Chemistry Reference
In-Depth Information
entropy losses due to the uniform swelling of a randomly branched polymer, up to
R
2
k
B
T
=
F
conf
/
(24)
(
Nn
)
1
/
2
a
2
where
R
ideal
=
1
/
4
is the unperturbed, Gaussian, size of an ideal, randomly
lar Coulomb repulsions and/or the osmotic pressure of counterions leads to:
a
(
Nn
)
aN
5
/
6
2
l
B
1
/
3
n
1
/
6
2
)
−
2
(
α
/
a
)
,
N
/
n
(
α
(
l
B
/
a
)
R
random
=
(25)
aN
3
/
4
1
/
2
n
1
/
4
2
)
−
2
α
,
N
/
n
(
α
(
l
B
/
a
)
An interesting feature of (
25
) is the fractal dimension, specifying how the mass
N
of a randomly branched PE depends on its size
R
. For small clusters that release
many counterions into the bulk solution,
d
f
∼
=
/
6
5, whereas for large (osmotic) clus-
=
/
ters,
d
f
3. Again, the counterion localization threshold is set by the increase in
the electrostatic potential,
=
4
N
1
/
6
in
the regime of free counterions (small clusters). Another remarkable feature of (
25
)
is, that the counterion condensation threshold occurs when the number of branching
points
N
l
B
Q
/
R
which grows upon an increase in
N
as
∼
n
reaches the characteristic value
=
α
−
2
)
−
4
, which again is indepen-
/
(
l
B
/
a
dent of the spacer length
n
.
These results have been generalized further in [
31
], for charged polymeric frac-
tals with arbitrary connectivity characterized by spectral dimension,
d
s
, (the latter
relates the longest path in the fractal,
R
max
=
aN
1
/
d
s
, to its mass
N
) and arbitrary
fractal dimension
d
f
(in the absence of ionic charges) in
d
-dimensional space. For
ideal (Gaussian) fractals
d
f
=
∼
2
d
s
/
(
2
−
d
s
)
. For charged fractals:
⎧
⎨
d
s
d
s
d
f
+(
d
−
2
)(
d
f
−
d
s
)
2
d
f
−
d
s
d
s
d
f
+(
d
−
2
)(
d
f
−
d
s
)
d
f
−
2
l
B
/
N
∗
aN
(
α
a
)
,
N
R
fractal
=
(26)
d
f
−
d
s
d
s
d
f
⎩
aN
1
/
d
s
N
∗
α
,
N
Here, the threshold value for the number of monomers, corresponding to the onset
of charge renormalization, is specified as:
d
s
d
f
−
(
d
−
2
)(
d
f
−
d
s
)
d
s
N
∗
=
α
(
l
B
/
a
)
(27)
d
f
(
d
−
2
−
d
s
)
(
d
−
2
−
d
s
)
As follows from (
26
) charge renormalization effects occur only when
d
<
d
s
+
2.
4.3
Polyelectrolyte Cylindrical (Molecular) Brushes
Molecular brushes are polymers composed of a long main chain (backbone), onto
illustration. The graft-copolymers are classified as molecular brushes, provided the
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