critical adsorption conditions within the variational trial function approach [228]
(see also [52]). The main conclusion is that the critical surface charge density grows
with decreasing size of the adsorption patch; however, it always remains larger than
that for a uniformly charged surface and approaches this limit for larger enough
charged patches. The effect of patterns on the adsorbing polyelectrolyte, and some
degree of commensurability with the patchiness on the surface, are interesting issues
for investigation in the future. Moreover, our model does not account for possible
charge regulation effects on adsorbing surfaces and on polymer chains themselves
caused by an approaching polyelectrolyte. Such effects are discussed for the adsorption of polyelectrolytes onto a titratable spherical particle [98]. Other effects such as
a physical roughness of the adsorbing surface might also nontrivially alter the
position of the adsorption–desorption threshold [229].
Decoupling of Surface Potential and Polymer Surface Density—This does not
allow us to predict the amount of adsorbed polymer or the degree of surface charge
compensation by the weakly adsorbed polyelectrolytes. In a more realistic model, the
amount of already adsorbed polyelectrolyte should limit the adsorption of subsequent
segments of the chain. Namely, the effective surface charge density of the adsorbing
surface is renormalized by the adsorbed part of the polymer and thus the critical
adsorption parameter has to grow in order to adsorb the entire chain. The
corresponding eigenvalue equation is then, however, only amenable to a numerical
solution [152, 226]. In the current model, the adsorbed polyelectrolyte layer effectively overcharges the surface because the charge density on the interface prior to the
polyion adsorption is exactly balanced by small mobile counter- and co-ions from the
electrolyte and the ionic distributions stay unchanged upon polyelectrolyte adsorption.
Polymer Length—We restrict ourselves to consideration of the ground state for
the adsorption problem. For polymers shorter than a dozen Kuhn segments, the
contributions of excited states to the Green function have to be taken into account.
The value of the critical surface charge density is then expected to grow.
Linear Versus Nonlinear Theory—We apply the linear Debye–Huckel model to
compute the electrostatic potential emerging near a charged surface. It is known,
however, particularly at low salt concentrations, that the surface potential for the
planar and curved interfaces can exceed the 25 mV allowed by this theory. In this
limit, the full nonlinear Poisson–Boltzmann theory must be implemented, both for
the surface [154, 230] and for the polyelectrolyte [231, 232]. Remarkably and quite
unfortunately, the changes in scaling for the critical surface charge density
predicted by our adsorption model occur in this limit of low salt. The electrostatic
potentials in this limit are expected to be high and polyelectrolyte adsorption is
therefore rather strong. Also, in the model the polyelectrolytes are treated as weakly
charged, with the linear charge density below Manning’s counterion condensation
threshold.
Low-Dielectric Interfaces—The presence of a low-dielectric material beneath
the adsorbing boundary has considerable implications for adsorption. In particular,
in the proximity of such an interface the polyelectrolyte experiences a repulsive
force from the image charges that effectively displace the polymer from the
interfacial region of a high attractive potential. This image repulsion grows
48
R.G. Winkler and A.G. Cherstvy
(see also [52]). The main conclusion is that the critical surface charge density grows
with decreasing size of the adsorption patch; however, it always remains larger than
that for a uniformly charged surface and approaches this limit for larger enough
charged patches. The effect of patterns on the adsorbing polyelectrolyte, and some
degree of commensurability with the patchiness on the surface, are interesting issues
for investigation in the future. Moreover, our model does not account for possible
charge regulation effects on adsorbing surfaces and on polymer chains themselves
caused by an approaching polyelectrolyte. Such effects are discussed for the adsorption of polyelectrolytes onto a titratable spherical particle [98]. Other effects such as
a physical roughness of the adsorbing surface might also nontrivially alter the
position of the adsorption–desorption threshold [229].
Decoupling of Surface Potential and Polymer Surface Density—This does not
allow us to predict the amount of adsorbed polymer or the degree of surface charge
compensation by the weakly adsorbed polyelectrolytes. In a more realistic model, the
amount of already adsorbed polyelectrolyte should limit the adsorption of subsequent
segments of the chain. Namely, the effective surface charge density of the adsorbing
surface is renormalized by the adsorbed part of the polymer and thus the critical
adsorption parameter has to grow in order to adsorb the entire chain. The
corresponding eigenvalue equation is then, however, only amenable to a numerical
solution [152, 226]. In the current model, the adsorbed polyelectrolyte layer effectively overcharges the surface because the charge density on the interface prior to the
polyion adsorption is exactly balanced by small mobile counter- and co-ions from the
electrolyte and the ionic distributions stay unchanged upon polyelectrolyte adsorption.
Polymer Length—We restrict ourselves to consideration of the ground state for
the adsorption problem. For polymers shorter than a dozen Kuhn segments, the
contributions of excited states to the Green function have to be taken into account.
The value of the critical surface charge density is then expected to grow.
Linear Versus Nonlinear Theory—We apply the linear Debye–Huckel model to
compute the electrostatic potential emerging near a charged surface. It is known,
however, particularly at low salt concentrations, that the surface potential for the
planar and curved interfaces can exceed the 25 mV allowed by this theory. In this
limit, the full nonlinear Poisson–Boltzmann theory must be implemented, both for
the surface [154, 230] and for the polyelectrolyte [231, 232]. Remarkably and quite
unfortunately, the changes in scaling for the critical surface charge density
predicted by our adsorption model occur in this limit of low salt. The electrostatic
potentials in this limit are expected to be high and polyelectrolyte adsorption is
therefore rather strong. Also, in the model the polyelectrolytes are treated as weakly
charged, with the linear charge density below Manning’s counterion condensation
threshold.
Low-Dielectric Interfaces—The presence of a low-dielectric material beneath
the adsorbing boundary has considerable implications for adsorption. In particular,
in the proximity of such an interface the polyelectrolyte experiences a repulsive
force from the image charges that effectively displace the polymer from the
interfacial region of a high attractive potential. This image repulsion grows
48
R.G. Winkler and A.G. Cherstvy