Elsevier

Solid State Ionics

Volume 188, Issue 1, 22 April 2011, Pages 104-109
Solid State Ionics

Harnessing evolutionary programming for impedance spectroscopy analysis: A case study of mixed ionic-electronic conductors

https://doi.org/10.1016/j.ssi.2010.10.004Get rights and content

Abstract

A modified Genetic Programming (GP) method has been developed for the analysis of impedance spectroscopy data. It gives a functional form of the distribution function of relaxation times (DFRT) in the sample. The evolution force is composed of lowering the discrepancy between the model's prediction and the measured data, while keeping the model simple in terms of the number of free parameters. The DFRT that the program seeks for has the form of a peak or a sum of several peaks. All the peaks are known mathematical functions (e.g., Gaussians). The user can let the program search for many types of peaks or to limit the search. Finding a functional form of the underlying DFRT has two main assets. (a) DFRT is unique and (b) a functional form makes it possible to develop a physical model and compare it to the function. In addition, if more than one peak is present and each peak can be related to a different phenomenon, the peaks can be directly separated for further analysis. The analysis method is demonstrated using synthetic data as well as experimental data of Gd0.1Ce0.9O1.95 (GDC).

Introduction

Impedance spectroscopy is an effective tool for the analysis of the electrical response of different materials. The challenge using this characterization tool lies within the analysis method, and the number of methods is large [1]. The most common approach for analyzing impedance spectroscopy in electroceramics is constructing an equivalent circuit. It is typically composed of series of RC elements, with many variations on this theme [2]. Equivalent circuit solutions can be categorized as a “direct approach” analysis, since the user compares the experimental data to a prediction of a given model. The model is a parametric representation of the system, and the Holy Grail becomes now to find physical meanings to the various parameters. In a non-parametric analysis, one can find the underlying distribution function of the relaxation times (DFRT) directly using, for example, Fourier transforms with some filtering [3], [4], [5]. The problem can be put into a Fredholm equation (of the first or second kind), which is a classical ill-posed inverse problem. Both of these approaches have their advantages and disadvantages. Construction of equivalent circuits may seem simpler but as the system is more complex, the number of free parameters grows. Consequently, the equivalent circuits in almost all real systems have a number of possible structures with different parameters that can yield exactly the same impedance spectrum. Some a-priori assumptions must be made in order to choose the appropriate circuit for the system examined. The main advantage of the “inverse approach” is that less a-priori assumptions about the system have to be made. However, the latter analysis techniques provide a point-by-point, non-parametric evaluation of the DFRT, rather than a functional form of it. Since the ultimate goal of extracting a DFRT from the data is to gain some physical interpretation of the system, a non-parametric DFRT comes short as compared to a parametric function in this regard [6].

One can, however, enjoy the traits of both approaches and seek a DFRT that is based on known functions. Using additional a-priori knowledge, e.g. that the distribution function is positive, smooth and drops to zero at (plus and minus) infinity, the DFRT can be constructed from a series of peaks. Based on this approach, evolutionary programming techniques were introduced in order to seek for the best distribution function that its convolution fits the experimental data [7], [8].

Our new analysis method is based on evolutionary programming techniques which are inspired by biological evolution. The evolutionary programming technique we have developed here is a modified Genetic Programming (GP). This approach includes a population of candidate solutions (models) to an optimization problem that evolves towards a better solution over generations. In each generation, each one of the proposed models in the population is assigned with parameter values that generate its best possible fit. The evolution pressure that changes the model population is based on (a) the fitness of a model's convolution with a known kernel to the data, and (b) the penalty for complexity, as manifested by the number of free parameters in the model. The impedance is written as:Z(ω)=Rpol0G(τ)K(ω,τ)dτwhere G(τ) is the DFRT and K(ω,τ) is a known kernel. We use the Debye kernel which is defined as:K(ω,τ)=11+iωτ.

Representing the impedance in the logarithmic scale for convenience:Z(log(ω))=RpolΓ(log(τ))1+iωτd(log(τ)).

The “fitness function” measures the difference between the calculated and experimental Z, avoiding over-fitting in a way that will be described below. This function ranks the resulting solutions and the most optimal ones are allowed to evolve and form the next population. In each generation, one random model is also allowed to evolve to the next generation, to increase the survival chances of mutations. The fitness function is designed to assess both the best fitted solutions and the complexity of the model, which here means preferential for model functions with less free parameters. To avoid over-fitting, the user has to enter two sets of data, taken at identical conditions. The program finds the parameters for each model using one set of data, but computes the fitness function using a linear combination of the two sets. Hence over-fitting to one set does not increase the calculated fitness.

Previous work was based on a very general approach where the model was built from a selection of basic mathematical functions and terminals [8]. By putting together the basic functions and terminals a DFRT can be built and evolved. The problem that arose using the previous version was that in some cases it created functions that could not be separated into several simple parts, i.e., the resulting distribution functions could have few parameters (low complexity) but a complex functional form. Thus, an updated version has been developed where the basic model is based on assemblage of known functions of peaks. One of the benefits of this analysis method is to separate easily different processes, described by known functions that can be treated analytically [8], [9].

The updated version of our Impedance Spectroscopy Genetic Programming (ISGP) program is based on selection of symmetric and non symmetric mathematical peaks. The ISGP program contains a “bank” of several known distribution functions commonly used in impedance spectroscopy analysis, including: Gaussian, Lorentzian, Hyperbolic Secant, Kirkwood–Fuoss and Cole–Cole (all with 3 free parameters per peak); Pearson VII and Havriliak–Negami (with 4 free parameters per peak) and Pseudo Voigt with 5 free parameters. It is also possible to choose an asymmetric form of some peaks. A linear combination of the peaks is used to create the DFRT. Each mathematical form of a peak can have a different physical nature. By putting together a series of peaks, an analytic distribution function can be built. The evolution is created by adding or subtracting a peak, by changing the peak type or by matching better parameters to the same model.

The first stage of the analysis procedure is checking the validity of the data using Kramers–Kronig (K–K) transforms [10], [11] . Since complex impedance data can be distorted by experimental artifacts, using K–K transforms should give valuable input on the validity of the data. A filter can be introduced in the program that ranks the results close to the high and low frequency limits with less weight to the overall fitness. This has been discussed broadly elsewhere [7].

After checking the validity of the data, it is first normalized by the maximum value of real Z (Z′) in order to fulfill threshold requirements of the program. The next stage is to let the ISGP code run for some generations (the number of which is chosen by the user) and then analyze the resulting model. Another feature of the program is the presentation of a discrepancy–complexity plot of all the generated models over the generations [12]. The discrepancy is determined by the (log) variance of the experimental data as compared to the theoretical data convoluted from the DFRT. The complexity is defined, as mentioned above, as the total number of free parameters. The program is designed to find the best model that fits the data with minimum parameters. The discrepancy–complexity plot demonstrates that starting from a certain complexity, adding more parameters does not improve the fitness. Therefore, the program will usually choose a model of this certain complexity, rather than adding another peak which will increase the complexity. We try below to demonstrate the abilities of this novel technique using a case study of Gd doped ceria (Gd0.1Ce0.9O1.95, GDC).

The resulting DFRT holds the information about the sample. However, many researchers are accustomed to look at the resistive and reactive parts of the sample's response to the ac stimulus. Here are the guidelines on how to translate the DFRT results into this alternative view. The DFRT is comprised of several peaks; each corresponds to a process in the material. The i-th peak is around a central relaxation time τi; has an area Si and a characteristic peak width of ΔτI, see Fig. 1 for an illustration. Additionally, the data have been normalized by Rmax. Then RmaxSi corresponds to the resistive part of the i-th process; τi/RmaxSi corresponds to the reactive part of the i-th process; and, using the simplest error propagation scheme, Δτiτi corresponds to (ΔRiRi)2+(ΔCiCi)2 (where “R” and “C” here denote the resistive and reactive parts respectively). It is not possible to determine from the results of a single IS experiment alone, which part is more responsible for the expansion of the peak, i.e., which “element” is more “distributed”.

Diffusion phenomena are usually analyzed by equivalent circuits with Warburg [13] or Warburg-like elements [14], [15], [16]. The impedance response of mixed ionic-electronic conductors (MIEC) commonly results in a half-tear-drop-shaped arc in the Cole–Cole plot. This unique shape is commonly related to diffusion processes. Other systems that exhibit similar behavior are anodes in SOFCs [4]. The impedance expression for a simple Warburg element is given by:Zw=Rwtanhiωτwiωτwwhere Rw is the Warburg resistance and τw is the Warburg diffusion process parameter. The τw corresponds to ld2/Di with ld the effective diffusion thickness and Di the effective diffusion coefficient of species i.

A simulation of the theoretical Warburg element with added white noise was carried out using our analysis program. The white noise is generated independently for both the real and imaginary parts. It contains an additive contribution and a multiplicative contribution that are calculated independently for each data point. The additive contribution is distributed randomly between (typically) plus and minus 0.15% of the maximum of real Z, and the multiplicative contribution is distributed randomly between (typically) plus and minus 0.5% of the value of each data point. This procedure gives quite realistic noise, similar to what we get in real measurements.

The DFRT of the Warburg element with the above mentioned white noise (Fig. 2a) is characterized by two closely positioned peaks (Fig. 2b). The unique shape of the DFRT can help identify diffusion processes even when the impedance spectra do not show it clearly. In addition, since the resulting DFRT suggests the presence of the two processes, each process can be analyzed individually.

Section snippets

Experimental procedure

Gd doped ceria (GDC) was prepared via Glycine Nitrate Process (GNP) [17]. The GNP is a self combustion synthesis used mainly to produce nano-sized metal oxide powders. Commercial powders of the metal nitrates were purchased from Sigma Aldrich Ltd: Gd2(NO3)∙6H2O (99.99%, Fluka, USA), Ce(NO3)∙6H2O (≥ 99.0%, Aldrich, France), glycine (99%, Sigma, Korea). Proper amounts of the metal nitrates were dissolved in distilled water. The mole ratio between the glycine and the total nitrates was set to 0.5.

Results and discussion

Experimental impedance results of GDC at various temperatures and different oxygen partial pressures did not show clearly Warburg shape impedance, probably since other processes overlap the main one. However, the resulting DFRTs do show clearly two closely positioned peaks that change central frequency and relative strength (i.e., peak area) between them as a function of temperature and oxygen partial pressure. The next goal would be to determine whether each peak is a characteristic of a

Conclusions

We have developed a novel analysis technique for impedance spectroscopy measurements, that finds a functional (parametric) form of the distribution of relaxation times. This is a modification of our recently published Genetic Programming technique, where here we limit the program to search for functions that are combined of known mathematical peak functions. It is inherently safe against over-fitting. The novel ISGP approach has been used for a MIEC. This has been done using both synthetic data

Acknowledgements

Partial financial support of the Goldberg Fund for Energy Research and of the Charles Krown Research Fund is gratefully acknowledged. S.H. wishes to thank the Marcus Foundation. S.B. wishes to thank the Center for Absorption in Science- Ministry of Immigrant Absorption for their support.

References (21)

  • B.A. Boukamp

    Solid State Ionics

    (2004)
  • J. Jamnik et al.

    Electrochim. Acta

    (1999)
  • L.A. Chick et al.

    Mater. Lett.

    (1990)
  • J.R. Macdonald

    Impedance Spectroscopy

    (1987)
  • H. Schichlein et al.

    J. Appl. Electrochem.

    (2002)
  • A. Leonide et al.

    J. Electrochem. Soc.

    (2008)
  • J. Banys et al.

    Phys. Rev. B

    (2002)
  • Y. Tsur et al.

    Rare Met. Mater. Eng.

    (2006)
  • S. Hershkovitz et al.
  • A.B. Tesler et al.

    J. Electroceram.

    (2010)
There are more references available in the full text version of this article.

Cited by (82)

View all citing articles on Scopus
View full text