Direct Structure Analysis of Small-Angle Scattering Data from Polydisperse Colloidal Particles

The internal structure of colloidal particles of spherical, rod-like or lamellar symmetry can be determined from small-angle scattering data by means of the well established `indirect Fourier transformation' technique followed by the numerical deconvolution of the pair distance distribution function. The method works without any preliminary knowledge of the particles under investigation except a rough estimate of their maximal size. However, its use has hitherto been restricted to monodisperse or very weakly polydisperse systems. In this paper a new method is presented that is capable of determining the radial scattering length density distribution of particles in strongly polydisperse samples and provides a parameter that quantifies the degree of polydispersity present in the system. The only additional information necessary is the function type which describes the size distribution of the particles under investigation in the best way possible. However, the method is limited to polydisperse systems which may be characterized by only one linear size parameter. In addition, it is now possible to represent the density profile by smooth spline functions, i.e. the technique is no longer restricted to step functions as basis functions. The mathematical procedure is outlined in detail and numerical results of simulations as well as of experimental data from small-angle neutron scattering and small-angle X-ray scattering measurements of surfactant aggregates in water and water/oil systems are given.