Time-Fractional Phase Field Model of Electrochemical Impedance

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.

Temperature-Dependent Fractional Dynamics in Pseudo-Capacitors with Carbon Nanotube Array/Polyaniline Electrodes

“…Here, M is the ambipolar mobility, and µ a is the effective (ambipolar) chemical potential (see details in [16]). The corresponding impedance models were denoted as Z C−CH and Z RL−CH , respectively.…”

“…In [ 16 ], a generalized diffusion impedance model for materials with a subdiffusion phase transition is proposed. The model is based on the fractional Cahn–Hilliard equation with fractional time derivatives.…”

“…A one-dimensional cell with reflecting and absorbing boundaries is considered. Phase-field generalizations of anomalous diffusion models AD-Ib and AD-Ia presented in [ 31 ] are described by the following time-fractional equations [ 16 ] with Caputo and Riemann–Liouville derivatives: Here, M is the ambipolar mobility, and is the effective (ambipolar) chemical potential (see details in [ 16 ]).…”

“…The form of depends on boundary conditions. For a cell with reflecting boundary, in [ 16 ], it was obtained …”

“…As is known, fractional-order technique using fractional calculus and fractional equivalent circuits [ 3 ] is effective to describe the dynamics of supercapacitors [ 6 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 ]. Recently, the time-fractional phase-field model has been applied to describe PANI/MWCNT pseudo-capacitors [ 16 ]. Here, the simplified representation of a fractional circuit model is used to describe the impedance spectra, cyclic voltammograms, and charging–discharging in potentiostatic mode.…”