Nonlinear equilibrium spherical dust structures I: Basic equations and criterion for the existence of self-organized structures

Basic equations for dust structures are formulated that account for the balance of the forces, plasma fluxes, and grain charges with allowance for nonlinearity in the screening of individual grains and possible violation of quasineutrality due to the interaction of collective fields with plasma fluxes. A theory of non-linear drag forces exerted by plasma fluxes on dust grains is developed for moderate drift flux velocities, higher than the mean ion thermal velocity but much lower than the acoustic speed. It is shown that equilibrium dust structures have finite sizes and negative charges and that they can exist only in a certain range of intensities of external fluxes on their surfaces. When there is no additional volume ionization, the size of the structures is determined by the intensity of the external flux. A study is made of a weakly ionized dusty plasma in which the interaction of its components with neutral gas atoms plays a major role. The ion, electron, and dust density distributions, as well as the distributions of the dust grain charges and plasma fluxes, are calculated self-consistently as functions of the distance from the center of a structure.     

Nonlinear screening of dust grains and structurization of dusty plasma

A review of theoretical ideas on the physics of structurization instability of a homogeneous dusty plasma, i.e., the formation of zones with elevated and depressed density of dust grains and their arrangement into different structures observed in laboratory plasma under microgravity conditions, is presented. Theoretical models of compact dust structures that can form in the nonlinear stage of structurization instability, as well as models of a system of voids (both surrounding a compact structure and formed in the center of the structure), are discussed. Two types of structures with very different dimensions are possible, namely, those smaller or larger than the characteristic mean free path of ions in the plasma flow. Both of them are characterized by relatively regular distributions of dust grains; however, the first ones usually require external confinement, while the structures of the second type can be self-sustained (which is of particular interest). In this review, they are called dust clusters and self-organized dust structures, respectively. Both types of the structures are characterized by new physical processes that take place only in the presence of the dust component. The role of nonlinearities in the screening of highly charged dust grains that are often observed in modern laboratory experiments turns out to be great, but these nonlinearities have not received adequate study as of yet. Although structurization takes place upon both linear and nonlinear screening, it can be substantially different under laboratory and astrophysical conditions. Studies on the nonlinear screening of large charges in plasma began several decades ago; however, up to now, this effect was usually disregarded when interpreting the processes occurring in laboratory dusty plasma. One of the aims of the present review was to demonstrate the possibility of describing the nonlinear screening of individual grains and take it into account with the help of the basic equations for the equilibrium between plasma components when analyzing equilibrium structures. The effect of plasma screening nonlinearity on both the diffusion processes and the forces of dust drag by plasma fluxes is analyzed. It is shown how self-organized dust structures form in these processes. In the limit of very small dust grain charges, the forces acting on the dusty plasma components and the set of basic equations for stationary dust structures (with allowance for nonlinear screening) take a standard form. New qualitative effects, such as the suppression of diffusion due to ion scattering from dust grains and the formation of structures of different configurations, are described. A detailed comparison with previous results is performed. It is shown that the solution of basic nonlinear equations for dust structures yields new qualitative effects. A number of new effects to be studied in future dusty plasma experiments with the formation of structures in spherical chambers are predicted (it is assumed that diffusion will play a significant role under microgravity conditions). Recent ground-based experiments, as well as experiments carried out onboard the International Space Station, directly confirm the nonlinear character of screening and the significant role played by this nonlinearity in the structurization of dusty plasma. Experiments on the formation of structures consisting of smaller dust grains within structures formed of larger grains are discussed. It is shown that those experiments can be interpreted only using the concept of nonlinear screening.     

Self-consistent quasineutral dust structures

Distributions of the dusty plasma parameters (electron, ion, and dust densities; dust grain charge; and ion drift velocity) in quasineutral dust structures whose dimensions are much greater than the mean free path of the ions in their interactions with neutral particles are calculated numerically under conditions such that ionization sources are located outside the structures. Planar, cylindrical, and spherical structures are investigated. It is shown that static equilibrium structures are governed by a single (basic) parameter: the electrostatic potential drop between the center of the structure and its boundary. It is found that the maximum value of the basic parameter (in energy units) does not exceed the electron temperature. The basic parameter also determines the total number of dust grains in the structure and the power of external ionization sources that are necessary to sustain this structure. The fact that the basic parameter varies within a limited range allows one to consider all the possible structures with a given dimensionality (planar, cylindrical, and spherical).     

Processes accompanying the charging of dust grains in the ionospheric plasma

The influence of the neutral component of the dusty ionospheric plasma on the process of dust grain charging is analyzed. Microscopic ion fluxes onto a dust grain are calculated with allowance for the interaction with the neutral components of the ionospheric plasma for both negatively and positively charged dust grains. For the latter case, which takes place in the presence of intense UV or X-ray solar radiation, the electron heating caused by the photoelectric effect is also investigated. It is found that the efficiency of electron heating depends on the density of neutral particles. The altitudes at which these effects appreciably influence the charging of different types of nano- and microscale dust grains are determined. It is shown that these effects should be taken into account in describing noctilucent clouds, polar mesosphere summer echoes, and physical phenomena involving grains of meteoric origin.     

Development of a self-consistent model of dust grain charging at elevated pressures using the method of moments

A model of dust grain charging is constructed using the method of moments. The dust grain charging process in a weakly ionized helium plasma produced by a 100-keV electron beam at atmospheric pressure is studied theoretically. In simulations, the beam current density was varied from 1 to 10⁶ μA/cm². It is shown that, in a He plasma, dust grains of radius 5 μm and larger perturb the electron temperature only slightly, although the reduced electric field near the grain reaches 8 Td, the beam current density being 10⁶ μA/cm². It is found that, at distances from the grain that are up to several tens or hundreds of times larger than its radius, the electron and ion densities are lower than their equilibrium values. Conditions are determined under which the charging process may be described by a model with constant electron transport coefficients. The dust grain charge is shown to be weakly affected by secondary electron emission. In a beam-produced helium plasma, the dust grain potential calculated in the drift-diffusion model is shown to be close to that calculated in the orbit motion limited model. It is found that, in the vicinity of a body perturbing the plasma, there may be no quasineutral plasma presheath with an ambipolar diffusion of charged particles. The conditions for the onset of this presheath in a beam-produced plasma are determined. __________ Translated from Fizika Plazmy, Vol. 29, No. 3, 2003, pp. 214–226. Original Russian Text Copyright ? 2003 by Filippov, Dyatko, Pal’, Starostin.     

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.     

Heat transfer in dust structures in an RF discharge plasma

Results are presented from experimental studies of heat transfer in liquid-like plasma-dust structures. The experiments were performed with aluminum oxide grains ～3–5 μm in size in an RF discharge plasma. The heat capacity of the dust grains in plasma is measured. The thermal conductivity and thermal diffusivity of liquid-like plasma-dust structures are deduced under the assumption that the observed temperature gradients and the propagation of a thermal perturbation in a dusty plasma are related to heat conduction within the dust component. The measured temperature dependences of the thermal conductivity and thermal diffusivity are in qualitative agreement with the results of numerical simulations performed in the model of a simple single-atom liquid. It is shown that quantitative discrepancy between the experimental and numerical results is related to the energy loss of dust grains in their collisions with the neutral particles of the ambient gas.     

Dynamics of dust grains in a two-component dusty plasma induced by solar radiation under microgravity conditions

Results are presented from experimental studies of the dynamics of dust grains charged via photoemission under microgravity conditions. The experiments are performed with bronze grains exposed to solar radiation on board the Mir space station. The velocity distribution, temperature, mean charge, and friction and diffusion coefficients of dust grains are determined. An analysis of the data obtained shows that the polarization caused by the separation of opposite charges can significantly affect the transport processes in a two-component dusty plasma consisting of dust grains and the electrons emitted by them.     

Dust grain charges in a nuclear-track plasma and the formation of dynamic vortex dust structures

Results are presented from Monte Carlo calculations of the electric charge of dust grains in a plasma produced during the slowing down of the radioactive decay products of californium nuclei in neon. The dust grain charging is explained for the first time as being due to the drift of electrons and ions in an external electric field. It is shown that the charges of the grains depend on their coordinates and strongly fluctuate with time. The time-averaged grain charges agree with the experimental data obtained on ordered liquidlike dust structures in a nuclear-track plasma. The time-averaged dust grain charges are used to carry out computer modeling of the formation of dynamic vortex structures observed in experiments. Evidence is obtained of the fact that the electrostatic forces experienced by the dust grains are potential in character.     

Positive column of a glow discharge in neon with charged dust grains (a review)

The effect of charged micron-size dust grains (microparticles) on the electric parameters of the positive column of a low-pressure dc glow discharge in neon has been studied experimentally and numerically. Numerical analysis is carried out in the diffusion-drift approximation with allowance for the interaction of dust grains with metastable neon atoms. In a discharge with a dust grain cloud, the longitudinal electric field increases. As the number density of dust grains in an axisymmetric cylindrical dust cloud rises, the growth of the electric field saturates. It is shown that the contribution of metastable atoms to ionization is higher in a discharge with dust grains, in spite of the quenching of metastable atoms on dust grains. The processes of charging of dust grains and the dust cloud are considered. As the number density of dust grains rises, their charge decreases, while the space charge of the dust cloud increases. The results obtained can be used in plasma technologies involving microparticles.     

Nonlinear screening of dust grains and structurization of dusty plasma: II. formation and stability of dust structures

The second part of the review on dust structures (the first part was published in Plasma Phys. Rep. 39, 515 (2013)) is devoted to experimental and theoretical studies on the stability of structures and their formation from the initially uniform dusty plasma components. The applicability limits of theoretical results and the role played by nonlinearity in the screening of dust grains are considered. The importance of nonlinearity is demonstrated by using numerous laboratory observations of planar clusters and volumetric dust structures. The simplest compact agglomerates of dust grains in the form of stable planar clusters are discussed. The universal character of instability resulting in the structurization of an initially uniform dusty plasma is shown. The fundamental correlations described in the first part of the review, supplemented with effects of dust inertia and dust friction by the neutral gas, are use to analyze structurization instability. The history of the development of theoretical ideas on the physics of the cluster formation for different types of interaction between dust grains is described.     

One-dimensional nonlinear self-organized structures in dusty plasmas

Dusty plasmas, which are open systems, can form stable one-dimensional self-organized structures. Absorption of plasma by dust particles results in the plasma flux from the plasma regions where the dust is absent. It is found that, in a one-dimensional dust layer, this flux is completely determined by the number of dust particles per unit area of the layer surface. This number determines all of the other parameters of the steady-state dust structure; in particular, it determines the spatial distributions of the dust density, dust charge, electron and ion densities, and ion drift velocity. In these structures, a force and electrostatic balance is established that ensures the necessary conditions for confining the dust and plasma particles in the structure. The equilibrium structures exist only for subthermal ion flow velocities. This criterion determines the maximum possible number of dust particles per unit area in the steady-state structure. The structures have a universal thickness, and the dust density changes sharply at the edge of the structure. The structures with a size either less than or larger than the ion mean free path with respect to ion-neutral collisions, quasi-neutral and charged structures, and soliton-and anti-soliton-like structures are investigated. Laboratory experiments and observations in extraterrestrial plasma formation are discussed in relation to dust structures.     

Nonlinear Wave Structures and Plasma−Dust Effects in the Earth’s Atmosphere

The interaction of charged dust grains with nonlinear vortical structures in the Earth’s atmosphere is analyzed. Certain aspects of the atmosphere?ionosphere interaction, in particular, mechanisms for the appearance of dust grains at ionospheric altitudes, are discussed. It is shown that, at certain altitudes, there are regions in the wavenumber space in which conditions leading to the excitation of acoustic?gravity waves are satisfied. The interaction of nonlinear acoustic?gravity waves with dust grains of meteoric origin at ionospheric altitudes, which leads to the mixing and redistribution of dust grains over the region where vortices exist, is investigated. The possibility of formation of vertical and horizontal dust flows in dusty ionospheric plasma as a result of modulational instability is analyzed. The dynamics of dust grains in dust devils frequently arising in the atmosphere above well-heated surfaces is modeled. The vortical structure of such a dust devil is characterized by a reduced pressure in the center, which facilitates the lifting of small dust grains from the surface. The formulated model is used to calculate the trajectories of dust grains in dust devils with allowance for the influence of the electric field generated in the vortex by colliding dust grains. The calculations show that dust devils play an important role in the transport of dust grains.     

Formation of quasi-two-dimensional dust structures in an external electric field

Conditions are considered under which quasi-two-dimensional extended structures are formed consisting of charged dust grains that are suspended in a gravitational field by an external electric field. Formulas are derived that describe the relationships between the parameters of the intergrain interaction potential, the number of dust grains, and the gradients of the linear electric field of the device. A criterion is proposed that determines the onset of a new dust layer in a quasi-two-dimensional dust system.     

Numerical simulations of ion and electron transport in a low-temperature dusty plasma

Charged particle transport and kinetic processes in a low-temperature dusty plasma are numerically simulated. Dust grains are represented as spheres with a given radius. The self-consistent electric field in the plasma surrounding a charged dust grain is calculated taking into account the perturbations of plasma quasineutrality near the grains. It is shown that applying an external electric field leads to a rearrangement of the plasma space charge and a break of the spherical symmetry of the electron and ion density distributions around the grain. The mutual influence of two identical charged dust grains is considered, and the energy of the electrostatic interaction between the grains is calculated. It is shown that this energy has a minimum at a certain finite distance between the grains.     

Charging of dust grains in a nonequilibrium plasma of a stratified glow discharge

A theoretical model is presented that describes the charging of dust grains in the positive plasma column of a stratified glow dc discharge in argon. A one-dimensional self-consistent model is used to obtain axial profiles of the electric field, as well as the electron energy distribution function along the axis of the discharge tube. Radial profiles of the electric field are determined in the ambipolar diffusion approximation. It is assumed that, in the radial direction, the electron distribution function depends only on the total electron energy. Two-dimensional distributions of the discharge plasma parameters are calculated and used to determine the potential and charge of a test dust grain at a certain point within the discharge and the electrostatic forces acting on it. It is shown that the grain charge distribution depends strongly on the nonequilibrium electron distribution function and on the nonuniform distribution of the electric field in a stratified glow discharge. A discussion is presented on the suspension of dust grains, the separation of grains by size in the discharge striations, and a possible mechanism for the onset of vortex dust motion at the edge of a dust cloud.     

Ion-acoustic solitons in dusty plasma

The dynamics of dust ion-acoustic solitons is analyzed in a wide range of dusty plasma parameters. The cases of both a positive dust grain charge arising due to the photoelectric effect caused by intense electromagnetic radiation and a negative grain charge established in the absence of electromagnetic radiation are considered. The ranges of plasma parameters and Mach numbers in which ??conservative?? (nondissipative) solitons can exist are determined. It is shown that, in dusty plasma with negatively charged dust grains, both compression and rarefaction solitons can propagate, whereas in plasma with positively charged dust grains, only compression solitons can exist. The evolution of soliton-like compression and rarefaction perturbations is studied by numerically solving the hydrodynamic equations for ions and dust grains, as well as the equation for dust grain charging. The main dissipation mechanisms, such as grain charging, ion absorption by dust grains, momentum exchange between ions and dust grains, and ion-neutral collisions are taken into account. It is shown that the amplitudes of soliton-like compression and rarefaction perturbations decrease in the course of their evolution and their velocities (the Mach numbers) decrease monotonically in time. At any instant of time, the shape of an evolving soliton-like perturbation coincides with the shape of a conservative soliton corresponding to the current value of the Mach number. It is shown that, after the interaction between any types of soliton-like perturbations, their velocities and shapes are restored (with a certain phase shift) to those of the corresponding perturbations propagating without interaction; i.e., they are in fact weakly dissipative solitons.     

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.     

Analysis of the formation of ordered dust-grain structures in a thermal plasma

The possibility is studied of the formation of ordered dust-grain structures in a low-temperature thermal plasma consisting of electrons, ions, and micron-sized charged dust grains. The range of the required values of the coupling parameter Γ defining the degree to which the plasma is nonideal is calculated using the results of diagnostic measurements carried out in a plasma consisting of combustion products of propane in air with grains of different materials. The results obtained show that the most favorable conditions for the formation of strongly correlated grain structures (for both positively and negatively charged grains) take place at the maximum grain number density and a plasma temperature close to the minimum flame temperature (∼1600 K). In this case, the optimum grain radii lie in the range 4–10 μm and the maximum value of the parameter Γ is less than 200. Since the calculated values of Γ give an upper estimate, liquidlike ordered structures are most likely to form in a thermal plasma. Based on the results of the analysis, it is stated that an increase in the parameter Γ and, accordingly, the formation of plasma-crystal structures in a thermal plasma can only occur for positively charged grains. __________ Translated from Fizika Plazmy, Vol. 26, No. 7, 2000, pp. 626–632. Original Russian Text Copyright ? 2000 by Samarian, Vaulina, Nefedov, Petrov.     

Effect of dust size distribution on ion-acoustic solitons in dusty plasmas with different dust grains

Theoretical studies are carried out for ion acoustic solitons in multicomponent nonuniform plasma considering the dust size distribution. The Korteweg?de Vries equation for ion acoustic solitons is given by using the reductive perturbation technique. Two special dust size distributions are considered. The dependences of the width and amplitude of solitons on dust size parameters are shown. It is found that the properties of a solitary wave depend on the shape of the size distribution function of dust grains.