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Fragility in mixed alkali tellurites
Journal of Non-Crystalline Solids 337 (2004) 254–260 www.elsevier.com/locate/jnoncrysol
Fragility in mixed alkali tellurites Karl Putz, Peter F. Green
*
Department of Chemical Engineering, The University of Texas at Austin, Texas Materials Institute, Speedway, 26th Street, Austin, TX 78712 1062, USA Received 7 October 2003
Abstract Linear viscoelastic stress relaxation and calorimetric measurements were performed on a series of mixed alkali tellurite glasses of composition 0:3ð½xNa2 O þ ð1 xÞLi2 OÞ þ 0:7TeO2 at temperatures near and above the glass transition temperature, Tg . The stress relaxation data were well described by the stretched exponential function, GðtÞ ¼ G0 exp½ðt=sÞb , where s is the relaxation time, b is the distribution of relaxation times and G0 is the high frequency modulus. The fragility, determined from the temperature dependence of s, exhibited a minimum in the middle of the mixed alkali composition. A possible connection between the kinetic and the thermodynamic dimensions of this system was established, wherein the heat capacity change at the Tg , DCp ðTg ), and the fragility are correlated. 2004 Elsevier B.V. All rights reserved.
1. Introduction Mixed alkali oxide network glasses continue to be of interest to researchers due largely to the unusual dynamic processes which occur in these systems. Typically two, or more, types of alkali oxides are added to the oxide of a network former (SiO2 , GeO2 , P2 O5 , TeO2 , etc.) to create a mixed alkali glass. The effect of an alkali oxide on the local structure is to ‘depolymerize’ the three dimensional covalent network of the glass; oxygen anions that formerly created bridges between the network forming species become non-bridging, each associated with an alkali cation. The unusual dynamic processes that occur in these systems are commonly identified as mixed alkali effects (MAEs). The most well documented manifestation of a mixed alkali effect is the non-linear dependence of the ionic conductivity on the relative composition of the dissimilar alkali [1–3]. Specifically, the ionic conductivity decreases orders of magnitude as one type of alkali ion is gradually substituted for another (while the total alkali content remains fixed). The magnitude of the depression
*
Corresponding author. Tel.: +1-512 471 3188; fax: +1-512 471 7060. E-mail address: [email protected] (P.F. Green). 0022-3093/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2004.04.009
of the conductivity, which occurs when the number of dissimilar alkali cations is comparable, decreases with increasing temperature. While properties associated with ion dynamics in mixed alkali glasses exhibit highly nonlinear changes upon exchanging dissimilar cations, static properties, such as the density and index of refraction, do not exhibit changes beyond that which could be accounted for by an effective medium approximation. The mechanical relaxation (MR) spectra for mixed alkali glasses are substantially different from those of their single alkali analogs [4–13]. The spectra of mixed alkali glasses reveal the existence of appreciable structural relaxations, by comparison. If a material exhibits dissipative behavior, such as an elasticity or viscoelasticity, its response to a mechanical stress is not immediate and is characterized by a phase lag, d. For an elastic solid d ¼ 0, whereas for a truly dissipative (viscous) response, d ¼ 90. TanðdÞ, the ratio of the energy dissipated per cycle to the energy stored per cycle, is measured by mechanical relaxation experiments [14,15]. The MR spectrum of a single alkali glass exhibits a single peak at temperatures far below the glass transition temperature. In mixed alkali glasses, on the other hand, a much larger peak in tanðdÞ occurs at higher temperatures, close to the glass transition temperature, while the low temperature (single alkali) peak is diminished considerably, or has disappeared completely. The
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size of the peak generally increases to its maximum value when the number of dissimilar ions is comparable [1–3]. In mixed alkali systems, alkali ion dynamics have been discussed almost exclusively within the context of dynamic processes (MR and conductivity) that occur below the glass transition, Tg . In this paper we are interested in the dynamics of mixed alkali tellurites in the temperature range in the vicinity of, and above, Tg . Komatsu et al. showed that the kinetic fragility, m (the change in viscosity, g, with 1=T at T ¼ Tg ), and the glass transition temperature went through a minimum in the middle of the mixed alkali composition for a series of composition 0:2f½xNa2 O þ ð1 xÞLi2 Og þ 0:8TeO2 glasses [16–18]. On the other hand they indicated that thermodynamic fragility, determined by the change in the configurational heat capacity with temperature in the vicinity at Tg , remained constant, independent of temperature. In mixed alkali phosphates, by contrast, Green et al. [19,20] showed that both kinetic and thermodynamic fragilities exhibit minima in the middle of the mixed alkali composition regime. In this paper we use stress relaxation experiments to examine primary relaxations in mixed alkali tellurites. We use higher concentrations of alkali oxides in these tellurites than those used in the Komatsu studies, 0:3f½xNa2 O þ ð1 xÞLi2 Og þ 0:7TeO2 (the total fraction of the alkali oxides is fixed at 0.3 and the mole fraction of Na2 O is denoted by x). Three important results arise from this study: (1) The kinetic and the thermodynamic fragilities exhibit minima in the middle of the mixed alkali composition range. (2) The minimum in the fragility is connected to a minimum in the configurational heat capacity and that the fragility is correlated with the distribution of relaxation times. (3) Often a relaxation time of 100 s is used as a rule-ofthumb to denote the glass transition temperature in oxide glasses. While this works reasonably well in some systems, typically silicates, we show that this procedure fails on tellurites.
2. Experimental procedure The series of alkali tellurites used in this study were of the composition 0:3ð½xNa2 O þ ð1 xÞLi2 OÞ þ 0:7TeO2 , where x is the fraction of the total alkali ions that are sodium. They were prepared by melting the appropriate mixtures of Na2 CO3 (Aldrich, 99.995%), Li2 CO3 (Aldrich, 99.99%), and TeO2 (Aldrich, 99+%) in a ceramic at 800 C for 20 min. The samples were cast into a steel mold to create bars of dimension 3 · 12.5 · 50 mm3 , which were then annealed at 200 C for 4 h and allowed to cool slowly down to room temperature. Tg and DCp ðTg Þ were determined by differential scanning calorimetry (DSC) at a heating rate of 20 C/min.
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Relaxations that occur at or near the glass transition temperature were determined by performing a series of stress relaxation experiments at different temperatures. An advanced Rheometrics expansion system (ARES) Rheometer, by Rheometric Scientific, in the torsion rectangular geometry was used to perform the measurements. In a stress relaxation tests the sample is deformed rapidly to a final strain, c0 , where c0 < cc
0:005%, lower than the onset of a non-linear response. The, stress relaxation modulus, GðtÞ, in then measured, GðtÞ ¼ rðtÞ=c0 , where rðtÞ is the material response (stress) to the deformation. The data were fit to a stretched exponential function b
GðtÞ ¼ G0 exp½ðt=sÞ ;
ð1Þ
where G0 is the high frequency modulus, s is the average relaxation time, and 0 < b < 1 is a measure of the breadth of relaxation times. The full width at half maximum (in decades) in the distribution of relaxation times 1=b. The viscosity may also be determined from the stress relaxation test: Z 1 g¼ GðtÞdt: ð2Þ 0
The data were fit in the regime where the strain, c0 , was constant. The following precautions were taken in these experiments. In order to relieve any fictive temperature effects, the samples were allowed to equilibrate for up to two days at the measurement temperature. When the samples were at equilibrium, the samples could be strained both clockwise and counterclockwise in succession and the resulting data from consecutive tests would completely overlap. The time to relieve fictive temperature effects decreased significantly as the temperature was increased.
3. Results All the stress relaxation data could be accurately described by a stretched exponential, Eq. (2), for GðtÞ. A typical plot of GðtÞ is shown in Fig. 1 for a mixed alkali glass of x ¼ 0:8, from the series 0:3f½xNa2 O þ ð1 xÞLi2 Og þ 0:7TeO2 . These data were determined from measurements performed at 238 C, using a strain of 0.002%. The relaxation time determined from this data is s ¼ 1:3 s. A value of b ¼ 0:5 was extracted from these data indicating a broad distribution of relaxation times, on the order of just over two decades. The high frequency modulus, G0 , of this sample is relatively temperature insensitive, and has a value of 6.7 · 1010 dyn/cm2 . The high frequency shear moduli in these mixed alkali systems are affected by slight changes in x. Fig. 2 shows
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Fig. 1. The temperature dependence of the stress relaxation modulus is shown here for the glass melt of composition ½0:24Na2 O þ 0:06Li2 O þ 0:7TeO2 at 238 C and a strain of 0.002%. The solid line is a fit to the data.
Fig. 3. The temperature dependencies of s, for mixed alkali tellurites of composition 0:3f½xNa2 O þ ð1 xÞLi2 Og þ 0:7TeO2 are shown here. Each data set was normalized using a Tg that was defined as the temperature at which s ¼ 100 s.
4. Discussion
Fig. 2. High frequency modulus, G0 , is shown to deviate from linear additivity.
The temperature dependence of the viscosities of network forming oxide glasses, like other glass-forming liquids, can be characterized as ‘strong’ or ‘fragile’ depending on the viscosity, g, temperature, T , relation in the vicinity of Tg (kinetic fragility) [21]. If the log g 1=T dependence is linear (Arrhenius), then the behavior is ‘strong’. Conversely, the dynamics are identified as ‘fragile’ if the log g 1=T dependence is highly nonlinear. Purely tetrahedral, covalently bonded, network glasses, such as SiO2 and GeO2 , exhibit behavior that is ‘strong’, whereas systems in which the bonding is characterized as dispersive, or non-directional, exhibit ‘fragile’ behavior. If the network structure SiO2 or GeO2 , for example, is depolymerized with the addition of alkali oxides, rendering the bonding more ionic in character, then the g T relation becomes ‘fragile’. The ‘fragility’ is characterized by an index, m, [22] 1 d logs m¼ ; ð3Þ T dð1=T Þ g
that there is a significant deviation from linear additivity in the high frequency shear modulus. We also reported a similar observation in mixed alkali phosphates [5], implying that this observation may not be specific to this system. The temperature dependencies of the relaxation times for samples with different Na/Li ratios are shown in Fig. 3, where s is plotted as a function of Tg =T . The data for each sample in Fig. 3 are normalized by a glass transition temperature defined by the temperature at which the relaxation is 100 s, in accordance with the convention commonly adopted [21].
T ¼Tg
where s, which is proportional to the viscosity, g, has a temperature dependence that is reasonably well described by the Vogel–Fulcher equation s ¼ s0 eB=ðT T1 Þ :
ð4Þ
In Eq. (4), B and T1 are constants. Typically, the glass transition is operationally defined as the temperature at which the viscosity is 1013 P or where the relaxation time is 100 s [21]. The increase in fragility with increasing depolymerization is often associated with an increase in the excess entropy. Through the Adam– Gibbs [13] equation a connection is made between the
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excess entropy of the glass beyond that of the liquid, in the vicinity of Tg , and the kinetic fragility, m. Indeed this implies, at least in inorganic network glasses, a connection between the kinetic and thermodynamic features of the glass. An analysis of the relaxation time data indicates that the fragility exhibits a minimum in the middle of the composition regime, as shown in Fig. 4. It turns out that while in this system the glass transition temperature defined by a constant relaxation time at Tg can deviate appreciably from the Tg determined using differential scanning calorimetry, the overall conclusions, the minimum in m, do not change. The Tg s determined using DSC are plotted in Fig. 5 they exhibit a minimum. The
Fig. 4. The compositional dependence of the fragility index, m, is shown here. One set of indices was determined using the calorimetric Tg while the other was determined using the Tg defined where s ¼ 100 s.
Fig. 5. Calorimetric Tg s of the mixed alkali tellurites, determined using DSC with a heating rate of 20 C/min, are shown here.
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Tg s of other mixed alkali glasses such as phosphates also exhibit a minimum. The fact that this phenomenon is observed in other mixed alkali glasses strongly indicates that the minimum in Tg may indeed be universal and an explanation would not rely on structural details specific to the system. In a later section we will discuss a connection between m, Tg and the configurational heat capacity at Tg . In the meantime we examine the magnitude of the relaxation times, determined from the stress relaxation experiments, at the glass transition temperatures of the samples determined using differential scanning calorimetry (DSC). Fig. 6 shows a plot of the relaxation times at the calorimetric Tg for mixed alkali tellurites with different Li/Na ratios. It is clear from this figure that the relaxation times not only exhibit a maximum but they vary by many orders of magnitude from one composition to another. A recent analysis of a wide range of glasses shows that the viscosities of the calorimetric Tg s for a wide range of glasses also vary by orders of magnitude [23]. For the purposes of this study the point of emphasis is that if the fragility is calculated using the DSC Tg instead of the operationally defined Tg , the basic conclusion remains the same; the fragility exhibits a minimum when the Li/Na ratio is approximately 1, indicating that the samples containing single alkali cations are most fragile. This is very clear from the data in Fig. 4 that include m calculated using both sets of Tg s. In single alkali glasses, changes in the fragility have been connected to the heat capacity change at the glass transition [21]. The increase in the heat capacity change at Tg signifies an increase in the configurational entropy of the system ((weaker) non-directional bonds). This connection is made through the Adam–Gibbs equation [13]. In this model spatial domains, each composed of n
Fig. 6. The values of s are shown here as a function of x at the relevant calorimetric Tg .
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molecules, are imagined to comprise the sample. Each domain is assumed to relax independently of the other. With decreasing temperature the liquid becomes denser and the barrier to rearrangement within these clusters increase. The barrier for rearrangement should scale as the size of the cluster. Since the relaxation time of the system should depend exponentially on the activation barrier, s ¼ s0 eC=TSc ;
ð5Þ
where C is a function of the cluster size and energy per barrier; s0 is an intrinsic relaxation time associated with the dynamics of the system and Sc is the configurational entropy of the system. To this end, the reduction in the viscosity, or equivalently, the longest relaxation time is due to the reduction in the number of configurations available to the system. In contrast, within the context of the Vogel–Fulcher equation, or equivalently the WLF equation, the increase in the viscosity is due to a decrease in the free volume. The connection between the kinetic fragility and the heat capacity change is generally made as follows. As the temperature decreases, the difference between the entropy of the liquid and that of the crystal, the excess entropy, Sex , diminishes and at a sufficiently low temperature the excess entropy vanishes, the Kauzmann temperature TK . This situation would in principle never be realized since the glass transition intervenes; TK is therefore a limiting value of Tg . It is well documented [21,24–28] that if one assumes that the configurational entropy accounts entirely for Sex , then Z Tg DCp ðT Þ dT ; ð6Þ Sc ¼ T TK
Fig. 7. Heat capacity change, DCp ðTg Þ, that occurs upon heating through Tg is shown here as a function of composition. The line drawn through the data is a guide to the eye.
the change in Tg with x. This is in direct contrast to Komatsu et al. [16] who showed that for the same system with a 20% alkali content there was no difference in the heat capacity change at the glass transition. This is due to the fact that the glasses used in this study are of a larger alkali composition and thus any effects of the mixed alkali nature of the glasses are more apparent. Since the change in fragility is related to changes in the configurational entropy, the configurational entropy (degrees of freedom) should be correlated with m in other ways. Specifically the connection to the distribution of relaxation times could be examined. Fig. 8 shows
where the configurational heat capacity is DCp . SecT ondly, if DCp ðT Þ ¼ Tg DCp ðTg Þ then Sc ¼ DCp ðTg ÞðTg =TK 1Þ:
ð7Þ
Thirdly, if one assumes that TK ¼ T1 , then an equivalence between the VTF and the Adam–Gibbs equation is established. Note further that the minimum value that the fragility index could possess would occur when T1 ¼ 0. Herewith, the fragility index m2 m ¼ mmin þ min Tg DCp ðTg Þ; ð8Þ C sðT Þ where mmin ¼ log s0g and C ¼ BðTg =T1 ÞDCp ðTg Þ. A comparison with our data in Fig. 3 indicates that the minimum in the fragility could be associated with minima in Tg (Fig. 5) and an expected minimum in DCp ðTg Þ. To examine the connection to DCp ðTg Þ in further detail, calorimetric measurements were performed on the materials. The heat capacity changes at Tg are plotted as a function of x Fig. 7. These data exhibit a minimum in DCp ðTg Þ where x ¼ 0:5. Overall our results show that in this system, there exists a connection between the fragility, the heat capacity change at Tg and
Fig. 8. Stretching exponent, b, is shown as a function of normalized temperature for two mixed alkali tellurites (x ¼ 0:5, 0.8) and a single alkali tellurite (x ¼ 0).
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the temperature dependence of b for glasses in the study, which indicates that b is constant over the range of temperatures in the vicinity of Tg . Komatsu et al. predicted that the resistance to crystallization would be the result of a larger distribution of relaxation times, or a smaller b, which is clearly not the case in this system [16]. The trends in b are associated with trends in the configurational entropy change at Tg in these materials. With this in mind, we might make further connections with correlations found between m and b in a variety of systems [22]. An empirical relation was established to show the connection between m and b, m c1 c2 b;
ð9Þ
where c1 and c2 are constants. It is clear from Fig. 9, that this correlation also provides a reasonable description of the data in our system. We now discuss the dynamics in further detail. An explanation for the mixed alkali effect has remained elusive for decades and only in recent years has there been considerable progress toward development of a model that could account for many of these observations. The amorphous structure of the glass precluded the development of ionic transport models that are similar to those describing transport in crystalline metals. Extended X-ray fine structure absorption measurements revealed that alkali cations occupied and maintained distinct structural environments within the glass structure [29–31]. To this end, the recent modified site relaxation model, based on the notion that ionic species occupied distinct sites and as the ions hopped
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from one site to another, was developed [32–38]. When an alkali cation (e.g. Aþ ) hops from one site to another it leaves an unoccupied site (near a non-bridging oxygen anion) and the configuration of that site begins to relax. The new location of the Aþ cation could be a site that had been just previously been occupied (site remains open for a sufficiently short period of time that it would not have relaxed to a new equilibrium configuration) by another type of cation, Bþ , or an Aþ cation. If the site had been vacant for a sufficiently long period of time, then it might have relaxed to a configuration different from that of either A-site or B-site. The arrival of the cation also triggers a reconfiguration of the new site. The probability that the hop is successful is much higher if the site is an A-site, in the case of a Aþ cation and vice versa for a Bþ cation. These constraints imposed on the dynamics in mixed alkali glasses have a profound effect on the time scales of ion dynamics. As the temperature increases the relaxation rates of the host, or main network, which control the viscosity, increase. The alkali ions hop at different rates, orders of magnitude faster. With increasing temperature the time scales of both processes increase. At sufficiently high temperatures the dynamics of both processes occur at comparable time scales, coupled dynamics. We suggest that the compositional dependence of s, and by extension g, perhaps reflects the fact that the constraints imposed on the dynamics of the mixed alkali glasses have a much more long-ranged effect on the dynamical process that control the viscosity than one might have previously imagined.
5. Concluding remarks
Fig. 9. b is plotted as a function of the kinetic fragility index, m, for the mixed alkali tellurites (). The (þ) other data points are from Boehmer et al. [22].
Dynamic and calorimetric measurements were performed on a series of mixed alkali tellurite glasses and glass melts of the composition 0:3f½ðxÞNa2 O þ ð1 xÞLi2 Og þ 0:7TeO2 where x denotes the mole fraction on Na2 O. The measurements enabled determination of the high frequency modulus of each sample, the longest relaxation time that characterizes the dynamics and the distribution of relaxation times. The fragility of the melts exhibited a minimum in the middle of the mixed alkali range. The dynamics in the mixed alkali system are slower than those which occur in the single alkali analogs. The slow dynamics are reminiscent of the dynamics in the mixed alkali glasses below Tg . This may reflect the fact that the dynamics of the cations continue to influence the rate of network relaxations in the vicinity of Tg , as discussed above. The minimum in the fragility was reconciled with an expected minimum in DCp ðTg Þ, through the Adam–Gibbs equation and the Vogel–Fulcher equations. Moreover, commensurate trends in the distribution of relaxation times accompanied trends in the fragility index.
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D.E. Day, J. Non-Cryst. Solids 21 (1976) 343. J.O. Isard, J. Non-Cryst. Solids 1 (1969) 235. M.D. Ingram, Phys. Chem. Glasses 28 (1987) 215. P.F. Green, D.L. Sidebottom, R.K. Brow, et al., J. Non-Cryst. Solids 231 (1998) 89. P.F. Green, E.F. Brown, R.K. Brow, J. Non-Cryst. Solids 255 (1999) 87. B. Roling, M.D. Ingram, J. Non-Cryst. Solids 265 (2000) 113. B. Roling, A. Happe, M.D. Ingram et al., J. Phys. Chem. B 103 (1999) 4122. B. Roling, M.D. Ingram, Phys. Rev. B 57 (1998) 14192. W.J.T. Van Gemert, H.M.J.M. Van Ass, J.M. Stevels, J. NonCryst. Solids 16 (1974) 281. H.M.J.M. Van Ass, J.M. Stevels, J. Non-Cryst. Solids 16 (1974) 27. H.M.J.M. Van Ass, J.M. Stevels, J. Non-Cryst. Solids 16 (1974) 267. H.M.J.M. Van Ass, J.M. Stevels, J. Non-Cryst. Solids 15 (1974) 215. H.M.J.M. Van Ass, J.M. Stevels, J. Non-Cryst. Solids 14 (1974) 131. J.D. Ferry, Viscoelastic Properties of Polymers, Wiley, New York, 1970. C.W. Macosko, Rheology: Principles, Measurements, and Applications, VCH, New York, NY, 1993. T. Komatsu, T. Noguchi, R. Sato, J. Am. Ceram. Soc. 80 (1997) 1327. T. Komatsu, T. Noguchi, Y. Benino, J. Non-Cryst. Solids 222 (1997) 206. T. Komatsu, R. Ike, R. Sato et al., Phys. Chem. Glasses 36 (1995) 216.
[19] P.F. Green, R.K. Brow, J.J. Hudgens, J. Chem. Phys. 109 (1998) 7907. [20] P.F. Green, E.F. Brown, R.K. Brow, J. Non-Cryst. Solids 263&264 (2000) 195. [21] C.A. Angell, J. Non-Cryst. Solids 131–133 (1991) 13. [22] R. Boehmer, K.L. Ngai, C.A. Angell et al., J. Chem. Phys. 99 (1993) 4201. [23] E. Rossler, K.U. Hess, V.N. Novikov, J. Non-Cryst. Solids 223 (1998) 207. [24] C.A. Angell, J. Phys. Chem. Solids 49 (1988) 863. [25] C.A. Angell, J. Non-Cryst. Solids 73 (1985) 1. [26] R.K. Sato, R.J. Kirkpatrick, R.K. Brow, J. Non-Cryst. Solids 143 (1992) 257. [27] R.J. Kirkpatrick, R.K. Brow, Solid State Nucl. Mag. Res. 5 (1995) 9. [28] G. Adam, J.H. Gibbs, J. Chem. Phys. 43 (1965) 139. [29] G.N. Greaves, K.L. Ngai, Phys. Rev. B 52 (1995) 6358. [30] J. Swenson, A. Matic, C. Karlsson et al., Phys. Rev. B 63 (2001) 132202/1. [31] E. Ratai, J.C.C. Chan, H. Eckert, Phys. Chem. Chem. Phys. 4 (2002) 3198. [32] M.D. Ingram, Physica A 266 (1999) 390. [33] M.D. Ingram, Glass Sci. Technol. 67 (1994) 151. [34] M.D. Ingram, P. Maass, A. Bunde, Chim. Chronika 23 (1994) 221. [35] A. Bunde, M.D. Ingram, P. Maass, J. Non-Cryst. Solids 172–174 (1994) 1222. [36] A. Bunde, K. Funke, M.D. Ingram, Solid State Ionics 105 (1998) 1. [37] A. Bunde, K. Funke, M.D. Ingram, Solid State Ionics 86–88 (1996) 1311. [38] K. Funke, C. Cramer, B. Roling et al., Solid State Ionics 85 (1996) 293.
Fragility in mixed alkali tellurites Karl Putz, Peter F. Green
*
Department of Chemical Engineering, The University of Texas at Austin, Texas Materials Institute, Speedway, 26th Street, Austin, TX 78712 1062, USA Received 7 October 2003
Abstract Linear viscoelastic stress relaxation and calorimetric measurements were performed on a series of mixed alkali tellurite glasses of composition 0:3ð½xNa2 O þ ð1 xÞLi2 OÞ þ 0:7TeO2 at temperatures near and above the glass transition temperature, Tg . The stress relaxation data were well described by the stretched exponential function, GðtÞ ¼ G0 exp½ðt=sÞb , where s is the relaxation time, b is the distribution of relaxation times and G0 is the high frequency modulus. The fragility, determined from the temperature dependence of s, exhibited a minimum in the middle of the mixed alkali composition. A possible connection between the kinetic and the thermodynamic dimensions of this system was established, wherein the heat capacity change at the Tg , DCp ðTg ), and the fragility are correlated. 2004 Elsevier B.V. All rights reserved.
1. Introduction Mixed alkali oxide network glasses continue to be of interest to researchers due largely to the unusual dynamic processes which occur in these systems. Typically two, or more, types of alkali oxides are added to the oxide of a network former (SiO2 , GeO2 , P2 O5 , TeO2 , etc.) to create a mixed alkali glass. The effect of an alkali oxide on the local structure is to ‘depolymerize’ the three dimensional covalent network of the glass; oxygen anions that formerly created bridges between the network forming species become non-bridging, each associated with an alkali cation. The unusual dynamic processes that occur in these systems are commonly identified as mixed alkali effects (MAEs). The most well documented manifestation of a mixed alkali effect is the non-linear dependence of the ionic conductivity on the relative composition of the dissimilar alkali [1–3]. Specifically, the ionic conductivity decreases orders of magnitude as one type of alkali ion is gradually substituted for another (while the total alkali content remains fixed). The magnitude of the depression
*
Corresponding author. Tel.: +1-512 471 3188; fax: +1-512 471 7060. E-mail address: [email protected] (P.F. Green). 0022-3093/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2004.04.009
of the conductivity, which occurs when the number of dissimilar alkali cations is comparable, decreases with increasing temperature. While properties associated with ion dynamics in mixed alkali glasses exhibit highly nonlinear changes upon exchanging dissimilar cations, static properties, such as the density and index of refraction, do not exhibit changes beyond that which could be accounted for by an effective medium approximation. The mechanical relaxation (MR) spectra for mixed alkali glasses are substantially different from those of their single alkali analogs [4–13]. The spectra of mixed alkali glasses reveal the existence of appreciable structural relaxations, by comparison. If a material exhibits dissipative behavior, such as an elasticity or viscoelasticity, its response to a mechanical stress is not immediate and is characterized by a phase lag, d. For an elastic solid d ¼ 0, whereas for a truly dissipative (viscous) response, d ¼ 90. TanðdÞ, the ratio of the energy dissipated per cycle to the energy stored per cycle, is measured by mechanical relaxation experiments [14,15]. The MR spectrum of a single alkali glass exhibits a single peak at temperatures far below the glass transition temperature. In mixed alkali glasses, on the other hand, a much larger peak in tanðdÞ occurs at higher temperatures, close to the glass transition temperature, while the low temperature (single alkali) peak is diminished considerably, or has disappeared completely. The
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size of the peak generally increases to its maximum value when the number of dissimilar ions is comparable [1–3]. In mixed alkali systems, alkali ion dynamics have been discussed almost exclusively within the context of dynamic processes (MR and conductivity) that occur below the glass transition, Tg . In this paper we are interested in the dynamics of mixed alkali tellurites in the temperature range in the vicinity of, and above, Tg . Komatsu et al. showed that the kinetic fragility, m (the change in viscosity, g, with 1=T at T ¼ Tg ), and the glass transition temperature went through a minimum in the middle of the mixed alkali composition for a series of composition 0:2f½xNa2 O þ ð1 xÞLi2 Og þ 0:8TeO2 glasses [16–18]. On the other hand they indicated that thermodynamic fragility, determined by the change in the configurational heat capacity with temperature in the vicinity at Tg , remained constant, independent of temperature. In mixed alkali phosphates, by contrast, Green et al. [19,20] showed that both kinetic and thermodynamic fragilities exhibit minima in the middle of the mixed alkali composition regime. In this paper we use stress relaxation experiments to examine primary relaxations in mixed alkali tellurites. We use higher concentrations of alkali oxides in these tellurites than those used in the Komatsu studies, 0:3f½xNa2 O þ ð1 xÞLi2 Og þ 0:7TeO2 (the total fraction of the alkali oxides is fixed at 0.3 and the mole fraction of Na2 O is denoted by x). Three important results arise from this study: (1) The kinetic and the thermodynamic fragilities exhibit minima in the middle of the mixed alkali composition range. (2) The minimum in the fragility is connected to a minimum in the configurational heat capacity and that the fragility is correlated with the distribution of relaxation times. (3) Often a relaxation time of 100 s is used as a rule-ofthumb to denote the glass transition temperature in oxide glasses. While this works reasonably well in some systems, typically silicates, we show that this procedure fails on tellurites.
2. Experimental procedure The series of alkali tellurites used in this study were of the composition 0:3ð½xNa2 O þ ð1 xÞLi2 OÞ þ 0:7TeO2 , where x is the fraction of the total alkali ions that are sodium. They were prepared by melting the appropriate mixtures of Na2 CO3 (Aldrich, 99.995%), Li2 CO3 (Aldrich, 99.99%), and TeO2 (Aldrich, 99+%) in a ceramic at 800 C for 20 min. The samples were cast into a steel mold to create bars of dimension 3 · 12.5 · 50 mm3 , which were then annealed at 200 C for 4 h and allowed to cool slowly down to room temperature. Tg and DCp ðTg Þ were determined by differential scanning calorimetry (DSC) at a heating rate of 20 C/min.
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Relaxations that occur at or near the glass transition temperature were determined by performing a series of stress relaxation experiments at different temperatures. An advanced Rheometrics expansion system (ARES) Rheometer, by Rheometric Scientific, in the torsion rectangular geometry was used to perform the measurements. In a stress relaxation tests the sample is deformed rapidly to a final strain, c0 , where c0 < cc
0:005%, lower than the onset of a non-linear response. The, stress relaxation modulus, GðtÞ, in then measured, GðtÞ ¼ rðtÞ=c0 , where rðtÞ is the material response (stress) to the deformation. The data were fit to a stretched exponential function b
GðtÞ ¼ G0 exp½ðt=sÞ ;
ð1Þ
where G0 is the high frequency modulus, s is the average relaxation time, and 0 < b < 1 is a measure of the breadth of relaxation times. The full width at half maximum (in decades) in the distribution of relaxation times 1=b. The viscosity may also be determined from the stress relaxation test: Z 1 g¼ GðtÞdt: ð2Þ 0
The data were fit in the regime where the strain, c0 , was constant. The following precautions were taken in these experiments. In order to relieve any fictive temperature effects, the samples were allowed to equilibrate for up to two days at the measurement temperature. When the samples were at equilibrium, the samples could be strained both clockwise and counterclockwise in succession and the resulting data from consecutive tests would completely overlap. The time to relieve fictive temperature effects decreased significantly as the temperature was increased.
3. Results All the stress relaxation data could be accurately described by a stretched exponential, Eq. (2), for GðtÞ. A typical plot of GðtÞ is shown in Fig. 1 for a mixed alkali glass of x ¼ 0:8, from the series 0:3f½xNa2 O þ ð1 xÞLi2 Og þ 0:7TeO2 . These data were determined from measurements performed at 238 C, using a strain of 0.002%. The relaxation time determined from this data is s ¼ 1:3 s. A value of b ¼ 0:5 was extracted from these data indicating a broad distribution of relaxation times, on the order of just over two decades. The high frequency modulus, G0 , of this sample is relatively temperature insensitive, and has a value of 6.7 · 1010 dyn/cm2 . The high frequency shear moduli in these mixed alkali systems are affected by slight changes in x. Fig. 2 shows
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Fig. 1. The temperature dependence of the stress relaxation modulus is shown here for the glass melt of composition ½0:24Na2 O þ 0:06Li2 O þ 0:7TeO2 at 238 C and a strain of 0.002%. The solid line is a fit to the data.
Fig. 3. The temperature dependencies of s, for mixed alkali tellurites of composition 0:3f½xNa2 O þ ð1 xÞLi2 Og þ 0:7TeO2 are shown here. Each data set was normalized using a Tg that was defined as the temperature at which s ¼ 100 s.
4. Discussion
Fig. 2. High frequency modulus, G0 , is shown to deviate from linear additivity.
The temperature dependence of the viscosities of network forming oxide glasses, like other glass-forming liquids, can be characterized as ‘strong’ or ‘fragile’ depending on the viscosity, g, temperature, T , relation in the vicinity of Tg (kinetic fragility) [21]. If the log g 1=T dependence is linear (Arrhenius), then the behavior is ‘strong’. Conversely, the dynamics are identified as ‘fragile’ if the log g 1=T dependence is highly nonlinear. Purely tetrahedral, covalently bonded, network glasses, such as SiO2 and GeO2 , exhibit behavior that is ‘strong’, whereas systems in which the bonding is characterized as dispersive, or non-directional, exhibit ‘fragile’ behavior. If the network structure SiO2 or GeO2 , for example, is depolymerized with the addition of alkali oxides, rendering the bonding more ionic in character, then the g T relation becomes ‘fragile’. The ‘fragility’ is characterized by an index, m, [22] 1 d logs m¼ ; ð3Þ T dð1=T Þ g
that there is a significant deviation from linear additivity in the high frequency shear modulus. We also reported a similar observation in mixed alkali phosphates [5], implying that this observation may not be specific to this system. The temperature dependencies of the relaxation times for samples with different Na/Li ratios are shown in Fig. 3, where s is plotted as a function of Tg =T . The data for each sample in Fig. 3 are normalized by a glass transition temperature defined by the temperature at which the relaxation is 100 s, in accordance with the convention commonly adopted [21].
T ¼Tg
where s, which is proportional to the viscosity, g, has a temperature dependence that is reasonably well described by the Vogel–Fulcher equation s ¼ s0 eB=ðT T1 Þ :
ð4Þ
In Eq. (4), B and T1 are constants. Typically, the glass transition is operationally defined as the temperature at which the viscosity is 1013 P or where the relaxation time is 100 s [21]. The increase in fragility with increasing depolymerization is often associated with an increase in the excess entropy. Through the Adam– Gibbs [13] equation a connection is made between the
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excess entropy of the glass beyond that of the liquid, in the vicinity of Tg , and the kinetic fragility, m. Indeed this implies, at least in inorganic network glasses, a connection between the kinetic and thermodynamic features of the glass. An analysis of the relaxation time data indicates that the fragility exhibits a minimum in the middle of the composition regime, as shown in Fig. 4. It turns out that while in this system the glass transition temperature defined by a constant relaxation time at Tg can deviate appreciably from the Tg determined using differential scanning calorimetry, the overall conclusions, the minimum in m, do not change. The Tg s determined using DSC are plotted in Fig. 5 they exhibit a minimum. The
Fig. 4. The compositional dependence of the fragility index, m, is shown here. One set of indices was determined using the calorimetric Tg while the other was determined using the Tg defined where s ¼ 100 s.
Fig. 5. Calorimetric Tg s of the mixed alkali tellurites, determined using DSC with a heating rate of 20 C/min, are shown here.
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Tg s of other mixed alkali glasses such as phosphates also exhibit a minimum. The fact that this phenomenon is observed in other mixed alkali glasses strongly indicates that the minimum in Tg may indeed be universal and an explanation would not rely on structural details specific to the system. In a later section we will discuss a connection between m, Tg and the configurational heat capacity at Tg . In the meantime we examine the magnitude of the relaxation times, determined from the stress relaxation experiments, at the glass transition temperatures of the samples determined using differential scanning calorimetry (DSC). Fig. 6 shows a plot of the relaxation times at the calorimetric Tg for mixed alkali tellurites with different Li/Na ratios. It is clear from this figure that the relaxation times not only exhibit a maximum but they vary by many orders of magnitude from one composition to another. A recent analysis of a wide range of glasses shows that the viscosities of the calorimetric Tg s for a wide range of glasses also vary by orders of magnitude [23]. For the purposes of this study the point of emphasis is that if the fragility is calculated using the DSC Tg instead of the operationally defined Tg , the basic conclusion remains the same; the fragility exhibits a minimum when the Li/Na ratio is approximately 1, indicating that the samples containing single alkali cations are most fragile. This is very clear from the data in Fig. 4 that include m calculated using both sets of Tg s. In single alkali glasses, changes in the fragility have been connected to the heat capacity change at the glass transition [21]. The increase in the heat capacity change at Tg signifies an increase in the configurational entropy of the system ((weaker) non-directional bonds). This connection is made through the Adam–Gibbs equation [13]. In this model spatial domains, each composed of n
Fig. 6. The values of s are shown here as a function of x at the relevant calorimetric Tg .
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molecules, are imagined to comprise the sample. Each domain is assumed to relax independently of the other. With decreasing temperature the liquid becomes denser and the barrier to rearrangement within these clusters increase. The barrier for rearrangement should scale as the size of the cluster. Since the relaxation time of the system should depend exponentially on the activation barrier, s ¼ s0 eC=TSc ;
ð5Þ
where C is a function of the cluster size and energy per barrier; s0 is an intrinsic relaxation time associated with the dynamics of the system and Sc is the configurational entropy of the system. To this end, the reduction in the viscosity, or equivalently, the longest relaxation time is due to the reduction in the number of configurations available to the system. In contrast, within the context of the Vogel–Fulcher equation, or equivalently the WLF equation, the increase in the viscosity is due to a decrease in the free volume. The connection between the kinetic fragility and the heat capacity change is generally made as follows. As the temperature decreases, the difference between the entropy of the liquid and that of the crystal, the excess entropy, Sex , diminishes and at a sufficiently low temperature the excess entropy vanishes, the Kauzmann temperature TK . This situation would in principle never be realized since the glass transition intervenes; TK is therefore a limiting value of Tg . It is well documented [21,24–28] that if one assumes that the configurational entropy accounts entirely for Sex , then Z Tg DCp ðT Þ dT ; ð6Þ Sc ¼ T TK
Fig. 7. Heat capacity change, DCp ðTg Þ, that occurs upon heating through Tg is shown here as a function of composition. The line drawn through the data is a guide to the eye.
the change in Tg with x. This is in direct contrast to Komatsu et al. [16] who showed that for the same system with a 20% alkali content there was no difference in the heat capacity change at the glass transition. This is due to the fact that the glasses used in this study are of a larger alkali composition and thus any effects of the mixed alkali nature of the glasses are more apparent. Since the change in fragility is related to changes in the configurational entropy, the configurational entropy (degrees of freedom) should be correlated with m in other ways. Specifically the connection to the distribution of relaxation times could be examined. Fig. 8 shows
where the configurational heat capacity is DCp . SecT ondly, if DCp ðT Þ ¼ Tg DCp ðTg Þ then Sc ¼ DCp ðTg ÞðTg =TK 1Þ:
ð7Þ
Thirdly, if one assumes that TK ¼ T1 , then an equivalence between the VTF and the Adam–Gibbs equation is established. Note further that the minimum value that the fragility index could possess would occur when T1 ¼ 0. Herewith, the fragility index m2 m ¼ mmin þ min Tg DCp ðTg Þ; ð8Þ C sðT Þ where mmin ¼ log s0g and C ¼ BðTg =T1 ÞDCp ðTg Þ. A comparison with our data in Fig. 3 indicates that the minimum in the fragility could be associated with minima in Tg (Fig. 5) and an expected minimum in DCp ðTg Þ. To examine the connection to DCp ðTg Þ in further detail, calorimetric measurements were performed on the materials. The heat capacity changes at Tg are plotted as a function of x Fig. 7. These data exhibit a minimum in DCp ðTg Þ where x ¼ 0:5. Overall our results show that in this system, there exists a connection between the fragility, the heat capacity change at Tg and
Fig. 8. Stretching exponent, b, is shown as a function of normalized temperature for two mixed alkali tellurites (x ¼ 0:5, 0.8) and a single alkali tellurite (x ¼ 0).
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the temperature dependence of b for glasses in the study, which indicates that b is constant over the range of temperatures in the vicinity of Tg . Komatsu et al. predicted that the resistance to crystallization would be the result of a larger distribution of relaxation times, or a smaller b, which is clearly not the case in this system [16]. The trends in b are associated with trends in the configurational entropy change at Tg in these materials. With this in mind, we might make further connections with correlations found between m and b in a variety of systems [22]. An empirical relation was established to show the connection between m and b, m c1 c2 b;
ð9Þ
where c1 and c2 are constants. It is clear from Fig. 9, that this correlation also provides a reasonable description of the data in our system. We now discuss the dynamics in further detail. An explanation for the mixed alkali effect has remained elusive for decades and only in recent years has there been considerable progress toward development of a model that could account for many of these observations. The amorphous structure of the glass precluded the development of ionic transport models that are similar to those describing transport in crystalline metals. Extended X-ray fine structure absorption measurements revealed that alkali cations occupied and maintained distinct structural environments within the glass structure [29–31]. To this end, the recent modified site relaxation model, based on the notion that ionic species occupied distinct sites and as the ions hopped
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from one site to another, was developed [32–38]. When an alkali cation (e.g. Aþ ) hops from one site to another it leaves an unoccupied site (near a non-bridging oxygen anion) and the configuration of that site begins to relax. The new location of the Aþ cation could be a site that had been just previously been occupied (site remains open for a sufficiently short period of time that it would not have relaxed to a new equilibrium configuration) by another type of cation, Bþ , or an Aþ cation. If the site had been vacant for a sufficiently long period of time, then it might have relaxed to a configuration different from that of either A-site or B-site. The arrival of the cation also triggers a reconfiguration of the new site. The probability that the hop is successful is much higher if the site is an A-site, in the case of a Aþ cation and vice versa for a Bþ cation. These constraints imposed on the dynamics in mixed alkali glasses have a profound effect on the time scales of ion dynamics. As the temperature increases the relaxation rates of the host, or main network, which control the viscosity, increase. The alkali ions hop at different rates, orders of magnitude faster. With increasing temperature the time scales of both processes increase. At sufficiently high temperatures the dynamics of both processes occur at comparable time scales, coupled dynamics. We suggest that the compositional dependence of s, and by extension g, perhaps reflects the fact that the constraints imposed on the dynamics of the mixed alkali glasses have a much more long-ranged effect on the dynamical process that control the viscosity than one might have previously imagined.
5. Concluding remarks
Fig. 9. b is plotted as a function of the kinetic fragility index, m, for the mixed alkali tellurites (). The (þ) other data points are from Boehmer et al. [22].
Dynamic and calorimetric measurements were performed on a series of mixed alkali tellurite glasses and glass melts of the composition 0:3f½ðxÞNa2 O þ ð1 xÞLi2 Og þ 0:7TeO2 where x denotes the mole fraction on Na2 O. The measurements enabled determination of the high frequency modulus of each sample, the longest relaxation time that characterizes the dynamics and the distribution of relaxation times. The fragility of the melts exhibited a minimum in the middle of the mixed alkali range. The dynamics in the mixed alkali system are slower than those which occur in the single alkali analogs. The slow dynamics are reminiscent of the dynamics in the mixed alkali glasses below Tg . This may reflect the fact that the dynamics of the cations continue to influence the rate of network relaxations in the vicinity of Tg , as discussed above. The minimum in the fragility was reconciled with an expected minimum in DCp ðTg Þ, through the Adam–Gibbs equation and the Vogel–Fulcher equations. Moreover, commensurate trends in the distribution of relaxation times accompanied trends in the fragility index.
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