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

General equation

The Warburg diffusion element (Template:Math) 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:

$Z_{\mathrm{W}}=\frac{A_{\mathrm{W}}}{\sqrt{\omega }}+\frac{A_{\mathrm{W}}}{j\sqrt{\omega }}$

$|Z_{\mathrm{W}}|=\sqrt{2}\frac{A_{\mathrm{W}}}{\sqrt{\omega }}$

where

• Template:Math is the Warburg coefficient (or Warburg constant);
• Template:Mvar is the imaginary unit;
• Template:Mvar 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:

$Z_{\mathrm{O}}=\frac{1}{Y_0}\tanh \left(B\sqrt{j\omega }\right)$

where $B=\frac{\delta }{\sqrt{D}},$

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

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

$Z_{W_{\mathrm{S}}}=\frac{A_{\mathrm{W}}}{\sqrt{j\omega }}\tanh \left(B\sqrt{j\omega }\right)$

Warburg Open (W_O)

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

$Z_{W_{\mathrm{O}}}=\frac{A_{\mathrm{W}}}{\sqrt{j\omega }}\coth \left(B\sqrt{j\omega }\right)$

References

Template:Electrochem-stub