# Cause of imaginary part of Warburg element in EIS

In electrochemical impedance spectroscopy experiments (EIS) the diffusion of electroactive species at flat, macroscopic electrodes is commonly described by the Warburg element with the impedance expression

$$Z_W = \frac{σ}{\sqrt{w}} - i\frac{σ}{\sqrt{w}}$$

This means that the values have a phase angle of $$-45^\circ$$.

What is an intuitive way to understand the fact, that the Nernst-diffusion layer contributes both a real and imaginary part to the impedance of the system?

I find that it is relatively straightforward to understand that the more the diffusion layer grows, the shallower the concentration gradient gets and therefore that the current decreases. This would explain the real part. I am unsure however, where the imaginary part, or energy-storing component, fits into this picture.

It is very hard to find any information in this regard, as practically all literature treats this just as a result of the mathematical derivation of the faradaic impedance, but does not go into any kind of intuitive explanation.

• Could it be the double layer capacitance that is responsible for the imaginary part? Impedance of capacitors decreases with frequency just as the 2nd term in the given equation – ChemEng Oct 16 '19 at 22:40
• @ChemEng Thanks for the idea, but no, the double-layer is modeled separately. A simple model for an electrochemical reaction is the Randles circuit, which places the double-layer capacitance in parallel with the Faradaic impedance (series connection of charge transfer resistor and Warburg element). – Jens Oct 17 '19 at 7:12

The case of Warburg element (which is a particular case of CPE : http://www.consultrsr.net/resources/eis/cpe1.htm) can be related to dissipative energy phenomena and storage energy phenomena $$\textbf{homogeneously}$$ distributed. A possible equivalent circuit can be made of repeated identical pattern of resistor-capacitor. This is a particular case of network for example of the kind represented page 10, figure 7(a) in the paper : https://fr.scribd.com/doc/71923015/The-Phasance-Concept , only with the part noted $$P_\varphi$$, without $$R_p$$. Of course different interpretations and equivalent circuits can be proposed, depending on the kind of electrochemical system. This is briefly discussed page 26 in the above referenced paper.