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Dielectric modulus analysis of mixed alkali LixRb1−xPO3 glasses
Journal of Non-Crystalline Solids 307–310 (2002) 1012–1016 www.elsevier.com/locate/jnoncrysol
Dielectric modulus analysis of mixed alkali Lix Rb1x PO3 glasses C. Karlsson a b
a,*
€rjesson , A. Mandanici b, A. Matic a, J. Swenson a, L. Bo
a
Department of Applied Physics, Chalmers University of Technology, S-412 96 G€oteborg, Sweden Dipartimento di Fisica and INFM, Universit a di Messina, Sal. Sperone 31, 98166 Messina, Italy
Abstract Dielectric data of mixed alkali Lix Rb1x PO3 glasses have been analysed in the modulus formalism. The imaginary part of the complex modulus was fitted with a KWW function. The stretching parameter b is larger in the mixed glasses as compared to the single alkali glasses, which indicates that the mixed alkali glasses behave as single alkali glasses of effectively lower concentrations. The finding is consistent with the random ion distribution model, which assumes that the two kinds of alkali ions are moving in distinctly different conduction pathways. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 63.30.Hs
1. Introduction The mixed alkali effect (MAE) in glasses, that is the non-linear deviation from additivity of diffusion-related properties as one cation A is replaced by another B, is a longstanding problem [1,2]. A number of models for the MAE have been proposed over the years [1–7]. Of these the most well known is probably the dynamic structure model (DSM) [3]. The DSM assumes that the different cations have different local environments, which means that A ions do not easily jump to B sites and vice versa. This leads to the formation of preferred A and B conduction paths. A (B) sites may change to B (A) sites through local structural
*
Corresponding author. E-mail address: [email protected] (C. Karlsson).
relaxation on short time scales. However, no unambigous evidence of such a local site relaxation have been found. Recently, it has been proposed with the random ion distribution model (RDM) that the MAE is mainly of structural origin, and that local relaxation is not necessary for the MAE [8]. Diffraction experiments in combination with reverse Monte Carlo (RMC) simulations of mixed alkali phosphate glasses showed that the cations are randomly distributed between the phosphate chains, and that the local environment for A and B cations are distinctly different [8,9]. Furthermore, it was shown that the environment of each cation species is essentially the same in glasses of different composition. However, EXAFS results have shown that alkali sites are not concentration invariant, but the effects are small and may not be visible in diffraction experiments or RMC modelling [10]. Thus, in the RDM the MAE is assumed
0022-3093/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 2 ) 0 1 5 6 6 - 1
C. Karlsson et al. / Journal of Non-Crystalline Solids 307–310 (2002) 1012–1016
to be a consequence of the random mixture of ions and the different local environments for different ions. Although the RDM is the first model based on experimental evidence, one should note that the ideas in the RDM have earlier been put forward, in a slightly different way, by Maass [6], who used computer simulations to show that the composition dependence of activation energies and diffusion coefficients could semi-quantitatively be accounted for by an model partly based on the DSM, but not including any local relaxation. The difference between the RDM and DSM is that the RDM is a simple, purely qualitative model based on the real structure of glasses. This is important since the aim is to explain the MAE on the basis of experimental findings, using the simplest possible model. In this work we have performed dielectric spectroscopy on Lix Rb1x PO3 glasses. Dielectric data can be presented in several ways, either as relative permittivity ^ðf Þ ¼ 0 i00 , conductivity r^ðf Þ, imb ðf Þ, or modulus M b ¼ M 0 þ iM 00 ¼ 1=^ pedance Z . In this paper we shall mainly use the modulus formalism. A more extensive conductivity study of the present glasses is to be published elsewhere. The dielectric modulus can be written as [11] Z 1 dU b ðxÞ ¼ M1 1 M expð ixtÞ dt ; dt 0 ð1Þ where UðtÞ describes the relaxation of the electric field E after the application of a step in the displacement D, that is EðtÞ ¼ Eð0ÞUðtÞ. The relaxation function is often approximated by a stretched exponential, or KWW function, b
UðtÞ ¼ f exp½ðt=sÞ ;
concentration of alkali ion A, the relative concentration of alkali ion B is hence 1 x). However, Tomozawa et al. found that this was the case only at high total alkali cation concentrations [12]. At low total alkali concentrations they found a minimum of b close to x ¼ 0:5. There is a wider range of studies of single alkali glasses [17,18], and the general trend is that b decreases with increasing alkali ion concentration. In this paper, we perform modulus analysis on Lix Rb1x PO3 glasses where x ¼ 0, 0.25, 0.5, 0.75 and 1.0. The b parameter is extracted from the modulus spectra by using a new fitting function proposed by Bergman [19]. We show that the width of the modulus (M 00 ) peak decreases, and equivalently b increases, in the mixed glasses as compared to the single alkali glasses.
2. Experimental The samples were prepared by ordinary melt quenching, as described in [9]. Dielectric measurements were performed on thin, circular samples coated with silver paint, using a Novocontrol Alpha High Resolution Dielectric Analyzer in the frequency range 102 –106 Hz. The temperature could be varied in between 110 and 450 K with an accuracy of 0.1 K by means of a nitrogen cooled cryostat. The measured quantity is the complex impedb ðf Þ ¼ U b =bI , where U b is the ance of the sample, Z b applied voltage and I the resulting current. From the complex impedance the complex modulus is calculated as b ¼ M 0 þ iM 00 ¼ i2pfC0 Z b; M
ð2Þ
where 0 < b 6 1 is the stretching parameter, which quantifies the deviation from Debye behaviour (where b ¼ 1). There are a few dielectric studies in the literature of mixed alkali, or more generally mixed mobile ion, glasses where the data is analysed in the modulus formalism [12–16]. In general, the b parameter tends to increase with mixing, and exhibit a maximum close to x ¼ 0:5 (x is the relative
1013
ð3Þ
where f ¼ x=2p is the frequency and C0 is the empty cell capacitance given by C0 ¼ 0 A=d, where 0 is the vacuum permittivity; A, the sample area and d, the sample thickness.
3. Results Fig. 1 shows the imaginary part M 00 ðf Þ of the modulus of RbPO3 (x ¼ 0) at different
1014
C. Karlsson et al. / Journal of Non-Crystalline Solids 307–310 (2002) 1012–1016
Fig. 1. Imaginary part of the dielectric modulus M 00 ðf Þ for RbPO3 (x ¼ 0). Solid curves are fits to Eq. (4) as discussed in the text.
temperatures. Solid curves are fits, which will be discussed later. In order to compare the shape of the modulus curves, the data points can be superimposed on each other by rescaling the axes. Fig. 2 shows experimental modulus curves for the compositions x ¼ 0, 0.5, and 1.0 scaled to such master curves. The scaling was done by dividing the M 00 and frequency scale with Mmax and the peak frequency f , respectively. Data taken at different temperatures are shown for each composition. It is clear that the mixed composition (x ¼ 0:5) exhibit narrower peaks. The width of the modulus peak, and its decrease in the mixed alkali glasses, can be quantified by the stretching parameter b in Eq. (2), if it is assumed that the relaxation function is described by the KWW function. The b parameter can be found by fitting Eqs. (1) and (2) to experimental data. This is, however, numerically difficult. The difficulties can be avoided by using an approximate frequency representation of the KWW function, which allows fitting directly in the frequency domain. Such a representation has been proposed by Bergman [19]. The imaginary part of the general susceptibility in the frequency domain due to a KWW relaxation function has been found to be well approximated by (for b P 0:4) [19]
Fig. 2. Master plot of experimental data for compositions (a) x ¼ 0, 0.5 and (b) x ¼ 0:5, 1.0. Note that several temperatures are shown for each composition. For x ¼ 0 the temperatures shown are 320; 340; . . . ; 440 K (20 K step), for x ¼ 0:5 the temperatures are 400–500 K (20 K step), and for x ¼ 1 the temperatures are 300–400 K (20 K step).
X 00 ðxÞ ¼
Xp00 b
b 1 b þ 1þb ½bðxp =xÞ þ ðx=xp Þ
:
ð4Þ
The parameters in Eq. (4) are related to the KWW parameters according to b b, Xp00 ðf =2Þb, xp 1=½sðb1 Cðb1 ÞÞ1=2 . Eq. (4) was originally devised for fitting the imaginary part of a general susceptibility X ðxÞ to a KWW function. However, it can equally well be used to give a frequency domain representation of the KWW function for the imaginary part of the modulus. The b values, resulting from fits of Eq. (4), are shown in Fig. 3. Some of the fitted curves are
C. Karlsson et al. / Journal of Non-Crystalline Solids 307–310 (2002) 1012–1016
1015
Fig. 4 shows the same data as in Fig. 1 in a log– log plot. It is clear that the high frequency points, in particular at lower temperatures, deviate from KWW behaviour. The deviations may be due to the constant loss, that is the Af 1:0 term in the general expression for the conductivity: r0 ðf Þ ¼ p r0 ½1 þ ðf =f0 Þ þ Af 1:0 . This could also explain why the curves of a given composition in Fig. 2 do not overlap completely on the high frequency flank. The discrepancy between the data and the KWW fits is hardly due to the use of Eq. (4), since the equation has been shown to work down to at least b ¼ 0:4 [19].
4. Discussion Fig. 3. Average values of the stretching parameter b, obtained from curve fitting, for different compositions.
shown in Fig. 1. During the fitting procedure it was noticed that b depends on the frequency interval chosen for fitting. This introduces uncertainties in the determinations of b. Although high frequency points were excluded in the fitting procedure, b of the same glass varied slightly for different temperatures. The error limits in Fig. 3 are estimated from this variation of b.
Fig. 4. Imaginary part of the dielectric modulus M 00 ðf Þ for RbPO3 plotted in a log–log plot. Solid curves are fits to Eq. (4).
It has been shown, by various authors, that the modulus peak width decreases, and hence the stretching parameter b increases, as the alkali concentration decreases in single alkali glasses [17,18]. We have here seen that the b parameter increases with mixing in mixed alkali glasses. A possible interpretation of the data is therefore that the mixed alkali glasses behave similar to single alkali glasses of lower cation concentrations, an idea that earlier has been put forward by Greaves and Ngai in their semi-empirical approach to the MAE, where they also made quantitative estimations of the dc conductivity in mixed alkali glasses [4]. That is, the x ¼ 0:75 composition essentially behaves as a ðLi2 OÞ0:75 –P2 O5 glass with respect to properties related to the cation diffusion. This can be understood qualitatively in the light of the RDM, which gives a more complete structural picture than available before, and which suggests that, regarding ionic conduction, a mixed alkali glass Ay B1y G effectively behaves like two diluted glasses Ay G and B1y G (A and B denotes alkali species) where the conduction takes place in distinctly different pathways for the two kinds of ions [8]. This should be valid in a first approximation, but, in order to obtain the proper curvature of the composition dependence of the cation diffusion coefficients, Coulomb interactions between the cations should be taken into account [6]. We would like to emphasize that the conclusions above should be valid regardless of the
1016
C. Karlsson et al. / Journal of Non-Crystalline Solids 307–310 (2002) 1012–1016
appropriateness of using the modulus formalism, since it is based on comparisons with earlier modulus analyses made in similar ways. There is currently a debate whether the conductivity or modulus formalism is the most appropriate way to present data from dielectric measurements [20–22], but we believe that the arguments given here are valid irrespective of whether one prefers the modulus approach or not. Although we have seen above that it is plausible that the increase of the b parameter in mixed alkali glasses is because the mixed glasses behave like diluted single alkali glasses (due to the experimental fact that the two kinds of ions have distinctly different local environments), we have, so far, not provided any explanation for why b changes with composition in single alkali glasses. One explanation, as provided by the coupling model, is that the coupling or degree of cooperativity, which is reflected in the coupling parameter n ¼ 1 b, between ions decrease when the concentration decreases [18,23,24]. Thus, decreased coupling would then be responsible for the modulus peak narrowing, and the corresponding increase in b in mixed alkali glasses. We may finally note that the idea of decreased coupling as x approaches 0.5 should be consistent with the observation that the typical cation jump distance tends to increase in mixed compositions [25]. This finding is also in agreement with [4], where increased jump distances in mixed alkali glasses were proposed to be the main reason for the MAE.
5. Conclusions By analyzing the modulus spectra of Li–Rb mixed alkali phosphate glasses we have shown that the b parameter increases with mixing. This indicates that the mixed alkali glasses behave like diluted single alkali glasses. The observation is qualitatively in agreement with the RDM, in which the two kinds of cations have distinctly different local environments and the diffusion is assumed to
mainly take place in separate pathways for the two kinds of ions.
Acknowledgements We would like to thank Professor K.L. Ngai for valuable discussions. This work was financially supported by the Swedish Research Council.
References [1] D.E. Day, J. Non-Cryst. Solids 21 (1976) 343. [2] M.D. Ingram, Glastech. Ber. Glass Sci. Technol. 67 (1994) 151. [3] A. Bunde, M.D. Ingram, P. Maass, J. Non-Cryst. Solids 172–174 (1994) 1222. [4] G.N. Greaves, K.L. Ngai, Phys. Rev. B 52 (1995) 6358. [5] B.M. Schulz, M. Dubiel, M. Schulz, J. Non-Cryst. Solids 241 (1999) 149. [6] P. Maass, J. Non-Cryst. Solids 255 (1999) 35. [7] R. Kirchheim, J. Non-Cryst. Solids 272 (2000) 85. [8] J. Swenson, A. Matic, C. Karlsson, L. B€ orjesson, C. Meneghini, W.S. Howells, Phys. Rev. B 63 (2001) 132202. [9] J. Swenson, A. Matic, A. Brodin, L. B€ orjesson, W.S. Howells, Phys. Rev. B 58 (1998) 11331. [10] G.N. Greaves, A.V. Chadwick, M.C. Jermy, M.E. Smith, R. Zhu, J. Non-Cryst. Solids 235–237 (1998) 766. [11] K.L. Ngai, C. Le on, Phys. Rev. B 60 (1999) 9396. [12] M. Tomozawa, J.M. Hyde, J.F. Cordaro, M. Yoshiyagawa, Phys. Chem. Glasses 33 (1992) 69. [13] G.N. Greaves, K.L. Ngai, Phys. Rev. B 52 (1995) 6358. [14] W.C. Huang, H. Jain, J. Non-Cryst. Solids 212 (1997) 117. [15] D.L. Sidebottom, J. Non-Cryst. Solids 255 (1999) 67. [16] A.R. Kulkarni, P. Lunkeheimer, A. Loidl, Mater. Chem. Phys. 63 (2000) 93. [17] S.W. Martin, C.A. Angell, J. Non-Cryst. Solids 83 (1986) 185. [18] K.L. Ngai, S.W. Martin, Phys. Rev. B 40 (1989) 10550. [19] R. Bergman, J. Appl. Phys. 88 (2000) 1356. [20] K.L. Ngai, C. Le on, Solid State Ionics 125 (1999) 81. [21] B. Roling, J. Non-Cryst. Solids 244 (1999) 34. [22] D.L. Sidebottom, B. Roling, K. Funke, Phys. Rev. B 63 (2000) 024301. [23] K.L. Ngai, R.W. Rendell, H. Jain, Phys. Rev. B 30 (1984) 2133. [24] K.L. Ngai, J.N. Mundy, H. Jain, O. Kanert, G. BelzerJollenbeck, Phys. Rev. B 39 (1989) 6169. [25] C. Karlsson, A. Mandanici, A. Matic, J. Swenson, L. B€ orjesson, in press.
Dielectric modulus analysis of mixed alkali Lix Rb1x PO3 glasses C. Karlsson a b
a,*
€rjesson , A. Mandanici b, A. Matic a, J. Swenson a, L. Bo
a
Department of Applied Physics, Chalmers University of Technology, S-412 96 G€oteborg, Sweden Dipartimento di Fisica and INFM, Universit a di Messina, Sal. Sperone 31, 98166 Messina, Italy
Abstract Dielectric data of mixed alkali Lix Rb1x PO3 glasses have been analysed in the modulus formalism. The imaginary part of the complex modulus was fitted with a KWW function. The stretching parameter b is larger in the mixed glasses as compared to the single alkali glasses, which indicates that the mixed alkali glasses behave as single alkali glasses of effectively lower concentrations. The finding is consistent with the random ion distribution model, which assumes that the two kinds of alkali ions are moving in distinctly different conduction pathways. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 63.30.Hs
1. Introduction The mixed alkali effect (MAE) in glasses, that is the non-linear deviation from additivity of diffusion-related properties as one cation A is replaced by another B, is a longstanding problem [1,2]. A number of models for the MAE have been proposed over the years [1–7]. Of these the most well known is probably the dynamic structure model (DSM) [3]. The DSM assumes that the different cations have different local environments, which means that A ions do not easily jump to B sites and vice versa. This leads to the formation of preferred A and B conduction paths. A (B) sites may change to B (A) sites through local structural
*
Corresponding author. E-mail address: [email protected] (C. Karlsson).
relaxation on short time scales. However, no unambigous evidence of such a local site relaxation have been found. Recently, it has been proposed with the random ion distribution model (RDM) that the MAE is mainly of structural origin, and that local relaxation is not necessary for the MAE [8]. Diffraction experiments in combination with reverse Monte Carlo (RMC) simulations of mixed alkali phosphate glasses showed that the cations are randomly distributed between the phosphate chains, and that the local environment for A and B cations are distinctly different [8,9]. Furthermore, it was shown that the environment of each cation species is essentially the same in glasses of different composition. However, EXAFS results have shown that alkali sites are not concentration invariant, but the effects are small and may not be visible in diffraction experiments or RMC modelling [10]. Thus, in the RDM the MAE is assumed
0022-3093/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 2 ) 0 1 5 6 6 - 1
C. Karlsson et al. / Journal of Non-Crystalline Solids 307–310 (2002) 1012–1016
to be a consequence of the random mixture of ions and the different local environments for different ions. Although the RDM is the first model based on experimental evidence, one should note that the ideas in the RDM have earlier been put forward, in a slightly different way, by Maass [6], who used computer simulations to show that the composition dependence of activation energies and diffusion coefficients could semi-quantitatively be accounted for by an model partly based on the DSM, but not including any local relaxation. The difference between the RDM and DSM is that the RDM is a simple, purely qualitative model based on the real structure of glasses. This is important since the aim is to explain the MAE on the basis of experimental findings, using the simplest possible model. In this work we have performed dielectric spectroscopy on Lix Rb1x PO3 glasses. Dielectric data can be presented in several ways, either as relative permittivity ^ðf Þ ¼ 0 i00 , conductivity r^ðf Þ, imb ðf Þ, or modulus M b ¼ M 0 þ iM 00 ¼ 1=^ pedance Z . In this paper we shall mainly use the modulus formalism. A more extensive conductivity study of the present glasses is to be published elsewhere. The dielectric modulus can be written as [11] Z 1 dU b ðxÞ ¼ M1 1 M expð ixtÞ dt ; dt 0 ð1Þ where UðtÞ describes the relaxation of the electric field E after the application of a step in the displacement D, that is EðtÞ ¼ Eð0ÞUðtÞ. The relaxation function is often approximated by a stretched exponential, or KWW function, b
UðtÞ ¼ f exp½ðt=sÞ ;
concentration of alkali ion A, the relative concentration of alkali ion B is hence 1 x). However, Tomozawa et al. found that this was the case only at high total alkali cation concentrations [12]. At low total alkali concentrations they found a minimum of b close to x ¼ 0:5. There is a wider range of studies of single alkali glasses [17,18], and the general trend is that b decreases with increasing alkali ion concentration. In this paper, we perform modulus analysis on Lix Rb1x PO3 glasses where x ¼ 0, 0.25, 0.5, 0.75 and 1.0. The b parameter is extracted from the modulus spectra by using a new fitting function proposed by Bergman [19]. We show that the width of the modulus (M 00 ) peak decreases, and equivalently b increases, in the mixed glasses as compared to the single alkali glasses.
2. Experimental The samples were prepared by ordinary melt quenching, as described in [9]. Dielectric measurements were performed on thin, circular samples coated with silver paint, using a Novocontrol Alpha High Resolution Dielectric Analyzer in the frequency range 102 –106 Hz. The temperature could be varied in between 110 and 450 K with an accuracy of 0.1 K by means of a nitrogen cooled cryostat. The measured quantity is the complex impedb ðf Þ ¼ U b =bI , where U b is the ance of the sample, Z b applied voltage and I the resulting current. From the complex impedance the complex modulus is calculated as b ¼ M 0 þ iM 00 ¼ i2pfC0 Z b; M
ð2Þ
where 0 < b 6 1 is the stretching parameter, which quantifies the deviation from Debye behaviour (where b ¼ 1). There are a few dielectric studies in the literature of mixed alkali, or more generally mixed mobile ion, glasses where the data is analysed in the modulus formalism [12–16]. In general, the b parameter tends to increase with mixing, and exhibit a maximum close to x ¼ 0:5 (x is the relative
1013
ð3Þ
where f ¼ x=2p is the frequency and C0 is the empty cell capacitance given by C0 ¼ 0 A=d, where 0 is the vacuum permittivity; A, the sample area and d, the sample thickness.
3. Results Fig. 1 shows the imaginary part M 00 ðf Þ of the modulus of RbPO3 (x ¼ 0) at different
1014
C. Karlsson et al. / Journal of Non-Crystalline Solids 307–310 (2002) 1012–1016
Fig. 1. Imaginary part of the dielectric modulus M 00 ðf Þ for RbPO3 (x ¼ 0). Solid curves are fits to Eq. (4) as discussed in the text.
temperatures. Solid curves are fits, which will be discussed later. In order to compare the shape of the modulus curves, the data points can be superimposed on each other by rescaling the axes. Fig. 2 shows experimental modulus curves for the compositions x ¼ 0, 0.5, and 1.0 scaled to such master curves. The scaling was done by dividing the M 00 and frequency scale with Mmax and the peak frequency f , respectively. Data taken at different temperatures are shown for each composition. It is clear that the mixed composition (x ¼ 0:5) exhibit narrower peaks. The width of the modulus peak, and its decrease in the mixed alkali glasses, can be quantified by the stretching parameter b in Eq. (2), if it is assumed that the relaxation function is described by the KWW function. The b parameter can be found by fitting Eqs. (1) and (2) to experimental data. This is, however, numerically difficult. The difficulties can be avoided by using an approximate frequency representation of the KWW function, which allows fitting directly in the frequency domain. Such a representation has been proposed by Bergman [19]. The imaginary part of the general susceptibility in the frequency domain due to a KWW relaxation function has been found to be well approximated by (for b P 0:4) [19]
Fig. 2. Master plot of experimental data for compositions (a) x ¼ 0, 0.5 and (b) x ¼ 0:5, 1.0. Note that several temperatures are shown for each composition. For x ¼ 0 the temperatures shown are 320; 340; . . . ; 440 K (20 K step), for x ¼ 0:5 the temperatures are 400–500 K (20 K step), and for x ¼ 1 the temperatures are 300–400 K (20 K step).
X 00 ðxÞ ¼
Xp00 b
b 1 b þ 1þb ½bðxp =xÞ þ ðx=xp Þ
:
ð4Þ
The parameters in Eq. (4) are related to the KWW parameters according to b b, Xp00 ðf =2Þb, xp 1=½sðb1 Cðb1 ÞÞ1=2 . Eq. (4) was originally devised for fitting the imaginary part of a general susceptibility X ðxÞ to a KWW function. However, it can equally well be used to give a frequency domain representation of the KWW function for the imaginary part of the modulus. The b values, resulting from fits of Eq. (4), are shown in Fig. 3. Some of the fitted curves are
C. Karlsson et al. / Journal of Non-Crystalline Solids 307–310 (2002) 1012–1016
1015
Fig. 4 shows the same data as in Fig. 1 in a log– log plot. It is clear that the high frequency points, in particular at lower temperatures, deviate from KWW behaviour. The deviations may be due to the constant loss, that is the Af 1:0 term in the general expression for the conductivity: r0 ðf Þ ¼ p r0 ½1 þ ðf =f0 Þ þ Af 1:0 . This could also explain why the curves of a given composition in Fig. 2 do not overlap completely on the high frequency flank. The discrepancy between the data and the KWW fits is hardly due to the use of Eq. (4), since the equation has been shown to work down to at least b ¼ 0:4 [19].
4. Discussion Fig. 3. Average values of the stretching parameter b, obtained from curve fitting, for different compositions.
shown in Fig. 1. During the fitting procedure it was noticed that b depends on the frequency interval chosen for fitting. This introduces uncertainties in the determinations of b. Although high frequency points were excluded in the fitting procedure, b of the same glass varied slightly for different temperatures. The error limits in Fig. 3 are estimated from this variation of b.
Fig. 4. Imaginary part of the dielectric modulus M 00 ðf Þ for RbPO3 plotted in a log–log plot. Solid curves are fits to Eq. (4).
It has been shown, by various authors, that the modulus peak width decreases, and hence the stretching parameter b increases, as the alkali concentration decreases in single alkali glasses [17,18]. We have here seen that the b parameter increases with mixing in mixed alkali glasses. A possible interpretation of the data is therefore that the mixed alkali glasses behave similar to single alkali glasses of lower cation concentrations, an idea that earlier has been put forward by Greaves and Ngai in their semi-empirical approach to the MAE, where they also made quantitative estimations of the dc conductivity in mixed alkali glasses [4]. That is, the x ¼ 0:75 composition essentially behaves as a ðLi2 OÞ0:75 –P2 O5 glass with respect to properties related to the cation diffusion. This can be understood qualitatively in the light of the RDM, which gives a more complete structural picture than available before, and which suggests that, regarding ionic conduction, a mixed alkali glass Ay B1y G effectively behaves like two diluted glasses Ay G and B1y G (A and B denotes alkali species) where the conduction takes place in distinctly different pathways for the two kinds of ions [8]. This should be valid in a first approximation, but, in order to obtain the proper curvature of the composition dependence of the cation diffusion coefficients, Coulomb interactions between the cations should be taken into account [6]. We would like to emphasize that the conclusions above should be valid regardless of the
1016
C. Karlsson et al. / Journal of Non-Crystalline Solids 307–310 (2002) 1012–1016
appropriateness of using the modulus formalism, since it is based on comparisons with earlier modulus analyses made in similar ways. There is currently a debate whether the conductivity or modulus formalism is the most appropriate way to present data from dielectric measurements [20–22], but we believe that the arguments given here are valid irrespective of whether one prefers the modulus approach or not. Although we have seen above that it is plausible that the increase of the b parameter in mixed alkali glasses is because the mixed glasses behave like diluted single alkali glasses (due to the experimental fact that the two kinds of ions have distinctly different local environments), we have, so far, not provided any explanation for why b changes with composition in single alkali glasses. One explanation, as provided by the coupling model, is that the coupling or degree of cooperativity, which is reflected in the coupling parameter n ¼ 1 b, between ions decrease when the concentration decreases [18,23,24]. Thus, decreased coupling would then be responsible for the modulus peak narrowing, and the corresponding increase in b in mixed alkali glasses. We may finally note that the idea of decreased coupling as x approaches 0.5 should be consistent with the observation that the typical cation jump distance tends to increase in mixed compositions [25]. This finding is also in agreement with [4], where increased jump distances in mixed alkali glasses were proposed to be the main reason for the MAE.
5. Conclusions By analyzing the modulus spectra of Li–Rb mixed alkali phosphate glasses we have shown that the b parameter increases with mixing. This indicates that the mixed alkali glasses behave like diluted single alkali glasses. The observation is qualitatively in agreement with the RDM, in which the two kinds of cations have distinctly different local environments and the diffusion is assumed to
mainly take place in separate pathways for the two kinds of ions.
Acknowledgements We would like to thank Professor K.L. Ngai for valuable discussions. This work was financially supported by the Swedish Research Council.
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