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:

${\displaystyle {Z_{\mathrm {W} }}={\frac {A_{\mathrm {W} }}{\sqrt {\omega }}}+{\frac {A_{\mathrm {W} }}{j{\sqrt {\omega }}}}}$

${\displaystyle {|Z_{\mathrm {W} }|}={\sqrt {2}}{\frac {A_{\mathrm {W} }}{\sqrt {\omega }}}}$

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:

${\displaystyle {Z_{\mathrm {O} }}={\frac {1}{Y_{0}}}\tanh \left(B{\sqrt {j\omega }}\right)}$

where ${\displaystyle B={\tfrac {\delta }{\sqrt {D}}},}$

where ${\displaystyle \delta }$ 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:

${\displaystyle Z_{W_{\mathrm {S} }}={\frac {A_{\mathrm {W} }}{\sqrt {j\omega }}}\tanh \left(B{\sqrt {j\omega }}\right)}$

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:

${\displaystyle Z_{W_{\mathrm {O} }}={\frac {A_{\mathrm {W} }}{\sqrt {j\omega }}}\coth \left(B{\sqrt {j\omega }}\right)}$