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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
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
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Worth to remember
R: resistance, unit: W
r: resistivity, W cm
C: capacitance, F
e: permittivity, F cm-1
L
R r
A
A
Ce
L
A
L
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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
t: time
f: frequency
w: angular frequency = 2pf
wt: phase angle
..and the following current is given
I = I0 sin (wt + q)
q: phase shift
I, U
according to
q: phase shift
time
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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
Z*(w) = Z’ + j Z’’ = ZRe + jZIm = R + j X
Z
j  1
q
Nyquist plot / Cole-Cole plot
R
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Admittance
• Instead of impedance, we may use the inverse, i.e.
admittance
Z: impedance
Y: admittance
R: resistance
G: conductance
X: reactance
B: suceptance
R
G
G2  B2
G
R
R2  X 2
X
B
G2  B2
B
X
R2  X 2
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Z*(w) = R + j X
Y*(w) = G + j B
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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The capacitor
• The capacitor, C, can store
electrical charges
A
A
C  e  e0e r
L
L
e: permittivity
e0: permittivity of free space
7
1.5x10
er: relative dielectric constant
7
1.0x10
-X / W
• Only contributes to the
imaginary part of the
impedance
-6
C = 10 F
Increasing
frequency
6
5.0x10
0.0
1

)
X   wC
0.0
6
5.0x10
7
1.0x10
R/W
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7
1.5x10
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
X  wL
0
-6
L = 10 H
-X / W
-2
Increasing
frequency
-4
-6
0
2
4
R/W
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6
8
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”):

ZQ  Y  jw)
– If n = 1, the CPE represents an ideal capacitor
– If n = 0, the CPE represents a resistor

n 1
– If n = -1, the CPE represents an inductor
– If n = 0.5 the CPE represents a Warburg element
Peak frequency: w0 = (RC)-1
-X / W
100
50
n=1
n = 0.9
n = 0.8
0
0
50
100
R/W
150
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200
Constant phase element
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
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Real impedance spectra
0.20
0.006
6
0.004
0.15
-X / MW cm
5
0.002
0.010
0.10
0.012
0.014
3
0.05
4
2
1
6
0.00
0.00
0.05
0.10
0.15
0.20
R / MW cm
The spectrum can be fitted by using:
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0.25
0
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)
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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)
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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
Mainframe
• DC bias: up to 40 V
• AC amplitude: 0.1 – 3 V (rms)
• Prize (2008): ~ 35 k€
ZG4 test interface
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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: ??
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Softwares for fitting of impedance spectra
• ZView (Scribner Associates)
• EqC for Windows (Bernard Boukamp / WisseQ)
• Others??
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Grain boundaries in ionic conductors
-X / MW cm
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
R / MW cm
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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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Grain boundaries in ionic conductors
-X / MW cm
0.6
0.4
0.2
0.0
0.0
R2
R1
0.2
0.4
0.6
0.8
1.0
R / MW cm
• The ratio R2 to R1 is dependant on both physical and
microstructural properties
NorFERM-2008, Gol
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
w0,bulk
w0,gb

r gb
r bulk
 bulk

 gb
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Grain boundaries in ionic conductors
• Assuming a sample with ”normal” microstructure (G >> g)
• In the case of two semi-circles: bulk > gb
– Transport in grains is preferred, but the perpendicular grain
boundaries are unavoidable
 bulk
1

R1
 gb 
1 g
1 C1

R2 G R2 C2
Cbulk  C1
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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
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Grain boundaries in ionic conductors
 gb
bulk

G
2g
Transport will be preferred along parallel grain
boundaries compared to that through grain interiors
C1 ~ Cbulk
R1 ~ Rgb||
2
R 2  R gb ,perpendicular
g
g
 2R gb ||    2R 1  
G
G
2
C 2  C gb ,perpendicular
G
G
1
 C gb ||    C1  
2
g
g
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2
Grain boundaries in ionic conductors
G
 bulk   gb   bulk
2g
C1 ~ Cbulk
R 2  R gb ,perpendicular
R1 ~ Rbulk
g bulk
 R1
G gb
2
C 2  C gb ,perpendicular
G
G
1
 C gb ||    C1  
2
g
g
NorFERM-2008, Gol
Grain boundaries in ionic conductors
Summary:
•
Two arcs are observed  bulk > gb
Then bulk = 1 and gb ~ 2C1/C2
•
One arc is observed
The resistance associated with this arc may
correspond to the bulk, the parallel grain
boundaries or a combination
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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 0.1 - 2 V
• For electrodes: typically tens of mV
U
• It is also possible to study electrode responses under DC bias
time
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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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Finite length diffusion elements
Finite length Warburg element
(Short terminus)
Finite space Warburg element
(open terminus)
-X/W
40
20
20
0
0
20
40
R/W
60
80
0
100
0
40
-X/W
-X/W
40
20
40
60
R/W
80
100
Warburg element:
CPE with n =0.5
20
0
0
20
40
R/W
60
80
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100
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.
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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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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
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Quiz
• In this room at 19:00
– Interesting bonus question!!!
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