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

Deconvolution of immittance spectroscopy (IS) data into a not directly measurable distribution of relaxation times (DRT) of polarisation processes occurring in the measured system provides useful information on the system. It primarily regards the number of such processes identifiable as relaxation time constants. Upon changing the conditions of the measurement, it also regards the variation particularly of the position and intensity of the relaxation time in the DRT spectrum. These information to be revealed by numerical inversion commonly by constrained least squares (LS) minimisation or fast Fourier transformation (FFT) helps to construct an equivalent electrical circuit (EEC) model for subsequent complex nonlinear LS (CNLS) fitting of the IS data to simulate system behaviour. The inverse problem of the deconvolution of 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 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) taken to match the immittance of the studied system. As a novelty, we derive from a SIE of the second kind the hitherto concealed complex valuedness of the DRT using the Hilbert integral transform (HT) and DRT properties for Kelvin-Voigt type RLC networks. We emphasis 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.

MALKOW Thomas; 

2019-04-10

ELSEVIER SCIENCE SA

JRC115248

1572-6657 (online),   

https://publications.jrc.ec.europa.eu/repository/handle/JRC115248,   

10.1016/j.jelechem.2019.01.057 (online),