Adsorption of Polyelectrolytes onto Charged Surfaces 33homogenous layers [55, 56]. However, this naive approach is rather limited
and can be only used to describe the adsorption of homopolymers in which the
charge distribution along the chain is homogeneous [55, 57]. Some extensions
of the bare theory have been developed to account for more complex polymer
architectures [58, 59]. The mean-field theory is based in the fact that the
conformation of a polymer chain in a mean-field potential, U(z), is affected by
other chains due to their mutual interactions and by the interaction with the
surface/interface. These issues can be accounted by an order parameter,
dependent on the local monomer concentration c(z) at a distance z from the
surface
(z) c(z) (1)
The order parameter is described by the Edward’s equation in the long
chain limit
22
2ad
(U(z) E) 0
6 dz
(2)
where a is the size of the monomer, and E representes the free energy
associated with the ground state of the system, which can be calculated from
the differences between the chemical potential of the polymer in the bulk and
adsorbed onto the surface. The mean-field potential included contributions
associated with both electrostatic interactions and excluded volume one, being
possible to give a definition for U(z) considering a Θ solvent as follows
w c (z)^22
U(z) f (z)
2
(3)
with f and w^2 being the free energy and the third virial coefficient,
respectively. The latter provides information about the repulsive interactions
occurring between the monomers. ψ is a dimensionless potential accounting
for the electrostatic interactions. Considering that the role of the electrostatic
potential can be considered negligible (ψ = 0) far from the surface, and
assuming that the generalized Poisson-Boltzman is satisfied by the potential
used, it is possible to write