Conditions and approximations for retrieving the complex permittivity and conductivity from the effective permittivity

The electromagnetic characterization of materials, also known as dielectric spectroscopy, plays a key role in many applications. Its aim is to return either the effective permittivity or the effective conductivity over frequency for a sample of the material under investigation. Effective permittivity and effective conductivity contain the same information amount and therefore only the former is considered in the following, being the same considerations applicable even to the latter with minor changes. Usual materials are lossy and therefore their phasor-domain effective permittivity is complex-valued. Two different physical mechanisms contribute to losses, namely, movable free charges (conductivity) and polarizability due to bounded charges (permittivity). These two contributes sum up to return the effective permittivity. In practical applications, however, it would be of interest to distinguish between permittivity and conductivity from the effective permittivity. Recently, authors have shown in [1] that the retrieval of the static conductivity and the high-frequency permittivity from the effective permittivity can always be accomplished but that it is generally unfeasible to distinguish between permittivity and conductivity from the effective electric permittivity. However, despite this general statement, there is space to analyze and make research on the conditions and approximations under which this retrieval is possible. For instance, based on reasonable hypotheses, one could consider a conductivity varying slowly with frequency. This approach is implicitly considered in [2], though no clear framework about applicability conditions is provided. Even, in case the real part of the conductivity varies slowly with frequency, the imaginary part of conductivity can be considered equal to zero in first approximation. This could enable an iterative solution of the problem aimed at finding a good approximation for the conductivity. Additionally, the retrieval of conductivity and permittivity could be combined with measurements of effective dielectric permittivity and theoretical models for both conductivity and permittivity to improve the last ones. Among all, an intriguing and exciting application field are graphene-polymer nanocomposites [3], where a frequency dependent theory of electrical conductivity and dielectric permittivity lead to a much better agreement between measurements and theoretical models. These and possible other fields of application deserve further studies. As investigative tool the Hilbert transform is intended to be used. This is a thoughtfully analyzed transform of which the Kramers-Kronig transform is a special case. However, the latter is less studied and less properties for it are known. To validate the proposed methods, authors intend to apply the retrieval of the conductivity and permittivity firstly to synthetic data and secondly to measurements data available in the literature. [1] G. Giannetti and L. Klinkenbusch, "Interpretation and separability of the effective permittivity in case that both permittivity and conductivity are complex," in 2023 URSI International Symposium on Electromagnetic Theory. IEEE, 2023. [2] M. Bakry and L. Klinkenbusch, "Using the Kramers-Kronig transforms to retrieve the conductivity from the effective complex permittivity," Advances in Radio Science, vol. 16, 2018. [3] X. Xia, Y. Wang, Z. Zhong and G. J. Weng, "A frequency-dependent theory of electrical conductivity and dielectric permittivity for graphene-polymer nanocomposites," Carbon, vol. 111, 2017.