Application of the paracrystal model to GISAXS analysis of the 3D self-assembled nanoparticle crystals

Abstract

An original diffraction model for the evaluation of the grazing-incidence small-angle X-ray scattering (GISAXS) of the three-dimensional (3D) supported self-assembled nanoparticle crystals is presented. The model utilizes the formalism of traditional crystallography that is applied to nanoparticles as scattering entities. The infinite paracrystal concept was adopted in order to include cumulative disorder that is typical for the self-assembled nanoparticle structures. Azimuthal averaging around a chosen crystallographic direction allows incorporation of texture effects. A transformation of the SAXS paracrystal equations to comply with the GISAXS geometry was employed to describe the scattering process in the limit of the kinematical approximation which facilitates calculations. The model was developed for the face-centered cubic (FCC) symmetry, however, its extension to any other cubic symmetry is straightforward. The model was tested on a 3D nanoparticle assembly formed on silicon from a dried drop-cast colloidal solution of monodisperse silver nanoparticles of 5.8 ± 0.6 nm diameter. Presence of the FCC nanoparticle crystal with the [111] texture along the substrate normal was evidenced from evaluation of the synchrotron GISAXS data. The degree of paracrystal disorder is 2.5% and 3.6% parallel and perpendicular to the substrate, respectively, suggesting the anisotropic character of the self-assembling process. A SAXS measurement of the nanoparticle crystal formed from the same solution in a capillary revealed a randomly oriented FCC structure with a slightly larger paracrystal disorder of 4%. Obviously, different conditions of the nanoparticle crystal formation affect its azimuthal and translational disorder.