# Broadband Complex Permittivity Spectra: Cole–Cole vs Circuit Models

## TL;DR

The paper compares Cole–Cole and circuit models for analyzing electrical properties of materials, focusing on separating electrode and material polarization effects.

## Contribution

A novel method is introduced to estimate the constant phase element exponent using permittivity-frequency data.

## Key findings

- A parallel resistor and constant phase element model better separates material and electrode polarization effects.
- The electrode polarization frequency correlates with material conductivity and electrode properties.
- Kaolinite chargeability aligns with its fundamental definition using model-derived conductivity values.

## Abstract

Hydraulic properties such as porosity, water, and clay
content
can be inferred from electrical parameters like permittivity, conductivity,
and resistivity. Spectral data enhance this analysis by revealing
features such as pore size and clay type in wet particulate media.
In liquid samples, electrode polarization is clearly observed, as
orientational polarization occurs only at higher frequencies (MHz
to sub-GHz). In contrast, particulate media exhibit electrode polarization
artifacts that obscure spatial polarization peaks within the Hz–MHz
range, especially in highly conductive materials like wet clayey soils,
making the Cole–Cole model insufficient for distinguishing
these effects. Therefore, a general circuit model using a parallel
form of a resistor and a constant phase element configuration more
effectively separates inherent material polarization from electrode
polarization. The electrode polarization limiting frequency (f
EP) correlates with both material conductivity
and electrode properties, even with low-polarization electrodes like
Ag/AgCl. A novel method is introduced to estimate the effective constant
phase element exponent (
η~
) using the slope of log permittivity vs
log frequency. Finally, the chargeability of kaolinite (m = 0.83–0.86), derived from the ratio of critical frequencies
between the Cole–Cole and Pelton models, aligns with its fundamental
definition: m = (σ∞ –
σ0)/σ∞, where σ0 is the DC conductivity and σ∞ is
the high-frequency conductivity.