4 Weak Adsorption: WKB Approximation
Exact solutions for critical adsorption of polyelectrolytes have been presented for
planar and spherical surfaces. Adsorption onto a cylindrical surface can no longer
be treated analytically. Here, a suitable principle for an approximate solution is
desirable. We have proposed the WKB approximation as such a scheme [48].
4.1 WKB Approximation Scheme
The WKB scheme is typically used in quantum mechanics to find an approximate
solution of a time-independent one-dimensional Schro ¨dinger equation of the form:
@
2 cðxÞ
@x 2 þ QðxÞ
2 cðxÞ ¼ 0
(32)
in the vicinity of the simple zero x 0 of the potential Q(x 0 )
2
¼ 0 [61–64]. More
precisely, the WKB method yields a solution of the more general equation:
@
2 cðxÞ
@x 2 þ ½QðxÞ
2 À OðxފcðxÞ ¼ 0
(33)
in the form:
cðxÞ ¼ SðxÞ½ax
1=3 J À1=3 ðxÞ þ bx
1=3 J 1=3 ðxފ;
(34)
with the definitions OðxÞ ¼ SðxÞ
À1 @
2 SðxÞ=dx
2
; SðxÞ ¼ QðxÞ
À1=2 xðxÞ
1=6 and a, b
being constant coefficients. x is related to Q according to [64]:
xðxÞ ¼
Ð x 0
x Qðx
0
Þdx
0 for x < x 0 in the region QðxÞ
2 > 0
e
À3pi=2
Ð x
x 0
Qðx
0
Þ
j
jdx
0 for x > x 0 in the region QðxÞ
2 < 0
(
:
(35)
Note, that we exchanged the interval over which Q
2 is positive and negative,
respectively, compared to [64]. In the limit OðxÞ
j
j(
ffiffiffiffiffiffiffiffiffiffiffiffi
QðxÞ
j
j
p
, the general solution
of (34) yields an approximate solution of (32).
Using the properties of Bessel functions, c(x) can be expressed in terms of
Airy functions Ai of negative and positive arguments, respectively [153]. Then, we
obtain the uniformly valid Langer solution [48, 64]:
Strong and Weak Polyelectrolyte Adsorption onto Oppositely Charged Curved. . .
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