Small-angle scattering of dense, polydisperse granular porous media: Computation free of size effects

Abstract

Small-angle x-ray and neutrons scattering is a widespread experimental tool for the investigation of the microstructure of random heterogeneous materials. Validation of (computer-generated) model microstructures often requires the numerical computation of the scattering intensity, which must be carried out with great care due to finite size effects. In this paper, a new method for this computation is presented. It is superior to previously existing methods for three reasons: First, it applies to any type of microstructure (not necessarily granular). Second, closed-form expressions of the size effects inherent to the proposed method can be rigorously derived and removed (in this sense, our method is free of size effects). Third, the complexity of the new algorithm is linear and the computation can easily be updated to account for local changes of the microstructure, while most existing algorithms are quadratic and any change of the microstructure requires a full recomputation. The present paper provides full derivation and validation of this method. Application to the computation of the scattering intensity of dense, polydisperse assemblies of spheres is then presented. A new, simple algorithm for the generation of these dense configurations is introduced. Finally, the results are critically reviewed in the perspective of hardened cement pastes.

Mesoscale texture of cement hydrates

“…as opposed to the pure C-S-H of the simulations. At large q values (i.e., q > 1 nm −1 ), we find, in agreement with experiments, a dependence IðqÞ ∼ q −4 typical of a Porod regime (29), indicating that the C-S-H surfaces have a subnanometric roughness (30). The Porod regime is followed at smaller q by the same q −3 dependence detected in the experiments that extends over more than one order of magnitude in length and that has been reported and discussed in the literature (4,13,27,31).…”

“…Having computed the two-point fluctuation autocorrelation function η 2 ðrÞ, the IðqÞ was deduced in absolute scale using the following relation: IðqÞ = −½2πðΔρÞ 2 =q ½dðReðη 2 ðqÞÞ=dq , where Δρ is the scattering length density contrast, andη 2 ðqÞ is the 1D Fourier transform of η 2 ðrÞ (36). We also computed IðqÞ using the projection theorem (30) and the digitized projections of the 3D images in x, y, and z directions. Both computations were in good agreement.…”

Soft X‐ray Ptychographic Imaging and Morphological Quantification of Calcium Silicate Hydrates (C–S–H)

“…One can perform a similar computation using the reconstructed projective phase images provided by a ptychographic imaging technique. The main interest is twofold: first, to localize at the nm scale the region of interest (ROI) where the SAS will be computed; and second, to capture the correlative properties of the ROI according to simple algebraic laws, as found already in numerous SAS experiments on cement and polydisperse granular media …”

Growth of sheets in 3D confinements — a model for the C–S–H meso structure

“…A powder average was computed by averaging over 200 orientations. In this first study, the removal of domain size effects was not considered as discussed, for example, by Brisard and Levitz [14]. Nevertheless, the interfacial region between the outer product and the inner product/cement grain is not an artefact because it is certainly also seen by experimental small-angle scattering.…”

“…They state that C-S-H cannot be described by monodisperse particle packings and diffusion limited aggregation. They suggest an Apollonian sphere packing with a power-law distribution of particle sizes, or alternatively a power-law distribution of pore sizes [14]. However, recent NMR experiments by Muller et al [15] indicate a rather narrow, bi-model distribution of pores in C-S-H, consisting of the interlayer and the gel pore space.…”