Defining Warburg Conductivity for Binary Electrolytes to Simplify Concentration Overpotential Calculation
Authors: Choudhury Basudev , Jangale Akash Suhas , Suthar BharatKumar 
Published: February 2024

Ionic conductivity is an essential and easy-to-use concept to find the potential drop across a separator when no concentration gradient exists. For a separator having thickness d, area A, tortuosity τ, porosity ε, filled with an electrolyte having conductivity κ, the resistance can be expressed as Rion=d/(Aκε/τ). The potential drop across the separator when current I[A] is flowing can be expressed as ΔV=IRion, which is valid when no concentration gradient exists. For the case when the concentration gradient is present (i.e., t+≠1) during the flow of current, the potential drop across the separator will require solving of mass conservation partial differential equations (coupled PDEs) along with MacInnes equation (modified Ohm’s law for electrolytes) [1]. For binary electrolytes, apart from conductivity, these equations will require additional parameters such as diffusion coefficient (D), transference number (t+), and thermodynamic factor (TDF). Various works to find out concentration and temperature-dependent D, t+, and TDF have become available recently [2–5].
To make the potential drop calculation convenient and to avoid solving PDEs, we can extend the conductivity concept for diffusion effect by defining Warburg conductivity κW. The κW effectively combines the effect of D, t+, and TDF and provides a straightforward way to estimate the extent of additional potential drop due to the concentration effect. For porous materials, just like effective conductivity κeff=κε/τ, we can then define the effective Warburg conductivity κW,
eff=κWε/τ and use it conveniently.
Using κW,
eff, we can define the Warburg resistance of the separator as RW=d/(AκW,
eff). Such RW can then be directly used to approximate the additional potential drop (apart from IRion) in the separator as ΔV=IRW. Figure 1 shows κ data from Landesfeind et al. [5] and the calculated κW using the D, t+, and TDF data (also from Landesfeind et al. [5]) for the electrolyte EC:DMC (1:1 w:w) at different concentrations and temperatures. From Figure 1, it can be seen that for the electrolyte EC:DMC (1:1 w:w), the κW is always lower than the κ for a given temperature and concentration, which means the net potential drop across the separator (after the concentration profile is fully established) will be I(Rion+RW) where the RW values will be higher than Rion. For example, a thick glass fiber separator at 20°C filled with 1M LiPF6 EC:DMC (1:1 w:w) with thickness d=200μm, porosity ≈ 1, tortuosity ≈ 1, and having areal current of 1mA/cm2 will have initially a potential drop of ≈ 1.86 mV (ΔV = IRion = 1 mA/cm2 × 200 μm / (1.076 × 1/1)) when the concentration profile is flat and later when the concentration profile is fully established the net potential drop across separator can be calculated (using κW = 0.450 S/m from Figure 1) as ≈ 6.3 mV (ΔV=I(Rion+RW)=1.86 mV+1 mA/cm2 × 200 μm / (0.450 × 1/1)). References:

Newman, J., & Thomas-Alyea, K. E. (2012). Electrochemical systems. John Wiley & Sons.

Valøen, L. O., & Reimers, J. N. (2005). Transport properties of LiPF6-based Li-ion battery electrolytes. Journal of The Electrochemical Society, 152(5), A882.

Nyman, A., Behm, M., & Lindbergh, G. (2008). Electrochemical characterisation and modelling of the mass transport phenomena in LiPF6–EC–EMC electrolyte. Electrochimica Acta, 53(22), 6356-6365.

Ma, Y., Doyle, M., Fuller, T. F., Doeff, M. M., De Jonghe, L. C., & Newman, J. (1995). The measurement of a complete set of transport properties for a concentrated solid polymer electrolyte solution. Journal of the Electrochemical Society, 142(6), 1859.

Landesfeind, J., & Gasteiger, H. A. (2019). Temperature and concentration dependence of the ionic transport properties of lithium-ion battery electrolytes. Journal of The Electrochemical Society, 166(14), A3079.

Figure 1