1. Impedance spectroscopy - with emphasis on applications towards grain boundaries and electrodics
Harald Fjeld
Department of Chemistry, University of Oslo, FERMiO, Gaustadalléen 21, NO-0349 Oslo, Norway
NorFERM-2008, Gol

2. Outline What is impedance?
Passive electrical circuit elements and their characteristics
Impedance spectroscopy
Tools of the trade
Impedance spectrometers
Softwares for fitting of data
Applications
Grain boundaries in ionic conductors
Electrodics
NorFERM-2008, Gol

3. Worth to remember R: resistance, unit: W r: resistivity, W cm
C: capacitance, F
e: permittivity, F cm-1 A L
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4. What is impedance?
Impedance is a general expression for electrical resistance, mostly used for alternating currents
For a sinusoidal current, the voltage is given according to
U = U0 sin wt
..and the following current is given
according to
I = I0 sin (wt + q)
t: time
f: frequency
w: angular frequency = 2pf
wt: phase angle
q: phase shift
NorFERM-2008, Gol

5. What is impedance?
Impedance is a general expression for electrical resistance, mostly used for alternating currents
From Ohm’s law, the impedance is given by the ratio of voltage and current. This equals the magnitude of the impedance, Z, when represented in a two-dimensional room spanned by real and imaginary vectors. In addition, we also want to know the phase shift (q) X R q Z
Z*(w) = Z’ + j Z’’ = ZRe + jZIm = R + j X
Nyquist plot / Cole-Cole plot
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6. Admittance
Instead of impedance, we may use the inverse, i.e. admittance
Z: impedance Y: admittance
R: resistance G: conductance
X: reactance B: suceptance
Z*(w) = R + j X
Y*(w) = G + j B
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7. Passive electrical circuit elements
An alternating current can be phase shifted with respect to the voltage
The phase shift depends on what kind of sample the current passes
To describe the response from a sample on the alternating current, we introduce 3 passive circuit elements (R, C and L)
The current and voltage through a resistor, R, is not phase shifted  the impedance is not dependant on frequency
A resistor only contributes to the real part of the impedance
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8. The capacitor The capacitor, C, can store electrical charges
e: permittivity
e0: permittivity of free space
er: relative dielectric constant
Only contributes to the imaginary part of the impedance
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9. The inductor
As opposed to the capacitor, which is an ideal isolator, the inductor is an ideal conductor
Only contributes to the imaginary part of the impedance
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10. The (RQ) circuit
Constant phase elements (CPE) may be regarded as non-ideal capacitors defined by the constants Y and n, and their impedance is given according to
The CPE is very versatile (“a very general dispersion formula”):
If n = 1, the CPE represents an ideal capacitor
If n = 0, the CPE represents a resistor
If n = -1, the CPE represents an inductor
If n = 0.5 the CPE represents a Warburg element
Peak frequency: w0 = (RC)-1
Constant phase element
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11. Impedance spectroscopy in solid state ionics
What: A technique for studying the conductivity of ionic conductors, mixed conductors, electrode kinetics and related phenomena
Features:
Eliminates the need for non-blocking electrodes
The impedance due to grain interiors, grain boundaries and different electrode properties can be measured independently
How:
A small AC voltage (e.g. 10 mV – 1 V) is imposed on the sample over a wide range of frequencies (e.g. 1 MHz – 0.1 Hz), and the complex impedance is measured
NorFERM-2008, Gol

12. Real impedance spectra
The spectrum can be fitted by using:
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13. Tools of the trade Solartron 1260 Freq. range: 10 µHz – 32 MHz
Input impedance: 1 MW
DC bias: up to 41 V
AC amplitude: 5 mV – 3 V (rms)
Prize (2008): ~ 40 k€
Considered as the state-of-the-art impedance spectrometer
Options: can be combined with a potentiostat (1287) or a high impedance interface (1296)
NorFERM-2008, Gol

14. Tools of the trade
HP 4192A
Out of production since 2001, replaced by 4294A (4192A has been observed for sale at ebay)
Freq. range: 5 Hz – 13 MHz
Input impedance: 1 MW
DC bias: up to 40 V
AC amplitude: 5 mV – 1.1 V (rms)
NorFERM-2008, Gol

15. Tools of the trade Novocontrol alpha-A
Can be equipped with different test interfaces for different purposes (in Oslo: ZG4)
Freq. range: 30 µHz – 20 MHz
Input impedance: 1 TW
DC bias: up to 40 V
AC amplitude: 0.1 – 3 V (rms)
Prize (2008): ~ 35 k€
Mainframe
ZG4 test interface
NorFERM-2008, Gol

16. Tools of the trade Hioki 3522-50
A cheap, but OK alternative for ”standard tasks”?
Freq. range: 1 mHz – 100 kHz (+DC)
Input impedance: 1 MW??
DC bias: up to 10 V
AC amplitude: 10 mV – 5 V (rms)
Prize: ??
NorFERM-2008, Gol

17. Softwares for fitting of impedance spectra
ZView (Scribner Associates)
EqC for Windows (Bernard Boukamp / WisseQ)
Others??
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18. Grain boundaries in ionic conductors
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19. Grain boundaries in ionic conductors
The brick layer model
S.M. Haile, D.L. West, J. Campbell, Journal of Materials Research 13 (1998) 1576
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20. Grain boundaries in ionic conductors
The ratio R2 to R1 is dependant on both physical and microstructural properties
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21. Grain boundaries in ionic conductors
Criteria for two distinguishable arcs:
R1 and R2 are comparable in magnitude
The characteristic frequencies of the two arcs are significantly different
w0 = (re)-1
Assuming ebulk = egb leads to
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22. Grain boundaries in ionic conductors
Assuming a sample with ”normal” microstructure (G >> g)
In the case of two semi-circles: sbulk > sgb
Transport in grains is preferred, but the perpendicular grain boundaries are unavoidable
NorFERM-2008, Gol

23. Grain boundaries in ionic conductors
In the case of only one semi-circle:
The resistance associated with this arc may correspond to the bulk, the parallel grain boundaries or a combination
NorFERM-2008, Gol

24. Grain boundaries in ionic conductors
Transport will be preferred along parallel grain boundaries compared to that through grain interiors
C1 ~ Cbulk R1 ~ Rgb||
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25. Grain boundaries in ionic conductors
C1 ~ Cbulk R1 ~ Rbulk
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26. Grain boundaries in ionic conductors
Summary:
Two arcs are observed  sbulk > sgb
Then sbulk = s1 and sgb ~ s2C1/C2
One arc is observed
The resistance associated with this arc may correspond to the bulk, the parallel grain boundaries or a combination
NorFERM-2008, Gol

27. Electrodics
The capacitances associated to the electrode processes are much higher than those of bulk and grain boundaries
In order to investigate electrodes, one should apply “small” amplitudes of the probe signal
For bulk and gb: typically V
For electrodes: typically tens of mV
It is also possible to study electrode responses under DC bias
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28. Possible electrode procesess
Charge transfer
Presuambaly happening on the triple phase boundaries
Dissociative adsorption of H2 and/or O2
Gas diffusion impedance
Gas conversion impedance / gas concentration impedance
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29. Finite length diffusion elements
Finite length Warburg element (Short terminus)
Finite space Warburg element (open terminus)
Warburg element:
CPE with n =0.5
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30. Electrodics: a case study of a complete fuel cell
A large number of different contributions (many parameters to fit)
Some constraints must be given to fit the data to the model
R. Barfod, Fuel Cells 6 (2006) 141.
NorFERM-2008, Gol

31. Limitations of impedance spectroscopy
Many parameters to fit: sufficient amount of data is necessary
Overlapping processes in the frequency-plane may not be separated
In theory, an indefinite number of equivalent circuits can be used to explain a recorded spectrum
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32. Literature and acknowledgments
The impedance course at Risø is acknowledged for inspiration
R. Barfod, A. Hagen, S. Ramousse, P.V. Hendriksen, M. Mogensen, Fuel Cells 6 (2006) 141.
S.M. Haile, D.L. West, J. Campbell, Journal of Materials Research 13 (1998) 1576
NorFERM-2008, Gol

33. NorFERM-2008, Gol

34. Quiz In this room at 19:00 Interesting bonus question!!!
NorFERM-2008, Gol