Dielectric Relaxation in Alkali Metal Oxide ...

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Dielectric Relaxation in Alkali Metal Oxide Conductive Glasses Studied by Complex Impedance Measurements BY D. RAVAINE,J. P. DIARDAND J. L. SOUQUET* Laboratoire de Cinitique Electrochimique Minkrale (ERA no 384), Ecole Nationale Sup6rieure d’Electrochimie et d’Electromktallurgie, B.P. 44 Domaine Universitaire, 38401 Saint Martin d’Heres, France Received 4th February, 1975 The measured impedance of vitreous materials plotted in the complex plane permits the representation of their electrical properties by an analytical equation of this impedance as a function of frequency and three characteristicparameters which can be determined with accuracy. Electrical properties of the same glasses under the same experimental conditions have already been studied and in this paper the value of impedance diagrams as compared to the usual representations of experimental results is studied. These results are interpreted by comparing the material to a simple parallel combination of a capacitor and a pure resistance. It is shown that such an arbitrary equivalent circuit gives a complex permittivity whose frequency dependence, in the case of ionic conductors, cannot be deduced with accuracy from the experimental data. In particular it is shown that the existence of a low frequency relaxation deduced from these conventional representations is entirely dependent on the accuracy of measurements.

The complex permittivity E* of a dielectric material is given by E* = c’ -jd’ where is the relative permittivity and E” the dielectric loss. Oxide glasses containing alkali metal oxides are ionic conductors. The electric current is carried entirely by alkali metal ions as has been shown by several author^.^-^ The usual method of studying the dielectric properties of vitreous ionic conductors is to subtract a contribution ( T / C O E ~from E”, where (T is the d.c. conductivity and E~ the permittivity of free space. The frequency dispersion of the remaining part of the complex permittivity E, = E’ -j(c” - ( T / o E ~ is ) used in characterising the dielectric properties. Experimental study of E’ and of E: = E ” - O / O E ~ , has led several authors 4-10to measure dielectric relaxation at audio and radio frequencies (generally between 1 kHz and 100 kHz). Numerous correlations between the dielectric relaxation parameters and the d.c. conductivity of glasses have produced general agreement that in the silicate glasses dielectric relaxation and conductivity arise from the same ionic motion.4*6* 11-14 For this reason the bulk electrical properties of glasses have recently been treated in terms of formalisms which avoid artificially dividing the E” data into a conductive part and a dielectric part. These formalisms are the complex electric modulus M* 15-17 and the impedance formalism 2 * , 1 8 the latter having been used previously for the study of solid polycrystalline ionic conductors. * 9-21 Both formalisms are complementary and very similar and the results obtained in the impedance representation are easily reduced to the modulus representation since Z* = M*/joC, where C,,is the capacitance of the cell without the sample.22 E’

COMPLEX IMPEDANCE OF GLASSY ELECTROLYTE

In an earlier paper l8 we have shown that the complex plane representation of the impedance of symmetrical cells of the type metallalkali metal conductive glass1 1935

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DIELECTRIC R E L A X A T I O N I N G L A S S E S

metal can be used to characterize the electrical properties of glassy electrolytes by overcoming the effect of electrode polarization. For frequencies within the range 5 Hz to 500 kHz and for temperatures between 50°C and 450°C,the experimental points, plotted from measurements at a given temperature, form a portion of a circle which is characteristic of the electrolyte. When these arcs are not deformed by the superposition of curves representing electrode polarization, the experimental points can be fitted to a circle in the following two ways (i) by calculating the mean deviation of the experimental points from the nearest corresponding points on thearc; the analysis of 20 experimental curves has shown that the ratio of this mean deviation to the circle radius was less than 0.9 % and (ii) starting from two experimental points, taken as a base, we have measured the angles defined by the segments joining these base points to other experimental points. These angles were found to be equal to within 1.7 % showing that the curve between the two base points is in fact a circular arc. In both cases, the relative deviation can be attributed to the error of the measuring apparatus (Alcatel Impedancemeter type 2531). If these arcs are extrapolated to high frequencies, they cross the origin within the same limits of accuracy of measurement. The circle radius which extends to the origin defines an angle orn/2 with the real axis. Fig. I gives an illustration of the results obtained for a borosilicate glass.

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I Z l C O s 4 (0) FIG.1 .-Experimental curves obtained for different temperatures with the cell : Pt10.19K20-0.37 B203-0.44 Si021Pt. Frequencies are expressed in kHz. *, 400°C; 0, 370°C ; 0 , 352°C.

The analytical equation for such an arc in the complex plane is : where cu is the angular frequency and Zo is the intercept on the real axis of the zero frequency extrapolation of the experimental arcs. Its value is the ohmic resistance of the electrolyte and can be used for determination of the conductivity G of the electrolyte.'** 23 The analysis of distribution in frequency of the experimental points over any isothermal circular arc was carried out by a geometrical method identical to that applied by Cole and Cole to dielectric constants.24 This analysis shows that f(co)

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D . R A V A I N E , J . P . D I A R D AND J . L . S O U Q U E T

is equal to the expression (co/wO)'-' where coo is the angular frequency corresponding to the top of the portion of circle. The analytical expression for the complex impedance of the electrolyte is therefore Z* = Z,/[l -t- (jo/coo)l-aJ. (2) Apart from illustrating relationship (2), this geometrical method affords a precise determination of the angular frequency coo and the parameter a. For our experimental conditions the value of r appears to be temperature independent. In contrast 0 and oohad activation energies of similar size.'' The values of the characteristic parameters defined for different glasses are represented in table 1. TABLE 1.--PROPERTIES

OF THE GLASSES STUDIED

(4 mole fraction composition

0.33 L i 2 0 0.40 Na20 0.33 Na20 0.30 Na20 0.25 Na,O 0.22 Na20 0.21 Na20 0.14 Na,O 0.33 K2O 0.25 K20 0.20 KzO

o!R-l cm-1 at 100cC

C Ln -

2

0.66 SiO, 0.60 SiOz 0.66 SiO, 0.70 SiO, 0.75 SiO, 0.78 SiO, 0.79 SiO, 0.86 SiOz 0.66 SiOz 0.75 SiOz 0.80 S i 0 2

9" 10.5"

16" 15" 21" 12" 16.5" 12" 16.5" 18" 20.5"

4.68 x 10-7 3 . 0 8 ~1W6 10--7 5.86~ 2 . 6 3 ~10--7 3.93 x 1.57x lops 1.40~

7 . 3 4 ~10-9 3.07 x 10-7 2.11 x I .05 x

tuo/kHz at 100cC

224 1680 333 164 19.1 10.2 7.0 6.7 128

(b) mole fraction composition

0.19 K 2 0 0.19 K 2 0 0.19K20

0.19 B 2 0 3 0.37 B 2 0 3 0.41 B 2 0 3

0.62 SOa 0.44 Si02 0.40Si02

an 2

o/R-1 cm-1

13.5" 13"

2 . 2 7 ~lo-' 1 . 2 9 lo-* ~ 3.1 x lo-'

10"

at 300°C

9.71

0.69

activation energy of' conduction1 kcal mol-1

activation energy o f kcal rnol-1

15.3 13.9 14.5 15.3 16.2 17.1 17.1 16.8 14.8 16.3 19.0

14.2 15.2 15.8 16.9 17.1 17.6 17.1 14.5 16.7 19.6

15.6

activation activation energy of energy of oJo/kHz conduction/ OO/ at 300°C kcal rnol-1 kcal rnol-1

16.6 11.2 23.9

23.5 26.7 24.3

23.6 26.9 24.3

M E A S U R E M E N T O F D I E L E C T R I C P E R M I T T I V I T Y OF V I T R E O U S I O N I C CONDUCTORS

Two techniques of measurement are used for the determination of E$ defined above. (1) A sinusoi'dal excitation is applied to the sample and its conductance G and susceptance B are measured. A d.c. measurement gives the resistance R of the glass. The real and imaginary parts of the complex permittivity are obtained by :

This interpretation of the measurements assumes that the glass can be treated as a simple parallel combination of a capacitor with complex relative permittivity E: and a pure resistance R,

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DIELECTRIC R E L A X A T I O N I N GLASSES

(2) When a constant potential is applied to the material, the transient current is measured against time (charging or discharging current). Here again, the separation, for the interpretation of the measured charging current, into an adsorption current (representative of dielectric relaxation processes) and a steady-state current (representative of conduction processes) means that the material is considered as an electrical circuit with a resistance and a capacitance in parallel. The interpretation of the desorption current by means of dielectric relaxation processes alone is based on the same equivalence. On this basis the approximate method of Hamon 2 5 or more accurate transform methods 2 6 * 27 can be used to determine G (charging current) or LO Coe:(w) (discharging current). For glasses, several authors 4-10 have observed that the plots of 8: against w or log co show a maximum characteristic of dielectric absorption. However, it has been noted 4 * 6 * 8 * 2 8 that the measurements of cz are subject to appreciable inaccuracy close to or below this maximum. In some cases the expected maximum is not found 6*2 9 or the value of R has been adjusted to obtain a maximum.29 The analytical equation (2) for complex impedance of the material which we propose may be used to recalculate the previously published results on the variation of dielectric permittivity of glass against frequency. We describe this and compare our results with results published elsewhere. In the Conclusion is a criticism of the use of the dielectric permittivity representation applied to conducting glasses. We propose briefly a new method for studying the electrical properties of such conductors. This method has been described more fully in an earlier paper.lg C A L C U L A T I O N OF E' A N D E: F R O M THE A N A L Y T I C A L I M P E D A N C E E Q U A T I O N

We use relationships (3) and (4) to compare the dielectric permittivity representat ion with the impedance representation. By introducing the analytical impedance equation (2) which we have shown to be in accurate agreement with a.c. results we obtain : = a sin an/2 E : ( O ) = -(G-$) 1 WCO

Eo

c o p

where 0 is the conductivity deduced from Zo. The curves oft$ against log o,calculated from the analytical expression for values of the parameter az/2 deduced from the measurements (table 1) show no maximum (fig. 2, curves a and a'). However, the determination of 2, (in our case) or of electric resistance R (by direct current measurement) contains, as for all measurements, a certain degree of inaccuracy. If we call the measured value of 2, (Zreas = 2, & AZo) &:(a) = 1 oc, Two cases exist. (1) When < Zo, a maximum value of i$ is found correspondingto an increasing value of frequency as the relative error increases. Curves b, c, d, e on fig. 2 illustrate the plot of &$(a) against log w / w o with a negative relative difference between ZCeas and Z o of between 0.5 % and 2 %. The angular frequency w,,, of the maximum can be defined by the analytical equation of these curves :

Zreas

-(.-)'.

,reas

Zreas

(7)

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D . R A V A I N E , J . P . D I A R D A N D J . L. SOUQUET

When h&;(U))


--

- 0, we have

(2) 1-a

1

=

1939

AZO

X-

6co a sin an12 zFeaS' The curves on fig. 2 and 3 show how a very slight variation of Z f e a smodifies the

values of ~2 significantly. It is obvious that all the expressions containing this term will be influenced by the sensitivity of ~2 to the magnitude of A&. Thus on fig. 3 the variations of t$j against &'(a)(Cole and Cole diagram) are shown. The analytical equation for these curves is obtained by cancelling co between relationships (5) and (6) (curve a) or between relationships (6) and (7) (curves b, c, d, e). (2) When Zgeas 2 Zo, E& becomes

1% W l W O FIG.2.-Curves a and a' : E; against log w plotted from relationship (2). Curve a :uv/2 = 15' ; curve a' :ar/2 = 20". Curves b, c, d and e : the curve a at different negative values of relative error AZo/Zo = 0.5 %; 1 %; 1.5 %; 2 %. 1

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5

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1s

25

20

30

35

c'

FIG.3 . 4 o l e and Cole diagrams : curves a, b, c, d, e correspond to the curves a, b, c, d, e, on fig. 2.

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DIELECTRIC R E L A X A T I O N I N GLASSES

It is easily seen that the curves &$(a), a sum of two decreasing functions of u), do not show any maximum and the values of e$(w) decrease monotonically with increasing frequency. COMPARISON W I T H P U B L I S H E D R E S U L T S

(1) The previous section enables us to understand why a maximum in the plots of ,$(log ~ r ) ) is not always found. This fact was experimentally noticed by Isard and Prod’homme.* We have seen that the existence or not of a maximum is dependent on the accuracy of measurement of the d.c. conductivity. (2) In the case Zomca< s Zo, curves b, c, d and e plotted on fig. 2 and 3 show great similarity with those published by authors who find a maximum. We observe in particular : (a) a significant broadening of the peak in the range of frequencies studied, generally interpreted by a wide dispersion of relaxation times; (b) an asymmetry of these curves with reference to the maximum frequency and characterized by rapid falling away at low frequencies ; (c) in the Cole and Cole diagrams, the curves shown can be roughly compared to flattened arcs, with shapes particularly sensitive to the magnitude of A&. The curves shown on fig. 3 for different values of AZo entirely confirm the observations of Prod’homme and G ~ i d e e . The ~ ~ identification of these experimental curves with circular arcs in a Cole and Cole diagram is very approximate. It by no means allows an easy verification of the Cole and Cole equations which, in any case, are not compatible with the asymmetry of the experimental curves €:(log a). Neither is there a good fit to the experimental data for any value of p in the DavidsonCole equation.3o CONCLUSION

(1) The calculation shows that the determination of E:(u) from our measurements with alternating currents is particularly sensitive at the absorption frequency and lower frequencies to the accuracy of the d.c. measurement of the resistance of the electrolyte. This fact is explained by the expression used to obtain d-j:

When the frequency decreases, G(u) approaches 1/R. Consequently, when o deis the product of an increasing function in terms of u)-l and a decreasing creases, &;(a) function [G(w)- 1/ R ] . Inaccuracy in the second term will therefore be amplified to an increasing extent. Several authors have noticed this problem 6 * 34* 35 which also exists for measurements of transient currents. In that case, the inaccuracy which affects the measurement of absorption or desorption current (decreasing function) for long durations is amplified by the term 0 - l (increasing function) and gives rise to the same deformations as illustrated on fig. 2 and 3. Consequently, it would be out of place to formulate a theoretical interpretation of the variation of E$(CO) and thus to establish comparisons with curves that are so sensitive to experimental errors. The justification of this remark is confirmed by the great variety of models deduced from this one type of repre~entation.~. 1 1 * l2. 31-33 (2) We must stress the arbitrary nature of the equivalence established between the electrolyte and a circuit with pure resistance and capacitance in parallel. It leads to a separation of conductive and dielectric phenomena and governs the choice of the manner in which the results are presented and the interpretation that follows. The formalism of complex electric modulus and complex impedance avoid this difficulty. (3) The results which were presented in the original impedance representation and the characteristic parameters which we defined are independent of any presumed 43

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D . R A V A I N E , J. P . D I A R D A N D J . L. S O U Q U E T

1941

electrical diagram. We have also shown that the analytical equation for complex impedance has been determined with good accuracy for a large number of samples (see table 1). For this reason we have proposed in an earlier paper a theoretical model which can be used to obtain this equation without having recourse to the usual hypothesis of independence of electric and dielectric phenomena. It is not the purpose of this paper to discuss this model, nevertheless we can say it is derived from (2) for which an equivalent electric circuit can be set up composed of an infinite number of circuits in series, each made up of a resistance and capacitance in parallel. Each of these micro-circuits has a time constant z and the constant a is characteristic of the distribution of the time constant z around the most probable value ro = l/oo. The symmetry of the distribution, expressed as a function of the variable term log z/z, as well as knowledge of the microscopic structure of glasses, enables us to interpret this dispersion assuming a distribution of activation energies of conduction. This interpretation suggests that these glasses can be considered, from the point of view of their electrical properties, as homogeneous dielectric materials and heterogeneous conductors. The authors thank Dr. H. J. L. Trap of the International Commission on Glass for supplying borosilicate glass samples and the Society Sovirel for supplying sodosilica glasses. C. A. Kraus and E. J. Darby, J. Amer. Chem. Soc., 1922,44,2783. K. K. Evstropev and G. A. Pavlova, Trudy Leningr. Tekh. Inst. Im Lensoveta, 1958,46,49.

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