Analytical Solution for the Impedance of a Porous Electrode

235

195

The short diffusion lengths in insertion battery nanoparticles render the capacitive behavior of bounded diffusion, which is rarely observable with conventional larger particles, now accessible to impedance measurements. Coupled with improved geometrical characterization, this presents an opportunity to measure solid diffusion more accurately than the traditional approach of fitting Warburg circuit elements, by properly taking into account the particle geometry and size distribution. We revisit bounded diffusion impedance models and incorporate them into an overall impedance model for different electrode configurations. The theoretical models are then applied to experimental data of a silicon nanowire electrode to show the effects of including the actual nanowire geometry and radius distribution in interpreting the impedance data. From these results, we show that it is essential to account for the particle shape and size distribution to correctly interpret impedance data for battery electrodes. Conversely, it is also possible to solve the inverse problem and use the theoretical "impedance image" to infer the nanoparticle shape and/or size distribution, in some cases, more accurately than by direct image analysis. This capability could be useful, for example, in detecting battery degradation in situ by simple electrical measurements, without the need for any imaging.2

Section: Application and Discussionmentioning

confidence: 99%

Effects of Nanoparticle Geometry and Size Distribution on Diffusion Impedance of Battery Electrodes

Song¹,

Bazant

2012

235

195

“…The corresponding complex impedance in the frequency domain ͑variable ͒ is evaluated by substituting s = j. 10 The assumption s = j also eliminates the transient part of the response in the mathematical solution but includes the sinusoidal steady-state output response.…”

Section: Mathematical Methods and Solution Techniquementioning

confidence: 99%

Analytical Expression for the Impedance Response for a Lithium-Ion Cell

Sikha

White

2008

An analytical expression to predict the impedance response of a dual insertion electrode cell (insertion electrodes separated by an ionically conducting membrane) is presented. The expression accounts for the reaction kinetics and double-layer adsorption processes at the electrode-electrolyte interface, transport of electroactive species in the electrolyte phase, and insertion of species in the solid phase of the insertion electrodes. The accuracy of the analytical expression is validated by comparing the impedance response predicted by the expression to the corresponding numerical solution. The analytical expression is used to predict the impedance response of a lithium-ion cell consisting of a porous LiConormalO2 cathode and mesocarbon microbead anode. A qualitative graphical method to identify the co-existence of solid and solution phase transport limitations in the impedance spectra of insertion electrodes is also discussed in the paper.

“…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…”

mentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

Analytical Expression for the Impedance Response of an Insertion Electrode Cell

Sikha

White

2007

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...