On fractional Poincaré inequalities

We show that fractional (p, p)-Poincaré inequalities and even fractional Sobolev-Poincaré inequalities hold for bounded John domains, and especially for bounded Lipschitz domains. We also prove sharp fractional (1,p)-Poincaré inequalities for s-John domains.     

Ultrasonic degradation of aqueous carboxymethylcellulose: effect of viscosity, molecular mass, and concentration

It is well established that prolonged exposure of solutions of macromolecules to high-energy ultrasonic waves produces a permanent reduction in viscosity. It is generally agreed as well and also this study proved the hydrodynamic forces to have the primary importance in degradation. According to this study the sonolytic degradation of aqueous carboxymethylcellulose polymer or polymer mixtures is mainly depended on the initial dynamic viscosity of the polymer solution when the dynamic viscosity values are in the area range enabling intense cavitation. The higher was the initial dynamic viscosity the faster was the degradation. When the initial dynamic viscosities of the polymer solutions were similar the sonolytic degradation was dependent on the molecular mass and on the concentration of the polymer. The polymers with high molecular mass or high polymer concentration degraded faster than the polymers having low molecular mass or low polymer concentration. The initial dynamic viscosities were adjusted using polyethyleneglycol.     

Approach to Equilibrium for the Phonon Boltzmann Equation

We study the asymptotics of solutions of the Boltzmann equation describing the kinetic limit of a lattice of classical interacting anharmonic oscillators. We prove that, if the initial condition is a small perturbation of an equilibrium state, and vanishes at infinity, the dynamics tends diffusively to equilibrium. The solution is the sum of a local equilibrium state, associated to conserved quantities that diffuse to zero, and fast variables that are slaved to the slow ones. This slaving implies the Fourier law, which relates the induced currents to the gradients of the conserved quantities. Partially supported by the Belgian IAP program P6/02. Partially supported by the Academy of Finland.     

Glass‐embedded silver nanoparticle patterns by masked ion‐exchange process for surface‐enhanced Raman scattering

Glass‐embedded silver nanoparticle patterns were fabricated by masked silver–sodium ion‐exchange process followed by etching to reveal the particles for surface‐enhanced Raman scattering (SERS). The intensity of the enhanced Raman signal is comparable to that of the fluorescence, and the detection limit of 1 nM for Rhodamine 6G has been achieved. Raman images at different etching depths and corresponding morphological images are compared to find optimal SERS signal. Our results demonstrate that silver nanoparticle patterns embedded in glass can be used as SERS‐active substrates. Nanoparticles can be formed in a glass of high optical quality and have potential to be integrated with optical waveguides for a sensor chip. Copyright © 2010 John Wiley & Sons, Ltd.     

On Copper(I) Fluorides,the Cuprophilic Interaction,the Preparation of Copper Nitride at Room Temperature,and the Formation Mechanism at Elevated Temperatures

Our attempts to synthesize the hitherto unknown binary copper(I) fluoride have led to first successes and a serendipitious result: By conproportionation of elemental copper and copper(II) fluoride in anhydrous liquid ammonia, two copper(I) fluorides were obtained as simple NH₃ complexes. One of them presents an example of ligand‐unsupported “cuprophilic” interactions in an infinite [Cu₂(NH₃)₄]²⁺ chain with alternating Cu–Cu distances. We discovered that both copper(I) fluorides can easily be converted into Cu₃N at room temperature, just by applying a vacuum. Additionally, we investigated the formation mechanism of the classical synthesis route of Cu₃N that starts with CuF₂ and flowing NH₃ in the temperature range between ambient and 290 °C by means of thermal analysis and in situ neutron diffraction. The reaction proceeds at elevated temperatures through the formation of a blue and amorphous ammoniate Cu(NH₃)₂F₂, the reformation of CuF₂, and finally the redox reaction to form Cu₃N.     

The [U2F12]2− Anion of Sr[U2F12]

The D_2h‐symmetric dinuclear complex anion [U₂F₁₂]^2? of pastel green Sr[U₂F₁₂] shows a hitherto unknown structural feature: The coordination polyhedra around the U atoms are edge‐linked monocapped trigonal prisms, the U^V atoms are therefore seven‐coordinated. This leads to a U–U distance of 3.8913(6) Å. A weak U^V–U^V interaction is observed for the dinuclear [U₂F₁₂]^2? complex and described by the antiferromagnetic exchange J_exp of circa ?29.9 cm^?1. The crystalline compound can be easily prepared from SrF₂ and β‐UF₅ in anhydrous hydrogen fluoride (aHF) at room temperature. It was studied by means of single crystal X‐ray diffraction, IR, Raman and UV/VIS spectroscopy, magnetic measurements, and by molecular as well as by solid‐state quantum chemical calculations.     

Icosahedral Au(72): a predicted chiral and spherically aromatic golden fullerene

Quantum chemical calculations demonstrate the spherical aromaticity and high thermodynamic stability of Au(72), a predicted I-symmetric golden fullerene.