J. CHEM. SOC. FARADAY TRANS., 1994, 90(14), 2065-2070 Conductivity and Dielectric Relaxation in Hydrated Fused Salts G. P. Johari, D. A. Wasylyshyn and S. K. Jain Department of Materials Science and Engineering, McMaster University, Hamilton, Ontario, Canada L8S 4L7 The dielectric properties of Ca(NO,), . 2.9H20,Cd(NO,), -2.37H20and (Ca + Cd)(NO,), -2.92H20have been measured in the supercooled liquid state immediately above their respective calorimetric glass-transition tem- peratures. Over the frequency range 12-105 Hz, their permittivity and loss spectra are featureless. By subtrac- ting the contributions from electrode impedance.'and dc conductivity, the remaining E*, which is due to a polarization process, is described well by a Davidson-Cole relaxation function with /? in the range 0.26-0.40 (f0.05).The activation energies for conduction are 200-220 kJ mol-'.p decreases with decreasing tem- perature, as does the contribution to permittivity from orientation polarization. The latter, which differs from the behaviour of most liquids, indicates a decrease in the number densities of the molecular and ion-pair dipoles and/or their dipole moments with decreasing temperature. Analyses of the data in E* and M* ( =E*-') formal-isms give the same results. The former is direct, although it requires knowledge of the exponent for the elec- trode impedance. The data also fit an asymmetric distribution of conductivity relaxation times at least as well as in the literature, but this fit appears to be misleading. In the basic macroscopic theory of dielectrics, Cole' has important in determining their behaviour.The data have also reminded us that electromagnetic measurements alone do not been described by a distribution of conductivity relaxation distinguish between conduction and polarization currents, as time^,^.^'^ but this significantly underestimates M* at high only the total current Jtotal= J, + aP/at, appears in frequencies. Part of the data on Ca(NO3),.2.9H2O and Maxwell's equation. Any separation of conduction current J, Cd(NO,), * 2.37H20 has been briefly reported before in a dif- and polarization current aP/dt should be made in other ways, ferent context," where an analysis in terms of a combination by, for example, justifiably describing the free electric charges of conductivity and polarization processes was deferred.' (ions) as mutually exclusive to bound ones (dipoles) on a Most of the dielectric data are new.certain timescale. (The timescale distinguishes a permanent ion pair from a transient ion-pair dipole.) The procedure thus introduces a dc conductivity, go, with J, = a,E and a polar- Experimental ization P = xoE where xo is the electric susceptibility and E The details of the equipment and measurement technique the electric field. Thus E/Jtota,= (l/a,)/(l + icoxo/ao),with a have been described before.".' Briefly, the permittivity, E', Maxwell's relaxation time formally equal to xo/o,, and co as and loss, E", were measured by means of a GenRad 1689 the angular frequency of the electric field. Variation in the digibridge, over the frequency range 12-10' Hz.The glass- form of the denominator of this equation leads to various transition temperature, %, was measured by differential scan- shapes of the spectra. Virtually all analyses of the dielectric ning calorimetry at a heating rate of 20 K min-'. The data of ionic solids2 (inorganic, organic or polymeric) are cur- uncertainties in the measurement of E' and E" are 0.1% and rently done without dividing the Jtotalinto its components. 0.5% respectively, of temperature is kO.1 K and of Tp is f1 This has led to a general description of the dielectric data, K. even for materials containing molecular dipoles, in terms of a All chemicals were AnalaR (BDH) grade. The electrolytes distribution of conductivity relaxation times arising from a were heated to a temperature 20-40°C above their melting stretched exponential decay of the electric field under the points and kept for 2 h.This caused a loss of water of crys- constraint of a constant displacement tallization. Their water content was determined gravimet- In recognition of the presence of molecular dipoles in ionic rically and by Karl-Fischer titration. The samples used in substances, and of the requirement from the law of chemical this study were transparent and remained in their super- equilibrium that ions and dipolar ion pairs coexist in fused cooled state at room temperature for several days without salts,' the analysis of M* is more appropriately done in terms crystallization.of two processes.6 It has been demonstrated6 that the dielec- tric properties of ionic solids are attributed to: (i) a Maxwel- Results and Data Analysis lian conductivity, i.e. with a single relaxation time and (ii) a dipolar relaxation with a Davidson-Cole type asymmetric Representative data on conductivity, 0, and permittivity, E', distribution of relaxation times.' Nevertheless, complex per- spectra of Ca(NO,), -2.9H20 at different temperatures are mittivity, &*, can be directly analysed by computing the inter- shown in Fig. 1, of Cd(NO,), 2.37H20 in Fig. 2 and of (Ca facial or electrode polarization effects and the dc conductivity + CdNNO,), -2.92H20 in Fig. 3. None of the spectra show a from the measured data.8.9 For several network oxide low-frequency plateau or shoulder in E' that could be associ- this analysis gave the same results as the M* ated with the limiting, low-frequency value, E~ The dc con- gla~ses,~ .analysis.6 ductivity plateau is discernible as an intermediate-frequency Here we report the dielectric properties of three electrolytes shoulder for some temperatures (as for example at 251.0 K in containing water (of crystallization) as molecular dipoles and Fig. 2; it is more clearly discernible on a linear scale plot of interpret these properties in terms of both E* and M* formal-a). Contributions from electrode polarization effects domi- isms. These substances are weak electrolytes, i.e. they contain nate at frequencies below this shoulder, and contributions dipolar ion pairs in thermal equilibrium with ions.The from conductivity and/or dipolar relaxation effects dominate results show that both conduction and polarization are at frequencies above it. , +\ +, +\ (c) -9 -5 -6 h-I E q -7 b Y Q,-0 -8 , v I I/(111 I 1 1 III'I' ' '')Ill" ' ''1IIII I 1"1111 i r88111a i iilllll I isr,sA-910 100 1000 10000 100000 frequency/Hz Fig. 2 Variation of ac conductivity and permittivity against fre- quency for Cd(NO,), .2.378,0 at different temperatures/K: (a) 239.1, (b)243.7, (c) 247.1,(d)251.0 -5 -6 rh I E q -7 b v Q, --8 -9 I 10 100 1000 10000 100000 frequency,/Hz Fig. 3 Variation of ac conductivity and permittivity against fre- quency for (Ca + CdXNO,), 2.92H ,O at different temperatures/K: e (a)234.5,(bl 238.3,(cl 242.1 Following Cole and co-w~rkers,'*~,~ we represent the elec- trode impedance, Zel,as a complex function, Z,*,= Zo(io)-", where 2, is the characteristic of the electrode/dielectric inter- face, and the exponent n is equal to 0.5 in the present case.This is equivalent to a 'constant-phase element' in series with the bulk dielectric properties of the material. For an electrode impedance, Zel, in series with a dielectric sample, the measured conductance, G,,,, in S, and permittivity, &beasare given byY8 Gmeas (GIC,) + im' (1)CO + IWE;,,, = 1 + Z,*,[(G/C,) + ico~']C, (this follows from two admittances, YT = G + ios'C, and Y; = l/Z,*,, being taken in series).In eqn. (l), C, is the capacitance of the empty cell (its value, which differed for dif- ferent measurements, was between 12 and 15 pF), o the angular frequency, E' the true or bulk permittivity (from dipolar reorientation) of the sample and G its true or bulk conductance, which is the sum of the dc and ac (or dipolar) contributions. For frequencies when G % OE'C,and lZ,*,I= 2, Go-" 4 1, eqn. (1) is expanded as a Taylor series, and after truncating at the first order it may be written as, co co and (3)L \ 1 L0-l J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 These are transformed to the equations, used here, EL,,, = &ic+ Eiip where the conductivity, a = a, + adip = e,G/C,. Here, e, is the permittivity of vacuum (=8.8514 pF m-I), a, and adip are the dc and dipolar contributions to a and &iC and &Gip are those to E".The last terms on the right-hand side of eqn. (2)-(6) represent the contribution from interfacial impedance (the units of Z, are formally i2 Hz" or S-' s-"). At fre- quencies low enough for ddip to be negligible and &Lip to be equal to the equilibrium value, E,, a,,,, varies as w-", and EL,,, as m-("+'), and for n = 0.5, as found here and in earlier eqn. (4) becomes : Before proceeding further, the validity of three conditions, namely, (i) G >> we'C,, (ii) 2, Gw-" + 1 and (iii) the impor- tance of terms higher than first order [these terms have been neglected in order to obtain eqn. (2) and (3) by Taylor expan- sion of eqn.(l)] was examined for the sets of data used for the calculations. For the lowest-temperature data sets, the conditions were satisfied by only one or two lowest-frequency data. Eqn. (1) was also Taylor-expanded to fourth-order terms to see if the inclusion of such terms would increase the number of data points needed to determine a. and 2,.Cal-culations with simulated (not measured) data showed that this did not increase the number of low-frequency data points needed, and that the effect of inclusion of such terms remained insignificant up to the frequency range of 1 MHz, which is higher than the frequency used here. The calcu- lations were done as follows. For a pair of a,,,, values at frequencies miand wj (j > i), ameas(mi) -ameas(mj) lr co= { 2, a;[ cos( -) -]}(w; 1/2 -0;'j2) (8)4 eo For four sets of wi or oj,there are six pairs of mi, wj.From these, an average value of 2,'; was calculated by using eqn. (8). For the same values of mi, a. was then calculated by substituting the Zoo; value into eqn. (7). 2, was then deter- mined by substituting go into Zoo;. The calculations were refined by independently adjusting a. and 2, by infinitesimal amounts until a. in eqn. (4) and &bip in eqn. (6)became constant with decreasing frequency at the low-frequency end of the spectrum. It should be pointed out that according to eqn. (6), as w -0, E' from electrode impedance, or (EL,,, -&hip), approaches co for all positive values of n. Thus, relatively small errors in the estimated a, and 2, will cause -&iip)to become positive (instead of remaining zero) as a+ 0.It did become positive in some cases, but only after a plateau value of &kip or E, had been reached. Few analytical procedures could entirely avoid this occurrence. We also realize that the measured values as such can be fitted to eqn. (1) by using computational fitting algo- rithms, a step that needs to be considered and justified in a manner that removes the subjectivity of parameter fitting. We chose to separate the contributions so that the physical meaning of each parameter could be brought to the fore in an old-fashioned manner. The data obtained from ca. 30 sets of isothermal spectra, particularly those which gave the needed information, are summarized in Table 1.Representative plots for the analysis of a,,,, into adc, dintand adipcontributions, and of E' into &Lt and &iipcontributions are shown in Fig. 4. Similarly, repre- sentative plots for the analysis of EL,,, and EL,,, into dc, inter- facial and dipolar contributions are shown in Fig. 5. The &, ~~and E& values obtained from the analysis are plotted in a complex plane in Fig. 6. These show a skewed arc or a depen-dence according to the empirical Davidson-Cole relaxation function,' Eo -E, (9)E* = E, + (1 + ioz,)B with values of e0, E,, p and fo (&"-peak frequency) sum- marized in Table 1. Other empirical stretched exponential relaxation functions such as the KWW Havriliak-Negami's empirical eq~ation'~with its extra parameter 1 -a, and Jonscher's equation" also seemed to fit the data.In view of the large errors associated with the resolved dipolar contributions (e.g. needing to subtract 100.09 from 101.00 to obtain the dipolar contribution to E" at 20 Hz), it seemed inappropriate to use the data for a comparison of the relative merits of the various equations. For simplicity we use eqn. (9). Fig. 7 shows an Arrhenius plot of the dc conductivity and the dielectric relaxation rate of eqn. (9) for the various glasses. The plots of a. are linear over the entire temperature range, but thefo data deviate from a straight line considerably at temperatures wheref, <lo0 Hz. The seven sets of E' and E" data at frequencies < 100 Hz have consider- able measurement errors and were not as reliable for deter- mining &iipand &iipafter subtraction of the very large contributions from electrode impedance and dc conduction particularly at the low-frequency end of the complex-plane plots where fo lies.The values of fo < 100 Hz are therefore meaningless. The values of AE are also approximate, but /3 values at such temperatures are precise. Practical corrections Table 1 Dielectric relaxation and other parameters and T, of several fused electrolytes" 252.9 Ca(NO,), * 2.9H,O 5410 7.8 (T, = 237 K) 32.0 5.0 0.28 249.2 1490 7.7 29.0 0.9 0.28 246.0 373 7.6 27.4 0.25 0.30 242.6 85.0 7.5 23.3 0.10 0.36 239.1 20.0 7.4 15.8 0.07 0.30 236.5 8.45 7.1 12.7 0.05 0.26 Cd(NO,), .2.37H20 (T, = 233 K) 25 1.o 1240 5.0 16.0 0.8 0.40 247.1 189 4.7 14.8 0.2 0.38 243.7 37.0 4.7 11.6 0.10 0.42 239.1 6.35 4.5 7.5 0.05 0.30 (Ca + Cd)(N0,)2* 2.928,O (7''= 235 K) 246.7 3910 7.4 26.0 5.0 0.30 242.1 663 7.3 24.0 0.7 0.28 238.3 115 7.2 21.6 0.15 0.28 234.5 25.0 7.0 16.0 0.07 0.28 ;* Errors are: *5% in 0,; kO.1 in E~ k0.5 in c0; k0.05 in fl and 20% info (= 1/2nz,). 2068 J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 3020 1 1 -lo{ I -121 I 1 I111111100 1 I (""" ' ' 'I'"" ' '")IIJ10 lo00 10000 100000 f req uency/H z Fig. 4 Analysis of measured ac conductivity (0)into dc (-), interfacial (+) and dipolar ( x ) contributions and measured permit- tivity (0)into interfacial (+) and dipolar (x) contributions for Ca(NO,), .2.9H20 at 239.1 K for an electrode impedance by an arbitrary (unmeasurable) quantity 2, become difficult whenf, approaches a range of frequency where the E' and E" data are less accurate. Discussion Dielectric properties of ionic materials and polymer electro- lytes are generally attributed to charge transport a10ne,~-~ and are analysed in terms of M*. ColeI6 has summarized the difficulties one encounters in such an analysis. Nevertheless, the analysis offers the advantage of conveniently reducing the overwhelmingly large contribution from interfacial polariza- tion and dc conductivity as it suppresses the features at the low-frequency side of the M* spectrum and accentuates those at the high-frequency side.For ionic materials and polymer electrolytes the M* analysis has led to two conclu-sions:2-4,6,10 (a) the dc conductivity is non-Maxwellian, i.e. instead of a single relaxation time it involves an asymmetric distribution of conductivity relaxation times according to a stretched exponential relaxation function,'2*'3 and (b)a faster relaxation process contributes to the dielectric properties, for the stretched exponential relaxation function underestimates the high-frequency data.3y4'6 Our analysis of the data is strictly in terms of a dc conduc- tion and dielectric or polarization process. It contrasts the commonly used approach, based on the KWW parameterPKWWfor dc conduction, in which a polarization process is assumed to be absent from the dielectric behaviour observed at 10-3-108 Hz, and/or Odc is not distinguished from ddip .2-4 It also differs from an earlier procedure6 in which the inter- facial polarization contributions to CT and E' were reduced to I" 9-\LU8-7-$ 6-yI3-\ ,+..' a h ooo frequency/Hz Fig.5 Analysis of measured E' and E" (0)into dc (A),interfacial ( +) and dipolar ( x ) contributions for (Ca + CdXNO,), * 2.928,O at 238.3 K virtually zero by converting E* into M*, although the results obtained there6 are identical to those obtained by the present method. The data can be analysed also in (total) complex imped- -7-(4 6-5-4-3-2-A-1-##--'+ + 0 A-I I I I 1 I I I I I I I I I -0451'") "i'"3 ++yt4--+-+.+2-1-/N+ \-'t 0 I I 1'1 I I I I I 1 I 'I I I I 1 I I 4 5 6 7 8 9 101112131415161718192021 2223 & Fig.6 Cole-Cole plots for (c) Ca(NO3),.2.9H,O (239.1 K), (b) Cd(N03),.2.37H,0 (243.7 K) and (a) (Ca + CdXN0,),.2.92H20 (238.3 K) J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 4, A + 3.95 4.00 4.05 4.10 4.15 4.20 4.25 4.30 103 KIT Fig. 7 Arrhenius plots of the dc conductivity and relaxation rate for (a) Ca(NO,), .2.9HZ0, (b) Cd(NO,), .2.37HZ0 and (c) (Ca+ CdXNO,), * 2.92HZ0.Data for relaxation rates below 100 Hz are meaningless as discussed in the text. ance, Z* (= 2'-iZ), formalism. A complex plane plot of Z* for substances shows several depressed circular areas.All three representations are derivable from a single set of measurements, but the identification and interpretation of the frequency of maxima or other features with characteristic times led to quite different answers. This raises questions on the relevance of an intercomparison of the derived quantities against other characteristic times (e.9. NMR, mechanical, calorimetric) in the same material, and whether a self-consistent interpretation of these quantities is meaningful in terms of molecular processes. An example of this is the dielec- tric behaviour of 14.2mol.o/o LiC1-propylene glycol mixtures, to which both conduction and dipolar processes contribute, and yet the data in M* formalism can be mistaken for con- duction process alone with a temperature-independent PKWW parameter." In this study also we anticipate a dipolar contri- bution to E* from water dipoles, yet the data can be fitted to dc conduction alone with a distribution of conductivity relax- ation times" implicit in the formalism for the stretched expo- nential decay of electric field due to ionic diffu~ion.~ The data in Table 1 and Fig.7 give an activation energy of 210 f10 kJ mol-' for dc conduction. The Arrhenius equa- tion should describe the dielectric relaxation rate as well, but deviations from this equation may occur, as evident in Fig. 7, owing to at least two effects: (1) analytical errors associated with the use of limited data, particularly at the low-frequency end, cause large errors in the calculations offo at low tem- peratures, and (2) the possibility that the effective size of ion-pair dipoles becomes smaller as the density increases on decreasing the temperature.This reduction in size would increase the relaxation rate. In the absence of independent information on the size of ion pairs, it is difficult to resolve the importance of the second effect. In our experience, the dc conductivity can be estimated reasonably accurately from the data over the limited frequency range. Subtraction of very large contributions to E' and E" from electrode polarization and dc conductivity, as mentioned earlier, prevents an accu- rate determination of the dielectric relaxation time. The dipolar relaxation may be seen as an a process, in con- trast with the modes of localized diffusion observed in silicateg.'' and other glasses6 (i.e.far below their TJ,whose E* has been similarly analysed.E~ decreases with temperature, which is opposite to that required by the Curie law, AE (or E~ -E,) -C/(T-To). E, also decreases with temperature and is attributed to the decrease in both the strength of the high-frequency, or low-temperature, relaxation already observed" and the infrared contribution to E' for ionic solids.' The distribution parameter P decreases with decreasing temperature, or equivalently the half-width of the spectrum increases. Most of the changes in the dielectric parameters seen in Table 1 are not unusual, but the decrease in the ALEwith tem- perature is remarkable.One framework for this analysis is based on the Kirkwood-Frohlich equation,20.2 where Ndis the number density of molecular dipoles, k, T the thermal energy, po is the dipole moment of an isolated mol- ecule and g is the dipolar orientational correlation factor. A decrease of AEwith temperature is caused by either an over- whelmingly large decrease in N, (which usually increases on cooling) or a decrease in g or both. There is also a further effect here which is related to the ion-pair dipole formation in weak electrolytes. In such cases po decreases with the distance of separation between the ions in an ion pair, which depends upon the density of the material. Furthermore, concentration of such permanent ion pairs may decrease or increase depending on the magnitude of the parameters for the ion s ion-pair association constant,22 KA=-4x~a~exp( Z,Z, 2) 3000 a&g k, T where Z1 and 2, are the ionic charges, e the electronic charge, N the Avogadro number, and a the ion-size param- eter.KA , or equivalently, the concentration of ion-pair dipoles, is sensitively dependent upon the product go T, since a is expected to remain constant with changing temperature. Without further information on the po of ion-pair dipoles, we tentatively conclude that the decrease in AEwith temperature is due to a combination of these effects. To calculate KA , it is necessary to know E~ of different melts, which can be deter- mined more accurately when the concentration of water is high. In that case the strength of the dipolar relaxation associated with ion pairs would become an independent measure of their concentration.Our aim here is to suggest that the effects observed should occur in most ionic melts. Finally, it is instructive to compare the three types of analyses of the dielectric data represented only in the M* for-malism, namely: (i) with a distribution of conductivity relax- ation times using the KWW parameter,2-4s'0 (ii) with a Maxwellian conductivity and dipolar rela~ation~**.~ and (iii) with a distribution of conductivity relaxation times,23 but using the Davidson-Cole parameter.' Fig. 8 shows represen- tative spectra of M" for the fused electrolytes fitted to: (i) KWW asymmetric distribution of ionic conductivity relax- ation timeslo and (ii) dc conductivity and dipolar relaxation J.CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 analysed, the electrode impedance distorts the shape of the low-frequency part of the M’ and M” spectra. M” O.O4I0.03 n 0.06 0.05 0.04 M” 0.03 0.02 0.01 C 0.04 1 I log (frequency/Hz) Fig. 8 M” spectrum showing ine Maxwellian conductivity relax- ation and dipolar relaxation in (a)Ca(NO,), * 2.9H20 (246.0 K), (b) Cd(NO,), .2.37H20 (243.7 K), and (c) (Ca + Cd)(NO,),.2.928,0 (238.3 K). Rectangles are the measured values. The low-frequency peak is due to Maxwellian conductivity (A)and the high-frequency peak due to a polarization process ( x ). The calculated values for a distribution of conductivity relaxation times are shown by a contin- uous line.(a) M, = 0.125 and PKWW= 0.50, (b) M, = 0.203 and BKWW= 0.52 and (c) M, = 0.132 and PKWW= 0.50. Note that the last calculations underestimate the values at high frequencies. process with parameters given in Table 1 (fit of the Davidson-Cole equation to the M” data23 is not given, for although it fits the data better than the KWW parameter, it too implies a distribution of ionic conductivity relaxation times and ignores polarization effects from dipolar reorientation). As is generally ob~erved,~-~.~ the former fit is unsatisfactory at high frequencies. The latter seems to provide a satisfactory description of the dielectric data. We recommend that for separating ionic from polarization cur- rents in dielectric materials, analyses can be done either in the E* (Fig.4-6) or M* (Fig. 8) formalism, bearing in mind that each has its weaknesses: the analysis of E* data requires an assumed value of n, and the M* data conceal the effects of electrode impedance. When simulated (not real) data are Conclusions The dielectric properties of several fused salts containing water of hydration have contributions from both the ionic and dipolar diffusion, or polarization of water and ion-pair dipoles. Although the data can be described also in terms of a distribution of conductivity relaxation times without regard to dipolar diffusion, such a description seems inadequate and misleading. Direct analysis of the dielectric properties by separating the conduction and polarization effects is less concealing of the various processes than the analysis done after conversion of the E* data into an electric modulus formalism.For inter-preting the dielectric properties of ion-containing materials, it seems preferable to use E* directly so that the effects of elec- trode polarization can be separated and meaningfully inter- preted. References 1 R. H. Cole, J. Noncryst. Solids, 1991, 131-133, 1125. 2 For reviews of the experimental data, see C. A. Angell, Chem. Rev., 1990, 40,523; Annu. Rev. Phys. Chem., 1992, 43, 693; J. Wong and C. A. Angell, Glass, Structure by Spectroscopy, Marcel Dekker, New York, 1976, ch. 11. 3 P. B. Macedo, C.T. Moynihan and R. Bose, Phys. Chem. Glasses, 1972,13, 171. 4 C. T. Moynihan, L. P. Boesch and N. L. Laberge, Phys. Chem. Glasses, 1973, 14, 122. 5 R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butter-worths, London, 1959. 6 G. P. Johari and K. Pathmanathan, Phys. Chem. Glasses, 1988, 29, 219. 7 D. W. Davidson and R. H. Cole, J. Chem. Phys., 1951,19, 1484. 8 J. F. Johnson and R. H. Cole, J. Am. Chem. Soc., 1951,73,4536. 9 R. H. Cole and E. Tombari, J. Noncryst. Solids, 1991, 131-133, 969. 10 S. K. Jain and G. P. Johari, Phys. Chem. Glasses, 1989,30, 135. 11 G.P. Johari, Polymer, 1986, 27, 866. 12 R. Kohlrausch, Ann. Phys., 1887,12, 393. 13 G.Williams and D. C. Watts, Trans. Faraday SOC.,1970,66,80. 14 S. Havriliak and S. Negami, Polymer, 1967,8, 161. 15 A. K. Jonscher, Nature (London), 1977, 267, 673; J. Muter. Sci., 198 1, 16, 2037; Dielectric Relaxation in Solids, Chelsea Dielec- trics Press, London, 1983. 16 R. H. Cole, Annu. Rev. Phys. Chem., 1989, 40,1; J. Noncryst. Solids, 1991, 131-133, 11 19. 17 K. Pathmanathan and G. P. Johari, J. Chem. Phys., 1991, 95, 5990. 18 P. N. Huang and G. P. Johari, J. Mol. Liquids, 1993,56,225. 19 R. P. Lowndes and D. H. Martin, Proc. R. SOC.London, Ser. A, 1970,316,351. 20 J. G.Kirkwood, J. Chem. Phys., 1939,7,911. 21 H. Frohlich, Theory of Dielectrics, Oxford University Press, Oxford, 2nd edn., 1958. 22 C. W. Davies, Zon Association, Butterworths, London, 1962. 23 K. Pathmanathan and J. R. Stevens, J. Appl. Phys., 1990, 68, 5128. Paper 4/00182F; Received 12th January, 1994