EIS, Diffusion, and Multi-Layer Coatings!

In the last issue of the Gamry newsletter, we talked about the Porous Bounded Warburg impedance and its application to the Rotating Disk Electrode. This issue we are going to talk more about equivalent circuits representing diffusion.

We recently came across two articles (Ref 3, 4) that talked about EIS measurements on an electrode coated with a multilayer film. The effect of diffusion of a reactant through the two-layer film was discussed. The conclusions presented by the articles surprised us, and they may surprise you also!

Warburg Diffusion Elements - a Quick Review

The Warburg Circuit Elements for Diffusion

Most equivalent circuit modeling programs include the three basic "Warburg" or Diffusion circuit elements. In the Gamry Echem Analyst^TM we call them the Warburg, Bounded Warburg, and Porous Bounded Warburg (from left to right.) You may also see them called the W, T, and O circuit elements. There is a short description of these elements in the last issue of this Newsletter, and in Gamry's "EIS Primer" or in Reference 1 or 2.

The Warburg ( W, left ) element represents diffusion from a nearly infinite cell to a flat electrode. The Bounded Warburg ( T, center ) represents diffusion within a thin layer of electrolyte, such as electrolyte trapped between a flat electrode and a glass microscope slide. The Porous Bounded Warburg ( O, right ) represents diffusion through a thin layer of electrolyte, such as electrolyte trapped between an electrode and a permeable membrane covering it. It is also called a Bounded or Finite Warburg with an Open Boundary.

The Problem

Both articles (Ref 3, 4) consider the geometry shown to the left. The two layers can be permeable films, paint, or perhaps a thin layer of liquid. Transport through the films is assumed to be by diffusion, and each layer will have its own unique thickness and diffusion coefficient. Freger ( Ref 3 ) assumed that the bulk solution was homogeneous, while Diard, et. al. ( Ref 4 ) explicitly considered diffusion in the bulk solution as well as in the layers.

Prior to the publication of these articles, we assumed, as did many others, that since the two layers were applied serially to the electrode, we could simply place the equivalent circuit for Layer 1 in series with that for Layer 2. What the authors point out is that the boundary condition ( at the Layer 2-Layer 1 interface ) for solving the diffusion problem in Layer 2 involves the diffusion of material through Layer 1! There can be an interaction and a synergy for some values of the thickness of the layers and the relative diffusion coefficients! The only way material can enter Layer 2 from the Bulk Solution is by diffusing through Layer 1!

The Solution, and a Model

The equations quickly become quite complicated, with hyperbolic tangent functions and nested fractions. They will not be repeated here! Fortunately, Freger drew a simple model that involves only the Bounded Warburg circuit elements (

, T and Porous Bounded Warburg

, O ), but with a twist.

The Equivalent Circuit of the 2-Layer Model of Freger

Usually the parameters of each of the components in a circuit are not related to each other. In Freger's model, however, the parameters are related to each other, as is shown in the model on the right. Each of bounded Warburg elements requires two parameters, a Y parameter [ units: S-s^(1/2) ] that controls the magnitude of the impedance and a B parameter [units: s^(1/2) ] that reflects the frequency at which the bounded Warburg impedance deviates from a 45° line on the Nyquist plot. Freger's model only uses 4 parameters total, not the 8 (4 Y parameters and 4 B parameters) that you might expect from the model. The Y parameter for the upper left Bounded Warburg is a combination of Y1 and Y2 values.

Version 5.0 of the Echem Analyst^TM added a script that allows you to simulate the response of a equivalent circuit model. (Ver 4 users may download it here at no charge!) This script was used to simulate Freger's model, with the parameters identified as shown in the schematic, above. We were very pleased to find out that if two parameters in a model are given the same name, they are treated as the very same parameter! Gosh, our programmers are good! But just remember that you must give parameters distinct and different names if you wish them to be treated independently!

The model shown above has a total of 5 independent parameters: Y1, Y2, B1, B2, and "Y2*Y2/Y1". It can be shown that K²*D in Freger's equation 10 is the same parameter as the Y used in the calculation of Z for a Bounded Warburg.

The last parameter, "Y2*Y2/Y1", is just a name -- the Echem Analyst does not do the calculation for you. You must do the calculation yourself and enter the value of "Y2*Y2/Y1". Although Freger's model can be used for simulation, it can not be used to fit a data set because the calculation is not done automatically. We have certainly put that on our "wish list," however!

A Disbonded Coating

We simulated a disbonded coating using Freger's model for diffusion through the 2-layer film in parallel with the capacitance of the intact coating. The coating (Layer 1) was assumed to be 20 µm thick and to have a coating capacitance of 300 pF-cm^-2 (dielectric constant= 6.78). A water layer (Layer 2) with a thickness of 2 µm was assumed to lie under the coating.

We assumed that the actual water content of the coating was negligible, so that the dielectric constant (and thus the capacitance) remained unchanged. We also made the assumption that oxygen was at a low concentration (1x10^-3 Mol/L), and that the actual concentration in the coating was only 1/100-th of that in the water layer and in the bulk of the solution. The coating was assumed to be fairly "tight" so that the diffusion coefficient in the coating was only 0.0001 of that in water. The Table, below, shows the properties and the B and Y parameters of each of the layers.

Parameters Used to Simulate Freger's Model
Layer	Layer Thickness	Diffusion Coefficient	Concentration	B	Y
	µm	m²-s^-1	mol-L^-1	s^1/2	S-s^1/2
1, coating	20	9x10^-14	0.01x10^-3	67	1.15x10^-8
2, water	2	9x10^-10	1x10^-3	0.067	1.15x10^-4

We used the new simulation capability of the Echem Analyst Ver. 5.03 to generate the EIS spectrum from 10 mHz to 100 kHz. This simulated data is shown in the plot, below. You can see on this Bode plot that the phase remains close to 90 deg over most of the spectrum. The magnitude shows a hint of a plateau at about 1 Mohm, where the phase peaks at about 1 kHz. Except for these hints, the curve looks pretty much like that of a capacitor.

Simulated data for our "Disbonded Coating."
The equivalent circuit for the REAP2CPE model is shown in the inset.

The Fit to Another Equivalent Circuit

Had we obtained this data for a coating in the laboratory, we would probably tried to fit the data to the pseudo-standard "paint" model. We used the Gamry "REAP2CPE" model, which is identical to the "paint" model except that the capacitors in the model have been replaced with a more general "Constant Phase Element," or CPE. The schematic for the REAP2CPE model is shown in the inset, above.

The results of the fit are also shown on the plot. The fit to the Bode magnitude is excellent. The phase (green) is quite good also, but shows a little bit of deviation. The parameter values from the fit are shown at the right. The "Goodness of Fit" number makes the fit quite believable. The average error in the data is about 2.5%, based on the Goodness of Fit number. The uncertainties in the fit parameters is generally acceptable also. Both CPEs are really quite close to capacitors (the exponents are VERY close to 1.0!) Had we obtained this data in the laboratory, we would be quite satisfied with the fit to this equivalent circuit.

A Possible Mis-Interpretation

Given the fit to the REAP2CPE model, above, how might we interpret the fit results? One common interpretation for the change in coating capacitance from 300 pF initially to 369 pF (from the fit, above) is that water had permeated the coating and changed the effective dielectric constant. The Brasher-Kingsbury equation ( ref 5 ) can be used to calculate the volume percentage of water in the coating (%v):

where C is the coating capacitance after exposure (369 pF) and C_O is the initial capacitance (300 pF). In this equation, 80 is the dielectric constant of water. From this change in capacitance, we calculate about a 4.7% (v/v) water uptake in the coating, based upon the Brasher-Kingsbury equation. In reality, the model used to simulate these data presumed NO water uptake by the coating, just a slow permeation of water through the coating, and collecting in a thin layer underneath the coating. This gives weight to Freger's assertion that examples in the literature exist where a two-layer model may have been misinterpreted as simple water uptake by the coating.

Summary

We have used Freger's model for a two-layer coating to simulate the EIS response of the coating. We found that the presence of water layer under and otherwise intact coating DID change the EIS response, contrary to our prior conceptions. We also found that a commonly used, equivalent circuit model fit the data to an acceptable degree, making it easy to misinterpret the EIS results. Although EIS is a powerful tool, it can not be used alone to answer all coating questions.

Electrochemical Impedance of Multi-Layer Electrodes