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 vs. log ω ) exists with a slope of value –1/2.

General equation

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 AW is the Warburg coefficient (or Warburg constant); This equation assumes semi-infinite linear diffusion, that is, unrestricted diffusion to a large planar electrode.

Finite-length Warburg element

If the thickness of the diffusion layer is known, the finite-length Warburg element is defined as: where where \delta 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: