Chemistry Reference
In-Depth Information
Fig. 8 Cell model of Lifson
and Katchalsky [
74
] showing
radii
a
and
R
monomers (that tend to stretch the polyelectrolyte) and the tendency of the poly-
electrolyte to increase its entropy by forming globular or entangled coils (at low or
high polymer concentrations, respectively). As shown in Fig.
8
, the polyelectrolyte
backbone is modeled as a stretched cylinder of radius
a
and length
h
. That cylinder
is surrounded by another cylindrical cell (radius
R
and length
h
). The electrical
charge on the backbone is approximated by a uniform charge on the surface of the
inner cylinder. The counterions are dissolved in the cylindrical space between radii
a
and
R
, where they form an ionic cloud. The radius
R
depends on the concentration
of the polyelectrolyte. It is low in highly concentrated solutions and increases with
decreasing concentration to reach infinity in an infinitely diluted solution. The
electrostatics in that cloud are described by the Poisson Boltzmann equation. In a
manner analogous to the Debye H¨ckel theory, the electrostatic potential caused
by the interactions between the stretched backbone on one side and the surrounding
counterions on the other side is calculated by solving the Poisson Boltzmann
differential equation. The electrostatic potential
'
(
r
) in the cylindrical space
between the radii
a
and
R
(
a
r
R
) is:
kT
e
2
l
b
r
2
sinh
2
'ð
r
Þ¼
ln
½
b
ln
Ar
ðÞ
;
(29)
2
ð
R
2
a
2
Þ
where
l
is a (dimensionless) charge density parameter that describes the charge
density on the polyelectrolyte's backbone. When the repeating unit is a 1:1 electrolyte,
that parameter becomes:
l
B
b
;
l ¼
(30)
where
l
B
is the Bjerrum length:
e
2
4
pee
0
kT
;
l
B
¼
(31)
which characterizes the solvent through its relative dielectric constant
e
. Parameters
b, e,
and
e
0
are the length of that repeating unit, the elementary charge, and the
permittivity of vacuum, respectively, The two other parameters
A
(which is an
