consist of many chains and many colloidal spheres. A total charge close to zero
of this polyelectrolyte–colloidal assembly is often a necessary condition for
higher-order aggregation and phase separation in polyelectrolyte–colloid mixtures.
The critical density was shown to be nearly independent of the polyelectrolyte
molecular weight and of the concentration in solution. Experiments have shown
that, as the salt concentration in solution increases, the value of js c j grows and
scales for spherical colloidal particles as [121, 128, 130]:
js c j $ k
b l with b l ¼ 1 À 1:8:
(54)
The exponent b l depends also on the polyelectrolyte linear charge density r and
on the polyelectrolyte stiffness [130], which make it a nonuniversal characteristic
for the used polyelectrolytes. Typically, polyelectrolytes with smaller r reveal a
stronger dependence of s c on k. The polyelectrolyte–sphere binding affinity was
shown to decrease with the polymer persistence length and to increase with the
polyelectrolyte linear charge density. Note that for complexation of polyelectrolytes of different r with the spherical dimethyl dodecylamine oxide (DMDAO)
micelles, a modified dependence on the critical micelle charge density has been
suggested, namely js c j $ k
1:8 r
À0:6 [128].
The size of some micelles, e.g., DMDAO, as well as their shape can change
with k. At large salt content, cylindrical rather than spherical micelle shapes are
observed. For cylindrical micelles, larger scaling exponents are usually measured
than for spherical micelles of the same composition, e.g., from b l % 1.4 for PVASDMDAO to b l % 2.5 for P(AMPS/AAm)-DMDAO complexes [128, 129].
DMDAO micelles also form complexes with quite persistent polyelectrolytes
such as double-stranded DNA, with an exponent b l % 1.6 [129]. This weaker k
dependence on the polyelectrolyte–sphere critical adsorption is consistent with our
theoretical result in the low-salt limit. Moreover, although the micelle surface
charge density is proportional to the degree of micelle protonation, which is
controlled in experiments via the pH value, the exact relation for every micelle
type is not known. The surface charge density has therefore been obtained for some
systems from the pH titration data via calculation of the surface potential of the
micelle using the Debye–Hu ¨ckel solution (15) on the sphere surface.
Experimental values of s c (k) extracted from several studies on polyelectrolyte–sphere complexation performed by Dubin and coworkers have been presented [59].
The experimentally determined exponents b l are typically smaller than those obtained
theoretically over the relevant range of ka values; the prediction of the model in this
range is already close to the regime |s c | ~ k
3
. However, the electrostatic part of the
polyelectrolyte persistence length l
el
p decreases with k, where the scaling relation l
el
p
$ k
À1 is observed experimentally for flexible polyelectrolytes, e.g., polyacrylamides
such as P(AMPS/AAm) [172]. Theoretical calculations [41, 149, 163] and computer
simulations [148, 161] predict:
l
el
p $ k
À4=5
À k
À6=5
;
(55)
Strong and Weak Polyelectrolyte Adsorption onto Oppositely Charged Curved. . .
29
of this polyelectrolyte–colloidal assembly is often a necessary condition for
higher-order aggregation and phase separation in polyelectrolyte–colloid mixtures.
The critical density was shown to be nearly independent of the polyelectrolyte
molecular weight and of the concentration in solution. Experiments have shown
that, as the salt concentration in solution increases, the value of js c j grows and
scales for spherical colloidal particles as [121, 128, 130]:
js c j $ k
b l with b l ¼ 1 À 1:8:
(54)
The exponent b l depends also on the polyelectrolyte linear charge density r and
on the polyelectrolyte stiffness [130], which make it a nonuniversal characteristic
for the used polyelectrolytes. Typically, polyelectrolytes with smaller r reveal a
stronger dependence of s c on k. The polyelectrolyte–sphere binding affinity was
shown to decrease with the polymer persistence length and to increase with the
polyelectrolyte linear charge density. Note that for complexation of polyelectrolytes of different r with the spherical dimethyl dodecylamine oxide (DMDAO)
micelles, a modified dependence on the critical micelle charge density has been
suggested, namely js c j $ k
1:8 r
À0:6 [128].
The size of some micelles, e.g., DMDAO, as well as their shape can change
with k. At large salt content, cylindrical rather than spherical micelle shapes are
observed. For cylindrical micelles, larger scaling exponents are usually measured
than for spherical micelles of the same composition, e.g., from b l % 1.4 for PVASDMDAO to b l % 2.5 for P(AMPS/AAm)-DMDAO complexes [128, 129].
DMDAO micelles also form complexes with quite persistent polyelectrolytes
such as double-stranded DNA, with an exponent b l % 1.6 [129]. This weaker k
dependence on the polyelectrolyte–sphere critical adsorption is consistent with our
theoretical result in the low-salt limit. Moreover, although the micelle surface
charge density is proportional to the degree of micelle protonation, which is
controlled in experiments via the pH value, the exact relation for every micelle
type is not known. The surface charge density has therefore been obtained for some
systems from the pH titration data via calculation of the surface potential of the
micelle using the Debye–Hu ¨ckel solution (15) on the sphere surface.
Experimental values of s c (k) extracted from several studies on polyelectrolyte–sphere complexation performed by Dubin and coworkers have been presented [59].
The experimentally determined exponents b l are typically smaller than those obtained
theoretically over the relevant range of ka values; the prediction of the model in this
range is already close to the regime |s c | ~ k
3
. However, the electrostatic part of the
polyelectrolyte persistence length l
el
p decreases with k, where the scaling relation l
el
p
$ k
À1 is observed experimentally for flexible polyelectrolytes, e.g., polyacrylamides
such as P(AMPS/AAm) [172]. Theoretical calculations [41, 149, 163] and computer
simulations [148, 161] predict:
l
el
p $ k
À4=5
À k
À6=5
;
(55)
Strong and Weak Polyelectrolyte Adsorption onto Oppositely Charged Curved. . .
29