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:
Z W = A W ω + A W j ω {\displaystyle {Z_{\mathrm {W} }}={\frac {A_{\mathrm {W} }}{\sqrt {\omega }}}+{\frac {A_{\mathrm {W} }}{j{\sqrt {\omega }}}}}
| Z W | = 2 A W ω {\displaystyle {|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, 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 is defined as:
Z O = 1 Y 0 tanh ( B j ω ) {\displaystyle {Z_{\mathrm {O} }}={\frac {1}{Y_{0}}}\tanh \left(B{\sqrt {j\omega }}\right)}
where B = δ D , {\displaystyle B={\tfrac {\delta }{\sqrt {D}}},}
where δ {\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 (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. It is described by the following equation:
Z W S = A W j ω tanh ( B j ω ) {\displaystyle 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. It is described by the following equation:
Z W O = A W j ω coth ( B j ω ) {\displaystyle Z_{W_{\mathrm {O} }}={\frac {A_{\mathrm {W} }}{\sqrt {j\omega }}}\coth \left(B{\sqrt {j\omega }}\right)}
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