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 ( $Z W$ ) 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

$A W$ 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 ( $W S$ ) for a transmissive boundary, and the Warburg Open ( $W O$ ) for a reflective boundary.

Warburg Short (W_S)

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 (W_O)

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" (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".

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[1] "Equivalent Circuits - Diffusion - Warburg". 22 September 2023.

[2] "Electrochemical Impedance Spectroscopy (EIS) - Part 3 – Data Analysis" (PDF). Archived from the original (PDF) on 2015-09-15. Retrieved 2023-11-12.

[3] "EIS Spectrum Analyser Help. Equivalent Circuit Elements and Parameters".

[4] "EIS Spectrum Analyser Help. Equivalent Circuit Elements and Parameters".

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