Geoscience Reference
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F.R/ D e ˇE 0 r 1
a 2
R 2
The equation for determining R takes the form (nonsingular case)
R 2 ˇ j U.R/ j 0 R C .1 C 2ˇ. j U.a/ j j E 0 j / ;
˛ o .a;R/
2DR D
1
q 1
e ˇ a 2
a 2
R 2
(3.172)
where
a 2
D j U.R/ E 0 j D
a 2 . j U.a/ j j U.R/ j /:
R 2
Now our task is to find ˛ o . A trivial but tedious algebra yields
˛ o .a;R/ D a 2 v T e ˇ 1 C ˇ j U.a/ j C ˇ j E 0 j C
1 ˇ
R 2
a 2 e ˇ
Once the matching distance is known as a function of the particle size, it is easy to
find the charging efficiencies for any potential. We therefore begin with the analysis
of the dependencies of R D R.a/ and then present the results on the dependence of
the charging efficiencies on particle sizes for the potentials given by Eq. 3.102 .
The equation describing the dependence of the matching distance on the particle
size for U.r/ D 0 has the structure,
q R o .a/ C a 2 ;
R.a/ D
(3.173)
with
2D
v T :
R 0 .a/ D
(3.174)
The value of R o .a/ is independent of a , so at very small particle size the matching
distance is of the order of the molecular mean free path, as has been expected. At
large particle size a l the difference R.a/ a / l.
When the ion-particle interaction is turned on, the analysis becomes more
complex. It can be done only numerically, but first we analyze the behavior of the
function R.a/ at small a l;l c . In our analysis we assume that U.a/ !1 as
a ! 0.
Let us begin with the attractive potentials. At small particle size, ˛ 0 a 2 v T
ˇ j U.a/ j
(the leading term in U.a/ is retained). The term of the same order of
magnitude on the right-hand side of Eq. 3.172 is 2ˇ j U.a/ j a 2 =R 2 . Equation 3.172
then gives
4D
v T :
R.a/
(3.175)      Search WWH ::

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