CuBr by impedance spectroscopy

.__ __ EB

CQa ELSEWER

SOLID STATE IONICS

Solid State Ionics 83 (1996) 191-198

Study of polycrystalline CuBr and the interface Cu 1CuBr by impedance spectroscopy S. Villain, M.-A. Desvals, G. Clugnet, P. Knauth

*

EDIFIS (URA CNRS 443), Faculte’ des Sciences de Marseille-St J&&me, Case 511, F-13397 Marseille Cedex 20, France Received 29 August

1995; accepted

10 November

1995

Abstract The electrical properties of polycrystalline copper(I) bromide were investigated between 20 and 430°C by impedance spectroscopy with copper electrodes. Extrinsic and intrinsic regions in y-CuBr and a domain of fast ionic conduction in B-CuBr are separated. Enthalpies of migration of copper interstitials (2.5 kJ/mol) and copper vacancies (50 kJ/mol) and the enthalpy of formation of Frenkel defects (160 kl/mol) are deduced. The phase boundary copper/copper(I) bromide can be represented by a parallel circuit of an interfacial resistance and a constant phase-angle (CPA) element. The interfacial resistance depends exponentially on temperature and is practically negligible above 350°C. Contributions of charge transfer resistance and space charge resistance are discussed. Prefactors of CPA elements depend also exponentially on temperature; they are compared with interfacial capacitances. Keywords: Cuprous bromide; Impedance; Interfaces; Frenkel defects

1. Introduction The copper(I) halides are well-known solid Cu+ ion conductors [l-3], but their electrical properties remain relatively scarcely investigated compared with the silver halides, perhaps because of the sensitivity to oxidation of copper(I) compounds. Recently, we reported the electrical properties of copper(I) iodide and of the phase boundary with copper electrodes [4]. We found exponential temperature dependencies for the resistance and capacitance of the phase boundary. The interfacial resistance was negligible for the high temperature (Y- and P-phases. Safadi et al. [51 investigated the electrical conductivity of copper(I) bromide by four-point experi-

* Corresponding

author. Fax: + 33-9128-8556.

0167-2738/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0167-2738(95)00237-5

ments to separate bulk and interfacial effects. Surprisingly, they reported the existence of a contact resistance with copper electrodes only for the high temperature p- but not for the low-temperature yphase. The electrical conductivity of the P-phase attained values higher 1 0-l cm- ‘, comparable to fast-ion conducting a-AgI. This is one order of magnitude

higher

than

for

WCUI

[4] and

than

the

to P-CuBr by Wagner and Wagner [l] from measurements at constant ac frequency (1000 Hz). On the other hand, tracer diffusion studies on copper(I) bromide published recently by Johansson et al. [6,71 showed only slightly higher diffusion coefficients for P-CuBr compared with CL-CuI. In this study, we adopt a complementary approach using impedance spectroscopy as a powerful tool to separate bulk and interfacial contributions. Several questions should be clarified concerning (i) the elecdata reported

S. Villain et al./Solid

192 12

34

5

State Ionics 83 (1996) 191-198

6

7

9

8

Fig. 1. Three-electrode cell (see text).

trical properties of polycrystalline copper(I) bromide, especially of the fast ion-conducting B-phase; (ii) the electrical properties of the phase boundary between copper electrodes and copper(I) bromide, especially the interfacial resistance and its temperature dependency; (iii) the thermodynamics of point defects in the bulk and near interfaces [8].

density before annealing ((4.90 &-0.05) g/cm3), determined by measuring the dimensions and weighing the pellets was more than 95% of the literature value

m. The experimental set-up was previously described [4]. Several cycles of evacuation and filling with dry argon were performed at room temperature, 110°C and 350°C to remove adsorbed traces of oxygen and moisture. All experiments were then performed under a stationary atmosphere of high-purity argon. The sample pellets were annealed in situ at 350°C for at least two hours with massive spring loaded copper electrodes (99.999% Strem) before the measurements. Impedance was measured between 20 and 430°C with an EG & G Electrochemical Impedance Analyzer Model 6310. The frequency range investigated was generally between 10-l and lo4 Hz, but the

2. Experimental Commercial products of different purity (CuBr: > 99.0% Fluka, 99.999% Aldrich) were used. Control X-ray diffractograms were in excellent agreement with ASTM data [9]. Sample pellets of 1.3 cm diameter and 0.1 to 0.4 cm thickness were prepared by compression under 7500 bar at room temperature; the pellets were then slightly polished. The specific

450.0

t

-I

350.0

i t

*

t,

t

,+

t

lt

t

+ .+

50.0

.-

. I 100.0

I

I 300.0

500.0 Zre

Fig. 2. Impedance parameters

kR,

t 1 900.0



of Cu/CuBr/Cu at 129°C without (+I and with CM> a dc bias (100 mV). Left arc (10
spectrum

R = 52.5

700.0

< 104): fit

S. Villain et al./Solid

domain of lower frequencies was explored in several experiments (e.g. Fig. 2). The ac amplitude was 5 mV rms. Two-point dc experiments were performed with a high-impedance electrometer (Keithley 617) and a constant current source (Keithley 220). We used two different electrochemical cells: (i) The two-electrode cell with copper electrodes of about 0.7 cm2 area was described previously [4]. A mirror-like electrode surface was obtained with different abrasives down to 0.25 pm. (ii) The threeelectrode cell is shown in Fig. 1. The counter electrode (1) of pure copper had a large surface and was fixed in the sample holder (2) in contact with the copper(I) bromide pellet (3). The sample holder was designed from a high-temperature resistent and electrically insulating alumino-silicate ceramic with low dielectric loss (stumatite). The reference electrode (4) and the working electrode (51, placed side by side, were from pure copper and had a contact radius of (0.05 ? 0.02) cm. The ratio of the contact radii of working and counter electrode is lower than l/12 and the deviation of the equipotential lines around the working electrode from spherical symmetry is thus about 25% [l 11. The temperature was measured with a chromel-alumel thermocouple (7) placed near the sample. A slight pressure was applied through an alumina tube (6) loaded with a stainless-steel spring placed outside the furnace (8) and held in position with a push-button (9). DC bias applied between working and reference electrode in impedance measurements was below 100 mV. 3. Results and discussion 3.1. Impedance

193

State Ionics 83 (1996) 191-198

spectra

A typical impedance spectrum of CuBr obtained with the three-electrode cell is presented in Fig. 2. The high-frequency semi-circle depends on the thickness of the sample pellet; it can thus be attributed unambiguously to the electrolyte bulk. In addition to the bulk arc, a second low-frequency semi-circle is observed, which does not change with the pellet thickness, but depends on the area of the electrolyte/electrode contact. The low-frequency arc can therefore be attributed to the interface copper/copper(I) bromide. Application of a dc bias (50-100 mV) between working and reference elec-

trode does not change the impedance spectrum significantly. The slight increase of the right arc shown in Fig. 2 is not confirmed in other experiments. Similar impedance spectra with two semi-circles are obtained with the two-electrode cell. The complex plane impedance diagram is typical for a solid electrolyte with one mobile species contacted by two non-blocking metal electrodes [12]. 3.2. Equivalent

circuit

In accordance with these results, the equivalent circuit of the Cu ]CuBr ICu cell can be described as a series switching of two parallel circuits of a resistance and a constant phase-angle element (CPA). The complex impedance of a CPA-element Q can be written with a constant prefactor Y and an exponent 12 [ 131: Ze = Y- ’ (j~)-~. The first parallel element (R, Q) represents the bulk of the solid electrolyte. The second parallel element of a resistance (R,,) and a CPA (Q,,) is characteristic of the interface copper electrode/copper(I) bromide. The electrical parameters of bulk and phase boundary electrolyte/electrode are determined using the NLLS fit program of Boukamp [ 141. The CPA characteristic of the bulk is near an ideal capacity (n > 0.91, does not depend on temperature and is slightly higher than the ideal geometrical value. The CPA-element representative of the interface electrode/electrolyte is less ideal (0.6 < n < 0.8), possibly because of the residual roughness of the electrodes. In two-point dc control experiments, metallic copper deposits with dendritic shape were formed between the electrodes and the electrolyte pellets. The resistances R(,,) measured in these experiments corresponded generally quite well to the sum of the bulk resistance R,,,_,,) and interfacial resistance Rintcacj determined from ac measurements, e.g. two-point experiments with a pellet of 0.266 cm thickness at room temperature: Rcdcj = 15.28 Ma, Rbulk+) = 13.13 Ma, Rintcacj= 2.11 Ma. 3.3. Conductivity mide

of polycrystalline

copper(l)

bro-

According to the different geometry of the two electrochemical cells used, the bulk conductivity u

S. Villain et al./Solid

194

State Ionics 83 (1996) 191-198

was calculated from the electrical resistance R using two different equations. With coplanar electrodes and rectilinear equipotentials [ 1.51:

4

3

*. i

g=Z/(

RA),

(1)

t 2

where Z is the thickness of the sample and A the area of the electrode/electrolyte contact. In experiments with three electrodes (cf. Fig. 1) and under the assumption of hemispherical equipotentials, the conductivity was calculated according to [15]: U= l/(R(2rrA)“‘)

t

t

(2)

with the electrode area A = 1/2(4~. r’). The uncertainty of the electrode radius I is at least 25%. The linear geometry was thus preferred for calculation of bulk conductivity. Values of conductivity of polycrystalline CuBr are plotted against reciprocal temperature in Fig. 3. The phase transition between the y- and the fast ion-conducting B-phase is indicated by a step increase of the conductivity. The transition temperature (376°C) is slightly lower than in the literature (385°C [l], 379°C [2], 384°C [lo]>, but the difference is near the experimental uncertainty. In the B-phase, the measured conductivity is in good agreement with the results of Safadi et al. [2] and Clemen and Funke [3] and higher than the data of Wagner and Wagner 111. We can conclude that in the B-phase, the conductivity is really higher than 1 a-’ cm- ‘. A kink in the plot at about 180°C marks the transition from extrinsic to intrinsic conductivity. This temperature is distinctly lower than in previous work. Enthalpies of activation of conduction are calculated from the slopes of the Arrhenius plot and are summarized and compared with the literature in Table 1. For P-CuBr, the calculation is not very accurate in view of the small number of experimental points, but the agreement with literature values is good. In the y-phase, the enthalpies of activation reported by Safadi et al. appear unrealistically high, especially in the “extrinsic” region. A comparison with the related copper(I) iodide [4] shows that the enthalpies of activation are nearly identical for the extrinsic region ( = 50 kJ/mol). It is interesting to notice that the value of B-CuBr, which has a wurtzite-type hexagonal structure, is distinctly smaller than for the comparable B-CuI structure (88

]‘.“;“.‘I”“:.‘.‘;““,

-6

0,001

0,0015

0,002

0,0025

0,003

0,0035

l/T (K-1) Fig. 3. Electrical conductivity of polycrystalfine CuBr. Two-elcctrode cell: (W ) > 99.0%; ( 0) 99.999%; three-electrode cell: ( l ) 99.999%.

kJ/mol [4]> Similar observations have been made in the tracer diffusion study of Johansson et al. [6]. The decisive factor is the existence of structural disorder in B-CuBr but not in B-CuI.

Table 1 Enthalpies literature

of activation

for conduction

Ha /(kJ/mol) P-CuBr

30

y-CuBr “intrinsic”

105

“extrinsic”

46

and comparison

Ref. 34 151 31 161 179 (270-385°C) 116[61 106 (165-270°C)

[51 [51

with

S. Villain et al./Solid

3.4. Thermodynamics

State Ionics 83 (1996) 191-198

of point defects

Thermodynamic data of point defects in copper(I) bromide can be deduced from the enthalpies of activation. In the highly disordered B-phase, we can assume that the enthalpy of activation (30 kJ/mol) is nearly identical to the enthalpy of migration of copper interstitials. This is somewhat higher than in copper iodide (20 kJ/mol [4]), perhaps reflecting the higher density of CuBr. We conclude that the enthalpy of migration of copper interstitials in copper halides is generally about 25 kJ/mol, thus clearly higher than for silver interstitials in silver halides (= 10 kJ/mol [161). Using this result, we can calculate the enthalpy of formation of Frenkel defects, because we can reasonably suppose that copper interstitials are also the most mobile species in intrinsic y-CuBr. Assuming that the enthalpy of migration of copper interstitials H,(Cuf) is identical in the B- and in the y-phase, about 25 kJ/mol, we can deduce the enthalpy of formation of Frenkel defects H,(Fr) in CuBr from the relation: H, = H,(Cuf)

+ O.SH,(Fr).

(3)

The calculated enthalpy of formation of Frenkel defects (160 kJ/mol) is higher than in CuI (135 kJ/mol [4]>. This reflects the greater difficulty to form interstitials in the denser CuBr structure, which lattice enthalpy is also higher [17]. From the enthalpy of activation in the extrinsic region, we can finally deduce the enthalpy of migration of copper vacancies, which are the majority ionic defect at low temperature: it is slightly less than 50 kJ/mol. This is very close to the value in copper(I) iodide [4] and higher than the enthalpy of migration of silver vacancies in the silver halides (5 30 kJ/mol [16l). 3.5. Bulk concentration

of ionic charge carriers

We can reasonably assume that B-CuBr is a purely ionic conductor (electronic transference number at 400°C: 2 X lop5 [l]). Therefore, we can use the Nemst-Einstein relation, Vi = Di - ci . P/(

RT)

(4)

between the ionic conductivity tri and the copper

195

diffusion coefficients Di [6] to calculate the concentration of mobile Cuf ions ci. At 420°C this concentration amounts to 0.058 mol/cm3. For comparison, we can determine the concentration of Cu+ ions from the molar density of CuBr at room temperature [lo]: 0.036 mol/cm3. This is the same order of magnitude; we can conclude that nearly all Cu+ ions in the fast-ion conducting Bphase are mobile. 3.6. Grain boundaries Generally, grain boundary regions in polycrystals are characterized by impurity segregation, dislocations, imperfect contact between grains, sometimes presence of a grain boundary phase. These features have a detrimental effect on the ion-transport properties of ceramics. Globally, the polycrystal is a complex superposition of grain boundaries “parallel” and ‘ ‘ perpendicular’ ’ to the current (brick-layer model [ 13,181). Theory shows that the impedance response of “parallel” interfaces, which are shortcircuiting the bulk, appears together with the bulk response as one high-frequency arc, whereas the arc corresponding to “perpendicular” interfaces is observed at lower frequencies [18]. In our case, we note the absence of a significant impedance response due to blocking (“perpendicular”) grain boundaries. We conclude that the amrealing of the polycrystalline samples at 350°C permits a sufficient structural relaxation of the grain boundary region in order to enable an easy copper-ion transport across the interface. Easy structural relaxation appears possible, because the cations are very mobile in copper(I) halides. A similar effect of annealing was reported by Maier for silver chloride [ 181. Concerning the “parallel” contributions, a “smooth” variation of slope is observed in the conductivity plot (Fig. 3) in the transition region from extrinsic to intrinsic behaviour; the Arrhenius curve shows no “knee”. As was discussed by Maier [18], this is characteristic for conduction through space charge layers. We conclude that the extrinsic conductivity is partly caused by highly conducting space charge regions in the polycrystal. Additional support for transport along “parallel” pathways comes from the bulk capacitances which are slightly larger than the ideal geometrical value [16].

196

3.7. Phase boundary

S. Villain et al./ Solid State Ionics 83 (1996) 191-198

‘T

copper / copper(I) bromide

t One should emphasize the good reproducibility of the electrical interface parameters obtained in different experiments with variation of the experimental conditions (planar and spherical electrodes, different preparation of the electrode surfaces). Therefore, interfacial resistance and capacitance will be discussed in the following, although preparation of phase boundaries by contacting different crystals is ambigous, especially in view of possible impurity segregation and the presence of dislocations and other non-equilibrium defects at the interface. Schmalzried [ 191 discussed recently the significance of phenomenological electrical parameters for the thermodynamics (inter-facial capacity) and kinetics (interfacial resistance) of interfaces. Interfacial resistance Rint and prefactors Y of the CPA-elements are plotted in Figs. 4 and 5 against reciprocal temperature. We notice an approximately exponential temperature dependency of the inter-facial resistance above about 180°C i.e. in the intrinsic region. There, the enthalpy of activation amounts to about 110 kJ/mol (in the same order of magnitude as for copper(I) iodide [4]). The interfacial resistance is practically negligible at the temperatures above 350°C and especially in the P-phase. This finding is in opposition with the work of Safadi et al. [5], who reported the existence of a particular interfacial resistance in this phase. Below lSO”C, the data are more scattered and the enthalpy of activation is about 50 k.I/mol. In accordance with a model of the interface metal/solid electrolyte under small ac signals [20], we assume that the interfacial resistance has two parts: the charge transfer resistance, which can be described by analogy with the Butler-Volmer theory [19], and the resistance of the space charge, due to changes of concentration of mobile charge carriers near the interface. The contact with the copper electrode leads to an increased concentration of copper interstitials, which can recombine with copper vacancies [19,20]. In the extrinsic domain of the ionic conductor, where the bulk conductivity is due to vacancies, the resulting vacancy depletion layer near the electrode gives a contribution to the inter-facial resistance (and capacitance). In the “intrinsic” domain, where interstitial conduction is assumed in the

6

--

6

--

4

--

2

--

1

--

0

--

L? E" d: Y z d B

-

. -1

.

t t -2

j',',:"":~",!""I"'ll

0.001

0,0015

0,002

l/l

0,0025

0,003

0,0035

(K-1)

Fig. 4. Resistance of the phase boundary Cu/CuBr: electrode cell.

(0)

three-

bulk, this contribution should vanish. The interfacial resistance is thus assumed to be a pure charge transfer resistance in this domain. Near equilibrium, the charge transfer resistance R,, is inversely proportional to the exchange current density i, [19]: R,,=RT/(F-i,).

(5)

In this interpretation, the exponential temperature dependency of R,, corresponds to a thermally activated flux of the mobile charge carriers, which is proportional to their concentration and mobility [ 191. The enthalpy of activation in the intrinsic region is therefore the sum of the enthalpies of formation and migration of copper interstitials, i.e. comparable to the enthalpy of activation of bulk conduction in good accordance with the experimental result. In the extrinsic region, the interfacial resistance is the sum of the charge transfer resistance and the

S. Villain et al./Solid

State Ionics 83 (1996) 191-198

197

mean “width” of the space charge layer near an interface [19]. In Eq. (6), .s is the dielectric constant in the interfacial region (we take for simplicity a value of 10, close to the bulk value of CuBr) and &a is the vacuum permittivity. At 420°C in the fast-ion conducting B-phase, we find a Debye length of about 0.03 nm, less than the radius of a Cuf ion. The “space charge” is at that temperature already a “surface charge”, as in other phases with high concentration of mobile charge carriers, in particular metals (‘ ‘Faraday cage’ ‘1. A capacitance per area C,, (Cu’ compact layer capacitance) can then be calculated in a Helmholtz model [19,20] using: c,

. MI D

,

, 0,001

.

0,0015

0,002 l/T

0,002s

I 0,003

0,0035

(K-l)

Fig. 5. Prefactors of CPA elements of the phase boundary Cu/CuBr: ( 0 ) three-electrode cell.

resistance of the vacancy depletion layer. The width of the space charge layer is fixed by the constant “extrinsic” bulk concentration of copper vacancies, due e.g. to impurities and structural defects. The temperature dependencies of both the “space charge resistance” and the charge transfer resistance are thus due to a change of mobility of the majority charge carrier. In the extrinsic domain, the enthalpy of activation of the interfacial resistance should thus correspond to the enthalpy of migration of copper vacancies. This is in agreement with the experiment. Using the concentration of mobile Cu+ ions ci at 420°C calculated above, we can estimate the Debye length h from the equation [151:

A2=

& * &g.

RT/( F2 . Ci)_

(6)

The Debye length, whose analog can be found in the theory of liquid electrolytes and in semiconductor theory, is an equilibrium quantity and represents a

= E. &o/h,

(7)

and one finds: C, = 0.3 mF/cm*. The prefactors Y plotted in Fig. 5, which can be considered as nonideal capacitance values, are larger above 200°C. A part of the observed difference can be related to the roughness of the interface. The interface area may be larger than the ideal geometrical value used for calculation, because CuBr softens at high temperature and can adapt to the residual rugosity of the electrode surface. A solid state “wetting” process of AgBr on graphite electrodes with drastic increase of the interfacial capacitance was previously described by Raleigh [21]. According to Eqs. (6) and (7), the Helmholtz capacitance is proportional to the square root of concentration of mobile charge carriers. Therefore, the enthalpy of activation for the interfacial capacitance in the intrinsic region should be about one fourth of the enthalpy of formation of Frenkel defects [221, relatively close to the experimental value (60 k.l/mol). In the extrinsic region, the Debye length and thus the interfacial capacitance should be constant, because the concentration of charge carriers does not depend on temperature in this region. Accurate measurements at low temperature are difficult, but the few experimental points (Fig. 5) may indicate the validity of this conclusion. Although these results seem in relatively good agreement with a space charge model, different artefacts must finally be discussed: (i> A simple contact problem as origin of the interfacial impedance arc can be excluded, because interfacial capacitance

198

S. Villain et al./Solid

State Ionics 83 (1996) 191-198

should in this case be much lower and approximately constant with temperature [22]. (ii) A non-equilibrium blocking of ionic current is characterized by a huge increase of the inter-facial impedance arc, if a dc bias is applied (concentration profiles due to polarization 1221). This was not found in our case. (iii) The formation of red copper(I) oxide at the phase boundary was not observed after experiments and by X-ray diffraction. However, a segregation of dissolved oxygen at the interface cannot of course be excluded. (iv) The presence of dislocations and other non-equilibrium defects, particularly grain boundaries in the polycrystalline samples, can perturb the space charge at the phase boundary.

4. Conclusion By impedance spectroscopy on polycrystalline CuBr with Cu electrodes, we could show that: (1) The complex plane impedance diagram is typical of a solid electrolyte with one mobile species and two non-blocking electrodes. (2) The bulk conductivity of fast-ion conducting P-CuBr is indeed higher than 1 a-’ cm- ’ . (3) The enthalpies of migration of copper interstitials and copper vacancies are comparable to those in CuI, but clearly higher than those in silver halides. The enthalpy of formation of Frenkel defects is higher than in CuI, in accordance with the density and lattice enthalpy. (4) The phase boundary resistance depends exponentially on temperature. For interpretation, we suppose two distributions to the interfacial resistance: one part is due to the charge transfer and the other to a space charge at the interface. The prefactors of CPA elements characteristic of the phase boundary depend also exponentially on temperature. They are compared with space charge capacitances. To confirm this qualitative description, electrochemical experiments and electron microscopic observations of the interface Cu/CuBr should be done

on monocrystalline samples. However, this is a difficult task, because of the difficulty of preparation of single crystals of CuBr [lo] and their sensitivity to all types of radiation.

References [l] J.B. Wagner and C. Wagner, J. Chem. Phys. 26 (1957) 1597. [2] W. Biermann and H.J. Oel, Z. Phys. Chem. (NF) 17 (1958) 163. [3] C. Clemen and K. Funke, Ber. Bunscnges. Physik Chem. 79 (1975) 1119. [4] S. Villain, J. Cabam?, D. Roux, L. Roussel and P. Knauth, Solid State Ionics 76 (1995) 229. [5] R. Safadi, I. Riess and H.L. Tuller, Solid State Ionics 57 (1992) 125. [6] J.X.M.Z. Johansson, K. SkGld and J.E. Jorgensen, Solid State Ionics 59 (1993) 297. [7] J.X.M. Zheng-Johansson, K. Skcld and J.E. Jorgensen, Solid State Ionics 70/71 (1994) 522. [S] J. Maier, Solid State Ionics 70/71 (1994) 43. [9] ASTM Data, NBS Circular No. 539 (1953). [lo] A. Neuhaus, K. Reeker and R. Schoepe, Z. Phys. Chem. (NF) 77 (1972) 127. [ll] S. Pizzini, M. Bianchi, A. Corradi and C. Mari, J. Appl. Electrochem. 4 (1974) 7. [12] R.D. Armstrong and M. Todd, in: Solid State Electrochemistry, P.G. Bruce, ed. (Cambridge University Press, Cambridge, 1995) p. 264. [ 131 J.R. MacDonald, Impedance Spectroscopy Emphasizing Solid Materials and Systems (Wiley, New York, 1987). [14] B.A. Boukamp, Equivalent Circuit, 2nd Ed. (University of Twente, The Netherlands, 1989). [15] C. Deportes et al., Eiectrochimie des Solides (Presses Universitaires de Grenoble, 1994) p. 257. [16] U. Lauer and J. Maier, Ber. Bunsenges. Physik. Chem. 96 (1992) 111. [17] H.D.B. Jenkins, in: Handbook of Chemistry and Physics, R.C. Weast, ed. 61st Ed, (CRC Press, Boca Raton, 1981) p. D88. [18] J. Maier, Ber. Bunsenges. Physik. Chem. 90 (1986) 26. [19] H. Schmalzried, Chemical Kinetics of Solids (VCH, Weinheim, 1995). [20] J. Jamnik, J. Maier and S. Pejovnik, Solid State Ionics 75 (1995) 51. [21] D.O. Raleigh,‘J. Electrochem. Sot. 121 (1974) 632 and 639. [22] U. Lauer and J. Maier, J. Electrochem. Sot. 139 (1992) 1472.