Analytical Expression for the Impedance Response of an Insertion Electrode Cell

Abstract

An analytical expression for the impedance response of an insertion cathode/separator/foil anode cell sandwich is presented. The analytical expression includes the impedance contributions from interfacial kinetics, double-layer adsorption, and solution-phase and solid-phase diffusion processes. The validity of the analytical solution is ascertained by comparison with the numerical solution obtained for a LiCoO 2 /polypropylene/lithium metal cell. The flexibility of the analytical solution is utilized to analyze various limiting conditions. An expression to estimate solid-phase diffusion coefficient of insertion species in a porous electrode influenced by the solution-phase diffusion process is also derived. Electrochemical impedance spectroscopy ͑EIS͒ technique has been extensively used in the analysis of lithium battery systems, especially to determine kinetic and transport parameters, 1-3 understand reaction mechanisms, 4 and to study degradation effects. [5][6][7] However, the mathematical interpretation of the impedance response of electrochemical systems is complicated by the processes occurring in the system. This drives researchers to adopt lumped circuit models [8][9][10] or finite transmission line models 11 to interpret impedance data. However, these types of models provide little information on the fundamental physical processes occurring in the cell. To gain more understanding of the physical processes, macrohomogenous models for porous electrodes have been used by some researchers. [12][13][14][15][16][17][18] These models primarily use porous electrode theory 19,20 to describe the porous nature of the electrode/separator and concentration solution theory to treat the transport processes in the electrolyte phase. The thermodynamics and kinetics of the reactions at the electrode/electrolyte interface are also described in these models in detail. Most of these models also account for the solidphase diffusion of the active species into the bulk. While such detailed models throw light on the impedance behavior of systems when complicated by transport and kinetic processes, the mathematical interpretation is not straightforward.In the list of comprehensive models developed to simulate the impedance behavior of lithium batteries, Doyle et al. 13 simulated the impedance response for a metal anode/separator/porous cathode system with all the above-mentioned details. Guo et al. 15 used a similar model to estimate the diffusion coefficient of lithium in carbon. Later, Dees et al. 18 included an electronically insulating oxide layer at the electrode/electrolyte interface to model a LiNi 0.8 Co 0.15 Al 0.05 O 2 cathode. However, these models use a numerical scheme to solve for the variables to obtain the frequency domain impedance spectrum. There are also some analytical models 14,16,17 available in the literature, but they are not as comprehensive as the numerical models. Meyers et al.14 presented an analytical solution to the impedance response of a porous electrode in the absence of solution-phase concentr...

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“…The magnitude and phase of the response can then be interpreted to give the battery's complex (real and imaginary) impedance as a function of frequency. This method has the limitation that it is a small-signal method that assumes a linear battery model [37] (this fact is actually an asset in the analytical electrochemical models of Sikha and White [55,56] mentioned at the end of Section 4). These models are not often used as a stand-alone simulation of a battery.…”

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“…At further lower frequencies ͑Ͻ10 −3 Hz͒, the phase angle approaches 90°indicating a capacitive-type response typical of insertion electrodes. 8,5,13 The impedance response of the separator is purely resistive above 10 Hz and also approaches a resistive behavior at very low frequencies ͑ϳ10 −4 Hz͒. Between these frequency ranges, the impedance of the separator reveals two closely overlapping time constants due to the electrolyte transport processes in the separator region.…”

“…This situation also complicates the accurate estimation of solid and solution phase diffusion coefficients using traditional techniques. 13 Most of the EIS experimental impedance data were analyzed using the complex plane impedance plot ͑Nyquist͒, which has good sensitivity to various processes that occur at different frequency ranges based on the …”

“…In our previous work, we presented a methodology to develop a closed-form analytical solution for the impedance response of a single-insertion porous electrode considering the reaction kinetics, double-layer charging, and charge and mass transport processes in solid and solution phases. 13 This work presents a comprehensive analytical model for a cell sandwich consisting of dual-insertion porous electrodes. The model accounts for the reaction kinetics and double-layer adsorption at the interface, potential distribution in the porous electrode and the solution phase, and mass transport in the solution ͑electrolyte͒ phase and solid ͑in-sertion electrodes͒ phase.…”

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“…Numerous experimental methods were developed to determine them, e.g., potentiostatic intermittent titration technique (PITT), [6][7][8][9][10] galvanostatic intermittent titration technique (GITT), [11][12][13] cyclic voltammetry (CV), 14 and electrochemical impedance spectroscopy (EIS). 15,16 Depending on the method, a wide range of solid diffusion coefficients is reported in the literature for a similar AM. 17 For each technique, a physical model (with either a numerical or an analytic solution) is used to fit the system response to an out-of-equilibrium perturbation and to adjust the parameters.…”