Chapter - 3

Electrical Relaxation Process Studies in NASICON Type Glasses

3.1 Introduction

Migration of ions through a disordered material has been a topic of interest to

researchers for many years because of its inlportance in technological applications. In this

direction, many new ionic conducting glasses have been synthesized and studied [I-:]. The

monovalent (~ i ' , ~ a ' . K', Agf, etc.) and divalent (pb2', etc.) ion conducting glasses which

find applications in electrochemical devices such as batteries, sensors, fuel cell, etc., have

been prepared and studied. Apart from the technical application studies, these glasses help to

understand how the structure of the host glass in which the ions present influence their

mobility. The drastic increase in the dc conductivity with increasing ion concentration is a

common feature observed in many glassy materials [4-71.

For several decades, researchers have striven to understand the dynamics of the

mobile ions in solid ion conductors by interpreting the frequency dependent features of their

electric and dielectric response [8-121. These materials share common features such as (i)

disordered arrangements of the mobile ions within an otherwise rigid matrix, and (ii)

thermally activated hopping processes of the mobile ions leading to dc conduction. Highly

conducting solid electrolytes include structurally disordered ionic crystals such as a-

RbAg415 and the p-aluminas as well as inorganic ionic glasses [9]. A remarkable feature of

these materials is that their response to an applied field as described by an ac conductivity

~ ( o ) [8,9,11]. The physical origin of ac conductivity behavior is not yet con~pletely

understood.

Dielectric spectroscopy [8,11,13,14] has been traditionally applied to investigate

dipolar relaxation in liquids and solids where reorientation of permanent dipoles gives rise to

characteristic frequency-dependent features of the complex permittivity, &*(a) = ~ ' ( o ) -

is"(o). In this case, &'(a) increases with decreasing frequency approaching a limiting value

E~ at low frequencies associated with the polarization resulting from alignment of the diploes

along the direction of the electric field. Concon~itantly, the imaginary part of &"(a) passes

through a maximum at a frequency which is temperature dependent and whose inverse is

commonly associated with the characteristic time required for dipoles to reorient. In the

dipolar situation, a dc conductivity is unexpected and when observed usually indicates the

presence of impurity ions in the materials. This unwanted contribution to the dielectric loss

is normally subtracted away in a trivial manner.

For the ion-conducting materials discussed above, one might at first consider a

similar subtraction of the dc contribution so as to treat the resulting permittivity as a

relaxation process akin to that occurring in the dipolar case [ l I] . At one level, this appears

to be reasonable, since the mobile ions and the oppositely charged matrix resemble an

assembly of dipoles at short times. However, in practice, the polarization is inseparable from

the eventual conduction process. The mobile ion, which creates polarization by reorienting

locally, is the same ion that later separates from its immediate neighborhood to produce

conduction at lower frequencies. In ion conducting materials polarization and conduction

are, therefore, integrated into a single, continuous process [15].

As a consequence, alternative representations of the dielectric response have been

explored in literature over the past 30 years in attempts to better characterize the ion

dynamics in these materials. One prominent representation is the ac conductivity [16], o*(o)

= iasos*(a), which through linear response theory can be related back to fundamental

statistical quantities of the ion hopping motions such as the mean squared displacement,

<r2(t)>hop [ 1 71

However, considerably more prominent in the literature is the electric n~odulus

representation ~ * ( o ) = ~/E*((o), which was developed in 1972 [18]. The modulus shares an

important analogy with shear stress relaxation measurements by mechanical spectroscopy in

solids and in the case of ion-conducting materials, the frequency dependent of ~ * ( o ) can be

related to a corresponding time-dependent evolution of the electric field resulting from ion

displacements.

Among the two representations which one offers better insight into the nature of

ionic relaxation mechanism is yet to resolve. The electric modulus formalism of the

Moynihan et al. is one of the methods to understand the electrical relaxation process of ion-

conducting materials [19,20]. However, the Moynihan et al. approach fails to account for the

high frequency wing of the imaginary part of modulus data [21]. Several researchers have

noted the discrepancies between interpretations drawn from the conductivity and electrical

modulus formalisms [21-251.

The present chapter describes the electrical relaxation process of different

NASICON type materials by using both the ac conductivity and the electrical modulus

formalism. The first method is the analysis of ac conductivity by using Jonscher's universal

dielectric response (UDR) function. The UDR characterizes the non-Debye relaxation

feature of the materials. There are several inodels available to account the physical picture of

the universal dielectric response. Here, two phenomenological models are applied to account

the UDR, like (i) anonlalous diffusion model and (ii) jump relaxation model. In the second

method electrical modulus formalism is employed to describe the dielectric relaxation

through the KWW decay function. The electrical modulus formalism does not fit well with

measured data at high frequency. The poor fitting at high frequency is eliminated by using

anomalous diffusion model.

3.2 Experimental Details

Preparation of different ion conducting NASICON type phosphate glasses and their

temperature and frequency dependent electrical conductivity (parallel conductance G and

capacitance C) measurements have already been described in Chapter - 2. The measured G

and C were used to calculate the o(o), E'(o), E"(w) of permittivity and M1(u), MU(w) of

electric modulus.

The ac conductivity is obtained from the relation:

o(w) = GdlA, (3.1)

where d is thickness of the sample and A is the area of cross section. The real and imaginary

parts of permittivity are obtained from the relation:

&'(a) = C ~ I A E ~ and ~ " ( o ) = GdlA~oo , (3.2)

where EO = 8.854~10-l4 Ficm is the permittivity of free space. The real and imaginary parts

of the electric modulus are obtained from the relation:

M1(o) = E"(W)/(E'(W)~+E"(~)~) and Mu(@) = E'(w)/(E'(w)~+E"(w)~) (3.3).

3.3 Results and Discussion

3.3.1 Ac Conductivity and Permittivity

Figures 3.l(a-f) show the log-log plots of o(o) versus o for different NASICON

type glasses at different temperatures. The ac conductivity o(o), generally shows frequency

dispersion. The study of this dispersion behavior offers an opportunity to gain insight into

the details of the ion migration process, particularly the interaction of the migrating ion

within the materials.

The following features are observed from the ac conductivity spectra. The ac

conductivity is almost independent of frequency at lower frequencies and approaches the dc

conductivity od, as frequency is decreased. The dc conductivity is caused by translational

diffusion of the mobile ions. However, with the increase in frequency, it shows dispersion,

i.e, around the hopping frequency, o, the ac conductivity sets in, which shifts to higher

frequencies with increasing temperatures and for frequency greater than hopping frequency.

The o(o) is close to a frequency power law with exponent < 1 and it characterizes the non-

Debye feature [26].

At high temperature, a decrease in conductivity is observed in the low frequency

region and it is related to blocking of ions between the electrode and samples (electrode

polarization) [27]. This electrode polarization is also discussed in IS analysis of Chapter - 2.

The low frequency conductivity plateau length is found to increase with the increase in

temperature. In a power law regime, the ac conductivity is much less temperature dependent

than the dc conductivity.

This has been reconfirmed by plotting ln(a(w)) versus 1 OOOIT in Fig. 3.2 for NACP

sample at frequencies 0.042 kHz, 0.1 kHz, 1.0 kHz, 10 kHz, 100 kHz, 500 kHz and 1 MHz.

From Fig. 3.2, one can clearly see that at high frequency region (power law region), the

conductivity is much less temperature dependent than the dc conductivity.

. . . ! O 321K

10": A 331K D 341K 0 351 K Q 361 K

0 D 371 K

\ 0 381 K

2 lo-8; * 391 K n 1 0 401 K 3 w

0 410K

lo-'; + 420 K >. 430 K

(b) % 440 K l o - 1 o ' , 9 . . . . . - 450 K

lo2 lo3 lo4 lo5 lo6 lo7

o [radls]

Figs. 3.l(a-c). Ac conductivity spectra of NACP, NTZP and LACP at different temperatures. The solid lines are the best fits to Eq. (3.9).

KACP ~ $ 4

w [radls]

Figs. 3.l(d-f). Frequency dependent conductivity spectra of LTZP, KACP and KBP at different temperatures. The solid lines are the best fits to Eq. (3.9).

ode obtained from IS -0- 0 042 kHz -c- 0 100 Khz

A 1000 kHz C. 10 00 kHz

- 0- 100 0 kHz 4 500 0 kHz

Fig. 3.2. In(a(w)) versus 1000/T plot for the NACP glass.

T11e dynamic collductivity is usually analyzed by using the Jonscher's empirical

relation [28]:

O(W) K o[(w/L~)~) ",-I +(w/w~) * I - ' 1, (3.4)

where n~ and n2 are empirical constants of the materials and o, is the characteristic hopping

frequency. For various materials. in the frequency-independent region of conductivity, Eq.

(3.4) has been approximated with nl = 0, and n2 = n. Therefore, Eq. (3.4) becomes, I-n n

O(W) K [ O ~ + O ~ ' - " C O ~ ] = K o ~ + K o ~ o , (3.5)

where K is the constant of proportionality. The parameters included in K may be obtained

from the dc conductivity [29],

ad, = Kop = [ ~ e ~ a * / k ~ ] ~ c ( l - c ) w ~ , (3 .6)

where -y is the geometrical factor, which includes correlation factor; c is the concentration of

mobile ions on N equivalent lattice sites per unit volume; a is the hopping distance: e is the

electronic charge; T is the absolute temperature; and k is the Boltzmann constant. Equation

(3.5) has a general form, consisting of the dc and ac conductivity [28]:

O(O) = o d c + ~ a n , (3.7)

where the coefficient of the second term is A which is found to be temperature dependent

and is also related to the dc conductivity by [30]

A = G ~ ~ / o ~ ~ , (3.8)

substituting Eq. (3.8) into Eq. (3.7) provides an expression for the frequency dependent

conductivity [30,3 11:

Equation (3.9) is called the universal power law (UPL) or Alnlond and West

conductivity formalism. Equation (3.9) is used to fit the ac conductivity data and the

parameters ode, up, and n are extracted. Low frequency plateau conductivity deviates at high

temperatures due to the electrode effect. There are two ways to eliminate the electrode effect

(i) by inserting additional parameters in the fitting equation or (ii) by omitting the data

points in the region where there is prominent electrode effect. Here the second approach has

been chosen in fitting.

A standard procedure for non-linear least square (NLLS) fitting is provided in

Microcal's ORIGIN software package and it is used for the analysis of ac conductivity data.

The fitting parameters are extracted by Levenberg-Marquardt method of NLLS fitting. All

parameters were iterated about initial values using NLLS fitting with statistical weighing

value to obtain the best fit to the data (statistical weighing minimum value is necessary,

since the data span several orders of magnitude, both in conductivity and in frequency) [32].

The fitted paranleters have becn tabulatcd for a few saillplcs and arc given in 'Tables 3.1 (a-

c).

Tables 3.l(a-c): Parameters obtained from the fits of ac conductivity data by using Eq. (3.9) for different NASICON type glasses.

(a) NACP glass

(c) KACP glass

In Figs. 3.l(a-f), the symbols represent the measured ac conductivity data at various

T (K) 3 5 8

temperatures and the continuous lines are the best fits to Eq. (3.9). Similar results are

obtained for other samples. The frequency exponent n is temperature independent for all the

ode (Slcm) I I w, (radls)

samples and is given in Tables 3.2(a-f). The universal power law behavior is obeyed by all

n

samples. A comparison has been made between the UPL and impedance spectroscopy fitted

1.80t0.03 x 1 0-9 / 6.7420.04 x 10' I 0.63i0.03 ;

result for dc conductivity, and is shown in Fig. 3.3 ( ~ ( 0 ) and od, parameters obtained from

IS and UPL respectively representing the dc conductivity).

Fig. 3.3. Dc conductivity versus temperature plots for the sample LACP, LTCP and LTZP glasses. The open symbols are the data obtained from UPL fit and the filled symbols are the

data obtained from IS.

Figure 3.3 shows temperature dependent dc conductivity of some lithium based

NASICON type glasses. The open symbols and filled symbols represent respectively the

UPL and IS results. Both UPL and IS provide the same magnitudes of dc conductivity.

Figures 3.4(a-f)&3.5(a-f) show the dc conductivity and the hopping frequency as

function of inverse of temperature respectively. The temperature dependent dc conductivity

and activation energy of the NFP, NTP, LFP, LTP and KFP glasses are already reported in

the literature [33] and agree with present results well within experimental error. Furthermore

(i) sodium based glasses NACP, KAZP, NTCP. KTZP, NAP and NBP are compared to NFP

and NTP, (ii) lithium based glasses LACP, LTCP and LTZP are compared to LFP and LTP

and (iii) potassium based glasses KACP, KTCP, KTZP, KAP. KBP and KBiP are compared

to KFP.

The NAZP, NTZP, NAP, NBP glasses are having higher dc conductivity values

compared to NFP glass (Fig. 3.4(a)). However, the N d based glasses NACP, NAZP, NTCP,

NTZP, NAP, NBP, and NFP have conductivity Iower than the NTP. The dc conductivity of

NBP glass is higher compared to NACP, NAZP, NTCP, NTZP, NAP and NFP glasses. The

higher dc conductivity value of NBP is due to the mixed anionic effect [34,35]. Similar

effect has been observed in KBP glass.

-8 Q KTCP

ov -16- V

K

NAZP 1 NAZPEu NAZPHo NAZPNd NAZPTb NAZPTm NAZPEr

-6 NACP NACPDy NACPEr NACPHo NACPNd NACPSm NACPTb NACPTm

1 000rr [K"]

Figs. 3.4(a-f). Arrhenius plots of In(od,T) versus 1000/T for different NASICON type glasses. The solid lines represent the Arrhenius fits.

16

- cn 3 12 - - 3= V

1I: - 8

2.4 2.8 3.2 4 1 000/T [K-'1 2.0 2.4 2.8 3.2

1 O O O r r [K"]

G KACP KTCP KT2 P KAP KBP KBiP

NACP NACPDy NACPEr NACPHo NACPNd NACPSm NACPTb NACPTm

Figs. 3.5(a-0. Reciprocal temperature dependence of the hopping frequency for different NASICON type glasses. The continuous lines are the best fits to Arrhenius Eq. (3.11).

The LACP. LTCP and LTZP glasses are having higher dc conductivity values

compared to LFP and LTP at high temperature (Fig. 3.4(b)). However, the ~ i ' based glasses

LACP. LTCP and LTZP have dc conductivity lower than the LTP at low temperature. The

KAP and KBP glasses have higher dc conductivity values compared to KFP (Fig. 3.4(c)).

The KACP and KTZP glasses are having dc conductivity values almost equal to that of

KFP. The KTCP and KBiP glass's dc conductivity values are lower compared to KFP. The

dc conductivity of KBP glass is higher compared to KACP, KTCP, KTZP. KAP and KFP

glasses. Similar variations have been observed in the hopping frequency of the glasses (Figs.

3.5(a-c)). The dc conductivity and hopping frequency do not vary in the lanthanide oxides

doped glasses (NACP + 2wt%Lnz03 and NAZP + 2wt%Ln203) and in NQ "-,Ti, j+(P04)3 3 - ~

glasses are clearly seen in Figs. 3.4(d-f)&3.5(d-0.

The dc conductivity, od,. and ]lopping frequency, a,, obtained from UPL are

temperature dependent and they are found to obey the Arrhenius equations:

odcT = Aoexp(-E,/kT), (3.10)

o, = ooexp(-E,lkT), (3.1 1)

where A. is the UPL dc conductivity pre-exponential factor, E, is the UPL dc conductivity

activation energy for mobile ions. wo is the pre-exponential of hopping frequency, and Ep, is

the activation energy for hopping frequency. Figures 3.4(a-f)&3.5(a-f) show ln(od,T) versus

10001T and ln(w,) versus lOOOIT respective;!. for different NASICON type glasses; the

continuous line is fitted to Arrhenius equations. Activation energies, E,, Ep, and pre-

exponential factors Ao, wo are determined using the Arrhenius equation by linear regression.

The activation energies and pre-exponential factors are shown in Tables 3.2(a-f). The

activation energy, Ep, of the hopping frequency is in agreement to the activation energy, E,,

for the dc conductivity. The variation falls within the experimental error. This indicates that

the charge carrier have to over come the same energy barrier while conducting as well as

relaxing.

The dc conductivity pre-exponential factor of NASICON type glasses is found to be

differing by many orders of magnitude. Furthermore, it is found that the magnitudes of pre-

exponential factors increase with increasing activation. The magnitudes of A. are related

with activation energies. E,, by the Meyer-Nedel (M-N) rule or the conlpensation law of the

form [36-381:

lnAo = CLE, + P: (3.12)

where a and p are constants. The M-N rule is examined for the A3TiB1P;O12 (A = Li, K and

B' = Cd, Zn) glasses, to account for the possible source of the large variation in magnitude

of A"; the result is shown in Fig. 3.6(a). The expression for ionic conductivity pre-

exponential factor, Ao, is derived from the theory of ionic conduction and is found to be

[3 91 :

A. = ~e2a2c(l-c)yk- 'wo]e~P(~/k) , (3.13)

where oo is the fundamental vibrational frequency of the mobile ions, S is entropy for ion

conduction and the other parameters having their usual meaning. All the parameters in the

square bracket that make up the prefactors in Eq. (3.13) are unlikely to vary much in the

case of A3TiB1P3012 (A = Li, K and B' = Cd, Zn) glasses. This confirms that the entropy

term in Eq. (3.13) is the only possible source of the large variation in the dc conductivity

pre-exponential factor. The principal factor controlling conduction is the activation energy

(enthalpy) and, of the terms which constitute the pre-exponential factor, the entropy term is

of prime significance. The hopping frequency provides further evidence for the significant

variation of the entropy effect and it is also shown in Fig. 3.6(b).

Fig. 3.6. (a) Plots of In(Ao) versus dc conductivity activation energy and (b) In(o0) versus hopping frequency activation energy for A3TiB1P3OI2 (A = Li, K and B' = Cd, Zn) glasses.

The solid line is a fit to Eq. (3.12).

1 type phosphate glasses 1 ' , . , . , . , . I 4 8 12 16 20

Fig. 3.7. in(od,) versus ln(o,) plots for various NASICON type glasses.

Both w, and od, which are proportional, and the constant of proportionality, which is

almost universal, vary weakly with temperature. Barton [40], Nakajima [41], and Namikawa

[42] (BNN), carried out a closer analysis of the proportionality and arrived at the following

equation to be valid for most of the ion conducting materials:

udc = P E o A & ~ ~ , (3.14)

where p is a numerical constant order of 1, EO is the free space permittivity, nr (=r,-E,) is

the pemiittivity change from the unrelaxed baseline (E,) to fully relaxed level (r,). The BNN

relation conveys the important information about ac and dc conduction, which are closely

correlated to each other and are having the same mechanism. The plot of Injod,) versus

In(w,) is shown in Fig. 3.7, where the values of ode and a, were obtained from the best fits

of Eq. (3.9). The dashed lines are the least-square straight-line fits. The slopes are found to

be almost equal to unity for NASICON type glasses. The od, versus o, fall within the lower

and upper dashed lines as shown in Fig. 3.7 and this band is due to the material dependent

variation of AC. The slopes imply that the de. and ac conductions are correlated with each

other and the BNN relation is obeyed.

Tables 3.2(a-0: Dc conductivity pre-exponential factor (Ao), dc conductivity activation energy (E,), hopping frequency Pre-exponential factor (ao), hopping frequency activation energy (E,), UPL exponent (n), KWW function exponent CIS) and conductivity relaxation time activation energy (E,) for different NASICON type glasses (The error quoted is the maximum value that is obtained).

(a) Sodium based NASICON type glasses

1 Sample I ln(Ao) 1 E, (eV) / In(oo) / ED (eV) 1 n 1 a 1 E, ( ev ) / I

NFP / 10.38 / 0.73 / 32.67 1 0.73 / 0.66 / 0.55 1 0.71 1

1~ I

I I I , NTP 12.55 1

NACP 14.09 NAZP 1 11.76

l NTZP NAP NBP

(b) Lithium based NASICON type glasses

/ Sample / ln(A0)

NTCP

0.62 0.63

0.82 0.76

0.85 0.76

10.96 1 1.69 12.24

36.66 35.55

0.74

0.84 0.78

33.53 i 0.74

0.63 0.60

0.73 33.97 1 0.71 0.70 i 34.76 / 0.69

0.63 --

0.59 1 0.76 0.66 0.69

0.54 0.72 0.52 0.69 I

(d) Na4AlCdP3012 + 2wt0/iLn203 Sample

NACPNd NACPSm NACPTb NA4CPDy NACPHo NACPEr NACPTm

(e) Na4A1ZnP3OI2 + 2wt0/oLn203

In(Ao) (S1crn)K 14.01 15.43 15.12 15.36 14.95 13.88 / 0.86 15.16 / 0.91

Sample

NAZPEu NAZPTb BAZPHo NAZPEr NAZPTm

The imaginary part of permittivity corresponding to Eq. (3.7) is the permittivity loss

36.19 / 0.83 37.72 1 0.88

ln(A0)

0.79 I 35.35 0.78 1 35.91

12.37 12.10

(f) Na4.7++TTi~.3-x(P04)3.3-a

~ " ( a ) and it is given by

&"(a) = [O(W)-O~~]/E~O = (A/ E g ) ~ ( ~ - l ) ,

The real part of dielectric permittivity &'(a) is found to be

&'(a) = &m+(A/ ~~)tan(nx/2)w("~),

ET (eV) + 0.02 0.85 0.90

Ea (ev) + 0.02 0.89 0.93

NAZPNd

n Ea (eV) ln(oo)

12.01 12.53 12.41

where ern is the high frequency value of cr(o).

I

0.63 / 0.70 0.63 j 0 68

Ep (eV)

Figures 3.8(a-f) show the real part of permittivity &'(a) for different NASICON type

0.91 0.91

0 85 0.87

1

0.60 1 0.77 0.57 1 0.79

0.78 0.80

0.78 ' 35.72 0.80 1 36.20 0.79 / 35.99

Compos -ition

n 0.03

0.65

glasses as function of frequency at different temperatures. In the high frequency region, the

well known non-Debye behavior o("-" is observed [28,43].

P .t 0.02 0.70 0.69

ln(oo) (radls) 36.66

1 38.02

rt 0.03 (S1cm)K i 0.02 1 (radls)

0.60 0.60

Ea (eV) i 0.02

In(A0) (S1cm)K

37.72 37.87 -----

JI 0.03

0.78 ' 0.60 0.80 / 0.60 0.81 1 0.60

P i 0.02 0.60

0.91 1 37.73

Ep (eV) ' n * 0.03 1 + 0.03

11.68 I 0.78 1 35.63

0.58 ) 0.77 0.62 0.79 0.58 0.80

ln(oo) (radls)

x = 0.0

ET (eV) ~ k 0.02 ,

0.57 i

0.87 0.90 0.89 0.89

0.81 , 0.60 I 0.57 I 0.76 ~

Ep (eV) i. 0.03

32.75 11.20 1 0.60

0.63 0.63 0.63 , 0.68 , 0.88 0.63 1 0.69 / 0.86 1

0.89

0.58

0.63 1 0.69 1 0.87 I

co [radls]

y- NTZP A

- x'x -

.. . . . . . . . . ----I . . . . . . . . . . . ...."I . o2 l o 3 l o 4 l o 5 l o 6 l o 7

co [radls]

I 465 K P . ..-

l o 2 l o 3 l o 4 l o 5 l o 6 l o 7 (lj [radls]

Figs. 3.8(a-c). Variation of €'(a) versus o for NACP, NTZP and LACP glasses at different temperatures.

co [radls]

"0 1 "02

KAC P I

I rl KBP

[radls]

Figs. 3.8(d-f). Variation of &'(a) versus c i ~ for LTZP, KACP and KBP glasses at different temperatures.

At low frequency. the sharp increase is undoubtedly due to the contribution of the

charge accuinulation at the electrode-sample interface [44]. In low frequency region at high

temperature the electrode effect was observed and the power law deviates in the low

frequency region. The elimination of this effect is very difficult in a small frequency

window at the low frequency region. At high frequencies due to high periodic reversal of the

field at the interface, the contribution of charge carriers towards the dielectric constant

decreases with increasing frequency. Hence: &'(a) decreases with increasing frequency.

Furthelmore, the universal dielectric response of the NASICON type glasses is

accounted by using two phenomenological models: (i) the anomalous diffusion model [45]

and (ii) the jump relaxation model [46]. The anomalous diffusion model is used to extract

the physical relevant parameters, namely. length and time scales involved in the diffusion

process. The jump relaxation model is used to obtain the energy involved in the forward and

backward hopping. These are described in the following section.

3.3.l(a) Anomalous Diffusion Model (ADM)

The ac part of the conductivity remains an enigma. As thc power law featurc is nlost

prominent at high frequencies, it presumably must represent short time motion that occurs

prior to the hopping of the ion past its barrier; i.e., motion of the ion within its potential well

or possibly reiterative pair wise hopping between adjacent sites [47].

Empirically, the ac part can be obtained by assuming that the response of the electric

displacement, D,, to increments of the electric field are described at short times by a Curie-

von Schweidler current [48]

j(t) = dD,idt =tm". (3.17)

Currently, two basic views on how this Curie-von Schweidler current arises.

According to the first view, the high frequency power law is assumed to represent the high

frequency wing of a relaxation process whose low frequency wing is covered by dc

conductivity [49]. This relaxation is presumed to result from hopping of ions over local

energy barriers at high frequencies and long-range excursions over multiple barriers at low

frequencies.

The second view interprets the frequency dependence of the conductivity as simply

the result of changes in the manner in which the ions diffuse [50]. At long time (low

frequency) the mean-square displacement of a diffusing ion is linear in time, reflecting a

constant coefficient of diffusion and hence also constant conductivity ad,. At shorter times,

the ion is strongly influenced by the local environment, including interactions with other

neighboring ions, and exhibits a mean-square displacement, which increases more slowly

(the many-particle ion displacement calculation and discussions are presented in next

chapter).

Number of models has been proposed based upon anomalous diffusion to account for

the ac part of the conductivity [51-541. Niinicrical simulations for thc ~ncan-square

displacement. <r2>, of an ion performing a random walk on a fractal lattice indicate

According to Sidebottonl approach [45,55], expressions for the real part of dielectric

permittivity ~ ' ( o ) and conductivity o(a) are derived from Eq. (3.18) and are given by

with a d c = .Kk20c, K = e2n,,/6k~ (3.21)

where g(n) = (1-n)T(l-n) = r(2-n), Q, = wlo,, n,, is number density of charge carrier, a,

establishes the frequency scale which separates the two regimes of diffusion and 5 is the

cross correlation length. Further, Eqs. (3.19)&(3.20) can be simplified by invoking the so-

called Maxwell relation; i.e., that c ~ ~ ~ / E , E ~ = a, is proportional to the crossover frequency

o, [7] and they are given by:

€'(a) = Em 1 + h(n) sin - (a/o,)"-' , L Ln:l '1 O(W) = C T ~ C 1 + h(n) cos - (olw, ) , L Ln;J n l

with h(n) = fng(n), and a, = fa,.

o [radls]

o [radls]

102 lo3 lo4 lo5 l o 6 lo7 o [radls]

lo2 lo3 lo4 lo5 lo6 l o 7 w [radls]

1 E-4 LTC P

o [radls] o [radls]

301 K 0 313< h 323 K

Figs. 3.9(a-0. €'(a) versus o and o(o) versus o for different NASICON type glasses. The solid lines are the best fits to Eq. (3.24) & Eq. (3.25) for &'(a) and o(w) respectively.

The Eq. (3.22) and Eq. (3.23) may be written as;

Equations (3.24)&(3.25) are used to fit the measured real part of permittivity and the

ac conductivity data by the non-linear least square fitting method where the parameters E, ,

adc, a, and n are extracted. At high temperature, low frequency &'(a) data are omitted in the

fitting approach. Both equations are fitted well with the ~ ' ( o ) and o(o) experimental data

and thc rcsults arc shown in I;igs.3.9(a-1') lor a I'cw NASlCON typc glasscs. A similar lilting

procedure is followed for the other samples. The dc conductivity. od,, and crossover

frequency, a,, are temperature dependent and are found to obey the Arrhenius relation.

Activation energies of the ode and o, are equal to the E, and Ep obtained from UPL.

The physically relevant length and time scales involved in the diffusion process are

obtained by using Eq. (3.21). The n,, is calculated using pN,,IM, where p is density, Ka, is

Avogadro's number and M is molecular weight of the compound. The 5 is calculated and

it's averaged over the temperature range investigated for some NASICON type samples.

These results are shown in Tables 3.3(a-c). The lanthanide oxides doped and composition

varied NASICON type glass's dc and ac conductivity do not change considerably and so

their ADM analysis results are not given in Tables 3.3(a-c). The ion cross correlation length

5 in A,B,(P03)4 system is higher when the B site occupancy with a single ion in tri, tetra or

pentavalent state e.g. NAP, NBP, NFP and NTP than when the B site occupancy with more

than single ion in tri, tetra or pentavalent state e.g. NACP, NAZPl NTCP and XTZP.

Further, this behavior has been confirmed from the ion displacement curve of many-particle

by Fourier transformation method and it is discussed in Chapter - 4.

The Moynihan et al. approach fails to account the high frequency wing of imaginary

part of modulus data. Exploiting the anomalous diffusion model eliminates the uncertainty

in electric modulus fitting in the high frequency wing will be discussed in section 3.3.2.

The n values 0.60-0.70 are observed for all the NASICON type glasses. These vaIues

were commonly found in a recent literature survey suggesting that they are universal values

for characterizing the non-Debye behavior of materials [56,57]. These particular values are

in the range that were obtained in the Monte Carlo simulation by Maass et al. [47] This

together with the small scale of 5, supports the Coulombic interaction-based diffusion

models such as the jump relaxation model of Funke [46] and the Coulomb-interaction lattice

gas model of Maass and coworkers [47,52].

Tables 3.3(a-c): Number density (n,,), ion cross correlation length ( k ) , jump relaxation model frequency exponent (p) and ac conductivity activation energy (E,,) for different NASICON type glasses (The error quoted is the maximum value that is obtained).

NTCP NTZP NAP NBP NFP NTP

0.325 0.273 0.248 1

3.37 x loL' 1 1.00 1 0.63 ' 3.75 x lo2'

3.74 x loL' 3.35 x 10" 3.32 x lo2' 3.64 x lo2 '

1.1 1 1.18

0.63 0.66

1 . 1 5 1.30 1.25

0.69 / 0.217 0.66 0.65

0.248 0.224

3.3.l(b) Jump Relaxation Model (JRM)

Conductivity dispersion can be understood in terms of Debye-Hiickel type

interactions between a hopping ion and the surrounding [46]. By construction. the jump-

relaxation spectra fulfill the relation in low frequency region (i.e.. 10 Hz - few MHz):

G ( o ) - D ~ ~ ( l + l / ~ t ~ ) ' ~ . (3.26)

where p < 1 is the slope in the power law regime and tl is the time constant, which is equal

to the onset of the relaxation process and is roughly equal to the rate of successful hops.

Equation (3.26) is used to find the p value. The p value is equal to n of the UPL. The p

values are listed in Tables 3.3(a-c). The JRM low frequency conductivity relation is almost

equal to the .4lmond-Wcst conductivily I'o~~malism. l'hc JRM rclation is used to lit

conductivity data and results are shown in Fig. 3.10 for KBP at selected temperatures.

1 E-6 KBiP

Fig. 3.10. JRM relation (Eq. (3.26)) for KBiP glass at selected temperatures

Fro111 the JRM, a given ion at site A can move to an adjacent vacancy at site B over

an activation barrier. In general, the cage-effect potential due to Coulombic interactions is

less favorable at site B than it was at site A, hence ion hopping is biased to return. However,

there will be a relaxation response of the surrounding environment, which tries to

accommodate the new location of the hopping ion. A11 this leads to a correlated forward-

backward hopping process, i.e., the occurrence of "unsuccessful jumps" that contribute only

to the ac, and not to the dc conductivity, and to the power law dispersion in the appropriate

frequency range. The exponent (1-p) indicates the mismatch between potential energies of

the two wells. The dc conductivity activation energy E, corresponds to hopping after the

surrounding environment has relaxed and is related to the ac conductivity activation energy

E,, by (1-p)E,. The calculated values of E,, for NASICON type glasses are shown in Tables

3.3(a-c).

3.3.2 Electric Modulus Analysis

An alternative approach to investigate the electrical response of materials is the

complex electric modulus ~ * ( o ) . The electrode polarization effect is suppressed in this

representation [19]. Figures 3.1 1 (a-c)&3.11 (d-f) show respectively the real and the

imaginary parts of electric n~odulus as a function of frequency for different NASICON type

glasses at selected temperatures. In Figs. 3.1 ](a-c), at all temperatures M1(o) reaches to a

constant value at high frequencies and tends to zero at low frequencies suggesting negligible

or absence of electrode polarization. The Mu(@) shows a slight asymmetric peak at each

temperature as seen in Figs. 3.1 1(d-0. The low frequency wing of the peak represents the

range of frequencies in which the ions can move over long distances, i.e. ions can perform

successful hopping from one site to the neighboring site. On the other hand, corresponding

to the high frequency wing of the MV(o) peak, the ions are spatially confined to their

potential wells and the ions can make only localized motion within the well. The peak is

positioned at around the center of the dispersion (Figs. 3.1 l(a-c)). The peak frequency shifts

towards higher frequencies with temperature. Similar features are observed for the

remaining glasses. The M"(o) is related to the energy dissipation taking place in the

irreversible conduction process.

w [radls] [radls]

LO [radls]

[radls]

16' lo3 I$ 16" 10716' o [radls]

o [radls]

Figs. 3.11(a-0. Frequency dependence of modulus isotherms of different NASICON type glasses.

The dielectric relaxation observed in glasses does not correspond to a single

relaxation time function. In these systems, the dielectric loss is not symmetrical about the

logarithm of frequency of the maximum loss and the Cole-Davidson function 1581 does not

give an adequate fit to the experimental data. The relaxation process is not exponential since

the spectrum is not a Lorentzian. Its skewed shape is well represented by the Laplace

transform of the Kohlrausch-Williams-Watts (KWW) [59,60] decay function as recognized

by Moynihan et al. [19,20],

#( I ) = exp[-(t I P 1 , (3 '27)

= M,[I -N*(w)], (3.28)

where M, = lls, is the inverse of the high-frequency dielectric permittivity and the function

40) is the time evolution of the electric field within the dielectric, T is defined as the

conductivity relaxation time, the exponent /3 characterizes the degree of non-Debye behavior

and is related to the full width at half maximum of MU(w) versus u curve and ~ * ( w ) is

defined as:

where L denotes a pure imaginary Laplace or a one sided Fourier transform. Simple analytic

expression for the Fourier transform given by Eq. (3.29), for the decay function #(tj is

available only for P = 0.5 and 1.0. For other values of P, Fourier transform of $(ti can be

obtained by approximating $(r) as sum of exponential terms.

where s is rnultipies of TO, the Fourier transform of ~ * ( w ) is then

The coefficients of g, are evaluated by a least squares fit of exp[-(tlz ) P ] in Eq.

(3.30) for a given value of p. A linear-least square fitting routine has been developed to fit

the experimental modulus data. The routine has been developed using the Mathenztica 4.1

package. Tables of (1-N") and N" are generated through Eq. (3.3 1 j. The values M' and M"

are then calculated using the g, coefficients and Eqs. (3.3 1)&(3.28).

Recently, Bergman [61] has made approximate frequency representation of the

KWW function, which allows fitting directly in the frequency domain. The imaginary part

of the M" in frequency domain due to KWW decay function has been well approximated by

(for p 2 0.4):

where Mu,, is peak maximum of imaginary part of n~odulus, and o,,,,, is the peak

frequency of imaginary part of modulus. The inverse of a,,, is equal to conductivity

relaxation time z.

A comparison has been made between the Moynihan et al. fitting approach of Eqs.

(3.27)&(3.28) and the Bergman fitting approach of Eq. (3.32). Figures 3.12(a)&(b) show

respectively the M"(a) versus o for KTZP at 397 K and K4P at 391 K along with

Moynihan et al. and Bergman fitting curves. As both fitting approaches are equal, the

Bergman fitting approach is used for further analysis.

[radls] o [radls]

Figs. 3.12(a)&(b). Frequency dependent of M" for the KTZP and the KAP glasses at 397 K and 391 K respectively. The solid line is a fit to Eqs. (3.27) and (3.28), the discontinuous line is a fit

to Eq. (3.32).

l o2 lo3 lo4 lo5 lo6 lo7 lo2 lo3 104 105 1 0 9 0 ~ o [radls] o [radls]

o [radls] o [radls]

Figs. 3.13(a-f). Imaginary part of electric modulus spectra of different NASICON type glasses at different temperatures. The solid lines are the best fits to Eq. (3.32)

1 O O O R [K']

I . . . . . . ' 2.0 2.4 2.8 3.2

1000iT [K"] 2:4 218 3:2

1000n- [K"]

Figs. 3.14(a-f). Arrhenius plots of z for different NASICON type glasses. The solid lines are the best fits to Eq. (3.33).

The peak maximum (M,,,), peak frequency (o,,,=lir) and stretching exponent ,O are

obtained from M"(a) data by using the Bergman fitting approach for all NASICON type

glasses and typical fitted curves are shown in Figs. 3.13(a-f). For given NASICON type

glass, the ,O is temperature independent and it is material dependent as shown in Tables

3.2(a-f). Value of ,b less than unity, shows the non-Debye behavior of the NASICON type

glasses. Generally, the high frequency tail of experimental data of MU(w) are poorly fitted in

KWW fitting approach (Figs. 3.12(a-b)&3.13 (a-0).

Conductivity relaxation time, t, for different NASICON type glasses are plotted as a

function of reciprocal temperature, 1000/T, in Figs. 3.14(a-f). The z is temperature

dependent and is found to obey the Arrhenius equation:

z = zoexp( E,IkT), (3.33)

where TO is the pre-exponential factor and E, is the activation energy for conductivity

relaxation time. The activation energy E, is listed in Tables 3.2(a-f) itself for the comparison

of and E, with n and E,.

A comparison of the numerical value n with those of stretch exponeiltial parameter P [62,63] (shown in Tables 3.2(a-0) of the conductivity relaxation model clearly show that

n # 1-P. The reasons for numerically different values of ,b and (1-n) may be (i) the limited

frequency window of measurement and, (ii) poor fitting method at high frequency wing. The

uncertainty fitting in MU(o) at the high frequency wing has been eliminated by using the

Sidebottom approach of ADM [45]. The ADM fitted &'(a) (Eq. (3.24)) and ~ ( w ) (Eq.

(3.25)) results are converted into the electrical modulus by using:

Mf(o) = ~ ' ( a ) / ( ~ ' ~ ( a ) i ( ~ ( a ) / a ) ~ ) , (3.34)

MV(o) = ( G ( ~ ) / ~ ) / ( € ' ~ ( u ) + ( ~ ( 0 ) / 0 ) ~ ) ; (3.35)

and the results are shown in Figs. 3.15(a-f) for different NASICON type glasses at selected

temperatures. It is clear from Figs. 3.15(a-f), that the ADM's Eqs. (3.34)&(3.35) fits well in

the high frequency wing of M"(a) and eliminates the uncertainty in the KWW fitting. It has

been widely acknowledged that fits using KWW typically miss this wing and underestimate

the actual data at these frequencies. The KWW fitting a,,,, values and ADM fitting o,

values are equal and they are shown in Fig. 3.16 for a few samples.

NACP (d) ,

w [radls] w [radls]

o [radls] w [radis]

o [radis]

Figs. 3.15(a-f). Isothermal modulus spectra for a few NASICON type glasses at selected temperatures. The solid cunres are according to Eqs. (3.34) & (3.35).

Fig. 3.16. Temperature dependence of conductivity relaxation frequency for a few NASICON type glasses.

The KWW and ADM fitting approaches provide the same quantitative information

regarding the conductivity relaxation frequency i.e., steady state ion transport process.

However, the ADM fitting is better than KWW fitting. It is evident from the analysis that

the ADM play an important role at the high frequency region and therefore the high

frequency wing of MU(u) has superior fitting.

3.4 Conclusions

Ac conductivity and permittivity behavior of various NASICON type glasses are

examined by employing universal power law. This conduction arises from the motion of

mobile ions within the material. The frequency independent conductivity found at low

frequencies represents the random process in which the ions diffuse throughout the network

by performing repeated hops between charge compensating sites. The power law dispersion,

however, indicates a non-random process, wherein the ion motion is correlated. Since the

frequency exponent n ranges between 0.60-0.70, this correlated motion is 'subdiffusive' and

indicates a preference on the part of the ion that has hopped away to return to where it

started. The BNN relation is found to be valid for all the NASICON type glasscs. The

variation of diclcctric pcrmitlivity with frcrlucncy is atlributcd to ion dill'usion and

polarization occurring in the NASICON type glasses.

Ac conductivity has been explained by using ADM and JRM. The ADM is used to

extract the physically relevant length and time scales involved in the diffusion process. The

jump relaxation model is used to obtain the energy involved in the forward and backward

hopping.

The modulus spectra are fitted to the KWW decay function. The shape of the

imaginary part of the modulus suggests that the relaxation processes of the present glasses

are non-Debye in nature. The high frequency tail of MU(u) are poorly fitted in both

Moynihan et al. and Bergman fitting approaches. Uncertainty in fitting M"(w) at the high

frequency wing has been eliminated by using the ADM. The KWW and ADM fitting

approaches provide the same quantitative information regarding the conductivity relaxation

frequency i.e., steady state ion transport process. However, the ADM fitting is improved

when compared with KWW fitting. It is evident from the analysis that the ADM play an

important role at the high frequency region and therefore the high frequency wing of M"(w)

has a superior fitting.

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