Approximation locale précisée dans des problèmes multi-échelles avec défauts localisés

We proceed here with our systematic study, initiated in [3], of multiscale problems with defects, within the context of homogenization theory. The case under consideration here is that of a diffusion equation with a diffusion coefficient of the form of a periodic function perturbed by an $L^r({\mathbb{R}}^d)$, $1<r<+\infty$, function modelling a localized defect. We outline the proof of the following approximation result: the corrector function, the existence of which has been established in [3], [4], allows us to approximate the solution to the original multiscale equation with essentially the same accuracy as in the purely periodic case. The rates of convergence may however vary, and are made precise, depending upon the $L^r$ integrability of the defect. The generalization to an abstract setting is mentioned. Our proof exactly follows, step by step, the pattern of the original proof of Avellaneda and Lin in [1] in the periodic case, extended in the works of Kenig and collaborators [12], and borrows a lot from it. The details of the results announced in this Note are given in our publications [2], [11].     

Nanoscopic Porous Iridium/Iridium Dioxide Superstructures (15 nm): Synthesis and Thermal Conversion by In Situ Transmission Electron Microscopy

Porous particle superstructures of about 15 nm diameter, consisting of ultrasmall nanoparticles of iridium and iridium dioxide, are prepared through the reduction of sodium hexachloridoiridate(+IV) with sodium citrate/sodium borohydride in water. The water-dispersible porous particles contain about 20 wt % poly(N-vinylpyrrolidone) (PVP), which was added for colloidal stabilization. High-resolution transmission electron microscopy confirms the presence of both iridium and iridium dioxide primary particles (1–2 nm) in each porous superstructure. The internal porosity (≈58 vol%) is demonstrated by electron tomography. In situ transmission electron microscopy up to 1000 °C under oxygen, nitrogen, argon/hydrogen (all at 1 bar), and vacuum shows that the porous particles undergo sintering and subsequent compaction upon heating, a process that starts at around 250 °C and is completed at around 800 °C. Finally, well-crystalline iridium dioxide is obtained under all four environments. The catalytic activity of the as-prepared porous superstructures in electrochemical water splitting (oxygen evolution reaction; OER) is reduced considerably upon heating owing to sintering of the pores and loss of internal surface area.     

Polymer Nanocomposites for Emulsion‐Based Coatings and Adhesives

Emulsion‐based coatings and adhesives are in growing demand due to an increased awareness of health and safety issues arising from solvent‐based polymer manufacturing processes. However, emulsion‐based techniques often require additional development to achieve equal or better application performance compared to solvent‐based processes. The inclusion of nanoparticles in emulsion‐based coatings and adhesives can be considered as a promising means to enhance performance. This paper reviews the current progress on the synthesis of emulsion‐based nanocomposites for coating and adhesive applications and addresses the principles and techniques for nanoparticle dispersions and their inclusion into polymer latexes. The effects of nanoparticle shape and size on the enhancement of nanocomposite properties are also highlighted. Among the reinforcing nanoparticles such as nanoclays, carbon nanotubes, and cellulose nanocrystals (CNCs), CNCs are promising due to their abundance, nontoxicity, and accessible surface hydroxyl groups, which facilitate their compatibility with polymer latexes via physical and chemical treatments.     

The concept of proof in the light of mathematical work

ZDM – Mathematics Education - Our article aims to show how illuminating mathematical work as a concept from didactics of mathematics is useful in understanding issues relating to proving and...