Size and concentration determination of colloidal nanocrystals by small-angle X-ray scattering

The accurate determination of the dimensions of a nano-object is paramount to the development of nanoscience and technology. Here we provide procedures for sizing quasi-spherical colloidal nanocrystals (NCs) by means of small-angle X-ray scattering (SAXS). Using 2.5 to 10 nm PbS NCs as a model system, the protocols outline the extraction of the net NC SAXS pattern by background correction and address the calibration of scattered X-ray intensity to an absolute scale. The NC size distribution is retrieved by fitting the corrected SAXS pattern either to parametrized analytical distributions or to a distribution constructed through a Monte Carlo approach. We compare the two methods and show that they yield nearly identical estimates of the NC diameter in the case of an NC ensemble with a monodisperse and monomodal size distribution. Extending the analysis to PbSe, CdSe, and CdS NCs, we provide SAXS-calibrated sizing curves ranging from 2.5 to 7 nm, which relate the NC diameter and the NC band-gap energy as determined using absorbance spectroscopy. In comparison with sizing curves calibrated by means of transmission electron microscopy (TEM), we systematically find that SAXS calibration assigns a larger diameter than TEM calibration to NCs with a given band gap. We attribute this difference to the difficulty of accurately sizing small objects in TEM images. To close, we demonstrate that NC concentrations can be directly extracted from SAXS patterns normalized to an absolute scale and that SAXS-based concentrations differ less than 10% from concentrations obtained using absorbance spectroscopy.