The properties of some aggregates “grown” on a computer by diffusion-limited aggregation have been investigated. Calculations showed that the intensity of the small-angle x-ray and neutron scattering from the aggregates was proportional to q−D for qL ≫ 1, where D > 0, L is a length that characterizes the large-scale structure of the aggregate, q = 4πλ−1 sin(θ/2), λ is the wavelength, and θ is the scattering angle. The magnitude of the exponent D was appreciably smaller than the fractal dimensions that many simulations have shown to be typical of the mass fractal aggregates grown by diffusion-limited aggregation. The calculations suggest that the aggregates have structure on two different characteristic-length scales.

The intensity I(q) of the small-angle x-ray or neutron scattering has been calculated for a system of randomly oriented, independently scattering pores with a number distribution of pore diameters which has the form of a power law. As has already been shown, [P. W. Schmidt, J. Appl. Cryst. 15, 567–569 (1982)], when the number distribution of the maximum diameters a of the pores is proportional to a−γ, I(q) is proportional to q−(7−γ), where q = 4πλ−1sin(θ/2), θ is the scattering angle, and λ is the wavelength. The coefficient of the power-law intensity has been expressed in terms of some of the constants which determine the diameter distribu-tion. Equations have been obtained for the scattered intensity I(q) at q values larger and smaller than those at which power-law scattering occurs. The intensity scattered by this system is compared with the intensity from a system of pores with fractal pore-boundary surfaces which have a fractal dimension D.

Small-angle x-ray scattering, nitrogen adsorption, and scanning tunneling microscopy show that a series of activated carbons host an extended fractal network of channels with dimension D(p) = 2.8-3.0 (pore fractal), channel width 15-20 A (lower end of scaling), network diameter 3000-3400 A (upper end of scaling), and porosity of 0.3-0.6. We interpret the network as a stack of quasiplanar invasion percolation clusters, formed by oxidative removal of walls between closed voids of diameter of approximately 10 A and held in registry by fibrils of the biological precursor, and point out unique applications.

This paper studies the static structure factor of a system of fractal aggregates at various degrees of densification. The system we use for this study is carbonaceous soot, which is composed of diffusion limited cluster aggregates with a fractal dimension of 1.8. The range of density is great, from the aerosol to a system of lightly touching clusters and then to ground and compressed samples. The data involve a combination of light scattering and small-angle x-ray scattering over a {ital q} range of 3{times}10{sup {minus}3}{le}q{le}6thinspnm{sup {minus}1}. We are able to explain all the features of the data with scaling arguments based on comparisons of the scattering length scale q{sup {minus}1}, where {ital q} is the magnitude of the scattering wave vector, and the various length scales of the system that are density dependent. {copyright} {ital 1998} {ital The American Physical Society}

Small-angle x-ray scattering has been used to investigate the structure of some carbon blacks, some silicas, and an alumina-silica catalyst carrier on length scales from about 5 to 10,000 Å. Equations developed for structural studies of fractal and non-fractal aggregates of primary particles have been employed to analyze the scattering data. From the intensity data, the average diameters of the primary particles could be calculated or estimated. Despite the very different origins of the samples and the fact that the average diameters of the particles varied from about 30 to over 1000 Å, the scattered intensities from the samples had many common features. The data showed that the primary particles had a uniform density and were bounded by smooth or fractal surfaces. On length scales greater than the diameters of the primary particles but not more than a few times larger than the average diameters of the aggregates, some of the aggregates were mass fractals, and others were surface fractals.

Contrast matching is often carried out using small angle neutron scattering (SANS) and adsorbed H2O/D2O mixtures in an effort to obtain additional pore structure information. In this work, we explore the use of single adsorbates with similar electron density to those of the matrix so that contrast-matching may be conducted with small angle x-ray scattering (SAXS). In particular, we have used a series of halogenated organic liquids and several porous and particulate silicas and a microporous carbon to demonstrate contrast matching with SAXS. Also, halogenated silylation compounds have been used which yield surface modification groups with electron density similar to silica. From analyzing the change in scattering as a function of adsorbate loading or surface modification, pore morphology, pore size distribution and/or surface texture information is obtained.

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 Å, 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.

Small-angle X-ray and neutron scattering are important techniques for studying the structure of fractals and other disordered systems on a scale of lengths from about 10 to 2000 angstrom. This review begins with a brief outline of some properties of fractals. The small-angle scattering from fractal systems is then discussed and the effect of polydispersity is considered. The intensity of small-angle scattering from fractals and other disordered systems is often proportional to a negative power of the quantity q = 4-pi-lambda-1sin(theta/2), where theta is the scattering angle and lambda is the X-ray or neutron wavelength. From the magnitude of the exponent that describes this type of scattering, which is often called power-law scattering, much important information can be obtained. Some situations in which power-law scattering can be expected are described. To illustrate the scattering from fractals and disordered systems, several experimental investigations of mass-fractal silicas and porous solids are reviewed and some calculations of the small-angle scattering from model fractal systems are outlined.

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.

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 Å 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 Å, 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 Å that was the material from which the other four gels were produced.

Analysis of small-angle x-ray-scattering data from a series of controlled-pore glasses (nominal pore sizes from 75 to 2000 A\r{}) and from Vycor porous glass has revealed the fractal nature of the surfaces of these materials. The controlled-pore glasses show a relatively low degree of surface irregularity, with a surface fractal dimension of D=2.20\ifmmode\pm\else\textpm\fi{}0.05, regardless of pore size. Vycor glass, the fractal properties of which have been under some debate, has a rougher surface, with D=2.40\ifmmode\pm\else\textpm\fi{}0.10. The D value for Vycor has been verified by small-angle neutron scattering. The fractal dimensions refer to surface irregularity in the range 10--100 A\r{}. The scattering data suggest that the carriers of the fractal surface properties are interconnected units (particles, clusters) with an average diameter of 300 and 450 A\r{} for the controlled-pore glasses of pore size 75 and 170 A\r{}, respectively, and units with an average diameter of 350 A\r{} for Vycor glass.

AbstractRecent progress in two fields of small‐angle scattering is reviewed: (a) New procedures have been developed by Kotlarchyk and Chen and by Triolo, Griffith, and Compere for calculating the intensity of the small‐angle scattering from polydisperse systems of interacting particles of different sizes. These techniques have significantly increased the quantitative information that can be obtained from the scattering data. (b) The pore boundaries in many porous solids have been found to be fractal surfaces. In a porous solid in which the pores have an average diameter ϵ and the pore boundary surfaces have a fractal dimension D, the scattered intensity for qϵ, >> 1 is proportional to q−(6‐D), where q = πλ−1sin(θ/2), θ is the scattering angle, and λ is the wavelength. Some small‐angle scattering studies of fractal porosity are outlined.

The fractal dimension $D$ of a mesoporous silica gel was determined by three independent approaches: (1) adsorption methods based on (a) tiling of the surface with molecules of different cross-sectional area and (b) analysis of the change in measured surface area as a function of the size of the silica particles; (2) analysis of one-step dipolar energy transfer between adsorbed rhodamine B and malachite green; (3) measurements of power-law small-angle x-ray scattering. All three techniques indicate that the surface is extremely rough and irregular, with a fractal dimension $D$ nearly equal to 3.

A convenient approximation has been obtained for the intensity of the small-angle X-ray or neutron scattering from an assembly of independently scattering spherical particles with constant density when the radius-distribution function is a Gaussian with a full width at half maximum which is not greater than 1.7 times the most probable radius. The approximation expresses the intensity in terms of elementary functions. Criteria are given for setting a bound on the error in the approximation.

Small-angle x-ray scattering investigations are performed on charcoal samples from black cherry wood which were heated to temperatures ranging from 600/sup 0/ to 2000/sup 0/C during pyrolysis. This report summarizes the results of these investigations which give information about the pore structure of black cherry wood. The authors also develop a general picture of the dependence of charcoal porosity on the temperature to which the wood was heated during pyrolysis.