IMPEDANCE SPECTROSCOPY:
DIELECTRIC BEHAVIOUR OF POLYMER
ELECTROLYTES
By
Ri Hanum Yahaya Subban Ph. D
Faculty of Applied Sciences/Institute of Science
UiTM Shah Alam
OUTLINE
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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
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
IS: BACKGROUND
L
• Resistance of sample
A
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
IS: PRINCIPLE
V(t)
Sine wave signal V(t) = Vo sin t
i(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 )
: phase shift
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
PRINCIPLE: IS
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’)
Complex Impedance
plane
Im. Z
Z”
Z

Magnitude of Z, Z =
[(Z’)2
+
(Z”)2]1/2
Z’
Real Z
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
Computer
Impedance spectrometer:
LCR meter/FRA
Electrode
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
IS PLOT OF SIMPLE CIRCUITS
a. Pure resistance R
R
Z = R for all values of  or f
Z = R and  = 0
Z’ = R and Z” =0
Z”
Z”
R
Impedance plot is a point on the real axis at Z’ = R
Z’Z’
IS PLOT OF SIMPLE CIRCUITS
b. Pure capacitance C
C
Z = 1 = -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” -ve Z”

Z’Z’
Impedance plot is a
straight line lying on
the Z” axis
IS PLOT OF SIMPLE CIRCUITS
c. R and C connected in series
R
C
The total impedance
Z=R- j
C
-ve Z”

R
Z’
With Z’ = R and Z” = -1
C
On complex plane the graph becomes
a straight line at Z’ = R, parallel to the Z” axis
Z’
IS PLOT OF SIMPLE CIRCUITS
d. R and C connected in parallel
1 = 1 + 1
= 1 + jC
Z
R
1/j C
R
Z= R
R
C
1 + j C
=
R(1 - jRC)
= R(1 - jRC)
1 + (RC)2
(1 + jRC) (1 - jRC)
=
R
-
jR2C
1 + 2R2C2
=
Z’
-
1 + 2R2C2
jZ”
with Z” = RC
Z’
On eliminating  : (Z’- R/2)2 + (Z”)2 = (R/2)2
 Equation of a circle
IS PLOT OF SIMPLE CIRCUITS
d. R and C connected in parallel
-ve Z”
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
where RC =   Time constant or
Relaxation time
m

R/2
R
Z’
Note: Z’ and Z” axes must have
the same scales to see the
semicircle
From m , C can be calculated for an unknown circuit
IS PLOT OF SIMPLE CIRCUITS
e. Combined circuits
Rs
C1
C1
-ve Z”
C2
-ve Z”
Rs + R1
2 = R2C2
1 = R1C1
1 = R1C1
Rs
R2
R1
R1
Z’
due to internal resistance of electrolyte/electrode interface
R1
R1+ R2
Z’
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

Sample
Rb= bulk resistance
Rb
Cb Cb (Cg) bulk capacitance
Expected impedance plot
-ve Z” m
Cb is related to vacuum capacitance Co;
Cb = Co

Z’
Rb/2
Rb
And Co = oA
d
A - area of cross section  - dielectric constant
d - thickness of sample o – permittivity of free space
IS PLOT OF MODEL SYSTEMS
a. Ionic solid with two non-blocking electrodes: experimental results
Rb

Cb
Cb(Cg) bulk capacitance

(a) Li6SrLa2Ta2O12 with Li electrodes
(b) Li6BaLa2Ta2O12
Thangadurai and Weppner Ionics 12 (2006) 81-92
Note: depressed/distorted semicircles
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”
R
m
C’dl

R/2
C
R
R

Cdl
+ + + -
Expected impedance plot
Equivalent circuit
C’dl
+ + + -
-
C
Cdl= effective double layer capacitance
Z’
Cdl will add a spike to the
Impedance plot
+
IS PLOT OF MODEL SYSTEMS
b. An ionic solid with two blocking electrodes: experimental
results
SS/PVC-LiCF3SO3/SS
Au/Li6BaLa2Ta2O12/Au
Thangaduarai and Weppner
Ionics 12 ( 2006) 81-92
Subban and Arof, Journal of New Materials for
Electrochemical Systems 6 (2003) 197-203
Note: depressed/distorted semicircles and slanted/curved spikes
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
-ve
Z”

Pt
Grain
boundary
Z’
Rb
Rgb
Cb
Equivalent circuit
Cgb
Pt
Bulk
Thickness of grain boundary is small
 large Cgb
Rgb is large - larger semicircle for GB
The overall σ :
is determined by Rb +Rgb
Rb+ Rgb
Rb
Electrode/electrolyte
interface
Grain
Expected impedance plot
Cdl
IS PLOT OF MODEL SYSTEMS
c. A polycrystalline solid with two blocking electrodes: experimental
results
100 Hz
10 kHz
R1
C1
From the values of the capacitances
different semicircles can be associated
with different conduction process in the
sample
R2
C2
High frequency semicircle (small C) bulk conduction
Low frequency semicircle (large C)  grain boundary
conduction
Cdl
SS/ Li1+xCrxSn2-xP3-yVyO12/SS
Norhaniza, Subban and Mohamed, Journal
of Power Sources 244 (2013) 300-305
Note: Slanted/curved spike and
depressed/distorted semicircles
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)
R1
C1
Bulk
•
•
•
R3
R2
C2
Grain boundary
C3
Different phases or Orientation
of crystal planes
Cdl
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
0.1-10 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.
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)
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 =
1
= 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
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
-Q-
• CPE alone gives an inclined
straight line (pink) at angle (n=90)
Q
R
n= 90°
(1-n)= 90°
C
Z’
• 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 
IS PLOT OF REAL SYSTEMS AND CPE
The general equivalent circuit of a solid electrolyte with non perfect
blocking electrodes may take the form
R1
R2
R3
CPE4
CPE1
-Z”
CPE2
CPE3
Resulting impedance plot will have depressed
semicircles and a slanted spike
Here processes are assumed to be well separated
Z’
R1
R1 + R2
R1 + R2 + R3
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
IMPEDANCE RELATED FUNCTIONS
A.C voltage applied to a sample v = Zi
Generally impedance Z = R + j X; R = resistance, X = reactance
1
Hence the current , i 
v  Yv
Z
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
Co 
oA
d
IMPEDANCE RELATED FUNCTIONS
Complex Admittance
Y ( ) 
1
Z ( )
Y '

1
Z ' ( )  jZ " ( )
Z'
Z ' Z"
2
Z ' Z"
2
Z
Y"
2
Z'

'2
(Z
2

jZ "
Z ' Z"
2
"
 Z
"2
)
Complex Permittivity
 
Z
'
Co (Z
"
'2
 Z
"2
"
)
Z
Co (Z
"
M   C o Z
'
M
M 
'
(
'2
 Z
"2
)
M ( )   Z ( )    C o Z  j  C o Z
Complex Electrical Modulus

'2
'
"
"
 CoZ
 )
'
M 
"
"2
"

'2
"
"2
(   )
'
2
Z, Y AND M PLOTS FOR SIMPLE CIRCUITS
C
R
M”
-ve Z”
Y”

m

Y’

1/Rb
C0/2C
R
Z’
M’
Z, Y AND M PLOTS FOR SIMPLE CIRCUITS
R
C
Y”
M”
-ve Z” m

R/2
M’


Co/2C
Z’
R/2
m
R
1/R
Y’
 u 


 s  v
SOME APPLICATION OF IS
DETERMINATION OF DC IONIC CONDUCTIVITY
: - Z” vs Z’
OF IONIC CONDUCTORS
1.20E+03
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
1.00E+03
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
8.00E+02
-Zi()
Z
i (Ω)
R
6.00E+02
Note: both Z‘ and Z‘’ axes must have the
same scale in order to see the semicircle .
4.00E+02
If only spike is present , it can be extended to
obtain the intercept
R
2.00E+02
p2
2
0.00E+00
0.00E+00 2.00E+02 4.00E+02 6.00E+02 8.00E+02 1.00E+03 1.20E+03
p1
2
Zr (Ω)
S.K. Deraman Ph.D thesis UiTM 2014
 is calculated from R by using
 =LA/Rb
L - thickness of sample
A - area of contact
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
R1
CPE2
CPE1
A0
A5
Semi-circle disappears
Only resistive component
A5
prevails at higher frequency as
NH4Br/I content increases
Equivalent circuit of PVA-NH4x
at high NH4x concentration
A25
A30
Hema et. al J. Non Crystalline Solids 355
(2009) 84-90
CPE3
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
Two time-constant impedance spectrum
-Z”
C
Z’
B
R1
R1 + R2
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
C1
R1
Cg
C2
R2
C3
R3
Can be
equivalent
to
CR
Rg
C2
RR
R2
a is more suitable for a polycrystalline sample
b is more suitable for a homogeneous material
SOME APPLICATION OF IS
DETERMINATION OF DIELECTRIC
: ’ vs log f (permittivity/
PARAMETERS OF IONIC CONDUCTORS
dielectric constant,
transport processes )
Low frequency:
Static dielectric
constant  10 4
High frequency:
Optical dielectric
constant at 104Hz 
between 3.5 and 5
Compared to pure
PVC film = 3
PVC(1-x)LiCF3SO3xLiPF6
Subban and Arof Ionics 9
(2003) 375-381
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
In ionic conductors:
Relaxation peaks usually not observed due to
large electrode polarisation effects
Alternative is M’ /M” or ac conductivity
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) 755-761
SOME APPLICATION OF IS
DETERMINATION OF DIELECTRIC
: M” vs log f (relaxation
PARAMETERS OF IONIC CONDUCTORS
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
Gondaliya et.al Materials Sciences
and Applications 2 (2011) 16391643
Relaxation time  =1/2fmax
SOME APPLICATION OF IS
DETERMINATION OF DIELECTRIC
PARAMETERS OF IONIC CONDUCTORS
:
Tan  = ”/‘
Tan  vs log f
(relaxation time and
nature of conductivity
relaxation)
•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.
 ( t )  exp[  (
t
M
)

],  
1 . 14
FWHM
Effect of temperature
Effect of concentration
MG30-LiCF3SO3
Yap et. al, Physica B 407(20120 2421-2428
SOME APPLICATION OF IS
Number density of charge
DETERMINATION OF IONIC
carriers h, mobility ,
TRANSPORT PARAMETERS IN IONIC :
diffusion coefficient D,
CONDUCTORS
transference number t ion etc,
s = hem
Chitosan-LiClO4-TiO2-DMC
æ 2(Ze2 ) ö
s =ç
÷h Eat exp(-Ea / kT )
è 3kTm ø
  Ea 

kT


   o exp 
gradient

 Ea
k
Muhammad et. al, Key Engineerinhg
Materials 594-595 (2014) 608-612
SOME APPLICATION OF IS
Number density of charge
DETERMINATION OF IONIC
carriers h, mobility ,
TRANSPORT PARAMETERS IN IONIC :
diffusion coefficient D,
CONDUCTORS
transference number t ion etc,
1/ 2
l  v
 2Ea 
v

 m 
Eg. For CMC
l  1 . 5 nm
 
CMC-NH4Br
l
v
 kT 
D  
2

 he




H+
CMC
Sample with
optimised
conductivity
Samsudin and Isa, J. of Applied Sciences
12 (2012) 174-179
SOME APPLICATION OF IS
DETERMINATION OF IONIC
CONDUCTION MODEL
: log () -  dc vs. log () (exponent
s)
 ac   o  r 
    

ac
 A
dc
s

ac
Gradient = s
Small Polaron
Hopping (SPH ) model
Samsudin and Isa, J. of Current
Engineering Research 1(2) (2011)
7-11
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