Deconvolution of immittance data: Some old and new methods

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Abstract

The background and history of various deconvolution approaches are briefly summarized; different methods are compared; and available computational resources are described. These underutilized data analysis methods are valuable in both electrochemistry and immittance spectroscopy areas, and freely available computer programs are cited that provide an automatic test of the appropriateness of Kronig–Kramers transforms, a powerful non-linear-least-squares inversion method, and a new Monte Carlo inversion method. The important distinction, usually ignored, between discrete-point distributions and continuous ones is emphasized, and both recent parametric and non-parametric deconvolution/inversion procedures for frequency-response data are discussed and compared. Information missing in a recent parametric measurement-model deconvolution approach is pointed out and remedied, and its priority evaluated. Comparisons are presented between the standard parametric least squares inversion method and a new non-parametric Monte Carlo one that allows complicated composite distributions of relaxation times (DRT) to be accurately estimated without the uncertainty present with regularization methods. Also, detailed Monte Carlo DRT estimates for the supercooled liquid 0.4Ca(NO3)2 · 0.6KNO3 (CKN) at 350 K are compared with appropriate frequency-response-model fit results. These composite models were derived from stretched-exponential Kohlrausch temporal response with the inclusion of either of two different series electrode-polarization functions.

Section snippets

Background

Both electrocatalysis and immittance spectroscopy data usually involve distributions of relaxation times or activation energies, but because methods of estimating such distributions, called deconvolution or inversion, are thought to be difficult to apply or not readily available, such techniques for aiding in understanding physico-chemical processes present in materials are often underutilized. A list of acronym definitions, including ones for fitting models, is included at the end of the

Least-squares deconvolution methods

The regularization method, essentially a non-parametric approach, involves a regularization parameter whose value is chosen to ameliorate inversion problems by a kind of smoothing process, one that necessarily introduces some inaccuracy in DRT estimation. Here emphasis is on the PLS approach for estimating an unknown DRT, g(τ) or a transformation of it [13]. It involves expressing the frequency-response data, I(ω), as an integral from zero to infinity over g(τ)/[1 + iωτ], or temporal data, f(t),

PLS inversion approaches

The 2002 PLS publication of Ref. [14] stated that methods of obtaining information about DRTs were not well developed and proposed “a method for identification, from impedance spectra, of the distribution of time constants associated with activation or relaxation processes.” Their approach, although not so mentioned, is just a version of the PLS one described much earlier by others and instantiated in the PLLS and PNLLS approaches included in the widely used LEVM program and its predecessor,

Acronym definitions

    General

    CNLS

    complex non-linear least squares

    DIA

    differential impedance analysis [7], [25]

    DRT

    distribution of relaxation times

    KK

    Kronig–Kramers transform relations

    LEVM

    CNLS fitting and inversion program [10]

    PLS

    parametric least squares

    PLLS

    parametric linear least squares

    PNLLS

    parametric non-linear least squares

    PWT

    proportional weighting [10]

    UWT

    unity weighting [10]

    Single and composite frequency-response fitting models

    CD

    Cole–Davidson response function defined at the impedance level (see Section 3.2.3.1)

    K1

    conductive-system Kohlrausch frequency-response model (see

Acknowledgements

We thank Dr. P. Lunkenheimer for providing the present CKN data and he and Dr. B.A. Boukamp for valuable comments and suggestions. The work of Dr. E. Tuncer was sponsored by the US Department of Energy, Office of Electricity Delivery and Energy Reliability, Superconductivity Program for Electric Power Systems, under Contract No. DE-AC05-00OR22725 with Oak Ridge National Laboratory, managed and operated by UT-Battelle, LLC.

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