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 NorFERM-2008, Gol

Worth to remember R: resistance, unit: W r: resistivity, W cm C: capacitance, F e: permittivity, F cm-1 A L NorFERM-2008, Gol

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

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 NorFERM-2008, Gol

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 NorFERM-2008, Gol

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 NorFERM-2008, Gol

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 NorFERM-2008, Gol

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 NorFERM-2008, Gol

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 NorFERM-2008, Gol

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

Real impedance spectra The spectrum can be fitted by using: NorFERM-2008, Gol

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

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

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

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

Softwares for fitting of impedance spectra ZView (Scribner Associates) EqC for Windows (Bernard Boukamp / WisseQ) Others?? NorFERM-2008, Gol

Grain boundaries in ionic conductors NorFERM-2008, Gol

Grain boundaries in ionic conductors The brick layer model S.M. Haile, D.L. West, J. Campbell, Journal of Materials Research 13 (1998) 1576 NorFERM-2008, Gol

Grain boundaries in ionic conductors 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 NorFERM-2008, Gol

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

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

Grain boundaries in ionic conductors Transport will be preferred along parallel grain boundaries compared to that through grain interiors C1 ~ Cbulk R1 ~ Rgb|| NorFERM-2008, Gol

Grain boundaries in ionic conductors C1 ~ Cbulk R1 ~ Rbulk NorFERM-2008, Gol

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

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 It is also possible to study electrode responses under DC bias NorFERM-2008, Gol

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 NorFERM-2008, Gol

Finite length diffusion elements Finite length Warburg element (Short terminus) Finite space Warburg element (open terminus) Warburg element: CPE with n =0.5 NorFERM-2008, Gol

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

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 NorFERM-2008, Gol

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

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