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

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.