Time-Fractional Phase Field Model of Electrochemical Impedance

Pavel E L’Vov,Renat T Sibatov,Igor O Yavtushenko,Evgeny P Kitsyuk

doi:10.3390/fractalfract5040191

Abstract

In this paper, electrochemical impedance responses of subdiffusive phase transition materials are calculated and analyzed for one-dimensional cell with reflecting and absorbing boundary conditions. The description is based on the generalization of the diffusive Warburg impedance within the fractional phase field approach utilizing the time-fractional Cahn–Hilliard equation. The driving force in the model is the chemical potential of ions, that is described in terms of the phase field allowing us to avoid additional calculation of the activity coefficient. The derived impedance spectra are applied to describe the response of supercapacitors with polyaniline/carbon nanotube electrodes.

Highlights

Electrochemical impedance spectroscopy (EIS) is one of the most informative methods for studying charge carrier transport characteristics in different types of materials [1,2].The basic approach of the method implies the measurement of impedance dependence on applied voltage frequency with low value of amplitude
The driving force is the chemical potential of ions that is described in terms of phase-field that allows us to avoid the additional calculation of the activity coefficient
We should obtain the bipolar modification of well known Cahn–Hilliard equation that is widely applied in simulation of first order phase transitions [13,36]

Summary

Introduction

Electrochemical impedance spectroscopy (EIS) is one of the most informative methods for studying charge carrier transport characteristics in different types of materials [1,2]. Most microscopic models of electrochemical power sources are based on a system of diffusion equations that do not take into account collective transport and the anomalous type of ion diffusion, and are unable to explain the nonlinear nature of the responses of the devices under consideration. Application of the phase-field theory to the analysis of intercalation and transport of ions in electrochemical systems was launched by Guyer et al [7,8] who developed the Fractal Fract. The frequency dependences of the generalized Warburg impedance are derived for the case of ion transport controlled by the subdiffusive Cahn–Hilliard equation with fractional derivatives.

Brief Survey on Fractional Cahn–Hilliard Equations

Cahn–Hilliard Equation for Ionic Transport and Its Fractional Generalization

Mono-Fractional Phase-Field Subdiffusion

Impedance of Fractional Phase-Field Diffusion

Conclusions