A theory of distribution functions of relaxation times for the deconvolution of immittance data

Abstract

Deconvolution of immittance spectroscopy (IS) data into a not directly measurable distribution of uncorrelated relaxation times (DRT) of polarisation processes occurring in the measured system provides useful information on the system. It primarily regards the identification of the number of such processes through their relaxation time constants, τ. Upon changing the conditions of the measurement, it also covers the variation particularly of the position and intensity of the relaxation time in the DRT spectrum.These information revealed by numerical inversion commonly using constrained least squares (LS) minimisation or fast Fourier transformation (FFT), help to construct an equivalent electrical circuit (EEC) model, which subsequently allows complex nonlinear least squares (CNLS) fitting of the IS data to simulate system behaviour.The inverse problem of deconvolving IS data into DRT is based on the principle of superposition of the immittances of passive (lumped) elements, such as a resistor, R, an inductor, L and a capacitor, C combined in parallel or series and forming branches of RLC electrical circuit networks of different types (Kelvin-Voigt, Maxwell-Wagner, Cauer, Foster, etc.). In the limit of an infinite number of branches, the network immittance convolved with the DRT function is mathematically described by a Fredholm singular integral equation (SIE) equated with the immittance of the studied system.As a novelty, from a SIE of the second kind, we derive the hitherto concealed complex valuedness of the DRT using the Hilbert integral transform (HT) and DRT properties for Kelvin-Voigt type RLC networks. We emphasise the necessity and usefulness of complex valued DRT for its numerical estimation by the FFT method and LS minimisation. We also provide for the derivation of theoretical DRT from rational immittances known analytically in closed form, e.g. assuming their representation by a HT compliant EEC model.