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Size of CaF2 Crystals Precipitated from Glasses in the Na2O/K2O/ CaO/CaF2/Al2O3/SiO2 System and Percolation Theory Roˆmulo Petrini Fogaca de Almeida,† Christian Bocker, and Christian Ru¨ssel* Otto-Schott-Institut, UniVersita¨t Jena, Fraunhoferstrasse 6, 07743 Jena, Germany ReceiVed May 25, 2008. ReVised Manuscript ReceiVed July 16, 2008
Glasses with the compositions (in wt %) xNa2O · 7.69K2O · 10.58CaO · 12.5CaF2 · 5.77Al2O3 · (63.45 x)SiO2 (with x ) 7.65-12.65) were thermally treated at 550 and 570 °C for 1-160 h. This resulted in the precipitation of crystalline CaF2. The crystallite sizes of this phase depended on neither the time nor the temperature of thermal treatment. During the crystallization process, the viscosity of the residual glass melt increased, and supposedly a viscous layer is formed around the growing crystals, which hinders further crystal growth. However, the chemical composition strongly affected the mean crystallite size that was calculated from XRD line broadening. With increasing Na2O concentration, the crystals get larger and their number decreases. This was explained by percolation theory. Here, a statistical distribution of network modifiers within the glass network is assumed. Above an average number of covalent bonds per network forming unit of 2.4, the network is rigid; however, within the rigid network, tiny floppy regions occur. The maximum size of a floppy region can be calculated from the respective glass composition and increases with increasing Na2O concentration. This runs parallel to the mean size of the crystallites.
1. Introduction Crystallization of glass is usually considered to be a twostep process, the nucleation and the subsequent crystal growth. Although crystal growth velocities usually exhibit a maximum slightly below (some 10 to 100 K) liquidus temperature, the maximum in the nucleation rate is slightly above the glass-transformation temperature.1 To determine nucleation rates, glasses are thermally treated near the maximum of the nucleation rate and then heated up to a temperature at which notable crystal growth occurs.2 Finally, the nuclei are counted using microscopic techniques. If the nuclei formed did not simultaneously grow, they would dissolve again during heating, because the critical radius of a nucleus increases with temperature, due to decreasing crystallization enthalpies. This proves that nuclei are not only formed at temperatures slightly above Tg but already show a certain crystal growth velocity. Fundamental studies on nucleation and crystal growth are mostly restricted to isochemical systems, i.e., to these, the crystalline phase formed has the same chemical composition as the glassy phase.1,3,4 The composition of the glassy phase then does not change during nucleation and the crystal growth velocity will not change with time. Systems, however, that provide interesting materials properties are usually multi* Corresponding author. E-mail:
[email protected]. † Current address: LAMAV Universidade Federal de Sao Carlos, Brasil.
(1) Gutzow, I.; Schmelzer, J. The Vitreous State; Springer: Berlin, 1995. (2) Tammann, G. Der Glaszustand; Leopold Voss: Leipzig, Germany, 1933. (3) Pewner, B. G.; Kluer, B. P. Proceedings of the XV International Congress on Glass, Leningrad, Soviet Union, July 3-7, 1989; Nauka: Moscow, 1989; Vol. 26, p 277. (4) Kelton, K. F. J. Non-Cryst. Solids 2000, 274, 147.
component systems (see for example refs 5, 6); hence the composition of the glassy phase, and especially that near the growing crystals, depends on time. It is of special scientific interest and technological potential if the change in the composition near the crystals leads to an increase in the viscosity and a decrease in the diffusivity of those components the crystal is formed.7 The crystal growth then decelerates with time, and the preparation of glass ceramics with large volume concentrations of crystals with sizes in the nanometer range is enabled. Such materials with crystallite sizes [Al2O3]).
2. Percolation Theory
3. Experimental Procedures
Glass forming liquids can widely be described using the random network model of Zachariasen. Here, glass melt components that form predominantly covalent bonds, such as SiO2 or P2O5, are regarded as network formers, whereas those that form ionic bonds, such as Na2O or CaO, are denoted as network modifiers. Hence, from the chemical composition, the mean number of bridging oxygens per network former can be calculated. According to the random network model, the covalent bonds are randomly distributed in the glass network. That means, the probability of a Qngroup (n ) number of bridging oxygens; i.e., Q2 ) chain structure) to link with another Qn group is solely given by the respective concentrations. According to the percolation theory of Phillips and Thorpe,14,15 a three-dimensional network becomes rigid if the number of covalent bonds per network forming unit, υ, exceeds the value υcr of 2.4. However, also in a rigid network, tiny nonrigid (“floppy”) regions occur whose size, R*, can be calculated from the percolation model.14-16
(
R/ ) d0
υ∞ υ - υCr
)
Glasses in the system Na2O/K2O/CaO/CaF2/Al2O3/SiO2 were melted from reagent-grade Na2CO3, K2CO3, CaCO3, CaF2, Al(OH)3, and SiO2 in batches of 200 g in a covered platinum crucible at 1480 °C. The melts were casted on a copper block and given to a furnace preheated to 450 °C. The furnace was then switched off and the sample allowed to cool. The samples were thermally treated at temperatures in the range of 550-570 °C. From powdered samples XRD patterns (Siemens D 5000) were recorded. The lower detection limit of crystals with simple cubic structures should be around 1%. Glasses and thermally treated samples were cut into pieces 6 × 6 × 15 mm3 and studied by a dilatometer (Netzsch 402 ES) using a heating rate of 10 K min-1. The sample compositions are summarized in Table 1. The fluoride concentrations of the samples were determined by wet chemical analysis as well as by energy-dispersive X-ray analysis using a scanning electron microscope Jeol 7100 equipped with an EDAX detector. The water concentrations were determined using NIR transmission spectroscopy of thin sections (100 µm) and an extinction coefficient of 40 L/(mol cm) at 2.8 µm.19
θ
(1)
with υ∞ ) number of covalent bonds per network former for a fully polymerized network () 4 for silicates), υ ) average number of covalent bonds per network former, according to the chemical composition, υCr ) average number of covalent bonds at the percolation threshold. R* ) radius of the floppy region, d0 ) size of a structural unit () (Vm/Na)1\3, with Vm ) molar volume, Na ) Avogadro’s number), and θ ) 0.85. In the past few years, percolation theory has been applied to describe nucleation in glass forming systems. Here, the (14) Thorpe, M. J. Non-Cryst. Solids 1983, 57, 355. (15) Philipps, J.; Thorpe, M. Solid State Commun. 1986, 53, 847. (16) Avramov, I.; Keding, R.; Ru¨ssel, C. J. Non-Cryst. Solids 2000, 272, 147.
4. Results The melted glasses were visually transparent and X-ray amorphous. The fluoride concentrations of the glassy samples were 4.6 ( 0.2 wt %, which means that 20 ( 3.5% of the fluoride was evaporated during melting. This quantity did not depend on the Na2O concentration of the samples. Wet chemical analysis and EDX were in agreement. The analytical alkali concentrations were identical with those according to the batch composition within the limits of error and hence evaporation losses were negligible. The analytical Na2O and (17) Keding, R.; Ru¨ssel, C. J. Non-Cryst. Solids 2005, 351, 1441. (18) Avramov, I.; Keding, R.; Ru¨ssel, C.; Kranold, R. J. Non-Cryst. Solids 2000, 278, 13. (19) Geotti-Bianchini, F.; Geiβler, H.; Kra¨mer, F.; Smith, I. H. Glass Sci. Technol. 1999, 72, 103.
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Figure 1. X-ray patterns of sample A tempered at 550 °C for different periods of time.
Figure 3. Quantity of CaF2 formed (area under the peak at 2θ ) 47.1°) as a function of time; sample A was heat treated at 550 °C.
Figure 2. Mean crystallite size as a function of time; sample A was heattreated at 550 °C.
Figure 4. Glass-transition temperature as a function of time; sample A was heat treated at 550 °C.
fluoride concentrations are shown in Table 1, columns 8 and 9, respectively. The water concentrations of the glasses as determined by NIR spectroscopy were in the range from 0.020 to 0.025 wt %, which is equivalent to 0.07 and 0.09 mol % H2O, respectively. Figure 1 shows XRD patterns of sample A tempered at 550 °C for different periods of time. The patterns exhibit lines attributed to the cubic crystal phase CaF2 (JCPDS 35-0816). Lines attributed to other crystal phases are not observed. The lines are notably broadened, which enabled the calculation of mean crystallite sizes from the Scherrer equation Gλ (2) Bcos θ with G ) 0.899 for a cubic system, λ ) wavelength of the radiation (CuKR: 0.154 nm), B ) full width at half-maximum, and θ ) Bragg angle of the XRD peak. Figure 2 shows the mean crystallite size of the sample A tempered at 550 °C for different periods of time. They are all in the range from 7.5 to 8.5 nm and hence do not depend on time within the limits of error. In Figure 3, the quantity of CaF2 formed (area under the peak at 2θ ) 47.1°) is shown as a function of the time, sample A was tempered at 550 °C. The quantity of crystalline phase formed shows a steep increase within the first 40 h. An even longer thermal treatment (within the limits of error) does not result in a further increase in the quantity of CaF2 formed. Glasses and thermally treated samples were also studied by dilatometry. The shapes of the dilatometric curves recorded are apd)
Figure 5. X-ray patterns of samples A-F tempered at 570 °C for 160 h.
proximately the same for tempered and untempered samples. Hence, also in the case of tempered samples, the glass transition temperature, Tg, can clearly be determined. In Figure 4, Tg is shown as a function of the time, sample A was tempered at 550 °C. Although the Tg of the glass is around 450 °C, it increases with the time of thermal treatment. After 40 h, the glass transition temperature was around 550 °C, i.e., the temperature at which the sample was thermally treated. The effect of the chemical composition of the samples upon the XRD-patterns is shown in Figure 5. In all samples, the same XRD lines all attributed to crystalline CaF2 are observed. The full width at half-maximum (fwhm) decreases with increasing Na2O concentration. Simultaneously, the line
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Figure 6. Mean crystallite size of samples tempered at 570 °C for 160 h as a function of the Na2O concentration.
Figure 7. Size of a floppy region (calcuted using eq 1) as a function of the Na2O concentration.
intensity decreases. This means that at small Na2O concentrations, a higher quantity of very small crystallites is formed, whereas at larger Na2O concentrations, a comparably small quantity of comparably large crystallites is formed. The effect of the glass composition on the crystallite size is shown in Figure 6. The error in the Na2O concentration was assumed to be equal to the scattering of the analytical values ((0.025 wt %). The error in the crystallite size was assumed to be caused by the error in the fwhm ((0.02°) from which the crystallite size was calculated. The error in the crystallite size hence increases with increasing mean size of the crystals. Depending on the Na2O concentration, the mean crystallite sizes are in the range from 7.5 to 50 nm. Hence, the comparably small variation of the Na2O concentration (7.65-12.65 mol %) results in a notable effect on the crystallite size. The diameters of the floppy regions according to percolation theory were calculated using eq 1. Here, the following was assumed: (1) silicon has the coordination number 4, (2) aluminum also possesses a coordination number of 4 and occurs as AlO4- tetrahedra, the charge of which is compensated by alkali ions. This assumption is justified in the compositions studied, because here the alkali concentrations are (much) larger than the alumina concentration.20 Further assumptions are (3) alkalis and alkaline earths either occur near AlO4 tetrahedra (to compensate their charge) or they give rise to nonbridging oxygens. In summary, silicon and aluminum act as network former, whereas the concentration of alkalis and alkaline earths exceeding the aluminum concentration leads to the formation of nonbridging oxygen. A further assumption is that (4) fluoride anions weaken the network and possess a structural role similar to that of network modifiers. It should be noted that according to MAS NMR spectroscopy21 as well as thermodynamic modeling, Al-F bonds should occur in the studied glasses.22 Furthermore, increasing the fluoride concentration in alkali aluminosilicate glasses leads to a decrease in viscosity,7 which proves the weakening of the network. According to the chemical analyses of the respective glasses, around 80 ( (20) Vogel, W. Glass Chemistry; Springer: Berlin, 1992. (21) Stebbins, J. R.; Zeng, Q. J. Non-Cryst. Solids 2000, 262, 1. (22) Shakchmatkin, B. A.; Vedishcheva, N. M.; Wright, A. C. J. NonCryst. Solids 2004, 345 & 346, 461.
Figure 8. Mean crystallite size of samples with different Na2O concentrations tempered at 570 °C for 160 h as a function of the size of a floppy region.
3% of the fluoride introduced in the batch is still present in the cooled glass. To calculate the diameters of the floppy regions, it was assumed that 80% of the fluoride is still present and acts as network modifier. The water concentrations of the glasses were in the range from 0.07 to 0.09 mol % and had a minor effect on the mean diameters of the floppy regions (0.06-0.8 structural units at Na2O concentrations of 7.65 and 12.65 mol %, respectively). In Figure 7, the calculated diameters of the floppy regions are shown as a function of the Na2O concentrations of the respective glasses. The diameters decrease with increasing Na2O concentrations and are in the range from 9 to 39 structural units. That means that at all compositions studied, the network is rigid and contains tiny floppy regions. A complete transformation from a rigid to a floppy network would take place at a Na2O concentration of 14.3 mol % (in Figure 7, this concentration is denoted as the percolation limit). Figure 8 presents the mean crystallite sizes obtained from XRD line broadening as a function of the calculated size of a floppy region. The error in the size of the floppy region is due to the error in the Na2O and fluoride concentrations and increases while approaching the percolation limit. Within the limits of error, a linear correlation of the mean crystallite size with the size of a floppy region is observed.
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5. Discussion
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assumption that Na2O quantitatively acts as network modifier.25 The distribution of the Qn groups determined at room temperature is closer to the model of a random distribution than to that of a binary distribution.25-27 Quantitative experimental studies on the high-temperature distribution of Qn groups, for example, by means of 29Si magic spinning (MAS) nuclear magnetic resonance (NMR) have not been reported in the literature up to now. It should, however, be assumed that while increasing the temperature, the Qn distribution further shifts toward a random distribution.27 In binary silicate glass melts, also the dependency of viscosity on the chemical composition could quantitatively be explained by assuming a random formation of Qn groups.27 Hence, the assumptions made in the following seem to be justified. The compositions studied are near the percolation threshold, which is reached at an Na2O concentration of 14 mol %. Hence, the maximum sizes of the floppy regions within the rigid network are comparably large and, depending on the Na2O concentration, in the range from 9 to 39 structural units d0. As already noted above, according to ref 16, nucleation may occur if the size of a floppy region is larger than that of a critical nucleus at the respective temperature. This has already been shown for the case of melts in the system BaO/TiO2/SiO2.16,17 Using the percolation model, it is assumed that inside a floppy region, the diffusivity is infinitely large and spontaneous nucleation takes place. In the present case, assuming the size of a structural unit of around 0.2 nm, the sizes of the floppy regions are in the range from 1.8 to 5.8 nm, slightly above the glass-transition temperature, and in any case should be larger than the critical diameter of a nucleus. Hence, nucleation inside the floppy region should occur. Once nuclei are formed, they should start to grow. In the course of the diffusion process, a layer around the nuclei formed should be depleted by those components the crystal consists. In the present case, CaF2 is formed, which if removed from the glass composition, leads to a decrease in the mean number of covalent bonds per network former. As already noted, this also results in an increase in the viscosity in the diffusion layer around the crystals. The size of the floppy regions should be in the range from 1.8 to 5.8 nm, which is notably smaller than that of the crystallites after thermal treatment (9.5 to 50 nm). This illustrates, that the crystals must have been grown during thermal treatment. Because within the time scale of some hours illustrated in Figure 2, the crystals precipitated from a given glass composition all possess the same size, the crystal growth should have occurred earlier.
As recently reported from the Na2O/K2O/CaO/CaF2/Al2O3/ SiO2 system, the glass-transition temperature, Tg, strongly decreases with increasing CaF2 concentrations of the glass. Hence, during the crystallization process, an increase in viscosity near the formed nuclei should occur and result in an increase of Tg of the residual glassy phase. In course of nucleation and subsequent crystal growth, Tg approaches that temperature, the sample was tempered. This is shown in Figure 4 for sample A during tempering at 550 °C. A similar behavior has already been reported for glass composition B in ref 7. The lines in the XRD patterns are notably broadened. In principle, X-ray line broadening might have two reasons: first the occurrence of small crystallites with sizes
