Developments in the understanding of the ionic conductivities of glasses

Materials Chemistry and Phq’sics, 23 (I 989) 2 1l--L23

211

F THF IONIC CONDUCTIVITIFS OF GI ASSFS

D. P. ALMOND School of Materials Science, University of Bath, Bath BA2 7AY (U.K.)

ABSTRACT The factors which determine the magnitude of ionic conductivity are presented with particular emphasis on the role of the mobility terms. The frequency and temperature dependences of the ac conductivities of glasses are discussed and it is shown how ac and dc conductivities may be related. It is shown that ac conductivity data may be analysed to provide estimates of a characteristic frequency which may be interpreted as an effective ion hopping rate. Ion hopping rate data for glasses are reviewed and it is concluded that they are significantly enhanced by an entropic term. A rationalization of conductivity prefactors is presented which is based on a simple explanation of the magnitude of the entropy of activation, It is shown that this leads to a Meyer-Neldel relationship between prefactors and activation energy. Data of fast ion conducting glasses are presented to illustrate the various characteristics discussed.

INTRODUCTION The objectives of this article are to review the progress that has been made in understanding the conductivities of what may be classified as ‘Fast Ion Conducting Glasses’.The ionic conductivities of both glassy and crystalline conductors have many features in common, however, and the understanding of each has benefitted from results obtained from the other, Consequently, this review will include ideas originally developed to explain the ionic conductivity found in crystalline materials which have proved valuable in the study of conducting glasses. Ionic conductivity is only rarely measured directly using ion conducting electrodes. In the vast majority of studies it is obtained from an analysis of ac conductivity which is measured employing readily deposited electronically conducting electrodes. AC conductivity, however, exhibits its own range of complex characteristics. To provide a full description of ionic conductivity it is necessary to include an explanation of both ac conductivity and dc conductivity. In fact, not surprisingly, there seems to be a close relationship between the two and a detailed analysis of ac conductivity provides additional information about the conduction process.

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212

IONIC CONDUCTION Ionic conduction in a solid is generally attributed to the thermally activated hopping of ions from site to site. The resulting conductivity, IS, is found to vary with temperature, T, as:

oT = Aexp(-E/kl)

(1)

in which E is an activation energy and A is known as the conductivity prefactor. The vast majority of studies of ion conducting glasses have gone no further than the determination of these two parameters, Of them, the activation energy is the more fundamental, being the mean energy barrier that must be surmounted by an ion as it hops from site to site. Numerous studies have shown it to vary in a systematic way with composition. The conduction prefactor, A, however, is related in a rather more complex way to the physical properties of the conductor. The expression for ionic conductivity, given below, has been derived many times before [l-3] using random walk theory:

UT = Ne*a*c(l -c)y k-lwoexp(-G!kT)

(2)

where exp(-G/kT) = exp(Qk) exp (-H&T)

and in which N is the number of sites available for occupancy by mobile ions, e is their electronic charge (assuming that they are singly charged), a is the mean jump distance, c is the fractional occupancy of the sites, 1: is a correlation factor, k, is Boltzman’s constant, wo, is the fundamental vibrational frequency of a mobile ion in the potential well at a site, G,S and H are the activation free energy, entropy and enthalpy for conduction.

The mobile ion concentration in most fast ion conductors is taken to be determined in large pan by their nonstoichiometry and as a result to be independent of temperature. This assumption will be examined again later. For such a conductor, eqn. [2] can be written as:

where K is the temperature independent carrier concentration term:

K = Ne*a*c (1 - c) y k-’ and wp is the ion hopping rate given by:

(4)

213

wp = we exp (-H/kT)

(5)

with the effective attempt frequency

we = w. exp (SnC)

(6)

Whilst the terms in the carrier concentration term, K, can usually be estimated with some certainty, those which determine the hopping rate are generally unknown. In particular, the hopping rate is strongly dependent on the entropy of activation which is a notoriously difficult quantity to predict. It is evident that an independent estimate of ion hopping rate would provide a valuable contribution to the explanation of a particular material’s ionic conductivity. A number of groups [4-9) have successfully employed ultrasonic and internal friction techniques to obtain ion hopping rates in glasses. This work has recently been reviewed [10] and will not be discussed here. Instead, an analysis of ac conductivity will be presented which also provides estimates of hopping rates.

AC CONDUCTIVITY As has been mentioned, the majority of studies of ionic conductivity employ an analysis of ac conductivity data to avoid the complication of developing ion conducting electrodes. The sample forms the dielectric of a capacitor whose plates are formed by metallic films deposited on two of its surfaces. The sample’s finite ionic conductivity affects the properties of the capacitor which can be measured using a conventional admittance bridge. The simplest electrical representation of an ionic conducting sample is an R-C series equivalent circuit. It is a simple matter to show that the effective conductivity of such a circuit element rises from zero with increasing measurement frequency to a frequency independent value of l/R. This ‘plateau’ value of conductivity, is taken to be equal to the true dc ionic conductivity. In high ionic conductivity materials or at high temperatures, this plateau regime may occur at frequencies outside the ranges of conventional admittance bridges [above about 5MHz). Ionic conductivity must then be deduced by an analysis of the frequency dependent part of the ac conductivity. In practice, the ac conductivity of ionic conductors is found to be considerably more complex than predicted by a simple R-C series equivalent circuit. An example of the frequency dependence of the conductivity

of a ‘real’ ion conducting glass isshown in Fig. la. These measurements of an AgI:Ag,MoO,

glass were obtained at low temperatures, where the thermally activated conductivity is low, and where a simple frequency independent conductivity plateau might be expected. There are numerous examples of similar sets of data in the literature: recent examples may be found in refs [I l-151. The behaviour of the conductivity is seen to deviate from the frequency independent plateau in two ways: i) at lowest temperatures and at higher frequencies a strong power law frequency dependence of conductivity is found to develop and ii) at lower frequencies the conductivity ‘plateau’ is found to have a small but

r

IO0

10'

(a)

(n)

Fig.1. (a) The frequency U%p%ndenc% of the ionic concfucttvityof AgMg2MoO4 glass, (b) the frequency dependence of the dielectrtc constant x’ and dielectric loss x” of the Agl:Ag2Mo04 glass measured at 120°C, (c) ionic conductivity aand ton hopping rate wp deduced from (a) as explained In the text. Data taken from 181.

significant power law frequency dependence. The presence of the latter ‘low frequency dispersion’ is pa~~lady evident in the -84 and -9&C data. The higher freque~

power law increase in fructify,

(high frequency dispersion) is widely accepted to be a character&tic of ionic coflducfors. Many workers have #ted that the frequency dependence of ac ~~~i~~

@wf appears to have th%form:

215 Q(W)= d

(0)+ Bw”

(7)

in which d (0) corresponds to the plateau value of dc conductivity, B is a temperature dependent parameter and n is an exponent whose value lies between zero and one. The presence of a low frequency dispersion, however, has not been noticed, or has been ignored, by the vast majority of workers in this fietd. It is true that in many materials this is a weak dispersion with a slope close to zero. Jonscher [16] first drew attention to this effect in studies of the dielectric properties of hollandite and wet sand. He demonstrated both the real and imaginary parts of the dielectric susceptibility, x’(w) and x”(w), to exhibit a power law frequency dependence:

which is consistent with o(w)= w” since x”(w) = O(W)/QW. In addition, he pointed out that a consequence of the Kramers-Kronig relations is:

_fW

= ax(nm-2)

(9)

x’(w)-x,

at all frequencies w for which the power law frequency dependence is adhered to

These characteristics were found in the Agl:Ag2MoOq glass dielectric data, shown in Fig.1 b. It is clear that x’ and x” have the same slope at low frequencies. An accurate value of the small exponent nf may be obtained from the magnitudes of x’ and x” and eqn.[9]. The relationships between x’ and x” also provide a means of demonstrating the presence of low frequency dispersion. The conductivity data of polycrystalline LiGa02 are presented in Fig.Pa as an example of data which appears to exhibit no low frequency dispersion. The display of the two components of its dielectric susceptibility Fig.Sb, however, shows that low frequency dispersion is present, with an admittedly low exponent of 0.055. The importance of low frequency dispersion is that it raises questions about the validity of the procedure of obtaining a measure of dc ionic conductivity from the ac conductivity plateau. As a frequency independent conductivity is not, in fact, exhibited, what value is to be taken as an estimate of the dc conductivity? A comprehensive

model of ionic conductivity needs to account for both high and low frequency

dispersion, their magnitudes, their temperature dependencies and their relationships to dc conductivity. There have been numerous attempts to parameterise data obtained for particular conductors by fitting to a wide range of complex equivalent circuits. In the theoretical sphere, attention has been paid, almost

216 exclusively,

to

the more evident high frequency dispersion [17-201. The more comprehensive analyses

of ac conductivity are a direct development of the work of Jonscher [16,21,22]. In fact, most of the electrical characteristics of real ionic conductors can be found in these publications. However, perhaps because they are expressed in dielectric terms, they appear to have been overlooked by many workers in this field. Nonetheless, it is a simple matter to convert to more familiar conductivity or admittance terms. Jonscher [16] found that the imaginary component of dielectric susceptibility took the form:

n2 - 1 os(w/wc)nl-l + (w/w3

x”(w) = cl@

(9)

Hence

O(W)=

(w/w,)“1 +(W/WC)“2

(10)

in which nf and n2 are the low and high frequency dispersion exponents and wc is a characteristic frequency. This characteristic frequency determines the frequency range over which the transition from the low frequency dispersive behaviour to the high frequency dispersive behaviour occurs. It is evident from data sets of ac conductivity, such as those shown in Fig.la, that this transition frequency range decreases with decreasing temperature.

-4

1

I

I

I

I 059K

-5-----”

656K

?S 959K

;‘,_

_

-0

/553K

t

2 lo1

1

I

I

I

I

3

4

5

6

7

lop$requencY

Wt)l

6

2 lb)

3

4

5

lo9,IfreclusncY

6

‘I

(HZ)1

Fig.2. (a) The frequency dependence of the ionic conductivity of LiGaO2.. (b) the frequency dependence of the dielectric constant x’ and dielectric bss x” of LlGaOp measured at 959K. Data taken from (261.

6

217

It was suggested by the author [23-261 that this characteristic frequency might be identified with the thermally activated ion hopping rate. In the limi! of nt --f 0, the low frequency dispersion term approximates to the dc conductivity plateau and a simple, testable expression for the high frequency dispersion term is obtained:

c(w) = K’wp + K’wp (w/wp)” where K’ = K/T =o(o)[l

(11)

+ (w/wp)“]

This expression was found to accurately conform to p-alumina data and has since been employed in the analysis of numerous other conductors [8,13,15,25,26].

If a finite low frequency dispersion is

included, the expression becomes:

o(w) = K’wp [wrWp)“’ + (w/w~)“~]

(‘2)

Thus for any values of the two exponents, the magnitude of the ac conductivity comprises two equal parts where w = wp &

o(w) = 2K’wp = 20(o) where w = wp

Thus the ion hopping rate, wp, and what has been taken to equate the dc conductivity may be obtained graphically or by the fitting of eqn.12 to data. The frequencies at which w = wp are shown in Fig.la and the deduced values of wp and o(o) are plotted in an Arrhenius fashion in Fig.lc. It is evident in Fig.lc that both hopping rate and conductivity have the same activation energy, indicating that the mobile ion concentration of Agl:Ag2 MoOa glass is temperature independent. Similar relationships between ion hopping rates and conductivities, have been found for a variety of ion conducting glasses [13,15,25] Although, the above analysis is purely phenomenological, it provides a simple means of unifying ac and dc conductivity and of estimating ion hopping rate. It provides no proof that the characteristic frequency is the hopping rate, but independent experiments on well characterfsed conductors, such as beta-alumina, indicate that it may be equated with hopping rate, or taking into account expenmental error, a value very close to it. It is difficult to imagine what other characteristic frequency might be found in a conventional

hopping ion conductor.

Earlier, the concept of a conductivity relaxation time, zo, was developed [27j by employing a parallel R-C equivalent circuit to model the frequency dependence of conductivity. The imaginary part of the complex electric modulus was shown to pass through a peak, like a Debye dielectric loss peak, where wrO = 1. This type of analysis has been performed on the data obtained from numerous ion conducting glasses. Problems associated with the presence of dispersions were dealt with by introducing distributions of relaxation times. The view taken here, which is the one which appears to be gaining in

218 acceptance, is that the dispersive effects are intrinsic to the ionic conduction mechanism. This will be fully discussed below. If an intrinsically dispersive element is included in the ionic conductor equivalent circuit, it has been shown [28,29] to throw doubt on the simple interpretation of electric modulus loss peaks. Dissado and Hill [30] have developed a detailed many-body microscopic theory of ionic conduction. In this, full account is taken of the interactions that occur between neighboudng ions subsequent to thermally activated ion hops. It is recognised that in reality the process cannot be that of particles moving independently of each other because they are charged particles and the density of them is high. Following the arrival of an ion at an atomic site, the neighbourfng cluster of ions respond in a complex fashion to their changed environment. The resulting spectrum of excitations leads to the power law increase in conductivity at high frequencies-high frequency dispersion. During this initial relaxation process the ions of the cluster remain at their respective atomic sites. After a time corresponding to the mean residence time, however, the relaxation becomes enhanced by ions hopping out of the cluster. This leads to a set of excitations which cause the low frequency dispersion.

The predicted behaviour of the ac conductivity at high and low frequencies is:

o(w) = [ANcNe(Ciea)2kTl wc (w/wJn’;

w < wc

and o(w) = [AN,N, (6ea)2Mj

wc (wAvr.Jn2 ; w > wc

(13)

in which A is a numerical constant, NC is the den&y of clusters, N, the average number of displaceable ions per cluster and (sea) is the effective charge displacement per ion with 6 less than unity. It will be noticed that these expressions are identical in form to eqn.12 which was obtained semi-empirfcally. The frequency wc corresponds to the characteristic relaxation rate of the cluster rather than the hopping rate of individual ions. Similarly, the charge transported is that associated with a cluster of ions rather than a single ion. This theoretical description is remarkably comprehensive and adds substantially to the simple phenomenological

analysis.

Recently, a number of workers [31,32] have fitted data showing high frequency dispersion to the Kohlrausch law, which has been obtained in a fundamental way by Palmer et - al.[33]. It has been pointed out [34] that the cluster development processes in the Dissado and Hill model are equivalent to the rather generalised hierarchical developments examined by Palmer et al. The former model, however, retains the advantages of dealing with both dispersions and of being couched in terms directly related to the ion conduction process. It should also be mentioned that Dyre [l;? has developed a simple model of ac conductivity, which appears to be particularly suitable for glasses,based on an exponential distribution of relaxation times.

219

RATIONALISATION OF CONDUCTIVITY PREFACTORS The values of ion hopping rate that have been obtained by ac conductivity analysis or ultrasonics, in an admittedly small number of glasses, indicate that the entropy of the activation term (eqns.2 and 6) has a significant effect on the magnitude of the conductivity prefactor. Empirical measurements of effective attempt frequencies (eqn.6) differ by one or two orders of magnitude from estimates of the true attempt frequency, obtained from the harmonic potential well expression [35]:

w. = (H/2ma2) f

(14)

where m is the mass of the mobile ion. A compilation of such data is presented in Table I. Whilst the contents of the table are restricted to fast ion conducting glasses, it should be noted that very similar differences between we and w. have been found in crystalline conductors [36]. In common with a range of other thermally activated phenomena, conductivity prefactors of groups of conductors have often been found to vary systematically with activation energy. A particularly clear example [8] is provided by the Agl-Ag oxysalt glasses in which the prefactor changes by some five orders

Fig.3. The conductivity prefactors of the Agl-Ag oxysalt glasses displayed logarithmically m

their

activation energies. The slope of the broken line through the data corresponds to a Dienes characteristic temperature To of 350°C which is similar to the glass transition temperature of these glasses. For sources and tabulation of data see [8].

220

Estimates of attempt frequency w, from eqn. 14, experimental mea~reme~s

Table I.

characteristic

temperatures

15 and glass transition temperature T,.

T, calculated from eqns. 6 and

glass

E(eV)

Temp Range w. (K)

of we,

We

To

Ts

HZ

HZ

(K)

CM

WAs$t$Q

0.27

119-163

1.2.xl0'2

5xltlf5

390

425

b7'4W4

0.26

130-185

1.2x1012

1x10=

450

340

(~f)0.5[(~2~f0,6(~203)0.5~0.50.335

SO-400

1.310'2

7.5x1013

990

*

&$.B203

0.56

100-480

l.hw~2

42x10~4

1200

-

tns')O.l(~~3)0.9

0.45

170-370

1.1x1012

1x1014

1200

440

of magnitude while activation energy covers the range 0.14 to 0.475eV. These prefactors are shown plotted logarithmically against activation energy in Fig.3. The implied linear relationship between IogA and E has been observed in the equivalent data of numerous other thermally activated conducting systems, both electronic and ionic, and is known as the Meyer NeMel Rule [37-391. Since all the data shown in Fig.3correspond to the same 75% proportion of A@, the enMnous variations in prefactor cannot be abbot entropy,

to changes in mobile ion ~~e~mtion.

however, appears in the exponent of the exponential

The activation

term in eqn. 2 for the conductivityPrefacb

A. Thus the variation of A with E, or more properly l-f, might be explained by an entropy of activation which increased linearly with E. Dimensionally, the cOrMant of such a proportionality has units of inVerSe temperature. Thus we may set: S=E/To

(15)

where To is a characteristic temperature. Uvarov and Hairetdinov [40] have, quite independently, come to an identical conclusion after finding Meyer-Neldei like behaviour in the conductivity data of over 200 ionic conductors. Dienes j41], many years ago, proposed the same relatio~hip activation energy to explain atomic Mouton data of metals. He i~e~et~

between entropy and

To as the melting

temperature, Tm, because at this temperature the free energy of the activated state becomes zero, &

G = E - TmS = 0. This interpretation was extended by Almond and West [36] who suggested that To

might correspond to an ion ordering temperature.lt is well known that stoichiomettic excesses in fast ion conductors, both vacancies and interstitials, have the tendency to form ordered arrays below specific temperatures. Above the ordering temperature the ions are ‘itinerant’ and all may participate in conduction whilst at low temperatures they are bound in an array and only a few are available. by thermal activation, for conductiOn. In this low temperature state, the conductor might be called a ‘fast ion conductof

as its ion hopping rates have been found 1361to be enhanced by the entropic term. The

reverse appears to be the case above To where we have an itinerant ion conductor

.

Nowick et al [42] have suggested that the Meyer-Neldel rule can be explained in an essentially identical fashion. They also introduce a low temperature state in which the ions become localised by traps to form an ‘associated state’. They point out that the activation energy for such a state must include both a trapping energy and an activation energy for migration. The net entropy of activation also includes two separate contributions which could explain the largeness of the entropic terms obtained for many fast ion conductors. The above interpretations appear to be in accord with expertmental data. It has been noticed [36] that unexpectedly low mobile ion concentrations are found in materials measured at low temperatures, in the fast ion state, together with anomalously high apparent entropies of activation. Whereas large mobile ion concentrations and small entropic terms characterfse materials in the higher temperature itinerant ion state. The line drawn through the data in Fig. 3 has a slope corresponding

to T, = 435 K, a temperature

remarkably close to the glass transition temperature Tg of these glasses and which is only a few tens of degrees below their melting temperature. The possibility of a simple link between Tg and To is explored in Table

I in which the value of To necessary to generate the difference between we and w. is tabulated

and compared, where possible, with values of Tg. The results are inconclusive. Whilst To and Tg values are similar in magnitude for the first two glasses listed, there is considerable difference in the remainder. However, several different methods were employed to estimate We, the validity of all which are yet to be proven. At this stage ft can only be stated that the resufts of the Agl-Ag oxysalt glasses are suggestive of a relationship between To and Tg which might be investigated in other systems.

DISCUSSION AND CONCLUSIONS In this review I have concentrated on the influence of the ion mobility term on the magnitude of ionic conductivity. Particular attention has been drawn to the relationship between ionic conductivity and ac response. The results of a number of investigations have indicated that the entropy of activation is a parameter which has a significant effect in determining ion mobility. Unfortunately, it is also the least understood and probably the most difficult factor to calculate in even the simplest crystalline conductors. In glasses where structure and conduction paths are seldom established with any certainty, the entropy of activation is still more difficult to interpret in a specific fashion. The appearance of Meyer-Neldel behaviour in some systems indicates both a linear dependence of entropy on activation energy and an upper characteristic temperature. It is not suggested that all systems will exhibit this behaviour. The conditions for its appearance are that all the other factors, carrier concentration etc, in eqn.2 for the conductivity are invariant in a group of compounds which have a range of activation energies and that measurements are made below an effective ordering temperature. These conditions will be violated in may complex glass systems and in these it is probable that carrier concentration effects will dominate. Similarly, the effect of composition, in mixed alkali glasses, on ion mobility will be more complex than that

222

discussed here. However, hopping rate data from ac conductivity analysis may provide helpful clues about the nature and properties of the mixed alkali effect. Despite the recent efforts of a small number of research groups, there is still far too little data which separates mobility and ion concentration. Only when such data has been obtained for a wide range of glasses will we be able to judge the value of the ideas presented here.

ACKNOWLEDGEMENTS It is a pleasure to acknowledge the many years of fruitful collaboration with Dr AR. West of the University of Aberdeen and to acknowledge his students and colleagues at Aberdeen who have obtained much of the experimental data upon which this work has been based.

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