# Fractional Calculus as a Tool for Modeling Electrical Relaxation Phenomena in Polymers

## TL;DR

This paper introduces a new fractional calculus model to better understand and predict dielectric relaxation in polymers used in electronics and energy storage.

## Contribution

A novel electrical fractional model (EFM) is proposed to simultaneously describe α-relaxation and interfacial polarization in polymers.

## Key findings

- The EFM successfully fits dielectric spectra for amorphous polymers like PEI, PVC, and PVB.
- The model captures both α-relaxation and interfacial polarization using fractional differential equations.
- The EFM provides improved interpretability over traditional integer-order models.

## Abstract

The dielectric relaxation behavior of polymeric materials is critical to their performance in electronic, insulating, and energy storage applications. This study presents an electrical fractional model (EFM) based on fractional calculus and the complex electric modulus (M*=M′+iM″) formalism to simultaneously describe two key relaxation phenomena: α-relaxation and interfacial polarization (Maxwell–Wagner–Sillars effect). The model incorporates fractional elements (cap-resistors) into a modified Debye equivalent circuit to capture polymer dynamics and energy dissipation. Fractional differential equations are derived, with fractional orders taking values between 0 and 1; the frequency and temperature responses are analyzed using Fourier transform. Two temperature-dependent behaviors are considered: the Matsuoka model, applied to α-relaxation near the glass transition, and an Arrhenius-type equation, used to describe interfacial polarization associated with thermally activated charge transport. The proposed model is validated using literature data for amorphous polymers, polyetherimide (PEI), polyvinyl chloride (PVC), and polyvinyl butyral (PVB), successfully fitting dielectric spectra and extracting meaningful physical parameters. The results demonstrate that the EFM is a robust and versatile tool for modeling complex dielectric relaxation in polymeric systems, offering improved interpretability over classical integer-order models. This approach enhances understanding of coupled relaxation mechanisms and may support the design of advanced polymer-based materials with tailored dielectric properties.