- Email: [email protected]
Frequency dependent conductivity of single alkali and mixed alkali phosphate glasses
Journal of Non-Crystalline Solids 345&346 (2004) 514–517 www.elsevier.com/locate/jnoncrysol
Frequency dependent conductivity of single alkali and mixed alkali phosphate glasses a,*
A. Mandanici a
, C. Karlsson b, A. Matic b, J. Swenson b, M. Cutroni a, L. Bo¨rjesson
b
Dipartimento di Fisica and INFM, Universita` di Messina, Salita Sperone, 31 98100 Messina, Italy Department of Applied Physics, Chalmers University of Technology, S-412 96 Go¨teborg, Sweden
b
Available online 6 October 2004
Abstract The frequency dependent conductivity r(f) of single alkali glasses LiPO3 and RbPO3 has been compared with the conductivity of the mixed alkali glass Li0.5Rb0.5PO3. The low frequency plateau value r0 of the conductivity drops by several orders of magnitude in the mixed alkali glass. In this work the effect of mixed alkali on the shape of the conductivity r(f) in the dispersive region has been analyzed. The slope of the conductivity curve, in the log r vs. log f plot, increases more rapidly in the single alkali glasses than in the mixed composition. The onset of the dispersive region is more gradual in the mixed alkali glass. The results indicate that the mixed alkali glass behaves as a dilute single alkali glass. Ó 2004 Elsevier B.V. All rights reserved. PACS: 66.10.Ed; 61.43.Fs; 66.30.Hs
1. Introduction
2. Experimental procedures
Many aspects of the conduction in ionic glasses are still to be understood [1,2]. One of these is the mixed alkali effect: in a glass containing two different mobile cationic species, the conductivity is lower than in the corresponding glasses containing only one of the conducting species. Recently the mixed alkali effect has been investigated in metaphosphate glasses with cations of different ionic radii [3]. The dc conductivity in the mixed alkali glass Li0.5Rb0.5PO3 is six orders of magnitude lower than the conductivity of single alkali glasses, LiPO3 and RbPO3 [3]. In this work the effect of mixed alkali on the shape of the conductivity r(f) of the metaphosphate glasses in the dispersive region has been studied. The observed behavior is analogous to that of single alkali glasses with very low concentration of mobile ions.
Samples were prepared by melt quenching as described in [3]. The complex impedance of thin, circular samples coated with silver paint, was measured using a Novocontrol Alpha High Resolution Dielectric Analyzer in the frequency range 102–107 Hz. Isothermal spectra were recorded at selected temperatures between 250 K and 500 K using a nitrogen cooled cryostat, with an accuracy of ±0.1 K. The conductivity was calculated from the measured impedance Z* using the equation r*(f) = r(f) ir00 (f) = e0/C0Z*, where C0 denotes the empty cell capacitance.
*
Corresponding author. Tel.: +39 90 676 5039; fax: +39 90 395 004. E-mail address: [email protected] (A. Mandanici).
0022-3093/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2004.08.076
3. Results Fig. 1 shows the frequency dependent conductivity for the LiPO3, RbPO3, and Li0.5Rb0.5PO3 glasses at 400 K. The frequency fpol corresponding to the onset of electrode polarization effects (marked by open circles
A. Mandanici et al. / Journal of Non-Crystalline Solids 345&346 (2004) 514–517 -4
10
10
-5
-6
LiPO 3 400 K - 250 K RbPO3 420 K - 280 K Li0.5Rb0.5PO 3 500 K - 400 K
10
3
10
2
10
1
10
0
-7
0
10
-8
10
(f)/
cm-1)
10
LiPO 3 RbPO 3 Li 0.50Rb 0.50PO 3
(
-1
4
400 K
10
515
-9
10
-10
10
-11
10
-12
10
-3
10
-2
10
-1
10
0
10
1
2
10
10
3
10
4
10
5
10
6
10
7
10
f (Hz)
in Fig. 1) was evaluated using complex impedance plots, where Z00 (f) is plotted vs. Z 0 (f). Only the frequency range f > fpol corresponding to the bulk properties of the samples was considered for further data analysis. The conductivity plateau value is much lower in the Li0.5Rb0.5PO3 glass than in the single alkali glasses. As the frequency increases the conductivity rises above its plateau value featuring a dispersive behavior. As proposed by Jonscher [4], a simple power law allows to describe the smooth transition from the constant value to the dispersive regime p
ð1Þ
The onset of the dispersive region is characterized by the frequency f0 (see the vertical bars in Fig. 1), which can be determined as the frequency at which the condition r(f) = 2r0 is fulfilled. The low frequency plateau value of the conductivity, r0 is usually considered equal to the dc conductivity rdc. The power law exponent p corresponds to limiting value of the slope of the conductivity curve in the dispersive region, in a plot of log r(f) vs. log f. When experimental data are available up to high frequencies and low temperatures, eventually the slope of the conductivity curve in the dispersive region can achieve a value very close to 1. In these cases, a further conductivity contribution, known as nearly constant loss [5–7], has to be considered, rðf Þ ¼ r0 ½1 þ ðf =f0 Þp þ Af :
6
10
7
10
8
10
9
10
[ f / (Hz)] / [
Fig. 1. Frequency dependent conductivity of Li1xRbxPO3 glasses (x = 0, 0.5, 1) at 400 K in the frequency range between 0.01 Hz and 10 MHz. The open symbols correspond to the onset fpol of electrode polarization effects at low frequencies. The vertical bars denote the frequency f0 which appears in Eq. (1).
rðf Þ ¼ r0 ½1 þ ðf =f0 Þ :
10
ð2Þ
In order to compare the shape of the conductivity response the following scaling technique was adopted: conductivity data for each composition at different temperatures were plotted as log(r/rdc) vs. log(f/rdcT). If the shape of the conductivity dispersion does not depend on temperature, scaled conductivity data at different temperatures superimpose in a common curve.
10
10
11
10
12
10
13
10
14
10
15
T . ( cm / K)] 0
Fig. 2. Summerfield scaling of conductivity for the single LiPO3, and RbPO3 glasses and the mixed alkali Li0.50Rb0.50PO3 glass.
This approach was first proposed by Summerfield for the conductivity of amorphous semiconductors [8]. Many glassy ionics exactly follow this scaling, although some exceptions have been proposed recently [9,10]. Results of the Summerfield scaling procedure for the samples investigated are shown in Fig. 2. Conductivity data for LiPO3 at different temperatures in the range 250– 400 K collapse on a single curve. A scaled conductivity curve is also obtained for the RbPO3 glass in the temperature range 280–420 K, and for the Li0.5Rb0.5PO3 glass in the temperature range 400–500 K. While the resulting curves for the single alkali glasses are closely similar, the mixed alkali glass shows a different behavior. The conductivity r(f) of the mixed alkali glass increases more slowly than the single alkali glasses as the frequency increases. The shape of the conductivity r(f) in the dispersive region can be analyzed using the slope of the conductivity curve in a plot of log r(f) vs. log f. Following the approach introduced by Schroder and Dyre [10] the slope of the conductivity curve was plotted versus r(f)/r0 as shown in Fig. 3, in order to enhance the difference in shape between the conductivity dispersion data of the different glasses. The approximate value of the slope D[log r(f)]/D[log(f)] at each frequency was estimated using the forward incremental ratio [log r(fi+1) log r(fi)]/[log(fi+1) log(fi)]. Fig. 3 shows that the mixed alkali glass behaves differently from the single alkali glasses at the onset of the dispersive region, whereas differences between LiPO3 and RbPO3 become relevant only at higher frequencies/shorter timescales. Fig. 4 shows the behavior of the slope D[log r(f)]/ D[log(f)] as a function of frequency, hence excluding any scaling parameter. It can be observed that in single alkali glasses the slope of conductivity curve increases almost abruptly above the low frequency plateau while in the mixed alkali glass the onset of dispersion is less marked and the increase of the slope is more gradual.
516
A. Mandanici et al. / Journal of Non-Crystalline Solids 345&346 (2004) 514–517
0.8
log ( f )
0.6
log ( f ) /
0.7
0.4
0.5
RbPO3 300 K LiPO 3 240 K Li0.5Rb0.5PO 3 420 K
0.3 0.2 0.1 0.0 0.0
0.5
1.0
1.5
2.0
(f)/
log [
2.5
3.0
3.5
] 0
Fig. 3. Approximate slope of the conductivity dispersion r(f) in Li1xRbxPO3 (x = 0, 0.5, 1) glasses as a function of the scaled conductivity r(f)/r0 [for each composition, instead of a master curve, we have reported conductivity data at a single temperature at which the largest part of the conductivity dispersion fell in the frequency range measured].
0.8
log ( f )
0.7 0.6 0.5
log
(f)/
0.4 0.3 0.2
RbPO3 300 K LiPO 3 240 K Li0.5Rb0.5PO 3 420 K
0.1 0.0 -0.1 -2 10
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
ferent approaches in Figs. 2–4. In previous work [3], it was shown that a much larger activation energy (about 1.7 times larger) characterizes the dc conductivity in the mixed alkali glass compared to the single alkali glasses, and also indications of a larger hopping distance was obtained, like in glasses with very low density of mobile ions. The deviation of the mixed alkali glass response from that of the single alkali glasses, shown in Figs. 2– 4, is similar to that firstly noticed by Roling and Martiny [11] in a mixed lithium–sodium borate glass. They also found that in (single alkali) sodium borate glasses the slope D[log r(f)]/D[log(f)] of the conductivity increases more slowly in the samples with lower concentration of sodium ions. The transition from the conductivity plateau to the dispersive region is more gradual in the glasses with lower alkali content [11]. Compared to the present investigation these results would suggest that, with respect to the conductivity, mixed alkali glasses behave as diluted single alkali glasses. This is in agreement with the conclusions drawn from a separate analysis in terms of electrical modulus [12]. It is not clear if the existence of different local environments in mixed alkali glasses, as expected for instance by some microscopic model [13], could bring distinctive features in the frequency dependence of conductivity of mixed alkali glasses compared to the r(f) of dilute single alkali glasses. However, useful information could be provided from experimental studies at higher frequencies, to fully characterize the dispersive behavior up to the nearly constant loss regime. 5. Conclusion
6
10
7
f (Hz) Fig. 4. Frequency dependence of the approximate slope of conductivity for single and mixed alkali metaphosphate glasses Li1xRbxPO3 at selected temperatures.
4. Discussion The Li1xRbxPO3 mixed alkali system is interesting since the large size difference between the cationic species results in a large mixed alkali effect on conductivity [3]. In fact, the dc conductivity has been found to be six orders of magnitude lower in the mixed alkali glass than in LiPO3 and RbPO3 at 300 K. Aiming to compare also the frequency dependent behavior of the conductivity of Li1xRbxPO3 mixed alkali glasses (x = 0, 0.5, 1), in the present work the validity of Summerfield scaling is investigated. First of all it has been observed that the two single alkali glasses show a closely similar behavior. Secondly, the behavior of the frequency dependent conductivity in the mixed Li0.5Rb0.5PO3 glass is different from that of the single alkali glasses, as proved using dif-
The mixing of alkali mobile species in metaphosphate glasses leads to a remarkable decrease of the dc conductivity. Using different approaches we also show a change in the shape of the conductivity response r(f). The slope of the conductivity curve vs. frequency increases almost abruptly above the plateau in single alkali glasses. In the mixed alkali glass the onset of the dispersive region is more gradual, i.e. the increase of slope of the conductivity curve in a log–log plot is slower in the mixed than in the single alkali glasses. A similar behavior has been observed in glasses with low concentration of mobile ions, so the mixed alkali glass behaves like a single alkali glass with low ionic concentration. References [1] A. Bunde, K. Funke, M.D. Ingram, Solid State Ionics 105 (1997) 1. [2] C.A. Angell, K.L. Ngai, G.B. McKenna, P.F. McMillan, S.W. Martin, J. Appl. Phys. 88 (2000) 3113. [3] C. Karlsson, A. Mandanici, A. Matic, J. Swenson, L. Bo¨rjesson, Phys. Rev. B 68 (2003) 64202. [4] A.K. Jonscher, Nature (London) 267 (1977) 673.
A. Mandanici et al. / Journal of Non-Crystalline Solids 345&346 (2004) 514–517 [5] W.K. Lee, J.F. Liu, A.S. Nowick, Phys. Rev. Lett. 67 (1991) 1559; W.K. Lee, B.S. Lim, J.F. Liu, A.S. Nowick, Solid State Ionics 53–56 (1992) 831; A.S. Nowick, A.V. Vaysleyb, W. Liu, Solid State Ionics 105 (1998) 121. [6] K.L. Ngai, J. Chem. Phys. 110 (1999) 10576. [7] D.L. Sidebottom, Phys. Rev. B 61 (2000) 14507. [8] S. Summerfield, Philos. Mag. B 52 (1985) 9.
517
[9] S. Murugavel, B. Roling, Phys. Rev. Lett. 89 (2002) 195902. [10] T.B. Schroder, J.C. Dyre, Phys. Rev. Lett. 84 (2000) 310; J.C. Dyre, T.B. Schroder, Rev. Mod. Phys. 72 (2000) 873. [11] B. Roling, C. Martiny, Phys. Rev. Lett. 85 (2000) 1274. [12] C. Karlsson, A. Mandanici, A. Matic, J. Swenson, L. Bo¨rjesson, J. Non-Cryst. Solids 307–310 (2002) 1012. [13] J. Swenson, A. Matic, C. Karlsson, L. Bo¨rjesson, C. Meneghini, W.S. Howells, Phys. Rev. B 63 (2001) 132202.
Frequency dependent conductivity of single alkali and mixed alkali phosphate glasses a,*
A. Mandanici a
, C. Karlsson b, A. Matic b, J. Swenson b, M. Cutroni a, L. Bo¨rjesson
b
Dipartimento di Fisica and INFM, Universita` di Messina, Salita Sperone, 31 98100 Messina, Italy Department of Applied Physics, Chalmers University of Technology, S-412 96 Go¨teborg, Sweden
b
Available online 6 October 2004
Abstract The frequency dependent conductivity r(f) of single alkali glasses LiPO3 and RbPO3 has been compared with the conductivity of the mixed alkali glass Li0.5Rb0.5PO3. The low frequency plateau value r0 of the conductivity drops by several orders of magnitude in the mixed alkali glass. In this work the effect of mixed alkali on the shape of the conductivity r(f) in the dispersive region has been analyzed. The slope of the conductivity curve, in the log r vs. log f plot, increases more rapidly in the single alkali glasses than in the mixed composition. The onset of the dispersive region is more gradual in the mixed alkali glass. The results indicate that the mixed alkali glass behaves as a dilute single alkali glass. Ó 2004 Elsevier B.V. All rights reserved. PACS: 66.10.Ed; 61.43.Fs; 66.30.Hs
1. Introduction
2. Experimental procedures
Many aspects of the conduction in ionic glasses are still to be understood [1,2]. One of these is the mixed alkali effect: in a glass containing two different mobile cationic species, the conductivity is lower than in the corresponding glasses containing only one of the conducting species. Recently the mixed alkali effect has been investigated in metaphosphate glasses with cations of different ionic radii [3]. The dc conductivity in the mixed alkali glass Li0.5Rb0.5PO3 is six orders of magnitude lower than the conductivity of single alkali glasses, LiPO3 and RbPO3 [3]. In this work the effect of mixed alkali on the shape of the conductivity r(f) of the metaphosphate glasses in the dispersive region has been studied. The observed behavior is analogous to that of single alkali glasses with very low concentration of mobile ions.
Samples were prepared by melt quenching as described in [3]. The complex impedance of thin, circular samples coated with silver paint, was measured using a Novocontrol Alpha High Resolution Dielectric Analyzer in the frequency range 102–107 Hz. Isothermal spectra were recorded at selected temperatures between 250 K and 500 K using a nitrogen cooled cryostat, with an accuracy of ±0.1 K. The conductivity was calculated from the measured impedance Z* using the equation r*(f) = r(f) ir00 (f) = e0/C0Z*, where C0 denotes the empty cell capacitance.
*
Corresponding author. Tel.: +39 90 676 5039; fax: +39 90 395 004. E-mail address: [email protected] (A. Mandanici).
0022-3093/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2004.08.076
3. Results Fig. 1 shows the frequency dependent conductivity for the LiPO3, RbPO3, and Li0.5Rb0.5PO3 glasses at 400 K. The frequency fpol corresponding to the onset of electrode polarization effects (marked by open circles
A. Mandanici et al. / Journal of Non-Crystalline Solids 345&346 (2004) 514–517 -4
10
10
-5
-6
LiPO 3 400 K - 250 K RbPO3 420 K - 280 K Li0.5Rb0.5PO 3 500 K - 400 K
10
3
10
2
10
1
10
0
-7
0
10
-8
10
(f)/
cm-1)
10
LiPO 3 RbPO 3 Li 0.50Rb 0.50PO 3
(
-1
4
400 K
10
515
-9
10
-10
10
-11
10
-12
10
-3
10
-2
10
-1
10
0
10
1
2
10
10
3
10
4
10
5
10
6
10
7
10
f (Hz)
in Fig. 1) was evaluated using complex impedance plots, where Z00 (f) is plotted vs. Z 0 (f). Only the frequency range f > fpol corresponding to the bulk properties of the samples was considered for further data analysis. The conductivity plateau value is much lower in the Li0.5Rb0.5PO3 glass than in the single alkali glasses. As the frequency increases the conductivity rises above its plateau value featuring a dispersive behavior. As proposed by Jonscher [4], a simple power law allows to describe the smooth transition from the constant value to the dispersive regime p
ð1Þ
The onset of the dispersive region is characterized by the frequency f0 (see the vertical bars in Fig. 1), which can be determined as the frequency at which the condition r(f) = 2r0 is fulfilled. The low frequency plateau value of the conductivity, r0 is usually considered equal to the dc conductivity rdc. The power law exponent p corresponds to limiting value of the slope of the conductivity curve in the dispersive region, in a plot of log r(f) vs. log f. When experimental data are available up to high frequencies and low temperatures, eventually the slope of the conductivity curve in the dispersive region can achieve a value very close to 1. In these cases, a further conductivity contribution, known as nearly constant loss [5–7], has to be considered, rðf Þ ¼ r0 ½1 þ ðf =f0 Þp þ Af :
6
10
7
10
8
10
9
10
[ f / (Hz)] / [
Fig. 1. Frequency dependent conductivity of Li1xRbxPO3 glasses (x = 0, 0.5, 1) at 400 K in the frequency range between 0.01 Hz and 10 MHz. The open symbols correspond to the onset fpol of electrode polarization effects at low frequencies. The vertical bars denote the frequency f0 which appears in Eq. (1).
rðf Þ ¼ r0 ½1 þ ðf =f0 Þ :
10
ð2Þ
In order to compare the shape of the conductivity response the following scaling technique was adopted: conductivity data for each composition at different temperatures were plotted as log(r/rdc) vs. log(f/rdcT). If the shape of the conductivity dispersion does not depend on temperature, scaled conductivity data at different temperatures superimpose in a common curve.
10
10
11
10
12
10
13
10
14
10
15
T . ( cm / K)] 0
Fig. 2. Summerfield scaling of conductivity for the single LiPO3, and RbPO3 glasses and the mixed alkali Li0.50Rb0.50PO3 glass.
This approach was first proposed by Summerfield for the conductivity of amorphous semiconductors [8]. Many glassy ionics exactly follow this scaling, although some exceptions have been proposed recently [9,10]. Results of the Summerfield scaling procedure for the samples investigated are shown in Fig. 2. Conductivity data for LiPO3 at different temperatures in the range 250– 400 K collapse on a single curve. A scaled conductivity curve is also obtained for the RbPO3 glass in the temperature range 280–420 K, and for the Li0.5Rb0.5PO3 glass in the temperature range 400–500 K. While the resulting curves for the single alkali glasses are closely similar, the mixed alkali glass shows a different behavior. The conductivity r(f) of the mixed alkali glass increases more slowly than the single alkali glasses as the frequency increases. The shape of the conductivity r(f) in the dispersive region can be analyzed using the slope of the conductivity curve in a plot of log r(f) vs. log f. Following the approach introduced by Schroder and Dyre [10] the slope of the conductivity curve was plotted versus r(f)/r0 as shown in Fig. 3, in order to enhance the difference in shape between the conductivity dispersion data of the different glasses. The approximate value of the slope D[log r(f)]/D[log(f)] at each frequency was estimated using the forward incremental ratio [log r(fi+1) log r(fi)]/[log(fi+1) log(fi)]. Fig. 3 shows that the mixed alkali glass behaves differently from the single alkali glasses at the onset of the dispersive region, whereas differences between LiPO3 and RbPO3 become relevant only at higher frequencies/shorter timescales. Fig. 4 shows the behavior of the slope D[log r(f)]/ D[log(f)] as a function of frequency, hence excluding any scaling parameter. It can be observed that in single alkali glasses the slope of conductivity curve increases almost abruptly above the low frequency plateau while in the mixed alkali glass the onset of dispersion is less marked and the increase of the slope is more gradual.
516
A. Mandanici et al. / Journal of Non-Crystalline Solids 345&346 (2004) 514–517
0.8
log ( f )
0.6
log ( f ) /
0.7
0.4
0.5
RbPO3 300 K LiPO 3 240 K Li0.5Rb0.5PO 3 420 K
0.3 0.2 0.1 0.0 0.0
0.5
1.0
1.5
2.0
(f)/
log [
2.5
3.0
3.5
] 0
Fig. 3. Approximate slope of the conductivity dispersion r(f) in Li1xRbxPO3 (x = 0, 0.5, 1) glasses as a function of the scaled conductivity r(f)/r0 [for each composition, instead of a master curve, we have reported conductivity data at a single temperature at which the largest part of the conductivity dispersion fell in the frequency range measured].
0.8
log ( f )
0.7 0.6 0.5
log
(f)/
0.4 0.3 0.2
RbPO3 300 K LiPO 3 240 K Li0.5Rb0.5PO 3 420 K
0.1 0.0 -0.1 -2 10
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
ferent approaches in Figs. 2–4. In previous work [3], it was shown that a much larger activation energy (about 1.7 times larger) characterizes the dc conductivity in the mixed alkali glass compared to the single alkali glasses, and also indications of a larger hopping distance was obtained, like in glasses with very low density of mobile ions. The deviation of the mixed alkali glass response from that of the single alkali glasses, shown in Figs. 2– 4, is similar to that firstly noticed by Roling and Martiny [11] in a mixed lithium–sodium borate glass. They also found that in (single alkali) sodium borate glasses the slope D[log r(f)]/D[log(f)] of the conductivity increases more slowly in the samples with lower concentration of sodium ions. The transition from the conductivity plateau to the dispersive region is more gradual in the glasses with lower alkali content [11]. Compared to the present investigation these results would suggest that, with respect to the conductivity, mixed alkali glasses behave as diluted single alkali glasses. This is in agreement with the conclusions drawn from a separate analysis in terms of electrical modulus [12]. It is not clear if the existence of different local environments in mixed alkali glasses, as expected for instance by some microscopic model [13], could bring distinctive features in the frequency dependence of conductivity of mixed alkali glasses compared to the r(f) of dilute single alkali glasses. However, useful information could be provided from experimental studies at higher frequencies, to fully characterize the dispersive behavior up to the nearly constant loss regime. 5. Conclusion
6
10
7
f (Hz) Fig. 4. Frequency dependence of the approximate slope of conductivity for single and mixed alkali metaphosphate glasses Li1xRbxPO3 at selected temperatures.
4. Discussion The Li1xRbxPO3 mixed alkali system is interesting since the large size difference between the cationic species results in a large mixed alkali effect on conductivity [3]. In fact, the dc conductivity has been found to be six orders of magnitude lower in the mixed alkali glass than in LiPO3 and RbPO3 at 300 K. Aiming to compare also the frequency dependent behavior of the conductivity of Li1xRbxPO3 mixed alkali glasses (x = 0, 0.5, 1), in the present work the validity of Summerfield scaling is investigated. First of all it has been observed that the two single alkali glasses show a closely similar behavior. Secondly, the behavior of the frequency dependent conductivity in the mixed Li0.5Rb0.5PO3 glass is different from that of the single alkali glasses, as proved using dif-
The mixing of alkali mobile species in metaphosphate glasses leads to a remarkable decrease of the dc conductivity. Using different approaches we also show a change in the shape of the conductivity response r(f). The slope of the conductivity curve vs. frequency increases almost abruptly above the plateau in single alkali glasses. In the mixed alkali glass the onset of the dispersive region is more gradual, i.e. the increase of slope of the conductivity curve in a log–log plot is slower in the mixed than in the single alkali glasses. A similar behavior has been observed in glasses with low concentration of mobile ions, so the mixed alkali glass behaves like a single alkali glass with low ionic concentration. References [1] A. Bunde, K. Funke, M.D. Ingram, Solid State Ionics 105 (1997) 1. [2] C.A. Angell, K.L. Ngai, G.B. McKenna, P.F. McMillan, S.W. Martin, J. Appl. Phys. 88 (2000) 3113. [3] C. Karlsson, A. Mandanici, A. Matic, J. Swenson, L. Bo¨rjesson, Phys. Rev. B 68 (2003) 64202. [4] A.K. Jonscher, Nature (London) 267 (1977) 673.
A. Mandanici et al. / Journal of Non-Crystalline Solids 345&346 (2004) 514–517 [5] W.K. Lee, J.F. Liu, A.S. Nowick, Phys. Rev. Lett. 67 (1991) 1559; W.K. Lee, B.S. Lim, J.F. Liu, A.S. Nowick, Solid State Ionics 53–56 (1992) 831; A.S. Nowick, A.V. Vaysleyb, W. Liu, Solid State Ionics 105 (1998) 121. [6] K.L. Ngai, J. Chem. Phys. 110 (1999) 10576. [7] D.L. Sidebottom, Phys. Rev. B 61 (2000) 14507. [8] S. Summerfield, Philos. Mag. B 52 (1985) 9.
517
[9] S. Murugavel, B. Roling, Phys. Rev. Lett. 89 (2002) 195902. [10] T.B. Schroder, J.C. Dyre, Phys. Rev. Lett. 84 (2000) 310; J.C. Dyre, T.B. Schroder, Rev. Mod. Phys. 72 (2000) 873. [11] B. Roling, C. Martiny, Phys. Rev. Lett. 85 (2000) 1274. [12] C. Karlsson, A. Mandanici, A. Matic, J. Swenson, L. Bo¨rjesson, J. Non-Cryst. Solids 307–310 (2002) 1012. [13] J. Swenson, A. Matic, C. Karlsson, L. Bo¨rjesson, C. Meneghini, W.S. Howells, Phys. Rev. B 63 (2001) 132202.