A Warburg impedance element can be difficult to recognize because it is nearly always associated with a charge-transfer resistance (see charge transfer complex) and a double layer capacitance (see double layer (interfacial)), but is common in many systems. The presence of the Warburg element can be recognised if a linear relationship on the log of a Bode plot (log|Z| versus log(ω)) exists with a slope of value –1/2.
The Warburg diffusion element (ZW) is a constant phase element (CPE), with a constant phase of 45° (phase independent of frequency) and with a magnitude inversely proportional to the square root of the frequency by:
This equation assumes semi-infinite linear diffusion, that is, unrestricted diffusion to a large planar electrode.
Finite-length Warburg element
where is the thickness of the diffusion layer and D is the diffusion coefficient.
There are two special conditions of finite-length Warburg elements: the Warburg Short (WS) for a transmissive boundary, and the Warburg Open (WO) for a reflective boundary.
Warburg Short (WS)
This element describes the impedance of a finite-length diffusion with transmissive boundary. It is described by the following equation:
Warburg Open (WO)
This element describes the impedance of a finite-length diffusion with reflective boundary. It is described by the following equation:
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