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J. Phys. Chem. B 1999, 103, 4122-4127
Interrelation between Different Mixed Cation Effects in the Electrical Conductivity and Mechanical Loss Spectra of Ion Conducting Glasses† B. Roling,*,‡ A. Happe,‡ M. D. Ingram,§ and K. Funke‡ Institut fu¨r Physikalische Chemie, Westfa¨lische Wilhelms-UniVersita¨t Mu¨nster, Schlossplatz 4/7, 48149 Mu¨nster, Germany, and Department of Chemistry, UniVersity of Aberdeen, Aberdeen AB24 3UE, Scotland, U.K. ReceiVed: October 5, 1998; In Final Form: February 4, 1999
Several mixed cation effects in the electrical conductivity and mechanical loss spectra of mixed alkali and mixed alkali-alkaline earth glasses are identified, and their interrelation is analyzed. Apart from the classical mixed cation effects in the ionic diffusivities, we find characteristic effects in the frequency response of the complex electrical conductivity and in the magnitudes of mechanical loss peaks. We observe that the latter two effects are the more pronounced, the less the diffusivities of the faster and the slower ions differ. This suggests that these two mixed cation effects are fingerprints of an incomplete decoupling between the transport of the faster and the slower ions.
1. Introduction In mixed alkali glasses with a given overall alkali ion content, the diffusivities of both types of alkali ions increase with increasing relative concentration. There exists a diffusivity crossover point where the diffusivities are identical. This point is usually close to the point where the concentrations of the dissimilar ions are identical. On both sides of the diffusivity crossover point, the charge transport, i.e., the ionic conductivity, is dominated by the more mobile ionic species. Accordingly, the ionic conductivity is minimal at a composition close to the diffusivity crossover composition. These phenomenona are wellknown as the classical mixed alkali effect in glass.1-3 If the diffusivities of the dissimilar ions differ considerably, one observes two separate peaks in the mechanical loss spectra of mixed alkali glasses. The mechanical loss, tan δ, is usually measured as a function of temperature at constant frequency. In this case, the low-temperature peak is related to the faster ion transport process and the high-temperature peak to the slower ion transport process.4-8 The activation energies of the transport processes can be determined from the frequency dependence of the peak temperatures. The magnitude of the high-temperature peak is usually larger than that of the low-temperature peak even if the relative concentration of the slower ions is smaller than that of the faster ions. In mixed alkali glasses with almost equal concentrations of the dissimilar alkali ions, a lowtemperature peak is not detected. From the fact that the activation energy of the dc conductivity is lower than the activation energy of the high-temperature peak one can conclude that the faster transport process is operative but does not significantly contribute to the mechanical loss. This peak height effect is a second important mixed alkali effect in glass. A third effect is observed in the frequency response of the complex electrical conductiVity. At low frequencies, the real part of the conductivity is independent of frequency and identical to the dc conductivity. Above a characteristic onset frequency, the real part of the conductivity increases with frequency. In †
Dedicated to C. Austen Angell on the occasion of his 65th birthday. Westfa¨lische Wilhelms-Universita¨t. § University of Aberdeen. ‡
mixed alkali glasses, the crossover from dc to dispersive conductivity extends over a broader frequency range than in single alkali glasses. This is equivalent to a considerably higher static dielectric constant of mixed alkali glasses as compared to single alkali glasses.9-11 Most models describing the ion dynamics in mixed alkali glasses focus on the longsrange diffusivities of the ions. The reduced diffusivities in mixed alkali glasses as compared to single alkali glasses are attributed to interactions between dissimilar ions,3,12-15 to phase separations16 or to cation sites of different size.17,18 In the “dynamic structure model” of Bunde, Ingram, and Maass,19,20 the reduced ionic diffusivities in mixed alkali glasses are explained in terms of a “site relaxation/site memory” effect. In that model, each type of mobile ion is able to adapt the glassy network according to its spatial and chemical requirements. The hopping of each ion occurs readily via sites of its own type, while an additional activation energy is required to enter sites of a different type. When the number density of the ions of one type decreases, the number density of the corresponding favorable sites decreases as well. This leads to an increase of the activation energy required for the long-range transport. While all these models yield a qualitatively correct composition dependence of the ionic diffusivities, none of them is able to account for the mixed alkali effects in the mechanical loss peak heights and in the frequency response of the electrical conductivity. In our opinion, an important step toward a comprehensive theoretical understanding of all these mixed alkali effects would be achieved by a phenomenological description of the way these effects are interrelated. In this paper, we compare the electrical conductivity and mechanical loss spectra of several mixed alkali and mixed alkali-alkaline earth glasses. The spectra of mixed alkalialkaline earth glasses are particularly informative since the alkaline earth ions are always less mobile than the alkali ions even if the alkaline earth ions are present in high concentrations. We find a correlation between the difference of the activation energies of the faster and the slower ion transport process, ∆EA, and the mixed cation effects in mechanical loss peak height and in the frequency response of the electrical conductivity.
10.1021/jp9839652 CCC: $18.00 © 1999 American Chemical Society Published on Web 03/23/1999
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These effects are stronger, the smaller is ∆EA, i.e., the less the diffusivities of the faster and the slower ions differ. 2. Experimental Section 2.1. Glass Preparation. The following glasses have been prepared for our experiments. 0.3 (yNa2 O‚(1 - y)Li2O)‚0.7B2O3 with y ) 0, 0.2, 0.4, 0.6, 0.8, 1; 0.1144 (yNa2O‚(1 - y)Li2O)‚ 0.7116SiO2‚0.1617Al2O3‚0.0062TiO2‚0.0036B2O3‚0.0019As2O3‚ 0.0006K2O with y ) 0.041, 0.33, 0.5, 0.67, 0.959; and Na2O‚2 MO‚4 SiO2 with M ) Mg, Ca. For the preparation of the mixed Li/Na borate glasses, dry mixtures of Na2CO3, Li2CO3, and B2O3 were melted in a platinum crucible at 1000 °C for 1 h. The melts were then poured into a graphite mold and held inside the furnace for another 10 min. After cooling to room temperature the samples were annealed 20 K below their respective glass transition temperatures for 12 h. The aluminosilicate glasses and the mixed sodium-alkaline earth glasses were supplied by Pilkington plc. For their preparation, mixtures of Na2CO3, CaCO3, MgO, Al2O3, TiO2, B2O3, As2O3, K2CO3, and finely ground quartz were melted at 1500 °C for 1 h, stirred, held at 1450 °C for 1.5 h, and cast. The glasses were annealed 30 K below their respective glass transition temperatures for 1 h. 2.2. Electrical Conductivity Measurements. The samples for the electrical conductivity measurements were either cylindrical or rectangular in shape (area ≈ 1 cm2, thickness ≈ 1 mm). Gold electrodes were sputtered onto the sample surfaces. The frequency- and temperature-dependent conductivities of the mixed Li/Na borate glasses were measured in the frequency range from 5 Hz to 4 MHz using an HP 4192A impedance analyzer. The conductivities of the mixed Li/Na aluminosilicate glasses and of the mixed sodium-alkaline earth silicate glasses were measured in the frequency range from 1 Hz to 300 kHz using a Schlumberger SI 1260 impedance analyzer. 2.3. Mechanical Loss Experiments. Frequency- and temperature-dependent mechanical loss experiments were performed on the mixed Li/Na aluminosilicate glasses and on the mixed sodium-alkaline earth silicate glasses. The glasses were cut into bars (length ≈ 50 mm, width ≈ 5 mm, thickness ≈ 1.3 mm), and the sample surfaces were ground and polished. For our experiments we used a Rheometric Scientific Dynamic Mechanical Analyzer Mk III in the three-point bending mode. For details of the measurements see ref 21.
Figure 1. Activation energy of the dc conductivity as a function of glass composition for mixed Li/Na borate and aluminosilicate glasses.
Figure 2. Conductivity master curves of mixed alkali borate glasses of compositions 0.3(yNa2O‚(1 - y)Li2O‚0.7B2O3.
3. Results In Figure 1, the activation energy of the dc conductivity, EAdc, is shown as a function of composition for the mixed Li/Na borate and aluminosilicate glasses. The temperature-dependent dc conductivities of these glasses were taken from the lowfrequency plateaux of the conductivity isotherms. The values for EAdc were then obtained from a conventional Arrhenius plot. As seen from Figure 1, the borate glasses exhibit a much stronger mixed alkali effect in the activation energy of the conductivity than the aluminosilicate glasses. In Figures 2 and 3, master curves of the frequency response of the real part of the conductivity are presented for different single and mixed alkali borate and aluminosilicate glasses. Each master curve has been generated from different isotherms by plotting σ′/σdc as a function of ν/(σdcT).22-24 At high values of ν/(σdcT), the master curves of single and mixed alkali glasses are indistinguishable, but in the crossover regime from dc to dispersive conductivity, a pronounced mixed alkali effect is observed. Remarkably, this effect is almost equally pronounced
Figure 3. Conductivity master curves of mixed Li/Na aluminosilicate glasses.
in both systems. According to the Kramers-Kronig relations, these distinct frequency responses correspond to a higher static dielectric constant of mixed alkali glasses as compared to single alkali glasses. This is shown exemplarily for one single alkali and one mixed alkali borate glass in Figure 4. In Figure 5 we present mechanical loss spectra of the aluminosilicate glass with equal amounts of Li2O and Na2O, i.e., with y ) 0.5. The mechanical loss, tan δ, is plotted as a function of temperature at different frequencies. Only one loss
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Figure 4. Experimental spectra of the dielectric function of one single alkali and one mixed alkali borate glass.
Figure 7. Mechanical loss peak height above background as a function of glass composition for the mixed Li/Na aluminosilicate glasses.
Figure 5. Mechanical loss spectra of the mixed aluminosilicate glass with y ) 0.5
Figure 8. Mechanical loss spectrum of the glass Na2O‚2MgO‚4SiO2 at a frequency of 1 Hz.
Figure 6. Activation energies of the dc conductivity, EAdc, and of the mechanical loss, EAml, plotted versus glass composition for the mixed Li/Na aluminosilicate glasses.
peak is observed. Its peak position is shifted to higher temperatures as the frequency is increased. Plotting the frequency logarithmically as a function of the inverse peak temperature (Arrhenius plot), we obtain a value of EAml ) 0.98 eV for the activation energy of the mechanical loss peak. In the spectra of all five aluminosilicate glasses studied here, we find only one mechanical loss peak. In Figure 6 the activation energy of this peak and the activation energy of the dc
conductivity are plotted versus glass composition. Except for the glass with y ) 0.041, the activation energy of the mechanical loss is considerably higher than that of the dc conductivity. In Figure 7, the height of the mechanical loss peak at a frequency of 1 Hz, tan δ1Hz max, is plotted as a function of glass composition. Except for the glass with y ) 0.041, the peak magnitudes are clearly larger than the typical magnitudes found for the alkali loss peak in single alkali glasses.5,6,21,25-27 Remarkably, the widths of the mechanical loss peaks do not depend on glass composition. For all glasses, the widths are approximately three times larger than the width of a Debye peak. In Figures 8 and 9, we present for comparison the mechanical loss spectra of two mixed alkali-alkaline earth glasses, Na2O‚ 2MgO‚4SiO2 (NMS) and Na2O‚2CaO‚4SiO2 (NCS) at a frequency of 1 Hz. In both spectra, two mechanical loss peaks are clearly distinguished, one low-temperature peak related to the faster sodium ion transport and one high-temperature peak related to the slower alkaline earth ion transport. The loss spectra differ, however, in two respects. For the NMS glass the separation of the loss peaks on the temperature scale is larger than for the NCS glass. Furthermore, the height of the lowtemperature peak is much smaller for the NCS glass as compared to the NMS glass. In Figure 10 the conductivity master plots of these glasses are compared with the master plot of the single alkali glass 0.3Na2O‚0.7B2O3. Within the experimental error, the shape of the master curves of all three glasses are identical, i.e., the
Ion Conducting Glasses
Figure 9. Mechanical loss spectrum of the glass Na2O‚2CaO‚4SiO2 at a frequency of 1 Hz.
Figure 10. Conductivity master plots of the glasses 0.3Na2O‚0.7B2O3,Na2O‚2MgO‚4SiO2 and Na2O‚2CaO‚4SiO2.
presence of the alkaline earth ions does not obviously affect the frequency response of the conductivity. 4. Discussion Our experimental results on the mixed alkali borate and aluminosilicate glasses clearly show that the mixed alkali effects in the activation energies of the ionic diffusivities, in the frequency response of the conductivity and in the height of the mechanical loss peaks, are not interrelated in a simple way. If one of these effects is particularly strong it does not follow that so are the others. The maximum in the activation energy of the dc conductivity as a function of glass composition reflects the diffusivity crossover of the dissimilar ions. Thus, the relatively flat maximum of EAdc(y) in the case of the aluminosilicate glasses suggests that in contrast to the borate glasses, the diffusivity of the Li+ ions is only slightly affected by the presence of the Na+ ions and vice versa. Nevertheless, the mixed alkali effect observed in the frequency response of the conductivity is almost as pronounced as in the borate glass system. The much larger static dielectric constant of mixed alkali glasses in comparison to single alkali glasses implies that the ion transport in mixed alkali glasses generates larger dipole moments being in phase with the electrical field. From this it follows that the ions have to cover larger distances in order to overcome the percolation barriers. These are the highest free energy barriers on a percolation path through the glassy network. At times shorter than the average time the ions need to overcome those barriers,
J. Phys. Chem. B, Vol. 103, No. 20, 1999 4125 they are effectively trapped. This leads to a phase difference between current density and electrical field, i.e., to a dielectric polarization. As seen from Figures 2, 3, and 4, this dielectric polarization effect is particularly strong for glasses with almost equal amounts of dissimilar alkali ions, i.e., for glasses with compositions close to the diffusivity crossover composition. From this observation the following question arises. What is the condition necessary for the occurrence of this effect, similar concentrations or similar diffusivities of the dissimilar ions? Here, it is helpful to consider the conductivity master curves of the NMS glass and of the NCS glass, see Figure 10. In these glasses, the alkaline earth ions are present in high concentrations, but their diffusivity is much lower than that of the sodium ions. The absence of the dielectric polarization effect in these glasses suggests that similar diffusivities are an important requirement for the occurrence of the dielectric polarization effect. Let us now analyze the mechanical losses of the mixed cation glasses. In all glasses, we find an Arrhenius-type relation between peak temperature and applied frequency, ν ) νo exp(-EA/kBTpeak). The values for the preexponential factor ν0 are always of the order of 1014 Hz. From this it follows that in an experiment performed at constant frequency ν, the activation energy EA is proportional to the peak temperature Tpeak. We first compare the mechanical loss spectra of the NMS glass and the NCS glass, see Figures 8 and 9. In the case of the NMS glass, the difference between the peak temperatures, ∆T, and, thus, the difference between the activation energies of the sodium and the magnesium transport, ∆EA, is large. In this case, the sodium peak height is comparable to that found in single alkali glasses. On the other hand, in the NCS glass, ∆EA as well as the sodium peak height are clearly smaller. This kind of correlation between ∆EA and the height of the lowtemperature peak is also observed for many mixed alkali glass systems.5 The mechanical loss spectra of the aluminosilicate glasses with y ) 0.33, 0.5, 0.67, and 0.959 show large mechanical loss peaks with activation energies higher than those of the dc conductivities. Therefore, we think that these peaks are hightemperature peaks with activation energies determined by the transport of the slower alkali ion. The low-temperature peaks seem to be undetectable. It is helpful to compare these spectra with the spectra of mixed Li/Na silicate glasses as obtained by Shelby and Day.5 These authors find that the magnitude of the low-temperature Li peak in a Li silicate glass is reduced by 50% when 2% of Li is replaced by Na. For those silicate glasses, ∆EA is clearly larger than for the aluminosilicate glasses studied here. Thus, the observation that the low-temperature peaks are undetectable in the case of the aluminosilicate glasses with y ) 0.33, 0.5, 0.67, and 0.959 is well in line with the correlation between ∆EA and the magnitude of the low-temperature peak as found for other mixed alkali glasses. On the other hand, it is remarkable that in the case of the glass with y ) 0.041, the transport of the faster Li+ ions seems to determine the mechanical loss behavior. The reason for this kind of behavior is still unclear. According to linear response theory, the mechanical caused by an ion transport process is given by28
tan δ(ω) )
NV ω M′kBT
|ˆ (Q)|2 Scoh,mi(Q,ω) ∑ Q
(1)
Here, NV and M′ are the number density of the mobile ions and the real part of the complex mechanical modulus, respectively. Scoh,mi denotes the coherent dynamic structure factor of the
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mobile ions, while ˆ (Q) are the Fourier components of the coupling function (r) between mechanical deformation of the sample and the potential energy of the mobile ions. The change of the potential energy of an ion at position r due to an external mechanical strain u is given by ∆Epot(r) ) (r)u. Neglecting the weak frequency dependence of M′ and integrating eq 1 with respect to ω results in
NV
∫0∞tan δ(ω) d ln ω ) M′k T ∑|ˆ (Q)|2 Scoh,mi(Q) B
(2)
Q
Scoh,mi(Q) is the static coherent structure factor of the mobile ions. In ion conducting glasses, the shape of mechanical loss peaks on the ln ω scale is found to be almost independent of the nature of the underlying transport process.5,21 In this case, the lefthand side of eq 2 is proportional to the height of the tan δ peak, i.e.,
tan δmax ∝
NV
|ˆ (Q)|2 Scoh,mi(Q) ∑ M′k T Q
(3)
B
Therefore, mechanical loss peaks are expected to be the more intense, • the higher the number density of mobile ions, • the more the potential energy of the ions is affected by the mechanical strain, • the more easily density fluctuations of the mobile ions on length scales 2π/|Q| do occur in thermal equilibrium, i.e., in the absence of a mechanical field. As we have shown in recent papers,22,24 the number densities of mobile ions can be estimated from the conductivity master curves of glasses. A change in the number density is reflected in a shift of the conductivity master curve on the ν/(σdc‚T) scale. In the case of the mixed alkali borate and aluminosilicate glasses studied here, we do not observe such a shift (Figures 2 and 3). From this we may argue that the reduction of the magnitude of the low-temperature mechanical loss peak in mixed cation glasses is not caused by a reduction of the number density of the faster ions. Thus, either the coupling function between the potential energy of the faster ions and the mechanical field, (r), or the static structure factor of the faster ions, Scoh,fi(Q), must depend on the value of ∆EA. Static structure factors can in principle be determined by diffraction methods, but it is very difficult to extract the partial structure factors of the mobile ions from the overall structure factor. To the best of our knowledge, experimental data of the partial structure factors of alkali ions in mixed alkali glasses are presently not available. The large magnitudes of high-temperature peaks as compared to low-temperature peaks have been explained by the assumption that high-temperature peaks are caused by a coupled diffusion process of faster and slower ions.29,30 The time window of this process is determined by the diffusion of the slower ions. Within this time window, the faster ions can be considered as a kind of electrical fluid, i.e., the fast diffusion of these ions can instantaneously compensate for electrical dipole moments caused by jumps of the slower ions. Because of this, density fluctuations of the slower ions on various length scales are easily possible, and a large amount of mechanical energy is absorbed. We have shown that the dielectric polarization effect as well as the mechanical loss peak height effect become the more pronounced, the more the value of ∆EA is decreased. ∆EA can be regarded as a measure of the degree of decoupling between
the transport processes. When ∆EA is large, i.e., when the time scales of the transport processes differ considerably, the faster and the slower ions move independently, and the dielectric polarization effect as well as the peak height effect are absent. With decreasing ∆EA, the peak height effect sets in first. This can be concluded from the conductivity and mechanical loss spectra of the NCS glass and of other mixed alkali-alkaline earth glasses.21,27 In the case of the NCS glass, ∆EA seems to be small enough for the peak height effect to occur but too large for the dielectric polarization effect to be detected (Figures 9 and 10). At even smaller values of ∆EA both effects are present. This has been observed for many mixed alkali glasses. 5. Conclusions We have studied the interrelation between different mixed cation effects in the conductivity spectra and the mechanical loss spectra of mixed alkali and mixed alkali-alkaline earth glasses. Apart from the classical mixed cation effects in the diffusivities of the dissimilar ions, a dielectric polarization effect and a mechanical loss peak height effect have been observed. The latter two effects become more and more pronounced, as the difference between the activation energies of the faster and the slower transport processes is reduced. We, therefore, suggest that these effects are fingerprints of an interdependence or, in other words, of an incomplete decoupling between these processes. Previously,31,32 interdependences have been discussed between the motions of like ions undergoing apparently identical processes such as may occur, for example, in the diffusive migration of sodium ions in all-sodium silicate glasses. Our conclusion arrived at here, namely that there are strong interdependences between unlike ions which are undergoing dissimilar processes with different activation energies, is a new result which may have profound implications for the theory of the mixed alkali effect and of diffusion processes in glasses, generally. Further experimental and theoretical work is required. Acknowledgment. We thank B. Tilley, J. Greengrass, H. L. Eaves, and C. Bracegirdle from the Pilkington Technology Center for providing glasses for our experiments, and D. Mackay from the Central Workshop, University of Aberdeen for preparing samples for the mechanical loss and the electrical conductivity measurements. Financial support by the Fonds der Chemischen Industrie is also gratefully acknowledged. B.R. is indebted to the Deutsche Forschungsgemeinschaft for a research grant making possible his stay in Aberdeen. References and Notes (1) Isard, J. O. J. Non-Cryst. Solids 1969, 1, 235. (2) Day, D. E. J. Non-Cryst. Solids 1976, 21, 343. (3) Ingram, M. D. Phys. Chem. Glasses 1987, 28, 215. (4) Ro¨tger, H. Wiss. Z. Friedrich-Schiller-Univ. Jena, Mathematisch- Naturwiss. Reihe 1958, 7, 237. (5) Shelby, J. E.; Day D. E. J. Am. Ceram. Soc. 1969, 52, 169. (6) Shelby, J. E.; Day D. E. J. Am. Ceram. Soc. 1970, 53, 182. (7) McVay, G. L.; Day, D. E. J. Am. Ceram. Soc. 1970, 53, 508. (8) Fleming, J. W.; Day, D. E. J. Am. Ceram. Soc. 1972, 55, 186. (9) Tomozawa, M.; Yoshiyagawa, M. Glastechn. Ber. 1983, 56k, 939. (10) Hyde, J. M.; Tomozawa, M. Phys. Chem. Glasses 1986, 27, 147. (11) Tomozawa, M. J. Non-Cryst. Solids 1996, 196, 280. (12) Mazurin, O. V. Structure of glass, Vol. 4; Consultants Bureau: New York, 1965; p 5. (13) Hendrickson, J. R.; Bray, P. J. Phys. Chem. Glasses 1972, 13, 107. (14) Dietzel, A. H. Phys. Chem. Glasses 1983, 24, 172. (15) Sato, R. K.; Kirkpatrick, R. J.; Brow, R. K. J. Non-Cryst. Solids 1992, 143, 257. (16) Charles, R. J. J. Am. Ceram. Soc. 1965, 48, 432. (17) Lengyel, B.; Boksay, Z. Z. Phys. Chem. 1963, 223, 49.
Ion Conducting Glasses (18) Stevels, J. M. Handbuch der Physik, Band 20; Springer: Berlin, 1957. (19) Bunde, A.; Ingram, M. D.; Maass, P. J. Non-Cryst. Solids 1994, 172-174, 1222. (20) Bunde, A.; Maass, P. Physica A 1993, 200, 80. (21) Roling, B.; Ingram, M. D. Solid State Ionics 1998, 105, 47. (22) Roling, B.; Happe, A.; Funke, K.; Ingram, M. D. Phys. ReV. Lett. 1997, 78, 2160. (23) Roling, B.; Lange, M.; Funke, K.; Ingram, M. D. Phys. ReV. B 1997, 56, 13619. (24) Roling, B. Solid State Ionics 1998, 105, 185. (25) Day, D. E.; Stevels, J. M. J. Non-Cryst. Solids 1974, 14, 165.
J. Phys. Chem. B, Vol. 103, No. 20, 1999 4127 (26) Phalippou, J.; Masson, S.; Boyer, A.; Zarzycki, J. J. Non-Cryst. Solids 1974, 14, 178. (27) Roling, B.; Ingram, M. D. Phys. ReV. B 1998, 57, 14192. (28) Kno¨dler, D.; Stiller, O.; Dieterich, W. Philos. Mag. B 1995, 71, 661. (29) van Ass, H. M. J. M.; Stevels, J. M. J. Non-Cryst. Solids 1974, 16, 27. (30) Meyer, M. Dissertation, University of Giessen, 1995. (31) Ngai, K. L. J. Non-Cryst. Solids 1996, 203, 232. (32) Funke, K. Prog. Solid St. Chem. 1993, 22, 111. (33) *Corresponding author.
