Accurate Nanoparticle Size Distribution from SAXS Measurements

The extensive use of nanoparticles and the increasing attention to regulatory provisions for nanomaterials generally induce a huge interest in nanoparticle size characterization. A major challenge is associated with the metrological approach as addressed in the published ISO standard[1].


SAXS is now known as a dependable method for the investigation of nanoparticle size distribution[2]. This article discusses the capability to run reliable SAXS measurements in the lab on large size nanoparticles (> 100 nm) and to define accurately the size distribution even for a bi-modal population with close size values by using the expectation maximization (EM) technique for the analysis.

The EM framework[3] applied to the SAXS on spherical nanoparticles gives the solution for the particle size distribution offering the highest level of agreement with the experimental data, from the statistical point of view[4]. No assumption is made on the mathematical form of the distribution (form-free), which is a major benefit over other calculation procedures.

Xenocs instruments were used to perform SAXS measurements. Xenocs data reduction software (Foxtrot) was used for data processing as it enables easy reduction from 2D to 1D data and buffer contribution subtraction in complete intensity units. The data were then directly analyzed using the EM method.

The first sample comprises of an aqueous solution of large SiO2 particles[5] with a concentration of 1 mg/L. Particle size distribution is determined by fitting the SAXS curve using the EM method (Figure 1). The resultant size distribution curve is in line with the description as given by the provider[5].


Figure 1. Volume-weighted particle size distribution of a diluted solution of SiO2 nanoparticles in water (c = 1 mg/L) from 1D SAXS data fitting using the EM method. Nominal diameter = 340 nm. SAXS measurements run on a Xeuss HR at Sassenage, France.

A bi-modal mixture was prepared from SiO2 aqueous dispersions from Microparticles GmbH:

  • Sample A: Nominal size 115 nm
  • Sample B: Nominal size 148 nm

From the EM analysis, two distinct peaks of the size distribution are seen at 115 and 148 nm (Figure 2). The obtained volume-weighted size distribution is in line with those of the individual samples.

This result demonstrates that SAXS in the laboratory using a Xenocs system and the EM method enables the detection of a bi-modal nanoparticle suspension with high precision, even with mixture of particles being close in size.

Volume-weighted particle size distribution

Figure 2. Volume-weighted particle size distribution of a bi-modal SiO2 particle aqueous dispersion from 1D SAXS data fitting using the EM method. Nominal diameters = 115 and 148 nm. SAXS measurements run on a Nano-inXider at Sassenage, France.


[1] Nanotechnologies -- Vocabulary - Part 2: Nano-objects ISO/TS 80004-2:2015

[2] Particle size analysis - Small angle X-ray scattering method ISO 17867:2015

[3] M. Bertero and P. Boccacci, Introduction to inverse problems in imaging, Institute of Physics, Bristol, UK (1998)

[4] F. Benvenuto, H. Haddar and B. Lantz, SIAM J. Appl. Math., 2016, 76(1), 276-292

[5] Sample courtesy of Karlsruhe Institute of Technology, Germany.


This information has been sourced, reviewed and adapted from materials provided by Xenocs.

For more information on this source, please visit Xenocs.

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