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Connectivity of Q-species in binary sodium-silicate glasses
Available online at www.sciencedirect.com
Journal of Non-Crystalline Solids 354 (2008) 1133–1136 www.elsevier.com/locate/jnoncrysol
Connectivity of Q-species in binary sodium-silicate glasses Ondrej Gedeon b
a,*
, Marek Lisˇka b, Jan Macha´cˇek
a
a Department of Glass and Ceramics, Institute of Chemical Technology, Technicka 5, CZ-166 28 Prague, Czech Republic Vitrum Laugaricio – Joint Glass Center of Institute of Inorganic Chemistry SAS, Alexander Dubcek University of Trencin and RONA Lednicke Rovne, Studentska 2, SK-911 50 Trencin, Slovak Republic
Available online 19 November 2007
Abstract Series of sodium-silicate glasses were simulated by molecular dynamics. Glass structures were described in frame of Q-species. Connectivity among different Q-species decouples structure into alkali-associated and alkali-migration-blocking parts. Results strongly support modified random network theory. Ó 2007 Elsevier B.V. All rights reserved. PACS: 61.43.Fs; 02.70.Ns; 81.05.Kf Keywords: Molecular dynamics; Alkali silicates; Medium-range order
1. Introduction The silicate glass, as a prototype of the covalently bonded glass can be efficiently described by the continuous random network (CRN) Model [1]. The model is based on the assumption of the well ordered short-range arrangement around silicon atoms resulting in SiO4 tetrahedra, so called Q-units. The Q-units are interconnected via the bridging oxygen (BO), which in ideal case join two SiO4 tetrahedra. If the alkali species enter the covalently bonded glass, the silicate network includes non-bridging oxygen as a result of ion bonding between oxygen and alkali species. It was suggested, and now it is generally accepted, that the network incorporating the alkali species can be described by modified random network (MRN) Model [2]. The model predicts a non-homogeneous alkali distribution inside the silicate network. However, the substance of such non-ideal distribution is not fully understood. Questions arise about the shape of the alkali distribution in space, e.g. if alkali species tend to form 3-D clusters, or clusters of a lower dimensionality; if the clusters tend to *
Corresponding author. Tel.: +420 220443695; fax: +420 224313200. E-mail address: [email protected] (O. Gedeon).
0022-3093/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2006.11.028
percolate the silicate network or prefer to be separated from each other. One of the possible quantitative glass structure descriptions resides in quantifying the distribution of the so called Qn-units. Here, Q stands for quaternary, i.e. SiO4 tetrahedron in case of silicate glasses, and n denotes the number of bridging oxygens (BO). Q-units are well recognized in glass silicate structure, not significantly deviating from their crystalline counterparts. Hence, a well-defined shortrange ordering may be attributed to the strong covalent bonds between Si and O atoms. On the other hand, the alkali species are distributed much randomly in space, especially in shorter ranges. As a result, partial-pair radial distribution function (PP RDF) of O–Na disables to define Na coordination unambiguously [3] as the minimum of the PP RDF behind the first peak is located in a wide flat area so that determination of the minimum is rather uncertain. Moreover, the minimum does not reach zero value. Nevertheless, as each NBO is associated with an alkali species, Q-units can be advantageously used for the description not only of the NBO distribution but also for the distribution of alkali species. Today, glass characterization in frame of Q-units is well accessible by means of spectroscopic methods, e.g. Raman
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[4] and 29Si MAS NMR [5–9] spectroscopy. In addition, theoretical approaches, like molecular dynamics (MD) method or thermodynamic modelling [10–12] offer alternative approaches giving glass structure expressed in terms of Q-units. These approaches, however, are limited just to the Q-unit distribution or can give information about the connectivity among Q-units (short-range order of Q-units). On the other hand, closer inspection of the experimental results revealed serious problems in the interpretation of NMR results [13]. Hence, MD is the only method able to study the larger structural units. Structure snapshots obtained by MD simulations do not confirm MRN [14], i.e. MD predicts rather homogeneous distribution of alkali ions. On the other hand, alkali dynamics revealed preferential transport paths [15,16], contradicting so the static picture obtained by MD. The present work focuses to the large structures formed in binary glasses xNa2O Æ (1 x)SiO2, x = 0, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.33, 0.35, 0.40, 0.45, 0.50, 0.55, 0.60, 0.66, described in frame of Q-units. Interconnection of different Qn species decomposes glass structure into a few sub-networks. Q4-sub-network can be understood as space-barrier for alkali migration while the sub-network formed by the interconnection of other Qn-units enables alkali ions to migrate along with a significantly lower energy. Such network decoupling should bridge the static and dynamic approaches to space distribution of alkali ions. The evolution of alkali-associated glass sub-networks with increasing alkali concentration is presented here. 2. Computational details The MD simulations of the xNa2O Æ (1 x)SiO2 glasses, x = 0, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.33, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.66, were performed by DL_POLY software [17]. Interatomic interactions were described by Buckingham short-range potential and Coulomb long-range potential. BKS parameterization [18] was used; the potential parameters for Si–O, O–O can be found in [18], and for Na–Na, Na–O in [19]. A short-range cut-off was set to ˚ and the Coulomb part of the potential was summed 7.6 A using classic Ewald scheme. A cubic computational box with periodic boundaries was used. Newtonian equations of motion were numerically integrated by the leap-frog algorithm with time step of 2 fs. MD simulations were performed under constant pressure. The cooling regime comprised a step-by-step temperature decrease (temperature step was 100 K). The simulation started at 5000 K. Each temperature step consisted of the numeric control of the reached temperature and pressure (NPT, 2500 time steps) followed by equilibration (NVE, 7500 time steps). The cooling procedure was stopped at 300 K where structure analyses were performed. Non-bridging (NBO) and bridging oxygen (BO) atoms were identified from the snapshots by the following way. All oxygen atoms coordinated by one silicon atom within ˚ were considered as NBO. Oxygen the radius of 2.46 A
atom with more than one silicon atom within this radius ˚ corresponds to was identified as BO. The value 2.46 A the centre of the zero area behind the first peak of PP RDF (Si–O). 3. Results and discussion The structure of MD sodium-silicate glass is shaped by both the quality of the empirical potentials and by the thermodynamics imposed to the system during the cooling procedure, just to mention the most important effects. The direct comparison of experiment with MD (same potentials) can be found in [3,13,20]. However, it must be stressed that even the excellent coincidence of the experimental results (neglecting fact that they are not identical and differ with the increasing distance), i.e. RDF, distribution of Q-units, and Q-units connectivity, cannot assure the correct glass structure of larger space groups. On the other hand, space limitations as well the shape of the basic units are one of the limiting factors for the formation of larger structural groups. The distributions of Q-units, i.e. the number of Qn as a function of alkali concentration, include the essential part of thermodynamics in glass (Gibbs energy) as most of internal energy is deposited in Si–O–Si bonds. Results of the presented MD simulations [21] coincide with Random model [22], which is based on the random distribution of NBO’s (alkali ions) around silicon. Such randomness may be attributed to freezing of the structure at a relatively high temperature. At these temperatures, the silicon tetrahedra are well formed already but the temperature is still high enough to prevent the formation of the equilibrated structure units exceeding silicon vicinity. It worth stressing the randomness is not associated with space but with connectivity. Hence, the random connectivity among Q-units is a starting point for the further space analysis. Although the distribution of Q-units is fully random in MD simulations (in sense of the previous paragraph) and therefore alkali ions are distributed randomly with respect to silicon tetrahedra, it does not mean that alkali ions are randomly distributed in space. Moreover, even if the alkali ions were randomly distributed in space some preferential non-uniform transport pathways could exist as a result of the network surrounding around alkali ions, which is not fully shaped only by the alkali ion itself but must confirm requirements of more ions of the network. On the other hand, interconnected Qn-units (n = 1, 2, 3) define a natural sub-network, along which alkali ions can migrate easily. Hence, it is advantageous to keep track of connectivity of Q-units associated with alkali ions. Fig. 1 displays an evolution of alkali-associated sub-network for increasing concentration of Na. (Note, the number of Q0 is zero for the three lower concentrations and negligible – below 1% – for the x = 0.4.) As expected the alkali-associated subnetwork gets denser and denser with increasing alkali concentration. In 0.1Na2O Æ 0.9SiO2 and 0.2Na2O Æ 0.8SiO2 glasses only short paths are established and therefore alkali
O. Gedeon et al. / Journal of Non-Crystalline Solids 354 (2008) 1133–1136
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Fig. 1. Alkali-associated sub-network in 0.1Na2O Æ 0.9SiO2 (upper left cube), 0.2Na2O Æ 0.8SiO2 (upper right cube), 0.3Na2O Æ 0.7SiO2 (lower left cube), and 0.4Na2O Æ 0.6SiO2 (lower left cube) glasses: Interconnection of Qn-units (n = 1, 2, 3).
mobility is here mostly determined by the number of alkali ions and average activation energy. The last quantity is, however, determined by the thickness of silica sub-network barrier. Closer inspection shows that no percolation cluster is formed even for 0.3Na2O Æ 0.7SiO2 or 0.4Na2O Æ 0.6SiO2 glasses but the finite alkali clusters are now fairly dense resulting in the significant decrease of the average activation energy for alkali hopping. The continuous decrease of activation energy and increasing alkali mobility is therefore a result of the thinner silica sub-network (formed only by Q4) what explains why no abrupt change in diffusion coefficient/conductivity can be found. 4. Conclusion The structure of sodium-silicate glasses were described in frame of Q-units. Interconnection of different Q-units decomposes glass into silica sub-network and alkali-associated sub-network. The shape of alkali-associated sub-network strongly supports the modified random network theory. Alkali-associated Q-units do not tend to interconnect but prefer to form separated clusters. Acknowledgements This work was supported by the Slovak Grant Agency for Science through the Grant No. 1/0218/03 and by the
Grant Agency of the Czech Republic through the grant No 104/06/0202. This study was also a part of the research programme MSM 6046137302 preparation and research of functional materials and material technologies using microand nanoscopic methods. References [1] W.J. Zachariasen, J. Am. Ceram. Soc. 54 (1932) 3841. [2] G.N. Greaves, A. Fontain, P. Lagarde, D. Raoux, S.J. Gurman, Nature 293 (1981) 611. [3] J. Machacek, O. Gedeon, Phys. Chem. Glasses 44 (2003) 308. [4] V.N. Bykov, A.A. Osipov, V.N. Anfilogov, Phys. Chem. Glasses 41 (2000) 10. [5] H. Maekawa, T. Maekawa, K. Kawamura, T. Yokokawa, J. NonCryst. Solids 127 (1991) 53. [6] L. Olivier, X. Yuan, A.N. Cormack, C. Jager, J. Non-Cryst. Solids 293 (2001) 53. [7] H. Eckert, Prog. NMR Spectrocs 24, 1192, 159. [8] J.F. Stebbins, J. Non-Cryst. Solids 106 (1998) 359. [9] W.A. Buckermann, W. Muller-Warmuth, Glastech Ber. 65 (1992) 18. [10] S.J. Gurman, J. Non-Cryst. Solids 125 (1990) 151. [11] B.A. Shakhmatkin, N.M. Vedishcheva, M.M. Shultz, A.C. Wright, J. Non-Cryst. Solids 177 (1994) 249. [12] N.M. Vedishcheva, B.A. Shakhmatkin, M.M. Shultz, B. Vessal, A.C. Wright, B. Bachra, A.G. Clare, A.C. Hannon, R.N. Sinclair, J. NonCryst. Solids 192&193 (1995) 292. [13] J. Machacek, O. Gedeon, M. Liska, J. Non-Cryst. Solids 352 (2006) 2173.
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[14] P. Jund, W. Kob, R. Jullien, Phys. Rev. B 64 (2001) 134303. [15] E. Sunyer, P. Jund, W. Kob, R. Jullien, J. Non-Cryst. Solids 307 (2002) 939. [16] O. Gedeon, M. Liska, J. Machacek, Phys. Chem. Glasses 43 (5) (2002) 241. [17] W. Smith, T.R. Forester,, 2000.
[18] B. van Beest, G.J. Kramer, R.A. van Santen, Phys. Rev. Lett. 64 (1990) 1955. [19] X. Yuan, A.N. Cormack, J. Non-Cryst. Solids 283 (2001) 69. [20] J. Machacek, O. Gedeon, Ceramics-Silikaty 47 (2) (2003) 45. [21] H.R. Phillip, Phys. Chem. Solids 32 (1971) 1935. [22] O. Gedeon, J. Machacek, M. Liska, Glasstech. Ber. Glass Sci. Technol. 77C (2004) 224.
Journal of Non-Crystalline Solids 354 (2008) 1133–1136 www.elsevier.com/locate/jnoncrysol
Connectivity of Q-species in binary sodium-silicate glasses Ondrej Gedeon b
a,*
, Marek Lisˇka b, Jan Macha´cˇek
a
a Department of Glass and Ceramics, Institute of Chemical Technology, Technicka 5, CZ-166 28 Prague, Czech Republic Vitrum Laugaricio – Joint Glass Center of Institute of Inorganic Chemistry SAS, Alexander Dubcek University of Trencin and RONA Lednicke Rovne, Studentska 2, SK-911 50 Trencin, Slovak Republic
Available online 19 November 2007
Abstract Series of sodium-silicate glasses were simulated by molecular dynamics. Glass structures were described in frame of Q-species. Connectivity among different Q-species decouples structure into alkali-associated and alkali-migration-blocking parts. Results strongly support modified random network theory. Ó 2007 Elsevier B.V. All rights reserved. PACS: 61.43.Fs; 02.70.Ns; 81.05.Kf Keywords: Molecular dynamics; Alkali silicates; Medium-range order
1. Introduction The silicate glass, as a prototype of the covalently bonded glass can be efficiently described by the continuous random network (CRN) Model [1]. The model is based on the assumption of the well ordered short-range arrangement around silicon atoms resulting in SiO4 tetrahedra, so called Q-units. The Q-units are interconnected via the bridging oxygen (BO), which in ideal case join two SiO4 tetrahedra. If the alkali species enter the covalently bonded glass, the silicate network includes non-bridging oxygen as a result of ion bonding between oxygen and alkali species. It was suggested, and now it is generally accepted, that the network incorporating the alkali species can be described by modified random network (MRN) Model [2]. The model predicts a non-homogeneous alkali distribution inside the silicate network. However, the substance of such non-ideal distribution is not fully understood. Questions arise about the shape of the alkali distribution in space, e.g. if alkali species tend to form 3-D clusters, or clusters of a lower dimensionality; if the clusters tend to *
Corresponding author. Tel.: +420 220443695; fax: +420 224313200. E-mail address: [email protected] (O. Gedeon).
0022-3093/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2006.11.028
percolate the silicate network or prefer to be separated from each other. One of the possible quantitative glass structure descriptions resides in quantifying the distribution of the so called Qn-units. Here, Q stands for quaternary, i.e. SiO4 tetrahedron in case of silicate glasses, and n denotes the number of bridging oxygens (BO). Q-units are well recognized in glass silicate structure, not significantly deviating from their crystalline counterparts. Hence, a well-defined shortrange ordering may be attributed to the strong covalent bonds between Si and O atoms. On the other hand, the alkali species are distributed much randomly in space, especially in shorter ranges. As a result, partial-pair radial distribution function (PP RDF) of O–Na disables to define Na coordination unambiguously [3] as the minimum of the PP RDF behind the first peak is located in a wide flat area so that determination of the minimum is rather uncertain. Moreover, the minimum does not reach zero value. Nevertheless, as each NBO is associated with an alkali species, Q-units can be advantageously used for the description not only of the NBO distribution but also for the distribution of alkali species. Today, glass characterization in frame of Q-units is well accessible by means of spectroscopic methods, e.g. Raman
1134
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[4] and 29Si MAS NMR [5–9] spectroscopy. In addition, theoretical approaches, like molecular dynamics (MD) method or thermodynamic modelling [10–12] offer alternative approaches giving glass structure expressed in terms of Q-units. These approaches, however, are limited just to the Q-unit distribution or can give information about the connectivity among Q-units (short-range order of Q-units). On the other hand, closer inspection of the experimental results revealed serious problems in the interpretation of NMR results [13]. Hence, MD is the only method able to study the larger structural units. Structure snapshots obtained by MD simulations do not confirm MRN [14], i.e. MD predicts rather homogeneous distribution of alkali ions. On the other hand, alkali dynamics revealed preferential transport paths [15,16], contradicting so the static picture obtained by MD. The present work focuses to the large structures formed in binary glasses xNa2O Æ (1 x)SiO2, x = 0, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.33, 0.35, 0.40, 0.45, 0.50, 0.55, 0.60, 0.66, described in frame of Q-units. Interconnection of different Qn species decomposes glass structure into a few sub-networks. Q4-sub-network can be understood as space-barrier for alkali migration while the sub-network formed by the interconnection of other Qn-units enables alkali ions to migrate along with a significantly lower energy. Such network decoupling should bridge the static and dynamic approaches to space distribution of alkali ions. The evolution of alkali-associated glass sub-networks with increasing alkali concentration is presented here. 2. Computational details The MD simulations of the xNa2O Æ (1 x)SiO2 glasses, x = 0, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.33, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.66, were performed by DL_POLY software [17]. Interatomic interactions were described by Buckingham short-range potential and Coulomb long-range potential. BKS parameterization [18] was used; the potential parameters for Si–O, O–O can be found in [18], and for Na–Na, Na–O in [19]. A short-range cut-off was set to ˚ and the Coulomb part of the potential was summed 7.6 A using classic Ewald scheme. A cubic computational box with periodic boundaries was used. Newtonian equations of motion were numerically integrated by the leap-frog algorithm with time step of 2 fs. MD simulations were performed under constant pressure. The cooling regime comprised a step-by-step temperature decrease (temperature step was 100 K). The simulation started at 5000 K. Each temperature step consisted of the numeric control of the reached temperature and pressure (NPT, 2500 time steps) followed by equilibration (NVE, 7500 time steps). The cooling procedure was stopped at 300 K where structure analyses were performed. Non-bridging (NBO) and bridging oxygen (BO) atoms were identified from the snapshots by the following way. All oxygen atoms coordinated by one silicon atom within ˚ were considered as NBO. Oxygen the radius of 2.46 A
atom with more than one silicon atom within this radius ˚ corresponds to was identified as BO. The value 2.46 A the centre of the zero area behind the first peak of PP RDF (Si–O). 3. Results and discussion The structure of MD sodium-silicate glass is shaped by both the quality of the empirical potentials and by the thermodynamics imposed to the system during the cooling procedure, just to mention the most important effects. The direct comparison of experiment with MD (same potentials) can be found in [3,13,20]. However, it must be stressed that even the excellent coincidence of the experimental results (neglecting fact that they are not identical and differ with the increasing distance), i.e. RDF, distribution of Q-units, and Q-units connectivity, cannot assure the correct glass structure of larger space groups. On the other hand, space limitations as well the shape of the basic units are one of the limiting factors for the formation of larger structural groups. The distributions of Q-units, i.e. the number of Qn as a function of alkali concentration, include the essential part of thermodynamics in glass (Gibbs energy) as most of internal energy is deposited in Si–O–Si bonds. Results of the presented MD simulations [21] coincide with Random model [22], which is based on the random distribution of NBO’s (alkali ions) around silicon. Such randomness may be attributed to freezing of the structure at a relatively high temperature. At these temperatures, the silicon tetrahedra are well formed already but the temperature is still high enough to prevent the formation of the equilibrated structure units exceeding silicon vicinity. It worth stressing the randomness is not associated with space but with connectivity. Hence, the random connectivity among Q-units is a starting point for the further space analysis. Although the distribution of Q-units is fully random in MD simulations (in sense of the previous paragraph) and therefore alkali ions are distributed randomly with respect to silicon tetrahedra, it does not mean that alkali ions are randomly distributed in space. Moreover, even if the alkali ions were randomly distributed in space some preferential non-uniform transport pathways could exist as a result of the network surrounding around alkali ions, which is not fully shaped only by the alkali ion itself but must confirm requirements of more ions of the network. On the other hand, interconnected Qn-units (n = 1, 2, 3) define a natural sub-network, along which alkali ions can migrate easily. Hence, it is advantageous to keep track of connectivity of Q-units associated with alkali ions. Fig. 1 displays an evolution of alkali-associated sub-network for increasing concentration of Na. (Note, the number of Q0 is zero for the three lower concentrations and negligible – below 1% – for the x = 0.4.) As expected the alkali-associated subnetwork gets denser and denser with increasing alkali concentration. In 0.1Na2O Æ 0.9SiO2 and 0.2Na2O Æ 0.8SiO2 glasses only short paths are established and therefore alkali
O. Gedeon et al. / Journal of Non-Crystalline Solids 354 (2008) 1133–1136
1135
Fig. 1. Alkali-associated sub-network in 0.1Na2O Æ 0.9SiO2 (upper left cube), 0.2Na2O Æ 0.8SiO2 (upper right cube), 0.3Na2O Æ 0.7SiO2 (lower left cube), and 0.4Na2O Æ 0.6SiO2 (lower left cube) glasses: Interconnection of Qn-units (n = 1, 2, 3).
mobility is here mostly determined by the number of alkali ions and average activation energy. The last quantity is, however, determined by the thickness of silica sub-network barrier. Closer inspection shows that no percolation cluster is formed even for 0.3Na2O Æ 0.7SiO2 or 0.4Na2O Æ 0.6SiO2 glasses but the finite alkali clusters are now fairly dense resulting in the significant decrease of the average activation energy for alkali hopping. The continuous decrease of activation energy and increasing alkali mobility is therefore a result of the thinner silica sub-network (formed only by Q4) what explains why no abrupt change in diffusion coefficient/conductivity can be found. 4. Conclusion The structure of sodium-silicate glasses were described in frame of Q-units. Interconnection of different Q-units decomposes glass into silica sub-network and alkali-associated sub-network. The shape of alkali-associated sub-network strongly supports the modified random network theory. Alkali-associated Q-units do not tend to interconnect but prefer to form separated clusters. Acknowledgements This work was supported by the Slovak Grant Agency for Science through the Grant No. 1/0218/03 and by the
Grant Agency of the Czech Republic through the grant No 104/06/0202. This study was also a part of the research programme MSM 6046137302 preparation and research of functional materials and material technologies using microand nanoscopic methods. References [1] W.J. Zachariasen, J. Am. Ceram. Soc. 54 (1932) 3841. [2] G.N. Greaves, A. Fontain, P. Lagarde, D. Raoux, S.J. Gurman, Nature 293 (1981) 611. [3] J. Machacek, O. Gedeon, Phys. Chem. Glasses 44 (2003) 308. [4] V.N. Bykov, A.A. Osipov, V.N. Anfilogov, Phys. Chem. Glasses 41 (2000) 10. [5] H. Maekawa, T. Maekawa, K. Kawamura, T. Yokokawa, J. NonCryst. Solids 127 (1991) 53. [6] L. Olivier, X. Yuan, A.N. Cormack, C. Jager, J. Non-Cryst. Solids 293 (2001) 53. [7] H. Eckert, Prog. NMR Spectrocs 24, 1192, 159. [8] J.F. Stebbins, J. Non-Cryst. Solids 106 (1998) 359. [9] W.A. Buckermann, W. Muller-Warmuth, Glastech Ber. 65 (1992) 18. [10] S.J. Gurman, J. Non-Cryst. Solids 125 (1990) 151. [11] B.A. Shakhmatkin, N.M. Vedishcheva, M.M. Shultz, A.C. Wright, J. Non-Cryst. Solids 177 (1994) 249. [12] N.M. Vedishcheva, B.A. Shakhmatkin, M.M. Shultz, B. Vessal, A.C. Wright, B. Bachra, A.G. Clare, A.C. Hannon, R.N. Sinclair, J. NonCryst. Solids 192&193 (1995) 292. [13] J. Machacek, O. Gedeon, M. Liska, J. Non-Cryst. Solids 352 (2006) 2173.
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[14] P. Jund, W. Kob, R. Jullien, Phys. Rev. B 64 (2001) 134303. [15] E. Sunyer, P. Jund, W. Kob, R. Jullien, J. Non-Cryst. Solids 307 (2002) 939. [16] O. Gedeon, M. Liska, J. Machacek, Phys. Chem. Glasses 43 (5) (2002) 241. [17] W. Smith, T.R. Forester,
[18] B. van Beest, G.J. Kramer, R.A. van Santen, Phys. Rev. Lett. 64 (1990) 1955. [19] X. Yuan, A.N. Cormack, J. Non-Cryst. Solids 283 (2001) 69. [20] J. Machacek, O. Gedeon, Ceramics-Silikaty 47 (2) (2003) 45. [21] H.R. Phillip, Phys. Chem. Solids 32 (1971) 1935. [22] O. Gedeon, J. Machacek, M. Liska, Glasstech. Ber. Glass Sci. Technol. 77C (2004) 224.