On the Regular Behavior of Entire Functions of Zero Order along Curves of Regular Rotation

Mathematical Notes - We study the relationship between the existence of an $$upsilon$$ -density $$Delta^gamma (alpha,beta)$$ of roots along the curves of regular rotation of an entire function...     

Effect of the chemical nature of the ionogenic groups of ion-exchange membranes on the size of the electroconvective instability region in high-current modes

The size of the electroconvective instability region on the membrane-solution boundary at currents exceeding the limiting diffusion current was measured by laser interferometry. The influence of the chemical nature of the ionogenic groups of ion-exchange membranes on the development of electroconvective instability was studied. The thickness of the electroconvection region decreased as the catalytic activity of the ionogenic groups of commercial and pilot membrane samples with respect to the heterolytic water dissociation increased. The maximum size of the electroconvective instability region and the minimum currents at which it was recorded for the anion-exchange membranes under study were determined for the highly basic modified anion-exchange membrane MA-41M with an almost completely suppressed water dissociation function. A correlation was found between the size of the convective instability region and the characteristic points on the current-voltage curves.     

Effect of an ion-exchange filler on the deionized water quality in electrodialysis

Desalination channels, containing an inert separator and a monolayer of ionites AV-17 and KU-2 taken in various volume ratios, are studied while maintaining concentrations of all solution components invariant. It is shown that the composition of the ion-exchange filler of the desalination channel affects the rate of transport of salt ions through relevant membranes, pH, and specific resistance of desalinated solution. The behavior of membrane systems in an overlimiting state is explained by using notions about different mechanisms governing the so-called overlimiting current through anion-exchange and cation-exchange membranes     

A generalization of the Lindelöf theorem

We present a generalization of the Lindelöf theorem on conditions that should be imposed on the coefficients of the Taylor series of an entire transcendental function ? in order that the relation $ln M_f (r) - \tau r^\rho , r \to \infty , M_f (r) = \max \left\{ {\left| {f(r)} \right|:|z| = r} \right\}$ , be satisfied.     

The evolution of longitudinal and transverse acoustic waves in a medium with paramagnetic impurities

We study the bipartial interaction of longitudinal and transverse acoustic pulses with a system of paramagnetic impurities with an effective spin S=1/2 in a crystalline layer or on a surface in the presence of an arbitrarily directed external constant magnetic field. We derive a new system of evolution equations that describes this interaction and show that, in the absence of losses, for equal phase velocities of these acoustic components, and under the condition of their unidirectional propagation, the original system reduces to a new integrable system of equations. The derived integrable system describes the pulse dynamics outside the scope of the slow-envelope approximation. For one of the reductions of the general model that corresponds to the new integrable model, we give the corresponding equations of the inverse scattering transform method and find soliton solutions. We investigate the dynamics and formation conditions of the phonon avalanche that arises when the initial completely or incompletely inverted state of the spin system decays. We discuss the application of our results to describing the interaction dynamics of spins and acoustic pulses in various systems with an external magnetic field.     

Integrable models for the dynamics of a longitudinal-transverse acoustic wave in a crystal with paramagnetic impurities

We theoretically study the evolution of longitudinal-transverse acoustic pulses propagating parallel to an external magnetic field in a system of resonant paramagnetic impurities with an effective spin S=1/2. For equal group velocities of the longitudinal and transverse waves, the pulse dynamics is shown to be described by evolution equations. In limiting cases, these equations reduce to equations integrable in terms of the inverse scattering transform method (ISTM). For the most general integrable system of equations that describes the dynamics of acoustic pulses outside the scope of the slow-envelope approximation, we derive the corresponding ISTM equations. These equations are used to find a soliton solution and a self-similar solution. The latter describes the leading edge of the packet of acoustic pulses generated when the initial unstable state of a spin system decays. Analysis of our solutions and models indicates that the presence of a longitudinal acoustic wave leads not only to a change in the amplitude and phase of the transverse wave but also to a qualitatively new dynamics of sound in such a medium.     

Electrochemical Parameters of Heterogeneous Bipolar Membranes: Dependence on the Structure and Nature of Monopolar Layers

The current–voltage curves, transport numbers of ions, and electrochemical impedance spectra are obtained for pilot samples of heterogeneous bipolar membranes synthesized from cation- and anion-exchange membranes. In the synthesis, the composition of the monopolar layers of membranes was varied or a catalyst was introduced into them. The voltage drop and the frequency spectra at the pilot membranes are close to those for industrial membrane MB-3. The transport numbers of co-ions in the pilot membranes are smaller and those of the hydrogen and hydroxyl ions are larger than in MB-3. The effective water dissociation constants in the pilot membranes, calculated from the impedance spectra, are close to those of MB-3.     

Electroconvection in systems with heterogeneous ion-exchange membranes

The results of studying the surface morphology of heterogeneous cation-(MK-40) and anion-exchange (MA-40) membranes and calculating the structure of electroconvective vortices generated by the electric body force are shown. The body force and its distribution are estimated by taking into account real parameters of the membrane surface morphology. The calculations of vortices were carried out by solving the Navier-Stokes equation with the no-slip boundary condition and the preset body force distribution. It is shown that the body force induced by the flowing current can generate pairs of electroconvective vortices (electroosmosis of the second kind), where the size of induced vortices is comparable with the intermembrane gap in electrodialysis cells.     

Electric mass transport of sodium chloride through cation-exchange membrane MK-40 in dilute sodium chloride solutions: A rotating membrane disk study

The polarization properties of an electromembrane system consisting of an MK-40 membrane and a dilute sodium chloride solution are investigated with an experimental apparatus, which includes a rotating membrane disk with a horizontally positioned membrane. For the electrochemical systems of MK-40/0.01 M NaCl and MK-40/0.001 M NaCl, effective ion transport numbers and partial current-voltage curves are determined for sodium and hydrogen ions, and limiting-current densities and the diffusion-layer thickness are calculated as functions of the rotation rate of the membrane disk. The space-charge distribution in the diffusion layer and in the membrane is calculated for various current densities and rotation rates of the membrane. It is shown that when electric-current densities are greater than the limiting value, ion fluxes of the salt increase as a result of a decrease in the effective thickness of the diffusion layer. This decrease is caused by the development of space charge, electroconvection, water dissociation, and the exaltation effect in the region near the membrane. It has been established that in dilute solutions the limiting current is not purely electrodiffusive in nature.     

Universal relationships for the phonon spectra in BCC,FCC, and HCP crystals with a short-range interatomic interaction

The frequencies of the phonon branches that correspond to the vibrations of the close-packed atomic planes in bcc, fcc, and hcp crystals with short-range interatomic interaction are shown to be described by a universal relationship, which only contains two parameters for each branch, for any polarization λ. These phonon branches correspond to the (ξ, ξ, 0) direction in bcc crystals, the (ξ, ξ, ξ) direction in fcc crystals, and the (0, 0, ξ) direction in hcp crystals. This universal relationship can only be violated by long-range interactions, namely, the interactions outside the sixth coordination shell in a bcc crystal, the fifth coordination shell in an fcc crystal, and the eleventh or tenth coordination shell in an hcp crystal. The effect of these long-range interactions for each phonon branch can be quantitatively characterized by certain parameters Δ _nλ, which are simply expressed in terms of the frequencies of three phonons of the branch. The values of these parameters are presented for all bcc, fcc, and hcp metals whose phonon spectra are measured. In most cases, the proposed relationships for the frequencies are found to be fulfilled accurate to several percent. In the cases where the Δ _nλ parameters are not small, they can give substantial information on the type and scale of long-range interaction effects in various metals.