When analyzed according to the procedure outlined by Macdonald, the data set named small Montmorillonite 55 C produces the impedance spectrum plot shown below. In case you are not used to looking at impedance data plotted in this fashion, the analysis frequency starts at the lone point on the right hand side, and increases to the left.
This is not the best data set of the lot, but it is not the worst either. It was chosen to test the procedure based on the facts that it does display an arch and that there is a little non-reproducibility. The later is indicated by the pairs of values at the real impedance (Z') of about 45 Ohm. The dual-values are probably a result of splicing two data sets, taken under slightly different conditions. Because the data set has these minor imperfections, it serves as a test for the robustness of the expectation-maximization (EM) algorithm as well as a test for the 'linear combination of Debye dielectrics' model inferred.
The symmetry of the main arch (centered at Z'~25 W) indicates an impedance element with well-behaved dielectric properties. The real and imaginary components of the immittance (IZ=Z/R0) support this. Show below are the components of the complex immittance for the same data set. The imaginary component was chosen to be subject to relaxation time constant deconvolution (or inversion) using the expectation-maximization (EM) algorithm. The relaxation time-constant distribution produced an expectation, noted in the figure below as "fit". There is very little noticeable difference between the imaginary dielectric coefficient and the expectation step reproduction.
The resulting relaxation time-constant distribution is shown below. The distribution is flat at the high time-constant end (the left hand side of the figure), decreases to about zero around 1 ms, and displays a peak at about 120 ns (t=0.120 ms). The peak half-width is about 12 ns. The longer time constant distribution results from the low angular frequency (below 100 krad/s) trend seen in the above figure. The time constant distribution peak corresponds to the imaginary dielectric coefficient peak seen above around 10 Mrad/s.
If one can project that the line starting up from Z'~47 W in the complex-plane impedance plot (the first figure) is linear, then the process resulting in this structure is diffusion. Mass diffusion results in an complex impedance proportional to w-1/2. This inverse frequency behavior may also be seen in the low frequencies of the imaginary immittance plot, the second figure. Subsequently, the plateau of long relaxation time constants between 1 ms and 1 s is probably due to mass diffusion.
Link to more on modeling diffusion effects
Fitting the immittance model using the EM algorithm allows interpolation of the data to frequencies other than actually measured. Shown below is the data together with the "ideal" real and imaginary immittance. Model equations for the real (I') and imaginary (I") immittance components are given in the lower left hand corner of the figure. The real part of the complex immittance, I', does not result from fitting the model to the real component of the data. The model data were all derived from the a(t) amplitude spectrum shown above and was obtained by fitting the imaginary, I", component of the measurement data set. The amplitude of the model I' data was not altered but the data are offset by the model value at 108 rad s-1. The reason for having to apply this offset is not clear at this time. Nonetheless, the faithful reproduction of the I' and I" trends, even at frequencies not included in the measurement data set, is encouraging.
