1. ( electrochemical impedance spectroscopy, EIS)
Chapter 6 : Alternative current method
( electrochemical impedance spectroscopy, EIS)

2. 6.1 basic consideration 1. Probing electrochemical system
circuit analysis : black box grey box
for electrochemical system:
Rc t RS
Cd t
R.E
W.E
electric behavior of elements : resistor, capacitor, etc.

3. Equivalent circuit and circuit description code: CDC

4. 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

5. An expression analogous to Ohm's Law allows us to calculate the impedance of the system as:
阻抗 (Impedance) (Z)
导纳 (Admittance) (Y)

6. EIS models usually consist of a number of elements in a network
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

7. Impedances in Parallel Z1 Z2 Z3
For linear impedance elements in parallel, the equivalent impedance is

8. 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

9. 2. Display of impedance
Using Eulers relationship
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,

10. 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.

11. Nyquist Plots
A Nyquist plot is made up of a series of vectors representing the total magnitude of the resistance and capacitance components
Phase angle
Non Resistive Component

12. Bode impedance plot
Impedance
Rct
Solution resistance
Frequency →

13. 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)

14. 1) Electrolyte resistance (uncompensated resistance) (Rs, RU)
2) Double layer capacitance (Cdl)
3) Coating capacitance (Cc)
Conversion film, passivation film, polymeric coating, etc.
Typical Relative Electrical Permittivity
vacuum 1
water
80.1 ( 20 ℃ )
organic coating
4 - 8

15. 4) Warburg impedance: related to diffusion
At x = 0

16. If product is insoluble:
depends on ω. When ω→∞

17. 韦伯格系数（Warburg factor ）

18. If product is soluble:

19. 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:
 Is the thickness of the diffusion layer

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

21. 5) Charge transfer resistance (Rct)
Without concentration overpotential
At small overpotential
At higher overpotential
For medium overpotential and ==0.5

22. With concentration overpotential

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

24. 7) Virtual Inductor ZC = jL ZC=0 ZC = L
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.

25. Common Equivalent Circuit Models
1/  Y0( j)  
         Y0( j)      
Q (CPE)
O (finite Warburg)
1/Y0( j)1/2
     Y0( j)1/2           
W (infinite Warburg)
jL   
 1/  jL          L
   1/ jC        
jC    C
R      
1/R       R
Impedance
Admittance
Equivalent element
The dependent variables are R, C, L, Yo, B and a.

26. 6.3 Simplification of EC
For electrode with metal current collector，RCE→0，RWE→0
Z = 1/ jC       
Compare with CWE and CCE, CW-C is very small. Therefore, the above circuit can be simplified as

27. How can we further simplify this circuit?Z1
Cdl ,2 RL
Cdl ,1
How can we further simplify this circuit?
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

28. 2) When using reference electrode : RctZ1
Cdl ,2 RL
Cdl ,1
2) When using reference electrode : RS
Rct
W.E
1) No rxn , Rct  , RL
Cdl
Cdl
if 1/Cdl >> 0
ideal polarization electrode

30. 6.4 impedance measurement
variatory :  ~ 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 :
measure :
frequency  respond analyzer : z =z  zj z f
z z
Nyquit

31. EIS (Summary)
We start here at the high frequency

32. 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

33. Rs

34. RS
Cdl
2) For (RC)
Randles Cell

35. Capacitive impedance semicircle

36. 3) EC with and without diffusionWO WR
Rct
Cdl Rs
This circuit models a cell with polarization due to a combination of kinetic and diffusion processes
At low frequency

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

38. At higher frequency
When i0, Rct  0, no circle appears.

39. The whole spectrum

40. 4) For coated metal system
there are two well defined time constants in this plot

41. 6) RL and (RL) RL R
(RL)

42. 0<n<1
6) RQ and (RQ)
Q independence of 
(RQ) RQ

43. 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

51. 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