IMPEDANCE SPECTROSCOPY

IMPEDANCE SPECTROSCOPY By Dr. Mehran Javanbakht

Fundamentals of Electrochemical Impedance Spectroscopy

I.Introduction: physics & electrotechnics • Definition & properties of impedance • Simple RC circuits and their spectra • Measurement principles and graphical analyses

How to characterize a two-pole (an electrical system)? Case A: a steady state current-voltage curve, I(U), existsat any moment.

How to characterize a two-pole (an electrical system)? Case B: the current-voltage curve, I(U) depends also on time, t The system can be characterizedthrough the time-dependence of the current: I(t) – U(t) relations are analyzed. Possibilities: transient response (following a jump or pulse)ac methods, „frequency” response (sinusoidal perturbation) Passive, linear electrical „two-poles”; I(k*U)=k*I(U)) are considered.

Operational definition of impedance: Stimulus: U(t)= Uac∙sin(ωt) Response: I(t) =Iac∙sin(ωt+ φ) Impedanceis defined as Z (Uac/Iacand φ) (Since I(k*U)=k*I(U)), the Uac/Iac is not dependent on Uac.)

Impedance (at one frequency): • is defined as Z (Uac/Iacand φ), complex number • ZUac/Iac· eiφ =Zabscos (φ) + i· Zabssin(φ) Euler’s formula with ZabsUac/Iac and i -1 • admittance: Y1/Z(Yabs=1/Zabs and φY= -φZ) • immittance = common term for impedance and admittance

Impedance (as function of frequency, (ω=2πf )): • it is called a spectrum (typically 10-3/s <w< 10+7/s); • representations • r∙eiφ r∙(cos(φ)+i∙sin(φ)) = Re + i∙Im; Nyquist: Im(Z) vs Re(Z) • ln(r∙eiφ)  ln(r)+i∙φ • Bode: lg(Zabs) vs lg(f ) andφ vs lg(f )

Impedance of a network of RCL elements can be calculated just in the same way as the resistance of a network of resistors. • Impedance: • of serially connected elements: Zs = Z1+Z21/Ys= 1/Y1+1/Y2 • of parallely connected elements: Yp = Y1+Y21/Zp= 1/Z1+1/Z2 • Impedance: • of a resistor: ZRR • of a capacitor: ZC 1/(iwC) • of an inductor: ZL iwL

How do the spectra look like? Examples of simple circuits:

Resistance:

Capacitance:

Rs-Cs:

Cp||Rp Semicircle Characteristic frequency ω0=1/RpCp Time constant τ0=1/ω0

Symmetries:

Rs-Cp||Rp

(Rs-Cs)||Rp

Circuits of different topologies may have the same impedance function. There is no unique connection of circuit and spectrum.

Rp||Cp sequences yield semicircle sequences; they are merged if the RC time constants are close to each other.

Simplest way: using sine-wave voltages FRA, lock-in amplifier ac voltage source: sine-wave generator Umeas: U(t)= Uac∙sin(ωt) Imeas: I(t) =Iac∙sin(ωt+ φ)

Typical measurement setups: for resistive systems for capacitive systems (dielectric spectroscopy)

Simple way of analysis: plotting and determining characteristic values Cp=1/(ω0*Rp) Rs Rs+Rp Use various representations.

Nyquist vs Bode representations: ln(r∙eiφ)  ln(r)+i∙φ • Advantages (both are good): • Nyquist: „structures” are better seen • Bode: complete documentation of the data

Simplest way: using sine-wave voltages FRA, lock-in amplifier ac voltage source: sine-wave generator Umeas: U(t)= Uac∙sin(ωt) Imeas: I(t) =Iac∙sin(ωt+ φ)

Turnkey EIS systems are available for 15-60 k€.

Mechanistic studies and identification of processes • Goal: Identification of the appropriate model • (equivalent circuit AND the underlying physico-chemical processes) • Measure Z(ω) as function of E, ci, T, etc • Repeat • Construct model with reasonable assumptions; calculate its impedance function (perhaps expressed as an equivalent circuit, also as function of E, ci, T, etc) • Estimate the model’s parameters (e.g. by NLLS fitting) • Until • a. the measured and calculated Z(ω) spectra are similar to each other; • b. the E, ci, T, etc dependencies are correct (not self-contradictory).

Determination of values of parameters (when the model is already established) Cp: Interfacial capacitance Structure of the interface (double layer, adsorption) Bulk - interface Rs: Bulk conductivities General characterization Dielectric spectroscopy Rp: Charge transfer resistance Electrochemical kinetics Corrosion

Measurement modes: Multiple frequency mode: impedance spectrum measurement (at constant Edc) – followed by the determination of the parameters • Single frequency mode: with scanned Edc; examples: • ac voltammetry (for characterization of charge transfer); • capacitance measurements with (high) f and with (slowly) scanned Edcrequires the a priori knowledge of the equivalent circuit;

Bulk resistance • Resistance is determined through impedance spectrum measurements if: • a single resistance cannot be measured (only a network’s impedance); • typically high resistance materials which are difficult to be contacted; • jointly with the determination of permittivity; • polymer membranes, ionic conductors, porous structures.

Time evolution of the impedance spectra of a physically dryingstyrene-acrylateself-standing resin film in 0.1M KNO3 solution (100 kHz - 1 Hz), Lendvay-Győrik et al, 2007.

Interfacial capacitance 1. Calculation of adsorbate coverages: thus • 2. Determination of „zero points” (of space charge layers): • Metal / solution of a binary electrolyte of low concentrationHg (Au, Ag) in 1-100 mM NaF solution (two mobile charge carriers) • II: Metal / extrinsic semiconductor junction n – or p doped semiconductor metallized or a semiconductor electrode in aqueous solution (one fixed and one mobile charge carrier)

Interfacial capacitance Determination of „zero points” (of space charge layers): Model: The distribution of the mobile charges (ions or electrons or holes) is determined by the electrostatic potential and the thermal motion: Poisson - Boltzmann equation Expressed quantity: space charge layer capacitance vs potential.

Determination of „zero points”, A: Metal / solution of a binary electrolyte of low concentration (two mobile charge carriers): Capacitance has a minimum at the pzc (potential of zero charge - at which the ion accumulation nearby the metal, in the solution vanishes). „Gouy-Chapman minimum”, Hg in NaF solution, Grahame (1947)

Determination of „zero points”, B: Metal (or electrolyte) / extrinsic semiconductor junction (one fixed and one mobile charge carrier) Mott-Schottky plot (ZnO, Freund & Morrison, 1989) 1/C2 vs E: determination of n0 and Efb (flatband potential – at which the space charge layer in the semiconductor vanishes)

Parallel resistance • Parallel resistance – interpreted as a charge transfer resistance • exchange current density is calculated ( = kinetics information) • typical use: average corrosion rate is calculated • for details, see many application notes

Determination of Rp - corrosion tests Fe in H2SO4 (5..100mM) at corrosion potential (Lendvay-Győrik et al, 2000) TiCxNy film (on steel) in Na2SO4 (0.5M) at function of time (Senna et al, 2000)

Determination of Rp - corrosion tests, inhibitor studies Fe in 1M HCl with and without 1 mM oct-1-yn-3-ol (octynol, inhibitor), at corrosion potential (Lendvay-Győrik et al, 2003) c b a a: without 1-octynol b: with 1 mM 1-octynol c: with 1 mM 1-octynol, after an anodic treatment

Coating tests: • An ideal polymer, insulating coating is capacitive. • Corrosion + transport through the pores – causes a shunt term - C||R. EIS response of a pipeline coating in 5% NaCl at 65°C, L. Grayet al (2003) in: D. Loveday et al, JCT CoatingsTech, 2005

Technical issues: 1.EIS can be used for characterizing stable systems only.A good practice for testing stability is to repeat the measurements (e.g. with decreasing then increasing frequencies). Kramers-Kronig test may help. 2. Decrease noise. Use Faraday-cage. Use the preamplifier supplied with the potentiostat. Connect an oscilloscope to the E output of the potentiostat to monitor noise level. 3. Troubleshooting:Check the system by measuring the impedance spectra of resistors and dummy cells of similar characteristics to the systems studied.

4. Cells for high frequency (>1..10 kHz) impedance measurements: a. Low impedance reference electrode should be usedb. Avoid cells of low „feedback ratio” c. Ensure uniform current density distribution

Traditional, „clean” cells may have bad hf behaviour

a. Low impedance reference electrode: Avoid high resistance solution paths between cell and reference electrode. To shunt the high resistance paths, use a capacitively coupled auxiliary reference electrode (C≈1-10μF)

b. Low feedback ratio: Do not place the counter electrode „far away” from the W and R

c. Uniform current density distribution From www.mpmtechnologies.com, MPM Technologies, Inc., State College, PA, USA

5. Calibration:with dummy cells having Z(ω) similar to the system studied + with two resistors (approx. Rsol,1andRsol,2) for the hf accuracy.

Fitting of impedance spectrum: a demo; measurement: Ir(100) in 0.1M HCl, 0.1V vs SCE, model: □,◊ measured, x,+ calculated absolute values and phase angles

• Always plot the measured and fitted curves together in various representations (Bode is the best for this) • Calculate & plot the difference of the measured and fitted points – try to get rid of the systematic deviations • Inspect errors of parameters

How to present data? • 1. Raw data: measured impedance spectra • Nyquist (r∙eiφ r∙(cos(φ)+i∙sin(φ)) = Re + i∙Im): Im(Z) vs Re(Z), „structures” are better seen • Scale of Im(Z) and Re(Z) must be identical • Bode (ln(r∙eiφ)  ln(r)+i∙φ): • lg(|Z|) vs lg(f ) and φ vs lg(f ), for documentation of the data • Other representations like log(Im(Z)) vs log(Re(Z)): avoid • 2. Processed impedance data (with or without fitting) a. Normalize to unit area (Ohm•cm2)b. Subtract series (solution) resistance → interfacial Zic. Zi → interfacial admittance, Yi(ω) → interfacial capacitance, Ci(ω)Ci(ω) Yi(ω) /i ω=1/ [i ω(Z(ω)-Z(ω))]is also a complex function → Bode, Nyquistd. Plot together fitted and measured spectra

IMPEDANCE SPECTROSCOPY