Annu. Rev. Phys. Chern. 1992.43:693-717 Copyright © J<l92 hy Annual Reviews Inc. All right .. reserved

MOBILE IONS IN AMORPHOUS

SOLIDS

C. A. Angell

Department of Chemistry, Arizona State University, Tempe, Arizona 85287- 1604

KEY WORDS: superionic glasses, decoupling index, conductivity, correlation times

INTRODUCTION

Mobile ions in amorphous solids are interesting from both academic and practical perspectives. They are of academic interest, because of the challenge to explain how the structures of a wide variety of liquids can change during cooling so as to permit truly enormous differences in their particle mobilities (up to factors of 1014) to develop. In the fluid states of most nonpolymeric fast ion glass-forming liquids, the time scales for electrical relaxation (by the most mobile species) and mechanical relaxation (by the least mobile species) are within an order of magnitude ( 1 , 2). They are of practical interest, because these materials may serve as the solid electrolyte in a wide variety of electrical devices, such as primary and secondary batteries, sensors, and electro chromic displays. The field of high conductivity amorphous solids has been very active during the past two decades, as it has become an increasingly important part of the broader field of solid state ionics. With the urgent societal need for nonpolluting vehicle power sources now being a subject of legislation in many states, the importance of this subject area is destined to grow considerably.

The amorphous solid conductor field comprises two important subfields and shares a further subfield with the crystalline branch of the su bject. The subfields referred to are, respectively, fast ion-conducting ("superionic") glasses and salt-in-solvating-polymer solutions. The shared subfield is that of rotator phases and certain other highly disordered crystals that obey the Vogel-Tammann Fulcher (VTF) equation and show glass transitions.

693 0066--426X/92j1 1O 1 -0693$02.00

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694 ANGELL

These substances are crystalline with respect to the centers of mass of one of the major constituent particles, but amorphous with respect to most, or all, of their orientations, or with respect to the position coordinates of other constituents. The rotator phase may be (and usually is) an ionic solid with one or more quasi spherical components (e.g. BF.j, R4N+) or a molec­ular substance [e.g. CN(CH2)2CN] containing small amounts of dissolved ionic salt.

The salt-in-polymer class of conductor qualifies as a solid only when the chain polymer solvent molecular weight is large enough to induce rubbery behavior via the entanglement phenomenon (3), or when shorter chains have been chemically cross-linked to induce a rubbery condition. The rubbery condition implies that the relaxation time of the substance to macroscopic shearing forces is very long, although the response to micro­scopic shear and macroscopic electrical forces remains liquid-like. Indeed, much of the research on this class of substance is, for convenience of manipulation, performed on lower molecular weight polymer solutions, which are truly liquid under most conditions. It is, in essence, a branch of nonaqueous electrolyte solution chemistry; hence, it is given less attention in this review than the glassy solid subfield.

Each of these subfields has been extensively reviewed by many expert authors, both as select reviews (4-15) and as components of larger, edited works on the whole solid state ionies field ( 1 5-18). Earlier reviews of each subfield by Souquet (5), Angell (6) (glasses), Armand (7), and Pethrick & McLennaghan ( 1 5) (salt/polymers) tended to focus on classical transport properties (dc conductivity, transport numbers) and the structures respon­sible. More recent reviews, however, have been more phenomenologically oriented and more detailed regarding kinetic processes. There has been growing emphasis on frequency-dependent properties that delve into subtle aspects of the ion dynamics problem. Finally, as the field has grown, whole journal issues, which contain several specialized reviews of its component parts, have appeared ( 1 9, 20), and separate detailed reviews of its parts, e.g. of phosphate glasses (2 1), are also being published.

With such a wealth of information available, this review can only serve a useful purpose if, after a summary of the basic phenomenology, it focuses on the most current developments and contentious issues, and at least partly succeeds in providing a broad and unifying view. Wc devote special attention to the following controversial issues: factors limiting the maximum conductivity provided by these systems; conductivity pathways and microheterogeneity in superionic glasses; and problems in phenomen­ology of relaxation caused by mobile ions, such as dielectric relaxation versus conductivity relaxation in relation to nonexponential kinetics, the emerging enigma of NMR correlation time versus dielectric/conductivity

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MOBILE IONS 695

relaxation times discrepancies, and high frequency and/or low temperature excitations.

The plastic crystal subfield, however, is underdeveloped, despite its considerable promise. We discuss this subfield in the final section.

BASIC PHENOMENOLOGY

Amorphous solid conductors form when liquids or solutions containing ions cool from a high temperature liquid state (or dry by evaporation of a volatile component) without crystallizing. To be solid, some process that establishes a finite shear modulus must occur. For superionic glasses, this process is the glass transition at which the shear and enthalpy (overall structure) relaxation times become longer than ordinary experimental times [usually considered as 1 02 sec (23), but somewhat arbitrarily, as the two times are not necessarily the same, and for fragile glassformers they can differ by one to four orders of magnitude (24; M. Tatsumisago, in preparation)]. For salt/polymer systems, however, the shear modulus becomes finite at the liquid-to-rubber transition.

To some extent, the properties of the glass, and to a lesser extent those of the rubber, depend on the cooling rate at which the passage from the liquid to solid state is carried out. However, for the interesting systems with highest mobilities, the thermal history effect is not a very useful variable for mobility enhancement. Thus, except for noting the interest content of the study by Ingram et al (25), which showed that quenched, but not anillealed, AgI-Ag2Mo04 glasses had non-Arrhenius conductivity plots, we bypass this issue. (For the general case, see Refs 26-28b; for the fast ion conductor case, see Refs. 5, 1 0, 28a,b.)

During cooling, all systems show enormous increases in viscosity (or shear relaxation time), but favorable systems show only small increases in conductivity relaxation time. Thus, a high conductivity is preserved in the glassy statc�. This decoupling of conductivity modes from structural modes is illustrated in Figure 1 and can be quantified by the ratio of shear relaxation time to conductivity relaxation time, first utilized for this pur­pose by Moynihan et al (29) and now generally called the decoupling index � (6,8, 1 4),

1 .

Because the shear relaxation times at and above T g are only available in a few cases, we have found it more practical, for discussion of superionic glass phenomology, to use the enthalpy relaxation time, 't'H, in place of't's in Equation 1. LH is obtained directly from the conventional measurement of the glass transition temperature Tg by differential scanning calorimetry

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696 ANGELL

2 1

...........

....

0 ·1

's -2

"" -3

� -4

-- -5 0>-. -{i ... .s; -7 ..

-8 u ::I -9 "0 c -10 0 "" -II Oli -12 � ·13

(a)

·14 .-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.... -.-.-�.-.-.-.-.-.-.-.-.-.,.".-.-... -." ..... -... -.-...

-15 -16

0 2 5 6 7

A NaSi glass •

Tg/T

OJ 35AgC1.45Ag1.20CsCI

• 70Ag1.30AhMo<4

• 60Agl.40(Ag,O.2B20a:l

• 57AgzS.40Ag135B,S3

+ 44LiI.56Li4P2. S7 o 22.5(LiCl)z27.5LizO.50Bl03

C NaSi glass .. Na2°.3SiOz .. 40Ca(NO�)2..60KN03

• H3Uo}Q,.4HzO

• H,Zn2C1 .12HzO

8

1035 660 323 259 738

ref 5 4

2 143 84

66 59 57

162 36

163

Figure I (a) Arrhenius plots of various interesting vitreous ionic conductors. The data points indicate the lowest temperature studied and the conductivity at T. (sometimes by extrapolation). In a few cases. additional data have been obtained in the liquid state, and these are included. Note the proton conducting glass indicated by dashed line: Plot appears Arrhenius but the Arrhenius law preexponent <To value is unphysical. Dotted lines indicate that a familiar Uo (10-'_10-3 0-1 em-I) is common to both excellent and poor conductors. Arrow on conductivity axis indicates maximum conductivity for Liel aqueous solution at ambient temperature. Dot-dashed line at 10-14 Q-I em-I is conductivity at Tg expected for fully coupled systems. (b) Tg-reduced Arrhenius plots for a subset of part (a) systems, which show how the systems may be ordered according to efficiency of decoupling of conductivity modes from structural relaxation modes, by Tg scaling.

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MOBILE IONS 697

(DSC) or differential thermal analysis (DT A), and is 2 ± I x 102 s at Tg when the latter is determined at 10 K/min scan rate (23). To characterize the pToperties of a given amorphous phase conductor with respect to the freedom of mobile ions from matrix interference, we are mainly interested in the decoupling index at the point in temperature at which the structure becomes fixed on cooling. This is closely related to Tg (depending on how quickly relative to 10 K/min the glass was formed). Thus, the R, in which we are interested for purposes of material property definition is R, (Tg), hereafter called the decoupling index Ri:

* LH(Tg) 2 x 102 R, = -- = ----- .

Lu{Tg) T:iTJ

Because this quantity varies between 1014 and 10-3 for different systems, it is usually sufficient to have approximate values of Lu at Tg• Therefore, it is convenient to choose the simple Maxwell relaxation time, an average of a distribution, defined by (30)

9+5xlO-13

(ru) = eOf.co/(Jdc � - 2.

(Jde where (J is measured in n-1 /cm - 1. The material property Ri can, therefore, be defined simply in terms of the dc conductivity at the reference tem­peratureT* = Tg, as determined by scanning calorimetry at 10 K min - 1.

*_ 2x102 • _ 14 Rt -9

>< 10

-13 (Jdc(Tg) - 2± 1 x 10 (Jdc(Tg) 3.

or

10g Ri == 14.3+1og(JPg'

At the glass transition temperature, the temperature dependence of the conductivity relaxation time changes to an Arrhenius form of lower slope than that above Tg• Thus, the conductivity at room temperature (which is the property of most practical interest) is determined by both the decou­pIing index (which is the quantity of greatest phenomenological interest) and the temperature interval between Tg and ambient. These features are illustrated in Figure 1b, which is the reduced form used earlier by Angell (13) and Kawamura & Shimoji (2), and is now updated to include recent data on some new, highly decoupled systems (3 1-34) and on some proton conducting glasses (35, 36). Strategies that have been used to increase the ambient temperature conductivity therefore aim either at diminishing the mobile ion-to-matrix coupling, or at decreasing the glass transition tem­perature. Their implementations are considered further below.

In the case of salt/polymer solutions, there is another feature to be

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698 ANGELL

considered: Most of the configurational (liquid-like) degrees of freedom remain active as the rubbery (solid) state sets in; consequently, a glass transition occurs only at much lower temperature. At the glass transition, the rubbery character completely disappears. In this case, however, the conductivity at the glass transition has become almost immeasurably small « 1 0- 14 n-I cm- I), because although the conducting modes are com­pletely decoupled from the macroscopic (entanglement-based) viscosity, they are very strongly coupled to the microviscosity. In fact, if the same mechanical relaxation process that was used in defining the decoupling index for superionic glasses is used to define mechanical relaxation for the salt polymer systems, then the salt/polymer decoupling indices are subunity-in fact, as low as 1 0- 3 (37, 3S). Strategies for increasing the conductivity of such systems must focus on understanding and eliminating the source of this supercoupling, and on decreasing the glass transition temperature for the salt/solvent as far below ambient as possible. The supercoupling phenomenon has only recently been recognized (37), and only one systematic study has been reported (39). Thus, solutions to the problem are still far away. It appears to be a different problem from that of ion pairing [which has long been recognized (7, 9, 40-42) and characterized ( I , 43-45)], because of the opposite com posi ti on dependences (39). The primary problem of decreasing Tg (6) has been given some attention (15, 46a,b), and recent progress has been reviewed (12, 14).

It is of both academic and practical importance to know which ionic species is most responsible for an observed high conductivity. In many cases, e.g. Li2S-SiS2 glasses (32-34), it is the Li+ ions. However, in such glasses as LiF-PbF 2-AIF 3, it is by no means obvious whether Li + or F­are the primary carriers. This issue is particularly important in the case of salt/polymer systems, such as NaCI04 in polyethylene oxide and poly­propylene oxide systems, in which it was initially supposed that the Na + ion was the mobile species. It is very desirable, for the practical purpose of avoiding polarization problems in salt/polymer-based batteries, that the cation be the main current-carrying species, but mounting evidence now indicates that the cation transport t+ number is very low, except at the highest salt concentrations ( 1 2, 47, 4S). Even at a high concentration, t+ is still considerably below 0.5 (IS, 49-51), unless it is deliberately con­strained to be unity by attaching the anions to the polymer backbone (52a-c). In the latter case, the ambient temperature conductivity is unfortunately very low (52a-c) unless crown ether additions are made to free the cations (52b), or unless special high polarity side chains are attached to the polar backbone (52c). In the case of the ionene polymers, in which the cations are in the polymer backbone, the anionic transport number is clearly unity (53), often involving a Grotthus mechanism.

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FACTORS LIMITING THE AMBIENT

TEMPERATURE CONDUCTIVITY

MOBILE IONS 699

The theoretical limit to ionic conductivity is necessarily far below that for electronically conducting systems for reasons associated with the very different masses and, hence quantum mechanical wavelengths, of the mov­ing particle:s. Even in the case of protons, no equivalent of the Bloch waves in metals can be identified in practice. The theoretical limit for ionic conductivity is set by the current flowing instantaneously because of the ions oscillating in their quasi lattice sites; this can be measured by far infrared (FIR) absorption or reflection spectroscopy (54, 55). This limit is closely approached by hot ionic liquids, e.g. molten LiCI (see Ref. 8, Figure 10), but the best ambient temperature conducting glasses so far identified have a conductivity almost two orders of magnitude smaller (see Figure la), and one of them has the disadvantage of being unstable against crystallization.

Figure Ib ranks the conductivity of different glasses against their struc­tural relaxation times by normalizing the temperature scale to a tem­perature of equal structural (not shear) relaxation time, viz. the calori­metric glass transition temperature. This comparison shows that the greatest progress in the effort to free the mobile ions from the frictional effects of their more slowly relaxing neighbors, remains that made in 1983 by ChiodelIi et al (56) and Susman et al (57), with their respective developments of the silver iodoborate and iodoborate-phosphate sys­tems, and complex sodium zirconium phospho silicate glasses (NaSi glass). Although these are extremely conductive at their Tgs, they are not the best ambient temperature conductors, because the Tgs are relatively high.

Adding alkali halides to a base glass almost always increases the amb­ient temperature conductivity (58-60), primarily because of the "plasticiza­tion effect" (so named because of the analogy to the effect oflow molecular weight solvents on polymer viscosities and glass transition temperatures). The T g in these cases is lowered in proportion to the plasticizing alkali halide mole fraction, but the decoupling remains the same. A good example is provided by the early data of Malugani & Robert (58) on LiPOrLiX (X = CI, Hr, I), in which the iodide-doped glass with lowest Tg had the highest conductivity at room temperature, but in which the conductivities at Tg were: the same almost within experimental error, i.e. the decoupling index was unaffected. This is demonstrated in Figure 2, which shows conductivity Arrhenius plots for compositions at each extreme of several families and the locus of conductivities at Tg for all members of each family. Figure 2 also shows that the plasticization effect dominates in the

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700 ANGELL

0 0 LiP03 1 4 • 33Lil.67LiPO 3

-I • 33LiBr.67LiP03 13 • 33LiC1.67LiPO�

'e -2 '" 12

� -3 :& � AgP03 II

:� A lOAg1.90AgPO�

\:;�;�;�:\!: R-r .... -4 '" • 20Agl.80AgPO 3 10 :::I

"CI + 30Agl.70AgPOs c -5 0 II 40Agl.60AgP03 '" 9 Oii .. 50Ag1.50AgPO� .g -6 .. 55Agl.45AgPO,

8 III AgI-TlPOl \i -7

0.000 0.001 0.002 0.003 0.004

1fT Figure 2 Arrhenius-plots of conductivities of end members of families of vitreous conduc­

tors. For intermediate members, only conductivity at Tg is shown. Locus of ITT, values is

given by solid line for each family. For Li+ conductors, conductivity at Tg is unchanged by halide additions. For AgI-containing systems ITT,' hence decoupling index Ri', increases after

short "incubation" interval.

more highly conducting Li halide + Li thiophosphate family (60). In the AgPOrAgI system, on the other hand, the early study of Malugani et al (61b) shows that both decoupling and plasticization effects contribute to the very high conductivity of the AgI-rich glasses (see Figure 2), once XAg1 > 0. 1 . The suggestion from the data that there is an initial plasticization­only range seems to be supported by the more recent glass and liquid study of Pathmanathan et al (61 a), and this is interesting in view of some structural studies discussed later. The combination of plasticization and decoupling effects also seems to be true for thc AgI-containing systems reviewed by M inami & Tanaka (4) and for the iodoborate systems studied by ChiodelIi et al (56). Finally, Figure 2 illustrates the case of TII-AgP03 glasses (62, 85), in which the conductivity initially increases despite decreases in O'T (hence, in decoupling index), until at 20% TIl the decou-g JPling effect suddenly starts to increase sharply to the glass-forming com-JPosition limit. This must be related to the Tl + -Ag+ exchange equilibrium (between 1- and - p-o- sites discussed elsewhere) (62, 85).

Mixing of the glassformer components has frequently enhanced con­ductivity (63-65b). From the data of Magistris et al (64), this is clearly caused by an enhancement of the decoupling, because O'T increases with g second glassformer content despite increases in Tg• Pradel and Ribes

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MOBILE IONS 701

(65a,b) have suggested that this effect results from a tendency to micro­scopically demix, with enhancement of mobile ion content (hence, of conductivity) in one of the continuously connected microdomains.

Major improvements in room temperature conductivity have long been known to occur in glasses when the oxide ion is replaced by sulfide or selenide (5, 31 a,b), and such systems have recently been under intensive study (32·-34, 68, 69). This improvement was thought to be due to the easier passage ofthe mobile ion past the more polarizable nonmetal species. However, this notion would imply decreased friction specifically for the cation, hence an improved decoupling index, whereas examination of the conductivity at Tg shows that scarcely any change in aTg (hence in Rn

occurs as a result of the sulfide substitution. The ambicnt temperature conductivity increase, in other words, is primarily caused by the large Tg

decrease, which reflects the loosening of the entire structure, rather than of the particular mobile ion-matrix interaction. In contrast, when selenide replaces sulfide, the ambient temperature conductivity does not increase, despite Tg decreases of some 50 K (69). Thus, in this case R1 has decreased, making it evident that the decoupling is greatest in the more open structures.

The greatest loosening, or Tg reduction, effects occur when the anion­network, chain, or oligomeric structure is dismantled and when the anions combine a low effective charge density with low polarizability. Thus, in AgI-containing systems, the value of Tg decreases as metaphosphate is replaced by pyrophosphate, when pyrophosphate is replaced by lower­charged pyrosulfate or dichromate (70), and when dichromate is replaced by chromate, which brings Tg to room temperature. Finally, it can go to well below room temperature, if all complex ions are removed and only halide anions remain (71). Such systems can have a conductivity of almost 10-1 n-I cm- I at ambient temperature. Replacement of Ag+ by the smaller Cu+ in AgI-rich compositions gives lower conductivities in some cases and higher conductivities in others (72; T. Minami, private com­munication).

Likewise, if the silicate chains of Li20· Si02 glass (Tg = 700 K) are broken down to orthosilicate ions, Tg decreases to 500 K (73); replacing SiO�- by CI04 or CIO] can yield Tg values well below room temperature. However, somewhere in the sequence the high decoupling index becomes lost. An extreme case is the high LiI content binary (LiI + organic cation iodide) system studied by Cooper & Angell (74), in which R� falls to about 102. Interestingly, in the related AgI-organic cation systems recently reported by Kawamura & Hiyama (75), the conductivity remains very high, 10-2 n-I cm-I at 25°C, and R: is �lO'2. This result presumably reflects the: essential difference between AgI-containing and alkali-halide­containing systems discussed earlier, viz. the AgI enhances the conductivity

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702 ANGELL

by providing new and more decoupled pathways, whereas the alkali salts only plasticize the system to enhance existing pathways of constant de­coupling index. If these do not pre-exist, then Lil addition leads to poor conductors.

Recently, an extreme case of conductivity enhancement by AgI involved actual precipitation of ct-AgI in microcrystals, which were stabilized against the IX -+ f3 transition by the small size of crystals produced during very rapid quenching in an AgB03 matrix (76).

In polymeric systems, different strategies must be followed; however, we only summarize the principles involved. In the usual salt/polymer solutions, the conductivity maximum observed around M: 0 = 1: 16 increases in magnitude with decreases in solution Tg and with increases in solution fragility. Fragility (67) is defined by the parameter D (fragil­ity oc D- ' ) of the modified VTF equation

4.

where To is related to the glass tranSItIOn temperature Tg by (67) Tg/To = I +0.0255D. Near the composition of maximum conductivity in these systems, the system is supercoupled with R, ;::::::; 10- 3, almost inde­pendent of temperature (37, 39). In the ionene polymers, however, there is considerable conductivity at room temperature (53), despite Tg of;::::::; 100oe, because of the decoupling of the anion motions, and R; may be of order 107• These polymers, which are effectively polymerized molten salts, pro­vide one of the two conceptual bridges between polymeric and superionic amorphous solid electrolytes. The other bridge is realized by the pro­gressive addition of salt to polymer until the latter becomes the minority component (39), as has been partially carried out by Przyluski & Wieczorek (77), and now completed by Angell and Liu (in preparation).

CONDUCTIVITY PATHWAYS AND STRUCTURAL MICROHETEROGENEITY IN SUPERIONIC GLASSES

The structures responsible for the high conductivity obtained in superionic glasses, have been of great interest and much controversy, particularly in the case of the Ag+ -conducting glasses. Early in this subject's history, the tendency of AgI-containing glasses to extrapolate conductivity (5) and other properties (7) to the values for superionic crystalline ct-AgI had led to the supposition that a-AgI-like clusters exist in the glass and that these overlap increasingly with increasing AgI content. The existence of some sort of inhomogeneity has been argued directly from small angle neutron scattering (SANS) studies (79). These studies have suggested a dimension

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MOBILE IONS 703

of some loA for the dispersed phase. An early calorimetric study on AgI­AgP03 (80) has been used ( 1 7) to argue that, at least at low AgI content, the tendenc:y is to order AgI within the AgP03 matrix, rather than to form clusters. However, the partial molar heat of mixing of AgI seems to reverse sign near XAg1 = 0.3, which implies clustering at higher XAg1• Indeed, the opaque yellow appearance of slowly cooled AgI-AgP03 glasses of XAg1 > 0.50 strongly implies that a macroscopic phase separation can occur.

More recently, a new light has been thrown on these systems by the elastic neutron scattering studies of Tachez et al (8 1 ) and Borjesson et al (82-83). These authors have shown that, on doping a silver oxide-modified oxide glass like AgP03 with AgI, a new sharp diffraction peak arises at very low Q ( � 0.7 A-I) with intensity proportional to AgI content. According to Borjesson (83), this occurs not only in the complex P205- and B20rbased systems (8 1-82b), but also when there is no extended chain or network structure present at all, viz. in the mixed salt glass (AgIh.75(Ag2Mo04)o25 (see Figure 3). In the latter case, however, the low Q peak is distinctly

smeared out. Borjesson also shows that the peak is essentially absent from AgCl-doped glasses and from the corresponding alkali halide (LiCI- and

... .9 u � ., .. = t: = .. 'OJ

(b)

°O�--+---"�O--�'5--�20 QrA_X (a) (e)

3.0

'.5 (i)

'.0

1.5 (II)

1.0 (iii) 0.5

0.0 0 , Q/A,-'

�.7. •• ��.�.2�'--�O'�'f--�O� .• O� Agi concentration X

Figure 3 (a) Reduced structure factors for four highly structured single-component glasses, which show prominent prepeaks at � 2 A - ' , and for silver iodoborate glass with prepeak at 0.7 A - '. (b) Structure factors showing low-Q prepeaks in three very different AgI-Ag oxanion glasses: (i) Agl-Ag,O-2B ,0 3. (ii) (AgI)o 75' (Ag,MoO 4)025. (iii) AgI-AgPO 3. contrasted with (iv) prepeakles.s structure factor for pure AgP03 (curves for i, ii, and iii displaced upward for clarity). (c) Intensity of low-Q peak in (AgI)x' (Ag,O' 2B,03) ,-x as function of AgI content. Note how peak tends to vanish at XAg, = 0. 1 . (From Ref. 83, with permission of the Materials Research Soci'ety.)

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704 ANGELL

NaBr-doped) systems. Noting that the peakless systems are also relatively poorer conductors, Borjcsson links the low Q peak and the associated intermediate range order to the high conductivity known for the AgI­doped systems. We would add that the systems with strongly increasing prepeaks are those that owe their increasing conductivity to increased decoupling, as well as to decreasing Tg (see earlier discussion of Figure 2). Whether the low Q peak is equally prominent in the compound-related glasses AgI-AgCI-CsCl [related to CsAg4Is (70)] and Ag2S-AgI-As2S3 [related to Ag}SJ (20)], or in the mixed cation systems AgT-NaP03 (85) and Til' AgP03 (62) is not yet known.

Sharp low Q diffraction peaks have long been recognized as indicators of intermediate range order in the classical glassformers (86), although their origin has been a subject of controversy (86, 87). The AgI-superionic glass peak is at the lowest Q value observed for any nonmolecular system. The authors observe that its value, �0.7 A, and width, � 0 . 5 A, imply an intermediate range ordering of repeat length, 8-lO A, and correlation length, 15 A, which is not inconsistent with the earlier SANS results. However, Borjesson points out the difficulty raised by the failure of small angle x-ray scattering, which should be sensitive to electron-rich AgI, to detect a small angle peak (16 1). Most recently, Borjesson & McGreevy (88) have tackled this problem by application of McGreevy's reverse Monte Carlo computer simulation method and concluded that the low Q peak is due to "local density fluctuations In the network of P04 tetrahedra caused by the requirement to maintain connectivity while decreasing the average density." From the form of the P-P radial distribution function, they further suggest that there is a weakly fractal character in this expanded network structure.

Before discussing how these notions may support alternative models of the conduction mechanism, we draw attention to an interesting, and probably diagnostic, correlation. Borjesson points out that the intensity of the prcpeak extrapolates to zero not at XAgl = 0, but rather at the point X Agl = 0.1. In discussing the difference between effects of alkali halide additions to LiP03 and AgI addition to AgP03, we noted earlier (Figure 2) that only in the latter case did the decoupling increase with added salt. We now further observe that the increase in decoupling index (or conductivity at Tg) in AgI-AgP03 like the AgI activity coefficient only begins to rise beyond X = 0. 1 (see Figure 2). This would appear to connect the structure behind the prepeak (and the break in FIR spectral shifts discussed below) with the decoupling enhancement in AgI-based super­ionics. Finally, we note that in the case of TII-AgPO 3 glasses (62), in which an exchange of Ag between 1- neighbors and -O-P- neighbors can occur, the increase in uT.' hence in decoupling index, only commences above 20%

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MOBILE IONS 705

TII, although the conductivity increases continuously. Clearly, the pre­diction to test would be the absence of a prepeak at 20% Tii and a sharp increase thereafter. A further test would be to see if the prepeak increases with increasing [W04]/[W04+S04] at constant XAg1 in the system AgI­Ag2S04-Ag2W04 (89), in which Tg, but not (J250, is a strong function of the oxyanion ratio (90) (which implies that Ri can change at constant XAg1).

Finally, we must consider the input into the structural character of superionic glasses from vibrational spectroscopy. Many studies of glasses containing polarizable components, like AgI, have been made by Raman spectroscopy (97, 98). However, because the FIR bands represent the attempt frequencies for activated conducting motions, we focus attention on the FIR spectra in the quest for new indications of the structures connected most directly to the conducting pathways. Alkali phosphates were studied by Exarhos & Risen (9 1). Alkali silicates have been described by Exarhos et al (92) and Gervais et al (93), and alkali borates have been subject to detailed FIR study by Kamitsos and coworkers (94) and Liu & Angell (54). Finally, Nazri et al (in preparation) and Martin et al (in preparation) have examined AgI-AgP03 glasses.

Kamitsos and coworkers (94) observed asymmetry and structure in their alkali metal borate FIR spectra, and Ingram et al (100) have deconvoluted these into separate components, which they assign to distinct sites represent­ing "cluster" and "tissue" components of the glass. In Ingram's "cluster­bypass" model ( 10 1), the.conducting paths are conjectured to be localized within a tissue component, which [in accord with concepts advanced first by Tammann (102) and later Hayler & Goldstein (103) and Goodman (104)] forms after denser clusters have impacted during cooling. The cluster bypass model has the advantage of predicting that there should be more tissue, hence higher conductivity, in quenched glasses, as is the case in practice. It also seems compatible with the interpretation of the sharp prepeak in AgI-containing glasses preferred by Borjesson & McGreevy (88). These: authors see the AgI filling out the low density fluctuations in the oxyanion matrix, rather than forming distinct clusters that overlap or percolate in the more common "cluster overlap" models ( 105, 106). An element of confusion enters here, however, when the FIR spectra of AgI­AgP03 glasses are examined because the most definitive evidence for different oscillation sites might be expected in these cases. Although low frequency Raman spectra show structure consistent with the presence of Ag-halide like sites (97), the FIR spectrum, rather than developing new low frequency components, and then moves systematically back to lower frequencies, without any broadening, as XAg1 increase to 0.5 (Martin et ai, in preparation). There is no obvious explanation for the absence of separate

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706 ANGELL

Ag site frequencies. However, the finding is consistent with the absence of separate Ag+ jumping internal friction peaks ( 1 07). The only comparable alkali halide-doped case, Button's ( l 08) study of LiCI-Li20�B203 glasses, shows a simple and progressive shift to longer wavelengths, again without obvious band broadening, as LiCI is added to lithium metaborate.

FREQUENCY-DEPENDENT ION DYNAMICS AND

CORRELATION FUNCTIONS

The subject of relaxation kinetics, with its current emphasis on quantitative treatment of nonexponential relaxation, is alive with controversy. This follows a period in which many workers, including myself, had feIt matters were reaching some consensus status, insofar as a certain ubiquity was being recognized ( l 09-l l lb) for the stretched exponential (or KWW) relaxation function ( 1 09):

8(t) = exp [ - (t/r)/l]. 5.

Its application was helping rationalize long-standing discrepancies between nuclear spin-lattice relaxation (SLR) and conductivity relaxation (5, 17, 1 1 1 - 1 1 3) on the one hand, and quasielastic neutron scattering diffusivities and conductivity ( 1 14) on the other. In addition, the treatment of electrical properties of conducting solids by use of the complex modulus formalism (30), which considerably simplified data manipulation, seemed to have won broader acceptance, and the resulting spectral widths seemed con­sistent with those from other spectroscopies. However, any tendency to unwarranted complacency was halted by the confluence of experts pro­moted by the 1 989 Heraklion conference organized by Ngai and Wright and published in three volumes of the Journal of Non-Crystalline Solids (131/133). In the following, we encapsulate some of the now troublesome issues.

The first issue concerns the question of the appropriateness of the modulus formalism both for electrical and mechanical relaxation processes, and the consequences to nonexponentiality parameters if not. Both Cole (116) and Kremer ( 1 1 7a) point out that only in the complex susceptibility or complex conductivity formalisms are the quantities directly defined by the fundamental Maxwell equations. Furthermore, common relaxation times are only found for multilaterally. bonded liquids, like glycerol, for which common relaxation functions are expected when susceptibilities, not moduli, are compared ( 1 1 8, 1 1 9). A procedure for combining the virtues of the modulus analysis (to suppress electrode capacitance effects) with the notion that the electrical response of a con-

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MOBILE IONS 707

ductor should be treated as a dc conductance in parallel with a relaxing dielectric, was suggested by Johari & Pathmanathan ( 120), who applied it to several different glass systems. More recently, Cole & Tombari ( 1 2 1 ), who used a direct susceptibility approach, found that their resistance-capa­citance data on the NazO' 3SiOz glass previously studied by Moynihan et al ( 1 22a,b) when first treated by removal of a Warburg surface im­pedance, gave a good fit to a Davidson-Cole (DC) dielectric relaxation function after subtraction of a dc conductance. The best fit DC parameter, IX = 0.36, implies an equivalent KWW fJ parameter of 0.47, which is con­siderably smaller than that of Moynihan et ai's KWW fit to the electrical modulus spectrum (fJ = 0.6) ( l 22a,b). This treatment, therefore, modifies the electrical response spectral shape considerably and throws into question, or at least modifies, some correlations made recently, e.g. between conductivity and nuclear spin-lattice relaxation correlation func­tions ( 1 1 1 b, 1 12) and between nonexponentiality parameter fJ and decou­piing index ( 13) . On the other hand, this treatment tends to reconcile those cases in wlhich large differences have been noted, like the extremely broad log TI versus l /Tcurve observed for 109 Ag nuclear spin relaxation in AgI­AgzO-Bz03 glasses studied by Chung et al ( 123). For instance, an analysis by Niklassen et al ( 124), based on a complex capacitance representation of the AgIx-(Ag20' 2B203)I-x data almost equivalent to that of Cole & Tombari (121 ), also gave best accord with a DC expression, the shape parameter of which was compatible with the SLR findings. The detailed comparison with SLR, however, raises a new question, which we discuss below.

Another recent treatment of conductivity in fast ion glasses has been the compkx conductivity analysis of Kremer et al ( 1 1 7b) for the interesting case of anion conduction in polymeric organic salt glasses, viz. the poly­ionenes [(CHr-CH2)1O-NHi].Br';- (or [ZnBr:lln)' This has the advantage of casting the real part of the conductivity in the form that has become well known for its unification oflow frequency conductivity and high frequency resonance absorption (optical phonon) regimes (see Ref. 1 2 1 , Figure 4, and Ref. 13, equivalent diagrams for both AgI-based glasses and electrical and mechanical absorption processes). For further details on the complex conductivity analysis, refer to the discussion by Funke ( 125).

We now turn to a second problem area: the relation between conductiv­ity and SLR. The manner in which the failure of the classical Bloembergen, Purcell, and Pound (BPP) theory ( 1 26) for SLR in glasses and vis­cous liquids is rectified by replacing the BPP exponential relaxation assumption by the nonexponential kinetics characteristic of slow processes, has become well recognized ( 109, 1 27a-c). In 1 986, we noted (8) that this recognition is all that is necessary to explain the large and often-discussed

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708 ANGELL

difference between the activation energy usually observed for SLR in the usual multi-MHz experiments, and that seen in conductivity. This has since been explained in detail (17, 112, 129). Quite simply, the nonexponential relaxation leads to frequency-dependent conductivity and frequency­dependent NSR for frequencies above that of the system's internal relax­ation frequency. For conductivity measurements conducted at constant frequency, high temperature measurements "see" the frequency-independent response and give the dc conductivity. At a temperature where the (fixed) bridge frequency and the (temperature dependent) intrinsic relaxation frequency coincide, there is a crossover to a lower activation energy regime. This has rarely been discussed in the conductivity literature, although it is obvious enough from the many figures of log a(O) versus log (frequency) that have been presented (30, 122a,b, 130). lf the frequency of the bridge is raised increasingly toward values typical of the SLR experiment, the crossover temperature moves to temperatures, approaching or surpassing the glass transition temperature. For most glasses, measurements made at high SLR spectrometer frequencies see only the frequency-dependent (lower activation energy) regime. This lower activation energy depends on the departure from exponentiality of the dynamic process, approximately according to the relation

6.

This relation was proposed on theoretical grounds by Ngai ( 109) and has been supported by a variety of experiments ( 1 32a,b). Dyre ( 1 31) shows that it is consistent with other empirical relations for glasses. Such a regime has, for instance, been observed in the recent work on LizS' SiSz glass of IPradel & Ribes (133), who deliberately extended the ac measurements to low temperatures far into the frequency-dependent regime (see Figure 4). However, other reports ( 17, 1 12, 134, 135) have suggested that Ea (ac) increases with decreasing frequency. The general case remains to be clari­lied. A feature of all these studies, however, is that the Arrhenius pre­exponents of the plots in the frequency-dependent regime are unphysically low. Because the activation energies are determined by the departure from {�xponential relaxation, their interpretation must be intimately connected to the model used to explain nonexponential relaxation, of which there are now a considerable number ( 1 36-1 39).

Although a reconciliation of NSR with conductivity data, based on nonexponential relaxation functions, has now been reported by several laboratories (124, 133, 134, 139) and interpreted microscopically by Elliot & Owens ( 140), there seems to be a problem that has not yet been discussed. There is a tacit assumption that the correlation function for the spin­relaxing and electrical field-relaxing fluctuations is the same. In the Elliott-

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10 kHz (0). 100 kHz (.).1 MHz (C). 10 MHz (e).

3 5 8

MOBILE IONS 709

\r' ----------------------, ,;

3 4 5

Figure 4 Arrhenius plots for electrical conductivity at various fixed frequencies, for the systems (AgI)o.37,(O.5Ag,SiS,)o 625 and (Li,S)o 5' (GeS,)o.,. Insert shows temperature depen­dences of spin-lattice relaxation times [log (TI'ls) versus liT], from which the characteristic time for SLR fluctuations at T(T" min) can be obtained. (Adapted from Ref. \33, with permission of J. Non-Cryst. Solids.)

Owens treatment, this assumption is explicit. However, based on the data discussed below, we suggest that this may be a poor assumption for superionic glasses and that the agreement claimed between SLR and con­ductivity behavior is only qualitative in nature. The problem is discussed in more detail in a current paper ( l35), but the essential point is that in superionic glasses, the fluctuations responsible for relaxing the weakly coupled nuclear spin system seem to have a lower characteristic frequency than those that relax the electric field. Furthermore, the opposite situation appears to hold in polymer/salt solutions. It is probably more than coinci­dence that the decoupling indices for conductivity (represented as log­arithms), have opposite signs in these two cases. By way of confirmation, we find that in the case of an aqueous lithium chloride solution, LiCI· DH20, the relaxation of 8Li studied by the method of {3-radiation detected NMR (]4 1 ) occurs on almost the same time scale as conductivity relaxation (29, ]30). The decoupling index for this system (for the discussion of which the rir:" ratio was first invented) is close to unity over the whole temperature range of study (29).

To document the above, we consider the data of Refs. 123, 1 28, l33, 134, and 141 , some of which are seen in Figure 4. Chung et al (123) noted that their Tl minimum for 109 Ag relaxation in an iodoborate glass implied

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710 ANGELL

a slower process (by approximately one order of magnitude) than for a mechanical relaxation associated with Ag+ jumping, but attributed this to failure of BPP theory. However, if we compare the simple Maxwell relax­ation time (an average value) for conductivity in this glass (143), obtained from the relation (r.,.) = eoe",,/(Jde, with their 109 Ag Le at the temperature of the TI minimum (where WeLe � 1), we obtain Le/<L.,.) = 10-7.8/ 10-9.3 = 101.5; This is much greater than any difference between (t.,.) and La (most probable time) that could arise from any of the common re­laxation functions for fast ion relaxation used to correct for the BPP exponential relaxation assumption. The difference is even greater in the silver iodogermanate case studied by Pradel & Ribes (133), Le/<L.,.) == 10-7.8/10-10= 102.2, but about the same for their Li2S'SiS2 glass (133), Lc/(r.,.) = 10-8.0/10-9.62 = 101.62, as can be confirmed from data given in Figure 4 by using fL = 10.2 MHz and 1 5.8 MHz for 109Ag and 7Li, respectively. The most detailed cases available, however, are those reported by Martin et ai, in which several spectrometer frequencies have been applied. They studied the cases of (Li20)x-(SiS2) I-x (134) and (LiCl)x-(LizO· 2B203)I_x glasses ( 1 28) by using samples identical to those studied by ac conductivity (134, 135). These studies not only confirm the sort of differences seen in the above cases, but show that they are t,emperature-dependent; the NMR correlation time has the larger acti­vation energy, consistent with its longer relaxation time. This is, of course, the opposite of the once-prevalent idea that SLR in conducting glasses detects a fast low activation energy process.

In the case of salt-in-polymer electrolyte solutions, on the other hand, comparison of data on 23Na relaxation reviewed by Greenbaum et al (144) shows that Le/<L.,.) � 10-8.7/10-8.0 = 10-0.7• In the latter cases, which are effectively liquids rather than glasses, the time scale for relaxation of the matrix in which the ions move is about 102 shorter than the conductivity relaxation time, as indicated by (composition-dependent) decoupling indexes Rt = 10-3-10-I (37-39). These values contrast strongly with the Rt values of � 1012, which characterize the superionic glasses. This implies that the fluctuations (in electric field gradient, etc.) that couple most effectively to the spin system (hence, that minimize TI/Le) are not those determined by the frequency of ionic jumping, unless there is some error in the common view that the conductivity relaxation time and mobile ion jump time are very similar when many ions are present. The nature of these less common but more effective fluctuations is, of course, of interest in connection with migration models for glasses. For instance, it could be argued that the larger activation energy for Le implies the presence of a class of deep traps in which the electric field gradient is very different from that in shallow traps in which the ions spend most of their time. To

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MOBILE IONS 7 1 1

understand the domination of conductivity by shallow-trapped ions, one must suppose that such traps are highly degenerate. Because of the field gradient effect, however, the less frequent visits to the deeper traps would dominat,e the SLR. An increase in shallow trap depth in poorer conducting glasses would explain the tentative correlation of tc/tq with Rr•

This picture has much in common with that of the weak electrolyte model ( 1 1 5). The emphasis on a trapping topology over an energy land­scape provides an analogy with models of electron transport in glasses and helps rationalize their common features (S. Sridhar et aI, in preparation). Clearly, iit would be of value to follow both conductivity and SLR behavior up into the liquid state, because the decoupling index there changes rapidly towards unity. If, as we expect, tc then varies in a non-Arrhenius fashion with changing temperature, then the phenomenon to which we have drawn attention would be identified as another participant in the serial decoupling phenomenon discussed recently by several authors ( 142a,b).

We now briefly address a third area of contention in frequency­dependcnt behavior of fast ion conducting systems. This concerns fixed temperature/very high frequency, or fixed frequency/very low temperature, behavior. It deals with the problem of connecting resonance and relax­ational modes of motion in amorphous materials and crossovers between different power-law domains for frequency-dependent properties. It is most actively studicd by NMR ( 1 39, 1 4 1 , 1 64), in which spin relaxation by two-level systems (ADWS) is invoked (1 32a,b, 141), but has also been studied by light-scattering ( 146a,b), ultrasonic relaxation ( 147), and dielec­tric/conductivity relaxation ( 1 3, 148a), in which the role of fractal exci­tations in geometrically non fractal amorphous Euclidean systems has been suggested ( 13). Strom et al ( 1 49) provide a good overview. In all cases, the question of interest concerns the source and nature of the apparently universall excitations responsible for an excess absorption over that expected from anharmonicity, and also over that predicted by most relax­ation functions (165).

PLASTIC CRYSTALS AND RELATED SYSTEMS

We hav(: long known that some crystals with asymmetric constituent particles can exhibit heat capacity anomalies on cooling (and subsequent reheating) that are close analogues of the glass transition discussed earlier (150, 1 5 1) . The motions being frozen out at Tg are, in most plastic crystal cases, reorientational motions. Frequently associated with these reorien­tational motions are diffusional modes, which, if the diffusing particles are charged, lead to a high ionic conductivity in the solid state. Several well­known fast ion conducting systems, such as Li2S04 ( 1 52, 1 53) and

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Li2MgCl4 ( 1 54), fit this description with respect to conductivity, but never t:xhibit the glass transition at lower temperatures because of the prior intercession of an ordering phase transition. We restrict our discussion here to the more limited range of systems that either exhibit the glass transition directly or carry the other signature of typical glass-forming systems, viz. non-Arrhenius transport behavior in the mobile state above Tg•

We limit the scope in the latter manner so that we can relate the obser­vations to the same parameters that have proven useful in discussing superionic glasses on the one hand, and salt-in-polymer solutions on the other. With these restrictions, this section can be very short, because the number of systems investigated in the fast ion conductor context is very small. It is, in any case, an open question whether many of the rotator phases studied by solid state ionists (e.g. Refs. 1 52-1 54) could be quenched t.o orientationally disordered glasses if specialized rapid quenching tech­niques, such as those used in some glassy state studies (73), were employed. For instance the threefold rotation of NO) in NaN03 cannot be quenched in, and the transition being approached on cooling is not of the glass transition type, but is a A. transition.

These materials are interesting because, like salt-in-polymer systems, they have a mechanism for providing solid-like behavior above Tg (in this case it is the crystal lattice); however, unlike the salt-in-polymer systems, they have no obvious source of the supercoupling that depresses the salt­in-polymer conductivity so seriously-indeed, they are more closely akin to the ionene polymers, in which the conducting species motions are decoupled, R� � 1 07 ( 1 1 7b). The first examples described were double salts of LiBF 4, LiCIO 4, and LiI with tetraalkyl ammonium type cations, except for having one R group methoxylated ( 1 55). They gave Tg values lower than those of the best conducting PPO- or PEO-based salt-in-polymer solutions, but yielded conductivities at 25°C that were only comparable, 1 0 - 5 a- I cm- I , probably because of a lower fragility ( 1 5 1 ). Since then, a related compound has been shown to be about one order of magnitude more conductive.

An alternative approach is to use a molecular plastic crystal, or a solid solution of plastic crystals, as solvent for small amounts of ionic solute. The advantage of this approach is that the Tg value of the room-tem­perature-solid solvent can be very low, e.g. l I S K, for succinnonitrile­malanonitrile mixtures ( 1 57). Because it is difficult to confirm that con­ductivity occurs wholly within the large grains of the rotator phase crystals, solute concentrations have been kept very small, 0.1 molar; nonetheless, very high conductivities, � 10- 3 n- I cm - I, have been observed. Opti­mization of this approach could clearly yield interesting materials. These

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MOBILE IONS 7 1 3

systems show highly non-Arrhenius conductivities and dielectric relax­ations that fit the VTF equation well (To :::::: 0 .8Tg) ( 1 57; D. List and C. A. Angell, in preparation). Conductivity and dielectric relaxations are well separated in frequency, as expected ( 1 58), and dielectric relaxations are nonexponential, with P(KWW) = 0.7-0.8. In studies at different solvent compositions (succinnonitrile : malanonitrile or succinnonitrile : glutaro­nitrile ratios), some differences in VTF parameters may be seen; however, values from conductivity (mobile species) and dielectric (matrix species) relaxation are comparable, which implies coupled relaxation mechanisms, as in pollymer/salts. The actual conductivity relaxation time is longer because of the low ion concentration.

Anoth(�r crystalline solid case in which VTF behavior is observed involves Ag7GeSesI, as reported recently by Ribes and coworkers (159). In an Arrhenius plot, the conductivity of this compound shows strong curvature that can be linearized by use of Equation 4 with To = 90 K. This is 50 K bdow the temperature of an anomaly in the DSC scan (not shown

in Ref. 1 59), which was assumed to be a glass transition. The authors analyze the data for this and several related compounds by using Equation 4 in the form

7.

with Ea in electron volts. Ifwe convert their data from Equation 7 to Equation 4 using D = 1 . 1 6 X

1 04 Ea/To, we find D values ranging from 5.92 to 33.6, which covers almost the whole range of fragilities observed previously for plastic crystals ( 1 5 1 , 157). A similar behavior has long been known for the case o f the compound Ag3SI ( 1 60), but has never been analyzed in these terms. Clearly, the source of non-Arrhenius behavior, and of excess heat capacity in these cases, cannot lie in reorientations, but must lie somehow in the configurational entropy associated with cation disordering. On the other hand, the relation between what is jumping and what is freezing out is not clear, because the DSC anomaly occurs at a temperature where (J :::::: 1 0 3 Q 1 cm - t, which corresponds to a L" value of only 1 0 - 9 s, cf. 102 s for relaxation at a glass transition. A glass transition due to freezing of mobile ions could only be detected by DSC when (J :::::: 1 0 - 1 4 Q- I cm- I . The detailed study of these compounds as a new class of plastic crystals, as well as an interesting class of fast ion conductors, seems warranted.

ACKNOWLEDGMENTS

This work has been supported by the Department of Energy under grant No. DE-F902-89ER4535398. The author thanks H. Senapati for assistance with data and diagrams and for a critical reading of the manuscript.

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