Pitfalls in Simulating the Diffusion Potential of Li-Ion Cells
Authors: Bizeray Adrien Mathieu , Howey David Alastair , Monroe Charles W 
Published: February 2020

Variations of liquid-phase composition in electrochemical devices induce overpotentials referred to as ‘diffusion potential’ or ‘concentration potential’, which can have large effects for high current operation. Recently, an apparent discrepancy was observed in the electrochemical modelling literature when accounting for the effect of liquid-phase concentration on electrochemical devices’ voltage drop [1].

Liquid-phase potential is modelled using a modified form of the Ohm’s law known as the ‘MacInnes equation’ that accounts for the diffusion potential. The diffusion potential term in this equation presents a pre-factor involving the cation transference number t+
. The pre-factor 2(1-t+) is mostly used [2–5] but sometimes (1-2t+) is found in the literature [1, 6–8]. We argue that this apparent disagreement arises from the use of distinct albeit confusing formalisms when deriving the MacInnes equation [9].

The pre-factor (1-2t+) emerges when the Nernst–Planck theory (dilute solution) is used to derive the MacInnes equation, whereas the pre-factor 2(1-t+) appears when using the Onstager–Stefan–Maxwell theory (concentrated solution). However, we show that the definition of liquid-phase electric potential differs in these theories. In the concentrated solution theory, a voltage measured by a cation-reversible reference electrode is used providing an unambiguous connection to a measurement process. We show that the apparent discrepancy is resolved when relating Nernst–Planck voltage to a properly referenced voltage and demonstrate that both theories agree when considering the Onstager–Stefan–Maxwell theory in the limit of infinite dilution.

Finally, we present a case study for lithium-ion batteries that shows the impact on simulation results of using the Nernst–Planck voltage instead of a measurable voltage. Although the battery terminal voltage remains relatively unaffected, this has a significant impact on the predicted heat generated by the battery (see Figure 1).

Acknowledgements

This work was funded by the Joint Center for Energy Storage Research, an energy storage hub supported by the United States Department of Energy. A.M. Bizeray was supported by Samsung Electronics Co. Ltd. through a Global Research Outreach program in collaboration with the Samsung Advanced Institute of Technology.

References

[1] A. M. Ramos, Applied Mathematical Modelling, 40, 115–125 (2016).

[2] P. M. Gomadam, J. W. Weidner, R. A. Dougal and R. E. White, J. Power Sources, 110(2), 267-284 (2002).

[3] K. Smith and C-Y Wang, J. Power Sources, 160(1), 662-673 (2006).

[4] N. Chaturvedi, R. Klein, J. Christensen, J. Ahmed and A. Kojic, IEEE Control Syst. Mag., 30(3), 49-68, (2010).

[5] G-H Kim, K. Smith, J. Ireland and A. Pesaran, J. Power Sources, 210, 243-253 (2012).

[6] M. W. Verbrugge and P. Liu, J. Electrochem. Soc., 152(5), D79-D87 (2005).

[7] G. Richardson, G. Denuault and C. P. Please, J. Eng. Math., 72, 41-72 (2012).

[8] A. M. Bizeray, S. Zhao, S. R. Duncan and D. A. Howey, J. Power Sources, 296, 400-412 (2015).

[9] A. M. Bizeray, D. A. Howey and C. W. Monroe, “Resolving a discrepancy in diffusion potentials, with a case study for Li-ion batteries” Submitted to the Journal of the Electrochemical Society.

Figure 1