4
0.3
0.2
0.2
0 -0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
E − E° / V
0.1 -0.2
0 -0.3
-0.2
-0.1
0
0.1
0.2
0.3
E − E° / V
-0.4
Fig. 1. Comparison of the LS voltammograms for reaction (1) calculated with the aid of Eq. (16) (line) and numerical step-function method (circles) [21]. For the stepfunction method, the potential increment was dE = 0.1 mV. The dimensionless cupffiffiffiffi pffiffiffiffiffiffiffiffi. rrent is defined as W ¼ nFAcI RT vDnF Ox
integration has been carried out with the standard operations for numerical integration of the software package MATHCAD [24]. For numerical integration, MATHCAD uses the Romberg method as the default, where the interval of integration is divided into equally spaced subintervals [25]. The features of the voltammogram agree well with those simulated with other methods [21]. Fig. 1 also shows a comparison between voltammograms calculated with Eq. (16) (solid line) and simulated by the standard step-function numerical method [21,22] (circles). For numerical simulations, the step potential was dE = 0.1 mV. The agreement between the two methods is obvious. The average relative difference between the dimensionless current values is 0.025%. The general applicability of Eq. (16) to more complex voltammetric techniques is illustrated by considering the case of squarewave voltammetry. It has been recently proposed that the potential form of the ramped SWV can be represented by the following Fourier series [26]:
EðtÞ ¼ Ei vt þ
K 4Esw X sin½ð2n 1Þx t 2n 1 p n¼1
with K ¼ 1;
ð17Þ
where x is the angular frequency and Esw is the amplitude of the square-wave. For K P 20, Eq. (17) gives an excellent approximation to the real ramped square-wave voltage. Note that in this version of the technique, square-wave is superimposed on a dc ramp, rather than on a staircase voltage as frequently used in the standard Osteryoung-type SW voltammetry [27]. Thus, the general solution (14) can be easily adopted for ramped SWV recognising that K d nF 4xEsw X uðt sÞ ¼ h þ cos½xð2n 1Þðt sÞ dt RT p n¼1
ð18Þ
Fig. 2 shows square-wave voltammogram calculated with this approach. Its properties are in excellent agreement with theoretical voltammograms simulated with other methods [27].
Fig. 2. Dimensionless square-wave voltammogram for reaction (2) calculated by combining Eqs. (14), (17), and (18). The dimensionless current is defined as I pffiffiffiffi W¼ . The conditions of the calculation are: frequency of the square-wave nFAcOx
Df
f = 10 Hz, amplitude Esw = 50 mV, scan rate of the dc ramp v = 0.1 V s1, and the angular frequency is defined as x ¼ 2pf . Other conditions are the same as for Fig. 1.
2.2. EC0i mechanism (reversible catalytic electrode mechanism) An EC0i electrode mechanism is represented by the following reaction scheme: ð19Þ The electrode reaction is assumed to be reversible, whereas the chemical reaction is irreversible, attributed with a first order rate constant k in units of s1. The integral equation describing reaction scheme (19) at a stationary planar electrode reads [14] Z t cOx 1 ekðtsÞ pffiffiffiffi ¼ IðsÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds ð20Þ uðtÞ 1þe pðt sÞ nFA D 0 kt
, where k(0) = 1. The In this case, kernel function is kðtÞ ¼ epffiffiffi pt other auxiliary functions are defined as in the previous section. Applying convolution theorem to (20) yields: b GðpÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi IðpÞ pþk
ð21Þ
Considering (7), Eq. (21) is transformed into to the following form: pffiffiffiffiffiffiffiffiffiffiffi pþk Q ðpÞ ¼ GðpÞ ð22Þ bp The latter equation can be easily rearranged to ! 1 1 k pffiffiffiffiffiffiffiffiffiffiffi GðpÞ þ pffiffiffiffiffiffiffiffiffiffiffi GðpÞ Q ðpÞ ¼ b pþk p pþk
ð23Þ
The inverse Laplace transform of (23) is pffiffiffiffi Z qðtÞ ¼ cOx nFA D
t 0
pffiffiffi eks ds pffiffiffiffiffi þ k uðtsÞ ps 1 þ e
Z 0
t
! pffiffiffiffiffi erfð ksÞ ds 1 þ euðtsÞ
ð24Þ
V. Mircˇeski, Zˇ. Tomovski / Journal of Electroanalytical Chemistry 619–620 (2008) 164–168
Taking into account (12) and (13), the final solution for the current function is ! pffiffiffiffi Z t eks euðtsÞ d ðuðt sÞÞ ek t 1 dt pffiffiffiffiffi p ffiffiffiffiffi ffi IðtÞ ¼ cOx nFA D ds þ ps p t 1 þ euð0Þ ð1 þ euðtsÞ Þ2 0 Z t pffiffiffiffiffiffiffi pffiffiffiffiffi euðtsÞ d ðuðt sÞÞ dt erfð ksÞ ds þ cOx nFA D k ð1 þ euðtsÞ Þ2 0 pffiffiffiffiffi 1 þerfð k t Þ ð25Þ 1 þ euð0Þ
1.8
Under conditions of LSV, the solution reads ! pffiffiffiffi Z t eks eðui hðtsÞÞ h ekt 1 pffiffiffiffiffi p ffiffiffiffiffi ffi IðtÞ ¼ cOx nFA D ds þ ps ½1 þ eðui hðtsÞÞ 2 p t 1 þ eui 0 Z t pffiffiffiffiffiffiffi pffiffiffiffiffi eðui hðtsÞÞ h erfð ksÞ ds þ cOx nFA D k ½1 þ eðui hðtsÞÞ 2 0 pffiffiffiffiffi 1 þerfð kt Þ 1 þ eui
0.8
1.6
167
5
1.4 1.2 1 4
0.6 3
0.4 0.2
2 1
ð26Þ
It can be readily shown that for k = 0 the solution (26) is identical with the solution for a simple charge transfer reaction given by Eq. (16). Fig. 3 depicts several LS voltammograms calculated with the aid of Eq. (26) for various rate constants. By increasing the rate of the regenerative chemical reaction the voltammogram transforms into a sigmoid curve, as expected for a steady-state voltammetric experiment [14].
0 -0.25 -0.2 -0.15 -0.1 -0.05
0
0.05 0.1 0.15 0.2 0.25
E − E° / V Fig. 3. Dimensionless LS voltammograms for catalytic EC0i reaction mechanism (reaction 19) calculated on the basis of Eq. (26) for catalytic rate constant k = 1 (1); 3.16 (2); 10 (3); 31.62 (4); and 100 s1 (5). The scan rate is v = 1 V s1. Other conditions are the same as for Fig. 1.
2.3. Electrode mechanism with adsorption of both reactant and product
OxðaqÞ ¼ OxðadsÞ
Here, a concern is to a complex electrode mechanism at a stationary planar electrode in which both reactant and product adsorb on the electrode surface obeying a linear adsorption isotherm law. The adsorption equilibria are attributed with a common adsorption constant b in units of cm (reaction scheme 27).
RedðaqÞ ¼ RedðadsÞ
n
OxðaqÞ þ ne ¼ RedðaqÞ n
ð27Þ
n
The integral equation describing the electrode mechanism is [27] pffiffi Z t 2 pffiffiffiffiffiffiffiffiffiffi cOx cOx ea t erfcða tÞ a 2 p ffiffiffi ffi ¼ IðsÞea ðtsÞ erfcða t sÞds 1 þ euðtÞ 1 þ euðtÞ nFA D 0 ð28Þ
1 0.6
4 3
0.8
0.5
2
0.4
0.6
1
0.3 0.4 0.2 0.2 0.1
0 -0.3
-0.2
-0.1
0
E − E° / V
0.1
0.2
0.3
0 -0.3
-0.2
-0.1
0
0.1
0.2
0.3
E − E° / V
Fig. 4. (A) Comparison of the LS voltammograms calculated with the aid of Eq. (16) (solid line), referring to simple reaction of a dissolved redox couple (reaction 1), and Eq. (32) (circles), referring to adsorption complicated electrode mechanism (reaction 27) for very week adsorption characterized with log(b cm1) = 3.5. (B) Effect of the adsorption constant on LS voltammograms calculated with the application of Eq. (32) for log(b cm1) = 3.5 (1); 2.5 (2); 2 (3); and 1 (4). The other conditions of the calculation for both (A) and (B) are: v = 1 V s1, D = 5 106 cm2 s1, T = 298 K, and n = 1.
V. Mircˇeski, Zˇ. Tomovski / Journal of Electroanalytical Chemistry 619–620 (2008) 164–168
168
pffiffi with a kernel kðtÞ ¼ expða2 tÞerfcða tÞ, where a is an auxiliary pffiffiffi adsorption parameter, a ¼ bD. The functions g(t), q(t), GðpÞ; IðpÞ; and Q ðpÞ are defined as in the previous cases. In addition, pffi 2
c ea t erfcða t Þ
and let FðpÞ to be the Lawe define the function f ðtÞ ¼ Ox 1þeuðtÞ 1 pffiffi place transform of f(t). The function KðpÞ ¼ pffiffipðaþ is the Laplace pÞ transform of k(t) [8]. Applying Laplace transform to (28) yields IðpÞ GðpÞ FðpÞ ¼ ab pffiffiffi pffiffiffi pða þ pÞ
ð29Þ
any voltammetric technique, providing the potential-time function of the particular technique is analytically given. To the best of our knowledge, we provide for the first time analytical solutions of integral equations of EC0i catalytic mechanism and mechanism complicated by adsorption of the redox couple under conditions of LS voltammetry. Though all solutions are derived without involving any previous approximation, they involve an integral than needs finally to be numerically evaluated.
Taking into account Eq. (7) and rearranging one gets Q ðpÞ ¼
1 1 ðGðpÞ FðpÞÞ þ pffiffiffi ðGðpÞ FðpÞÞ ba b p
The inverse Laplace transform of (30) is pffi 2 Rt cOx c ea t erfcða t Þ 1 ffi qðtÞ ¼ ba Ox 1þeuðtÞ þ 1b 0 p1ffiffiffi ps 1þeuðtÞ Rt
p1ffiffiffiffi 0 ps
1b
2 cOx ea ðtsÞ
pffiffiffiffiffi erfcða tsÞ
1þeuðtsÞ
cOx 1þeuðtsÞ
ð30Þ
ds ð31Þ
ds
Acknowledgements V. Mircˇeski acknowledges gratefully the financial support of A.v. Humboldt-Stiftung and the Ministry of Education and Science of the Republic of Macedonia. Zˇ. Tomovski acknowledges with gratitude the financial support of the ICP, University of Stuttgart, Germany, as well as the support of Dr. Rudolf Hilfer.
The first derivation of function q(t) yields the current, as follows:
0 1 pffiffi pffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 2 2 2 aea t ða p t a p t erfða t Þ ea t Þ ð1 þ euðtÞ Þ ea t erfcða tÞeuðtÞ u0 ðtÞ cOx @ euðtÞ u0 ðtÞ pffiffiffiffi pt A IðtÞ ¼ ba ð1 þ euðtÞ Þ2 ð1 þ euðtÞ Þ2 Z
c þ Ox b
t 0
1 euðtsÞ u0 ðt sÞ 1 1 pffiffiffiffiffi ds þ pffiffiffiffiffiffi ps ð1 þ euðtsÞ Þ2 p t ð1 þ euð0Þ Þ
Z 0
t
1 Jðt sÞ 1 1 pffiffiffiffiffi ds pffiffiffiffiffiffi ps ð1 þ euðtsÞ Þ2 p t 1 þ euð0Þ
where
! ð32Þ
References
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffiffiffiffi ða pðt sÞea ðtsÞ erfcða t sÞ aÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ euðtsÞ Þ Jðt sÞ ¼ pðt sÞ pffiffiffiffiffiffiffiffiffiffi 2 ea ðtsÞ erfcða t sÞeuðtsÞ u0 ðt sÞ 2
and u0 ðtÞ ¼ dtd uðtÞ. derived from
The solution for (32) by
LSV can be easily recognising that . When b ? 0, the solution (32) converges to (14), i.e., the electrode mechanism with adsorption of the redox couple simplifies to the simple charge transfer reaction of a dissolved redox couple, which supports the correctness of the solution (32). This is confirmed by the results shown in Fig. 4A, where the current functions for LSV calculated with formula (16) and (32) are compared, the latter referring to very week adsorption, i.e., log(b) = 3.5. In addition, Fig. 4B displays the effect of the increasing adsorption strength on the shape of LS voltammograms calculated by application of Eq. (32). The peak current increases, whereas the diffusion tail of the voltammogram descends by enhancing the strength of adsorption, being in agreement with the expected properties of the considered electrode mechanism. 3. Conclusion
We have demonstrated that integral equations describing reversible electrode processes at a stationary planar electrode under voltammetric conditions can be effectively solved by applying Laplace transforms. A rigorous mathematical procedure is developed upon definition of an auxiliary charge function Rt qðtÞ ¼ 0 IðsÞds and its first derivation that yields the final current–time function. The presented procedure is applicable to
[1] S.W. Feldberg, Digital simulation of electrochemical surface boundary phenomena: multiple electron transfer and adsorption, in: J.S. Mattson, H.B. Mark, H.C. MacDonald (Eds.), Electrochemistry: Calculations, Simulation and Instrumentation, vol. 2, Marcel Dekker, New York, 1972, p. 185. [2] B. Speiser, Numerical simulation of electroanalytical experiments: recent advances in methodology, in: A.J. Bard (Ed.), Electroanalytical Chemistry, vol. 19, Marcel Dekker, New York, 1996, p. 1. [3] L.K. Bieniasz, Computer Chem. 21 (1997) 1. [4] L.K. Bieniasz, Electrochem. Commun. 3 (2001) 149. [5] F.G. Cottrell, Z. Physik. Chem. 42 (1902) 385. [6] D. Ilkovicˇ, Collect. Czech. Chem. Commun. 6 (1934) 498. [7] Z. Karaoglanoff, Z. Elecktrochem. 12 (1906) 5. [8] M.R. Spiegel, Laplace Transforms, Schaum Publications Co., New York, 1965. [9] K.B. Oldham, J. Spainer, The Fractional Calculus, Academic Press, New York, 1974. [10] K.B. Oldham, J. Electroanal. Chem. 430 (1997) 1. [11] P.J. Mahon, K.B. Oldham, J. Electroanal. Chem. 445 (1998) 179. [12] P.J. Mahon, K.B. Oldham, J. Electroanal. Chem. 464 (1999) 1. [13] P.J. Mahon, K.B. Oldham, Electrochim. Acta 46 (2001) 953. [14] R.S. Nicholson, I. Shain, Anal. Chem. 36 (1964) 706. [15] J.C. Myland, K.B. Oldham, J. Electroanal. Chem. 153 (1983) 43. [16] A.C. Ramamurthy, S.K. Rangarajan, Electrochim. Acta 26 (1981) 111. [17] L.B. Bieniasz, J. Electroanal. Chem. 347 (1993) 15. [18] L.B. Bieniasz, Computer Chem. 16 (1992) 311. [19] Y.P. Gokhshtein, Dokl. Akad. Nauk. SSSR 126 (1959) 598. [20] H. Matsuda, Y. Ayabe, Z. Electrochem. 59 (1955) 494. [21] M.L. Olmstead, R.S. Nicholson, J. Electroanal. Chem. 16 (1968) 145. [22] R.S. Nicholson, M.L. Olmstead, in: J.S. Mattson, H.B. Mark, H.C. Macdonald (Eds.), Electrochemistry: Calculations Simulation and Instrumentation, vol. 2, Marcel Dekker, New York, 1972, pp. 120–137. [23] V. Mircˇeski, J. Electroanal. Chem. 545 (2003) 29. [24] MATHCAD, MathSoft Inc., Cambridge, MA, 1998. [25] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes, Cambridge University, New York, 1992. [26] J. Zhang, Si.-X. Guo, A.M. Bond, M.J. Honeychurch, K.B. Oldham, J. Phys. Chem. B 109 (2005) 8935. [27] V. Mircˇeski, Š. Komorsky-Lovric´, M. Lovric´, Square-wave voltammetry, Theory and Application, Springer, 2007.