Method of matched asymptotic expansions versus intuitive approaches: calculation of space-charge sheaths

A comparison is performed of results predicted by the method of matched asymptotic expansions, by the Child-Langmuir model, and by patching in three problems of the theory of space-charge sheaths: a collisionless steady-state sheath, a matrix sheath, and a collisionless RF sheath. In each problem, results are compared between themselves and with the exact solution. In all the cases, the Child-Langmuir model and patching provide results which are accurate to the first approximation in the sheath voltage but not to the second one, irrespective of details of patching. For a steady-state sheath and a matrix sheath, accuracy of the asymptotic solution is exponential. The asymptotic solution for an RF sheath is accurate to the second approximation and can be further improved, if necessary. Comparison of numerical values shows that the method of matched asymptotic expansions indeed provides a considerably higher accuracy.