It is also interesting to note that such a regular decrease in the power of k
resembles the change in the laws of electrostatic repulsion between the particles of
corresponding geometry in simple-salt solutions. Namely, at large interparticle
separations R, kR ) 1, two uniformly charged planes, cylinders, or spheres repel
each other according to the dependencies $ e
ÀkR , $ e
ÀkR
ðkRÞ
À1=2 , and $ e
ÀkR
ðkRÞ
À1 [168].
5.2 Comparison Between Theory and Experiment
An alternative technique to the described approaches is the variational principle to
determine critical parameters [41, 56, 57]. The quality of the result, however,
depends crucially on the applied trial function. Such an approach has been pursued
for cylindrical and spherical surfaces [41, 56, 57]. For adsorption on a cylindrical
surface, this method yields very good quantitative agreement with the WKB
result for the critical adsorption condition s c . By contrast, for adsorption onto a
sphere, the results of van Goeler and Muthukumar [56] disagree with the scaling
predictions from the WKB model as well as from the approximate solution
employing the Hulthe ´n potential. Namely, in the limit of ka ( 1, the variational
calculation predicts the dependence js c j ~ (ka)
2 , both for the rod and sphere
situations. Note that this variational technique employs a specific dependence of
trial functions on the eigenvalues. Variational calculations have also been used to
study polyelectrolyte adsorption onto a sphere [57]. As a result, the same dependence on s c has been found as for the solution with the Hulthe ´n potential and the
WKB approach, but the numerical factor is different.
The complex formation between flexible and semiflexible, both biological and
synthetic, polyelectrolytes with oppositely charged spherical colloidal particles,
cationic or nonionic micelles, and dendrimers has been systematically studied
experimentally by Dubin and coworkers [43, 44, 121–126, 128, 129]. The considered cationic micelles possess a homogeneously charged surface, which can be
tuned continuously by addition of charged and uncharged groups up to several e per
1,000 A ˚ 2 . Typically, the sphere diameter is 20–40 A ˚ ; however, considerably larger
particles have also been studied. The polyelectrolyte persistence length l p ¼ l/2 is
on the order of the sphere radius or smaller (approximately 30 A ˚ for NaPSS, AMPS/
AAm copolymers, and PDADMAC; 40 A ˚ for hyaluronic acid; and 12 A ˚ for PAA)
[44, 130]. More flexible and more rigid polyelectrolytes have also been considered.
The polyelectrolytes were in the intermediate charge density regime, typically
below the threshold for the Manning counterion condensation [116, 163, 169–171].
Experiments have shown that no polyelectrolyte–micelle complexation occurs
when the micellar surface charge density is below the critical charge density |s c |.
Above this density, the turbidimetric titration curves reveal a dramatic increase in
turbidity that indicates complexation, because the average molecular mass of the
complexes is much larger than that of polyelectrolytes alone. The complexes may
28
R.G. Winkler and A.G. Cherstvy
resembles the change in the laws of electrostatic repulsion between the particles of
corresponding geometry in simple-salt solutions. Namely, at large interparticle
separations R, kR ) 1, two uniformly charged planes, cylinders, or spheres repel
each other according to the dependencies $ e
ÀkR , $ e
ÀkR
ðkRÞ
À1=2 , and $ e
ÀkR
ðkRÞ
À1 [168].
5.2 Comparison Between Theory and Experiment
An alternative technique to the described approaches is the variational principle to
determine critical parameters [41, 56, 57]. The quality of the result, however,
depends crucially on the applied trial function. Such an approach has been pursued
for cylindrical and spherical surfaces [41, 56, 57]. For adsorption on a cylindrical
surface, this method yields very good quantitative agreement with the WKB
result for the critical adsorption condition s c . By contrast, for adsorption onto a
sphere, the results of van Goeler and Muthukumar [56] disagree with the scaling
predictions from the WKB model as well as from the approximate solution
employing the Hulthe ´n potential. Namely, in the limit of ka ( 1, the variational
calculation predicts the dependence js c j ~ (ka)
2 , both for the rod and sphere
situations. Note that this variational technique employs a specific dependence of
trial functions on the eigenvalues. Variational calculations have also been used to
study polyelectrolyte adsorption onto a sphere [57]. As a result, the same dependence on s c has been found as for the solution with the Hulthe ´n potential and the
WKB approach, but the numerical factor is different.
The complex formation between flexible and semiflexible, both biological and
synthetic, polyelectrolytes with oppositely charged spherical colloidal particles,
cationic or nonionic micelles, and dendrimers has been systematically studied
experimentally by Dubin and coworkers [43, 44, 121–126, 128, 129]. The considered cationic micelles possess a homogeneously charged surface, which can be
tuned continuously by addition of charged and uncharged groups up to several e per
1,000 A ˚ 2 . Typically, the sphere diameter is 20–40 A ˚ ; however, considerably larger
particles have also been studied. The polyelectrolyte persistence length l p ¼ l/2 is
on the order of the sphere radius or smaller (approximately 30 A ˚ for NaPSS, AMPS/
AAm copolymers, and PDADMAC; 40 A ˚ for hyaluronic acid; and 12 A ˚ for PAA)
[44, 130]. More flexible and more rigid polyelectrolytes have also been considered.
The polyelectrolytes were in the intermediate charge density regime, typically
below the threshold for the Manning counterion condensation [116, 163, 169–171].
Experiments have shown that no polyelectrolyte–micelle complexation occurs
when the micellar surface charge density is below the critical charge density |s c |.
Above this density, the turbidimetric titration curves reveal a dramatic increase in
turbidity that indicates complexation, because the average molecular mass of the
complexes is much larger than that of polyelectrolytes alone. The complexes may
28
R.G. Winkler and A.G. Cherstvy