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 (log |Z| vs. log ω) exists with a slope of value –1/2.
General equation
[edit]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);
- j is the imaginary unit;
- ω is the angular frequency.
This equation assumes semi-infinite linear diffusion,[1] that is, unrestricted diffusion to a large planar electrode.
Finite-length Warburg element
[edit]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.
Warburg Short (WS)
[edit]This element describes the impedance of a finite-length diffusion with transmissive boundary.[3] It is described by the following equation:
Warburg Open (WO)
[edit]This element describes the impedance of a finite-length diffusion with reflective boundary.[4] It is described by the following equation:
References
[edit]- ^ "Equivalent Circuits - Diffusion - Warburg". 22 September 2023.
- ^ "Electrochemical Impedance Spectroscopy (EIS) - Part 3 – Data Analysis" (PDF). Archived from the original (PDF) on 2015-09-15. Retrieved 2023-11-12.
- ^ "EIS Spectrum Analyser Help. Equivalent Circuit Elements and Parameters".
- ^ "EIS Spectrum Analyser Help. Equivalent Circuit Elements and Parameters".