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

Electrochemical
Impedance Spectroscopy
Part 1: Polarization Resistance:
Familiar parameter measured in a new way
June 6, 2008
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.
2
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.
3
Copyright © M. L. Taylor `
Why would I want to use EIS? (I have DC)
• DC methods drive the system from equilibrium
• Reaction parameters are timedependent 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
Voltammetry
E
Potentiodynamic
Scan E E
log(I)
Galvanostatic/ I
Pulsed Current
4
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
Rm
F M
C
• electrochemical charge
Cm
transfer reactions
F
• 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
5
Copyright © M. L. Taylor `
Polarization Resistance
• Useful for predicting a corrosion rate under steadystate 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.
η
Rp
Inet =
Rp
7
Copyright © M. L. Taylor `
AC Determination of Polarization Resistance
• Impedance has real and imaginary Rp
components, both of which depend on
Rs
the AC frequency.
• Solution resistance is measured
Cdl
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
8
Copyright © M. L. Taylor `
Simple Passive Circuit Elements
Element Impedance Symbol
Ri
Resistor ZRi (ω) = Ri
Cdl
j
Capacitor ZCi (ω) = −
ωC
√
Note: j = −1
9
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
Z1
1
Parallel Zparallel =
Z2
1
Z1 + 1
Z2
10
Copyright © M. L. Taylor `
Impedance of the Randles Cell
Rp
• consider the capacitor and Rp in parallel
Rs
1
Zp = 1
Rp + 1
−j
Cdl
ωCdl
• A little complicated, but we can multiply by the complex conjugate
to of the bottom to seperate real and imaginary (multiplied by j)
terms.
2
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
have:
Rp 2
ωCdl Rp
Z = Rs + Z =−
1 + ω 2 CRp
2
1 + ω 2 Cdl Rp
2 2
11
Copyright © M. L. Taylor `
Continued
• 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, !
RP
RS
log(f), hz f = 2πω
12
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
Z'', Imaginary Impedance, !
• Capacitance may be determined
"max
1
ωmax = .
y
uenc
Low Fre
Rp Cdl
Freq
quenc
High
y
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.
13
Copyright © M. L. Taylor `
Electrochemical
Impedance Spectroscopy
Part 2: Advanced Circuit Elements
June 20, 2008
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/CleanUp
• Electrochemical systems RP (Ω)
Corrosion Rate
(mm/y)
Example
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
16
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 13761388 `
17
Advanced Passive Circuit Elements
Element Impedance Symbol
Pseudo
Inductor ZL = jωL Li
√
ZW = σ × (1 − j) ω
!i
Warburg
1 (Qi,!i)
Constant Phase Z= 1−α
Element
(jωA)
√
Note: j = −1
18
Copyright © M. L. Taylor `
Pseudo Inductance
• No physical basis for inductance
in an electrochemical system
• Yet it is observed!
• More on this later
19
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.
z''
45º
Z'
√
ZW = σ × (1 − j) ω Randles Cell with Warburg
Impedance Element Included
20
Copyright © M. L. Taylor `
Constant Phase Element
• A Generalized Impedance Element
1 (Qi,!i)
Z=
• For values of α=0, the CPE behaves as (jωA)
1−α
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, 1α 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
•
20
Surface roughness
10
• uneven distribution of charge 0
0 20 40 60 80 100
• Observed more often than perfect effect of α
capacitance, but easily misinterpereted.
21
Copyright © M. L. Taylor `
Linear Systems Theory
• Criteion for a linear system:
• The system is described by linear
functions
• 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
log(I)
• Reversing the singal to the starting
point gives no hysteresis
• The system is Finite e0
E
• No Inﬁnite Values of impedance are
allowed
22
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).
Macdonald. Reﬂections on the history of
electrochemical impedance spectroscopy.
Electrochimica Acta (2006) vol. 51 (8/9) pp.
23
Copyright © M. L. Taylor `
13761388
XY 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
function
Current
#e
• (as seen on the Gamry
insturments in our lab) Potential e0
• Geometric examination Lissajous ﬁgure
extracts parameters to
calculate immittance (Y) ∆e Z = Zcos (φ)
Z =
• Immittance is the inverse of ∆i Z = Zsin (φ)
impedance.
∆i α×β
sin (φ) = =
∆i ∆i∆e
24
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 dualbeam
oscilloscope
• The phase angle can be directly observed by measuring peak to peak.
b
a
Rs e (jω) 
Z =
eR (jω) 
eR
Z = Zcos (φ) e
Z = Zsin (φ)
input (potential)
response (potential drop)
25
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.
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. 18231826.
27
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
Z''
requirements of measuring at low
frequencies
• Curve Fits are questionable because an
excellent ﬁrst guess is required for a Z'
proper ﬁt. '#####$
(#####$
•
)#####$
Geometric method
#$
#$ %#####$ )######$ )%#####$ (######$ (%#####$ '######$ '%#####$ &######$
%######$
!)#####$
*+,*.,$/,01,$
&######$ 2*1*$
!(#####$
•
'######$
works well for one time constant.
!'#####$
+,+.,/01$
!&#####$
(######$
+02/03$
!%#####$ #$
•
4,/,$
!'######$ #$ '######$ %######$ )######$ *######$
See attached excel sheet for automated
!"#####$ !(######$ 567$
!'######$ 82/0390+/9$
extrapolation !&######$
!%######$
!"######$
Extrapolated data for a real system
28
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. 463506
• 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. 13761388
30
Copyright © M. L. Taylor `
Reading: Solartron Analytical
TB/ANALYTICAL/001
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
TB/ANALYTICAL/2 By W. J. McCarter
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 TaiChin 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 MultiCell 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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