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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 METHODA.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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Resistance of sample R = L A IS: BACKGROUND A L = 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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PRINCIPLE: IS Sine wave signal V(t) = V o sin t of low amplitude is applied to a sample V o = maximum voltage = 2 f, angular frequency The resulting current i(t) = i o (sin t + ) = phase difference between i(t) and V(t) (current is ahead of voltage by ) The impedance Z = V(t) = V o sin ( t) i(t) i o (sin t + ) Z is a function of frequency and has magnitude Z = V o = Z o and a phase angle i o Both Z and are frequency dependent quantities : phase shift i(t) V(t) IS: PRINCIPLE

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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” Z’Z’ Im. Z Real Z ZZ Complex Impedance plane

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

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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

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IS PLOT OF SIMPLE CIRCUITS R Z’Z’ Z”Z” Z”Z” Z’Z’ R 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 = 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’Z’ Z”Z” -ve Z” Z’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 C Z’Z’ -ve Z” Z’Z’ C c. R and C connected in series R 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 1= = 1 + j C Z R 1/j C R Z = R 1 + j C = R(1 - j RC) = R(1 - j RC) (1 + j RC) (1 - j RC) 1 + ( RC) 2 = R - j R 2 C 1 + 2 R 2 C 2 R 2 C 2 = 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 m RC = 1 m = 1 RC -ve Z” Z’Z’ R mm 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 RsRs R1R1 R2R2 R1R1 C1C1 C2C2 C1C1 -ve Z” Z’Z’RsRs 1 = R 1 C 1 R s + R 1 -ve Z” Z’Z’ R1R1 R 1 + R 2 2 = R 2 C 2 1 = R 1 C 1 e. Combined circuits due to internal resistance of electrolyte/electrode interface

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R b = bulk resistance IS PLOT OF MODEL SYSTEMS a. Ionic solid with two non-blocking electrodes Cell arrangement R and C connected in parallel (equivalent circuit) (assume no electrode resistance) Eg: Ag/AgI/Ag (Ag + mobile, I - immobile) No ion accumulation at the electrodes RbRb CbCb C b is related to vacuum capacitance C o ; C b = C o And C o = o A d C b ( C g ) bulk capacitance Sample Electrodes -ve Z” Z’Z’ mm RbRb R b /2 Expected impedance plot 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 RbRb CbCb C b ( C g ) bulk capacitance (a)Li 6 SrLa 2 Ta 2 O 12 (b)Li 6 BaLa 2 Ta 2 O 12 with Li electrodes Thangadurai and Weppner Ionics 12 (2006) Note: depressed/distorted semicircles

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

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b. An ionic solid with two blocking electrodes: IS PLOT OF MODEL SYSTEMS Thangaduarai and Weppner Ionics 12 ( 2006) Subban and Arof, Journal of New Materials for Electrochemical Systems 6 (2003) Au/Li 6 BaLa 2 Ta 2 O 12 /Au SS/PVC-LiCF 3 SO 3 /SS Note: depressed/distorted semicircles and slanted/curved spikes experimental results

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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) Grain boundary Grain Electrode/electrolyte interface C dl R gb C gb RbRb CbCb Equivalent circuit -ve Z” Z’Z’ RbRb R b + R gb Bulk Thickness of grain boundary is small large C gb R gb is large - larger semicircle for GB The overall σ : is determined by R b +R gb System = crystalline grain + grain boundaries + electrode/electrolyte interface Pt Expected impedance plot

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IS PLOT OF MODEL SYSTEMS c. A polycrystalline solid with two blocking electrodes: SS/ Li 1+x Cr x Sn 2-x P 3-y V y O 12 /SS Norhaniza, Subban and Mohamed, Journal of Power Sources 244 (2013) Note: Slanted/curved spike and depressed/distorted semicircles From the values of the capacitances different semicircles can be associated with different conduction process in the sample 10 kHz 100 Hz C1C1 R1R1 C2C2 R2R2 C dl experimental results High frequency semicircle (small C) bulk conduction Low frequency semicircle (large C) grain boundary conduction

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IS PLOT OF MODEL SYSTEMS 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) R2R2 C2C2 R1R1 C1C1 R3R3 C3C3 C dl 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 valuesPhenomenon responsible 2-20 pFBulk(main phase) 10 pF Second phase, orientation etc nFGrain 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. TYPICAL C VALUES

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IS PLOT OF REAL SYSTEMS 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) IS plot of real systems and devices are usually complicated

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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: Z CPE = 1 = Z o (jω) -n, 0 ≤ n ≤ 1 Y 0 (jω) n When n = 0, Z is frequency independent and Z o R, CPE ≅ pure Resistance When n = 1, Z = 1 /jωY 0. Hence Y o 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” Z’Z’ R n= 90° -Q- (1-n)= 90° R Q 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 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 R2R2 CPE2 R1R1 CPE1 R3R3 CPE3 CPE4 R1R1 R 1 + R 2 R 1 + R 2 + R 3 -Z” 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) 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 Generally referred to as ‘immitances’

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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 Where ImmitanceSymbolRelationComplex Form ImpedanceZ-Z’ – jZ” AdmittanceYY = Z -1 Y’ + j Y” Permittivity = 1/j C o Z = Y/ j C o ’ - j ” Electric modulus M M = -1 = j C o Z M’ + j M” Hence the current,

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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 R C R -ve Z” Z’Z’ Y” Y’Y’ mm 1/R b C 0 /2C M”M” M’M’

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Z, Y AND M PLOTS FOR SIMPLE CIRCUITS R C -ve Z” Z’Z’ mm R R/2 1/R Y”Y” Y’Y’ M”M” M’M’ mm C o /2C

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SOME APPLICATION OF IS S.K. Deraman Ph.D thesis UiTM 2014 In general IS plot consists of a depressed semicircle with a tilted spike and intercept on the real axis corresponds to R b R b 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= R b Note: both Z‘ and Z‘’ axes must have the same scale in order to see the semicircle. is calculated from R by using =LA/R b L - thickness of sample A - area of contact If only spike is present, it can be extended to obtain the intercept R p22p22 p12p12 R PVC-NH 4 CF 3 SO 3 -Bu 3 MeNTf 2 N - Zi( ) DETERMINATION OF DC IONIC CONDUCTIVITY OF IONIC CONDUCTORS :- Z” vs Z’

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SOME APPLICATION OF IS Hema et. al J. Non Crystalline Solids 355 (2009) A0A5 A25A30 A5 Semi-circle disappears Only resistive component prevails at higher frequency as NH 4 Br/I content increases PVA-NH 4 x (x = Cl, Br, I) CPE2 R1R1 CPE1 Equivalent circuit of PVA-NH 4 x at low NH 4 x concentration CPE3 Equivalent circuit of PVA-NH 4 x at high NH 4 x concentration DETERMINATION OF DC IONIC CONDUCTIVITY : - Z’’ vs. Z’ plot OF IONIC CONDUCTORS

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SOME APPLICATION OF IS ANALYSIS OF IS PLOT: CHOOSING EQUIVALENT CIRCUITS Choosing the correct equivalent circuit can be difficult An example: R1R1 R 1 + R 2 -Z” Z’Z’ Two time-constant impedance spectrum A B B C Some possible equivalent circuits Softwares do not give a unique equivalent circuit (model) for a particular IS plot but may suggest a number of complicated circuits (multiple models)

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SOME APPLICATION OF IS 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. ANALYSIS OF IS PLOT: CHOOSING EQUIVALENT CIRCUITS C1C1 C2C2 C3C3 R1R1 R2R2 R3R3 Can be equivalent to CgCg CRCR C2C2 R R2R2 RgRg b a An example: The circuits below can give 3 distinct semi-circles in the IS plot if their time constants are well separated a is more suitable for a polycrystalline sample b is more suitable for a homogeneous material

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SOME APPLICATION OF IS 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 Subban and Arof Ionics 9 (2003) In ionic conductors: Relaxation peaks usually not observed due to large electrode polarisation effects Alternative is M’ /M” or ac conductivity PVC(1-x)LiCF 3 SO 3 xLiPF 6 Low frequency: Static dielectric constant 10 4 High frequency: Optical dielectric constant at 10 4 Hz between 3.5 and 5 Compared to pure PVC film = 3 DETERMINATION OF DIELECTRIC PARAMETERS OF IONIC CONDUCTORS ’ vs log f (permittivity/ dielectric constant, transport processes ) :

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SOME APPLICATION OF IS Woo et. al Materials Chemistry and Physics 134 (2012) Same trend: variation with concentration DETERMINATION OF DIELECTRIC PARAMETERS OF IONIC CONDUCTORS : ’ vs concentration PCL-NH 4 SCN

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SOME APPLICATION OF IS DETERMINATION OF DIELECTRIC PARAMETERS OF IONIC CONDUCTORS : M” vs log f (relaxation time, transport processes ) Relaxation peaks Relaxation time =1/2 f max Relaxation peak is responsible for fast segmental motion which reduces the relaxation time and increase the transport properties Gondaliya et.al Materials Sciences and Applications 2 (2011) f max PEO- AgCF 3 SO 3

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SOME APPLICATION OF IS Tan = ”/ ‘ MG30-LiCF 3 SO 3 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. DETERMINATION OF DIELECTRIC PARAMETERS OF IONIC CONDUCTORS : Tan vs log f (relaxation time and nature of conductivity relaxation) Yap et. al, Physica B 407( Effect of concentration Effect of temperature

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SOME APPLICATION OF IS DETERMINATION OF IONIC TRANSPORT PARAMETERS IN IONIC CONDUCTORS : Number density of charge carriers , mobility , diffusion coefficient D, transference number t ion etc, Muhammad et. al, Key Engineerinhg Materials (2014) Chitosan-LiClO 4 -TiO 2 -DMC

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

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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: f l to f h 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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