Linear stability of WENO schemes coupled with explicit Runge–Kutta schemes

This paper focuses on the results of the linear stability analysis of the finite‐difference weighted essentially non‐oscillatory (WENO) schemes with optimal weights. The standard WENO schemes between the third and 11th order, the order‐optimised WENO schemes of the sixth and eighth order and the bandwidth‐optimised WENO schemes of the third and fourth order are considered. Several explicit Runge–Kutta schemes including the recently published strong stability‐preserving explicit Runge–Kutta schemes are considered for time discretisation. The stability limits as well as dissipation and dispersion properties dependent on the Courant–Friedrichs–Lewy number are presented for a hyperbolic model equation. The different combinations of space and time discretisation schemes are compared in terms of their accuracy and efficiency. For a parabolic model equation, the viscous term is discretised with high‐order central differences. The stability limits for the parabolic problem are presented as well. Numerical results of linear test cases are shown. Copyright © 2011 John Wiley & Sons, Ltd.     

A Cucurbit[8]uril 2:2 Complex with a Negative pKa Shift

A viologen derivative carrying a benzimidazole group ( V-P-I ²⁺; viologen–phenylene–imidazole V-P-I ) can be dimerized in water using cucurbit[8]uril (CB[8]) in the form of a 2:2 complex resulting in a negative shift of the guest pK_a, by more than 1 pH unit, contrasting with the positive pK_a shift usually observed for CB-based complexes. Whereas 2:2 complex protonation is unclear by NMR, silver cations have been used for probing the accessibility of the imidazole groups of the 2:2 complexes. The protonation capacity of the buried imidazole groups is reduced, suggesting that CB[8] could trigger proton release upon 2:2 complex formation. The addition of CB[8] to a solution containing V-P- I ³⁺ indeed released protons as monitored by pH-metry and visualized by a coloured indicator. This property was used to induce a host/guest swapping, accompanied by a proton transfer, between V-P-I ³⁺ ⋅ CB[7] and a CB[8] complex of 1-methyl-4-(4-pyridyl)pyridinium. The origin of this negative pK_a shift is proposed to stand in an ideal charge state, and in the position of the two pH-responsive fragments inside the two CB[8] which, alike residues engulfed in proteins, favour the deprotonated form of the guest molecules. Such proton release triggered by a recognition event is reminiscent of several biological processes and may open new avenues toward bioinspired enzyme mimics catalyzing proton transfer or chemical reactions.     

Solvato Modulation of the Magnetic Memory in Isotopically Enriched Erbium Polyoxometalate

Single-Molecule Magnet (SMM) property is by essence molecular, while commonly measured in solid crystalline state. Solvent crystallization molecules are usually neglected in the analysis and interpretation of solid-state properties. The solvation/desolvation process in the polyoxometalate(POM)-based Na₉[Er(W₅O₁₈)₂] ⋅ 35 H₂O SMM demonstrates that the dehydrated form relaxes more than 1000 times faster than the initial state, while the rehydration process allows the quasi complete recovering of the initial magnetic behaviour. This dehydration process is monitored by thermogravimetric analysis (TGA) and temperature-dependent X-ray powder diffraction, and rationalized by periodic quantum chemical calculations evidencing the tremendous role of the labile water molecules in the stability of the edifice. Ab-initio calculations highlight that sodium ions localization in the structure drive the magnetic responses. Isotopic enrichment with nuclear spin free (¹⁶⁶Er, I=0) Er^III ions shows that the relaxation dynamics in the quantum regime depends on the nuclear spin.