Warburg Element

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

General equation

The Warburg diffusion element () 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:

:$= \frac+\frac$

:$= \sqrt\frac$

where
* is the Warburg coefficient (or Warburg constant);
* is the imaginary unit;
* is the angular frequency.

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:

:$= \frac \tanh\left(B \sqrt\right)$

where $B=\tfrac,$

where $\delta$ is the thickness of the diffusion layer and is the diffusion coefficient.

There are two special conditions of finite-length Warburg elements: the Warburg Short () for a transmissive boundary, and the Warburg Open () for a reflective boundary.

Warburg Short (W_S)

This element describes the impedance of a finite-length diffusion with transmissive boundary. It is described by the following equation:

:$Z_ = \frac \tanh \left(B \sqrt\right)$

Warburg Open (W_O)

This element describes the impedance of a finite-length diffusion with reflective boundary. It is described by the following equation:

:$Z_ = \frac \coth\left(B \sqrt\right)$

References

Electrochemistry

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