IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 31, NO. 4, AUGUST 2003
Method of Matched Asymptotic Expansions
Versus Intuitive Approaches: Calculation of
Mikhail S. Benilov
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.
Child–Langmuir sheath, matrix sheath, patching,
RF sheath, sheaths.
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.,
–) 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
–. However, discussion still continues (e.g., –)
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