The universal low frequency responses of ionic conductors: A review of data and a unified model interpretation

Solid State lonics 5 (1981) 27 34 North-Holland Publishing Company

THE UNIVERSAL LOW FREQUENCY RESPONSES OF IONIC CONDUCTORS: A REVIEW OF DATA AND A UNIFIED MODEL INTERPRETATION

K. L. N ga l

Naval Research Laboratory Washington, D. C. 20375

Dielectric relaxation, AC conductivity, ultrasonic attenuation, complex impedance, internal friction and spin-relaxation measurements on NaB-alumina and their model interpretations that have been proposed are reviewed. Remarkable interrelations between the different data pieces are shown to exist, and all the data conform to the predictions of a unified and universal model proposed by the author for condensed matter in general. I.

INTRODUCTION

Considerable interest in fast ionic conductors has motivated a variety of measurements [1-24] on physical properties that can shed more light on the underlying physics and mechanisms that either control or are connected to the high ion conductivity. The measurements of interest here in this review include: (I) Dielectric relaxation [2-6], '(2) AC conductivity [3-5], (3) ultrasonic attenuation [10], (4) microwave absorption [Ill, (5) complex impe@ance [7,8], (6) internal friction [15-18] or Q-~, (7) NMR [18-23] and EPR [24] relaxations• These measurements performed by a number of workers in different laboratories have provided valuable information that should lead eventually to an understanding of fast ion transport mechanism and associated effects that are important for applications as solid electrolytes. These measurements are at low frequencies of ',~<10 GHz. The measurements involve several very different techniques and 1I cover the vast frequency range ~f 10<~
0 167 2738/81/0000

If we consider also other fast ionic conductors such as Li^Ti^O., Hollandite [12,13] and Li TiS~, • ~ J I . x z t h e n a d d i t i o n a l m o d e l s s u c h a s (g) i o n t r a n s f e r across crystal-defect-induced random b a r r i e r s b e t w e e n i d e a l , u n p e r t u r b e d c h a n n e l [ 1 3 ] ; (h) breakdown of absolute rate theory for ionic hopping [29,30]; and (i) effects of low dimensionality [31] have also been proposed to explain similar low frequency response phenomena. 2.

DATA AND MODELS

For concise presentation of a large quantity of the quoted experimental data on Na~-A£209, I have reproduced a number of them in Fig. I. The AC conductivity data and discussions can be found in an accompanying paper by Strom and Ngai in this conference, and w ~ l not b~ dealt with here. __ 7 Figures la-lc are the Na and A£ nuclear spinlattice relaxation times T 1 measurements for Na~-A~20 as a function of temperature reported 3 by Walstedt et al. [19] and by Bjorkstam and Villa [20]. Walstedt et al. have discussed their T~ data in terms of relaxation processes generated by activated hopping motion with distribution of barrier energies. On the other hand, Villa and Bjorkstam have related the asymmetric T 1 minimum to a two-dimensional continuum diffuslon model of uncorrelated diffusion with the well-defined activation energy from conductivity results. Wolf [25] has suggested that the T 1 asymmetry comes from the predominance of different relaxation mechanisms at high and low temperatures. This suggestion follows from his comprehensive model on Na~-A~203. A~ low temperatures he found that all excess Na ions form associated complexes around o x y g e m i n t e r s t i t i a l s , with formation enthalpies ~. and ~ for the two 1 2 types of sites in the assoclated complexes. With increasing+temperature freely mobile interstitial excess Na ion are created thermally from the associated complexes. From comparison of the calculated temperature dependence of the Haven ratio with experimental data, Wolf has found a best fit characterized by ~ ~ 0.05 eV and ~0.13 eV. The T data at t~mperatures belowZthat ~ 1 of the T minimum has an activation energy of about 0.645 eV which is near H~.. This leads Wolf to suggest the possibility tha~ the low temperature nuclear spin-relaxation behavior is domi-

0000/$02.75 © North-Holland Publishing Company

28

K.L. Ngai / Low frequency responses o f ionic conductors

nated by t~e temperature dependence of the lifetime of Na ions trapped in one of the associated c o m p l e x e s . While at high temperatures he considered the spin-relaxation is dominsted by the long-range diffusive motion of Na ions with the consequence that the activation energy found agrees rather well with the activation energies determined from the ionic conductivity and the tracer diffusivity. I have pointed out the interrelation between dielectric loss spectra (plots of g" vs. log f) or equivalently the complex modulUSl(M*=M'+iM") spectra [6] where M*=(g'+ig") and nuclear spin-relaxation data and applied my unified and universal model of low frequency fluctuation dissipation and relaxation properties of condensed matter [28] to spin-relaxation of Na~-A£ )0q. Not only have I explained the asymmetry an~ ~requency dependence as observed in spin-relaxation data, but also emphasized that whatever mechanism that is responsible for the observed T1-minimum it should also manifest itself in predictable ways in dielectric loss g", electric modulus M" as well as in internal friction Q measurements. The model is microscopic in nature and in simple words it essentially says that all dissipation and relaxation processes at low frequencies (or long times) will necessarily be accompanied by excitation and deexcitation of low energy modes of the system under study. These low energy modes turn out to have ubiquitous charadteristics and lead to well-defined ~odifications of the response function from I" exp(-t/l ) for a relaxation .o process with relaxation tlme I to @(t) = ( e / I ) -n l-n n o . o (E t) exp(-ct /(l-n)I E ). Here E is the c c e upper cut-off energy of ~hese low energy modes, n has been called the infrared divergence exponent that can range from zero to one and c = exp(ny) with y = 0.577. The details can he found in Ref. 28. The mechanism contained in the model is of such generality that it applies to relaxation of permanent dipoles or localized hopping motion of a charged particle, and to electric field relaxation [6,32] due to migration of mobile charged particles such as ions in ionic conductors. In each case the decay function is given [28] by ~(t) = exp(-ctl-n/ (l-n)I En). At the last international conference on fas°tCionic conduction in Wisconsin, I have considered only the spin-relaxation data of Ref. 21. In this work, I consider all the available spin-relaxation [20,21] (Fig. la-lc), electric modulus [6] (Fig. Id), dielectric loss [2] (Fig. le) and internal friction [15-17] (Fig. Ig-lh) data on Na~-AJ~^03. In principle these measurements probe both t~e long range ionic diffusion process, or localized ionic motion and localized ionic relaxation. In the latter category, possible candidates include reorientation of the split interstitial within the cell that does not contribute to transport suggested by Simmons et al., and the generation and "recombination" of frsely mobile Na interstitials from trapped Na in the associated region suggested by Wolf. Although in certain instances such as the electric modulus M" data Fig. Id clearly indicates the cause as due to long range ionic motion, yet for other sets of data the situation is not that clear.

These are best discussed in terms of the Arrhenius plots of log u vs I/T for all the measurements. This plot (Fig. 2) is designed in the same manner as that of Ingrams' with his M" data plotted as (O) but I have here added the dielectric2~oss g" data of Ref. (2) (A)~7the T I data of Na ~ of Ref. (19) (V) and of A~-- of Ref. (20) (~) and the ultrasonic attenuation data of longitudinal waves propagating parallel to the c-axis (+) of Ref. (10). Most of the data lie on two straight lines in the Arrhenius plots each with the same activation energy of 0.165 eV, suggesting two different processes. However, this dichotomy is far from being clear at this time and in some cases even leads to contradiction. For example, the M' and M" data of Ingram for the electric field parallel to the conduction planes are correctly interpreted as long range ionic motion (see the equivalent g" vs log f plot in Ref. (6) which increases monotonically Lowards low frequencies). The mean conductivity relaxation frequencies ~(M ) lle on the same straight max line in the Arrhenlus plot as the loss peak frequencies v(g" ) obtained from dielectric loss g" data of R ~ (2) (taken also with c axis perpendicular to the electric field) which, exhibiting a dielectric loss peak, should he interpreted as caused by localized ionic motion or some dipolar-like relaxation. The same upper straight line goes through the internal friction data of (a) Simmons et al. (e) which they considered as due to localized reorientation of the split interstitial, and of (b) Walstedt et al. (O) which they interpreted as due to a distribution of barrier heights for activated ionic diffusion motion proposed also by Walstedt and coworkers in spin-relaxation studies [21]. Not only these two interpretations are incompatible with each 23 other, but also the Na T. data of both Refs. 1 (19) and (20) fall completely out of the upper line, lying instead right on the lower straight line which goes through the ~(M" ) for M" data .max taken when the electric field is applled perpendicular to the conduction planes. Moreover, the data (+) of the ultrasonic attenuation peaks 1000/T(~ ) vs applied frequency of Almond et mx al. [I0] ~or longitudinal waves propagating parallel to the c-axis when plotted in Fig. 2 are scattered about the upper line and far above the lower line where, at first thought, they should lie. Hense a clear identification of the nature of the Na ion motion or relaxation responsible for each individual measurement is not at hand at this writing. •

,

in spite of this uncertainty in the experimental situation, we can nevertheless make considerable progress in the understanding of these low frequency data with my unified model [28]. In Fig. I the model predictions for log M" vs log f data (Fig. Id), for the log T 1 vs 1000/T data (Fig. la-le), and for the Q-I vs T or ]000/T data (Fig. lg-lh) are depicted by the solid curves labelled n=0.65. All theoretical curves are for the same infrared divergence exponent n, the only parameter that determines the spectral dispersion in M", g", T I or Q versus frequency plots at

K.L. Ngai I Low frequency responses o f ionic conductors

29

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K.L. Ngai / Low lkequency responses o/ionic conductors

constant T. All plots in Fig. 1 against T or 1000/T at a fixed frequency are obtained from plots of the corresponding quantity versus frequency through t~e t~ansformation of log f - l o g Vma x = E~ ( T - - T m a x ) / 2 . 3 0 3 that relates log f to IO00/T. H e r e ".. and T are the fremax . ax . ,, quency a~d temperature locatlon o~ maxrmum M , g" or Q-- value, and E~ is the activation energy that governs the shift of .' with temperature. • ax . This procedure is meanlngfuT provzded n does not change with temperature. There is some evidence [33] that n does change below 100K for Na~-A~ 203 . The small but systematic deviation of the Q data of Figs. Ig-lh from the model prediction for T < 100K may be due to this variation in n. For Na~-alumina, Fig. 2 suggests the value of 0.165 eV for E*. The procedure for obtaining the theoretica~ curve for log T I versus 1000/T has been described elsewhere [28]. It is important to emphasize that there is only one adjustable parameter n in all the model predictions for W', g", T and Q-The fact that, I for the same choice of n=0.65, good fits to seven sets of data are simultaneously achieved is worth emphasis. These results imply that there is some common underlying mechanism for the spectral dispersion of H", g", Q and T I.

In comparison, the barrier-height distribution model [19] with the Gaussian distribution parameters as used by Walstedt et ~i. to fit their spin-relaxation data [19] yields at T=-I60°C results in marked disagreement with M" vs. log f data of Ingram, (see Fig. Id); the g" data of Fig. le and the internal friction data of Figs. lg-lh. The 2D c o n t i n u u m d i f f u s i o n mode] o f V i l l a a n d B j o r k s t a m [ 2 0 ] m a k e s no p r e d i c t i o n on dielectric loss and internal friction unless one adds other assumptions or ingredients such as the "defect diffusion" model proposed by Glarum and developed by others. Even with these addenda, t h e 2D c o n t i n u u m d i f f u s i o n model yields predict i o n s that do not agree with W', g" and Q- data. The most serious difficulty of the 2D continuum diffusion model has ~een pointed out by Walstedt et al. [21]. For Li spin-relaxation in (Na,Li) ~-alumina [21], the T 1 minimum is very nearly symmetrical, suggesting that "two-dimensional" effects are absent. Further it is hard to reconcile continuum diffusion with interstitialcy-type diffusion involving hopping between sites [25,27]. It is also important to point out that the specific spectral dispersion (i.e., distribution of relaxation times) of the relaxation process that is observed in Na~-Ai20_ is not unique to Na?alumina or to ionic conductors. Previously we have shown that both dielectric and mechanical relaxation for condensed matter of different physical and chemical types have spectral dispersion as predicted by the unified model through ~(t) or its related ¢(t) for n that lies between zero and one. I have included in Fig. If M" data for Ag 14AsO4, another ionic conductor, and in Fig. 1~ internal friction data for amorphous Nb Ge, a superconductor. Each is accompanied by 3 the theoretical prediction for a choice of n

value indicated i n the figures. These two exampies are among thousands of cases we have analyzed that reinforce the universality of low frequency responses of condensed matter, and the necessity of a fundamental mechanism such as the one that I have proposed [28]. Recent data taken in Li fl-alumina by Walstedt eL al. [21] has revealed a broad minimum accompanied by a "shoulder" at higher temperatures. The shoulder has been interpreted as an unresolved T I minimum caused by the diffusion barrier hopprng process. The low temperature T minimum is 1 idsntified as caused by some form of localized Li+ ion motion. By implication similar localized Na ion motion that leads to spin-lattice relaxation is then also present in Nafl-allmfina. Hence despite the initial divergent points of view on the spin-relaxation mechanism in Nafl-A£20 , more • 3 data has led to some reconciliation among them• Wolf has considered one mode of localized ionic mo~ion as the thermally activated generation of Na in t~e associated region with formation energy H- ~ 0.05 eV. This particular local i motion may be responszble for some of the measurements of Fig. 1. Interestingly, with the observed (Fig. 2) value of Eff=O.165 eV, if a [o~al motion is responsible ~or any of the ~", Q-- and T l data, then according to the unified model the local motion should have a microscopic activation energy EA=0.058 eV which is close to the H~ value. However, in view of the aforementione~ still confusing experimental situation, it is premature to ascertain the importance of Wolf's mechanism in the data of Fig I. The t r e " 27 . quency dependence [201 8f T for A~ splnrelaxation is TlCOaJ .35± .7 ~etween l~0nand 170°K. The unified model predicts [281 T, ' .: , which • i for n=0.65 turns out to be identical to that observed. The attempt frequency is modified from the microscopic I to I~ = [(]-n)exp(nT) gni ]l/(1-n). I f Ec ' ~ >7 1, [ ~ c a n be much c ~ l o n g e r t h a n t m. T h i s f e a t u r e of the unified model is consistent with the observation of low p r e f a c t o r s (l~)for the ionic anomalously hopping rate as deduced from spin-relaxatio, data [23,29-31] for ionic conductors other than Na~-A~203 Complex impedance Z measurements of Na~-alumina a n d N a ~ " - a l u m i n a h a v e b e e n made b y E n g s t r o m a n d coworkers [7]. I n b o t h c a s e s , ReZ a n d ImZ b e h a v e -n as w at low frequencies. This behavior is equivalent to the real and imaginary part of the dielectric susceptibility havin~ the frequency dependence of g'(m), g"(m)o.'., n-* which is also predicted by the unified model [28]. Similar impedance frequency dependence has also been observed [8] for the ionic conductor ~-PbF 2.

That M" spectrum Fig. Id of Na~-aiumina can be well accounted for by electric fieh] relaxation [6,32] via ionic motion with the decay function ¢(t3 derivable from the unified model with 0
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K.L. Ngai ' Low frequency responses o f ionic conductors

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I quote the detailed measurements of Howell et al. [32] on a glass forming 40 mol % Ca(NOR)2-60 mol % KNO 3 molten salt. Its M" vs frequency data at a number of temperatures from 25.3°C (the glass-liquid transition temperature T is around 60°C) are well accounted for at each t~mperature by the decay function ~(t) with n values which are now a function of temperature T, Fig. 3. l-n The unified model gives @(t)=exp(-ct / (l-n) IoEn),c which is now rewritten as exp[-(t/ ~p)l-n] with Ip=[(l-n) exp(ny)Enlo ]I/(l-n). For lo=~=exp(EA/kT) with E A being a constant microscopic activation energy, variation of n with T will lead to non-Arrhenius behavior of with activation energy EA/[I-n(T)] a function o~ T. In Fig. 3 1 have plo]fted also f =-I/I Vs. IO00/T from the location of the H"maXximaPof the M" vs. log f data sets. It is indeed nonArrhenius. The solid curve is the prediction of the unified model from the temperature variation of n alone• Thus the unified model not only can account for spectral dispersion at any temperature, but the spectral dispersion (i.e. n) variation with T according to the unified model, determines in a well-defined manner the shift of the H" peak position f with T. The good max . agreement between the experlmental d a t a (X) w i t h the model prediction (solid curve) on I/I is another example of the predictive power of Pthe unified model. To further emphasize the universal pattern, dielectric loss X" data taken for several polymers above and near their glass transition temperatures are shown in Figs. 5-7. In Figs. 5 and 6 the solid curves are from my model prediction for n values as indicated. Note the good fit to data at all temperatures, and the decrease in n as T increases above T and the polymer evolves from a glassy to a g rubbery state. The peak frequency v(X" max ) f max shifts with I/T in a non-Arrhenius fashzon (see also Fig. 4) referred to as Vogel-WiF empirical law. Such empirical law can be derived quantitatively from our model through T-dependences of

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Another connection of the electrical relaxation data of 0.4Ca(NO3)2-0.6KNO q melt with that of another solid ionic conductor Hollandite [12,13] is of interest to point out. The H" data of the melt can be replotted as log o vs. log (frequency) at various temperatures. Here o is the ac conductivity (see Fig. 2 of Ref. 32). The variations of log o with log f at several temperatures for a limited log f range of two decades (say 102 to 104 Hz) look like that of the frequency dependent conductivity data of Hollandite presented [12] also for two decades in f. Indeed this type of behavior of log o is widespread, not unique to ionic conductors including Na~-alumina and occurs in many different materials. On the other hand a model [13] that has been proposed for Hollandite is based on

K.L. Ngai / Low frequeno' responses o f ionic conductors

its one-dimensional nature and the assumption of a distribution of activated barrier energies A of a very special form of W(A)=W exp(-5/kT ) for < O . A
[8]

[9] [I0] [11] [12] [13]

[14] [15]

[16]

[17]

[18] [19]

[20]

[21]

[22] [23] [24] [25] [26] [27] [28] [29]

References [1]

[2] [3]

[4] [5] [6]

[7]

Fast ion transport in solids, Vashista, P., Munday, J.N. and Shenoy, G.K. (eds.), (NorthHolland, N.Y., 1979). Radzilowski, R.H., Yao, Y.F. and Kummer, J.T., J. Appl. Phys. 40 (1969) 4716. Strom, U., Taylor, P.C., Bishop, S.G., Reinecke, T.L. and Ngai, K.L., Phys. Rev. BI3 (1976) 3329. Anderson, A.C., Ref. 1, p. 255. Grant, R.J., Hodge, l.M., Ingram, M.D. and West, A.R., Nature 266 (1977) 42. Ingram, M.D., J. Am. Ceramic Soc. 63 (1980) 248; Grant, R.J. and Ingram, M.D., J. Electroanal. Chem. 83 (1977) 193. Engstrom, H., Bates, J.B. and Wang, J.C., Solid State Comm. 35 (1980) 543.

[30] [31] [32]

[33] [34] [35]

33

Jonscher, A.K. and Reau, J.M., J. Mater. Sci. 13 (1978) 563; see also Fontanella, J.J., et al. (this conference). Strom, U., von Schickfus, M. and Hunklinger, S., Phys. Rev. Lett. 4!I (1978) 910. Almond, D.P. and West, A.R., Phys. Lett. 69A (1978) 130. Barker, A.S., Ditzenberger, J.A. and Remeika, J.P., Phys. Rev. BI4 (1976) 535. Deori, K.L. and Jonscher, A.K., J. Phys. C, 12 (1979) L289. Bernasconi, J., Beyeler, H.U., Str~ssler, S. and Alexander, S., Phys. Rev. Lett. 42 (1979) 819. Jonscher, A.K., Deori, K.L. Reau, J.M. and Haoli, J., J. Mater. Sci. 14 (1979) 1308. Barmatz, M. and Farrow, R., in 1976 Ultrasonics Symposium Proceedings, de Klerk, J. and MeAvoy, B. (eds.) (IEEE, N.Y., 1976). Simmons, J.H., Franklin, A.D., Young, K.F. and Linzer, M., J. Am. Ceramic Soc. 63 (1980) 78. Walstedt, R.E., Berg, R.S., Remeika, J.P., Cooper, A.S., Prescott, B.E. and Dupree, R., Ref. I, p. 355. Boilet, J.P., Zippiroli, L., Delplanque, G. and Jerome, L., Phil. Hag. 32 (1975) 343. Walstedt, R.E., Dupree, R., Remeika, J.P. and Rodriques, A., Phys. Rev. B15 (1977) 3442. Villa, M. and Bjorkstam, J.L., Phys. Rev. B22 (1980) 5044; Ref. I, pp. 289-296; Mag. Resonance Rev. 6 (1980) 1. Walstedt, R.E., Berg, R.S., Remeika, J.P. and Dupree, R., Solid State Comm. 37 (1981) 209. Bjorkstam, J.L., Ferloni, P. and Villa, M., J. Chem. Phys. 73 (1980) 2932. Berthier, C., Ref. 1, p. 171. Kurtz, S.R. and Stapleton, H.J., Ref. I, p. 319. Wolf, D., J. Phys. Chem. Solids 40 (1979) 757. McWhan, D.B., Varma, C.M., Hsu, F.L.S. and Remeika, J.P., Phys. Rev. BI5 (1977) 553. Wang, J.C., Gaffari, M. and Choi, S., J. Chem. Phys. 63 (1975) 772. Ngai, K.L., Ref. I, p. 203; Comments Solid State Phys. 9 (1980) 127-147. Huberman, B.A. and Boyce, J.B., Solid State Comm. 25 (1978) 759. Boyce, J.B., Mikkelsen, Jr., J.C. and Huberman, B.A., Solid State Comm. 29 (1979) 507. Riehards, P.M., S o l i d State Comm. 25 (1978) 1019. Howell, F.S., Bose, R.A., Macedo, P.B. and Moynihan, C.T., J. Phys. Chem. 78 (1980) 639. Strom, U. and Ngai, K.L., this conference. Patterson, G.D., Stevens, J.R. and Lindsey, C.P., J. Macromol. Sci.-Phys. BI8 (1980). Bucaro, J.A., Dardy, H.D. and Corsaro, R.D., J. Appl. Phys. 46 (1975) 741; Lai, C.C., Macedo, P.B. and Montrose, C.J., J. Amer. Ceramic Soc. 58 (1975) 120.