Paul W. Schmidt

Small‐angle x‐ray scattering from the surfaces of reversed‐phase silicas: Power‐law scattering exponents of magnitudes greater than four

The small‐angle x‐ray scattering from fully and partially derivatized porous silicas has been studied. Power‐law‐scattering exponents of magnitude greater than 4 have been found in all cases. The magnitudes of the exponents increased with the alkyl chain length and with the degree of surface derivatization. In a preliminary model to explain these observations, a power‐law‐scattering exponent with magnitude greater than 4 is related to a ‘‘fuzzy’’ pore boundary, in which the density varies continuously at the pore boundary instead of changing discontinuously from a value of zero in the empty pore to the essentially constant density characteristic of the bulk silica, as is usually assumed in analyses of the small‐angle scattering from porous silicas.

X‐Ray Study of Critical Opalescence in Argon

The small‐angle x‐ray scattering from argon has been investigated at the critical pressure and at four other pressures over a range of temperatures above and below the liquid—vapor transition temperature for each pressure. For three of these pressures the angular distribution of the scattering was analyzed in terms of the Ornstein—Zernike theory of critical opalescence and found to be in good agreement with this theory. Values of the short‐range correlation length and relative values of the isothermal compressibility were determined for each condition of pressure and temperature. For the conditions for which compressibilities can be found by numerical differentiation of the isotherms of Michels, Levelt, and De Graff, the compressibility values calculated by numerical differentiation are in good agreement with those obtained from the scattering curves.

Small‐Angle X‐Ray Scattering Determination of Particle‐Diameter Distributions in Polydisperse Suspensions of Spherical Particles

A scheme is developed for using small‐angle x‐ray‐scattering data to find the distribution of the particle diameters in dilute colloidal suspensions of noninteracting spherical particles. No assumptions need to be made about the form of the distribution function. An investigation is made of the errors likely to be introduced in the numerical operations on the data. The method was tested on theoretical scattering curves for which the calculations could be done both numerically and analytically. After these tests showed the feasibility of the calculation, diameter distributions were computed from the x‐ray small‐angle scattering data from some polydisperse colloidal samples composed of spherical particles. The results suggest that this technique may often be a useful procedure for analysis of small‐angle x‐ray‐scattering data.

Some Fundamental Concepts and Techniques Useful in Small-Angle Scattering Studies of Disordered Solids

Some concepts important for the analysis of small-angle x-ray and neutron scattering data are reviewed. After the scattering cross sections are discussed, some procedures for calculating the scattered intensity are described. Special attention is devoted to the evaluation of the correlation function of a scatterer, and some important properties of this function are pointed out. Approximate expressions for the scattered intensity for qξ ≪ 1 and qξ ≫ 1 are developed from the general equations for the scattered intensity. [Here ξ is the diameter of the scatterer; q = 4πλ−1sin(θ/2);λ is the scattered wavelength; and θ is the scattering angle.] These concepts and results are applied in a review of some small-angle scattering studies of fractals and disordered solids.

Small‐Angle X‐Ray‐Scattering Determination of Diameter Distributions

Test calculations have been carried out to develop a practical method of using small‐angle x‐ray‐scattering data to determine the particle‐diameter distribution in polydisperse suspensions of noninteracting spherical particles. A new method has been developed for extrapolating the scattering curve for scattering angles beyond the largest angles at which data can be obtained. This extrapolation technique is shown to be an improvement over an earlier method [J. H. Letcher and P. W. Schmidt, J. Appl. Phys. 37, 649 (1966)] by being less sensitive to uncertainties in the data and to choices of the constants used in the extrapolation.

Small Angle X‐Ray Scattering from Montmorillonite Clay Suspensions. II

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 A for the 10% concentration to 13 A 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 A 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 A. 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 A and with 2 or 3 platelets per aggregate.

An analysis of the fractal properties of the surfaces of globular proteins

The fractal properties of the surfaces of ten globular protein molecules have been investigated by calculations that made use of crystallographic data on the atomic coordinates of these proteins. On length scales from about 1.5 to at least 7.5 A, the fractal dimensions Ds of the protein surfaces were found to lie between 2.10 and 2.17 and thus were not much greater than the value Ds=2 characteristic of a smooth surface. The calculated fractal dimensions Ds did not depend on the molecular mass, molecular diameter, biological function, or origin of the proteins.

Calculation of the Intensity of Small‐Angle X‐Ray Scattering at Relatively Large Scattering Angles

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.

Small‐angle x‐ray scattering study of the fractal morphology of porous silicas

Small‐angle x‐ray and neutron scattering measurements have shown that on a length scale smaller than the average pore diameter but larger than the diameters of atoms or small molecules, the pore surfaces in four commercial porous silica gels with average pore diameters ranging from approximately 200 to 2500 A are fractal and have a fractal dimension D=2.15±0.10. When these gels were manufactured, the nonequilibrium micropore structure was relaxed by thermal methods. The scattering data indicate that in the gels with average pore diameters of about 200 and 500 A, and perhaps also in the two gels with larger average pore diameters, the relaxation process leads to a pore structure nearly identical in form but on a larger scale than the structure in a gel with an average pore diameter of 60 A that was the material from which the other four gels were produced.

Determination of particle-diameter distributions in silica and gold suspensions

Abstract A small angle X-ray scattering method for determining particle-diameter distributions in dilute polydisperse suspensions of spherical particles with average diameters under 1,000 A has been tested by using this technique to find the particle-diameter distribution in three Ludox4 silica suspensions and two gold sols. The results of the X-ray determinations were checked by comparing them with diameter distributions obtained from electron micrographs. Within experimental uncertainty, the distributions obtained by the two methods were in agreement. The results of this investigation suggest that the X-ray method is a practical way for finding diameter distributions in suspensions of spherical particles with average diameters under about 1,000 A. The accuracy, while not high, should be sufficient for most purposes. This technique is applicable even to particles too small to permit diameter distributions to be obtained from electron micrographs.