Trapped ion effect on shielding, current flow, and charging of a small object in a plasma

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1 PHYSICS OF PLASMAS VOLUME 10, NUMBER 5 MAY 2003 rpped ion effect on shielding, current flow, nd chrging of smll object in plsm Mártin Lmpe ) Plsm Physics Division, Nvl Reserch Lbortory, Wshington, D.C Rjiv Goswmi Institute for Reserch in Electronics nd Applied Physics, University of Mrylnd, College Prk, Mrylnd Zoltn Sternovsky nd Scott Robertson Physics Deprtment, University of Colordo, Boulder, Colordo Vleriy Gvrishchk Science Applictions Interntionl Corportion, McLen, Virgini Guruds Gnguli nd Glenn Joyce Plsm Physics Division, Nvl Reserch Lbortory, Wshington, D.C Received 15 August 2002; ccepted 28 Jnury 2003 he problem of electrosttic shielding round smll sphericl collector immersed in nonflowing plsm, nd the relted problem of electron nd ion flow to the collector, dte to the origins of plsm physics. Clcultions hve typiclly neglected collisions, on the grounds tht the men free pth is long compred to the Debye length. However, it hs long been suspected tht negtive-energy trpped ions, creted by occsionl collisions, could be importnt. his pper presents self-consistent nlytic clcultions of the density nd distribution function of trpped nd untrpped ions, the potentil profile, the ion nd electron current to the collector, nd the floting potentil nd chrge of the collector. Under typicl conditions for dust grins immersed in dischrge plsm, trpped ions re found to dominte the shielding ner the grin, substntilly increse the ion current to the grin, nd suppress the floting potentil nd grin chrge, even when the men free pth is much greter thn the Debye length Americn Institute of Physics. DOI: / I. INRODUCION When smll object is immersed in plsm, both electrons nd positive ions flow to the object nd re bsorbed on its surfce. If the object is electriclly floting, it will cquire negtive chrge, due to more rpid bombrdment by electrons thn by ions. In recent yers, there hs been gret del of interest in the physics of dusty plsms, i.e., plsms tht contin mny prticultes dust grins with rdii tht re smll compred to the Debye length. In typicl lbortory experiments, prticulte sizes re 1 10 m, nd the grin chrge is on the order of thousnds of electron chrges. A vriety of interesting collective behviors occur becuse of the very strong plsm-medited interction between dust grins. However the most fundmentl issue of dusty plsm physics is the response of the plsm to the presence of single dust grin, i.e., the shielding round the chrged grin, the electron nd ion current to the grin, nd the stedy stte floting potentil nd chrge on the grin. Anlyses of dusty plsm 1 8 hve drwn on the theories developed in erlier times for Lngmuir probes nd spcecrft chrging his body of work comprises n enormous literture, beginning with Lngmuir nd collbortors 9 in the 1920s, followed by Electronic mil: lmpe@ppd.nrl.nvy.mil mny clssic ppers. heoreticl work on dusty plsm hs most often ssumed tht if there is no plsm streming reltive to the dust grins the chrge nd shielding of ech grin re given by the orbitl-motion-limited OML theory, 1 6,9,11 16 or by simpler pproximtions to the OML result such s the Debye-shielded potentil. OML theory, nd nerly ll of the theoreticl tretments dting bck to Lngmuir, 9 neglect collisions in treting the plsm response ner the object. his would seem to be quite resonble ssumption, since the men free pths for collisionl processes re typiclly long compred to the Debye length the chrcteristic length of the shielding cloud round the object in lbortory, spce, nd strophysicl plsms. If the plsm is collisionless, then ions coming in towrd the object from the mbient plsm will either contct the object in which cse it is usully ssumed they re bsorbed, or miss the object nd fly bck out to the mbient plsm. If the plsm potentil is tken to be zero, ll of these ions hve positive energy nd cnnot be confined ner the object. However, Bernstein nd Rbinowitz 11 commented in 1959 tht if there re occsionl collisions ner the object, ions cn lose energy nd be unble to escpe from the negtive potentil well. he density of these trpped ions cn thus X/2003/10(5)/1500/14/$ Americn Institute of Physics

2 Phys. Plsms, Vol. 10, No. 5, My 2003 rpped ion effect on shielding, current flow slowly build up until it reches n pprecible level, which could ply very importnt role in the dynmics. Since they found the popultion of trpped ions to be determined by collisions nd most difficult to clculte, Bernstein nd Rbinowitz crefully specified tht in order to obtin trctble problem their clcultion would be restricted to objects lrger thn specified size, for which trpped ions cnnot occur. Down through the yers, number of other uthors commented tht trpped ions could be importnt, but tht the trpped ion problem would probbly not be trctble. 2,11,17,18 Collisionless theories were thus generlly used for dusty plsm, even though trpped ions should be importnt for the smll dust grins. 18 In 1992, Goree 17 mde most remrkble observtion. He noted tht once trpped ion is creted, it will orbit the grin nd remin in the potentil well until it hs nother collision which either kicks it out of the well, or cuses it to fll onto the grin nd be bsorbed. Since the cretion rte of trpped ions is proportionl to the collision frequency, nd the loss rte is lso proportionl to, the density of trpped ions must be independent of in stedy stte. Goree lso confirmed in Monte Crlo simultion tht the totl number of trpped ions cn be quite significnt. In 2000 Zobnin et l. 18 performed more detiled Monte Crlo simultion, ctully clculting the trpped ion density profile n t (r) nd the self-consistent potentil (r). hey lso found tht n t (r) is indeed lrge. In recent Letter, 19 we sketched out fully nlytic method for clculting the distribution of trpped s well s untrpped ions, nd solved self-consistently for n t (r), (r), nd the untrpped ion density n u (r). We showed tht under typicl conditions the inner prt of the shielding cloud is mde up primrily of trpped ions, nd tht (r) is thus different from the results of the collisionless OML theory. In the present pper, we give the complete derivtion of those results, nd we clculte in ddition the ion distribution function nd the collisionl ion current to the grin. We find tht the collisionl current is usully dominnt, even in regimes of firly low collisionlity, becuse in stedy stte the trpped ion density is very lrge nd essentilly ll trpped ions eventully fll onto the grin fter sequence of collisions. Since collisionlity substntilly increses the ion current, the negtive floting potentil f of the grin is reduced to s little s 50% of the widely used OML result. he grin chrge is proportionl to f, nd thus cn lso be substntilly smller thn the OML result. he outline of the pper is s follows: In Sec. II, we introduce the model nd its ssumptions, nd derive the equtions tht determine n t (r), n u (r), (r), nd the ion distribution function f i (r,v). In Sec. III we derive the collisionl contribution to the ion current, nd show how to clculte f fully self-consistently. In Sec. IV, we give the results for some specific cses, nd discuss the generl nture of the solutions. In Sec. V we summrize nd discuss some future directions for reserch. In the Appendix we evlute the vlidity of one of our key ssumptions, the neglect of centrifugl potentil brriers to the rdil motion. II. SELF-CONSISEN CALCULAION OF HE POENIAL AND HE RAPPED ION DISRIBUION A. Model nd ssumptions We consider stedy stte consisting of single sttionry sphericl grin of rdius, immersed in plsm, with no mgnetic field. he grin is ssumed to be smll compred to the Debye length D 4n 0 e 2 ( 1 e 1 ) 1/2. he plsm consists of positive ions, ssumed for convenience to be singly-chrged, electrons, nd neutrl molecules. In the mbient plsm, ll species re ssumed to be Mxwellin, with tempertures e for electrons nd for both ions nd neutrls. he clcultion cn esily be extended to the cse where the ion nd neutrl tempertures re not equl. We ssume tht none of the plsm species re flowing. he present clcultion thus pplies to dust grins in bulk plsm, e.g., grins tht my be slowly settling through dischrge, or in microgrvity sitution grins which permnently reside in the plsm. It should be noted tht in typicl dusty plsm lbortory experiments, the dust grins levitte t the edge of sheth, where strong electric field blnces grvity. In this region, ions strem by the dust t velocity of the order of the ion sound speed, nd this ion flow hs importnt consequences. he present clcultion does not pply here, lthough we re looking into the possibility of extending it to this cse. We consider wekly-ionized dischrges, where the dominnt types of ion collision re normlly chrge-exchnge nd elstic collisions with neutrls. We include in our model only chrge-exchnge collisions, which we define s collisions tht trnsfer n electron from the neutrl to the ion, without ny exchnge of momentum. hus chrge-exchnge collision ner the grin simply replces fst incoming ion with slow ion whose velocity is chosen from the neutrl molecule distribution. hese collisions re prticulrly effective in creting trpped ions, or in cusing trpped ion to fll onto the grin. We shll neglect ion ion collisions nd elstic ion neutrl collisions. 20 We ssume tht the chrgeexchnge collision frequency is energy-independent. 21 his is n importnt simplifiction which enbles us to develop n nlytic model. Furthermore, we ssume tht is smll, in the sense tht the probbility of collision is smll during the time for n untrpped ion to trverse the potentil well, or for trpped ion to mke one rottion in its orbit. Roughly speking, this is equivlent to the ssumption tht the men free pth mfp D. Finlly, we mke n ssumption tht hs been widely used s the bsis of the orbitl-motion-limited OML theory. 1 6,11 15 Our system is sphericlly symmetric bout the grin, so in between collisions the energy nd ngulr momentum J of n ion re conserved. In sphericl coordintes, the rdil eqution of motion for n ion cn be written where mr Ur, J2 Urer 2mr 2 1 2

3 1502 Phys. Plsms, Vol. 10, No. 5, My 2003 Lmpe et l. rpped ions re creted by ion-neutrl chrge-exchnge collisions. Every time collision occurs, the old ion disppers, nd new ion is creted whose velocity is chosen t rndom from the neutrl molecule distribution function exp(mv 2 /2). An ion from the mbient plsm is ccelerted s it flls into the negtive potentil well surrounding the grin, so the result of collision within the potentil well is to replce fst ion by much slower ion, which probbly cnnot escpe from the well. he new ion will either be lost promptly by flling onto the grin, or will become trpped ion which orbits the grin. We begin our clcultion by prtitioning the trpped ion popultion into seprte clsses depending on the loction where the previous collision occurred. Consider the clss of trpped ions which were creted by collisions t rdil loction r, nd let h(r,v,;r) be the phse-spce distribution function of these ions. Here r is the present loction nd v(v,) is the present velocity of the ion; is the ngle between r nd v. Becuse of sphericl symmetry, h(r,v,;r) does not depend on the ngulr coordintes of r nd r, nor on the zimuthl coordinte of v. If trpped ions were collisionless, then the stedy-stte Vlsov eqution would tell us tht the distribution function h(r,v,;r), for given birthplce r, is function only of the constnts of the motion, energy 2mv 1 2 e(r) nd ngulr momentum Jmvr sin. Here we hve used the ssumption tht U(r) hs no mxim, so tht there re trjectories connecting ll phse-spce points tht re ccessible to n ion with specified nd J. Since the ions re born Mxwellin, this distribution must be of the form, hr,v,;rcrexp mv2 2 er. 4 FIG. 1. Possible shpes for the rdil effective potentil U(r). is n effective potentil energy for rdil motion, including centrifugl force term. he first term of 2 is ttrctive, while the centrifugl force term is repulsive. Depending on the vlue of J nd the detils of (r), U(r) cn hve no extrem, one minimum, or one minimum nd one mximum, s shown in Fig. 1. Our ssumption is tht U(r) hs no mximum. It follows tht, in the bsence of collision, the trjectory of n ion psses through ll vlues of r such tht J2 er 2mr 2. 3 For positive-energy ions, this mens ll vlues of r tht lie between some r min nd. Negtive-energy ions, on the other hnd, re trpped between some r min nd r mx. he ssumption tht U(r) hs no mximum is not exctly correct for ll ions. 5 However it hs been shown 6 tht it is good pproximtion, becuse mximum in U(r) occurs only for ions in smll rnge of J, nd the mximum is lwys so low tht it blocks the trjectories of only smll number of ions. We shll elborte further on this ssumption in the Appendix. B. Distribution of ions creted t single loction r Actully, trpped ions do undergo chrge-exchnge collisions, nd we do not wnt to neglect this. If collision occurs t r, the ion is lost from h(r,v,;r), nd new ion is dded to different clss h(r,v,;r). But we hve ssumed tht the collision frequency is energy-independent, so there is no correltion between the vlue of v nd the probbility tht collision hs occurred. Furthermore, we hve ssumed tht the time between collisions 1 is long compred to the orbit period of trpped ion. hus there is essentilly no correltion between the vlue of rr nd the probbility tht n ion creted t r hs hd nother collision before it gets to r. It follows tht ny ion in h(r,v,;r) is eqully likely to hve been lost to collision. herefore h(r,v,;r) must be of the form 4, even with collisions. However, the Mxwellin distribution 4 is not populted by trpped ions for every vlue of v nd. Severl conditions must be stisfied. First, the ion must hve negtive totl energy, i.e., 1 2mv 2 er; 5 otherwise it cn escpe to r nd is not trpped ion. A second condition is tht the totl energy must be greter thn e(r), since the ion ws born t r with positive kinetic energy, i.e., erer 1 2mv 2. 5b A third condition is tht the ion must hve enough ngulr momentum so tht its trjectory does not intercept the grin rdius. Since we ssume tht the orbitl period is short compred to the collision time, we tret ions whose trjectory intercepts the grin s if they re lost immeditely, nd just delete them from the trpped ion distribution. Using conservtion of energy nd ngulr momentum, it is esy to show tht if the ion is to miss the grin, it must hve t lest minimum kinetic energy specified by ere 2 r mv2. 5c If the ion does not stisfy 5c, it does not hve enough energy to void flling onto the grin. But even if 5c is stisfied, the ion must hve enough perpendiculr velocity to void flling onto the grin. Agin using conservtion of energy nd ngulr momentum, this leds to requirement on,

4 Phys. Plsms, Vol. 10, No. 5, My 2003 rpped ion effect on shielding, current flow... r1 2ere mv 2 sin 0 r,vsin. 5d Note tht 0sin 0 1if5c is stisfied. We cn combine 5 5c into condition where v 0 2 r,rv 2 v 1 2 r, 1 2mv 2 1 rer, nd v 0 (r,r) is the lrger of the two minim specified by Eqs. 5b nd 5c. o summrize, Eq. 6 sttes tht trpped ion which ws creted t r but is now t r must hve i t lest s much energy s it gined by flling from r to r; ii t lest enough energy to void flling onto the grin; but iii not so much energy tht it cn escpe from the potentil well. It is lso possible to show tht if r is smll enough, condition i determines the minimum velocity v 0, but if r is lrger it is condition ii tht is the controlling fctor. Specificlly, if r0rr, 1 2 mv 0r,rerer, 2 2 ere r 2 2, if rr 0 r, 8 where r 0 is function of r defined s the solution of 6 7 where hr,v,;rcrexp mv2 2 er x 1, r 1 rr 1 rv 1 rv vv 0 r,rsin sin 0 r,v, 12 if x0, 0, if x0. C. Clcultion of the coefficient C r We cn use the stedy stte condition to determine the coefficient C(r). As first step, we define quntity g(r) such tht in stedy stte 4r 2 g(r)dr is the totl number of trpped ions which were born in the volume element between r nd rdr. g(r) is thus the integrl of h(r,v,;r) over r, v, nd, grcrr 1 r r 14drr 2 e er/ v 1 r v 0 r,r 2dvv 2 e mv2 /2 00 r,v 0 r,v d sin 16 2 Crr 1 r r 1drr 2 Gr,r, r 0 r2 r 2 r We note tht 6 cn only be stisfied if v 0 2 (r,r)v 1 2 (r), which requires tht ere or equivlently r 2 r 2. 2 r 2 2 er, Eqution 11 is esily stisfied for smll grins ( D ) nd smll vlues of r D, where (r) is roughly the bre Coulomb potentil, nd we recll tht 0. But for r D shielding becomes strong, nd eventully Eq. 11 fils for r greter thn some rdius which we shll cll r 1. rpped ions cnnot exist for rr 1, becuse in this region the potentil well is very shllow, nd negtive-energy ions cnnot hve enough ngulr momentum to escpe flling onto the grin. 22 hus there re no orbiting trpped ions if the grin size is very lrge, r 1. king ccount of ll of the constrints 6 11, we see tht h(r,v,;r) is function tht is Mxwellin in v nd in e(r), but with mny voids in phse spce where h(r,v,;r)0. he presence of these voids cn be mde explicit by including pproprite step functions in the definition of h(r,v,;r), i.e., by rewriting Eq. 4 s where v 1 r Gr,r 1 2 eer/ dvv v 2 e mv2 /2 0 r,r 0 r,v d sin 0 r,v v 1 r e er/ v 0 r,r dvv 2 e mv2 /2 1 2 r 2 22 ere r 2 mv 2 2 3/2 m r 2 2 r 2 u 1 r u0 r,r duu 2 e u2. exp r2 er 2 e r In the lst step of Eq. 14, the integrl is reduced to n error function form by using the new vrible, u mv2 2 2 ere r 2 2, 15 nd u 0 (r,r), u 1 (r) re given by Eq. 15 with v 0 (r,r), v 1 (r) substituted for v. An ion is lost from g(r) every time one of these ions hs collision, i.e., the loss rte is

5 1504 Phys. Plsms, Vol. 10, No. 5, My 2003 Lmpe et l. dgr dt gr. loss 16 In stedy stte, this loss rte must be equl to the rte t which ions re dded to g(r) by collisions t r. he rte of collisions t r is simply n u (r)n t (r), nd ech of these collisions cretes new ion. But the new ion is trpped ion which contributes to g(r) only if conditions 5d, 6, nd 11 re stisfied, with rr. hus the cretion rte is dgr dt r 1 rn u rn t r cretion v 1 r v0 dvv 2 e mv2 /2 0 r,v r,r 0 r,v d sin 0 dvv 2 e mv2 /2 0 d sin 4 2 3/2 1/2 m r 1 r n u rn t re er/ Gr,r. 17 Using Eqs , we cn solve for C(r), giving hr,v,;r /2 5/2 m n u rn t re mv2 /2 e ercr/ Gr,r r 1 drr 2 Gr,r r 1 rr 1 rv 1 rv vv 0 r,rsin sin 0 r,v. 18 D. Clcultion of the trpped ion density nd the potentil We cn now clculte the trpped ion density n t (r) by integrting h(r,v,;r) over (v,;r). Since n t (r) lso ppers s source term on the RHS of Eq. 18, this procedure ctully yields liner integrl eqution for n t (r), n t r r 1drKr,rnt r r 1drKr,rnu r, where Kr,r 4 2 3/2 1/2 m r2 e er/ Gr,rGr,r r 1 drr 2 r 1 r. Gr,r o complete the clcultion, it is necessry to solve Eq. 19 self-consistently with Poisson s eqution, 1 r 2 d dr r2 d dr 4en urn t rn e r. 21 In Eq. 21, the trpped ion density is determined from the self-consistent solution of 19. he electron density n e (r) cn be ccurtely pproximted by Boltzmnn fctor, 14 n e rn 0 exper/ e. 22 he untrpped ion density n u (r) is specified s n explicit functionl of (r) by the OML theory, 4 n u r n 0 2 exp er 11 2 r 2 dtt 2 e t2 er/ 1 er t 2 2 eer/ dtt 2 e t2 er/ 2 2 r2 r 2 exp r2 er 2 e r 2 2 dtt 2 e t2, r 2 er 2 e/r where the first integrl is tken only over vlues of t such tht the rgument of the squre root is positive. Finlly, it is necessry to specify boundry conditions for Eq. 21 t r nd r. he boundry condition is t r is of course ()0. he boundry condition t r depends on the physicl sitution. If the collector is probe bised to specified potentil 0, the boundry condition is simply () 0. If the collector is dust grin, () is set equl to the floting potentil f, i.e., the potentil for which the electron flux F e to the grin is equl to the ion flux F i, F e F i. 24 king velocity moment over the Mxwell Boltzmnn distribution, the electron flux is found to be F e n 0 e 2m e e e f / e. 25 In the limit 0, F i is entirely due to untrpped ions, nd the ion flux is given by OML theory, 8 F OML n 0 2m e f Using 25 nd 26 in 24, we obtin the well-known OML result tht f is the solution of 1 e f m e. 27 exp e f e 1/2 m e ypiclly Eq. 27 gives e f e to 3 e, depending on the ion mss. However, we shll see tht for finite vlues of

6 Phys. Plsms, Vol. 10, No. 5, My 2003 rpped ion effect on shielding, current flow , F i is usully dominted by trpped ions, nd is often substntilly lrger thn F OML from 26. hus it is necessry to evlute F i from the trpped ion distribution nd solve 24 numericlly for f. his will be discussed in Sec. III below. E. Solution procedure A procedure for solving Eqs is s follows: i Rewrite Poisson s eqution, together with the boundry condition 0, in the form of Guss lw, rze 1 r r drq t rq u rq e r r 2, 28 where Ze is the grin chrge, ZeQ t (r) is the trpped ion chrge contined within rdius r, ZeQ u (r) is the devition of the untrpped ion chrge contined within rdius r from the mbient vlue (4/3)r 3 n 0 e, nd ZeQ e (r) is the devition of the electron chrge within rdius r from the mbient vlue. ii Begin with the OML solution 4 for (r) nd n u (r), i.e., the solution of Eqs , with boundry condition 27. his includes no trpped ions. Cll these (0) (r) nd n (0) u (r). Use (0) (r) in Eqs. 14 nd 20 to evlute K(r,r). iii Clculte first pproximtion n (1) t (r) to the trpped ion density from Eq. 19, neglecting the first term on the RHS. n (1) t (r) cn be interpreted s the popultion of first genertion trpped ions creted by collisions of untrpped ions. Integrte n (1) t (r) to obtin Q (1) t (r). iv Reclculte (r) from Eq. 28, choosing Z so tht the boundry condition 24 is stisfied. In the limit of smll, this just mens f must stisfy 27. For finite, itis necessry to explicitly clculte the electron nd ion flux to the grin nd choose Z so tht they re equl. his will be discussed in Sec. III. Using the new (1) (r), reclculte G(r,r), K(r,r), n u (r), nd n e (r) from Eqs. 14, 20, 22, 23. v Clculte second iterte n (2) t (r) by using n (1) t (r) in the first term on the RHS of 19. n (2) t (r) cn be regrded s the popultion of trpped ions creted by either the collision of n untrpped ion, or of first-genertion trpped ion. vi Go bck to step iv nd proceed with this itertion scheme to convergence. o prevent overshoots in the itertion process, it is sometimes useful to subdivide the itertion dding in only frction of the correction to n t (r) t ech itertion step, but in prctice the solution converges fter only few itertions. In Sec. IV, results will be shown in vriety of cses for n t (r), n u (r), (r), nd for the ion flux F i nd the grin potentil f s function of collision frequency. F. Distribution function of trpped ions After solving for (r), we cn write down n explicit expression for the trpped ion distribution function f t (r,v,) by integrting h(r,v,;r) over the source loction r. king ccount of ll of the step functions in Eq. 18 for f t (r,v,), nd both requirements on v 0 (r,r) from 8, this gives f t r,v,r 1 rv 1 rv 1 2 mv2 2 ere r 2 2 sin sin 0 r,v m r 1 drr 2 3/2 2 e mv2 /2 1 2 mv2 erer n urn t rgr,r r 1 drr 2 e erer/. Gr,r 29 he results for f t (r,v,) will be shown in Sec. IV. III. COLLISIONAL CONRIBUION O HE ION CURREN Eqution 26 from OML theory gives the collisionless flux of untrpped ions to the grin, F OML. More precisely, OML theory ssumes tht the men free pth is so lrge tht essentilly ll collisions occur fr from the grin, where the potentil is zero. It is further ssumed tht the ion distribution is Mxwellin in the mbient plsm, i.e., t sufficiently lrge r. As Bernstein nd Rbonowitz 11 pointed out, there must be collisions which mintin the Mxwellin distribution, but it is not necessry in OML theory to tke these collisions explicitly into ccount, since the ions which deposit on the grin re ssumed to come in from the Mxwellin mbient plsm without hving ny dditionl collisions. In relity n ion my hve chrge-exchnge collision ner the grin, on its wy in from the mbient plsm. If chrge-exchnge collision occurs, the energy of the newly creted ion is on verge less thn tht of the ion tht it replces, nd therefore the new ion is sttisticlly more likely to fll onto the grin. hus collisions increse the ion flux to the grin. It is esy to see tht this increse cn be very substntil, even when the men free pth is quite lrge. o estimte the collisionl effect, it is useful to think of the rdius r, such tht e(r )3/2, s the outer edge of the sheth round the grin. Normlly, r is in the vicinity of D to 2 D. If n incoming untrpped ion undergoes collision within rr, fst ion which probbly would not hve hit the grin is removed, nd slow ion is creted which probbly cnnot escpe from the potentil well. his ion my fll onto the grin immeditely if it hppens to hve low ngulr momentum. If the ion hs enough ngulr momentum, it will orbit the grin rther thn contcting it, but eventully will hve nother chrge-exchnge collision. On verge, ech collision brings the ion to lower-energy stte closer to the grin, nd eventully the resulting ion will fll onto the grin. hus, essentilly every collision of n untrpped ion within rr results in n ion depositing on the grin. he collection re of the sheth is r 2, which is very lrge compred to the cross-section 2 of the grin. How-

7 1506 Phys. Plsms, Vol. 10, No. 5, My 2003 Lmpe et l. ever this re must be weighted by the probbility tht n ion neutrl chrge-exchnge collision occurs while the ion is in the sheth. Roughly speking this probbility is of order r / mfp, where mfp is some verge men free pth. hus the ion flux to the grin, due to chrge-exchnge collisions t rr,is F coll r 3 2 F th, 30 mfp where F th is the therml flux in the mbient plsm, F th n 0 (/2m i ) 1/2. his should be dded to the OML flux, to obtin n pproximte expression for ion flux to the grin tht is ccurte to first order in r / mfp, F i 1 e f r 3 2 mfp F th 31 for ions of energy. An pproximte version of this expression not including the effect of shielding ws given by Ntnson 23 in For dusty plsms with D, the collisionl contribution 30 is usully lrger. It should be noted tht Eq. 30 is ctully n underestimte of the collisionl deposition, becuse collision which occurs on the fringes of the potentil well t rr lso slightly increses the probbility tht the ion will deposit on the grin. he cumultive effect of these distnt collisions over lrge volume lso contributes to the collisionl deposition. We shll now clculte the ion flux F i to the grin. Our clcultion includes both the OML flux nd the collisionl flux in unified wy; we shll show explicitly tht the OML flux is the result of collisions within the mbient plsm, while the collisionl flux is the result of collisions within the potentil well round the grin. Assuming tht the plsm dimensions re lrge compred to the men free pth, ny ion tht reches the grin will hve hd collision t some time in its pst. Let r be the plce t which the lst collision occurred. We cn write F i s n integrl over the rte t which collisions occur t point r, multiplied by the probbility p(r) tht the new ion creted by collision t r will deposit on the grin without hving nother collision, F i dr4r n t rn u rpr. 32 o clculte p(r), let us first consider n ion creted t r with velocity v, on trjectory tht will intersect the grin. he probbility tht this ion will rech the grin without hving nother collision is P coll r,vexp r dr/vr r, 33 where v r (r) is the ion s rdil velocity when it is t position r. Exct evlution of P coll (r,v) would require numericl clcultion of ll of the phse spce trjectories, but fortuntely simple nd ccurte pproximtion is possible. P coll (r,v) plys n importnt role t lrge r (r mfp ), where it prevents divergence due to multiple counting of ions which undergo mny collisions in the lrge volume outside the potentil well. But P coll (r,v) is close to unity for collisions within the potentil well, rr, since the plsm hs been ssumed to be wekly collisionl, i.e., r D mfp. hus it is sufficient to write n pproximte expression for P coll (r,v) which becomes exct in the lrge-r limit, nd goes to unity t r mfp. o do this, we note tht n ion tht strts t r mfp, nd whose trjectory intersects the grin without collision, will necessrily follow trjectory tht is nerly rdil. 24 he length of the trjectory will thus be close to r, nd in fct it is good enough to pproximte it s r. Furthermore, lmost ll of the time on the trjectory will be spent t rr, where v is close to its initil vlue. hus it is sufficiently ccurte to pproximte the energy-dependent men free pth s mfp v/. We cn then write P coll r,ve r/v. 33b Next, we note tht, ccording to Eqs. 5c nd 5d, n ion s trjectory will intersect the grin if it stisfies ny of the following four conditions: Incoming ions with low initil kinetic energy: mv r 2 2 ere, b Incoming ions with higher kinetic energy but low ngulr momentum: 2 r 2 2 ere mv2 2, 0. 34b c Outgoing trpped ions with low initil kinetic energy: mv r 2 2 ere, mv 2 2 er, c d Outgoing trpped ions with higher kinetic energy but low ngulr momentum: 2 r 2 2 ere mv2 2 er, 0. 34d In Eq. 34, 0 is given by Eq. 5d. he probbility p(r) is thus given by A dvv 2 d sin exp mv2 2 r v pr, 35 0 dvv 2 e mv2 /2 0 d sin where A is the re of v spce which stisfies conditions 34. After doing the -integrl nd collecting terms, the double integrl in Eq. 35 becomes

8 Phys. Plsms, Vol. 10, No. 5, My 2003 rpped ion effect on shielding, current flow... pr 2 duu 2 0 exp r u 2 v th u 2 r22 r r 2 r 2 er/ duu 2 0 exp r u 2 duu 2 ere/r 2 2 exp r u 2 v th uu 2 2 ere r 2 2 r2 2 r 2 v th u er/ duu 2 ere/r exp u 2 r v th uu 2 2 ere r 2 2, 36 where v th (2/m) 1/2. Equtions 32 nd 36 specify the ion current to the grin. Results will be shown in Sec. IV. We find tht for typicl smll but finite vlues of, the dominnt contribution to Eq. 32 is from collisions tht occur within the potentil well, t rr, nd tht F i is lrger thn F OML. his is becuse the density of trpped ions is very lrge ner the grin, nd chrge-exchnge collision ner the grin is very likely to yield n ion tht flls immeditely onto the grin. hus, for relistic vlues of it is necessry to clculte the floting potentil f by setting F i from 32, 36 equl to F e from 25. Since the ion current is substntilly lrger thn the OML vlue, the floting potentil f is significntly reduced s compred to the OML result 27. However, Eqs. 32 nd 36 lso reduce to the usul OML result in the limit 0. In this limit, the dominnt contribution to the integrl in 32 is from collisions tht occur in the rnge rv th /, i.e., one or more men free pths wy from the grin. Let us seprte the integrl into rnge from to s, nd rnge from s to, where s is some point such tht r sv th /, 37 es, 37b 2 s c Clerly, in Eq. 32 the contribution to F i from the rnge from rs vnishes s 0, nd thus in this limit F i dr4r n t rn u rpr. 38 s he inequlities 37 re stisfied for ll rs. Expnding Eq. 36 in ll of the smll prmeters, nd dditionlly using the fct 11,15,4 tht (r)r 2 s r, we find tht to lowest order pr 2 duu 2 r 2 0 exp u 2 r v th u 2 e du exp u 2 r. 39 r 2 0 v th u Inserting 39 in 32, reversing the order of integrtion, nd using n(r)n 0 for rr, we find F i n 0 2 du u2 u 2 e 0 e dr exp s r v th u n 0 2 v th duu u 2 e 0 e exp u2 s n 0 2 v th duu u 2 e 0 e u2 v th u n 0 2m 1 e, 40 which is the OML flux. Notice tht the coefficient cnceled out of Eq. 38, becuse the integrl in 38 is itself proportionl to 1 s 0. he lower bound s of the integrl, which ws chosen somewht rbitrrily, lso drops out to lowest order. In essence, the OML flux rises from collisions tht occur in region of the plsm where (r)0, i.e., where the presence of the grin hs no influence. Our work extends the theory to first order in by including collisions which occur ner the grin, where (r)0, but still requiring tht D mfp. IV. RESULS AND DISCUSSION A. Nerly collisionless limit he theory developed in Secs. II nd III depends on three dimensionless prmeters, / e, / D, nd mesure of the collisionlity which we choose to be D /v th. Note tht we hve ssumed tht is constnt, nd therefore the men free pth v/ is proportionl to v. he rtio of D to the men free pth is thus of order D /v th in the mbient plsm, but within the potentil well the ion velocities re much lrger, nd thus the men free pth is much lrger. hus D /v th is ctully n overestimte of the collisionlity of the plsm. We shll present solutions for vriety of prmeter choices, listed in ble I. We consider first the sitution in the limit of very smll but nonzero D /v th. hen the ion current is given by F OML of Eq. 26, the floting potentil is given by Eq. 27,

9 1508 Phys. Plsms, Vol. 10, No. 5, My 2003 Lmpe et l. ABLE I. Prmeters for the numericl solutions. For Cse 1, the exct vlue of D /v th does not mtter in Figs. 2 7, s long s it is smll, but for Figs we use the explicit vlue Cse number Figure numbers / e / D D /v th , , nd n t (r) nd (r) re given by the self-consistent solution of Eqs he solution for floting dust grin, for Cse 1: 0 nd / e 0.01, / D 0.015, 41 is shown in Figs Figure 2 shows the density n t (r) of trpped ions solid curve; the devition of the untrpped ion density from the mbient vlue, n u (r)n u (r)n 0 dshed curve; nd the negtive of the devition from the mbient electron density, n e (r)n 0 n e (r) dotted dshed curve. Notice tht n t n u n e ner the grin, i.e., trpped ions dominte the shielding ner the chrged grin. In Fig. 3 we show the integrted chrge densities Q t (r), Q u (r), nd Q e (r), respectively the integrls of en t (r), en u (r), nd en e (r) from to r, ech scled to the grin chrge Ze. Note tht t r2 D, the grin chrge is 68% neutrlized, with 41% due to trpped ions, 27% due to untrpped ions, nd 1% due to electrons. At r, 47% of the shielding is due to trpped ions, 52% to untrpped ions, nd 1% to electrons. In Fig. 4, we plot r(r). On this semilog plot, the unshielded Coulomb potentil would pper s horizontl stright line, nd the Debye-shielded potentil would pper s the oblique dotted line. We plot the complete solution s the solid curve, nd the OML solution with no trpped ions s the dshed curve. Note tht the inclusion of the trpped ions increses the shielding nd FIG. 3. Integrted trpped ion chrge from r to r Q t (r), solid, devition of the integrted untrpped ion chrge from mbient Q u (r), dshed, nd devition of the integrted electron chrge from mbient Q e (r), dotted dshed, ll scled to the chrge on the grin, for Cse 1. brings the potentil to within 25% of the Debye-shielded potentil for r5 D. For lrge r, (r) is proportionl to r 2, s hs been discussed previously. 11,15,4 In Fig. 5, we show ion distribution functions t three loctions, 5 r2, close to the grin; 5b r10, midwy in the sheth; 5c r D 66.6, close to the outer limit of the sheth. he distribution function consists of trpped ions for 1 2mv 2 (r)e(r) negtive totl energy nd untrpped ions for 1 2mv 2 (r)e(r) positive totl energy. Both the trpped nd untrpped portions of the distribution function f i (r,v,) re isotropic functions f (r,v), except tht there re voids for certin rnges of, representing ions whose trjectories intersect the grin. In Fig. 5 we hve plotted the isotropic function f (r,v) s solid curve in the trpped ion region, nd s dshed curve in the untrpped ion region. hese curves re on n rbitrry scle. We hve lso plotted the criticl ngle 0 (r,v) which defines the voids, s will be explined below. his is the dotted curve, nd it is on the scle shown t the left of the figure. We note FIG. 2. rpped ion density n t (r), solid, devition of untrpped ion density from mbient n u (r), dshed, nd devition of electron density from mbient n e (r), dotted dshed, ll scled to mbient density n 0, for Cse 1: / e 0.01, / D 0.015, 0. FIG. 4. Plot of (r/)e(r)/ e for Cse 1: self-consistent potentil including trpped ions solid, potentil with trpped ions neglected dshed, Debye potentil dotted.

10 Phys. Plsms, Vol. 10, No. 5, My 2003 rpped ion effect on shielding, current flow tht in ll cses there is discontinuity in f (r,v) t the zero-energy point 1 2mv 2 (r)e(r), nd tht the nture of the distribution is very different for the trpped nd untrpped ions. Consider first the untrpped ions. Since the mbient ion distribution is Mxwellin, Liouville s eqution indictes 4 tht the untrpped prt of the ion distribution t loction r is lso Mxwellin f r,vconstexp mv2 2 er, except tht outgoing ions whose trjectories hve lredy intersected the grin re removed from the distribution. his forms void in the Mxwellin for 0 (r,v), where 0 (r,v) is monotoniclly decresing function of both r nd v which is defined by Eq. 5d. For 1 2mv 2 (r)e(r) we hve the trpped ion prt of the distribution, given by Eq. 29. Outgoing ions whose trjectories hve lredy intersected the grin re removed from the trpped ion distribution; hence there is void for 0 (r,v). But in ddition, incoming ions whose trjectories will intersect the grin re removed from the trpped ion distribution. As explined in Sec. II B, this is done becuse these ions hve very short lifetime s compred to the orbiting trpped ions. hus, for negtive-energy ions, there is lso void in the distribution for 0 (r,v). Note tht f (r,v) hs spike for trpped ions with slightly negtive totl energy. here re mny such ions becuse they cn be creted by collisions tht occur nywhere in the lrge volume t the edge of the potentil well, where (r) is slightly negtive. Smller velocities in Fig. 5 correspond to more strongly trpped ions, which re creted by collisions deeper within the sheth. he volume vilble for these collisions is smller, but on the other hnd there re mny collisions t smll r becuse the ion density there is lrge see Fig. 2.As result of the blnce between these opposing trends, f (r,v) generlly decreses t smll vlues of v, but there is gentle pek t moderte vlues of v for the cses r2 nd r 10. Notice lso tht for very smll v, 0 (r,v) becomes lrger thn /2. his mens tht the void hs eten up the entire distribution, nd there re no trpped ions t ll with these smll vlues of v. he condition for this is tht inequlity 5c is violted. he results shown in Figs. 2 5 pper to present prdox: we hve ssumed tht the collision frequency is vnishingly smll, nd yet trpped ions, creted only by collisions, re dominnt feture of the solution. Indeed, (r), n t (r), nd f i (r,v,) do not depend on the vlue of, in this limit of smll. How cn this be possible, since there re no trpped ions if 0? he explntion of the 0 limit is tht for ny smll but nonzero vlue of the cretion rte of trpped ions is proportionl to nd the loss rte is lso proportionl to, so the stedy stte is independent of. However, the time necessry to rech stedy stte is inversely proportionl to, so stedy stte is never reched for 0. In prctice, is lwys very fst compred to mcroscopic times such s the lifetime of the dischrge, or times chrcterizing the motion of dust grin, so stedy stte tretment is indeed pproprite. rpped ions clerly re dominnt plyer in the shielding round the grin, for Cse 1. One my sk more generlly under wht conditions the density of trpped ions is lrge? he essentil requirement is tht nerly ll of the new ions creted by chrge-exchnge collisions within the sheth become trpped ions. here re two conditions for this: tht the new ion does not escpe to r, nor does it fll onto the grin. he potentil f t the grin is of the order of e, while new ions re born with energy of the order of. hus, very few of the new ions cn escpe from the well, if / e 1. Provided this is true, the sheth cn be regrded s extending roughly to the point r typiclly bout D to 2 D ) where e(r )3/2. Equtions 34 specify the conditions under which the new ion creted t point r r will fll onto the grin. hese equtions indicte tht this is unlikely to hppen if 2 /r 2 / e. When both of these conditions re stisfied, 2 /r 2 / e 1, 42 the stedy stte trpped ion popultion within the potentil well gretly exceeds the untrpped ion popultion, since FIG. 5. he ion distribution function f (r,v,) is n isotropic function f (r,v) which hs voids for certin rnges of, s discussed in the text. We show f (r,v) for Cse 1 t r2, b r10, c r D f (r,v) is shown s solid curve for the rnge mv 2 /2e(r) trpped ions, nd s dshed curve for mv 2 /2e(r) untrpped ions. hese curves re on rbitrry scle. he quntity 0 (r,v) which chrcterizes the voids in the distribution is shown s the dotted curves, with scle t left. For v less thn criticl vlue defined by Eq. 5c, 0 (r,v)/2, which mens the distribution function is entirely void.

11 1510 Phys. Plsms, Vol. 10, No. 5, My 2003 Lmpe et l. FIG. 8. F i /F OML s function of collisionlity index D /v th. FIG. 6. rpped ion density n t (r), scled to n 0, for Cse 1 (/ D 0.015, / e 0.01, solid curve, Cse2(/ D 0.1, / e 0.01, dshed curve, nd Cse 3 (/ D 0.015, / e 0.04, dotted curve. In ll three cses, 0. nerly every collision of n untrpped ion t rr results in the cretion of trpped ion, but only smll frction of the collisions of trpped ions result in the disppernce of trpped ion. If / e 2 /r 2, the trpped ion popultion flls off becuse mny of the newly born ions hve low ngulr momentum nd immeditely fll onto the grin. In the opposite limit where / e pproches unity, the trpped ion popultion gin flls off, becuse mny of the newly born ions hve enough energy to escpe to r. Figure 6 shows the trpped ion density for Cse 1 nd for two cses with 0 tht test the limits 42. In Cse 2, / D 0.1 nd / e 0.01, nd in Cse 3 / D nd / e Note tht the trpped ion density in these cses is pprecibly smller thn in Cse 1, where / D nd / e In Fig. 7 we plot r(r) for ech of the three cses. In Cses 1 nd 3, / D is quite smll, nd (r) is very close to the simple Debye-shielded potentil, out to r 5 D. For lrger r, (r)r 2. his behvior t lrge r is due to ion bsorption on the grin. 11,4 In Cse 2, where / D 0.1 is lrger, the potentil still shows Debye-type form out to r5 D, but with shielding length tht is bout 20% lrger thn D. In generl, the old OML theory which omits trpped ions shows Debye-type shielding out to r 5 D, but with shielding length tht is longer thn Debye. Including trpped ions in our theory decreses the shielding length, nd brings the potentil closer to Debye. But for lrge grins, the OML shielding is significntly weker thn Debye, 2,4 nd even with trpped ions remins noticebly weker thn Debye. Actully, it is bit of mystery s to why the Debye form works s well s it does, especilly when rr. he usul derivtion of Debye shielding proceeds by linerizing n ssumed Boltzmnn form for the ion density, n i rn 0 exp er 43 n 0 1 er 43b nd inserting Eq. 43b nd the equivlent expression for electrons into the Poisson eqution 21. However, the nonliner Boltzmnn form 43 is grossly wrong for rr ;it gives n i (r)e 100 n 0 ner the grin! he lineriztion is lso completely invlid, s n i (r)n 0 nd e(r) ner the grin. Nonetheless, the linerized form 43b works pretty well in deriving the shielded potentil from Poisson s eqution. FIG. 7. r(r) for the three cses of Fig. 6. Cse 1: solid curve. Cse 2: dshed curve. Cse 3: hevy dotted curve. he Debye potentil is shown s the light dotted line. B. Smll but finite collision frequency For nonzero collision frequency, we must determine the floting potentil f by setting the ion flux to the grin F i, from 32 nd 36, equl to the electron flux F e from 25. he determintion of f is done self-consistently with the solution for n t (r). Figure 8 shows F i /F OML s function of the collisionlity mesure D /v th, for / D 0.015, / e We see tht collisionlity increses F i substntilly even when D /v th is smll. he grin potentil f is shown s function of D /v th in Fig. 9. he floting potentil is suppressed by up to 50%, s result of the collisionl increse in ion current to the grin. o very good

12 Phys. Plsms, Vol. 10, No. 5, My 2003 rpped ion effect on shielding, current flow FIG. 9. Floting potentil f, s function of collisionlity index D /v th. pproximtion, the chrge on the grin is Ze f ; thus Z is lso reduced by up to 50% s compred to the usully ccepted OML vlue. It should be noted tht there re two contrry effects t work in determining the dependence of F i on. For specified grin potentil f, collisions ner the grin strongly increse F i. But collisionlity lso decreses f, nd this in turn reduces F i somewht. In Fig. 8 we show F i scled to the OML ion flux F OML from Eq. 26. Here F OML mens the OML flux to grin whose potentil is the OML floting potentil given Eq. 27. In some erlier work, 25 we compred F i to n intermedite benchmrk, the OML flux from 26, to grin whose potentil is the full self-consistent vlue of f, including the effect of collisions. his difference in the definition of F OML explins the pprent difference between Fig. 8 nd some of the results quoted in Ref. 25. In Fig. 10, we show the trpped ion density n t (r) for severl vlues of D /v th : Cse 1, D /v th 0; Cse 4, D /v th 0.037; Cse 5, D /v th In ll cses, / e 0.01 nd / D Curiously enough, the trpped ion density is seen to decrese s increses. his my seem prdoxicl since trpped ions re creted by collisions. he FIG. 11. Potentil profile (r) for Cse 1, D /v th solid curve; Cse 4, D /v th dshed curve; Cse 5, D /v th 0.47 dotted curve. explntion is tht f decreses with incresing collisionlity, i.e., the potentil well becomes shllower nd hence trps fewer ions. We show the complete potentil profile (r) for Cses 1, 4, 5 in Fig. 11. In ll of these cses n t (r) significntly exceeds n u (r) ner the grin, so trpped ions re dominnt fctor in shielding. In Fig. 12, we plot the integrnd of Eq. 32, I(r) 4r 2 n t (r)n u (r)p(r). his plot shows the distribution of loctions r where n ion hd its lst collision before hitting the grin. he result is shown for Cse 1, D /v th i.e., essentilly zero, solid curve; Cse 4, D /v th dshed curve; nd Cse 5, D /v th 0.47 dotted curve. his is log log plot, so tht we cn disply both the sptil scle r D of the sheth nd the much longer collisionl scle rv th /. Note tht in ll cses there is pek in I(r) tht occurs t r D. his is due to the occsionl collisions tht occur within the sheth. he pek occurs for two resons: he trpped ion density is very lrge in this rnge of r, nd if collision occurs within the sheth, the resulting ion is quite likely to hit the grin. he FIG. 10. rpped ion density n t (r) for Cse 1, D /v th solid curve; Cse 4, D /v th dshed curve; Cse 5, D /v th 0.47 dotted curve. FIG. 12. he integrnd I(r) ofeq.32, for / e 0.01, / D 0.015, nd D /v th solid curve, D /v th dshed curve, D /v th 0.47 dotted curve. I(r) is the reltive contribution to the ion flux to the grin, from collisions t r.

13 1512 Phys. Plsms, Vol. 10, No. 5, My 2003 Lmpe et l. shpe of the pek is roughly independent of nd the mplitude of the pek is roughly proportionl to ; hence the contribution to F i is roughly proportionl to. For the cses D /v th nd 0.47, this is the dominnt contribution to F i. In ddition, I(r) hs brod plteu tht extends out to severl times the men free pth v th /, nd then flls off exponentilly becuse if collision occurs t such lrge rdius, the resulting ion is likely to hve nother collision before it cn rech the grin. his contribution to I(r) hs n mplitude tht is proportionl to, nd sptil rnge tht is inversely proportionl to ; hence its contribution to F i is independent of. his is in fct the OML current. For the cses D /v th nd 0.037, the collisionl contribution from the pek t r D is clerly distinguished from the OML contribution from the plteu t rv th /, but for the cses D /v th 0.47 the two regimes re beginning to merge. his cse pushes the limits of vlidity of our theory, which ssumes to be smll. V. CONCLUSIONS We hve found tht trpped ions creted by chrgeexchnge collisions cn dominte both the shielding round chrged grin nd the ion current to the grin, which determines the floting potentil. In the limit where the collision frequency 0, the trpped ion density in the shielding cloud round the grin cn be over n order of mgnitude lrger thn the untrpped ion density. he conditions for the trpped ion density to be lrge re e, so tht nerly ll the newly creted ions re trpped in the potentil well, nd 2 / D 2 / e, so tht very few newly creted ions fll immeditely onto the grin. For finite but smll, the ion current F i to the grin increses. his increse is roughly proportionl to, but nonetheless very lrge. he incresed ion flow to the grin suppresses the floting potentil f. In the bsence of collisions, e f is typiclly e to 3 e, but we hve seen trpped ion/collisionl effects reduce e f by s much s 50%. Self-consistently including this reduction in the well depth reduces the density of trpped ions slightly; thus the trpped ion density is ctully lrgest in the limit 0. he presence of lrge popultion of trpped ions profoundly chnges the nture of the interction of grin with other grins, nd with externl forces. We hve previously rgued 4 tht shielding by untrpped ions cnnot led to net ttrctive electrosttic force between negtively-chrged grins. But grin with its trpped ion cloud cn behve like clssicl tom; the trpped ion cloud cn be polrized, thereby shielding the grin from electric fields, 2,17 nd possibly leding to vn der Wls type ttrctive forces between grins. We re studying these effects t the present time. his pper hs been concerned with trpped ion nd collisionl effects for sphericl dust grin which is t floting potentil. It hs recently been shown, 26,27 both theoreticlly nd experimentlly, tht the sme effects cn lso be importnt in connection with the ion current to cylindricl Lngmuir probe, either bised or floting. ACKNOWLEDGMENS M.L. wishes to thnk Dr. Wllce Mnheimer for contributing some good ides erly in the work. he work of M.L., R.G., V.G., G.G., nd G.J. ws supported by the Office of Nvl Reserch nd NASA. he work of Z.S. nd S.R. ws supported by the Office of Fusion Energy Sciences of the Deprtment of Energy. APPENDIX: VALIDIY OF HE OML ASSUMPION NO POENIAL BARRIERS If the effective potentil U(r;J), defined in Eq. 2, hs mximum U mx 0 which occurs t rr m, then phse spce for ions with ngulr momentum J nd energy U mx is prtitioned into two regions: trpped ions for r r m, nd untrpped ions for rr m. here re no collisionless trjectories connecting the two regions, so it is possible tht the distribution function hs different form in ech region. We hve voided significnt mthemticl complictions by ssuming tht there re no such brriers, or more precisely tht if there is mximum in U(r) it is so low or occurs in such limited rnge of J tht it ffects negligible number of ions. It follows then tht trpped ions re simply synonymous with negtive-energy ions. he neglect of centrifugl potentil brriers hs long history, nd is one of the key ssumptions of the orbitlmotion-limited OML theory of probes. here hs been renewed interest recently in the question of vlidity of this ssumption. Bernstein nd Rbinowitz 11 showed long go tht in the limit D, there re no potentil brriers if the mbient ion distribution is monoenergetic. It ws generlly ssumed, over the yers, tht potentil brriers could be neglected if D, even if the distribution is Mxwellin. However, Allen, Annrtone nd de Angelis 5 showed recently tht for Mxwellin distribution, there re lwys some ions subject to potentil brriers, even if / D is smll. We subsequently clculted 6,7 the ctul mgnitude of the potentil brriers, under the ssumption tht (r) hs the Debye form, r f r er/ D, A1 t lest for r out to severl D. As we hve seen in Sec. IV, Eq. A1 ppers to be good qulittive pproximtion, so it forms resonble bsis for estimting the effect of potentil brriers. We concluded in Ref. 6 tht the most significnt potentil brriers occur for J in the rnge, 1 2 D f J2 m 3 4 D f, A2 where U(r) does indeed hve mximum U mx nd the vlue of the mximum is in the rnge 0.01 e f D U mx 0.02 e f D. A3 Only frction of order 0.01(/ D )e f / of those ions whose ngulr momentum stisfies A2 re stopped by the potentil brrier. Although e f / is lrge quntity ( f

14 Phys. Plsms, Vol. 10, No. 5, My 2003 rpped ion effect on shielding, current flow e to 3 e, which cn be 100, / D is typiclly smll in dusty plsms, nd in most cses only smll frction of the ions re reflected t the brrier. here is n dditionl considertion tht limits the number of trpped ions tht cn be reflected by potentil brrier. Consider n ion newly creted by collision t r. he typicl vlue of J 2 is mr 2, nd thus condition A2 cn be estimted s 1 e f 2 D r2 2 3 e f D 4 D, A4 U mx is lrge enough to reflect substntil number of ions only in cses where e f 20. A5 D In these cses, Eq. A4 reduces pproximtely to 3 r 4. A6 D So even in those unusul cses where the brrier is firly strong, its primry effect is to trp few ions with slightly positive energy, creted in limited sptil region. In essence, the potentil brrier slightly lters the perimeter of the potentil well. Neglecting the effect of potentil brriers generlly ppers to be quite resonble pproximtion for dusty plsms. 1 E. C. Whipple, Rep. Prog. Phys. 44, J. E. Dugherty, R. K. Porteus, M. D. Kilgore, nd D. B. Grves, J. Appl. Phys. 72, C. K. Goertz, Rev. Geophys. 27, M. Lmpe, G. Joyce, G. Gnguli, nd V. Gvrishchk, Phys. Plsms 7, J. E. Allen, B. M. Annrtone, nd U. deangelis, J. Plsm Phys. 63, M. Lmpe, J. Plsm Phys. 65, ; J. E. Allen, J. ech. Phys. 43, M. Lmpe, G. Joyce, G. Gnguli, nd V. Gvrishchk, Phys. Scr., 89, J.-P. Boeuf nd C. Punset, in Dusty Plsms, edited by A. Bouchoule Wiley, New York, 1999, Chp. 1, p H. Mott-Smith, Jr. nd I. Lngmuir, Phys. Rev. 28, J. E. Allen, R. L. Boyd, nd P. Reynolds, Proc. Phys. Soc. London, Sect. B 70, I. B. Bernstein nd I. N. Rbinowitz, Phys. Fluids 2, F. F. Chen, J. Nucl. Energy, Prt C 7, J. G. Lfrmboise, University of oronto, Institute For Aerospce Studies, Report No Copies re vilble from the U-IAS librry. 14 J. G. Lfrmboise nd L. W. Prker, Phys. Fluids 16, J. E. Allen, Phys. Scr. 45, B. M. Annrtone, M. W. Allen, nd J. E. Allen, J. Phys. D 25, noted tht if the grin is lrge enough, collisions t distnce of order one men free pth from the grin cn cuse ions to fll directly onto the grin, thereby effecting trnsition from the OML theory to the rdil inflow theory of Allen, Boyd, nd Reynolds Ref. 10. See lso Ref. 12. his effect is included in the present work, s well s the dditionl effect of orbiting trpped ions creted by collisions. 17 J. Goree, Phys. Rev. Lett. 69, A. V. Zobnin, A. P. Nefedov, V. A. Sinel shchikov, nd V. E. Fortov, JEP 91, M. Lmpe, V. Gvrishchk, G. Gnguli, nd G. Joyce, Phys. Rev. Lett. 86, Properly speking, chrge-exchnge collisions re elstic sctterings by 180 in the center-of-mss frme. For fst ions, there is pronounced pek in the differentil scttering cross section t 180, s shown by M. L. Vestl, C. R. Blkley, nd J. H. Futrell, Phys. Rev. A 17, hus chrge-exchnge is clerly the dominnt ion collision. he differentil cross section is less peked for ion energy below 1 ev, nd it becomes more difficult to distinguish chrge-exchnge from other elstic scttering. However, for purposes of creting nd destroying trpped ions, clerly chrge-exchnge collisions hve the gretest effect. Elstic nd ion ion collisions lso result in energy loss nd pitch-ngle scttering of n incoming fst ion, nd we believe tht including these types of collisions would in effect slightly increse the collision frequency without qulittively chnging the results. A comprehensive discussion of the role of these different types of collisions is given by A. V. Phelps, J. Appl. Phys. 76, he energy dependence of chrge-exchnge cross sections is given by R. Hegerberg, M.. Elford, nd H. R. Skullerud, nd summrized by M. A. Liebermn nd A. J. Lichtenberg, Principles of Plsm Dischrges nd Mterils Processing Wiley, New York, 1994, p. 78. In generl depending on the specific gs, the energy dependence is somewhere between constnt cross-section nd constnt collision frequency. 22 We recll here tht we hve ssumed U(r) hs no mxim, nd consequently tht ll trpped ions hve negtive energy. In relity, U(r) hs very wek mxim tht permit smll number of ions with smll positive energy to be trpped t rr 1. he neglect of these trpped ions is usully of little consequence, s discussed in the Appendix. 23 G. L. Ntnson, Zh. ekh. Fiz. 30, Sov. Phys. ech. Phys. 30, An ion tht hits the grin will hve energy of order e t impct. herefore its ngulr momentum cn be no lrger thn order (2 e /m) 1/2. If this ion strts t r mfp with therml energy of order, ngulr momentum conservtion indictes tht sin 2 (/ mfp ) 2 ( e /). Normlly / mfp 10 2,so1. 25 M. Lmpe, G. Gnguli, V. Gvrishchk, R. Goswmi, nd G. Joyce, Collisionl nd nonliner effects on grin chrge nd intergrin force, in Proceedings of the hird Interntionl Conference on Physics of Dusty Plsms, Durbn, South Afric, My 2002, AIP Conf. Proc. 649, Z. Sternovsky nd S. Robertson, Appl. Phys. Lett. 81, Z. Sternovsky, S. Robertson, nd M. Lmpe, Phys. Plsms 10,