34 Sara Llamas, Laura Fernández-Peña, Ana Mateos-Maroto et al.
2
22
2 Bd
sinh 4 l f
dz
(4)
where κ and lB are the inverse of the Debye and the Bjerrum lengths,
respectively. The resolution of equation (4) requires the definition of two
boundary conditions. The first one considering that the surface is fixed, being
the so called electrostatic boundary condition, defined as follows
B
0d
4l
dz
(5)
with σ being the charge density of the surface. The second boundary condition
can be written as
1
0d1
dz d
(6)
where d is the extrapolation length, which depends on the specific nature of
the interaction between polymer and surface. For those cases in which the
interaction is mainly attractive, d assumes positive values in the same range
than the molecular size. On the other side, when the interactions are mainly
repulsive it assumes negative values. For soft interpenetrable surfaces, such as
those found during the deposition of multilayers onto a pre-adsorbed
polyelectrolyte layers, d assumes values close to 0 [60].
One of the main issues to consider for describing the adsorption of
polyelectrolytes onto oppositely charged surfaces is the charge inversion
phenomena. Thus, the adsorption of polyelectrolytes onto a charged surface
occurs beyond the charge neutralization threshold, and a certain degree of
overcompensation has been found experimentally [61, 62]. For surfaces with
low charge and under low ionic conditions, the role of the excluded volume
interactions can be considered almost negligible. Thus, the thickness of the
layer assumes values close to that corresponding to the thickness of a single
chain, δ. These conditions always lead to a charge inversion by the polymer
adsorption that can be described as follows