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Constrained Diffusion - The T Element

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The Warburg circuit element is a good place to start thinking about diffusion, but often semi-infinite diffusion is not a reasonable model. The O and T circuit elements are useful models for diffusion when a thin film ( and therefore, finite diffusion ) is involved.


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Diffusion Circuit Elements - T

The T circuit element is characteristic of another type of film -- a film which contains a fixed amount of electroactive substance. The classical "thin layer electrochemistry" cell is an example of such a system. Batteries or supercapacitors also may share this behavior. The common feature is the fixed amount of electroactive material present. Once it has been consumed, it can not be replenished.

In the Gamry Echem Analyst™ software it is called the "bounded Warburg."  This name is more descriptive of what the element really stands for.

Nyquist Plot for a T Element
A Nyquist plot for the T element

The figure to the left shows the Nyquist plot for the T diffusion element. Like the O element, the T element is characterized by two parameters, an "admittance" parameter, Yo, and a "time constant" parameter, B (units: sec ). At high frequency ( f > 2 / B2 ) the T circuit element is indistinguishable from a Warburg impedance! This frequency range is shown in red in the figure. Since the time for a molecule to diffuse across the thin layer is much longer than the period of the AC stimulus applied the electrode does not 'see' that the film or coating is of finite thickness.

At low frequency, the T element looks like an R and a C in series, with R=(B / Yo) / 3.

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Equations for the T element

The equations for the complex admittance ( Y(omega) and complex impedance ( Z(omega) ) are given by the equations below. The T circuit element gets its name from the hyperbolic  tangent  ( tanh[] ) admittance response.

Equations for Y and Z for a T element

Yo has the same definition as for the Warburg impedance. Yo can be used to calculate a diffusion coefficient for the mobile species within the film, coating, or thin layer cell using the same equations. For large values of the argument (the red region of the Nyquist plot, above), the tanh and coth functions both approach unity and the impedance has the same omega dependence as the Warburg.  This region can be used to estimate Yo.

If the thickness of the thin layer is delta, then the constant B is related to this thickness and the diffusion coefficient, D. The parameter B characterizes the time it take for a reactant to diffuse from one side of the layer to the other.

B=delta / sqrt(D)

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The Warburg Circuit Element
Calculating Diffusion Coefficients
Fitting
EIS Data

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REFERENCES
(1) Boukamp, B., "Equivalent Circuit (EQUIVCRT.PAS) Users Manual", Univ. Twente, Enschede, the Netherlands, 1989.
(2) "Diffusion Impedance for Planar, Cylindrical, and Spherical Geometry", T. Jacobsen, K West, Electrochimica Acta, 40(1995), 255.
 

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