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SAXS analysis of textures formed by phase separation and crystallization of Al2O3–SiO2 glasses
Journal of Non-Crystalline Solids 282 (2001) 265±277
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SAXS analysis of textures formed by phase separation and crystallization of Al2O3±SiO2 glasses Takahiro Takei a,b,*, Yoshikazu Kameshima a, Atsuo Yasumori a, Kiyoshi Okada a, Nobuhiro Kumada b, Nobukazu Kinomura b a b
Department of Metallurgy and Ceramics Science, Graduate School of Science and Engineering, Tokyo Institute of Technology, O-okayama, Meguro, Tokyo 152-8552, Japan Institute of Inorganic Synthesis, Faculty of Engineering, Yamanashi University, 7 Miyamae, Kofu, Yamanashi 400-8511, Japan Received 17 May 2000; received in revised form 2 October 2000
Abstract Textures formed by phase separation and crystallization of Al2 O3 ±SiO2 glasses with 15, 25 and 50 mol% Al2 O3 compositions were investigated by small angle X-ray scattering (SAXS) and TEM. From the Porod analysis of the SAXS patterns, droplet structures were found in glasses with 15 and 50 mol% Al2 O3 compositions, whereas an interconnected structure was formed by spinodal decomposition in the glass with 25 mol% Al2 O3 . In the initial-stage crystallization of mullite, nucleation was considered to occur in the Al2 O3 -rich regions. The crystallites of mullite were of similar size to the phase separated region. The relationship between the annealing time and the size of the interconnected texture formed by spinodal decomposition was analyzed using Cahn's theory, giving good agreement with the result obtained by the Porod method. The distance distribution of the phase separated grains was obtained from the radial distribution function calculated by Fourier transformation of the SAXS patterns. The calculated intergrain distances increased with increasing annealing time and were in good agreement with those obtained from TEM observation. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 61.43.Fs; 64.75.+g; 61.10.Eq
1. Introduction Mullite Al42x Si2 2x O10 x , the only thermodynamically stable compound in the Al2 O3 ±SiO2 system at atmospheric pressure, is found in various aluminosilicate ceramics. For example, mullite is a major crystalline phase in porcelain and plays an important role in the development of its mechan-
* Corresponding author. Tel.: +81-55 220 8616; fax: +81-55 254 3035. E-mail address: [email protected] (T. Takei).
ical strength. In recent years, mullite have been identi®ed as possible high temperature structural materials because of their high mechanical strength and creep resistance at high temperature. A variety of crystallization reactions have been reported in the preparation of high-purity mullite from various raw materials. The crystallization process of mullite falls into two categories [1]: direct crystallization of mullite from an amorphous phase and crystallization via a spinel phase. The former type of crystallization was reported when the precursor is homogeneous at the molecular level such as in glasses and monophasic gels
0022-3093/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 1 ) 0 0 3 1 5 - 5
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[2], and the latter type was reported when the precursor is inhomogeneous at the molecular level as in diphasic gels [3±5], rapidly hydrolyzed gels [6] and kaolinite [7]. Although the crystallization processes of mullite from various raw materials have been studied in detail as described above, the crystallization kinetics of mullite have not been investigated in detail, especially in glasses. Furthermore, the metastable immiscibility reported in this system should in¯uence the crystallization of mullite. We have reported [8] that the nucleation of mullite is in¯uenced by the phase separated texture formed prior to crystallization because the activation energy for nucleation of mullite from the glass within the immiscibility composition is smaller than from the glass in the miscible region. The relationship between the immiscibility texture and the mullite crystallization texture is, however, not yet clearly understood. In order to clarify this relationship, it is necessary to investigate the phase separation and crystallization processes precisely. Small angle Xray scattering (SAXS) is a useful technique to investigate these phenomena because the expected size of the texture ranges around 1±100 nm, an appropriate size for SAXS analysis. In this paper, the textures formed by phase separation of Al2 O3 ±SiO2 glasses and the crystallization of mullite from these glasses were examined by SAXS and TEM. 2. Experimental procedure The starting materials were tetraethylorthosilicate (TEOS) and aluminum nitrate nonahydrate. Three compositions of TEOS and aluminum nitrate nonahydrate were prepared, containing 15, 25 and 50 mol% Al2 O3 (designated R15, R25 and R50, respectively). These compositions were mixed in absolute ethanol for 3 h, then gelled slowly in an oven at 60°C for 1 month [6]. The resulting gels (SH gel [6]) were calcined at 500°C for 6 h and 800°C for 24 h to remove organic components, pressed into rods at 98 MPa in a CIP, and sintered at 1300°C for 12 h. Glasses were prepared by melting and ultraquenching these Al2 O3 ±SiO2 sintered rods using an
arc image furnace. The rods were melted at 2100°C and ultra-quenched between twin rollers rotating at 1500 rpm. The estimated quenching rate was higher than 105 K/s [9]. The resultant ultra-quenched glasses were in the form of ¯akes. The glasses were annealed at 900°C for 1±48 h in an electric box furnace to induce phase separation and mullite crystallization. The crystalline phase formed in the annealed glasses was determined by powder X-ray diractometry (XRD) using monochromatic CuKa radiation. SAXS was measured using monochromatic MoKa radiation with a position sensitive proportional counter. The X-rays were collimated by a Kratky U-slit with incident slit of 0.03 mm and height limiting slit of 10 mm. The voltage and current of the X-ray source were 50 kV and 200 mA. The SAXS measurements were carried out in the angular range )0.5 to 7.5 with step intervals of 0.002° and a ®xed time of 3 h. The microstructure of the heat-treated glasses was observed using a transmission electron microscope to con®rm the results of SAXS analysis. 3. Calculations Since the measured SAXS intensity was scattering from slit-collimated X-rays, it was necessary to correct the intensity to pinhole collimated Xrays. The correction was carried out using general theory [10]. Since the existence of even slight ¯uctuations of electron density in the sample in¯uences the analysis result, Luzzati et al. [11] reported a calibration method to counteract eect of the ¯uctuation by adding a speci®c constant to the SAXS intensity to satisfy the proportional decrease against S 4 . This calibration method was mentioned by Kratky [12] that the intensity calibrated thus is satisfactory for use with Porod's method. Therefore, this method was adopted in the present work. In SAXS analyses, the sample is generally assumed to consist of two phases with dierent electron densities. Since the immiscibility and crystallization textures were considered to ®t this condition, their texture sizes were evaluated using the Porod method [13,14], in which the transversal
T. Takei et al. / Journal of Non-Crystalline Solids 282 (2001) 265±277
lengths, la and lb , corresponding to the sizes of the phases a and b are calculated from the measured intensity, I S, by the following equation: lr
4Q ; p lim S 4 I S S!1
la
lr 1
w
; lb
lr ; w
1
where Q is invariant, S is the scattering vector represented by S 4p sin h=k, and w is the volume fraction of phase a. In this study, w was calculated from the fraction of Al2 O3 -rich phase, the composition of which was evaluated from the immiscibility region in Al2 O3 ±SiO2 system calculated by molecular dynamics simulation method [15]. The resulting fractions of the Al2 O3 -rich phases were 12, 34 and 83 vol% in R15, R25 and R50, respectively. The value of Q is calculated from Eq. (2), Z 1 2 Q S 2 I SdS 2p2 Ie V Dq w 1 w; 2 0
where Ie is Thomson's constant, V the irradiated sample volume and Dq is the dierence of electron density between phases a and b. The textures formed by phase separation and crystallization were also investigated by determining the radial distribution function of the grains. In these calculations, the grains corresponded to the dispersed phase in the binodal texture, the crystallized mullite grains and the diameter of the interconnected texture in the spinodal decomposition. The radial distribution function was calculated as follows. The observed scattering intensity, I S, can be represented by two scattering terms [16]: I S Ip S Ii S:
3
Here, Ip (S ) and Ii (S ) are scattering intensities by one and two grains, respectively. The value of Ip (S ) can be calculated from basal scattering formula. This is generally expressed using a scattering function, W S, by the following equation: 2
Ip S Nn Ie
3 sin Sr 3 S r3
Nn2 Ie W2 S;
2 Sr cos Sr 4
267
where N is total number of grains in V and n is total number of electrons in one grain. On the other hand, Ii S is expressed in terms of the number density of the grains, Ii S Nn2 Ie W2 S
Z
P r exp ir SdV ZV 1 sin Sr dr 4pr2 P r Nn2 Ie W2 S Sr 0 Z 1 sin Sr Nn2 Ie W2 S dr 4pr2 fP r P0 g Sr 0 Z 1 sin Sr dr 4pr2 P0 Sr 0 Z 1 sin Sr 2 2 2 dr I0 S; Nn Ie W S 4pr fP r P0 g Sr 0 5
where r is a vector between arbitrary points in dierent grains, P r is a function of grain number density, P0 is the mean number density of grains and I0 S is the zero-order scattering. The zeroorder scattering is generally negligible in these calculations because its intensity is very small except at very low angles. Therefore, the total scattering intensity, I S, can be expressed as I S Ip S Ii S Nn2 Ie W2 S Z 1 4pr2 f P r 1 0
P0 g
sin Sr dr : Sr
6
The dierential radial distribution function of the grains is obtained by Fourier transformation of Eq. (6): Z 2r 1 4pr2 f P r P0 g Si S sin Sr dS; 7 p 0 i S
I S Nn2 Ie W2 S
1:
8
In these calculations, the scattering function, W(S), was obtained from Eq. (4). However, the W(S) curve shows heavy hunting over a relatively large angular range for genuine spheres with uniform electron density. In order to obtain a smooth W(S) curve, Zeng et al. [17] used an approximation derived from a combination of the theories of
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Porod and Guinier. According to the Porod theory, scattering intensity over a relatively large angular range decreases proportionally with scattering vectors to the minus fourth power. On the other hand, the scattering intensity at small angles range was theoretically approximated to a Gaussian function by Guinier. This combination method gave a smooth W(S) curve without hunting, although it is dicult to smoothly connect the calculated Porod and Guinier curves. Therefore, the following method was used in this study. Particles are assumed to have a Gaussian type size distribution and the function X(S) was used instead of Nn2 Ie W2 S, though W(S) is generally used for spherical particles of similar size. The X(S) is expressed by the following equation: ( ) Z 1 1 p r r0 2 2 2 X S Nn Ie W S exp dr f f 1 2 Z 1" 2r3 Q 3 sin Sr Sr cos Sr w S 3 r3 1 3p 1 ( )# 1 p r r0 2 exp dr; 9 f f
intensity / a.u.
102
100
10-2
10-4 10-2
Ω(S) Nn2 IeΨ2 (S) Guinier approximation Porod approximation 10-1
100 -1
S / nm
Fig. 1. The X S; Nn2 Ie W2 S, Guinier approximation and Porod approximation as a function of scattering vector S.
where w is the volume fraction of the particles, r0 is the mean particle radius and f is the integral width of the particle size distribution. In the calculation, r0 was calculated from the transversal length, l , by the relation r0 3=4l and f was ®xed as f r0 =2: The calculated curves of X S; Nn2 Ie W2 S, Porod and Guinier approximations are shown in Fig. 1. The X S line was used in this work for further calculations rather than Nn2 Ie W2 S. 4. Results and discussion 4.1. Porod analysis Fig. 2 shows the XRD patterns of R15, R25 and R50 samples as-quenched and annealed at 900°C for various times. The XRD patterns of all the asquenched glasses showed only a halo at 2h 20°± 25°. The XRD patterns of the glasses annealed at 900°C showed mullite crystallization after annealing R15, R25 and R50 for 48, 24 and 12 h, respectively. Fig. 3 shows the SAXS curves of these samples. In all samples, the single peak observed in each pattern becomes more intense and shifts toward smaller angles with increasing annealing time. Since the peak is observed even before mullite crystallization, it is considered to be generated by immiscibility texture formed prior to crystallization. Since the only peak appearing in a dilute system is at 0°, these textures are considered to arise from a dense system. Fig. 4 shows relationship between annealing time and intensity of these peaks. The peak top intensity was found to become high with increase of Al2 O3 content in chemical composition of samples. The logarithmic intensities increase with slope of 1/6±1 in all samples. The small slope of 1/6 in R50 sample was plausibly considered that it was attributed to volume shrinkage by mullite crystallization because Al2 O3 rich phase plays major phase only in R50 sample. In all samples, these slopes are similar values of 2/3 in up to 4 h, whereas the other slopes are not similar. Since the slopes have various values depending on samples, the intensity plots are insucient to analyze phase separated texture. Therefore, the transversal length was then evaluated for the
T. Takei et al. / Journal of Non-Crystalline Solids 282 (2001) 265±277
Fig. 2. XRD patterns of R15, R25 and R50 samples annealed at 900°C for various times.
Fig. 3. SAXS patterns of R15, R25 and R50 samples annealed at 900°C for various times.
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log (Intensity)
-3
1/1
R15 R25 R50
1/6
3/4 1/2
-4
2/3
2/3
-5
slope ≈ 2/3
-1
0 1 log (firing time) / h
2
2.0 phase separation crystallization (12h) D=0.03 3 = 1/ 0. 31
:
1.5
D=
R50 :
= .47 =0
R25 0 D=
= .50
-1
0 D=
2/5
3 = 1/ crystalli0. 32 phase separation zation (60h)
:
0.5
0.0
= .39
:
D
1/2
crystallization phase separation (24h) D=0.04 1/5 .20= D=0 D=0.13 :
1.0
:
log(transversal length of Al2 O3 -rich phase) / nm
Fig. 4. Logarithmic plots for change of intensity in the three samples as a function of annealing time.
D=
1/2
R15 0
log t / h
1
2
Fig. 5. Logarithmic plots for change of transversal lengths of Al2 O3 -rich phase in the three samples as a function of annealing time.
two separated phases using the Porod method. Fig. 5 shows the relationship between annealing time and transversal length of all the samples. The transversal lengths increase in all the samples and the curve slopes change by three steps with in-
creasing annealing time. The slope change was considered to arise from dierences in the growth mechanisms of the immiscibility and mullite crystallization textures. Table 1 shows these slopes, D1 ; D2 and D3 , and the in¯ection times, t12 and t23 , obtained by the Porod method. The slopes in R15 and R50 samples were close to 1/2 and 1/3 in the ®rst and second steps of annealing time, respectively. Burnett and Douglas [18] reported that the size of phase separated grains formed by a nucleation-growth mechanism in the binodal region increases proportionally with time to the 1/2 power up to the equilibrium volume fraction of the grains and to the 1/3 power after reaching the constant volume fraction. From this coincidence in the slope ratios, we consider that the binodal structure is formed by a nucleation-growth mechanism in R15 and R50, but not in R25. On the other hand, the slopes of R25 did not correspond well with slopes of 1/2 and 1/3. Since the composition of R25 is intermediate between R15 and R50, the former is suggested to correspond to the spinodal region. It is therefore plausible that an interconnected structure is formed in this sample by spinodal decomposition. The growth process for this sample is discussed in the following section in terms of Cahn's theory. In the third step (crystallization of mullite), the slopes of all samples become very small compared with those of the ®rst and second steps. To compare the sizes of the phase separated and crystallized textures, the crystallite size of mullite in R25 and R50 was evaluated from the integral width of the XRD 121 re¯ection using Sherrer's equation. The crystallite size of R15 could not, however, be calculated because the intensity of its 121 mullite re¯ection was too weak. Fig. 6 shows the changes in the crystallite size of the mullite and the Porod grain size of the Al2 O3 -rich phase as a function of annealing time. Here, the Porod grain size was calculated from the transversal lengths assuming spherical shape using the relation (Porod grain size) 3=2 transversal length. In the case of R50, the Al2 O3 -rich phase corresponds to the matrix; the calculated size therefore pertains, to the region rather than the grain size. The crystallite size of the mullite is in good agreement with the Porod grain size of the Al2 O3 -rich phase of the
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271
Table 1 Slopes of logarithmic plots of transversal lengths in three samples
a b
Sample
D1 a
R15 R25 R50
1/2 2/5 1/2
t12 b (h) 5 3.4 1.3
D2
t23 (h)
D3
Crystallization time (h)
1/3 1/5 1/3
12 24 6.7
0.13 0.04 0.03
48 24 12
Dm indicates slope of logarithmic plots of transversal length in the mth stage. tmn indicates in¯ection time between the mth and nth stages.
50
crystallite or grain size / nm
40
R50 R50 grain size R50 crystallite size
30
R25 grain size R25 crystallite size 20
10 10
R25 50 100 firing time / h
Fig. 6. Change of crystallite sizes of R25 and R50 as a function of annealing time.
phase separated texture in R25, however, by comparison with the size of the Al2 O3 -rich phase in R50, grain size is smaller than the Porod grain size. From these results, it is concluded that the mullite in R25 crystallizes from the Al2 O3 -rich regions in the phase separated texture because the sizes of Al2 O3 -rich phase and crystallite are similar. On the other hand, the mullite which crystallizes in R50 has a larger grain size than R25 because the Porod grain size of the Al2 O3 -rich phase in R50 is much larger than in R25, but is much smaller than the dimensions of the Al2 O3 rich phase which constitutes the matrix from which the mullite crystallizes. Fig. 7 illustrates schematically the mechanisms of immiscibility and mullite crystallization in
Al2 O3 ±SiO2 glasses. In R15, Al2 O3 -rich grains are formed as droplets by a nucleation-growth mechanism in the SiO2 -rich matrix, whereas SiO2 -rich grains are formed as droplets in the Al2 O3 -rich matrix by the same mechanism in R50, for which the Porod analysis suggests their compositions were a composition in the binodal region. Crystallization of mullite occurs from Al2 O3 -rich phase but the size indicates very sluggish growth even after prolonged annealing times. This may be due to the low diusion coecient of the Al2 O3 ±SiO2 system [19]. On the other hand, the interconnected texture formed in R25 by spinodal decomposition constitutes the matrix from which the mullite crystallizes. 4.2. Cahn's analysis Since many works demonstrated the eectiveness of Cahn's theory [20±23] to investigate the texture formed by spinodal decomposition, the texture of R25 was examined by this method. In Cahn's theory [24,25], the texture is explained by random superposition of sinusoidal waves with characteristic wavelengths, km . The wavelength corresponds to the typical size of the largest ampli®cation of composition ¯uctuation. The wavelength is determined only by the initial annealing time because the composition ¯uctuation occurs very quickly in spinodal decomposition. The SAXS curves of R25 samples ®red up to 8 h are shown in Fig. 8. Up to 4 h, the SAXS curves show two peaks at S 0.8±0.9 and 1.1 nm 1 . The peak at 0:8±0:9 nm 1 grew up depending on ®ring time, whereas that at 1:1 nm 1 grew down and was hid in 0:8±0:9 nm 1 peak for longer than 4 h. Since only the peak at 0:8±0:9 nm 1 dominated further
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Fig. 7. Schematic illustration of the development of immiscibility and mullite crystallization textures in Al2 O3 ±SiO2 glasses.
R(S) / h-1
0.2
Sm
0.1
0.0
-0.1
0
1
2 S / nm-1
Fig. 8. SAXS patterns of R25 samples annealed at 900°C up to 8 h.
longer the ®ring time of 4 h, it was found that the peak at 0:8±0:9 nm 1 was generated by the composition ¯uctuation in the spinodal decomposition, while that at 1:1 nm 1 was by another texture mentioned afterward. The initial stage of spinodal decomposition in R25 samples was considered to occur during the ®rst 4 h, because the peak grew up at 0:8±0:9 nm 1 and the slope ratio of the
Fig. 9. Change of ampli®cation factor, R(S), as a function of S in R25 sample.
transversal length changes at this time (t12 shown in Table 1 above). The wavelengths km were then calculated for various S values from the slope ratio for the plots of ln I S vs annealing time. Fig. 9
T. Takei et al. / Journal of Non-Crystalline Solids 282 (2001) 265±277
shows relationship between S and the ampli®cation factor R S calculated from km . Since the maximum ampli®cation factor (Sm ) was observed at around 0.8 nm 1 in Fig. 9, km was calculated to be about 8 nm by the equation km 2p=Sm . In order to examine suitability of the resulting value of km 8 nm, the texture of spinodal decomposition was computed using this km . The texture pattern was calculated from 100 randomly oriented sinusoidal waves, which had a Gaussian distribution of amplitudes and ¯at distributions of phase angles and orientation. The ratio of the Al2 O3 ± and SiO2 -rich phases in this texture was w/(1)w) 34/66, for which the computed texture is shown in Fig. 10. The texture has the appearance of the two interconnected phases generally observed in spinodal decomposition.
273
The transversal lengths of the two phases were evaluated from the computed texture and were compared with the lengths obtained from the Porod analysis. The transversal lengths from Cahn's method were calculated by counting numbers of arbitrary lengths in the computed texture. Table 2 shows the transversal lengths of R25 obtained from the Porod and Cahn methods. These values showed very good agreement. In the previous section, the transversal length evaluated from the Porod method was shown to increase proportionally with annealing time to the 2/5 power up to 4 h, and to the 1/5 power at annealing times from 4 to 24 h as shown in Fig. 5 and Table 1. On the basis of Cahn's theory, the slope ratio in the initial stage D1 should be zero because spinodal decomposition in this stage occurs only amplifying ¯uctuation of composition. On the
Fig. 10. Cross-section of computed spinodal texture for R25 sample.
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Table 2 Transversal lengths calculated from the Porod and Cahn methods in R25 samples annealed at 900°C for 4 h Transversal length (nm) Al2 O3 -rich phase SiO2 -rich phase
Porod
Cahn
5.3 10.0
5.2 10.0
other hand, the slope ratio D1 obtained from the Porod analysis is 2/5 (Table 1). This dierence in the slope ratio is considered to occur for the following reasons : an ultra-®ne texture was formed during quenching process, the average size of which is expressed by the transversal length derived in the Porod method. The ultra-®ne texture was shown as the peak at 1.1 nm 1 in Fig. 8. Both this and the spinodal decomposition texture are ampli®ed depending on km . Annealing at 900°C causes the ultra-®ne texture formed during quenching to disappear, with the formation of a phase separated texture characterized by km 8 nm. The resulting value of 2/5 for D1 was generated from both disappearing of ultra-®ne texture and amplifying of spinodal decomposition texture depending on km . 4.3. Grain radial distribution function In colloid studies, the radial distribution function (RDF) for the colloid grains can be used to provide information on intergrain distances [26,27]. This technique was therefore adopted in this study. The grain RDF can be calculated by Fourier transformation of the SAXS intensity expressed in Eq. (7). This calculation method is essentially similar to the RDF used for structural analysis of amorphous substances, but diers in that only scattering intensity data are used for the calculations. Calculations of usual atomic RDF require atomic scattering factors, whereas the scattering function of the grains is necessary for calculation of grain RDF. In this study, the weighted scattering function X S calculated from Eq. (9) was used as the scattering function W S of grains. Fig. 11 shows the calculated grain differential RDF (DRDF) of R15, R25 and R50 samples. In these grain DRDF curves, the distance of the nearest neighbor grains is represented in
Fig. 11 by arrow marks. The ®rst neighbor distance was found to increase with annealing time in all samples, corresponding to the growth of phase separated texture. The shape of the ®rst peaks gradually changes and the peaks splitted with longer annealing time. These splittings correspond to the crystallization of mullite in these samples. In the grain DRDF of R15 and R50, the peak heights are low in the samples annealed for 1 h, whereas the peaks in R25 are large even at short annealing times. Since the area of the ®rst peak represents the coordination number of the grains, these numbers are seen to increase with annealing time in R15 and R50 but are almost constant in R25 sample. These results are compatible with dierences in the formation speed of the textures in these samples. In R15 and R50 these textures are of droplet structure formed by a nucleation-growth mechanism, while in R25 the interconnected texture is formed by spinodal decomposition at a much faster rate than the formation of the droplet structure. In R15 and R50, the ®rst neighbor distances are either similar to or somewhat larger than the grain size of the phase separated texture evaluated by the Porod analysis. The obtained DRDF results are thus in good agreement with the Porod analysis. In R25, the ®rst neighbor distance is almost the same with the grain size evaluated by the Porod analysis. Since the distance in the sample annealed for 6 h is in good agreement with the Cahn analysis of km 8 nm, it is concluded that the DRDF results can be used to analyze the immiscibility texture of the present samples. To further examine the results of the SAXS analysis, the textures were observed by TEM. Fig. 12 shows the TEM photographs and electron diraction of selected areas of R15, R25 and R50. The dierence in the electron density of Al2 O3 -rich and SiO2 -rich phases is small because Al and Si have similar atomic numbers, resulting in the low
T. Takei et al. / Journal of Non-Crystalline Solids 282 (2001) 265±277
275
crystallization 8
8
900oC 48h
900o C 48h
6
6 24h 4π r2 (ρ(r)-ρ0 )
4π r2(ρ(r)-ρ0 )
24h
4 12h
4 12h
2
2
6h
6h 0
0 1h -2
(a)
crystallization
0
5
10
15
20
1h -2
25
0
5
8
15
20
25
r / nm
900o C 48h
6 4π r2 (ρ(r)-ρ0 )
10
(b)
r / nm
24h
4
crystallization
12h
2 6h
0 1h
-2
(c)
0
5
10
15
20
25
r / nm
Fig. 11. Grain DRDF curves of the three samples; (a) R15, (b) R25, (c) R50.
contrast of the two phases. In the photographs, the dark and thin parts correspond to the Al2 O3 rich and SiO2 -rich phases, respectively. In R15 annealed at 900°C for 1 h, very ®ne dark spots are distributed in the matrix. The electron diraction shows a halo pattern, indicating that this droplet texture corresponds to a binodal phase separated region. A similar droplet texture of pale spots dispersed in a dark matrix was observed in R50, also corresponding to a binodal microtexture with
an Al2 O3 -rich matrix. In this sample, the texture clearly changes on annealing for 24 h to the crystallized texture of mullite. In R25, the interconnected texture observed after annealing at 900°C for 1 h grows at longer annealing times. The interconnected texture is in good agreement with the texture simulated by the Cahn analysis (Fig. 10). Since the electron diraction shows only a halo, these textures are formed by spinodal decomposition. The ®ne texture observed in all the
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Fig. 12. TEM photographs and electron diraction of the three samples annealed at 900°C for various times; (a) R15, (b) R25, (c) R50.
as-quenched samples is considered to be formed by phase separation during quenching and/or by small composition ¯uctuations in the molten state. The intergrain distances of some samples were calculated from the TEM photographs and com-
pared with the DRDF results. Fig. 13 shows the ®rst neighbor intergrain distance obtained from the TEM photographs together with the DRDF curves. The distances obtained from the DRDF curves are in good agreement with those from the
Fig. 13. Nearest neighbor distances of grains evaluated from TEM observation and DRDF curve.
T. Takei et al. / Journal of Non-Crystalline Solids 282 (2001) 265±277
277
TEM photographs. We can therefore say that the grain DRDF calculation can be used to analyze phase separated textures.
Zealand Institute for Industrial Research and Development for critical reading and editing of this manuscript.
5. Conclusion
References
The phase separation and crystallization textures formed in Al2 O3 ±SiO2 glasses were examined by the Porod, Cahn and radial distribution function methods using SAXS intensities and comparing the results with TEM observations. The following results were obtained. 1. The Porod analysis indicated the presence of droplet textures in R15 and R50 but an interconnected texture is formed in R25 by spinodal decomposition. 2. Cahn analysis shows that the size of the interconnected texture formed in the initial stage is 8 nm, in good agreement with the transversal length by Porod analysis. 3. Grain radial distribution function analysis shows that the ®rst neighbor intergrain distance increases with longer annealing times, in good agreement with TEM observations. 4. TEM observations indicate droplet textures corresponding to binodal regions in R15 and R50 while interconnected textures formed by spinodal decomposition were observed in R25. These results are in good agreement with the SAXS analysis. 5. The crystallization of mullite is considered to occur from an Al2 O3 -rich phase previously formed by phase separation in the glasses.
[1] H. Schneider, K. Okada, J.A. Pask, Mullite and Mullite Ceramics, Wiley, Chichester, 1994. [2] D.X. Li, W.J. Thomson, J. Am. Ceram. Soc. 73 (1990) 964. [3] J.S. Lee, S.C. Yu, Mater. Res. Bull. 27 (1992) 405. [4] W.C. Wei, J.W. Halloran, J. Am. Ceram. Soc. 71 (1988) 581. [5] D.X. Li, W.J. Thomson, J. Mater. Res. 5 (1990) 1963. [6] K. Okada, N. Otsuka, J. Am. Ceram. Soc. 69 (1986) 652. [7] A. Fualtieri, M. Bellotto, G. Artioli, S.M. Clerk, Phys. Chem. Miner. 22 (1995) 215. [8] T. Takei, Y. Kameshima, A. Yasumori, K. Okada, J. Am. Ceram. Soc. 82 (1999) 2876. [9] H.S. Chen, C.E. Miller, Rev. Sci. Instrum. 41 (121) (1970) 1237. [10] A. Guinier, G. Fournet, Small-Angle Scattering of X-rays, Wiley, New York, 1955. [11] V. Luzzati et al., J. Molec. Biol. 3 (1961) 367. [12] O. Kratky, Pure Appl. Chem. 12 (1966) 483. [13] G. Porod, Kolloid. Z. 124 (1951) 83. [14] G. Porod, Kolloid. Z. 125 (1952) 51. [15] T. Takei, Y. Kameshima, A. Yasumori, K. Okada, J. Mater. Res. 15 (2000). [16] I. Nitta. X-Sen Kessho Gaku No.2, Maruzen, Tokyo, 1961. [17] Y.W. Zeng, P. Riello, A. Benedetti, G. Fagherazzi, J. NonCryst. Sol. 185 (1995) 78. [18] D.G. Burnett, R.W. Douglas, Phys. Chem. Glasses 11 (1970) 125. [19] R.F. Davis, J.A. Pask, J. Am. Ceram. Soc. 55 (10) (1972) 525. [20] J.W. Cahn, J. Chem. Phys. 42 (1965) 93. [21] J.W. Cahn, Acta Metall. 9 (1961) 795. [22] C.M. Jantzen, D. Schwahn, J. Schelten, H. Herman, Phys. Chem. Glasses 22 (1981) 122. [23] G.F. Neilsen, Phys. Chem. Glasses 10 (2) (1969) 54. [24] J. Zarzycki, F. Naudin, J. Non-Cryst. Solids 1 (1969) 215. [25] M. Tomozawa, R.K. MacCrone, H. Herman, Phys. Chem. Glasses 11 (1970) 136. [26] G.W. Brady, J. Chem. Phys. 19 (1951) 1547. [27] O. Glatter, O. Kratky, Small Angle X-ray Scattering, Academic Press, London, 1982.
Acknowledgements We are grateful to Associate Professor T. Yano of the Tokyo Institute of Technology for permission and teaching to use TEM. We are also grateful to Dr Kenneth J.D. MacKenzie of New
www.elsevier.com/locate/jnoncrysol
SAXS analysis of textures formed by phase separation and crystallization of Al2O3±SiO2 glasses Takahiro Takei a,b,*, Yoshikazu Kameshima a, Atsuo Yasumori a, Kiyoshi Okada a, Nobuhiro Kumada b, Nobukazu Kinomura b a b
Department of Metallurgy and Ceramics Science, Graduate School of Science and Engineering, Tokyo Institute of Technology, O-okayama, Meguro, Tokyo 152-8552, Japan Institute of Inorganic Synthesis, Faculty of Engineering, Yamanashi University, 7 Miyamae, Kofu, Yamanashi 400-8511, Japan Received 17 May 2000; received in revised form 2 October 2000
Abstract Textures formed by phase separation and crystallization of Al2 O3 ±SiO2 glasses with 15, 25 and 50 mol% Al2 O3 compositions were investigated by small angle X-ray scattering (SAXS) and TEM. From the Porod analysis of the SAXS patterns, droplet structures were found in glasses with 15 and 50 mol% Al2 O3 compositions, whereas an interconnected structure was formed by spinodal decomposition in the glass with 25 mol% Al2 O3 . In the initial-stage crystallization of mullite, nucleation was considered to occur in the Al2 O3 -rich regions. The crystallites of mullite were of similar size to the phase separated region. The relationship between the annealing time and the size of the interconnected texture formed by spinodal decomposition was analyzed using Cahn's theory, giving good agreement with the result obtained by the Porod method. The distance distribution of the phase separated grains was obtained from the radial distribution function calculated by Fourier transformation of the SAXS patterns. The calculated intergrain distances increased with increasing annealing time and were in good agreement with those obtained from TEM observation. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 61.43.Fs; 64.75.+g; 61.10.Eq
1. Introduction Mullite Al42x Si2 2x O10 x , the only thermodynamically stable compound in the Al2 O3 ±SiO2 system at atmospheric pressure, is found in various aluminosilicate ceramics. For example, mullite is a major crystalline phase in porcelain and plays an important role in the development of its mechan-
* Corresponding author. Tel.: +81-55 220 8616; fax: +81-55 254 3035. E-mail address: [email protected] (T. Takei).
ical strength. In recent years, mullite have been identi®ed as possible high temperature structural materials because of their high mechanical strength and creep resistance at high temperature. A variety of crystallization reactions have been reported in the preparation of high-purity mullite from various raw materials. The crystallization process of mullite falls into two categories [1]: direct crystallization of mullite from an amorphous phase and crystallization via a spinel phase. The former type of crystallization was reported when the precursor is homogeneous at the molecular level such as in glasses and monophasic gels
0022-3093/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 1 ) 0 0 3 1 5 - 5
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[2], and the latter type was reported when the precursor is inhomogeneous at the molecular level as in diphasic gels [3±5], rapidly hydrolyzed gels [6] and kaolinite [7]. Although the crystallization processes of mullite from various raw materials have been studied in detail as described above, the crystallization kinetics of mullite have not been investigated in detail, especially in glasses. Furthermore, the metastable immiscibility reported in this system should in¯uence the crystallization of mullite. We have reported [8] that the nucleation of mullite is in¯uenced by the phase separated texture formed prior to crystallization because the activation energy for nucleation of mullite from the glass within the immiscibility composition is smaller than from the glass in the miscible region. The relationship between the immiscibility texture and the mullite crystallization texture is, however, not yet clearly understood. In order to clarify this relationship, it is necessary to investigate the phase separation and crystallization processes precisely. Small angle Xray scattering (SAXS) is a useful technique to investigate these phenomena because the expected size of the texture ranges around 1±100 nm, an appropriate size for SAXS analysis. In this paper, the textures formed by phase separation of Al2 O3 ±SiO2 glasses and the crystallization of mullite from these glasses were examined by SAXS and TEM. 2. Experimental procedure The starting materials were tetraethylorthosilicate (TEOS) and aluminum nitrate nonahydrate. Three compositions of TEOS and aluminum nitrate nonahydrate were prepared, containing 15, 25 and 50 mol% Al2 O3 (designated R15, R25 and R50, respectively). These compositions were mixed in absolute ethanol for 3 h, then gelled slowly in an oven at 60°C for 1 month [6]. The resulting gels (SH gel [6]) were calcined at 500°C for 6 h and 800°C for 24 h to remove organic components, pressed into rods at 98 MPa in a CIP, and sintered at 1300°C for 12 h. Glasses were prepared by melting and ultraquenching these Al2 O3 ±SiO2 sintered rods using an
arc image furnace. The rods were melted at 2100°C and ultra-quenched between twin rollers rotating at 1500 rpm. The estimated quenching rate was higher than 105 K/s [9]. The resultant ultra-quenched glasses were in the form of ¯akes. The glasses were annealed at 900°C for 1±48 h in an electric box furnace to induce phase separation and mullite crystallization. The crystalline phase formed in the annealed glasses was determined by powder X-ray diractometry (XRD) using monochromatic CuKa radiation. SAXS was measured using monochromatic MoKa radiation with a position sensitive proportional counter. The X-rays were collimated by a Kratky U-slit with incident slit of 0.03 mm and height limiting slit of 10 mm. The voltage and current of the X-ray source were 50 kV and 200 mA. The SAXS measurements were carried out in the angular range )0.5 to 7.5 with step intervals of 0.002° and a ®xed time of 3 h. The microstructure of the heat-treated glasses was observed using a transmission electron microscope to con®rm the results of SAXS analysis. 3. Calculations Since the measured SAXS intensity was scattering from slit-collimated X-rays, it was necessary to correct the intensity to pinhole collimated Xrays. The correction was carried out using general theory [10]. Since the existence of even slight ¯uctuations of electron density in the sample in¯uences the analysis result, Luzzati et al. [11] reported a calibration method to counteract eect of the ¯uctuation by adding a speci®c constant to the SAXS intensity to satisfy the proportional decrease against S 4 . This calibration method was mentioned by Kratky [12] that the intensity calibrated thus is satisfactory for use with Porod's method. Therefore, this method was adopted in the present work. In SAXS analyses, the sample is generally assumed to consist of two phases with dierent electron densities. Since the immiscibility and crystallization textures were considered to ®t this condition, their texture sizes were evaluated using the Porod method [13,14], in which the transversal
T. Takei et al. / Journal of Non-Crystalline Solids 282 (2001) 265±277
lengths, la and lb , corresponding to the sizes of the phases a and b are calculated from the measured intensity, I S, by the following equation: lr
4Q ; p lim S 4 I S S!1
la
lr 1
w
; lb
lr ; w
1
where Q is invariant, S is the scattering vector represented by S 4p sin h=k, and w is the volume fraction of phase a. In this study, w was calculated from the fraction of Al2 O3 -rich phase, the composition of which was evaluated from the immiscibility region in Al2 O3 ±SiO2 system calculated by molecular dynamics simulation method [15]. The resulting fractions of the Al2 O3 -rich phases were 12, 34 and 83 vol% in R15, R25 and R50, respectively. The value of Q is calculated from Eq. (2), Z 1 2 Q S 2 I SdS 2p2 Ie V Dq w 1 w; 2 0
where Ie is Thomson's constant, V the irradiated sample volume and Dq is the dierence of electron density between phases a and b. The textures formed by phase separation and crystallization were also investigated by determining the radial distribution function of the grains. In these calculations, the grains corresponded to the dispersed phase in the binodal texture, the crystallized mullite grains and the diameter of the interconnected texture in the spinodal decomposition. The radial distribution function was calculated as follows. The observed scattering intensity, I S, can be represented by two scattering terms [16]: I S Ip S Ii S:
3
Here, Ip (S ) and Ii (S ) are scattering intensities by one and two grains, respectively. The value of Ip (S ) can be calculated from basal scattering formula. This is generally expressed using a scattering function, W S, by the following equation: 2
Ip S Nn Ie
3 sin Sr 3 S r3
Nn2 Ie W2 S;
2 Sr cos Sr 4
267
where N is total number of grains in V and n is total number of electrons in one grain. On the other hand, Ii S is expressed in terms of the number density of the grains, Ii S Nn2 Ie W2 S
Z
P r exp ir SdV ZV 1 sin Sr dr 4pr2 P r Nn2 Ie W2 S Sr 0 Z 1 sin Sr Nn2 Ie W2 S dr 4pr2 fP r P0 g Sr 0 Z 1 sin Sr dr 4pr2 P0 Sr 0 Z 1 sin Sr 2 2 2 dr I0 S; Nn Ie W S 4pr fP r P0 g Sr 0 5
where r is a vector between arbitrary points in dierent grains, P r is a function of grain number density, P0 is the mean number density of grains and I0 S is the zero-order scattering. The zeroorder scattering is generally negligible in these calculations because its intensity is very small except at very low angles. Therefore, the total scattering intensity, I S, can be expressed as I S Ip S Ii S Nn2 Ie W2 S Z 1 4pr2 f P r 1 0
P0 g
sin Sr dr : Sr
6
The dierential radial distribution function of the grains is obtained by Fourier transformation of Eq. (6): Z 2r 1 4pr2 f P r P0 g Si S sin Sr dS; 7 p 0 i S
I S Nn2 Ie W2 S
1:
8
In these calculations, the scattering function, W(S), was obtained from Eq. (4). However, the W(S) curve shows heavy hunting over a relatively large angular range for genuine spheres with uniform electron density. In order to obtain a smooth W(S) curve, Zeng et al. [17] used an approximation derived from a combination of the theories of
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Porod and Guinier. According to the Porod theory, scattering intensity over a relatively large angular range decreases proportionally with scattering vectors to the minus fourth power. On the other hand, the scattering intensity at small angles range was theoretically approximated to a Gaussian function by Guinier. This combination method gave a smooth W(S) curve without hunting, although it is dicult to smoothly connect the calculated Porod and Guinier curves. Therefore, the following method was used in this study. Particles are assumed to have a Gaussian type size distribution and the function X(S) was used instead of Nn2 Ie W2 S, though W(S) is generally used for spherical particles of similar size. The X(S) is expressed by the following equation: ( ) Z 1 1 p r r0 2 2 2 X S Nn Ie W S exp dr f f 1 2 Z 1" 2r3 Q 3 sin Sr Sr cos Sr w S 3 r3 1 3p 1 ( )# 1 p r r0 2 exp dr; 9 f f
intensity / a.u.
102
100
10-2
10-4 10-2
Ω(S) Nn2 IeΨ2 (S) Guinier approximation Porod approximation 10-1
100 -1
S / nm
Fig. 1. The X S; Nn2 Ie W2 S, Guinier approximation and Porod approximation as a function of scattering vector S.
where w is the volume fraction of the particles, r0 is the mean particle radius and f is the integral width of the particle size distribution. In the calculation, r0 was calculated from the transversal length, l , by the relation r0 3=4l and f was ®xed as f r0 =2: The calculated curves of X S; Nn2 Ie W2 S, Porod and Guinier approximations are shown in Fig. 1. The X S line was used in this work for further calculations rather than Nn2 Ie W2 S. 4. Results and discussion 4.1. Porod analysis Fig. 2 shows the XRD patterns of R15, R25 and R50 samples as-quenched and annealed at 900°C for various times. The XRD patterns of all the asquenched glasses showed only a halo at 2h 20°± 25°. The XRD patterns of the glasses annealed at 900°C showed mullite crystallization after annealing R15, R25 and R50 for 48, 24 and 12 h, respectively. Fig. 3 shows the SAXS curves of these samples. In all samples, the single peak observed in each pattern becomes more intense and shifts toward smaller angles with increasing annealing time. Since the peak is observed even before mullite crystallization, it is considered to be generated by immiscibility texture formed prior to crystallization. Since the only peak appearing in a dilute system is at 0°, these textures are considered to arise from a dense system. Fig. 4 shows relationship between annealing time and intensity of these peaks. The peak top intensity was found to become high with increase of Al2 O3 content in chemical composition of samples. The logarithmic intensities increase with slope of 1/6±1 in all samples. The small slope of 1/6 in R50 sample was plausibly considered that it was attributed to volume shrinkage by mullite crystallization because Al2 O3 rich phase plays major phase only in R50 sample. In all samples, these slopes are similar values of 2/3 in up to 4 h, whereas the other slopes are not similar. Since the slopes have various values depending on samples, the intensity plots are insucient to analyze phase separated texture. Therefore, the transversal length was then evaluated for the
T. Takei et al. / Journal of Non-Crystalline Solids 282 (2001) 265±277
Fig. 2. XRD patterns of R15, R25 and R50 samples annealed at 900°C for various times.
Fig. 3. SAXS patterns of R15, R25 and R50 samples annealed at 900°C for various times.
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T. Takei et al. / Journal of Non-Crystalline Solids 282 (2001) 265±277
log (Intensity)
-3
1/1
R15 R25 R50
1/6
3/4 1/2
-4
2/3
2/3
-5
slope ≈ 2/3
-1
0 1 log (firing time) / h
2
2.0 phase separation crystallization (12h) D=0.03 3 = 1/ 0. 31
:
1.5
D=
R50 :
= .47 =0
R25 0 D=
= .50
-1
0 D=
2/5
3 = 1/ crystalli0. 32 phase separation zation (60h)
:
0.5
0.0
= .39
:
D
1/2
crystallization phase separation (24h) D=0.04 1/5 .20= D=0 D=0.13 :
1.0
:
log(transversal length of Al2 O3 -rich phase) / nm
Fig. 4. Logarithmic plots for change of intensity in the three samples as a function of annealing time.
D=
1/2
R15 0
log t / h
1
2
Fig. 5. Logarithmic plots for change of transversal lengths of Al2 O3 -rich phase in the three samples as a function of annealing time.
two separated phases using the Porod method. Fig. 5 shows the relationship between annealing time and transversal length of all the samples. The transversal lengths increase in all the samples and the curve slopes change by three steps with in-
creasing annealing time. The slope change was considered to arise from dierences in the growth mechanisms of the immiscibility and mullite crystallization textures. Table 1 shows these slopes, D1 ; D2 and D3 , and the in¯ection times, t12 and t23 , obtained by the Porod method. The slopes in R15 and R50 samples were close to 1/2 and 1/3 in the ®rst and second steps of annealing time, respectively. Burnett and Douglas [18] reported that the size of phase separated grains formed by a nucleation-growth mechanism in the binodal region increases proportionally with time to the 1/2 power up to the equilibrium volume fraction of the grains and to the 1/3 power after reaching the constant volume fraction. From this coincidence in the slope ratios, we consider that the binodal structure is formed by a nucleation-growth mechanism in R15 and R50, but not in R25. On the other hand, the slopes of R25 did not correspond well with slopes of 1/2 and 1/3. Since the composition of R25 is intermediate between R15 and R50, the former is suggested to correspond to the spinodal region. It is therefore plausible that an interconnected structure is formed in this sample by spinodal decomposition. The growth process for this sample is discussed in the following section in terms of Cahn's theory. In the third step (crystallization of mullite), the slopes of all samples become very small compared with those of the ®rst and second steps. To compare the sizes of the phase separated and crystallized textures, the crystallite size of mullite in R25 and R50 was evaluated from the integral width of the XRD 121 re¯ection using Sherrer's equation. The crystallite size of R15 could not, however, be calculated because the intensity of its 121 mullite re¯ection was too weak. Fig. 6 shows the changes in the crystallite size of the mullite and the Porod grain size of the Al2 O3 -rich phase as a function of annealing time. Here, the Porod grain size was calculated from the transversal lengths assuming spherical shape using the relation (Porod grain size) 3=2 transversal length. In the case of R50, the Al2 O3 -rich phase corresponds to the matrix; the calculated size therefore pertains, to the region rather than the grain size. The crystallite size of the mullite is in good agreement with the Porod grain size of the Al2 O3 -rich phase of the
T. Takei et al. / Journal of Non-Crystalline Solids 282 (2001) 265±277
271
Table 1 Slopes of logarithmic plots of transversal lengths in three samples
a b
Sample
D1 a
R15 R25 R50
1/2 2/5 1/2
t12 b (h) 5 3.4 1.3
D2
t23 (h)
D3
Crystallization time (h)
1/3 1/5 1/3
12 24 6.7
0.13 0.04 0.03
48 24 12
Dm indicates slope of logarithmic plots of transversal length in the mth stage. tmn indicates in¯ection time between the mth and nth stages.
50
crystallite or grain size / nm
40
R50 R50 grain size R50 crystallite size
30
R25 grain size R25 crystallite size 20
10 10
R25 50 100 firing time / h
Fig. 6. Change of crystallite sizes of R25 and R50 as a function of annealing time.
phase separated texture in R25, however, by comparison with the size of the Al2 O3 -rich phase in R50, grain size is smaller than the Porod grain size. From these results, it is concluded that the mullite in R25 crystallizes from the Al2 O3 -rich regions in the phase separated texture because the sizes of Al2 O3 -rich phase and crystallite are similar. On the other hand, the mullite which crystallizes in R50 has a larger grain size than R25 because the Porod grain size of the Al2 O3 -rich phase in R50 is much larger than in R25, but is much smaller than the dimensions of the Al2 O3 rich phase which constitutes the matrix from which the mullite crystallizes. Fig. 7 illustrates schematically the mechanisms of immiscibility and mullite crystallization in
Al2 O3 ±SiO2 glasses. In R15, Al2 O3 -rich grains are formed as droplets by a nucleation-growth mechanism in the SiO2 -rich matrix, whereas SiO2 -rich grains are formed as droplets in the Al2 O3 -rich matrix by the same mechanism in R50, for which the Porod analysis suggests their compositions were a composition in the binodal region. Crystallization of mullite occurs from Al2 O3 -rich phase but the size indicates very sluggish growth even after prolonged annealing times. This may be due to the low diusion coecient of the Al2 O3 ±SiO2 system [19]. On the other hand, the interconnected texture formed in R25 by spinodal decomposition constitutes the matrix from which the mullite crystallizes. 4.2. Cahn's analysis Since many works demonstrated the eectiveness of Cahn's theory [20±23] to investigate the texture formed by spinodal decomposition, the texture of R25 was examined by this method. In Cahn's theory [24,25], the texture is explained by random superposition of sinusoidal waves with characteristic wavelengths, km . The wavelength corresponds to the typical size of the largest ampli®cation of composition ¯uctuation. The wavelength is determined only by the initial annealing time because the composition ¯uctuation occurs very quickly in spinodal decomposition. The SAXS curves of R25 samples ®red up to 8 h are shown in Fig. 8. Up to 4 h, the SAXS curves show two peaks at S 0.8±0.9 and 1.1 nm 1 . The peak at 0:8±0:9 nm 1 grew up depending on ®ring time, whereas that at 1:1 nm 1 grew down and was hid in 0:8±0:9 nm 1 peak for longer than 4 h. Since only the peak at 0:8±0:9 nm 1 dominated further
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T. Takei et al. / Journal of Non-Crystalline Solids 282 (2001) 265±277
Fig. 7. Schematic illustration of the development of immiscibility and mullite crystallization textures in Al2 O3 ±SiO2 glasses.
R(S) / h-1
0.2
Sm
0.1
0.0
-0.1
0
1
2 S / nm-1
Fig. 8. SAXS patterns of R25 samples annealed at 900°C up to 8 h.
longer the ®ring time of 4 h, it was found that the peak at 0:8±0:9 nm 1 was generated by the composition ¯uctuation in the spinodal decomposition, while that at 1:1 nm 1 was by another texture mentioned afterward. The initial stage of spinodal decomposition in R25 samples was considered to occur during the ®rst 4 h, because the peak grew up at 0:8±0:9 nm 1 and the slope ratio of the
Fig. 9. Change of ampli®cation factor, R(S), as a function of S in R25 sample.
transversal length changes at this time (t12 shown in Table 1 above). The wavelengths km were then calculated for various S values from the slope ratio for the plots of ln I S vs annealing time. Fig. 9
T. Takei et al. / Journal of Non-Crystalline Solids 282 (2001) 265±277
shows relationship between S and the ampli®cation factor R S calculated from km . Since the maximum ampli®cation factor (Sm ) was observed at around 0.8 nm 1 in Fig. 9, km was calculated to be about 8 nm by the equation km 2p=Sm . In order to examine suitability of the resulting value of km 8 nm, the texture of spinodal decomposition was computed using this km . The texture pattern was calculated from 100 randomly oriented sinusoidal waves, which had a Gaussian distribution of amplitudes and ¯at distributions of phase angles and orientation. The ratio of the Al2 O3 ± and SiO2 -rich phases in this texture was w/(1)w) 34/66, for which the computed texture is shown in Fig. 10. The texture has the appearance of the two interconnected phases generally observed in spinodal decomposition.
273
The transversal lengths of the two phases were evaluated from the computed texture and were compared with the lengths obtained from the Porod analysis. The transversal lengths from Cahn's method were calculated by counting numbers of arbitrary lengths in the computed texture. Table 2 shows the transversal lengths of R25 obtained from the Porod and Cahn methods. These values showed very good agreement. In the previous section, the transversal length evaluated from the Porod method was shown to increase proportionally with annealing time to the 2/5 power up to 4 h, and to the 1/5 power at annealing times from 4 to 24 h as shown in Fig. 5 and Table 1. On the basis of Cahn's theory, the slope ratio in the initial stage D1 should be zero because spinodal decomposition in this stage occurs only amplifying ¯uctuation of composition. On the
Fig. 10. Cross-section of computed spinodal texture for R25 sample.
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T. Takei et al. / Journal of Non-Crystalline Solids 282 (2001) 265±277
Table 2 Transversal lengths calculated from the Porod and Cahn methods in R25 samples annealed at 900°C for 4 h Transversal length (nm) Al2 O3 -rich phase SiO2 -rich phase
Porod
Cahn
5.3 10.0
5.2 10.0
other hand, the slope ratio D1 obtained from the Porod analysis is 2/5 (Table 1). This dierence in the slope ratio is considered to occur for the following reasons : an ultra-®ne texture was formed during quenching process, the average size of which is expressed by the transversal length derived in the Porod method. The ultra-®ne texture was shown as the peak at 1.1 nm 1 in Fig. 8. Both this and the spinodal decomposition texture are ampli®ed depending on km . Annealing at 900°C causes the ultra-®ne texture formed during quenching to disappear, with the formation of a phase separated texture characterized by km 8 nm. The resulting value of 2/5 for D1 was generated from both disappearing of ultra-®ne texture and amplifying of spinodal decomposition texture depending on km . 4.3. Grain radial distribution function In colloid studies, the radial distribution function (RDF) for the colloid grains can be used to provide information on intergrain distances [26,27]. This technique was therefore adopted in this study. The grain RDF can be calculated by Fourier transformation of the SAXS intensity expressed in Eq. (7). This calculation method is essentially similar to the RDF used for structural analysis of amorphous substances, but diers in that only scattering intensity data are used for the calculations. Calculations of usual atomic RDF require atomic scattering factors, whereas the scattering function of the grains is necessary for calculation of grain RDF. In this study, the weighted scattering function X S calculated from Eq. (9) was used as the scattering function W S of grains. Fig. 11 shows the calculated grain differential RDF (DRDF) of R15, R25 and R50 samples. In these grain DRDF curves, the distance of the nearest neighbor grains is represented in
Fig. 11 by arrow marks. The ®rst neighbor distance was found to increase with annealing time in all samples, corresponding to the growth of phase separated texture. The shape of the ®rst peaks gradually changes and the peaks splitted with longer annealing time. These splittings correspond to the crystallization of mullite in these samples. In the grain DRDF of R15 and R50, the peak heights are low in the samples annealed for 1 h, whereas the peaks in R25 are large even at short annealing times. Since the area of the ®rst peak represents the coordination number of the grains, these numbers are seen to increase with annealing time in R15 and R50 but are almost constant in R25 sample. These results are compatible with dierences in the formation speed of the textures in these samples. In R15 and R50 these textures are of droplet structure formed by a nucleation-growth mechanism, while in R25 the interconnected texture is formed by spinodal decomposition at a much faster rate than the formation of the droplet structure. In R15 and R50, the ®rst neighbor distances are either similar to or somewhat larger than the grain size of the phase separated texture evaluated by the Porod analysis. The obtained DRDF results are thus in good agreement with the Porod analysis. In R25, the ®rst neighbor distance is almost the same with the grain size evaluated by the Porod analysis. Since the distance in the sample annealed for 6 h is in good agreement with the Cahn analysis of km 8 nm, it is concluded that the DRDF results can be used to analyze the immiscibility texture of the present samples. To further examine the results of the SAXS analysis, the textures were observed by TEM. Fig. 12 shows the TEM photographs and electron diraction of selected areas of R15, R25 and R50. The dierence in the electron density of Al2 O3 -rich and SiO2 -rich phases is small because Al and Si have similar atomic numbers, resulting in the low
T. Takei et al. / Journal of Non-Crystalline Solids 282 (2001) 265±277
275
crystallization 8
8
900oC 48h
900o C 48h
6
6 24h 4π r2 (ρ(r)-ρ0 )
4π r2(ρ(r)-ρ0 )
24h
4 12h
4 12h
2
2
6h
6h 0
0 1h -2
(a)
crystallization
0
5
10
15
20
1h -2
25
0
5
8
15
20
25
r / nm
900o C 48h
6 4π r2 (ρ(r)-ρ0 )
10
(b)
r / nm
24h
4
crystallization
12h
2 6h
0 1h
-2
(c)
0
5
10
15
20
25
r / nm
Fig. 11. Grain DRDF curves of the three samples; (a) R15, (b) R25, (c) R50.
contrast of the two phases. In the photographs, the dark and thin parts correspond to the Al2 O3 rich and SiO2 -rich phases, respectively. In R15 annealed at 900°C for 1 h, very ®ne dark spots are distributed in the matrix. The electron diraction shows a halo pattern, indicating that this droplet texture corresponds to a binodal phase separated region. A similar droplet texture of pale spots dispersed in a dark matrix was observed in R50, also corresponding to a binodal microtexture with
an Al2 O3 -rich matrix. In this sample, the texture clearly changes on annealing for 24 h to the crystallized texture of mullite. In R25, the interconnected texture observed after annealing at 900°C for 1 h grows at longer annealing times. The interconnected texture is in good agreement with the texture simulated by the Cahn analysis (Fig. 10). Since the electron diraction shows only a halo, these textures are formed by spinodal decomposition. The ®ne texture observed in all the
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T. Takei et al. / Journal of Non-Crystalline Solids 282 (2001) 265±277
Fig. 12. TEM photographs and electron diraction of the three samples annealed at 900°C for various times; (a) R15, (b) R25, (c) R50.
as-quenched samples is considered to be formed by phase separation during quenching and/or by small composition ¯uctuations in the molten state. The intergrain distances of some samples were calculated from the TEM photographs and com-
pared with the DRDF results. Fig. 13 shows the ®rst neighbor intergrain distance obtained from the TEM photographs together with the DRDF curves. The distances obtained from the DRDF curves are in good agreement with those from the
Fig. 13. Nearest neighbor distances of grains evaluated from TEM observation and DRDF curve.
T. Takei et al. / Journal of Non-Crystalline Solids 282 (2001) 265±277
277
TEM photographs. We can therefore say that the grain DRDF calculation can be used to analyze phase separated textures.
Zealand Institute for Industrial Research and Development for critical reading and editing of this manuscript.
5. Conclusion
References
The phase separation and crystallization textures formed in Al2 O3 ±SiO2 glasses were examined by the Porod, Cahn and radial distribution function methods using SAXS intensities and comparing the results with TEM observations. The following results were obtained. 1. The Porod analysis indicated the presence of droplet textures in R15 and R50 but an interconnected texture is formed in R25 by spinodal decomposition. 2. Cahn analysis shows that the size of the interconnected texture formed in the initial stage is 8 nm, in good agreement with the transversal length by Porod analysis. 3. Grain radial distribution function analysis shows that the ®rst neighbor intergrain distance increases with longer annealing times, in good agreement with TEM observations. 4. TEM observations indicate droplet textures corresponding to binodal regions in R15 and R50 while interconnected textures formed by spinodal decomposition were observed in R25. These results are in good agreement with the SAXS analysis. 5. The crystallization of mullite is considered to occur from an Al2 O3 -rich phase previously formed by phase separation in the glasses.
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Acknowledgements We are grateful to Associate Professor T. Yano of the Tokyo Institute of Technology for permission and teaching to use TEM. We are also grateful to Dr Kenneth J.D. MacKenzie of New