Particle Size Distribution of Nanospheres by Monte Carlo Fitting of Small Angle X-Ray Scattering Curves

Abstract

A method for recovering the size distribution of spherical particles from small angle scattering data by using a Monte Carlo interference function fitting algorithm is presented. The method is based on the direct simulation of the small angle scattering data upon the assumption of non‐interacting hard sphere ensembles (“dilute” solution approximation). The algorithm for retrieving the particle size distribution does not require any additional parameters apart from the input of the scattering data. The fitting strategy necessarily implies positive particle size distributions, while preserving the advantage of the indirect transformation method for data desmearing. Furthermore, the present approach does not use any regularisation procedures of the best fit solution and favours smooth particle size distributions. The Monte Carlo procedure has been tested against several simulated cases with various types of mono‐ and bi‐modal size distributions and different noise levels. In the special case of non‐interacting spheres, the Monte Carlo fitting algorithm had the same retrieving ability as the well assessed indirect transformation, structure interference and maximum entropy methods. Finally, the algorithm was applied to retrieve the distribution of spherical nanopowders produced by gas‐to‐particle conversion both as free powder and as reinforcing second‐phase agent in polymer nanocomposites.

Characterization and Analysis of Porosity and Pore Structures

“…A number of approaches have been suggested for calculating pore volume/size distributions. These include the smooth surface approach (Anovitz et al 2013a), the polydisperse hard sphere model (PRINSAS, Hinde 2004, Radlinski 2006, Maximum Entropy approaches (Jaynes 1983;Skilling and Bryan 1984;Culverson and Clarke 1986;Potton et al 1986Potton et al , 1988aHansen and Pedersen 1991;Jemain et al 1991;Semenyuk and Svergun 1991), regularization or maximum smoothness (Glatter 1977(Glatter , 1979Moore 1980;Svergun 1991;Pederson 1994), total non-negative least squares (Merrit and Zhang 2004;, Bayesian (Hansen 2000) and Monte Carlo (Martelli and Di Nunzio 2002;Di Nunzio et al 2004;Pauw et al 2013). There are also methods available based on Titchmarsh transforms for determining size distributions (Fedorova and Schmidt 1978;Mulato and Chambouleyron 1996;Botet and Cabane 2012), and the structure interference method (Krauthäuser et al 1996).…”

Copper and copper oxide nanoparticles in a cellulose support studied using anomalous small-angle X-ray scattering

“…The particle size distribution was retrieved from S bb (q) using a Monte Carlo fitting similar to the one described in [30]. The reason we chose the Monte Carlo method instead of other methods is the fact that it does not assume any predetermined shape for the distribution and is not very sensitive to noise in the data.…”

Structure of nickel nanoparticles in a microcrystalline cellulose matrix studied using anomalous small-angle X-ray scattering

“…The particle size distribution was retrieved from S bb ðqÞ using a Monte Carlo fitting procedure similar to that described by Martelli & di Nunzio (2002) with the modification that spheres of different radius were picked randomly from an array R n ¼ =q n . In order to reduce statistical error the PSFs were smoothed before fitting.…”