Electrochemistry Communications 000 (1999) 000–000
Parametric inverse problems of electrode kinetics Leonid L. Frumin a, Gleb V. Zilberstein b,* b
a Molecular Manipulation, Inc., Rehovot, Israel Impression CD, Soreq Technological Centre, 81800, Nahal Soreq, Yavne, Israel
Received 27 January 1999; received in revised form 26 February 1999; accepted 26 February 1999
Abstract The purpose of the present paper is the development of a methodical approach to determine kinetic parameters of electrochemical reactions from experimental results using relatively simple techniques for experimental data treatment, which should be of considerable interest for analytical applications. q 1999 Elsevier Science S.A. All rights reserved. Keywords: Inverse problems; Electrode kinetics; Analytical chemistry; Polarography
1. Introduction The issues of correct treatment of electrochemical measurement results are currently gaining major importance both in theoretical aspects and for practical application in quantitative analysis. The use of computers and PC-based instruments has greatly simplified data collection, improving accuracy and reliability of the experiment. However, the subsequent experimental data treatment and interpretation have mostly remained at the old pre-computer stage. The software packages of the PC-based instruments normally contain primitive data manipulation means, such as smoothing, integration, differentiation, etc. Likewise, the problem of adequate data treatment has not been solved in the numerous publications on the theory of the current–voltage curves or by the respective software packages for their computing or simulation. The reason is evident — all these works aim at solving the Direct Problem, when the semiempirical kinetic parameters, such as the exchange current i0, transfer parameter a and double-layer capacity K0 are given, the current–voltage curve being the desired calculation result. However, what is really of interest to an experimental scientist is the Inverse Problem — kinetic parameter calculation based on the experimentally measured voltammogram. An absence of accessible methods for the inverse problem solution precludes a complete use of information contained in a current–voltage curve for electrochemical analysis. Until recently, the transition part of the current–voltage curve has * Corresponding author. Fax: q972-8-943-47-98; e-mail:
[email protected]
been discarded and almost never used in analysis. Instead, separate small parts of the current–voltage curve were used, each to determine one of the kinetic parameters. The inaccuracies inherent to such an approach may be difficult or impossible to evaluate. Frequently, the experimentalist does not even suspect of the existing possibility to extract several parameters from a single experiment simultaneously, thus improving accuracy and reliability of the kinetic results obtained. The purpose of the present paper is the development of a methodical approach to determining kinetic parameters of electrochemical reactions from experimental results using relatively simple techniques for experimental data treatment, which should be of considerable interest for analytical applications [1].
2. Discussion For simplicity, let us limit ourselves to the classic problem of evaluation of the kinetic parameters (i0, a, K0) of the neutralization reaction (a first-order process) of binary electrolyte anions on the background of an inert buffer electrolyte of the same valence z based on the results of the measurements of the current–voltage curve on the mercury drop electrode. The first step of the data treatment is the transition to a set of dimensionless variables, which are more convenient in practical calculations. As the potential measurement unit, we shall employ kBT/e and, for the current unit, the value 8pRC0D, R being the electrode (mercury drop) radius, D the diffusion coefficient of the active anions, and C0 the total anion con-
1388-2481/99/$ - see front matter q 1999 Elsevier Science S.A. All rights reserved. PII S 1 3 8 8 - 2 4 8 1 ( 9 9 ) 0 0 0 1 4 - 4
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centration far from the electrode, including that of the buffer electrolyte anions. We consider as given the active ion concentration c10, defined as a fraction of the total anion concentration, and also the equilibrium potential of the neutralization reaction weq. All potentials will have the electrolyte potential for the reference point. The general form of the current– voltage curve, taking into account c1 the potential dependence on the reaction current and on the electrode potential w, has been obtained in our recent work [1]. Let us now review the main results obtained in [2,3]. The traditional diffuse double-layer model has been supplemented by matching the Boltzmann particle distribution in the double layer with the logarithmic distribution in the quasi-neutral diffusion region. As opposed to the well-established Chapman–Gouy model, this model is valid not only for small, but also for large current values, including the saturation current. The approach proposed is valid for Debye radius values small compared to the electrode radius, which is almost universally true for not-too-dilute electrolytes. To determine the kinetic parameters, we may limit ourselves to the kinetic and transitional parts of the cathode branch of the current–voltage curve, which may be represented as follows: i exp(bzc)si 0[c10y1q(1yi)2] exp[z(wyweq)]
(1)
where bs1-a, csw-c1. Eq. (1) determines the current–voltage curve, jointly with the equation for the potential c1:
wyc 1sA[(1yi) exp(zc 1/2)yexp(yzc 1/2)]
(2)
where i is the current to the electrode, w the electrode potential relative to the infinity, and A the dimensionless parameter defined as Asx2/(2pK 0 R d)), where Rd is the Debye radius, and C0 the unit area capacity of the double layer. Note that the expression obtained in [1] differs from the well-known Frumkin formula in that it has the (1-i) factor at the first exponential. The current–voltage curve may be obtained by solving numerically the system including Eqs. (1) and (2). Experimentally, an excess of buffer electrolyte is often employed, allowing simplification of the data treatment. In this case the potential c1 no longer depends on the active electrolyte concentration, being still dependent on the electrode potential, according to Eq. (2). An excess of buffer electrolyte has the consequence that the relative value of the electrode current expressed in the adopted dimensionless units becomes small compared to unity, the same being valid for the relative concentration c10 of the active anions. Thus, the experimentalist has a set of points of the current– voltage curve in(wn), ns1,2,...,N, where n is the point number, and N is the total number of the curve points. The traditional method to treat similar experimental results is to select the Tafel region of the curve, where the current is small. In this case Eq. (1) may be notably simplified: in exp(bzcn)si 0 c10 exp[z(wnyweq)]
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(3)
Then, the right and left parts of Eq. (3) are divided by the respective term, taken for a certain experimental point m: in exp[bz(cncm)]sexp[z(wnywm)] im
(4)
In this case, the exchange current i0 disappears from the equation, and the remaining unknown parameter b may be estimated after taking logarithms of the left and right parts of Eq. (4) from the tangent of the straight line obtained. Turning back to Eq. (3), the exchange current may also be estimated. The double-layer capacity has to be calculated from a separate preliminary measurement, required for the calculation of c1(w). Such an approach to the inverse problem solution may hardly be regarded as satisfactory, as only a part of the information contained in the current–voltage curve is used in the data treatment. Moreover, the double-layer capacity may also be estimated directly from the same set of experimental results without any need for extra measurements or estimates. Let us now describe the solution of this inverse problem in the general case, according to the novel approach proposed. We shall use Eqs. (1) and (2) to solve the direct problem. From the mathematical standpoint, we are dealing with a parametric inverse problem of nonlinear estimation of the parameter vector x¯ s{x m}s{a,i 0 ,K 0} [2,3]. The initial data for the problem include the experimental data set in,wn,ns1,2,..,N, and the theoretical model, insF(x,wn), describing the current as a function of the parameter vector and defined implicitly by Eqs. (1) and (2). In order to estimate the parameter vector, and taking into account the statistical character of the experiment, we shall use the least-squares technique (LST) and seek to minimize the sum of squares: [inyF(x,wn)]2 ´min{x} s n2 ns1 N
Ms 8
(5)
where s n2 is an estimate of the dispersion of the respective experimental point. Note that in the more general case, when a statistical correlation exists between separate experimental points, a covariance matrix W is normally introduced. Eq. (5) was obtained for the case of the experimentally most relevant case of statistically independent measurements, with a diagonal covariance matrix: Wijsdijsj-2, dij being the Kronecker symbol (a unitary matrix). Note that the parameter vector x enters nonlinearly in the dependence insF(x,wn); thus, the linear least-squares techniques used in polynomial approximations is inapplicable to the current problem. The solution has to be found by direct minimization of the LST functional M. Among the applied mathematical computational methods, an ample set of algorithms exists for direct functional minimization. The details on such algorithms are scattered in a vast number of special papers and monographs; hence, we find it useful to review briefly the main ideas of the most frequently used approach to solving this type of problem [4–10].
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The minimization of the functional M starts from finding an initial approximation for the parameter vector xsx0. Next, the function F(x,wn) is decomposed into a Taylor series in the vicinity of this initial approximation: ≠F(x,wn) (xjyxj0) ≠x j js1 m
F(x,wn)fF(x 0,wn)q 8
(6)
Substituting Eq. (6) into Eq. (5) and varying the LST functional, we obtain a linear least-squares problem for the parameter vector correction zsx-x0: Gzsf
(7)
where the linear operator G is called the static Fisher operator. Its components are found from the following relations: ≠F(x,wn) dF(x,wn) y2 sn ≠xj ≠x i ns1 N
Gijs 8
Table 1 gk values for bi- and triparametric recoveries
The right part f of Eq. (7) is given by the expression: ≠F(x,wn) [inyF(x 0,wn)]s ny2 ≠x i ns1
Fig. 1. Reconstructed current–voltage curve.
(8)
N
fis 8
(9)
Eq. (7) is solved by inverting the G matrix. Next, the second correction may be calculated, using the new initial approximation which takes into account the first correction, and thus repeating the iterations we may find the unknown parameter vector x. The iterative process convergence has to be studied specially, given the strongly nonlinear character of the problem. The convergence criterion of the iterative process may be represented in the form M-N. Here M is the least-squares functional (5), and N the number of experimental points. Note that the direct problem has to be solved several times at each iteration, as it is required to determine both the function F itself and its derivatives. Thus, of primary importance becomes the availability of an analytical expression for the current–voltage curve, or of a fast numerical algorithm for its calculation. This was the reason that led the authors to develop an asymptotic method for calculation of such curves [2,11], allowing the direct problem to be solved with small computational effort. An example of a test problem solution using the general techniques outlined is illustrated in Fig. 1 and Table 1. Modelling was performed using the scheme described below, called the closed-cycle scheme. Namely, the current–voltage curve is calculated for the given parameter values (i0,a,K0), and then, using a random number generator, a Gaussian random experimental uncertainty is simulated. In the final stage, the iterative parameter vector search method is implemented, using the Eqs. (7)–(9), and the relative uncertainty for each of the recovered parameters is evaluated. The uncertainties of the parameter estimates depend on the parameter values, on the number of the experimental points and on the experimental dispersion. For a quantitative estimate, it is useful to introduce the dimensionless values gk, which are the amplification factors of the relative uncertainty of the parameter recovery in the course of the solution of the inverse problem:
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Triparametric Biparametric
i0
a
K0
15 3.7
6 1.4
42
g ksGkky1≤i≤PNxky1N
(10)
The gk factor equals the relative uncertainty of parameter recovery divided by the experimental dispersion. The values of these factors for bi- and tri-parametric recoveries are presented in Table 1. The first line of Table 1 presents results for the problem with three unknown parameters, and the second line, for the bi-parametric problem, the double-layer capacity has to be determined by an independent measurement. As follows from the table, the worst data recovery is obtained in the tri-parametric problem. Moreover, the largest gk factor value is that of the double-layer capacity. The result of the inverse problem solution has also been presented in Fig. 1. ‘Experimental points’ are represented by crosses, and the resulting reconstructed current–voltage curve by the full line. What are the advantages of the proposed general approach to solving parametric inverse problems? First and foremost, this approach makes use of the complete information provided by the experimental results. Secondly, we are able to account for the statistical properties of the experiment and, last but not the least, objective uncertainty estimates of the recovered unknown parameters are provided. Further development of this approach will imply accounting for specific adsorption phenomena and the solution of inverse problems of polarography [12,13].
3. Query to the author References have been renumbered, please check.
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References [1] G.V. Zilberstein, L.L. Frumin, 216th American Chemical Society National Meeting, Analytical Division, Boston, MA, USA, Aug. 1998, Abstr. 113. [2] L.L. Frumin, G.V. Zilberstein, J. Electrochem. Soc. 144 (1997) 3458. [3] L.L. Frumin, G.V. Zilberstein, J. Electrochem. Soc. 145 (1998) 720. [4] Y. Bard, Nonlinear Parameter Estimation, Academic Press, New York, 1974 (or Russian edition: Statistika, Moscow, 1979, p. 360). [5] J.C. Nash, M. Walker-Smith, Nonlinear Parameter Estimation, Marcel Dekker, New York, 1987. [6] G.A.F. Seber, C.J. Wild, Nonlinear Regression, Wiley, New York, 1989. [7] N.R. Draper, Applied Regression Analysis, Wiley, New York, 1981.
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[8] D.M. Trujillo, H.R. Busby, Practical Inverse Analysis in Engineering, CRC Press, Boca Raton, FL, 1997. [9] H.W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordecht, 1996. [10] E. Hensel, Inverse Theory and Applications for Engineers, PrenticeHall, Englewood Cliffs, NJ, 1991. [11] L.L. Frumin, G.V. Zilberstein, Comput. Technol. 4 (1995) 258 (in Russian). [12] L.L. Frumin, G.V. Zilberstein, V.L. Varand, International Meeting of Computers and University Education, Novosibirsk, Russia, May 1997. [13] G.V. Zilberstein, L.L. Frumin, International Workshop of Chemical Analysis and Modern Calculation Methods, Beer-Sheva, Israel, May 1998.
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