which defines l 0 via the boundary condition AiðÀð3xðaÞ=2Þ
2=3 Þ ¼ 0 or xðaÞ ¼
2ai
3=2
1 =3 and:
ð r 0
a
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
6
l
y
K 0 ðkrÞ
K 1 ðkaÞ
À jl 0 j
s
dr ¼
2
3
ai
3=2
1 :
(43)
For l 0 ¼ 0, this equation yields the condition:
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
24pjrs c j
ek B Tk 3 lK 1 ðkaÞ
s
ð 1
ka
ffiffiffiffiffiffiffiffiffiffiffi
K 0 ðxÞ
p
dx ¼
2
3
ai
3=2
1
(44)
for the critical charge density.
As displayed in Fig. 10, s c reveals a crossover from a planar-like scaling
s c $ ðkaÞ
3 for ka ) 1 to the dependence s c $ ðkaÞ
2 for ka ( 1. The latter relation
follows by an asymptotic expansion of the Bessel functions K 1 ðkaÞ for small
arguments, which yields:
js c j ¼
ai
3
1 Ek B Tl
54pjrja
ð 1
0
ffiffiffiffiffiffiffiffiffiffiffi
K 0 ðxÞ
p
dx
À2
k
2
;
(45)
with
Ð 1
0
ffiffiffiffiffiffiffiffiffiffiffi
K 0 ðxÞ
p
dx ¼ 2:12.
In the adsorbed state, the eigenfunctions c follow from (36). In particular,
the asymptotic radial dependence of the eigenfunction in the regime r > r 0 can
be determined. Here, x is given by:
xðrÞ
j
j ¼
1
k
ð kr
kr 0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
6
l
jl 0 j À y
K 0 ðkrÞ
K 1 ðkaÞ
s
dr;
(46)
which yields for kr 0 ) 1 the relation xðrÞ
j
j ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
6jl 0 j=l
p
r. Then, expansion of the
Airy function yields:
c 0 ðrÞ ¼ N c
lk
6jl 0 jr 2
1=4
e
Àr
ffiffiffiffiffiffiffiffiffi ffi
6jl 0 j=l
p
(47)
for r ) r 0 . We emphasize the rather strong dependence of the decay length in this
equation on the eigenvalue l 0 .
The radial polyelectrolyte density profile follows from:
PðrÞ ¼
rc 0 ðrÞ
2
Ð
rc 0 ðrÞ
2 dr
:
(48)
24
R.G. Winkler and A.G. Cherstvy