Modelling plasma discharges at high electronegativity

Abstract

Macroscopic models for the equilibrium of a three-component electronegative gas discharge are developed. Assuming the electrons and the negative ions to be in Boltzmann equilibrium, a positive ion ambipolar diffusion equation is derived. Such a discharge can consist of an electronegative core and may have electropositive edge regions, but the electropositive regions become small for the highly electronegative plasma considered here. In the parameter range for which the negative ions are Boltzmann, the electron density in the core is nearly uniform, allowing the nonlinear diffusion equation to be solved in terms of elliptic integrals. If the loss of positive ions to the walls dominates the recombination loss, a simpler parabolic solution can be obtained. If recombination loss dominates the loss to the walls, the assumption that the negative ions are in Boltzmann equilibrium is not justified, requiring coupled differential equations for positive and negative ions. Three parameter ranges are distinguished corresponding to a range in which a parabolic approximation is appropriate, a range for which the recombination significantly modifies the ion profiles, but the electron profile is essentially flat, and a range where the electron density variation influences the solution. The more complete solution of the coupled ion equations with the electrons in Boltzmann equilibrium, but not at constant density, is numerically obtained and compared with the more approximate solutions. The theoretical considerations are illustrated using a plane parallel discharge with chlorine feedstock gas of p = 30, 300 and 2000 mTorr and , corresponding to the three parameter regimes. A heuristic model is constructed which gives reasonably accurate values of the plasma parameters in regimes for which the parabolic profile is not adequate.