03_nemerical1 - 678 IEEE TRANSACTIONS ON PLASMA SCIENCE VOL...

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678 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 31, NO. 4, AUGUST 2003 Method of Matched Asymptotic Expansions Versus Intuitive Approaches: Calculation of Space-Charge Sheaths Mikhail S. Benilov Abstract— A comparison is performed of results predicted by the method of matched asymptotic expansions, by the Child–Lang- muir 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 asymp- totic 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. Index Terms— Child–Langmuir sheath, matrix sheath, patching, RF sheath, sheaths. I. INTRODUCTION T HE PROBLEM of contact of a plasma with a surface rep- resents a classical example of a problem in which regions with essentially different properties appear: while the bulk of the plasma is quasi-neutral, there is a thin layer (sheath) adja- cent to the surface in which deviations from quasi-neutrality are essential. The method of matched asymptotic expansions (e.g., [1]–[6]) is a standard tool for dealing with such problems and represents a powerful alternative to intuitive approaches. This method was applied to the sheath theory as early as in the 1960s [7]–[10]. However, discussion still continues (e.g., [11]–[16]) as to whether an intuitive approach based on patching provides a more appropriate description than the method of matched asymptotic expansions, especially in the cases of collisionless and moderately collisional sheaths. Some authors argue that the method of matched asymptotic expansions leads to physical inconsistencies, while others be- lieve that results provided by patching are grossly inaccurate. Neither of these reasonings is considered here. One should note, however, that the alleged inconsistencies are based on misinter- pretation of asymptotic results; on the other hand, it is difficult to believe that results obtained by able scientists in the course of years by means of patching are totally incorrect. Manuscript received December 11, 2002; revised April 6, 2003. This work was performed within activities of Projects 32 411/99 of FCT and FEDER and NNE5/2001/282 of the EC and of the action COST 529 of the EC. The author is with Departamento de Física, Universidade da Madeira, 9000
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This note was uploaded on 06/08/2011 for the course ELECTRICAL 124 taught by Professor Ghjk during the Spring '11 term at Institute of Technology.

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03_nemerical1 - 678 IEEE TRANSACTIONS ON PLASMA SCIENCE VOL...

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