Physical Interpretation of the Warburg Impedance Available to Purchase

The physical origin of the capacitive behavior of the Warburg impedance (Z_W) and adsorption pseudocapacitance (C_ø) is discussed. A simple derivation of Z_W, based only on the Nernst diffusion layer thickness (δ = (πDt]^1/2), is presented. An alternate equivalent circuit is proposed in which Z_W is represented by a series combination of a resistor (the Warburg pseudoresistance [R_W]) and a capacitor (the Warburg pseudocapacitance [C_W]). The values of these new circuit elements are independent of potential. On the other hand, they are proportional to ω^−1/2, leading to the so-called “constant phase element” that is observed experimentally. In contrast, the usual elements employed in equivalent circuit models to describe the voltage-current characteristics of the interphase (e.g., the faradaic resistance [R_F], double-layer capacitance [C_dl], and adsorption pseudocapacitance [C_ø]) are independent of frequency but may change with potential over many orders of magnitude. It is shown that the frequency dependence of C_W and R_W follows from the well-known dependence of δ on the square root of time.