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Understanding Double Layer in Electrodes Using Scaling Analysis - Applications to Battery Modeling

*Abstract:*Batteries are the most popular energy storage devices in the recent decades. Development of a complete mathematical model for batteries is very vital in the prediction of cell performance, optimizing parameters and also for scale up. To improve the predictive capability of first principle models, a fundamental understanding of all the processes occurring inside the battery is required which includes the capacity fade mechanisms and the side reactions. A few major failure modes that exist in batteries are thermal runaway, solvent oxidation side reaction, aging, and breakdown of electrodes with over-discharge resulting in release of oxygen ^{[1]}.

In literature, to relate the current density to electric potential of an electrode in any electrochemical device, usually two approaches are used. The simplest of those is to use Butler Volmer equation which is derived from electrode surface reaction kinetics ^{[2]}. The second approach is to assume an electric double layer (EDL) structure^{ [3], [4], [5]} for interface and incorporate Frumkin correction to the Butler Volmer equation. EDL structure was initially proposed based on theoretical grounds and its existence has been verified experimentally in the literature. Assuming the existence of the double layer, Bazant ^{[6]} has derived the expression for Debye length using scaling. In this work, EDL structure was obtained from the first principle model and rate equations, with no assumption on the structure a priori. This provides a mathematical understanding for the existence of a double layer at interface purely based on scaling arguments. Debye length is obtained as a scale for diffuse layer thickness out of the scaling arguments. Unlike conventional approach where Boltzmann distribution of ions is an assumption made to derive the equation of Debye length, this approach results in Boltzmann distribution for ion concentration near interface as a leading order solution to the differential equation.

This can be further extended to include the kinetics of the side reactions into the model. Thus an improved cell model which includes the failure modes can be obtained and can be used to get a better prediction of concentration and potential variations inside the electrochemical cell.

**References:**

[1] Ramadass, P., et al. "Development of first principles capacity fade model for Li-ion cells." *Journal of the Electrochemical Society* 151.2 (2004): A196-A203.

[2] Mayneord, W. V. "John Alfred Valentine Butler. 14 February 1899-16 July 1977." *Biographical Memoirs of Fellows of the Royal Society* (1979): 145-178.

[3] Helmholtz, H. von. "Ueber einige Gesetze der Vertheilung elektrischer Ströme in körperlichen Leitern mit Anwendung auf die thierisch‐elektrischen Versuche." *Annalen der Physik* 165.6 (1853): 211-233.

[4] Stern, O. Z.Electrochem, 30, 508 (1924)

[5] D. C. Grahame, Chem. Rev., 41 (1947) 441

[6] Bazant, Martin Z. "10.626 Electrochemical Energy Systems (Massachusetts Institute of Technology: MIT OpenCourseWare), h ttp." *ocw. mit. edu, License: Creative Commons BY-NC-SA* (2011).