Theory of a cavity around a large floating sphere in complex (dusty) plasma
S. Khrapak, P. Huber, H. Thomas, V. Naumkin. V. Molotkov, A. Lipaev
aa r X i v : . [ phy s i c s . p l a s m - ph ] A p r Theory of a cavity around a large floating sphere in complex (dusty) plasma
Sergey Khrapak, ∗ Peter Huber, and Hubertus Thomas
Institut f¨ur Materialphysik im Weltraum, Deutsches Zentrum f¨ur Luft- und Raumfahrt (DLR), 82234 Weßling, Germany
Vadim Naumkin, Vladimir Molotkov, and Andrey Lipaev
Joint Institute for High Temperatures, Russian Academy of Sciences, 125412 Moscow, Russia (Dated: May 1, 2019)In the last experiment with the PK-3 Plus laboratory onboard the International Space Station,interactions of millimeter-size metallic spheres with a complex plasma were studied [M. Schwabe etal. , New J. Phys. , 103019 (2017)]. Among the phenomena observed was the formation of cavities(regions free of microparticles forming a complex plasma) surrounding the spheres. The size of thecavity is governed by the balance of forces experienced by the microparticles at the cavity edge.In this article we develop a detailed theoretical model describing the cavity size and demonstratethat it agrees well with sizes measured experimentally. The model is based on a simple practicalexpression for the ion drag force, which is constructed to take into account simultaneously the effectsof non-linear ion-particle coupling and ion-neutral collisions. The developed model can be useful fordescribing interactions between a massive body and surrounding complex plasma in a rather wideparameter regime. I. INTRODUCTION
Understanding fundamental interactions between anobject and surrounding plasma is an exceptionally im-portant problem with application to astrophysical top-ics [1, 2], plasma technology [3], plasma medicine [4],complex (dusty) plasmas [5–7] and fusion related prob-lems [8]. Considerable progress on the interaction ofmicron-size plastic particles with weakly ionized plasmamedium has been achieved thanks to complex plasma re-search program under microgravity conditions onboardthe International Space Station (ISS). This particularlyconcerns particle charging, the ion drag force, interparti-cle interactions, linear and non-linear wave phenomena,see for instance Ref. [9] for a recent review.Here another related problem is addressed, namelyhow a bigger object interacts with surrounding complexplasma. New information about these interactions hasbeen obtained from the last experimental campaign withPK-3 Plus laboratory onboard ISS [10]. In these ex-periments the metallic spheres of one millimeter diam-eter were injected into a low-temperature rf dischargetogether with microparticles forming a complex plasma.Various phenomena were observed, including motion ofspheres through a complex plasma cloud, generation ofbubbles, “repulsive attraction”, and excitation of low-frequency waves [10].It was also observed that when a sphere passes througha complex plasma cloud, it is surrounded by a cavity of afew millimeter in diameter, where no microparticles arepresent. It is the size of the cavity which is the mainobject of interest here. The size of the cavity is relativelyeasy to measure and it contains important informationabout the system parameters. The size obviously de-pends on the balance of forces acting on the particles ∗ [email protected] located at the cavity edge. The main forces identifiedare the short-range electric repulsion from the highlycharged sphere and the long-range attraction triggeredby the ion flow (the ion drag force), which is directedtowards the sphere surface [10]. In this article we firstpropose a simple practical expression for the ion dragforce for the conditions relevant for the experiment. Inparticular, this expression allows us to take into accountsimultaneously the effects of non-linear ion-particle cou-pling and ion-neutral collisions. Then, using this expres-sion, we formulate the force balance condition and esti-mate theoretically the cavity diameter. We show thatthe estimated diameter agrees well with the results ofexperimental measurements.The theoretical approximation developed here shouldbe applicable (possibly with some modifications) to othersituations corresponding to the interaction of large ob-jects with complex plasmas, such as for instance probe-induced voids and particle circulations [11–16], as well asthe formation of boundary-free clusters [17].
II. EXPERIMENT
The experiment to be discussed is the last experimentof the PK-3 Plus laboratory, which operated onboard theISS in 2006-2013 [9, 18, 19]. The experiment is describedin detail in Ref. [10]. Here we provide only the briefsummary, necessary for the understanding of this article.The PK-3 Plus laboratory consisted of a radio-frequency plasma chamber with two electrodes of 6 cm indiameter separated by a distance of 3 cm. The electrodeswere surrounded by grounded guard rings, see sketch inFig. 1. Dispensers mounted in the guard rings were usedto introduce microparticles of various sizes into the gasdischarge. Highly charged microparticles formed largesymmetric three-dimensional clouds in the plasma bulk.Typically, these clouds contained a central particle-free
FIG. 1. Sketch of the PK-3 Plus discharge chamber. Adaptedfrom [18]. region – the so-called “void” – attributed to the actionof the ion drag force, pushing the particles to the pe-riphery [9, 20–24]. Strong interparticle interactions be-tween microparticles resulted in structures typical for thefluid and solid states. Structural and dynamical proper-ties of the particle component were studied at the mostfundamental kinetic level, providing new insight into thephysics of a new plasma state of soft condensed mat-ter [25].Many important fundamental phenomena were stud-ied using PK-3 Plus laboratory, including, for instance,equilibrium and non-equilibrium phase transitions [26–29], lane formation [30], wave excitation [31], instabili-ties [32, 33], Mach cones [34, 35], etc.In the experiment discussed here interactions betweenmillimeter size metallic spheres and complex plasmas un-der microgravity conditions were studied. Previously,penetration of complex plasma clouds by fast chargedprojectiles was already investigated under microgravityconditions [36, 37]. In particular, the dynamics of the for-mation of an elongated cavity in the projectile’s wake wasanalyzed in detail. Present work deals with much largerand slower objects interacting with complex plasma.For the purpose of the experiment, the dispensers wereshaken so strong that the metallic spheres of 1 mm diame-ter that were present inside the dispensers broke throughthe sieve and entered the bulk plasma region togetherwith the microparticles remaining in the dispensers [10].Furthemore, the cosmonaut Pavel Vinogradov, who per-formed the experiment, shook the experimental containerto impact momentum on the spheres. The shaking hadlittle effect on the plasma and microparticles, but accel-erated spheres by collisions with chamber walls. As aresult, the spheres experienced an almost force-free mo-tion inside the discharge chamber [10].
FIG. 2. Experimental video-images showing a metallic spheresurrounded by complex plasma in an argon discharge. In(a) the pressure is 17.5 Pa, the particles interacting with thesphere have a diameter of 1 . µ m, the diameter of the cavityis ≃ . ≃ . The analysis of the motion of spheres through com-plex plasma clouds was reported previously [10]. Herethe main interest is to the size of the cavities that arecreated around the spheres. The size can be relativelyeasily estimated for events when the spheres cross theplane formed by the laser sheet used to illuminate com-plex plasma. Two such exemplary events are shown inFig. 2. About twenty such events have been analyzed andthe sizes of the cavities have been estimated as follows.The crossing of a laser sheet by the sphere corresponds toseveral video frames. From these frames a single framewith the largest cavity size is selected. After correctingthe aspect ratio of the video frame, it has been verifiedthat the observed cross sections of the cavities have ashape close to a circle. In a graphical editor, a circle hasbeen selected that fits most accurately into the cavity ob-served on the frame. The diameter of this circle is usedas an estimate of the cavity diameter. The relative errorin estimating the diameter does not exceed 5 %. Moreaccurate evaluation of the cavity size and shape can bemade from a careful analysis of three-dimensional trajec-tories of the spheres, as described in Ref. 10 for a singlecrossing event, but this is not necessary for the presentpurpose.The sizes of the cavities have been measured for dif-ferent neutral gas pressures (in the range between ≃ ≃
30 Pa) and for situations where the spheres wereinteracting with microparticles of different sizes (com-plex plasma cloud consisted of particles with diameters of1.55, 2.55, 3.42, 6.8, 9.19, and 14.9 µ m as well as their ag-glomerates; this mixture was heterogeneous with smallerparticles located closer to the discharge center and big-ger particles pushed further to the periphery, see Fig 2).It has been observed that the cavity size increases withpressure, but is practically insensitive to the microparti-cles size with which sphere is interacting. These trendscorrelate well with the results of theoretical considerationperformed below. III. THEORY
When a sphere is immersed into a plasma it starts tocollect electrons and ions on its surface, just as smallermicroparticles do. As a result both spheres and micropar-ticles are charged negatively, the surface (floating) po-tential being roughly of the order of the electron tem-perature, which ensures that the ion and electron fluxesto the surface can balance each other. If the sphere issurrounded by a complex plasma, the particles experi-ence the following forces. At short distances there is astrong electrostatic repulsion of negatively charged par-ticles from the negatively charged sphere. At sufficientlylong distances from the sphere, the ion drag force asso-ciated with the ion flux towards the sphere surface canovercome the electrostatic repulsion. There can be alsothe pressure force exerted by the microparticle cloud, di-rected towards the metallic sphere. This, however, wasshown to be numerically small for typical experimentalconditions [10] and will not be considered.Our main assumption is that the cavity boundary po-sition is mainly determined by the balance between theelectric repulsion at short distances and the ion-drag-mediated attraction at long distances. In the followingwe consider the force balance for an individual micropar-ticle located at an equilibrium position, where both forcescompensate each other. In this way we neglect (i) the ef-fect of particles on the distribution of the electrostaticpotential around the sphere and (ii) some reduction ofthe ion drag force in dense dust clouds [38]. Both as-sumptions are reasonable in not too dense microparticleclouds as those observed in the experiment.The ratio of the ion drag to the electric forces, F i /F el ,is known to be approximately constant for subthermalion flows and then to decrease relatively fast in thesuper-thermal regime [39]. For this reason, the cavityboundary should be roughly located at a position where M = u/v T i ∼
1, where u is the ion drift velocity, v T i is the ion thermal velocity, and M is the ion thermalMach number. This implies that the perturbations cre-ated by a large floating metallic sphere at the positionof the boundary are relatively small (much smaller thanin the sheath region formed around the sphere surface,where the ion drift is super-sonic). This suggests to fo-cus on the long-range asymptote of the electric poten-tial generated by a large floating body and not on theplasma properties in its immediate vicinity. In this com-paratively far region the effects associated with plasmaabsorption on the sphere govern the distribution of theelectric potential and this simplifies considerably the con-sideration, as we will see below. The first step, however,is to develop an appropriate model for the ion drag force. A. Ion drag force
In the parameter regime investigated, the character-istic length scale of ion-particle interactions exceeds theplasma screening length, indicating that ion-particle in-teractions are non-linear. Several theoretical approacheshave been developed for this regime, mostly using binarycollision approximation [40–44]. However, these purelycollisionless treatments are not very appropriate for ourpurpose, because ion-neutral collisions can be importantin the pressure range investigated [45]. Collisional effectscan be incorporated into kinetic or hydrodynamic calcu-lations using the linear plasma response formalism [46–49]. Unfortunately, as we have just discussed, the lin-ear approximation is not justified in present conditions(as well as in most other complex plasma experiments),because of significant non-linearities in ion-particle in-teractions. An approach, which accounts for both non-linearity in ion-particle interactions and the effect of ion-neutral collisions is required.Recently, the ion drag force has been calculated self-consistently and non-linearly using particle in cell codes,taking into account ion-neutral collisions [50]. These cal-culations demonstrated that the magnitude of the forceis sensitive to the ion velocity distribution function forsuperthermal ion flows. It was shown that the finite col-lisionality initially enhances the ion drag force up to afactor of 2 relative to the collisionless result. Larger col-lisionality eventually reduces the ion drag force, whichcan even reverse sign in the continuum limit [49, 51–54],but this regime is too far from typical experimental con-ditions. Most important for our present purpose is thatthe collisional drag enhancement can be represented byan almost universal function of scaled collisionality andflow velocity, for which simple fits are available [50].We pursue the following strategy. First, an ad hoc simple practical expression for the collisionless ion dragforce, based on our earlier theoretical results from thebinary collision approach, is derived. It is demonstratedto be in good agreement with the non-linear collisionlesssimulation results of Ref. [50]. Then a correction factor,expressing the influence of ion-neutral collisions on theion drag force, as suggested in [50], is added to the colli-sionless expression. This provides us with a new practicalexpression for the non-linear ion drag force in the colli-sional regime, which will be then used to estimate thesize of the cavity around the metallic sphere.We start with an expression for the ion drag force de-rived for the regime of intermediate non-linearity [40] F i = (cid:16) √ π/ (cid:17) a n i m i v T i u (cid:18) zτ z τ (cid:19) , (1)where Λ is the modified Coulomb logarithmΛ = 2 Z ∞ e − x ln (cid:18) λx/a + zτ x + zτ (cid:19) dx. (2)Other notation is as follows: a is the particle radius, n i , m i , T i , v T i = p T i /m i are the ion density, mass, tem-perature, thermal velocity, z = e | φ s | /T e is the particlesurface potential ( φ s ) expressed in units of the electrontemperature T e , τ = T e /T i is the electron-to-ion tem-perature ratio, and λ is the effective plasma screeninglength.This expression applies to subthermal ion flows, u . v T i . It can be considered a generalization of the standardCoulomb scattering theory, by taking into account theimpact parameters beyond the plasma screening length:all ions which approach the grain closer than λ are in-cluded in the consideration. Therefore, it is sometimesreferred to as the modified Coulomb scattering approach.Quantitatively, the approach has been originally pro-posed for the regime β = zτ ( a/λ ) .
5, where β is knownas the scattering parameter [41]. In the regime β ≪
1, itreduces to the conventional Coulomb scattering theory.We can further simplify Eqs. (1) and (2) as follows.We neglect the collection part of the momentum transfer[first two terms in brackets of Eq. (1)]. In the expressionfor the modified Coulomb logarithm we make use of thetypical condition zτ ≫ ≃ Z ∞ e − x ln (1 + 2 x/β ) dx. Thus, the modified Coulomb logarithm depends mainlyon β , and it is easy to demonstrate (by way of directnumerical integration) that for β & ≃ . /β ).In the non-linear regime considered this becomes simplyΛ ≃ . /β . This allows us to write F i ≃ . a n i T i M zτ ( λ/a ) . (3)This represents an expression for the non-linear iondrag force in the collisionless regime to be compared withnumerical results from Ref. [50]. In that numerical in-vestigation the particle surface potential as well as theelectron-to-ion temperature ratio were fixed to z = 2and τ = 100, respectively. The ratio λ D e /a varied in therange from 10 to 200, where λ D e = p T e / πe n e is theelectron Debye radius. The ion drag force was expressed in units of n e T e a . To simplify the comparison we canrewrite Eq. (3) as( F i /n e T e a ) ≃ . λ D e /a )( u/c s ) , (4)where c s = p T e /m i is the ion sound velocity. In arrivingto Eq. (4) we assumed quasineutrality, n e = n i = n , andused the dominance of ion screening, λ ≃ λ D i = λ D e / √ τ .Note also that u/c s = M/ √ τ . The obtained formula (4)demonstrates very close agreement with the numericalresults presented in Figs. 8, 9 and 10(a) of Ref. [50].Thus, the region of validity of the approximation (3), β ∼ O (10), is somewhat expanded in the non-linearregime compared to the original approach (1) designedfor β ∼ O (1). Moreover, detailed comparison shows thatit is reliable not only for the subthermal regime, but alsofor near-thermal and slightly superthermal ion flows (inthe regime where difference in ion velocity distributionfunctions does not lead to considerable variations in theion drag force). Further insight comes from the carefulanalysis of the data shown in Fig. 10(b) of [50], whichdemonstrates that in the collisionless limit Eq. (4) re-mains accurate even at M = 2 ( u/c s = 0 . λ D e /a . F i ≃ . a n i T i M zτ ( λ/a ) F (˜ ν ) , (5)with F (˜ ν ) = 1 + A ˜ ν B ˜ ν + C ˜ ν , (6)where ˜ ν = νr c /c s is the reduced collisionality and r c isthe non-linear shielding cloud radius, derived in Ref. [50].The latter is approximately r c ≃ . λ D e (cid:18) aλ D e T i T e (cid:19) / . The coefficients provided in Ref. [50] for the drift distri-bution of ion velocities (which is more appropriate forions drifting through the stationary background of neu-trals under the action of electric force, compared to aconventional shifted Maxwellian distribution) are A = 7 + 3 M, B = 1 . M, C = 0 . A. Let us now compare the magnitudes of the electrostaticand ion drag forces in the limit of a weak electric field E ,when the ion drift is subthermal. The ion drift velocityis expressed u = eEm i ν eff , (7)where ν eff is the effective collision frequency, which isfield-dependent in general, but constant in the subther-mal drift regime (weak electric field) [55, 56]. The elec-trostatic force is F el = QE, (8)where Q is the particle charge. The ratio of the collision-less ion drag force, Eq. (3), to the electric force, Eq (8),is then | F i /F el | ≃ . ω p i /ν eff ) = 0 . ℓ i /λ ) , (9)where ω p i = p πe n i /m i is the ion plasma frequency, ℓ i = v T i /ν eff is the ion mean free path with respect tocollisions with neutrals. In deriving Eq. (9) it was as-sumed that screening is mostly associated with the ioncomponent and, hence, λ ≃ v T i /ω p i . For the particlecharge we used | Q | ≃ z ( aT e /e ). Equation (9) is verysimilar to that derived earlier in Ref. [40]. It can now beimproved by taking ion-neutral collisions into account.An obvious modification reads | F i /F el | ≃ . ω p i /ν eff ) F (˜ ν ) . (10)The necessary condition of particle attraction to thesphere at long distances is | F i /F el | > | F i /F el | drops to unity will determine the cavity radiusin this approximation.Equations (5) and (6) represent an important inter-mediate result, providing new simple practical tool toevaluate the ion drag force under typical experimentalconditions. We have a good opportunity to test it bycomparing the predicted size of cavities with those ob-served experimentally. B. Electric potential around sphere
At sufficiently long distances from the sphere, the elec-tric potential distribution is dominated by ion absorptionon the sphere surface. The ion flux conservation allowsto obtain the electric potential in the weakly perturbedquasi-neutral region. For a large sphere ( R s ≫ λ ) andcollision-dominated ion flux to its surface ( R s ≫ ℓ i ) sim-ple expressions for the potential and electric field are [57] φ ( r ) ≃ − ( T e /e )( R s /r ) , E ( r ) ≃ − ( T e /e )( R s /r ) , (11)where R s is the sphere radius. C. Cavity radius
The radius of the cavity is found as follows. We ap-proximate the effective collision frequency with ν eff = ν (cid:16) γM + p γ M + 1 (cid:17) , (12)where for argon ions in argon gas ν ≃ . × P Pa ( P Pa is the neutral gas pressure expressed in Pa), and γ ≃ .
23 [56]. The physics behind Eq. (12) is as fol-lows. In a weak electric field, the ion drift velocity isdirectly proportional to the field, u ∝ E , and, thus, theeffective collision frequency is constant ν eff ≃ ν . In a strong field, however, the drift velocity scales approxi-mately as the square root of the field, u ∝ √ E . Thisimplies ν eff ∝ √ E ∝ ν M . Equation (12) is constructedto reproduce these two limiting regimes and provides areasonable interpolation between them using experimen-tal information on drift velocities of Ar + ions in argongas (see Appendix A for a comparison). We substitutethis in Eq. (10) and find the critical Mach number M ∗ corresponding to the condition | F i /F el | = 1. Then using M = ( eE/mv Ti ν eff ) together with the long-range asymp-tote of the electric field (11) we finally obtain for thecavity radius R cav ≃ R s (cid:18) T e T i ℓ i R s M ∗ (cid:19) / , (13)where we have used ℓ i ν eff = v T i and m i v i = T i . Theprocedure only applies to sufficiently slow drifts, M ∗ .
2, so that equation (5) for the ion drag force remainsadequate.
A priori it is difficult to predict correctly the depen-dence of the cavity size on the neutral gas pressure. If, asone may expect intuitively, M ∗ is nearly constant (about M ∗ ∼ D. Numerical estimates
For the conditions relevant for the experiments on theinjection of milimeter-size metallic spheres in PK-3 Plusfacility we adopt the following plasma parameters, basedon our previous simulations with the SIGLO-2D code [18,28, 29]. The central plasma density depends linearly onpressure and, to a reasonable accuracy, described by n ≃ (1 .
20 + 0 . P Pa ) × , where n is in cm − . The electrontemperature decreases very weakly with pressure and inthe range investigated we can take a fixed value T e ≃ T i ∼ T n ∼ .
03 eV.The force balance model developed is almost indepen-dent of the size of the microparticles forming the complexplasma. The only point where the dependence on theparticle radius a appears explicitly is when defining thenon-linear shielding cloud radius r c . Furthermore, thisdependence is extremely weak, r c ∝ a / . For this rea-son we take a fixed “average” radius a = 3 µ m, providinga relevant “logarithmic” length scale for the mixture ofparticles present in the experimental chamber (diametervaries from 1 .
55 to 14 . µ m [10]).With the specified parameters, a numerical calculationis easy to perform. We have first verified that the nec-essary condition | F i /F el | > M = 0 is satisfied inthe regime investigated. We then estimated M ∗ and thecavity size as described in Sec. III C. The resulting depen-dence of the cavity diameter on the neutral gas pressure
10 15 20 25 30 350246810 Experiment Theory D i a m e t e r ( mm ) Pressure (Pa)
FIG. 3. Dependence of the cavity diameter on the neutralgas pressure. Circles are experimental measurements (sym-bol’s size is comparable to experimental uncertainty), thesolid curve corresponds to the theoretical calculation. is shown in Fig. 3. The agreement with experimentalresults is reasonable.Note that on the low-pressure side, the cavity diametercan be underestimated, because the critical velocity fromEq. (10) exceeds 2 (at P .
15 Pa). This is where themodel developed overestimates the ion drag force andhence pushes microparticles closer to the sphere. Theactual cavity size can be larger than the theory predicts,as we indeed see in the experiment.
IV. DISCUSSION
The experimentally measured cavity size and its de-pendence on the neutral gas pressure have been demon-strated to be in good agreement with the theoretical ap-proximation developed. It is appropriate to discuss sev-eral issues related to limitations and generalizations ofthe theoretical model.The cavity size is predicted to be independent (or, atleast, very weakly dependent) on the size of micropar-ticles interacting with the big sphere. This is, however,true only when the non-linear model for the ion drag forceis appropriate, that is for sufficiently large microparticleswhen the condition β & V. CONCLUSION
Interactions between millimeter size floating spheresand a complex plasma have been studied in the PK-3Plus laboratory onboard ISS. One of the manifestationsof these interactions represents the formation of cavities(regions free of microparticles) around the spheres. Thecavity size is dictated by the balance of forces acting onthe particles at the cavity edge, most important forcesbeing the electric repulsion at short distances and the ion-drag-mediated attraction at long distances. In this articlewe have proposed a simple practical approach to estimatethe ion drag force for experimentally relevant conditions(with the main point to account simultaneously for non-linear ion-particle ineractions and ion-neutral collisions).This has resulted in a simple theoretical approximationfor the force balance condition and allowed us to esti-mate the size of the cavity and its dependence on plasmaparameters. The results of theoretical calculation havebeen demonstrated to agree well with the experimentalresults. In addition, generalization of the model for theregime of collisionless ions has been made (see below inthe Appendix). The theoretical approach reported canbe useful in situations when large objects interact withcomplex plasmas. Experimental data Formula (12) M = u / v T E/N (Td)
FIG. 4. Reduced drift velocity of Ar + ions in Ar gas at T =300 K as a function of E/N – the ratio of electric field strengthto the neutral gas density. The latter is measured in Townsend(Td) units; 1 Td = 10 − V cm . Symbols correspond to theexperimental data [58]. The curve is calculated using Eqs. (7)and (12). ACKNOWLEDGMENTS
We thank Mierk Schwabe and Erich Z¨ahringer for care-ful reading of the manuscript. The microgravity researchis funded by the space agency of the Deutsches Zentrumf¨ur Luft- und Raumfahrt e.V. (DLR) with funds fromthe federal ministry for economy and technology accord-ing to a resolution of the Deutscher Bundestag undergrant No. 50WP0203 and 50WM1203. Support fromROSCOSMOS of the PK-3 Plus project is also acknowl-edged.
Appendix A: Mobility of Ar + ions in Ar gas Figure 4 shows the comparison between experimentaldata on Ar + ion mobility in Ar gas [58] and the approx-imation of Eqs. (7) and (12). For subthermal ( M <
Appendix B: Cavity size in the collisionless regime
Let us consider a hypothetical situation of a floatingsphere in the collisionless regime for the ion component.This corresponds to the regime ℓ i ≫ R s , which can berealized at very low pressures. This situation can alsobe of some relevance and interest in the context of astro-physical plasmas. In this case the long-range asymptote of the electrostatic potential around a sphere is again dic-tated by the ion absorption on the sphere surface. Quitegenerally, the potential can be estimated from φ ( r ) ≃ − T i e J J ( r ) , (B1)where J is the flux of ions on the sphere surface and J ( r )is their influx into the spherical surface of radius r [25].(This consideration works also in the collisional case, butin that case we made use of already existing expressionsfor the potential and electric field [57]). In the case ofthin collisionless sheath around a large sphere we have J ≃ πR n B c s , (B2)where n B ≃ n e − / ≃ . n is the plasma densityat the sheath edge and c s = p T e /m i is the ion soundvelocity. In the weakly perturbed region sufficiently farfrom the sphere the influx J ( r ) is simply J ( r ) ≃ √ πr n v T i . (B3)This yields φ ( r ) ≃ − . √ τ ( T i /e )( R s /r ) . (B4)In the case of a smaller object (e.g. microparticle), theorbital motion theory (OML) [6, 59, 60] can be appliedto give J = √ πR n v T i (1 + zτ ) , (B5)and in this regime φ ( r ) = − ( T i /e )( R s /r ) (1 + zτ ) ≃ QR s /r . (B6)The well known long-range ∝ r − asymptote is repro-duced [61]. We identify the main difference from thecollisional regime: The potential drops faster as ∝ r − instead of ∝ r − decay [62]. The location of the boundarycan be estimated from the condition M ∗ ≃
2, because inthe collisionless regime the ratio | F i /F el | decreases veryquickly with M . The energy conservation then simplyyields − eφ ( r ) = m i u m i v i = 2 T i . (B7)Combining (B4) and (B7) we finally get for the collision-less regime R cav ≃ . R s ( T e /T i ) / . (B8)This describes the cavity size in the collisionless limit.For τ ∼
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