B. Markovsky, markovb@mail.biu.ac.il

Summer Courses, Bar-Ilan University, September 2014

(<1% of the entire topic!)

Basics of Impedance

Spectroscopy

The main goal of this

presentation is a brief

“Introduction” to EIS

Electrochemical Techniques

50 V/s

40 30

20

10 3.69 V

3.76 V

t

T=300C

CVs of LiMO2 electrode.

Dynamic technique.

10 V/s

Time domain. Current = f(time)

Relaxation: Current decays with time

An Electrochemical Impedance Spectrum

Frequency domain from high to low frequencies

100 kHz

1 µHz

The number of papers on EIS has

doubled every 4 – 5 years !

AC vs. DC methods

time

Potential

time

Once we apply DC methods, the cell is totally changed. Surface and volume changes

Phase transitions

Electrolyte oxidation/reduction

time time

For AC methods, very small perturbation is applied.

Nearly non-destructive! Cell is unchanged!

Potential

Current

Current

Alternating current (ac) methods: the

merits

• To grasp the entire features of the system:

Sometimes we need

X-ray

to see the inside of our

body.

Sometimes we need ac

methods

to see the inside of an

electrochemical cell.

Electrochemical Impedance Spectroscopy (EIS) is actually a

special case among electrochemical techniques.

It is based on the perturbation of an equilibrium state, while the

standard techniques are dynamic (e.g. CV) or are based on the

change from an initial equilibrium state to a different, final state

(e.g. potential step, chronocoulometry).

Hence, EIS is a small-signal technique where, in the

analysis of the impedance spectra, a linear current-voltage

relation is assumed.

ELECTROCHEMICAL IMPEDANCE

SPECTROSCOPY

ELECTROCHEMICAL IMPEDANCE

SPECTROSCOPY

Impedance spectroscopy is a non-destructive technique and so

can provide time dependent information about the properties of

a system but also about ongoing processes such as:

- corrosion of metals,

- discharge and charge of batteries,

- electrochemical reactions in fuel cells,

capacitors or any other electrochemical

process.

Everyone knows about the concept of electrical resistance.

What is resistance?

It is the ability of a circuit element to resist the flow of electrical

current.

Ohm's law defines resistance R in terms of the ratio between

voltage, E, and current, I. ( )

( )

E tR

I t

Resistance. Ideal Resistor

Ideal Resistor

This relationship is limited to only one circuit element ----

the ideal resistor !

An ideal resistor has several simplifying properties:

• It follows Ohm's law at all current and voltage levels.

• It's resistance value is independent on frequency.

• AC current and voltage signals through a resistor are in

phase with each other.

( )

( )

E tR

I t

In a Real World: Circuit elements exhibit much more complex behavior. In place of

resistance, we use impedance, which is a more general circuit

parameter.

Like resistance, impedance is a measure of the ability of a

circuit to resist the flow of electrical current.

Electrochemical Impedance is normally measured using a

small excitation signal (3 – 10 mV). This is done so that the cell's

response is pseudo-linear.

In a pseudo-linear system, the current response to a

sinusoidal potential will be a sinusoid at the same frequency

but shifted in phase.

Sinusoidal Current Response in a Linear System

E is the amplitude of the signal, and is the radial (angular)

frequency. (in radians/second) and frequency f (in Hertz (1/sec)

are related as: =2 f

Phase-shift Phase-shift

time

time

A purely sinusoidal voltage: Et=E0 sin t

Phasor (Vector)

diagram

for an ac-Voltage

- phase angle

E

I

Phasor (Rotating Vector) diagram

Response dI of dE from the Current / Potential

relation:

We can disturb an electrical element at a

certain potential E with a small

perturbation dE and we will get at the

current I a small response perturbation

dI.

In the first approximation, as the

perturbation dE is small, the response dI

will be a linear response as well.

An oval is plotted. This oval is known as a "Lissajous figure".

In Electrical Engineering to add together resistances, currents or DC

voltages “real numbers” are used.

But real numbers are not the only kind of numbers we need to use

especially when dealing with frequency dependent sinusoidal

sources and vectors.

Complex Numbers were introduced to allow complex

equations to be solved with numbers that are the

square roots of negative numbers, √-1.

Complex Numbers

i=-1

In electrical engineering, √-1 is called an “imaginary number”

and to distinguish an imaginary number from a real number the

letter “ j ” known commonly in electrical engineering as the

j-operator, is used.

The letter j is placed in front of a real number to signify its imaginary

number operation. Examples of imaginary numbers are:

j3, j12, j100 etc.

A complex number consists of two distinct but very much

related parts, a “Real Number ” plus an “Imaginary Number”.

Complex Number =

Real number + Imaginary number

Complex Numbers represent points in a two-dimensional

complex plane that are referenced to two distinct axes.

The horizontal axis is called the “Real Axis” while the

vertical axis is called the “Imaginary Axis”.

The real and imaginary parts of a complex number,

Z are abbreviated as Re(z) and Im(z).

Complex Numbers. Complex Plane

Negative Imaginary Axis

Two Dimensional Complex Plane (Four Quadrant

Argand Diagram)

Z= -8 – j5

Z = 5 + j0

Z = 0 + j4

Imaginary axis

Real axis

Complex Numbers. Complex Plane

i=-1

The plane of complex numbers spanned by the vectors 1

and i, where i is the imaginary number. Every complex

number corresponds to a unique point in the complex plane

(Argand or Gauss plane).

Complex writting Using Euler’s relationship

it is possible to express the impedance as a complex function.

The potential is described as,

and the current response as,

The impedance is then represented as a

complex number:

exp( ) cos sini i

0( ) exp( )E t E j t

0( ) exp( )I t I i t i

0 0exp( ) (cos sin )E

Z Z i Z iI

00

0

cos( )( ) cos( )( )

( ) cos( ) cos( )

E tE t tZ t Z

I t I t t

The expression for Z() is composed of a real and an

imaginary part. If the real part is plotted on the X axis and the

imaginary part on the Y axis of a chart, we get a "Nyquist

plot“. (Harry Nyquist, 1889-1976).

0 0exp( ) (cos sin )E

Z Z i Z iI

The Nyquist plot results from

the RC circuit. The semicircle

is characteristic of a single

"time constant".

1 1 1

Z R i C

Data Presentation: Nyquist Plot with Impedance Vector

R

C

Real part

Ima

gin

ary

part

Et=E0 sin t (1) It=I0 sin (t + ) (2)

General formulae of

a circle:

X 2 + Y 2=r2

(r is the radius)

R

C

Semicircle in Nyquist plot

The impedance of an ohmic resistance R and a capacitance C

in parallel can be written as follows:

Another popular presentation method is the "Bode plot". The

impedance is plotted with log frequency (log ) on the X-axis

and both the absolute value of the impedance (|Z| =Z0 ) and

phase-shift on the Y-axis.

Unlike the Nyquist plot, the Bode plot explicitly shows frequency

information.

The Bode Plot

A parallel R-C combination

The parallel combination of a resistance and a capacitance, start in the admittance representation:

Transform to impedance representation:

A semicircle in the impedance plane!

1( )Y j C

R

2

2 2 2 2 2

1 1 1/( )

( ) 1/ 1/

1

1 1

R j CZ

Y R j C R j C

R j R C jR

R C

R

C

General formulae of

a circle:

X 2 + Y 2=r2

(r is the radius)

The semicircle is characteristic of a single

“RC-constant”.

Electrochemical impedance plots often contain

several semicircles.

Often only a portion of a semicircle is seen.

Semicircles in Nyquist plots

Bode plot (Zre, Zim)

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06

frequency, [Hz]

Zre

al, -

Zim

ag

, [

oh

m]

Zreal

Zimag

Bode plot: absolute (Z), phase vs. frequency

1.E+02

1.E+03

1.E+04

1.E+05

1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06

Frequency, [Hz]

ab

s(Z

), [

oh

m]

0

15

30

45

60

75

90

Ph

as

e (

de

gr)

abs(Z)

Phase (°)

Zreal

Zimag

Different Bode representations

Other representations

Bode graph. “Double log” plot

1;R RZ R Y

R

1( ) /Z R R j C

j C

2 2

2 2 2 2 2 2

1( )

/

1 1

YR j C

C R Cj

C R C R

Semi- circle

‘time constant’:

= RC

The rate of an electrochemical reaction: charge-transfer,

diffusion…

Whenever diffusion effects completely dominate the

electrochemical reaction mechanism, the impedance is called the

Warburg Impedance.

For diffusion-controlled electrochemical reaction, the current is

45 degrees out of phase with the imposed potential.

In this case, (450) , the real and imaginary components of the

impedance vector are equal at all frequencies.

In terms of simple equivalent circuits, the behavior of Warburg

impedance (a 450 phase shift) is midway between that of a

resistor (a 00 phase shift) and a capacitor (900 phase shift).

Warburg Impedance

Warburg impedance

Diffusion: Warburg element

Semi-infinite diffusion, Flux (current) : First Fick’s Law Potential :

ac-perturbation:

Second Fick’s Law :

Boundary condition :

0

ln

x

CJ D

x

RTE E C

nF

( ) ( )C t C c t

2

2

C CD

t x

( , )x

C x t C

PITT – Small potential steps E from Eeq, I vs. t is measured.

Kinetic limitations other than diffusion are ignored.

= l2/D

D(E)=[1/2l It1/2/QmX(E)]2= [(1/2l (It1/2/E)/ Cint(E)]2

EIS – the semi-infinite (Warburg) domain for finite-space diffusion

response: Z’’ vs. Z’ at the low frequency is analyzed.

D = 0.5 l2 [CintAw]-2

PITT and EIS for the same electrode potential, should provide the constant:

Aw [(It1/2)/E]= (2)-1/2

This is a proof that the measurements are correct.

-1/2

It1/2 is the time invariant at t<< (short-time domain)

PITT and Impedance spectroscopy

Real thin (1500 ) cathode LixV2O5

450, Warburg element

M. Levi, Z. Lu, D. Aurbach, JPS, 2001, 97, 482

Diffusion time . Li+ diffusion coefficient in LiV2O5

Li+ diffusion coefficient in thin film graphite electrodes

M. Levi, D. Aurbach, J. Phys. Chem., 1997, 101, 4641

Li+ intercalation cathode LixCoO2

Li+ diffusion in LixCoO2 at low frequencies

Thin Na-V2O5 electrodes, 3 , 1 – 2 mg/cm2

450, Warburg element

Thin Na-V2O5 electrodes. Li+ diffusion coefficient

45°

Equivalent Circuit Concept

R Rsol

Rsol

Before starting the analysis and modeling of the experimental

results one should be certain that the impedances are valid.

There is a general mathematical procedure

(Kramers-Kronig), which allows for the verification

of the impedance data.

The impedance measured is valid provided that the following

4 criteria are met: linearity, causality, stability, finiteness.

Analysis and Modeling. Data Validation

1. Linearity: A system is linear when its response to a sum of individual input

signals is equal to the sum of the individual responses.

Electrochemical systems are usually highly non-linear

and the impedance is obtained by linearization of equations

for small amplitudes.

For the linear systems the response is independent of the

amplitude.

It is easy to verify the linearity of the system:

if the obtained impedance is the same when the

amplitude of the applied ac-signal is halved

then the system is linear.

Electrochemical systems are, in general,

not linear.

A very small portion

of the I vs. V curve

appears to be

linear (pseudo-linear)

In normal EIS practice, a small (1 to 10 mV) AC signal is applied to the cell.

With such a small potential signal, the system is pseudo-linear.

A typical Lissajous plot for

a linear system

A Lissajous plot showing

a non-linear response

AC

Cu

rren

t / A

AC

Cu

rren

t / A

AC Potential / V

Lissajous plot

2. Causality:

The response of the system must be entirely

determined by the applied perturbation.

The impedance measurements must also be

stationary.

The measured impedance must not be time

dependent !

3. Stability:

The stability of a system is determined by its response to

inputs. The system is stable if its response to the impulse

excitation approaches zero at long times.

The measured impedance must not be time dependent.

This condition may be easily checked by

repetitive recording of the impedance spectra;

then the obtained Bode plots should be identical.

Measuring an impedance spectrum takes time

(minutes - hours).

The system being measured must be at a steady-state

throughout the time required to measure the spectrum.

A common cause of problems in EIS measurements and

analysis is drift in the system being measured.

Standard EIS analysis tools may give wildly inaccurate

results on a system that is not at steady-state.

Steady-State Systems

Possible tests for the validity of EIS data

Kronig-Kramers (KK) test

The Kronig-Kramers (KK) relations are mathematical properties

which connect the real and imaginary parts of any complex

function.

During the KK test, the experimental data points are fitted using

a special model circuit which always satisfies the KK relations.

Is the impedance data stable?

• Application of K-K test for system

stability

4. Finiteness:

The real Zreal and imaginary Zim components of the

impedance must be finite-valued over the entire

frequency range 0 < ω < ∞.

In particular, the impedance must tend to a constant

real value for ω → 0 and ω → ∞.

Instrumentation:

3-electrode cell in a thermostat

Potentiostat

Frequency Response Analyzer

40 years ago…..

From Prof. B. Boukamp’s lecture, Intern. Symp. on EIS, 2008

Measuring impedance by means of

oscilloscopes, Sept. 1960

I

Impedance analysis in the old days

FRA PC

BTU

CE

RE

WE

Pouch-cell

Frequency response analyzer (FRA)

Cell

∫ Z" R(t) cos(t)

∫ Z' R(t) sin(t)

cos(t)

Vo sin(t)

osc. t

sin(t)

)(noise)sin()sin()( 0 ttkAtItRk

kk

This is necessary.

S/N increase by repeated measurements

.

..

Harmonic components

Vanishes by orthogonality

)(2

)(cos2

d sin)(1

re

o

2

oo

o

2

o

0int

int

ZV

IZ

V

ItttR

T

T

)(2

)(sin2

d cos)(1

im

o

2

oo

o

2

o

0int

int

ZV

IZ

V

ItttR

T

T

64

∫

More applications for Li-Batteries

Z’ / Ohm

-Z’’

/ O

hm

Z’ / Ohm

Z’ / Ohm

Impedance spectra of Lithium electrodes

LiAsF6 1 M LiAsF6 0.25 M LiAsF6 1 M

+200 ppm H2O

D. Aurbach, E. Zinigrad, A. Zaban, J. Phys. Chem., 100, 1996, 3091

3 hours aging

6 days aging

2D Graph 1

Z' / Ohm

0 20 40 60 80 100

Z" / O

hm

-60

-40

-20

0

2D Graph 2

0 100 200 300 400

Z" / O

hm

-200

-100

0

0 100 200 300 400

Z" / O

hm

-400

-300

-200

-100

0

2D Graph 4

Z' / Ohm

0 100 200 300 400

-200

-100

0

2D Graph 5

0 100 200 300 400

-200

-100

0

0 200 400 600 800

-400

-200

0

Initial state, 300C After aging 4 weeks at 600C

4.6 V

4.7 V

20 Hz 158 mHz

5 mHz

20 Hz

20 Hz

5 mHz

5 mHz

5 mHz

5 mHz

32 mHz

32 mHz

12.6 Hz 2.5 Hz

Li-Intercalation Electrodes Li[Mn-Ni-Co]O2

1-st

Semi

circle

2-nd

Semi

circle

W R1C1 R2C2

4.4 V

50 kHz

50 kHz

50 kHz

-3000

-2000

-1000

0

0

1000

2000

3000

010

2030

40

Z''

/ O

hm

Z' /

Ohm

Cycle number

3D Graph 2

293-A15-2imp,Z' vs Col 17 vs 293-A15-2imp,Z''

293-A15-3imp,Z' vs Col 18 vs 293-A15-3imp, Z''

293-A15-4imp,Z' vs Col 19 vs 293-A15-4imp, Z''

293-A15-5imp, Z' vs Col 20 vs 293-A15-5imp,Z''

293-A15-6imp, Z' vs Col 21 vs 293-A15-6imp,Z''

-1500

-1000

-500

0

0

500

1000

1500

010

2030

40

Z"

/ O

hm

Z' /

Ohm

Cycle number

3D Graph 3

2imp,Z' vs Col 17 vs 2imp,Z''

3imp,Z' vs Col 18 vs 3imp, Z''

4imp,Z' vs Col 19 vs 4imp, Z"

5imp,Z' vs Col 20 vs 5imp,Z''

6imp, Z' vs Col 21 vs 6imp, Z"

2D Graph 1

Cycles

0 10 20 30 40 50

Rs

f / O

hm

.cm

2

25

50

75

100

125

150

175

Uncoated

AlF3-coated

Uncoated material AlF3-coated material

50 Hz 20 Hz

20 Hz 20 Hz

20 Hz 50 Hz

5 mHz 5 mHz 5 mHz

5 mHz

5 mHz

5 mHz 5 mHz

5 mHz

5 mHz

Li[Ni-Mn-Co]O2

F. Amalraj, B. Markovsky et al., JES, 160, A2220, 2013

Thin-film Li[Ni-Mn-Co]O2 electrode (no PVdF, CB),

cycled, E=4.3 V. Impedance data fitting ZView

Warburg element,

Li+ Solid-state diffusion

1-st 2-nd

Slope=-1.076

Equivalent circuit

• Why we use it?

– Intuitive

– Practical

– Relatively easy to model

• Disadvantages

– Ambiguities

– Does not tell the mechanism of the reaction

– Explanation is difficult !

Rel

CPEsf

RctRsf

CPEctZwRel

CPEsf

RctRsf

CPEctZw

A Typical Equivalent Circuit

W

Solution resistance

Surface films

Interfacial charge transfer

Solid state diffusion

Intrcalation capacitance

Intercalation electrodes can be described by

the following equivalent analog

Building blocks of equivalent circuits

Fuel cell: Equivalent circuit analog

Conclusions-1

• Impedance is not a physical reality

– It is an alternating current technique.

– It is in the frequency domain.

– It is rigorously generalized concept that

contains whole features of an electrochemical

system.

75

Conclusions-2

• In many cases data fitting is not

needed.

– Exact plotting is essential.

– Graphical analysis is very useful.

• Experience is needed for data fitting.

• Anyway, enjoy EIS measurements!

– It is non-destructive technique.

– Good to understand your system

systematically.

So, What is Electrochemical Impedance

Spectroscopy?

“Probing an electrochemical system with

small ac-perturbation over a range of

frequencies”.

The main question:

1. Barsukov, E. and Macdonald, J. R. 2005. Impedance

Spectroscopy, 2nd ed. Wiley-Interscience, New York.

2. Conway, B. E. 1999. Electrochemical Supercapacitors, Kluwer

Academic/Plenum, New York.

3. A. Bard and L. Faulkner, Electrochemical Methods.

4. Orazem, M. and Tribollet, B. 2008. Electrochemical Impedance

Spectroscopy (The ECS Series of Texts and Monographs)

Wiley-Interscience, New York.

5. Solartron Analytical Frequency Response Analyzer (FRA).

Available at http://www.solartronanalytical.com/Pages/

1260AFRAPage.htm.

6. Lectures by Prof. Bernard Boukamp, University of Twente, Dept. of

Science &Technology, Enschede, The Netherlands.

Literature

……..Sunrise in Ein-Gedi, Dead Sea……