Plasma charge force near the dust particles chain levitated in a gas-discharge

. In the current paper a numerical study of the forces acting of a one-dimensional chain of three dust particles levitating in the near-electrode layer of a gas discharge plasma is presented. In the described model dust particle motion is calculated in consideration of the action of gravity, external electric field, the Coulomb repulsion and the force induced by plasma space charge. The dependences of the dust particles charges and their position in space on the mean value of external electric field were calculated. The investigation showed the effect of discharging dust particles in the chain due to ion focusing. The spatial distribution of forces acting on the dust particle chain has been studied. It is noted that the Coulomb repulsion force relative to the center of the chain loses its symmetry with an increase in the mean electrostatic field. It is shown that the displacement of a chain of dust particles is determined by the force induced by the plasma space charge.


Introduction
Dust particles entering the gas-discharge plasma are charged to large values of the negative charge.Clouds of ions accumulate around them, which are distorted, giving rise to an effect called ion focusing.Ion focusing contributes to the formation of oscillations in the plasma and a dipole moment in the "ion cloud-dust particle" system.In a system of several dust particles, these effects lead to the phenomenon of self-organization -the appearance of structures resembling crystals in a gasdischarge dusty plasma.The simplest type of dust structures is a dust chain [1].
There are several ways to study these structures numerically.The first is a DiP3D model [12][13][14][15].In this model the calculation of plasma flow around the structures of dust particles is performed selfconsistently.It takes into account elastic and inelastic collisions of electrons and ions with neutral atoms [15].
The arrangement of dust particles in the chains was also considered in [11,16] with the help of the DRIAD (Dynamic Response of Ions And Dust) model, in which, using the MAD (molecular asymmetric dynamics) approximation, the simultaneous motion ion-dust was simulated.The main disadvantage of the DRIAD is its computational noise which does not allow to calculate the spatial distribution of the plasma potential with decent accuracy [11].
In this paper, a numerical model for studying the structural properties of a one-dimensional dust particles chain is presented.The results consist of a selfconsistent calculation of the dust particle charges and dust particle position depending on the configuration of the external electric field.Using the presented model, the dependences of the main forces (the Coulomb force, the electrostatic field force, and the force of the plasma space charge) on the external electric field are calculated.

Model
The presented model is thoroughly presented in [17], therefore, the following chapter contains only a brief summary of the model's main elements.The model is divided into two blocks.Block 1 calculates the selfconsistent spatial distributions of the space charge n(r) and plasma potential U(r) around a stationary dust particle chain.Block 2 calculates the equilibrium positions of each dust particle according to the forces present in the computational domain.
In this paper, the one-dimensional chain of three dust particles placed on the z axis (x = y = 0) is considered.In the computational domain the external electric field E = (0, 0, E) and the gravitational field g = (0, 0, g) are present.The problem is cylindrically symmetric.Therefore spatial distributions of the space charge n(x, y, z) and plasma potential U(x, y, z) are also cylindrically symmetric and should be defined as functions n(ρ, z) and U(ρ, z).
In the model the dimensionless quantities are used.All variables denoting length are normalized to the ion Debye length λi = (kTi/4πe 2 n0) 1/2 , where n0 is the unperturbed plasma density.The latter is chosen as a normalization parameter for the ion and electron densities spatial distributions.All variables denoting energy are normalized to the ion thermal energy kBTi, velocity variables are normalized to the ion thermal velocity VTi = (kBTi/mi) 1/2 , and time variables are normalized to λi/VTi.The dimensionless forms of the dust particle charge and the external electric field are defined as: In subsequent descriptions, the strokes will be omitted.The external electric Е(z) is set as a linear function: In the Block 1, the spatial distributions of the space charge and plasma potential near a fixed chain of dust particles are calculated.Calculations in Block 1 are divided into two consecutive parts: the calculation of ion density spatial distribution and the calculation of selfconsistent plasma potential.
In the first part the trajectories of ~ 10 6 ions are calculated simultaneously according to Newton's motion equations.The initial velocity of each ion is determined according to the Maxwellian velocity distribution.At the time t = 0 the total potential spatial distribution is determined analytically as a superposition of the Debye-Hückel potential of three dust particles and the potential of the external electric field: where k is the order number of a dust particle in a chain of N particles, and zk is the position of the k th dust particle on the z axis.
The ion motion is calculated for a constant value of the external electric field Em since it is assumed that the ion flow has sufficient kinetic energy not to be affected by the slope coefficient Ek.
The ion density spatial distribution ni(ρ, z) is defined through the residence time of ions in cylindrical segments of space (ρ, z).The electron density spatial distribution ne(ρ, z) is determined according to the Boltzmann distribution.The space charge spatial distribution n(ρ, z) is defined as: ( , ) ( , ) ( , ), In the course of their movement, ions can collide with a dust particle or with a neutral atom.The average time between ion-atom collisions is determined by the mean free path.The collision of ions with dust particles is determined directly.In the second part of Block 1, the charges of each dust particle are modified by the formula: where Ii,k is the ion flux to the kth dust particle, calculated from the number of collisions of ions with the dust particle, and Ie,k is the electron flux to the kth dust particle, which is determined according to [18].
From the obtained space charge spatial distribution n(ρ, z), a new self-consistent plasma potential spatial distribution U(ρ, z) is determined according to the Poisson equation: (6) Which is solved using the Jacobi method.The electric potential calculated by ( 6) modifies the total potential spatial distribution in the computational domain: where rk is the distance from the point to the k th dust particle.
Steps 1 and 2 are consecutively calculated in Block 1 multiple times by an iterative cycle.Iterative algorithm of Block 1 can be described as follows: 1. Calculation of ion trajectories in the potential (3) or (7). 2. Calculation of space charge spatial distribution n (ρ, z) (4).3. Calculation of dust particles charges (5). 4. Calculation of self-consistent plasma potential spatial distribution U(ρ, z) (6). 5. Transition to step 1.The iterative execution of the Block 1 continues until the electric potential in the system stops changing.
Block 2 calculates the positions of the dust particles according to the determined forces.The following forces are considered: -Coulomb repulsion force Fq.
-Plasma space charge force Fpl, calculated in Block 1. -Electrostatic force FE.
-Friction with the neutral gas Ffr.The sum of named forces determines the total force Fk acting on a single kth dust particle: In accordance with total force, the motion equations for dust particles are solved: (9) These equations are calculated in a constant plasma potential spatial distribution until total force acting on each dust particle becomes zero.In this case dust particles positions correspond to equilibrium.
Blocks 1 and 2 are executed consequently.The complete algorithm is as follows: 1.The initial system parameters are set.2. Block 1 is executed (with the algorithm described above).3. The equilibrium positions of dust particles are redetermined.4. Transition to step 2. Blocks 1 and 2 are executed iteratively until the values of dust particles charge and their positions stop changing.

Results
In this paper the dust particle chain of three is considered with radius of each one being equal r0 = 0.01.The ion mean free path was set to value li = 5, which corresponds to neutral gas pressure p = 5 Pa.The case of balance between the gravitational force and the electrostatic force is considered, i.e.Fg + FEm = 0.
Figures 1 and 2 show the dependencies of the dust particles coordinates and charges equilibrium values on the mean electrostatic field strength Em. Figure 1 shows that the generally symmetrical chain of dust particles undergoes restructuring and becomes asymmetrical as Em increases.An increase in Em also results in a change in the dust particles charges.The charges of the first and second dust particles grow linearly, while the charge of the last dust particle remains almost unchanged over the entire range of the chosen parameters.Such a dependence of the charge, as well as the fact that the charge of the first dust particle turns out to be greater than the charges of the second and third particles, is explained by the discharge phenomenon, also described in [12][13][14][15]17].This phenomenon consists in the fact that behind the first dust particle there is an ion focusing region, entering which the second dust particle begins to experience frequent collisions with said ions.As a result, according to formula (5), the total charge of such particle decreases.Figure 3 shows the dependence of the electrostatic force value acting on dust particles in the chain on the electrostatic field strength Em.In this paper, the external electrostatic field is aligned with the z axis.Its effect on dust particles is such that it pushes them in the direction of field weakening.The force of gravity, on the contrary, shifts dust particles towards larger electrostatic fields.In each experiment, the gravitational force was set such that Fg = FEm.Figure 3 shows that the electrostatic force value changes linearly with increasing Em.Which means that despite the increase in Em, the dust chain, as a whole, is displaced against the action of the electrostatic field.Figure 5 shows the dependence of the plasma space charge force value induced by the plasma acting on dust particles in the chain on the value of Em.It can be seen that this force grows for all three dust particles, while this growth is fastest for the third dust particle, for which, for small values of Em, the force turns out to be negative.For large values of Em, this force acts in the opposite direction from the strength of the external electrostatic field and displaces the chain in the direction of the gravitational force.Due to the fact that this force is greater in magnitude for the first and second dust particles, these particles are closer to each other than the second and third particles of the chain.As a result, the Coulomb force becomes asymmetric, which partly compensates the effect of the plasma space charge force.
Based on the results obtained in Figures 3-5, it can be concluded that the main role in the restructuring and displacement of the chain as a whole is played by the electrostatic field, gravity and the plasma space charge.
The total forces from these three sources, acting on the entire chain, are shown in Figure 6.In the event that the plasma space charge force would be equal to zero, the established gravitational force would be equal to the electrostatic field value.However, as Em increases, ion focusing occurs, which carries the dust particle away in the direction of increasing external field E, until this total force is compensated by the slope of Ek.The plasma space charge force is also significant for small fields and reaches 80% of the gravitational force.As the value of Em increases and the relation Fg = FEm is maintained, the Fpl becomes equal to ~35% Fg.  Figure 7 shows the ratio of the total plasma space charge force to the total external electrostatic field.This result shows that for small values of Em, the fraction of the force Fpl compansating the action of the electrostatic field turns out to be constant and averages ~37.5%.As the field increases, this fraction drops sharply to 31%.This is due to the fact that as Em increases, it becomes more and more difficult for ions to orbit around a dust particle and they are carried away by the external field easier.This phenomenon was considered in the work of the authors [19] for the case of a isolated dust particle.According to [19], as Em increases, the ion cloud ionizes, at which the density of ions orbiting around a dust particle decreases, and, as a result, the effect of ion focusing decreases, which leads to a decrease in the Fpl value.

Conclusion
In this work, the dependence of the forces acting on a chain of three dust particles on the strength of an external electrostatic field is studied.Said chain was placed in a gravitational and external electric field and oriented along them.The problem under study considers the levitation of particles in the near-electrode region of an RF discharge or in the stratum of a DC discharge.In this case the electric field can be approximated by a linear dependence.
To solve this task, recently developed model was used.This model is based on a mean field approximation for an ion-plasma interaction and the sequential calculation of the motion of ions and dust particles.Dust particles charges were determined from the equilibrium of ions and electrons flows towards their surfaces.Positions of the dust particles in the chain were calculated taking into account: the gravitational force, the electrostatic force, the Coulomb force, and the force from the plasma space charge.
The dependences of the dust particles charges and their position in space in the mean value of external electric field were calculated.The charge of the first dust particle in the chain increase linearly with an increase of the external electric field.The effect of discharging of the second and third dust particles due to ion focusing was demonstrated.
It is shown that for a small value of mean external electrostatic field, the Coulomb force is symmetric with respect to the centre of the chain, and the plasma space charge force is approximately equal to the gravitational force established for each mode.As the field increases, the symmetry in the Coulomb force is broken, as a result of the growth of the ion focusing effect and the restructuring of the dust particles chain.
It is shown that the displacement of dust particles as a whole, when the total gravitational force is equal to the force from the mean electrostatic field, is determined by the force induced by the plasma space charge.
It is shown that the ratio of the total force induced by the plasma space charge to the total electrostatic force acting on the dust chain decreases with an increase in the mean electrostatic field, which is a consequence of the ionization of the ion cloud orbiting around the ordered structure of particles.

Fig. 1 .
Fig. 1.Dependence of dust particles equilibrium positions Zk on the strength of the external electrostatic field Em.

Fig. 2 .
Fig. 2. Dependence of the dust particles equilibrium charges Qk on the strength of the external electrostatic field Em.

Fig. 3 .
Fig. 3. Dependence of the electrostatic force value FE acting on dust particles in the chain on the strength of the external electrostatic field Em.

Figure 4
Figure 4 shows the dependence of the Coulomb force value, which acts on dust particles in the chain, on the value of the electrostatic field strength Em.The data presented in this figure confirm what was said earlier that for small values of the field, the chain turns out to be almost symmetrical.As the field increases, the center of the chain is displaced against the ion flow direction, closer to the first dust particle.It should also be noted that the value of the Coulomb force decreases with increasing field.Both observations seem

Fig. 4 .Fig. 5 .
Fig. 4. Dependence of the Coulomb force value Fq acting on dust particles in the chain on the strength of the external electrostatic field Em.

Fig. 6 .
Fig. 6.Dependence of the sum of the Fpl, Fq and FE acting on dust particles in the chain on the strength of the external electrostatic field Em.

Fig. 7 .
Fig. 7. Dependence of the ratio of the total plasma space charge force value Fpl to the total external electrostatic field FE on the strength of the external electrostatic field Em.