Small-Angle Scattering Analysis of Empty or Loaded Hierarchical Porous Materials

Abstract

Small-angle scattering of X-rays (SAXS) or neutrons (SANS) is one of the few experimental methods that can in principle be used for the in situ study at the mesoscopic scale of physicochemical phenomena occurring inside nanoporous solids. However, the potential of the method is often limited by the lack of suitable data analysis methods to convert scattering data into real-space structural information. This is notably the case for most porous materials of practical interest, which exhibit a hierarchical structure with micro, meso, and macropores, with often a secondary material confined in the pores, such as in supported catalysts, as well as fuel-cell and battery materials. In the present paper, we propose a general analysis of X-ray scattering by this type of material. Assuming that each structural level is statistically independent from the others and has a distinct characteristic length scale, compact mathematical expressions are derived for the scattering of the entire hierarchical structure. The results are particularized to the SAXS analysis of SBA-15-ordered mesoporous silica loaded with copper nitrate as well as of supported catalysts obtained after heat treatment of that material. The SAXS data analysis shows that the nitrate fills both the micro- and mesopores of the material, while the metallic copper obtained after heat treatment is found only in the mesopores. Moreover, the mesoscopic-scale spatial distribution of the metal depends on the heat treatment, in line with earlier electron tomography studies. The main ideas underlying the SAXS data analysis were presented in a recent communication (Gommes Angew. Chem., Int. Ed. 2015, 54, 11804−11808). Here, we generalize the approach and provide a comprehensive discussion of how any level in a hierarchical structure contributes to its overall scattering pattern. The results, as well as the general modeling methodology, will be of interest to anyone interested in the quantitative analysis of small-angle scattering data of empty or loaded porous solids and more generally of any type of hierarchical material

Small-angle scattering and scale-dependent heterogeneity

“…Using that value of a and A ðlÞ PS ¼ 252 m 2 cm À3 leads to the mesopore radius R ¼ 45 Å . This value accords beautifully with the estimated radius from nitrogen adsorption of 40 Å as mentioned earlier, as well as with the value R ¼ 43 Å obtained via a thorough modelling of the scattering pattern (Gommes et al, 2016).…”

“…The relevant density differs according to the specific type of scatteringelectron or scattering length densities for small-angle scattering of X-rays (SAXS) or neutrons (SANS), respectivelybut the mathematics are the same. The apparent simplicity of that statement hides a wealth of difficult questions related notably to the numerical evaluation of correlation functions from scattering data (Glatter, 1977), the determination of nontrivial structural information (Tchoubar & Mé ring, 1969), the development and adjustment of structural models to scattering patterns (Pedersen, 1997;Svergun, 1999), the SAXS analysis of hierarchical structures (Gupta et al, 2006;Gommes et al, 2016), the enumeration of structures compatible with a given correlation function (Gommes et al, 2012a) etc. Any new, albeit mathematically equivalent, perspective on smallangle scattering may shed new light on all these questions as well.…”

A novel methodology to study nanoporous alumina by small-angle neutron scattering

“…Combining these direct imaging methods with scattering techniques can overcome these issues by giving access to averaged structural information over a larger volume (Gommes et al, 2016) and can also provide information about the pore orientation in both transverse and longitudinal directions. X-rays (via scattering or diffraction) have been mainly used to characterize the pore arrangement and longrange ordering (Dore et al, 2002;Engel et al, 2009;Napolskii et al, 2010;Waheed et al, 2011;Roslyakov et al, 2013Roslyakov et al, , 2016 by tilting the sample along the vertical or horizontal axis normal to the beam.…”

Tailoring the 3D Structure of Pd Nanocatalysts Supported on Mesoporous Carbon for Furfural Hydrogenation

“…To clarify the effect of nanoparticle location on catalytic properties and to design desired catalyst, the precise determination of the location of nanoparticle is essential. Small‐angle X‐ray scattering (SAXS) technique has been applied to characterize the average porosity and metal loading for catalyst particles supported on porous support ,. Nevertheless, (scanning) transmission electron microscopy ((S)TEM) is one of the most employed techniques to provide spatial resolution at the nanoscale and even atomic scale, as well as the particle distribution homogeneity on individual support pieces.…”

“…Small-angle X-ray scattering (SAXS) technique has been applied to characterize the average porosity and metal loading for catalyst particles supported on porous support. [14,15] Nevertheless, (scanning) transmission electron microscopy ((S)TEM) is one of the most employed techniques to provide spatial resolution at the nanoscale and even atomic scale, as well as the particle distribution homogeneity on individual support pieces. However, to resolve the actual location and distribution of the nanoparticle in a 3D volume quantitatively with high accuracy is still challenging.…”