Conformations and Solution Properties of Star-Branched Polyelectrolytes - Self Organized Nanostructures of Amphiphilic Block Copolymers - page 20

4 Localization of Counterions in Salt-Free Solutions

of Branched Polyelectrolytes: Effect of the Polyion Topology

The localization of counterions in the intramolecular volume is a common feature
manifested in dilute salt-free solutions of branched polyions of different topolo-
gies, including dendritic (star-burst), randomly (hyper)branched PEs, PE molecular
brushes, etc. The physical reason for this phenomena is the same as outlined for PE
stars: a strongly charged, branched, macroion creates a high electrostatic potential,
which attracts counterions and retains them in the intramolecular volume, in spite
of a significant loss in the translational entropy. The effect is most pronounced in a
dilute solution, where the concentration of counterions in the bulk is extremely low.
Remarkably, linear PEs of arbitrary large N cannot induce a similar effect. This
is because both the charge, Ne , and the size, R =

3 ,ofastretched
polyion in a dilute salt-free solution, scale proportionally to N [ 91 , 92 ], so that their
ratio remains

2 l B / 1 /

α aN ( α a )

1 / 3

1. Note that we do not discuss here the effect of Manning
condensation, which occurs when the distance between two neighboring charges
along the chain is smaller than l B . A detailed discussion of this case can be found
in [ 93 ].
A theoretical analysis of the effect of counterion localization in a dilute solution
of weakly charged branched polyions of different topologies [ 31 - 33 ] and ionic mi-
crogels [ 34 , 35 ] , was performed on the basis of a cell model, similar to that used
here for a star-like PE. The elastic term in the free energy that accounts for the
conformational entropy of a uniformly swollen branched macromolecule, has to be
specified depending on the polyion topology. The shape of the cell might also be
modified. For example, in the case of a molecular PE brush, a cylindrical instead of
spherical cell should be used.
Similarly to the case of a quenched star-like PE, two regimes of ion distribu-
tion were distinguished. A fairly uniform distribution of “free” counterions is found
at low degree of branching, whereas a strong localization of counterions in the in-
tramolecular volume is expected for a high degree of branching of the polyion. In
the latter case only a minor fraction of the counterions is released to the exterior
volume of the cell.
We recall that for a star-like PE, the transition from the regime of a barely charged
polyion, to the osmotic regime, occurs at a characteristic number of branches, p =
α − 1 / 2

∼ α ≤

2 ,
and is independent of the length N of an arm. Below we briefly summarize the
results obtained for branched polyions of different topologies.

) − 1 . The latter depends on the combination of the parameters

( l B / a α ( l B / a )

4.1 Ionic Dendrimers (Star-Burst Polyelectrolytes)

The conformations of charged (regular) star-burst polymers (flexible ionic
dendrimers), were analyzed theoretically in [ 33 ]. Referring to Fig. 4 a , relevant
architectural parameters for a star-burst polymer are the number of generations,