Grain in a Plasma in the Presence of External Electric Field: Kinetic Calculation of Effective Potential and Ionic Drag Force

Kinetic calculations of the effective grain potential are presented for the case of weakly-ionized plasma in the external electric field. The drag force associated with the ionic drift in the external field is found. It is shown that the absorption of electrons and ions by the grain can cause the ch...

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Datum:	2010
Hauptverfasser:	Zagorodny, A.G., Rogal, I.V., Momot, A.I., Schweigert, I.V.
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Sprache:	English
Veröffentlicht:	Відділення фізики і астрономії НАН України 2010
Schlagworte:	Плазма і гази
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spelling	irk-123456789-132832010-11-04T14:36:12Z Grain in a Plasma in the Presence of External Electric Field: Kinetic Calculation of Effective Potential and Ionic Drag Force Zagorodny, A.G. Rogal, I.V. Momot, A.I. Schweigert, I.V. Плазма і гази Kinetic calculations of the effective grain potential are presented for the case of weakly-ionized plasma in the external electric field. The drag force associated with the ionic drift in the external field is found. It is shown that the absorption of electrons and ions by the grain can cause the change of the direction of the drag force. Представлено кiнетичнi розрахунки ефективного потенцiалу порошинки для випадку слабоiонiзованої плазми у зовнiшньому електричному полi. Знайдено силу опору, яка пов’язана з дрейфом iонiв у зовнiшньому полi. Показано, що поглинання електронiв та iонiв порошинкою може привести до змiни напрямку сили опору. 2010 Article Grain in a Plasma in the Presence of External Electric Field: Kinetic Calculation of Effective Potential and Ionic Drag Force / A.G. Zagorodny, I.V. Rogal, A.I. Momot, I.V. Schweigert // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 29-35. — Бібліогр.: 22 назв. — англ. 2071-0194 PACS 52.27.Lw http://dspace.nbuv.gov.ua/handle/123456789/13283 en Відділення фізики і астрономії НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Плазма і гази Плазма і гази
spellingShingle	Плазма і гази Плазма і гази Zagorodny, A.G. Rogal, I.V. Momot, A.I. Schweigert, I.V. Grain in a Plasma in the Presence of External Electric Field: Kinetic Calculation of Effective Potential and Ionic Drag Force
description	Kinetic calculations of the effective grain potential are presented for the case of weakly-ionized plasma in the external electric field. The drag force associated with the ionic drift in the external field is found. It is shown that the absorption of electrons and ions by the grain can cause the change of the direction of the drag force.
format	Article
author	Zagorodny, A.G. Rogal, I.V. Momot, A.I. Schweigert, I.V.
author_facet	Zagorodny, A.G. Rogal, I.V. Momot, A.I. Schweigert, I.V.
author_sort	Zagorodny, A.G.
title	Grain in a Plasma in the Presence of External Electric Field: Kinetic Calculation of Effective Potential and Ionic Drag Force
title_short	Grain in a Plasma in the Presence of External Electric Field: Kinetic Calculation of Effective Potential and Ionic Drag Force
title_full	Grain in a Plasma in the Presence of External Electric Field: Kinetic Calculation of Effective Potential and Ionic Drag Force
title_fullStr	Grain in a Plasma in the Presence of External Electric Field: Kinetic Calculation of Effective Potential and Ionic Drag Force
title_full_unstemmed	Grain in a Plasma in the Presence of External Electric Field: Kinetic Calculation of Effective Potential and Ionic Drag Force
title_sort	grain in a plasma in the presence of external electric field: kinetic calculation of effective potential and ionic drag force
publisher	Відділення фізики і астрономії НАН України
publishDate	2010
topic_facet	Плазма і гази
url	http://dspace.nbuv.gov.ua/handle/123456789/13283
citation_txt	Grain in a Plasma in the Presence of External Electric Field: Kinetic Calculation of Effective Potential and Ionic Drag Force / A.G. Zagorodny, I.V. Rogal, A.I. Momot, I.V. Schweigert // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 29-35. — Бібліогр.: 22 назв. — англ.
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first_indexed	2025-07-02T15:13:21Z
last_indexed	2025-07-02T15:13:21Z
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fulltext	GRAIN IN A PLASMA IN THE PRESENCE OF EXTERNAL ELECTRIC FIELD GRAIN IN A PLASMA IN THE PRESENCE OF EXTERNAL ELECTRIC FIELD: KINETIC CALCULATION OF EFFECTIVE POTENTIAL AND IONIC DRAG FORCE A.G. ZAGORODNY,1 I.V. ROGAL,1 A.I. MOMOT,2 I.V. SCHWEIGERT3 1Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine (14b, Metrolohichna Str., Kyiv 03143, Ukraine; e-mail: AZagorodny@ bitp. kiev. ua ) 2Kyiv National Taras Shevchenko University, Faculty of Physics (2, Academician Glushkov Str., Kyiv 03680, Ukraine; e-mail: momot@ univ. kiev. ua ) 3Institute of Theoretical and Applied Mechanics (4/1, Institutskaya Str., Novosibirsk 630090, Russia; e-mail: ischweig@ yahoo. com ) PACS 52.27.Lw c©2010 Kinetic calculations of the effective grain potential are presented for the case of weakly-ionized plasma in the external electric field. The drag force associated with the ionic drift in the external field is found. It is shown that the absorption of electrons and ions by the grain can cause the change of the direction of the drag force. 1. Introduction Theoretical description of various phenomena observed experimentally in dusty plasma (dusty structure forma- tion, excitation of dust-acoustic waves, existence of spa- tial domains free of dust particles (voids), etc.) requires the knowledge of the explicit form of effective grain potentials. The numerical solution of the appropriate boundary-value problem, as well as numerical simula- tions of such a potential, does not give us necessary an- alytical expressions. Thus, one has to use approximate relations obtained within the framework of various mod- els. Obviously, the more consistent description of plasma processes is used, the more fine details concerning the ef- fective grain potentials are known. That is why it is very important to have the kinetic theory of grain screening in a plasma. The additional essential requirement is that the theory should take the electron and ion absorption by grains into account. The kinetic theory of the effective grain potential with regard for the grain charging by a plasma current was proposed for the first time in Refs. [1–3]. With this pur- pose, the point sinks were introduced into the kinetic equations. Later on, the point sink model was substan- tiated on the basis of a consistent microscopic theory of dusty plasma [4, 5]. The proposed kinetic description turns out to be also efficient for theoretical studies of the effective potentials of moving grains [6–9]. The presence of the external force fields can be easily taken into account as well. This gives the possibility to generalize the kinetic description of the grain screening in a weakly ionized plasma exposed to an electric field, which was done in Ref. [10,11] disregarding the influence of charging currents, to the case of absorb- ing grains. It is possible to expect that the absorption of plasma particles by grains can influence the asymptotic behavior of the potential (as it is observed in the case of immovable grains in a plasma without external electric field [1–3]) and lead to the qualitative changes of the po- larization forces (as in the case of a grain moving in the Maxwellian plasma [6–9]). Obviously, the description of these effects within the kinetic model will give more details about the grain effective potential than that ob- tained in the fluid approximation [12]. In turn, this will provide more consistent calculations of the ionic drag force which are needed, in particular, for studies of the grain dynamics in the sheath region. The purpose of the present paper is to give a kinetic description of the effective grain potential in a weakly ionized plasma under the presence of an external electric field taking the electron and ion absorption by grains into account. The paper is organized in the following order. The basic set of equations is formulated in Section 2. Plasma dynamics is described by the Bhatnagar–Gross–Krook (BGK) kinetic equation [13], in which the point sinks of plasma particles are introduced. Such sinks naturally appear in the kinetic equations in the course of their derivation on the basis of the microscopic treatment [4, 5]. The formal solution of the problem is given in Section 3. In Section 4, we present specific calculations related ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 29 A.G. ZAGORODNY, I.V. ROGAL’, A.I. MOMOT, I.V. SCHWEIGERT to a weakly ionized plasma in an external electric field. The qualitative influence of the external electric field, as well as some analytical relations describing such an influence, are presented in Sec. 4. 2. Basic Set of Equations The consistent derivation of the kinetic equations for plasma particles in the presence of grains gives the fol- lowing result [4, 5]:{ ∂ ∂t + v ∂ ∂r + eα mα E(r, t) ∂ ∂v + 1 mα Fext α (r,v, t) ∂ ∂v } × ×fα(r,v, t) = −fα(r,v, t) ∫ dv′ ∫ dq′σα(q′,v − v′)× ×\|v − v′\|fg(r,v′, q′, t) + Iα, (1) where fα(r,v, t) and fg(r,v, q, t) are the one-particle distribution functions normalized by volume for plasma particles and grains, respectively, α = e, i (electron and ion), Iα is the collision term describing the elastic scat- tering of electrons and ions by a grain and neutrals (if present), Fext α (r,v, t) is the external force field, σα(q, v) is the charging cross-section which is given for the colli- sionless plasma by σα(q, v) = πa2 ( 1− 2eαq mαv2a ) θ ( v2 − 2eαq mαa ) , q is the grain charge which is an additional variable and, in the general case, can depend on time. Now let us ap- ply Eq. (1) to the calculations of the effective grain potentials. In the case of a single immovable grain, fg(r,v′, q′, t) = δ(r)δ(v′)δ(q′ − q), Eq. (1) reduces to{ ∂ ∂t + v ∂ ∂r + eα mα E(r, t) ∂ ∂v + 1 mα Fext α (X, t) ∂ ∂v } × ×fα(X, t) = Iα − vσα(q, v)fα(X, t)δ(r), (2) where X denotes (r,v). For the sake of simplicity, we do not use, in what fol- lows, the collision term calculated in terms of correlation functions of microscopic fluctuations. Instead, we use a simple version of the model collision integral (simple Bhatnagar–Gross—Krook model) proposed in Ref. [13], namely Iα = −να ( fα(r,v, t)− Φα(v) ∫ dv′fα(r,v′, t) ) . (3) Here, να is the effective collision frequency, Φα(v) is the distribution function generated in the course of plasma particle collisions. In view of the fact that the plasma particle absorp- tion considerably suppresses the influence of a nonlin- earity [14, 15], we can suggest that the sinks cause a small perturbation of the effective electric field and thus fα(X, t) = f0α(v) + δfα(X, t). The linearized version of Eq. (2) reads{ ∂ ∂t + v ∂ ∂r + 1 mα Fext α (X, t) ∂ ∂v } δfα(X, t)− − eα mα ∇Φ(r, t) ∂f0α(v) ∂v = −S(0) α (v, t)δ(r)− −να { δfα(X, t)− Φα(v) ∫ dvδfα(X, t) } , (4) where S(0) α (v, t) = vσα(q(t), v)f0α(v) is the intensity of the plasma particle sink, and f0α(v) is the unperturbed distribution function. The potential Φ(r, t) is governed in this case by the Poisson equation ΔΦ(r, t) = −4πq(t)δ(r) − 4π ∑ α eαn0α ∫ dvδfα(X, t). (5) Here, n0α is the unperturbed plasma particles density. 3. Effective Potential of Charged Grains (General Relations) The solution of Eq. (4) is given by δfα(X, t) = eα mα t∫ −∞ dt′ ∫ dX ′Wα(X,X ′; t− t′)× ×∂Φ(r′, t′) ∂r′ ∂f0α(v′) ∂v′ − − t∫ −∞ dt′ ∫ dX ′Wα(X,X ′; t− t′)S(0) α (v′, t′)δ(r′), (6) 30 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 GRAIN IN A PLASMA IN THE PRESENCE OF EXTERNAL ELECTRIC FIELD where Wα(X,X ′; t− t′) satisfies the equation{ ∂ ∂t + v ∂ ∂r + 1 mα Fext α (X, t) ∂ ∂v } Wα(X,X ′; τ) = = −να { Wα(X,X ′; τ)− Φα(v) ∫ dvWα(X,X ′; τ) } (7) with the initial condition Wα(X,X ′; 0) = δ(X −X ′). (8) As is seen from Eqs. (7) and (8), the quantity Wα(X,X ′; τ) is the phase density of probability of the particle transition from the phase point X ′ to the point X during the time period τ = t − t′ in the system with no self-consistent particle interaction through the elec- tric field. Substituting Eq. (6) into Poisson equation (5) and performing the Fourier transformation yield Φkω = 4πqω k2ε(k, ω) − 4π k2ε(k, ω) × × ∑ α eαn0α ∫ dv ∫ dv′Wαkω(v,v′)S(0) αω(v′), (9) where ε(k, ω) is the dielectric response function ε(k, ω) = 1− i ∑ α 4πe2αn0α k2mα × × ∫ dv ∫ dv′Wαkω(v,v′)k ∂f0α(v′) ∂v′ , (10) Wαkω(v,v′) = ∫ dRe−ikR ∞∫ 0 dτeiωτWα(X,X ′, τ), R = r− r′. In the stationary case where q(t) = q, Eq. (9) reduces to Φk = 4πq k2ε(k, 0) − 4π k2ε(k, 0) ∑ α eαn0α× × ∫ dv ∫ dv′Wαk(v,v′)S(0) α (v′). (11) Here, Wσk(v,v′) = Wσkω(v,v′) ∣∣ ω=0 , ε(k, 0) = 1 + k2 D k2 , k2 D = ∑ α k2 α, k2 α = 4πe2αn0α Tα . (12) Thus, we have obtained the general relations describ- ing the effective macroparticle potentials with regard to the electron and ion absorption by grains and the col- lisions of plasma particles with neutral gas molecules. These relations make it possible to recover all known analytical results for the effective grain potential. For example, in the case of an isotropic plasma with no ex- ternal field, we have Wαkω(v,v′) = iδ(v − v′) ω − kv + iνα − − ναΦα(v) (ω − kv + iνα)(ω − kv′ + iνα) × × [ 1− iνα ∫ dv Φα(v) ω − kv + iνα ]−1 , (13) which yields the stationary grain potential given by Φ(r) = qe−kDr r + i ∑ α 4πeαn0α× × ∫ dk (2π)3 eikr k2 + k2 D ∫ dv vσα(q,v)f0α(v) kv−iνα 1 + iνα ∫ dv Φα(v) kv−iνα . (14) In the collisionless limit (να → 0), this relation is sim- plified to Φ(r) = qe−kDr r + i ∑ α 4πeαn0α× × ∫ dk (2π)3 eikr k2 + k2 D ∫ vσα(q, v)f0α(v) kv − i0 dv, (15) and thus we have Φ(r) = qe−kDr r − Q̃ r g(kDr), (16) where g(x) = e−xEi(x)− exEi(−x), Q̃ = 2π kD ∑ α eαn0α ∞∫ 0 dv v2σα(q, v)f0α(v). (17) ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 31 A.G. ZAGORODNY, I.V. ROGAL’, A.I. MOMOT, I.V. SCHWEIGERT At kDr � 1, Eq. (16) yields the well-known result Φ(r) ' − 2Q̃ kDr2 , i.e. the effective potential is of the dipole type. In the strongly collisional limit (να � ksα, sα =√ Tα/mα), Eq. (14) reduces to a superposition of the screened and unscreened potentials Φ(r) = (q + S̃) e−kDr r − S̃ r , (18) which is in agreement with the results obtained in terms of the drift-diffusion approximation [1, 16]. Here, S̃ = ∑ α S̃α = ∑ α eαn0α k2 DDα ∫ dvS(0) α (v), Dα = Tα/(mανα) is the diffusion coefficient. When de- riving (18), we have put Φα(v) = f0α(v). Thus, we see that, in the case of the dusty plasma which can be treated as an open system, the stationary screened po- tential considerably depends on the details of the plasma dynamics, in contrast to the case of a screened potential for the ordinary plasma. 4. Effective grain potential in the weakly ionized plasma exposed to an external electric field If an external electric field E0 = (0, 0, E0) is present, then Eq. (2) generates an equation for the unperturbed distribution function, i.e. eα mα E0 ∂f0α(v) ∂v = −να { f0α(v)− Φα(v) ∫ dv′f0α(v′) } , (19) and Wα (X,X ′; τ) [see Eq. (7)] satisfies the equation{ ∂ ∂t + v ∂ ∂r + eα mα E0 ∂ ∂v } Wα (X,X ′; τ) = = −να { Wα (X,X ′; τ)− Φα(v) ∫ dvWα (X,X ′; τ) } . (20) Integrating Eq. (19) results in f0α(v) = vz∫ −∞ dv′zβαΦα (v⊥, v′z) exp(−βα(vz − v′z)), βα > 0, (21) f0α(v) = − ∞∫ vz dv′zβαΦα (v⊥, v′z) exp(−βα(vz − v′z)), βα < 0 , (22) where βα = 1 vα , vα = eαE0 mανα . The solution of Eq. (20) for vα > 0 is given by Wαkω(v,v′) = βα να { δ(v⊥ − v′⊥)θ(vz − v′z)× × exp [−iψα(v⊥, v′z)] + ναWαkω(v′)× × vz∫ −∞ dv′′z Φα(v⊥, v′′z ) exp [−iψα(v⊥, v′′z )] } exp[iψα(vz)] , (23) where ψα(v⊥, vz) = βα να { (ω − k⊥v⊥ + iνα)vz − kzv 2 z 2 } , (24) Wαkω(v′) = βα να ∞∫ −∞ dvz exp [iψα(v′⊥, vz)− iψα(v′⊥, v ′ z)]× ×θ(vz − v′z) { 1− βα ∫ dv vz∫ −∞ dv′′z Φα(v⊥, v′′z )× × exp [iψα(v⊥, vz)− iψα(v⊥, v′′z )] }−1 . (25) For vα < 0, we have Wαkω(v,v′) = −βα να { δ(v⊥ − v′⊥)θ(v′z − vz)× × exp [−iψα(v⊥, v′z)] + ναWαkω(v′)× 32 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 GRAIN IN A PLASMA IN THE PRESENCE OF EXTERNAL ELECTRIC FIELD × ∞∫ vz dv′′z Φα(v⊥, v′′z ) exp [−iψα(v⊥, v′′z )] } exp[iψα(vz)], (26) where Wαkω(v′)=−βα να ∞∫ −∞ dvz exp [iψα(v′⊥, vz)− iψα(v′⊥, v ′ z)]× ×θ(v′z − vz) { 1 + βα ∫ dv ∞∫ vz dv′′z Φα(v⊥, v′′z )× × exp [iψα(v⊥, vz)− iψα(v⊥, v′′z )] }−1 . (27) Substituting Eqs. (21)–(27) in Eq. (10) for the dielec- tric response function ε(k, ω) yields ε(k, ω) = 1 + ∑ α k2 α κ2 α(k)Gα(k, ω) × × ∞∫ 0 dy exp(−y)W ( ω + iνα − kzvαy κα(k)sα ) , (28) where Gα(k, ω) = 1 ω + iνα [ ω + iναW ( ω + iνα κα(k)sα )] , κ2 α(k) ≡ κ2 α = k2 + ikzvανα s2α , (29) W (z) = 1− z exp ( −z 2 2 ) z∫ 0 dy exp ( y2 2 ) + +i (π 2 )1/2 z exp ( −z 2 2 ) . (30) With the accuracy up to the notation, Eqs. (28) and (29) recover the result obtained in Ref. [10]. For \|kzvα\| � να, Eq. (28) is simplified to ε(k, ω) = 1 + ∑ α k2 α κ2 αGα(k, ω) W ( ω + iνα κα(k)sα ) = = 1 + ∑ α k2 α κ2 α (ω + iνα)W ( ω+iνα καsα ) ω + iναW ( ω+iνα καsα ) . (31) In the drift-diffusion approximation (ω � να), we have ε(k, ω) ' 1 + i ∑ α ω2 pα να(ω − kzvα + ik2Dα) , (32) where ω2 pα = 4πe2αnα/mα. In the stationary case (ω = 0), ε(k, 0) = 1− i ∑ α ω2 pα να(kzvα − ik2Dα) . (33) For the further calculations, we need to know the ex- plicit forms of the quantities describing the contribu- tion of the absorption processes to the effective poten- tial, namely, to the second terms in Eqs. (9) and (11). We denote these by Φ(s) kω and Φ(s) k , i.e., Φ(s) kω = 4πQ(s) ω k2ε(k, ω) , (34) where Q(s) ω = ∑ α eαn0α ∫ dv ∫ dv′ Wαkω (v,v′)S(0) αω(v′) . (35) With regard for the explicit form of Wαkω (v,v′), one obtains Q(s) ω = −i ∑ α eαn0α Gα(k, ω) ∫ dv S (0) αω(v) ω − kv + iνα × × [ 1−W ( ω − kv + iνα√ ikzvανα )] , Re (√ ikzvανα ) > 0. (36) Notice that, in the case of collisional plasma, the prob- lem of the appropriate approximation for the charging cross-section σα(q, v) is not yet solved (rigorously speak- ing, by introducing any cross-section, we assume that the incoming particle flux in course of its movement from the infinity to the scattering center is disturbed by this cen- ter only). Therefore, the description of the grain charg- ing is usually done in terms of charging currents [17, 18] or in self-consistent kinetic simulations [19, 20]. With ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 33 A.G. ZAGORODNY, I.V. ROGAL’, A.I. MOMOT, I.V. SCHWEIGERT the dependence of charging currents on the collision fre- quency and other plasma parameters being known, it is possible to propose reasonable approximations for the integral terms which include the charging cross-sections. In particular, such an idea was used in Ref. [21] in order to calculate the effective grain potential in the case of a weakly collisional regime of the grain charging. Namely, it was assumed in [21] that∫ dvvσi(q, v)F (v) = σ0i ∫ dvvF (v), (37) where F (v) is an arbitrary (but regular) function, and σ0i is the effective cross-section taken from the relation for the charging current Ji = ni ∫ dvvσi(v)f0i(v) = niσ0i ∫ dvvf0i(v). (38) A similar approximation can be used in the case under consideration. Moreover, as follows from the compari- son of the consistent calculations of the effective grain potential in the case of the low-collisional regime with the appropriate estimates on the basis of the relations for charging currents, the approximation of the type∫ dvvσα(q, v)F (v) = Jα nα ∫ dvF (v) (39) is even more efficient. Here, Jα is the charging current generated by particles of the α species. So doing, we are led to the following relations for the effective charge associated with the grain charging: Q(s) ω = i ∑ α eαJαω ∞∫ 0 dy 1 Gα(k, ω) exp(−y) ω − kzvαy + iνα × × [ 1−W ( ω − kzvαy + iνα κα(k)sα )] . (40) As follows from Eq. (36) and (40) for να � ksα, kzvα, we have Q(s) ω = −i ∑ α eα Gα(k, ω) 1 (ω + iνα) × × [ 1−W ( ω + iνα√ ikzvανα )] Jαω, Jαω = nα ∫ dvS(0) αω(v). (41) In the stationary case (ω = 0), Eq. (41) reduces to Q(s) = − ∑ α eα ναGα(k, ω) [ 1−W (√ iνα kzvα )] Jα, Gα(k, ω) 'W ( iνα κα(k)sα ) . (42) If the condition \|κα(k)sα\| � να is satisfied, then Gα(k, 0) ' k2s2α + ikzvανα ν2 α . (43) Obviously, for kzvα � να, we have Q(s) ' − ∑ α eα ναGα(k) Jα. (44) Thus, in the drift-diffusion approximation (να � ksα, kzvα), we obtain Φk = 4πq k2ε(k, 0) − 4π k2ε(k, 0) ∑ α eα ναGα(k) Jα . (45) Neglecting the field influence on electrons (ve = 0), i.e. assuming that electrons can be described by the Boltzmann distribution, we can put ε(k, 0) = 1 + k2 e k2 + k2 i k2 + ikzviνi/s2i , Ge(k) = k2s2e ν2 e . (46) With the potential Φk being known, we can calculate the force acting on the grain due to the polarization of the medium F = −q ∂Φ(r) ∂r ∣∣∣∣ r=0 = −iq ∫ dk (2π)3 kΦk . (47) If the external field is absent, then Φk depends on the squared k, and the force acting on the particle is equal to zero. The situation crucially changes if particle fluxes occur in the plasma. Taking the explicit form of ε(k, 0) and Gi(k) into account, we find Fz = − q 2π2 ∫ dk ikz k2ε(k, 0) [ q − ∑ α eα ναGα(k) Jα ] . (48) For small velocities vi, Eq. (48) reduces to Fz = qvi 2π2Di ∫ dk k2 z k2(k2 + k2 D) × × [ k2 DS̃i k2 + k2 i k2 + k2 D ( q − k2 DS̃ k2 )] . (49) In terms of Jα, S̃ = ∑ S̃α = ∑ eαJα/(k2 DDα). 34 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 GRAIN IN A PLASMA IN THE PRESENCE OF EXTERNAL ELECTRIC FIELD After the integration, we have Fz = qvi 6DikD { k2 i (q − S̃) + 2k2 DS̃i } . (50) We see that if there is no absorption by the grain (S̃α = 0), then the drag force is positive, i.e. it acts in the direction of the ion flux velocity. However, if the condition k2 i ( q − S̃ ) + 2k2 DS̃i > 0 (51) is satisfied and q < 0, then the force becomes negative, as it was found in kinetic simulations in Ref. [22]. Equa- tions (50)–(51) reproduce the results obtained earlier in Refs. [7, 9]. 5. Summary and Conclusions The constituent kinetic theory of dusty plasma is used to calculate the effective grain potential for the case of a weakly ionized plasma in an external electric field. The grain charging by plasma currents is taken into account with regard for elastic collisions of plasma particles with neutrals within the framework of the BGK collision term. The drag force associated with the ionic drift in the external field is found. It is shown that the absorption of electrons and ions by the grain can cause the change of the direction of the drag force. This work is partially supported by joint NASU-RFFR grant. 1. A.G. Zagorodny, A.V. Filippov, A.F. Pal’ et al., Prob- lems of Atomic Science and Technology, Series: Plasma Physics 12, 99 (2006). 2. A.G. Zagorodny, A.V. Filippov, A.F. Pal’ et al., Proc. of 2nd Intern. Conf. “Dusty Plasmas in Applications” (Odessa, 2007) p.176. 3. A.V. Filippov, A.G. Zagorodny, A.F. Pal et al., JETP Letters 86, 761 (2007) 4. A.G. Zagorodny, A.V. Filippov, A.F. Pal’ et al., Prob- lems of Atomic Science and Technology, Series: Plasma Physics 14, 70 (2008). 5. A.G. Zagorodny, Theor. Math. Phys. 160, 1100 (2009). 6. A.G. Zagorodny, A.V. Filippov, A.F. Pal’ et al., J. Phys. Studies 11, 158 (2007). 7. A.V. Filippov, A.G. Zagorodny, and A.I. Momot, JETP Lett. 88, 24 (2008). 8. S.V. Vladimirov, S.A. Khrapak, M. Chaudhuri, and G.E. Morfill, Phys. Rev. Lett. 100, 055032 (2008). 9. A.V. Filippov, A.G. Zagorodny, A.I. Momot et al., JETP 108, 497 (2009). 10. I.V. Schweigert, V.A. Schweigert, and F.M. Peeters Phys. of Plasmas 12, 113501 (2005). 11. V.A. Schweigert, Plasma Phys. Rep. 27, 997 (2001). 12. M. Chaudhuri, S.A. Khrapak, and G.E. Morfill, Phys. Plasmas 14, 022102 (2007). 13. P.L. Bhatnagar, E.P. Gross, and M. Krook, Phys. Rev. 94, 511 (1954). 14. O. Bystrenko and A. Zagorodny, Phys. Lett. A 299, 383 (2002). 15. O. Bystrenko and A. Zagorodny, Phys. Rev. E 67, 066403 (2003). 16. A.V. Filippov, A.G. Zagorodny, A.I. Momot et al., JETP 104, 147 (2007). 17. L.G. D’yachkov, A.G. Khrapak, S.A. Khrapak, and G.E. Morfill, Phys. Plasmas 14, 042102 (2007). 18. L.G. D’yachkov, S.A. Khrapak, and A.G. Khrapak, Ukr. J. Phys. 53, 1053 (2008). 19. I.V. Shveigert and F.M. Peeters, JETP Letters 86, 572 (2007). 20. I.V. Schweigert, A.L. Alexandrov, D.A. Ariskin et al., Phys. Rev. E 78, 026410 (2008). 21. S.A. Khrapak, B.A. Klumov, and G.E. Morfill, Phys. Rev. Lett. 100, 225003 (2008). 22. I.V. Schweigert, A. Alexandrov, and F.M. Peeters, IEEE Trans. on Plasma Science 32, 623 (2004). Received 05.10.09 ПОРОШИНКА В ПЛАЗМI У ПРИСУТНОСТI ЗОВНIШНЬОГО ЕЛЕКТРИЧНОГО ПОЛЯ: КIНЕТИЧНИЙ РОЗРАХУНОК ЕФЕКТИВНОГО ПОТЕНЦIАЛУ ТА IОННОЇ СИЛИ ОПОРУ А.Г. Загороднiй, I.В. Рогаль, А.I. Момот, I.В. Швейгерт Р е з ю м е Представлено кiнетичнi розрахунки ефективного потенцiалу порошинки для випадку слабоiонiзованої плазми у зовнiшньо- му електричному полi. Знайдено силу опору, яка пов’язана з дрейфом iонiв у зовнiшньому полi. Показано, що поглинання електронiв та iонiв порошинкою може привести до змiни на- прямку сили опору. ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 35

Grain in a Plasma in the Presence of External Electric Field: Kinetic Calculation of Effective Potential and Ionic Drag Force

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