Nonuniform Dichotomies via Lyapunov Functions

For a nonautonomous linear equation x′ =  A(t)x we show how to characterize a nonuniform exponential dichotomy using strict Lyapunov functions. In particular, the stable and unstable subspaces are obtained from invariant families of cones determined by each Lyapunov function. We also obtain converse theorems, constructing explicitly a family of strict Lyapunov functions for each nonuniform exponential dichotomy. We emphasize that nonuniform exponential dichotomies include as a very particular case (uniform) exponential dichotomies.     

Understanding the Deactivation Pathways of Iridium(III) Pyridine-Carboxiamide Catalysts for Formic Acid Dehydrogenation

The degradation pathways of highly active [Cp*Ir(κ²-N,N-R-pica)Cl] catalysts (pica=picolinamidate; 1 R=H, 2 R=Me) for formic acid (FA) dehydrogenation were investigated by NMR spectroscopy and DFT calculations. Under acidic conditions (1 equiv. of HNO₃), 2 undergoes partial protonation of the amide moiety, inducing rapid κ²-N,N to κ²-N,O ligand isomerization. Consistently, DFT modeling on the simpler complex 1 showed that the κ²-N,N key intermediate of FA dehydrogenation ( I_NH ), bearing a N-protonated pica, can easily transform into the κ²-N,O analogue ( I_NH2 ; ΔG^≠≈11 kcal mol⁻¹, ΔG ≈−5 kcal mol⁻¹). Intramolecular hydrogen liberation from I_NH2 is predicted to be rather prohibitive (ΔG^≠≈26 kcal mol⁻¹, ΔG≈23 kcal mol⁻¹), indicating that FA dehydrogenation should involve mostly κ²-N,N intermediates, at least at relatively high pH. Under FA dehydrogenation conditions, 2 was progressively consumed, and the vast majority of the Ir centers (58 %) were eventually found in the form of Cp*-complexes with a pyridine-amine ligand. This likely derived from hydrogenation of the pyridine-carboxiamide via a hemiaminal intermediate, which could also be detected. Clear evidence for ligand hydrogenation being the main degradation pathway also for 1 was obtained, as further confirmed by spectroscopic and catalytic tests on the independently synthesized degradation product 1 c . DFT calculations confirmed that this side reaction is kinetically and thermodynamically accessible.     

Pseudopeptide‐Based Hydrogels Trapping Methylene Blue and Eosin Y

We present herein the preparation of four different hydrogels based on the pseudopeptide gelator Fmoc‐l ‐Phe‐d ‐Oxd‐OH (Fmoc=fluorenylmethyloxycarbonyl), either by changing the gelator concentration or adding graphene oxide (GO) to the water solution. The hydrogels have been analysed by rheological studies that demonstrated that pure hydrogels are slightly stronger compared to GO‐loaded hydrogels. Then the hydrogels efficiency to trap the cationic methylene blue (MB) and anionic eosin Y (EY) dyes has been analyzed. MB is efficiently trapped by both the pure hydrogel and the GO‐loaded hydrogel through π–π interactions and electrostatic interactions. In contrast, the removal of the anionic EY is achieved in less satisfactory yields, due to the unfavourable electrostatic interactions between the dye, the gelator and GO.     

A Carbocationic Triarylmethane-Based Porous Covalent Organic Network

A thermally stable carbocationic covalent organic network (CON), named RIO-70 was prepared from pararosaniline hydrochloride, an inexpensive dye, and triformylphloroglucinol in solvothermal conditions. This nanoporous organic material has shown a specific surface area of 990 m² g⁻¹ and pore size of 10.3 Å. The material has CO₂ uptake of 2.14 mmol g⁻¹ (0.5 bar), 2.7 mmol g⁻¹ (1 bar), and 6.8 mmol g⁻¹ (20 bar), the latter corresponding to 3 CO₂ molecules adsorbed per pore per sheet. It is shown to be a semiconductor, with electrical conductivity (σ) of 3.17×10⁻⁷ S cm⁻¹, which increases to 5.26×10⁻⁴ S cm⁻¹ upon exposure to I₂ vapor. DFT calculations using periodic conditions support the findings.