Chemistry Reference
In-Depth Information
inverse length) and
b
(which is dimensionless) are determined from the condition of
electroneutrality in the cylindrical cell of radius
R
and from the condition that at
r
(
r
) has to reach an extreme (for symmetry
reasons). As long as the charge density parameter
l
is “small” parameter
b
is a
real number (between zero and one), whereas it is an imaginary number (between
zero and 1.0i) for “large” charge densities. The distinction between “small” and
“large” depends on the polyelectrolyte concentration. When
b
becomes imaginary,
b
has to be replaced by
¼
R
, the electrostatic potential
'
in (
29
). Consequently there are two different regions
where the remaining parameters (
A
and
b
) have to be determined:
When
b
is real:
jj
2
1
b
l ¼
Þ
;
(32)
1
þ b
coth
bg
LK
ð
1
þ b
coth
b
ln
AR
½
ðÞ
¼
0
:
(33)
When
b
is imaginary:
2
1
þ jj
l ¼
Þ
;
(34)
1
þ jj
cot
ð
j g
LK
b
ln
A
þ jj
ln
R
þ
arctan
jj¼
0
;
(35)
where
g
LK
is another dimensionless parameter:
ln
R
a
g
LK
¼
(36)
that is related to the volume fraction
f
p
of the polyelectrolyte in the aqueous
solution:
R
2
a
f
p
¼
ln
:
(37)
Unfortunately, there is no analytical solution to determine
A
and
b
. But at infinite
dilution (i.e., when
g
LK
!1
) one finds from (
32
) and (
33
):
b ¼
1
:
0
l
for
l
1
and
b ¼
0
:
0
for
l >
1
:
(38)
Figure
9
shows the results for
b
(
l
,
g
LK
) as calculated from (
33
)to(
35
).
Lifson and Katchalsky [
74
] determined the influence of the electrostatic poten-
tial
(
r
) on the thermodynamic properties of an aqueous solution of a single
polyelectrolyte through an expression for the change of the Helmholtz energy
D
'
F
LK
that is due to the presence of the electrostatic potential by:
