Plasma-Wall Interaction: Sheath and Pre-Sheath

Plasma-Wall Interaction: Sheath and Pre-sheath Under most conditions, a very thin negative sheath appears in the vicinity of walls, due to accumulation of electrons on the wall. This is in turn necessitated by the need to reduce the electron flux well below the un-inhibited thermal flux ne c̄e /4, which would lead to very high negative current densities toward the wall. In the limit of an insulating wall, we must have Γe = Γi  ne c̄e /4, and the flux equality imposes a specific sheath potential drop (see later). But even where there is net negative current (electron capture) its magnitude is still low enough compared to the thermal value that a negative sheath forms (somewhat against intuition). Any ion arriving at the sheath’s edge will be accelerated by the sheath and captured by the wall. This non-zero ion flux requires, by continuity, a means of accelerating ions from conditions far from the wall, unless these conditions already imply a strong wall-directed ion flux. Thus, there is normally a Pre-sheath where the plasma is still neutral, but the potential is already falling toward the wall. The specifics of the pre-sheath vary depending on the situation, and in most cases conditions are 2D or 3D. In contrast, the sheath is almost always so thin as to be 1D, and its physics is simpler and universal, largely independent of outside conditions. For this reason we analyze first the sheath, and it will turn out that its entry condition provides the required wall boundary condition to use for the much larger scale of the pre-sheath. The Sheath Because the sheath thickness (a few Debye lengths) is usually much less than any collisional  not nearly mean free path, we ignore collisions in it. With no magnetic field (or with B parallel to the wall), the electron force balance is between pressure and potential gradients, leading to Boltzmann equilibrium: ne = nes e e(φ−φs ) kTe (1) s is the sheath’s edge. Actually, this balance is also valid in the pre-sheath, although Where  less rigorously. For ions with no scattering or ionizing effects: ni ui = nis uis mi u2i 2 + eφ = mi u2is 2 (2) + eφs (3) and eliminating ui between (2) and (3): nis ni =  s −φ) 1 + 2e(φ mi u2 is The net charge density ρch = e(ni − ne ) can then be used in Poisson’s equation −ρch d2 φ = : dx2 0 1 (4)   e(φ−φs ) d2 φ n e is nes e− kTe −  = 2e(φs −φ) dx2 0 1 + mi u2 (5) is and, actually, nes = nis since this must connect with outside, neutral zone. We can simplify notation by defining non-dimensional variables,  e(φs − φ) 0 kTe x uis  ϕ = ≥ 0 ; ξ = ; λDs = ; M = (6) is kTe λDs e2 nes kTe mi This yields, d2 ϕ 1 −ϕ  = −e + dξ 2 1+ (7) 2ϕ 2 Mis and integration of the RHS with A first integral of (7) is obtained by multiplication times dϕ dξ respect to ϕ:   2 2 M 1 dϕ 2ϕ = +e−ϕ + is 2
1 + 2 − C (8) 2 dξ 2
Mis The constant of integration C is obtained by imposing zero slope at s (but only on the magnified sheath scale!). At s we also have ϕ = 0, so C = 1 + Mis2 (9) When taking the square root in (8), the (−) sign is appropriate since √ dϕ = − 2 dξ  Mis2 1+ 2ϕ −1 m2is − (1 − e−φ ) dφ dξ < 0 in the sheath: (10) Unfortunately, a second integration is not analytically feasible, but important information can be obtained by examining the radicand of (10) near the sheath’s edge, i.e. when ϕ  1:     2 3  ϕ 1 2ϕ 2ϕ 1 2ϕ 2 −φ 2 1 + 2 − 1 − (1 − e ) = Mis 2 − + ··· (11) Mis Mis Mis 8 Mis2 16 Mis2        1 1 3 1 1 ϕ2 ϕ3 2 + ··· = 1− 2 ϕ + ϕ + ··· − − ϕ− 2Mis4 2 6 2 Mis 6 The leading term (in ϕ2 ) must be positive or zero for real solutions near the sheath edge: Mis2 ≥ 1 (12) and this is called the Bohm sheath criterion. Supersonic ion entry is uncommon, so we will accept the weaker criterion. Mis = 1 (13) When this is satisfied, the next term in (11) is 13 ϕ3 > 0, and the sheath potential profile is real. Thus, the ions enter the sheath at their (isothermal) speed of sound,  kTe uis = (14) mi 2 and this is the boundary condition needed for the outer (pre-sheath) solution. Once ϕ exceeds unity (φs − φ ≥ kTe ), the expression (10) simplifies (roughly) to  √ dϕ 2ϕ = − 2 Mis
2  = 23/4 ϕ1/4 (15) dξ Mis  ↑ Mis = 1 which can be integrated: 1 −ϕ− 4 dϕ = 23/4 dξ 4 − ϕ3/4 = 23/4 ξ + D 3 where, using ϕ = 0 at s, D = −23/4 ξs ξs − ξ = 25/4 3/4 ϕ 3 (16) In particular, at the wall ξ = ξw , ϕ = ϕw , so the sheath thickness is ξs − ξw = In physical variables, 3/4  xs − xw 25/4 eΔφsh = λDs 3 kTe 25/4 3/4 ϕw . 3 (17) so the sheath increases in thickness with the 3/4 power of the sheath potential drop. Putting also in (17) the definition of λDs , we can reorganize it in the form ji = enes uis √  3/2 4 2 e Δφsh = 0 9 mi (xs − xw )2 (18) which is the Child-Langmuir equation for space-charge limited current. The Collisional Pre-sheath An alternative derivation of the Bohm criterion could be arrived at in the following way. Start with steady 1D flow and impose continuity of ion flow and quasineutrality, d (ne ui ) = 0 dx and ne = ni And include momentum conservation for both ions and electrons, n e m i ui dui dpi + = ne eEx − mi ne νin ui dx dx  dpe e du + = −ne eEx − me ne νen ue ne m u i e dx dx  Adding these equations and combining with continuity, we can write, d ( ( (mi ne u2i + pi + pe ) = 0 − mi ne νin ui − ( m( e n( e ν( en ue    dx ne k(Te +Ti ) 3 or, with Te  Ti , d kTe   (mi ui + )  −mi νin (n e ui ) dx ui So, we get the same sheath entry velocity  kTe uis = = uBohm mi   (n  e ui ) Particle and energy flux to a wall (A) Random particle flux across a plane Assume a Maxwellian distribution: 2 w    2 2 2  m 3/2 − m(wx + wy + wz ) 2kT e f =n 2πkT (19) The forward directed flux across a plane (say, the yz plane) is, ˆ∞ ˆ∞ Γx = dwz dwy wy =−∞ ˆ∞ wz =−∞ wx f (w  )dwx dwy dwz (20) 0 It is actually better to use spherical velocity conditions, where wx = wcosθ and dwx dwy dwz = d3 w = 2πw2 sinθdwdθ: π ˆ∞ ˆ2 Γx = f (w)2πw3 sinθcosθdθdw (21) w=0 θ=0 ˆπ/2 ˆπ/2 sin2 θ 1 ) = Integrate first on θ : sinθcosθdθ = d( 2 2 0 ˆ∞ Γx = π 0 m 3/2 ) f (w)w dw = πn( 2πkT 3 w=o Change variables: mw2 2kT = ζ, wdw = ˆ∞ mw2 e− 2kT w3 dw 0 kT dζ m m 3/2 2kT kT Γx = πn( ) m m 2πkT  ˆ∞ 1 ζe−ζ dζ = √ n 2π 0    1 4 kT m (22)  Γx = n kT 2πm (23) This is usually written in terms of the mean value of the velocity c̄ ≡ 1 n which, with a similar calculation gives,  c̄ = ´∞ ´π 0 0 wf (w)2πw2 sinθdθdw, 8 kT π m (24) By division of (5) and (6), we obtain Γx = nc̄ 4 (note x is an arbitrary direction in this case) (25) (B) Random flux of kinetic energy across a plane We now want to calculate: ΓE ˆ∞ ˆπ/2 1 = ( mw2 )f (w)(wcosθ)2πw2 sinθdθdw 2 (26) w=0 θ=0 π = m 2 ˆ∞ m 3/2 π ) w5 f (w)dw = m n( 2 2πkT w=0 ˆ∞ mw2 w5 e− 2kT dw (27) w=0 Using the name change of the variable of before, ΓE  3/2  3 π m 1 2kT = m n 2 2πkT 2 m ˆ∞ 0 ζ 2 e−ζ dζ    ∞ ∞  ´ ´ −ζ  +2 ζe dζ=2 −ζ 2 e−ζ  ΓE = 2kT and comparing to (6) ,  0 0 kT 2πm ΓE = Γx 2kT (28) (29) So the average particle that crosses the plane carries an energy 2kT . This is more than the average energy per particle ( 32 kT ) because faster particles cross more often. (C) Particle flux to a repelling wall 5 This could be electrons approaching a negatively charged wall, with a sheath in front. We do the calculation outside the sheath, and, if there are no collisions, the same flux will cross any plane, including the wall plane. The calculation is like in part (A), but now we notice that not all particles reach the wall: they must have enough x-directed energy to overcome the repulsion. If qφw > 0 (for example, electrons, with q = −e to a negative wall), we need, m(wcosθ)2 ≥ qφw 2 and for a fixed w,  θM AX = cos −1 (30) 2qφw mw2 (31) and so the inner integration (on θ) now gives, θˆ M AX θM̂ AX sinθcosθdθ = 0 0       −cos2 θ 1 2qφw 1 2 = 1 − cos θM AX = 1− d 2 2 2 mw2  and for the integration (on w), wM IN = 2qφw , m √ 2qφw from (14)  ˆ∞ Γ = (32) f (w)w 3  2qφw dw 1− mw2 m  m = πn 2πkT 3/2 ˆ∞ √ 2qφw 2 − mw 2kT e  w 3  2qφw dw 1− mw2 (33) m Using again ζ = mw2 2kT :  3/2  2 ˆ∞  1 2kT qφw 1 m −ζ e ζ 1− dζ Γx = πn 2 m 2πkT kT ζ    qφw  kT nc̄ 4 For the last integral, integrate by parts (e−ζ dζ = −d(e−ζ )), nc̄ Γ = 4  ∞  ˆ∞ qφw −ζ  −ζ − ζ− e  + e dζ qφw kT kT    qφkTw    0   e− ∴ Γ = nc̄ − qφw nc̄ e kT ≤ 4 4 6 qφw kT (34) Clearly, the wall repulsion reduces the flux (strongly if qφw > kT ). (D) Energy flux to a repelling wall The same considerations apply now. We start with (8), but with the limits modified as in part C. The algebra is straightforward, although tedious, and the result is: ΓE = nc̄ − qφw e kT (2kT + qφw ) 4    (35) Γ So now the average crossing particle carries an energy 2kT + qφw . The extra qφw per particle is what is required for it to actually reach the wall, while the 2kT part is what is would carry in thermal form with no repulsion. 7