  Understanding the effect of inhomogeneity on the charge regulation and
dielectric properties, and how it depends on the conformational characteristics
of the macromolecules is a long-standing problem. In order to address this
problem, we have developed a field-theory to study charge regulation and local
dielectric function in planar polyelectrolyte brushes. The theory is used to
study a polyacid brush, which is comprised of chains end-grafted at the
solid-fluid interface, in equilibrium with a bulk solution containing
monovalent salt ions, solvent molecules and pH controlling acid. In particular,
we focus on the effects of the concentration of added salt and pH of the bulk
in determining the local charge and dielectric function. Our theoretical
investigations reveal that the dipole moment of the ion-pairs formed as a
result of counterion adsorption on the chain backbones play a key role in
affecting the local dielectric function. For polyelectrolytes made of monomers
having dipole moments lower than the solvent molecules, dielectric decrement is
predicted inside the brush region. However, the formation of ion-pairs (due to
adsorption of counterions coming from the dissociation of added salt) more
polar than the solvent molecules is shown to increase the magnitude of the
dielectric function with respect to its bulk value. Furthermore, an increase in
the bulk salt concentration is shown to increase the local charge inside the
brush region.
