Contrast variation in small-angle scattering experiments on polydisperse and superparamagnetic systems: basic functions approach

The development of the basic functions approach [Stuhrmann (1995). Modern Aspects of Small-Angle Scattering, edited by H. Brumberger, pp. 221-254. Dordrecht: Kluwer Academic Publishers] for the contrast variation technique in small-angle scattering from systems of polydisperse and superparamagnetic non-interacting particles is presented. For polydisperse systems the modified contrast is introduced as the difference between the effective mean scattering length density (corresponding to the minimum of the scattering intensity as the function of the scattering length density of the solvent) and the density of the solvent. Then, the general expression for the scattering intensity is written in the classical way through the modified basic functions. It is shown that the shape scattering from the particle volume can be reliably obtained. Modifications of classical expressions describing changes in integral parameters of the scattering (intensity at zero angle, radius of gyration, Porod integral) with the contrast are analyzed. In comparison with the monodisperse case, the residual scattering in the minimum of intensity as a function of contrast (effective match point) in polydisperse systems makes it possible to treat the Guinier region of scattering curves around the effective match point quite precisely from the statistical viewpoint. However, limitations of such treatment exist, which are emphasized in the paper. In addition, the effect of magnetic scattering in small-angle neutron scattering from superparamagnetic nanoparticles is considered in the context of the basic functions approach. Conceptually, modifications of the integral parameters of the scattering in this case are similar to those obtained for polydisperse multicomponent particles. Various cases are considered, including monodisperse non-homogeneous and homogeneous magnetic particles, and polydisperse non-homogeneous and homogeneous magnetic particles. The developed approach is verified for two models representing the main types of magnetic fluids - systems of polydisperse superparamagnetic particles located in liquid carriers.