How Fast can a Subharmonic Function Decay Along a Ray?

Much is known about the connection between the growth and decayof subharmonic functions. The results indicate that there isa general principle: asubharmonic function cannot decay ‘toofast’ relative to its growth.Three theorems are provedwhich, together with work previously published elsewhere, givea fairly complete account of how this principle works out fora subharmonic function having extremal decay along a ray. 1991Mathematics Subject Classification: 30D20, 31A05.     

The influence of ionic and nonionic surfactants on aggregative stability and electrical surface properties of aqueous suspensions of titanium dioxide

The influence of concentration of nonionic TRITON X-100 and anionic ATLAS G-3300 surfactants, and pH of medium on the size and zeta-potential of TiO2 particles in the water suspensions has been studied. Suspensions have been prepared by mixing of the titanium dioxide in the suitable mediums at 10 min and 6 h correspondingly. It was established, that the duration of mixing of the suspensions has an essential influence on the dependence of zeta-potential and size of particles versus concentration of surfactant. However, the duration of mixing does not influence the dependence of electrical conductivity and pH of the suspensions on concentration of surfactant. It is shown that anionic ATLAS G-3300 surfactant is more effective stabilizator of aqueous suspensions of titanium dioxide, than nonionic surfactants of TRITON X-100. It is found that hydrophobic interaction has important role in the processes of stabilization of suspensions for nonionic surfactant, and for anionic surfactant--moving of psi(delta)-planes into solution's depth.     

Hydrophobic interactions between polymethacrylic acid and sodium laureth sulfate in aqueous solutions

The role of hydrophobic interaction in the development of associative processes is demonstrated, based on the concentration dependences of the viscosity and pH of binary solutions of polymethacrylic acid as an anionic polyelectrolyte and sodium laureth sulfate as an anionic surfactant. It is found that the inflection point on the dependence of the difference between the pH values of binary solutions of polymethacrylic acid and sodium laureth sulfate on the polyelectrolyte concentration is a criterion for determining the predominant contribution from hydrophobic interaction, as is the inflection point on the dependence of pH of individual solutions of polymethacrylic acid on the polyelectrolyte concentration.     

Nature of some features in Raman spectra of hydroxyapatite‐containing materials

A very broad vibrational band ranging from 1000 up to 4000 cm⁻¹ and two relatively sharp bands at 5000 and 5027 cm⁻¹ are found in the Raman scattering spectrum of hydroxyapatite‐containing films obtained by gas detonation spray method. We developed a theoretical model that interprets the broad band as a result of strong interaction between the high‐frequency hydrogen bond vibrations and lattice phonons. Both sharp bands around 5000 cm⁻¹ are assigned to the overtones of v‐OH vibrations. Copyright © 2016 John Wiley & Sons, Ltd.     

Effects of surfactants and pH of medium on zeta potential and aggregation stability of titanium dioxide suspensions

The effects of benzethonium chloride, sodium dodecylbenzenesulfonate, and 4-(1,1,3,3-(tetramethylbutyl)phenyl poly(ethylene glycol) on the zeta potential and aggregation stability of aqueous rutile-form titanium dioxide suspensions are studied in the pH range of 2–12. It is shown that the nonionic surfactant does not affect significantly the zeta potential and aggregation stability of the suspensions. The influence of ionic surfactants on the aggregation stability of the suspensions considerably depends on the pH of a medium. At pH values above the isoelectric point of titanium dioxide suspensions (pH₀ = 6.2), the suspensions demonstrate a high aggregation stability in the presence of the anionic surfactant, sodium dodecylbenzene-sulfonate (irrespective of its content), while, at pH < pH₀, the aggregation stability of the suspensions markedly increases with the surfactant concentration. In the presence of the cationic surfactant, benzethonium chloride, the aggregation stability of the suspensions is independent of the surfactant concentration at pH < pH₀, whereas, at pH > pH₀, it increases with the surfactant concentration.     

Fermi resonance and strong anharmonic effects in the absorption spectra of the ν-OH (ν-OD) vibration of solid H- and D-benzoic acid

The vibrational spectra of polycrystalline benzoic acid (BA) and its deuterated derivative were studied over the wide frequency region 4000–10 cm⁻¹ by IR and Raman methods. A theoretical analysis of the hydrogen bond frequency region and calculations at the B3LYP/6-311++G(2d, 2p) level for the benzoic acid cyclic dimer in the gas phase were made. In order to study the dynamics of proton transfer two formalisms were applied: Car–Parrinello Molecular Dynamics (CPMD) and Path Integrals Molecular Dynamics (PIMD). It was shown that the experimentally observed very broad ν-OH band absorption is the result of complex anharmonic interaction: Fermi resonance between the OH-stretching and bending vibrations and strong interaction of the ν-OH stretching with the low frequency phonons. The theoretical analysis in the framework of such an approach gave a good correlation with experiment. From the CPMD calculations it was confirmed that the O–HO bridge is not rigid, with the OO distance being described by a large amplitude motion. For the benzoic acid dimer we observed stepwise (asynchronous) proton transfer.