1

Steffi Krause,

School of Engineering and Materials Science,

s.krause@qmul.ac.uk

Materials Research Institute

Training Event:

Impedance measurements

mailto:s.krause@qmul.ac.uk

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Content

1. Introduction to impedance

2. Equivalent circuits of common electrode systems

3. Practical aspects of impedance measurements

– The experimental setup (two-, three- and four-electrode measurements)

– Use of interdigitated electrodes

– Validity of impedance data

4. Application examples

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1. Introduction 1. What is impedance? • Apply a constant voltage (dc), measure the current ratio of the applied voltage to the current (V/I) = resistance of the material.

• Apply alternating voltage (ac), measure the current ratio V/I = impedance of the material.

• In many materials, the impedance changes with the frequency of the applied voltage due to the properties of the liquid or solid.

dc

dc

I

VR

I

VZ

3

Definition

Sinusoidal perturbation:

Current response:

Impedance:

tVtV sin)( 0

ΦtItI sin)( 0

)(

)(

tI

tVZ

tV o

r I

V

I

/

f2

Repeat measurement at a range of different frequencies Electrochemical Impedance Spectroscopy (EIS)

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What information can impedance

spectroscopy provide?

Examples:

• Conductivity of media (solutions, polymer layers)

• Capacitance and therefore dielectric constants of materials

• Electrolyte uptake of materials

• Defects in films (pores, incomplete coverage)

• Thickness of films (binding of molecules to a surface, dissolution of polymer films etc.)

• Surface area of electrodes

• Double layer capacitance of electrodes.

• Surface roughness or porosity of electrodes

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The advantages of impedance measurement

over other techniques include:

• Rapid acquisition of data (often within minutes)

• Accurate, repeatable measurements

• Non-destructive

• Highly adaptable to a wide variety of different applications.

• Interrogates relaxation phenomena whose time constants range over

several orders of magnitude as impedance spectroscopy uses a

large range of frequencies

• e.g. frequency range of 0.1 Hz to 100 kHz investigate processes

that can have time constants 10s to 10 µs.

Can investigate bulk processes (e.g. diffusion, mobility of charge

carriers) and Interface processes (e.g. charge transfer)

tf

1

6

Impedance presentation in the complex plane

• The impedance has a

magnitude ( ) and a

phase () and is thus a

vector quantity. It is

therefore convenient to

present impedance in

complex notation

Z”

Z’

Z 0

f

0

0

0I

VZ

"')sin(cos0 jZZjZZ

1jWhere , Z’ is the real part and Z” the imaginary

part of the impedance.

jeZZ 0or

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Impedance response of selected circuit

elements

RZ 0

0

,1

C

j

CjZ

CZ

10

2

Z'

-Z"

Z'

-Z"

R C

(a) (b)

RZ

8

… and their combinations

Z'

-Z" R

C

(d)

Z'

-Z"

R C

(c)

C

tjIRtItV

)()()(

22

2

0

1

CRZ

C

jRZ

CR

1arctan

j

tCV

R

tVtI

)()()(

1

1

Z11

j

C

Rj

C

RZ

1 2

1-

22

20

C

RZ

)arctan( CR

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2. Equivalent circuits

Impedance spectra can be complex depending on

how complicated the system is you look at. One

simple way of handling the data is to fit impedance

spectra to networks of electrical circuit elements,

so-called equivalent circuits.

Caution: Equivalent circuits are only useful if they

are based on the physical and chemical properties

of the system and do not contain arbitrarily chosen

circuit elements.

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Equivalent circuit of a metal electrode in an

electrolyte solution

Three quantities need to be taken into account.

a The resistance of the electrolyte solution between reference and working electrodes Re,

b The electrical double layer which can be expressed as the double-layer capacitance Cdl, and

c The impedance of the charge transfer process also called Faradaic impedance Zf.

Z f

C dl

R e

11

Z f

C dl

R e

12

(a) Three-electrode cell:

The three electrode cell

is used in conjunction

with a potentiostat.

A potentiostat controls

the potential applied to

the working electrode

and permits the

measurement of the

current it passes.

The reference electrode is connected to the potentiostat through a high

resistance circuit that draws no current from it.

As the current cannot flow through the reference electrode, a current

carrying auxiliary (counter) electrode is placed in the solution to

complete the current path.

A V

13

Z f

C dl

R e

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(b) The electrical double

layer at the electrode

solution interface:

Electrodes carry a charge, which

can be controlled by the potential

applied to the electrode.

This results in electrostatic

interactions between the electrode

and the ions in the solution.

SM qq

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(b) Capacitance of the electrical double layer

• Under certain conditions, the electrical double

layer resembles an ordinary parallel plate

capacitor:

Ad

C rdl0

+

+

+

+

-

-

-

-

f

x

16

Z f

C dl

R e

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(c) The charge transfer process

The Faradaic impedance has to be considered in the presence of

electroactive species.

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Equivalent circuit of a metal electrode in an

electrolyte solution

The Faradaic impedance can often be described by a simple resistor Rct at sufficiently high frequencies.

In the absence of any electroactive species, the Faradaic impedance becomes infinitely large, and the equivalent circuit simplifies to a serial combination of electrolyte resistance and double layer capacitance.

R ct

C dl

R e

Z f

C dl

R e

Re Cdl

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Nyquist plot for a simple electrochemical cell

without diffusion

AC impedance measurements at a series of different frequencies can be used to identify and separate the different circuit components. This can be done graphically from the Nyquist plot.

The frequency at the top of the semicircle, where the imaginary part of the impedance reaches its maximum, is

From this the time constant for the Faradaic process can be defined as

Z'

-Z"

R e

R ct

low

frequencies

high

frequencies

R ct

C dl

R e

dlctCR

1max

dlctCR

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Other forms of data presentation: Nyquist Plot

versus Bode Plot

R ct

C dl

R e

Z'

-Z"

R e

R ct

low

frequencies

high

frequencies

log f

log

Z0

(a)

log f

log

Z'

log

Z"

(b)

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Faradaic processes and diffusion

related phenomena

Previously, the Faradaic process was described in terms of a simple charge transfer resistance neglecting the diffusion of electroactive species.

The Faradaic impedance Zf can be presented by the charge transfer resistance and the mass transfer impedance ZW, also called the Warburg impedance.

R ct Z W

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The Warburg impedance ZW

where

Since real and imaginary parts of

the Warburg impedance ZW have

the same value, a plot of ZW”

versus ZW’ would show a straight

line with a phase angle of 45o.

2/12/1

jZW

*2/1*2/122

11

2 RROO cDcDAFn

RT

Z'

-Z"

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The Randles circuit for an electrochemical

cell with diffusion

The straight line with a slope of 1

is due to the Warburg impedance

and indicates a purely diffusion

controlled reaction at the low

frequency limit.

A medium diffusion coefficient for

oxidised and reduced species can

be calculated from the coefficient

of the Warburg impedance

Z'

-Z"

ZW

Cdl

ReRct

Distance from electrode

1t

Concentrationt

0

t2 t

t

3

4

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Behaviour of real systems

Equivalent circuit are made up of circuit elements, which are related to the physical processes in the system under investigation. In many cases, ideal circuit elements such as resistors and capacitors can be applied.

Mostly, however, distributed circuit elements are required in addition to the ideal circuit elements to describe the impedance response of real systems adequately.

Deviation from ideal behaviour can be observed if the electrode surfaces are rough or one or more of the dielectric materials in the system are inhomogeneous.

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The constant phase element

Constant phase element (CPE)

, )('

1jC

ZCPE

where C’ and are frequency independent parameters and 10

In the complex plane diagram, a CPE would appear as straight

line at a constant phase angle of . For ,

the CPE describes an ideal resistor, and for , it describes

an ideal capacitor.

)90( 0

1

Z'

-Z"

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Nyquist plot for a simple electrochemical cell

with a rough electrode surface

Z'

-Z"

R ct

CPE

R e

Frequency dispersion in

electrochemical systems

leads to a depressed

semicircle in the complex

impedance plane,

shown for different values of

the exponent of the CPE

□ = 0.9,

= 0.8,

= 0.7

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3. Practical aspects of impedance

measurements

– The experimental setup (two-, three- and four-

electrode measurements)

– Use of interdigitated electrodes

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Typical measurement setup with an FRA and a potentiostat

Pol

x1

x0.01

+reject dc

I/V convert

IR Compx1 / x10

reject dc

x1 / x10 RE

FRA

ECI

CE RE1 RE2 WE

I out

V outP/G stat

DC ref

Sweep

The polarisation potential and the ac perturbation are added together and applied to the

electrochemical cell at the counter electrode (CE terminal). The voltage difference

between the two reference electrodes RE1 and RE2 is measured and fed back to the

control loop, which corrects the voltage applied to the counter electrode until the

required potential difference between RE1 and RE2 is established. The voltage

measured between RE1 and RE2 and the current measured at the working

electrode are amplified by the potentiostat and fed into the FRA as voltage signals.

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Two-, three- and four-electrode

measurements

• Two electrodes: impedance measured

includes that of the counter electrode, the

electrolyte solution and the working

electrode

• Three electrodes: the impedance

obtained will only be influenced by the

properties of the working electrode and the

properties of the electrolyte solution

between working and reference electrodes

• Four electrodes: The impedance

measured depends purely on the

properties of the electrolyte or membrane

between the two reference electrodes

CE RE1 RE2 WE

CE RE1 RE2 WE

(a)

(b)

CE RE1 RE2 WE

(c)

The impedance is always measured between RE1 and RE2

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Interdigitated electrodes

Parallel plate sensor Interdigitated sensor (cross section)

http://implicit.che.utah.edu/~choi/dielectric_spectroscopy.pdf

http://implicit.che.utah.edu/~choi/dielectric_spectroscopy.pdf

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thin film

Problems:

• Contribution of substrate to measurement,

• Electrical field of interdigitated electrodes decays exponentially with distance from the dielectric sensor interface. The penetration depth is roughly a third of the electrode spacing.

http://implicit.che.utah.edu/~choi/dielectric_spectroscopy.pdf

substrate

http://implicit.che.utah.edu/~choi/dielectric_spectroscopy.pdf

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Validity of experimental data

Impedance is only properly defined as a

transfer function when the system under

investigation fulfils the conditions of

(i) linearity,

(ii) causality and

(iii) stability

during the measurement.

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(i) Linearity

Electrochemical systems are non-linear.

Non-linearity mainly affects the low-frequency part of the spectrum, which is determined by the Faradaic current.

The high-frequency part of the spectrum determined by the electrolyte resistance and the double layer capacitance shows approximately linear behaviour

Solution: Non-linear effects can generally be overcome by making the

amplitude of the perturbation signal small enough to approach quasi-linear

conditions. This can be achieved by measuring impedance spectra at

different perturbation amplitudes. The amplitude can be regarded sufficiently

small when it has no significant effect on the impedance spectrum measured.

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(ii) Causality

Deviation from causality can arise when the

response is not caused by the input but for

example by a concentration, current or

potential relaxation upon departure of the

system from equilibrium. Causality can also

be disturbed as result of instrument artefacts

or noise.

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(iii) Stability

The stability of an electrochemical system is usually not guaranteed when continually changing systems such as corroding electrodes or batteries are investigated.

Solution: Establish whether a system has changed during the course of an impedance measurement by repeating the experiment and comparing both sets of data.

Eliminate instabilities by adjusting the measuring conditions accordingly.

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Characterisation of changing

systems It may also be desirable to characterise systems, which are known to change over time or even to measure changes of the complex impedance in real time.

As long as the measurement time is short compared to the time constant of relaxation of the system, valid impedance data can be obtained.

Since the measurement time is greater for low measurement frequencies, high frequency data can be valid even when low frequency data are affected by non-stationary behaviour of the electrochemical system.

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Kramers-Kronig relations have been used as a diagnostic

tool for the validation of impedance data to establish, which

frequency range contains consistent data.

Originally developed for optical applications, they can also

be applied to electrochemical impedance spectroscopy.

The Kramers-Kronig relations are a series of integral

equations, which govern the relationship between the real

and imaginary parts of complex quantities for systems

fulfilling the conditions of linearity, causality and stability.

Kramers-Kronig Relations

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dxx

ZxZZ

0

22

)(')('2)("

dxx

ZxZx

ZZ

0

22

)(")("2)0(')('

0

22

0 )(ln2)( dxx

xZ

dxx

ZxxZZZ

0

22

)(")("2)(')('

The imaginary part of the impedance can be calculated from the real part of the

impedance from

If the high frequency limit of the real part of the impedance is known, the real part of the

impedance can be obtained from the imaginary part of the impedance using

If the zero frequency limit of the real part of the impedance is known, the real part of the

impedance can be obtained at any frequency from the imaginary part using

The relationship between phase angle and modulus of the impedance is

To assess whether experimental data fulfil the Kramers-Kronig relations, one part of the

impedance is calculated from the other part of the impedance, which has been

experimentally determined.

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Summary: Validation methods

• Measure impedance spectra with different perturbation

amplitudes. The amplitude is sufficiently small when it

has not influence on the impedance spectrum

measured.

• Repeat experiment and compare both sets of data to

establish whether a system has changed during the

course of an impedance measurement

• Kramers-Kronig relations to establish which frequency

range contains consistent data.

4. Sensor applications

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Example 1: Multiple Sclerosis (MS)

monitoring:

• MS is a progressive disease of the central nervous

system that frequently leads to disability.

• Inflammation precedes the onset of clinical attacks or

relapses causing axonal damage.

• Therapies for suppressing inflammation are only

partially effective.

• MMP-9 is useful to detect subclinical disease

progression treatment can be modified to limit

neuronal damage and prevent disability.

Enzyme

Generic sensor materials for

protease detection based on

peptide cross-linked hydrogels

substrate substrate

hydrogel film protease

buffer

Peptide degraded peptide

Degradation has

been monitored using

QCM and impedance

measurements.

MMP-9 sensor LGRMGLPGK

cross-linked dextran hydrogel

MMP-9 addition

Impedance spectra of hydrogel coated

IDEs before (spectrum 1) and after

degradation by MMP-9 (spectrum 2)

Impedance changes at 100 Hz before

and after the addition of MMP-9 to

charge transfer buffer, pH 7.5

Biosensors and Bioelectronics 68 (2015) 660-667

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Example 2: Affinity biosensors based on

impedance measurements

Unlabelled DNA and protein

targets can be detected by

monitoring changes in

surface impedance when a

target molecule binds to an

immobilized probe.

http://www.pharmaco-

kinesis.com/tech_biosensor_tech.php

Analyst, 2012, 137, 819

Example 2: Label-free aptamer-based electrochemical

impedance biosensor for 17β-estradiol

Analyst, 2012, 137, 819

(A) Impedance spectra

(Nyquist plots) of electrodes

incubated

with different concentrations of

17β-estradiol:

(a) 1x10-11 mol L-1;

(b) 5x10-11 mol L-1;

(c) 1x10-10 mol L-1;

(d) 5x10-9 mol L-1;

(e)1x10-8mol L-1.

(B) Linear relationship

between the change of

impedance (DI)

and concentration of 17β-

estradiol.

Summary

• Impedance spectroscopy is a versatile technique for the

determination of electrical parameters of materials and

electrodes over a wide range of frequencies.

• It is non-destructive.

• Equivalent circuits are useful for the modelling of

impedance spectra.

• There is a wide range of sensor applications such as

label-free affinity biosensors.

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