Graphical Conversion between Compliance and Modulus, Permittivity and Electric Modulus, and Impedance and Admittance

Masahiro Nakanishi

doi:10.1155/2014/538206

Abstract

Spectrometries probing relaxation and retardation phenomena, such as dielectric, mechanical, and impedance spectroscopies, often require the analyses with both susceptibilities spectra z and its reciprocals 1/z (e.g., complex permittivity and electric modulus, mechanical compliance and mechanical modulus, and impedance and admittance). In the present paper, the geometric relation between z and 1/z is derived and the procedure to convert z into 1/z on a Cole-Cole diagram is proposed. This method helps us to relate them intuitively and yields clearer understanding on their interrelations. Moreover, it opens the new route for the geometric approach to derive many mathematical properties of spectra. The relation between peak position of z spectrum and that of 1/z spectrum and the shape of spectra are discussed on the basis of this method.

Highlights

In spectrometries focusing on relaxation or retardation phenomena, such as dielectric spectroscopy [1], dynamic mechanical spectroscopy [2], and impedance spectroscopy [3], frequency dependent susceptibilities and their reciprocals are often subject to consideration
For example, the complex permittivity is usually employed to discuss the dynamic nature of localized charges, such as the rotational diffusion of dipoles, while its reciprocal, electric modulus, is employed to assess the dynamics of delocalized charges, such as the migration of ions [4,5,6]
General properties of modulus grids on the Cole-Cole diagram for compliance were studied in detail

Summary

Introduction

In spectrometries focusing on relaxation or retardation phenomena, such as dielectric spectroscopy [1], dynamic mechanical spectroscopy [2], and impedance spectroscopy [3], frequency dependent susceptibilities and their reciprocals are often subject to consideration. For example, the complex permittivity is usually employed to discuss the dynamic nature of localized charges, such as the rotational diffusion of dipoles, while its reciprocal, electric modulus, is employed to assess the dynamics of delocalized charges, such as the migration of ions [4,5,6] This is the case for the mechanical spectroscopy. A Cole-Cole diagram [11, 12] is one of the most frequently used representations for relaxation spectra Speaking, it is a plot of a complex quantity z (or its complex conjugate, International Journal of Spectroscopy depending on the definition) on the complex plane, where complex permittivity, compliance, and impedance are often chosen as z. It opens the route to the geometric method to assess the mathematical properties of spectra

Theory

Examples for Applications

Conclusions