Complex models in Electrochemical Impedance Spectroscopy

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Complex models in Electrochemical Impedance
Spectroscopy
From a solid state perspective

• Bernard A. Boukamp

41th Heyrovský Discussion 8th Symposium on
Electrochemical Impedance Spectroscopy
Tret, 15-19 June, 2008.
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33 Years ago

• Stanford, Li-ion conducting oxides
• Ionic conductivity
• Grain boundary resistance

RM gtgt ZSample
RM
Osc.
Sample
Two-phase lock-in amplifier
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Analysis in the old days.
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Outline (optional)

• 1. New developments in data analysis
• 2. Do we really need Data Validation
• 3. Old strategy that still works
• 4. Complex systems with simple solutions
• 5. Simple systems turning complex
• With the help of a little time domain
• Mixed conductivity model
• Where simple CNLS-analysis doesnt work
• Putting all your data in one bag
• Wake up call (conclusions)

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Advances since then

• Analysis developments
• Joop Schoonman, multi parameter fit (U.
  Utrecht)
• J. Ross Macdonald, LEVM
  (U. Chapel Hill)
• Yaw-Wen Hu, on a PDP-8 lab-E (8k memory!)
• Marquardt-Levemberg algorithm, (RQ)Q
  (Stanford)
• Equivalent Circuit (main frame system, 1981)
• EquivCrt, MS-DOS with pre-analysis (1985) ?
  EqCwin
• Simplex (Nelder-Mead)
• Genetic Algorithm (1998) and more

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Analysis methods

• Marquardt Levenberg algorithm (seventies)
• powerful but complex tool (derivatives!)
• yields error estimates
• requirement good starting values
• NelderMead or Simplex algorithm (see e.g.
  1)
• simple programming, only function evaluation
• no real requirement on starting values
• local minimum problem, restarts necessary
• Issue of proper choice of weight factors
• No weight factor is a weight factor choice too

1 J-L. Dellis and J-L. Carpentier, Solid State
Ionics 62 (1993) 119.
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General object function
Function to be minimized with Z(?,?j) the
model function andZk-2 the weight factor.
Set of (good) starting parameters? Genetic
Algorithm! genes, generations, mating, crossover,
mutations combination with Gauss-Newton
minimization
2 T.J. VanderNoot, I.Abrahams, J. Electroanal.
Chem., 448 (1998) 17. 3 M. Yang, X. Zhang, X.
Li, X. Wu, J. Electroanal. Chem., 519 (2001) 1.
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Data analysis, recent developments

• Genetic algorithm
• Starting point large group of individuals
  (genes)
• using randomly created parameter sets, Xj (?)
• parameters restricted to a predetermined range
• New generations created by cross-over of the
  genes
• Selection rules mostly to partly random
• Best individuals are kept in a quality gene
  pool (niche)
• New individuals by random mutations
• After each generation test of G-N convergence
• for the optimum set Xopt(?) (best individual).

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Genetic algorithm

• Genetic algorithm, continued
• Standard object function, S(?), to be minimized
• Fitness of an individual Fj 1/(Sj (?) 1)
• Cross-over operations
• Mutation rate critical for rapid success, Pm ?
  0.3 0.4
• Selected parameter(s) xk in Xopt(?) replaced by
  random number xk
• Failure in improvement Fopt after number of
• generations
• ? construct new generation (except for
  niche)

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Recent improvement in GA
4 J. Yu, H. Cao, Y. He, A new tree structure
code for equivalent circuit and evolutionary
estimation of parameters, Chemometrics Int.
Lab. Syst. 85 (2007) 27
Example of tree structure, but what is the
difference with the standard CDC? R1(C1R2(C2R3
Q1 ) )
Claim better than LEVM, but it takes a Hydra
supercomputer at 1.2 TFlops, average solution
time (for 999 runs) 2 min.!
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Literature search
Web of Science ISI Key words Genetic
Algorithm AND impedance First result 161
hits Excluding anten 98 hits Actually EIS
related 4 hits All four publications on
demonstration of programs !!
For now, we continue with CNLS-analysis
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2.
Do we really need Data Validation ?
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Analyze this!
?2CNLS 1.3 10-5
Simple (RQ)(RQ) ??
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Data analysis ?
J. Ross Macdonald The un-analysed data is not
worth generating.
Is the un-validated data worth
analyzing? Bernard A. Boukamp
But
The Kramers-Kronig check!
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Data validation
Kramers-Kronig relations (old!)
Real and imaginary parts are linked through the
K-K transforms
Response only due to input signal
Response scales linearly with input signal
State of system may not change during measurement
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Putting K-K in practice
Relations, Real ? imaginary Imaginary ? real
Problem Finite frequency range extrapolation of
dispersion ? assumption of a model.
5 M. Urquidi-Macdonald, S.Real D.D.
Macdonald, Electrochim.Acta, 35 (1990) 1559. 6
B.A. Boukamp, Solid State Ionics, 62 (1993) 131.
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Linear KK transform
Linear set of parallel RC circuits
?k Rk?Ck
Create a set of ? values ?1 ?max-1 ?M
?min-1 with 7 ?-values per decade
(logarithmically spaced).
If this circuit fits the data, the data must be
K-K transformable!
7 B.A.Boukamp, J.Electrochem.Soc, 142 (1995)
1885
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Actual test
Fit function simultaneously to real and imaginary
part Set of linear equations in Rk, only one
matrix inversion! Display relative residuals
It works like a K-K compliant flexible
curve
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Analyze this again!
?2KK 6.5 10-8
PZT, Pb(Zr0.53Ti0.47)O3
?2CNLS 1.3 10-5
Simple (RQ)(RQ) ??
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3.
Old strategy that still works! or The art of
de-convolution.
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Fitting to the Extreme
PbZr0.53Ti0.47O3//gold electrodes N2 (1.2 Pa O2),
580C, 0.1 Hz 65 kHz
?2CNLS 6.2?10-8 ?2K-K 5.8?10-8
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Subtraction of (Qdiel Rel)
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The hidden R(RQ)
With electrode response subtracted - (CRW)
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After subtraction of Cdl
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The proof of the pudding
One of the possible equivalent circuits
CDC (QRR(RQ)(CRW)) ?ps2 6.2?10-8
err. Q1 3.2?10-9 S.sn 7.9 -n1 0.886
- 0.6 R2 2800 ? 0.01 R3 2290
? 1.7 R4 3420 ? 1.8 Q5 2.88
?10-7 S.sn 6.0 -n5 0.668 - 1.0 C6
1.74 ?10-8 F 0.5 R7 68100 ? 1.7 W8
4.19 ?10-7 S.s1/2 1.6
Electrode response (Randles circuit) in analogy
with liquid electrolyte with redox couple.
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Consistency of model
Results of two samples with different thickness
Arrhenius graph of Cdl and Wdiff (ionic
electrode response) . N2 (Po2 1.2 Pa). Above
TC Warburg is activated with EAct 65?3 kJ.mol-1.
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Pure PZT at 600?C
Ionic conductivity is constant, electronic
conductivity ? (p O2)1/4
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Electronic conduction
Air measurements from 600C down, N2 (1 Pa O2)
rapid cool-down to RT, then going to 600C Eact
110?10 kJ/mol
Decrease by a factor 100 from air to N2 in the
low temperature range!
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Ionic conduction
High temperatures, (almost) independent of
Po2. Low temperature decrease of factor 10 from
20 kPa to 1 Pa O2
Note the good correspondence between the two
samples!
8 B.A. Boukamp, M.T.N. Pham, D.H.A. Blank,
H.J.M. Bouwmeester, Sol.Stat.Ionics 170 (2004) 239
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Dielectric response
Dielectric response is not a pure capacitor. It
shows CPE behaviour with 0.85 ? n ? 1
Curie peak is not affected by reduction.
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4.
Complex systems with simple solutions.
Porous La(1-x)SrxCo(1-y)FeyO3-? cathodes.
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Impedance of LSCF electrode
La0.6Sr0.4Co0.2Fe0.8O3-? Screen printed symmetric
cell design with yttria doped ceria interlayers
on TZ3Y. Cells prepared by ECN (master thesis by
Arjen Giesbers, 2004) N2 ? 10-4 atm O2.
Gerischer!
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Gerischer impedance

• What is the Gerischer Impedance?
• Semi-infinite diffusion with side reaction
• But finite dc-value!
• Distinctly different from FLW, low frequency
  side
• Semi-infinite diffusion Warburg, high
  frequency side
• Already postulated in 1951 Heinz Gerischer,
• Why is it important for SOFC anodes and cathodes?

9 H. Gerischer, Z. Phys. Chem. 198 (1951) 216.
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Simple Gerischer
Model of a CEC electrochemical reaction in a
liquid electrolyte with an inert electrode.
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Solid state ionics analogue
Possible mechanisms leading to a Gerischer type
response.
The ALS model 10
Slow adsorption diffusion 11
Fick-2
10 S.B.Adler, J.A. Lane B.C.H. Steele,
J.Elchem.Soc. 143 (1996) 3554. 11 R.U.
Atangulov and I.V. Murygin, Solid State Ionics 67
(1993) 9-15.
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Alternative explanation
Leaky transmission line. Series resistance, r1,
in ?/m. Parallel capacitance, c, in F/m.
Parallel resistance, r2, in ?m. For semi-infinite
circuit
With k (r2c)-1 and Z0 (r1/c)1/2
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Comparison FLW - Gerischer
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Gerischer parameters
ALS r0/cv 0.64
ALS cv -0.30
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SOFC anode Gerischer
pO2 (pH2/pH2O) dependence of parameters. R ?
(pO2)0.15, consistent with n-type k ? (pO2)-0.36
Thin porous ceramic anode (La0.7Ca0.3Cr0.2Ti0.8O3-
?) on YSZ. 20?m thick, n-type
conductivity. Simulation with double fractal
Gerischer.
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Full expression of DFG
adding a new twist
Double fractal Gerischer
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Ti-YSZ/Ni cermet
Cooperation with Peter Holtappels, EMPA
(Switzerland) Ni/YSZ-Ti20 Cermet electrode
(symmetrical cel). Shape similar to Gerischer
in low frequency part,
Thin active layer!
but in-between Gerischer and FLW. Subtract high-f
dispersion. Analyses of the resulting
dispersion FL-Gerischer?
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Finite Length Gerischer
Slope n
Slope n ? ?
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Mathematics of FLG
Finite Length Gerischer
Double fractal Gerischer
Necessary math
Tanh in real and imag.
Can be easily analysed in a spread-sheet program
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FLDFG vs. DFG
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pH2, pH2O dependence
Problem sample not very stable
Note B L/?D
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5.
Simple systems turning complex Li-diffusion in
LixCoO2 Well-defined layers made with PLD
RF-sputtering
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Real cathode LixCoO2
Expected response
RF-sputtered layer of LixCoO2
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Thin film LixCoO2
Current response of a 0.75?m RF-film to
sequential 50mV potential steps from 3.80V to
4.20V.
IS of a RF-film electrode (?) fresh (?)
charged (?) intermediate SoCs. () CNLS-fit.
Freq. range 0.01 Hz 100 kHz.
Peter J. Bouwman, Thesis, U.Twente 2002.
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6. With a little help from
Time domain measurement
Discrete Fourier transform
Correction / simulation for t??
X0 leakage current.
Impedance
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Step Experiment
Sequence of 10 mV step Fourier transformed
impedance spectra, from 3.65 V to 4.20 V at 50 mV
intervals. Fmin 0.1 mHz
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CNLS-fit of FT-data
R1 550 0.5 R2 49 10 Q3, Y0
6.8?10-3 12 ,, n 0.96 8 O4, Y0
0.047 1.5 ,, B 30 2.4 T5, Y0
0.028 2.9 ,, B 5.9 2.9
Circuit Description Code R(RQ)OT Fit
result ?2CNLS 3.7?10-5
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Bode Graph
Double logarithmic display almost always gives
excellent result !
Bode plot, Zreal and Zimag versus frequency in
double log plot
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6. Mixed conductivity model
Jamnik Maier Model System

• Discretisation of transport in solid and across
  interfaces
• Transport coordinate
• Reaction coordinate
• Chemical capacitance

Jamnik Maier, Phys. Chem. -Chem. Phys. 3 (2001)
1668
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Intercalation model
Arguments of tanh (FSW) and coth (FLW)
functions are equal
Time constant
J. Jamnik, Sol. Stat. Ionics 157 (2003) 19
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6.
Where simple CNLS-analysis doesnt work
Estimate the Distribution of Time Constants
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Distribution of Time Constants
No model involved! Just solve with
Group of Prof. Ellen Ivers-Tiffée (U.
Karlsruhe), use imaginary part Substitution of
variables
Fourier Transform
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FT - deconvolution
Discretization and Fourier transform leads to
Use a filter window to minimize end problems
compare with Zre!
Inverse Fourier
There is a hitch extension of frequency range
needed, use single time constant model.
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Example of transformation
H. Schichlein, A.C. Müller, M. Voigts, A. Krügel
and E. Ivers-Tiffée J. Appl. Electrochemistry 32
(2002) 875-882.
Part-II 21/36
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Advantage of DRT
H. Schichlein et al.,
Relaxation distributions (a) calculated from
impedance data measured under open circuit
conditions and (b) simulated from the physical
submodel both for variation of oxygen partial
pressure at the cathode.
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9.
Putting all your data in one bag

• Combining all data sets in one optimization
  routine with as second external parameter, e.g.
• Temperature
• Partial pressure
• Polarisation
• Concentration
• Etc.

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Multi CNLS-fit
J.R. Dygas , K. Pietruczuk, W. Bogusz, F.
Krok Joint least-squares analysis of a set of
impedance spectra Electrochimica Acta 47 (2002)
2303

• Analysis of impedance of polymer compounds.
• Problem scatter of n-values of CPE-elements.
• Simultaneous fit of T-data with constraints for
  CPEs, e.g.
• fixed value
• linear T-dependence
• Significant reduction in parameter noise.

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Multivariate analysis
Thilo Hilpert (Baden-Baden) Multivariate
analysis of multiple impedance spectra Solid
State Ionics 177 (2006) 15771581
Reducing fit-noise with incomplete
dispersions, extended temperature relations
(Arrhenius, etc.)
Example, resistance and derivatives
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10. Wake up call !

• Conclusions
• New analysis methods are coming, but advantage?
• Data quality check essential for complex
  systems!
• Simultaneous data fits can improve parameters
• Time domain measurement-frequency domain
  analysis!
• Good theory mixed conductors, but exceptions?
• Alternative DRT analysis

• Watch also these developments (in progress)
• Finite element approach for complex structures
• Well-defined micro-electrodes (Prof. Jürgen
  Fleig)
• ????

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Acknowledgement
My thanks to all those scientists who are
improving our understanding of impedance
spectroscopy and who are finding new ways and new
methods for analysis and interpretation.
University of Twente
But specially to Prof. Henny Bouwmeester, Mr.
Arjen Giesbers, Dr. Peter Holtappels, Mr. Maarten
Verbraeken, Prof. Ellen Ivers-Tiffée, Dr. Andre
Weber
This presentation was made possible through
financial support from the Dutch Ministry of
Economic Affairs through the EOS-SOFC programme.
The sunken land of Driene
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