Chapter 6 : Alternative current method

( electrochemical impedance spectroscopy, EIS)

circuit analysis : black box grey box

electric behavior of elements : resistor, capacitor, etc.

for electrochemical system:

Rc t

RS

Cd t

R.E

W.E

6.1 basic consideration

1. Probing electrochemical system

Equivalent circuit and circuit description code: CDC

Cd

Rr

RL

Cad

Rad

RL(Cd(Rr(RadCad)))

Sinusoidal Current Response in a Linear System

The excitation signal is a function of time

E0 is the amplitude of the signal, and is the radial frequency

In a linear system, the response signal, It, is shifted in phase () and has an amplitude of I0

0( ) sinE t E t

0( ) sin( )I t I t

An expression analogous to Ohm's Law allows us to calculate the impedance of the system as:

00

0

sin( )( ) sin( )

( ) sin( ) sin( )

E tE t tZ Z

I t I t t

阻抗 (Impedance) (Z)

导纳 (Admittance) (Y) ZY

1

EIS models usually consist of a number of elements in a network. Both serial and parallel combinations of elements occur.

Impedances in Series   

Z1 Z2 Z3

For linear impedance elements in series, the equivalent impedance is

1 2 3totalZ Z Z Z

Impedances in Parallel 

Z1

Z2

Z3

For linear impedance elements in parallel, the equivalent impedance is

1 2 3

1 1 1 1

totalZ Z Z Z

1 2 3Y Y Y Y

If we plot the sinusoidal signal on the X-axis of a graph and the sinusoidal response signal I(t) on the Y-axis, an oval is plotted. Analysis of oval figures on oscilloscope screens was the method of impedance measurement prior to the lock-in amplifiers and frequency response analyzers

Using Eulers relationship

exp( ) cos sinj j it is possible to express the impedance as a complex function. The potential is described as,

and the current response as

The impedance is then represented as a complex number,

0( ) exp( )E t E j t

0( ) exp( )I t I j t j

0 0exp( ) (cos sin )E

Z Z j Z jI

2. Display of impedance

0 0exp( ) (cos sin )E

Z Z j Z jI

The expression for Z() is composed of a real part (Z’) and an imaginary part (Z’’). If the real part is plotted on the Z’ axis and the imaginary part on the -Z’’ axis of a chart, Nyquist plot is gotten.

Nyquist PlotsNyquist Plots

A Nyquist plot is made up of a series of vectors representing the A Nyquist plot is made up of a series of vectors representing the total magnitude of the resistance and capacitance componentstotal magnitude of the resistance and capacitance components

Non Resistive Component

Phase angle

Bode impedance plot

Impedance

Frequency →

Solution resistanceRct

6.2 Electrochemical elements:

1) Electrolyte resistance (uncompensated resistance) (Rs, RU)

2) Double layer capacitance (Cdl)

3) Coating capacitance (Cc)

4) Warburg impedance (related to diffusion) (W)

5) Polarization resistance/Charge transfer resistance (Rct i0)

6) Constant phase element (Q)

7) Virtual inductor (L)

1) Electrolyte resistance (uncompensated resistance) (Rs, RU)

2) Double layer capacitance (Cdl)

3) Coating capacitance (Cc)

Conversion film, passivation film, polymeric coating, etc.

0 r AC

d

Typical Relative Electrical Permittivity

vacuum 1

water 80.1 ( 20 ℃ )

organic coating 4 - 8

4) Warburg impedance: related to diffusion

O Rne 0

0

sin OO

x

cI I t nFD

x

0

exp sin42 / 2 /

O

O O O

I x xc t

nF D D D

At x = 00

sin4

O

O

Ic t

nF D

0

sin4

R

R

Ic t

nF D

0~

0~

~~ 1lnlnO

sO

O

sO

c

c

nF

RT

c

c

nF

RT平

4sin

022

0

0~

~

tDcFn

RTI

c

c

nF

RT

OOO

sO

depends on ω. When ω→∞~ ~ 0

0 2 2 0 2 2 0

1f

O O O O

RT RTZ

n F c D n F c DI

If product is insoluble:

0

0 2 2 0 2 2 0

1f

O O O O

RT RTZ

n F c D n F c DI

韦伯格系数（ Warburg factor ）

2 2 0O O

RT

n F c D

1/ 2( ) (1 )WZ j

2 2 0 0Re Re

1 1

2 Ox Ox d d

RT

n F A c D c D

If product is soluble:

2 2 0(1 )

2W

i i

RTZ j

n F c D

This form of the Warburg impedance is only valid if the diffusion layer has an infinite thickness. Quite often this is not the case.

If the diffusion layer is bounded, the impedance at lower frequencies no longer obeys the equation above. Instead, we get the form:

1/ 21/ 2

0 (1 ) tanhj

Z jD

Is the thickness of the diffusion layer

This impedance depends on the frequency of the potential perturbation.

At high frequencies the Warburg impedance is small since diffusing reactants don't have to move very far. At low frequencies the reactants have to diffuse farther, thereby increasing the Warburg impedance.

2 2 0(1 )

2W

i i

RTZ j

n F c D

←Frequency

On a Nyquist plot the infinite Warburg impedance appears as a diagonal line with a slope of 0.5.

5) Charge transfer resistance (Rct)

At small overpotential0

1EC ct

RTZ R

nF i

At higher overpotential EC

RTZ

nFI

For medium overpotential and ==0.5

1

0

2exp exp

2 2EC

RT nF nFZ

nFi RT RT

Without concentration overpotential

With concentration overpotential

0 00

0 0

s sO R

ECO R

RT RT c RT cZ

nFi nFc nFcI I

6) Constant Phase Element

Capacitors in EIS experiments often do not behave ideally. Instead, they act like a constant phase element (CPE) .

1A

C

For an ideal capacitor, the constant A = 1/C (the inverse of the

capacitance) and the exponent = 1.

For a constant phase element, the exponent is less than one

and of no definite physical meaning.

0

1( ) n

QZ jY

7) Virtual Inductor

The impedance of an electrochemical cell can also appear to be

inductive.

Some authors have ascribed inductive behavior to adsorbed

reactants. Both the adsorption process and the electrochemical

reaction are potential dependent. The net result of these

dependencies can be an inductive phase shift in the cell

current .

Inductive behavior can also result from nonhomogeneous

current distribution, which lead inductance and potentiostat

non-idealities. In these cases, it represents an error in the EIS

measurement.

ZC = jL ZC=0 ZC = L

1/  Y0( j)            Y0( j)       Q (CPE)

O (finite Warburg)

1/Y0( j)1/2      Y0( j)1/2            W (infinite Warburg)

jL     1/  jL          L

   1/ jC         jC    C

R       1/R       R

ImpedanceAdmittanceEquivalent element

0 cothY j B j 0/Tanh B j Y j

Common Equivalent Circuit Models

The dependent variables are R, C, L, Yo, B and a.

6.3 Simplification of EC

辅 研

参

R辅

Cd辅

Zf辅RL

Cd研

Zf研R研

Cd研、辅

界面 界面

For electrode with metal current collector ， RCE→0 ， RWE→0

Compare with CWE and CCE, CW-C is very small. Therefore, the above circuit can be simplified as

4

SC

k d

Cd辅

Zf辅RL

Cd研

Zf研

Z = 1/ jC       

How can we further simplify this circuit?

Z1Z1

Cdl ,2

RL

Cdl ,1

1)When using electrode with large effective area and exchange current . Cdl very large 1/Cd very small

RL

used for measurement of conductivity of solution

RS

Rct

W.E

1) No rxn , Rct ,

ideal polarization electrode

if 1/Cdl >> 0

2) When using reference electrode :

RLCdl

Z1Z1

Cdl ,2

RL

Cdl ,1

Cdl

辅 研

参

Cd

Rr

ZwRL

Cs Rs

6.4 impedance measurementvariatory : 106 ~ 10-3 Hz

single generator : from 105 ~ 10-3 Hz

5 ~ 10 point /decade

v0 = 5 mV for high impedance system :10 mV

lock in amplifier : 0 sinv v t measure : sin( )i i t

frequency respond analyzer : z =z zj

z

f

z

zNyquit

EIS (Summary)EIS (Summary)

We start here at the high frequency

6.5 Impedance characteristics of processes

1) Ideally Polarizable Electrode

An ideally polarizable electrode behaves as an ideal capacitor

because there is no charge transfer across the solution-electrode

boundary.

Circuit code: RsCdl

sdl

jZ R

C

sdl

jZ R

C

Rs

Randles Cell 2) For (RC)

1Y j C

R

2

2 2

1

1 1 ( ) 1 ( )

R R R CZ j

Y j CR RC RC

2'

1 ( )

RZ

RC

2

2''

1 ( )

R CZ

RC

2 2' ' '' 0Z RZ Z 2 2

2' ''2 2

R RZ Z

RS

Cdl

2 22' ''

2 2

R RZ Z

Capacitive impedance semicircle

WO WR

Rct

Cdl

Rs

1

1/ 2 1/ 2

1s dl

ct

Z R j CR j

At low frequency 1/ 2

1/ 2 2

'

'' 2

s ct

dl

Z R R

Z C

2'' ' 2s ct dlZ Z R R C

This circuit models a cell

with polarization due to a

combination of kinetic

and diffusion processes

3) EC with and without diffusion

2'' ' 2s ct dlZ Z R R C

;

'

'' 2s ct

dl

Z R R

Z C

1/ 2

1/ 2 2

'

'' 2

s ct

dl

Z R R

Z C

''Z

'Z

the Warburg Impedance appears as a straight line with a slope of 45°

At higher frequency

2 2 2'

1ct

sdl ct

RZ R

C R

2 2

2' ''2 2ct ct

s

R RZ R Z

* 1

dl ctC R

When i0, Rct 0, no circle appears.

2

2 2 2''

1dl ct

dl ct

C RZ

C R

The whole spectrum

4) For coated metal system

there are two well defined time constants in this plot

6) RL and (RL)

RLR

(RL)

6) RQ and (RQ)

0

1( ) n

QZ jY

'

0

cos( )2Q

n nZ

Y

''

0

sin( )2Q

n nZ

Y

0<n<1

Q independence of

RQ (RQ)

7) Uniqueness of Models

This spectrum can be modeled by any of the equivalent circuits You cannot assume that an equivalent circuit

that produces a good fit to a data set represents an accurate physical model of the cell

0 50 100 150 2000

50

100

150

200

-Z''/

cm-2

Z'/ cm-2

0 rpm 100 rpm 500 rpm 1000 rpm 2000 rpm 5000 rpm

0.01 0.1 1 10 100 1000 100001000000

10

20

30

40

50

60 0 rpm 100 rpm 500 rpm 1000 rpm 2000 rpm 5000 rpm

Phas

e an

gle

/ o

f/Hz

0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

3000

40 60 80 100 120 140 160 1800

10

20

30

40

50

60

70

80

90

100 0 rpm 100 rpm 500 rpm 1000 rpm 2000 rpm 5000 rpm

-Z

''/

cm-2

Z'/ cm-2

a: 10 mg l-1

-Z

img/

(.c

m-2)

Zre/(.cm-2)

0 rpm 100 rpm 500 rpm 1000 rpm 2000 rpm 5000 rpm

0.01 0.1 1 10 100 1000 100001000000

10

20

30

40

50a: 10 mg l-1

0 rpm 100 rpm 500 rpm 1000 rpm 2000 rpm 5000 rpm

Phas

e an

gle

/ o

f/Hz

0 1000 2000 3000 4000 5000 6000 70000

1000

2000

3000

4000

5000

6000

7000(A)

-Z''

/ c

m2

Z' / cm2

-0.45 -0.35 -0.25 0.00 +0.15 +0.29

-9000 -6000 -3000 0 3000

-2000

-1000

0

1000

2000 (B)

-Z

'' /

cm

2

Z' / cm2

0.4 V 0.5 V 0.53 V 0.57 V 0.62 V

0 200 400 600 800 1000 1200 1400 1600

0

200

400

600

800

1000

1200

1400

1600

(C)-Z

'' /

cm

2

Z' / cm2

0.67 V 0.70 V 0.80 V 1.00 V 1.10 V 1.20 V 1.30 V

elements CoPc FePc FeCoPc2

L/ H·cm-2 7e-7 5.788e-7 7.851e-7

Rs/ ·cm-2 1.677 2.024 1.598

Rct/ ·cm-2 3.05 1.966 0.6907

Rc1/ ·cm2 0.6397 0.9987 0.4835

Rc2/ ·cm-2 2.238 2.204 0.4143

Rtotal/ ·cm-2 7.6047 7.1927 3.1865