Post on 26-Dec-2015
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