• Electrochemical

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    • Abstract: ElectrochemicalImpedance SpectroscopyPart 1: Polarization Resistance:Familiar parameter measured in a new wayJune 6, 2008 Objective• The purpose of this lecture series is to generate a set of notes on

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Impedance Spectroscopy
Part 1: Polarization Resistance:
Familiar parameter measured in a new way
June 6, 2008
• 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.
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
• 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.
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)
Excitation System Response
DC Technique Input Output
Potentiostatic / E I
Pulsed Potential
Cyclic E I
Scan E E
Galvanostatic/ I
Pulsed Current
Copyright © M. L. Taylor `
Introduction of the Equivalent Circuit
• Describe a system in terms Analogous Systems
of impedance
Mechanical Electrical
• not a model, but an analog
• Possible to describe a
M Rm
variety of systems
• acoustics
• electrochemical charge
transfer reactions
• fluids in a container
Mass Inductor
• mechanical materials
systems Spring Capacitor
• Practically any system has
Normal Force Electrical Potential
an impedance associated
with it
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
• Our model circuit / system for consideration
Copyright © M. L. Taylor `
DC determination of Polarization Resistance
• Measure Rp (Polarization Resistance)
by scanning potential away from
• 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.
Inet =
Copyright © M. L. Taylor `
AC Determination of Polarization Resistance
• Impedance has real and imaginary Rp
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
Ionic Metal

Soluion Bulk
Measured by applying a small Double Layer (conductor)
amplitude AC signal over a range of
appropriate frequencies
Figure: Randles Cell
Copyright © M. L. Taylor `
Simple Passive Circuit Elements
Element Impedance Symbol
Resistor ZRi (ω) = Ri
Capacitor ZCi (ω) = −

Note: j = −1
Copyright © M. L. Taylor `
Rules for Circuit Analysis
• Impedances in series add directly
• Impedances in parallel add inversely
Combination Circuit Equation
Series Z1 Z2
Zseries = Z1 + Z2
Parallel Zparallel =
Z1 + 1
Copyright © M. L. Taylor `
Impedance of the Randles Cell
• consider the capacitor and Rp in parallel
Zp = 1
Rp + 1
• A little complicated, but we can multiply by the complex conjugate
to of the bottom to seperate real and imaginary (multiplied by j)
Rp ωCdl Rp
Zp = −j
1+ω 2 CR2
p 1 + ω 2 Cdl Rp
2 2
• Adding Rs and then Separating into Z’ (real) and Z’’ (imaginary), we
Rp 2
ωCdl Rp
Z = Rs + Z =−
1 + ω 2 CRp
1 + ω 2 Cdl Rp
2 2
Copyright © M. L. Taylor `
• Z’ and Z’’ are related to |Z| by the pythagorean theorem, and to ϕ by
the tangent function
Z = Z + jZ Z
φ = tan−1
|Z| = Z 2+Z 2 Z
• 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.
|Z|, !
log(f), hz f = 2πω
Copyright © M. L. Taylor `
Graphical Evaluation of the Randles Cell
• The high frequency intercept is equivalent to the solution resistance,
• The low frequency intercept is equivalent to the polarization
resistance plus the solution resistance
-Z'', Imaginary Impedance, !
• Capacitance may be determined
ωmax = .
Low Fre
Rp Cdl
Z', Real Impedance, !
• ωmax is the frequency where -Z’’ is at its maximum
• This geometrical - mathematical technique can be applied to any
combination of RC circuits.
Copyright © M. L. Taylor `
Impedance Spectroscopy
Part 2: Advanced Circuit Elements
June 20, 2008
• 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
Copyright © M. L. Taylor `
• Electrochemical systems RP (Ω)
Corrosion Rate
can be described by analog
circuits 20 14 Mild Steel /
Strong Acid
• Impedance is a complex
500 0.6
Mild Steel /
measure of resistance to Natural water
change in the system,
related by ohm’s law Mild Steel /
10,000 0.03
Inhibited Water
• Individual processes may be
delineated from one 1,000,000 0.0003 Passive Metal
another using EIS
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.
Macdonald. Reflections on the history of
electrochemical impedance spectroscopy.
Electrochimica Acta (2006) vol. 51 (8/9) pp.
Copyright © M. L. Taylor 1376-1388 `
Advanced Passive Circuit Elements
Element Impedance Symbol
Inductor ZL = jωL Li

ZW = σ × (1 − j) ω
1 (Qi,!i)
Constant Phase Z= 1−α

Note: j = −1
Copyright © M. L. Taylor `
Pseudo Inductance
• No physical basis for inductance
in an electrochemical system
• Yet it is observed!
• More on this later
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.

ZW = σ × (1 − j) ω Randles Cell with Warburg
Impedance Element Included
Copyright © M. L. Taylor `
Constant Phase Element
• A Generalized Impedance Element
1 (Qi,!i)
• For values of α=0, the CPE behaves as (jωA)
a perfect Capacitor and A=C
• For α=2.0, the CPE behaves as a α−1
perfect inductor and A=L Cdl = Adl ωmax
• Sometimes, 1-α is written as α, so it is adjust for true capacitance
importatnt to pay attention, as it changes
the actual capacitance measured. -50
• Explained by inhomogeneities in the -40
surface. -30

Surface roughness
• uneven distribution of charge 0
0 20 40 60 80 100
• Observed more often than perfect effect of α
capacitance, but easily misinterpereted.
Copyright © M. L. Taylor `
Linear Systems Theory
• Criteion for a linear system:
• The system is described by linear
• The system must be causal
output (i)
• Response is generated only due to an
imposed stimulus
• The system is reversible excitation (E)
• 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 e0
• No Infinite Values of impedance are
Copyright © M. L. Taylor `
• 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).
Macdonald. Reflections on the history of
electrochemical impedance spectroscopy.
Electrochimica Acta (2006) vol. 51 (8/9) pp.
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)
#i' #i
• Plot V(t) vs. I(t) as a parametric
• (as seen on the Gamry
insturments in our lab) Potential e0
• Geometric examination Lissajous figure
extracts parameters to
calculate immittance (Y) ∆e Z = |Z|cos (φ)
|Z| =
• Immittance is the inverse of ∆i Z = |Z|sin (φ)
∆i α×β
sin (φ) = =
∆i ∆i∆e
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
• The phase angle can be directly observed by measuring peak to peak.
Rs |e (jω) |
|Z| =
eR (jω) |
Z = |Z|cos (φ) |e|
Z = |Z|sin (φ)
input (potential)
response (potential drop)
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
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
• 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.
3 circuits with two time constants
• Even worse than degenerate cases which can be equally well fit to an
experimental spectrum
are circuits consisting of too many
elements. Adding enough elements
will fit practically ANY curve. 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.
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
• Curve Fits are questionable because an
excellent first guess is required for a Z'
proper fit. '#####$

Geometric method
#$ %#####$ )######$ )%#####$ (######$ (%#####$ '######$ '%#####$ &######$
&######$ 2*1*$

works well for one time constant.
!%#####$ #$

!'######$ #$ '######$ %######$ )######$ *######$
See attached excel sheet for automated
!"#####$ !(######$ 567$
!'######$ 82/0390+/9$
extrapolation !&######$
Extrapolated data for a real system
Copyright © M. L. Taylor `
End of Part II
Copyright © M. L. Taylor `
• 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
Copyright © M. L. Taylor `
Reading: Solartron Analytical
High frequency, high current impedance spectroscopy: Experimental protocols enabling By F Mansfeld
measurement up to 1MHz at high current densities Get it here...
By John Harper, Mike Rust, Brian Sayers and Andrew Savage
Get it here... Technote 29
The Application of Impedance Spectroscopy to Cementitious Systems
Use of auxiliary channels for impedance analysis: Detecting failure mechanisms within a fuel Get it here...
cell / battery stack
By J Harper and B Sayers Technote 31
Get it here... Electrochemical Impedance Spectroscopy (EIS) for Battery Research and Development
By Hong Shih and Tai-Chin Lo
TB/ANALYTICAL/3 Get it here...
Parallel Channel Combinations
By J Harper and B Sayers Technote 33
Get it here... The Potentiodynamic Polarization Scan
By D G Evans and L L Scribner
TB/ANALYTICAL/004 Get it here...
Solartron CellTest® System
Impedance measurement techniques 1470 Tech
Brian Sayers Instrumentation for the Characterization of Energy Storage Devices and Multi-Cell Systems
Get it here... By B Sayers and A Hinton
Get it here...
Technote 04
Identification of Electrochemical Processes by Frequency Response Analysis Coatings Tech
By C Gabrielli Determination of Coating Adhesion Using Electrochemical Impedance Spectroscopy
Get it here... By A Hinton
Get it here...
Technote 06
An Introduction to Electrochemical Impedance Measurement Advanced Instrumentation for Solid State Applications
By N D Cogger and N J Evans By A Hinton and B Sayers
Get it here... Get it here...
Technote 10 Beyond the Limits: 1296 Dielectric Interface
Frequency Response Analysis By A Hinton and B Sayers
By N D Cogger and R V Webb Get it here...
Get it here...
Advanced Instrumentation for Civil Engineering Applications
Technote 17 By A Hinton and B Sayers
Understanding Electrochemical Cells Get it here...

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