Particle size distributions from small-angle scattering using global scattering functions

Control and quantification of particle size distribution is of importance in the application of nanoscale particles. For this reason, polydispersity in particle size has been the focus of many simulations of particle growth, especially for nanoparticles synthesized from aerosols such as fumed silica, titania and alumina. Single-source aerosols typically result in close to a log-normal distribution in size and micrograph evidence generally supports close to spherical particles, making such particles ideal candidates for considerations of polydispersity. Small-angle X-ray scattering (SAXS) is often used to measure particle size in terms of the radius of gyration, R_g, using Guinier's law, as well as particle surface area, S/V, from the Porod constant B and the scattering invariant Q. In this paper, the unified function is used to obtain these parameters and various moments of the particle size distribution are calculated. The particle size obtained from BET analysis of gas adsorption data directly agrees with the moment calculated from S/V. Scattering results are also compared with TEM particle-counting results. The potential of scattering to distinguish between polydisperse single particles and polydisperse particles in aggregates is presented. A generalized index of polydispersity for symmetric particles, PDI = BR_g⁴/(1.62G), where G is the Guinier prefactor, is introduced and compared with other approaches to describe particle size distributions in SAXS, specifically the maximum-entropy method.