Chemistry Reference
In-Depth Information
In the absence of charges, or at a low degree of ionization of the arms, the star
conformation is controlled by a balance between two first terms in (
7
) (i.e., the short-
range interarm repulsions and the conformational entropy of stretched arms). As a
result, the star size is given by (
4
). That is, the power law dependencies, obtained on
the basis of the blob model, are recovered. The physical reasons why there is a match
of the star size as obtained by the scaling and in the mean field approximations are
3.2
Polyelectrolyte Star Conformation in a Dilute Salt-Free
Solution
The box-like model allows for a straightforward analysis of the counterion localiza-
tion, which is essential for understanding the specific properties of salt-free solutions
of branched PEs. In the case of a PE star, the first two terms in the free energy in
(
7
) are complemented by a contribution due to Coulomb interactions between all
the charges (charged monomers and mobile ions) in the cell,
F
Coulomb
, and by the
translational entropy of all mobile ions,
F
ions
. Following the line of arguments of
only mobile (monovalent) counterions, which compensate the net charge of the star
polyion. We assume that
Q
∗
≤
Q
counterions are localized in the outer volume of
Q
∗
)
the cell,
R
≤
r
≤
D
, whereas the remaining (
Q
−
counterions are retained inside
the star volume (0
. In the framework of the box-like model, the counterion
concentration is assumed to have constant (but different) values inside and outside
the star:
c
(
in
)
ions
≤
r
≤
R
)
and
c
(
out
)
ions
Q
∗
)
/
R
3
3
Q
∗
/
D
3
R
3
=
3
(
Q
−
4
π
=
4
π
(
−
)
, respectively. The
entropic term in the free energy is, therefore, given by:
ln
c
(
in
)
Q
∗
ln
c
(
out
)
ions
Q
∗
)
F
ions
/
k
B
T
=(
Q
−
ions
+
(10)
and the Coulomb interaction term is given by:
l
B
Q
∗
F
Coulomb
/
k
B
T
=
R
ϑ
(
R
/
D
)
(11)
e
2
where
l
B
=
is a rational function of
x
,
star size,
x
/
k
B
T
is the Bjerrum length and
ϑ
(
x
)
5. The minimization of the free energy, (
7
), (
8
),
(
9
), (
10
), and (
11
), results in equilibrium values of the star size,
R
, and that of the
uncompensated charge,
Q
∗
. The latter is of special interest and can be found from
the equation:
=
R
/
D
→
0,
ϑ
(
x
)
→
3
/
ln
Q
1
D
3
1
R
l
B
1
Q
∗
=
Q
∗
−
R
3
−
(12)
2
ϑ
(
R
/
D
)
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