Sheath governing equations in computational weakly-ionized plasmadynamics

Abstract

To date, fluid models of plasma sheaths have consisted of the coupling of the electric field potential equation obtained through Gauss’s law to the charged species conservation equations obtained through the drift–diffusion approximation. When discretized using finite-difference stencils, such a set of equations has been observed to be particularly stiff and to often require more than hundreds of thousands of iterations to reach convergence. A new approach at solving sheaths using a fluid model is here presented that reduces significantly the number of iterations to reach convergence while not sacrificing on the accuracy of the converged solution. The method proposed herein consists of rewriting the sheath governing equations such that the electric field is obtained from Ohm’s law rather than from Gauss’s law. To ensure that Gauss’s law is satisfied, some source terms are added to the ion conservation equation. Several time-accurate and steady-state cases of dielectric sheaths, anode sheaths, and cathode sheaths (including glow and dark discharges) are considered. The proposed method is seen to yield the same converged solution as the conventional approach while exhibiting a reduction in computational effort varying between one-hundred-fold and ten-thousand-fold whenever the plasma includes both quasi-neutral regions and non-neutral sheaths.