THE INFLUENCE OF SCREENING EFFECTS ON THE GRAIN CHARGE IN A THERMAL DUSTY PLASMA

The thermal dusty plasma characterized by the equality of the temperatures of all components occurs mainly under the terrestrial conditions in the combustion products of various fuels shukla . One of the important problems of the theory of a dusty plasma is the problem of determination of the charge of dust grains fortov_ufn ; morfill_rev . This problem is solved with the use of various methods considering its specificity with that or other degree of adequacy. Below, we discuss the most typical works.

In work einb , the existence of an analogy between the process of thermoionization of dust grains and the process of ionization of atoms was used for the determination of the charge of dust grains. The charge of atoms at the ionization varies by at most several values of electron charge. Therefore, it is possible to neglect the screening effect and to consider that the electrons pass a sufficiently large (infinite) path, by leaving the ions emitting them. The same approach was applied in work einb to the determination of the charge of identical dust grains of radius $r_{\text{p}}$. Within this method, the analog of the ionization energy $I$ in the Saha equation is the work function of electrons leaving the surface of grains, which have charge $z$: $I\Rightarrow W_{0}+ze^{2}/r_{\text{p}}$, where $W_{0}$ is the work function for the surface of neutral grains, and the term $ze^{2}/r_{\text{p}}$ ($z$ is the charge in units of the electron charge $e$) has meaning of the work needed to move an electron from the surface of a grain to infinity. It was shown that, under the made assumptions, the mean charge $Z$ of dust grains for $Z\gg 1$ is given by the expression

where $n_{\text{es}}=2(2\pi m_{\text{e}}kT/h^{2})^{3/2}\exp(-W_{0}/kT)$ is the equilibrium density of electrons near the surface of an emitting grain, and $n_{\text{e}}$ is the mean density of electrons in a plasma.

Within the approach developed in einb , which will be called quasichemical, works smith ; armus1 ; armus2 considered the possibility to form a negative charge and constructed the distribution function over charges of grains. It was shown that the coefficient $kTr_{\text{p}}/e^{2}$ is the distribution variance. We note that a small number of dust grains can be negatively charged, but, on the average, the dust grains in a plasma of combustion products have a positive charge.

In work armus3 , the charge of dust grains of the same size $r_{\text{p}}$ is determined from the condition of balance of currents: the thermoemission current is balanced by the current of electrons, which recombine with charged dust grains. The thermoemission current is determined, like that in einb , by the Richardson–Dushman equation. It is assumed that all electrons colliding with a dust grain recombine. The recombination rate is estimated in the approximation of binary collisions as $-\alpha n_{\text{e}}n_{\text{p}}$, where $\alpha=\sigma v$, $\sigma$ is the cross-section of a dust grain ($\sigma\approx\pi r_{\text{p}}^{2}$), $v$ is the mean thermal velocity of relative motion, and $n_{\text{p}}$ is the mean density of dust grains in a plasma. The balance equation for flows allows one to establish the connection between the mean charge $Z$ and the mean density of electrons $n_{\text{e}}$: $Z=kTr_{\text{p}}/e^{2}\,\ln n_{\text{es}}/n_{\text{e}}$. In fact, this result coincides with $\eqref{eq:einb}$. In the same work, the value of $n_{\text{e}}$ was taken from experimental data (in principle, it can be calculated theoretically), which gave the estimate of the value of charge. For example, for $n_{\text{e}}=3.47\times 10^{10}$ cm^-3, $n_{\text{p}}=2\times 10^{7}$ cm${}^{-3},$ $T=3285$ K, and for the grains with the radius $r_{\text{p}}=10^{-4}$ cm, the charge $Z$ was calculated to be 1700.

There is no basic difference between the above-mentioned approaches, which was indicated in sodha1 ; sodha2 , where the formula of equilibrium constants was obtained by the kinetic way. This was further used in einb ; smith ; armus1 ; armus2 .

Works samuy1 ; samuy2 ; samuy3 ; samuy4 ; zimin_etal present the attempts to consider the inhomogeneous distribution of electrons near a dust grain and the influence of the appropriate electric field on the recombination. In samuy1 , the screening effects were considered in the linear approximation. Work zimin_etal dealt with a separate dust grain, and the distribution of the electric field in its neighborhood was found as a solution of the Poisson equation in the Debye–Hückel approximation.

A certain contradiction is inherent in the above approaches: the introduction of the notion ‘‘the mean density’’ is not proper, because it was assumed that the thermoemission electrons go to infinity.

Therefore, some different approaches were naturally developed. There, it was considered that the thermoemission electrons remain in a bounded region including the dust grain gibson ; zhuh_etal ; maren1 ; maren2 ; dragan_cell . This region (we will call it the cell) is electrically neutral. The electrons can pass from one cell to another one, but the mean number of electrons in each cell under the given conditions remains fixed. Such an approach allows one to consider only one single dust grain in a cell, i.e., the problem becomes one-particle. We note that, in the frame of cell approaches, the introduction of the mean density of emitted electrons is quite proper.

The system composed from identical spherical dust grains, electrons emitted by them, and a buffer gas, was considered in gibson . Each dust grain is located inside an electrically neutral spherically symmetric cell of radius $r_{\text{c}}$. The size of a cell is determined from the geometric reasoning as half the mean distance between dust grains:

The dust grains have the same charge $Ze$. The electric field inside a cell is determined by the numerical solution of the Poisson equation under relevant boundary conditions. In gibson , it was accepted that, on the boundary of a cell, the potential and the strength of the electric field are zero. For the cases of weak and strong screenings of a grain by the electron cloud, the approximation formulas for the electric field potential were obtained. In work zhuh_etal , it was shown that the solution of the electrostatic problem in the case of the weak screening ($r_{\text{D}}\gg r_{\text{p}}$) transfers in (1), if the subsystem of dust grains is rarified.

In maren1 ; maren2 , the electric field inside a cell was determined by the solution of a linearized Poisson equation supplemented by two boundary conditions, which establish: (i) the equality of the electric field strength on the boundary of a cell to zero and (ii) the connection of the electric field strength on the surface of a grain with its charge.

We note that the boundary condition (i) is trivial (it follows from the Ostrogradskii–Gauss theorem) and gives no new results. Moreover, the influence of the electric field on the charging of dust grains was not considered in the proper way.

Work dragan_cell , which was also based on the cell approach to the description of properties of a dusty plasma, used the notion ‘‘the plasma potential’’ (‘‘bulk plasma potential’’) introduced earlier in dragan1 ; dragan2 . It was proposed to reckon the electric field potential $\phi$ from this ‘‘plasma potential’’. The authors commented on such a choice of the reference point for the potential that, only in this case, one can set $\phi(r)|_{r\rightarrow\infty}\rightarrow 0$. In the previous works, ‘‘the plasma potential’’ was defined so that the Boltzmann distribution was consistent with the condition of the electric neutrality of a plasma. This proposition is not proper, since the process of thermoelectronic emission from an isolated grain is nonstationary, and the use of the Boltzmann distribution is impossible.

The consideration of the processes of charging and the influence of the screening effects on them has become especially actual after the discovery of ordered structures in the thermal dusty plasma nefedov_ufn . This problem was analyzed, in particular, in works zagor1 ; zagor2 ; zagor3 , where the influence of screening effects on the charging of a separate grain was studied. In work zagor1 , the behavior of the screened field of a dust grain, which was described with a linearized Poisson equation, was studied by numerical methods. It was shown that the essential deviation from a linear theory of screening should be expected for grains, whose radius is of the order of the Debye one. The dynamics of the charging of a dust grain in the presence of external sources of ionization with regard for the photoemission from its surface was numerically studied in zagor2 . The value of charge was determined from the condition of equality of flows. The densities of electrons and ions in the vicinity of a dust grain and the charge of a grain as a function of the time were calculated. In work zagor3 , the influence of boundaries on the screening of a point-like dust grain was studied.

In addition, the Saha equation became again used as a tool to describe the properties of a dusty plasma in the recent works sodha_etal1 ; sodha_etal2 (see also dragan1 ; dragan2 ). This is explained by the simplicity and the physical clearness of this approach. Therefore, the necessity to comprehensively study the possibility of the use of the quasichemical approach seems obvious.

In the present work, we study the value and the variance of the mean charge of dust grains in the thermal plasma. To this end, we use: (i) the cell approximation and (ii) an approach based on the quasichemical model of the charging of dust grains. No buffer gas is considered.

Here, we will formulate the basic positions of the cell model and will determine the energy of the electrostatic field created by a charged dust grain. The limiting case of a weakly charged grain will be analyzed as well.

In this section, we recall the main results of works einb ; smith ; armus1 ; armus2 and generalize the approach developed in them. For this purpose, we use the results obtained in the previous section. We will construct the distribution function of grains over charges, like that in armus2 but with regard for the performed corrections, and will determined the variance $Z_{\text{D}}$ of this distribution.

Here, we have presented the results of theoretical studies of the influence of the screened electric field on the charging of dust grains in a thermal plasma. We have applied the following approximations: (i) the cell model of dusty plasma for the description of the distribution of the electrostatic field; (ii) the quasichemical approach for the determination of the mean charge of dust grains, distribution of grains over charges, and variance of this distribution.

The introduction of an electrically neutral cell containing a grain allowed us to: (i) consider the circumstance that the emitted electrons remain in the vicinity of a dust grain, rather than move to infinity; (ii) describe satisfactorily the distribution of the potential of the electrostatic field of a charged grain. Due to the introduction of a cell, we obtained a closed system of equations (14), (16), and (26) for the determination of the mean charge $Z$.

The influence of the inhomogeneity of the distribution of the electrostatic field is significant for $g(z)<1$. In Fig. 6, we demonstrate the line corresponding to $g(z)=0.9$ at the temperature $T=3000$ K and the radius of grains $r_{\text{p}}=10^{-4}$ cm. Above this line (region I), we deal with densities $n_{\text{p}}$ and charges $z$, for which it is necessary to use the corrected formulas for the determination of the mean charge of dust grains. Below this line (region II), the influence of the screening effect can be neglected.

We have shown that the screening causes an increase of the mean charge of dust grains as compared with the prediction of the theory omitting the screening effects. As the size of a cell increases, i.e., as the mean distance between dust grains increases, result (26) is transformed into the well-known Einbinder–Smith–Arshinov–Musin formula (23).

For the grains of the radius $r_{\text{p}}=10^{-5}$ cm, the differences in the predictions of the two approaches are slight. This is explained by the fact that the grains of such sizes have very small charges ($\sim 1$), and the screening plays no significant role.

In the experiments fortov_etal with grains $\mathrm{CeO_{2}}$ ($W_{0}=2.75$ eV), the mean density of grains $n_{\text{p}}$ varied in the limits (0.2–5.0) $\times 10^{7}$ cm^-3, and the temperature $T$ was changed in the interval (1700–2200) K. The measured mean density of electrons $n_{\text{e}}$ was in the limits (2.5–7.2) $\times 10^{10}$ cm${}^{-3},$ the mean radius of grains $r_{\text{p}}=4\times 10^{-5}$ cm, and the lower experimental bound of the mean charge $Z\approx 500$.

For the grains with the indicated radius and the density $n_{\text{p}}=2\times 10^{6}$ cm${}^{-3},$ the dimensionless radius of a cell $\varsigma\approx 62$. At the temperature $T=2000$ K, the mean charge and the mean density of thermoemission electrons are, respectively, $Z\approx 442$ and $n_{\text{e}}\approx 6.91\times 10^{9}$ cm^-3 according to (26) and (14); the charge distribution variance is $Z_{\text{D}}\approx 49$ according to (30). At the temperature $T=2200$ K, we have determined $Z\approx 560$, $n_{\text{e}}\approx 8.76\times 10^{9}$ cm${}^{-3},$ and $Z_{\text{D}}\approx 55$.

The calculated mean charge is close to the experimental result ($Z\approx 500$). However, the calculated densities of electrons are less than experimental ones by one order. This fact can be apparently explained by the neglect of the influence of an ionized buffer gas in the calculations. Therefore, the comparison of the theoretical and experimental results is not quite proper.

The study of the influence of the screening of a buffer gas will be performed separately.

We note also that the essential excess of the mean charge $Z$ over $Z_{0}$ is accompanied by the violation of the condition of applicability of the linearized Poisson equation. The error of the calculated mean charge $Z$ increases with $Z$, because, in this case, the contribution of the region, where the nonlinear effects are of importance, increases as well. This will be studied in further work.

The author thanks Prof. M.P. Malomuzh and Dr. V.I. Zasenko for the fruitful discussion of the results. The present work was partially supported by the Ministry of Education and Science, Youth and Sport of Ukraine (grant No. 0112U001739909).