1 daria vladikova iees - bas, 10 acad. g. bonchev st., 1113 sofia, bulgaria centre of excellence...

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Daria VladikovaIEES - BAS, 10 Acad. G. Bonchev St., 1113 Sofia, BULGARIA

Centre of Excellence “Portable and Emergency Energy Sources”

E-mail: [email protected]

IEES-BAS

Centre of Excellence

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IEES-BAS


“a,b,c” Impedance is an introductory course, which aims at giving basic knowledge in the field of electrochemical impedance spectroscopy. The course offers general information concerning: principle of the impedance spectroscopy; basic definitions; structural modelling – main electrical and electrochemical elements and their physical meaning, models of basic electrochemical phenomena.

“a,b,c” Impedance can be regarded as a necessary prerequisite for the next group of lectures on advanced impedance techniques (non-stationary and differential impedance analyses).

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1. ntroduction5

2. What is Electrochemical Impedance Spectroscopy 10

3. Impedance of Electrochemical Systems 13

3.1. Basic Hypotheses 14

3.2. Impedance Presentation and Monitoring

3.3. Advantages and disadvantages of Electrochemical Impedance Spectroscopy 20

4. Main Steps in the Classical Impedance Investigation 21

5. Impedance Models 24

5.1. Impedance Elements 25

5.1.1. Lumped Elements 26

Resistance26

Capacitance28

Inductance 30

5.1.2. Frequency Dependent Elements 32

Warburg Element 32

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Bounded Constant Phase Element 42

5.2. Simple Calculations 46

..Basic Electrochemical elements 50

5.3.1. Main Structures of the Electrochemical Models 50

5.3.2. Model Description Conventions 52

5.3.3. Models without Diffusion Limitations 53

Ideally Polarizable Electrode 53Modified Ideally Polarizable Electrode 54

Polarizable Electrode 55

Modified Polarizable Electrode 58

Faradaic Reaction with One Adsorbed Species 59

5.3.4. Models with Diffusion Limitations 60

Randles Model 60

Modified Randles model 62

5

2001

Established

European Impedance Internet Centre

withbase organization

IEES - BAS

2001 Workshop – ITALY

2002 Workshop – SZECHIA

2003 Sofia Impedance Days

CHAINS EVENTS

Internat. Mycrosymp on EIS

(every 3 years)

Ist – 1987 -RUSSIA

IInd - 1990 - BULGARIA

IIIrd - 1993 - BULGARIA

IVth – 1996 -POLAND

Vth – 1999 - HUNGARY

VIth - 2002 - CZECHIA

VIIth – 2005 – CZECHIA

Internat. Symp. on EIS

(every 3 years)Ist –1989 -

FRANCE

IInd –1992 - USA

IIIrd – 1995 - BELGIUM

IVth – 1998 - BRAZIL

Vth – 2001 - ITALY

VIth – 2004 - USA

VIIth – 2007 - FRANCE

IEES-BAS

Centre of Excellence1. INTRODUCTION

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UNIQUE POSSIBILITY to separate different steps in the total process under investigation

EASY PERFORMANCE of experiments with accessible digital instrumentation

MATURITY in the software exploitation

EASY PERFORMANCE of VIRTUAL impedance data analysis (e- data analysis)

From SCIENTIFIC point of view From APPLIED point of view

IMPEDANCEIMPEDANCEOFFERS IMPORTANT ADVANTAGES

• COVERS WIDE RANGE OF OBJECTS

• SERVES & UNITES a great variety of research & applied areas

- batteries - fuel cells& materials- semiconductors - ceramics- biosensors - biological objects- nano-particles - materials testing- corrosion & inhibitors - crystallization etc

IEES-BAS

Centre of Excellence1. INTRODUCTION

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BASIC IMPEDANCE LITERATUREBASIC IMPEDANCE LITERATURE

IEES-BAS


1. INTRODUCTION

D. C. Graham, Chem. Rev., 1947, 41, 441. P. Delahay, New Instrumental Methods in Electrochemistry, 1965, Wiley-Interscience, New York. P. Delahay, Double Layer and Electrode Kinetics, 1965, Wiley-Interscience, New York. D. E. Smith, Electroanalytical Chem. 1966, 1,1(Eds. A. J. Bard, Marcel Dekker), New York. M. Sluyters-Rehbach and J. H. Sluyters in On the impedance of galvanic cell. The potential dependence of the faradaic parameters for electrode processes with coupled homogeneous chemical reactions, Electroanalytical Chem. 1970, 4,1(Eds. A. J. Bard, Marcel Dekker), New York. J. R. Macdonald in Superionic Conductors, (Eds. G. D. Mahan, W. L. Roth), Plenum Press, New York, 1976, p.81. J. R. Macdonald in Electrode Processes in Solid State Ionics, (Eds. M. Kleitz and J. Dupuy), Reidel, Dordrecht, Holland, 1976, p.149. M. C. H. McKubre and D. D. Macdonald in A Comprehensive Treatise of Electrochemistry, (Eds. J. O’M Bockris, B. E. Conway and E. Yeager), Plenum Press, New York, 1977. D. D. MacDonald, Transient Techiques in Electrochemistry, Plenum Press, New York, 1977. R. D. Armstrong, M. F. Bell and A. A. Metcalfe, Electrochem. Chem. Soc. Spec. Rep. 1978, 6, 98. W. I. Archer and R. D. Armstrong, Electrochem. Chem. Soc. Spec. Rep. 1980, 7, 157. C. Gabrielli, Identification of Electrochemical Process by Freguency Response Analysis, Monograph Reference 004 /83, Solartron Instr.Group, Farnsborough, England, 1980. D. D. Macdonald and M. C. H. McKubre, Electrochemical Impedance Technigues in Corrosion Science: Electrochemical Corrosion Testing, STP 272, ASTM, Philadelphia, PA, 1981. J. R. Macdonald, IEEE Trans. Electrical Insulation EI-15, 1981, 65. D. D. Macdonald and M. C. H. McKubre, Modern Aspects of Electrochemistry, (Eds. J. O’M Bockris, B. E. Conway and R. E. White), Plenum Press, New, 1982, 14, 61. M. Sluyters-Rehbach and J. H. Sluyters in Comprehensive Treatise of Electrochemistry, (Eds. E. Yeager, J. O.’M. Bockris, B. E. Conway and S. Sarangapani), Plenum Press, New York, 1984, p. 177.

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BASIC IMPEDANCE LITERATUREBASIC IMPEDANCE LITERATURE

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1. INTRODUCTION

C. Gabrielli, Identification of Electrochemical Processesby Frequency Respose Analysis, Technical Report № 004, Solartron, Hampshire, 1984.(can be dounloaded from http://accessimpedance.iusi.bas.bg) J. R. Macdonald (Ed.), Impedance Spectroscopy - Emphasizing Solid Materials and Systems, Wiley-Interscience, New York, 1987. C. Gabrielli, Use and Applications of Electrochemical Impedance Tecniques, Technical Report № 024, Solartron, Hampshire, 1990. Z. Stoynov, B. Grafov, B. Savova-Stoynova and V. Elkin, Electrochemical Impedance, 1991, Publishing House Science, Moscow (in Russian). D. D. Macdonald in Tecniques for Characterization of Electrodes and Electrochemical Processes, (Eds. H. R. Varma and J. R. Selman, J.Wiley&Sons), New York, 1991, p.515. F. Mansfeld and W. J. Lorenz in Tecniques for Characterization of Electrodes and Electrochemical Processes, (Eds. H. R. Varma and J. R. Selman, J.Wiley&Sons), New York, 1991, p.581. C. M. A. Brett and A. M. Oliveira Brett, Electrochemistry, Principles, Methods and Applications, 1993, Oxford University Press. A. Lasia, Electrochemical Impedance Spectroscopy and Its Applications, Modern Aspects of Electrochemistry, B. E. Conway, J. Bockris, and R. White, Edts., Kluwer Academic/Plenum Publishers, New York, 1999, Vol. 32, p. 143-248. http://www.wkap.nl/prod/b/0-306-45964-7 Second International Symposium on Electrochemical Impedance Spectroscopy, Electrochimica Acta, 38, 14, 1993. Third International Symposium on Electrochemical Impedance Spectroscopy, Electrochimica Acta, 41, 7/8, 1996. EIS’98 Proceedings – Impedance Spectroscopy” Electrochimica Acta, 44, 24, 1999. Fifth International Symposium on Electrochemical Impedance Spectroscopy, Electrochimica Acta, 47, 13/14, 2002.R. Cottis and St. Turgoose, Electrochemical Impedance and Noise, Eds. B. C. Syrett, NACE International, 1440, South Greek Drive, Houston, TX77084, 1999.

http://www.wkap.nl/prod/b/0-306-45964-7

http://www.sciencedirect.com/science?_ob=IssueURL&_tockey=%23TOC%235240%231996%23999589992%2318707%23FLT%23Volume_41,_Issues_7-8,_Pages_957-1399_(6_May_1996)&_version=1&_urlVersion=0&_userid=0&md5=640b43b8fa1728b7d7fc82c4de34aed9

http://www.sciencedirect.com/science?_ob=IssueURL&_tockey=%23TOC%235240%231999%23999559975%23111111%23FLA%23Volume_44,_Issue_24,_Pages_4113-4464_(31_July_1999)&_version=1&_urlVersion=0&_userid=0&md5=2bbba246c6f785bf1a6caf47797af103

http://www.sciencedirect.com/science/journal/00134686

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2.2. WHAT IS ELECTROCHEMICAL WHAT IS ELECTROCHEMICAL IMPEDANCE SPECTROSCOPYIMPEDANCE SPECTROSCOPY

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The Electrochemical Impedance Spectroscopy is based on the classical method of the TRANSFER FUNCTION (TF)

Linear System Sinwave input

x( i) = A sin t

Sinwave output

y( i) = B sin (t

Principle:

1. If the system under investigation is LINEAR (LS),

• LS is perturbed with sinwave input x(i) and the response y( i) is measured;

• The response y( i) is also sin wave with the same freqiency and different amplitude and phase;

2. The ratio output / input signal determines the complex transfer coefficient for the corresponding frequency:

k( i) = y( i) / x( i)

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2.2. WHAT IS ELECTROCHEMICAL IMPEDANCE WHAT IS ELECTROCHEMICAL IMPEDANCE

SPECTROSCOPYSPECTROSCOPY

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k( i) = y( i) / x( i)

Linear System

Principle:

3. If the input is composed by sin wave signals X (ii) and the output Y (ii) – by the same set of frequencies (1- n), the ratio between the two vectors

H(ii) = Y (ii) / X (ii)

is the Transfer Function H(ii)

TF describes the frequency dependence of the transfer coeffcient k(i)

4. The transfer from the time-domain to the frequency domain is performed by LAPLAS transform. For steady state linear systems it is replaced by FOURIER transform

Complex number (Re; Im)

Depends on and the object’s properties

X (ii) = Sinwave input (1- n) Y (ii) = Sinwave output (1-

n)

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2.2. WHAT IS ELECTROCHEMICAL IMPEDANCE WHAT IS ELECTROCHEMICAL IMPEDANCE

SPECTSCOPYSPECTSCOPY

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Principle:

5. TF is impedance H(ii) = Z (ii)

when the input signal is current (I) and

the output signal is voltage (U)

6. TF is admittance H(ii) = Y (ii) = Z-1 (ii)

when the input signal is voltage (U) and

the output signal is current (I)

7. When the frequency range (1- n)is large and covers all the properties of the system, the system is observable, otherwise S is partially observable.

Conclusion:

The Transfer Function H(ii) describes totally a linear, steady-state and observable system.

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3.3. IMPEDANCE OF ELECTROCHEMICAL IMPEDANCE OF ELECTROCHEMICAL SYSTEMSSYSTEMS

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Electrochemical systems behave as big, non-linear, non-steady state, semi-irreversible systems with distributed parameters in macro-and micro scales. During operation or investigation processes of mass- and energy transfer take place.

Obviously the application of the TF approach needs a number of simplifications and assumptions. They are generalized in few

BASIC WORKING HYPOTHESES

They can be divided in 2 groups:

1. Working hypotheses from system analysis point of view

2. Working hypotheses from electrochemical point of view.

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3. 1.3. 1. Basic Working Hypotheses Basic Working HypothesesIEES-BAS


2.1.1. Working hypotheses from system analysis point of view

I. Linearity: This requirement is fulfilled if the input perturbation signal is small enough to keep the state of the investigated system unchanged. The requirement for a small signal covers the potential, the current, as well as the quantity of electricity for half a period

(very important at low frequencies!).

•Verification of the hypothesis for linearity: the measured impedance should not depend on the amplitude of the signal.

•Experimental verification: Consecutive impedance measurements in the full frequency range with decreasing amplitude and analysis of the weighted differences.

•Small signal : - depends on the investigated system; in some cases 3-5 mV, in others 50-100 V

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II. Causality: This requirement means that all the changes in the investigated system are caused by the perturbation signal, i.e. the output signal is a result of only the input signal and can not appear if there is no input signal.

III. Single input, single output: This requirement could be achieved if the rest of the parameters (temperature, concentration, d.c. signal, pH etc.) are kept constant by passive or active conditioning.

IV. Observability: This requirement postulates, that all the phenomena under study can be observed in the measured frequency range.

3. 1.3. 1. Basic Working Hypotheses Basic Working Hypotheses

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V. Lack of memory effects: This requirement means that the investigated system dos not “remember” the history of the experiment. That means that the result does not depend on the order of the measurements.

This property could be expected form entirely reversible systems.

•Experimental verification: Performance of two consecutive impedance measurements - the one with scanning from high to low frequencies and the other – on the opposite – from low towards high frequencies, followed by and analysis of the weighted differences.


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2.1.1. Working hypotheses from electrochmical point of view

• Additiveness of the Faradaic current and the charging current of the double layer

• Electrical neutrality of the electrolyte – the total density of the charges in every point of the solution is zero

• Lack of convection and migration – i.e. there are no changes in the local concentration of the electrolyte

• Lack of lateral mass and charge fluxes at the electrode surface.

Conclusion:

• The performance of a precise impedance investigation strongly depends on the correctly organized experimental setup, experimental conditions and measuring technique and on the careful preparation of the object.

• Some of the electrochemical simplifications help for the construction of the models


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3. 2.3. 2. Impedance Presentation and MonitoringImpedance Presentation and MonitoringIEES-BAS


3.2.1. Impedance presentation

Z (ii) = Y (ii) / X (ii) = U (ii) / I (ii) complex number

Presentation in Cartesian coordinates:

Z (ii) = Rei + iImi

i = (-1)1/2; i = 1, 2 ,….n – denotes the frequency range

Nd – frequency density (measured frequencies in one decade; 3-5 for screening; 10-15 for precise measurements;

Down scanning – from high to low frequencies

Presentation in Polar coordinates:

Z (ii) = =Z

Z= (Rei2 + iImi

2)1/2 - modulus ; i = Arc tan Imi/Rei - phase

ije

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3. 2.3. 2. Impedance Presentation and MonitoringImpedance Presentation and MonitoringIEES-BAS


3.2.1. Impedance monitoring (graphical visualization)

The problem of impedance monitoring comes from the 3-dimensional nature of the data, which should be plotted in a 2-dimensional pattern.

The most common presentations are the complex plane (Nyquist) plot (in Cartesian coordinates) and Bode plots (in polar coordinates).

Bode plots

Recalculated 3Dset of data:D3 [ i, Zi, i i = 1,

2, ..nCoordinates:

xi = lg i ; y1i = Zi;, y2i = i

Complex plane (Nyquist) plot

Experimental 3D set of data:D3 [ Rei, Imi,ii = 1,

2,..n

Coordinates:xi = Re; yi = - Im-Im

ReR1 R

2

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3.3.3.3. ADVANTAGES AND DIADVANTAGES OF ADVANTAGES AND DIADVANTAGES OF ELECROCHEMICAL IMPEDANCE SPECTROSCOPYELECROCHEMICAL IMPEDANCE SPECTROSCOPY

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From one side the impedance (or admittance) functions contain all the information for the investigated system (if the working hypotheses are fulfilled at the selected working point).

From another side this information has to be extracted from the data, i.e. the data analysis is an identification procedure.

Advantage: The electrochemical impedance has the unique possibility to separate the kinetics of the different steps involved in the total process under investigation, because as a transfer function it is a local, linear an full description of the system under study. A number of processes are taking place, caused by the perturbation signal. The impedance, however, does not measure them, i.e. it is not a physical reality, but information property of the object.

Disadvantage: Since impedance is not a physical reality, the interpretation of the experimental data is based on the construction of a working model, following a preliminary working hypothesis, which should be identified. This introduces a subjective component in the analysis.

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Centre of Excellence4. MAIN STEPS IN THE CLASSICAL IMPEDANCE 4. MAIN STEPS IN THE CLASSICAL IMPEDANCE

INVESTIGATIONINVESTIGATION

I STAGE

DATAMONITORING

MEASUREMENT D3 [ Rei, Imi, i]

-Im

ReR1 R

2

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II STAGE – DATA ANALYSIS

Choice of a Hypothetical Model

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4. MAIN STEPS IN THE CLASSICAL IMPEDANCE 4. MAIN STEPS IN THE CLASSICAL IMPEDANCE INVESTIGATIONINVESTIGATION

P = Par.Ident. {i, Rei, Imi, I M [S] }

estimated parameters

supposed model

Parametric Identification

(CNLS)

1. Zi = Simulation {M [S,P] I i, } given

2. Choice of measure for proximity (distance between measured and estimated data)

3. Evaluation of the distance for “Best fit”

Model Validation

Data Analysis will be given in the lectures on

the Workshop

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( D3 [i, Rei, Imi] ) -Im

Z

ReZ

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4. MAIN STEPS IN THE CLASSICAL IMPEDANCE 4. MAIN STEPS IN THE CLASSICAL IMPEDANCE INVESTIGATION - INVESTIGATION - ExampleExample

I. Measurement &Data Monitoring

II. Data Analysis

R1

CPE

R2

3.

R1

CPE

R21.

C

R2

C1

R3

C2

R12.

1. Choice of Hypothetical models

P(S) = R1, R2, C

(estimated values)

P(S) = R1, R2,

C1, R3, C2

P(S) = R1, R2, CPENO

NO

R1

CPE

R2

3.

R1

CPE

R21.

C

R2

C1

R3

C2

R12.

3. Models Validation2. Parametric Identification

(CNLS)

YES

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Centre of Excellence5. IMPEDANCE MODELS5. IMPEDANCE MODELS

There are few approaches for presentation of the impedance models. The electrical circuit modelling approach is very convenient for impedance studies of electrical properties. In this case the electrical circuit has a response identical to that obtained from the measurement of the investigated system.

The electrical circuit can be regarded as a construction of different electrical and electrochemical elements (structural elements) connected under given laws.

If the model is not formal, the values of its elements could give a significant contribution to the physical understanding of the investigated system.

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Centre of Excellence 5.1. IMPEDANCE ELEMENTS5.1. IMPEDANCE ELEMENTS

Impedance elements are described with one or more parameters, which determine their dimensions.

Impedance elements can be divided it 2 basic groups:

• Lumped elements: resistance R; capacitance C; inductance L. They are directly adopted form electrotechniques, i.e. they are electrical elements and can describe homogeneous systems.

•Frequency dependent elements – they describe frequency unhomogeneity. They are developed for descrption of some electrochmical processes, i.e. they are electrochemical elements.

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5.1.1. LUMPED ELEMENTS 5.1.1. LUMPED ELEMENTS

RESISTANCE RESISTANCE R R

R is the simplest modelling element

1. Modelling in the time (t) domain – follows Ohm’s Law:

UR=R.I

(UR - voltage drop; I – current)

Dimensions: ohm (Ω) = VA-1 = m2kgA-2s-3

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RESISTANCE RESISTANCE R R

3. Physical meaning:• description of: energy losses; dissipation of energy;

potential barrier; electronic conductivity or conductivity of very fast carriers

• Electrolyte resistance - Zs(i) = Rs - for water based electrolytes

• Ohmic resistance - RRsRm (Rm - R of metallic leads)

2. Modelling in the frequency () domain:

ZR (i) = R only real part (Re=R;

Im = 0)

RR

0 100 200 300 4000

100

200

-Im

/

Re /

Impedance diagram in the frequency range 105-10 Hz

0 100 200 300 4000

100

200

-Im

/

Re / 0 100 200 300 400

0

100

200

-Im

/

Re / 0 100 200 300 400

0

100

200

-Im

/

Re /

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CAPACITANCE CAPACITANCE CC

)t(Udt)t(iC

U oC

t

tC

o

1

1. Modelling in the time (t) domain – Capacitance C can be regarded as a proportionality coefficient between the voltage Uc and the integral of the current i running through the capacitance :

Dimensions: F = sΩ-1

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CAPACITANCE CAPACITANCE CC

3. Physical meaning:

• modelling of : mass and charge accumulation, dielectric polarization, integral relation between parameters;

•Double layer capacitance Cdl . The impedance of the double layer has a capacitive character.

0 5 10

0

5

-Im

/

Re /

C C 1 1 = 1E-4= 1E-4

C C 2 2 = 1E-3= 1E-3


CC

0 5 10

0

5

-Im

/

Re / only imaginary part (Re = 0) The impedance decreases with the increase of the frequency.


11C C)i(C)(i)(iZ

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INDUCTANCE INDUCTANCE LL

1. Modelling in the time (t) domain –

The Inductance L can be regarded as a proportionality coefficient between the voltage UL and the derivative of the current i:

Dimensions: H = Ωs

dt

tdiLU L

)(

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INDUCTANCE LINDUCTANCE L

3. Physical meaning:

• Modelling of : self inductance of the connecting cables, the measuring cell and investigated objects, self inductance of current flow or of charge carriers movement;

• accumulation of magnetic energy;


0 5 10-5

0

-Im

/

Re /

LLL L 1 1 = 1E-4= 1E-4

L L 2 2 = 1E-3= 1E-3

0 5 10-5

0

-Im

/

Re / only imaginary part (Re = 0) The impedance increases with the increase of the frequency.


L)(iZL i .

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5.1.2. FREQUENCY DEPENDENT ELEMENTS 5.1.2. FREQUENCY DEPENDENT ELEMENTS

WARBURG ELEMENT WWARBURG ELEMENT W

1. Modelling in the time (t) domain

Warburg element (1896 year) is the first electrochemical element introduced for impedance description of linear semi-infinite diffusion, which obeys the second Fick’s low:

)/(/ 22 xcDtc

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0 6 120

6

12

-Im

/ k

Re / k

Impedance diagram in the frequency range 105-10 Hz with W = 100 ohm.s1/2

WW


Re = Im - phase shift = 450 and frequency independent

W is a proportionality coefficient known as Warburg

coeffcient

Dimensions: Ωm2s1/2

)i(W)i(W)i(Z //W 12121

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REMARK: The applied frequencies should ensure conditions at which the sin wave should not reach the end of the diffusion layer.

Thus Warburg impedance is a one port element – it has only one input. This property does not allow the introduction of another element after Warburg impedance.

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CONSTANT PHASE ELEMENT CPECONSTANT PHASE ELEMENT CPE

1. Modelling :

• CPE represents an empirical relationship;

• CPE describes frequency dependent impedance caused by surface roughness or non-uniformly distributed properties of the irregular electrode surface.

A – proportional factor [Ωm-2sn];

n – exponential coefficient (CPE exponent) corresponding to the phase angle n п/2

n1CPE )(iA)(iZ

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0 20 400

20

-Im

/

Re /

nn = 0.45= 0.45

nn = 0.2= 0.2

nn = 0.8= 0.8

nn = 0.9= 0.9

CPECPE

Impedance diagram in thefrequency range 105-10 Hz

0 20 400

20

-Im

/

Re / 0 20 40

0

20

-Im

/

Re / 0 20 40

0

20

-Im

/

Re /

2. CPE is a generalized element . Its properties depend on the value of n.• for n = 0.5 .CPE

corresponds to diffusion with deviations from the second Fick’s law;

• for n = 0 CPE models distorted resistance( n < 0 is related to inductive energy accumulation);

• for n = 1 – CPE models distorted capacitance;

• for n = -1 + CPE gives distorted inductance.

• For integer values of n ( n = 1, 0, -1) CPE models respectively the lumped elements C, R and L.

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Centre of Excellence5.1.2. FREQUENCY DEPENDENT ELEMENTS 5.1.2. FREQUENCY DEPENDENT ELEMENTS


3. Physical meaning of CPE.

CPE may have direct physical meaning :• the generalized resistance n = 0 - 0.2 may model conductance of

ionic clouds or conductance connected with accumulation of magnetic or electrostatic energy;

• the generalized capacitance n = 0.8 - 1 may model surface roughness of the electrode or distribution of the charge carrier density, i.e. a double layer with complicated stricture;

• The generalized Warburg n = 0.4 - 0.6 may present non-ideal geometry of the diffusion layer; presence of migration or convection; diffusion connected with energy loses or accumulation of charges; constrains of the host matrix to the diffusion of species,unhomogeneous diffusion;

CPE may be also used for formal better modelling of an external similarity with the measured impedance.

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3. Physical meaning of CPE.

REMARK: In general CPE is semi-infinite element. It models the impedance of homogeneous semi-infinite layer, i.e. of a layer with a thickness bigger than the penetration depth of the perturbation signal. Thus the CPE has only an input with the exception in the cases when n = 1, 0, -1 and CPE has the features of lumped elements.

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BOUNDED ELEMENTSBOUNDED ELEMENTS

In real systems very often at low frequencies the perturbation signal penetrates to the end of the layer, which behaves as a layer with a finite thickness. For more precise modelling of such systems bounded electrochemical elements are introduced.

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BOUNDED WARBURG ELEMENT BWBOUNDED WARBURG ELEMENT BW

1. Modelling

Bounded Warburg element describes the impedance of a linear diffusion in a homogeneous layer with finite thickness:

R0 is the total resistance[Ω] of the layer at = 0

At high frequencies ( ) BW behaves as

Warburg element.

2

1

2

2

02

1

BW

Rith((i)(iZ )W

W

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BOUNDED WARBURG ELEMENT BWBOUNDED WARBURG ELEMENT BW

0 100 200 300 4000

100

200

-Im

/

Re / 0 100 200 300 400

0

100

200

-Im

/

Re / 0 100 200 300 400

0

100

200

-Im

/

Re / 0 100 200 300 400

0

100

200

-Im

/

Re /

BWBW

R R 0 0 = 100= 100

R R 0 0 = 200= 200

R R 0 0 = 300= 300

R R 0 0 = 400= 400

W = 0.01W = 0.01


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BOUNDED BOUNDED CONSTANT PHASE ELEMENT BCPCONSTANT PHASE ELEMENT BCP

1. Modelling

BCP represents the impedance of a bounded homogeneous layer with CPE behaviour of the conductivity in the elementary volume and a finite conductivity R0 at d.c. ( ):

n and A are the CPE coefficients.

0

))A(ith(R)(iA)(iZ n0

n1BCP

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4. Properties of BCP element – the most generalized element

0 100 200 300 4000

100

200

-Im

/

Re / 0 100 200 300 400

0

100

200

-Im

/

Re / 0 100 200 300 400

0

100

200

-Im

/

Re / 0 100 200 300 400

0

100

200

-Im

/

Re /

R R 0 0 = 100= 100

R R 0 0 = 200= 200

R R 0 0 = 300= 300

R R 0 0 = 400= 400

nn = 0.45 = 0.45A = 0,01A = 0,01

BCPBCP


•For high enough frequencies a BCP

tends to the classical CPE:

The error is small.

• For frequencies below a given limit b

the element displays a behavior of pure

resistance R0:

The frequency limits a and b are obtained

with a relative error 1%:a =(2.7A-1 R0

-1)n-1

b =(0.14A-1 R0-1)n-1

Z)(Z CPEBCP a

R)(Z0bBCP

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•criterion for verification of BCP:

( and are the angles of the diagram’s asymptotes respectively at low and high frequencies.

4. Properties of BCP element

200 400

200

b

-Im

/

Re /

a

R R o o = 400= 400

A = 0.01A = 0.01n = 0.3n = 0.3

BCPBCP


= 2n/2)

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BOUNDED BOUNDED CONSTANT PHASE ELEMENT CONSTANT PHASE ELEMENT BCPBCP

Remark:

BCP can be applied for n = 0 – 0.6 because of the initial assumption that the investigatd object is regarded as a conductor.

Obviously at higher values for n the system demonstrates capacitive behaviour.

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5.2. SIMPLE CALCULATIONS5.2. SIMPLE CALCULATIONS

EXAMPLE ONEXAMPLE ON R R ANDAND C C ELEMENTS ELEMENTS

The impedance calculations of the combinations of elements follows some rules:

1. When the elements are connected in series, their impedance are added to each other:

2. When the elements are connected in parallel, their admitances, i.e. the reciprocals of the impedance are added :

ZIPE (i) = ZR(i) + ZC(i)

1/Z PE(i) =1/ ZR(i) +1/ ZC(i)

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Connections between elements:

R + C

ZC(i) = -iC)-1ZR(i) = R

C

R

Series

connection:

Z (i) = ZR(i) + ZC(i)

RCParallel

connection:

1/Z (i) =1/ ZR(i) +1/ ZC(i)

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C

R

Series connection:

Z (i) = ZR(i) + ZC(i)

Z (i) = ZR (i) + ZC (i)

= R + (iC)-1 = R - i(C)-1

100

100

-Im

Z, O

hm

ReZ, Ohm

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)ωi(Z

1

)ωi(ZR

1 +

)ωi(ZC

1

Z (i) = + i 221 Tω

ωRT

221 Tω

R

0 2000

200

10Hz

- Im

, Re,

1Hz

Paralel connection:

1/Z (i) =1/ ZR(i) +1/ ZC(i)ZC

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5.5.33.. BASIC ELECTROCHEMICAL MODELSBASIC ELECTROCHEMICAL MODELS5.3.1. 5.3.1. MAIN STRUCTURES OF MAIN STRUCTURES OF

ELECTROCHEMICAL MODELSELECTROCHEMICAL MODELS

1. Voigt’s Structure – consists of meshes with impedances Zk (i), connected in series.

)i(Z)i(Z k

The flowing current is equal for all meshes. The phenomena modelled by each mesh start instantaneously. Their rates depend on their own time-constants.

Voigt’s model structure is applied for impedance description of solid state samples

R1

C1

R2

C2

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5.5.33. BASIC ELECTROCHEMICAL MODELS. BASIC ELECTROCHEMICAL MODELS5.3.1. 5.3.1. MAIN STRUCTURESMAIN STRUCTURES OF OF ELECTROCHEMICAL MODELS ELECTROCHEMICAL MODELS

1. Ladder Structure – consists of a number of kernels corresponding to the modelled phenomena. The modelled phenomena occur consequently.

The model has a typical “ladder”structure

2. Application – description

of processes at the electrode interface.

C1

C2

R2

R1

Z(i) = Z1(i) + {Z2(i) + [Z3(i) + Z4(i) + …)-1]-1}-1

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5.5.33. BASIC ELECTROCHEMICAL MODELS. BASIC ELECTROCHEMICAL MODELS5.3.2. 5.3.2. MODEL DESCRIPTION CONVENTIONS MODEL DESCRIPTION CONVENTIONS

1. Structures:

La: - ladder; Vo: - Voigt

2. Elements: R, C, L, W, BW, CPE, BCP

3. Connections: “ “ – in series; “/” – in parallel

4. Parameters: dimensions in SI ;

delimiters – “;”; multiple parameters separator – “\”

Example: La: R R/CPE par: 120; 200; 0.01\0.9

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5.3.3. 5.3.3. MODELS WITHOUT DIFFUSION MODELS WITHOUT DIFFUSION LIMITATIONS LIMITATIONS – IDEALLY POLARIZABLE – IDEALLY POLARIZABLE

ELECTRODE (IPE)ELECTRODE (IPE)

0 10 20 30 40 500

10

20

-Im

/ k

Re / k0 10 20 30 40 50

0

10

20

-Im

/ k

Re / k0 10 20 30 40 50

0

10

20

-Im

/ k

Re / k0 10 20 30 40 50

0

10

20

-Im

/ k

Re / k

CCDLDL=0.01=0.01

RRSS=10E+3=10E+3

RRSS=20E+3=20E+3

RRSS=30E+3=30E+3

RRSS=40E+3=40E+3

CDL

RS


IPE describes a case when there is absence of any process at the electrode surface.

1. Structure: La: Rs Cdl

The structural elements have a direct physical meaning and correspond to the electrochemical parameters:

• the electrolyte impedance is presented as resistance (Rs)

• the double layer is presented as simple capacitance Cdl

2. Impedance:

ZIPE (i) = Rs – i(Cdl)-1

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5.3.3. 5.3.3. MODELS WITHOUT DIFFUSION MODELS WITHOUT DIFFUSION LIMITATIONS LIMITATIONS – MODIFIED IDEALLY – MODIFIED IDEALLY POLARIZABLE ELECTRODE (MIPE)POLARIZABLE ELECTRODE (MIPE)

The modelling of the double layer with pure capacitance is a simplification, reasonable for concentrated

electrolytes.1. Modified Structure: La: Rs CPEdl

• When the electrode surface is inhomogeneous or rough, the impedance diagram is deformed because of geometrical factors;

• For more complicated structures of the double layer Cdl is presented as CPE of capacitive nature

2. Impedance:

ZMIPE (i) = Rs + A-1 (i)-n

0 1000

100

-Im

/

Re / 0 100

0

100

-Im

/

Re /

RRSS= 100= 100

AA = 0.01 = 0.01 nn = 1= 1

nn = 0.8= 0.8

CPE

RS


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5.3.3. 5.3.3. MODELS WITHOUT DIFFUSION MODELS WITHOUT DIFFUSION LIMITATIONS LIMITATIONS – POLARIZABLE ELECTRODE – POLARIZABLE ELECTRODE

(PE) (PE)

The model of polarizable electrode, known also as simple Faradaic reaction gives a simple impedance description of an electrochemical reaction at the electrode surface.

1. Structure: La: Rs Cdl/ZF

• the current corresponding to the reaction is treated as additive to the current of the double layer charging(working hypothesis);

CDL

RS

ZF

• the model includes additional (Faradaic) impedance in parallel to Cdl

• the model works in the absence of diffusion limitation and the presence of a single step electrochemical reaction. As a result ZF is simplified to a resistance, called charge transfer resistance Rct and the model becomes:

La: Rs Cdl/ Rct

Rct

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(PE) (PE)

0 100 200 300 400 500 6000

100

200

300

-Im

/

Re /

RRss = 100 = 100

CCDLDL= 1E-4= 1E-4

R R ct ct = 200= 200CDL

RS

Rct


2. Impedance diagram:

• Geometrically the impedance diagram is presented as an ideal semicircle with a diameter Rct;

• for the semi-circle intercepts the real axis in Rs

• for the intercept is in a point with value Rs + Rct

• the imaginary component reaches a maximum at the so called characteristic frequency 0

0

0 = (CdlRct) –1 = T-1 (T is the time-constant)

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(PE) (PE)

0 100 200 300 400 500 6000

100

200

300

-Im

/

Re / 0 100 200 300 400 500 600

0

100

200

300

-Im

/

Re / 0 100 200 300 400 500 600

0

100

200

300

-Im

/

Re / 0 100 200 300 400 500 600

0

100

200

300

-Im

/

Re /

RRSS = 100 = 100

CCDLDL= 1E-4= 1E-4

R R ct ct = 200= 200

R R ct ct = 300= 300

R R ct ct = 400= 400

R R ct ct = 500= 500

CDL

RS

Rct


4. Physical meaning – the structural parameters have direct physical meaning (Rs, Rct,Cdl)

• for partially reversible charge transfer reaction at equilibrium

Rct = (RT/nF)(1/I0) (I0 - exchange current)

• Rct depends on the rate of reaction, which is potential dependent and thus Rct vary with the potential, i.e.the diameter of the semi-circle changes.

3. Impedance:

ZPE(i) = Rs + Rct(1 + )-1 -iRctT (1 + )-1

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5.3.3. 5.3.3. MODELS WITHOUT DIFFUSION MODELS WITHOUT DIFFUSION LIMITATIONS LIMITATIONS – MODIFIED POLARIZABLE – MODIFIED POLARIZABLE

ELECTRODE (MPE)ELECTRODE (MPE)

1. Structure: La: Rs CPEdl/ Rct

2. Application: one of the most applied model structures, which describes the depression of the semicircle often observed in real systems.

3. Physical meaning: the application of the MPE model may be a better, but formal description of the investigated system, or it may have a physical meaning – description of the electrode’s surface roughness.

CPE

RS

Rct

0 100 200 3000

100

200

-Im

/

Re /

RRElSElS= 100= 100

R R ct ct = 200= 200

AA = 0.01 = 0.01 nn = 1= 1

nn = 0.8= 0.8


0 100 200 3000

100

200

-Im

/

Re /

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5.3.3. 5.3.3. MODELS WITHOUT DIFFUSION MODELS WITHOUT DIFFUSION LIMITATIONS LIMITATIONS – FARADAIC REACTION WITH – FARADAIC REACTION WITH

ONE ADSORBED SPECIESONE ADSORBED SPECIES

0 200 4000

100

200

-Im

/

Re / 0 200 400

0

100

200

-Im

/

Re / 0 200 400

0

100

200

-Im

/

Re / 0 200 400

0

100

200

-Im

/

Re /

RR1 1 = 50= 50

RR2 2 = 100= 100

CC11 = 1E-3 = 1E-3

RR3 3 = 200= 200

CC22 = 1E- = 1E-22

CC22 = = 33E-3E-3

CC22 = 1E-3 = 1E-3

CC22 = = 33E-E-44


R1

C1

C2

R3

R2

1. Structure: La: R1 C1/R2 C2/R3

The model describes a heterogeneous reaction occurring in two steps with adsorption of the intermediate product and absence of diffusion limitations:

B X + e step I

X P + e step II

2. Physical meaning: the structural parameters have direct physical meaning: R1 = RS; C1 = CDL; R2 =

Rct; C2 = Cad

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5.3.4. 5.3.4. MODELS WITH DIFFUSION MODELS WITH DIFFUSION LIMITATIONS LIMITATIONS – RANDLES MODEL– RANDLES MODEL

1. Structure: La: RS Cdl/Rct W

The model is based on the assumptions of the polarizable electrode with account of the diffusion limitations.

2. Impedance:

ZRNS(i) = RS + [iCdl +

(Rct + W1 iW111

0 50

5

-Im

/ k

Re / k0 5

0

5

-Im

/ k

Re / k0 5

0

5

-Im

/ k

Re / k0 5

0

5

-Im

/ k

Re / k

CDL

RS Rct

W

RRSS=100=100

RRctct= 5E+3= 5E+3

W = 100W = 100CCDLDL=3E-4=3E-4

CCDLDL=1E-3=1E-3

CCDLDL=3E-3=3E-3

CCDLDL=1E-2=1E-2


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5.3.4. 5.3.4. MODELS WITH DIFFUSION MODELS WITH DIFFUSION LIMITATIONS LIMITATIONS – RANDLES MODEL– RANDLES MODEL

3. Relation between the structural parameter W and the electrochemical parameters:

W = Rct[kf(DO)-1/2 + kb (DR)-1/2]

(kf, kb – reaction rates of the “forward”and “backward” reactions; DO and DR – diffusion coefficients of the species)

The structural model has 4 parameters, which can be determined from the impedance (RS, Rct, Cdl, W), while the electrochemical impedance model has 7 parameters (RS, Cdl, I0, kf, kb, DR, DO), which can not be determined directly.

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5.3.4. 5.3.4. MODELS WITH DIFFUSION MODELS WITH DIFFUSION LIMITATIONS LIMITATIONS – MODIFIED RANDLES – MODIFIED RANDLES

MODEL (MRN)MODEL (MRN)

1. Sructure: La: RS Cdl/Rct CPE

MRN describes geometrical or activation inhomogeneity of the surface or deviations from the linear diffusion process. That happens very often when the diffusion occurs in a diluted solution or in case that the diffusion does not obey Fick’s law.

0 200 4000

100

200

-Im

/ k

Re / k0 200 400

0

100

200

-Im

/ k

Re / k0 200 400

0

100

200

-Im

/ k

Re / k0 200 400

0

100

200

-Im

/ k

Re / k

CDL

RS Rct

W CPE

RRSS=20=20

RRctct= 150= 150

CCDLDL=1E-2=1E-2

AA = 0.1 = 0.1nn = 0.5 = 0.5nn = 0.45 = 0.45nn = 0.4 = 0.4nn = 0.3 = 0.3


CDL

RS Rct

W CPECPECDL

RS Rct

W CPECPE BCP

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