1. IMPEDANCE SPECTROSCOPY: DIELECTRIC BEHAVIOUR OF POLYMER ELECTROLYTESBy
Ri Hanum Yahaya Subban Ph. D
Faculty of Applied Sciences/Institute of Science
UiTM Shah Alam

2. OUTLINE IMPEDANCE SPECTROSCOPY (IS) BACKGROUND IS PRINCIPLE
IS TECHNIQUE
IS PLOT OF SIMPLE CIRCUITS
IS PLOT OF MODEL SYSTEMS
IS PLOT OF REAL SYSTEMS
CONSTANT PHASE ELEMENT (CPE)
IS PLOT OF REAL SYSTEMS AND CPE
IMPEDANCE RELATED FUNCTIONS
Z, Y AND M PLOTS FOR SIMPLE CIRCUITS
SOME APPLICATION OF IS
SOME PRACTICAL DETAILS FOR IS

3. IS: BACKGROUND D. C METHOD A.C METHOD
R=V/I cannot be used due to polarisation of charges
- at electrode-electrolyte interface
at defect regions inside the sample
(grain boundaries, phase boundaries etc. )
Polarisation effects are avoided and impedance (Z) is measured
- Since Z changes with applied signal frequency, Z must be measured as a function of frequency and resistance of sample evaluated
Also known as
 AC Impedance Spectroscopy
 Complex Impedance Spectroscopy
 Electrochemical Impedance Spectroscopy
(when applied to electrochemical systems)
Popular use of IS:
 To determine electrical conductivity of ionic conductors
 To identify different processes that contribute to the total
conductivity: bulk contribution, grain boundary contribution,
diffusion, etc.
 Through identifying an equivalent circuit for the impedance
plot involved

4. IS: BACKGROUND Resistance of sample R =  L A
 = resistivity of the material
L = length of the sample
A = area of cross-section of the sample
Conductivity  = 1 = L/A
 R
By measuring R,L and A,  can be calculated

5. IS: PRINCIPLE PRINCIPLE: IS Sine wave signal V(t) = Vo sin t
i(t)
V(t)
Sine wave signal V(t) = Vo sin t
of low amplitude is applied to a sample
Vo = maximum voltage
 = 2f, angular frequency
The resulting current i(t) = io (sin t + )
= phase difference between i(t) and V(t)
(current is ahead of voltage by )
The impedance Z = V(t) = Vo sin (t)
i(t) io (sin t + )
Z is a function of frequency and has magnitude Z = Vo = Zo and a phase angle  io
Both Z and  are frequency dependent quantities
: phase shift
PRINCIPLE: IS

6. IS: PRINCIPLE
Since ac impedances are frequency dependent quantities they are represented by Z()
Z() can be considered as a complex quantity with a real component Z’() and imaginary component Z”()
Z() = Z’() + j Z”() , j =-1
where real impedance = Z’ = Z cos( )
imaginary impedance = Z” = Z sin( )
with a phase angle  = tan-1 (Z”/ Z’)
Magnitude of Z, Z = [(Z’)2 + (Z”)2]1/2 Z” Z’
Im. Z
Real Z 
Z
Complex Impedance
plane

7. IS: TECHNIQUE
Small ac signal (V  10 mV) is applied to sample over a wide range of frequency (mHz to MHz)
Liquid sample
Solid sample
Sample holder
Electrode
Sample
Impedance spectrometer:
LCR meter/FRA
Electrode
Computer

8. IS: TECHNIQUE
Measure Z(f) as a function of f(=2f) over a wide range of frequency (mHz to MHz)
Plot Z(f) versus f in the form of -Z’’(f) vs Z’(f) for various f (Cole-Cole plot/ Complex impedance plot/Nyquist plot)
Useful to evaluate :
-electrical parameters such as conductivity of ionic
conductors(solid or liquid), mixed conductors
- electrode-electrolyte interfacial effects and related phenomena
- electrochemical parameters/processes of the system under study
Also used for studying dielectric behaviour of materials

9. IS PLOT OF SIMPLE CIRCUITSZ’ Z”
a. Pure resistance R
Z = R for all values of  or f
Z = R and  = 0
Z’ = R and Z” =0
Impedance plot is a point on the real axis at Z’ = R

10. IS PLOT OF SIMPLE CIRCUITS
Z = = -j
jC C
Z’ = 0 and Z” = -1 C
Z = Z” varies with frequency
As  increases, Z decreases
Z points lie along the Z” axis  Z’ Z”
-ve Z” C
b. Pure capacitance C
Impedance plot is a straight line lying on the Z” axis

11. IS PLOT OF SIMPLE CIRCUITS
The total impedance
Z = R - j C
With Z’ = R and Z” = -1  Z’
-ve Z” C
c. R and C connected in series R
On complex plane the graph becomes
a straight line at Z’ = R, parallel to the Z” axis

12. IS PLOT OF SIMPLE CIRCUITS
= = jC
Z R /j C R
Z = R
1 + j C
= R(1 - jRC) = R(1 - jRC)
(1 + jRC) (1 - jRC) (RC)2
= R jR2C
1 + 2R2C 2R2C2
= Z’ jZ” with Z” = RC Z’
On eliminating  : (Z’- R/2)2 + (Z”)2 = (R/2)2
d. R and C connected in parallel C R
 Equation of a circle

13. IS PLOT OF SIMPLE CIRCUITS
d. R and C connected in parallel
Impedance plot is a semicircle with centre (R/2, 0) on the Z’ axis
Maximum point on the semicircle corresponds to mRC = 1
 m = 1 RC 
-ve Z” Z’ R m
where RC =   Time constant or
Relaxation time
From m , C can be calculated for an unknown circuit
Note: Z’ and Z” axes must have the same scales to see the semicircle
R/2

14. IS PLOT OF SIMPLE CIRCUITS
e. Combined circuits R1 R1 R2 Rs C1 C1 C2
-ve Z” Z’ Rs
1 = R1C1
Rs + R1
-ve Z” Z’ R1
R1+ R2
2 = R2C2
1 = R1C1
due to internal resistance of electrolyte/electrode interface

15. IS PLOT OF MODEL SYSTEMS
a. Ionic solid with two non-blocking electrodes
Eg: Ag/AgI/Ag (Ag+ mobile, I- immobile)
No ion accumulation at the electrodes
Cell arrangement  R and C connected in parallel (equivalent circuit)
(assume no electrode resistance)
Electrodes 
Rb= bulk resistance
Sample Rb Cb
Cb (Cg) bulk capacitance
Expected impedance plot
-ve Z” Z’ m  Rb
Rb/2
Cb is related to vacuum capacitance Co;
Cb = Co
And Co = oA d
A - area of cross section  - dielectric constant
d - thickness of sample o – permittivity of free space

16. IS PLOT OF MODEL SYSTEMS
a. Ionic solid with two non-blocking electrodes: experimental results Rb Cb 
Cb(Cg) bulk capacitance
Li6SrLa2Ta2O12
Li6BaLa2Ta2O12
with Li electrodes 
Thangadurai and Weppner Ionics 12 (2006) 81-92
Note: depressed/distorted semicircles

17. IS PLOT OF MODEL SYSTEMS
b. Ionic solid with two blocking electrodes
Egs: AgI with Pt electrodes, Ag+ mobile and I- immobile
Ions cannot enter the electrodes , get accumulated at the electrodes - +
two double layer of charges at electrode/electrolyte interfaces
two double layer capacitances at the interfaces ( C’dl)
-ve Z” Z’ R
R/2  m
Cdl will add a spike to the Impedance plot
Expected impedance plot
Equivalent circuit R C
C’dl R C
Cdl 
Cdl= effective double layer capacitance

18. IS PLOT OF MODEL SYSTEMS
b. An ionic solid with two blocking electrodes:
experimental results
SS/PVC-LiCF3SO3/SS
Au/Li6BaLa2Ta2O12/Au
Subban and Arof, Journal of New Materials for Electrochemical Systems 6 (2003)
Thangaduarai and Weppner Ionics 12 ( 2006) 81-92
Note: depressed/distorted semicircles and slanted/curved spikes

19. IS PLOT OF MODEL SYSTEMS
c. Polycrystalline solid with two blocking electrodes
Conduction will occur inside the grain (intra grain-bulk conduction) and along the grain boundaries (inter grain conduction)
System = crystalline grain + grain boundaries + electrode/electrolyte interface
Electrode/electrolyte interface
Expected impedance plot
Grain
-ve Z” Z’ Rb
Rb+ Rgb  Pt Pt
Grain boundary
Bulk
Thickness of grain boundary is small  large Cgb
Rgb is large - larger semicircle for GB
The overall σ :
is determined by Rb +Rgb
Cdl
Rgb
Cgb Rb Cb
Equivalent circuit

20. IS PLOT OF MODEL SYSTEMS
c. A polycrystalline solid with two blocking electrodes:
experimental results
High frequency semicircle (small C) bulk conduction
Low frequency semicircle (large C)  grain boundary
conduction
100 Hz
10 kHz
From the values of the capacitances different semicircles can be associated with different conduction process in the sample C1 R1 C2 R2
Cdl
SS/ Li1+xCrxSn2-xP3-yVyO12/SS
Norhaniza, Subban and Mohamed, Journal of Power Sources 244 (2013)
Note: Slanted/curved spike and depressed/distorted semicircles

21. IS PLOT OF MODEL SYSTEMS
TYPICAL C VALUES
In general, a number of processes can contribute to the total conduction and an ideal equivalent circuit (hypothetical ) may be represented by the following simple circuit (various circuits possible) R2 C2 R1 C1 R3 C3
Cdl
Bulk
Grain boundary
Different phases or Orientation of crystal planes
Double layer capacitance
at the electrode
R and C values ,particularly C values differ for different processes
Each transport process may give a semicircle to the Impedance plot
From the approximate C values different processes may be identified
Approximate C values
Phenomenon responsible
2-20 pF
Bulk(main phase)
 10 pF
Second phase, orientation etc.
1-10 nF
Grain boundary
Fcm-2
Double layer/surface charge
0.2 mFcm-2
Surface layer at electrode/adsorption
Actual identification of different processes must be based on dependence on temperature, pressure, etc.

22. IS PLOT OF REAL SYSTEMS
IS plot of real systems and devices are usually complicated
Deviate from ideal behaviour due to :
Distorted semicircles may arise due to
- Overlap of semicircles with various time constants
Depressed semicircles may arise due to
- Electrolyte is not homogeneous
- Distributed microscopic properties of the electrolyte
Slanted or curved spikes may arise due to
- Unevenness of electrode/electrolyte interfaces
- Charge transfer across the electrode/electrolyte interface, diffusion of
species in the electrolyte or electrode
The deviation from ideal behaviour of Impedance plot is explained in terms of a new circuit parameter called Constant Phase Element (CPE)

23. CONSTANT PHASE ELEMENT (CPE)
In general CPE has the properties of R and C (equivalent to a leaky capacitor)
Mathematically impedance of a CPE is given by the
Complex quantity:
ZCPE = = Zo (jω)-n , 0 ≤ n ≤ 1
Y0(jω) n
When n = 0, Z is frequency independent and Zo  R, CPE ≅ pure Resistance
When n = 1, Z = 1 /jωY0 . Hence Yo  C, CPE ≅ pure Capacitance

24. CPE When 0 < n < 1, CPE acts as intermediate between R and C
Can show that R and CPE in parallel gives a circular arc in the impedance plane as shown
Usually CPE is denoted by the circuit element Q
-Z” R
CPE alone gives an inclined straight line (pink) at angle (n=90)
CPE // R gives a tilted semicircle with its centre (C) depressed so that the plot appears as an arc (green)
- The diameter of the semicircle is inclined at (1-n) = 90 
-Q- Q R
n= 90° Z’
(1-n)= 90° C

25. IS PLOT OF REAL SYSTEMS AND CPE
The general equivalent circuit of a solid electrolyte with non perfect blocking electrodes may take the form R2
CPE2 R1
CPE1 R3
CPE3
CPE4 R1
R1 + R2
R1 + R2 + R3
-Z” Z’
Resulting impedance plot will have depressed semicircles and a slanted spike
Here processes are assumed to be well separated

26. IMPEDANCE RELATED FUNCTIONS
There are several other measured or derived quantities related to impedance (Z) which often play important role in IS:
Admittance (Y)
Dieletric/Permittivity (ε)
Modulus (electric) (M)
Generally referred to as ‘immitances’
The four different formalisms give the same information in different ways
However each formalism highlights different features of the system
Thus it may be worthwhile to plot the data in more than one formalism in
order to extract all possible information from the results
Z plot gives prominence to most resistive elements
M plot gives prominence to smallest capacitance
Eg: To study grain boundary effects , Z plot is good
To study bulk effects M plot is good

27. IMPEDANCE RELATED FUNCTIONS
A.C voltage applied to a sample v = Zi
Generally impedance Z = R + j X; R = resistance, X = reactance
Hence the current ,
Immitance
Symbol
Relation
Complex Form
Impedance Z -
Z’ – jZ”
Admittance Y
Y = Z-1
Y’ + j Y”
Permittivity 
 = 1/jCoZ = Y/ jCo
’ - j ”
Electric modulus M
M =  -1 = jCo Z
M’ + j M”
Where

28. IMPEDANCE RELATED FUNCTIONS
Complex Admittance
Complex Permittivity
Complex Electrical Modulus

29. Z, Y AND M PLOTS FOR SIMPLE CIRCUITS
C0/2C M” M’ R
-ve Z” Z’ Y” Y’ m 
1/Rb  

30. Z, Y AND M PLOTS FOR SIMPLE CIRCUITS
Co/2C
-ve Z” Z’ m  R
R/2
R/2 

31. SOME APPLICATION OF IS
DETERMINATION OF DC IONIC CONDUCTIVITY OF IONIC CONDUCTORS :
- Z” vs Z’ R
p2 2
p1
PVC-NH4CF3SO3-Bu3MeNTf2N
In general IS plot consists of a depressed semicircle with a tilted spike and intercept on the real axis corresponds to Rb
Rb may be determined graphically by drawing the best semicircle OR by fitting R//C circuit with suitable values of R and C. Here the value R= Rb
-Zi()
Note: both Z‘ and Z‘’ axes must have the same scale in order to see the semicircle .
If only spike is present , it can be extended to obtain the intercept
is calculated from R by using
=LA/Rb
L - thickness of sample
A - area of contact
S.K. Deraman Ph.D thesis UiTM 2014

32. SOME APPLICATION OF IS
DETERMINATION OF DC IONIC CONDUCTIVITY : - Z’’ vs. Z’ plot
OF IONIC CONDUCTORS
PVA-NH4x (x = Cl, Br, I)
Equivalent circuit of PVA-NH4x at low NH4x concentration
CPE2 R1
CPE1 A0 A5
Semi-circle disappears
Only resistive component prevails at higher frequency as NH4Br/I content increases A5
Equivalent circuit of PVA-NH4x at high NH4x concentration
CPE3
A25
A30
Hema et. al J. Non Crystalline Solids 355 (2009) 84-90

33. SOME APPLICATION OF IS
Analysis Of IS Plot: Choosing equivalent circuits
Choosing the correct equivalent circuit can be difficult
Softwares do not give a unique equivalent circuit (model) for a particular IS plot but may suggest a number of complicated circuits (multiple models)
Some possible equivalent circuits A
An example: B R1
R1 + R2
-Z” Z’
Two time-constant impedance spectrum C B

34. SOME APPLICATION OF IS
Analysis Of IS Plot: Choosing equivalent circuits
The model chosen should not only fit the IS data but also must be verifiable through other experiments, theories and justifiable through other known facts , etc.
An example: The circuits below can give 3 distinct semi-circles in the IS plot if
their time constants are well separated b a Cg CR C2 RR R2 Rg C1 C2 C3 R1 R2 R3
Can be equivalent to
a is more suitable for a polycrystalline sample
b is more suitable for a homogeneous material

35. SOME APPLICATION OF IS
DETERMINATION OF DIELECTRIC PARAMETERS OF IONIC CONDUCTORS
’ vs log f (permittivity/ dielectric constant, transport processes ) :
Low frequency:
Static dielectric constant  10 4
ε’ :
a measure of a material’s polarisation
associated with capacity to store charge and
represents the amount of dipole alignment in
a given volume
related to dielectric relaxation
High frequency:
Optical dielectric constant at 104Hz  between 3.5 and 5
Compared to pure PVC film = 3
In ionic conductors:
Relaxation peaks usually not observed due to large electrode polarisation effects
Alternative is M’ /M” or ac conductivity
PVC(1-x)LiCF3SO3xLiPF6
Subban and Arof Ionics 9 (2003)

36. SOME APPLICATION OF IS
DETERMINATION OF DIELECTRIC PARAMETERS OF IONIC CONDUCTORS :
’ vs concentration
Same trend: variation with concentration
PCL-NH4SCN
Woo et. al
Materials Chemistry and Physics 134 (2012)

37. SOME APPLICATION OF IS
DETERMINATION OF DIELECTRIC PARAMETERS OF IONIC CONDUCTORS
M” vs log f (relaxation time , transport processes ) :
PEO- AgCF3SO3
Relaxation peaks
Relaxation peak is responsible for fast segmental motion which reduces the relaxation time and increase the transport properties
fmax
Relaxation time  =1/2fmax
Gondaliya et.al Materials Sciences and Applications 2 (2011)

38. SOME APPLICATION OF IS
DETERMINATION OF DIELECTRIC PARAMETERS OF IONIC CONDUCTORS
Tan  vs log f (relaxation time and nature of conductivity relaxation) :
Tan  = ”/‘
Single well defined resonance peak is an indication of long range conductivity relaxation in good ionic conductors
From FWHM value can find out Debye conformation (=1.14) or otherwise.
Effect of temperature
Effect of concentration
MG30-LiCF3SO3
Yap et. al, Physica B 407(

39. SOME APPLICATION OF IS
Number density of charge carriers h, mobility , diffusion coefficient D, transference number t ion etc,
DETERMINATION OF IONIC TRANSPORT PARAMETERS IN IONIC CONDUCTORS :
Chitosan-LiClO4-TiO2-DMC
Muhammad et. al, Key Engineerinhg Materials (2014)

40. SOME APPLICATION OF IS
Number density of charge carriers h, mobility , diffusion coefficient D, transference number t ion etc,
DETERMINATION OF IONIC TRANSPORT PARAMETERS IN IONIC CONDUCTORS :
CMC-NH4Br
Eg. For CMC H+
CMC
Sample with optimised conductivity
Samsudin and Isa, J. of Applied Sciences 12 (2012)

41. SOME APPLICATION OF IS DETERMINATION OF IONIC CONDUCTION MODEL
log () -  dc vs. log () (exponent s ) :
Gradient = s
Small Polaron Hopping (SPH ) model
Samsudin and Isa, J. of Current Engineering Research 1(2) (2011) 7-11

42. SOME PRACTICAL DETAILS FOR IS
Frequency window limitation: the available equipment have limited frequency range: fl to fh
Only part of IS spectrum is obtained (depends on R and C value)
Changing the temperature may show different parts of the full spectra provided no new conduction processes comes into play at different temperatures
Curve fitting is needed to see full spectrum