On bifurcation and symmetry of solutions of symmetric nonlinear equations with odd-harmonic forcings

In this work we study existence, bifurcation, and symmetries of small solutions of the nonlinear equation Lx = N(x, p, epsilon) + mu f, which is supposed to be equivariant under the action of a group OHm, and where f is supposed to be OHm-invariant. We assume that L is a linear operator and N(., p, epsilon) is a nonlinear operator, both defined in a Banach space X, with values in a Banach space Z, and p, mu, and epsilon are small real parameters. Under certain conditions we show the existence of symmetric solutions and under additional conditions we prove that these are the only feasible solutions. Some examples of nonlinear ordinary and partial differential equations are analyzed. (C) 1995 Academic Press, Inc.

Small angle X-ray scattering and IR spectroscopy study of metal carbonyl complexes immobilized on a silica gel surface chemically modified with piperazine

The carbonyl complexes [WCl(CO)(3)(bipy) (HgCl)] (1), [Fe(CO)(4)(HgCl)(2)] (2) and W(CO)(6)] (3) were immobilized on a silica gel surface organofunctionalized with piperazine groups. The products obtained were studied by IR spectroscopy and small angle X-ray scattering (SAXS) techniques. The IR data show that the immobilization of heterobimetallic compounds 1 and 2, on the functionalized surface, occurred through the mercury atom, while for 3 the displacement of one CO group by the nitrogen of a piperazine molecule was observed. The data obtained from SAXS indicate that particles have a uniform size and reveal suitable modifications on the functionalized surface after immobilization of metal carbonyl complexes. The average intermolecular distance (l(ij)) for piperazine ligands on support is 8.7 Angstrom, for the metal carbonyl complex 1 it is 18.8 Angstrom, for complex 2 it is 16.2 Angstrom and for complex 3 it is 15.3 Angstrom. Copyright (C) 1996 Elsevier B.V. Ltd

Small-angle X-ray scattering study of the smart thermo-optical behavior of zirconyl aqueous colloids

The smart thermo-optical systems studied here are based on the unusual thermoreversible sol-gel transition of zirconyl chloride aqueous solution modified by sulfuric acid in the molar ratio Zr/SO4:3/1. The transparency to the visible light changes during heating due to light scattering. This feature is related to the aggregates growth that occurs during gelation. These reversible changes can be controlled by the amount of chloride ions in solution. The thermoreversible sol-gel transition temperature increases from 323 to 343 K by decreasing the molar ratio Cl/Zr from 7.0 to 1.3. In this work the effect of the concentration of chloride ions on the structural characteristics of the system has been analyzed by in situ SAXS measurements during the sol-gel transition carried out at 323 and 333 K. The experimental SAXS curves of sols exhibit three regions at small, medium and high scattering vectors characteristics of Guinier, fractal and Porod regimes, respectively. The radius of primary particles, obtained from the crossover between the fractal and Porod regimes, remains almost invariable with the chloride concentration, and the value (4 Angstrom) is consistent with the size of the molecular precursor. During the sol-gel transition the aggregates grow with a fractal structure and the fractal dimensionality decreases from 2.4 to 1.8. This last value is characteristic of a cluster-cluster aggregation controlled by a diffusion process. Furthermore, the time exponent of aggregate growth presents values of 0.33 and 1, typical of diffusional and hydrodynamic motions. A crossover between these two regimes is observed.

Small-angle X-ray and nuclear-magnetic resonance study of siloxane-PMMA hybrids prepared by the sol-gel process

Transparent siloxane-polymethylmethacrylate (PMMA) hybrids were synthesized by the sol-gel process through hydrolysis of methacryloxyproyltrimethoxysilane (TMSM), tetramethoxysilane (TMOS) and polymerization of methylmethacrylate (MMA) using benzol peroxide (BPO) as catalyst. These composites have a good chemical stability due to the presence of covalent bonds between the inorganic (siloxane) and organic (PMMA) phases. The effects of siloxane content, pH of the initial sol and BPO content on the structure of the dried gels were analyzed by small-angle X-ray scattering (SAXS). SAXS results revealed the presence of an interference (or correlation) peak at medium q-range for all compositions, suggesting that siloxane groups located at the ends of PMMA chains form isolated clusters that are spatially correlated. The average intercluster distance - estimated from the q-value corresponding to the maximum in SAXS spectra - decreases for samples prepared with increasing amount of TMSM-TMOS. This effect was assigned to the expected increase in the number density of siloxane groups for progressively higher siloxane content. The increase of BPO content promotes a more efficient polymerization of MMA monomers but has no noticeable effect on the average intercluster distance. High pH favors polycondensation reactions between silicon species of both TMOS and TMSM silicon alcoxides, leading to a structure in which all siloxane clusters are bonded to PMMA chains. This effect was confirmed by Si-29 nuclear-magnetic resonance (NMR) measurements.

Bifurcation of limit cycles from a centre in R-4 in resonance 1:N

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Small angle X-ray scattering study of structural changes in silica gel modified with organofunctional groups

Silica gel surfaces, organofunctionalized with 2-mercaptobenzimidazole, iminosalicylaldehyde and imidazole groups were examined using the small angle X-ray scattering technique (SAXS). From the scattering intensity data it was concluded that particles have a uniform size after the coupling reaction. The chemical treatment of the silica gel leads to an attachment of the organofunctional groups on the solid-pore interface of the silica with an increase of the mean size of the solid phase and some coalescence of the pores. © 1989.

Experimental analysis of heat recoveries with heat pipes operating in low inclination angle

Heat recovery devices are important in the optimization of thermal systems, since they can be used to reduce thermal losses to the environment. The use of heat pipes in these types of equipment can provide heat recoveries of higher efficiency, since both fluid flows are external and there are less contamination risks between the hot and cold fluids. The objective of this work is to study a heat recovery unit constructed with heat pipes and mainly, to analyze the influence of the inclination of the heat pipes on the performance of the equipment. For this analysis, a heat recovery unit was constructed which possesses 48 finned heat pipes in triangular geometry, the evaporator and condenser being of the same length. This unit was tested in an air-air system simulating a heat recovery process in which heat was supplied to the hot fluid by electrical resistances. The results have shown that there exists an inclination at which the system has a better performance, but for higher inclinations there is no significant increase of the efficiency of the system. This paper also presents the influence of inclination of heat pipes on effectiveness and NTU parameters which are important in heat exchanger design.

Pore structure characterization of kaolin, metakaolin, and their acid-treated products using small-angle X-ray scattering

Pore structure of dealuminated kaolin and metakaolin was studied by small-angle X-ray scattering (SAXS). Both parent kaolin and metakaolin have about 10% of the total pore volume provided by globular pores with 105 Å mean pore size. Their surface area is about 14 m2/g. Acid dealumination of kaolin causes an increase of its globular pore volume without an appreciable change in the mean pore size, its surface area increasing up to about 90 m2/g. Acid dealumination of metakaolin enhances the globular pore volume, although there is generation of slit-shaped pores with a narrow thickness distribution whose mean value is 14 Å. This interlayer spacing causes an increase in surface area of about 190 m2/g by SAXS. © 1994.