Analytical expression for the sheath edge around wedge-shaped cathodes

The sheath is the boundary layer separating a quasi-neutral plasma from a material electrode. Understanding the sheath is important for numerous applications, including plasma-based ion implantation, plasma etching of semiconductors, plasma assisted electrostatic cleaning, and Langmuir probes. In a 1D planar geometry, the Child-Langmuir (CL) law describes the sheath when the bias on a negative electrode, i.e., a cathode, is much greater than the electron temperature. In this case, the sheath width $s$ is an eigenvalue of the problem. In 2D, the sheath edge is an unknown line (an ``eigen-boundary") which is determined by a set of coupled, nonlinear, partial differential equations. I have found an expression for the sheath edge around a 2D wedge-shaped cathode with included angle $\theta_w$. In polar coordinates $\left(r,\theta\right)$, the sheath edge is a solution of $r\sin\left(a\theta\right)=as$ where $s$ is the planar sheath width far from the corner and $\theta_w=2\pi - \pi/a$, so that $a=1/2$ gives a knife edge, while $a=2/3$ gives a square corner. This result is verified by comparison with the numerical solutions of Watterson [P. A. Watterson, J. Phys. D $\bf{22}$, 1300 (1989)].