The Warburg diffusion element is an equivalent electrical circuit component that models the diffusion process in dielectric spectroscopy. That element is named after German physicist Emil Warburg.
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, 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| vs. 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:
where
This equation assumes semi-infinite linear diffusion, [1] that is, unrestricted diffusion to a large planar electrode.
If the thickness of the diffusion layer is known, the finite-length Warburg element [2] is defined as:
where
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.
This element describes the impedance of a finite-length diffusion with transmissive boundary. [3] It is described by the following equation:
This element describes the impedance of a finite-length diffusion with reflective boundary. [4] It is described by the following equation:
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