Conformations and Solution Properties of Star-Branched Polyelectrolytes - Self Organized Nanostructures of Amphiphilic Block Copolymers - page 7

Short-range interactions (van der Waals attraction and hard-core repulsion)
between monomers can be accounted for using a virial expansion. As long as
the volume fraction of monomers in a star polymer is significantly below unity,
only pairwise monomer-monomer interactions, with

second virial coefficient

a 3 a 6 , υ ∼ ( 1 − 2 χ ( T )) , or ternary interactions, with third virial coefficient w ∼

are relevant. The former depends on the Flory-Huggins parameter χ and is positive

under good solvent ( 2).

In a good solvent, binary interactions are dominated by the repulsive part of the
monomer-monomer interaction potential (hard-core repulsion), whereas in a poor
solvent, binary interactions are attractive (due to the van der Waals forces). A
special case

χ < 1 / 2) and negative under poor solvent conditions ( χ > 1 /

0 (vanishing net binary interactions) corresponds to theta-solvent
conditions, where weak attraction between monomers is exactly compensated by
their excluded volume.
Subsequent theoretical studies [ 19 , 20 ] have incorporated short-range monomer-
monomer interactions using the mean field approximation, but systematically un-
derestimated conformational entropy losses in the stretched arms. These theories
thus overestimate the star size.
The first theories that implemented a proper balance of intramolecular inter-
actions and conformational elasticity of the branches were developed by Daoud
andCotton[ 21 ] and by Zhulina and Birshtein [ 22 - 24 ] . These theories use scaling
concepts (the blob model), originally developed by de Gennes and Alexander to de-
scribe the structure of semidilute polymer solutions [ 64 ] and planar polymer brushes
[ 65 , 66 ]. Here, the monomer-monomer interactions were incorporated on the level
of binary or ternary contacts (corresponding to good and theta-solvent conditions,
respectively), and both dilute and semidilute solutions of star polymers were con-
sidered. Depending on the solvent quality and the intrinsic stiffness of the arms, the
branches of a star could be locally swollen, or exhibit Gaussian statistics [ 22 - 24 ].

υ =

2.1 Star Polymer Conformation in a Dilute Solution

According to the blob model, a flexible neutral star polymer can be envisioned as
an array of concentric shells of closely packed blobs. For a visualization of the
blobs, see Fig. 1 a . The chain ends are assumed to be localized at the edge (i.e.,
within the outermost blobs), and each chain contributes one blob to each shell. The
chain segment inside a blob remains unperturbed by the interactions with other
branches and, therefore, exhibits Gaussian or excluded-volume statistics under
theta- or good solvent conditions, respectively. For transparency, we consider first
athermal, υ = a 3 , and theta-solvent,

0, conditions. The blob size at distance r
from the star center is equal to the average interchain separation =

υ =

p 1 / 2 ,which r /

coincides with the local correlation length, ξ ( r ) . The latter is related to the local

polymer concentration, c p ( r ) , by the same scaling law as in a semidilute polymer

) = a 3 ] − ν / ( 3ν − 1 ) ,where solution, ξ ( r a [ c p ( r ) ν is the Flory exponent ( ν ≈ 3 / 5and