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**IMPEDANCE SPECTROSCOPY: DIELECTRIC BEHAVIOUR OF POLYMER ELECTROLYTES**

By Ri Hanum Yahaya Subban Ph. D Faculty of Applied Sciences/Institute of Science UiTM Shah Alam

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**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

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**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

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**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

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**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 = 2f, 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

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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

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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

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IS: TECHNIQUE Measure Z(f) as a function of f(=2f) 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

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**IS PLOT OF SIMPLE CIRCUITS**

Z’ 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

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**IS PLOT OF SIMPLE CIRCUITS**

Z = = -j jC 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

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**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

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**IS PLOT OF SIMPLE CIRCUITS**

= = jC Z R /j C R Z = R 1 + j C = R(1 - jRC) = R(1 - jRC) (1 + jRC) (1 - jRC) (RC)2 = R jR2C 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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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.

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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)

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**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

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**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

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**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

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**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

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**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/jCoZ = Y/ jCo ’ - j ” Electric modulus M M = -1 = jCo Z M’ + j M” Where

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**IMPEDANCE RELATED FUNCTIONS**

Complex Admittance Complex Permittivity Complex Electrical Modulus

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**Z, Y AND M PLOTS FOR SIMPLE CIRCUITS**

C0/2C M” M’ R -ve Z” Z’ Y” Y’ m 1/Rb

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**Z, Y AND M PLOTS FOR SIMPLE CIRCUITS**

Co/2C -ve Z” Z’ m R R/2 R/2

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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

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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

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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

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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

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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)

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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)

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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 =1/2fmax Gondaliya et.al Materials Sciences and Applications 2 (2011)

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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(

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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)

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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)

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**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

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**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

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