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 electrolyte having conductivity κ, the resistance can be expressed as R_ion[Ω]=d/(Aκε/τ). The potential drop across the separator when current I [A] is flowing can be expressed as ΔV = IR_ion, 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 (PDEs) along with MacInnes equation (modified Ohm’s law for electrolytes) [1]. For binary electrolyte, 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 κ_W. The κ_W effectively combines the effect of D, t₊ and TDF and provide a straightforward way to measure the extent of additional potential drop due to concentration effect. For porous materials, just like effective conductivity κ_eff = κε/τ, we can then define the effective Warburg conductivity κ_W,eff = κ_Wε/τ and use it conveniently.

Using κ_W,eff, we can define Warburg resistance of the separator as R_W ≡ d/(Aκ_W,eff). Such R_W can then be directly used to approximate the additional potential drop (apart from IR_ion) in the separator as ΔV = IR_W. Figure 1 shows κ data from Landesfeind et al. [5] and the calculated κ_W 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 κ_W 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(R_ion+R_W) where the R_W values will be higher than R_ion. For example, a thick glass fiber separator at 20°C filled with 1 M LiPF₆ EC:DMC (1:1 w:w) with thickness d = 200 μm, porosity ≈ 1, tortuosity ≈ 1, and having areal current of 1 mA/cm² will have initially a potential drop of ≈ 1.86 mV (ΔV = IR_ion = 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 κ_W = 0.450 S/m from Figure 1) as ≈ 6.3 mV (ΔV = I(R_{ion +} R_W)= 1.86mV + 1 mA/cm²×200 μm/(0.450×1/1)).

References:

1. Newman, J., & Thomas-Alyea, K. E. (2012). Electrochemical systems. John Wiley & Sons.
2. Valøen, L. O., & Reimers, J. N. (2005). Transport properties of LiPF6-based Li-ion battery electrolytes. Journal of The Electrochemical Society, 152(5), A882.
3. 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.
4. 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.
5. 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.