where z is the charge of the mobile ion; F , R, and T retain their usual meanings; j is the imaginary unit; ω is the angular frequency; L is the diffusion length; D1 and D2 are two solid-state diffusion coefficients; θ and 1 -θ are the surface coverage of D1 and D2 , respectively; Γ is the Thermodynamic Factor, C eq is the equilibrium Li + concentration, and A is the active surface area. Details of the derivation of Equation 1 can be found in Appendix A: Theory of Parallel-diffusion Warburg Model. It is important to note that the PDW is derived based on the framework of solid-state Fickian diffusion and finite-length Warburg . Therefore, the application of the PDW is predicated on the following assumptions: (i) the electrode potential is a measure of the lithium activity at the electrode/electrolyte interface; (ii) the mass transport is governed only by one-dimensional Fickian diffusion;(iii) the kinetic effects and mass transport effects are well-separated by frequencies;