Analytical inversion of complex-valued DRT in impedance spectroscopy by integral transforms

The distribution of uncorrelated relaxation times (DRT) approach has become a routine tool for analysing impedance data, providing valuable information about the relaxations occurring within a system. However, the ambiguity in the value of the DRT and the limitations of analytical expressions derived from the Fuoss-Kirkwood relation necessitate the development of new methods for deriving single-valued DRT. Such expressions enable the use of a unique equivalent real formulation in the numerical estimation of complex-valued DRT, which is crucial for enhancing the understanding of the impedance response of measured biochemical and electrochemical systems beyond current practices. In this theoretical account, we employ integral transforms to invert a prototype impedance equation with complex-valued DRT, deriving previously unavailable analytical expressions for such single-valued DRT as a function of impedance with imaginary frequency argument. These expressions are essential for validating yet to be developed software codes that use model impedances explicitly known in closed form, and for deriving properties of the DRT allowing to constrain its numerical estimation, thereby facilitating the attainment of more accurate DRT spectra. To demonstrate the significance of complex-valued DRT, we present two examples, using model impedance data to derive simulated modulus and phase spectra of such DRT. These spectra can provide new insights into systems, which are practically inaccessible from the spectrum of a real-valued DRT. We conclude with several complementary recommendations for future developments necessary to advance the theoretical foundations of the DRT. This is intended to facilitate synergistic integration with established analysis techniques in impedance spectroscopy, thereby enabling enhanced consistency and accuracy in the analysis of valid frequency response data from measured biochemical and electrochemical systems featuring different immittance levels. • Dependency of the value of the distribution of relaxation times (DRT) on the nature of the impedance data is revealed. • Complex-valued DRT using the Hilbert transform circumvents value dependency on the nature of the impedance data. • Mellin space pendants of the Hilbert transformed real and imaginary parts of the impedance is crucial for DRT inversion. • Inversion by Mellin and Fourier transforms of the DRT impedance equation allows deriving analytical expressions in terms of the impedance with imaginary frequency. • Derivation of DRT properties (limits, derivatives, moments) useful as additional numerical solutions constraints for the DRT is made possible by the use of analytical expressions.