Helical structures in complex plasma I: Noncollective interaction

The conditions for the formation and stability of helical quasi-crystals in a complex plasma containing dust grains of equal size are investigated. A study is made of both the confinement of such helical structures in a direction transverse to the cylinder axis by means of an external parabolic potential well and the possibility of their self-confinement. Computer simulations of the helical dust structures were carried for two cases: for a structure of infinite length along the symmetry axis (or a closed structure in toroidal geometry) and for a structure of finite length. The dust grains were assumed to interact through a potential that is a superposition of the non-Debye nonlinear screened potential and the nonscreened noncollective attractive potential (the Lesage effect). Molecular dynamics simulations showed that, in the presence of dissipation, any initial random distribution of the dust grains interacting through such a potential in cylindrical geometry evolves to an equilibrium helical structure. When the external control parameter was varied smoothly, the pitch angle of the helix was observed to bifurcate (i.e., to undergo sharp jumps). The structure of the helix was also observed to bifurcate when the external parameter was varied: a helix changed into two interwoven helices, which then changed into three interwoven helices, etc. The smaller the confinement parameter (and, accordingly, the larger the radius of the helical structures) and the stronger the attractive forces between the grains, the larger the number of bifurcations. The results of analytical calculations of the parameters of the equilibrium structures and of their energies are in complete agreement with numerical results. It is also shown that noncollective attraction between dust grains makes it probable that helical structures will exists when the external confinement parameter is zero or even when it is negative. Bifurcations in such systems may provide the possibility of creating new types of memory elements.     

Sedimentation equilibrium measurements of the intermediate-size tobacco mosaic virus protein polymers

Short-column sedimentation equilibrium methods have been applied for the first time to tobacco mosaic virus (TMV) protein (0.1 M ionic strength orthophosphate) at pH 6.5 and at pH 7.0 to estimate molecular weights. Previous sedimentation velocity experiments at pH 6.5, 20 degrees C have led to the conclusion that the major boundary with an S0(20),w value of 24.4 +/- 0.1 S consists of a distribution of polymers which are mainly three-turn, 48-51-subunit helical rod aggregates. The directly measured z-average molecular weights together with sedimentation velocity data are entirely consistent with this assignment of a three-turn aggregate. Molecular weights have also been determined under two conditions where a large mass fraction of the protein sediments with an S0(20),w value of 20.3 +/- 0.2 S. At pH 6.5, 6-8 degrees C, the aggregates in this boundary are metastable and correspond to 50-60% of the preparation. At pH 7.0, 20 degrees C at equilibrium, 65-75% of the protein sediments at 20.3 S. The 20.3S boundary is very similar under both conditions and is interpreted as being composed of a distribution of protein aggregates centered about 39 +/- 2 subunits. This result is important in the interpretation of previous kinetic measurements of TMV self-assembly. The current view is that the 34-subunit structure of TMV protein, in the form of a cylindrical disk which is made up of two 17-subunit layers and has been characterized in single-crystal X-ray diffraction studies, plays a central role in the initial binding steps with RNA. The present results are not consistent with the view that there is a significant concentration of the TMV protein disk structure in solution under the usual conditions of TMV self-assembly.     

Tearing instability in a tokamak with a noncircular cross section

Using a straight-column model to describe tokamak plasma with a noncircular cross section, it is shown how to (i) find the boundary of tearing instability from the condition of existence of a magnetohydrodynamic plasma equilibrium different from that of a straight cylinder by solving a two-dimensional linear boundary-value problem with a second-order equation with respect to the flux coordinate and (ii) find the spatial structure of the tearing mode and the corresponding effective Δ' when there is only one resonance magnetic surface in the plasma for a given axial wavenumber by solving some kind of a boundary-value problem for the perturbation. The proposed approach is illustrated by numerical calculations for the case of an elliptical cross section as an example.     

Some problems of axisymmetric toroidal equilibria in orthogonal flux coordinates

Formulations of axisymmetric equilibrium problems are presented in orthogonal flux coordinates with boundary conditions that do not assume fixation of the boundary shape for both levitron-type traps with a plasma-embedded conductor and conductor-free configurations, including a tokamak. Illustrative examples of numerical solutions of these problems are presented. For traps with a conductor, it is demonstrated how the geometry of the equilibrium with an isodynamic magnetic surface can be found using flux coordinates.     

Boundary conditions for the electromagnetic field in a waveguide with a thin-walled annular plasma in the context of application to plasma microwave electronics

Effective boundary conditions for the electromagnetic field of the slow surface waves of a thinwalled annular plasma in a metal waveguide are derived and justified. With the boundary conditions obtained, there is no need to solve field equations in the plasma region of the waveguide, so that the dispersion properties of plasma waveguides can be investigated analytically for an arbitrary strength of the external magnetic field. Examples are given that show how to use the effective boundary conditions in order to describe surface waves with a normal and an anomalous dispersion. The boundary conditions are then employed to construct a theory of the radiative Cherenkov instabilities of a thin-walled annular electron beam in a waveguide with a thinwalled annular plasma. The single-particle and collective Cherenkov effects associated with low-and high-frequency surface waves in an arbitrary external magnetic field are studied analytically. The method of the effective boundary conditions is justified in the context of application to the problems of plasma relativistic microwave electronics.     

Helical structures in complex plasma II: Collective interaction

The conditions for the formation and stability of quasi-crystal helical structures formed of charged dust grains of equal size in a complex plasma are investigated. A study is made of both the transverse confinement of such structures by means of an external parabolic potential well and the possibility of their self-confinement or confinement by the field of the smaller background grains. It is shown that, in the presence of dissipation, any initial random distribution of dust grains in a cylindrical external potential evolves into an equilibrium helical structure with a constant pitch angle. It is found that, when the external control parameter is smoothly varied, the pitch angle of the helix changes abruptly and the structure of the quasi-crystal undergoes bifurcations, during which the number of interwoven helices changes. The smaller the confinement parameter and the weaker the attractive forces between the grains, the larger the number of bifurcations. In addition, it is found that, because of the attraction (both noncollective and collective) between the dust grains, stable helical structures may exist not only at zero value of the external confinement parameter, but also at a negative one, and that the collective interaction between the grains not only increases the number of bifurcations but also changes their character.     

Variational problems for the canonical profiles

A noncontradictory formulation of the variational problem for a canonical profile is proposed that refines the problem posed by B.B. Kadomtsev for a circular plasma cylinder. The results are generalized to a toroidal plasma with an arbitrary cross section. For the problem in toroidal geometry, boundary conditions are proposed with which to single out the Kadomtsev-like solution (the canonical profile) from the solutions to the Euler equation. Canonical profiles for the L-and H-modes are constructed. For a number of interesting examples, it is numerically shown that the second variation of the magnetic energy functional is positive. The canonical profile transport model is outlined, and the relationship between the canonical, numerical, and experimental profiles in tokamaks is briefly discussed.     

Plasma equilibrium in axisymmetric poloidal magnetic field configurations in flux coordinates

A simple derivation is given of equilibrium equations in flux coordinates in the general case of an anisotropic-pressure plasma. The issue of how to formulate the boundary conditions for these equations is discussed for two types of configurations—a straight system and a system with an internal conductor. Examples of numerical solutions to the equilibrium problem for these configurations are presented.     

Identification of the plasma boundary shape and position in the Damavand tokamak

A series of experiments and numerical calculations have been done on the Damavand tokamak for accurate determination of equilibrium parameters, such as the plasma boundary position and shape. For this work, the pickup coils of the Damavand tokamak were recalibrated and after that a plasma boundary shape identification code was developed for analyzing the experimental data, such as magnetic probes and coils currents data. The plasma boundary position, shape and other parameters are determined by the plasma shape identification code. A free-boundary equilibrium code was also generated for comparison with the plasma boundary shape identification results and determination of required fields to obtain elongated plasma in the Damavand tokamak.     

Combined ICR heating antenna for ion separation systems

A combination of one- and two-wave antennas (one and two turns of conductors around a plasma cylinder, respectively) is proposed. This combined antenna localizes an RF field within itself. It is shown that spent nuclear fuel processing systems based on ICR heating of nuclear ash by such a combined antenna have high productivity. A theory of the RF field excitation in ICR ion separation systems is presented in a simple and compact form.     

Quantum-mechanical theory of CD for infinite helical polymers

Quantum-mechanical equations are derived that are particularly well suited to actual computations of the CD for helical polymers. They make use of cyclic boundary conditions and helical symmetry, so that only two matrices with a size equal to the number of transitions considered need be diagonalized. The final equations are expressed directly in terms of monomer properties and helical parameters to invite the same input as earlier calculations, and are given as a rotational strength times a shape function for ease of comparison with the earlier work. The shape of the helix term is expressed as a derivative with respect to ω and depends on the distance between monomers along the helix axis. Other terms involving two electric transition dipoles depend on the distance from the helix axis to the transition center. These equations are directly comparable to the classical equations derived for cyclic boundary conditions and helical symmetry. We present an outline of the derivation and enough intermediate steps to clarify how the equations arise.     

Analysis of the distribution of flowing erythrocytes in a model vessel under an inhomogeneous magnetic field

Changes in the distribution of flowing erythrocytes in a straight cylinder were studied under an inhomogeneous magnetic field. The magnetic field was applied perpendicular to a cylinder, which had a 90° side vessel at the end (oriented towards the magnetic field) to detect changes in the erythrocyte distribution within the cylinder. (1) The attraction of paramagnetic erythrocytes by the magnetic field was demonstrated by an increase in the concentration (or number) of erythrocytes drawn into the side vessel. The flow of diamagnetic, oxygenated erythrocytes was unaffected. (2) The degree of attraction of the paramagnetic erythrocytes was proportional to ``(magnetic susceptibility)' and to ``(magnetic flux density) × (magnetic field gradient)' up to 10 T²/m, but it saturated at high magnetic field. The onset of the saturation depended on the magnetic susceptibility of the erythrocytes. (3) The degree of attraction depended on the hematocrit of the flowing erythrocyte suspension, with a maximum value at a low hematocrit. These phenomena are explained on the basis of the balance between the paramagnetic attractive force of the magnetic field and the collision rate between erythrocytes. Received: 2 May 1996 / Accepted: 1 July 1996     

Sedimentation of generalized systems of interacting particles. I. Solution of systems of complete Lamm equations.

A very general approach to the chemical equilibria between many interacting molecules during sedimentation (boundary, band, or active enzyme) taking into account boundary conditions, cell geometry, equilibrium constants, diffusion, enzyme kinetics, etc., is presented. Through a Fortran program, the method has been applied to two very simple but typical cases. With only minor adjustments, the method presented here for sedimentation studies can be extended to all sorts of problems in which “pools” of various species are interacting with each other.     

Some factors affecting oxygen uptake by red blood cells in the pulmonary capillaries

In this study we investigate the equations governing the transport of oxygen in pulmonary capillaries. We use a mathematical model consisting of a red blood cell completely surrounded by plasma within a cylindrical pulmonary capillary. This model takes account of convection and diffusion of oxygen through plasma, diffusion of oxygen through the red blood cell, and the reaction between oxygen and haemoglobin molecules. The velocity field within the plasma is calculated by solving the slow flow equations. We investigate the effect on the solution of the governing equations of: (i) mixed-venous blood oxygen partial pressure (the initial conditions); (ii) alveolar gas oxygen partial pressure (the boundary conditions); (iii) neglecting the convection term; and (iv) assuming an instantaneous reaction between the oxygen and haemoglobin molecules. It is found that: (a) equilibrium is reached much more rapidly for high values of mixed-venous blood and alveolar gas oxygen partial pressure; (b) the convection term has a negligible effect on the time taken to reach a prescribed degree of equilibrium; and (c) an instantaneous reaction may be assumed. Explanations are given for each of these results.     

Large deformation analysis of orthodontic appliances

The deformations of orthodontic appliances used for space closure are large so that any mathematical analysis will require a nonlinear approach. Existing incremental finite element and finite difference numerical methods suffer from excessive computational effort when analyzing these problems. An accurate segmental technique is proposed to handle these difficulties in an extremely efficient fashion. The segmental technique starts by assuming that an orthodontic appliance is composed of a number of smaller segments, the ends of which undergo small relative rotation. With an appropriate choice of local coordinate system the equilibrium equations for each segment are linearized and solved in a straightforward manner. The segments are then assembled using geometric and force compatibility relations similar to the transfer matrix method. Consequently, the original nonlinear boundary value problem is solved as a sequence of linear initial value problems which converge to the required boundary conditions. As only one segment need be considered at a time, the computations can be performed accurately and efficiently on a PC type computer. Although an iterative solution is used to match the boundary conditions, the time required to solve a given problem ranges from a few seconds to a couple of minutes depending on the initial geometric complexity. The accuracy of the segmental technique is verified by comparison with an exact solution for an initially curved cantilever beam with an end load. In addition, comparisons are made with existing experimental and numerical results as well as with a new set of experimental data. In all cases the segmental technique is in excellent agreement with the results of these other studies.     

Effect of surface roughness on slip flows in hydrophobic and hydrophilic microchannels by molecular dynamics simulation

The influences of surface roughness on the boundary conditions for a simple fluid flowing over hydrophobic and hydrophilic surfaces are investigated by molecular dynamics (MD) simulation. The degree of slip is found to decrease with surface roughness for both the hydrophobic and hydrophilic surfaces. The flow rates measured in hydrophobic channels are larger than those in hydrophilic channels with the presence of slip velocity at the walls. The simulation results of flow rate are correlated with the theoretical predictions according to the assumption of no slip boundary condition. The slip boundary condition also strongly depends on the shear rate near the surface. For hydrophobic surfaces, apparent fluid slips are observed on smooth and rough surfaces. For simple fluids flowing over a hydrophobic surface, the slip length increases linearly with shear rate for both the smooth and rough surfaces. Alternately, the slip length has a power law dependence on the shear rate for the cases of hydrophilic surfaces. It is observed that there is a no-slip boundary condition only when shear rate is low, and partial slip occurs when it exceeds a critical level.     

Mechanism of self-assembly of tobacco mosaic virus protein. I. Nucleation-controlled kinetics of polymerization.

The polymerization of tobacco mosaic virus protein has been found to proceed through metastable states under conditions where initially one of the two polymerization-linked protons is bound. These metastable polymers have been characterized and are found to be helical rods, which resemble the structure of equilibrium helical rods that form when both polymerization-linked protons are bound. At pH 6.5 and 20 °C the true equilibrium distribution of these helical rods has been shown to consist of sedimenting species that are much smaller, 24 to 34 S, than described previously, 100 to 200 S. The larger, non-equilibrium rods are produced by an overshoot in polymerization that results from the slow formation of 20 S nuclei followed by a very rapid elongation reaction. Generally, this sequence of rate processes is sensitive to the rate at which a reaction is initiated. In the present case it is the rate of heating or the rate of change of the pH that determines the reaction path and therefore the rate of attainment of equilibrium. In addition to the formation of metastable helical rods during polymerization overshoot, metastable 20 S aggregates can form when either equilibrium or non-equilibrium helical rods are depolymerized by cooling to 5 to 7 °C at pH 6.5. These 20 S aggregates are presumably two-turn disks or helices and can serve as nuclei for helical rod formation in subsequent polymerization reactions. Both helical rod and 20 S metastability are extremely sensitive to pH but, under carefully controlled conditions, the metastability is quite reproducible and reproducible nucleation-controlled polymerization kinetics can be observed even when polymerization-depolymerization cycling is carried out between branches of a hysteresis loop. Temperature- or pH-induced polymerization of tobacco mosaic virus protein can be made to proceed by the slow formation of 20 S, two-turn helix, nuclei followed by the rapid addition of one or more species comprising the 4 S protein. These results confirm a previously proposed kinetic mechanism for the non-equilibrium polymerization reaction (Scheele &; Schuster, 1974).     

Determination of reaction volumes and polymer distribution characteristics of tobacco mosaic virus coat protein

A method that allows the quantitative determination of reaction volumes from sedimentation velocity experiments in an analytical ultracentrifuge is presented. Combined with a second method for detecting pressure-induced depolymerization, general characteristics of polymer distributions may be probed. We show that it is possible to determine if a sample is in an equilibrium or metastable state of subunit association. Our approach to probe macromolecular aggregation systems by small pressure perturbations is not restricted to the use of centrifuges. This method has been applied to characterize certain aspects of the polymerization of tobacco mosaic virus coat protein (TMVP). There are at least two helical polymer conformations in RNA-free coat protein rods. The smaller, helix I, polymers are limited to sizes below about 70 subunits (four to five helical turns) and undergo some kind of cooperative conformational change before further subunits may be added indefinitely. In contrast to helix I, the larger helix II polymers occur as broader and skewed size distributions. Under moderately strong polymerization conditions, the equilibrium state can contain both types of helical rods. The reaction volume for the addition of trimers is -220 ml/mol for both types of helical polymers. These results are compared with the results of previous thermodynamic analyses of TMVP polymerization.     

Models for the 3(10)-helix/coil,pi-helix/coil,and alpha-helix/3(10)-helix/coil transitions in isolated peptides.

Models for the 3(10)-helix/coil and pi-helix/coil equilibria have been derived. The theory is based on classifying residues into helical or nonhelical (coil) conformations. Statistical weights are assigned to residues in a helical conformation with an associated helical hydrogen bond, a helical conformation with no hydrogen bond, an N-cap position, a C-cap position, or the reference coil conformation. The models for alpha-helix formation and 3(10)-helix formation have also been combined to describe a three-state equilibrium in which alpha-helical, 3(10)-helical, and coil conformations are populated. The results are compared with the modified Lifson-Roig theory for the alpha-helix/coil equilibrium. The comparison accounts for the experimental observations that 3(10)-helices tend to be short and pi-helices are not favored for any length. This work may provide a framework for quantitatively rationalizing experimental work on isolated 3(10)-helices and mixed 3(10)-/alpha-helices.     

The linear model and experimentally observed resonant field amplification in tokamaks and reversed field pinches

A review is given of the experimentally observed effects related to the resonant field amplification (RFA) and the Resistive Wall Mode (RWM) instability in tokamaks and reversed field pinches (RFPs). This includes the feedback rotation of RWM in RFX-mod RFP, dependence of the RWM growth rate on the plasma-wall separation observed in JT-60U, appearance of the slowly growing RWM precursors in JT-60U and similar phenomena in other devices. The experimental results are compared with theoretical predictions based on the model comprising the Maxwell equations, Ohm’s law for the conducting wall, the boundary conditions and assumption of linear plasma response to the external magnetic perturbations. The model describes the plasma reaction to the error field as essentially depending on two factors: the plasma proximity to the RWM stability threshold and the natural rotation frequency of the plasma mode. The linear response means that these characteristics are determined by the plasma equilibrium parameters only. It is shown that the mentioned effects in different devices under different conditions can be described on a common basis with only assumption that the plasma behaves as a linear system. To extend the range of the model validation, some predictions are derived with proposals for experimental studies of the RFA dynamics.