Quasistatic simulation of microwave discharges in a spherically symmetric electrode system in nitrogen

Results are presented from one-dimensional quasistatic simulations of steady microwave discharges in a spherically symmetric electrode system in nitrogen at pressures of 1–8 Torr. The computational model includes the equation for calculating the electric field strength in the quasistatic approximation, Poisson’s equation, the balance equations describing the kinetics of charged and neutral plasma particles, and the time-independent homogeneous Boltzmann equation for electrons. The processes involving vibrationally excited particles are taken into account by the familiar analytic expression for the vibrational distribution of molecules in the diffusion approximation. It is shown that, because of the electric field nonuniformity, the physical properties (in particular, the plasma ion composition) are different in different discharge regions.     

Self-consistent modeling of radio-frequency plasma generation in stellarators

A self-consistent model of radio-frequency (RF) plasma generation in stellarators in the ion cyclotron frequency range is described. The model includes equations for the particle and energy balance and boundary conditions for Maxwell’s equations. The equation of charged particle balance takes into account the influx of particles due to ionization and their loss via diffusion and convection. The equation of electron energy balance takes into account the RF heating power source, as well as energy losses due to the excitation and electron-impact ionization of gas atoms, energy exchange via Coulomb collisions, and plasma heat conduction. The deposited RF power is calculated by solving the boundary problem for Maxwell’s equations. When describing the dissipation of the energy of the RF field, collisional absorption and Landau damping are taken into account. At each time step, Maxwell’s equations are solved for the current profiles of the plasma density and plasma temperature. The calculations are performed for a cylindrical plasma. The plasma is assumed to be axisymmetric and homogeneous along the plasma column. The system of balance equations is solved using the Crank-Nicholson scheme. Maxwell’s equations are solved in a one-dimensional approximation by using the Fourier transformation along the azimuthal and longitudinal coordinates. Results of simulations of RF plasma generation in the Uragan-2M stellarator by using a frame antenna operating at frequencies lower than the ion cyclotron frequency are presented. The calculations show that the slow wave generated by the antenna is efficiently absorbed at the periphery of the plasma column, due to which only a small fraction of the input power reaches the confinement region. As a result, the temperature on the axis of the plasma column remains low, whereas at the periphery it is substantially higher. This leads to strong absorption of the RF field at the periphery via the Landau mechanism.     

Study of Bound Water of Poly-Adenine Using High Frequency Dielectric Measurements

The high frequency dielectric constant of poly-adenine (poly-A) was measured between 1 MHz and 1 GHz. The purpose of these experiments was to investigate the state of water molecules that are bound to the charged groups of the poly-A molecule. Analysis of the data using the Maxwell's mixture equation revealed the dielectric constant of bound water higher than we expected. Using Onsager's internal field in Debye's equation, we calculated the dielectric constant of water in the vicinity of a charged ion. The result of this computation demonstrates that the dielectric constant of bound water is much smaller than the normal value only in the immediate proximity of charged ions (within 2 Å). The dielectric constant increases rapidly to the normal value as the distance increases from 2 to 4 Å. This observation indicates that charged sites of polyions have only short range interactions with the surrounding water molecules. However, this conclusion pertains only to rotary diffusion of bound water since dielectric measurement is unable to detect translational diffusion.     

Theory of heat transport in a magnetized high-temperature plasma

The transport of charged particles across a strong magnetic field with a small random component is studied in the double diffusion approximation. It is shown that the density of the particles whose initial distribution is stretched along the field satisfies a subdiffusion equation with fractional derivatives. A more general initial particle distribution is also considered, and the applicability of the solutions obtained is discussed.     

Planar,cylindrical and spherical electrical double layers in biological systems. The effect of counterion size

The effect of counterion size on the electrical properties of an electrolyte solution in contact with charged planar, cylindrical and spherical surfaces is considered. Electrostatic interaction is considered by means of the mean electrostatic field, while the finite size of particles constituting the electrolyte solution is considered via the excluded volume effect within the lattice statistics. Different sizes of counterion are described by different values of the lattice constant. It is shown that the excluded volume effect considerably decreases the calculated number density of counterions near the charged surface. This effect is more pronounced in cylindrical geometry than in spherical geometry, and less pronounced than in planar geometry.     

An Ion Displacement Membrane Model

The usual assumption in treating the diffusion of ions in an electric field has been that the movement of each ion is independent of the movement of the others. The resulting equation for diffusion by a succession of spontaneous jumps has been well stated by Parlin and Eyring. This paper will consider one simple case in which a different assumption is reasonable. Diffusion of monovalent positive ions is considered as a series of jumps from one fixed negative site to another. The sites are assumed to be full (electrical neutrality). Interaction occurs by the displacement of one ion by another. An ion leaves a site if and only if another ion, not necessarily of the same species, attempts to occupy the same site. Flux ratios and net fluxes are given as functions of the electrical potential, concentration ratios, and number of sites encountered in crossing the membrane. Quantitative comparisons with observations of Hodgkin and Keynes are presented.     

Kinetic theory of the positive column of a low-pressure discharge in a transverse magnetic field

The influence of a transverse magnetic field on the characteristics of the positive column of a planar low-pressure discharge is studied theoretically. The motion of magnetized electrons is described in the framework of a continuous-medium model, while the ion motion in the ambipolar electric field is described by means of a kinetic equation. Using mathematical transformations, the problem is reduced to a secondorder ordinary differential equation, from which the spatial distribution of the potential is found in an analytic form. The spatial distributions of the plasma density, mean plasma velocity, and electric potential are calculated, the ion velocity distribution function at the plasma boundary is found, and the electron energy as a function of the magnetic field is determined. It is shown that, as the magnetic field rises, the electron energy increases, the distributions of the plasma density and mean plasma velocity become asymmetric, the maximum of the plasma density is displaced in the direction of the Ampère force, and the ion flux in this direction becomes substantially larger than the counter-directed ion flux.     

Electric field-induced concentration gradients in planar supported bilayers.

A simple method of generating electric field-induced concentration gradients in planar supported bilayers has been developed. Gradients of charged, fluorescently labeled probes were visualized by epifluorescence microscopy and could be observed at field strengths as low as 1 V/cm. Steady-state concentration gradients can be described by a simple competition between random diffusion and electric field-induced drift. A model based on this principle has been used to determine the diffusion coefficient of the fluorescent probes. This technique achieves a degree of electrical manipulation of supported bilayers that offers a variety of possibilities for the development of new molecular architectures and the study of biological membranes.     

Kinetic Theory Model for Ion Movement through Biological Membranes: III. Steady-State Electrical Properties with Solution Asymmetry

An electrodiffusion model for plasma membrane ion transport, which takes into account the influence of high electric field strengths and ion-membrane molecule interactions, is presented and analyzed. A generalized Nernst-Planck equation for steady-state situations is derived which has electric field-dependent mobility and diffusion coefficients. Under the assumption of a constant electric field within the membrane, this equation is integrated to give a more general form of the Goldman equation. Based on this equation numerical computations of ionic chord conductance as a function of applied electric field strength were carried out for several permeant ion concentration ratios. The model is capable of yielding significantly larger rectification ratios than is the Goldman equation. Further, high field asymptotes to the current vs. electric field strength curve do not generally intersect at the origin.     

Selective Inhibition of Absorption and Long Distance Transport in Relation to the Dual Mechanisms of Ion Absorption in Maize Seedlings

The influence of several uncouplers of oxidative phosphorylation and inhibitors of terminal electron transport was studied on absorption and long distance transport of both K and C1 at concentrations within each range of the dual isotherm typical of ion uptake by maize roots. At low concentrations in the range of system 1, the system considered to implement ion movement through the plasma membrane, root absorption and long distance transport are equally inhibited by a given inhibitor. In the high range of system 2, the system considered to mediate ion passage through the tonoplast, long distance transport is markedly less sensitive to inhibitors than is absorption. The observations are in accord with the hypothesis that only system 1 is involved in the uptake of ions from the external solution into the symplast, and hence into the xylem. At high concentrations, entrance into the symplasm is deemed to be largely by diffusion and therefore less inhibitor sensitive.With respect to absorption by the roots, the plasma membrane system is more inhibitor sensitive than is the tonoplast system. It is suggested that the difference in sensitivity is real, and not the consequence of an inequality of inhibitor concentration in the vicinity of the plasma membrane and tonoplast respectively.     

Theory for carrier-mediated zero-current conductance of bilayers extended to allow for nonequilibrium of interfacial reactions,spatially dependent mobilities and barrier shape

Summary A generalized form of the electrodiffusion equation, allowing for any shape of symmetrical energy barrier and any spatial dependence of the diffusion coefficient, is used to deduce theoretically the carrier-mediated conductance for thin (e.g., bilayer) membranes in the limit of low applied current. Both the Nernst-Planck and the Eyring single-barrier treatments are special cases of this more general approach, which allows for the effect of non-uniform properties of the lipid and non-uniform profiles of the forces acting within the membrane interior. Two independent mechanisms for ions to cross the membrane-solution interfaces are considered; namely, (1) the reaction at the interface between ions from solution and carriers from the membrane, and (2) the partition across the interfaces of complexes already formed in the solution. The rates of these reactions are taken into account using the rate equations of chemical kinetics; and the Poisson-Boltzmann equation is integrated in the aqueous solutions to evaluate the effect of charged polar head groups of the lipid. The analysis leads to an expression for the conductance, which, in the approximation of constant field, is an explicit function of such experimentally variable parameters as the concentrations and types of permeant ions and carriers in the aqueous phases, the total ionic strength and the nature of the polar head groups of the lipid. The functional relationship observable in an unknown membrane can, in principle, enable one to deduce such information as the mechanism of ion permeation across the interfaces, the magnitude of the surface charge, and the degree of ion-carrier complexation in the aqueous solutions.     

Effects of static magnetic fields on diffusion in solutions

Static magnetic fields affect the diffusion of biological particles in solutions through the Lorentz force and Maxwell stress. These effects were analyzed theoretically to estimate the threshold field strength for these effects. Our results show that the Lorentz force suppresses the diffusion of charged particles such as Na+, K+, Ca2+, Cl-, and plasma proteins. However, the threshold is so high, i.e., more than 10(4) T, that the Lorentz force does not affect the ion diffusion at typical field strengths (a few Tesla at most). Since the threshold of gradient fields for producing a change in ion diffusion through the Maxwell stress is more than 10(5) T2/m for paramagnetic molecules (FeCl3, O2) and plasma proteins, their diffusion would be unaffected by typical gradient fields (100 T2/m at most) and even by high gradient fields (less than 10(5) T2/m) used in magnetic separation techniques. In contrast, movement of deoxygenated erythrocytes and FeCl3 colloids (more than 10(3) molecules) is influenced by the usual gradient fields due to a volume effect.     

Classical transport in a field reversed configuration

The transport of charged particles in a field reversed configuration (FRC) was previously considered to be turbulent because it is much faster than classical predictions. Classical transport has mainly been developed for plasmas in which the gyroradii of particles are small compared to the scale lengths of the variation of the density and magnetic field. This assumption is quite inappropriate for an FRC where the magnetic field vanishes on a surface within the plasma. Classical theory has been extended to include large ion gyroradii. A classical loss-cone process is revealed that is consistent with the transport experiments in which the ion gyroradii were comparable in size to the plasma radius.     

Physico-chemical model of a voltage-gated sodium channel

A permeant ion is known to create in the channel pore a local electrical field, the intensity of which exceeds the intensity of an electrical field produced by the membrane potential. In our study, we consider a sodium channel model, in which the effects of a permeant ion, an inactivating particle, and pharmacological agents on mobile charged groups of the channel are semi-phenomenologically taken into account by using motion equations for a generalized structural variable. Stationary solutions for the equation correspond to “open,” “closed,” and “inactivated” channel states. Because of this, the channel free energy profile, as a function of the structural variable, has three local minima. The three energy values of these states depend both on the electrical field applied externally and on the near-membrane concentrations of permeant ions and acting pharmacological agents. Sodium channel activation and inactivation kinetics are considered resulting from relative changes of the free energy typical of the above three states of the channel. The results we obtained in the course of channel activation and inactivation modeling and their voltage dependence are qualitatively consistent with the commonly known experimental data. The proposed model allows one to qualitatively predict the dependence of the sodium channel kinetic characteristics on the concentrations of permeant ions and pharmacological agents.     

Heterogeneity of membrane transport quantified by the analysis of a unidirectional flux transient of charged tracer

A planar mosaic membrane consists of patches, each with a given area, diffusion coefficient, and mobility of charged tracer; a common electric field, constant in space and time, lies across all the patches. Given the properties of the patches, the transient of the total unidirectional flux (summed over the patches) is predictable. Here we deal with the inverse problem: Given only the observed transient of the total unidirectional flux (as defined experimentally by Ussing), the unknown transport heterogeneity of the mosaic membrane is to be analyzed. Results obtained previously for uncharged tracers are generalized to include effects of the field. In particular, the ratio of the arithmetic and harmonic means (both area-weighted) of the diffusion coefficients, evaluated over the membrane, is expressed in terms of only the observed transient and the field strength and is used to characterize the heterogeneity; and the unique exact solution of the inverse problem for two kinds of patches is recovered at any field strength. If the mosaic consists of n distinct kinds of patches, a sweep of the field strength from low to high values reveals (at most) n steplike shapes in the time course of the total unidirectional flux (normalized to its final steady value), which permit an approximate analysis of the heterogeneity by elementary means.     

Mathematical model of the interaction of cations and anions in a channel

A mathematical model of the ionic channel permeable both to anions and cations is considered. The model takes into account the electrostatic interaction between oppositely charged ions and does not suppose single-file movement. An equation for zero-current potential is derived, which leads to the Goldman equation in the limit of low ion concentrations. The model is used to describe concentration relationships of zero-current potentials on a lipid bilayer with amphotericin B channels which cannot be described on the basis of the independence principle.