What is EIS - Tutorial

Research Institute for Nanotechnology

University of Twente, Dept. of Science & Technology, Enschede, The Netherlands

Electrochemistry of Materials

Bernard A. Boukamp

Solid State Ionics-17Toronto, June/July 2009.

Impedance Spectroscopy

SSI-17 Workshop

28 June 09My where abouts

E-mail: b.a.boukamp utwente.nl@

Address:

University of Twente

Dept. of Science and Technology

P.O.Box 217

7500 AE Enschede

The Netherlands

www.ims.tnw.utwente.nl

SSI-17 Workshop

28 June 09Electrochemical techniques

Time domain (incomplete!): Polarisation, (V I ) Potential Step, (V I (t) ) Cyclic Voltammetry, (V f(t)- I(V ) ) Coulometric Titration, (V - I dt ) Galvanostatic Intermittent Titration (Q V (t) )

Frequency domain:

Electrochemical Impedance Spectroscopy(EIS)

steady state

relaxation

dynamic

relaxation

transient

perturbation of equilibrium state

Next slide

SSI-17 Workshop

28 June 09Time or frequency domain?

1.E-07

1.E-06

1.E-05

1.E-04

0 1000 2000 3000

Time, [sec]

Cur

rent

, [A

]

SSI-17 Workshop

28 June 09Advantages of EIS:

System in thermodynamic equilibrium

Measurement is small perturbation (approximately linear)

Different processes have different time constants

Large frequency range, Hz to GHz (and up)

Generally analytical models available

Evaluation of model with Complex Nonlinear Least Squares (CNLS) analysis procedures (later).

Pre-analysis (subtraction procedure) leads to plausiblemodel and starting values (also later)

Disadvantage: rather expensive equipment, low frequencies difficult to measure

SSI-17 Workshop

28 June 09Black box approach

Assume a black box with two terminals (electric connections).

One applies a voltage and measures the current response (or visa versa). Signal can be dc or periodic with frequency f, or

angular frequency =2f ,with: 0 <

Phase shift and amplitude changes with !

SSI-17 Workshop

28 June 09So, what is EIS?

Probing an electrochemical system with a smallac-perturbation, V0ejt, over a range of frequencies.The impedance (resistance) is given by:

The magnitude and phase shift depend on frequency.

Also: admittance (conductance), inverse of impedance:

[ ]0 0( )0 0

( )( ) cos sin( )

j t

j t

V e VVZ jI I e I

+

= = =

[ ]( )

0 0

0 0

1( ) cos sin( )

j t

j t

I e IY jZ V e V

+

= = = +

real +j imaginary.

= 2fj = -1

SSI-17 Workshop

28 June 09

Impedance resistance

Admittance conductance:

hence:

Complex plane

2 2

1( )( )

1

re im

re im

re im

re im re im

Z jZYZ Z Z

Z jZZ jZ Z jZ

= =

+

= +

2 2

1( )( )

re im

re im

Y jYZY Y Y

= =

+Representation of impedance value, Z = a +jb, in the complex plane

SSI-17 Workshop

28 June 09

Adding impedances and admittances

1 2 3 ...total nn

Z Z Z Z Z= + + + =

1 2 3 ...total nn

Y Y Y Y Y= + + + =

A linear arrangement of impedances can be added in the impedance plane:

A ladder arrangement of admittances (inverse impedances) can be added in the admittance plane:

SSI-17 Workshop

28 June 09Simple elements

The most simple element is the resistance:

(e.g.: electronic- /ionic conductivity, charge transfer resistance)

Other simple elements:

Capacitance: dielectric capacitance, double layer C,adsorption C, chemical C (redox)

Inductance: instrument problems, leads,negative differentialcapacitance !

1;R RZ R Y R= =

See next page

SSI-17 Workshop

28 June 09

Take a look at the properties of a capacitor:

Charge stored (Coulombs):

Change of voltage resultsin current, I:

Alternating voltage (ac):

Impedance:

Admittance:

Capacitance?

Q C V=

d dd dQ VI Ct t

= =

00

d( )d

j tj tV eI t C j C V e

t

= =

0ACd

=

( ) ( ) 1( )CVZI j C

= =

( ) 1( )CY Z j C = =

SSI-17 Workshop

28 June 09Combination of elements

What is the impedance of an -R-C-circuit?

Admittance?

1( ) /Z R R j Cj C

= + =

2 2

2 2 2 2 2 2

1( )/

1 1

YR j C

C R CjC R C R

= =

+

+ +

Semi-circle

time constant: = RC

time constant: = RC

SSI-17 Workshop

28 June 09

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

Transform to impedance representation:

A semicircle in the impedance plane!

A parallel R-C combination

1( )Y j CR

= +

2

2 2 2 2 2

1 1 1/( )( ) 1/ 1/

11 1

R j CZY R j C R j CR j R C jR

R C

= = =

+

=

+ +

Plot next slide

R

C

SSI-17 Workshop

28 June 09Impedance plot (RC)

0.0E+00

2.0E+04

4.0E+04

6.0E+04

8.0E+04

0.0E+00 2.0E+04 4.0E+04 6.0E+04 8.0E+04 1.0E+05 1.2E+05

Zreal, [ohm]

-Zim

ag, [

ohm

]

1 MHz 1 Hz

518 HzR = 100 k

C = 3 nF

fmax = 1/(6.3x310-9x105)=530 Hz

SSI-17 Workshop

28 June 09Limiting cases

What happens for > ?

> :

This is best observed in a so-called Bode plotlog(Zre), log(Zim) vs. log(f ), orlog|Z| and phase vs. log(f )

22 2

1( )1jZ R R j R R j R C =

+

2 2 2 2 2 2

1 1 1( )1j R RZ R j j

RC C

= +

Next slides

SSI-17 Workshop

28 June 09Bode 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]

Zrea

l, -Z

imag

, [o

hm]

ZrealZimag

-1

-2

SSI-17 Workshop

28 June 09Bode, abs(Z), phase

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]

abs(

Z), [

ohm

]

0

15

30

45

60

75

90

Phas

e (d

egr)

abs(Z)Phase ()

SSI-17 Workshop

28 June 09

Capacitance: C() = Y() /j for an (RC) circuit:

Dielectric: () = Y() /jC0 C0 = A0/d

Modulus: M() = Z() j

Other representations

1 1( ) ( ) / /C Y j j C j C jR R

= = + =

0 0

( ) ( ) iondY jA

= =

2 2

2 2 2( ) ( ) 1CR j RM Z j

C R +

= =+

SSI-17 Workshop

28 June 09Simple model

Example: an ionically conducting solid,e.g. yttrium stabilized zirconia,

Zr1-xYxO2-x .Apply two ionically blocking electrodes,

in this case thick gold.

Measure the resistance (impedance)as function of frequency:

Equivalent circuit: (C[RC])12

1( ) 11g

ionint

Zj C

Rj C

= +

+

Schematic arrangement of sample and electrodes.

SSI-17 Workshop

28 June 09Low & high f - response

Low frequency regime, series combination Rion-Cint:

Straight vertical line in impedance plane.

High frequency regime,parallel combination of Rion//Cgeom:

Semicircle through the origin.

12( ) /ion intZ R j C =

2

2 2 2 2 2 2( ) 1 1ion geomion

ion geom ion geom

R CRZ jR C R C

=

+ +

SSI-17 Workshop

28 June 09Debije model:

An ionic conductor between two blocking electrodes:

Impedance representation Admittance representation

1( )( )

YZ

=

SSI-17 Workshop

28 June 09

Zreal

Zimag

Other representations

Bode representation

Different Bode representation

SSI-17 Workshop

28 June 09Diffusion, Warburg element

Semi-infinite diffusion,Flux (current) :(Fick-1)

Potential :

ac-perturbation:

Fick-2 :

Boundarycondition :

0

ln

x

CJ DxRTE E CnF

=

=

= +

( ) ( )C t C c t= +

2

2

C CDt x

=

( , )x

C x t C

=

Solution through Laplace transform: next page

Li-battery cathode

SSI-17 Workshop

28 June 09Diffusion, Warburg element

Semi-infinite diffusion,Flux (current) :(Fick-1)

Potential :

ac-perturbation:

Fick-2 :

Boundarycondition :

0

ln

x

CJ DxRTE E CnF

=

=

= +

( ) ( )C t C c t= +

2

2

C CDt x

=

( , )x

C x t C

=

Solution through Laplace transform: next page

Li-battery cathodeRedox on inert electrode.

SSI-17 Workshop

28 June 09Warburg element, cont.

2

2

( , )( , ) C x pp C x p Dx

=

Laplace transform:

Transform of Fick-2:

General solution:

Transform of V (t):

Transform of I (t):(Fick-1)

( ) ( , )RTE p C x pnFC

=

( , ) ( , )c x t C x p

( , ) cosh / sinh /C x p A x p D B x p D= +

0

( , )( )x

C x pI p nFDx =

=

Boundary condition:Boundary condition:

( , ) 0x

C x p

=

SSI-17 Workshop

28 June 09

Define impedance in Laplace space!

Warburg impedance

2

( )( )( ) ( )E p RTZ pI p nF C D p

= =

1/2 1/202

( ) ( )( )

RTZ Z jnF C j D

= =

Take the Laplace variable, p, complex: p = s + j .

Steady state: s 0, which yields the impedance:

0 2( ) 2RTZ

nF C D=

with:

SSI-17 Workshop

28 June 09Transmission line

Real life Warburg, semi-infinite coax cable with r /m and c F/m:

( )WrZj c

=

Combination: Electrolyte resistance, Relyte Double layer capacitance, Cdl Charge transfer resistance, Rct Warburg (diffusion) impedance, Wdiff

Equivalentcircuit

SSI-17 Workshop

28 June 09Equivalent Circuit Concept

semicircle

45

SSI-17 Workshop

28 June 09Instruments

Measurement methodsBulk, conductivity:

two electrodes pseudo-four electrodes true four electrodes

Electrode properties: three electrodes

SSI-17 Workshop

28 June 09Respect for old scientists!

Image from the past:

Hand balanced bridges:

1.

2sample comp

RZ ZR

=

Very accurate, but limited range

& time consuming!

SSI-17 Workshop

28 June 09Ionic & MIEC conductivity

I+

I-

V+

V-

I+, V+

I-, V-

Two-electrode measurement,

influence of cables! For 100 kHz & 10 H:5 52 10 10 6.3cableZ j L j

= + =

Pseudo 4-point, inductance eliminated, but still additional capacitance of coax cables.

Twist cables in cell for low inductance!

I+

I-

V+V-

True 4-point measurement.Voltage probes for mixed conductors!Ionic and/or electronic probes.Spurious capacitances still possible.

SSI-17 Workshop

28 June 09Potentiostat, electrodes

General schematic

Vpwr.amp = A k VkA= amplification

Vwork Vref =Vpol. + V3 + V4

Current-voltage converter provides virtual ground for Work-electrode.

Source of inductive effects

SSI-17 Workshop

28 June 09

Cell morphology and reference electrodes

Thin cell geometry is very unreliable for 3-electrode impedance measurement.

Even with equal size electrodes, a small mismatch can cause the reference electrode to feel the counter electrode dispersion.(Nice treatment of problem with FEMLAB by Stuart Adler in J.El.chem.Soc.)

SSI-17 Workshop

28 June 09Solutions for reference el.

Solutions for optimal placing of the reference electrode.Left: two compartment cell. Right: single compartment cell.

SSI-17 Workshop

28 June 09Data validation

Kramers-Kronig relations (old!)

Kramers-Kronigconditions:

causality linearity stability (finiteness)

Real and imaginary parts are linked through the K-K transforms:

Response only due to input

signalResponse

scales linearly with input

signal

State of system may not change

during measurement

SSI-17 Workshop

28 June 09Putting K-K in practice

2 20

( ) ( )2( ) re reimZ x ZZ dxx

=

2 20

( ) ( )2( ) im imrexZ x ZZ R dx

x

= +

Problem:Finite frequency range: extrapolation of dispersion assumption of a model.

[1] M. Urquidi-Macdonald, S.Real & D.D. Macdonald,Electrochim.Acta, 35 (1990) 1559.

[2] B.A. Boukamp, Solid State Ionics, 62 (1993) 131.

Relations,

Real imaginary:

Imaginary real:

not a singularity!

SSI-17 Workshop

28 June 09Linear KK transform

[3] B.A.Boukamp, J.Electrochem.Soc, 142 (1995) 1885

Linear set of parallel RC circuits:

Create a set of values: 1 = max-1 ; M = min-1with ~7 -values per decade (logarithmically spaced).

k = RkCk

If this circuit fits the data, the data must be K-K transformable!

SSI-17 Workshop

28 June 09

Fit function simultaneously to real and imaginary part:

Set of linear equations in Rk, only one matrix inversion!

Display relative residuals:

Actual test

=

+

+=M

k ki

kikiKK

jRRZ1

2211)(

Shortcut to KKtest.lnk

It works like a K-K compliant

flexible curve

i

iimKKiimimag

i

ireKKirereal Z

ZZZZZ )(

,)( ,,,, =

=

SSI-17 Workshop

28 June 09No information in Rk values

SSI-17 Workshop

28 June 09Smooth between data points

SSI-17 Workshop

28 June 09Example K-K check

Impedance of a sample, not in equilibrium with the ambient.

2KK = 0.910-42CNLS = 1.4 10-4

SSI-17 Workshop

28 June 09Finite length diffusion

Particle flux at x=0:

Ficks 2nd law:

But now a boundary condition at x = L.

2

2

d),(d~

d),(d

xtxCD

ttxC=

0d),(d~)(

=

=xt

txCDtJ

Activity of A is measured at the interface at x=0. with respect to a reference, e.g. Amet

SSI-17 Workshop

28 June 09Finite length diffusion

0d

),(d=

=lxxtxC

[ ]llxlx

CtxCkxtxC

==

=

),(d

),(d

( )0),( CCtxC llx ===

Impermeable boundary at x =L:

Ideal source/sink with C = CL (=C0):

General expression for permeable boundary:

0),(),( CtxCtxc =

Replace concentration by its perturbation:

FSW

FLW

General!

SSI-17 Workshop

28 June 09FLD, continued

0000 ),(

lndlndln)(

==

==

xx txc

Ca

nFCRT

aa

nFRTtE

000

0

0 1lnlnln aa

aa

aaa

aa

+=

+=

0

00

d dlnd dln ( , )

x

a a a a aC C C C c x t

=

= =

Voltage with respect to reference C0 (a0):

Assumption: a

SSI-17 Workshop

28 June 09Up to the Frequency Domain!

Laplace transformation of E(t) and I(t) gives the complex impedance (with p=j):

DpxB

DpxApC ~sinh~cosh)( +=

Laplace space solution of Fick-2:

Djl

Dj

ZIEZ ~

coth~)()()( 0==

Djl

Dj

ZIEZ ~

tanh~)()()( 0==

FSW

FLW

=

=

=

dd

lndlnd

220

EnFSV

ca

SFnVTRZ

m

m

with:

SSI-17 Workshop

28 June 09Dispersions

1

dd

=

EVnFVCM

sint

Impedance representation of FSW and FLW.

High frequencies:

Z() = Z0 (j )-1/2

= Warburg diffusion

Low frequency limit:

FSW = capacitive

FLW = dc-resistance

SSI-17 Workshop

28 June 09

General finite length diffusion

0

coth()

coth

+=

+

jD

jD

j D l kZZj D k l j D

Generic finite length diffusion:

If k =0 then blocking interface FSW

If k = then ideal passing interface FLW

Plot for different values of k.

SSI-17 Workshop

28 June 09A Simple Example

H.J.M. Bouwmeester

AgxNbS2 is a layered structure, consisting of two-dimensional NbS2layers. Insertion and extraction of Ag+ions goes in an ideal manner (see graph).

Isostatically pressed and sintered sample. Some preferential orientation (in the proper direction!) will occur.

Simple cell design for EIS measurements.

SSI-17 Workshop

28 June 09Circuit Description Code

The Circuit Description Code presents an unique way to define an equivalent circuit in terms suitable for computer processing.

Elements: R, C, L, W

Finite length diffusion:

T = FSW = Tanhyp (Adm.) Cothyp (Imp.)

O = FLW = Cothyp (Adm.) Tanhyp (Imp.)

Constant Phase Element:

Q = CPE = Y0(j )n (Adm.) Z0(j )-n (Imp.)

SSI-17 Workshop

28 June 09The CPE

Constant Phase Element:

YCPE = Y0 n {cos(n /2) + j sin(n /2)} n = 1 Capacitance: C = Y0 n = Warburg: = Y0 n = 0 Resistance: R = 1/Y0 n = -1 Inductance: L = 1/Y0

All other values, fractal?

Non-ideal capacitance, n < 1 (between 0.8 and 1?)

SSI-17 Workshop

28 June 09Non-ideal behaviour

Deviation from ideal dispersion:

Constant Phase Element (CPE),

(symbol: Q )

0 0( ) cos sin2 2n n

CPEn nY Y j Y j = = +

General observations: Semicircle (RC ) depressed vertical spur (C ) inclined Warburg less than 45

n = 1, , 0, -1, ?n = 1, , 0, -1, ?

SSI-17 Workshop

28 June 09The Fractal Concept

How to explain this non-ideal behaviour?

1980s: Fractal behaviour (Le Mehaut)

= fractal dimensionality

i.e.: What is the length of the coast line of England?

Depends on the size of the measuring stick!

Self similarity

SSI-17 Workshop

28 June 09Fractals

Fractal line

Sierpinski carpet

Self similarity!Self similarity!

SSI-17 Workshop

28 June 09Fractal electrode

Cantor bararrangement

R aR a2R a3R

Impedance of the network:

2

1( ) 21

2...

Z Rj C

aRj C

a R

= + +

+ +

+

( ) 21

2...

aZ Rj aC

Rj aC

aR

= + +

+ +

+

( ) 2( )

aZ Rj aC

Z

= + +

( )( ) 2

aZZ Ra j CZ = + +

SSI-17 Workshop

28 June 09Arriving at the CPE

( )( ) 2

aZZ Ra j CZ = + +

( )2aZ Z

a =

( ) ( ) nZ A j =

S.H. Liu and T. Kaplan, Solid State Ionics 18 & 19 (1986) 65-71.

Frequency scaling relation:

In the low frequency limit this reduces to:

Which is satisfied by the formula:

with n = 1 ln2/lnaFractal dimension of Cantor bar, d = ln2/lnaHence: n = 1 d

SSI-17 Workshop

28 June 09Where we are at:

Unfortunately, experiments with rough electrodesshowed no relation between microstructure and fractal dimension from EIS measurements.

For bulk response at high frequency (dielectric response):

Theory by Klaus Funke (University of Mnster):

Concept of mismatch and relaxation

(See literature).

General assumption:

one dimensional transport with no lateral interaction!

General assumption:

one dimensional transport with no lateral interaction!

SSI-17 Workshop

28 June 09Different presentation

( ) ( ) nnn YYjYjYY /10000 with,)( ===Sometimes useful to rephrase the CPE:

PZT, dielectric response:

AYd

r

=0

0

SSI-17 Workshop

28 June 09Circuit Descriptions Code

1: ( C [ ] )2: ( C [ ( ) ( ) ] )3: ( C [ ( Q [ ] ) ( C [ ] ) ] )4: ( C [ ( Q [ R ( ) ] ) ( C [ R Q ] ) ] )5: ( C [ ( Q [ R ( R Q ) ] ) ( C [ R Q ] ) ] )

Breaking it down from the outside.

Find the parallel and series boxes.

Fill in with increasing nesting level.

0 = series

1 = parallel

2 = series, etc.

SSI-17 Workshop

28 June 09Reading the CDC

SSI-17 Workshop

28 June 09Equivalencies

Equivalent circuits with identical frequency dispersion (spectra):

Ra = RdRe/(Rd + Re)Rb = Re2/(Rd + Re)Cc = Cf [Rd/(Rd + Re)]2 Two circuits with identical dispersion

S. Fletcher, J.El.chem.Soc., 141 (1994) 1823

But many more combinations exist,

not only exactly identical responses,

but also identical within the experimental error!

But many more combinations exist,

not only exactly identical responses,

but also identical within the experimental error!

Physical meaning! Change experimental conditions: (P, T, , )

SSI-17 Workshop

28 June 09CNLS data analysis

Model function, Z(,ak), or equivalent circuit.Adjust circuit parameters, ak, to match data,Minimise error function:

with: (weight factor)

Non-linear, complex model function!

S w Z Z Z Zii

n

re i re i im i im i= + =

1

2 2, ,( ) ( ) { }

Effect of minimisation

w Z Z ai i i k= 2 2( , )

d 0d kS

a=for k = 1 ..M

SSI-17 Workshop

28 June 09Non-linear systems

Function Y (a1..aM) is not linear in its parameters, e.g.:

Linearisation: Taylor development around guess values, ajo:

Derivative of error sum with respect to aj :

....)..,()..,()..,(..

111

1

+

+= j

j aaj

MMM aa

aaxYaaxYaaxYM

( )[ ] ),,,,()( 00 =+= kZZjkZZ

j

Mi

i kk

k

MiMiii

j aaxYa

aaxYaxYyw

aS

+==

),(),(),(20 ..1..1..1

SSI-17 Workshop

28 June 09

A set of M simultaneous equations, in matrix form:

a = , solution: a = -1 =

With:

and:

Derivatives are taken in point ao1..M.

Iteration process yields new, improved values: aj = aoj + aj.

NLLS-fit

k

Mi

i j

Miikj a

axYaaxYw

= ),(),( ..1..1,

[ ]

=i k

MiMiiik a

axYaxYyw ),(),( ..1..1

SSI-17 Workshop

28 June 09

Having the derivatives is essential!

best method, calculate the derivatives on basis of the function: accuracy and speed.

Second best: numerical evaluation* (for properderivatives we have to calculate F(xi,a1..M)2M +1 times!!

The derivatives!

j

MjjiMjjiMi

j aaaaaxFaaaaxF

axFa

+=

2),....,,(),....,,(

),( 11..1

* This is actually an approximation

SSI-17 Workshop

28 June 09Marquardt-Levenberg

Analytical search: fast and accurate near true minimumslow far from minimum (and often erroneous)

Gradient search or steepest descent (diagonal terms only):fast far from minimumslow near minimum

Hence, combination!Multiply diagonal terms with (1+).

> 1, gradient search

Successful iteration:Snew < Sold

decrease (= /10).Otherwise increase

(= 10)

Successful iteration:Snew < Sold

decrease (= /10).Otherwise increase

(= 10)

Bottom line: good starting parameter estimates are essential!

SSI-17 Workshop

28 June 09Error estimates

For proper statistical analysis the weight factors, wi, should be established from experiment.

Other (dangerous) method:

Step 1: set weight factors, wi = g i-2

Step 2: assume variances can be replaced by parent

distribution, hence 2 1 (with = N M 1)Step 3:

Hence proportionality factor, g = S/.

[ ] ==

==i iii i

Mii

gS

wSaxYy 111),(1 22

2..1

22

SSI-17 Workshop

28 June 09Error analysis NLLS-fit

Based on this assumption we can derivethe variances of the parameters:

Error matrix, , also contains the covariance of the parameters:

g j,k 0, no correlation between aj and ak.g j,k 1, strong correlation between aj and ak.

Only acceptable for many data points ANDrandom distribution of the residuals

[ ] kkkka ggk ,1,2 ==

kjaa gkj , =

SSI-17 Workshop

28 June 09

Weight factors and error estimates

Errors in parameters: estimates from CNLS-fit procedure assumption: error distribution equal to parent distribution only valid for random errors, no systematic errors allowed!

rere i re i

iim

im i im i

i

Z ZZ

Z ZZ

=

=, ,( )

( ),

( )( )

Residuals graph:

Large error estimates:strongly correlated parameters (+ noise).Option: modification of weight factors.

SSI-17 Workshop

28 June 09Two different CNLS-fits

2 3.810-3R1 1290 4%R2 4650 2.7%C3 2.3810-103.8%R4 6580 2.6%C5 6.0710-9 7.3%

2 2.410-5R1 999 0.8%R2 4000 1.7%Q3 1.0310-9 7%-n3 0.898 0.6%R4 8020 0.9%Q5 1.0310-7 3.6%-n5 0.697 0.7%

CDC: R(RQ)(RQ)

CDC: R(RC)(RC)Values seem O.K. but look at the residuals!

Example of correct error estimates:

And of incorrect error estimates:

R(RC)(RC)

SSI-17 Workshop

28 June 09Residuals plot!

Systematic deviation,

Trace, bad fit

Good fit (not bad for a straight simulation!)

SSI-17 Workshop

28 June 09Subtraction procedure

Partial CNLS-fit of recognizable structure

Semicircle

Straight line (CPE, Cap., Ind.)

Subtract dispersion as series- or parallel component

Repeat steps until garbage is left

Be aware of errors due to consecutive subtractions

Sometimes restart and do a partial fit of a largergroup of parameters

SSI-17 Workshop

28 June 09Fingerprinting

Classification of capacitancesource approximate value

geometric 2-20 pF (cm-1)grain boundaries 1-10 nF (cm-1)double layer / space charge 0.1-10 F/cm2surface charge /adsorbed species 0.2 mF/cm2(closed) pores 1-100 F/cm3chemical capacitancesstoichiometry changes large (~1000 F/cm3)

Modified after: Peter Holtappels, TMR symposium Alternative anodes..., Jlich, March 2000.

SSI-17 Workshop

28 June 09

Capacitance of gas volume (e.g. O2):

Capacitance: or:

O2 produced:

Nernst:

Combination:

Gas phase capacitance

=ddEi Ct

=d

di tCE

Example:air, 700C, Vol. = 10 mm3Cox = 0.456 F !

PV=nRTPV=nRT

+= =

2d d ( ) dd ln4 4 4RT P P RT P RT nE

F P F P F V

=d 4 di t F n

=

24ox

FC V PRT

SSI-17 Workshop

28 June 09Conclusions on fitting

Many parameter, complex systems modelling:

Use Marquardt-Levenberg when quality starting values are available

Simplex (or Genetic Algorithm) for optimisation of rough guess starting values, as input for M-L NLSF

Check residuals when calculating Error Estimates

Look for systematic error contributions, remove if feasible.

Provide error estimates in publications!

Its human to err, its dumb not to include an error estimate with a number result

SSI-17 Workshop

28 June 09Fitting to the Extreme

PbZr0.53Ti0.47O3//gold electrodes

N2 (1.2 Pa O2), 580C, 0.1 Hz 65 kHz

PbZr0.53Ti0.47O3//gold electrodes

N2 (1.2 Pa O2), 580C, 0.1 Hz 65 kHz

2CNLS =

6.210-8

2K-K =

5.810-8

SSI-17 Workshop

28 June 09Subtraction of (Qdiel Rel)

SSI-17 Workshop

28 June 09The hidden R(RQ)

With electrode response subtracted

- (C[RW])

With electrode response subtracted

- (C[RW])

SSI-17 Workshop

28 June 09After subtraction of Cdl

SSI-17 Workshop

28 June 09The proof of the pudding

CDC: (QR[R(RQ)(C[RW])])

ps2 = 6.210-8 err. [%]

Q1 3.210-9 S.sn 7.9

-n1 0.886 - 0.6

R2 2800 0.01

R3 2290 1.7

R4 3420 1.8

Q5 2.88 10-7 S.sn 6.0

-n5 0.668 - 1.0

C6 1.74 10-8 F 0.5

R7 68100 1.7

W8 4.19 10-7 S.s1/2 1.6

Electrode response (Randlescircuit) in analogy with liquid electrolyte with redox couple.

One of the possible equivalent circuits

SSI-17 Workshop

28 June 09Consistency of model

Arrhenius graph of Cdl and Wdiff(ionic electrode response) .

N2 (Po2 ~1.2 Pa). Above TC Warburg is activated with EAct =653 kJ.mol-1. Parameters have been scaled by the geometric factor!

SSI-17 Workshop

28 June 09

Impedance & mixed conductivity (solid state)

Generally accepted Equivalent Circuit for mixed conducting oxides.

Electrode dispersion causes problem in analysis.

Simplified (but wrong) model, assuming electrodes impermeable for oxygen. (Au/MIEC/Au)

SSI-17 Workshop

28 June 09

Jamnik & Maier Model System

Discretisation of transport in solid and across interfaces Transport coordinate Reaction coordinate Chemical capacitance

Jamnik & Maier, Phys. Chem. -Chem. Phys. 3 (2001) 1668-78.

SSI-17 Workshop

28 June 09Jamniks intercalation model

Arguments of tanh(FSW) and coth (FLW) functions are equal:

2

1

tanh( )14

coth1 ( )4

i e Li

i e Li

i e

i e

i e

RR RZR R j R C

jR RR R j

jR R

j

+

+

= +

+ +

+

+

+ +

21 1.4 4/ ( ).i e chemL D R R C = = +

Time constant:

J. Jamnik, Solid State Ionics, 157(2003) p.19-28

SSI-17 Workshop

28 June 09Simplification for Re 0

( ) ( )2 2 2 22 2 2 2

tanh coth1 1( )4 4 4

2 24 4

coth22

j j j ji

i i j j j j

j j j j j ji i

j j j j

i

j j R e e e eZ R Re e e ej j j

e e e eR R e ee e e ej j

R jj

+ = + = + +

+ + + = =

=

For Re 0 the first term disappears. We focus now on the 3rd and 4th term (second term deals with processes at te interface). Setting Re=0 yields:

( )coth /( )

chem

L L j DZ

C j D

=

Inserting expressions for and Ri:

SSI-17 Workshop

28 June 09Gerischer in electrodes

electrolyte

Oad, Had, OHad

H2Oad

electrolyte

Oad, Had, OHad

H2Oad

electrolyte

Oad

electrolyte

Oad

Possible mechanisms leading to a Gerischertype response.

The ALS model #)

Slow adsorption& diffusion *)

#) S.B. Adler, J.A. Lane and B.C.H. Steele, J.Elchem.Soc. 143 (1996) 3554-64.*) R.U. Atangulov and I.V. Murygin, Solid State Ionics 67 (1993) 9-15.

Fick-2:2

2

d ( , ) d ( , ) ( , )d d

c x t c x tD k c x tt x

=

SSI-17 Workshop

28 June 09Alternative explanation

Z ZY

rr j c

Zk j

s

p( )

/

= =

+=

+1

2

0

1

Leaky transmission line. Series resistance, r1, in /m. Parallel capacitance, c, in F/m. Parallel resistance, r2, in m.

For semi-infinite circuit:

With k = (r2c)-1 and Z0 = (r1/c)1/2

Transmission line with side branches can lead to fractal response!I.D. Raistrick in: Impedance Spectroscopy, ed. J. Ross Macdonald, (Wiley 1987)

SSI-17 Workshop

28 June 09YSZ-Tb example

Example: mixed conducting YSZ-TbStabilized zirconia: Zr1-x-yYxTbyO2-0.5x-Zr = 4 +

Y = 3 +, hence oxygen vacancies:

Tb = 3 + / 4 +, hopping conduction (+ vacancies for Tb3+)

Electronic conductivity depends on pO2.

]Y2[][VO =

SSI-17 Workshop

28 June 09

Au/YSZ-Tb27.3/Au dispersions

pO2 = 0.9 bar, Au-electrodes0.01 Hz - 65 kHzpO2 = 0.9 bar, Au-electrodes0.01 Hz - 65 kHz

SSI-17 Workshop

28 June 09

?

A possible model?

Assumption: electrodes impermeable for oxygen.At electrodes:(+) oxidation and (-) reduction of Tb-sites,e.g. at the cathode (-):TbZr, OO TbZr, VO

Ficks 1st:(c = excess concentration)Ficks 2nd:

J Dd cdx

iF

J Dd cdxx L

ex L= =+= = = +

~ , ~ 2

d cdt

Dd cdx

k c = ~2

2

The k c term is related to a reaction of the excess concentration.

The k c term is related to a reaction of the excess concentration.

+e-O

SSI-17 Workshop

28 June 09Model development

In YSZ the following reactions can occur:

VO + MZr [VO MZr]

and: [VO MZr] + MZr [MZr VO MZr] ,

where MZr is either Y3+ or Tb3+.

Hence, we have a Faradaic diffusion process coupled with a chemical reaction!Hence, we have a Faradaic diffusion process coupled with a chemical reaction!

V RTF

a aa

RTFa

a RTFcW cx L= =

+= =

2 2 2

0

0 0 0ln

Relation voltage-concentration, with :W d ad c

=lnln

SSI-17 Workshop

28 June 09

There is life in Laplace space!

Laplace transformations:

Ficks second law:

General solution:

Boundary conditions (symmetry):

Impedance:

sc s Dd c sdx

kc s( ) ~ ( ) ( )= 2

2

c s A x k s D B x k s D( ) cosh ( ) / ~ sinh ( ) / ~= + + +

A c s B L k s Dx L= = +=0, ( ) sinh ( ) /~

Z s Vi

RTWF c

L k s D

k s D( )

tanh ( ) / ~

( ) ~= =

+

+

4 2 0

Which is not yet the Gerischer !!

SSI-17 Workshop

28 June 09

Resulting impedance:

Fractal form, n < 0.5:

Double fractal form: ( 1)

Resolving the complications

0( )( )

nZZ

k j =

+

Step 1: replace s by: + j. For steady state, 0.Step 2: tanh(x) 1 for large x, hence for L2k/D > 100,

or: D < 0.01 L2k.~~

Z RTW

F c Dk j( ) ~ = +

4 2 01

2b g

Z Z k j n( ) ( ) = + 0 Compare with the dielectric Havriliak-Negami function:

Compare with the dielectric Havriliak-Negami function:

0

11 ( )j

= +

+

SSI-17 Workshop

28 June 09Some results for YSZ-Tb

Lower Eact for e at higher temperatures

X0 Eact [kJ/m]eT 160 [SK/cm] 44G-Y0 6.210-7 [Ssn] -90K 4.9107 [s-1] 125

YSZ-Tb27.3YSZ-Tb27.3

YSZ-Tb27.3YSZ-Tb27.3

SSI-17 Workshop

28 June 09Polarisation dependence

Slopes of Rdiff. are close to F/RT .W-Y0 shows exponential dependence on V.

Slopes of Rdiff. are close to F/RT .W-Y0 shows exponential dependence on V.

Polarisation dependence of major diffusion sub-circuit, BiCuVOx/Ag,O2 electrode.

SSI-17 Workshop

28 June 09

Emerging model from data analysis of electrode dispersions.

Electrolyte surface is active !

Electrode model for BiCuVOx

SSI-17 Workshop

28 June 09Praise of the time domain

Intercalation cathode.

Change of potential = change of aA at the interface, hence A-diffusion:

00,ln)(

A

xA

aa

nFRTtE ==

Voltage-activity relation:0

d ( , )( )dA

Ax

C x tJ t Dx =

=

Fick 1 & 2, boundary conditions+ Laplace transform:

0( )() coth( )

ZV jZ lI Dj D

= =

SSI-17 Workshop

28 June 09Real cathode: LixCoO2

Z' [k]

0.0 2.5 5.0 7.5 10.0 12.5 15.0

Z" [k

]

0.0

2.5

5.0

7.5

10.0

3.78V

4.06V

3.69V

Rsl Rct W Re

Qsl Cdl

20Hz

27Hz

110Hz

MeasurementSimulation

3.84V

3.88V

36Hz

63Hz

Z' [k]

0.0 2.5 5.0 7.5 10.0 12.5 15.0

Z" [k

]

0.0

2.5

5.0

7.5

10.0

3.78V

4.06V

3.69V

Rsl Rct W Re

Qsl Cdl

20Hz

27Hz

110Hz

MeasurementSimulation

3.84V

3.88V

36Hz

63Hz

IS of a RF-film electrode: () fresh; () charged; ( ) intermediate SoCs. (+) CNLS-fit. Range: 0.01 Hz 100 kHz.

Peter J. Bouwman, Thesis, U.Twente 2002.

LiCoO2, RF film on silicon.

SSI-17 Workshop

28 June 09Diffusive part?

The lithium diffusion process is found at lower frequencies!

Compare the potential-step response time with lowest frequency of EIS experiment:

teq. >> 3000 s (~ 0.3 mHz) fmin ~ 10 mHz

Time [s]0 1000 2000 3000

Cur

rent

[A

.cm

-2]

0

1

2

3

4

5

6

7

8

9

10

3.80V3.85V

3.90V

3.95V

4.05V

4.10V

4.15V

4.20V

4.00V

Time [s]0 1000 2000 3000

Cur

rent

[A

.cm

-2]

0

1

2

3

4

5

6

7

8

9

10

3.80V3.85V

3.90V

3.95V

4.05V

4.10V

4.15V

4.20V

4.00V

Current response of a 0.75m RF-film to sequential 50mV potential steps from 3.80V to 4.20V.

MEASURE RESPONSE IN THE TIME DOMAIN!

MEASURE RESPONSE IN THE TIME DOMAIN!

SSI-17 Workshop

28 June 09Fourier transform

0

( ) ( ) j tX X t e dt

=

Fourier transform of a temporal function X (t):

Impedance:( )( )( )VZI

=

E.g. with a voltage step, V0: 0( )VVj

=

Model function: Laplace transform of transport equations and boundary conditions, with p = s +j . Set s = 0: impedance

SSI-17 Workshop

28 June 09Fourier Transform

Two problems with F-T:

Data is discrete: approximate by summation (X =at + b)

Data set is finite (next slide)

Very Simple Summation Solution (VS3):

X X t X t a t t

j X t X t a t t

i i i i i ii

N

i i i i i ii

N

( ) sin sin cos cos

cos cos sin sin

= + LNMOQP +

LNMOQP

=

=

1 1 11

1

1 1 11

1

b g

b g

XiXi -1

titi -1

X(t )=at + b

SSI-17 Workshop

28 June 09Simple exponential extension

Q t Q Q e t( ) /= + 0 1

X X t Q t jQ

Q e e tj tt

t

t

j tN

N

( ) [ ( ) ] /

= +

z z00

01e d d

Q e e t Q et t

jt tt

t

j t t N N N N

N

N1 1

1

2 2

1

2 2

z = + + ++RST

UVW/ / cos sin cos sin

d

Assume finite value, Q0, for t , this value can be subtracted before total FT.

Fit exponential function to selected data set in end range:

Full Fourier Transform:

Analytical transform of exponential extension:

SSI-17 Workshop

28 June 09Simple exponential decay

The effects of truncation can be studied using a simple exponential decay:

Fourier transform (from t = 0 to t = ):

( ) exp( / )( 0) 0 and ( 0) 1I t tV t V t

= < = =

2 2

1 1 1( ) , ( ) , ( ) 11jI V Z j

j

= = = +

Compare with truncated transform at tmax.

SSI-17 Workshop

28 June 09

-10

-8

-6

-4

-2

0

2

4

6

8

10

0.01 0.1 1 10 100

Frequency, [Hz]

re, i

m,

[%]

Delta-RealDelta-Imag tmax = 100 s, = 20 sec, Y (tmax) = 0.67%

Effect of truncation

-10

-8

-6

-4

-2

0

2

4

6

8

10

0.01 0.1 1 10 100

Frequency, [Hz]

re, i

m,

[%]

Delta-RealDelta-Imag tmax = 100 s, = 30 sec, Y (tmax) = 3.6%

,( ) ( )( )

real re trre

Z ZZ

=

,( ) ( )( )

imag im trim

Z ZZ

=

-10

-8

-6

-4

-2

0

2

4

6

8

10

0.01 0.1 1 10 100

Frequency, [Hz]

re, i

m,

[%]

Delta-RealDelta-Imag

tmax = 100 s, = 40 sec, Y (tmax) = 8.2%

SSI-17 Workshop

28 June 09Fourier transformed data

/10)(

teXXtX +=

( )

=

=

N

k kk

kk

ttj

tjttttXtX

tetXXN

1 1

1

0

sincos)()(

d)()(

Simple discrete Fourier transform:

Correction / simulation for t:

X 0 = leakage current.

Impedance:( )( )( )V tZI t

=

SSI-17 Workshop

28 June 09V-step experiment

Sequence of 10 mV step Fourier transformed impedance spectra, from 3.65 V to 4.20 V at 50 mV intervals. Fmin = 0.1 mHz

SSI-17 Workshop

28 June 09CNLS-fit of FT-data

R1 : 550 0.5 %R2 : 49 10 %Q3, Y0 : 6.810-3 12 %,, n : 0.96 8 %

O4, Y0 : 0.047 1.5 %,, B : 30 2.4 %

T5, Y0 : 0.028 2.9 %,, B : 5.9 2.9 %

Circuit Description Code:

R(RQ)OT *)Fit result:2CNLS = 3.710-5

*) O = FLW

T = FSW

SSI-17 Workshop

28 June 09Bode Graph

Double logarithmic display almost always gives excellent result !

Bode plot, Zreal and Zimag versus frequency in double log plot

SSI-17 Workshop

28 June 09More Fourier transform

Method Martijn Lankhorst:

fit polynomials to small setsof data points (sections):

analytical transformation to frequency domain:

More general extrapolation function (stretched exponential):

P t A tm tt

kk

mk

q

r( ) ==

0

P Aii k

t e t e

jtt

ik

i

i

mqi k j t

ri k j t

kqr

q r

( )( )!

( )! ( )

=

=

+

=

+1

1

0

1 1

1

11

piece wiseintegrationpiece wiseintegration

( ) 10,)( /10 += teQQtQ

(Fourier transform complicated, can be done numerically)

SSI-17 Workshop

28 June 09Non linear effects

-0.1

-0.05

0

0.05

-0.2 -0.1 0 0.1 0.2

Polarisation, [V]

Curr

ent,

[A]

-0.1

-0.05

0

0.05

-0.2 -0.1 0 0.1 0.2

Polarisation, [V]

Curr

ent,

[A]

(1 )

0

a aF FRT RTI I e e

=

Electrode response based on Butler-Vollmer:

When the voltage amplitude is too large, the current response will contain higher harmonics (i.e. is not linear with V).

I0 = 1 mAa = 0.4T = 23C

2 2 3 3 2 2 3 3

0

2 2 2 3 3 3

0

1 ... 1 ...2! 3! 2! 3!

( ) ( )( ) ...2! 3!

a a b bI I a b

a b a bI a b

= + + + + + + +

+

= + + + +

Substituting a = aF/RT, b = (1-c)F/RT and a serial expression for exp(), we obtain:

SSI-17 Workshop

28 June 09

At zero bias, with the perturbation voltage, ejt, this equation yields:

This clearly shows the occurrence of higher-order terms. When the polarization current is symmetric the even terms will drop out as a = b. At a dc-polarization the response is more complex:

Higher-order terms

2 2 3 32 3

0( ) ( )( ) ( ) ...

2! 3!j t j t j ta b a bI t I a b e e e

+= + + + +

}

3 3 22 2

0

2 2 3 3 3 32 3

( )( ) ( ) ( ) ...2!

( ) ... ... ...2! 2! 3!

j t

j t j t

a bI t I a b a b e

a b a b a be e

+ = + + + + + + +

+ + + + + +

SSI-17 Workshop

28 June 09Conclusions

Electrochemical Impedance Spectroscopy:

Powerful analysis tool

Subtraction procedure reveals small contributions

Presents more visual information than time domain

Almost always analytical expressions available

Equivalent Circuit approach often useful

Data validation instrument available (KK transform)

Also applicable to time domain data (FT: ultra low frequencies possible)

Able to analyse complex systemsUnfortunately, analysis requires experience!

SSI-17 Workshop

28 June 09Not just electrochemistry!

Data analysis strategy is applicable to any system where:

a driving force

a flux

can be defined/measured.

Examples:

mechanical properties, e.g. polymers: G () or J () & catalysis, pressure & flux, e.g. adsorption

rheology

heat transfer, etc.

No need to measure in the frequency domain!

SSI-17 Workshop

28 June 09Last slide

SSI-17 Workshop

28 June 09Active shielding

1xcoax

1xcoax

I+

I-

Diff.Amp.

System to eliminate added capacitances from cables.

Used by e.g. Solartron 1286/1287

SSI-17 Workshop

28 June 09

Frequency Response Analyser

Multiplier:

Vx(t)sin(t) &

Vx(t)cos(t)

Integrator: integrates multiplied signals

Display result:

a + jb = Vsign/VrefImpedance:

Zsample = Rm (a + jb) But be aware of the input impedance of the FRA!

SSI-17 Workshop

28 June 09

Influence of input impedances

Input impedances can seriously influence the measurements.

For the Solartron FRA:

R = 1 M

C = 47 pF (add cablecapacitance to this!)

SSI-17 Workshop

28 June 09Alternate set-up

Simple set-up for high impedance samples.

Improvement: use pre-amplifiers / impedance transformers.

Assignment:

Work out expression for Zsample in terms of:

a + jb = V1/V2(assume R1 = R2)

Match cable length for left and right

SSI-17 Workshop

28 June 09Dealing with x =

Develop in Taylor series around x = , e.g. for Zreal:

2 2

22

2

( ) ( )

( ) ( )( )( ) ( ) (...) ( )2!

( )( )

im im

im imim im

xZ x Zx

dZ x d Z xxxZ x x x Zdx dx

x x

=

+ + +

+

In the limit of x this reduces to:

2 2

( ) ( ) ( ) ( )1lim2

im im im im

xx

xZ x Z Z dZ xx dx =

= +

In discrete KK transforms, just ignore the point x = !

SSI-17 Workshop

28 June 09Uses for NLS-fit

Virtually all non-linear functions, provided that good initial guesses are available:

Complex impedance/admittance (CNLS-fit)

Conductivity relaxation

Polarisation (I-V ) measurements Permeation measurements

What ever you can think of

SSI-17 Workshop

28 June 09Mixed conductivity Tb-YSZ

Zr0.909-xY0.091TbxO2-gold electrodes

pO2 = 0.9 barpO2 = 0.9 bar

Eact for electronic conduction drops at high temperatures

Act. energies (kJmol-1):x = 0.091 126 (ionic)

67 (electr.)x = 0.182 130 (ionic)

64 (electr.)x = 0.273 56 (electr.)x = 0.364 52 (electr.)

SSI-17 Workshop

28 June 09Results YSZ-Tb36.4

YSZ-Tb364YSZ-Tb364

YSZ-Tb364YSZ-Tb364

X0 Eact [kJ/m]eT 460 [SK/cm] 39G-Y0 910-3 [Ssn] -33K 1.3106 [s-1] 82

SSI-17 Workshop

28 June 09Determining the CDC

( C [ ( Q [ R ( R Q ) ] ) ( C [ R Q ] ) ] )

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