Topological degree of a family of convex functions and localization of nash equilibrium points

We define the degree of a family of convex functionsf _x, wherex , and prove that, if this degree is different from zero, one has at least one Nash equilibrium point in . This is a criterion of localization of such equilibrium points.     

Complexation of Ni, Cu, Zn, and Cd by DOC in some metal-impacted freshwater lakes: a comparison of approaches using electrochemical determination of free-metal-ion and labile complexes and a computer speciation model, WHAM V and VI

Complexation of Ni(II), Cu(II), Zn(II), and Cd(II) by dissolved organic carbon (DOC) in some freshwater lakes in Rouyn-Noranda, Québec, Canada, where they were impacted by effluents from a nearby copper smelter, was measured by kinetic and equilibrium methods using cathodic and anodic stripping voltammetry. The measured free-metal-ion and labile metal-complex concentrations were compared with the predictions made by a widely-used computer speciation model, the Windermere Humic Aqueous Model (WHAM): WHAM V and its improved version WHAM VI. If it is assumed that 65% of the DOC is “active”, i.e. behaving as isolated humic substances such as fulvic acid, both versions of WHAM are able to predict the labile and free-metal-ion concentrations of Ni, Zn, and Cd reasonably well; however, both underestimate the free-copper-ion concentration by one to two orders of magnitude. WHAM VI is generally better than or equal to WHAM V for successfully predicting most of the free-metal-ion concentrations. The modelled competition by Al(III) and Fe(III) in the lake surface waters showed that in most cases Cu(II) was most affected by this competition. WHAM VI predicts a larger effect from the Al(III) and Fe(III) competition than does WHAM V.     

Study of poly(bisphenol A carbonate) relaxation kinetics at the glass transition temperature

In this work, the variations of the relaxation times are investigated above and below the glass transition temperature of a model amorphous polymer, the polycarbonate. Three different techniques (calorimetric, dielectric and thermostimulated currents) are used to achieve this goal. The relaxation time at the glass transition temperature was determined at the temperature dependence convergence of the relaxation times calculated with dynamic dielectric spectroscopy (DDS) for the liquid state and thermostimulated depolarisation currents (TSDC) for the vitreous state. We find a value of τ(T_g) = 110 s for PC samples. The knowledge of the temperature dependence, τ(T), and the value τ(T_g) enables to determine the glass-forming liquid fragility index, m. We find m = 178 ± 5.     

Phase equilibria in polymer/solvent systems. Part V: Thermodynamic theory of the swelling pressure equilibrium of a crosslinked substance with a solvent in various phases

A thermodynamic theory has been developed to define the swelling pressure equilibrium between a homogeneous gel and a pure solvent, where phase transitions of the solvent, such as evaporation and crystallization can occur. It is shown that the equilibrium curve, which describes the temperature dependence of the composition in the gel phase under the condition of a constant swelling pressure, has distinct bends at the transition temperatures. These bends are related to the enthalpies of transition of the pure solvent at the transition temperatures. As a consequence of the phase transition of the solvent the swelling pressure-temperature curve at constant composition of the gel shows a discontinuous behavior at the transition point. Numerical calculations with a modified Flory-Huggins expression, based on results of swelling and deswelling measurements of the system crosslinked PEG/water, are presented.The discussion includes natural systems, which are in the gel state, where water may crystallize in the extracellular space.     

Dynamic kinetic resolution of secondary alcohols with a readily available ruthenium-based racemization catalyst

An easy to handle and stable racemization catalyst for secondary alcohols is obtained by an in situ mixture of readily available [Ru(cymene)Cl₂]₂ with chelating aliphatic diamines. Optimization of the reaction revealed that N,N,N′,N′-tetramethyl-1,3-propanediamine as ligand racemizes aromatic alcohols completely within 5 h. This easy to handle and stable catalytic system is combined with a lipase-catalyzed resolution to provide an efficient dynamic kinetic resolution of secondary alcohols.     

Application of a dynamic method of minimisation in the study of reaction surfaces

This paper describes the application of Snyman's dynamic minimisation method to a fitted potential surface of H₃. Comparisons are made with conventional algorithms. A method is described to extend Snyman's method so that it will find only a particular kind of stationary point. It is emphasized that this method enables saddle points to be found without having to resort to approaches based on trial and error.     

On the determination of dynamic surface tensions by means of a modified bubble pressure method

A method for the determination of the dynamic surface tension of surfactant solutions is presented which allows to cover adsorption times down to 10 seconds. This method is based on the determination of the pressure inside two communicating bubbles. There is no deformation of the solution/air interface during the experimental procedure. Hence, in evaluating the kinetic data no surface area enlargement has to be taken into account. An automatically operating procedure should allow to cover adsorption times down to approximately one second and should improve the measuring accuracy substantially. Experimental investigations with aqueous n-decanoic acid solutions using the method proposed provided evidence that decanoic acid is adsorbed by a diffusioncontrolled mechanism.     

Growth of silica nanoparticles in methylmethacrylate-based water-in-oil microemulsions

The mechanism of silica particle formation in monomer microemulsions is studied using dynamic light scattering (DLS), atomic force microscopy, small-angle X-ray scattering (SAXS), and conductivity measurements. The hydrolysis of tetraethylorthosilicate (TEOS) in methylmethacrylate (MMA) microemulsions (MMA = methylmethacrylate) is compared with the formation of SiO₂ particles in heptane microemulsions. Stable microemulsions without cosurfactant were found for MMA, the nonionic surfactant Marlophen NP10, and aqueous ammonia (0.75 wt%). In the one-phase region of the ternary phase diagram, the water/surfactant ratio (R _w) could be varied from 6 to 18. The DLS and SAXS measurements show that reverse micelles form in these water-in-oil (w/o) microemulsions. The minimum water-to-surfactant molar ratio required for micelle formation was determined. Particle formation is achieved from the base-catalyzed hydrolysis of TEOS. According to atomic force microscopy measurements of particles isolated from the emulsion, the particle size can be effectively tailored in between 20 and 60 nm by varying R _w from 2–6 in heptane w/o microemulsions. For MMA-based microemulsions, the particle diameter ranges from 25 to 50 nm, but the polydispersity is higher. Tailoring of the particle size is not achieved with R _w, but adjusting the particle growth period produces particles between 10 and 70 nm.     

Application of a chemical equilibrium model to thermodynamic functions of transfer of alcohols from water to aqueous surfactants

In ternary aqueous solutions, hydrophobic solutes such as alcohols tend to aggregate with surfactants to form mixed micelles. These systems can be studied by meas of the functions of transfer of hydrophobic solutes from water to aqueous solutions of surfactant. These thermodynamic functions often go through extrema in the critical micellar concentration (CMC) region of the surfactant. A simple model based on interactions between surfactant and hydrophobic solute monomers, on the distribution of the hydrophobic solute between water and the micelles and on the shift in the CMC induced by the hydrophobic solute, can simulate the magnitude and trends of the transfer functions using parameters which are mostly derived from the binary systems. In order to check the model more quantitatively, volumes and heat capacities of transfer of alcohols from water to aqueous solutions of a nonionic surfactant, octyldimethylamine oxide, were measured. A quantitative agreement was achieved with three adjustable parameters. Good fits are also obtained for the transfers to the ionic surfactants, octylamine hydrobromide and sodium dodecylsulfate. When the equilibrium displacement contribution is small, the distribution constants and the partial molar properties of the alcohols in the micellar phase agree well with the parameters obtained with similar models.     

Synthesis, characterization and Schlenk equilibrium studies of methylmagnesium compounds with O- and N-donor ligands - the unexpected behavior of [MgMeBr(pmdta)] (pmdta = N,N,N′,N″,N″-pentamethyldiethylenetriamine)

MgMe₂ (1) was found to react with 1,4-diazabicyclo[2.2.2]octane (dabco) in tetrahydrofuran (thf) yielding a binuclear complex [{MgMe₂(thf)}₂(μ-dabco)] (2). Furthermore, from reactions of MgMeBr with diglyme (diethylene glycol dimethyl ether), NEt₃, and tmeda (N,N,N′,N′-tetramethylethylenediamine) in etheral solvents compounds MgMeBr(L), (L = diglyme (5); NEt₃ (6); tmeda (7)) were obtained as highly air- and moisture-sensitive white powders. From a thf solution of 7 crystals of [MgMeBr(thf)(tmeda)] (8) were obtained. Reactions of MgMeBr with pmdta (N,N,N′,N″,N″-pentamethyldiethylenetriamine) in thf resulted in formation of [MgMeBr(pmdta)] (9) in nearly quantitative yield. On the other hand, the same reaction in diethyl ether gave MgMeBr(pmdta) · MgBr₂(pmdta) (10) and [{MgMe₂(pmdta)}₇{MgMeBr(pmdta)}] (11) in 24% and 2% yield, respectively, as well as [MgMe₂(pmdta)] (12) as colorless needle-like crystals in about 26% yield. The synthesized methylmagnesium compounds were characterized by microanalysis and ¹H and ¹³C NMR spectroscopy. The coordination-induced shifts of the ¹H and ¹³C nuclei of the ligands are small; the largest ones were found in the tmeda and pmdta complexes. Single-crystal X-ray diffraction analyses revealed in 2 a tetrahedral environment of the Mg atoms with a bridging dabco ligand and in 8 a trigonal-bipyramidal coordination of the Mg atom. The single-crystal X-ray diffraction analyses of [MgMe₂(pmdta)] (12) and [MgBr₂(pmdta)] (13) showed them to be monomeric with five-coordinate Mg atoms. The square-pyramidal coordination polyhedra are built up of three N and two C atoms in 12 and three N and two Br atoms in 13. The apical positions are occupied by methyl and bromo ligands, respectively. Temperature-dependent ¹H NMR spectroscopic measurements (from 27 to −80 °C) of methylmagnesium bromide complexes MgMeBr(L) (L = thf (4); diglyme (5); NEt₃ (6); tmeda (7)) in thf-d₈ solutions indicated that the deeper the temperature the more the Schlenk equilibria are shifted to the dimethylmagnesium/dibromomagnesium species. Furthermore, at −80 °C the dimethylmagnesium compounds are predominant in the solutions of Grignard compounds 4-6 whereas in the case of the tmeda complex7 the equilibrium constant was roughly estimated to be 0.25. In contrast, [MgMeBr(pmdta)] (9) in thf-d₈ revealed no dismutation into [MgMe₂(pmdta)] (12) and [MgBr₂(pmdta)] (13) even up to −100 °C. In accordance with this unexpected behavior, 1:1 mixtures of 12 and 13 were found to react in thf at room temperature yielding quantitatively the corresponding Grignard compound 9. Moreover, the structures of [MgMeBr(pmdta)] (9c), [MgMe₂(pmdta)] (12c), and [MgBr₂(pmdta)] (13c) were calculated on the DFT level of theory. The calculated structures 12c and 13c are in a good agreement with the experimentally observed structures 12 and 13. The equilibrium constant of the Schlenk equilibrium (2 9c ? 12c + 13c) was calculated to be K_gas = 2.0 × 10⁻³ (298 K) in the gas phase. Considering the solvent effects of both thf and diethyl ether using a polarized continuum model (PCM) the corresponding equilibrium constants were calculated to be K_thf = 1.2 × 10⁻³ and K_ether = 3.2 × 10⁻³ (298 K), respectively.