Adsorption of Polyelectrolytes onto Charged Surfaces 39electrostatic attraction monomer-surface and the short-range repulsion
between monomers, with the thickness of the layer increasing as σ1/3 under Θ
solvent conditions. For the highest surface density, ion, the counterions
on the surface dominate the screening of the potential associated with the
surface inside the region defined by the layer thickness h. For z = h, the
effective surface charge becomes in σion. It is worth mentioning that the
density of polymer within the layer remains almost constant. For length scales
between h and h au 1/ 2f3/ 4, the layers assumes self-similar structures
analogous to those described in references [71, 73], being the thickness
defined by au1/ 2f 3/ 4. The above described theory proposed by Dobrynin and
Rubinstein [71-73] is an analysis of the adsorption of polyelectrolytes onto
solid surfaces based in scaling laws that identifies until six different regions
for the dependence of the layer thickness on the charge density σ.
Another model to describe the adsorption of polyelectrolytes onto charged
surfaces was proposed by Cohen-Stuart [77]. The bare assumption of this
model is that upon adsorption the polyelectrolyte chains lay on a flat
conformation onto the surface, being the role of the loops and tails on the
adsorption layers almost negligible. This model considers that the charge of
the adsorbed segments defined as the product between the number of adsorbed
segment θsp and the charge of these segments zsp must be compensated by the
formation of a diffuse layer of counterion, thus providing the bases for the
maintenance of the electroneutrality beyond the Debye length. The use of the
Gouy-Chapman approach provides a value for the charge density of the layer
defined by r sinh(y / 2), where εr and y are the relative dielectric constant
and the dimensionless potential of the layer adsorbed onto the surface,
respectively. Considering that the model proposed by Cohen-Stuart [77] is a
lattice model, the above equation can be rewritten in terms of a constant with
value 0.67 which is related to the characteristics of the lattice and the dielectric
constant of the water at room temperature, and the volume fraction of salt in
the solution defined by salt. This leads to the following expression
0.67 saltsinh(y / 2) (20)
The adsorbed amount is obtained as a balance between the electrostatic
and non-electrostatic affinity defined by the parameter χs. This leads to an
expression for the number of adsorbed segment as