CHAPTER N

A C CONDUCTMTY AND ELECTRIC MODULUS ST'UDIES OF

SCV, SCP, SCA AND SBS SAMPLES

4.1. A.C. THEORY

4.1.1. Impedance Spectroscopy

4.1.2. Dielectric Permittivity

4.1.3. Electric Modulus

4.1.4. Jonscher's Power Law

4.2. RESULTS AND DISCUSSION

4.2.1 . Complex Impedance

4.2.2. Frequency Dependence of Conductixit).

4.2.3. Dielectric Permittivity

4.2.4. Electric Modulus

4.3. CONCLUSIONS

REFERENCES

CHAPTER IV

kc. CONDUCTIVITT AND ELECIRIC MODULUS STUDIES OF

8CV, SCP, BCA AND SBS SAMPLE8

One of the important techniques to characterize many of the electrical

and electrochemical properties of fast ionic conductors is the conductivity

spectroscopy 11-31, In general, conductivity measurements are made by

applying the d.c. bias a m the material, but it results in the polarization at

the electnxlr-electrolyte interface, which opposes the applied field and hence,the

ionic current falls with time. To overcome this problem, the four-probe or two

rrversibl~ elpcbde methods are adopted to measure d.c. conductivity of the

materials 121. later, the single frequency measurements are canied out, but

thew methods are found to be madquate for understanding the complete

electncal bhavior of the materials, since the electrochemical processes

repmnted by banous md~mdual elements (K, C & L) are frequenw dependent

16 71 I J r r~c (~ , the a L technique has t w n de\eloprd ro rneasur m w a n c e

ober a range of frequencies to estunate the exact hulk conduchwtv and

frequrnof depndrnt conducmw m FICs From the unpedanm study, one can

o h m not only the bulk conduch~ity but also gram b u n d a y effects, loruc

transport, double layer formation at the electmdc-/electrolyte mterface, etc

Hence, m m n t years, Impedana- Spectmscup\ (IS] has become a powerful

techruque for charac temg the electncal properties of the FIC rnatenals and

thev mterfaces wth electron~caly conductmg electrodes Also, the IS

measurements can pmvide the dparmc pmperties to understand the

microscopic nature of the FIC materials 12, 5,6j.

Impedance Specw'opy is also d d Immittance S p e c m p y (IS] and

it is used to measure complex impedance Z', complex admittanw Y', complex

dielectric permittivity E' and complex modulus M' I2,9]. This chapter briefly

describes the a.c, theory, measurement & analysis of impedance data, a.c

conductivity and electric modulus. It also gives the detailed discussion of

temperature dependent of conductivity, ac conductivity, dielectric permittimh

and electric modulus studies to understand ion dynamics in the glassy

systems.

4.1. kc. THEORY

4.1.1. Impedance-

The complex unpedance Z/o) is defined as ratio of voltage V(t) to current

l(t) in the time domain on appl.yng a sinuso~dal signal of IOU arnphtude acmss a

solid electrolyte

whex I Z I is magnitude and ($1 is phase angle.

Impedance or lmrmttance spectroscopy means any one of the lollowing four

quantities

Complex impedance Z' = Z' -jZ"

Camplex Admittance Y* = Y ' + J Y " =

Complex permittjvity c' = c' - jt" = 11 ,jcK'J*

Complex ~ O ~ U ~ U S M* = M '+ jMW = jCQA'

Wher j= 4- 1, 6 is the vacuum capac~tancr and t~27rf is thr angul;v frequency

The admittance and permittivity are parallel functions characteristic at low

frequencies whereas the impedance and modulus are series functions at high

frequencies ( l ,6 ] .

The complex plane is used to represent the electrical quantities of real

and imaginary parts of complex impedance (2'). complex admittance (Y'),

complex permittivity (E*) and complex modulus (M*) 15, 71. The impedance and

admittance representations for the parallel & series combinations of an id& RC

circuit elements are discussed.

In series: The complex impedance for the series combination of

rsistance R and capacitance C, is given by

Z* = Z'-IZ" 4.6

u hms Z'- H, and 4 7

Fig. 4.1.a shows the resultant unpedance for R & C m series is

rcpmnted by a vertical line parallel to the irnggaq a d s lnrerszcting the real

;ids at R. The companding admittance plot @yes the semicircle intersecting

the rPal axis at the origin and at a point l / R as shown m f i . 4.1. b.

In parallel. The complex unpxlnntr for rhc parallel combmauon of % and C, 1s gven by

I I - = - t 4 0 Z * K,

Z' = Z'- /Z"

where,

1 ~ g . 4.1. a) Z vs. Z &, b) B vs.(; for xncs and c] Z" vb. 2' 8 d] H vs.G for pmllr! combinations of RC curuits

The complex impedance for a parallel combination of R, & C, represents

a semicircle, as shown in the Fig.4.1. c, mterseaing the rPal axis at origin & at a

point R. The difference between the two intercepts gives the bulk resistance [&I of the material. The equation of a circle uilh radius R/2 and center at (R/2.0) is

obtained from equations 4.10 & 4.11.

The corresponding complex admittance plot shorn in fig. 4.1.d gives the

straight line parallel to the irnaginq ads mtersecting the real a d s at l/R. The

~r.klionsh~p k twwn thc pariilkl and sc.nc.5 (ompnrnts is obtainrd frum the

equations 4.7,4.8,4.10 & 4.11

F g 4 2 a & b show the equnalenr cvcu~ts & the ~rnpedance plots

rspectlvely for an ]deal and ral sohd electml\?es From the fig 4 2 a, lt IS

observed a vemcal stcaght h e and a perfect srm~cucle in the unpedanw plot

for an ideal sobd electrohze From the fig 4 2 b. ~t IS observed that the inched

s t r z t h e 1s due to the presence of double layer capaatancp of elecbude

electrolyte interface. and the depressed senucucle cornponds to the parallel

combinahon of reslstanw and capautancr The angle of lnclinabon of the

s w t h e and angle of depression of the s e n u d e are due to the dsmbuted

81

la)

(b)

Fig. 4.2. Impedance spectra of a) ideal, b) real & c) polycn.;tallinc solid rl?i!roi!.rt.a

mimscopic material properties, termed as constant phase elemenr (CPE) (8.91.

The complex impedance of the real solid electrolyte is ispresented as

where 0 2 a I- 1. When a = 0, Z is frequency independent and K is the

resistance R, where as, when a = 1, Z = - jK/m, the CPE corresponds to

capacitance C. The fractional power law dependence on frequency is referred to

a s a constant-phase-angle element or non-Debye capacitor 110, 111. Fmm the

fig 4.2.b, the equivalent circuit for real solid electrolyte is given by the parallel

combination of CPE and bulk resistance in series with CPE of the interface.

Rg 4 2 c shows the unpedance spectrum for a polycnstaYlne sample

wth the two semnrcles, one due to resistance w~thm the grams of the matenals

and other due to the pamal or complete blockmg of charge camers at g m

houndanes From the fig 4 2 c the q u n almt clrcu~t for the polvcmtallme

samples IS given bv the senes of parallel combnabon of CPE & bulk reslstance,

CPE & gram boundary reslstance and the CPE of the nterface The gram

boundaries may act as a hindrance to the lomc transport or mav also

contribute to the h@ conduchon path due to the presence of large defect

density m the mterfaual w o n [7, 8, 12, 131

The complex impedance analysis has been performed using

A.Boukamp's equivalent cixuit softwan, in which non-linear least square

(NLLS) fittmg p d u r e is adopted (141. The NLLS fitting is pelformed in two

steps, i) by the pamal NILS fit, where the suitable points are selected on the

stra@t line part of the impedance data through which a tangential line is

drawn. The intersection of the line dth the real axis gives the bulk mistance.

ii) by choosing a set of three data points in the high frequency depressed

semicircle, a serniclrcle is fitted through origin and the W a l N I L 5 fit gives the

84

values for the resistance and CPE parameters. The entire impedance

distribution is simultaneously fined to the NLLS fit to obtain a set of parameters

(e.g. circuit elements) for the equivalent circuit. The most of systematic errors

are reduced and the parameters of the complex equivalent circuit are obtained.

If a varying field V(t) is applied to a matelial, then the polarization and

i n d u d charge Q are related as 1151

C) = c * I ,<, exp( /d) 4.16

where E* is the complex dielectric constant. The frequency dependent complex

dielectric permittivity is given by

(L'c) -1 4 17 (C i ) ( I ~ J O T )

uhere h and ~a arr the low and high frqurnry d~elertnc constants respecbvel\.

(1 = 2x1 is the angular fnquency, r is thc tune constant The E* is #\!en b)

C* = c8-,c,' 4.18

where E' is the relative permittivity or delrctric constant, 6" is the dielectric loss.

Separating the real and imaginary parts of the equation 4.17,

log f

Fig 4.3. Dielectric parametem a) c', b] E" & c) Ian fi as funcuon of frequenn

The above eqns. 4.19 to 4.21 are called Debye equations and 11. 4.3.a, b

& c show respectively E', E" & tan 6 as a function of frequericy for an deal

material 1151. In fig. 4.3 a, , E' is independent of frequency at low frequencies

and with increasing frequency, it passes an inflection at o = l/r and also, at

hlgher frequencies, it becomes asymptotic to %. In the fig. 4.3 b, the E" exhibits

a maximum at on = 1, where the oscillating charges are coupled with the

oscillating field & absorbs a maximum electrical energy. From the fig. 4 . 3 . ~ it is

found that tan6 exhibits a symmetric Debye behavior with a loss tangent

maximum at on =(t. / &)1/2.

Many real materials show a deviation from an ~dral equahons and exhibit

a non-Debye dielectric behavior. The non-Debpe dielectric response can be

described using Cole-Cole, Davidson-Cole and empirical expression proposed by

Iiavriliak-Negami (H-N) (16 - 181.

The dlelectnr prrmltt~wty c' IS rplnted to complex ~mwdancp Z* h\

1 C* --- 4.22

l(d',Z *

o 1s the angular frequency and 6 = t / A o b 1s the capacitance of free space

In the present study, the real and unaginanr parts of the dlektnc

W m t ~ w t v are calculated uslng the above quatlons 4 23 & 4 24, pellet

dlmens~ons and the measured unpedance data

The complex electric modulus is defined by the reciprocal of the complex

permittivity [I] .

where M' is the complex modulus, E' is the dielectric permittivity, M' is the real

and M" is the imaginary parts of modulus.

The complex elmmc modulus spectrum represents the measurr of the

dlstnbubon of Ion enewes or config~ratlons m the structure and it also

descnbes the electrical mlaxab~n and nucroscop~c prupertles of lonlc glasses

(19, 20) The modulus formalism has been adopted as ~t suppresses the

p o l m t l o n effects at thr eiet trodcirlect~'~l\lc interface Hence, the con~ple-s

electric modulus M (o) \pettra rflectb the d\narmc properneb of the sample

alone

For parallel combination of RC element, the lpal and imaginary parts of

the modulus are given by

F'lg 4 4 a shows the r a l M' t s magmn M" parts of ~ ~ r n p l e \ modulus

Spectrum for the parallel ro~nbindbon of RC tn a senucurle u l t e m 11ng the d

$7

a d s at 0 and b/C. In the fig. 4.4.b, for an ideal sohd electrolyte, the modulus

M" vs. log f represents the Debyr cutve with a single relaxation time and a

madmum at or = 1. In fig. 4.4.c, the impedance Z" vs. log f give the Debye peak.

the FWHM is found to be 1.44 decade and the corresponding p value is 1,

whose peak maxima coincide the peak of M vs. log f plot.

In case of real solid electrolyte, the modulus spectra exhibit a broad and

asymmetric non-Debye nature wth distribution of relaxation times. The

variation of relaxation time is continuous and is represented by normah&

function for the distribution of r~laxation times g(rl 12 11. Then the relaxation

function ~ ( t ) in time domain 1s given by

ifthr rrl~ualion tlmr r is thrmall~. ;~rtivatcti. rhrn

w h r r F, 1s the relaxabon acuvatlon ? n e w , k 1s ~ h r HOltzmann constant. T 1s

Ihr absolutr t e m p m t u r In thc frvquenm doman, the reiaxatlon funcnon is

exprssd m electnc modulus as

log i

Fig. 4.4. a) M vs. M', b) M" vs. log f and c) Z" vs. log f plots of lded RC riernrnts

where $(t] is the decay function given by

40) = exp[- ( f l r ~ ] 4.34

p is the stretched exponent, r is the relaxation time. The b(t) is the stretched

exponential KWW relavation function and is written as sum of exponential

terms

The g, coeniclents are evaluated by hrar least square fit of the above equahons

4 35 & 4 36 The lnput parameten p, M s & r for the fit are obtained from the

measured modulus data and wen? fitted to the h W funchon followmg the

method desmkd by Moyn~han et all2 11

4.1.4. Jorucher's Power Law

A typical frequency dependrnce t:onductiviv spectrum that shown in the

fg. 4.5 evhiblt three distinguish rgime a] low frequency dispersed b) an

intermediate plateau and c) conductivity d ~ s ~ r s i o n at hlgh frequency 115, 22 -

24). The variation of conductivity in thr low frequency rrgion mav be amibuted

to the polarization effects at thr el~trodeelectmlyte interface. At very low

frequenaes, morp charge accumulation oecurs and hence, drop in conductivity.

In the intermediate frequency @on, conductivity is almost found to be

frequency independent 8s equal to dc conductivity and at the hlgh hquency

region, the cpnductivlty increases with frequency. The frequency dependence of

electrical conductivity of solid electrolytes is explained by a simple expression

given by Jonscher's power law. The power law relates the frequency dependent

conductivity or the so-called universal d.wamic response (UDR) of lonlc

conductivity and frequency by 1'25,261

XY

plateau region

___--- 8

/ polarisation region

log f

Ftg. 1 5 Schematic representauon of log conciticu\;t\ s f rquenc?

where a , is the ac conductivity, 00 is the limiting zero frequency conductiiq

(udc), A is a pre-exponential constant, o = 27rf is the angular frequency and s 1s

the power law exjmnent where 0 < s < 1.

4.2. RESULTS AND DISCUSSION

4.2.1. Compla Impedance

The real and imaginary parts of complex impedance measurements were

made as a function of Erequency as well as at low temperatures (120 to 300k)

on the pressed pellets of type

conducting elechDde (silver) / sample / conducting electrode (sllverj

for formers, mdifier to formers and dopant compositions of SCV, SCP &. SCA

glassy samples. The measurd impedance data are anal@ using the

equivalent circuit software developed by A.Boukamp. All the circuit parameters

are adjusted simultaneously, m order to fit the measured data through N U

procedure to obtain the bulk resistance and the equivalent circuit of the

materials.

Fig. 4.6 a , b & c show the real (Z') and imaginary (2'7 parts of the

impedance spectra obtained at various low temperatures (120 to 300K) for thr

hgh conducting dopant salt composition of SCV, SCP 5, SCA samples

respectively. I b m the fig. 4.6 a, b & c, it is observed that with increase in

temperature in the lower range 120K to 200K, the formation of complete

dep& semicircle occurs and on further increase above 200K, the deprpsed

~emiclrcle starts disappear with a formation of an inched straight line. F'rom

the above fgures, it is observed that the intersection of th t semicirclr with thc

00

real axis shifts towards the origin with inrrease m temperature and the

frequency of the intersection with real axis also increase with temperature.

Similar behavior is observed for the low temperature dependence of the

impedance for former, modifier to formers and dopant composition respectively

of SCV, SCP & SCA glassy samples. The analysis of the spectra by the NLLS fit

yielded the values of the bulk resistance Rb from the intercept of the semidrcle

with real ads . The electrical behavior of the samples 1s represented as the

parallel combination of bulk resistance and distributed elements CPE due to

the depressed semicircle.

The bulk conductivity is calculated using the bulk resistance obtained

from the analyzed impedance data at the various temperatures (120K to 300q

and pellet dimensions for all the former, modifier to former and dopant salt

compositions of SCV, SCP & SCA glassy samples. Fig. 4.7 a, b & c show the log

(oTJ vs. 1000/T plots for all the dopant compositions of SCV, SCP & SCA glassy

samples respectively. The temperature dependence of conductidty obeyed the

Arrhenlus relation & activation ? n e w ('4 1s obtained from the slopes of the

linear fit. Fig. 4.8 a , b & c show mpectively the variahon of F_g with dopant salt

composition of all the SCV, SCP & SCA systems.

korn the fig. 4.6 a, b & c, it is observed that the depressed semicircle in

h~gh frequency w o n represents parallel combmauon of the bulk resistance

and bulk capacitance of the glassy materials (27 - 291. The center of the

depressed semicircle displaced below the real axis and it may be due to the

prrsencr of distribution of various elements at the elecb'olyteelectnxle system,

which indicates that the relaxation time r is not a single valued but distributed

cuntinuously around a mean value 11, 131. Hence, the distribution of various

elements in the system is represented by the constant phase w e element

CPE. The equivalent circuit is representd by the parallel cumbmauon of the Rb

and CPE for each glassy sample 1 I, 30.3 I ] .

Fig. 4.6. a impedance spectra for lugh conducting mmposluon of SCV glassy samples at Merent temperdtuRs

' f, *.

D 0 10 70 30 10 70

Zrd, [ohn~! (xlW'31 ,

Fig. 4.6, a Impedance spectra for high c o n d u m g co:n?oslrnn of SC'J glassy samples at different temperdturos

F!g. 4.6, b Impedance spectra for h ~ g h conductmgcaIlipos!tion of SCI'g:;is\y samplcs at dflerent ternperdtum

I:lg. 4.6, b impedance spectra for hgh conductvlg colnposiuoi> ol SCP glassy samples at diflerent temperatures

Fig 4.6, c, Impedance spectra for h~gh conducting composttion of SC, g!ass! samples at dinerent temperatures

+ 188K

\ x \;:;:

b . - - -

10 20 Z.r-1, [ohm] ( x l O A ? ) -->

Fig. 4.6. c, Impedance spectra for high conducung composition of SC.4 glassy samples at different temperatures

Fig. 4.7 Log(fl) vs. 1000/T plots of various dopant compositions of a) SCV, b) SCP and c) SCA samples

I I I

40 50 60 70

Dopant composit~on

Dopant composition

dopant composition

Fig. 4.8 Activation energy for different various dopant cornpositlons of a] S e , b) SCP and c) SCA samples

4.2.2. Frequency Dependence d Conductivity

The ac conductivity is calculated from the measured real & imaginary

parts of the impedance data and pellet dimensions for all the formers, modifier

to formers & dopant compositions of the SCV, SCP & SCA (120 to 300K) and

formers & modifier to formers compositions of SBS (300 to 473K) glassy

samples. Fig. 4.9 a, b, c & d show respectively the frequency dependence of

conductivity obtained at various temperatures for all the dopant compositions of

SCV, SCP & SCA and modifier to former compositions of SBS glassy samples.

From the fg 4.9 a, b, c & d, it is o b s e ~ e d that the conductivity spectra show the

hvO distinct regimes, within the measured frequency window, 1) plateau and 2)

dispersive regions. The plateau region corresponds to frequency independent

conductivity or known as dc conductivity. The a, is obtained on extrapolation to

loner frequencies and found to be in good ageement with the dc conductivity

o1)tained from the impedance analysis. The frequency dependence of

conductivity in the dispersive rgions for all Ihr compositions of the SCV, SCP,

SCA & SBS samples at various temperatures were analyzed using the Universal

Jonscher's power law.

The conductivity data were fitted to power law using non-linear

regression procedure. From the analysis, the fitting parameters 00, A & s were

obtained and table 4.1 gives 0, & s for the high conducting SCV, SCP, SCA &

SBS samples. Fig. 4.10 a, b, c & d respectively show the temperature

dependence of s for the dopant compositions of SCV, SCP bSCA and modifier

to former compositions of SBS samples. Fig. 4.1 1 a, b, c & d show the log (o/o,)

vs. log (a/%) plots at various temperatures, where o, is the characteristic

frequency at which om = 20~.

Fig. 4.9 Log (o) vs. log (a) plots for the hghest conducting dopant composition ofa) SCV & bj SCP samples at diRerent temperatures

a l , . 2 3 4 5 6

log (ol I

Fig. 4.9 Log (a) vs, log ((0) plots for the highest conducting c) dopant composition of SCA & dj m/f composition of SBS samples at different temperatures

Table 4.1. S, 4 8a 00 at temp (4 for high conducting SCV, SCP.

SCA and SBS samples

1 SCP / 165K / 0.643 ) 371.5 / 2.6073 x10" 1 1 A 1 l68K 1 0652 1 94.2 1.7476x107

SBS 413K 0548 62.8 1 177x10'

In the fig. 4.9 a,b,c & d of logo vs. logo plots, the frequency at which the

dispersion region starts from the dc conductivity plateau can be defied as the

characteristic kquency (%!,where the relaxation effects of the ions occur. Thls

characteristic frquency is termed as hopping rate ((+I or cross over frequency

and it occurs at a, = 20,1241. The relation between the dc conductivity and the

t~opping rate is given by a, = &, where k 1s the empmc~l constant, which

dcpends on the concentration of mobile ions and details of the conduction

mechanism. Fmm the Eg 4.9 a, b, c & d, it is observed that the dispersion

region decreases with increase in temperature, where characteristic frequency

PI, shift towards the high frequencies, and disappears on further increase of

irmperature in the measured frequency window 132,331. The ion-hopping rate

is less at low temperature a s compared to that at high temperature. It is also

found that the % is themially activated with the same e n e w a s ooT.

In the fig. 4.10 a, b, c & d, the variation of s uith temperature can be

mlated to the existence of distribution of relaxation parameters (34,351. From

the fig.4.11 a, b, c & d, the superimpossability of log (O/oO) vs, log (o/%] at

various temperatures suggests the conductivity relaxation mechanism is found

10 be temperature independent 1361. 91

Fig. 4.10 s a s a function of temperature for the vanous dopant compositions of a) SCV, b] SCP & c)SCA and m/f compositions of d] SBS samples

Fig. 4.11 Log (o/oo) vs. log (01%) plots for the hlghest conducting dopant composition of a) SCV and b) SCP samples at different temperaturrs

Fig. 4.1 1 Log (o/oo) vs. log (44) plots for the highest conducting cl dopant composition of SCA and d) m/f composition of SBS samples at different temperatures

As obsewed in the conductivity spectra, the plateau region of

conductivity may be due to transport of mobile ions in response to the applied

electric field. The long-range transport of ion result in oniy successful jumps

and their contribution yields d.c. conductivity (a,). The dispersive conductivity

region with frequency can be explained with respect to the predicted theoretical

Diffusion Controlled relaxation (DCR) model 135-37. In this model, it is

assumed that the glass forming matrix at the non-bridging oxygen (Si-O-, B-0 , V-O~, P-O- & As-O-) which is associated with a negatively charged ions to be a

site and it can be doubly occupied by the cation (Ag+). When a silver ion

diffuses to a singly occupied site, the positions of the original ion on those sites

excite and possess the higher e n e m configuration. At that time, the ions

mutually rearrange within the region to achieve a new lowest energy site result

m the former ion relaxation at that site is presumed to occur almost

instantaneously. The excited ion can relax back to its ori@ site or can diffuse

to another adjacent site. This indicates the cation hopping and diffusion

mechanism of the mobile silver ion at the local structures of the glass samples.

Thr lower s value obtained h m the plurr law r.uprssion can be correlated

with the higher successful jumps, which result in the hgher dc conductivity

obtained for the former, m/f and dopant varied compositions of the SCV, SCP &

SCA and former & m/f varied compositions of SBS samples.

4.2.3. Melectrif Permittivity Studies

From the measured 2' & 2" impedance data, E' and E" imaginaq

parts the dielectric permittivity were calculated according to the equations 4.23

& 4.24 for all the formers, m/f & dopant compositions of SCV, SCP & SCA and

formers & m/f compsitiohs of SBS systems.

Fig. 4.12 a, b, c & d show the real part of dielectric permittivity E' vs, log f

for the dopant compositions of SCV, SCP & SCA and modifier to former

Fig. 4.12 8' vs. log (freqi plots for the hlghest conducting dopant composition of a) SCV and b) SCP samples at various temperatures

Fig. 4.12 E' vs, log (fred plots for the highest conducting c) dopalt composition of SCA & d) m/f composition of SBS samples at various trmperatures

compositions of the SBS samples respectively. Fmm the fig. 4.12 a , b, c & d, it

is observed that with increase in frequenq, E' decreases and attains a constant

value at high frequencies. At high frequencies, due to the high periodic reversal

of the a.c field, there is no charge accumulation at the interface and hence the E'

remains constant, which can be explained in terms of the ion ~ s i o n

mechanism. At low frequencies, the charges get accumulated at the interfacial

region that lead to a net polarization of the ionic medium result in the fonnation

of space charge region at electrode-electrolyte interface which in turn increases

the dielectric constant E'. 124, 38-42]. In s" vs. log f, the loss peak could not be

observed in the measured frequency window and hence, the electric modulus

formalism is used for further analysis.

4.2.4. Electric Modulus

The electric modulus data is cdculated from the real & imaginary parts

of the measured impedance data and pcllet dimensions respectively using the

rquauons 4.27 & 4.28 in section 4.1.3 for all thr formers, m/f & dopant

compositions of SCV, SCP & SCA and formen & m/f compositions of the SBS

samples. The relaxation behavior is ana?vzed using the complex electric

modulus (M' = M' + j M ) formalism and the decay function 4 given by

Kohlrausch-Wk-watts ( k v . Fig 4.13 a, b, c Pu, d respechvely show the

unaginary part of electric modulus M vs. log f for the high conducting dopant

compositions of SCV, SCP & SCA samples and high conducting m/f

compositions of SBS samples measured at various temperatures.

R-om the fig 4.13 a, b, c & d, it is obsenred that the shape of the curves

are asymmetric with a long tad extending in the higher fquency region

exhibiting non-Debye behavior and it also o b s e ~ e d that the peak relaxation

frequency, f,, shfts towards the hlgher h q u e n q region with temperature.

The M curves a s a function of frequency werp fitted to equation 4.3 1 using the

9 5

procedure of Moynihan et al. by a least s q u m iterative software developed m

Fortran 77 as discussed in section 4.1.3 for all the dopant compositions of SCV,

SCP, SCA & m/f composition of SBS samples 12 11. The initial parameters M,, r

and fl are used in the fitting pmedure to obtain the best fit. The M, value are

obtained from the hlgh frequency limit of the M' vs, log f plots. The r & P are the

inverse of the peak relaxation frequency at M", and the full width half

maximum (FWHM) of the M vs. log f c u ~ e s respectively. The continuous lines

represent the fitted values of M , whereas the symbols represent the

experimental data. The value of P is graphically tuned such that the simulated

modulus curve could be superimposed on the experimentally obtained modulus

curves. Fig 4.14 a, b, c & d shows the temperature dependence of value for all

dopant compositions of SCV, SCP & SCA & rn/f compositions of SBS samples.

Fig 4.15 a, b, c & d show the log r vs. 1/T plots of all the dopant SCV, SCP &

SCA and modifier to former compositions of SBS samples. The log T vs. 1/T

plots are fitted to

by the linear least square fit and the relaxation activation energy is obtained

from the slope of the fit. The obtained Er, b, f,,% and !j values are gven in table

4.2 for the high conducting composition SCV, SCP, SCA & SBS systems. Fig

4.16 a, b, c & d show the normalized M"/M"t, vs. log f/f- obtairied at

different temperatures respectively for all the dopant compositions of SCV, SCP

& SCA and m/f compositions of SBS samples.

According to the obtained results, the formers, m&er to formers &

dopant compositions of SCV, SCP & SCA and former & m/f compositions of

SBS samples, the M curves are related to the energy dissipation in the

hversible conduction p m s s and they exhibit a non-exponential character of

decay p m s s .

pig. 4.13 M vs. log jfreq) plots for the highest conducting dopant composition of a) SCV and b) SCP samples at different temperatures

4 - + 1 2 3 4 5

log ifrrq)

Fig. 4.13 M" vs. log (freq) plots for the highest conducting c) dopant compsiuon of SCA & d) m/f composition of SBS samples at different temperatures

Rg. 4.14 P a s a function of temperature for the various dopant compositicr~s of a) SCV, b) SCP & c)SCA & m/f co~npositions of dj SBS samples

Pig. 4.15 Log (7) vs. 1000/T plots for the various dopant compositions of aj SCV, b) SCP & c) SCA & m/f compositions old) SBS samples

Table 4.2. P & f- at temp(K) and Er & bfor high conducting SCV, SCP,

SCA and SBS samples

-- / ~ i g h I T e m ~ / I Relaxation 1 1

cond&ng System

SCV

The broad non-Debye modulus spectra were interpreted in terms of

distribution of relaxaflon times r, which in turn were related to the distributions

of energy baniers Ea [15, 361. The M" vs. log f curves for all the samples

showed a deviation over the simulated data at higher frequencies much larger

than the peak frequency f-, whjch could be explained by considering the

contribution of asyrnmetnc double well potential (ADWP) confguration to the

conductivity 133, 43, 441. Fmm the above results, it could k concluded that

the relaxation time r is not a single valued, but is distributed over a mean value

140, 451.

1 ,

From the fig. 4.13 a, b, c & d, with increase in temperature the

broadness of the curves remain almost same, where P remains almost constant

would imply that all elements of the distributions have about the same

activation energy. However, the shift in the fm, with temperature could be

explained based on the distribution of attempt frequencies for the barner

crossover or a distribution of jump or ll@t distances following the crossover.

The broadness of the M" vs. log f curves is interpreted in terms of the

distribution of relaxation times for distinguishable physical processes. Fig 4.14

a , b, c & d showed that the P value is almost constant with temperature

b 1 I

1.5523~10-'~ / activation energy (E)

SCP 1 165K 1

! I

70 1 0.232i0.01 1 1.642 x10-l3

showing that the relaxation processes are temperature independent. The 97

168K ( 0.68

0.54

0.6

SBS t'"" 413K

5.89

44 0.3145i0.02

0.54 i x . 8 0.6212i0.03 1 9.562 x10-l3

0.260610.02 1 7.585 x10-13 I

Fig. 4.16 Mn/M" max vs. log (f/f-) plots for the highest conducting dopant composition of a) SCV and b) SCP samples at various temperatures

Fig. 4.16 M"/M" , vs. log (f/f-) plots for the highest conducting c] dopant composition of SCA and d) m/f composition of SBS samples at various temperatures

activation energies for the relaxation Er obtained from the slopes of the log r vs.

1000/T plot in fig. 4.14 a, b, c & d are slightly higher than thermal activation

energy, Ea. In the fig.4.lb a, b, c & d, the supeninpossability of normaked of

M"/MWmax vs. log (f/f-) spectra revealed that the distribution of relaxation time

is temperature independent.

Fig. 4.17a, b, c & d show the Z" & Mot vs. log f plots obtained at two

different temperatures for the high conducting compositions of SCV, SCP &

SCA and m/f compositions of SBS glassy samples respectively. From the fg

4.17 a, b, c & d, the observed broadening of Z" and M" spectra is due to the

existence of a distribution of relaxation timrs of the intrinsically dispersive

elements of the materials. The peak madma of Z" and M" spectra do not

coincide, a s there exist a dispersion effect due to the distribution of relaxation

times [46,47].

Fig 4.18 a, b, c PL d show the M" & log o vs. log f plots obtained at two

diiferent temperatures for the h~gh conductmg composition of SCV, SCP & SCA

and m/ f compositions of SBS glassy samples respectively. From the fig 4.18 a,

b, c & d, it is observed that the f,, from the M vs. log f modulus curve and the

% fmm log o spectra occur at the same frequency and with increase in

temperature both f,, & 4 shifts to higher frequencies. The hopping rate (+

with respect to the conductivity spectra and also the f- in the M" plot could be

a sensitive function of ion concentration and also may k due to some other

material property 1331. As observed from the impedance, power law and

modulus analysis respective&, the frequen~y at which the intersection occurs

with real axis, the frequeng at which relaxation effects hgin and the peak

relaxation frequency of M"-, shift towards the higher frequencies with increase

in temperature. From these observations, the conduction mechanism by which

the Ag ions preclude the matrix are comlated with ion hopping diffusion [48].

Fig. 4.17 Nonnalised M & Z" vs. log (freq) plots for the hghest conducting dopant composition of a) SCV & b) SCP samples at various temperatures

Fig. 4.17 NormaJkd M" & Z" vs. log (freq) plots for the highest conducting c) dopant composition of SCA & d) m/f ratio of SBS samples at vanous temperatures

pig. 4.18 M & log (0) vs. log (freqi plots for the highest conducting dopant composition of a) SCV & b) SCP samples at various temperatures

Fig. 4.18 M &, log (a) vs, log (freqj plots for the highest conducting c] dopant composition of SCA & d) m/f ratio of SBS samples at various trmperatures

4.3. CONCLUSIONS

The impedance measurements were made for all the formers, m/f &,

dopant compositions of the quaternary SCV, SCP & SCA samples at low

temperature (120 to 300K) in the frequency mnge of 40Hz to 100kHz. The a.c.

conductivity, dielectric permittivity and electric modulus were calculated from

the measured real & imaginary parts of the impedance data and pellet

dimensions, using their corresponding inter related formalisms. In the analyzed

impedance spectra, with increase in temperature, the intersection of the

semicircle with the real axis shills towards the ongin, in turn Increase in

conductivity. 'Rum the Arrhenius linear least square fit of the log (a] vs. 1/T

plot, the activation energy of the mobile charge carriers were calculated.

The conductivity with frequency dependence spectra were fitted to the

Jonscher's power law expression and obtained the fit parameters 00, s & A. The

obtained 00 is in good agreement with the ode. The conductivity spectral results

are explained with the eldsting theoretical Dflusion Contmlled Relaxation (DCR)

model. The modulus spectra were fitted to the KWW decay function and

obtained the stretched exponent B, the relaxation time r and the shape of the

spectra is non-Debye conlimed that the samples are ionic in nature. For

various temperaturs, the p value is found to be almost sarne. The

superimpossability of the n o m a h d M" spectra at different temperatures were

ascribed to temperature independent mechanisms of relaxation. The

conductivity and the irnaginaq pa^? of modulus M with a function of frequency

compared & idenhlied that the peak frequency f,, & the o, occurs at the sarne

frequency, which determines the material response in the vicinity of the

conductivity.relaxation time. The analysis by impedance, power law & modulus

formalisms suggest that the Ag' are the mobile cations & the mechanism by

which they migrate are correlated with the DCR model to e x p h the Ion

hopping diffusion of @ ions in the glassy matrix.

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