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graphite inhomogeneous interfaces
Solid State Ionics 24 (1987) 273-280 North-Holland, Amsterdam
THE ELECTRICAL RESPONSE
OF RbAg,&/GRAPHITE
INHOMOGENEOUS
INTERFACES
s.L&Y1 Instituteof Physics,EPRC, SlovakAcademy of Sciences, DtibravsMcesta 9, CS-842 28 Bratislava.Czechoslovakia
and J. TUCEK Research Instituteof Radiocommunications,Novodvorsti 994, CS-142 21 Praha. Czechoslovakia Received 5 January 1987; accepted for publication 21 April 1987
The ac impedance of the cell graphite/graphite + RbAg.,I,/RbA&I&raphite + RbAg.+I,/graphite was investigated in the frequency range from 10m4to 3~ lo5 Hz. A model assuming the graphite+RbA&IS mixture to represent a porous interface could explain the observed dependences in nearly the whole frequency range with the exception of the lowest frequencies. Attempts to improve the agreement presuming adsorption, discharge of charge carriers or diffusion of neutral species failed. An excellent fit could be achieved assuming a frequency dependent ac conductivity of the interface, approximately proportional to the frequency, as in the bulk of many solids with low conductivity, such as amorphous semiconductors or dielectrics.
1. Introduction
In the last decades there has been a steady interest in the properties of solid ionic conductor/electronic conductor interfaces. If there are no ionic species, which could transport charge through the interface by means of electrochemical reactions without a significant change of concentration, i.e. which were present in sufficient quantities in both materials, such interfaces are expected to be blocking, or at the best temporarily reversible [ l-31. In many papers, both theoretical [ 4-101 and expermental [9-l 1] the expected and/or the observed properties of such structures were demonstrated, however, with partial success only. One of the puzzling details is the frequency dependence of the interface capacitance and conductance, or of the impedance. In the case of ideal blocking at frequencies sufficiently low compared with the inverse of the dielectric relaxation time the capacitance is expected to be constant and the conductance, shunting it, to be zero. If charge blocking is incomplete a non-zero conducting path should be observed, which for materials
with unipolar conduction e.g. fast ionic conductors may be independent of frequency in the case of adsorption or slow reaction or proportional to w - “* in the case of slow diffusion of discharged reaction products. In all these cases the non-zero conductance is accompanied by a frequency dependence of the capacitance, either by a gradual step (asymptotically approaching w -* if adsorption takes place), or of the form c,w-“2 in the case of diffusion (Warburg impedance). In a complex plane the impedance of a completely blocking interface at low frequencies ought to be a straight line, parallel to the imaginary axis. Adsorption and slow electrochemical reaction should yield a semicircle, the Warburg impedance a line with the slope n/4, eventually going over to a circular arc at still lower frequencies [ 121, as a result of finite electrode thickness or separation [ 131. The contribution of interface impedance to the overall impedance is frequently not large enough to allow its good separation. In spite of that it is evident, that the frequency dependences differ from the expected ones, the observed arcs are usually not semicircles, and straight lines are not perpendicular
0 167-2738/87/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
214
3.Lrinyi,J.
Tukk/RbAg,I,/graphite
to the real axis. The interface capacitances and conductances show frequency dependences, which differ from asymptotical o-* and W* ones, respectively. The results can be at least formally treated as caused by a distribution of relaxation times, which is a plausible explanation if the origin of the distribution is clarified. As a possibility surface inhomogeneity or roughness was proposed [ 141, assuming the appropriateness of the model developed by de Levie [ 151 for liquid electrolytes. Some recent treatments used fractal geometry [ 161. From an experimental point of view the investigation of blocking interfaces in ionic crystals is troublesome. They may be studied at very low frequencies only, unless the bulk resistance is not lowered enough by elevated temperatures, which in turn make the complete charge blocking questionable. In this work we have utilized the high conductivity of RbAg&, which renders such investigation possible already at sufficiently low temperatures. In the material only the silver ions are mobile, which makes it comparatively simple. Graphite has been used to prepare blocking electrodes. The interfaces were inhomogeneous and the results were analyzed considering a model of porous electrodes. At low frequencies, at which the interfaces should behave as planar ones, systematic deviations were found from predictions of models of complete blocking, adsorption, slow reaction or diffusion controlled electrode processes. A nearly perfect agreement in the whole frequency range resulted from a heuristically introduced assumption, that the interface conductance is approximately proportional to the frequency.
2. Experimental procedure Pellets 1 mm thick with the diameter of 8 mm were pressed from RbAg& powder, prepared by the method described in [ 171. X-ray powder diffractograms did not reveal any phases besides RbAgJ, and the obtained structure was in agreement with published data [ 18,191. The dominant grain size, estimated visually from SEM pictures was 1 to 10 pm. Electrodes prepressed from the mixtures of the same powder with specpure graphite (lamellae of diameter u 10 pm, about 1 pm thick) were pressed to both sides of the electrolyte tablets in a PMMA case at 200
MPa. The thickness of the electrodes varied between 0.15 and 8 mm. Graphite covered silver discs with soldered leads served as contacts to the electrodes. ac impedance of the resulting cells graphite /graphite + RbAgJJ RbAg&/ graphite+ RbAgJ,/ graphite was measured usually at room temperature (in some cases at temperatures up to 420 K) in the frequency range 1O-4 to 3 x 10’ Hz using sinusoidal voltage with amplitudes ranging from 1 to 5 mV. At the lowest frequencies ( 10-4-10-2 Hz) the current flowing through the sample was recorded and its amplitude and phase shift were determined. At intermediate frequencies ( 1O-3 to lo3 Hz) a semibridge, similar to that described in [ 91 was used and at high frequencies (20 to 3 x lo5 Hz) the method of three voltmeters was utilized [ 191. Prolonged heating of the samples (at 400-420 K) [ 19,201 caused an evident degradation, increasing irreversibly the conductivity at the lowest frequencies. The study of such effects was out of the scope of our work, therefore only the room temperature results are presented in this paper.
3. Results Figs. 1 and 2 show the impedance of samples with the same composition ratio but differing thickness of the mixed electrodes. Subtracting the bulk impedance (essentially resistance) we obtain the impedance of the two interfaces connected in series. The true bulk resistance could not be measured in the available frequency range - it is approximately the highest frequency limit of the curves ( _ 1 Q). Transformation of the interface impedance to components of a parallel RC equivalent network yields the interface capacitance (fig. 3) and conductance (fig. 4). Although this representation does not bring new information, it makes some important features of the results more evident. In figs. 5 and 6 results obtained on structures with other electrolyte/graphite ratio are seen.
4. Theoretical analysis and discussion It was shown recently [ 2 l-231 that the model [ 151, originally developed for porous electrodes may be
3. Ldnyi, J. TuEek/RbAg,I,/graphite
5
‘c’ I Y
200
13
c Y
x” I
100
215
c
z
1
lo.ool Hz
<
3
100
I
11
2
50 -J 1
,
0
?' 0.01Hz '0.01Hz
0
1
0
2
100
R,[kOl
R,[O]
200
4
Fig. I. The impedance of samples with different thickness of electrode tablets: (I 1) t=4.66 mm, (13) t = 1.84mm and (19) t=O. I mm. The electrolyte/graphite volume ratio 1:1.
representative also for thick electrodes of less ideal e.g. sponge-like, or some other shapes and it may be successfully extended to encompass also rather thin mixed electrodes [ 191. For the purpose of the present analysis the conclusions of the paper [ 15 ] are
quite appropriate. The components of the impedance of a cylindrical pore &J= (1 -j)(RI20Cd)“* xcoth[(l
+j) L(oRC,I2)“*]
,
where R is the resistance of the electrolyte cylinder of unit length in the pore, C, the double layer capacitance of its surface, L the length and w the angular 1
t IO4
c z
lo
X
L
1
0.5
u” IO0
0
0
0.5
R,[MOl
1
Fig. 2. The impedance of sample (19) at the lowest frequencies. It is approximately a circular arc, indicating a conducting path across the interface.
2
10-2
10-3
100
f[Hzl
lo3
lo6
Fig. 3. The frequency dependence of the parallel capacitance of the samples from fig. 1.
s. Ldnyi, J. TuCek/RbAg,l,/graphite
216
1o-6t A_
10-s
I__L_1-
100
_
f
[Hz1
I
i_~
IO3
-_LI_
106
Fig. 4. The frequency dependence of the parallel conductance of the samples from fig. 1.
frequency, reduce below a limiting depending on pore diameter, to
frequency,
CO,= C, L , RoS= RLl3 , i.e. ZO=RL/3+(j&dL)-’
.
This is an impedance, represented by series combination of a resistance (part of the electrolyte resistance) and of the capacitance of the whole surface of the pore. While the original pore has been treated as a transmission line, the low frequency response cor-
Fig. 6. The frequency dependence of the parallel conductance of samples from fig. 5.
responds to that of a lumped element circuit and the complicated geometry simply increases the area of the interface. The quasi-planar behaviour corresponds to the frequency range of nearly constant capacitance in figs. 3 and 5 [ 191. It is evident, that shorter pores may behave as a transmission line only at higher frequencies as the longer ones, hence with more massive electrodes the limiting frequency is lower, provided the whole thickness of the electrode tablets is active, i.e. the conducting paths are not interrupted too frequently by additional electrolyte/graphite interfaces. For complete charge blocking and frequencies low compared to the inverse of the dielectric relaxation time the interface impedance should be Z=R,+CjwC,)-
2_($)“2+&_g)“’)
’
lo.2
a3
100
f[Hzl
IO3
lo6
Fig. 5. The frequency dependence of the parallel capacitance of samples with different thickness of electrodes: (1) t=4.2 mm, (2) t=2.93mm, (3) t=1.96mmand(7) t=0.16mm.Theelectrolyte/graphite volume ratio 4: 1.
where e is the charge of the proton, cr the conductivity, E the permittivity, k the Boltzmann constant, T the temperature and p the concentration of the mobile charge carriers [9]. It represents a series combination of a constant capacitance and resistance, which will be practically negligible. The total impedance is Z=Zi+Rb.
211
3. Ldnyi, J. Tu?ek/RbAg,I,/graphite
Since Rb is much greater than the real part of Z’, the measured admittance is approximately Y=Gp+joCp=w2~R,(1+w2C;R~)-’ +joC,(l
+o’~R;)-’
,
where k,, kr are the rate constants of forward and backward reactions and lo is the thickness of the layer of adsorbed reaction products. The fairly general expression given in [ 241 Zs=Rb+ljOCl
where c, = ie( &T)“2
+[R2+(jwC2)-‘+(1-j)
.
Its real part would asymptotically
approach an w2 dependence at low frequencies and a constant value at the higher ones. The capacitance is constant at low frequencies and fall as o2 at the higher ones. The limiting frequency with Rbm 1 a, and C- lo-’ F is about m 1 MHz. Therefore with planar electrodes the measured capacitance would have to be constant and the conduciance proportional to o2 in our frequency range. With electrodes not completely blocking a conducting path would have to exist, limiting the drop of the measured conductance toward the lowest frequencies. In series equivalent network representation the resistance would be of the form R,=Rb(RL+Rb)-‘[RL(Rbc)-‘+a21 x[C;~(RL+R’,-~+W*]-‘,
(1)
where RL stands for the shunting resistance. From this relation the a-2 asymptotic increase of the polarization resistance to a higher low frequency value is evident. The capacitance Cs=1+02~(R,+R,)2(02CsR~)-’
(2)
Ww-“2]-‘}-’
(3)
reduces at low frequencies in the case of free discharge and diffusion of reaction products to Z,=(l-j)
Woe”2
and if adsorption and diffusion takes place to Z,=R,f(jwC,)-‘+(1-j)
Ww-“2.
Cl is the double layer capacitance, C2, R2 are the components of the impedance associated with adsorption and W is called the Warburg constant 1241. We do not intend to discuss the consequences of electrode porosity in the present paper in detail, therefore we shall focus our attention to low frequencies, i.e. to the range of quasi-planar behaviour (below about 1 to 10 Hz). As it is evident from figs. 3 and 5, the capacitance is nearly constant and the conductance (figs. 4 and 6) follows an w2 dependence above 10-2-10-’ Hz. However, it becomes proportional to frequency at still lower frequencies. The same is seen in fig. 7 - the resistance below 0.1 Hz is inversely proportional to the frequency. In a complex plane the impedance consisting of the components (1) and (2) represents a semicircle,
increases towards the lowest frequencies as w2 lim C,= (02CsRi)-’
.
w-0
IO5
1
Were the electrodes not completely blocking but the discharge products would be adsorbed at the interface, the resistance would have a similar dispersion as in (1)) however, the capacitance would increase only to a finite value
4
10” 10-3
100 f [Hz
]
lo3
106
Fig. 7. The frequency dependence of the series resistance of the samples from fig. 5.
3. Lcinyi, J. TuEek/RbAg,l,/graphite
c d
LLI
/‘?
lo-3
.//.__A.___
100f,
103 f IHzl
lo6
Fig. 9. The approximation of experimental points using the present model. From the capacitance in the neighbourhood of the knee the characteristic frequency f, was obtained, which was used for computation of the dashed curves [ 191.
I
10-L
10-l
f
[Hz1
102
Fig. 8. Approximation of R, versus w and G, versus COusing expression ( 3). Over a limited range of frequencies a G, 5 w or R, - w - ’ could be obtained. The formula failed in reproducing the shape of impedance plot and its applicability requires the presence of two types of mobile charges.
with centre on the real axis. In the limit of complete blocking its radius increases to infinity, i.e. the impedance will be a line, perpendicular to the real axis. As it is seen, the low frequency part of impedance diagrams in fig. 1 does not fit this picture, nor is it the behaviour expected for a diffusion controlled process, for which the expected slope would be 1. An attempt to fit the expression (3) to our experimental data is demonstrated in fig. 8. In a limited frequency range approximately GPw o and Rp - w - ’ dependences could be obtained, however, at lower frequencies the deviation from experimental points increased and with parameters giving a better fit of conductance (dashed lines) the fit of resistance was worse and vice versa (dash-dotted lines). The low frequency parts of the GPversus w and R, versus w dependences suggest, that both the assumption of constant polarization resistance or the presence of a conducting path, shunting the interface
capacitance is not appropriate for the present results. On the other hand, if we assume an interface resistance approximately inversely proportional to frequency, or an interface conductance approximately proportional to frequency, we obtain a satisfactory agreement with the experiment. This assumption is not merely pragmatic. It is well known, that ac conductivity of the form CI,,c w*, with s usually between 0.75 and 1 is observed in the bulk of many solids having electronic or ionic conductivity [ 251. Jonscher [ 261 has suggested, that such type of conduction should take place in depletion layers of various materials as well. The step we undertook was to postulate the possibility, that the phenomenon occurs in blocking interfaces, regardless of whether they are depleted of charge carriers or not. Fig. 9 shows computed capacitance, conductance and series resistance versus frequency dependences. The parameters used for computation came from the analysis of the capacitance at intermediate frequencies, made assuming the electrodes consisted of identical equivalent pores [ 191. f. is a characteristic frequency, below which the interfaces behave as planar ones. However, the expected properties depend on the appropriateness of this approximation most sensitively in the same frequency range, which is used to estimatef,, what makes this analysis vulnerable. Not overestimating the role off0 a nearly perfect fit of all three curves with experimental points may be obtained, at least up to about lo4 Hz, with the exception of the slight depression on R, near 1
3. Llinyi, J. Tufek/RbAg,l,/graphite
Hz. The simplification was unavoidable at present, because we do not know the actual distribution of effective pore lengths and cross section but as we shall show later, it turned out to be realistic for our samples. The frequency dependent interface conductance causes a deviation of the complex plane plot of the impedance from the completely blocking case by a constant angle at low frequencies but with little or no effect at the higher ones. Such impedance is usually called the constant phase angle impedance and its origin is considered unknown [ 271. Depending on their structure, the pore-like properties of electrodes may extend to rather low frequencies and the deviation of the conductance from the o2 slope and even the w - ’ dependence of the resistance may not appear in the available frequency range [ 24,281. It may be worth to mention briefly some properties of equivalent pores, which were obtained using our model. They are summarized in table 1 [ 191. It is interesting to note, that for electrode pellets of the same composition the effective pore length is nearly proportional to their thickness and the pore diameter is approximately constant. This suggests, that the conducting paths are not interrupted too frequently by additional electrolyte/graphite interfaces and the concept of equivalent pores is not a bad assumption. As it is seen in figs. 2 and 4 to 7, samples 7 and 19 did not tit the general picture. The thickness of their electrode tablets was rather small (0.16 and 0.1 mm respectively) and their behaviour corresponds more or less to a leaky planar interface, probably as a result of some accidental electrolyte/silver contact. We treat the interface conductance N w and resist-
219
ante ~c0 - * parallelly as two possible options. However, such dependence should not hold at the same time over a considerable frequency range, unless the loss angle is rather small. In our case tan 6 was between 0.02 and 0.045, therefore the impedance measurement alone could not give preference to any of the two representations. Physically the frequency dependent conductance (or a small imaginary component of the capacitance) is more plausible. As to the origin of the frequency dependent conductance, we must admit, that this is an open question, not solved yet unambiguously even for the bulk of materials [ 251. The hypothesis of Jonscher and coworkers [ 291, that it is a unique property of solids, resulting from interaction of charge carriers with the rigid matrix, in which they move, seems to be promising for the interfaces as well. The magnitude of the frequency dependent conductance was sensitive to the preparation of samples and it seems to reflect the actual state (we do not know whether chemical or physical) of the interface.
5. Conclusions Our measurements have shown, that the impedance of the solid electrolyte/electrode interface has a component, which may be either a resistance, inversely proportional to the frequency, or a conductance, approximately proportional to the frequency. Such behaviour was suggested for depletion layers in the vicinity of electrodes. In our case, the interface is not in a state of depletion. However, it is possible that the blocking electrodes suppress the
Table 1 Size of equivalent pores calculated from experimental data. Sample
1
2 3 4 5 6 11 13
Electrode thickness t (mm)
Equivalent pore radius r (Pm)
Equivalent pore length L (mm)
Lit
Composition
4.2 2.93 1.96 0.91 0.85 0.7 4.66 1.84
6.4 7.4 6.8 7.1 7.6 7.5 &7 4.7
5.6 4.19 3.2 1.72 1.74 1.24 35.1 13.1
1.33 1.43 1.63 I .89 2.05 I .77 7.53 7.12
4:l 4:l 4:l 4:l 4:l 4:1 1:1 I:1
280
3. Ldnyi, J. Tufek/RbAg,l,/graphite
dc conductivity and thus reveal the frequency dependent ac conductivity of the interface region, which otherwise would be observable only at much higher frequencies.
References [ 1 ] J.R. Macdonald, Phys. Rev. 92 (1953) 4.
[ 21 T.A. Kriiger, The chemistry of imperfect crystals (NorthHolland, Amsterdam, 1964).
[ 3 ] s. Lanyi and K. Hricovini, J. Phys. Chem. Solids 44 ( 198 1) 905.
[ 41 H. Chang and G. Jaffe, J. Chem. Phys. 20 (1952) 107 1. [S] R.J. Friauf, J. Chem. Phys. 22 (1954) 1329. [6] J.H. Beaumont and P.W.M. Jacobs, J. Phys. Chem. Solids 28 (1967) 657. [ 71 D.O. Raleigh, Phys. Status Solidi (a) 4 (197 1) 2 15. [ 81 R. Michel, M. Maitrot and R. Madru, J. Phys. Chem. Solids 34 (1973) 1039. [9] S. Lbnyi, J. Phys. Chem. Solids 36 (1975) 775. [lo] J.R. Macdonald and D.R. Franceschetti, J. Chem. Phys. 68 (1978) 1614. [ 111D.O. Raleigh and H.R. Crowe, Solid State Commun. 8 (1970) 955. [ 121 J.R. Macdonald, J. Electroanal. Chem. 53 (1974) 1. [ 131 D.O. Raleigh, in: Electrode processes in solid state ionics, eds. M. Kleitz and J. Dupuy (Reidel, Dordrecht, Holland, 1976) p. 119. [ 141 D.R. Franceschetti and J.R. Macdonald, J. Electroanal. Chem. 101 (1979) 307. [ 151 R. de Levie, in: Advances in electrochemistry and electro-
chemical engineering, eds. P. Delahay and C.W. Tobias, Vol. 6 (Interscience, New York, 1967) p. 329. [ 161 J.C. Wang and J.B. Bates, Solid State Ionics 18119 (1986) 224. [ 171 J.N. Bradley and P.D. Greene, Trans. Faraday Sot. 63 (1967) 2516. [ 181 B. Scrosati, J. Electrochem. Sot. 1 I8 (197 1) 899. [ 191 J. TuEek, Thesis (Faculty of Electrical Engineering, Czech. Techn. Univ., Prague 1983). [20] J. TuEek and S. L&tyi, in: Transport properties of solids, eds. E. Majkova and E. Mariani (Inst. Phys. EPRC Slovak Acad. Sci., Bratislava, 1982) p. 215. [ 2 1 ] H. Kaiser, K.D. Beccu and M.A. Guthiar, Electrochim. Acta 21 (1976) 539. [22] J.P. Candy, P. Fouilleux, M. Keddam and H. Takenouti, Electrochim. Acta 26 (198 1) 1029. (231 J.P. Candy, P. Fouilleux, M. Keddam and H. Takenouti, Electrochim. Acta 27 (1982) 1585. [24] N.G. Bukun, E.A. Ukshe and V.G. Goffman, Elektrokhimiya 18 (1982) 653. [ 251A.E. Owen, in: Proc. XIth Int. Congr. on Glass, Glass ‘77, Prague 1977, ed. J. GGtz (Czech. Sci.-Techn. Sot. Prague, 1977) p. 372. [26] A.K. Jonscher, J. Mat. Sci. 16 (1981) 2037. [27] P.H. Bottelberghs, in: Solid electrolytes, eds. P. Hagenmuller and W. van Cool (Academic Press, New York, 1978) p. 145. [28] R.D. Armstrong, in: Electrode processes in solid state ionits, eds. M. Kleitz and J. Dupuy (Reidel, Dordrecht, Holland, 1976) p. 261. [29] R.M. Hill and A.K. Jonscher, Contemp. Phys. 24 (1983) 75. [ 301 S. L&iyi and J. TuEek, to be published.
THE ELECTRICAL RESPONSE
OF RbAg,&/GRAPHITE
INHOMOGENEOUS
INTERFACES
s.L&Y1 Instituteof Physics,EPRC, SlovakAcademy of Sciences, DtibravsMcesta 9, CS-842 28 Bratislava.Czechoslovakia
and J. TUCEK Research Instituteof Radiocommunications,Novodvorsti 994, CS-142 21 Praha. Czechoslovakia Received 5 January 1987; accepted for publication 21 April 1987
The ac impedance of the cell graphite/graphite + RbAg.,I,/RbA&I&raphite + RbAg.+I,/graphite was investigated in the frequency range from 10m4to 3~ lo5 Hz. A model assuming the graphite+RbA&IS mixture to represent a porous interface could explain the observed dependences in nearly the whole frequency range with the exception of the lowest frequencies. Attempts to improve the agreement presuming adsorption, discharge of charge carriers or diffusion of neutral species failed. An excellent fit could be achieved assuming a frequency dependent ac conductivity of the interface, approximately proportional to the frequency, as in the bulk of many solids with low conductivity, such as amorphous semiconductors or dielectrics.
1. Introduction
In the last decades there has been a steady interest in the properties of solid ionic conductor/electronic conductor interfaces. If there are no ionic species, which could transport charge through the interface by means of electrochemical reactions without a significant change of concentration, i.e. which were present in sufficient quantities in both materials, such interfaces are expected to be blocking, or at the best temporarily reversible [ l-31. In many papers, both theoretical [ 4-101 and expermental [9-l 1] the expected and/or the observed properties of such structures were demonstrated, however, with partial success only. One of the puzzling details is the frequency dependence of the interface capacitance and conductance, or of the impedance. In the case of ideal blocking at frequencies sufficiently low compared with the inverse of the dielectric relaxation time the capacitance is expected to be constant and the conductance, shunting it, to be zero. If charge blocking is incomplete a non-zero conducting path should be observed, which for materials
with unipolar conduction e.g. fast ionic conductors may be independent of frequency in the case of adsorption or slow reaction or proportional to w - “* in the case of slow diffusion of discharged reaction products. In all these cases the non-zero conductance is accompanied by a frequency dependence of the capacitance, either by a gradual step (asymptotically approaching w -* if adsorption takes place), or of the form c,w-“2 in the case of diffusion (Warburg impedance). In a complex plane the impedance of a completely blocking interface at low frequencies ought to be a straight line, parallel to the imaginary axis. Adsorption and slow electrochemical reaction should yield a semicircle, the Warburg impedance a line with the slope n/4, eventually going over to a circular arc at still lower frequencies [ 121, as a result of finite electrode thickness or separation [ 131. The contribution of interface impedance to the overall impedance is frequently not large enough to allow its good separation. In spite of that it is evident, that the frequency dependences differ from the expected ones, the observed arcs are usually not semicircles, and straight lines are not perpendicular
0 167-2738/87/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
214
3.Lrinyi,J.
Tukk/RbAg,I,/graphite
to the real axis. The interface capacitances and conductances show frequency dependences, which differ from asymptotical o-* and W* ones, respectively. The results can be at least formally treated as caused by a distribution of relaxation times, which is a plausible explanation if the origin of the distribution is clarified. As a possibility surface inhomogeneity or roughness was proposed [ 141, assuming the appropriateness of the model developed by de Levie [ 151 for liquid electrolytes. Some recent treatments used fractal geometry [ 161. From an experimental point of view the investigation of blocking interfaces in ionic crystals is troublesome. They may be studied at very low frequencies only, unless the bulk resistance is not lowered enough by elevated temperatures, which in turn make the complete charge blocking questionable. In this work we have utilized the high conductivity of RbAg&, which renders such investigation possible already at sufficiently low temperatures. In the material only the silver ions are mobile, which makes it comparatively simple. Graphite has been used to prepare blocking electrodes. The interfaces were inhomogeneous and the results were analyzed considering a model of porous electrodes. At low frequencies, at which the interfaces should behave as planar ones, systematic deviations were found from predictions of models of complete blocking, adsorption, slow reaction or diffusion controlled electrode processes. A nearly perfect agreement in the whole frequency range resulted from a heuristically introduced assumption, that the interface conductance is approximately proportional to the frequency.
2. Experimental procedure Pellets 1 mm thick with the diameter of 8 mm were pressed from RbAg& powder, prepared by the method described in [ 171. X-ray powder diffractograms did not reveal any phases besides RbAgJ, and the obtained structure was in agreement with published data [ 18,191. The dominant grain size, estimated visually from SEM pictures was 1 to 10 pm. Electrodes prepressed from the mixtures of the same powder with specpure graphite (lamellae of diameter u 10 pm, about 1 pm thick) were pressed to both sides of the electrolyte tablets in a PMMA case at 200
MPa. The thickness of the electrodes varied between 0.15 and 8 mm. Graphite covered silver discs with soldered leads served as contacts to the electrodes. ac impedance of the resulting cells graphite /graphite + RbAgJJ RbAg&/ graphite+ RbAgJ,/ graphite was measured usually at room temperature (in some cases at temperatures up to 420 K) in the frequency range 1O-4 to 3 x 10’ Hz using sinusoidal voltage with amplitudes ranging from 1 to 5 mV. At the lowest frequencies ( 10-4-10-2 Hz) the current flowing through the sample was recorded and its amplitude and phase shift were determined. At intermediate frequencies ( 1O-3 to lo3 Hz) a semibridge, similar to that described in [ 91 was used and at high frequencies (20 to 3 x lo5 Hz) the method of three voltmeters was utilized [ 191. Prolonged heating of the samples (at 400-420 K) [ 19,201 caused an evident degradation, increasing irreversibly the conductivity at the lowest frequencies. The study of such effects was out of the scope of our work, therefore only the room temperature results are presented in this paper.
3. Results Figs. 1 and 2 show the impedance of samples with the same composition ratio but differing thickness of the mixed electrodes. Subtracting the bulk impedance (essentially resistance) we obtain the impedance of the two interfaces connected in series. The true bulk resistance could not be measured in the available frequency range - it is approximately the highest frequency limit of the curves ( _ 1 Q). Transformation of the interface impedance to components of a parallel RC equivalent network yields the interface capacitance (fig. 3) and conductance (fig. 4). Although this representation does not bring new information, it makes some important features of the results more evident. In figs. 5 and 6 results obtained on structures with other electrolyte/graphite ratio are seen.
4. Theoretical analysis and discussion It was shown recently [ 2 l-231 that the model [ 151, originally developed for porous electrodes may be
3. Ldnyi, J. TuEek/RbAg,I,/graphite
5
‘c’ I Y
200
13
c Y
x” I
100
215
c
z
1
lo.ool Hz
<
3
100
I
11
2
50 -J 1
,
0
?' 0.01Hz '0.01Hz
0
1
0
2
100
R,[kOl
R,[O]
200
4
Fig. I. The impedance of samples with different thickness of electrode tablets: (I 1) t=4.66 mm, (13) t = 1.84mm and (19) t=O. I mm. The electrolyte/graphite volume ratio 1:1.
representative also for thick electrodes of less ideal e.g. sponge-like, or some other shapes and it may be successfully extended to encompass also rather thin mixed electrodes [ 191. For the purpose of the present analysis the conclusions of the paper [ 15 ] are
quite appropriate. The components of the impedance of a cylindrical pore &J= (1 -j)(RI20Cd)“* xcoth[(l
+j) L(oRC,I2)“*]
,
where R is the resistance of the electrolyte cylinder of unit length in the pore, C, the double layer capacitance of its surface, L the length and w the angular 1
t IO4
c z
lo
X
L
1
0.5
u” IO0
0
0
0.5
R,[MOl
1
Fig. 2. The impedance of sample (19) at the lowest frequencies. It is approximately a circular arc, indicating a conducting path across the interface.
2
10-2
10-3
100
f[Hzl
lo3
lo6
Fig. 3. The frequency dependence of the parallel capacitance of the samples from fig. 1.
s. Ldnyi, J. TuCek/RbAg,l,/graphite
216
1o-6t A_
10-s
I__L_1-
100
_
f
[Hz1
I
i_~
IO3
-_LI_
106
Fig. 4. The frequency dependence of the parallel conductance of the samples from fig. 1.
frequency, reduce below a limiting depending on pore diameter, to
frequency,
CO,= C, L , RoS= RLl3 , i.e. ZO=RL/3+(j&dL)-’
.
This is an impedance, represented by series combination of a resistance (part of the electrolyte resistance) and of the capacitance of the whole surface of the pore. While the original pore has been treated as a transmission line, the low frequency response cor-
Fig. 6. The frequency dependence of the parallel conductance of samples from fig. 5.
responds to that of a lumped element circuit and the complicated geometry simply increases the area of the interface. The quasi-planar behaviour corresponds to the frequency range of nearly constant capacitance in figs. 3 and 5 [ 191. It is evident, that shorter pores may behave as a transmission line only at higher frequencies as the longer ones, hence with more massive electrodes the limiting frequency is lower, provided the whole thickness of the electrode tablets is active, i.e. the conducting paths are not interrupted too frequently by additional electrolyte/graphite interfaces. For complete charge blocking and frequencies low compared to the inverse of the dielectric relaxation time the interface impedance should be Z=R,+CjwC,)-
2_($)“2+&_g)“’)
’
lo.2
a3
100
f[Hzl
IO3
lo6
Fig. 5. The frequency dependence of the parallel capacitance of samples with different thickness of electrodes: (1) t=4.2 mm, (2) t=2.93mm, (3) t=1.96mmand(7) t=0.16mm.Theelectrolyte/graphite volume ratio 4: 1.
where e is the charge of the proton, cr the conductivity, E the permittivity, k the Boltzmann constant, T the temperature and p the concentration of the mobile charge carriers [9]. It represents a series combination of a constant capacitance and resistance, which will be practically negligible. The total impedance is Z=Zi+Rb.
211
3. Ldnyi, J. Tu?ek/RbAg,I,/graphite
Since Rb is much greater than the real part of Z’, the measured admittance is approximately Y=Gp+joCp=w2~R,(1+w2C;R~)-’ +joC,(l
+o’~R;)-’
,
where k,, kr are the rate constants of forward and backward reactions and lo is the thickness of the layer of adsorbed reaction products. The fairly general expression given in [ 241 Zs=Rb+ljOCl
where c, = ie( &T)“2
+[R2+(jwC2)-‘+(1-j)
.
Its real part would asymptotically
approach an w2 dependence at low frequencies and a constant value at the higher ones. The capacitance is constant at low frequencies and fall as o2 at the higher ones. The limiting frequency with Rbm 1 a, and C- lo-’ F is about m 1 MHz. Therefore with planar electrodes the measured capacitance would have to be constant and the conduciance proportional to o2 in our frequency range. With electrodes not completely blocking a conducting path would have to exist, limiting the drop of the measured conductance toward the lowest frequencies. In series equivalent network representation the resistance would be of the form R,=Rb(RL+Rb)-‘[RL(Rbc)-‘+a21 x[C;~(RL+R’,-~+W*]-‘,
(1)
where RL stands for the shunting resistance. From this relation the a-2 asymptotic increase of the polarization resistance to a higher low frequency value is evident. The capacitance Cs=1+02~(R,+R,)2(02CsR~)-’
(2)
Ww-“2]-‘}-’
(3)
reduces at low frequencies in the case of free discharge and diffusion of reaction products to Z,=(l-j)
Woe”2
and if adsorption and diffusion takes place to Z,=R,f(jwC,)-‘+(1-j)
Ww-“2.
Cl is the double layer capacitance, C2, R2 are the components of the impedance associated with adsorption and W is called the Warburg constant 1241. We do not intend to discuss the consequences of electrode porosity in the present paper in detail, therefore we shall focus our attention to low frequencies, i.e. to the range of quasi-planar behaviour (below about 1 to 10 Hz). As it is evident from figs. 3 and 5, the capacitance is nearly constant and the conductance (figs. 4 and 6) follows an w2 dependence above 10-2-10-’ Hz. However, it becomes proportional to frequency at still lower frequencies. The same is seen in fig. 7 - the resistance below 0.1 Hz is inversely proportional to the frequency. In a complex plane the impedance consisting of the components (1) and (2) represents a semicircle,
increases towards the lowest frequencies as w2 lim C,= (02CsRi)-’
.
w-0
IO5
1
Were the electrodes not completely blocking but the discharge products would be adsorbed at the interface, the resistance would have a similar dispersion as in (1)) however, the capacitance would increase only to a finite value
4
10” 10-3
100 f [Hz
]
lo3
106
Fig. 7. The frequency dependence of the series resistance of the samples from fig. 5.
3. Lcinyi, J. TuEek/RbAg,l,/graphite
c d
LLI
/‘?
lo-3
.//.__A.___
100f,
103 f IHzl
lo6
Fig. 9. The approximation of experimental points using the present model. From the capacitance in the neighbourhood of the knee the characteristic frequency f, was obtained, which was used for computation of the dashed curves [ 191.
I
10-L
10-l
f
[Hz1
102
Fig. 8. Approximation of R, versus w and G, versus COusing expression ( 3). Over a limited range of frequencies a G, 5 w or R, - w - ’ could be obtained. The formula failed in reproducing the shape of impedance plot and its applicability requires the presence of two types of mobile charges.
with centre on the real axis. In the limit of complete blocking its radius increases to infinity, i.e. the impedance will be a line, perpendicular to the real axis. As it is seen, the low frequency part of impedance diagrams in fig. 1 does not fit this picture, nor is it the behaviour expected for a diffusion controlled process, for which the expected slope would be 1. An attempt to fit the expression (3) to our experimental data is demonstrated in fig. 8. In a limited frequency range approximately GPw o and Rp - w - ’ dependences could be obtained, however, at lower frequencies the deviation from experimental points increased and with parameters giving a better fit of conductance (dashed lines) the fit of resistance was worse and vice versa (dash-dotted lines). The low frequency parts of the GPversus w and R, versus w dependences suggest, that both the assumption of constant polarization resistance or the presence of a conducting path, shunting the interface
capacitance is not appropriate for the present results. On the other hand, if we assume an interface resistance approximately inversely proportional to frequency, or an interface conductance approximately proportional to frequency, we obtain a satisfactory agreement with the experiment. This assumption is not merely pragmatic. It is well known, that ac conductivity of the form CI,,c w*, with s usually between 0.75 and 1 is observed in the bulk of many solids having electronic or ionic conductivity [ 251. Jonscher [ 261 has suggested, that such type of conduction should take place in depletion layers of various materials as well. The step we undertook was to postulate the possibility, that the phenomenon occurs in blocking interfaces, regardless of whether they are depleted of charge carriers or not. Fig. 9 shows computed capacitance, conductance and series resistance versus frequency dependences. The parameters used for computation came from the analysis of the capacitance at intermediate frequencies, made assuming the electrodes consisted of identical equivalent pores [ 191. f. is a characteristic frequency, below which the interfaces behave as planar ones. However, the expected properties depend on the appropriateness of this approximation most sensitively in the same frequency range, which is used to estimatef,, what makes this analysis vulnerable. Not overestimating the role off0 a nearly perfect fit of all three curves with experimental points may be obtained, at least up to about lo4 Hz, with the exception of the slight depression on R, near 1
3. Llinyi, J. Tufek/RbAg,l,/graphite
Hz. The simplification was unavoidable at present, because we do not know the actual distribution of effective pore lengths and cross section but as we shall show later, it turned out to be realistic for our samples. The frequency dependent interface conductance causes a deviation of the complex plane plot of the impedance from the completely blocking case by a constant angle at low frequencies but with little or no effect at the higher ones. Such impedance is usually called the constant phase angle impedance and its origin is considered unknown [ 271. Depending on their structure, the pore-like properties of electrodes may extend to rather low frequencies and the deviation of the conductance from the o2 slope and even the w - ’ dependence of the resistance may not appear in the available frequency range [ 24,281. It may be worth to mention briefly some properties of equivalent pores, which were obtained using our model. They are summarized in table 1 [ 191. It is interesting to note, that for electrode pellets of the same composition the effective pore length is nearly proportional to their thickness and the pore diameter is approximately constant. This suggests, that the conducting paths are not interrupted too frequently by additional electrolyte/graphite interfaces and the concept of equivalent pores is not a bad assumption. As it is seen in figs. 2 and 4 to 7, samples 7 and 19 did not tit the general picture. The thickness of their electrode tablets was rather small (0.16 and 0.1 mm respectively) and their behaviour corresponds more or less to a leaky planar interface, probably as a result of some accidental electrolyte/silver contact. We treat the interface conductance N w and resist-
219
ante ~c0 - * parallelly as two possible options. However, such dependence should not hold at the same time over a considerable frequency range, unless the loss angle is rather small. In our case tan 6 was between 0.02 and 0.045, therefore the impedance measurement alone could not give preference to any of the two representations. Physically the frequency dependent conductance (or a small imaginary component of the capacitance) is more plausible. As to the origin of the frequency dependent conductance, we must admit, that this is an open question, not solved yet unambiguously even for the bulk of materials [ 251. The hypothesis of Jonscher and coworkers [ 291, that it is a unique property of solids, resulting from interaction of charge carriers with the rigid matrix, in which they move, seems to be promising for the interfaces as well. The magnitude of the frequency dependent conductance was sensitive to the preparation of samples and it seems to reflect the actual state (we do not know whether chemical or physical) of the interface.
5. Conclusions Our measurements have shown, that the impedance of the solid electrolyte/electrode interface has a component, which may be either a resistance, inversely proportional to the frequency, or a conductance, approximately proportional to the frequency. Such behaviour was suggested for depletion layers in the vicinity of electrodes. In our case, the interface is not in a state of depletion. However, it is possible that the blocking electrodes suppress the
Table 1 Size of equivalent pores calculated from experimental data. Sample
1
2 3 4 5 6 11 13
Electrode thickness t (mm)
Equivalent pore radius r (Pm)
Equivalent pore length L (mm)
Lit
Composition
4.2 2.93 1.96 0.91 0.85 0.7 4.66 1.84
6.4 7.4 6.8 7.1 7.6 7.5 &7 4.7
5.6 4.19 3.2 1.72 1.74 1.24 35.1 13.1
1.33 1.43 1.63 I .89 2.05 I .77 7.53 7.12
4:l 4:l 4:l 4:l 4:l 4:1 1:1 I:1
280
3. Ldnyi, J. Tufek/RbAg,l,/graphite
dc conductivity and thus reveal the frequency dependent ac conductivity of the interface region, which otherwise would be observable only at much higher frequencies.
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