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A normal power-law distribution of local resistivity with a uniform dielectric constant was found to be consistent with the constant-phase element ͑CPE͒. An analytic expression, based on the power-law resistivity distribution, was found that relates CPE parameters to the physical properties of a film. This expression worked well for such diverse systems as aluminum oxides, oxides on stainless steel, and human skin. Good values for film thickness were obtained, even when previously proposed expressions could not be used or yielded incorrect results. The power-law model yields a CPE impedance behavior in an appropriate frequency range, defined by two characteristic frequencies. Ideal capacitive behavior is seen above the upper characteristic frequency and below the lower characteristic frequency. A symmetric CPE response between the characteristic frequencies can be obtained by adding a parallel resistive pathway.Deterministic mathematical models for impedance spectroscopy are intended to provide information concerning physical processes such as kinetics, mass transfer, and reaction mechanisms, as well as material properties, such as permittivity and conductivity. The constant-phase element ͑CPE͒ model, which is purely a mathematical description, may adequately represent impedance data, but it gives no insight into the physical processes that yield such a response. However, capacitance values are often extracted from CPE data using such expressions asdeveloped by Brug et al. 1 or C HM = Q 1/␣ R f ͑1−␣͒/␣ ͓2͔ presented by Hsu and Mansfeld. 2 The parameters R e and R t in Eq. 1 represent the Ohmic and charge-transfer resistances, respectively, and the parameter R f in Eq. 2 represents the film resistance. The subscripts B and HM refer to the authors of the respective papers. Often, the capacitance values obtained are used to estimate the thickness of dielectric layers. Recently, Hirschorn et al. 3 associated these expressions unambiguously with either surface or normal time-constant distributions and demonstrated the importance of using the correct formula that corresponds to a given type of distribution. Limitations of Eq. 2 for predicting layer thickness were discussed. For a blocking system, where the film resistance is infinite, Eq. 2 cannot be used as it would yield an infinite capacitance. When the distribution of physical properties is broad, Eq. 2 may yield an inaccurate estimate of capacitance. Their work demonstrated the importance of developing physically reasonable models that can result in CPE behavior. Normal distributions of time-constants can be expected in systems such as oxide films, organic coatings, and human skin. Such normal time-constant distributions may be caused by distributions of resistivity and/or dielectric constant. Hirschorn et al. 4 used a powerlaw distribution of resistivity ␦where the parameters 0 and ␦ are the boundary values of resistivity at the interfaces, to show that, for Ͻ ͑ ␦ ⑀⑀ 0 ͒ −1 , an analytic expression for the impedance could be obtained asThe function g was evaluate...

Constant-phase element ͑CPE͒ behavior may be attributed to the distribution of physical properties in films, in the direction normal to the electrode surface. Numerical simulations were used to show that under assumption that the dielectric constant is independent of position, a normal power-law distribution of local resistivity is consistent with the CPE. An analytic expression for the impedance corresponding to a power-law distribution of resistivity was developed and used to establish a relationship between parameters for the CPE and physical properties of the film. The resulting impedance response was compared to that obtained from the Young resistivity distribution.

Constant-Phase Elements (CPE) are often used to fit impedance data arising from a broad range of experimental systems. Four approaches were used to interpret CPE parameters associated with the impedance response of human skin and two metal oxides in terms of characteristic frequencies and film thickness. The values obtained with each approach were compared against independent measurements. The power-law model developed recently by Hirschorn et al. 1,2 provided the most reliable interpretation for systems with a normal distribution of properties. Readers are cautioned that the CPE parameter Q does not provide an accurate value for capacitance, even when the CPE exponent α is greater than 0.9.Electrical circuits invoking constant-phase elements (CPE) are often used to fit impedance data arising from a broad range of experimental systems. The impedance for a film-covered electrode showing CPE behavior may be expressed in terms of ohmic resistance R e , a parallel resistance R || , and CPE parameters α and Q aswhere f is the frequency in units of Hz. When α = 1, the system is described by a single time-constant, and the parameter Q has units of capacitance; otherwise, Q has units of s α / cm 2 or F/s (1−α) cm 2 . 3 Under conditions that (2πf ) α R || Q 1,

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