Electrochemical Impedance Spectroscopy

Part 1: Polarization Resistance:Familiar parameter measured in a new way

June 6, 2008

Copyright © M. L. Taylor `

Objective

• The purpose of this lecture series is to generate a set of notes on Impedance Spectroscopy which are easy to use yet powerful enough that a researcher will be able to decide whether or not to pursue the technique more in depth.

• All figures are drawn by Matthew Taylor of the Georgia Institute of Technology, except where noted.

• This work may be used for any purpose, so long as reproduced figures are attributed to the author.

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Copyright © M. L. Taylor `

What is Electrochemical Impedance Spectroscopy?

• Electrochemical reactions are those that involve electron transfer.

• Corrosion Scientists are primarily interested in this type of phenomena.

• Impedance is a measure of resistance to response to an outside stimulus, such as a fluctuation in potential, and has both real and imaginary parts

• A Spectrum of data is collected by collecting measured impedances at individual frequencies and combined to form a spectrum of data.

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Copyright © M. L. Taylor `

Why would I want to use EIS? (I have DC)

• DC methods drive the system from equilibrium

• Reaction parameters are time-dependent and may be altered during DC. Observing the system tends to change it.

• There is more data avaliable (Rp as opposed to Rp+Rs can be measured)

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Excitation System Response

E I

E I

E

E E

log(I)

I �

DC Technique Input Output

Potentiostatic / Pulsed Potential

Cyclic Voltammetry

Potentiodynamic Scan

Galvanostatic/Pulsed Current

Copyright © M. L. Taylor `

Introduction of the Equivalent Circuit

• Describe a system in terms of impedance

• not a model, but an analog

• Possible to describe a variety of systems

• acoustics

• electrochemical charge transfer reactions

• fluids in a container

• mechanical materials systems

• Practically any system has an impedance associated with it

5

Analogous Systems

M

F

C Rm

M

Rm

Cm

F

Mechanical Electrical

Mass Inductor

Spring Capacitor

Normal Force Electrical Potential

Dashpot Resistor

Copyright © M. L. Taylor `

Polarization Resistance

• Useful for predicting a corrosion rate under steady-state conditions

• Commonly determined from Taefel slopes on the potentiodynamic scan

• Our model circuit / system for consideration

6

Copyright © M. L. Taylor `

DC determination of Polarization Resistance

• Measure Rp (Polarization Resistance) by scanning potential away from Ecorr

• In DC, the circuit describing the corroding surface (for a simple case) is nothing but a resistor. The Impedance being measured is entirely real.

• From Ohm’s law, we apply an overpotential and measure the current change, deriving Rp

• Any capacitance in the system may not be measured, as a capacitor acts as an open circuit under DC conditions!

• The solution Resistance is measured as a part of Rp and must be accounted for.

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Inet =!

Rp

Rp

Copyright © M. L. Taylor `

AC Determination of Polarization Resistance

• Impedance has real and imaginary components, both of which depend on the AC frequency.

• Solution resistance is measured separately from polarization resistance

• Capacitance of the ionic double layer may be measured

• Measured by applying a small amplitude AC signal over a range of appropriate frequencies

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Rp

Rs

Soluion BulkIonic

Double LayerMetal

(conductor)

Cdl

Figure: Randles Cell

Copyright © M. L. Taylor `

Simple Passive Circuit Elements

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ZRi (!) = RiRi

ZCi (!) = ! j

!C

Cdl

Element Impedance Symbol

Resistor

Capacitor

j =!"1Note:

Copyright © M. L. Taylor `

Rules for Circuit Analysis

• Impedances in series add directly

• Impedances in parallel add inversely

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Z1

Z2 Zseries = Z1 + Z2

Z1

Z2

Zparallel =1

1Z1

+ 1Z2

Combination Circuit Equation

Series

Parallel

Copyright © M. L. Taylor `

Impedance of the Randles Cell

• consider the capacitor and Rp in parallel

• A little complicated, but we can multiply by the complex conjugate to of the bottom to seperate real and imaginary (multiplied by j) terms.

• Adding Rs and then Separating into Z’ (real) and Z’’ (imaginary), we have:

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Rp

Rs

Cdl

Zp =1

1Rp

+ 1!j

!Cdl

Zp =Rp

1 + !2CR2p

! j!CdlR2

p

1 + !2C2dlR

2p

Z !! = !!CdlR2

p

1 + !2C2dlR

2p

Z ! = Rs +Rp

1 + !2CR2p

Copyright © M. L. Taylor `

Continued

• Z’ and Z’’ are related to |Z| by the pythagorean theorem, and to ϕ by the tangent function

• Plotting |Z| versus frequency yields a plot, from which graphical extraction of analog parameters may be performed. Along with ϕ vs. f, this is known as a Bode Plot.

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|Z| =!

Z !2 + Z !!2! = tan!1

!Z ""

Z "

"Z = Z ! + jZ !!

log(f), hz

|Z|,

! RP

RS

f = 2!"

Copyright © M. L. Taylor `

Graphical Evaluation of the Randles Cell

• The high frequency intercept is equivalent to the solution resistance, Rs

• The low frequency intercept is equivalent to the polarization resistance plus the solution resistance

• Capacitance may be determined

• ωmax is the frequency where -Z’’ is at its maximum

• This geometrical - mathematical technique can be applied to any combination of RC circuits.

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!max =1

RpCdl.

Z', Real Impedance, !

-Z'',

Im

agin

ary

Im

pedance, !

Low Freq

uency

High

Freq

uenc

y

"max

Electrochemical Impedance Spectroscopy

Part 2: Advanced Circuit Elements

June 20, 2008

Copyright © M. L. Taylor `

Overview

• Part 1: Dive in with polarization resistance and simple circuit analysis

• DC vs. AC techniques

• Simple analog circuits

• Resistance Polarization by AC methods

• Part 2: The heavy basis for EIS: Linear Systems Theory and how we actually measure spectra

• Measurement of Impedance

• Advanced Circuit elements

• Geometrical Extrapolation*

• Part 3: History of EIS, experimental and analysis considerations and pitfalls, and data validation (KK transforms, stability diagrams)

• Part 4: Reaction Mechanism Determination

15

Copyright © M. L. Taylor `

Refresher/Clean-Up

• Electrochemical systems can be described by analog circuits

• Impedance is a complex measure of resistance to change in the system, related by ohm’s law

• Individual processes may be delineated from one another using EIS

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RP (Ω)Corrosion Rate

(mm/y) Example

20 14 Mild Steel / Strong Acid

500 0.6 Mild Steel / Natural water

10,000 0.03 Mild Steel / Inhibited Water

1,000,000 0.0003 Passive Metal

Copyright © M. L. Taylor `

Why EIS? (II)

• The power of EIS arises from:

• (i) it is a linear technique and hence the results are readily interpreted in terms of Linear Systems Theory;

• (ii) if measured over an infinite frequency range, the impedance (or admittance) contains all of the information that can be gleaned from the system by linear electrical perturbation/response techniques;

• (iii) the experimental efficiency(amount of information transferred to the observer comparedto the amount produced by the experiment) is extraordinarily high;

• (iv) the validity of the data is readily determined using integral transform techniques (the Kramers–Kronig transforms) that are independent of the physical processes involved.

17

Macdonald. Reflections on the history of electrochemical impedance spectroscopy. Electrochimica Acta (2006) vol. 51 (8/9) pp. 1376-1388

Copyright © M. L. Taylor `

Advanced Passive Circuit Elements

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j =!"1Note:

ZL = j!L Li

ZW = ! ! (1" j)#

"!i

Z =1

(j!A)1!!

(Qi,!i)

Element Impedance Symbol

Pseudo-Inductor

Warburg

Constant Phase Element

Copyright © M. L. Taylor `

Pseudo Inductance

• No physical basis for inductance in an electrochemical system

• Yet it is observed!

• More on this later

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Copyright © M. L. Taylor `

Warburg Diffusion Impedance Element

• Discovered by Warburg

• Describes diffusion controlled electron transfer reactions

• σ can be used to calculate the diffusivity of the element in question, knowing more about the system.

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Randles Cell with Warburg Impedance Element Included

Z'

-z''

45º

ZW = ! ! (1" j)#

"

Copyright © M. L. Taylor `

Constant Phase Element

• A Generalized Impedance Element

• For values of α=0, the CPE behaves as a perfect Capacitor and A=C

• For α=2.0, the CPE behaves as a perfect inductor and A=L

• Sometimes, 1-α is written as α, so it is importatnt to pay attention, as it changes the actual capacitance measured.

• Explained by inhomogeneities in the surface.

• Surface roughness

• uneven distribution of charge

• Observed more often than perfect capacitance, but easily misinterpereted.

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(Qi,!i)

Z =1

(j!A)1!!

Cdl = Adl

!!

!!

max

"!!1

0 20 40 60 80 100

-50

-40

-30

-20

-10

0

effect of α

adjust for true capacitance

Copyright © M. L. Taylor `

Linear Systems Theory

• Criteion for a linear system:

• The system is described by linear functions

• The system must be causal

• Response is generated only due to an imposed stimulus

• The system is reversible

• Removal of stimulus causes the system to relax to its previous state

• Reversing the singal to the starting point gives no hysteresis

• The system is Finite

• No Infinite Values of impedance are allowed

22

E

log(I)

e0

excitation (E)

output (i)

Copyright © M. L. Taylor `

OR

• Constraints of Linear Systems Theory

•(i) the response of the system must be described by linear (differential) equations and hence the superposition principle must hold

•(ii) the system must be stable, i.e., upon removal of the perturbation the system should relax to its initial state

•(iii) the system must be causal, that is, the system must not produce a response before t = 0 (the time at which the perturbation is applied)

•(iv) the impedance must be finite (physical systems cannot contain singularities in the evolution of their properties).

23

Macdonald. Reflections on the history of electrochemical impedance spectroscopy. Electrochimica Acta (2006) vol. 51 (8/9) pp. 1376-1388

Copyright © M. L. Taylor `

X-Y Single Beam Oscilloscope Method

• Potentiostatic EIS

• Choose potential of interest

• Δe is chosen small for LST (around 10 mV)

• Plot V(t) vs. I(t) as a parametric function

• (as seen on the Gamry insturments in our lab)

• Geometric examination extracts parameters to calculate immittance (Y)

• Immittance is the inverse of impedance.

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Potential

Current

! "

#e

#i#i'

e0

|Z| =!e

!i

sin (!) =!i!

!i=

"! #

!i!e

Lissajous figure

Z ! = |Z|cos (!)Z !! = |Z|sin (!)

Copyright © M. L. Taylor `

Dual Beam Oscilloscope Method

• Measure the potential drop across a “resistor”, Rs

• Display input potential and output potential response on a dual-beam oscilloscope

• The phase angle can be directly observed by measuring peak to peak.

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|Z| =Rs|e (j!) |eR (j!) |

a

input (potential)response (potential drop)

b

|e|eR

Z ! = |Z|cos (!)Z !! = |Z|sin (!)

Copyright © M. L. Taylor `

Curve Fitting

• Complex Nonlinear Least Squares

• Change a parameter until “goodness of fit” decreases.

• Search for the closest achievable fit

• Caution: Fits are not always what they seem!

• Excellent (within 1 order of magnitude) first guess

• Degenerate Circuits are possible

• Local solutions may be found, which are not the global solution!

• Know thy system

• Make your model physical

26

Copyright © M. L. Taylor `

Degenerate Circuits

• Analogs must be based on physical phenomena, otherwise, BAD assumptions can be made

• A circuit without physical basis is worthless from a prediction standpoint.

• Degenerate circuits exist for all systems, where data will fit equally well to any of the (incorrect) degenerate cases as it will to the correct one.

• Even worse than degenerate cases are circuits consisting of too many elements. Adding enough elements will fit practically ANY curve.

27

Fletcher, S., Tables of Degenerate Electrical Networks for Use in the Equivalent- Circuit Analysis of Electrochemical Systems. Journal of The Electrochemical Society, 1994. 141(7): p. 1823-1826.

3 circuits with two time constants which can be equally well fit to an experimental spectrum

Copyright © M. L. Taylor `

Advanced Extrapolation Techniques

• Passive materials with very high polarization resistances may stymie completion of the capacitive semicircle seen in the Randles circuit due to time requirements of measuring at low frequencies

• Curve Fits are questionable because an excellent first guess is required for a proper fit.

• Geometric method

• works well for one time constant.

• See attached excel sheet for automated extrapolation

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!"#####$

!%#####$

!&#####$

!'#####$

!(#####$

!)#####$

#$

)#####$

(#####$

'#####$

#$ %#####$ )######$ )%#####$ (######$ (%#####$ '######$ '%#####$ &######$

*+,-*.,$/,01,-$

2*1*$

!"######$

!%######$

!&######$

!'######$

!(######$

#$

(######$

'######$

&######$

%######$

!'######$ #$ '######$ %######$ )######$ *######$

+,-+.-,/01$

+02/03$

4,/,$

567$

82/0390+/9$

Extrapolated data for a real system

Z'

-Z''

Copyright © M. L. Taylor `

End of Part II

29

Copyright © M. L. Taylor `

Reading• Book Chapters

• Mansfeld, Florian Analytical Methods in Corrosion Science and Engineering (2005) Ch. 13 pp. 463-506

• Books

• Cottis et al. Corrosion Testing Made Easy: Electrochemical Impedance and Noise. (1999)

• Barsoukov and J.R. MacDonald, Impedance Spectroscopy: Theory, Experiment, and Applications. (2005) pp. 608

• Baboian. Electrochemical Techniques for Corrosion Engineering. (1986)

• Peer Reviewed Papers

• Macdonald, Digby D. Reflections on the history of electrochemical impedance spectroscopy. Electrochimica Acta (2006) vol. 51 (8/9) pp. 1376-1388

30

Copyright © M. L. Taylor `

Reading: Solartron Analytical

31

TB/ANALYTICAL/001 High frequency, high current impedance spectroscopy: Experimental protocols enabling measurement up to 1MHz at high current densitiesBy John Harper, Mike Rust, Brian Sayers and Andrew SavageGet it here...

TB/ANALYTICAL/2Use of auxiliary channels for impedance analysis: Detecting failure mechanisms within a fuel cell / battery stackBy J Harper and B SayersGet it here...

TB/ANALYTICAL/3Parallel Channel CombinationsBy J Harper and B SayersGet it here...

TB/ANALYTICAL/004Solartron CellTest® SystemImpedance measurement techniquesBrian SayersGet it here...

Technote 04Identification of Electrochemical Processes by Frequency Response AnalysisBy C GabrielliGet it here...

Technote 06An Introduction to Electrochemical Impedance MeasurementBy N D Cogger and N J EvansGet it here...

Technote 10Frequency Response AnalysisBy N D Cogger and R V WebbGet it here...

Technote 17Understanding Electrochemical CellsBy A M KaufmannGet it here...

Technote 24Use and Applications of Electrochemical Impedance TechniquesBy C GabrielliGet it here...

Technote 26Analysis and Interpretation of EIS Data for Metals and Alloys

By F MansfeldGet it here...

Technote 29The Application of Impedance Spectroscopy to Cementitious SystemsBy W. J. McCarterGet it here...

Technote 31Electrochemical Impedance Spectroscopy (EIS) for Battery Research and DevelopmentBy Hong Shih and Tai-Chin LoGet it here...

Technote 33The Potentiodynamic Polarization ScanBy D G Evans and L L ScribnerGet it here...

1470 TechInstrumentation for the Characterization of Energy Storage Devices and Multi-Cell SystemsBy B Sayers and A HintonGet it here...

Coatings TechDetermination of Coating Adhesion Using Electrochemical Impedance SpectroscopyBy A HintonGet it here...

Advanced Instrumentation for Solid State ApplicationsBy A Hinton and B SayersGet it here...

Beyond the Limits: 1296 Dielectric InterfaceBy A Hinton and B SayersGet it here...

Advanced Instrumentation for Civil Engineering ApplicationsBy A Hinton and B SayersGet it here...

Advanced Instrumentation for Bioimpedance MeasurementsBy A Hinton and B SayersGet it here...

ImpedTech 1Impedance Measurement Techniques: Sine CorrelationBy B Sayers and A HintonGet it here...

Copyright © M. L. Taylor `

Reading: Gamry Instruments

32

Electrochemical Applications

Basics of EIS

Equivalent Circuit Modeling in EIS

EIS on Painted Samples

Multiplexed EIS on Painted Samples

DC Corrosion Techniques

Determination of Double-LayerCapacitance from a CPE

Rapid Electrochemical Assessment of Paints (REAP)

Electrochemical Instrumentation

Potentiostat Fundamentals

Compliance Voltage - How Much is Enough?

Reference Electrodes

Care of Vycor® Porous Glass Frits

Trouble-Shooting Your Gamry Potentiostat with the Universal Dummy Cell 3

Measurement of Small Electrochemical Signals

Accuracy Contour Plots

EIS Measurement of a Very Low Impedance Lithium Ion Battery

Verification of Low Impedance EIS Using a 1 mOhm Resistor NEW!

Open Source Scripting

Understanding iR Compensation

You can also find a list of Frequently Asked Questions here.