self-consistent field theory, with a coupling of polymer and ionic distributions next
to the interface, has been presented [151, 152].
2.2 Density Distribution Function
The density distribution P(r) of a point r(s) along the polymer contour (0 < s < L)
follows from:
PðrÞ ¼
ð
hdðr À rðsÞÞi ds
(4)
with:
d r À rðsÞ
ð
Þ
h
i$
ð
Gðr L ; Ljr
0
; sÞ dðr À r
0
ÞGðr
0
; sjr 0 ; 0Þ d
3 r L d
3 r 0 d
3 r
0
¼
ð
Gðr L ; Ljr; sÞGðr; sjr 0 ; 0Þ d
3 r L d
3 r 0 :
(5)
Using the ground-state Green function Gðr; sjr
0
; s
0
Þ ¼ c
Ã
0 ðr
0
Þc 0 ðrÞ e
Àl 0 jsÀs
0 j (2), we
obtain (note that c 0 (r) is a real function):
PðrÞ $ c 0 ðrÞ c 0 ðrÞ e
Àl 0 L
;
(6)
or, with the normalization
Ð
P(r) d
3 r ¼ 1:
PðrÞ ¼
c 0 ðrÞ
2
Ð
c 0 ðrÞ
2 d
3 r
:
(7)
3 Weak Adsorption: Exactly Solvable Models
In this section, we will discuss the adsorption behavior on planar and spherical
surfaces. Figure 3 illustrates the various geometries.
3.1 Planar Surface
Analytical results for the Green function are obtained for a planar geometry only.
For a planar surface, the linearized Poisson–Boltzmann equation yields the
screened potential:
8
R.G. Winkler and A.G. Cherstvy