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Crystallization statistics. A new tool to evaluate glass homogeneity
L.A. Souza
a
, M.L.G. Leite
a
, E.D. Zanotto
a,*
, M.O. Prado
b
a (^) Universidade Federal de Sa˜o Carlos, Vitreous Materials Laboratory, Rod. Washington Luiz, km 235, Sa˜o Carlos, SP 13.560-950, Brazil b (^) Comisio´ n Nacional de Energı´a Ato´ mica, Centro Ato´ mico Bariloche, Av. Ezequiel Bustillo, km 9.5, (8400) S. C. de Bariloche, Rı´o Negro, Argentina
Received 22 February 2005; received in revised form 27 September 2005 Available online 26 October 2005
Abstract
We propose and test a new method to evaluate the chemical homogeneity of glasses based on statistical analyses of the volume dis- tribution of crystals developed through thermal treatments. The method is based on the fact that each volume element of a glass piece subjected to a proper thermal treatment should exhibit a certain number of crystals, which is dictated by its chemical composition. We performed numerical simulations to interpret the experimental results obtained for some glasses, and demonstrate that this new method is adequate to determine the degree and scale of heterogeneity of glasses that display volume crystallization. Ó 2005 Elsevier B.V. All rights reserved.
PACS: 61.43.Fs; 81.40.z; 81.70.q
- Introduction
The chemical and physical differences of the starting materials used in glass production give rise to complex reactions during the first stages of melting. While some components melt, others continuously dissolve in the liquid. Moreover, due to molecular weight differences, some elements tend to sink to the bottom of the crucible while others partially volatilize. Diffusion processes are necessary to completely dissolve and homogenize the batch materials, but glass-forming melts are quite viscous, which impairs diffusion and mixing. Therefore, depending on the melting conditions (the materialÕs particle size, tempera- ture, time, mixing, etc.), the result is quite often a heteroge- neous glass. Knowledge of the variations in the chemical composition from one volume element of the glass to another (degree of homogeneity) and of the characteristic length (scale of homogeneity) of such heterogeneities is important for a number of applications. The standard technique to measure glass homogeneity in industries and laboratories was developed by Christiansen–
Shelybskie [1]. This technique is based on the fact that the transmission of visible light of an inhomogeneous material, for instance, a transparent glass powder immersed in a liquid, depends on the refractive index of both materials. The major advantage of this method is that it expresses the degree of homogeneity (r^2 ) and scale of the heterogene- ities, which is inferred by varying the size of the glass grains. However, some works [2,3] strongly criticize this method. The problem in question involves the equations used by Christiansen–Shelybskie, which disregarded sev- eral important contributions to the transmittance losses. Hoffmann [3] compared data from the literature obtained by the Christiansen–Shelybskie method and demonstrated incoherent results. For example, the degree of homogeneity of optical glasses and ordinary bottles measured by this method was found to be of the same order of magnitude! This is clearly inconsistent, because glasses produced on a large scale cannot be as homogeneous as optical glasses, which are produced by sophisticated stirring processes. To produce glasses of optical quality, the melt has to be stirred continuously, but on a laboratory scale, due to its simplicity, the most common homogenization process is to repeatedly melt and crush the glass. Although the use of this procedure is widespread, the number of times this
0022-3093/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2005.09.
- (^) Corresponding author. Tel.: +55 16 3351 8556; fax: +55 16 3361 5404. E-mail address: [email protected] (E.D. Zanotto).
www.elsevier.com/locate/jnoncrysol
Journal of Non-Crystalline Solids 351 (2005) 3579–
process is repeated and the melting times are completely empirical. Most researchers use from one to five melting/ crushing cycles, but rarely test the homogeneity of the resulting glass due to the practical difficulties of the Chris- tiansen–Shelybskie method. In this paper we propose a new method, based on a statis- tical analysis of the volume distribution of crystals that are developed by heat treatment above Tg , to characterize the degree and scale of the homogeneity of glasses that crystallize in the volume. Internal crystallization may occur spontane- ously above Tg , as in the glasses described by Fokin et al. [4], or it may be induced by adding nucleating agents. We then produce glasses with different degrees of homogeneity by varying the melting temperatures and the number of melt- ing/crushing cycles, and test the proposed method.
- Theoretical fundaments of the new method
The new method is based on the fact that, after a certain nucleation time tN at a given temperature TN , each volume element of a homogeneous glass has the same probability to nucleate a crystal. By the development of an appropriate thermal treatment (growth time tG at a temperature TG > TN ), crystalline nuclei may grow on an observable scale, allowing a sample cross-section to be examined by optical or electron microscopy. This procedure yields an average number of crystals N per examined field, and a correspond- ing standard deviation rN. If the condition of perfect homogeneity is met, according to PoissonÕs statistics, the average number of crystals and its standard deviation satisfy the condition rN/N = 1/N1/^. The value of N depends on the optical field area l^2 , which in turn is related to the microscopeÕs magnification, M, through a constant k, (l^2 = kM^2 ), and on the volume density of crystals N (^) v. Crystals generated by two-step (nucleation and growth) heat treatments lead to an N that depends on the product of the crystal nucleation rate I(TN ) and time tN , as well as on the product of the crystal growth rate U(TG ) and growth time, tG. A thermal nuclei, i.e., nuclei that formed during glass preparation, also grow upon heating to TG. From geometrical considerations, the following rela- tionship holds for the average number of spherical crystals N appearing on a surface area l^2 , after a proper heat treatment:
N ¼ 2 N (^) v l^2 U ðT (^) GÞtG; ð 1 Þ
where N (^) v = I(TN ) · tN is the volume density of crystals. For a set of measurements with pre-selected values of N (^) v, U(TG ) and tG:
N 1 =^2 / l. ð 2 Þ
A rN/N vs. 1/N1/2^ plot for a crystal distribution in a per- fectly homogeneous glass obtained with different microscope magnifications should show a straight line with slope 1 and
zero intercept. In the case of an inhomogeneous glass, how- ever, different volume elements exhibit different nucleation probabilities, and a r/N vs. 1/N1/2^ plot will not display a straight line of slope 1 with zero intercept. If the plot shows a transition from homogeneous to inhomogeneous behavior for a given N value, we can use Eq. (2) to assign a typical l that characterizes the scale of heterogeneity. For a given N (or l), the ratio (rN/N)homo /(rN/N) (^) measured is a measure of the degree of homogeneity. When this quo- tient is 1, homogeneity is maximum; values of the ratio lower than 1 indicate lower homogeneity. Let us explain this method in more detail. If one raises the temperature of a internally crystallizing glass for some time above Tg , crystalline nuclei may grow, for example, to a few micrometers. Then, by observing a polished cross- section of the sample with an optical or electron micro- scope, it is possible to evaluate the number of crystals Ni present in each observed field. The field area depends on the magnification M. N is thus the average of the N (^) i values obtained for various optical fields with the same M. In a homogeneous glass in the nucleation stage, before any significant crystal growth, the probability of finding a nucleus in each volume element is the same throughout the sample; in other words, the birth of one nucleus does not interfere with the appearance of another. The probabil- ity p(Nvi) of finding N (^) vi nuclei in a given sample volume is described by Poisson statistics [5], and is summarized by Eq. (3):
pðN (^) viÞ ¼
N Nv vieN^ v N (^) vi!
ðN (^) vi ¼ 0 ; 1 ; 2 ;... ; nÞ; ð 3 Þ
where N (^) v is the average crystal number for that volume. From now on, based on stereological considerations, we consider that the aforementioned analysis is also valid for a sample plane cut. In other words, Nvi is replaced by the number of crystals N (^) i counted in an optical field of the sample cut, and Nv is replaced by the average of a set of NiÕs, namely N (the average number of crystals in the glass surface in a field of size l^2 ). The standard deviation of the Ni values obtained with the same magnification M is given by Eq. (4a):
r ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P iðN^ i^ ^ N^ Þ
2
s 1
s
. ð4aÞ
Thus, for a number s of optical fields of size l^2 , N can be determined, enabling one to calculate the standard devia- tion of the distribution with Eq. (4a). If the glass is chemically homogeneous, then Eq. (4a) can be written as [5]
r ¼
ffiffiffiffi N
p
. ð4bÞ
For each magnification M, a corresponding (r, N) pair is obtained for a perfectly homogeneous glass. A plot of the
some data corresponding to heterogeneous samples. The data corresponding to the sample in Fig. 3 with slight inho- mogeneity deviates at a larger N than those of the sample having striae of higher crystal density. However, an analy- sis of the inhomogeneity size using Eq. (2) reveals the same scale of inhomogeneity for both. Moreover, the (rN/N) (^) homo /(rN/N) (^) measured ratios are different for the two samples, indicating a clear difference in the degree of heter- ogeneities. The same procedure was then used to analyze the experimental data for real glasses.
- Experimental procedure
To experimentally test the proposed method, two glass systems were evaluated: a glass with composition 23.26Na 2 O–
23.26CaO–47.38SiO 2 –6P 2 O 5 (mol%), which will be dubbed SSP6, and a 45Li 2 O–20Nb 2 O 5 –35SiO 2 (mol%) glass, which will be denominated LNS. These glasses were chosen because they display internal crystallization easily and have volatile elements that can lead to significant degrees of heterogeneity. We prepared the SSP6 glasses by melting appropriate quantities of SiO 2 (crushed Brazilian quartz) with analyti- cal grade Na 2 CO 3 (Mallinckrodt), CaCO 3 (Mallinckrodt) and Na 2 HPO 4 (Mallinckrodt). After weighing these reagents, they were mixed in a planetary mill for 1 h. Melt- ing was carried out in Pt crucibles in an electric furnace at 1350 °C for several hours. To obtain glasses with different degrees of homogeneity, we varied the number of melt- ing/crushing cycles. At first, we melted 100 g of glass for 1 h and poured it on a steel plate. Some of this glass was separated and a second part was broken up into 2 mm pieces, melted for another hour and poured onto a steel plate. We repeated this procedure and a third part was melted for another hour. All these glasses were then sub- jected to nucleation treatments for 8, 10, 12, 16 and 20 h at 570 °C and subsequently to a ÔdevelopmentÕ treatment at 720 °C for 10 or 20 min to allow for crystal growth. Glasses containing extra OH^ were also evaluated. After polishing with CeO 2 to an optical degree, measurements of N (^) i (number of crystals/unit area) were conducted with an optical microscope at 100, 250 and 500· (it is important to stress that the analyzed sample surfaces were, in fact, cross-sections of the crystallized volume). The same sam- ples were also prepared for SEM analysis and the measure- ments of N (^) i were carried out at 1000 and 1500·. The LNS glasses were melted by the conventional method, using Li 2 CO 3 (Synth, PA), Nb 2 O 5 (CBMM, opti- cal grade) and SiO 2 (crushed Brazilian quartz). After these chemicals were carefully weighed, they were mixed in a planetary mill for 30 min. Melting was carried out in Pt crucibles for 2 h in an electric furnace at 1250 °C. At first, we melted 100 g of glass for 1 h and poured it onto a steel plate. The glass was re-melted three times at 1250 °C for 30 min (A) and then divided into three parts. One part was further divided into parts which were melted at 1150 °C for 1 h (B) and 2 h (C and D), at 1300 °C for 2 h (E) and at 1200 °C for 2 h (F). The second part was melted at 1200 °C for 1 h (G) and, finally, the third part was melted at 1300 °C for 1 h (H). Fig. 5 shows a schematic diagram of the melting procedure. These glasses were heated at 650 °C to allow crystal growth from a thermal nuclei. We then polished several samples with CeO 2 and counted the number of crystals Ni in each optical field with an optical microscope, using 50, 100, 200, 320 and 500· magnification. Only sample (D) – which was melted at 1150 °C for 2 h – was nucleated at 555 °C during 24 h and annealed at 650 °C for 30 min for crystal growth. This sample was analyzed by SEM at 5000 · magnification. The use of different optical magnifications allowed us to clearly detect heterogeneities and to identify on which scale they were significant. After obtaining the experimental N (^) i,
0.0 0.2 0.4 0.6 0.8 1.
Unit Length
Unit Length
Fig. 3. Simulation of 1 mm length of vertical striae 100 lm width centered at x = 0.2, 0.4, 0.6 and 0.8. The matrix density is 120 crystals/mm^2 and the striae density is 6120 crystals/mm^2.
-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.
-0.
σ /N
Homogeneous Striae with 1.5x more crystals Striae with 50x more crystals
(ΝΝ)–1/
Fig. 4. Statistical analysis performed on computer-simulated crystal distributions corresponding to a homogeneous matrix and two inhomo- geneous matrices with vertical striae having different crystal densities. The straight line (slope 1 and 0 intercept) corresponds to the (r, N) pairs obtained for a perfectly homogeneous glass.
we calculated the corresponding N and r and plotted them as r/N vs. 1/N1/2^ to analyze the adherence of these experi- mental data to PoissonÕs statistics.
- Results
5.1. SSP6 glasses
We adopted the legends 0 M, 1 M and 2 M for glasses melted for 1 h, crushed and melted once more, and crushed and melted twice, respectively. We adopted this procedure expecting to vary the degree and scale of chemical hetero- geneity. All these glasses were heat-treated simultaneously, in the same furnace, to minimize temperature variations. The degree of crushing before melting is also important for homogenization. However, for this system, the particles were only crushed into coarse sizes of about 2 mm; there- fore, and thus homogeneity had to be achieved by atomic diffusion during melting. Fig. 6 shows the results for samples 0 M, 1 M and 2 M annealed at 570 °C for 10 h, as well as the results of a sim- ulation with a matrix having a crystal density similar to the experimentally evaluated glass (120 crystals/mm 2 ). Fig. 6 shows that the experimental data are close to the simulated results, and that, in this particular case, further crushing/melting does not significantly influence the glass homogeneity on a scanning electron microscope scale. We carried out the same test for other groups of samples, with results similar to those presented in Fig. 6. Therefore, subsequent melting is not necessary to homogenize the SSP6 glass, which has a low viscosity. This is an important conclusion, since re-melting increases the time and cost, can lead to the uptake of impurities and the loss of some components of high vapor pressure, such as P and Na.
Fig. 7 shows the results of N/l^2 for the three types of sam- ples. There were no significant changes in N/l^2 over time. The value of N would have been different had there been losses of P and Na. Due to the match between the actual and expected values of N (considering the experimental error), we conclude that, if any losses did occur, they were negligible. To evaluate an actual heterogeneous glass, similar mea- surements were conducted with a SSP6 glass melted in a humid atmosphere. This glass was quite heterogeneous, as is visually shown by its crystallization pattern (Fig. 8). The crystallization statistics results are plotted in Fig. 9.
Li 2 CO 3 +Nb 2 O 5 +SiO (^2) Melting at 1250˚C/2h + 3 re-melting (Sample A )
1150˚C/1h (Sample B )
1150˚C/2h (Samples C/D )
1300˚C/2h (Sample E )
1200˚C/2h (Sample F )
1200˚C/1h (Sample G )
1300˚C/1h (Sample H )
Li 2 CO 3 +Nb 2 O 5 +SiO (^2) Melting at 1250ºC/2h + 3 re-melting (Sample A )
1150ºC/1h (Sample B )
1150ºC/2h (Samples C/D )
1300ºC/2h (Sample E )
1200ºC/2h (Sample F )
1200ºC/1h (Sample G )
1300ºC/1h (Sample H )
Fig. 5. Re-melting schedule of the LNS glasses.
Fig. 6. Poisson test. Experimental data for 0 M (*), 1 M (s), 2 M (j) glasses annealed at 570 °C for 10 h – compared with the calculated data for a homogeneous matrix having the same crystal density: (h) simulation with the same number of windows utilized in the experiments, (m) simulation with 500 windows. Optical microscopy (OM); scanning electron microscopy (SEM). The straight line (slope 1 and 0 intercept) corresponds to the (r, N) pair obtained for a perfectly homogeneous glass.
LNS glasses exhibit a richer variety of behaviors. Fig. 11 shows glasses that became either homogeneous (D) or het- erogeneous (G). Overlapping the statistical results for glasses A, B and H (from Fig. 10) indicates that the large scale inhomogeneities found in glass A were eliminated when we prepared glasses B or H through a fourth re-melt-
ing at a different temperature. Moreover, the density of crystals at, for example, 50·, varied drastically from glass to glass. This variation could be due to homogenization, or else to different redox conditions of the Nb ions in the melt. But the similarity of statistical behaviors of B and H glasses was naturally due to homogenization. This new
0.0 0.2 0.4 0.6 0.8 1.
0.0 0.2 0.4 0.6 0.8 1.
0.0 0.2 0.4 0.6 0.8 1.0 0.0^ 0.2^ 0.4^ 0.6^ 0.8^ 1.
0.0 0.2 0.4 0.6 0.8 1.0 1.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.
500x 320x 200x
100x 50x
σ/N σ/N
σ/N σ/N
σ/N σ/N
σ/N^ σ/N
1/N 1/
(A)
500x
320x 200x
100x 50x
1/N 1/
(B)
500x
320x
200x
50x 100x
1/N 1/
(C)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.
5000x SEM
320x500x 200x 100x
1/N 1/
(D)
500x 320x
200x 100x
50x
1/N 1/
(E)
500x
320x
200x 50x 100x
1/N 1/
(F)
500x
320x 200x
100x 50x
1/N 1/
(G)
500x
320x 200x 100x
50x
1/N 1/
(H)
Fig. 10. Statistical test for the LNS glass. The straight line (slope 1 and 0 intercept) corresponds to the (r, N) pair obtained for a perfectly homogeneous glass.
method of evaluating chemical heterogeneity offers two advantages: it indicates the presence of heterogeneity and allows one to obtain information about their typical scale and degree.
- Conclusions
We propose a new method to infer the degree and scale of heterogeneity in glasses that crystallize in the volume on laboratory time scales. The method is based on statistical analyses of the crystal distribution in sample cuts, and characterizes glass homogeneity through r/N vs. 1/N1/ plots. These plots are straight lines with slope = 1 and intercept = 0 for perfectly homogeneous glasses, but devi- ate from linearity when the glass is heterogeneous. The value of N at which the system deviates from a homoge- neous sample indicates the size of the heterogeneity and the quantity (rN/N) (^) homo /(rN/N)measured gives information about the degree of heterogeneity. Experimental determi-
nations of crystallization statistics in several partially crystallized glasses using this new methodology have demonstrated its usefulness.
Acknowledgements
We are deeply indebted to FAPESP, CNPq, PRONEX and CAPES (Brazil) for funding this research.
References
[1] K. Ho¨gerl, G.H. Frischat, Glastech. Ber. Glass Sci. Technol. 67 (6) (1994) 145. [2] H.J. Hoffmann, R. Steinhart, Glastech. Ber. Glass Sci. Technol. 71 (11) (1998) 319. [3] H.J. Hoffmann, Glastech. Ber. Glass Sci. Technol. 72 (4) (1999) 99. [4] V.M. Fokin, E.D. Zanotto, J.W.P. Schmelzer, J. Non-Cryst. Solids 321 (1–2) (2003) 52. [5] A.M. Mood, Introduction to the Theory of Statistics, McGraw-Hill, New York, 1974.
Fig. 11. Micrographs of glass samples D – homogeneous and G – heterogeneous.
