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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 Sao Carlos, Vitreous Materials Laboratory, Rod. Washington Luiz, km 235, Sao Carlos, SP 13.560-950, Brazil b Comision Nacional de Energıa Atomica, Centro Atomico Bariloche, Av. Ezequiel Bustillo, km 9.5, (8400) S. C. de Bariloche, Rıo Negro, Argentina

Received 2 February 2005; received in revised form 27 September 2005 Available online 26 October 2005


We propose and test a new method to evaluate the chemical homogeneity of glasses based on statistical analyses of the volume distribution 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. 2005 Elsevier B.V. All rights reserved.

1. 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 material s particle size, temperature, time, mixing, etc.), the result is quite often a heterogeneous 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 (r2) and scale of the heterogeneities, 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 several 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

02-3093/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2005.09.021

* Corresponding author. Tel.: +5 16 3351 856; fax: +5 16 3361 5404. E-mail address: dedz@power.ufscar.br (E.D. Zanotto).

w.elsevier.com/locate/jnoncrysol Journal of Non-Crystalline Solids 351 (2005) 3579–3586 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 Christiansen–Shelybskie method.

In this paper we propose a new method, based on a statistical analysis of the volume distribution of crystals that are developed by heat treatment above Tg, to characterize the degreeandscaleofthehomogeneityofglassesthatcrystallize 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 byvaryingthemeltingtemperaturesandthenumberofmelting/crushing cycles, and test the proposed method.

2. 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 Poisson s statistics, the average number of crystals and its standard deviation satisfy the condition rN/N =1 /N1/2. The value of N depends on the optical field area l2, which in turn is related to the microscope s magnification, M, through a constant k,( l2 = kM2), and on the volume density of crystals Nv. 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 l2, after a proper heat treatment:

N ¼ 2Nvl2UðTGÞtG; ð1Þ where Nv = I(TN) · tN is the volume density of crystals. For a set of measurements with pre-selected values of

A rN/N vs. 1/N1/2 plot for a crystal distribution in a perfectlyhomogeneousglassobtainedwithdifferentmicroscope magnifications should show a straight line with slope 1 and zero intercept. In the case of an inhomogeneous glass, however, 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 atransition from homogeneousto inhomogeneous behavior foragivenNvalue,wecanuseEq.(2)toassignatypicallthat 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 microscope, 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 Ni values obtained for various optical fields with the same

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 Nvi nuclei in a given sample volume is described by Poisson statistics [5], and is summarized by

ðNvi ¼ 0; 1; 2;; nÞ; ð3Þ

where Nv 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 Ni counted in an optical field of the sample cut, and Nv is replaced by the average of a set of

Ni s, namely N (the average number of crystals in the glass surface in a field of size l2).

The standard deviation of the Ni values obtained with the same magnification M is given by Eq. (4a):

s . ð4aÞ

Thus, for a number s of optical fields of size l2, N can be determined, enabling one to calculate the standard deviation of the distribution with Eq. (4a).

If the glass is chemically homogeneous, then Eq. (4a) can be written as [5] r ¼ ffiffiffiffiNp . ð4bÞ

For each magnification M, a corresponding (r,N) pair is obtained for a perfectly homogeneous glass. A plot of the

3580 L.A. Souza et al. / Journal of Non-Crystalline Solids 351 (2005) 3579–3586

(r,N) pairs obtained for different M should plot as a straight line with slope 1 and zero intercept. We have chosen a plot of the type r/N vs. 1/N1/2, which yields the same function, to normalize for different glasses and to let the x-axis vary from optical to electron microscopy scale from left to right.

We thus propose a statistical test to evaluate glass homogeneity by plotting r/N vs. 1/N1/2 using experimental crystal densities measured in cross-sections of any previously heat-treated glass sample. A random behavior (line with slope = 1 and 0 intercept in a r/N vs. 1/N1/2 plot) is expected for magnifications M for which the optical fields are significantly larger than the typical heterogeneity scale. Deviations from the Poisson statistics are likely to be found when the optical field is smaller than the heterogeneity scale, since different fields correspond to different glass compositions.

One can now distinguish three different magnification levels for sampling a heterogeneous glass:

1. Low magnifications: each optical field covers a large area and the sample is homogeneous in that large scale. Thus, the experimental points in a r/N vs. 1/N1/2 plot will draw a straight line with slope 1 and 0 intercept; 2. Medium magnifications: the chosen field distinguishes areas with different crystal densities and the experimental points will appear above the straight line; 3. Very high magnifications (for example, SEM or AFM): the small field does not distinguish areas with different crystal densities, the number of crystals typically varies between 0 and 1 in each field, and the experimental r/N points appear below the straight line. This can be explained as follows: r is a measure of the distribution frequency width. However, when the magnification is sufficiently large, only two Ni values are possible: 0 (no crystals in the field) and 1 (1 crystal in the field) and this does not characterize a Poisson distribution; hence, the experimental r is underestimated.

Magnifications can be considered low, medium or very high according to the scale of heterogeneity. Depending on the experimental conditions, all the above possibilities or a particular one can be observed. Since the crystal nucleation and growth kinetics strongly depend on the glass chemical composition and, consequently, on its chemical homogeneity, different volume elements of a heterogeneous glass present different nucleation probabilities. Thus, for a heterogeneous glass, a measure of crystallization statistics plotted as r/N vs. 1/N1/2 will not be a straight line with inclination 1 and 0 intercept.

The aim of the method proposed here is therefore to quantify chemical heterogeneity in the glass volume, but crystallization statistics are normally obtained by counting crystals on polished cross-sections. Hence, the method relies on the property that a spatially random volume distribution of crystals generates a random distribution of crystals on a flat polished sample cross-section.

3. Computational test of the method

We performed a computational simulation that emulated surface cuts of homogeneous samples having randomly distributed crystals (Fig. 1) and surface cuts of inhomogeneous samples including striae having a high crystal density (Figs. 2 and 3). Using 500 randomly positioned windows for each magnification (corresponding to 100, 250 and 500· in an optical microscope, and to 1000 and 1500· in a SEM), the number of points per field, Ni, was computed and the average value N and its correspond- ing rN were calculated. Fig. 4 shows that, in the computer-generated homoge- neous specimen, r/N vs. 1/N1/2 points lie on a straight line of slope 1 and 0 intercept, while deviations are exhibited by

Fig. 1. Computational simulation of a 1 m cross-section surface with randomly distributed points. The surface density of points is 120 crystals/ mm2.

Unit Length

Unit Length

Fig. 2. Simulated sample cross-section of 1 · 1m m2 with vertical striae of high crystal density of 100 lm width centered at x = 0.2, 0.4, 0.6 and 0.8. The matrix density is 120 crystals/mm2 and the striae density is 300 crystals/mm2.

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some data corresponding to heterogeneous samples. The data corresponding to the sample in Fig. 3 with slight inhomogeneity deviates at a larger N than those of the sample having striae of higher crystal density. However, an analysis 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.

4. Experimental procedure

To experimentally test the proposed method, two glass systemswere evaluated:a glasswithcomposition23.26Na2O–

23.26CaO–47.38SiO2–6P2O5 (mol%), which will be dubbed

SSP6,anda45Li2O–20Nb2O5–35SiO2(mol%)glass,whichwill be denominatedLNS. Theseglasseswerechosenbecausethey displayinternalcrystallizationeasilyandhavevolatileelements thatcan lead to significantdegreesof heterogeneity. We prepared the SSP6 glasses by melting appropriate quantities of SiO2 (crushed Brazilian quartz) with analytical grade Na2CO3 (Mallinckrodt), CaCO3 (Mallinckrodt) and Na2HPO4 (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 melting/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 m 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 subjected to nucleation treatments for 8, 10, 12, 16 and 20 h at 570 C and subsequently to a development treatment at 720 C for 10 or 20 min to allow for crystal growth. Glasses containing extra OH were also evaluated. After polishing with CeO2 to an optical degree, measurements of Ni (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 samples were also prepared for SEM analysis and the measure- ments of Ni were carried out at 1000 and 1500·. The LNS glasses were melted by the conventional method, using Li2CO3 (Synth, PA), Nb2O5 (CBMM, opti- cal grade) and SiO2 (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.

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