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.
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:
{ZW
{|ZW|}=\sqrt{2}
| AW | |
| \sqrt{\omega |
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:
{ZO
where
B=\tfrac{\delta}{\sqrt{D}},
where
\delta
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.
This element describes the impedance of a finite-length diffusion with transmissive boundary.[3] It is described by the following equation:
| Z | |
| WS |
=
| AW | |
| \sqrt{j\omega |
This element describes the impedance of a finite-length diffusion with reflective boundary.[4] It is described by the following equation:
| Z | |
| WO |
=
| AW | |
| \sqrt{j\omega |