Memory Effect and Fractional Differential Dynamics in Planar Microsupercapacitors Based on Multiwalled Carbon Nanotube Arrays

Abstract

The development of portable electronic devices has greatly stimulated the need for miniaturized power sources. Planar supercapacitors are micro-scale electrochemical energy storage devices that can be integrated with other microelectronic devices on a chip. In this paper, we study the behavior of microsupercapacitors with in-plane interdigital electrodes of carbon nanotube array under sinusoidal excitation, step voltage input and sawlike voltage input. Considering the anomalous diffusion of ions in the array and interelectrode space, we propose a fractional-order equivalent circuit model that successfully describes the measured impedance spectra. We demonstrate that the response of the investigated micro-supercapacitors is linear and the system is time-invariant. The numerical inversion of the Laplace transforms for electric current response in an equivalent circuit with a given impedance leads to results consistent with potentiostatic measurements and cyclic voltammograms. The use of electrodes based on an ordered array of nanotubes reduces the role of nonlinear effects in the behavior of a supercapacitor. The effect of the disordering of nanotubes with increasing array height on supercapacitor impedance is considered in the framework of a distributed-order subdiffusion model.

Dispersive Transport Described by the Generalized Fick Law with Different Fractional Operators

“…Note that representations ( 12), (28), and (30) are quite general and they can be used to solve other fractional generalizations of the diffusion-advection equation. In addition to disordered semiconductors, electrode materials and electrolytes of supercapacitors [32] and lithium-ion batteries [33] can be considered as possible applications of the proposed generalizations of Fick's law and solutions of FFP equations.…”

Time-Fractional Phase Field Model of Electrochemical Impedance

“…The fractional differential model for planar supercapacitor proposed in [44] successfully describes the measured impedance spectra. Charge storage mechanism in PMSC implies the EDL formation, and the model used in [44] is based on the linear fractional diffusion equation. Here, we apply fractional phase-field diffusion model to pseudo-capacitors with PANI/MWCNT electrodes.…”

“…Here, B is a frequency-independent parameter. Based on fractional Equations ( 5) and ( 6), different generalizations of diffusion impedances can be obtained for different geometries and boundary conditions (see in [30,[41][42][43][44] and references therein). In recent paper [45], effects produced by ion subdiffusion in lithium-ion cell are described within the fractional-order generalization of the single-particle model.…”

“…In [44], we studied the behavior of planar microsupercapacitors (PMSC) with electrodes of MWCNT array under sinusoidal excitation, step voltage input and linear voltage input. The fractional differential model for planar supercapacitor proposed in [44] successfully describes the measured impedance spectra. Charge storage mechanism in PMSC implies the EDL formation, and the model used in [44] is based on the linear fractional diffusion equation.…”

Applications of Distributed-Order Fractional Operators: A Review

“…A similar application was presented in [ 312 ], where DO operators were introduced into the Letokhov model of photon diffusion to model non-resonant random lasers. Very recently, the effect of disordering of nanotubes within an electrode, on the impedance of a supercapacitor, was modeled using the DO subdiffusion model in [ 313 ]. All these applications highlighted the ability of the DO diffusion formulation to accurately capture highly anomalous diffusion behavior arising out of the presence of multiple temporal and/or spatial scales.…”