Chemistry: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
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:
- [math]\displaystyle{ {Z_\mathrm{W}} = \frac{A_\mathrm{W}}{\sqrt{\omega}}+\frac{A_\mathrm{W}}{j\sqrt{\omega}} }[/math]
- [math]\displaystyle{ {|Z_\mathrm{W}|} = \sqrt{2}\frac{A_\mathrm{W}}{\sqrt{\omega}} }[/math]
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:
- [math]\displaystyle{ {Z_\mathrm{O}} = \frac{1}{Y_0} \tanh\left(B \sqrt{j\omega}\right) }[/math]
where [math]\displaystyle{ B=\tfrac{\delta}{\sqrt{D}}, }[/math]
where [math]\displaystyle{ \delta }[/math] 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:
- [math]\displaystyle{ Z_{W_\mathrm{S}} = \frac{A_\mathrm{W}}{\sqrt{j\omega}} \tanh \left(B \sqrt{j\omega}\right) }[/math]
Warburg Open (WO)
This element describes the impedance of a finite-length diffusion with reflective boundary.[4] It is described by the following equation:
- [math]\displaystyle{ Z_{W_\mathrm{O}} = \frac{A_\mathrm{W}}{\sqrt{j\omega}} \coth\left(B \sqrt{j\omega}\right) }[/math]
References
- ↑ "Equivalent Circuits - Diffusion - Warburg". http://www.consultrsr.net/resources/eis/diffusion.htm.
- ↑ "Electrochemical Impedance Spectroscopy (EIS) - Part 3 – Data Analysis". http://www.ecochemie.nl/download/Applicationnotes/Autolab_Application_Note_EIS03.pdf.
- ↑ "EIS Spectrum Analyser Help. Equivalent Circuit Elements and Parameters". http://www.abc.chemistry.bsu.by/vi/analyser/parameters.html.
- ↑ "EIS Spectrum Analyser Help. Equivalent Circuit Elements and Parameters". http://www.abc.chemistry.bsu.by/vi/analyser/parameters.html.
 |
