A Linear Kronig-Kramers Transform Test for Immittance Data Validation

A method is described with which immittance data can be tested for Kronig-Kramers compliance. In contrast with other procedures, this method is l inear in nature and is based on a predetermined set of relaxation times. The model contains as many parameters (or less) as there are data sets. Three modes of operation are described, the first two are based on a linear fit of the model function to the imaginary part or to the real part of the data set. With the fit parameters the corresponding real or imaginary dispersion can be calculated and compared with the actual measurement. In the third mode a complex model function is fitted to the complete data set. As the model function does comply with (a relaxed set of) the Kronig-Kramers (K-K) rules, it will not be able to reproduce the data set satisfactory in the case of nonK-K behavior, as can be observed from the residuals plot. Due to its linear nature, no starting values are needed for the data validation. The main l imitation of this procedure is the size of the matrix and the accuracy of the matrix inversion. Introduction Electrochemical impedance spectroscopy (EIS) has become an important research tool within the entire electrochemical research community, with significant applications in corrosion research, solid-state electrochemistry, and aqueous and nonaqueous electrochemistry, as well as in electronics. Its application ranges from fundamental investigations to very applied uses such as product quality monitoring. The large advancement in EIS has been brought about by the development of powerful data analysis programs which have become generally available within the last decade. By now the best known and most used programs are LEVM by Macdonald ~-5 and EQUIVALENT CIRCUIT (EQUIVCRT) by the~ author. 6-s Both programs are based on a powerful nonlinear least squares fit algorithm developed by Levenberg 9 and Marquardt. ~~ Both complex nonl inear least squares (CNLS) programs are based on the use of an equivalent circuit (EqC) as a modeling function. The nonlinear fit procedure does require an adequate set of starting values for the adjustable parameters of the modeling function. For CNLS-fits with a large number of adjustable parameters, the speed of convergence critically depends on the quality of the starting values. Reasonable values generally can be obtained through graphical means. The software package EQUIVALENT CIRCUIT employs a special subroutine which provides a "rough" deconvolution of the immittance spectra, thus yielding a probable equivalent circuit together with a set of appropriate starting values. This subroutine has the potential for unveiling small contributions to the frequency response that are buried in the overall frequency dispersion. How well the modeling function reproduces the actual data set can best be observed in a graph of the relative residuals, A~,~ and h~., vs. log r where co is the radial frequency (2wf). The residuals are defined by X ~ XAco~) Ximi -XiAco0 hre,i ~'lX(coi)] and him,i = 'lX(coi)l [1] with X~o,~ and X~,j the real and imaginary parts of the i th data set (at frequency ~o~) and X~e(COi) and X~(co~) the real and imaginary parts of the modeling function for coi. IX(m~)l is the vector length (absolute value) of the modeling function. Besides impedance and admittance, X may also represent the modulus or the dielectric response. An optimum fit is obtained when the residuals are spread randomly around the log co axis. When the residuals show a systematic deviation from the horizontal axis, e.g., by R~ forming a "trace" around, above, or below the log co axis, the CNLS fit is not adequate. This can be caused by several factors, which can be classified into two categories, (i) the * Electrochemical Society Active Member. J. Electrochem. Soc., Yol. 142, No. 6, June 1995 9 The Electrochemical Society, Inc. data contain systematic errors; these can be due to the measuring setup and equipment, aging of the sample, slow change in the sample temperature, etc., and (ii) the chosen modeling function is inappropriate; this can be due to a wrong selection and/or arrangement of the dispersive elements, or it may be that the data require a nonideal transfer function (i.e., one that cannot be built up by a set of simple dispersion elements or transfer functions). It is important to be able to distinguish between cases (i) and (ii), so that no time is wasted on the interpretation of "bad" data. Here the Kronig-Kramers transforms can be used to indicate whether the data are at fault or the EqC is inadequate. The Kronlg-Kramers relations, which are based on the principle of causality, n-la dictate that the real and imaginary part of any immittance function are interdependent, provided that the following conditions are met: (i) causality: the response must be related to the excitation signal only; (ii) linearity: only the first-order term must be present in the response signal. For inherently nonlinear systems (e.g., electrode processes) this implies the use of small excitation voltages, e.g. <10 mV; (iii) stability: the system may not change with time, nor continue to oscillate when the excitation signal is removed, which requires the system to be passive; and (iv) finite: for all values of ~, including co ~ 0 and co -> =. For practical application of the K-K transforms, this last condition is not critical. The stability condition, however, is the key in the data validation process. The interdependence between the real and imaginary parts of the dispersion is presented in the Kronig-Kramers transform integrals. When the imaginary part of the dispersion is known, the real part can be obtained through the K-K transform integral. In the impedance representation Zre(co) -~ Rw 42 fo ~ XZjm(X ) coZim(fD ) ~ ~r x2 _ • ~ [2] while the imaginary part can be obtained from