Pitt Modelling of Concurrent Ion-Insertive and Capacitive Storage Using Laplace Domain Representations of Impedance
Authors: Hall Charles Aram , Jiang Yu , Lim Sean , Burr Patrick , Ignjatovic Aleksandar , Lennon Alison 
Published: February 2020

Recently, there has been great deal of interest in high rate electrochemical energy storage. We are now at a point where the distinction between high rate Li-ion batteries and high energy electrochemical capacitors has begun to blur1, 2. There is hence an interest in elucidating the nature of charge storage in these high rate systems, with the aim of rationally engineering high rate materials. If more than one storage mechanism is present, for example a bimodal storage system with equal parts Li-Ion insertion and double layer storage, then this process of analysing storage kinetics becomes more complex.
A number of methods have been proposed for separating the capacitive and insertive charge storage present in an electrochemical system. Amongst these, b-value analysis3-5 and step potential electrochemical spectroscopy6, 7 are widely used. Similarly potentiostatic/galvanostatic intermittent titration techniques (PITT/GITT) are popular choices to probe the reaction kinetics. However, these models are not widely transferable, as the underlying assumptions are strictly valid only for specific electrochemical system and their use in unsuitable systems may lead to incorrect conclusions. For example, in analysing fast Faradaic storage, it is often erroneous to assume that Cotrellian behaviour can predict the kinetics of a fast Faradaic process8.
An alternative approach is to inform the models through electrochemical impedance spectroscopy (EIS) insomuch as storage processes tend to be more clearly intuited in the frequency domain than the time domain. Building on previous modelling work, including the reports of Montella8, Churikov et al9, this paper presents an analytical PITT model from an EIS-informed equivalent circuit, via an inverse Laplace transform, that captures both a fast Faradaic storage process and a double layer capacitance (Figure 1A-C).
Modelling shows that even when the capacity is dominated by either ion insertion or double layer storage (viz QEDLC : QIns of 1000:1 or 1:1000), there is still a noticeable deviation from the single-storage-mechanism model, due to the time scales over which these processes act. Trust regimes are presented for where a single-storage-mechanism model, be it that proposed by Montella for ion-insertion-only storage, or that for capacitance-only storage, can be used in place of the bimodal storage model (Figure 1D).
Finally, the model is used for an exemplar case study, that of charge storage in amorphous TiO2 nanotubes, (Figure 1E) which exhibit fast and reversible lithium insertion. The nature of this storage is shown to be primarily due to fast insertion charge storage processes. Reasonable agreement is obtained between the characteristic system parameters between PITT and EIS measurements (Figure 1F) thereby demonstrating the validity of the approach. In future work, the model will be used to analyse the charge storage mechanisms of other materials and to guide the material engineering required for high-rate electrochemical charge storage systems.

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Figure 1