Ion transport in superionic conducting glasses

] O U R N A L OF

Journal of Non-Crystalline Solids 172 174 (19941 1315-1323

ELSEVIER

Ion transport in superionic conducting glasses A. Pradel*, M. Ribes Laboratoire tie Physicochimie des Mat~riaux Solides, URA D0407 CNRS CC3, Universitk de Montpellier 11, 34095 Montpellier ckdex 5, France

Abstract

Models of the dependence of conductivity with modifier content in glasses are reviewed in this paper. It is shown that the recent 'site memory effect' model provides very good fits of experimental data by predicting a power law dependence of dc conductivity or a logarithmic dependence of activation energy of conductivity on modifier content. The consequence of the mixed alkali effect (MAE) on frequency-dependent properties is emphasized. The coupling factors (for nuclear magnetic resonance-spin lattice relaxation, ns, and for electrical conductivity relaxation, n,) are shown to be constant and independent of composition. This may indicate that the mobile ion concentration remains practically constant when one cation is replaced by another. The MAE would then be caused by a change in the mobility of mobile ions.

1. Introduction

The high ionic conductivity of superionic conducting glasses makes these materials very interesting for applications in microionics (e.g., as components in microbatteries, microsensors or smart windows) as well as for basic research on ion transport phenomena in disordered matter. Despite the considerable efforts devoted to understanding ion motion in glasses, several aspects of the problem still remain obscure. For example, the strong dependency of conductivity, a, upon the content of modifier (the conductivity is typically increased by an order of magnitude when the modifier content doubles) is still a matter for controversy. Some

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author.

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intriguing phenomena such as the mixed alkali effect (MAE) or the mixed former effect observed in glasses for many years are not understood to date. The mixed alkali effect is related to non-linear variations of physical properties (e.g., high minimum conductivity) observed in a family of glasses when the relative proportion of two modifiers (e.g., Li20 and NaEO) is varied while the total modifier concentration is kept unchanged. In the mixed former effect, one or two maxima for conductivity are observed in a glass when a former (e.g., B 2 0 3 ) is progressively replaced by another former (PzOs), the modifier content being kept unchanged. The interpretations proposed to date for explaining conductivity behaviour in glasses can be classified in two groups depending upon whether they are phenomenological or concern microscopic aspects of ion transport. The characteristics of dc conductivity are the main basis for theoretical approaches

0022-3093/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved. SSD1 0 0 2 2 - 3 0 9 3 ( 9 4 1 0 0 0 7 9 - 3

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belonging to the first group, whereas microscopic models are built taking as a basis the results of investigations of ion dynamics (e.g., experimental data from ac conductivity, nuclear magnetic resonance (NMR) (spin lattice relaxation rate), quasielastic neutron or light scattering experiments). See Refs. [1-3] for reviews on the subject. The paper reviews studies on modifier concentration dependence of dc conductivity in oxide and chalcogenide glasses and describes a study of the ion dynamics in a family of chalcogenide glasses showing the mixed alkali effect: ac conductivity and nuclear spin lattice relaxation rates of 0.5[xNa2S-(1 - x)Li2S]-0.5SiS2 glasses were studied by impedance spectroscopy and 7Li NMR.

2. Composition-dependence of conductivity, O'dc, and activation energy Edc, in glasses Simple glasses comprising only a modifier M and a former F and corresponding to the general formula xM-(1 - x)F are discussed in this section. In all ionic conducting glasses, dc conductivity for temperatures lower than the glass transition temperature, Tg, obeys Arrhenius' law: ado = aO exp( -- E a c / R T ) .

to break ionic bonds and to bring an ion to a distance equal to half the hopping distance and a second term to account for the elastic distortion of the glassy matrix during ion displacement. This model is still under discussion and several improvements have been proposed to improve its accuracy [6]. The weak electrolyte theory [7] is based upon the fact that dc conductivity, aac, varies as the square root of modifier thermodynamic activity, a~2x. Large variations in ad~ are thus attributed to large variations in aM2x. Using this hypothesis, variations of aa~ and of Ed~ with modifier content, x, were modelled using a quasi-chemical description of M2X YX, mixture (YX, is a former) and assuming that three configurations were possible for X, i.e., Y - X - Y , Y - X - M and M - X - M [8]. As an example, fits of experimental data for xLi2S(1 - x)SiS2 glasses using this model are shown in Fig. 1. It can be seen that the model accounts for the considerable variations of conductivity with modifier content. However, experimental and calculated curves curve in opposite directions, indicating that the model as such lacks some accuracy. Elliott [9] recently proposed a new model for calculating activation energy of conductivity which includes the most recent data on glass structure and

(1)

The pre-exponential factor, ao, is generally very similar in glasses belonging to the same family. The variations in activation energy are thus responsible for variations in conductivity observed both when the nature or the concentration of mobile ion are changed. For example, the activation energy for x L i 2 S e - ( 1 - x)SiSez glasses falls from 0.4 eV to 0.3 eV when the Li2Se content (tool %) is increased from 0.2 to 0.6. Room temperature conductivity variations as great as three orders of magnitude (from 1 0 - 4 S m -1 to 10 -1 S m -1) are observed at the same time. Such large changes observed for all glasses led some authors to refer to the 'anomalous' dependence of conductivity on modifier content [4]. Anderson and Stuart [5] were the first to build a model to calculate activation energies. They proposed an expression for Ea, with two terms: a Coulombic term to account for the energy needed

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Fig. 1. Variation of conductivity at room temperature with composition for xLi2S-(1 - x)SiS2 glasses. The dashed line is a guideline for experimental data, whereas the solid line is a theoretical fit using the weak electrolytetheory based upon a quasi-chemicaldescription [8].

A. Pradel, M. Ribes / Journal of Non-Crystalline Solids 172-174 (1994) 1315-1323

The environment of B must relax to adapt to the new incoming ion A. It requires more energy than that needed for cation A to jump to a vacant site. Therefore, EAB = EAA + AEAB, AEAB being the excess energy. Monte Carlo method computer simulation showed that conductivity should obey a power law of the form

more precisely on the nature of the environment of mobile ions as established by experiments such as EXAFS. This model also includes an interaction potential that is more realistic than that proposed earlier by Stuart and Anderson. This potential comprises terms accounting for repulsion energy and polalization effects as well as the usual Coulombic term. The model predicts a large value for activation energy dominated by the Coulombic effect for low modifier content, x. For x larger than a critical value, xc, the activation energy decreases and the polarization term dominates. Finally, Ed¢ becomes a constant independent of x at a high modifier content. At about the same time, Rao proposed a similar approach [10]. His model predicts an important role for polarization energy when hopping distances become short, i.e., at high modifier contents. A new model for ionic transport in ionic conducting glasses called 'site memory effect' was recently proposed by Bunde and co-workers [4,11]. It is based on the experimental evidence that cations in glass create and maintain their own characteristic environments. The authors define two different jump probabilities: (i) 0 9 A A : V A A e X p ( - - E A A / k T ) , related to the probability of jump of cation A to a vacant nearest-neighbour site A; (ii) CbAB= VABexp(--EAB/kT), related to the probability of jump of cation A to a neighbouring interstitial site B.

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o~c = Ax ~

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and activation energy with a logarithmic dependence o n x: Edc = B + C Lnx.

(3)

Expressions for parameters A and ~ can be derived from Eqs. (1)-(3): A = a o e x p ( - B / k T ) and =

-

C/kT.

This model was used to fit experimental data obtained on several families of glasses (mainly chalcogenide types) studied in our laboratory. The fits of conductivity and activation energy plots for x L i 2 X - ( 1 - x)SiX2 (X = O, S, Se) are shown in Fig. 2 as examples. One can note the excellent agreement between experimental and calculated plots without the previous curve concavity problem (Fig. 2(a) for X = S). Similar fits were made successfully on several other glass systems. The corresponding calculated values for parameters of Eqs. (1) (3) are given in Table 1. Parameter C is smaller for chalcogenide glasses compared with that for oxide glasses. On the other hand, parameter B does not seem to be very sensitive to the

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Fig. 2. Variation of (a) conductivity, ado, at room temperature and (b) activation energy, Ea¢, with composition for xLi2~ (1 - x)SiX2 glasses. Lines are theoretical fits of experimental data using the 'site memory effect' model.

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,4. Pradel, M. Ribes / Journal of Non-Crystalline Solids 172-174 (1994) 1315 1323

Table 1 Calculated values for the pre-exponential factor, a0, and for parameters B and C as defined in the 'site memory effect' model

Glass system

Log ao (Sm 1)

C (eV)

B (eV)

Li2Se SiSe2 Li2S-SiS2 Li20-SiO2 Na2S-SiS2 Na2S-GeS2 Ag2S-GeS2 Li20 P4/sO2 Li20-B2/302 Li2S-GeS2

4.19 3.39 5.12 5.75 -

0.096 0.181 0.239 0.099

2.55 4.59 3.39

0.757 0.467 0.181

0.262 0.243 0.390 0.337 0.379 0.293 0.220 0.240 0.230

nature of glass. These results show that the recent 'site memory effect' model accounts for the variations of conductivity and activation energy with modifier content in many ionic conducting glasses.

3. The mixed alkali effect: the 0.5[xNazS(1-x)Li~S]-0.5SiS~ glasses as an example Non-linear variations of several physical properties are observed in a glass when the relative proportion of two alkali ions is varied while their total concentration is maintained constant. As already noted in the Introduction, the phenomenon is called the 'mixed alkali effect' (MAE). It has been reported many times for oxide glasses (cf. Refs. [1, 12] for reviews) but there have been no publications on the observation of such an effect in superionic conducting chalcogenide glasses. Glasses corresponding to the general formula 0.5[xNazS-(1 - x)Li2S]-0.5SiS2 were recently prepared by a fast quenching technique (twin roller quenching) in our laboratory. The glasses were studied by impedance spectroscopy and 7Li NMR. Experimental procedures are described in Ref. [ 13]. A classical MAE commonly reported for oxide glasses was observed for the first time in a chalcogenide family. Dependence of electrical characteristics (dc conductivity, a~, and activation energy, Ed~) and glass transition temperature, Tg, on composition, x, is shown in Fig. 3. Both Tg and ad¢ display a high minimum for x ,~ 0.5, whereas

Ed¢ displays a maximum for the same value of x. Although the consequences of the mixed alkali effect on dc properties are well known, very little work [14,15] has been carried out to date on the 'frequency dependence' of the mixed alkali effect. In fact, the most promising models for the description of microscopic processes of ionic transport in simple glasses (containing a former and a modifier only) are based upon experimental study on relaxation phenomena of ions at different frequencies. The same type of study might also help in gaining a better understanding of mixed alkali effect.

3.1. Electrical conductivity relaxation ( ECR) Numerous ac response functions can be derived from the observed complex impedance, Z*bs. Those most frequently used are the complex conductivity, a*(to) = [(S/e) Z*bs] - 1 - io~e~, and the complex electrical modulus, M* (o~) = e*(to)- 1 = [a*(to)/io9 + e~] - 1, in which ~o is the angular frequency, e~ is the upper frequency limit of permittivity and S/e is the geometrical factor of the sample. In this work, we chose to use complex conductivity formalism and to represent the real part, O'ac, of a*(~O) as a function of frequency. The variations of aac with frequency can be described by the empirical equation: aac = ado + Ao~~ in which aac, ado and w have their usual meanings, A is a temperature-dependent parameter and s has no physical significance a priori. Experimental data were used both to calculate the values of parameter s by fitting the plots 'log aa~ vs.f' ( f = to/2n) and to plot the isochronal Arrhenius diagrams of conductivity. A typical result of this type of plot is shown in Fig. 4 for 0.25Li2S-0.25Na2S-0.5SiSz glass (x = 0.5). The variations of O'ac at different frequencies led to a group of straight lines (in the intermediate temperature range) from which Ea¢ c a n be derived. However it should be noted that the linear part on 'log aac vs. 1/T' curves corresponds to a fairly limited temperature range. Independence of E~ with regard to frequency within the limits of experimental errors (often considerable) was observed in all cases, and average activation energy, (E~¢), could thus be determined for each glass. (E~c) is 0.23 eV for 0.25Li2S-0.25Na2S-0.5SiS2 glass. We had already observed this independence of E~¢ from

A. Pradel, M. Ribes / Journal of Non-Crystalline Solids 172 174 (1994) 1315 1323

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Fig. 3. Variation of glass transition temperature, Tg, of conductivity, ado , at r o o m temperature and of activation energy of conductivity, Edc, for 0.5[xNa2S (1 - x)LizS]-0.5SiS 2 glasses.

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Fig. 4. lsochronal d i a g r a m of conductivity, or, plotted in Arrhenius coordinates for 0.25 Na2S 0.25 Li2S-0.5SiS2 (x = 0.5). (a) 10 M H Z ; (b) 1 M H z ; (c) 100 kHz; (d) 10 kHz; (e) 1 kHz.

frequency in other glasses [1 3] and such behaviour is predicted by both the j u m p relaxation model of F u n k e [16] and the coupling model of Ngai [17]. The isochronal plots can in fact be separated in three d o m a i n s depending upon the temperature range observed. The 'high t e m p e r a t u r e d o m a i n ' (I) corresponds to variations of ado. The second region (II) corresponds to the temperature range where linear parts of tr,~ are observed and where Eac is calculated. Finally, in the lowest temperature range

(III), a,¢ is weakly thermally activated. Further interpretation of these isochronal curves requires the use of models to describe transport properties in ionic conducting glasses. We consider the coupling model which stipulates that, beyond a critical time, to, correlated processes m a k e relaxation rates time-dependent with power law dependence W(t) ~ t - " (0 < n < 1). This leads to the empirical Kohlrausch, Williams and W a t t ( K W W ) function: ~b(t) = exp( - t/r') #, fl = 1 - n and n = s. When t < t¢, the relaxation processes are not correlated and so W(t) is a constant. W(t) oc (ro)and n = 0. r' and ro are thermally activated with the following respective activation energies: E', and Ea. Thus, E , = ( l - n ) E ' a or E , = f i E ' , . In this model, Ea and not E', is considered as a true (primitive) microscopic activation energy of relaxation processes (in our notation E'a = Ed¢ and Ea = E,c), Let us consider the relaxation time for conductivity, %, calculated at the m a x i m u m on modulus plot or at the crossover from dc plateau to dispersive region on 'log tr,~ vs. l o g f ' plot where ~o% = 1. The isochronal plots can then be fitted: ~or,<< 1

domain I

o9%>> 1

f d o m a i n II ) d o m a i n III

a ~: exp - [ E a c / (1 - n)kT] = cra~; a oc exp - [Eac/kT], a oc co~ T ~ with ~ ~> 1 and }' ~ 1.

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A. Pradel, M. Ribes / Journal of Non-Crystalline Solids 172-174 (1994) 1315-1323

In the third domain (III), at very low temperatures, conductivity relaxation could be due to contributions of asymmetric double well potential [18,19].

-2

-2.5 3.2. 7Li N M R spin lattice relaxation

g Fig. 5 shows Arrhenius plots of spin lattice relaxation time data (log Ti-~ vs. T -~) for different 0.5[xNa2S-(1 - x)Li2S]-0.5SiS2 glasses. The 'low temperature' branch of T~ variations only were observed in the temperature range investigated at the Larmor frequency of experiments, i.e., ~L = 77.7 MHz. If we consider the NMR relaxation time, rs, calculated at the maximum on a log Ti- ~ vs. T- 1 plot where tOLZs= 1, the experimental results observed correspond to a domain where tOLZ~>>1. Variation of T~ magnitudes with glass composition did not show non-linear behaviour which is usually the signature of a mixed alkali effect. Activation energies, E~LT, were then calculated from the experimental data for each glass composition. Fig. 6 shows variations of these activation energies E~LTwith composition, x. A weak mixed alkali effect (larger, however, than experimental error fluctuations) was observed. Similar results, with a less marked maximum, were recently reported by Elliott [15] in a comparable study of oxide glasses with the general formula xNa20-(1 - x)Li20-2SiO2.

-3.5

-4

x=0.9 -4.5 2.5

Several approaches have been used to attempt to explain the electrical behaviour of the mixed alkali effect. The oldest ones [12] examined it from the structural angle: cations of different sizes are not interchangeable at their respective sites and/or interaction between alkali cations and glassy network gives rise to 'complexes'. No quantitative verifications have confirmed these approaches to date. Other more phenomenological models gave a better description of the intensity of the phenomenon. They are: (a) the weak electrolyte theory which assumes the existence of mixed interstitial pairs of low mobility and which is based upon a regular solution model [20];

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3.5

4 1000/T(K)

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Fig. 5. Arrhenius plots of 7Li N M R spin lattice relaxation rates, 1/T 1, for 0.5[xNa2S-(1 - x)Li2S]-0.5SiS2 glasses.

0.3

0.2

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3.3. Discussion

-3

0.1

0

I

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x Fig. 6. Composition dependence of activation energy, E,LT, calculated from Arrhenius plots of I/T1 for 0.5[xNazS(1 -- x)Li2S]-0.5SiS 2 glasses.

(b) the same regular solution model and the existence of negative mixing enthalpy between the two alkalis enabled the description of variations of dielectric relaxation strength [21]. Very recent models generally based upon computer simulation shed new light on mixed alkali effect. To

A. Pradel, M. Ribes /' Journal of Non-Crystalline Solids 172 174 (1994) 1315-1323

date, no investigation except that of Elliott cited above [15] and our study on LiEO-NaEO-P205 glasses [14] have been carried out to compare data on electrical conductivity relaxation with those on N M R spin lattice relaxation. A similar study is described below. First, parameters s (or n in Ngai's formalism, or more generally no) were calculated by fitting 'log a,¢ vs. f ' plots with equation aac -- aa~ + Ato~ for all glasses investigated, n~ can also be calculated from parameter fl~ = Eac/Edc via n~ = 1 - fl~. The calculated values for n~ are shown in Fig. 7. n~ is a constant within experimental errors and does not depend upon the Li/Na substitution ratio ((n~) = 0.55). The lack of dependence of n~ on x was reported in our earlier work on the mixed alkali effect on L i 2 0 - N a 2 0 - P 2 0 5 glasses [14], where (n~) was equal to 0.45. In the case of N M R experiments, parameter fl can be defined by fl = fl~ = EsL'r/EsHT, where E,,nT is the activation energy calculated from the 'high temperature' branch of'log Ti- l vs T - t, plots. Experimental determination of fls becomes possible as soon as the 'high temperature' branch of 'log Ti-1 vs. T -1' plots can be measured. An example of such a determination can be found in our earlier work on lithium-conducting glasses of the Li2S-SiS2 family [22]. The superionic character of these glasses and the relatively low experimental Larmor frequency (15.8 MHz) helped in

0.9 n

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Fig. 7. Composition dependence of parameters n. and n~ for 0.5[xNa2S-(l - x)Li2S]-0.5SiS2 glasses.

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observation of the high temperature part of the curves. In the case of oxide glasses, and more generally in the case of low conducting glasses, it is usually impossible to observe the high temperature branch which would correspond to temperatures higher than the Tg. Owing to the high experimental Larmor frequency of the current experiment, we were not able to observe the high temperature branch of the curves for the glasses investigated. However, it is known that the high temperature branch of 7"1 variations is frequency-independent, and when it was measured in simple glasses, its activation energy Esm was found to be equal to activation energy, Eao of O'dc [223. Let us suppose that this would apply to 0.5[XNaES-(l - x ) L i E S ] - 0 . 5 S I S 2 glasses where two types of ion are mobile (Li + and Na+). We then implicitly suppose that both mobile ions contribute to relaxation of 7Li. The appearence of a weak mixed alkali effect on the 'low temperature activation energies, gsLT, VS. composition' plots points in this direction (on the other hand, no mixed alkali effect appears in the diffusion coefficient experiment where only one mobile ion is probed). A similar hypothesis (i.e., EsHT = Eat)justified by the predictions of the 'diffusion controlled relaxation model' was also put forward by Elliott [-15] in a study of x L i 2 0 - ( 1 - x ) N a 2 0 - 2 S i O 2 glasses. One point called the hypothesis into question: in a study of this family of glass, G6bel et al. [23] were able to measure EsHTfor x = 0.5 and did not find it equal to gdc but equal to E~nv for x = 0. This results was only obtained on one composition and can be regarded with caution. Within the framework of the hypothesis (to be confirmed), one can calculate fl or fl~ with fl~ = EsLT/Ed~. By comparison with n~, one can define a parameter, ns, such as ns = 1 - fls. Variations of Edc and EsL T with x are shown in Fig. 8 for all glasses. Values for n~ calculated from these activation energies are shown in Fig. 7. ns is a constant within experimental errors and does not depend upon the Li/Na substitution ratio. A similar result was reported by Elliott [153 in his study of L i E O - N a z O - P 2 0 5 glasses. A constant value of n~ when the glass composition changed was said by the latter author to be indicative of a change in mobility rather than of a change in concentration in mobile ions, in contradiction with the weak electrolyte theory hypothesis.

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A. Pradel, M. Ribes / Journal of Non-Crystalline Solids 172-174 (1994) 1315-1323

0.7 0.6 0.5 ~" > 0.4 a~ 0.3( 0.2 0.1 I

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0.2

0.4

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X Fig. 8. Comparison of composition dependence of conductivity activation energy, Ed¢, and NMR activation energy, EsLT.

A similar interpretation of constant values of ns observed when the modifier content was changed in simple xLi20-(1 - x ) G e O 2 glasses had been put forward by Kanert et al. [24]. Comparison of calculated values for ns and n, reported in Fig. 7 indicates that n~ is larger than n, ((n~) = 0.72, (n~) = 0.55). More precise measurements and mainly experimental determination of EsHT would be needed to confirm this result. If it was so, it would back up the latest developments of the coupling model which predict that different coupling parameters n~and n, should be found with n~ < n~ [18]. It would also confirm recent comments by Tatsumisago et al. [25], i.e., conductivity relaxation time, %, is some two orders of magnitude shorter than the spin lattice relaxation time, zs, and has a significantly lower activation energy. It is noted that these comments were recently questioned by Elliott [26].

4. Conclusion

An attempt to review the latest developments on two intriguing behaviours of ion transport in glasses, i.e., dependence of conductivity with modifier content in glasses containing one modifier and mixed

alkali effect, was made in this paper. Studies on superionic conducting glasses have been emphasized. Models have been proposed recently to describe the dependence of activation energy of conductivity with modifier content, and include potentials that are more realistic than the original potential proposed by Stuart and Anderson for describing 'ionglassy network' interactions. The site memory effect model predicts power law dependence of dc conductivity or, in a similar way, logarithmic dependence of activation energy of conductivity on modifier concentration. This model allows excellent fits of experimental data for many glass systems but the model suffers from the fact that the different parameters have no physical meaning. This model also predicts the variations of conductivity with composition observed in case of mixed alkali effect. The consequence of the mixed alkali effect on frequency-dependent properties has been emphasized here. The coupling factors (ns and n,) or the fl exponent in the KWW formalism were shown to be constant and independent of composition. It may indicate that the mobile ion concentration remains constant when one mobile ion is replaced by another. The mixed alkali effect would then be due to a change in mobility of mobile ions. The coupling parameter for conductivity was shown to be smaller than the coupling parameter for NMR but, owing to the hypothesis required for their determination here, these values should be treated with caution. References [1] M.D. Ingram, Phys. Chem. Glasses 28 (1987) 215; Ber. Bunsenges. Phys. Chem. 96 (1992) 1592. [2] D. Brinkmann, Progr. NMR Spectrosc. 24 (1992) 527. [3] C.A. Angell, Ann. Rev. Phys. Chem. 43 (1992) 693. [4] A. Bunde and P. Maass, Physica A191 (1992) 415. i-5] O. Anderson and P. Stuart, J. Am. Ceram. Soc. 37 (1954) 573. [6] D. McElfresh and D. Owitt, J. Am. Ceram. Soc. 69 (1986) C-237. 1-7] D. Ravaine and J.L. Souquet, Phys. Chem. Glasses 18 (1977) 27. i-8] A. Pradel, F. Henn, J.L. Souquet and M. Ribes, Philos. Mag. B60 (1989) 741. [9] S.R. Elliott, J. Non-Cryst. Solids. 160 (1993) 29; these Proceedings, p. 4099J.

A. Pradel, M. Ribes / Journal t~f Non-Crystalline Solids 17~174 (1994) 1315 1323 [10] K.J. Rao, C. Estournes, A. Levasseur, MC.R. Shastry and M. Menetrier, Philos. Mag. B67 (1993) 389. [!1] P. Maass, A. Bunde and M. lngram, Phys. Rev. Lett. 68 (1992) 3064. [12] D,E. Day, J. Non-Cryst. Solids 21 (1976) 343. [13] A. Pradel and M. Ribes, Mater. Chem. Phys. 23 (19891121; J. Solid State Chem. 96 (1992) 247. [14] R. Chen, R. Yang, B. Durand, A. Pradel and M. Ribes, Solid State lonics 53-56 (1992) 1194. [15] S.R. Elliott, J. Phys. (Paris) Colloq C2 2 (1992) 51. [16] K. Funke, Progr. Solid State Chem. 22 (1993) 111 and references herein. [17] K.L. Ngai, J. Phys. IParis) Colloq. C2 2 (1992) 61 and references herein. [18] K.L. Ngai, J. Chem. Phys. 98 (1993) 6424.

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[19] K.L. Ngai, U. Strom and O. Kanert, J. Phys. Chem. Glasses 33 (1992) 109. [20] A. Kone, J.C. Reggiani and J.L. Souquet, Solid State Ionics 9&10 (19831 709. [21] M. Tomazawa and V. McGahay, J. Non-Cryst. Solids 128 (1991) 48. [22] A. Pradel and M. Ribes, J. Non-Cryst. Solids 131 133 (19911 1063. [23] E. G6bel, W. Mfiller-Warmuth, H. Olyschl/iger and H. Dutz, J. Magn. Res. 36 (1979) 371. [24] O. Kanert, M. Kloke, R. Kiichler, S. Riickstein and H. Jain, Bet. Bunsenges. Phys. Chem. 95 (19911 1061. [25] M. Tatsumisago, C.A. Angell and S.W. Martin, J. Chem. Phys. 97 (1992) 6868. [26] S.R. Elliott, submitted to J. Chem. Phys.