Detection of microheterogeneity in monolithic oxide glasses

Journal of Non-Crystalline Solids 52 (1982) 573-580 North-Holland Publishing Company

DETECTION OF MICROHETEROGENEITY GLASSES

573

IN M O N O L I T H I C

OXIDE

Teruo SAKAINO Kogakuin University, Shinjuku-Ku, Tokyo, Japan

Akio MAKISHIMA National Institute for Researches in Inorganic Materials, Sakura-rnura. lbaraki-Ken. 30.5 Japan

A microheterogeneity has been detected in monolithic oxide glasses, such as SiO2- and GeOz-glasses by means of small angle X-ray scattering (SAXS). The heterogeneity in question was estimated to be a few nanometers in size and considered to be small domains with electronic densities a little different from that of the matrix. The SAXS is a useful method to determine the size of particles ranging from about 2 to hundreds of nm, but for glasses it has weak points, for example: (I) the scattering intensity is very weak because of the small difference in the electron densities between the small domains and the matrix; and (2) the intrinsic SAXS intensity curve, in many cases, is masked by the skirt of the main scattering intensity curve the peak of which is observed in the larger angle region. The authors tried to express the skirt by a theoretical equation, and estimated the heterogeneities to be 1.7 and 1.9 nm for S i O 2- and GeO2-glasses, respectively.

1. Introduction Microheterogeneities of a few n a n o m e t e r s in size have been detected by m e a n s of small angle X - r a y scattering (SAXS) in some oxide glasses, particularly in p u r e m o n o l i t h i c glasses, such as SiO 2- a n d GeO2-glasses. T h e heterogeneity in question is considered to be small d o m a i n s with electron densities a little different from that of the matrix. SiO2-glasses d o p e d with G e O 2 are widely used as a m a j o r p a r t of optical waveguides, p l a y i n g the leading role in d e t e r m i n i n g the light-loss. F o r very p u r e waveguide glasses, the loss is d o m i n a t e d by Rayleigh scattering which has an intensity inversely p r o p o r t i o n a l to the fourth p o w e r of the wavelength [1]. Consequently, the heterogeneity in the glasses is c o n c l u d e d to be closely related to the loss. S a k a i n o et al. [2], from this p o i n t of view, first investigated some s o d a - l i m e glasses, a sheet glass, an optical glass a n d 48 M g O . 52 P2Os glass b y means of SAXS, detecting microheterogeneities of a b o u t 2 n m in size for each glass of a single phase. T h e y d e t e r m i n e d the size of the heterogeneities by S A X S using the following a s s u m p t i o n s : (1) The intensity of the scattered a n d diffracted X-rays (for convenience, to 0022-3093/82/0000-0000/$02.75

© 1982 N o r t h - H o l l a n d

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T. Sakaino, A. Makishima / Microheterogeneity in monolithic oxide glasses

4J H

.,'4

4J U O3

S c a t t e r i n g Angle (Small Angle Region)

Fig. 1. Schematic representation of eq. (I), SAXS(obs)=SAXS(int)+XS(netw).

be referred to as "scattering" hereafter) in the small-angle region from glasses of silicate and of analogous structures can be divided into SAXS(int) and XS(netw), where SAXS(int) denotes the intrinsic or net small-angle scattering intensity attributable to the microheterogeneities closely dispersed in the matrix and XS(netw) the scattering intensity resulting from only the uniform continuous network structure proposed by Warren [3]. In the small-angle region, consequently, he scattering intensity observed, SAX(obs), consists of two parts, SAXS(obs) = sgxS(int) + XS(netw),

(1)

where XS(netw) denotes the skirt portion of the main scattering intensity curve the maximum peak of which appears in the larger angle re#on (see fig. 1) [4]. (2) In the small angle region, the logarithm of the XS(netw) intensity is proportional to the square of the scattering angle. This relation was held with accuracy in the region below 4 degrees for Mo-Ka (35 kV).

2. Determination of microheterogeneity Although SAXS intensities have been observed for SiO 2 [5] and G e O 2 [6] glasses by several research workers, no result shows the scattering intensity curve with a negative inclination called the Guinier slope. It is likely that the SAXS intensity curves observed, SAXS(obs), usually include the skirt of the main scattering intensity curve with its peak in the larger angle region stated above. According to eq. (1), it is necessary first to determine the curve of the skirt, XS(netw), in order to obtain the SAXS(int) to be analyzed. The authors tried to derive a theoretical equation to express the skirt, XS(netw). Details of the deriving procedure are given below. Warren [3,6] derived the following equations, obtaining good agreement

T. Sakaino, A. Makishima / Microheterogeneity in monolithic oxide glasses

575

"~ 70 O

.~. 60 a; 50

m

4o -,-t ttl

c

30

¢-,

20 •~

10

ID

0 r.O

1 2 3 4 (sinO)/l X 102

5 6 (A-l)

Fig. 2. X-ray scattering for SiO2-glass, solid line, the skirt computed on the basis of Warren's structure model; open circles, values calculated by Bienenstock et al. [9]; triangles, observed by Levelut et al. [5]; squares, observed by Renninger et al. [5]; full circles, observed by Weinberg [5].

between calculated and observed values of the X-ray scattering intensity for Si02-glass in the larger angle region. (angstroms are used in the equations)

I / N =fs~[fsi

+ 4fo(sin 1.60S)/1.60S + 4fsi(sin 3.20S)/3.20S + 6fo(sin 4.00S)/4.00S + 12f(sin 5.20S)/5.20S

- 17fQ)6.05S)] + 2 f o [ f o + 2fs,(sin 1.60S)/1.60S + 3fo(sin 2.62S)/2.62S + 6f(sin 4.00S)/4.00S

- 8fQ(4.55S)1,

(2)

f=fsi + 2fo(sin 1.60S)/1.60S,

(3)

Q(x ) = ( 3 / x 2)[(sin x )fix - cos x],

(4)

S = (4 sin O)/X,

(5)

where fsi and fo are atomic scattering factors for Si and 0 atoms respectively and the Si-O distance is assumed to be 0.160 nm. On the other hand, the authors computed the scattering intensity curve applicable to the small-angle region; that is, the skirt which is denoted as XS(netw) here, as follows.

576

7". Sakaino, A. Makishima / Microheterogeneity in monolithic oxide glasses

120

o

80

L9

40

o

I 0

O.l

0.2

4~sine/l

0.3

(A-l)

Fig. 3. Small-angle X-ray scattering intensity curve for GeO2-glass: (A) observed by Pierre et al. [6] (B) computed by the authors [12].

Using the method of least squares, first, the authors determined a polynomial function which expresses'the atomic scattering factor for Si 4÷ ion as a function of (sin 8 ) / t . As the authors were interested in computing the X-ray scattering intensity in the small-angle region, the polynomial function was determined for small values of (sin 8 ) / 1 , ranging from 0 to 0.2, using data tabulated in International Tables for X-Ray Crystallography [7]: f4+ = 15.982Z 3 _ 23.300Z 2 _ 0.0253Z + 10.002,

(6)

where Z = (sin 0)/?~.

(7)

The limit of errors in eq. (6) is smaller than 0.92%, small enough for the present computations. The atomic scattering factor for the O 2- ion was also determined by an equation obtained by Tokonami [8] as follows: fo2- = 2.755 e x p ( - 3.949Z 2) + 5.907

exp(-ZO.64Z 2) + 1.269.

(8)

In 1969 Mozzi and Warren [8] reported a structural model of SiO2-glass. They obtained a new radial distribution function using pair functions by analyzing well refined X-ray scattering data of SiO2-glass. According to their structure model each silicon atom is tetrahedrally surrounded by four oxygen atoms, and each oxygen atom is bonded to two silicon atoms. The S i - O - S i bond angle is distributed between 120 and 180 ° with the maximum at 144 ° . Taking this structure model into account, the authors computed the X-ray intensity scattered in the small-angle region for SiO2-glass by varying the Si-Si interatomic distance in accordance with the Warren's bond angular distribution and by using the polynomial functions for atomic scattering factors of

12 Sakaino, A. Makishima / Microheterogeneity in monolithic oxide glasses

577

2.0

C~

0

"1.5

o

1.0

I 1 Q2 X 10 4

Fig. 4. Guinier plot of SAXS(int) for GeOz-glass computed by the authors. Si 4÷ and 0 2-. From these computations, it was found that the scattering intensity almost monotonically decreases with the decreasing value of (sin 0)/~,. The authors also determined polynomial functions of atomic scattering factors for Si and O atoms, respectively and computed the scattering intensity up to the small-angle region in the same way as the computation performed for atomic scattering factors for Si 4+ and O 2- ions. Agreement between observed and computed values was better for Si 4+ and 0 2- ions than for Si and O atoms, although both of the computed intensity curves showed the same trend. In fig. 2, open circles denote values of the scattering intensity calculated by Bienenstock et al. [9] from the scattering curve for SiO2-glass observed by Warren et al. [10] and the solid line shows the skirt computed by the authors using scattering factors of Si 4+ and 0 2- ions, on the basis of Warren's structure model, that is, a uniformly continuous random network structure which does not contribute to the scattering in the small-angle region. It is very likely, therefore, that the solid line includes no small-angle scattering. Full circles, triangles and squares also denote the small-angle scattering intensity curves for SiO2-glasses observed by Weinberg [5], Levelut et al. [5] and Renninger et al. [5], respectively. Contrary to the tendency of the skirt of the scattering curve, the SAXS intensity curves observed, corresponding to the SAXS(obs) shown in fig. 2, have no negative inclination. According to eq. (1), the SAXS(int) necessary to determine the Guinier plot can be obtained by subtracting the skirt, corresponding to the XS(netw) from the SAXS(obs) shown in fig. 2. In 1972 for a GeO2-glass, an X-ray scattering intensity curve in the small-angle region was observed by Pierre et al. [6],which also has no negative inclination as shown in fig. 3. In the same way as for SiO2-glass, the authors [12] tried to derive a theoretical equation expressing the skirt of the scattering

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T. Sakaino, A. Makishima / Microheterogeneity in monolithic oxide glasses

intensity curve which also has a peak in the larger angle region. According to eqs. (2), (3), (4) and (5), the skirt of the scattering intensity curve for the GeO2-glass in the small angle region was computed, using the value, 0.175 nm, as the atomic distance between Ge and O given by Leadbetter [11]. In the computing procedure the authors determined a polynomial, eq. (9), to express the atomic scattering factor for Ge 4+ as a function of (sin 8).2,, using the method of least squares. f4+ = 66.047Z 3 _ 85.213Z 2 + 0.329Z + 27.997.

(9)

As for the scattering factor of O 2- , eq. (8) is also available for the GeO2-glass. The curve corresponding to the skirt thus computed, that is, the XS(netw) for he GeO2-glass, is shown in fig. 3 as (B), which shows good agreement with both the position of the intensity peak and the shape of the curve observed in the larger angle region. The next procedure is to subtract each XS(netw) from each corresponding SAXS(obs) in order to obtain SAXS(int) both for SiO2- and GeO2-glasses. For example the result for the GeO2-glass is shown in fig. 4, which can be expressed by a straight line with a negative inclination called the Guinier plot, approaching the same value observed by Pierre et al. [6], about 49 e.u. mol-1 at 0 °. According to Guinier's equation, the scattering intensity in the small-angle region can be approximately expressed by the following equation [13]: I ( 0 ) = K V R 3 exp(--4~r2R202/3~2),

(10)

where K denotes a constant, V a total volume concerning with the X-ray scattering, R the inertial radius of the particles and ~ the wavelength of the X-rays. Taking logarithms of both the sides of eq. (10), a straight line with a negative inclination is obtained for the 02 axis, which is called a Guinier plot. From the the inclination, -Rz(4qr2/3~2), the size of the particles, R, can be determined. Making the Guinier plot from the SAXS(int) computed both for SiO2- and GeOz-glasses, microheterogeneities were determined to be 1.7 and 1.9 nm in size, respectively.

3. Discussion

Discussing the microheterogeneity thus detected in the monolithic oxide glasses, we must consider first the density fluctuation caused by thermal agitation in a melt in equilibrium. The average square deviation of the density fluctuation is given by the following equation [14] ( ( A d ) 2) = g r ( d g k T / V ) ,

(11)

where K r denotes the isothermal compressibility, d o the average density, k the Boltzmann constant, T the temperature and V the volume which a fluctuation occupies. Assuming the V in SiOz-glass to be (2 nm) 3, we obtain about 1% as

T. Sakaino, A. Makishima / Microheterogeneity in monolithic oxide glasses

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{ ( A d ) 2 ) / d 2 o . Finally the small-angle scattering intensity is computed to be about 25 e.u. per SiO 2 by substituting the value, 1%, into an equation expressing the SAXS intensity. The intensity, 25 e.u., is likely taking values proposed by Seward III et al. [15] and Zarzycki [14] into account. Pierre et al. [6] theoretically obtained 7.5 × 10 ~3/V-I/2 as the density fluctuation in the GeO2-glass, on the other hand, the authors computed {(Ad)2)/d20 to be about 1.3% using the size, 1.9 nm. After all, it may be concluded that there is no contradiction among these values considering the obscurity in the frozen-in temperature in the cooling process of the melts. Furthermore the existence of the clusters or embryos can be considered as precursors of the nucleus formation. In fact, Bando et al. [16] found small domains with a reflection a little higher than the matrix in a very pure SiO2-glass by means of an electron microscope of high resolution power, determining their size to be about 1.7 nm based on diffraction theory. On the other hand, Nukui et al. [17] obtained radial distribution functions for a SiO 2 melt at 1750 and 1850°C, concluding structures both of quartz and cristobalite to be present in the ratio of 1 : 2 in volume. This result also suggests the existence of microstructures in the melt and these are considered to be able to cause a SAXS when cooling.

4. Summary Microheterogeneities were detected not only in multicomponent oxide glasses but also in monolithic oxide glasses, such as SiO 2- and GeO2-glasses by means of SAXS and they were determined to be about 2 nm in size. The origin of the heterogeneity was discussed from standpoints of the density fluctuation, the structure of melts and electron microscopic observation.

References [1] R.D. Maurer, J. Non-Crystalline Solids 42 (1980) 197. [2] T. Sakaino, M. Yamane and F. Sato, Proc. Xth Int. Congr. Glass, Kyoto, 12 (1974) 16; T. Sakaino, A. Makishima and Y. Fujita, Rep. Asahi Glass Foundation for Ind. Tech. 24 (1974) 31; T. Sakaino, M. Yamane and F. Sato, J. Mat. Sci. Soc. Jap. 11 (1974) 318; T. Sakaino, M. Yamane and A. Makishima, Cent. Glass Ceram. Res. Bull. 22 (1975) 108. [3] B.E. Warren, J. Am. Ceram. Soc. 17 (1934) 249; Z. Krist. 86 (1933) 349. [4] T. Sakaino and A. Makishima, Proc. XIth Int. Congr. Glass, Prague, I (1977) p. 21. [5] D.L. Weinberg, J. Appl. Phys. 33 (1062) 1012; A.M. Levelut and A. Guinier, Bull. Soc. Franc. Mineral. Cryst. 90 (1967) 445; A.L. Renninger and D.R. Uhlmann, J. Non-Crystalline Solids 16 (1974) 325. [6] A. Pierre and D.R. Uhlmann, J. Appl. Cryst. 5 (1972) 216. [7] J.A. Ibes and W.C. Hamilton, International Tables for X-ray Crystallography, IV (The Kynoch Press, Birmingham, 1974) p. 74. [8] R.L. Mozzi and B.E. Warren, J. Appl. Cryst. 2 (1969) 164. [9] A. Bienenstock and B.G. Bagley, J. Appl. Phys. 37 (1966) 4840.

580 [10] [11] [12] [13] [14] [15]

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B.E. Warren, H. Krutter and O. Morningstar, J. Am. Ceram. Soc. 19 (1936) 202. A.L. Leadbetter and A.C. Wright, Z. Krist. 7 (1972) 37. A. Makishima and T. Sakaino, J. Chem. Soc. Japan, Chem. Ind. Chem. 10 (1981) 1684. A. Guinier, Ann. Phys. 12 (1939) 161; J. Chem. Phys. 40 (1943) 133. J. Zarzycki, Proc. Xth Int. Congr. Glass, Kyoto 12 (1974) 28. T.P. Seward III and D.R. Uhlmann, eds., R.W. Douglas, B. Ellis, Amorphous Materials (Wiley-Interscience, New York, (1972) p. 237. [16] Y. Bando and K. Ishizuka, J. Non-Crystalline Solids 33 (1979) 375. [17] A. Nukui, H. Tagai, H. Morikawa and S. Iawi, J. Am. Ceram. Soc. 61 (1978) 174.