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Central European Journal of Physics

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CEJP 2(1) 2004 35-66

Interactions of charged dust particles in clouds of charges

Vladimir A. Gundienkov1, Sergey I. Yakovlenko1 *,

General Physics Institute, Russian Academy of Science, Vavilova St. 38, Moscow, 119991, Russia

Abstract: Two charged dust particles inside a cloud of charges are considered as Debye atoms forming a Debye molecule. Cassini coordinates are used for the numerical solution of the Poisson-Boltzmann equation for the charged cloud. The electric force acting on a dust particle by the other dust particle was determined by integrating the electrostatic pressure on the surface of the dust particle. It is shown that attractive forces appear when the following two conditions are satisfied. First, the average distance between dust particles should be approximately equal to two Debye radii. Second, attraction takes place when similar charges are concentrated predominantly on the dust particles. If the particles carry a small fraction of total charge of the same polarity, repulsion between the particles takes place at all distances. We apply our results to the experiments with thermoemission plasma and to the experiments with nuclear-pumped plasma. © Central European Science Journals. All rights reserved.

Keywords: dust particle, Debye atom, Debye molecule, Poisson-Boltzmann equation, Cassini coordinates, thermoemission plasma, nuclear pumped plasma PACS (2000): 52.27

1 Introduction

The study of a plasma in which charged particles of micrometer size play a significant role (so-called dust plasma) is interesting from the fundamental and applied points of view [16]. Of special interest is the observation of collective effects caused by dust coupling. A number of experiments show that micron size particles can form spatial-ordered structures in thermoemission plasma [3] in gas-discharge plasma and in nuclear-pumped plasma [4]. The properties of strong-coupled plasma are often considered in the framework of the

* E-mail: syakov@kapella.gpi.ru

Received 5 November 2002; accepted 6 November 2003

so-called one-component model (see, for example, the review by [10]. According to this model, one of the charged components is treated as homogeneous in space. Polarization effects are taken into account in the form of corrections, in some cases.

Apparently, the physics of processes occurring in dust plasma differs from the one-component model. A dust particle surrounded by a shell (or cloud) of charges (with masses much smaller than the mass of the dust particle) should be the object of detailed consideration, first of all. A charged dust particle surrounded by a cloud of charges of the opposite sign is an analogue of an atom in gas kinetics.

In general, the charged cloud of such a "dust atom" may not be in thermodynamic equilibrium. However, we shall consider here the situation in which the charges in a cloud are Boltzmann-distributed. It is natural to call such a dust atom a Debye atom [15] in contrast to a Thomas-Fermi atom, in which a charged cloud is a degenerate electronic gas. Similarly, we can introduce the concept of a Debye molecule [17] and a Debye crystal. The Boltzmann distribution and the Poisson equation (that is, the Poisson-Boltzmann equation) describe mathematically the properties of such Debye systems.

It is natural to assume the presence of attractive forces caused by polarization of the charge shells of Debye atoms. However, reliable theoretical results demonstrating an attraction of Debye atoms do not yet exist. The exact solution of the Poisson-Boltzmann equation shows that the repulsion always takes place for the charged planes both in an electron cloud and in a plasma [2], [20]. Numerical simulation of Debye atoms interaction [17] were not quite reliable, as were the results of analytical calculations [5], [11].

The problem of particle interactions in dusty plasma is similar to the problem of colloidal particle interactions in electrolytes. The very concept of a Debye radius for plasmas was borrowed from the theory of electrolytes. The physics of colloid particle interactions in electrolytes has been investigated for a long time (see, for example, [2]. Until now, however, the problem of attraction forces has not been solved, at least for the case in which the colloid particle radius is smaller than the Debye radius.

Below, we attempt to reliably demonstrate the existence of polarization forces of attraction between Debye atoms and to determine the conditions under which attraction appears. This work differs essentially from other publications devoted to an analysis of charged dust particle interactions in plasmas and in electrolytes (see, for example, [1], [13] and [14]).

First, in contrast to a number of publications, we consider a situation in which the total charge of dust particles is not negligibly small compared with the total charge of the cloud particles of one sign. Moreover, we show here that the essential attraction takes place in an opposite limiting case, that is, when almost all the charge of one of sign is concentrated on dust particles, and the clouds consist of charges of only one (opposite) sign. (See [7-9] for preliminary results.)

Second, based on Debye molecule properties, a Debye atom has definite structure. The Debye atom has a core of a charged cloud close to the surface of a dust particle, when a dust particle has a high charge. In particular, the charge of a dust particle cannot, as a rule, be considered as an approximate delta function, even if its radius is much smaller

than the Debye radius.

Third, we calculate directly the resulting force on the dust particle from another particle and the charged cloud. The dependence of the potential energy of interaction on dust particle separation is calculated by integration of this force. In our case, the Poisson-Boltzmann equation is solved in an infrequently used coordinate system based on Cassini ovals. It allows a highly accurate calculation of an electric field near a small particle surface and reliably obtains the force of a particle interaction.

We apply our results to thermoemission plasma and to nuclear pumped plasma.

2 Formulation of the problem

2.1 The Poisson-Boltzmann equation

For the sake of definiteness, we shall consider thermoemission plasma, and speak about positively charged dust particles and the electronic cloud of a dust particle. However, basic results are also of use for dust plasma produced by the electrical discharge and for plasma ionized by an external source of hard radiation, when the particles are charged negatively, and the charged cloud consists mainly of positive ions. We discuss the nuclear pumped dust plasma below.

So, let us consider the case in which an electronic gas surrounding the charged dust particles is formed by the emission of electrons from dust particles at sufficiently high temperature T. In addition, the dust particles are surrounded by a partially ionized gas. In order to find the spatial distribution of the potential 0, the field intensity — V0, and the charge density p = e(Ni — Ne), we must solve the Poisson equation V(—V0) = 4np. The ions and electrons densities (Ni and Ne) appearing in this equation are determined by the Boltzmann distribution Ni = Ni0 exp(—0/T) and Ne = Ne0 exp(e0/T), where Ni0 and Ne0 are the ion and electron densities at the points of zero potential. Thus, the Poisson-Boltzmann equation takes the form:

A0 = 40e ^Na exp ^^ " Ni0 exp ^ Tty ^ . (1)

2.2 Dimensionless variables

We shall measure length in units of Debye radius rD = {T/4ne2Ne0)1=2 corresponding to electron density in points of zero potential. We use the dimensionless potential electric field intensity E, and electronic density ne:

' = 0e/T E = -V0 ■ erD/T ne = r3DNe = nD ■ exp('), (2)

where nD = rD Ne0 .

Eq. 1 is reduced to the following equation for dimensionless potential ' :

A' = exp(') — 5 exp(—'). (3)

Here 6 = Ni0/Ne0 is the parameter describing the additional ionization of gas. Since the plasma is quasi-neutral, one has 0 < 6 < 1.

For further estimations, we shall be guided by the experiments of [3], in which Ne0 = 2.5 • 1010cm~z and T = 0.146eV = 1700K. For characteristic values we have rD = 18 microns, T/e = 0.146 V, and T/erD = 80V/cm. The average radius of a dust particle was rp = 0.4 microns (r0 = rp/rD = 2.23 • 10_2) and its charge was Zpe = 500e. So, we have a field intensity on a particle surface Zpe/r2 = 4.5 • 104V/cm (E0 = E(r0) = 550).

2.3 Boundary conditions.

We will use the term "Debye atom" for a single charged dust particle surrounded by a cloud of lighter charges in thermodynamic equilibrium; two or more dust particles will be referred to as a Debye molecule. Formally, the analyses of a Debye atom and a Debye molecule differ only in the geometry of the problem. While analyzing a Debye atom, we can get by with the solution of the one-dimensional Poisson equation, assuming that the electron cloud is spherically symmetric. In an analysis of a diatomic Debye molecule, we can assume that the problem is symmetric about the x-axis connecting the nuclei (dust particles). Therefore, it is enough to consider the two-dimensional Eq. 3 in plane coordinates (x, y). When analyzing a Debye molecule, the problem is complicated considerably by the choice of boundary conditions.

In a real physical problem, the charge Zpe of a dust particle and its radius rp are specified. Hence, one boundary condition is the field intensity on the surface of dust particles S:

E0 = -Vj (4)

Thus, the charge of a dust particle is determined by the expression

Zp = f V^ds, zp = — f Eds. (5)

4ne J 4n J

Here z p is a dimensionless particle charge, connected to a particle charge in terms of an electronic charge Zp by the expression Zp = 4x • zp • nD; the area of the surface S is normalized with r2D.

The second boundary condition should be the zero-field value on a boundary surface S':

V's = 0. (6)

The zero-field intensity on the Debye atom or molecular boundary follows from quasi-neutrality of the system of charges. The basic purpose of Debye molecule consideration is to find resulting dependence of the particles' interaction force on the distance d between particles. In this case, it is more convenient to use other boundary conditions instead of Eq. 4, that is, to set a constant potential on a surface of dust particles,

'\s = '0 = const:

One can get the field intensity Eo on a surface of a dust particle by solving the Poisson-Boltzmann equation. The calculations with different values of '0 give the necessary value of Eo and charge value 2p (Eq. 5).

The resulting force of interaction of the dust particles is determined by integration of the electrostatic pressure on a surface of a dust particle. In one case the force is directed along an axis x, and its projection is determined by the expression

Here dsx is a projection of surface element ds on an axis x; the force F is connected to dimensionless force f by the expression F = (T 2/8ne2)f; the electric pressure is directed along the outward normal to the surface of dust particles.

3 Some properties of Debye atoms

3.1 The Debye atom

3.1.1 The Poisson-Boltzmann equation

The properties of a Debye molecule in some aspects are defined by properties of the Debye atoms forming this molecule. Therefore, we shall consider some properties of Debye atoms before beginning calculation of the force of dust particles interaction.

In the one-dimensional (that is, planar, cylindrically symmetric, or spherically symmetric) case, equation (3) and boundary conditions take the form

Here k = 0, 1, and 2 respectively for planar, cylindrically symmetric, and spherically symmetric cases; r = 0 corresponds to the beginning of a planar layer, center of the cylinder, or center of the sphere. One boundary condition sets the boundary of the Debye atom r = a0, on which the field is equal to zero.

The spherically symmetric case (k = 2) simulating a Debye atom and the flat case (k = 0), which allows us to study the potential variation near a dust particle surface, will be considered. In a spherically symmetric case, the convenient characteristic of a Debye atom is the dimensionless charge distribution (charge contained inside a sphere of radius r); it is defined by the expression z(r) = r2E(r).

3.1.2 Debye atom in a single-sign charge cloud

The case 5 = 0, in which the charged cloud consists of particles of one sign, corresponds, for example, to a thermoemission plasma [3] or a similar gas ionization process, in which the charges of one sign have completely concentrated on the dust particles [19]. Size a0 we choose equal to half of the average distance between dust particles a0=

— 1 /3

ap/rd = (Np /2rd), where Np is the density of dust particles (see Figure 1). The size ap =N¡1/3/2 is 24% less than the Wigner-Zeitz radius: rWZ = (4nNp/3)-1/3.

Consider the most interesting situation, when a dust particle radius rp is much less than the distance between dust particles r0= rp/rD ^ a0. In experiments [3] rp = 0.4 fim, Np = 5-107 cm-3, and ap = 13.6 fim; thus ap/rp= a0/r0= 34.

The results of equation (3) for the spherically symmetric case (k = 2) show that for the smaller charge zp = Zpe2/rD T < a3/3 of a small particle r0 ^ a0ithe charge, field, and potential distributions are given by expressions [6]:

*(r) = (a3/3) •

i-y a0

/ ^ *(r)

E (r) = -y'

'(r) = ( I0

/a|\ Oq_ 3 / r V V 3 ) r 2 y2ao)

(10a) (10b) (10c)

The expressions 10 are still of use for the points far from a dust particle surface (at r0+ 3r2/a3 > r < a0) when the charge is high zp > a0/3. The variation from these expressions takes place close to a surface (r0<r < r0+ 3r^/a0), where a sharp fall of z(r), E(r), and '(r) takes place (Figure 2). Otherwise, at the high charge of a dust particle, the Debye atom has some charged core close to a dust particle's surface. The charge of a dust particle together with the core is equal to zc this residual charge takes place at a large distance r ~ a0.

= a0/3. The screening of

The high charge condition zp = Zpe2/rD T

> zcor can be written in the form

7 > 7 =

Zp > 7cor = 6N

According to measurements of [3], the charge of dust particles was high:

Zp = 500 > Zcor = 262,

-p = 0.273 > -cor = 0.143.

However, calculations show (see Figure 2) that the dust particle charge in a thermal balance (Zp = 286, zp = 0.156) should be smaller then the value (Zp ° 500) measured by [3]. Hence, either the measurements of plasma parameters are not exact, or the charge of dust particles in experiments [3] is nonequilibrium (for details, see [18]).

3.1.3 Debye atom in plasma

In the case 6 = 0, when the charged cloud consists of particles of both signs, the Debye atom radius, as before, is determined as the distance (r = a0) at which the charge of a dust particle is completely compensated by plasma charges (E(a0)= 0). As in the case 6 = 0, the Debye atom radius is equal to half of the average distance between dust particles a0= ap/rD = (N-1/3/2rD). If 6 = 1, one isolated dust particle in an infinite volume of

plasma can be considered. If S ! 1 the Debye atom radius tends to infinity: a0 ! 1. This is because the finite charge of a particle 20 can be completely compensated by a quasineutral plasma only at its infinite sizes. If S < 1, the Debye atom radius is finite.

Electronic and ionic dimensionless charges contained in a charged cloud are determined by expressions:

The quantity 61 = 20i/z0e gives the relation of a free ion charge in Debye atom to an electron charge. Generally speaking, 61 should be a function of the parameters 6, a0 and '0. However, when the main contribution to integration (11) is the area of a small potential '(r) ^ 1, it is possible to put 61 ° 6.

Figure 3 illustrates the dependencies of z0e, z0i, and 61 on 6. In the results presented in Figure 3, the value of a0 for different values of 6 was chosen as large as possible for the radius of a dust particle corresponding to the experiments of Fortov et al. [3]: r0= rp/rd = 2.23-10"2. This was carried out by "test firing": when the value of a0 was chosen greater than that in Figure 3, the particle charge becomes infinitely large (z(r0) ! 1). The obtained dependencies z(r) and '(r) (see Figure 4) were used to determine z0= z(r0) and '0= '(r0) at r0 = 0.1 in the Debye molecule simulations presented below.

The number of both positive and negative charge, z0i and z0e, in the cloud grows with 6 because of the increase of the Debye atom volume (see Figure 3). At the same time, the uncompensated charge z0= z0e — z0i does not vary with changing 6. At the considered parameters, we have 61 ° 6.

As well as in the case 6 = 0, at the given value r0, the size a0 cannot be infinitely large when a particle charge z0 is infinitely high. The sharp fall of z(r), E(r), and '(r) as functions of r, caused by charge screening, takes place at distance (r — r0) ~ 1/E0 from a dust particle surface when value E0 = z0/r0 is high (see Figure 4). Thus, the size a0 is limited by some value a0max = a0(E0 ! 1). This limiting value increases logarithmically for 6 ! 1:

Since a Debye atom has a core screening the charge of the dust particle, we cannot ascribe the unscreened value of the charge to the dust particle while considering the interaction of Debye atoms.

3.2 About the character of dust particles' interaction 3.2.1 About interaction of the non-polarized Debye atoms

Let us imagine a naive situation in which charged clouds of Debye atoms do not interact with each other. Only repulsion forces will take place in this case. Really, for non-

polarized clouds the interaction force is expressed as

f (d) = .

Here d is the distance between dust particles and zeff (d)=E(d)-d2 is a total charge that is taking place inside a sphere of radius d around of a dust particle. This is an uncompensated charge of a dust particle. Due to quasineutrality of the Debye atom, one has zeff (r) > 0 at r > r0. The charges of the same sign repel each other: zeff (d)zp > 0.

The polarization of charged clouds is necessary for attractive forces. The number of negative charges should increase on the axis of a Debye molecule due to polarization if attractive forces take place.

3.2.2 The interaction of charged planes

The Poisson-Boltzmann equation (4) in a flat case (k=0) can be solved in quadratures. It obtains the interaction force of planes and obtains an accurate numerical solution of the Poisson - Boltzmann equation near the surface of a dust particle.

This shows that the charged planes (both planes surrounded by a cloud of charges of the same sign, and planes located in the plasma) repulse each other [2], [20]. For an illustration we shall consider the case 6 = 0 to get simple analytical expressions. It is useful for an estimation of the necessary accuracy of calculations of a field and a potential near the surface of a dust particle.

Consider the electrostatic pressure on the charged conducting plane, which is located between two conducting planes (left and right). The planes are under the potential '0 (see Figure 5). One of the planes can be removed to an infinite distance if necessary. The integration of the Poisson-Boltzmann equation for a plane case gives [18], [20]:

'(x) = ln(E2 + £?), E (x) = E1 ■ tan

The quantities E1 = exp('1/2) and '1 are connected to a0 by the expression:

ao = (Et) ■arctan (t) •

Here x is the distance from the central plane, which for simplicity is treated as infinitely thin; 'T is the potential value in a point x = a0, where the field intensity is equal to zero. Value a0 is equal to half the distance between planes if the density of charges on the planes is equal.

The potential at the left side and on the right side of the conducting plane is identical, '(—0) = '(+0) = '0. But a field intensity on a surface of the plane at the left side E(—0) = E01 and at the right side E(+0) = E02 differ because the distance from the central plane to the left plane 2a01 and to the right plane 2a02 differ. Thus an electrostatic pressure on a plane is:

p = (Eo22 - Eo2i).

(qq - x)ei) 2

The size a0 is the monotonously falling function of E1. If, for example, distance to the left plane 2a01 is more than the distance up to the right plane 2 a02;we have E01 > E02 and p < 0. Otherwise, the resulting pressure force is directed to the most removed plane. In particular, if we remove one of planes to an infinite distance, two planes will repulse.

Thus, the attraction of dust particles can arise only in a geometry that is not flat.

3.2.3 Accuracy of the potential calculation near the surface

In the numerical integration of the Poisson-Boltzmann equation, the value of the field intensity is determined in the grid points of a difference scheme. The value E0, determined approximately, corresponds to a field value some distance from a dust particle surface, of the order of a grid step. Let us estimate the error of calculated pressure. The relative difference of pressure determined at distances x and - x from a plane is given in flat geometry by the expression

Ap = jp - (E2(-x) - E2(x))| p p

As one can see in Figure 6, if the potential of a plane is not small ('0 ^ 1), even on small distances x ~ 0.01, the value Ap/p is in the approximate range of tens percents. At the same time, the difference of potentials at the left and on the right sides ('(—x) — '(x)) is practically equal to zero. Otherwise, the very high accuracy of calculation of the potential derivative near a dust particle surface requires numerical integration. Therefore, it demands a very small grid step near the surface.

Distances between dust particles much exceeding their diameter are of the most interest. At the same time, the method used for the numerical integration of the Poisson-Boltzmann equation should provide the maximal accuracy in the area near the surface of dust particles for an exact calculation of force on a dust particle. It is difficult to achieve sufficient accuracy in the calculation of force on small dust particles in the usual systems of coordinates.

4 The method of a two-center problem solution

4.1 Cassini coordinates

We used orthogonal coordinates constructed based on a known Cassini oval for a special case.

The relationship between variables u and v, specifying a point on Cassini oval with the Cartesian coordinates in quadrant x>0, y>0, is determined by the following expressions:

x(u,v) = —-\Zexp(2u) + 2exp(u) • cos(v) + 1 + exp(u) • cos(v) + 1, (13a) 2v2

y(u,v) = —v^"xp(2u) + 2exp(u) • cos(v) + 1 — exp(u) • cos(v) — 1. (13b) 22

The oval focus is located in point (d/2,0). Variable i>u>-1 is some analogue of a radial variable. At u<0, curves represent two independent ovals: at u=0 a coordinate line is a Bernoulli's lemniscate, that is, an oval with an infinitesimal waist. At 0.65>u>0 it represents an oval with a waist, and at u > 0.65 the oval has the ellipsoidal form. Variable n>v>0 is an analogue of a corner in polar coordinates. At v=0 points lay on a beam (d/2,i) on the x-axis, at v = n, the points come close to a corner formed by a line segment (0, d/2) on the abscissa and beam (0,i) on the ordinate. The character of coordinate lines is illustrated in Figure 7.

Use of coordinate (13) gives the following important advantages. First, the family of Cassini ovals qualitatively corresponds to an equipotential curve for two equally charged particles that are located in oval focuses. Second, the domain of the solution of equation (3) in these coordinates becomes rectangular. Third, the density of ovals is exponentially condensed to a surface of a dust particle. It makes an opportunity to use a uniform mesh even at the large distances between particles of small sizes.

4.2 On the method of numerical simulation

Without going into details, let us discuss the basic items of the numerical simulation method. The Cassini coordinates are especially convenient for use in a situation in which the radius of dust particles r0 is much less than the Debye radius r0 ^ 1, and the radius of the Debye atom r0 ^ a0. It is convenient to define the potential value '0 on small ovals close to circles. At the same time, the cloud of charges covering dust particles is described by an elliptical oval. It is convenient to set the field value to zero at this oval.

The surface of a dust particle and the surface corresponding to the boundary of a Debye molecule are described in coordinate (13) by constants:

Umin = ln^(d + 7-0)^ , umax = ln ^df (d + «0^ (14)

The boundary conditions (7) thus look like:

= 0 (15)

u—Umax

The Poisson-Boltzmann equation (3) with boundary conditions (14) and (15) was solved by a Gauss-Newton method of iterations with use of the software package MAT-LAB.

Figure 8 shows plots of an equipotential surface in two coordinate systems.

The three-dimensional coordinates formed by rotation of flat coordinates (13) around the x-axes are used to calculate the charge (5) and the interaction forces (8) of dust particles. The interaction energy of dust particles was calculated using the formula

U (d) = — f (x)dx + const. (16)

The constant usually was taken so that the potential energy was equal to zero in some point.

5 Results of calculations

5.1 Debye molecule in a cloud of charges of one sign (6 =0)

5.1.1 Parameters of calculations

The calculations were carried out for such parameters '0, r0, and a0 that correspond to an isolated Debye atom when d ^ a0. For this purpose the spherically symmetric problem (9) was solved and the potential '0on a particle surface for given r0 and a0was calculated. Then the two-centered problem for d =10a0 was solved using values '0, r0, and a0. The results for the spherically-symmetric problem and for the two-center problem coincided with high accuracy. Smaller values of d were used in the further series of calculations.

In a series of calculations shown in Figure 9, we were guided by plasma parameters of experiments by [3], and have put a0 = 0.755. The calculations show that the area at large distances d ~ 2 a0 is most interesting. Therefore, we have taken the radius of a dust particle r0 = 0.1 five times greater than in the experiment. Accordingly, potential '0= 1.16, taken from the one-center problem solution for r0 = 0.1, was smaller than the potential on a surface of a dust particle of small radius ('0 = 6.5 at r0 = rp/rD = 2.23-10_2). Otherwise, the small conducting ball was replaced by a conducting ball of greater size, with a charge partially compensated by charges of an electronic cloud. Such replacement is justified because the electrons situated near the dust particle surface are weakly polarized (see below).

5.1.2 Interaction force dependence on dust particle separation

A series of calculations with the given values '0, r0, and a0 were carried out for different values of d. The dust particle charge z0 is also a function of d in this case. Additional calculations were carried out with changed '0 or a0 to make the dust particle charge z0 not dependent on d.

The calculations have shown that the repulsion takes place at small distances between particles d~r0. It is not in accord with results of numerical calculations of [17], in which the dust particle attraction took place at d ~ r0. Apparently, there was some error in the calculations of electric field near the surface of the dust particle. The resulting force is very sensitive to such error (see 3.2). Actually, the charged cloud is weakly polarized close to the surface of a dust particle, so the repulsion force prevails over the polarizing attraction force at small distances.

Figure 9 show that the dust particles' interaction force have zero value at equilibrium point d = d0 ° 1.3 under the conditions of [3]. The position of the equilibrium point d = d0, in which a sign of interaction force changed, is less then the average distance between dust particles (2a0 = 1.5). The value d0 weakly depends on which quantity ('0, a0, or z0, a0) was kept constant in calculations at different d. The change a0 (at constant z0

and '0) influences the value of d0 some more. Apparently, it is better to make z0= const by changing the dust particle potential '0 = '0 (d).

It is impossible to consider a problem binary when d ^ a0. The essential repulsion from other dust particles takes place if the distance between dust particles is large (d > 2a0) (see a Figure 1). Therefore, we present the results of calculations only for d < 4a0.

One can estimate the electrostatic pressure compressing a dust particle gas as a function of an attraction force of dust particles F(2a0) at average distance 2a0,

Pe ° F(2ac) ■ Nf3 = ^i) ■ N2/3 ■ f (2a«). (17)

For surface tension in a "dust liquid" one has

( N1/3T2 \ ffE ° F ■ N2/' = [n^t) ■ f (2®0).

Comparing electrostatic pressure on dust particles with gas-kinetic pressure of dust particles and gas-kinetic pressure of electrons we have:

Pe/NPT = (T/8ne2Np/3) ■ f (2a0); Pe/NeT = (T/8ne2Ne) ■ N2p =3 ■ f (2a0).

In conditions of the experiments of Fortov et al. [3] one has |f (2a0)| ° 0.2; Pe = 9.740"7 ■ |f(2a0)! Top ° 2■10-7■ Top; ge = 3.5■10-9 ■ |f(2a0^H/M »740"10■ H/M; Pe/NPT ° 20; Pe/NeT ° 0.04.

Note, however, that the comparison of electrostatic pressure on dust particle gas with gas-kinetic electronic pressure does not allow one to make any essential conclusions. Electrons are not free; they are in an electrical field of dust particles. At the same time, it is possible to assume that the gas of Debye atoms in a mix with inert gas should show the tendency to compression under the conditions of the experiments. Such a situation was considered by [12]. Consideration of the influence of Debye atoms' interaction on the gas-kinetic property of dusty plasma is outside the framework of this paper.

5.1.3 Influence of the Debye atom size

The results of some series of calculations for various values a0 are presented on Figure 10. The calculations have shown that the attraction of the dust particles takes place only at a0 < 1. Already at a0 > 1.12, the equilibrium point d0 goes to large distance d0 > 4a0. It is possible to rewrite the condition a0= ap/rd < 1 for the dimensional quantity,

Ne0 > Necr = ~~o N^ ■ (18)

The electrostatic forces of compression become zero when d0 = 2a0, that is, when a0 = 1. Accordingly, condition a0 = 1 or Ne0 = Necr is a condition of pressure balance of the Debye atoms gas.

The condition of the high particle charge zp > 1/3 can be rewritten for a dust particle charge in terms of an electronic charge as follows

_ n Necr

Zp > Zecr =--.

p ecr 6 Np

We have Necr = 4.4-1010 cm-3 and Zecr = 460 for conditions of [3]. These values agree with values measured in the experiments: Ne0 ° 2.5-1010 cm-3, Zp ° 500.

The depth of a potential well cannot be determined based on two Debye atoms without the consideration of other dust particles. Therefore, we shall characterize the value of the interaction force of Debye atoms by its steepness in the equilibrium point:

e = f '(d)id=do = U//(d)id=do •

The oscillation frequency of dust particles around the equilibrium point d = d 0 is

! = 1611/2 - !0; !0 = — •

Here vT = (2T/mp)1/2is a thermal speed of dust particles and mp is their mass. Under the conditions of [3], we have mp ~ 2 10-12 gm, vT ~ 0.5 cm/sec, ap ~ 1.4-10-3 cm, and !0 = 357 sec-1, with 2n/!0 = 18 ms. Figure 10 shows that the strongest coupling takes place at 0.5 < a0 < 1. The gas of Debye atoms tends toward compression at these conditions.

For small-radius dust particles, the size of the Debye atom is also smaller. The results of calculations of the maximum value of the Debye atom radius a0max = a0(z0 ! 1), as a function of r0, can be approximated for convenience by this expression: a0max = 3- r0'3, or r0= (a0max/3)10/3 when r0 < 0.02. Consequently, the attraction takes place when the radius of a dust particle is not too large and not too small. At 0.5 < a0max < 1, we have a condition 2.5-10-3 < r0< 2.6-10-2. In the experiments of [3], r0= 2.23-10-2, and this condition is satisfied.

5.1.4 On the effect of a dust particle size

The small charged ball is replaced above with a ball of greater size and accordingly with a partially compensated charge. There is a natural question whether such a replacement is adequate. Some series of calculations were done with different values of r0 and corresponding values of '0. The change in the results of the calculations is insignificant if a dust particle radius is small in comparison with the radius of Debye atom a0.

For example, in a case a0 = 0.755 (see Figure 11) for r0 = 0.1¥0.2 (and for the choice of values of '0 corresponding to the given r0), the differences in the equilibrium point d0 = 1.28 is less than 2%, which corresponds to the available accuracy of calculations. The effect of the dust particle size becomes significant for r0 > 0.3a0. For r0 > 0.4, the polarization-induced attraction decreases to such an extent that the distance to the force sign-reversal point becomes larger than the mean distance between particles (d0 > 2a0). Therefore, it is possible to conclude that the electrons placed at distance r ° (0.3¥1)a0 are involved in polarization. In this connection, it is difficult to hope for an analytical evaluation of attraction forces.

5.2 A Debye molecule in a plasma (5 = 0)

5.2.1 Interaction force dependence on dust particle separation

As in the case of 5 = 0, the series of calculations were carried out to obtain the dependence of the interaction force of dust particles on distance d. The additional calculations were carried out with changed '0 or a0 to make the dust particle charge z0 independent of d. As in the case of 5 = 0, we chose the value of r0 greater than the radius of the atomic core, thus simulating a dust particle by a conducting sphere of a larger size, with a charge partially compensated by the free charges from the shell of the Debye atom. Thus, the polarization of the core was disregarded.

In the results shown in Figure 12, the size a0 for different values of 5 corresponds to the extremely large charge of a dust particle with radius rp/rD = 2.2340-2. This was done by test firing: when the value of a0 was greater than that given in Table 7, the particle charge obtained by solving Eq. (9) becomes infinitely large (z(rp/rD) ! 1). The obtained dependencies z(r) and '(r) were used for determining z0 = z(r0) and '0 = '(r0), at r0 = 0.1.

We did not get an evident attraction of dust particles at 1-5 ^ 1 in the range of parameter d < 2a0that corresponds to binary interaction (Figures 12a, 12b). The attraction arises when an appreciable share of a positive charge of plasma is carried with dust particles (at 5 < 0.7, see Figures 12c, 12d). The maximum attraction force and the maximum depth of a potential well arises when 5 = 0.

The decrease of an attraction force with growth of 5 has a simple explanation. As follows from the above calculations for 5 = 0, the attraction forces arise because electrons accumulate near the center between dust particles and provide an attraction of positively charged dust particles to the center of Debye molecule. This attraction force exceeds the repulsion force of dust particles because the Debye atom core screens the dust particle charge. At 1- 5 ^ 1 the effect of a charge screening by the core is the same. However, the attraction force essentially weakens because not only electrons but also positive charges are accumulated near the center of the Debye molecule.

In case of a small value of a plasma charge 5 ^ 1, the potential well depth is rather great. It is about several values of gas temperature. However, the binary consideration is limited in size of the order of magnitude of a diameter of Debye atom 2a0 (Np 1/3 > 2a 0r d ).

5.2.2 On the analytical approaches

The above conclusion concerning the absence of attraction for 5 ! 1 contradicts the results of recent approximate analyses by [5], [11] (see Figure 12a). It follows from these analyses that the attraction of dust particles takes place at 5 = 1 and at interparticle separation d > (31/2+1) /21/2 = 1.93 if the linearized Poisson-Boltzmann equation is used.

This result is surprising. In the linear approximation a potential in a point r is determined by the sum of the screened potentials of point charges located in points r1

and r2:

'r) = ©(|r-r1|) + ©(ir-r2i), ©(,)= -exp^), ,2-^1=,

According to simple reasons stated above in 3.2, the attraction force cannot take place in the absence of perturbation of a charged cloud of one dust particle under influence of other dust particles. The linear approximation should be

f (d) = -• 9©(x)dx Ix=d = zp2 ^ ^ ' J+ ^ • exp > 0,

which corresponds to repulsion.

Inaccuracy of the results of [5] and [11] is apparently associated with the following circumstance. Gerasimov and Synkevich [5] and Ivanov [11] sum the attraction force acting on an electronic cloud of the first dust particle from the second dust particle, and force (19) acting directly on a first dust particle. Such addition would be justified if the charged clouds of dust particles were rigidly connected with the dust particle through some other forces. However, there are no extraneous rigid forces in the problem under consideration. The presence of the attraction force of the electron shell of one charge to another charge only indicates that the given configuration of the charge shell is not in equilibrium. This force of attraction must lead to polarization of the charge shell. Nevertheless, the polarization was disregarded by [5] and [11]. There are no grounds to add this polarizing force to the force acting directly on the dust particle.

An analogous situation is the polarizing attraction of ordinary atoms. As is well known, for spherically symmetric atoms the polarizing interaction has no place in firstorder perturbation theory. It arises only in second-order perturbation theory, when the polarization of an electronic shell of one atom by charges of other atom is taken into account. An ordinary atom differs from a Debye atom only by the fact that its electrons move according to quantum-mechanical and not classical laws. The nature of polarization-induced forces is the same for an ordinary and a Debye atom.

6 Dust particles in nuclear-pumped plasma

6.1 Experimental results

Fortov et al. [4] reported on the collective phenomena observed in dust plasma formed because of dense gas ionization by nuclear fission fragments.

In one of these experiments, the plasma was excited by Cf252 fission fragments, and in the other by-products of the Ce141 ^-decay. We will concentrate on the latter data. The dust was composed of Ce02 particles with an average radius of rp = 0.5 fim. The gravity force was compensated by applying an external electric field with a strength of 10 V/cm. The system featured large regions of particles levitating over several minutes, exhibiting a short-range order in the spatial structure.

Measurements performed using a digitized video image of the structure of these zones shows the density of particles within a 150-^m-thick flat layer was 10~5^mp2. Accordingly, the volume density of dust particles was Np ~ 6-104 cmP'. The average charge of these particles, determined from the balance of gravitational and electrical forces, was Zp ° 400. The density of the charge of dust particles was ZpNp ~ 2.4-107 cmP'. The ion density, determined by measuring the current between electrodes and using the known ion drift velocity, was Ni ~ 108 cmP'.

The attraction of dust particles causes the collective phenomena in the nuclear-pumped plasma under consideration. As was stated above, the attraction of dust particles takes place if the charges of one sign are concentrated mainly on dust particles. Now we will check whether this condition is fulfilled [19].

6.2 The charge of dust particles

A negative charge on the dust particle surface may arise from a difference between average velocities of electrons and ions. This phenomenon is well known in physical electronics. Assuming the Maxwell velocity distribution and equating the flux of ions to the particle surface Ni-ui to that of electrons Ne-ue-exp(-e0/Te), one obtains:

= (TeN , / NmT_N _ Vp \2e) \N, me Te)

Here, 0p is the dust particle potential; ui = (T/4nmi)1/2 and ue = (Te/4nme)1/2are the average projections of the velocities of ions and electrons onto the axis perpendicular to particle surface; and Te and T are the electron and gas temperatures. Using this potential value, we may formally determine the charge of the particle:

° rjpip ° (N. ln (NmT y

p e \'2e2 ) \Ne me Te)

This estimate applies well to gas-discharge plasma, but may lead to considerable errors in the case of a plasma produced by a hard ionizing. Taking the electron temperature equal to the room temperature (Te ~ T = 300 K = 0.026 eV), we obtain Zp ~ 100. This estimate is about one-fourth of the value obtained from the experimental data (Zp ° 400). Apparently, the discrepancy is related to the fact that the secondary electron adheres to a dust particle before it is cooled in collisions with gas molecules.

6.3 Density of ions

The charge-balance equation and the quasi-neutrality condition describe the number densities of ions and electrons in the dust plasma. In the case under consideration, these relationships can be written as follows:

dN- = G - adNtNe - aLNiNp, Ne = N - ZpNp

Here ad is the dissociative recombination coefficient and aL is the Langevin recombination coefficient; G is the ionization rate per unit volume.

Under quasistationary conditions (dNi/dt = 0), we may solve the above quadratic equation and present the ratio of the ion density Ni to the charge density on a dust particle ZPNP in the following form:

N fa - 1\ 2 a - 1

ZN = V(—)+ ag " — ■ (20)

Here, a = ®L/(Zpad) is a parameter characterizing the ratio of the rates of the Langevin and dissociative recombination (for a > 1, recombination on the dust particles dominates), and g = G/(aLZpNp) is the reduced rate of ionization. Note an important circumstance: for an ionization rate satisfying the condition g = 1 or G = aL-Zp-Np, all the negative charge in the system is concentrated on the dust particles (Ne = 0, Nj= ZpNp) while the gas contains only positive ions.

In the experiments under consideration, the radioactive source provided ionization at a rate corresponding to 109/-decays per second in a volume of 20 cm3. Assuming that every 3-decay event liberates an energy of Ef = 138 keV, we obtain the following estimate for the ionization rate per unit volume:

' 109 -s"1 \ / Ef

G „ fiOL^ • (f „ 2 • 10" • ,-W

\20 • cm3) \EprJ

Here Epr = 36 eV is the energy necessary for the ion pair production in air.

The coefficient of ion recombination on dust particles according to Langevin is ®L = Zpe2bi ~ 0.064-cm3 s_1. This estimate is obtained for the ion mobility bi= 2/(mi-N-kia), where kia = (4/3)-4-10_16 cm2-(8T/x-mi)1=2 ~ 2.5-10"11 cm/s is the rate of collisions between ions and air molecules, considered as hard spheres with a cross section of 4-10"16 cm2.

The ion density and the share of the dust particles charge 6 = ZpNp/Ni can be estimated using (20). We use the dissociative recombination coefficient equal to ®d ~ 3-10"7 cm3/s and take into account that recombination on the dust particles dominates: a = aL/(ZPad) ~ 530. The estimated number density of ions Ni ~ 0.5-108 cm-3 agrees with the experimental values.

Moreover, expression (20) shows that, for the parameters under consideration, the negative charges are concentrated appreciably on dust particles, 6 = ZpNp/Ni ° 0.5. Thus, the attraction of dust particles can take place in these experiments.

7 Conclusion

Let us summarize the results of the above consideration.

(1) A Debye atom consists of a charged dust particle and shell (cloud of charges). For the large charge of the dust particle, the high-density region (core) of the electron cloud screens considerably the large charge of the dust particle near its surface. In

this connection, while considering the interaction of Debye atoms, we cannot ascribe the unscreened value of charge to a dust particle. The dust particle charge screened by the core has a universal value determined by the distance between dust particles. The electron shell of the Debye atom screens it.

(2) Attractive forces are associated with the polarization of charge shells of Debye atoms. The force of attraction is formed by polarization of a large fraction of electrons of the charge shell. The polarization of the core is insignificant.

(3) Forces of attraction between dust particles emerge at a comparatively large distance, approximately equal to the mean separation between dust particles. In this case, the Debye radius must be approximately equal to half the mean distance between dust particles.

(4) Attraction takes place if like charges are concentrated predominantly on dust particles. If dust particles carry a small fraction of the charge of some polarity, repulsion is observed at any distance.

(5) The electrostatic forces of interaction between dust particles vanish when a certain relation between the electron density and the density of dust particles converges. In this case, the Debye "liquid" is in equilibrium.

Since attractive forces appear at large distances, the problem of the formation of dust liquids and crystals can be solved correctly only if many-particle interactions are taken into account. However, we can draw the following two conclusions concerning the criteria for the emergence of collective phenomena based on the results presented by us here:

(a) in the case of a thermionic plasma, the electron density must be such that the Debye radius is approximately equal to half the mean value between dust particles;

(b) for a gas-discharge or a nuclear-excited plasma, the properties of the ionization source and the density of dust particles must be matched so that the main (usually negative) charge is carried by dust particles.

Acknowledgments

The authors are grateful to A.N. Tkachev for fruitful discussion of the results of the

present work and also Yu. I. Syts'ko for discussion of computational aspects of the

problem.

References

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Type of a curve <5 a o '0 2 0

dot a, b 0.999 4.1 2.455 0.282

dash a, b 0.9 1.71 2.426 0.283

dash-dot c, d 0.7 1.288 2.413 0.272

dot c, d 0.5 1.12 2.378 0.286

dash c, d 0.3 1.018 2.364 0.286

solid c, d 0.1 0.94 2.292 0.279

Table 1 Parameters used in calculations presented on Figure 12.

Debye molecule

d ~ 2ao

Charged cloud of Debye atom

Dust particle

Debye atom core

Fig. 1 The outline of Debye atoms and Debye molecule

Fig. 2 Dependence of a charge z(r) (solid curve), a field intensity E(r) (dotted curve), and potential '(r) (dashed curve) on distance r to the center of a dust particle S = 0. The radius of

Debye atom ao=

= AT-l/3

ap = N p

/2rd = 0.755 is taken from conditions of experiments of Fortov et

al (1997): T = 1700E, Np = 5-107cm rp/rD = 2.23-10-2.

Ne0= 2.5-1010 cm 3, rD = 18 eee, rp = 0.4 eee, r0=

Fig. 3 Interconnection of dust plasma parameters with Debye atom parameters: a) Dependence on parameter S of the Debye cloud, dimensionless electron chargezoe (solid curve) and ion chargezoi (dotted curve); b) Dependence on parameter S of parameter Si (solid curve) and radius of Debye atom ao (dotted curve). Parameters of a dust particle ro = 0.1, '(ro) = 2.4, z (r o) = 0.28.

r(r), E(r\ cp(r)

1 -ltr

I -10"

TO 1 1

1 1 \\ \

Fig. 4 Dependence of a charge z(r) (solid curve), a field E(r) intensity (dotted curve), and potential '(r) (dashed curve) on distance from the center of a dust particle, measured in terms of Debye radius, S = 0.999. Radius of a dust particle ro = 2.23 • 10~2 is taken from conditions of experiments of Fortov et al. (1997).

cp = cp0

cp = cpo

cp = cpo

eo2 e =0

left plane

central plane

right plane

Fig. 5 The outline of interacting charged planes.

Fig. 6 Dependence of an error in calculation of electric pressure on a conducting plane ^p =

——(E ( —E (solid curve) and difference of potentials '(-x) tg'(x) at the left and at the right of a plane (dotted curve) on distance x up to a plane. The considered plane is between two other charged planes; all planes are under potential 'o = 10. Half of distance up to the left plane ao = 6.27, up to right - ao = 2.08; thus p = 2.

Fig. 7 Gridline of Cassini coordinates.

U,V,q>

Fig. 8 A potential surface in the Cartesian coordinates '(x, y) (a) and in Cassini coordinates '(u, v) (b). The calculation is for this case: 5 = 0, r0 = 0.1, a0 = 0.755, '0 = 1.16.

i i i i i ' /\ \ :

2 ro 2ao / /

/ / / _

\ ¿1/

A / / /

- v\ / / -

l'\ / /

V'\ / / /

A / I /

V\ / /

\'.\ / / y

\',\ / / /

l\\ / / /

- l\\ / / /

\ \ i /''

\ \ ^ /''

1 1 1 1 1

Fig. 9 Dependence of a force projection f on an axis x (a) and potential energy of interaction of dust particles U (b) on distance between them d for a case J = 0. Normalization of potential energy U(d) is taken so that in a minimum U(d) equals zero. The solid curves correspond to constant potential on surfaces of dust particles '0 = 1.16. The dotted curves correspond to a constant charge of dust particles z0 = 0.156, achieved by fitting '0(d). The dash-dot curves correspond to a constant charge of dust particles z0 = 0.156, achieved by fitting a0(d) at '0 = 1.16. In Figure 9a the thin dash-dot curve gives the dependence of a dust particle charge z0(d) on distance d for a case of constant potential '0 = 1.16.

1 1 1 1 1 1 :/ ' 12

—--L." *0 - i i i i i i 1 1

05 0.6 0.7 0.8 0.9 1 1.1 1.2 1 3

Fig. 10 Dependence of the equilibrium-point coordinate d0 (thick curves) and steepness of force £ in a point do (thin curves with squares) on the size of Debye atom ao. The solid curves correspond 'o(d) = const; the dotted curves correspond to a zo(d) = const achieved by fitting 'o(d ).

1-e-1-1-1-1-1-

0.1 0.15 0.2 0.25 0.3 0.35 0.4

Fig. 11 Dependence of the coordinate do of the sign-reversal point for the force (equilibrium point) on ro/ao. Here, rocan be treated as the radius of the domain in which the polarization of the charge cloud is disregarded. It is possible to consider roas a radius of an area where the polarization of a charged cloud is neglected. Potential 'o for r = ro was determined for ao = 0.755.

f(d) U(d)

Fig. 12 Dependence of x component of the force f at plots r and c, and potential energy of interaction of dust particles U at plots b and d on distance d between them for different values S = 0. In all cases ro = 0.1. The thin solid curves in Figures a and b correspond to the analytical expressions of [11]: f (d) = const-(1/d)-(1+ d t 1/2d2)-exp(td); U(d) = const-(1/d2)-(1t 1 /2d )-exp(t d).