Small-angle scattering by supported nanoparticles: exact results and useful approximations

Abstract

In functional materials, nanoparticles are often dispersed in a porous support for the purpose of stabilizing them. This makes their characterization by small‐angle scattering challenging because the signal comprises contributions from the nanoparticles of interest, from the inert support and from their cross‐correlation. Exact analytical expressions for all three contributions are derived in the case of a Gaussian‐field model of the porous support, with nanoparticles randomly distributed over the surface. For low nanoparticle loading, the expressions simplify to the addition of properly scaled support and particle scattering. For higher loadings, however, the cross‐correlation cannot be ignored. Two approximations are introduced, which capture correlation effects in cases where the pores of the support are much larger or only slightly larger than the nanoparticles. The methods of the paper are illustrated with the small‐angle X‐ray scattering analysis of hollow metallic nanoparticles supported on porous carbon.

Stochastic models of dense or hollow nanoparticles and their scattering properties

“…The energy, detector distance and tilts were calibrated using a standard silver behenate powder and the 2D scattering patterns were reduced to 1D curves using the pyFAI software package (Ashiotis et al, 2015). The scattering by the empty carbon support was measured on a slightly different setup (Gommes et al, 2019). In all cases, the scattering data were corrected for the contribution of the empty sample Transmission electron micrographs of the hollow PtNi nanoparticles, in (a) the fresh and (b) the spent catalyst.…”

“…This approximation is valid in the case of low loading of the support, i.e. if the average electron density of the pore-filling phase (as if the particles were homogeneously dispersed in the pores) is much lower than that of the solid phase of the support (Gommes et al, 2019). As we shall discuss shortly, this applies to the catalysts of Section 2.…”

“…For that purpose, it is necessary to account first for the relative contribution of the particles and the porous carbon support to the measured scattered intensity. The total scattering is (Gommes et al, 2019) research papers J. Appl. Cryst.…”

“…where s is the volume fraction of the solid, l D is a characteristic length of the material, and the second contribution proportional to b accounts for the turbostratic structure of carbon (Perret & Ruland, 1968). The solid contribution was fitted to the SAXS data of the empty support, as thoroughly explained by Gommes et al (2019); the best fit is shown in Fig. 2 as a solid black line corresponding to parameters s = 0.17, l D = 158 Å and b = 1.85 Å .…”

“…The other material constants that enter equation 43are the electron densities of the support and the particles, s and p , and the particle concentration p . The electron densities are assumed to be s = 0.9 F cm À3 for the carbon support and p = 8.6 F cm À3 for the particles, assuming they are made up of pure Pt with traces of Ni (Gommes et al, 2019). The particle concentration p is calculated from the known metal loading and the average particle volume hVi.…”