The plasmasheath transition in low temperature plasmas: on the existence of a collisionally modified Bohm criterion

R P Brinkmann (1)

1 Institute for Theoretical Electrical Engineering

The plasma-sheath transition in stationary low temperature plasmas is investigated for arbitrary levels of collisionality. The model under study contains the equations of continuity and motion for a single ion species, Boltzmann's equilibrium for the electrons, and Poisson's equation for the field. Assuming that the electron Debye length λ D is small compared to the ion gradient length l = n i / ∂n i ∂x, a first order differential equation is established for the ion density n i as function of the transformed spatial coordinate q = n i dx. A characteristic feature of this novel 'sheath equation ' is an internal singularity of the saddle point type which separates the depletion-field dominated sheath part of the solution from the ambipolar diffusion controlled plasma. The properties of this singularity allow to define, in nonarbitrary way, a collisionally modified Bohm criterion which recovers Bohm's original expression in the collisionless limit but remains meaningful also when collisions are included. A comparison is made with the collisionally modified Bohm criteria proposed by Godyak [ V.A. Godyak, Phys. Lett. 89A, 80 (1982) ], Valentini [ H.-B. Valentini, Phys. Plasmas 3, 1459 (1996) ] and Chen [ X.P. Chen, Phys. Plasmas 5, 804 (1997) ] as well as with the approaches of Riemann [ K.-U. Riemann, J. Phys. D ; Appl. Phys. 24, 493 (1991) ] and Franklin [ R.N. Franklin, J. Phys. D: Appl. Phys. 36, 2821 (2003) ] who argued that the definition of a collisionally defined Bohm criterion is not possible. 1 The subdivision of a gas discharge into the plasma bulk and the sheath was introduced by Langmuir in 1928 [ 1 ]. Together with Tonks he also coined the term sheath edge for the separation of the two " rather distinct " zones [ 2 ]. (In the standard 1D geometry, bulk and sheath are intervals while the sheath edge is a point.) The connection of the two zones was first studied by Bohm who, in 1949, formulated his famous " criterion for a stable sheath " [ 3 ]: To ensure nonoscillatory solutions, ions must enter the sheath from infinity with a minimum speed (the 'Bohm speed ') v B = T e /m i. (Here, m i denotes the ion mass and T e the temperature of the electrons in energy units, i.e., k B ≡ 1.) The Bohm criterion |v i | ≥ v B has vexed the plasma community for more than four decades. One of its mysteries was that it could not only be derived from Bohm's original sheath equations (collisionless ions plus Poisson's equation), but also from a " presheath model " (collisional ions plus quasineutrality) ; there, however, from the properties of a singularity [ 4 ]. A systematic reconstruction of the Bohm criterion was given in 1991 by Riemann who investigated the plasma-sheath transition under the condition that the Debye length λ D is small compared to the mean free path λ [ 5 ]. Asymptotically matching the sheath (scale λ D) and the presheath (scale λ) by a transition layer (scale λ 4/5 D λ 1/5), he found a solution uniformly valid on all scales and showed that the Bohm criterion is marginally fulfilled. Riemann's analysis of the regime λ D /λ 1 motivated research into what happens when the scale ratio is large. It proved difficult: Some researchers who studied the plasma-sheath transition in the presence of collisions found that the Bohm criterion must be modified [ 6-10 ]. Others objected: Riemann stressed that " there is no reason and no basis to formulate a new modified Bohm criterion accounting for collisions in the sheath " and that the " various heuristic attempts " are " inconsistent and lead to unreasonable results " [ 11 ]. Franklin went along: " There is no such thing as a collisionally modified Bohm criterion " [ 12 ]. This manuscript aims to take part in that discussion by offering an alternative perspective. A standard sheath model will be analyzed, similar to the ones investigated in refs. [ 6-12 ]. As in [ 11 ] and [ 12 ], the Debye length λ D will be treated as small compared to a certain other scale of the dynamics. However, that 'other scale ' will not be the ion mean free path λ but the ion density gradient length l = n i /| ∂n i ∂x |. Employing arguments that are not heuristic, it will be shown that the assumption λ D l implies - for all ratios λ D /λ - the existence of a certain mathematical structure (a removable singularity) with properties that allow to unambiguously define a 'collisionally modified Bohm criterion '. 2 As stated, the model under study is quite conventional. A one-dimensional Cartesian geometry is assumed, described by an x-axis pointing from the electrode x E = 0 to the bulk. The field is related to the ion and electron densities via Poisson's equation, with the electrons obeying Boltzmann's relation with an electron temperature T e: − 0 ∂ 2 Φ ∂x 2 = 0 ∂E ∂x = e n i − n 0 exp eΦ T e . (1) Ionization is neglected so that the ion flux density Ψ i to the electrode is spatially constant. The equation of motion takes into account acceleration by the electrical field and friction due to charge exchange and elastic collisions with the neutrals: n i v i = −Ψ i, (2) m i v i ∂v i ∂x = eE − |v i | λ m i v i. (3) The mean free path λ depends on the velocity of the ions. In many gases, it scales like |v i | for small v i (Langevin interaction) and is constant for large v i (charge exchange interaction and 'hard sphere ' elastic collisions). In argon, literature results are well reproduced by λ(v i) = |v i |/n N σ 0 v 0 2 + v 2 i where n N is the gas density and σ 0 = 10 −18 m 2 and v 0 = 550 m/s. The system is completed by the conditions of neutrality and transport equilibrium for x → ∞

Mots clés: Physical Sciences

R P Brinkmann. The plasmasheath transition in low temperature plasmas: on the existence of a collisionally modified Bohm criterion. Journal of Physics D: Applied Physics, 2011, 44 (4), pp.42002. ⟨10.1088/0022-3727/44/4/042002⟩. ⟨hal-00589768⟩