Dispersive properties of a viscous numerical scheme for the Schrödinger equation

In this Note we study the dispersive properties of the numerical approximation schemes for the free Schrödinger equation. We consider finite-difference space semi-discretizations. We first show that the standard conservative scheme does not reproduce at the discrete level the properties of the continuous Schrödinger equation. This is due to spurious high frequency numerical solutions. In order to damp out these high-frequencies and to reflect the properties of the continuous problem we add a suitable extra numerical viscosity term at a convenient scale. We prove that the dispersive properties of this viscous scheme are uniform when the mesh-size tends to zero. Finally we prove the convergence of this viscous numerical scheme for a class of nonlinear Schrödinger equations with nonlinearities that may not be handeled by standard energy methods and that require the so-called Strichartz inequalities. To cite this article: L.I. Ignat, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

The kinetics of radiation-induced hydrogen abstraction by CCl3 radicals in the liquid phase: Cyclohexene

The kinetics of hydrogen abstraction from cyclohexene by CCl₃ radicals were studied in CCl₄ solution as a function of cyclohexene concentration and temperature in the range of 26–140°C. The CCl₃ radicals were produced both by radiolysis of CCl₄ and by photolysis of CCl₃Br. The rate constant for the reaction was found to be given by the equation where θ = 2.303 RT kcal/mol. This activation energy leads to C? H bond strength for the allylic hydrogen of 85 ± 1 kcal/mol, which means a resonance stabilization energy of 11 ± 1.5 kcal/mol for the C-C₆H₁₁ radical.     

Crystalline low band-gap alternating indolocarbazole and benzothiadiazole-cored oligothiophene copolymer for organic solar cell applications

A low band-gap alternating copolymer of indolocarbazole and benzothiadiazole-cored oligothiophene demonstrated balanced crystallinity and solubility; a solar cell combining this polymer with PC(61)BM in a preliminary test demonstrated power conversion efficiencies of 3.6%.     

Thermodynamics of interactions between lead(II) and cadmium(II) ions and azulene-based complexing polymer films

In order to understand the essential processes/interactions between the metal ions and modified electrodes which are based on complexing polymeric films, access to thermodynamic characteristics is compulsory. The paper enlarges the information concerning the sorption of metal ions within complexing polymer films, particularly based on azulene, which can be involved in metal detection sensors. Interactions between lead(II) or cadmium(II) ions and complexing polymer films have been studied using chemical preconcentration–anodic stripping method. The films have been obtained by controlled potential electrolysis in millimolar solutions of 4-azulen-1-yl-2,6-bis(2-thienyl)pyridine (L) in acetonitrile. PolyL films affinities towards these metal ions have been quantified at different temperatures by means of sorption isotherms. Parameters for sorption of lead(II) and cadmium(II) ions within polyL films have been calculated for Freundlich, Langmuir and Redlich–Peterson isotherms. The best fit was obtained when using Langmuir isotherm. The results evidence that lead ions are better sorbed than cadmium within polyL film. Thermodynamic parameters for the chemical sorption of lead(II) and cadmium(II) ions within polyL films have been calculated.     

Direct patterning of molecularly imprinted microdot arrays for sensors and biochips

We have used fountain pen microlithography to deposit arrays of molecularly imprinted polymer microdots on flat substrates. We visualize analyte binding to the dots by fluorescence microscopy with the aid of fluorescein as a model analyte. Elution and readsorption of the analyte to the MIP dots were possible if the porosity of the dots was improved by a sacrificial polymeric porogen. The imprinting effect was confirmed by using compounds structurally related to fluorescein. In addition, we show with another MIP specific to 2,4-D that, apart from the direct measurement of the binding of fluorescent compounds, a competitive immunoassay-type format can also be used to transduce the binding. We believe that this technique has a strong potential for the fabrication of biomimetic microchips and other types of integrated biosensors.     

4-(Azulen-1-yl) six-membered heteroaromatics substituted in 2- and 6- positions with 2-(2-furyl)vinyl, 2-(2-thienyl)vinyl or 2-(3-thienyl)vinyl moieties

The synthesis of pyrylium and pyridinium salts and pyridines with azulene-1-yl moieties in position 4 and two 2-heteroarylvinyl groups in positions 2 and 6 was accomplished. The pyrylium salts were obtained starting from pyranones and pyridines could be prepared from these salts by treating them with ammonium acetate. The general procedures for the synthesis of pyridinium salts, which occur with good results in less delocalized electronic systems, do not take place when applied to the above obtained pyrylium salts. Therefore, as starting material 4-(azulen-1-yl)-1-(n-butyl)-2,6-dimethylpyridinium perchlorate was used, which was condensed with heteroarylcarboxaldehydes. These compounds were completely characterized and some of their spectra were discussed. Their interaction with some metal ions was revealed, observing an affinity better than in the case of simple azulenepyridines. In the last part of the paper are presented redox potentials for several pyrylium salts and pyridines in comparison with those of the nonvinylogated derivatives.     

A dinuclear cobalt(II) complex of calix[8]arenes exibiting strong magnetic anisotropy

The solvothermal reaction of cobalt(II) acetate with p-tert-butylcalix[8]arene (calix) and triethylamine affords the compound (Et3NH)2 [CoII2(calix)2] (.2Et3NH) that shows a hydrogen bond bridged dinuclear complex [CoII2(calix)2]2- () with cobalt(II) ions in a tetrahedral geometry. The compound crystallises in the monoclinic, space group P2(1)/n with cell parameters a=14.89(1) A, b=20.90(2) A, c=30.87(4) A, beta=102.57(7) degrees, V=9376(16) A3, Z=2. The magnetic studies together with ab initio calculations are evidence of an important role of the geometry of the second coordination sphere of carbon and hydrogen atoms around the CoO4 core in quantifying the zero field splitting on cobalt sites. This results in strong magnetic anisotropies with a negative axial component on the cobalt fragments.     

Mechanisms of spin-mixing instabilities in antiferromagnetic molecular wheels

The microscopic theory of field-induced spin-mixing instabilities in antiferromagnetic molecular wheels CsFe8 is proposed. The basic features of magnetic torque measurements [O. Waldmann, Phys. Rev. Lett. 96, 027206 (2006)] are well explained by the interplay of three basic ingredients: the spin-mixing vibronic interaction with field-dependent vibronic constants, cooperative elastic interactions, and spin-mixing interactions independent from vibrations. The main contribution to spin mixing comes from second-order zero-field splitting mechanisms. At variance with previous interpretations, we find that the observed anomalies are not associated with a phase transition.     

A splitting method for the nonlinear Schrödinger equation

We introduce a splitting method for the semilinear Schrödinger equation and prove its convergence for those nonlinearities which can be handled by the classical well-posedness L²(Rd)-theory. More precisely, we prove that the scheme is of first order in the L²(Rd)-norm for H²(Rd)-initial data.