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Study of phase separation in Na2OB2O3SiO2 glasses by tem and optical diffraction
232
Journal of Non-Crystalline Solids 112 (1989) 232-237 North-Holland, Amsterdam
S T U D Y OF P H A S E S E P A R A T I O N IN N a 2 0 - B 2 0 3 - S i O 2 G L A S S E S BY T E M A N D O P T I C A L D I F F R A C T I O N F A N Xianping and C H E N Quanqing
Department of Materials Science and Engineering, Zhejiang University, Hangzhou, PR China As an extension of the small angle X-ray scattering and electron microscopy techniques for investigating phase separation in glasses, a method has been developed which involves taking electron microscopy phase separation morphology micrographs and then using an optical diffractometer (OD) to perform a Fourier transformation so as to obtain the small angle scattering intensity. By using this method, phase separation in NazO-B203-SiO 2 glasses was investigated. The results obtained are as follows: (1) The size of the interconnecting phase-separated particles measured by this method seemed to be proportional to the cube root of the heat treatment time. (2) The sequence of phase separation in the three liquid regions of the Na20-B203-SiO2 system is as follows: first, the glasses separate into a silica-rich phase and a sodium-borate-rich phase, and then the sodium-borate-rich phase separates into two other phases, one being borate-rich and the other sodium-rich.
1. Introduction Phase separation which occurs in many glassforming systems has been gradually investigated over the last sixty years and affects both the structure and the properties of glasses [1]. The recognition of the role of phase separation was one of the most important events in glass science in the past few decades. By controlling the degree of phase separation, it is possible to produce various kinds of materials of predicted microstructure. Small-angle X-ray scattering (SAXS) and electron microscopy (EM) are already established as powerful analytical tools for investigating phase separation. By electron microscope, the morphology of the phase separation in glasses can be obtained and the size of the phase-separated particles can be calculated from the micrographs by linear analysis. The average size of the phase-separated particles in a sample can also be calculated from SAXS data according to the Guinier equation [2], but no information about the morphology can be obtained. With all scattering techniques, for example Xray or visible light scattering, the scattering amplitude is related to the scattering particle density (e.g. the electron density for X-rays) by a Fourier transform. Thus, the characteristic angle at which 0022-3093/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
radiation is scattered and the size of the scattering object are inversely related. Therefore, a method has been developed [3] to study phase separation which involves taking micrographs and then performing a Fourier transformation using an optical diffractometer (OD), to obtain the optical scattering intensity I as a function of the scattering angle 0. Using this method, the morphology of the phase separation in materials can be obtained by electron microscope and at the same time the size of the phase-separated particles can be calculated from the optical scattering curve derived from the EM image negative. Using this method, Zarzycki and Naudin have investigated the phase separation in BEO3-PbO-A120 3 glasses [3]. The SAXS intensity from dispersed particles can be taken to a good approximation as
I ( k ) = Af21S( K ) 12,
(1)
where A f is the difference in scattering amplitude for the two phases, and S ( K ) is the structure factor for the particles. This can be written in the form
I( k ) = ( Ap )ZVU fo°C4~rr2a( r ) sin Kr d r ,
(2)
where V is the volume of a particle, N is the concentration of the particles and a ( r ) is the
233
Fan Xianping, Chen Quanqing / Phase separation in Na eO-B203-SiO 2 glasses
shape factor. For SAXS, it is the variation in shape which governs the scattering intensity. On Fourier transformation eq. (2) gives the expression for a( r ),
or(r)
1 fo ~ K sin K r I ( K ) dK. 21r2(hp)ar
(4)
OO
R 0 = (2~r/V~)f0 r4a(r) dr.
(5)
From eqs. (3), (4) and (5) we can get an approximate equation:
I( K ) = N V c V exp( - RZK 2/3).
(6)
Thus a plot of In I vs. K 2 should be linear with a slope of -R2o/3. If the phase separation particles are assumed to be spheres of radius R, then R = (5/3)~R0 .
S
LD
L2
(3)
The relationship between a(r) and the correlation volume, Vc, and radius of gyration, R 0, are as follows:
Vc = 4~rfo~r2a(r) dr,
Lc A
(7)
When the specimen contains a distribution of particle sizes, the logarithmic plot is no longer a straight line. If there are n particle sizes in the specimen, the scattering intensity curve is a sum of n straight lines which correspond to the different sized particles. An approximate graphical method to deal with this type of scattering curve has been used by Jellinek et al. [4]. By this method the particle size fraction can be determined.
Fig. 1. Diagram of the optical diffractometer. La, beam expander; Lc, collimating lens; A, aperture; S, specimen; L1, L2, lens systems.
and for the replica method, specimens were etched by H F after being polished. A JEOL 200cx electron microscope was used for observation and to record the micrographs. A schematic diagram of the OD is show in fig. 1. Before the objective plane, an aperture A is inserted, by which the irradiated area can be limited to a definite size. Behind the objective lens L1, another lens L2 is inserted to vary the magnification. The diffraction camera length L 0 was calibrated using a lattice image negative of known d-spacing. In this case L 0 = Dd/X, where D is the distance from the lst-order diffraction point to the central beam and ~ is the wavelength of the laser beam. The scattering angle 0 can be obtained from the relation 0 = r / L o, where r is the distance from the measuring point to the screen center. A photoelectric cell is located on the screen plane, which can be moved along the horizontal axis to measure the photo-intensity variation. If the phase-separated micrograph negative is put at the objective plane S and the photo-intensity is measured at each point along the radial direction, a plot of photo-intensity vs. scattering angle can be obtained.
2. Experimental 3. Results and discussion A glass of composition of 8 N a 2 0 . 2 7 B 2 0 3 • 65SiO2was melted at about 1500 ° C and then cast onto an iron plate. The resulting glass was heat treated at various temperatures for various periods of time to induce phase separation. The heat treatment conditions are as follows: 600°C, 1 h; 600 o C, 8 h; 600 o C, 24 h; 600 ° C, 168 h; 620 o C, 24 h; and 660°C, 24 h. The specimens were prepared by grinding, polishing and ion thinning
3.1. Compar&on of direct-transmission electron microscopy and replication techniques Figure 2 shows micrographs of specimens which were prepared by the replica method (a) and by ion thinning (b). As may be seen from fig. 2(a), the phase separation morphology in the micrograph obtained by the replica method is somewhat
234
Fan Xianping, Chen Quanqing / Phase separation in Na 20-B203-SiO 2 glasses
Fig. 3. TEM micrograph of a phase-separated specimen heat-treated for 168 h at 600 ° C.
Fig. 2. TEM micrographs obtained from a specimen by the replica method (a) and by ion thinning (b).
deformed due to serious surface etching so that there is a difference between the size of phase-separated particles calculated according to the micrograph morphology and those in the real specimen. Also sample surface polishing, heavy metal projection and the introduction of artifacts in reproducing a surface in the replica method affect the accuracy of a phase-separation micrograph. By using a direct-transmission electron microscope, a more dense phase can be identified and thus the modifier-rich phase can be distinguished from the glass-former-rich phase. Hence, if the electron beam sensitivity can be controlled, the phase-separation micrograph obtained by using the ionthinned specimens should be more detailed.
3.2. Optical diffraction Figure 3 shows the micrograph of a phase-separated specimen (heat treatment for 168 h at
600 ° C, magnification 3600 × ). Figure 4 gives the corresponding plot of the - I n I / I o vs . 0 2 obtained from the OD (curve 1) and its tangent which was derived at the tail of the curve. The other plots in fig. 4 were derived by using a graphical method which was suggested by Jellinek et al. [4]. From the slope of the tangent the size of the phase-separated particles could be calculated from eq. (6). Owing to the fact that the transmitted beam was very strong the scattering intensity measured near the central beam was inaccurate. The tangent used to calculate the size of the phase-separated particles must be carefully chosen in order that the influence of the transmitted
3 4
o
i/ /
r..J I
2
2'o 4'o 2 9 2 (XIO 7)
4
Fig. 4. Plot of - I n I / I o vs. 8 2.
Fan Xianping. Chen Quanqing / Phase separation in Na20-B203-SiO 2 glasses Table 1 The size of phase-separated particles from OD and TEM (in
A)
OD TEM
600 o C 8h
600 o C 24 h
600 ° C 72 h
600 o C 168 h
395+ 40 200+100
489+ 40 500+100
733:t: 40 700-t-100
1027+ 40 1000+ 100
beam is minimised. The calculated results of four phase-separated specimens (heat treatment for 8 h, 24 h, 72 h and 168 h at 600°C) obtained according to this method are listed in table 1 together with the size of the phase-separated particles directly measured from the EM micrographs. The data show that the results from the OD are consistent with the direct measurements from the EM micrographs. Because there is no slit in the OD, the slit correction required in SAXS can be omitted and it is much easier to analyse the scattering curve. In addition, according to the degree of phase separation, the magnification of the electron microscope, the size of the objective aperture and the camera length of the OD are easy to adjust to yield the optimum experimental conditions. Thus this method can be used for micro-regions as well as for bulk volumes.
3.3. Kinetics of phase separation Figure 5 is a plot of the mean radius of the phase-separate particles R vs. t 1/3 (t = heat treat-
I000
Boo ~ . 6oc r ~ 4oc
2~
f ib 2'0 t (rn inute)
Fig. 5. Radius of phase-separated particles vs. (time) ]/3.
235
ment time) obtained by the OD method. The results show that the size of the interconnecting phase-separated particles was proportional to the cube root of the heat treatment time. This relation is consistent with the results which have been obtained from isolated phase-separated particles. This indicates that whether the later phase separation occurs by nucleation and growth or by spinodal decomposition, the dimensions of the particles should increase as (time) 1/3.
3.4. Formation of three-liquid phases Figure 6 shows the micrographs of the phaseseparated specimens as a function of the heat treatment time. The specimens heat treated at 600 o C, for 1 h and 8 h were separated into two phases (silica-rich and sodium-borate-rich), while the specimens heat-treated for 24 h, 72 h and 168 h were separated into three phases. In the latter case the sodium-borate-rich phase was separated into two other phases, one being borate-rich and the other sodium-rich. It was also found that the size of the borate-rich particles increased gradually with time. Because the composition of the glass is in the three-liquid regions of the Na20-B203-SiO 2 system proposed by Hall et al. [5], we think that the sequence of the phase separation in this region is as follows: first, the sodium-borosilicate glasses separate into two phases, one being silica-rich and the other sodium-borate-rich, and then after a further time the sodium-borate-rich phase begins to separate into two other phases, one borate-rich and the other sodium-rich. Figure 7 shows micrographs of phase-separated specimens after heat treatment for 24 h at 620 °C (a) and at 660 °C (b). For the former the morphology of the borate-rich phase particles was an interconnected structure, which was produced by a spinodal decomposition mechanism, and for the latter the morphology of the borate-rich phase particles was droplet structure produced by nucleation and growth. Therefore, the formation process for the three-liquid regions may either be spinodal decomposition or nucleation and growth.
236
Fan Xianping~ Chen Quanqing / Phase separation in Na e O - B 2 0 7 S i O 2 glasses
.........
!
ii
Fig. 6. T E M micrographs of five phase-separated specimens heat-treated for 1 h (a), 8 h (b), 24 h (c), 72 h (d) and 168 h (e) at 600 o C.
4. Conclusions
Phase separation in glasses may be studied by taking a TEM micrograph and then using it to obtain an optical transform, from which the size of phase-separated particles can be calculated. The
morphology of the phase-separated structure can also be investigated. When the composition of the glass is in the three-liquid regions of the N a 2 0 - B 2 0 3 - S i O 2 system, the sequence of phase separation is as follows: first, the glasses separate into a silica-rich
Fan Xianping, Chen Quanqing / Phase separation in Na 20-B203-SiO 2 glasses
237
i! i i
Fig. 7. TEM rnicrographs of two phase-separated specimens heat-treated for 24 h at 620 ° C (a) and 660 ° C (b).
phase and a sodium-borate-rich phase, and then the sodium-borate phase separates into two other phases, one being borate-rich and the other sodium-rich. References [1] O.V. Mazurin and E.A. Porai-Koshits, Phase Separation in Glass (North-Holland, Amsterdam, 1984).
[2] A. Guinier and G. Fournet, Small-Angle Scattering of X-Rays (Wiley, New York, 1955). [3] J. Zarzycki and F. Naudin, J. Non-Cryst. Solids 1 (1969) 215. [4] M.H. Tellinek, E. Solomon and I. Fa~kcuchen, Ind. Eng. Chem. (Anal. Ed.) 18 (1946) 172. [5] W. Haller, D.H. Blackburn, F.E. Wagstaff and R.J. Charles, J. Am. Ceram. Soc. 53 (1970 34. [6] G. Thomas and M.J. Goringe, Transmission Electron Microscopy of Materials (Wiley, New York, 1979).
Journal of Non-Crystalline Solids 112 (1989) 232-237 North-Holland, Amsterdam
S T U D Y OF P H A S E S E P A R A T I O N IN N a 2 0 - B 2 0 3 - S i O 2 G L A S S E S BY T E M A N D O P T I C A L D I F F R A C T I O N F A N Xianping and C H E N Quanqing
Department of Materials Science and Engineering, Zhejiang University, Hangzhou, PR China As an extension of the small angle X-ray scattering and electron microscopy techniques for investigating phase separation in glasses, a method has been developed which involves taking electron microscopy phase separation morphology micrographs and then using an optical diffractometer (OD) to perform a Fourier transformation so as to obtain the small angle scattering intensity. By using this method, phase separation in NazO-B203-SiO 2 glasses was investigated. The results obtained are as follows: (1) The size of the interconnecting phase-separated particles measured by this method seemed to be proportional to the cube root of the heat treatment time. (2) The sequence of phase separation in the three liquid regions of the Na20-B203-SiO2 system is as follows: first, the glasses separate into a silica-rich phase and a sodium-borate-rich phase, and then the sodium-borate-rich phase separates into two other phases, one being borate-rich and the other sodium-rich.
1. Introduction Phase separation which occurs in many glassforming systems has been gradually investigated over the last sixty years and affects both the structure and the properties of glasses [1]. The recognition of the role of phase separation was one of the most important events in glass science in the past few decades. By controlling the degree of phase separation, it is possible to produce various kinds of materials of predicted microstructure. Small-angle X-ray scattering (SAXS) and electron microscopy (EM) are already established as powerful analytical tools for investigating phase separation. By electron microscope, the morphology of the phase separation in glasses can be obtained and the size of the phase-separated particles can be calculated from the micrographs by linear analysis. The average size of the phase-separated particles in a sample can also be calculated from SAXS data according to the Guinier equation [2], but no information about the morphology can be obtained. With all scattering techniques, for example Xray or visible light scattering, the scattering amplitude is related to the scattering particle density (e.g. the electron density for X-rays) by a Fourier transform. Thus, the characteristic angle at which 0022-3093/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
radiation is scattered and the size of the scattering object are inversely related. Therefore, a method has been developed [3] to study phase separation which involves taking micrographs and then performing a Fourier transformation using an optical diffractometer (OD), to obtain the optical scattering intensity I as a function of the scattering angle 0. Using this method, the morphology of the phase separation in materials can be obtained by electron microscope and at the same time the size of the phase-separated particles can be calculated from the optical scattering curve derived from the EM image negative. Using this method, Zarzycki and Naudin have investigated the phase separation in BEO3-PbO-A120 3 glasses [3]. The SAXS intensity from dispersed particles can be taken to a good approximation as
I ( k ) = Af21S( K ) 12,
(1)
where A f is the difference in scattering amplitude for the two phases, and S ( K ) is the structure factor for the particles. This can be written in the form
I( k ) = ( Ap )ZVU fo°C4~rr2a( r ) sin Kr d r ,
(2)
where V is the volume of a particle, N is the concentration of the particles and a ( r ) is the
233
Fan Xianping, Chen Quanqing / Phase separation in Na eO-B203-SiO 2 glasses
shape factor. For SAXS, it is the variation in shape which governs the scattering intensity. On Fourier transformation eq. (2) gives the expression for a( r ),
or(r)
1 fo ~ K sin K r I ( K ) dK. 21r2(hp)ar
(4)
OO
R 0 = (2~r/V~)f0 r4a(r) dr.
(5)
From eqs. (3), (4) and (5) we can get an approximate equation:
I( K ) = N V c V exp( - RZK 2/3).
(6)
Thus a plot of In I vs. K 2 should be linear with a slope of -R2o/3. If the phase separation particles are assumed to be spheres of radius R, then R = (5/3)~R0 .
S
LD
L2
(3)
The relationship between a(r) and the correlation volume, Vc, and radius of gyration, R 0, are as follows:
Vc = 4~rfo~r2a(r) dr,
Lc A
(7)
When the specimen contains a distribution of particle sizes, the logarithmic plot is no longer a straight line. If there are n particle sizes in the specimen, the scattering intensity curve is a sum of n straight lines which correspond to the different sized particles. An approximate graphical method to deal with this type of scattering curve has been used by Jellinek et al. [4]. By this method the particle size fraction can be determined.
Fig. 1. Diagram of the optical diffractometer. La, beam expander; Lc, collimating lens; A, aperture; S, specimen; L1, L2, lens systems.
and for the replica method, specimens were etched by H F after being polished. A JEOL 200cx electron microscope was used for observation and to record the micrographs. A schematic diagram of the OD is show in fig. 1. Before the objective plane, an aperture A is inserted, by which the irradiated area can be limited to a definite size. Behind the objective lens L1, another lens L2 is inserted to vary the magnification. The diffraction camera length L 0 was calibrated using a lattice image negative of known d-spacing. In this case L 0 = Dd/X, where D is the distance from the lst-order diffraction point to the central beam and ~ is the wavelength of the laser beam. The scattering angle 0 can be obtained from the relation 0 = r / L o, where r is the distance from the measuring point to the screen center. A photoelectric cell is located on the screen plane, which can be moved along the horizontal axis to measure the photo-intensity variation. If the phase-separated micrograph negative is put at the objective plane S and the photo-intensity is measured at each point along the radial direction, a plot of photo-intensity vs. scattering angle can be obtained.
2. Experimental 3. Results and discussion A glass of composition of 8 N a 2 0 . 2 7 B 2 0 3 • 65SiO2was melted at about 1500 ° C and then cast onto an iron plate. The resulting glass was heat treated at various temperatures for various periods of time to induce phase separation. The heat treatment conditions are as follows: 600°C, 1 h; 600 o C, 8 h; 600 o C, 24 h; 600 ° C, 168 h; 620 o C, 24 h; and 660°C, 24 h. The specimens were prepared by grinding, polishing and ion thinning
3.1. Compar&on of direct-transmission electron microscopy and replication techniques Figure 2 shows micrographs of specimens which were prepared by the replica method (a) and by ion thinning (b). As may be seen from fig. 2(a), the phase separation morphology in the micrograph obtained by the replica method is somewhat
234
Fan Xianping, Chen Quanqing / Phase separation in Na 20-B203-SiO 2 glasses
Fig. 3. TEM micrograph of a phase-separated specimen heat-treated for 168 h at 600 ° C.
Fig. 2. TEM micrographs obtained from a specimen by the replica method (a) and by ion thinning (b).
deformed due to serious surface etching so that there is a difference between the size of phase-separated particles calculated according to the micrograph morphology and those in the real specimen. Also sample surface polishing, heavy metal projection and the introduction of artifacts in reproducing a surface in the replica method affect the accuracy of a phase-separation micrograph. By using a direct-transmission electron microscope, a more dense phase can be identified and thus the modifier-rich phase can be distinguished from the glass-former-rich phase. Hence, if the electron beam sensitivity can be controlled, the phase-separation micrograph obtained by using the ionthinned specimens should be more detailed.
3.2. Optical diffraction Figure 3 shows the micrograph of a phase-separated specimen (heat treatment for 168 h at
600 ° C, magnification 3600 × ). Figure 4 gives the corresponding plot of the - I n I / I o vs . 0 2 obtained from the OD (curve 1) and its tangent which was derived at the tail of the curve. The other plots in fig. 4 were derived by using a graphical method which was suggested by Jellinek et al. [4]. From the slope of the tangent the size of the phase-separated particles could be calculated from eq. (6). Owing to the fact that the transmitted beam was very strong the scattering intensity measured near the central beam was inaccurate. The tangent used to calculate the size of the phase-separated particles must be carefully chosen in order that the influence of the transmitted
3 4
o
i/ /
r..J I
2
2'o 4'o 2 9 2 (XIO 7)
4
Fig. 4. Plot of - I n I / I o vs. 8 2.
Fan Xianping. Chen Quanqing / Phase separation in Na20-B203-SiO 2 glasses Table 1 The size of phase-separated particles from OD and TEM (in
A)
OD TEM
600 o C 8h
600 o C 24 h
600 ° C 72 h
600 o C 168 h
395+ 40 200+100
489+ 40 500+100
733:t: 40 700-t-100
1027+ 40 1000+ 100
beam is minimised. The calculated results of four phase-separated specimens (heat treatment for 8 h, 24 h, 72 h and 168 h at 600°C) obtained according to this method are listed in table 1 together with the size of the phase-separated particles directly measured from the EM micrographs. The data show that the results from the OD are consistent with the direct measurements from the EM micrographs. Because there is no slit in the OD, the slit correction required in SAXS can be omitted and it is much easier to analyse the scattering curve. In addition, according to the degree of phase separation, the magnification of the electron microscope, the size of the objective aperture and the camera length of the OD are easy to adjust to yield the optimum experimental conditions. Thus this method can be used for micro-regions as well as for bulk volumes.
3.3. Kinetics of phase separation Figure 5 is a plot of the mean radius of the phase-separate particles R vs. t 1/3 (t = heat treat-
I000
Boo ~ . 6oc r ~ 4oc
2~
f ib 2'0 t (rn inute)
Fig. 5. Radius of phase-separated particles vs. (time) ]/3.
235
ment time) obtained by the OD method. The results show that the size of the interconnecting phase-separated particles was proportional to the cube root of the heat treatment time. This relation is consistent with the results which have been obtained from isolated phase-separated particles. This indicates that whether the later phase separation occurs by nucleation and growth or by spinodal decomposition, the dimensions of the particles should increase as (time) 1/3.
3.4. Formation of three-liquid phases Figure 6 shows the micrographs of the phaseseparated specimens as a function of the heat treatment time. The specimens heat treated at 600 o C, for 1 h and 8 h were separated into two phases (silica-rich and sodium-borate-rich), while the specimens heat-treated for 24 h, 72 h and 168 h were separated into three phases. In the latter case the sodium-borate-rich phase was separated into two other phases, one being borate-rich and the other sodium-rich. It was also found that the size of the borate-rich particles increased gradually with time. Because the composition of the glass is in the three-liquid regions of the Na20-B203-SiO 2 system proposed by Hall et al. [5], we think that the sequence of the phase separation in this region is as follows: first, the sodium-borosilicate glasses separate into two phases, one being silica-rich and the other sodium-borate-rich, and then after a further time the sodium-borate-rich phase begins to separate into two other phases, one borate-rich and the other sodium-rich. Figure 7 shows micrographs of phase-separated specimens after heat treatment for 24 h at 620 °C (a) and at 660 °C (b). For the former the morphology of the borate-rich phase particles was an interconnected structure, which was produced by a spinodal decomposition mechanism, and for the latter the morphology of the borate-rich phase particles was droplet structure produced by nucleation and growth. Therefore, the formation process for the three-liquid regions may either be spinodal decomposition or nucleation and growth.
236
Fan Xianping~ Chen Quanqing / Phase separation in Na e O - B 2 0 7 S i O 2 glasses
.........
!
ii
Fig. 6. T E M micrographs of five phase-separated specimens heat-treated for 1 h (a), 8 h (b), 24 h (c), 72 h (d) and 168 h (e) at 600 o C.
4. Conclusions
Phase separation in glasses may be studied by taking a TEM micrograph and then using it to obtain an optical transform, from which the size of phase-separated particles can be calculated. The
morphology of the phase-separated structure can also be investigated. When the composition of the glass is in the three-liquid regions of the N a 2 0 - B 2 0 3 - S i O 2 system, the sequence of phase separation is as follows: first, the glasses separate into a silica-rich
Fan Xianping, Chen Quanqing / Phase separation in Na 20-B203-SiO 2 glasses
237
i! i i
Fig. 7. TEM rnicrographs of two phase-separated specimens heat-treated for 24 h at 620 ° C (a) and 660 ° C (b).
phase and a sodium-borate-rich phase, and then the sodium-borate phase separates into two other phases, one being borate-rich and the other sodium-rich. References [1] O.V. Mazurin and E.A. Porai-Koshits, Phase Separation in Glass (North-Holland, Amsterdam, 1984).
[2] A. Guinier and G. Fournet, Small-Angle Scattering of X-Rays (Wiley, New York, 1955). [3] J. Zarzycki and F. Naudin, J. Non-Cryst. Solids 1 (1969) 215. [4] M.H. Tellinek, E. Solomon and I. Fa~kcuchen, Ind. Eng. Chem. (Anal. Ed.) 18 (1946) 172. [5] W. Haller, D.H. Blackburn, F.E. Wagstaff and R.J. Charles, J. Am. Ceram. Soc. 53 (1970 34. [6] G. Thomas and M.J. Goringe, Transmission Electron Microscopy of Materials (Wiley, New York, 1979).