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Characterization of Colloidal Silica Particles by Ultra-Small-Angle X-ray Scattering Toshiki Konishi, Eiji Yamahara, and Norio Ise* Central Laboratory, Rengo Co., Ltd., 186-1, 4-Chome, Ohhiraki, Fukushima-ku, Osaka 553, Japan Received September 18, 1995. In Final Form: January 16, 1996
Introduction The small-angle X-ray-scattering (SAXS) method is one of the most reliable and widely used techniques for the structural study of polymeric materials. Using a SAXS apparatus with Kratky-type slits, the resolution at small angles was high enough to enable us to determine the radius of gyration of particles of about 100 Å. Recently, polymeric aggregates, colloidal dispersions, microphase separation of block copolymers, and so on, where the scale of density fluctuations was on the order of micrometers, have attracted much attention. Some of these structures can be studied by light-scattering techniques, but opaque or turbid samples cannot be resolved using this technique. Consequently, the ultra-small-angle X-ray-scattering (USAXS) technique is more versatile. In this laboratory, we constructed an USAXS apparatus based on the BonseHart principle1 and applied it to the structural study of colloidal dispersions.2,3 In the present paper, we determined the size and size distribution of colloidal silica particles using the USAXS technique, by curve-fitting to theoretical scattering curves. Furthermore, the Guinier plot was examined using scattering data at very low angles, and this plot was found to be satisfactory for the size determination of such large particles. Needless to say, our approach using the scattering data is an improvement on other methods, such as electron micrography, in the reliability of the results, since our method provides average values for incomparably larger populations of particles than electron micrography. Experimental Section Materials. The colloidal silica particles used (KE-P10W) were donated by Nippon Shokubai Co., Ltd., Osaka. The stated radius according to the manufacturer was 500 Å. The dispersion supplied (20 wt %) was dialyzed against Milli-Q water for 16 days; the silica concentration after dialysis was 8.28 vol %, assuming 2.2 g/cm3 as the specific gravity of silica. The dispersions used for USAXS measurement were prepared by diluting this stock dispersion with an aqueous 5 × 10-4 M NaCl solution. The addition of salt was necessary to prevent ordering of particles and peak scattering, in order to limit scattering to intraparticle scattering. USAXS Apparatus. The details of the USAXS apparatus were as described previously.2 Single crystals of Ge, which provided incident X-ray beams of higher intensities than crystals of Si, were used for curve-fitting with theoretical scattering curves, while crystals of Si were employed for Guinier analysis of scattering at very small angles. The measurements were carried out at room temperature (about 25 °C). The scattering data were corrected for the effect of beam size, assuming that the X-ray beam is of infinite height and zero width.4
Results and Discussion Fitting of the USAXS Curves with Intraparticle Scattering Factors. Figure 1 gives the USAXS curves (1) Bonse, U.; Hart, M. Z. Phys. 1966, 189, 151. (2) Konishi, T.; Ise, N.; Matsuoka, H.; Yamaoka, H.; Sogami, I. S.; Yoshiyama, T. Phys. Rev. B 1995, 51, 3914. (3) Konishi, T.; Ise, N. J. Am. Chem. Soc. 1995, 117, 8422. (4) Guinier, A.; Fournet, G. Small Angle Scattering of X-rays; Wiley: New York, 1955.
Figure 1. USAXS curves of colloidal silica particle dispersions at various concentrations. Sample: KE-P10W (stated radius: 500 Å). Monochromator: Ge. [particle]: 2, 3.92 vol %; 3, 3.18 vol %; ], 2.20 vol %; 0, 1.37 vol %.; 4, 0.845 vol %; O, extrapolated (c f 0); s, theoretical curve (Rz: 560 Å, σ: 8%). Intensities were divided by the silica concentration (c). The curves were shifted vertically by an order of 8.
at various particle concentrations (c). Distinct maxima in intensity characteristic of isolated spheres were observed at all particle concentrations. As the concentration of particles decreased, the scattering intensity was reduced at large q values [q ) 4π sin θ/λ, where 2θ is the scattering angle and λ is the wavelength of the X-ray (1.54 Å)], which made acquisition of accurate data difficult. On the other hand, at small q values and higher particle concentrations, the scattering profiles were presumably influenced by interparticle interference. Thus we decided that it was difficult to compare precisely the observed and theoretical scattering curves at finite particle concentrations. The extrapolation to zero particle concentration was effected by first-order least squares treatment of the reciprocal of the root of the scattering intensity (I) divided by the particle concentration, plotted against the particle concentration. Figure 2 gives such plots at five values of q. In Figure 1, in the region q > 1.31 × 10-2 Å-1, the data at low concentrations were not very reliable, so we used the averaged scattering intensity over this concentration range (data not shown in Figure 2). If the radius distribution is assumed to be Gaussian, the average scattering intensity I(q) of isolated spheres is given as
∫0∞p(R)R6Φs2(qR) dR
I(q) ) C
(1)
where C is a constant relating to the apparatus and proportional to the square of the difference of electron density between particles and solvent, p(R) equals (1/ 2πσ2)1/2 exp[-(R - Rn)2/2σ2] with Rn being the number average radius of the particles, and Φs2 is the particle scattering function of the sphere. The root of the z-average
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Figure 2. (I/c)-1/2 vs silica concentration (c) plots for KE-P10W at various scattering vectors. The magnitudes of the scattering vectors are 1.23 × 10-2, 9.05 × 10-3, 7.31 × 10-3, 6.52 × 10-3, and 3.16 × 10-3 Å-1 from top to bottom.
square radius Rz is given as
∫0∞p(R)R8 dR/∫0∞p(R)R6 dR]1/2
Rz ) [
(2)
Figure 3. USAXS curves of colloidal silica particle dispersions at various concentrations. Sample: KE-P10W (stated radius: 500 Å). Monochromator: Si. [particle]: O, 1.37 vol %; 3, 1.03 vol %; ], 0.845 vol %; 0, 0.491 vol %; 4, 0.373 vol %. The curves were shifted vertically by an order of 3.
For purposes of comparison with Guinier plots, we used Rz as the average radius. In Figure 1, the USAXS curve extrapolated to zero concentration and the theoretical curve with the best fit to the observation are given by the circles and the solid curve, respectively. From the parameter values used in the calculation, we obtained a value of Rz ) 560 Å and σ ) 8%. It is believed that the error associated with Rz is about 1%, since the q value at the peak can be determined fairly accurately. Next we compared the observed results with a randomly oriented ellipsoid having axes 2R, 2R, and 2νR with the assumption that the distribution of the radius is Gaussian and the scattering intensity is given by
I(q) ) C
∫0∞p(R)ν2R6Φe2(qR,ν) dR
(3)
where Φe2 is the particle scattering function of the ellipsoid. In the range ν ) 0.9-1.2, good agreements between the theory and experiment were obtained. From the above results, it was concluded that the colloidal silica particles under consideration are spherical or ellipsoids close to spheres. Guinier Plot of the Observed Scattered Intensity. It is well-known that, when interparticle interference is absent, the Guinier law holds at small q values:
I(q) ) I(q)0) exp(-Rg2q2/3)
(4)
where Rg is the root of the z-average square radius of gyration of the particle and I(q)0) is the scattering intensity at q ) 0. Figure 3 gives the USAXS curves obtained using crystals of Si. The extrapolation of (I/ c)-1/2 to zero concentration was carried out in a similar way, as shown in Figure 2. The scattering intensity thus extrapolated was plotted in Figure 4a according to the Guinier law. The Rg value obtained from the slope of the plot was 470 Å. For spheres of an average radius Rz, we obtain
Rz2 ) 5Rg2/3
(5)
The Rz value of the silica particle was 600 Å. This value
Figure 4. (a) Guinier plot of the extrapolated (c f 0) USAXS data of KE-P10W: O, experimental curve; s, initial slope calculated from the observed USAXS data at q2 ) 4.93 × 10-7 to 8.86 × 10-6 Å-2 by the least squares method. (b) Guinier plots of the observed scattering intensity and of the form factor: 0, observed intensity; 4, observed intensity corrected using the calculated structure factor (see text); s, theoretical initial slope for an isolated rigid sphere (radius: 560 Å).
agreed with the Rz determined by curve-fitting within 8%, which assures us of the validity of the Guinier method for extrapolation of USAXS data for large particles of diameters of 0.1 µm. Guinier Plot of the Intraparticle Scattering Factor. For monodisperse spheres, the scattering intensity
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is related to the intraparticle scattering factor P(q) and the structure factor S(q) by the following relationship:5
I(q) ) kP(q) S(q)
(6)
where k is a proportional constant. When we are able to estimate S(q), P(q) at finite particle concentrations is given by dividing I(q) by S(q).6 In colloidal dispersions, the S(q) value for a rigid sphere can be calculated as a function of the hard sphere diameter and the volume fraction, as described by Vrij et al.7 The Guinier plot of I(q) at 1.37 vol % is givin in Figure 4b, in which I(q)/S(q) (obtained using the S(q) determined by the procedure reported earlier6,7) was also plotted. It should be mentioned that the Guinier slope for an isolated sphere (radius 560 Å), which was obtained after extrapolation to zero concentration, does not fit the I(q)q2 plot (squares) nor the I(q)/S(q)-q2 plot (triangles), as shown in Figure 4b. This implies that correction of S(q) (5) Matsuoka, H.; Murai, H.; Ise, N. Phys. Rev. B 1988, 37, 1368. (6) Chu, B.; Li, Y.; Harney, P. J.; Yeh, F. Rev. Sci. Instrum. 1993, 64, 1510. (7) Vrij, A.; Jansen, J. W.; Dhont, J. K. G.; Pathmamanoharan, C.; Kops-Werkhoven, M. M.; Funaut, H. M. Faraday Discuss. Chem. Soc. 1983, 76, 19.
is not reliable and that the extrapolation of the scattering intensity to zero concentration is necessary if the correct radius value is to be obtained. The deviation from the linearity of I(q)/S(q), which was obtained at 1.37 vol % and [NaCl] ) 5 × 10-4 M, requires explanation. This might suggest either that interparticle interference is still present at this concentration of NaCl or that the rigid sphere model does not hold for the silica particles under consideration. If it is the case that interference is not negligible, it would mean that viscometric observations of Einstein behavior8 are inconsistent with our observation. On the other hand, it is also true that some researchers have concluded that silica particles are nonspherical.9 Further study is necessary before a definite conclusion on the observed deviation from the Guinier law can be given. Acknowledgment. Sincere thanks are due to Dr. J. Yamanaka, Polymer Phasing Project, ERATO, JRDC, Keihanna Plaza, Kyoto 619-02, Japan, for his suggestions and discussion. Our thanks are also due to Mr. Stephen Owen, Central Laboratory, Rengo Co., Ltd., Osaka, for his kind help in preparation of the manuscript. LA950777J (8) Yamanaka, J.; Hayashi, Y.; Ise, N.; Yamaguchi, T. Submitted. (9) Yates, D. E.; Healy, T. W. J. Colloid Interface Sci. 1976, 55, 9.
