On the Application of the Kramers-Kronig Relations to Evaluate the Consistency of Electrochemical Impedance Data

The use of the Kramers-Kronig (KK) relations to evaluate the consistency of impedance data has been limited by the fact that the experimental frequency domain is necessarily finite. Current algorithms do not distinguish between the residual errors caused by a frequency domain that is too narrow and discrepancies caused by a system which does not satisfy the constraints of the KK equations. A new technique is presented which circumvents the limitation of applying the KK relations to impedance data which truncate in the capacitive region. The proposed algorithm calculates impedance values below the lowest experimental frequency which "force" the data set to satisfy the KK equations. Internally consistent data sets yield low-frequency impedance values which are continuous at the lowest measured experimental frequency. A discontinuity between the calculated low-frequency values and the experimental data indicates inconsistency which cannot be attributed to the finite experimental frequency domain. To facilitate the interpretation of impedance measurements, an investigator should know whether the experimental data is characteristic of a linear and stable system. It has been suggested that the Kramers-Kronig (KK) relations can be employed to evaluate and analyze complex impedance data of electrochemical systems (1-3). These equations, developed for the field of optics, constrain the real and imaginary components of complex physical quantities for systems that satisfy the conditions of causality, linearity, stability, and finite impedance values at the frequency limits of zero and infinity (4-6). Bode (7) extended the concept to electrical impedance, and has tabulated various forms of these equations. Macdonald and UrquidiMacdonald (8) have demonstrated analytically that equivalent electrical circuits involving passive elements (R, C, L), the Warburg impedance, the pore diffusion model, R--C transmission lines, R--L transmission lines, and R--C--L transmission lines obey the KK relations. Consequently, experimental data that can be fitted to the analytical response of an equivalent circuit described above through nonlinear regression analysis satisfy the conditions of the KK relations. There is, however, controversy over the extent to which the KK relations can be used to validate electrochemical impedance data. The relationships are held to be valid, but Mansfeld and Shih (9-11) have argued that they are useless for data that do not include all the time constants for the system. Since the KK relations involved integrals over frequencies ranging from zero to infinity, valid data can appear to be invalid if the measured frequency range is insufficient. Macdonald et al. (2, 3, 12) have repeatedly emphasized the importance of collecting experimental * Electrochemical Society Active Member. data over a sufficiently wide frequency range in order to evaluate the KK relations with satisfactory accuracy, but this is not always possible. While the upper limit of modern frequency analyzers (65 kHz to 1 MHz) is sufficient for most electrochemical systems, the lower measureable frequency limit for systems exhibiting large time constants is often governed by noise. It is this value that currently restricts the utility of the KK transforms. There is, therefore, a need to address the problem of applying the KK relations to data sets with finite frequency domains which do not extend to a sufficiently low value. Once this problem is properly resolved then the sensitivity of the KK relations to evaluate experimental impedance data for violations of the causality, linearity, and stability conditions can be individually investigate d . Previous authors have approached the problem differently: Mansfeld and Shih (9-11) stated that the KK transforms yield valid results only when the impedance data have reached a dc limit within the experimental frequency domain. Application of a KK algorithm to valid data sets which truncate in the capacitive region result in discrepancies that may erroneously lead to the conclusion that the data sets are invalid. These discrepancies, however, are due to the neglected contributions to the integrals associated with the inaccessible frequency domain (12). In one specific case, they were able to validate the experimental data using a KK algorithm only after having extrapolated the data below the lowest measured frequency using the fitting parameters of an equivalent circuit (10). This shows the importance of performing the integration over the widest frequency domain possible. It should be pointed out that, since the impedance response of electrical circuits satisfy the KK relations, a "good fit" between the experimental data and the analytic response of an equivalent 68 J. Electrochem. Soc., Vol . 138, No. 1, J a n u a r y 1991 9 The Electrochemical Society, Inc. circuit (i.e., which exhibit randomly distributed residuals) indicates that the data set satisfies the constraints associated with the KK relations (8, 13). Macdonald et al. (3) suggested extrapolating the experimental data beyond the frequency extremes using polynomial expressions obtained from a regression analysis of the data set. Although this procedure minimizes the problem of the "the inaccessible 'tails' of the integrals" (12), extrapolation of higher order polynomials in this manner is dangerous and may be justifiable only within a narrow frequency domain. It is evident from the discussions of Shih and Mansfeld and Macdonald and Urquidi-Macdonald that the application of the KK relations to a broad range of experimental impedance data requires a method to eliminate errors associated with the unmeasured frequency domain. The object of this work was to develop such an algorithm that can distinguish between errors due to a finite frequency domain that is too narrow and discrepancies due to data that are truly inconsistent with the KK relations. This algorithm is intended to reduce the possibility for false rejection of a valid data set caused by an insufficient experimental frequency domain. The utility of the proposed technique is illustrated using synthetic data derived from equivalent circuits. Since the conditions of the KK relations are necessarily satisfied, the sensitivity of the technique for checking the violation of the stability criterion was investigated. The authors are currently investigating the application of this procedure to t ime-dependent processes such as the corrosion of copper, a luminum alloys, and hydrogenation of metal hydrides, and the extension of the method for application to purely capacitive systems. These will be addressed in subsequent papers. Zr(W) Concept Through the KK relations, the value of one impedance component (real or imaginary) can be calculated at any given frequency if the other component is known over the entire frequency range. However, experimental difficulties constrain the acquisition of data within a finite frequency domain ~ i , -< to < to . . . . The apparent lack of agreement between the experimental data and the corresponding KK transformations can be attributed to two factors: (i) the problem of residuals due to the unmeasured impedance values in the frequency ranges o~ < ~,.m and ~o > t o~ , and, (ii) to processes associated with a nonstationary, nonlinear, and/or noncausat system. The contributions to the integrals of the KK relations predominantly arise in the region near the frequency being evaluated (1). This means that if an experimental data Set truncates in the capacitive region the problem of residuals described by factor (i) will be much more significant at the low-frequency range than at the high-frequency range. This is evident from the examples given in Ref. (9-11) where the greatest discrepancies between the experimental data and the calculated impedance values occurred at low frequencies. The concept behind this work is that functions Z~(to) and Z~(~o) can be found for the low-frequency domain coo <to < to~, which, when appended to the experimental data set, force the data to satisfy the KK equations over the frequency range o~0 to ~O~,x. Residual errors at the high-frequency extreme do not influence this calculation since the major contribution to the integrals of the KK relations is from the lower frequency domain. The consequence of this assumption will be discussed in a later section. The frequency range to0-(Omax is chosen to satisfy the requirement that the impedance spectrum does not terminate in the capacitive region; i.e., the real component attains an asymptotic value and the imaginary component approaches zero as r --~ too. Internal consistency between the impedance components also requires that the calculated functions be continuous with the experimental data at torero. These requirements cannot simultaneously be satisfied for data from systems that do not satisfy the constraints of the KK relations. Discontinuities at tom~, can therefore be attributed to properties unrelated to the use of a finite frequency range in the collection of data. ] [ T I 92 t Ne o o o o o o o O O O o t ~g~.t s~ ~ S~gm~t : ~ I ol m Experimental 7 ~ .~. z, k ,l~ Frequency k-o . I ~ in terva l \ Low F requency Interval S~;ment Segment I ~ 8 ~ e ~ 4~ 1 I ' NC o o o o e