mixed alkali effect in nasicon glassesshodhganga.inflibnet.ac.in/bitstream/10603/5292/11/11_chapter...
TRANSCRIPT
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
References
[1] J. O. Isard, J. Non-Cryst. Solids 1 (1969) 235.
[2] D. E. Day, J. Non-Cryst. Solids 21 (1976) 343.
[3] A. H. Dietzel, Phys. Chem. Glasses 24 (1983) 172.
[4] M. Ingram, Glastech. Ber. 67 (1994) 151.
[5] J. F. Stebbins, Solid State Ionics 112 (1998) 137.
[6] R. M. Hakim and D.R. Uhlmann, Phys. Chem. Glasses 8 (1967) 174.
[7] R. M. Hakim and D.R. Uhlmann, Phys. Chem. Glasses 12 (1967) 132.
[8] O. L. Anderson and D. A. Stuart, J. Amer. Ceram. Soc. 37 (1954) 573.
[9] T. Uchino, T. Yoko, J. Phys. Chem. B 103 (1999) 1854.
[10] B. Rouse, P. J. Miller, and W. M. Risen, J. Non-Cryst. Solids 28 (1978) 193.
[11] A. C. Hannon, B. Vessal, and J. M. Parke, J. Non-Cryst. Solids 150 (1992) 97.
[12] J. Swenson, A. Matic, A. Brodin, L. Börjesson and W.S. Howells, Phys. Rev. B 58 (1998) 11331.
[13] J. Swenson, A. Matic, C. Karlsson, L. Borjesson, C. Meneghini, and W. S. Howells, Phys. Rev. B 63 (2001) 132202.
[14] B. Gee and H. Eckert, J. Phys. Chem. 100 (1996) 3705.
[15] F. Ali, A. V. Chadwick, G. N. Greaves, M. C. Jermy, K. L. Ngai and M. E. Smith, Solid State NMR 5 (1995) 133.
[16] T. Uchino, T. Sib, Y. Ogata, M. J. Iwasaki, J. Non-Cryst. Solids 146 (1992) 26.
[17] S. Balasubramanian and K. J. Rao, J. Non-Cryst. Solids 181 (1995) 157.
[18] J. Habasaki, I. Okada and Y. Hiwatari, J. Non-Cryst. Solids 208 (1996) 181.
[19] P. Maass, A. Bunde and M. D. Ingram, Phys. Rev. Lett. 68 (1992) 3064.
[20] P. Maass, J. Non-Cryst. Solids 255 (1999) 35.
[21] G. N. Greaves and K. L. Ngai, Phys. Rev. B 52 (1995) 6358.
[22] A. Bunde, M.D. Ingram, P. Maass, K.L. Ngai, J. Phys. A 24 (1991) 2881.
[23] R. Kirchheim, J. Non-Cryst. Solids 272 (2000) 85.
[24] D. P. Button, R. P. Tandon, C. King, M. H. Velez, H. L. Tuller, D. R. Uhlmann, J. NonCryst. Solids 49 (1982) 129.
[25] L. Abbas, L. Bih, A. Nadiri, Y. El Amraoui, D. Mezzane, B. Elouadi, Journal of Molecular Structure 876 (2008) 194.
[26] S. Mahadevan, A. Giridhar, A.K. Singh, J. Non-Cryst. Solids 88 (1986) 11.
[27] M. Saad, M. Poulin, Mater. Sci. Forum. 19&20 (1987) 11.
[28] A. Hurby, Czech. J. Phys. B 22 (1972) 1187.
112
[29] A. Hunt, J. Non-Cryst. Solids 220/1 (1997) 1.
[30] K. L. Nagi, J. Non-Cryst. Solids 203 (1996) 232.
[31] K. L. Ngai, G. N. Greaves and C. T. Moynihan, Phys. Rev. Lett. 80(1998) 1018.
[32] H. Aono, E. Sugimoto, Y. Sadaoka, N. Imanaka, G. Adachi, J. Electrochem. Soc. 136 (1989) 590.
[33] J. E. Kelly III, J.F. Cordaro, M. Tomozawa, J. Non-Cryst. Solids 41 (1980) 47.
[34] T. B. Schroder, J. C. Dyre, Phys. Rev. Lett. 84 (2000) 310.
[35] J. C. Dyre, T.B. Schroder, Rev. Mod. Phys. 72 (2000) 873.
[36] B. Roling, C. Martiny, Phys. Rev. Lett. 85 (2000) 1274.
[37] S. Vinoth Rathan, G. Govindaraj, Solid State Ionics (to be submitted)
[38] P. K. Dixon, L. Wu, S. R. Nagel, B. D. Williams, J. P. Carini, Phys. Rev. Lett. 65 (1990) 1108.
[39] P. K. Dixon, L. Wu, S. R. Nagel, B. D. Williams, J. P. Carini, Phys. Rev. Lett 66 (1991) 959.
[40] C. Leon, M. L. Lucia, J. Santamaria, and F. Sanchez-Quesada, Phys. Rev. B 57 (1998) 41.
[41] D. L. Sidebottom, P. F. Green, and R. K. Brown, Phys. Rev. B 56 (1997) 170.