polymers experimentally [236], is often a consequence of short-range
nonelectrostatic chain–surface attraction.
Nearly Gaussian Chains—For weak adsorption, The conformational properties
of adsorbing chains are assumed to be only weakly perturbed upon adsorption. This
might not be true well above the adsorption threshold, when quite dense polymer
layers are formed next to the interface. For critical adsorption conditions, however,
this approximation is expected to be valid.
Strong Adsorption Limit—Similarly to the weak adsorption limit, the inherent
limitations of the linearized electrostatic theory are to be remembered. Here, the
limitations might be even more severe because some well-defined polyelectrolyte
patterns are assumed in the model of polyelectrolyte–cylinder and polyelectrolyte–sphere complex formation, which require a strong polyelectrolyte–surface
attraction. Although the adopted approach does not predict overcharging in these
cases, dropping the assumption of an infinitely thin polyelectrolyte and including an
asymmetric charge neutralization along the DNA–sphere contact [237, 238]
reproduces the extent of nucleosome overcharging quite well [79]. In applications
to DNA–histone wrapping, our model of a uniformly bendable nonstretchable
worm-like chain should be severely modified [239, 240], if we are interested in
the precise energetics of DNA wrapping, with site-specific DNA–histone
interactions, sequence-specific curvature effects [241], and possible kinks in the
structure.
8 Conclusions
In this review, we have summarized theoretical concepts and recent advances
for the adsorption of linear polyelectrolyte molecules onto curved surfaces in the
weak and strong adsorption limit. A mean-field description is adopted, and the
interaction potentials between the polyelectrolyte and the surfaces are derived from
the linearized Poisson–Boltzmann equation for the corresponding geometries
(planes, cylinders, and spheres). The derivation of an exact analytical solution of
the adsorption problem for curved surfaces is a major challenge and is yet unsolved.
In the weak adsorption limit, we provide an exact analytical solution for
polyelectrolyte–sphere adsorption by replacing the Debye–Hu ¨ckel potential by
the Hulthe ´n potential. Other geometries require different approaches. As a generic
concept, we propose application of the WKB method of quantum mechanics, which
we adopted to electrostatic polyelectrolyte adsorption problems. We have demonstrate that this description provides valuable analytical solutions and resolves a
long-standing puzzle about the scaling properties of critical polyelectrolyte adsorption in curved geometries.
We have shown that the scaling behavior of the critical adsorption parameters
indeed changes dramatically in the limit of low salinity or large curvature of the
surface, e.g., for the critical surface charge density from |s c | ~ k
3 for a plane to
|s c | ~ (ka)
2 for a cylinder and to |s c | ~ (ka)
1 for the spherical surface. Maximal
entropic penalty for polyelectrolyte confinement near a sphere and, concurrently,
minimal energetic benefit from polyelectrolyte–surface attraction yield much larger
50
R.G. Winkler and A.G. Cherstvy
nonelectrostatic chain–surface attraction.
Nearly Gaussian Chains—For weak adsorption, The conformational properties
of adsorbing chains are assumed to be only weakly perturbed upon adsorption. This
might not be true well above the adsorption threshold, when quite dense polymer
layers are formed next to the interface. For critical adsorption conditions, however,
this approximation is expected to be valid.
Strong Adsorption Limit—Similarly to the weak adsorption limit, the inherent
limitations of the linearized electrostatic theory are to be remembered. Here, the
limitations might be even more severe because some well-defined polyelectrolyte
patterns are assumed in the model of polyelectrolyte–cylinder and polyelectrolyte–sphere complex formation, which require a strong polyelectrolyte–surface
attraction. Although the adopted approach does not predict overcharging in these
cases, dropping the assumption of an infinitely thin polyelectrolyte and including an
asymmetric charge neutralization along the DNA–sphere contact [237, 238]
reproduces the extent of nucleosome overcharging quite well [79]. In applications
to DNA–histone wrapping, our model of a uniformly bendable nonstretchable
worm-like chain should be severely modified [239, 240], if we are interested in
the precise energetics of DNA wrapping, with site-specific DNA–histone
interactions, sequence-specific curvature effects [241], and possible kinks in the
structure.
8 Conclusions
In this review, we have summarized theoretical concepts and recent advances
for the adsorption of linear polyelectrolyte molecules onto curved surfaces in the
weak and strong adsorption limit. A mean-field description is adopted, and the
interaction potentials between the polyelectrolyte and the surfaces are derived from
the linearized Poisson–Boltzmann equation for the corresponding geometries
(planes, cylinders, and spheres). The derivation of an exact analytical solution of
the adsorption problem for curved surfaces is a major challenge and is yet unsolved.
In the weak adsorption limit, we provide an exact analytical solution for
polyelectrolyte–sphere adsorption by replacing the Debye–Hu ¨ckel potential by
the Hulthe ´n potential. Other geometries require different approaches. As a generic
concept, we propose application of the WKB method of quantum mechanics, which
we adopted to electrostatic polyelectrolyte adsorption problems. We have demonstrate that this description provides valuable analytical solutions and resolves a
long-standing puzzle about the scaling properties of critical polyelectrolyte adsorption in curved geometries.
We have shown that the scaling behavior of the critical adsorption parameters
indeed changes dramatically in the limit of low salinity or large curvature of the
surface, e.g., for the critical surface charge density from |s c | ~ k
3 for a plane to
|s c | ~ (ka)
2 for a cylinder and to |s c | ~ (ka)
1 for the spherical surface. Maximal
entropic penalty for polyelectrolyte confinement near a sphere and, concurrently,
minimal energetic benefit from polyelectrolyte–surface attraction yield much larger
50
R.G. Winkler and A.G. Cherstvy