Relation between double layers and flux for a collisionless discharge with two negative components

The ion flux exiting a steady-state planar discharge with collisionless, kinetic positive ions and two negative Boltzmann components (i.e., a modified Tonks-Langmuir model) is calculated. For the plasma approximation, when the negative component temperature ratio exceeds 10.78, there are two solutions for the potential at the plasma edge over some range of the negative component density ratio. The existence of collisionless double layer solutions is confirmed by solving the plasma-sheath equation numerically for finite charge non-neutrality. A simple physical criterion for the existence of a double layer is given - a double layer is formed when the positive ion flux associated with the smaller (in absolute value) plasma edge potential is less than that associated with the larger edge potential. The flux calculated for finite nonneutrality is shown to agree with the flux calculated for the plasma approximation when the larger value of the flux is taken in the double-solution regime. Simple approximate expressions for the flux are derived.