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:
{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
- 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
If the thickness of the diffusion layer is known, the finite-length Warburg element[2] is defined as:
{Z_\mathrm{O}} = \frac{1}{Y_0} \tanh\left(B \sqrt{j\omega}\right)
where B=\tfrac{\delta}{\sqrt{D}},
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.[3] 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 (WO)
This element describes the impedance of a finite-length diffusion with reflective boundary.[4] 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
- ^ "Equivalent Circuits - Diffusion - Warburg". 22 September 2023.
- ^ "Electrochemical Impedance Spectroscopy (EIS) - Part 3 – Data Analysis". Archived 2015-09-15 at the Wayback Machine. Retrieved 2023-11-12.
- ^ "EIS Spectrum Analyser Help. Equivalent Circuit Elements and Parameters"
- ^ "EIS Spectrum Analyser Help. Equivalent Circuit Elements and Parameters"