Biomedical Engineering Reference
In-Depth Information
5.3
Polyelectrolyte theories
Some concepts necessary for describing characteristic behaviour of polyelectrolyte
solutions are given here. The Debye length
λ D , which represents a screening length,
giving a measure of the range of the potential around a point charge surrounded by other
point charges in such a way as to
'
'
screen
interactions, is written as
1 = 2
ε 0 ε 0 k B T
2e 2 I
λ D ¼
;
ð
5
:
1
Þ
2 P i
is the ionic strength (note that P i
1
z i n i
z i n i ¼
where I
¼
0 because of electro-
ε 0
neutrality), e is the elementary charge,
ε 0 is the dielectric
constant in vacuo, in i (r) is the local ion concentration, zi i is the valence of the ith species
and n i is the corresponding bulk concentration.
However, in many polyelectrolyte solutions the effect of counterions, always present
because of the requirement for electroneutrality, cannot be reduced to straightforward
Debye
is the dielectric coef
cient,
Hückel screening. In practice, it can be energetically favourable to have a
proportion of the counterions located in the vicinity of or
-
the surface of the macroion.
Then, the apparent charge of the polyelectrolyte will be reduced. The term counterion
condensation was used by Oosawa ( 1971 ) and Manning ( 1988 ) and re
'
at
'
ects the binding
of counterions to the charged chain. However, it is not appropriate to speak simply of
'
bound
'
and
'
free
'
ions, because even diffuse ions are bound, at least in a thermodynamic
sense.
What is the implication of chain stiffness? Even a very
flexible polyelectrolyte has a
'
'
tendency to
at low ionic strengths in order to maximize the separation of ionic
groups attached along the chain contour. The elastic and electrostatic contributions to
thefreeenergyofchaincurvaturehavebeencalculated for a polyelectrolyte chain of
contour length L = Na (N is the degree of polymerization and a is the length of a
monomer unit) consisting of Z units of elementary charge e separated by a contour
distance b = L/Z. The repulsion between charges was described by a screened Debye
stiffen
-
Hückel potential with screening length
λ D . From the free energy, the following expres-
sion for the total persistence length l p of long chains was derived (Odijk, 1977 ; Skolnick
and Fixman, 1977 ):
l p ¼
l pi þ
l pe ;
ð
5
:
2
Þ
with
l pe ¼ λ B λ D 2
4b 2
ξ M 2
=
Þ;
ð
5
:
3
Þ
where l pi
is the intrinsic persistence length in the absence of charges,
l pe is the
electrostatic persistence length,
ξ M =
λ B /b is the so-called Manning parameter and
λ B = e 2 /(4
πε 0 ε 0 k B T), the Bjerrum length, is the distance at which the Coulombic inter-
action between two unscreened elementary charges is equal to the thermal energy. This
shows how a polyelectrolyte chain becomes more
flexible in high ionic strength solvents,
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