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Small angle x-ray scattering has been used to study the interactions between clay platelets in a number of samples. Sodium montmorillonite clays have been studied in concentrations ranging from 5% to 85% by weight. The scattering from the samples with concentrations from 10% to 40% was analyzed by Fourier transformation to obtain the distribution functions for the platelet spacing. For the higher concentrations, the scattering curves were interpreted by fitting the data with theoretical curves for aggregates of platelets. The most probable platelet spacing ranged from 116 Å for the 10% concentration to 13 Å for the 85% concentration. If one plots platelet spacing as a function of grams of H2O per gram of clay for this series of samples, a reasonably linear relation is obtained. The effect of adding NaCl, CaCl2, and AlCl3 to 2% sodium montmorillonite suspensions was studied. Until the salt concentration is raised to a critical value, the scattering curve is identical to the curve obtained for dilute Na montmorillonite suspensions with no salt added. The latter scattering curve has been previously interpreted [R. Hight, Jr., W. T. Higdon, and P. W. Schmidt, J. Chem. Phys. 33, 1656 (1960)] as being due to independent montmorillonite layers 10 Å thick. The critical concentration decreased as the valence of the added salt was increased. Above the critical concentration, the scattering data show that aggregation takes place, with the aggregates consisting of about 6 to 8 platelets and with a platelet spacing of 19 to 21 Å. Calcium montmorillonite suspensions show essentially identical behavior. Fresh hydrogen montmorillonite samples show no aggregation, but in samples aged about four months, the scattering curves indicate the presence of aggregates with a platelet spacing of 22–24 Å and with 2 or 3 platelets per aggregate.

Suspensions of montmorillonite clay with concentrations of ½ and 2% have been investigated by small angle x-ray scattering, with two different centrifuge fractions being studied for each concentration. At a given concentration the scattering was observed to be the same for both centrifuge fractions. The scattered intensity was always proportional to the concentration. These results are what would be expected from dilute suspensions of thin platelets of identical thickness but different surface areas. When corrections for the finite height of the collimating slits are applied to the scattering curves, good agreement with the theoretical curves for thin platelets is obtained. From the deviation between the experimental and the theoretical curves at the largest observed scattering angles, the thickness of the platelets was calculated to be 9 A. An independent determination of the thickness can be made from a measurement of the zero angle scattered intensity from some material for which the ratio of the scattered intensity to the scattering from a single electron can be calculated. Two heavy gases, SF6 and C2Cl2F4 (Freon 114), were used for this purpose. The feasibility of using Ludox colloidal silica as a third standard was investigated. Within experimental accuracy, the thickness values calculated from these data agree with the values obtained from the shape of the scattering curves. Both values agree with the thickness determined from the light scattering experiments of M. B. M'Ewen and M. I. Pratt [Trans. Faraday Soc. 53, 535 (1957)]. Since the scattering curve for the clay is proportional to the inverse square of the angle at the smallest observed angles, the large dimensions of the particle must be at least 500 A.

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Since the small-angle x-ray scattering intensity can be expressed as a Fourier integral, the techniques of asymptotic expansion of Fourier integrals can be used to calculate the small-angle x-ray scattering at relatively large scattering angles. Some asymptotic expansion techniques which are often useful are described. The relation between the scattered intensity at relatively large angles and the characteristic function and its derivatives is discussed. The scattered intensity for both prolate and oblate ellipsoids of revolution is calculated to provide examples of asymptotic expansion methods, and the resulting expressions are evaluated numerically. The behavior of the scattered intensity at relatively large scattering angles for platelet particles of negligible thickness is described.