Integral method for the Stokes problemMéthode intégrale pour le problème de Stokes

We consider the bidimensional Stokes problem for incompressible fluids in stream function-vorticity formulation. For this problem, the classical finite elements method of degree one converges only in $\text{O}\text{(}\text{h}\text{)}$ for the quadratic norm of the vorticity, if the domain is convex and the solution regular. We propose to use harmonic functions obtained by a simple layer potential to approach vorticity along the boundary. Numerical results are very satisfying and we prove that this new numerical scheme leads to an error of order $\text{O}\text{(h)}$ for the natural norm of the vorticity and under more regularity assumptions from $\text{O}\text{(h}{}^{\mathrm{3/2}}\text{)}$ to $\text{O}\text{(h}{}^2\text{)}$ for the quadratic norm of the vorticity. To cite this article: T. Abboud et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 71–76     

Application of weigert effect to the pseudocoloration of a black and white signal

We propose here a method to transform the grey levels of a black and white transparency into different colours. The transparency to be processed, illuminated by a beam of linearly polarized white light, is twice recorded on a photograph silver-chloride emulsion. The direction of the light vector is rotated through π/2 between the exposures. The resulting plate is characterized by a spatially varying anisotropy. The main point is that the wavelength λ_B and λ_D, respectively corresponding to the cancellation of birefringence and dichroism, are function of the irradiance of the signal and of the successive exposure times. The plate is then illuminated in white light and observed between two polarizers P and A. It reconstructs an achromatic image of T when the respective axes of P and A are mutually perpendicular. A rotation of A from its setting crossed with P generate a colour distribution on the plate.     

Emission of electrons from mim systems: Energy distributions

In this part of our study we measured energy distributionsN(E) andN(E _X ) of electrons emitted from sandwich cathodes, and their basic temperature (at 300, 200 and 80 K) and voltage dependences. From the distributionN(E) at 80 K we calculated the mean free path of electrons in cathode and the effective temperature of distribution. The anomalous emission (beginning at small leakage voltages) was observed and its energy distribution measured, too. From the graphs it can be seen that some electrons acquire energy 2–5 eV during their transition through the sandwich cathode.     

Formation of a chiral NPPN ligand via metallation of acyclic NPNCN systems

A new type of bidentate N,N'-chelating ligand containing a chiral phosphorus centre has been synthesized via the metallation of an acyclic NPNCN species. The zwitterionic ligand backbone contains a phosphenium centre stabilised by an imido phosphine fragment.     

Mixed-valence MnIIIMnIV clusters [Mn7O8(O2SePh)8(O2CMe)(H2O)] and [Mn7O8(O2SePh)9(H2O)]: single-chain magnets exhibiting quantum tunneling of magnetization

The syntheses, structures, and magnetic properties of two new Mn7 complexes containing phenylseleninate ligands are reported. [Mn7O8(O2SePh)8(O2CMe)(H2O)] (1) and [Mn7O8(O2SePh)9(H2O)] (2) were both prepared by the reaction of 18 equiv of benzeneseleninic acid (PhSeO2H) with [Mn12O12(O2CMe)16(H2O)4] in MeCN. Complex 1 x 6MeCN crystallizes in the triclinic space group P, and complex 2 x 2CH2Cl2 crystallizes in the monoclinic space group P2(1)/m. Both compounds possess an unprecedented [Mn7O8]9+ core comprising a central [MnIII3(micro3-O)4]+ unit attached to [MnIV2(micro-O)2]4+ and [MnIV2(micro-O)(micro3-O)]4+ units on either side. In each cluster, the PhSeO2- groups function as bridging ligands between adjacent Mn centers. The structure reveals strong Se.O intermolecular contacts between Mn7 units to give a one-dimensional chain structure, with weak interchain interactions. Solid-state DC magnetic susceptibility measurements of complexes 1 and 2 reveal that they have very similar properties, and detailed studies on 1 by AC susceptibility measurements confirm an S = 2 ground-state spin value. In addition, out-of-phase AC signals are observed, suggesting slow magnetization relaxation. Magnetization versus DC field sweeps down to 0.04 K reveals hysteresis loops, but the temperature dependence of the coercivity is not what is expected of a single-molecule magnet. Instead, the behavior is due to single-chain magnetism, albeit with weak antiferromagnetic interactions between the chains, with the barrier to relaxation arising from a combination of molecular anisotropy and ferromagnetic intermolecular exchange interactions mediated by the Se...O contacts. An Arrhenius plot was constructed from the magnetization versus time decay data. The thermally activated region at > 0.5 K gave an effective relaxation barrier (Ueff) of 14.2 K. Below approximately 0.1 K, the relaxation is independent of temperature, which is characteristic of magnetization quantum tunneling through the anisotropy barrier. These Mn7 compounds are thus the first single-chain magnets to comprise polynuclear metal clusters and also the first for which the temperature-independent relaxation characteristic of tunneling has been identified. The work also emphasizes that out-of-phase AC signals for ostensibly molecular compounds are not sufficient proof by themselves of a single-molecule magnet.     

Bridge mediated two-electron transfer reactions: analysis of stepwise and concerted pathways

A theory of nonadiabatic donor (D)-acceptor (A) two-electron transfer (TET) mediated by a single regular bridge (B) is developed. The presence of different intermediate two-electron states connecting the reactant state D-(-)BA with the product state DBA-(-) results in complex multiexponential kinetics. The conditions are discussed at which a reduction to two-exponential as well as single-exponential kinetics becomes possible. For the latter case the rate KTET is calculated, which describes the bridge-mediated reaction as an effective two-electron D-A transfer. In the limit of small populations of the intermediate TET states D-B-A, DB-(-)A, D-BA-, and DB-A-, KTET is obtained as a sum of the rates KTET(step) and KTET(sup). The first rate describes stepwise TET originated by transitions of a single electron. It starts at D-(-)BA and reaches DBA-(-) via the intermediate state D-BA-. These transitions cover contributions from sequential as well as superexchange reactions all including reduced bridge states. In contrast, a specific two-electron superexchange mechanism from D-(-)BA to DBA-(-) defines KTET(sup). An analytic dependence of KTET(step) and KTET(sup) on the number of bridging units is presented and different regimes of D-A TET are studied.