88

Chapter IV

MIXED ALKALI EFFECT IN NASICON GLASSES

4.1 Introduction

The majority of the known methods for calculating the particular properties of

oxide glasses from composition are based on additive formulae that represent the

calculated property as a linear function of oxide concentrations. The evolution of the

glass structure according to the composition provides an overview of the behavior of each

species. However the physical properties of oxide glasses cannot generally be related to

the composition accurately by means of linear functions of the amounts of each

component. Linear factors may be used, to a first approximation, and many such sets of

aspects have been invoked for the guidance of glass technologists in developing or

modifying glass compositions to meet particular specifications [1]. One of the important

exceptions to this approximate linearity is the effect of changing the relative proportions

of the alkali oxides in glasses containing more than one alkali. When one alkali is

progressively substituted for another, the variation of physical properties with the amount

substituted is often so non-linear that the initial trend is later reversed, giving rise to a

maximum or a minimum. This extreme departure from linearity is called the mixed alkali

effect (MAE) [2-5].

The use of mixed alkalis has been exploited in many commercial compositions to

give glasses having superior combinations of properties that could be obtained with the

incorporation of any one alkali alone. This effect has a significant application [6-8] and

makes the mixed alkali glasses of special interest, for instance, low dielectric loss glasses

can easily be obtained by incorporating two different alkali. The challenge of the mixed

alkali effect arises from its universal occurrence and from the systematic way in which it

increases with the difference in sizes of the alkali ions. An adequate theory must be

applicable to any oxide glass, simple or complex, and must relate the effect only to the

ionic sizes. Many authors has put forward theories to explain the effect as far as a

particular property is concerned, more especially the electrical conductivity, but the

mixed alkali effect is noticeable on the majority of properties and it is essential for the

success of a theory that it agrees, at least qualitatively, with all the experimental facts.

89

The MAE in glasses gives rise to large changes in many dynamic properties,

particularly those related to ionic transport such as electrical conductivity, ionic diffusion,

dielectric relaxation and internal friction, when a fraction of the mobile ions is substituted

by another type of mobile ions [1, 2, 5, 9]. Macroscopic properties such as molar volume

and density, refractive index, thermal expansion coefficient, and elastic moduli usually

change linearly or only slowly with composition. Properties related to structural

relaxation, such as viscosity and glass transition temperature, usually exhibit similar

deviations from linearity as other mixed glass-forming systems which do not contain any

cations [2-4]. The reduced diffusivity in mixed alkali glasses as compared to single alkali

glasses cannot be explained by any major structural alteration upon the mixing of alkali

ions. Rather, experimental results show [10-13] that the alkali ions tend to preserve their

local structural environment regardless of the glass composition. Furthermore, the two

types of alkali ions are randomly mixed in the glass [13-15]. Similar conclusions have

been drawn from computer simulations of mixed alkali glasses [16-18].

Based on the experimental findings, a few theoretical models have also been

developed to understand the MAE [19-24]. These models consider either based upon

structural features e.g., conduction pathways [19, 21, 22] or based upon differing cation

interactions resulting from differences in the mass and/or size of the cation [23, 24].

However, these models are more or less unverified assumptions, such as site relaxation, a

selective hopping mechanism, or a crucial role of Coulomb interactions between the

mobile ions. The promising model which takes into account the two features of the MAE

is the dynamic structure model (DSM) reported by Bunde et al., and Maass et al., [19,

22]. In these models the reduced ion diffusivity in mixed alkali glasses has been

explained in terms of a site relaxation and memory effect, where each type of mobile

cation is able to adapt the glassy nature according to its spatial and chemical

requirements. Swenson et al., have predicted MAE and its relevant alkali conduction

pathways for the mixed alkali glass (LixRb1-xPO3) through reverse Monte Carlo structural

models by bonds valence model [12]. While all these models yield a qualitative

composition dependence of the ionic diffusivity, none of them is able to account for the

mixed alkali effect in the frequency response of the ionic conductivity. This present study

explores the conductivity and relaxation mechanism in mixed alkali NASICON glasses in

90

the system (LixNa(1-x))5TiP3O12 and (LixNa(1-x))4NbP3O12) in order to understand the

dynamics of charge carriers in such oxide systems. The ac conductivity and relaxation

mechanisms have been analyzed in the framework of the conductivity and the modulus

formalism. In the present work it has been shown that the conductivity formalism

accounts for the same qualitative variation of relaxation parameters with composition as

the modulus formalism. In this chapter the electrical properties of the glasses have been

studied for NASICON glasses with varying compositions in (LixNa(1-x))5TiP3O12

(LNTPx) and (Lix Na(1-x))4NbP3O12 (LNNbPx).

4.2 Synthesis and Characterization

The mixed alkali NASICON glasses were synthesized by the conventional melt

quenching method. Stoichiometric amount of analytical grade Li2CO3, Na2CO3, Nb2O5,

TiO2 and NH4H2PO4 were used as starting materials. All the compositions form glasses

when cast onto a steel mould; these glasses were subjected to X-ray diffraction studies

and no crystalline phases were detected. FTIR spectrum shows similar six main peaks at

~1200 , 1080, 983, 900, 741, 544 cm-1 for Niobium based glasses and five main peaks at

~ 1150, 1050, 920, 741, 571 for titanium based glasses. The assigns of these bands are

mostly from the contribution of various phosphate vibration and very few from Nb and Ti

vibration which has been discussed in chapter II. There is no deviation in vibration

frequency when alkali atom is replaced, which insists that there is no structural changes

in the glasses due to MAE.

The density (ρ) and the molar volume (V) for these glasses are shown in

Table 4.1. When Li2O is replaced by Na2O, it can be noted that the measured density as

well as the molar volume increases. These variation shapes are similar to those of mixed

Li2O and Na2O alkalis in the Li2O–Na2O–MoO3–P2O5 system [25]. Since the values of

the density and the molar volume are consistent with the ionic size, atomic weight of

lithium and sodium elements and their amount in these glasses, there is no MAE in these

parameters. Glassy nature was confirmed in DSC for all the samples. The glass transition

temperature Tg, the onset of the crystallization temperature Ts, the peak crystallization

temperature Tc, and melting temperature Tm, and the thermal stability parameters (∆T,S)

[26, 27] and Hurby’s parameter, Kgl [28] were determined and listed in Table 4.1. All the

91

critical temperature is low for x=0.6 insisting MAE in thermal properties of the sample.

The strength of the MAE in the composition for the glass transition temperature is

defined as,

∆Tg=Tg,lin – Tg (4.1)

where Tg,lin is the linear interpolation between the experimentally determined Tg values of

the two end members (the single alkali NASICON glasses) at the composition which

corresponds to Tg. The ∆Tg,min for (NaxLi(1-x))5TiP3O12 and (NaxLi(1-x))4NbP3O12 is 47

and 44 respectively.

Table 4.1: Glass transition temperature Tg in K, onset of crystalline temperature Ts

in K, crystalline temperature Tc in K, melting temperature Tm in K, thermal

stability parameters (∆T, S), Hruby parameter Kgl and strength of MAE ∆Tg for NASICON glasses.

Sample Tg Ts Tc1 Tc2 Tm ∆T S Kgl ∆Tg ρ Vm

Na5TiP3O12 699 774 791 842 1057 75 2.24 0.34 - 2.83 158.24

Na4Li1Ti P3O12 672 781 793 833 883 109 2.16 1.34 34 2.76 156.44

Na3Li2Ti P3O12 667*

737* 747* 775* 868* 70 1.20 0.66 47 2.73 152.31

Na2Li3Ti P3O12 676 801 806 852 970 125 0.96 0.79 44 2.71 147.54

Na1Li4Ti P3O12 680 790 803 845 1079 110 2.35. 0.44 48 2.69 142.51

Li5Ti P3O12 736 818 832 - 1104 82 1.82 0.35 - 2.61 140.83

Na4NbP3O12 693 - - - 1034 - - - - 2.91 161.44

Na3.2Li0.8Nb P3O12 664 784 799 - - - - - 30 2.9 157.56

Na2.4Li1.6Nb P3O12 652* 744* 792* - - - - - 44 2.87 154.74

Na1.6Li2.4Nb P3O12 664 754 804 - - - - - 33 2.85 151.32

Na0.8Li3.2Nb P3O12 669 786 838 - - - - - 29 2.83 147.85

Li4NbP3O12 701 914 933 - - - - - - 2.82 143.82 *denotes the minimum value

4.3 Impedance spectroscopy and dc conductivity analysis

Typical complex impedance plots for the glass at various temperature are shown

in Fig. 4.1. At low temperature, glasses show only one arc representing the bulk

properties and at high temperature, two arcs are found which represents the bulk and the

92

sample electrode interface effects. The impedance data are fitted using Boukamp

equivalent circuit and corresponding bulk resistance for particular temperature has been

calculated. The dc conductivity for each temperature was obtained from the bulk

resistances which follow Arrhenius behavior. The temperature dependence of the dc

conductivity obtained from the complex impedance plots are shown in Fig. 4.2 for

(NaxLi(1-x))5TiP3O12 glass compositions. It is noted that the variation of the conductivity

with temperature obeys Arrhenius equation σdcT=σ0exp(−Eσ/kBT), where σ0 is a

conductivity pre-factor and Eσ is the activation energy.

0 4 8 120

4

8

12

-Z''(

ω)x108 [

Ω]

Z'(ω)x108[Ω]

293K

303K

313K

323K

Boukamp fit

Fig. 4.1: Complex impedance plot for Na2Li3TiP3O12 systems at various

temperature.

2.0 2.4 2.8 3.2 3.6 4.0

-8

-6

-4

-2 (a)

log(σdcT) [S cm-1K]

1000/T [K-1]

NTP

NLTP0.8

NLTP0.6

NLTP0.4

NLTP0.2

LTP

Linear fit

2.0 2.4 2.8 3.2 3.6 4.0

-10

-8

-6

-4

-2

0

(b)

NNbP

NLNbP0.8

NLNbP0.6

NLNbP0.4

NLNbP0.2

LNP

Linear Fit

1000/T [K-1]

log(σdcT) [S cm-1K]

Fig. 4.2: Temperature dependent of dc conductivity observed from impedance

spectroscopy for composition variation of mixed alkali in (a) (NaxLi(1-x))5TiP3O12 and

(b) (NaxLi(1-x))4NbP3O12 glasses.

93

0.0 0.2 0.4 0.6 0.8 1.010

-14

10-12

1x10-10

1x10-8

1x10-6

1x10-4

273K

323K

373K

423K

x [Li/Na]

σdc [S cm-1]

0.5

0.6

0.7

0.8

0.9

1.0

Eσ

Fig. 4.3: Composition variation of dc conductivity and its respective activation

energy for (NaxLi(1-x))5TiP3O12.

0.0 0.2 0.4 0.6 0.8 1.0

10-11

1x10-10

1x10-9

1x10-8

1x10-7

1x10-6

1x10-5 273K

323K

373K

x [Li/Na]

σdc [Scm-1]

0.60

0.65

0.70

0.75

0.80

0.85

0.90

Eσ

Fig. 4.4: Composition variation of dc conductivity and its respective activation

energy for (LixNa(1-x))4NbP3O12.

The values of the activation energy Eσ were obtained from the least-squares

straight-line fits. The dependence of the conductivity at selected temperature (273K,

323K, 373K and 423K) and its corresponding activation energy on the relative

composition of NLTPx and NLNbPx NASICON glasses are shown in Figs. 4.3 and 4.4,

respectively. The 273K dc conductivity data for x=0.4, 0.6 and 0.8 of NLTPx has been

obtained from the extrapolated data of Arrhenius equation. These plots show a minimum

94

near x=0.6. It is worth to notice that this minimum is usually observed in mixed-alkali

glasses. This could be attributed to the maximum of the activation energy. Such behavior

is compatible with mixed-alkali effect. Similar trend associated with the glass transition

temperature. The conductivity and the glass transition temperature are expected to behave

in a similar manner since both properties are associated with the dynamics of the glass

system. The drop in conductivity related to the mixed-alkali effect is about five orders of

magnitude at 273K and four orders of magnitude at 323K compared to the original Li and

Na analogue glasses. Indeed, deep minimum in isotherm of the conductivity increases

with decreasing temperature as shown in Fig. 4.3 & 4.4. The magnitude of MAE in dc

conductivity at a particular temperature can be determined:

∆log(σdc)=log(σdc,lin)-log(σdc,min) (4.2)

where log(σdc,min) represents the minimum experimental value of log(σdc). The value of

log(σdc,lin) is obtained from the linear interpolation between the experimentally

determined logarithmic conductivity of the end members, at the composition which

corresponds to log(σdc,min). The calculated value of ∆log(σdc) by Eq. (4.2) at 323K, 373K

and 423K are 3.87, 3.23 and 2.69 for NLTPx and 1.73, 1.68 and 1.50 for NLNbPx

samples respectively. Both results show that the MAE becomes less pronounced as the

temperature increased. The disappearance of MAE with increase in temperature was

predicted by Hunt by applying the theory of percolation transport [29]. The MAE

strength in the dc conductivity of the two samples interpret that NLTPx shows stronger

MAE strength compare to the NLNbPx samples. Similar to the dc conductivity, the

strength of MAE in the activation energy is defined as:

∆Ea= Ea,max-Ea,lin, (4.3)

where Ea,max gives the maximum value of activation energy at x=0.6 composition and

Ea,lin is the activation energy corresponding to Ea,max obtained from the linear

interpolation between the activation energy of the two single alkali glasses. The

calculated values of ∆Ea by Eq. (4.3) are 0.37eV and 0.28eV for NLTPx and NLNbPx

samples respectively. The mixed alkali effect in the activation energy for mixed alkali

NLTPx glass system is stronger than NLNbPx glass systems.

The MAE observed in the NASICON glasses can be understood on the basis of

dynamic structure model (DSM) reported by Bunde et al., and Maass et al., [19,22], the

95

observed minimum of the conductivity in the glasses could be attributed to the distinctly

different local environment of the two alkali ions, which are preserved in the mixed

glasses. The pathway network is extended for the Li based glass than for Na based

glasses, explaining the higher conductivity of the former glass. In mixed alkali glasses Li

and Na ions have distinctly different conduction pathways and the pathway volume for

the Li ions is considerably larger than for the Na ions, which implies that Na conduction

gives only a minor contribution to the total conductivity. The argument is that the atomic

characteristics of Li and Na are very different and each cation may reside in a site formed

by a local environment in the single glasses as well as in the mixed-alkali compositions.

Generally in oxide glasses, lithium and sodium cations are normally connected with non-

bridging oxygen anions to satisfy the charge neutrality conditions. Since the activation

energy associated with Na-glass is larger than that of single Li-glass, one can predict that

the magnitude of the interactions and the polarization effects related to the alkali-

environment are different.

In a single alkali glass, an alkali ion moves into a site previously occupied by the

same type alkali, a sort of structural memory effect [19] favors its migration. However, in

the mixed alkali compositions the hopping dynamics of Li and Na cations are intimately

coupled with the structural relaxations of the glass network. For instance, to

accommodate the jump of Li cation into a site previously occupied by a sodium cation,

the latter must undergo a local relaxation, after which the cation can continue to migrate

through the matrix. As a matter of fact, the alkali cations have different local

environments; they occupy specialized sites with the possibility to retain the memory of

their original position before changing the site due to the conductivity. Since Li+ and Na+

ions are distinguishable, these sites form clusters of various sizes which are intertwined.

Note that any vacant site may become occupied by a cation of different nature. When that

happened the concerned cation becomes effectively trapped until site relaxation is

reconfigured to the newly occupying ion. This trapping mechanism causes a reduction in

the overall ionic diffusion, as a consequence of such trapping, a decrease in dc

conductivity is observed.

Statistically, it can be noted that a minimum may be pronounced for a

composition corresponding to a maximum disorder of alkali elements. Accordingly, the

96

conductivity decreases when substituting lithium by sodium and vice versa. In addition to

ionic conductivity, the glass transition temperature, Tg, which is not directly dependent

on ionic transport, shows a pronounced departure from linearity at intermediate mixed-

alkali ion compositions. It is also observed that the glass transition temperature is lower

for mixed glasses than for the original compositions (x=0, 1). Such behavior could also

be associated to the ’structural disorder’ imposed by the presence of two kinds of cations.

This kind of Tg variation was also reported for other vitreous mixed alkali glasses.

4.4 Ac conductivity analysis

The conductivity isotherm is investigated as a function of the frequency for all the

compositions and Fig. 4.5 shows the plot for Na2.4Li1.6NbP3O12. The dynamic

conductivity related to the real part of the complex conductivity showed a typical

behaviour: a frequency-independent plateau for low frequency range and a power-law

increase at high frequencies. The conductivity spectra have also been analyzed in the

framework of the Almond–West formalism. To get a clear picture of ac response of the

conductivity, frequency dependent conductivity for all the composition at 323K is shown

in Figs. 4.6 & 4.7. The conductivity plateau region is much lower in the mixed alkali

glasses than in the single alkali glasses. As the frequency increases the conductivity rises

above its plateau value featuring a dispersive behavior. At low temperature the plateau

features is not observed for the composition x=0.4 and 0.6. The rapid fall of the

conductivity at low frequencies for compositions x=0 and 1.0 is the well-known electrode

polarization phenomenon.

The movement of dissociated cations in the glass matrix can be described in the

conductivity representation framework by Jonsher’s universal power law relation

represented by Eq. (3.3). The temperature dependence of dc conductivity for various

composition obtained from the ac conductivity analysis is similar to that of the dc

conductivity of impedance spectroscopic studies. The hopping frequency obtained from

the Eq. (3.5) shows the trend similar to the dc conductivity. The ac conduction takes

place on the mixed alkali glass with complex subset of diffusion cluster or fat percolation

cluster. These clusters consist of two types of alkali glasses which are randomly mixed

and tend to attain the same local structure environment as in single alkali glasses with

97

different low dimensional conduction pathways. This results in a large energy mismatch

between the local potential of site Li+ and Na+ which reflects as high activation energy

for the ions to jump into the dissimilar energy sites.

102

103

104

105

106

107

10-12

10-11

1x10-10

1x10-9

1x10-8

1x10-7

1x10-6

ω[rad s-1]

σ'(

ω) [S cm-1]

273 K

283 K

293 K

303 K

313 K

323 K

333 K

343 K

353 K

363 K

373 K

383 K

393 K

403 K

AWM Fit

Fig. 4.5: Ac conductivity of Na2.4Li1.6NbP3O12 at different temperature. Solid lines

are fit to Almond West model.

102

103

104

105

106

107

1E-10

1E-9

1E-8

1E-7

1E-6

1E-5

σ'(

ω) [S cm-1]

ω[rad s-1]

NTP

N4L1TP

N3L2TP

N2L3TP

N1L4TP

LTP

Fig. 4.6: Ac conductivity plot of composition (NaxLi(1-x))5TiP3O12 at 323K. Solid lines

are fit to Almond West model.

98

2.1 2.4 2.7 3.0 3.3 3.6 3.9

2

3

4

5

6

7

8

9

1000/T [K-1]

log(ωp) [rad s-1]

NNP

N32NP

N24NP

N16NP

N8NP

LNP

Linear Fit2.1 2.4 2.7 3.0 3.3 3.6 3.9

1

2

3

4

5

6

7

8

log(ωp) [rad s-1]

1000/T [K-1]

LTP

N1L4TP

N2L3TP

N3L2TP

N4L1TP

NTP

Linear Fit

100 1000 10000 100000 1000000 1E7

1E-10

1E-9

1E-8

1E-7

1E-6

σ'(

ω) [S cm-1]

ω[rad s-1]

LNP

LNNP0.8

LNNP0.6

LNNP0.4

LNNP0.2

NNP

AWM fit

Fig. 4.7: Ac conductivity plot of composition (NaxLi(1-x))4NbP3O12 at 323K. Solid

lines are fit to Almond West model.

(a) (b)

Fig. 4.8: The temperature dependence of the cross-over frequency ωp for

composition variation of mixed alkali (a) (NaxLi(1-x))5TiP3O12 and (b)

(NaxLi(1-x))4NbP3O12 glasses.

99

Arrhenius behaviour of the cross-over frequency ωp obtained from the best fits for

all glass compositions is shown in Fig. 4.8. The values of the activation energy Eω for

cross-over frequency of charge carriers are obtained from the least-squares fits of data in

Fig 4.8 and are displayed in Table 4.2 .The dependence of the cross-over frequency at

various temperature (323 K, 373 K and 423 K) and its corresponding activation energy

for various samples with composition (NaxLi(1-x))5TiP3O12 is shown in Fig 4.9. It is

observed that cross-over frequency shows a minimum and the activation energy of the

cross-over frequency shows a maximum at a value of x=0.6, which strongly supports the

existence of the mixed alkali effect in the present NASICON glass compositions. But it is

to be noted from the Fig.4.9 that as the temperature increases, the strength of the hopping

frequency decreases between the single and the mixed alkali glasses.

0.0 0.2 0.4 0.6 0.8 1.010

0

102

104

106

108

log(ω

p)

323K

373K

423K

x (Li/Li+Na)

0.5

0.6

0.7

0.8

0.9

1.0

Eω [eV]

Fig. 4.9: Composition variation of hopping frequency at three different temperature

and its respective activation energy for (NaxLi(1-x))5TiP3O12.

100

0.0 0.2 0.4 0.6 0.8 1.01E-11

1E-10

1E-9

1E-8

1E-7

1E-6

σac[S cm-1]

σdc

1x105Hz

1x106Hz

x(Li/Li+Na)

Fig. 4.10: Variation of the dc conductivity at 323K and the ac conductivity, at 105

and 106Hz for (NaxLi(1-x))5TiP3O12 with lithium cation mole fraction.

0.0 0.2 0.4 0.6 0.8 1.0

1E-8

1E-7

1E-6

σ' [S cm-1]

σdc at 323K

σac at 10

5 Hz 323K

σacat 10

5 Hz 373K

x(Li/Li+Na)

Fig. 4.11: Composition dependence of dc conductivity at 373K and ac conductivity

at 323K and 373K with fixed frequency of 105

Hz for (NaxLi(1-x))5TiP3O12 glasses.

101

Fig. 4.10 shows the variation of the dc conductivity (473K) and the ac

conductivity, at 105 and 106Hz after subtracting the dc conductivity with the lithium-

cation mole fraction. It is interesting to note that the ac conductivity also goes through a

minimum at the same composition and in the same manner as in the dc conductivity.

Although, the magnitude of the effect seems to decrease with frequency, it is nevertheless

significant and indeed the ac conductivity exhibits MAE. This is a decisive result in

observing MAE in the ac conductivity. It is also evident from Fig. 4.11 that the MAE in

the ac conductivity is present at different frequencies at a fixed temperature and also at a

fixed frequency (105Hz) and two different temperature with significant depth. In both

cases, the depth decreases with increasing temperature or frequencies.

4.5 Electric modulus

An alternate method to analyze the ac electric response from the sample is electric

modulus. A typical modulus spectrum for one of the compositions is shown in Fig. 4.12

at different temperature. It may be noted that the spectrum is slightly asymmetric

suggesting a stretching behavior for the mixed alkali composition. In order to get a proper

description of the relaxation, data are fitted with Bergman’s approach which is an

approximate frequency representation of the KWW function, allowing direct fitting in the

frequency domain. The solid line curves in Fig. 4.12 are the fits to this equation and the

parameters M"max, ωmax and β are extracted from the fit. The modulus peak gets shifted to

higher frequency as the temperature is increased. An interesting feature observed in this

modulus representation is the relaxation peaks appear in lower frequency for mixed alkali

glasses compared to the single alkali glasses. This is due to the increase in relaxation time

when the single alkali glass is replaced by second alkali gradually, which is associated to

mixed alkali effect and indicates slowing down of the ionic motions both on local and

long ranges [25].

The width of the modulus peak can be quantified by the stretching parameter β.

During the fitting procedure it was noticed that β depends on the frequency interval

chosen for fitting. This introduces uncertainties in the determinations of β. Although high

frequency points were excluded in the fitting procedure, β of the same glass varied

slightly for different temperature. The error limits in Table. 4.2 are estimated from this

102

variation of β. The modulus peak width decreases and hence the stretching parameter β

increases, as the alkali concentration decreases in single alkali glasses. This increase in β

parameter in mixed alkali glasses is because the mixed alkali glass LixNa1-xG behaves as

two diluted glass LixG and Na1-xG, where G is the glass matrix and the conduction takes

place in distinctly different pathways for the Li and Na. According to the coupling model,

the coupling or degree of cooperation is reflected in the coupling parameter n=1-β,

between ions when the concentration decreases [21, 30, 31]. Therefore the increase in the

concentration of second alkali will decline the cooperation between two ions and the

corresponding increase in β observed in mixed alkali glasses. The typical cation jump

distance tends to increase in mixed composition which is proposed to be the main reason

for the MAE in glasses. The temperature dependent relaxation time dependence is shown

in Fig. 4.13 for NLTPx samples, which clearly shows the relaxation features in mixed

alkali glass. The activation energy extracted from the linear regression is given in

Table. 4.2

10110

210

310

410

510

610

7

0.000

0.008

0.016

M''(

ω)

ω [rad s-1]

323K

333K

343K

353K

363K

373K

383K

393K

403K

413K

423K

433K

Bergman Fit

103

Table 4.2: Activation energies of dc conductivity (Eσ), impedance peak (Eimp),

hopping frequency (Eω) and conductivity relaxation time (Eττττ), dc conductivity at 323

K and modulus stretching parameter β for the different NASICON type glasses.

Eσ Eω Eτ Samples

±0.02eV

(σdc± 0.04%) Scm-1 at 323K

β ±0.03

Na5TiP3O12 0.58 0.54 0.55 9.42 x10-8 0.60

Na4Li1Ti P3O12 0.84 0.83 0.82 4.46 x10-10 0.65

Na3Li2Ti P3O12 0.94 0.96 0.95 4.59 x10-11 0.68

Na2Li3Ti P3O12 0.91 0.89 0.89 1.85 x10-10 0.63

Na1Li4Ti P3O12 0.80 0.78 0.79 1.45 x10-8 0.62

Li5Ti P3O12 0.56 0.50 0.52 2.31 x10-6 0.60

Na4NbP3O12 0.61 0.58 0.58 7.51 x10-8 0.58

Na3.2Li0.8Nb P3O12 0.83 0.77 0.78 4.73 x10-10 0.60

Na2.4Li1.6Nb P3O12 0.89 0.86 0.87 9.16 x10-11 0.63

Na1.6Li2.4Nb P3O12 0.84 0.82 0.81 3.86 x10-10 0.63

Na0.8Li3.2Nb P3O12 0.79 0.78 0.77 6.79 x10-9 0.61

Li4NbP3O12 0.61 0.58 0.57 1.09 x10-6 0.61

2.1 2.4 2.7 3.0 3.3 3.6 3.9

-7

-6

-5

-4

-3

-2

-1

τ [s]

1000/T [K-1]

NTP

N4LTP

N3L2TP

N2L3TP

N1L4TP

LTP

Linear Fit

104

4.6 Scaling

4.6.1 Ac conductivity scaling

The ability to scale different conductivity isotherms so as to collapse all to one

common curve indicates that the process can be separated into a common physical

mechanism modified only by temperature scales. In this chapter scaling studies have been

performed in mixed alkali glasses in ac conductivity and electric modulus and the results

are discussed. In order to compare the shape of the conductivity response the scaling

technique proposed by Ghosh et al., and Summerfield are adopted which is explained in

chapter III. The ac conductivity curve for particular composition of NASICON glasses

collapse into a single master curve for different temperature. This is proved in both the

method of scaling insisting that the shape of the conductivity dispersion does not depend

on temperature. In this chapter both the methods are adopted to scale the ac conductivity

for varying composition.

Fig. 4.14 shows the results of Ghosh scaling procedure for the mixed alkali

glasses. In this, the conductivity axes of each conductivity isotherm for a particular glass

composition at different temperature has been scaled by the dc conductivity σdc and the

frequency axis by the crossover frequency ωp obtained from the fitting of conductivity

isotherms. Surprisingly, it has been found that the mixed alkali NASICON samples

collapse into single master curve, this simply means that the compositional independence

of the electrical relaxation mechanism. As the conductivity isotherms superpose on a

single master curve, this may imply that the relaxation mechanism is not only

independent of temperature but also independent of concentration and type (i.e.

concentration of Na+, Li+) of the ionic charge carriers. Therefore, the advantage of using

hopping frequency as the scaling frequency is that it is not specifically delimitated by the

composition range or the type of glass. The change in hopping length with composition is

manifested in the change in the hopping frequency which takes into account the

correlation effects between successive hops through the Haven ratio. Generally it is

observed that, Haven ratio increases when there is a decrease in mobile ion concentration

in single alkali glasses [33]. Since the mixed alkali glasses is similar to the dilute single

alkali glasses the Haven ratio takes into account of the mixed alkali glasses as one alkali

is replaced by the second. This insisted that the mobile ion concentration is not

105

necessarily need to be proportional to the cation concentration and so variation in mobile

ions (single and diluted single alkali glasses) can scale into single master curve in Ghosh

scaling approach.

1E-3 0.01 0.1 1 10 100 1000

1

10

100

α

α

αα

αα

αα

αα

αα

αα

αα

ααααααααα

β

β

ββ

ββ

ββ

ββ

ββ

ββ

βββββββββββ

Χ

Χ

ΧΧ

ΧΧ

ΧΧ

ΧΧ

ΧΧ

ΧΧΧΧΧΧΧΧΧΧΧΧΧ

ΓΓ

ΓΓ

ΓΓ

ΓΓ

ΓΓ

ΓΓΓΓΓΓΓΓΓΓΓΓΓΓΓΓΓ

ΩΩ

ΩΩ

ΩΩ

ΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩ

Ω

QQQQQQQQQQQQQQQQQQQQQ

QQQQQ

Q

BBBBBBBBBBBBBBBBBBBBBBBBBBBBB

BB

CCCCCCCCCCCCCCCCCCCCCCCCCCCCC

CC

DDDDDDDDDDDDDDDDDDDDDDDDDD

DDDD

E

EEEEEEEEEEEEEEEEEEEEEEEEEEEEEE

FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF

FFFFFFFFFFFFFFFFFFFFFFFFFFFFF

FF

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

ΟΟΟ

ΟΟ

ΟΟ

ΟΟ

ΟΟ

ΟΟ

ΟΟ

ΟΟ

ΟΟ

ΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟ

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

σ(ω

)/σdc

ω/ωp

Fig. 4.14: Scaling plots for the conductivity spectra of different mixed alkali

(NaxLi(1-x))5TiP3O12 NASICON type glasses at different temperature.

106

108

1010

1012

1014

1016

10-1

100

101

102

103

ω/(σdcT) [rad s

-1/ S cm

-1K]

σ'(

ω)/

σdc

x= 0, 1

Fig. 4.15: Summerfield scaling plots for the conductivity spectra of mixed alkali

(NaxLi(1-x))5TiP3O12 NASICON type glasses.

106

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.750.1

0.2

0.3

0.4

0.5

∆ log[σ'(

ω)]-∆ log(ω)

log(σ'(ω)/σdc)

Li5TiP

3O12

Na2Li3TiP

3O12

Na5TiP

3O12

Fig. 4.16: Approximate slope of the conductivity dispersion σ'(ω) in

(NaxLi(1-x))5TiP3O12 glasses as a function of the scaled conductivity σ(ω)/σdc.

103

104

105

106

107

108

0.2

0.4

0.6

∆ log[σ'(

ω)]/∆ log(ω)

ω [rad s-1]

Li5TiP

3O12

Na2Li3TiP

3O12

Na5TiP

3O12

Fig. 4.17: Frequency dependence of the approximate slope of conductivity for

single and mixed alkali (NaxLi(1-x))5TiP3O12 glasses.

107

Fig. 4.15 shows the results of Summerfield scaling procedure for the mixed alkali

glass samples. In this the conductivity data for different composition were plotted as

log(σ(ω)/σdc) vs. log(ω/σdcT). All the conductivity data collapse to single curve for

different temperature but when the conductivity of different composition is take into

account, the single alkali glasses are closely similar, whereas the mixed alkali glass

shows a different behavior. The conductivity σ'(ω) of the mixed alkali glass increases

slowly compare to the single alkali glasses as the frequency increases. The shape of the

ac conductivity σ'(ω) in the dispersive region can be analyzed using the slope of the

conductivity curve in a plot of logσ'(ω) against log(ω). In order to enhance the difference

in shape between the conductivity dispersion data of the different glasses the slope of the

conductivity curve was plotted against σ'(ω)/σdc as shown in Fig.4.16. This approach was

introduced by Schroder and Dyre [34, 35]. The approximate value of the slope

∆[log σ(ω)]/∆[log(ω)] at each frequency was estimated using the forward incremental

ratio [log(σ(ωi+1))-logσ(ωi)]/[log(ωi+1)-log(ωi)]. Fig. 4.16 shows that the mixed alkali

glass behaves differently from the single alkali glasses at the onset of the dispersive

region, whereas differences between single alkali glasses become relevant only at higher

frequencies/shorter timescales. Fig. 4.17 shows the behavior of the slope

∆[log(σ(ω))]/∆[log(ω)] as a function of frequency, hence excluding any scaling

parameter. It can be observed that in single alkali glasses the slope of conductivity curve

increases almost abruptly above the low frequency plateau while in the mixed alkali glass

the onset of dispersion is less marked and the increase of the slope is gradual. Fig. 4.17

insisted that the transition from the conductivity plateau to the dispersive region is more

gradual in the glasses with lower alkali content [36]. Compared to the present

investigation, these results would suggest that, with respect to the conductivity, mixed

alkali glasses behaves as diluted single alkali glasses. This is in agreement with the

conclusions drawn from an electrical modulus [37].

4.6.2 Electric modulus scaling

In order to compare the shape of the modulus curves, the data points can be

superimposed on each other by rescaling the axes with M"(ω) by M"max and the

frequency axis by ωmax. Fig. 4.18 shows the normalized modulus curves for all the

108

compositions. It is clearly seen that the lower frequency wing of the normalized modulus

curve superimpose into single curve but it does not happen in high frequency wing. This

is because that the mixed alkali glass have narrow curve compared to the single alkali

glass. This makes the stretching parameter β low for mixed glass. The result of modulus

scaling insist that the long range conduction process are same for various compositions,

whereas the relaxation process vary with composition, this makes the high frequency

curve not to collapse to single curve

10-4

10-2

100

102

104

0.0

0.3

0.6

0.9

1.2

x= 0.2,0.4, 0.6

x= 0, 1

M''(

ω)/M'' max

ω/ωmax

Fig. 4.18: Electric modulus scaling plots for mixed alkali (NaxLi(1-x))5TiP3O12 glass

systems.

The scaling for modulus described earlier clearly shows that KWW function can

not describe the relaxation process in the whole frequency and temperature range

particularly in the high frequency range. This is because that the full wave half maximum

width W varies significantly as single alkali is replaced by the other. Dixon et al., [38,

39] studied the universality by scaling the dielectric response of different glass formers

and shown the dielectric master curve. This scaling approach has been extended for

electric modulus and studied scaling for various oxide glasses.

109

-4 -2 0 2 4 6-6

-4

-2

0

W-1log(ω

pM''/

∆M

ω)

W-1(1+W

-1)log(ω/ω

p)

433K

443K

453K

463K

473K

483K

493K

Fig. 4.19 Dixon scaling plot of electrical modulus data for Na3Li2TiP3O12 glass at

different temperature

-4 -2 0 2 4 6-5

-4

-3

-2

-1

0

1

W-1log(ωpM''/

∆M

ω)

W-1(1+W

-1)log(ω/ω

p)

LTP

N1L4TP

N2L3TP

N3L2TP

N4L1TP

NTP

Fig. 4.20 Dixon scaling plot of electrical modulus data for mixed alkali

(NaxLi(1-x))5TiP3O12 glass systems.

110

In order to obtain a single curve that superimposes all the modulus, plots has been

constructed between W-1log(M''ωp/∆Mω) and W-1(1+ W-1) log(ω/ωp) where W is the

width of the modulus peak normalized to a Debye relaxation, ∆M is the modulus

relaxation strength, and ωp is the peak frequency for the maximum observed in M''. The

results are shown in Fig. 4.19, where the Dixon scaling approach is quite successful in

collapsing M'' for a range of temperature over which W changes substantially from 1.1 to

1.6 decades. Furthermore, the scaling curve obtained has exactly the same form as that

reported for other types of relaxation processes in structural glasses. It is also interesting

to see from Fig. 4.20 that the modulus scaling is successful for the mixed alkali glasses in

the composition variation which clearly indicates that Dixon scaling is excellent when

compare to the power law scaling which is also reported earlier [40]. It also reveals that

there are no intrinsic changes occurring in the ion motion and relaxation in the mixed

alkali glass [41].

111

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