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ROLE OF PbO IN LITHIUM ION TRANSPORT IN Li 2 O–PbO–B 2 O 3 GLASSES Munia Ganguli, M. Harish Bhat, and K.J. Rao* 1 Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore 560 012, India (Refereed) (Received November 9, 1998; Accepted November 17, 1998) ABSTRACT A wide range of compositions of glasses in the ternary Li 2 O–PbO–B 2 O 3 glass system was prepared, and dc and ac conductivity measurements were carried out on these glasses. The presence of lead leads to a decrease in dc conduc- tivities and an increase in the activation energies. This is likely to be due to the increase of the partial charges on the oxygen atoms and to the presence of the lone pair on the Pb atom; both of these factors impede lithium ion motion. The ac conductivity and dielectric behavior of these glasses support such a conjecture. © 2000 Elsevier Science Ltd KEYWORDS: A. glasses, D. dielectric properties, D. ionic conductivity INTRODUCTION Alkali borates are classical glass-forming systems [1–3]. The ability of boron to exist in three and four oxygen-coordinated environments and the high strength of the covalent B–O bonds it forms enable borates to form stable glasses. Lead is also known to be highly covalently bonded to oxygen in PbO [4]. Although PbO is not a good glass former by itself, it readily is incorporated into glasses [5–7]. PbO behaves both as a glass former and as a modifier in a variety of glass systems, such as silicates, phosphomolybdates, and phosphotungstates, and these aspects have been widely investigated in the literature [5–7]. *To whom correspondence should be addressed. Tel: 191-80-3092583. Fax: 191-80-3341683. E-mail: [email protected]. Materials Research Bulletin, Vol. 34, Nos. 10/11, pp. 1757–1772, 1999 Copyright © 2000 Elsevier Science Ltd Printed in the USA. All rights reserved 0025-5408/99/$–see front matter PII S0025-5408(99)00154-3 1757

Role of PbO in lithium ion transport in Li2O–PbO–B2O3 glasses

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ROLE OF PbO IN LITHIUM ION TRANSPORT IN Li 2O–PbO–B2O3 GLASSES

Munia Ganguli, M. Harish Bhat, and K.J. Rao*1Solid State and Structural Chemistry Unit, Indian Institute of Science,

Bangalore 560 012, India

(Refereed)(Received November 9, 1998; Accepted November 17, 1998)

ABSTRACTA wide range of compositions of glasses in the ternary Li2O–PbO–B2O3 glasssystem was prepared, and dc and ac conductivity measurements were carriedout on these glasses. The presence of lead leads to a decrease in dc conduc-tivities and an increase in the activation energies. This is likely to be due tothe increase of the partial charges on the oxygen atoms and to the presence ofthe lone pair on the Pb atom; both of these factors impede lithium ion motion.The ac conductivity and dielectric behavior of these glasses support such aconjecture. © 2000 Elsevier Science Ltd

KEYWORDS: A. glasses, D. dielectric properties, D. ionic conductivity

INTRODUCTION

Alkali borates are classical glass-forming systems [1–3]. The ability of boron to exist in threeand four oxygen-coordinated environments and the high strength of the covalent B–O bondsit forms enable borates to form stable glasses. Lead is also known to be highly covalentlybonded to oxygen in PbO [4]. Although PbO is not a good glass former by itself, it readilyis incorporated into glasses [5–7]. PbO behaves both as a glass former and as a modifier ina variety of glass systems, such as silicates, phosphomolybdates, and phosphotungstates, andthese aspects have been widely investigated in the literature [5–7].

*To whom correspondence should be addressed. Tel:191-80-3092583. Fax:191-80-3341683. E-mail:[email protected].

Materials Research Bulletin, Vol. 34, Nos. 10/11, pp. 1757–1772, 1999Copyright © 2000 Elsevier Science LtdPrinted in the USA. All rights reserved

0025-5408/99/$–see front matter

PII S0025-5408(99)00154-3

1757

We have recently examined the structural role of PbO in borate glasses [8]. As is wellknown, the nature of modification of the B2O3 glass structure is dependent on the concen-tration of the modifier oxide [1–3,9,10]. An alkali oxide such as Li2O first converts 3-coor-dinated trigonal borons (B3) to four-coordinated tetrahedral borons (B4) until theconcentrations of B3 and B4 are nearly equal in the glass structure (which corresponds to thediborate composition Li2Oz2B2O3). Above this concentration of Li2O, the ratio of B4/B3

decreases continuously until almost the limits of glass formation [9,10]. The present studywas focused on examining the effect of adding PbO to such a binary system. PbO itself hasthe tendency to enter the borate network, occupying 4-coordinated positions. In doing so,PbO puts a demand on the available modifier oxide, which otherwise would be taken upentirely for the modification of the borate structure. It was noted from a variety of spectro-scopic and other investigations that the oxygen from the modifier oxide coordinates to Pb inpreference to B, although when PbO concentration is low, partial use of O22 from Li2O forthe modification of B2O3 cannot be avoided for structural reasons [8]. Therefore, lithium leadborate glasses form a covalently bonded open network system over a wide range ofcompositions, and they can be expected to exhibit high lithium ion conductivity. In thisbackground, the dc and ac conductivity transport studies were performed on these glassesover a wide range of compositions, frequencies, and temperatures. Interestingly, it was foundthat the conductivities were low and the activation barriers high, and we discuss the possiblereason for this observation in the following sections.

EXPERIMENTAL

The nominal compositions of the glasses examined here and their designations are listedin Table 1. The method of preparation of these glasses by the conventional meltquenching method has been described elsewhere [8]. Electrical conductivity measure-ments were carried out on a Hewlett-Packard HP 4192A impedance-gain phase analyzerfrom 10 Hz to 10 MHz in the temperature range of 300 to 600 K. A homebuilt cellassembly (having a 2-terminal capacitor configuration and silver electrodes) was used forthe measurements. The sample temperature was measured using a Pt–Rh thermocouplepositioned very close to the sample. Annealed circular glass bits with parallel surfaceswere coated with silver paint on both sides to serve as electrodes. The samples wereabout 0.1 cm thick and about 1 cm in diameter.

The capacitance (Cp) and conductance (G) of all the samples were measured using animpedance analyzer. These were used to evaluate the real and imaginary parts of the compleximpedance using the relations

Z* 5 Z9 1 jZ0 5 1/(G 1 jvCp)

Z9 5 G/(G2 1 v2Cp2)

Z0 5 vCp/(G2 1 v2Cp

2)

The dc conductances were determined from the semicircular complex impedance (Z9 vs. Z0)plots by taking the value of intersection of the low frequency end of the semicircle with theZ9 axis. Two typical impedance plots are shown in Figure 1. The dc conductivity (s) of eachsample was estimated using the expressions 5 Gz(d/A), where d and A are the thickness andthe area of the sample, respectively.

1758 M. GANGULI et al. Vol. 34, Nos. 10/11

Arrhenius plots of the conductivities were drawn using the expression

s 5 s0exp[2Ea(dc)/kT]

where Ea(dc) is the dc activation energy and T is the temperature in K. Values of Ea(dc) wereestimated through linear regression analysis of logs vs. 1/T plots.

The dielectric constants, dissipation factor, and electric moduli have been determinedusing standard equations [11]. On the basis that M0 vs. log f plots are reflective of a stretchedexponential relaxation behavior, the value of the stretching exponentb was determined fromthe full-width at half-maximum (FWHM) of the M0 peaks by interpolation method. A plot ofFWHM vs. b was prepared for the purpose, using published work of Moynihan et al. [12].The ac activation energies [Ea(ac)] were determined from plots of logarithm of the loss peakfrequency vs. (1/T) using linear regression analysis.

RESULTS AND DISCUSSION

dc Conductivity Behavior of the Glasses.Arrhenius plots of the dc conductivities are givenin Figure 2 for the various glasses in the four different categories. Since the low temperatureresistivities were very high, as seen from Figure 2, conductivities were not measured belowthe laboratory temperature. In most of the glasses, only single linear variation of logs vs. 1/Twas observed. However, in glasses with higher resistivities, e.g., in LB3 and LB4, B3 and B4,

TABLE 1Nominal Compositions of the Glasses Investigated

in the Li2O–PbO–B2O3 Glass System

Code

Composition (mole fraction)

PbO Li2O B2O3

L seriesL1 0.00 0.50 0.50L2 0.05 0.50 0.45L3 0.10 0.50 0.40L4 0.15 0.50 0.35L5 0.20 0.50 0.30L6 0.25 0.50 0.25P1 0.20 0.50 0.30P2 0.20 0.40 0.40P3 0.20 0.30 0.50P4 0.20 0.20 0.60B1 0.00 0.70 0.30B2 0.10 0.60 0.30B3 0.20 0.50 0.30B4 0.30 0.40 0.30LB1 0.0 0.50 0.50LB2 0.10 0.45 0.45LB3 0.20 0.40 0.40LB4 0.30 0.35 0.35

1759BORATE GLASSESVol. 34, Nos. 10/11

and L6, a change of slope was observed. It was possible in some of these cases to obtain twoslopes and hence associate two activation energies for the two regions, although the lowertemperature data were insufficient. Values of the activation energies [Ea(dc)] determinedfrom the plots are given in Table 2 along with the room temperature conductivity [sdc(298K)]. P series of glasses exhibited the highest resistances and the two-slope phenomenon. Thechange of slopes also occurred at systematically higher temperatures in glasses with higherresistivities. The variation of the activation barriers has been plotted as a function ofconcentration of PbO in Figure 3. Although there is a significant scatter, it is very evident thatthe activation barriers increase with PbO content.

This behavior of the dc conductivity appeared quite surprising, because PbO is incorpo-rated into the network and, therefore, the resulting more open structure should have beenconducive to Li1 ion conduction. In addition, since [BO4/2]

2 or [PbO4/2]22 units would

appear to smooth out the negative charges, they were expected to assist lithium ion motion

FIG. 1Typical impedance plots for the glass samples (a) L2 at 432 K and (b) L6 at 495 K.

1760 M. GANGULI et al. Vol. 34, Nos. 10/11

through softening of the coulomb interaction. These expectations, however, may not havebeen correct for the following reasons. B–O and Pb–O, being heteroatom bonds, are bothsufficiently ionic. Therefore, oxygen atoms connected to B/Pb in the various structural unitscarry an effective negative charge. These partial charges on oxygen atoms have beencalculated approximately using Sanderson’s method [13] and are listed in Table 3. Further-more, although Pb is 4-coordinated, the lone pair on Pb generally tends to project away fromthe four Pb–O bonds as if it is an effective ligancy and therefore may substantially decreasethe available structural volume. Both of these factors can impair lithium ion motion because,as can be seen from Table 3, the partial charge carried by the oxygen atom attached to Pb issignificantly higher than that in different borate groups. This has the effect of decreasing the

FIG. 2Arrhenius plots of dc conductivity of (a) L series, (b) P series, (c) LB series, and (d) P seriesof glasses. In each series,‚ denotes the first glass;E, the second;3, the third;1, the fourth;h, the fifth; and✴, the sixth.

1761BORATE GLASSESVol. 34, Nos. 10/11

site energy for the lithium ions in the vicinity of Pb422 (5 [PbO4/2]

22) to an extent that ishigher than in a pure borate environment. In addition, the stereochemically active lone pairon Pb [14] would have the same effect on the Li1 ion trying to explore its passage in theregion of the lone pair. The Pb4

22 group thus has a “stickiness” for lithium ions. In PbO-freeor extremely PbO-poor glass compositions such as L1, L2, or L3, the activation barriers aresomewhat lower (0.4–0.5 eV) and a greater number of Li1 ions in these glasses aresurrounded by oxygen ions from borate groups. In other compositions, it is unlikely that thelithium ions can explore borate-rich pathways and bypass Pb4

22 groups. From Table 1(compositions) and Table 2 (activation energies), it can be seen that the PbO-poor glassesgenerally exhibit noticeably low activation barriers, while in glasses in which PbO content ishigher, the activation barriers are high. This is borne out more clearly in Figure 3. It ispossible to associate a cluster-tissue texture to the structure of these glasses [15–17]. SlightlyPbO-rich tissue regions appear to embalm the clusters, which are borate-rich regions. Thiseffectively removes the connectivities of the low activation barrier percolation paths for thelithium ion transport. The PbO-free compositions L1/LB1 and B1 have an activation barrierof '0.4 eV. These compositions are highly modified borate glasses in which we expect B4

2

units to have either partly broken down to B122 (as in B1) or reverted to trigonal B2

2 (as inLB1/L1). Because of the small percentage of B1

22, Li1 ions can easily explore the lowest

TABLE 2Room Temperature Conductivities and dc Activation Energies for the

Different Compositions

Samplesdc

a

(S/cm)

Ea(dc)

High temp. (eV) Low temp. (eV)

L1 2.503 1028 0.39 0.03L2 1.503 1028 0.46 —L3 1.603 1029 0.51 —L4 1.403 1029 0.51 —L5 7.903 10210 0.57 0.18L6 5.003 10210 0.56 0.19

P1 7.903 10210 0.57 0.18P2 4.003 10210 0.63 0.23P3 2.503 10210 (at 318 K) 0.63 0.17P4 1.003 10210 (at 332 K) — —

LB1 2.503 1028 0.39 0.03LB2 1.603 1029 0.51 —LB3 4.003 10210 0.63 0.23LB4 2.503 10210 (at 325 K) 0.45 0.17

B1 1.303 1027 0.43 —B2 8.303 1029 0.49 —B3 7.903 10210 0.57 0.18B4 6.303 10210 0.47 0.05

aAt 298 K, unless otherwise indicated.

1762 M. GANGULI et al. Vol. 34, Nos. 10/11

activation barrier ('0.4 eV) paths. This activation barrier is typical of lithium borate glassesof these compositions [18,19].

We also note that Li2O concentration itself has an effect on the activation barriers as in theP series. At very low levels of modification, there is substantial concentration of B3

0 speciesin addition to B4

2 and the partial charge on oxygen in B30 is lower than that in B4

2. However,as Li2O concentration is increased, B3

0 concentration is reduced and B22, in which the partial

charge on oxygen is the same as in B42, increases in concentration. First, this has the effect

of decreasing the openness of the structure because of the loss of B42, and second, increasing

the density of the oxygens carrying partial charge characteristic of B22/B4

2 units. This deepensthe potential wells of the lithium ions and increases their activation barriers as evident in theP series of glasses.

The low barrier regions at lower temperatures in various glasses in the Arrhenius plots of

FIG. 3Variation of dc activation energies [Ea(dc)] of all the glasses with PbO mole percent.

TABLE 3Different Borate and Plumbate Units Present in

Li2O–PbO–B2O3 Glasses, Along with TheirNotations and Partial Charges on O Atoms (dO)

in Them

Structural unit Notation dO

[BO3/2]0 B3

0 20.26[BO4/2]

2 B42 20.52

[BO2/2O]2 B22 20.52

[BO1/2O2]22 B1

22 20.71[PbO4/2]

22 Pb422 20.85

1763BORATE GLASSESVol. 34, Nos. 10/11

Figure 2 are often seen as a manifestation of cluster-tissue texture in glasses [15–17].However, such cluster-tissue texturing is more generally observed in glasses containingdiscrete anions rather than in glasses dominated by network structure [17]. Although thepresent glasses are covalent network structures, it would still be reasonable to assume that inLi2O-rich glasses, borate-dominated clusters would form, which are held together by a tissuewhose composition is PbO-rich. However, the degree of compositional differentiation re-ferred to here occurs over lengths of sub-nanoscale and is not to be treated as any interfacialphase separation. In the structure envisioned here, lithium ion motion in tissue is not onlysubjected to greater disorder-based scattering, but also a higher activation barrier. Theclusters, on the contrary, would contribute to much of the conductivity at ordinary temper-atures. Therefore, in the Arrhenius plots of Figure 2, the conductivities at higher temperaturesmay actually have substantial contribution coming from lithium ions associated with thetissue region. We speculate that this glass system would exhibit pressure-dependent conduc-tivities at '400 K. The conductivity values exhibit a sharp drop as pressure is increased,since the more compressible tissue changes its structure to one similar to that of the clusters[17].

ac Conductivity Behavior of the Glasses.As noted earlier, the ac conductivities of theglasses were studied between 300 and 600 K and over a frequency range of 10 Hz to 10 MHz.A typical example of the variation of ac conductivity is shown in Figure 4 as a function offrequency and temperature. The behavior of all the other glasses is qualitatively similar. Theac conductivities have been fitted to an Almond–West type of expression [20–22] with asingle exponent ofv (5 2pf):

s(v) 5 s(0) 1 Avs

FIG. 4Variation of logs with frequency at different temperatures for the glass sample L2.

1764 M. GANGULI et al. Vol. 34, Nos. 10/11

Typical fits of the conductivity data to the above equation are shown in Figure 5(a) and (b).It was found that the goodness of fit was satisfactory in all cases (see Figure 5 and Table 4).The values ofs(0), A, and s obtained for the various glasses at room temperature are listedin Table 4.

FIG. 5(a) Typical ac conductivity plots (at 400 K) of different glasses fitted to the power lawequations 5 s(0) 1 Avs. (b) Typical ac conductivity data of L5 glass at differenttemperatures fitted to the single exponent power law equation.

1765BORATE GLASSESVol. 34, Nos. 10/11

Several interesting features are evident from the data in Table 4. The values ofs(0)obtained from the Almond–West fits are similar in magnitude to the dc conductivities (sdc)obtained from impedance plots (see Table 2). Ordinarily it is suggestive of similar transportmechanisms being responsible for ac (nondiffusive) and dc (diffusive) conductivities. Mostof the glasses exhibited s values between 0.5 and 0.7, significantly lower than unity, exceptfor the least modified of the P series of glasses (P3 and P4). It is evident from an inspectionof Table 4 that s values are indeed strongly dependent on the concentration of the modifieroxide; higher Li2O leading to lower s values. Such behavior is often attributed to theinter-lithium ion interactions influencing the ac transport [23–25]. The values of s wereobtained at various temperatures in the range of 300 to 600K. These values of s are shownin Figure 6 as a function of temperature. Except in the case of L5, P4, and B3, there was noclear evidence of the occurrence of s minima [26–29] in this glass system. In fact, it issurprising that P4 exhibited a minimum in s because it is a glass with a dominant networkstructure. In this glass, s is also significantly high at low temperatures. Although the paucityof mobile ions at lower temperatures appears to be correlated with the high s values in P4,the origin of the high s values is not well understood. In general, we note that s valuesdecrease with increasing temperature and the lowest s values appear to be associated withhigh degree of modification.

Dielectric Relaxation Behavior of the Glasses.The dielectric behavior of these glasses wasinvestigated by measuring both dielectric constants and moduli. Since a very steep rise in the

TABLE 4Fitting Parameters for ac Conductivity Data for All the Glassesa

Sample s(0) A s x2

L1 2.173 1028 2.473 10211 0.65 0.0005L2 1.563 1028 5.643 10211 0.59 0.0016L3 (at 348K) 1.043 1028 1.923 10211 0.62 0.0001L4 1.093 1029 2.563 10212 0.69 0.0002L5 2.873 10210 1.633 10212 0.69 0.0020L6 9.943 10211 7.313 10213 0.72 0.0010

P1 2.873 10210 1.633 10212 0.69 0.0020P2 1.483 10210 3.583 10213 0.77 0.0007P3 (at 318 K) 8.473 10211 5.643 10215 1.07 0.0017P4 (at 332 K) 6.693 10211 1.253 10215 1.13 0.0018

LB1 2.173 1028 2.473 10211 0.65 0.0005LB2 1.193 1029 3.973 10212 0.68 0.0005LB3 1.483 10210 3.583 10213 0.77 0.0007LB4 5.953 10210 9.753 10215 0.94 0.0061

B1 1.143 1027 7.723 10211 0.58 0.0007B2 6.633 1029 1.823 10211 0.61 0.0009B3 2.873 10210 1.633 10212 0.69 0.0020B4 2.073 10210 7.653 10215 1.03 0.0030

aAt 298 K, unless otherwise mentioned.

1766 M. GANGULI et al. Vol. 34, Nos. 10/11

dielectric constants was noticed even as the frequencies were lowered to kHz range, whichis generally attributed to electrode polarization, the moduli were calculated and their behavioris shown in Figure 7 for a typical case. The behavior is qualitatively similar for all otherglasses examined here. The real part of the low frequency modulus (Fig. 7(a)) is indeed verylow, suggesting a high degree of dielectric susceptibility. It is not surprising since a networkof Pb4

22 and B42 is not only energetically weak compared to the silicates and phosphates, but

the presence of Pb makes it dielectrically more susceptible. Of greater interest is the behaviorof the imaginary part of the modulus with frequency as shown in Figure 7(b). The variationof M0/M0(max) of these glasses as a function of log(f/f0) where f0 is the frequency

FIG. 6Variation of the power law exponent s with temperature for (a) L series, (b) P series, (c) LBseries, and (d) B series of glasses. In each series,‚ denotes the first glass;E, the second;3,the third;1, the fourth;h, the fifth; and✴, the sixth.

1767BORATE GLASSESVol. 34, Nos. 10/11

corresponding to the peak maximum in M0 has been examined as shown in Figure 8 for thecase of L6. It is seen from Figure 8 that the superimposability of the above variations atdifferent temperatures is very good, suggesting a common relaxation mechanism in these

FIG. 7Typical plots of variation of (a) real (M9) and (b) imaginary (M0) parts of the electric moduluswith frequency for the L2 glass at different temperatures (‚, 329 K;3, 368 K;E, 402 K;1,447 K).

1768 M. GANGULI et al. Vol. 34, Nos. 10/11

FIG. 8Typical normalized plot of M0 vs. normalized frequency for the glass L6 at differenttemperatures (‚, 419 K;E, 478 K; 3, 525 K).

TABLE 5Activation Energies andb Values for the Different Compositionsa

Sample Ea(ac) (eV) Ea(dc) (eV) b

L1 0.39 0.39 0.48L2 0.47 0.46 0.49L3 0.53 0.51 0.54L4 0.53 0.51 0.59 (340 K)L5 0.54 0.57 —L6 0.59 0.56 —

P1 0.54 0.57 —P2 0.60 0.63 —P3 0.70 0.63 —P4 — — —

LB1 0.39 0.39 0.48LB2 0.51 0.51 0.63LB3 0.60 0.63 —LB4 0.45 0.45 —B1 0.38 0.43 0.56B2 0.46 0.49 0.63B3 0.54 0.57 —B4 0.65 0.47 —

aAt 298 K, unless otherwise indicated.

1769BORATE GLASSESVol. 34, Nos. 10/11

glasses for the given range of temperatures [11]. The FWHM values (.1.4 decades) suggestthat the relaxation mechanism is non-Debye like and that the use of a stretched exponentialfunction may be more appropriate. The FWHM of the M0 vs. log f plots (Fig.7(b)) were usedto calculate theb values in the stretched exponential (KWW) function for relaxation [30,31].Theseb values (at room temperature) are listed in Table 5 for all the glasses. The f0 valuesof the various compositions were plotted as a function of 1/T in a semilog plot and theactivation barriers were determined. These activation barriers Ea(ac) are also listed in Table5 along with the Ea(dc) values (for the high temperature region) obtained from the impedanceplots. The values for these two activation barriers were found to be very close. This could be

FIG. 9Variation of the stretched exponentb with temperature for (a) L series, (b) P series, (c) LBseries, and (d) B series of glasses. In each series,‚ denotes the first glass;E, the second;3,the third;1, the fourth;h, the fifth; and✴, the sixth.

1770 M. GANGULI et al. Vol. 34, Nos. 10/11

an indication that the ionic motions responsible for relaxation and for diffusive (dc) con-ductivity are the same. The values were in agreement with the similar values ofs(0) ands(dc).

Variation ofb was also examined as a function of temperature (Figure 9). In almost all theglasses, the value ofb was found to lie between 0.45 and 0.6, and fairly temperatureinsensitive. The relatively greater dependence of s on temperature may be contrasted with therather weak dependence ofb on the same. It is suggestive thatb and s are not related in anysimple complementarity such asb 1 s ' 1 [25].

An effort was made to examine whetherb is related to the activation barriers (Table 5) asshown in Figure 10. It was found that theb values are quite insensitive to the activationbarriers [11,23].

CONCLUSIONS

Li2O–PbO–B2O3 glasses are structurally very interesting because PbO extends the networkfeatures of the borate glasses, by getting itself incorporated into the network. In spite offorming an open structure, however, PbO does not help lithium ion motion because of theincreased effective negative charge on the oxygens attached to Pb and the presence of a lonepair on Pb. This results not only in poor lithium ion conduction, but also in unusual dielectricrelaxation behavior, wherebyb becomes virtually temperature independent. The ac conduc-tivity studies also suggest that the frequency exponent s in the conductivity expression andthe stretching exponentb of the KWW expression are not correlated.

FIG. 10Variation of the activation energies Ea(dc) (denoted byE) and Ea(ac) (denoted by‚) withb.

1771BORATE GLASSESVol. 34, Nos. 10/11

ACKNOWLEDGEMENTS

The authors are thankful to the Commission of the European Communities for financialsupport. One of the authors (M.G.) is grateful to the Council for Scientific and IndustrialResearch (CSIR), India, for a senior research fellowship.

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