Particle-In-Cell Simulations of a Current-Free Double Layer

Transcription

1 Particle-In-Cell Simulations of a Current-Free Double Layer S. D. Baalrud 1, T. Lafleur, C. Charles and R. W. Boswell American Physical Society Division of Plasma Physics Meeting November 10, Present address: CICART, University of New Hampshire APS-DPP November 10, 2010, p 1

2 Introduction Theories of current-free double layers make different assumptions about the electron velocity distribution function (EVDF) upstream: Stationary Maxwellian: Chen, POP (2006) Maxwellian plus half-maxwellian beam: Lieberman and Charles, PRL (2006) Counter streaming Maxwellian beams: Goswami et al POP (2008) Two temperature Maxwellians: Ahedo and Sánchez PRL (2009) We also develop a model for the EVDF Maxwellian depleted in density corresponding to losses The assumed EVDF has important consequences for double layer formation and potential drop We use PIC simulations to gain insight into what EVDFs to expect 1D in space, 3D in velocity phase space Simulates plasma expansion with a particle loss profile from B expansion The PIC simulations agree qualitatively with previous simulations and experiments (potential and density profiles, IVDFs, etc) EVDFs from PIC show depleted Maxwellians Can use the PIC results to refine the EVDF model APS-DPP November 10, 2010, p 2

3 Impetus: CFDL Thruster Experiments an electric field is measured at the source exit (double layer) Charles and Boswell Phys. Plasmas 2004 source chamber is the thruster expansion chamber simulates space APS-DPP November 10, 2010, p 3

4 Ion Beams Have Been Measured Ion beams have been measured E Equal electron and ion currents reach the downstream region All ions are accelerated by E Most electrons are reflected by E, but they are very hot: 50,000 o No need for an electron source Charles, et al, APL 2003 Sun, et al, PRL 2005 Charles, et al, POP 2004 Ion beams could be used to generate thrust APS-DPP November 10, 2010, p 4

5 φ DL can be calculated from current-free condition φ Δφ s2 Δφ DL Region 2 Region 1 Δφ s1 s2 DL2 DL1 s1 x Potential drops (sheaths and DL) calculated from current balance where c s,2 T e,2 /M i. dv x v x f e2 (x = DL 2 ) } {{ } Γ e,dl2 = n 2 e 1/2 c s,2 }{{} Γ i,dl2 (1) Similar expressions at x = s2 and x = s1, find φ s2, φ DL and φ s1 APS-DPP November 10, 2010, p 5

6 φ DL can be the floating potential ln(f e,dl2 ) ln(f e,dl1 ) eδφdl For a collisional upstream chamber, and infinite downstream chamber { 0 ; v x v c,dl f e2,x = n 2a πvt exp( v 2 a x /v2 T a ) ; v c,dl v x in which v c,dl 2e φ DL /m e is a truncation speed corresponding to φ DL Putting this into Eq. (1) yields φ DL = T e,a e ( ) ln n2 me 2π e 1/2 n 2a M i Since n 2a n 2, T e,a T e for v c,dl /v T e 1, this is the floating potential: φ DL T [ ( )] e Mi 1 + ln 2e 2πm e APS-DPP November 10, 2010, p 6

7 The upstream -e ΔφDL 0 wall causes depletion (b) ln(f e,s2 ) ln(f e,dl2 ) -e ΔφDL e Δφ s2 -e Δφ DL e Δφ s2 Accounting for depletion from the upstream wall, infinite downstream (c) ln(f e,s2 ) ln(f e,dl2 ) n b,dl2, v x < v DL f x,dl2 = e v2 x /v2 T e πvt e ln(f e,dl1 ) n a,dl2, v DL v x v s2 n c,dl2, v s2 < v x From symmetry: φ DL = φ s2. Putting this into Eq. (1) gives -e Δφ s2 e Δφ s2 -e Δφ s2 e Δφ s2 φ -e ( Δφ s1 + Δφ DL = T ( ) e DL ) e ln n 2 e 1/2 2πm e n -e ( Δφ s1 + Δφ ) c,dl2 n b,dl2 ln(f e,s1 ) Expect that n c,dl2 /n b,dl2 L/l s 1, so have φ DL T [ ( e n 2 )] c,dl2 M i 1 + ln 2e 2πm e n 2 2 M i -e Δφs1 -e Δφs1 APS-DPP November 10, 2010, p 7

8 Upstream wall determines pressure minimum The DL potential drop requires Expect that φ DL = T e e ln ) e 1/2 2πm e n c,dl2 ( n2 0 < n 2 e 1/2 2πm e < 1 n c,dl2 M i M i n c,dl2 n 2 { Lup /λ e n, L up < λ e,n 1, L up λ e,n Using λ e n = 1/(n n σ e n ) and n n = n o p where n o = m 3 and p in in mtorr, the less than one condition gives p[mtorr] e 1/2 2πm e /M i n o σ e n L up (2) For T e 1 ev in an argon plasma and L up = 5 cm, this gives p min 0.03 mtorr Can also use n c,dl2 /n 2 n o σ e n L up p[mtorr] to get scaling of φ DL with neutral pressure APS-DPP November 10, 2010, p 8

9 (b) ln(f e,s2 ) ln(f e,dl2 ) Simulation has reflection & depletion from both walls In the simulations, the downstream wall reflects most of the incident electron current φ s1, φ DL and φ s2 can still be calculated from current balance (but result is much more complicated) (c) -e ΔφDL e Δφ s2 ln(f e,s2 ) -e Δφ DL e Δφ s2 ln(f e,dl2 ) -e Δφ s2 e Δφ s2 -e Δφ s2 e Δφ s2 -e ( Δφ s1 + Δφ DL ) -e ( Δφ s1 + Δφ DL ) ln(f e,dl1 ) ln(f e,s1 ) -e Δφs1 -e Δφs1 ±e ( Δφ s2 - Δφ DL ) ±e ( Δφ s2 - Δφ DL ) APS-DPP November 10, 2010, p 9

10 PIC simulations conducted with phoenix phoenix is 1D in space, 3D in velocity phase-space Uses Monte Carlo to simulate electron-neutral and ion-neutral collisions A loss profile is imposed downstream to generate a DL Loss profile derived from volume expansion of a diverging solenoidal B Linear loss profiles have also been used (to connect with Meige et al) Electrons heated upstream in perpendicular direction using an inductive heating model Antenna placed in middle of upstream chamber for most simulations Perpendicular direction is heated (and EVDF tail repleted) by perpendicular-toparallel scattering Written in matlab Simulations take 2-5 days on a PC These simulate 10 5 macroparticles Intended to be identical to code of Meige et al, POP 12, (2005) Except that reference used a linear loss profile APS-DPP November 10, 2010, p 10

11 i + and e neutral cross sections from literature Simulate ion-neutral and electron-neutral collisions σ i,i : ionization collisions σ i,cx : charge-exchange collisions σ e,m : inelastic momentum transfer σ e,e : elastic collisions σ e,i : ionization collisions σ i,i σ i,cx σ i n [m 2 ] ε [ev] APS-DPP November 10, 2010, p 11

12 Expansion is simulated with particle loss Volume expansion: V /V o = (r/r o ) 2 = B/B o, so Vo 1 dv /dx = Bo 1 db/dx ν loss v 1 db B o dx Using solenoidal B upstream gives { 0 for 0 x x c ν loss (x) = 3R ν 4 (x x c ) o for x [R 2 +(x x c ) 2 ] 5/2 c x L Choose x c = 5 cm and R = 1.7 cm (R/L 0.17 in experiments), vary ν o v/r linear ν loss expansion ν loss νloss/νmax x [cm] APS-DPP November 10, 2010, p 12

13 Simulations capture double layer potential Qualitatively similar potentials as measured in experiments Linear and B-field expansion profiles give slightly different results DL potential drop is sharper for expansion Downstream potential is flatter for expansion Upstream potential is higher for linear Following parameters were used: Quantity Value Neutral pressure 1 mtorr Domain length 10 cm Number of grid cells 250 Time step s Total run time 25 µs Antenna frequency (ω o /2π) 10 MHz Antenna current density amplitude 100 A/m 2 q factor ν max s 1 φ [V] ne [m 3 ] x [cm] x [cm] linear ν loss expansion ν loss linear ν loss expansion ν loss 1 APS-DPP November 10, 2010, p 13

14 Ion beams are found in the simulations Simulations show acceleration of an ion beam from DL Qualitatively similar to the beams found in experiments Slow ion component forms downstream from ionization and charge exchange 0.25 x=2. 5 cm 0.2 x=6 cm [fi,x/ni]x 0.15 x=5 cm x=7 cm v x [km/s] Student Version of MATLAB APS-DPP November Student 10, Version 2010, of MATLAB p 14

15 Electrons and Ions Maxwellian in directions A very small population of ions is scattered from the beam in the DL and sheath regions The color bar has a log scale in density For the f e,y plot, blue is energy in +ŷ and red is in ŷ direction Student Version of MATLAB Student Version of MATLAB APS-DPP November 10, 2010, p 15

16 At low pressure wall effects can be seen in EVDF Pressure is 0.1 mtorr for these plots x = 1.5 cm x = 4 cm fe,s fe,dl [ev] [ev] x = 6 cm x = 9 cm fe,dl fe,s [ev] [ev] APS-DPP November 10, 2010, p 16

17 At higher pressure tail of EVDF gets filled in Pressure is 1 mtorr for these plots 1e14 1e14 f e, s2 1e12 1e10 f e, D L2 1e12 1e10 1e8 1e [ev] 1e8 1e [ev] f e, D L1 1e12 1e10 f e, s1 1e12 1e10 1e [ev] 1e [ev] APS-DPP November 10, 2010, p 17

18 Potential profiles depend on neutral pressure mtorr 0.1 mtorr 1 mtorr 10 mtorr 80 φ [V] x [cm] Find DL is indistinguishable at p =0.01 mtorr Next data point at 0.04 mtorr is nearly an arc This is consistent with our earlier estimate of 0.03 mtorr APS-DPP November 10, 2010, p 18

19 Summary The phoenix code has successfully simulated current free double layers in a 1-D geometry Simulations show an ion beam accelerated from the DL that agrees qualitatively with experimental measurements Have found EVDFs that agree with an analytic model based on depletion from losses to the plasma boundaries and repletion from e-n collisions The EVDF from simulations does not have an electron beam upstream, nor a two-temperature distribution (unlike the EVDF assumed in previous models) The minimum neutral pressure predicted from the model is consistent with that observed in the simulation results APS-DPP November 10, 2010, p 19