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A versatile software package in the form of a Python extension, named CDEF (computing Debye’s scattering formula for extraordinary form factors), is proposed to calculate approximate scattering profiles of arbitrarily shaped nanoparticles for small-angle X-ray scattering (SAXS). CDEF generates a quasi-randomly distributed point cloud in the desired particle shape and then applies the open-source software DEBYER for efficient evaluation of Debye’s scattering formula to calculate the SAXS pattern (https://github.com/j-from-b/CDEF). If self-correlation of the scattering signal is not omitted, the quasi-random distribution provides faster convergence compared with a true-random distribution of the scatterers, especially at higher momentum transfer. The usage of the software is demonstrated for the evaluation of scattering data of Au nanocubes with rounded edges, which were measured at the four-crystal monochromator beamline of PTB at the synchrotron radiation facility BESSY II in Berlin. The implementation is fast enough to run on a single desktop computer and perform model fits within minutes. The accuracy of the method was analyzed by comparison with analytically known form factors and verified with another implementation, the SPONGE, based on a similar principle with fewer approximations. Additionally, the SPONGE coupled to McSAS3 allows one to retrieve information on the uncertainty of the size distribution using a Monte Carlo uncertainty estimation algorithm.
The study described in this paper was conducted in the framework of the European nPSize project (EMPIR program) with the main objective of proposing new reference certified nanomaterials for the market in order to improve the reliability and traceability of nanoparticle size measurements. For this purpose, bimodal populations as well as complexly shaped nanoparticles (bipyramids, cubes, and rods) were synthesized. An inter-laboratory comparison was organized for comparing the size measurements of the selected nanoparticle samples performed with electron microscopy (TEM, SEM, and TSEM), scanning probe microscopy (AFM), or small-angle X-ray scattering (SAXS). The results demonstrate good consistency of the measured size by the different techniques in cases where special care was taken for sample preparation, instrument calibration, and the clear definition of the measurand. For each characterization method, the calibration process is described and a semi-quantitative table grouping the main error sources is proposed for estimating the uncertainties associated with the measurements. Regarding microscopy-based techniques applied to complexly shaped nanoparticles, data dispersion can be observed when the size measurements are affected by the orientation of the nanoparticles on the substrate. For the most complex materials, hybrid approaches combining several complementary techniques were tested, with the outcome being that the reliability of the size results was improved.
Most industrial nanoparticles have non-spherical shapes and also possess polydisperse size distributions, and due to their agglomeration/ aggregation state are difficult (or even impossible) to be addressed individually. Further, driven by regulatory purposes related to the identification of a material as a nanomaterial, the accurate measurement of the smallest dimension of a (nano)particulate material makes the analysis even more complex.
In the first phase of the EU Project nPSize - Improved traceability chain of nanoparticle size measurements (https://www.bam.de/Content/DE/Projekte/laufend/nPSize/npsize.html), the efforts are focused on synthesis of nanoparticles of well-defined, non-spherical shape. Following candidates of reference materials (CRM) with certifiable particle size (distribution) are under characterization with respect to their homogeneity and stability: (i) titania nanoplatelets (10-15 nm thickness x 50-60 nm lateral), (ii) titania bipyramides (~60 nm length x 40 nm width), (iii) titania acicular particles (100 nm length x 15-20 nm width; aspect ratio 5.5/6), (iv) gold nanorods (~10 nm width x 30 nm length), and (v) gold nanocubes (~55 nm x 55 nm x 55 nm).